- Free Bernheim Whinston Microeconomics Solutions. First 100 Words Bilingual Spanish Edition - First Aid For The Nbde Part Ii First Aid Series Pt 2 Jan 2th, 2019 Short Answer Questions Chapter 1-6. The SMART Journal Spring/Summer 2008 Volume 4, Issue 2 Page 60 Resistance, Both To Diversity Measures.
- Microeconomics (08 - Old Edition) by B. Douglas Bernheim available in Hardcover on Powells.com, also read synopsis and reviews. Bernheim and Whinstons Microeconomics focuses on the core principles of the intermediate.
ADVANCED MICROECONOMICS (56278) Dr. Keshab Bhattarai University of Hull Business School, Hull, England, UK. January 12, 2016
Abstract This monograph presents major elements of advanced microeconomic models for systematic thinking about the working of modern markets. Problems of consumers and producers are analysed concisely in partial, general equilibrium and game theoretic frameworks relating them to the micro level decision making processes with due consideration on the structure of markets and pulic policies. Exercises and assignments in workbook anticipate reading of relevant journal articles. JEL Classi…cation: D Keywords: microeconomic models H U 6 7 R X , H u ll,
U K . e m a il: K .R .B h a tta ra [email protected] hu ll.a c .u k
1
Contents 1 L1: 1.1 1.2 1.3
Axioms and optimisation Microeconomic Theory: Milestones . . . . . . . Axioms and Consumption Set . . . . . . . . . . Optimisation . . . . . . . . . . . . . . . . . . . 1.3.1 Consumer optimisation: . . . . . . . . . 1.3.2 Producer optimisation . . . . . . . . . . 1.4 A simple computable general equilibrium model 1.5 Methods for constructing Proofs . . . . . . . . 1.5.1 Direct method . . . . . . . . . . . . . 1.5.2 Converse and contrapositive . . . . 1.5.3 Equivalence . . . . . . . . . . . . . . . 1.5.4 Mathematical induction . . . . . . .
. . . . . . . . . . . . . . . with . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . labour . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . leisure . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . choice . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
2 L2: Consumption: Properties of a Utility Function 2.1 Properties of an indirect utility function . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Price elasticities and the elasticity of substitution . . . . . . . . . . . . . . . . 2.1.2 Consumer optimisation model: a numerical example . . . . . . . . . . . . . . 2.1.3 Econometric issues in estimation of demand functions and elasticities of demand 2.1.4 Restrictions in estimating a demand function: . . . . . . . . . . . . . . . . . . 2.2 Exercise 1: Consumer’s problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Marshallian demand functions from the CES preferences: . . . . . . . . . . . 2.2.2 Compensated and uncompensated demands . . . . . . . . . . . . . . . . . . . 2.2.3 Expenditure functions with the CES utility functions . . . . . . . . . . . . . . 2.3 Slutskey equation ( Decomposition of substituion and income e¤ects): Duality on consumer optimisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Comparative static analyis with matrix . . . . . . . . . . . . . . . . . . . . . 2.4 Exercise on Consumption and Demand . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Indirect utility and expenditure functions: Roy’s Identity . . . . . . . . . . . 2.4.2 Dual of the consumer’s optimisation problem . . . . . . . . . . . . . . . . . . 2.4.3 Shephard’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.4 Calibration of the constant elasticity of substitution (CES) demand function 2.4.5 Exercise 1.2: comparative static analysis of consumer choice . . . . . . . . . . 2.4.6 Exercise 2: Indirect utility function, Shephard’s Lemma and Roy’s Identity . 2.4.7 Duality in consumption and Slutskey decomposition . . . . . . . . . . . . . . 2.4.8 Problem 4: CES Demand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.9 Extra example on Shephard’s Lemma and Roy’s identity . . . . . . . . . . . . 2.4.10 Indierct utility function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.11 Expenditure function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.12 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.13 Shephard’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.14 Roy’s Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Revealed Preference Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Slutsky equation from the Revealed Preference Theory . . . . . . . . . . . . . 2.5.2 Further Developments in Consumer Theory . . . . . . . . . . . . . . . . . . .
2
8 8 9 10 10 11 12 14 14 15 15 15 15 16 17 18 19 19 21 22 23 23 24 26 29 30 30 31 32 35 36 37 38 39 40 40 40 41 43 44 44 45
2.5.3
Emprical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
3 L3: Production: Supply 47 3.0.4 Popular production functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.1 Supply function: an example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.1.1 Properties of a pro…t function . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.1.2 Production function and its scale properties . . . . . . . . . . . . . . . . . . . 52 3.1.3 Variable returns to scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.1.4 Cost function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.2 CES and Cobb-Douglas production functions . . . . . . . . . . . . . . . . . . . . . . 54 3.3 Comparative static: derivation of the CES cost function. . . . . . . . . . . . . . . . . 56 3.3.1 Exercise 5: cost minimisation . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.3.2 Properties of a cost function . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.3.3 Exercise 6 : CES and Cobb-Douglas supply functions . . . . . . . . . . . . . 62 3.3.4 Short run supply function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.3.5 Hotelling’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.3.6 Exercise 7: Minimising the cost with Cobb-Douglas and CES production function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.4 Consumer and producer surplus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.4.1 Pro…t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.4.2 Linear programming problem of a …rm . . . . . . . . . . . . . . . . . . . . . . 71 3.4.3 Duality in Linear Programming . . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.5 Input-Output Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.5.1 Numerical Example of Input Output Model . . . . . . . . . . . . . . . . . . . 76 3.5.2 Impact analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 3.5.3 Exercise 10: Input Output Model . . . . . . . . . . . . . . . . . . . . . . . . 80 4 L4: Markets: Perfect and Imperfect Competition 4.1 De…nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Cournot, Stackelberg and Cartel: which is better of consumer welfare? . . . 4.1.2 Price-Leadership by …rm 1 in Stackelberg equilibrium . . . . . . . . . . . . 4.2 Dixit-Stiglitz Model of Monopolistic Competition . . . . . . . . . . . . . . . . . . . 4.2.1 Monopolistic competition and Trade . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Monopolistic competition in an industry with two …rms . . . . . . . . . . . 4.3 Natural Monopoly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Bertrand Game of price competition . . . . . . . . . . . . . . . . . . . . . . 4.4 Price War: stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Monopolistic competition and trade . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Monoply, Oligopoly and tax . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Tripoly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Multinational Company: Microeconomic Theory of FDI . . . . . . . . . . . . . . . 4.8 Predatory pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.1 FDI under uncertainty: Dixit and Pindyk (1994) approach . . . . . . . . . 4.9 General equilibrium model of a multinational …rm:Batra and Ramachandran(1980) 4.9.1 Empirical evidence on growth e¤ects of FDI . . . . . . . . . . . . . . . . . . 4.9.2 Exercise 9: markets and competition . . . . . . . . . . . . . . . . . . . . . .
3
. . . . . . . . . . . . . . . . . .
83 83 87 88 90 91 92 95 97 98 102 103 106 110 114 114 116 119 119
4.9.3
General equilibrium with production . . . . . . . . . . . . . . . . . . . . . . . 122
5 L5: General Equilibrium Model and Welfare 5.1 What is a general equilibrium? . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Existence, uniqueness and stability of general equilibrium . . . . . . . . 5.2 Two fundamental theorems of welfare economics . . . . . . . . . . . . . . . . . 5.3 Pure exchange general equilibrium model . . . . . . . . . . . . . . . . . . . . . 5.3.1 Exercise 11: Ricardian trade model . . . . . . . . . . . . . . . . . . . . . 5.4 Simplest general equilibrium model . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 General equilibrium with production . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 A numerical example for the general equilibrium tax model . . . . . . . 5.6 Social Welfare Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Exercise 12: Social Welfare and General Equilibrium . . . . . . . . . . . . . . . 5.8 Two sector model of nessecity and luxury goods (income distribtuion) . . . . . 5.9 General equilibrium model of Trade: Ricardian Comparative Advantage Theory 5.9.1 Two Country Ricardian Trade Model . . . . . . . . . . . . . . . . . . . 5.9.2 Autarky or Trade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10 Exercise 12’: migration and factor mobility . . . . . . . . . . . . . . . . . . . . 5.11 General equilibrium with taxes . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.12 Exercise 13: Monopolistic Competition . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . .
122 122 124 127 129 133 133 134 137 143 146 149 153 153 154 162 163 168
6 L6: Game theory: Bargaining in Goods and Factors markets 171 6.1 Formal de…nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 6.1.1 Nash equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 . . . . . . . . . . . . .
. . . . . . . . . . . . .
. . . . . . . . . . . . .
. . . . . . . . . . . . .
174 175 176 179 183 187 190 191 191 192 194 196 199
7 L7: Game theory: Principal Agent and Mechanism Games and Auctions 7.1 Original Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 Full information scenario . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.2 Incomplete information scenario . . . . . . . . . . . . . . . . . . . . . . 7.1.3 Impacts of Assymetric (incomplete) Information on Markets . . . . . . . 7.1.4 Adverse Selection (hidden information) Problem . . . . . . . . . . . . . 7.1.5 Signalling and Incentives . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.6 Education Level- A Signal of Productive Worker . . . . . . . . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
200 200 201 201 205 206 207 209
6.2 6.3 6.4 6.5
6.6 6.7 6.8
6.1.2 Game of incomplete information: . . . . . . . . . . . . . . 6.1.3 Extensive form Game ( ) . . . . . . . . . . . . . . . . . . Story of GAME made easy . . . . . . . . . . . . . . . . . . . . . Types of games . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bargaining game . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Coalition and Shapley Values of the Game . . . . . . . . Pivotal player in a voting game in Nepal . . . . . . . . . . . . . . 6.5.1 Model of fruitless bargaining and negotiation . . . . . . . 6.5.2 Model of commitment, credibility and reputation . . . . . 6.5.3 Endogenous intervention: change in beliefs . . . . . . . . Equivalence of Core in Games and Core in a General Equilibrium Labour Market and Search and Matching Model . . . . . . . . . Exercise 14: Search Equilibrium . . . . . . . . . . . . . . . . . . .
4
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Model . . . . . . . .
. . . . . . . . . . . . .
. . . . . . . . . . . . .
. . . . . . . . . . . . .
7.2 7.3 7.4 7.5
7.6 7.7 7.8 7.9
Spence model of education . . . . . . . . . . . . . . . . . . . . . . . . . . Popular Principal Agent Games . . . . . . . . . . . . . . . . . . . . . . . Exercise 15: Principal Agent Problem . . . . . . . . . . . . . . . . . . . Mechanism Design for Price Discrimination: Low Cost Airlines Example 7.5.1 Mechanism for e¢ cient contract for a CEO . . . . . . . . . . . . 7.5.2 E¢ cient contracts of Land . . . . . . . . . . . . . . . . . . . . . . 7.5.3 Mechanism for Poverty Alleviation . . . . . . . . . . . . . . . . . Repeated Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Moral Hazard and Adverse Selection . . . . . . . . . . . . . . . . . . . . Auction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercise 16 :Optimal production of a multiproduct …rm . . . . . . . . . 7.9.1 A Microeconomic Model of FDI . . . . . . . . . . . . . . . . . . .
8 L8: Uncertainty and Insurance 8.1 Allais’paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1 Uncertainty of Good Times and Bad Times . . . . . . . . . . 8.1.2 Optimal Demand for Insurance . . . . . . . . . . . . . . . . . 8.2 Expected utility theory . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Measure of risk aversion . . . . . . . . . . . . . . . . . . . . 8.2.2 St Petersberg Paradox (Bernoulli Game) and Allais Paradox 8.2.3 Non-linear pricing Scheme . . . . . . . . . . . . . . . . . . . . 8.2.4 Job market applications . . . . . . . . . . . . . . . . . . . . . 8.2.5 Insurance market . . . . . . . . . . . . . . . . . . . . . . . . . 9 L9:Class Test
. . . . . . . . .
. . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
210 212 217 217 221 222 224 226 228 231 233 236
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
239 241 243 245 247 248 249 250 253 255 259
10 L10: Impact of Taxes and Public Goods in E¢ ciency, Growth and Redistribution: A General Equilibrium Analysis 263 10.1 First best principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 10.1.1 E¢ ciency in consumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 10.1.2 E¢ ciency in production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 10.1.3 E¢ ciency of Trade (Exchange) . . . . . . . . . . . . . . . . . . . . . . . . . . 264 10.1.4 A simple numerical example of optimal tax or optimal public spending . . . . 265 10.1.5 E¢ ciency in public goods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 10.1.6 Theory of second best . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 10.1.7 Externality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 10.1.8 Samuelson and Nash on Sharing Public Good . . . . . . . . . . . . . . . . . . 268 10.1.9 Sameulson’s Theorem on Public Good . . . . . . . . . . . . . . . . . . . . . . 269 10.1.10 Negative externality in production . . . . . . . . . . . . . . . . . . . . . . . . 270 10.2 Negative externality and Pigouvian tax . . . . . . . . . . . . . . . . . . . . . . . . . 271 10.3 Carbon Emmission in the UK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 10.3.1 A model of growth, …scal policy and welfare . . . . . . . . . . . . . . . . . . . 278 10.4 Fiscal Policy, Growth and Income Distribution in the UK . . . . . . . . . . . . . . . 280 10.4.1 Middle income hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286 10.4.2 Current Fiscal Policy Context . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 10.5 Features of Dynamic Tax Model of UK . . . . . . . . . . . . . . . . . . . . . . . . . . 291 10.5.1 Preferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 5
10.5.2 10.5.3 10.5.4 10.5.5 10.5.6 10.5.7 10.5.8 10.5.9
Production Technology . . . . . . . . . . . Trade arrangements . . . . . . . . . . . . . Government sector . . . . . . . . . . . . . . General Equilibrium in a Growing Economy Procedure for Calibration . . . . . . . . . . Data for the Benchmark Economy . . . . . Results on Redistribution . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
292 292 293 293 294 294 295 298
11 L11: Dynamic Computable General Equilibrium Model: Recent Developments 305 11.1 Capital market: models and issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310 11.1.1 Risk management in asset markets . . . . . . . . . . . . . . . . . . . . . . . . 313 11.1.2 Industrial regulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 11.1.3 IO Approach to pricing and industrial concentration . . . . . . . . . . . . . . 316 11.1.4 Signalling and Incentive Compatibility in the Financial Markets . . . . . . . 320 11.1.5 Moral hazards in the …nancial market . . . . . . . . . . . . . . . . . . . . . . 321 12 Assignment (optional): One in Four 12.1 General equilibrium and game theoretic analysis of …nancial 12.1.1 CGE Modelling of energy sector policies . . . . . . . 12.1.2 CGE Modelling of tax policies . . . . . . . . . . . . 12.1.3 Comparative Static Model . . . . . . . . . . . . . . . 12.2 Dynamic CGE model of the energy and emmission . . . . .
. . . . .
. . . . .
. . . . .
. . . . .
327 327 327 327 329 335
13 Regulation Theory and Practice 13.0.1 Theory of Regulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.0.2 Measures of concentration and performance . . . . . . . . . . . . . . . . 13.0.3 Regulation for solve the moral hazard problems in the …nancial markets 13.0.4 Regulation by mechanism design by banks . . . . . . . . . . . . . . . . . 13.0.5 Participation and incentive compatible constraints . . . . . . . . . . . . 13.0.6 Solving the mechanism design problem of a bank . . . . . . . . . . . . . 13.0.7 IO Approach to pricing and industrial concentration (HHI) . . . . . . . 13.0.8 Why regulation? Welfare e¤ects of monopoly . . . . . . . . . . . . . . . 13.0.9 Optimal advertising . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.0.10 Marginal productivity theory and tax credit . . . . . . . . . . . . . . . . 13.0.11 Capital stock with and without capital income tax . . . . . . . . . . . . 13.0.12 Technological development, human capital and tax rules . . . . . . . . . 13.0.13 Dixit-Stiglitz Model of Monopolistic Competition . . . . . . . . . . . . . 13.0.14 Market under imperfect competition and average cost pricing . . . . . . 13.0.15 Krugman (1980): Trade and scale economy and regulation . . . . . . . . 13.0.16 Regulation by non-linear pricing Mechanism . . . . . . . . . . . . . . . . 13.1 Articles and Texts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1.1 Best twenty articles in 100 years in the American Economic Review . . 13.1.2 Ten Best articles in the Journal of European Economic Association . . . 13.1.3 Best 40 articles in the Journal of Economic Perspectives . . . . . . . . .
. . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . .
338 338 339 340 341 341 342 343 344 344 345 346 346 348 349 351 352 357 357 358 359
6
sector . . . . . . . . . . . . . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
14 Real Analysis 14.1 Methods for constructing Proofs 14.1.1 Convergence . . . . . . . 14.1.2 Boundedness . . . . . . 14.1.3 Convex Hull . . . . . . 14.1.4 Correspondence . . . . 14.1.5 Fixed Point Theorems . . 14.2 SETS . . . . . . . . . . . . . . . 14.2.1 Relations and functions . 14.3 Limits . . . . . . . . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
368 369 370 370 371 371 372 372 373 373
15 Computation and software 15.1 GAMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2 MATLAB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3 Econometric and Statistical Software . . . . . . . . . . . . . . . 15.3.1 Quality ranking of journals in Economics . . . . . . . . 15.4 Core texts in Economic Theory and Equivalent reading
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
374 374 378 381 382 384
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
16 Schedule 386 16.1 Sample class test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388 16.2 Sample …nal exam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391 17 Tutorials in Advanced Microeconomics 17.1 Tutorial 1:Consumers’problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2 Tutorial 2: Dual of the consumer problem . . . . . . . . . . . . . . . . . . . . . 17.3 Tutorial 3: Dual of the producer’s problem . . . . . . . . . . . . . . . . . . . . 17.4 Tutorial 4: Markets, Price War and Stability Analysis . . . . . . . . . . . . . . 17.5 Tutorial 5: Ricardian General Equilibrium Trade Model . . . . . . . . . . . . . 17.6 Tutorial 6: General equilibrium with production . . . . . . . . . . . . . . . . . 17.7 Tutorial 7:Monopoly and monopolistic competition and taxes . . . . . . . . . . 17.8 Tutorial 8:Moral Hazard and Insurance . . . . . . . . . . . . . . . . . . . . . . . 17.9 Tutorial 9:Coalition, Bargaining, Signalling, Contract, Auction and Mechanism 17.10Tutorial 10: E¢ ciency and Social Welfare . . . . . . . . . . . . . . . . . . . . . 17.11Basic Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.11.1 Four rules of di¤erentiation . . . . . . . . . . . . . . . . . . . . . . . . . . 17.11.2 Unconstrained optimisation: using Hessian determinants . . . . . . . . . . 17.11.3 Constrained optimisation: Bordered Hessian Determinants . . . . . . . . 17.11.4 Linear Programming approach to input-output model . . . . . . . . . . .
7
. . . . . . . . . . . . . . .
. . . . . . . . . . . . . . .
396 396 396 398 399 400 401 403 405 407 409 410 410 410 412 413
1
L1: Axioms and optimisation
Rational economic agents use available resources to achieve their objectives in the best possible way. Microeconomics is about choices of these rational individuals who make decisions regarding the allocation of resources, particularly on how much to consume or invest, how to produce and how to interact in the markets remaining within the limits of resources they possess. It is also concerned about the consumption and saving or current or future consumption at the individual level and about the e¢ ciency welfare consequences of public policies that a¤ect them. Game theories have been applied increasingly in recent yeats to study strategic economic behavior of consumers and producers. Main elements of microeconomic theory thus consists of: 1. Consumer choice and demand for products and supply of labour and capital: demand functions. 2. Producer’s choice of products and demand for inputs: production, cost, pro…t and supply functions. 3. Analysis of prices in perfect and imperfect markets with complete and incomplete information. 4. General equilibrium analysis - determination of price system and optimal allocations. 5. Strategic analysis of decision making my consumers, producers, governments in a competitive global economy. 6. Competition and market power; game theory 7. Innovations and adoption new technologies, research and development. 8. Analysis of e¢ ciency and public policies; social welfare function, market failure, negative and positive externalities Good undestanding of microeconomic theories will lead to better policies and regulations for the e¢ cient functioning of the market economy. These policies particularly focus on competition, adoption of better technology, governance and information, correcting externality and good environment, social insurance, more equal distribution of income and identi…cation of cases for govermen intervention. For recent policies see relevant web page of the government such as in the Department for Business Innovation & Skills https://www.gov.uk/government/organisations/competition-andmarkets-authority. Analysis of all above are based on axioms or generally accepted truth about the behavior of economic agents that include axioms of completeness, transitivity, continuity, monotonicity and convexity.
1.1
Microeconomic Theory: Milestones
The existing knowledge in microeconomis is the result of hard work of many prominent economic thinkers: such as Smith (1776), Ricardo (1817), Cournot (1838), Bertrand (1883), Edgeworth (1925), Pareto (1896), Marshall (1890), Walras (1900), Veblen (1904), Slutskey (1915), Hicks (1939), Samuelson (1947), von Neumann and Morgenstern (1944), Nash (1950), Neumann (1957), Arrow (1953), Debreau (1959), Stigler (1961), Kuhn (1953), Shapley (1953), Robinson (1963),
8
Shelten ( 1965), Aumann (1966) Scarf (1967), Shapley and Shubik (1969), Harsanyi (1967), Spence (1974),Kahneman and Tversky (1979), Kreps (1990), Fundenberg and Tirole (1991) and Binmore (1992), Varian (1992), Osborne and Robinstein (1994), MasColell, Whinston and Green (1995), Starr (1997) Gravelle and Rees (2004), Rasmusen (2006), Snyder and Nicholson (2011). Studies by Cobb and Douglas (1928), Arrow (1963), Jorgenson (1963), Diamond and Mirrlees (1971), Alchian and Demsetz (1972), Ross (1973), Dixit and Stiglitz (1977), Deaton, and Muellbauer (1980), Krugman (1980), Shiller (1981), Grossman and Stiglitz (1980), listed in the best 20 articles published in AER in last 100 years, relate to microeconomic issues. http://www.eea-esem.com/eea-esem/2014/prog/list_sessions.asp http://editorialexpress.com/conference/MMF2014/program/MMF2014.html https://www.aeaweb.org/aea/2015conference/program/preliminary.php http://www.webmeets.com/RES/2013/prog/list_sessions.asp http://nobelprize.org/nobel_prizes/economics/laureates/; http://www.economicsnetwork.ac.uk/ http://cepa.newschool.edu/het/schools/game.htm; http://www.hull.ac.uk/php/ecskrb/Confer/research.html; http://homepage.newschool.edu/het/alphabet.htm http://editorialexpress.com/conference/GAMES2012/program/GAMES2012.html http://www.eea-esem.com/EEA-ESEM/2012/prog/list_sessions.asp
It is becoming sophisticated with the recent development of mathematical techniques and computational abilities. This monograph gradually develops these concepts so that all parts of the modern economies could be integrated into a dynamic general equilibrium model of modern economies towards end of this workbook. Axioms and a general summary is in this section. Then behavior of consumers and produced are analysed in the …rst two chapters followed by partial equilibrium analysis under perfectly and imperfectly competitive markets in section 4. Basic principles of general equilibrium models explained in section 5 followed by a short discussion of strategic models in chapters 6 and 7 and uncertainty and asymmetric information in section 8. Impacts of taxes on public goods and externalities are presented in section 10 followed by details of the dynamic general equilibrium model in section 11. Each section have takes problem solving approach to learning and contains exercises at the end. The last part contains details on the software used for empirical testing of various microeconomic theories with some listing of seminal articles and popular text books.
1.2
Axioms and Consumption Set
Let X = (x1; x2; :::::xn ) be quantities of n commodities in nonnegative orthant of X 2 Rn The consumption set X ful…lls following properties. These concepts date back to Pareto (1896), Marshall (1890), Slutskey (1915), Hicks (1939), Samuelson (1947), Debreau (1959) and others. 1. 0 6= X
Rn
2. X is closed 3. X is convex 4. 0
X
Let B 2 X be a feasible set such that x % x for all x 2 B: Axioms of Consumer Choice 9
Axiom 1: Completeness If x1 and x2; are both in X , x1; x2
X either x1 % x2 or x1 - x2 . Consumer can compare.
Axiom 2: Transitivity For x1; x2 ; x3
X
if x1 % x2 ; x2 % x3 then x1 % x3 . Consumer is consistent.
Axiom 3: Continuity Preference relations % xi
- xi are closed in Rn
Axiom 4: Monotonicity For x0 Rn and for all ' > 0 there exist some x 2 B' (x0 ) Rn such that x > x0 . More is prefered for less. Axiom 5: Convexity If x1 % x0 then tx1 + (1 t) x0 % x0 for all t 2 [0; 1] . Demand and supply of the market system are based on above axioms. Think of a simple problem of households and …rms in a market economy.
1.3
Optimisation
Linear and non-linear programming are applied in order to …nd the optimal solution subject to constraints. Objectives like the utility or pro…t or social welfare can be function of one or several variables. Constraints can be one or multiple. Linear programming is applied where the objective functions and constraints are of the linear form and non-linear optimisation techniques is applied when objectives or the constraints are non-linear. By duality theorem every maximisation problem has a corresponding minimisation problem, such as utility maximisation corresponds to expenditure minimisation to achieve a certain level of utility, pro…t maximisation corresponds of cost minimisation given the type technology of production. 1.3.1
Consumer optimisation:
Assuming above axioms are satis…ed, the major objective of a consumer is to maximise utility by consuming x commodities (u (x)) max u(x)
(1)
subject to the budget constraint assuming prices (p) and income (y) as given: p:x
y
Constrained optimisation (Lagrangian function) with price): L (x; ) = u(x) + [y
10
(2) as a Lagrange multiplier (or shadow
p:x]
(3)
First order conditions wrt each xi : @L (x; ) @u(x) = @x1 @x1
p1 = 0
::
(5)
@L (x; ) @u(x) = @xn @xn y
(4)
pn = 0
p:x = 0
(6) (7)
@u(x) @xi
@u(x) @xi
> 0 and pi > 0 =) = pi > 0. Here there are n + 1 …rst order conditions to solve for demand for x1; x2; :::::xn goods and . The maximum utility is obtained when all these optimal values are substituted in the utility function u (x ) : Thus the marginal rate of substitution between xj and xi should equal their price ratios in equilibrium: M RSj;i =
@L(x; ) @xj @L(x; ) @xi
=
pj M Uj M Ui ; = pi pj pi
(8)
Marginal utility of xi represents gain from consumption xi and pi represents pain to the consumer. Equilibrium psychologically is thus a point where the gain equals pain. 1.3.2
Producer optimisation
The major objective of producers is to maximise pro…t. They take prices of commodities (p) and inputs (w) as given: Pro…t function is a value function for for all input price w 0 and output levels y 2 Rn+ (p; w) = p:y
w:x
(9)
subject to f (x)
y
(10)
Constrained optimisation (Lagrangian) function: L (x; ) = py
w:x + [y
@L (x; ) = w1 @x1
f (x)]
f 0 (x1 ) = 0
:: @L (x; ) = wn @xn 11
(11) (12) (13)
f 0 (xn ) = 0
(14)
y
@f (x;) @xj @f (x;) @xi
f (x) = 0;
=
wj wi
(15)
A complete view of microeconomic process requires thinking about the general equilibrium in p the market. The relative price system pji for i = 1; ::; N and j = 1; ::; N determines the optimal allocation of resources in the economy. Consider the following example for this purpose.
1.4
A simple computable general equilibrium model with labour leisure choice
Consider an economy with two individuals, i = 1; 2 and two commodities x (goods) and y (services). Both households are endowed with given amount of capital stock k1 ; k 2 and time L1 ; L2 , which they spend either working or in the form of leisure. Households and …rms optimise taking prices of commodities (px ; py ) and factors (pL ; pk ) as given. Competition between suppliers and consumers or producers sets the equilibrium price of commodities and income of households (I1 ; I2 ) More speci…cally the problems of households and …rms can be stated as: Household’s problem: _
max U1 = xa1 1 y1b1 L1
g1
LS1
; a1 > 0; b1 > 0; g1 > 0:
(16)
subject to: _
_
I1 = px x1 + py y1 + pL L1
LS1 ;
_
I1 = pL L1 + pk K 1
(17)
_
x1
0; y1
0; L1
LS1
0: _
max U2 = xa2 2 y2b2 L2
LS2
g2
; a2 > 0; b2 > 0; g2 > 0:
(18)
subject to: _
_
I2 = px x2 + py y2 + pL L2
LS2 ;
_
I2 = pL L2 + pk K 2
(19)
_
x2 0; y2 0; L2 Firm’s problem:
LS2
max
0:
x
= px x
pk kx
pL LS1x
pL LS2x
(20)
subject to: x = kxx LS1x1x LS2x2x max
y
= py y
pk ky
pL LS1y
(21) pL LS2y
(22)
subject to: y = kyy LS1y1y LS2y2y 12
(23)
Equilibrium conditions: x = x1 + x2
(24)
y = y1 + y2
(25)
kx + ky = k1 + k 2
(26)
_
L1 + LS1x + LS1y = L1 ;
LS1 = LS1x + LS1y
(27)
_
L2 + LS2x + LS2y = L2 :::LS2 = LS2x + LS2y
(28)
Price normalisation: px + py + pL + pk = 1
(29)
Questions 1) Derive demand for x and y and leisure (or labour supply) by households 1 and 2 i. e. determine x1 ; x2 ; y1 ; y2 ; LS1; LS2. 2) Determine the demand for labour and capital by …rms supplying x and y, i,e, evaluate kx ; ky ; LS1x ; LS1y ; LS2x ; LS2y : 3) Compute the equilibrium relative price system for this economy that are consistent to optimisation problems of households and …rms. 4) What are the optimal allocations of resources in this economy? 5) Evaluate the demand for x and y and leisure by both households and …nd the optimal levels of their welfare. Is this Pareto optimal allocation? 6) Suggest tax and transfer scheme in this economy in order to improve the distribution system. 7) Explain notions of Hicksian equivalent and compensating variations in order to evaluate the welfare consequences of tax and welfare reforms proposed above. 8) Write GAMS code to solve the model and few simulation scenarios for comparative static analysis. 9) Propose reforms in the labour and capital markets for improving the e¢ ciency of allocations in this economy. GAMS programe: ge2by2.gms Bhattarai K. and J. Whalley (2003) Discreteness and the Welfare Cost of Labour Supply Tax Distortions, International Economic Review 44:3:1117-1133, August Bridel (2011) for a non-technical introduction to the general equilibrium modelling. Now solve this CGE (computable general equilibrium) model using GAMS. First assign values for behavioural parameters. Secondly, download gams at www.gams.com and install in your PC or laptop. Thirdly write model equations for numerical optimisation routine in GAMS (see GAMS programme …le ge2by2.gms and its result …le ge2by2.lst)
13
a1 0.5
Table 1: Parameters for the 2 by 2 model with _leisure _ a2 b1 b2 g1 g2 1x 1y L1 L1 x y 0.4 0.3 0.4 0.2 0.2 0.2 0.8 0.4 0.1 24 24 Table 2: E¢ cient allocation in the 2 by I1 I2 x1 x2 y1 y2 3.42 2.01 5.9 2.8 1.7 1.3 px py pk pl kx ky 0.289 0.599 0.047 0.064 35.3 14.7 ls11 6.7
ls12 8.9
ls21 6.7
ls22 8.9
_
_
L1 24
L2 24
k1 40
k2 10
2 model with leisure x y u1 u2 8.7 3.1 4.6 2.4 L1 L2 ls1 ls2 10.7 6.3 13.3 17.7 k1 40
k2 10
Fourth study the solution of the model systematically: _ _ _ _ I1 = pL L1 + pk K 1 (=0.064*24+0.047*40)=3.42; I2 = pL L2 + pk K 2 (= 0:064 24 + 0:047 10) = 2:01: Fifth check the equilibrium conditions; check that all equilibrium conditions are satis…ed: x = x1 + x2 = 5:9 + 2:8 = 8:7
(30)
y = y1 + y2 = 1:7 + 1:3 = 3:1
(31)
kx + ky = 35:3 + 14:7 = k1 + k 2 = 44 + 10 = 50
(32)
_
L1 + LS1x + LS1y = 10:7 + 6:7 + 6:7 = 24 = L1 ;
LS1 = LS1x + LS1y
(33)
_
L2 + LS2x + LS2y = 6:3 + 8:9 + 8:9 = 24 = L2 :::LS2 = LS2x + LS2y
(34)
Sixth, consider tax policy analysis a) introducing VAT in commodities x and y b) introducing taxes in labour and capital inputs c) set a revenue target and do equal yield tax reforms …nding model solution when all taxes are i) raised in VAT or ii) by labour income tax or iii) capital income tax or iv) equally by from these sources. Seventh, compute the optimal tax rates that maximise revenue (hint make tax rates endogenous and solve the maximisation routine). The reader is expected to study the real analysis section in the appendix at this point. It is important to understand these basic concepts in order to follow literature on economic theory or to develop concepts in economic theory.
1.5 1.5.1
Methods for constructing Proofs Direct method
a = b; b = c =) a = c: or a = b; c = d =) a:c = b:d For x; y; z 2 R prove that x + z = y + z =) x = y 14
1.5.2
Converse and contrapositive
A implies B A =) B if it converse B =) A is true then A () B here A and B are equivalent. If A person lives in Hull (A) then that person lives in Yorkshire (B). A =) B but converse is not true in this case B ; A and A < B Contrapositive implies not A implies not B A =) B 1.5.3
Equivalence
A () B Example Pythagorus theorem: h2 = p2 + b2 p2 2 b2 h2 + cos2 = 1 h2 = h2 + h2 () sin Revealed preference theory is equivalent to utility maximisation theory in deriving income and substitution e¤ects. 1.5.4
Mathematical induction
Example: Sum of the N natural numbers is : P (n) = 1 + 2 + 3 + :::: + n = n(n+1) 2 Check if this works for any integer k P (k) = 1 + 2 + 3 + :::: + k = k(k+1) ; now prove that P (k + 1) = (k+1)(k+2) : 2 2 Add and subtract k + 1 from both sides 1 + 2 + 3 + :::: + k + (k + 1) = k(k+1) + (k + 1) = (k + 1) k2 + 1 = (k+1)(k+2) =) P (k + 1) 2 2 Thus by mathematical induction P (k) and P (k + 1) are similar.
2
L2: Consumption: Properties of a Utility Function
Maximising the level of utility (satisfaction) from consumption of goods and services is the ultimate objective of all economic activities. Various speci…cations of utility functions are used to represent the level of welfare of households from consuming goods and services and leisure in an economy. From abstract functions to linear and non-linear utility functions are popular in the literature. The Cobb-Douglas and constant elasticity of substitution (CES) utility functions are very popular in the literature. There are also nested utility function for instance in a general equilibrium models with many goods one can consider of three levels of nests to capture the intra- period and inter temporal substitution between consumption and leisure based on relative prices and wage rates in the economy. The …rst level of nest aggregates the goods and services in composite consumption good, then the second level nest aggregates these composite goods with leisure. Then there is the nest of time separable utility functions to arrive at the life time utility for each household. The consumption shares of various goods are calibrated from the benchmark dataset (Blundell (2014), Deaton, and Muellbauer (1980), Barker, Blundell and Micklewright (1989) for more in depth study on demand side parameters of household demand functions). These utility functions are further modi…ed in order to represent positive or negative externalities in consumption. While recreation facilities in the neighbourhood generates positive externalities but pollutions reduce utility levles of households. In dynamic setting most studies apply the time separable utility functions. Aim of this section is to present popular models used in analysing preferences and demands of households for various commodities. It shows how to evaluate the welfare as well as the price and substitution
15
e¤ects of changes in prices and the elasticities of demand. Basics of revealed preference theory is reviewed at the end. Microeconomic theories are often tested econometrically to ascertain their validities (Houthakker (1950), Richter (1966), Afriat (1967), Kahneman and Tversky (1979),Varian (1982), McGuinness (1980), Bandyopadhyay (1988), Hey and Orme (1994), Lee and Singh (1994), Carey (2000), Deolalikar and Evenson (1989) Van Soest and Kooreman (1987), Blundell, and Preston (2008), Echinique (2011), Varian (2012), Vermeulen (2012), Schmeidler, D. (1989), Blundell (2014).). Where preference relations are complete, transitive, continuous, monotonous and convex then there exists a real valued utility function u : Rn+ =) R and this utility function has following properties: u (x) is strictly increasing in x if and only if % is strictly monotonic. u (x) is quasi-concave if and only if % is convex. u (x) is strictly quasi-concave if and only if % is strictly convex.
2.1 v:
Rn+
Properties of an indirect utility function =) R v (p; y) = max u(x) s.t. p:x
y
X 2 Rn
1. Continuous 2. Homegenous of degree zero in (p; y) 3. Strictly increasing in y 4. Decreasing in p 5. Quasiconvex in p and y. 6. Roy’s identity
0
xi p ; y
0
=
@v (p0 ;y 0 ) @pi @v(p0 ;y 0 ) @y
::::i = 1::m
(35)
Numerical Example: Derive demands for (x1 , x2 ) from ratios of marginal utilities (partial derivative of utility functions) given the prices [(p1 , p2 ) = (2; 4)] and income (a). max u = x1 x2 subject to 2x1 + 4x2 = a L (x1 ; x2 ) = x1 x2 + [a
2x1
4x2 ]
(36) (37)
@L (x1 ; x2 ) = x2 @x1
2 =0
(38)
@L (x1 ; x2 ) = x1 @x2
4 =0
(39)
16
From the
@L (x1 ; x2 ) = a 2x1 4x2 = 0 @ = 2: Then put this into the last FOC to get: x1 =
…rst two FOCs xx21 2 = x1 x2 = a4 a8 = a32 :
(40) a 4;
x2 =
a 8 ;,
a = 16 =) u By an envelop theorem evaluating the indirect utility function @L(x1 ;x2 ) a and Lagrange multiplier at the optimal solution: @u = : QED. If consumer @a = 16 = @a a 200 income a = 200 then x1 = a4 = 200 25 = 1250: 4 = 50; x2 = 8 = 8 = 25: Then u = x1 x2 = 50
2.1.1
Price elasticities and the elasticity of substitution
Now if p1 changes to 4 and p2 to 2 what will be elasticities, cross elasticities and elasticities of substition between x1 and x2 ? a 200 New demands: x1 = a8 = 200 8 = 25; x2 = 4 = 4 = 50: Utility is still 1250. Price elasticity of demand e1 =
dx1 =x1 dx1 p1 = = dp1 =p1 dp1 x1
25 2 = 2 50
0:5
(41)
1
(42)
e2 =
dx2 =x2 dx2 p2 25 4 = = = dp2 =p2 dp2 x2 2 50
e1 =
dx1 =x1 dx1 p2 = = dp2 =p2 dp2 x1
25 4 =1 2 50
(43)
e2 =
dx2 p1 25 1 dx2 =x2 = = = 0:5 dp1 =p1 dp1 x2 2 25
(44)
Cross price elasticity:
Elasticity of substituion bewteen x1 and x2 d =
x1 x2
=
p2 p1
=
x1 x2
=
25 50 2 4
= =
50 25 4 2
=1
(45)
e1;a =
dx1 =x1 dx1 a 1 200 = = =1 da=a da x1 4 50
(46)
e2;a =
dx2 =x2 dx2 a 1 200 = = =1 da=a da x2 8 25
(47)
e1;a =
dx1 =x1 dx1 a 1 200 = = =1 da=a da x1 8 25
(48)
e2;a =
dx2 =x2 dx2 a 1 200 = = =1 da=a da x2 4 50
(49)
d Income elasticities of demand
p2 p1
Before the change in price:
After the change in price:
17
2.1.2
Consumer optimisation model: a numerical example M ax U = X10:4 X20:6
(50)
Subject to p1 :X1 + p2 :X2 = 150
(51)
Lagrangian optimisation: L (X1 ; X2 ; ) = X10:4 X 0:6 + [150
p1 :X1
p2 :X2 ]
(52)
For base equilibrium assume that p1 = 3 and p2 = 2: Optimal demand for goods X1 X1 =
0:4 (150) 60 = = 20; p1 3
X2 = 0:4
U0 = X10:4 X20:6 = (20)
0:6 (150) 90 = = 45 p2 2 0:6
(45)
= 32:53
(53) (54)
Now assume that there is a subsidy in X1 of £ 1 and price reduces from 3 to 2; p1 = 2: Equivalent Variation What is the Hicksian Equivalent and compensating variations of price change? What are the income and substitution e¤ects of this price change? First …nd out how much money is required at new prices to guarantee the original utility by solving U0 = U0 =
0:4 (m0 ) 2
0:4
0:4 (m0 ) 2 0:4
0:6 (m0 ) 2
0:6
0:6 (m0 ) 2 0:6
; m0 =
= 32:53 2 (32:53) = 127:49 0:40:4 0:60:6
(55)
(56)
Equivalent variation (money to be taken away when prices fall) EV = 150
127 = 22:51
(57)
Compensating Variation For compensating variation …rst compute the demand in new prices and utility X1 =
0:4 (150) 60 = = 30; p1 2
X2 = 0:4
U1 = X10:4 X20:6 = (30) 0
U1 = @
0:4 m 3
00
10:4 A
18
0:6 (150) 90 = = 45 p2 2 0:6
(45)
0:6 (m00 ) 2
= 38:26
(58)
(59)
0:6
= 38:26
(60)
0
m0=
(38:26) 30:4 20:6 = 176:39 0:40:4 0:60:6
CV = 150
176:39 =
(61)
26:39
(62)
Summarising the Money Metric Utility Changes Due to Taxes Table 3: Summary Fall in Price Rise + -
EV CV
of Equivalent and Compensating Variation in Price Fall in Price Basis of evaluation 22.51 New Price-Old Utility + -26.39 OLD Price- New Utility
Substitution E¤ect : 2.5 =10-7.6; Income e¤ect:7.6=22.5/3 and total e¤ect: 10. 2.1.3
Econometric issues in estimation of demand functions and elasticities of demand
Measure of elasticity di¤ers by the functional forms used to estimate it. With data on quantiy (Y ) and price (X) ; in brief these can be stated as follows: Elasticity around the mean values of X and Y in a linear regression model , Yi = 1 + 2 Xi +ei is @Yi X @Yi de…ned as e = @X = 2 is it obtained as e = 2 X Then given estimate of the slope @X . In a log i Y i Y @Yi X dependent variable linear regression model of the form ln (Yi ) = 1 + 2 Xi +ei e = @X Y = 2X i Y @Yi 1 because @Xi Yi = 2 : Similarly elasticity in a log explanatory variable linear regression model: @Yi X @Yi 1 1 Yi = 1 + 2 ln (Xi ) + ei is given by e = @X = 2 X1i X Y = 2 Y * @Xi = 2 Xi . Then elasticity i Y @Yi X in a double log linear regression model, ln (Yi ) = 1 + 2 ln (Xi ) + ei is e = @Xi Y = 2 XYi X Y = @Yi 1 1 Elasticity in a regression model linear in reciprocal of an explanatory 2 Xi . 2 * @Xi Yi = @Yi X @Yi 1 variable, Yi = 1 + 2 X1i + ei is given by e = @X = 2 X1i Y1i * @X = 2 X 2 : In a a quadratic i Y i i
@Yi X regression model, Yi = 1 + 2 Xi + 3 Xi2 + ei the elasticity is e = @X = ( 2 + 2 3 Xi ) X Y i Y @Yi 2 = + 2 X : How to decide which one these two choose? First, should depend on the * @X 2 3 i i optimisation functions discussed in this chapter. Secondly choice between linear and log-linear models should be econometric tests such as MacKinnon, White and Davidson test.
2.1.4
Restrictions in estimating a demand function:
Suppose that you are interested in estimating the demand for beer in a country and consider the following multiple regression model: ln (Yi ) =
0
+
1
ln (X1;i ) +
2
ln (X2;i ) +
3
ln (X3;i ) +
4
ln (X4;i ) + 'i
i = 1 :::N
(63)
where Yi is the demand for beer, X1;i is the price of beer, X2;i is the price of other liquor products, X3;i is the price of food and other services, X4;i is consumer income. Coe¢ cients 0 , 1 , 2 , 3 ,and 4 are the set of unknown elasticity coe¢ cients you would like to estimate. Again assume that errors 'i are independently normally distributed, 'i N (0; 2 ). Given non-sample information on the relation between the price and income coe¢ cients as following:
19
1. (a)
i. sum of the elasticities equals zero: 1 + 2 + 3 + 4 = 0: ii. two cross elasticities are equal: 3 = 4 = 0 or 3 - 4 = 0 iii. income elasticity is equal to unity: 5 = 1
F-test can be applied to test the validity of such restrictions as: F =
(Rb
0
r) [Rcov (b) R0 ] J
1
(Rb
r)
(64)
Here J = 3 is the number of restrictions 2
1 R=4 0 0
0 1 0
2 3 2 3 3 b 0 0 1 6 7 0 5 ; b = 4 b2 5 ; r = 4 0 5 b 1 0 3
(65)
Thus empirical test of consumer behaviour whether purchase of x1 and x2 are proportionate to the changes in the level of income or prices are measures by income and price elasticities of demand. These are empirically estimated using the cross section and time series data on quantities and prices. These data can be obtained from various organisations1 Utility e¤ect of price changes will be higher for the commodity that is heavily weighted in the consumer’s consumption basket. In real life households vary by their income and have good varieties in consumption bundles as: Table 4: Consumption of households by sectors in UK, 2008 Deciles agri Prod Constr Dist Infcom Finins Rlest Prfspp Ghlthed H1 435 10520 201 3688 1071 1541 3915 432 1835 H2 671 16224 310 5688 1651 2377 6037 666 2831 H3 854 20663 395 7244 2103 3027 7689 848 3605 H4 1037 25073 479 8790 2552 3673 9330 1029 4375 H5 1223 29583 565 10372 3011 4334 11008 1214 5161 H6 1407 34031 650 11931 3463 4985 12663 1397 5937 H7 1676 40532 775 14210 4125 5938 15083 1663 7072 H8 1977 47829 914 16769 4868 7007 17798 1963 8345 H9 2358 57037 1090 19997 5805 8355 21224 2341 9951 H10 3864 93463 1786 32767 9512 13692 34779 3835 16307 Note: Constructed from the ONS data.
Othrsrv 1448 2233 2844 3451 4071 4683 5578 6582 7850 12863
Rich households with more income can consumer more goods and services and enjoy more utility than poor households. Governments apply commodity taxes (VAT of 20%) and income taxes in order transfer some income from the richer to poorer households. Such transfer may reduce the gap between the income of rich and poor but it is very di¢ cult to imagine a society with perfect equality. 1 Such as Food and Agriculture Organisation: http://faostat.fao.org/ or from the Department for Environment, Food & Rural A¤airsas https://www.gov.uk/government/statistical-data-sets/commodity-prices. In general consult government department web pages to …nd such data at https://www.gov.uk/government/organisations or for many other links in http://www.hull.ac.uk/php/ecskrb/Confer/research.html.
20
Table 5: Benchmark production tax and prices by sectors Deciles Leisure Consumption Income share Consshare Income tax rate H1 2577 38163 0.0281 0.0627 0.0 H2 7451 52401 0.0433 0.0552 0.32 H3 14230 66740 0.0551 0.0624 0.32 H4 21877 80983 0.0669 0.0850 0.32 H5 28269 95550 0.0789 0.0966 0.32 H6 35535 109917 0.0908 0.1067 0.32 H7 41156 130916 0.1081 0.1078 0.32 H8 46294 154484 0.1276 0.1323 0.32 H9 54041 178551 0.1521 0.1409 0.40 H10 73363 292582 0.2493 0.1945 0.50 Note: Constructed from the ONS data. More detailed estimates of price, income and cross elasticities of demand can be estimated from the survey data such as food and expenditure survey, travel and tourism survey, multiple household survey including the understanding society dataset that can be obtained from the data archive or could be constructed from the ONS. McFadden Daniel (1963) Constant Elasticity of Substitution Production Functions, Review of Economic Studies, 30, 2, 73-83 McFadden Daniel and Paul A. Ruud (1994) Estimation by Simulation, Review of Economics and Statistics, 76, 4 , 591-608 Stone R (1954) “Linear Expenditure System and Demand Analysis: An Application to the Pattern of British Demand”, Economic Journal 64:511-527.
2.2
Exercise 1: Consumer’s problem
Q1. Consider a utility maximisation problem of a consumer with the CES utility function on goods x1 and x2 : max u = (x1 + x2 )
1
x1; x2;
(66)
subject to the budget constraint with prices p1 and p2 and income y: p1 :x1 + p2 :x2 = y
(67)
Derive Marshallian demand functions x1 and x2 for and indirect utility function u x1; x2; . Prove that the v(p; y) is homegenous of degree zero in p and y. prove that it is increasing in y and decreasing in p. Prove Roy’s identity for this problem. 21
Q2. Show above properties in the following CES utility maximisation problem: max u = (x1 + x2 )
1
(68)
x1 ;x2
Subject to M = p1 x1 + p2 x2 2.2.1
(69)
Marshallian demand functions from the CES preferences: 1
L (x1 ; x2 ; ) = (x1 + x2 ) + [M @L 1 = (x1 + x2 ) @x1
1
1 @L 1 = (x1 + x2 ) @x2
p1 x1
p2 x2 ]
(70)
1
x1
1
p1 = 0
(71)
1
x2
1
p2 = 0
(72)
@L = M p1 x1 p2 x2 = 0 @ The marginal rate of substitution bewteen x1 and x2 1
x1 x2
=
M = p1 x2
p1 p2
1 1
p1 p2
(74)
1
p1 p2
x1 = x2
(73)
1
(75)
M = p1 x1 + p2 x2 ' # 1 h 1 p1 + p2 x2 = x2 p1 + p 2 = x2 p 1 p2
(76) 1
+ p2
1
i
1
p2
1
(77)
Properties of the CES demand functions 1
x2 = h
Now get the value of
p1
1
+ p2
1
M p2
Value function:
x1 = h p1
20
1
1
+ p2
1
i
1
1
M p2
1
i 1
1
p1 p2
1
=h 1
(78)
0
M p1 p1
1
1
+ p2
1
1
i 1 31
1 1 B M p2 6B M p1 C C 7 v (x1 ; x2 ) = [email protected] h iA + @h iA 5 1 1 1 1 p1 + p2 p1 + p2
22
(79)
(80)
If M and p1 and p2 increase by t it does not change the value function: v (x1 (tM; tp1 ; tp2 ) ; x2 (tM; tp1 ; tp2 )) = v (x1 ; x2 ) : @v(x1 ;x2 ) @v(x1 ;x2 ) @v(x1 ;x2 ) < 0 and < 0. > 0 and @M @p1 @p2 Roy’s Identity: @v (p0 ;M 0 ) @Pi @v(p0 ;y 0 ) @M
xi p0 ; M 0 =
::::i = 1::m
(81)
(Marshallian demand for xi equal negative of the ratio derivative of IUF wrt price and income). Note that the derivative of value function wrt price equals derivative of Lagranging function wrt price and this equals negative of lagrange multiplier times the demand for the product as: @L @V = = @pi @pi 2.2.2
x1 (p1 ; p2 ; m):
(82)
Compensated and uncompensated demands
Consumer’s primal problem is to maximise utility (U ) subject to budget constraints. When optimal demands for xi are substituted in the utility function it becomes indirect utility function (V ). By Roy’s identity Marshallian demand for xi equals the negavite of the ratio of the …rst derivative of V wrt pi to its …rst derivative wrt income (M ). Consumer’s dual problem is to minimise the expenditure (E) wrt a target utility U . When optimal values of xi are substituted in E it becomes an expenditure function. The …rst derivative of the expenditure function wrt pi is equal to the compensated demand function xi : This means given E(p1 ; p2 ; U ) compesated demand is 1 ;p2 ;U ) xci = @E([email protected] : While the compensated demand gives the pure substitution e¤ect of price change i and the Marshallian demand minus the compensated demand equals the income e¤ect of price change. Inverse of indirect utility function is the expenditure function. 2.2.3
Expenditure functions with the CES utility functions min E = p1 x1 + p2 x2
(83)
x1 ;x2
Subject to u = (x1 + x2 ) L (x1 ; x2 ; ) = p1 x1 + p2 x2 + @L = p1 @x1
1
@L = p2 @x2
1
(x1 + x2 ) (x1 + x2 )
1
h
(84) u
(x1 + x2 )
1
i
(85)
1
1
x1
1
=0
(86)
1
1
x2
1
=0
(87)
Expenditure Functions with the CES utility functions @L =u @
1
(x1 + x2 ) = 0
23
(88)
p1 = p2
'
1
x2 = u
'
p1 p2
1
1
+1
#
1
1
1
+ x2 h = u p1
#1
1
1 1
= x2 p1
p1 p2
u = (x1 + x2 ) = x2
(89)
1
p1 p2
x1 = x2
1
x1 x2
1
p2
= x2
1
+ p2
(90)
'
p1 p2
i
1
1
+1
( p2
1
)(
1
#1
(91)
)
Expenditure Functions with the CES utility functions h x2 = u p 1
1
+ p2
1
i
1
1
1
p2
=h p1
Putting x2 in x1 1
x1 = x2 p 1
1 1
1
p2
h x1 = u p 1
2.3
1
h = u p1
+ p2
1
i
1
+ p2
1
1
i
p1
1
+ p2
1
1
p2
1
1 1
p1
(92)
i1 1
1
p2
1
(93)
1
1 1
1
u p2
1
=h p1
u p1 1
1
+ p2
1
i1
(94)
Slutskey equation ( Decomposition of substituion and income effects): Duality on consumer optimisation L = p1 x1 + p1 x2 +
h
1
U
1 21 1 12 @L = p1 x x2 = 0 @x1 2 1 1 21 12 1 @L = p2 x x =0 @x2 2 1 2 1 1 @L = u x12 x22 = 0 @ p1 x2 p2 = > x1 = x2 p2 x1 p1 1 2
1 2
u = x1 x2 =
x2 =
1 2
p2 x2 p1 p1 p2
1 2
x2 =
1 2
1
u = up12 p2 24
1
x12 x22
p2 p1 1 2
i
(95) (96) (97) (98) (99)
1 2
x2
(100)
(101)
Then x1 = x2
p2 = p1
1 2
p1 p2
u
1 1 p2 = up1 2 p22 p1
(102)
Now the expenditure fucntion 1
1
1
E = p1 x1 + p2 x2 = p1 up1 2 p22 + p2 up12 p2
1 2
1
1
= 2up12 p22
(103)
E
u=
1
(104)
1
2p12 p22 Slutskey Equation: Total e¤ect of price change = (substituion e¤ect + income e¤ect) @x1 = @p1
@x1 @p1
Cmp
@E @x1 @p1 @E
(105)
Compensated demand
1 2
1 2
x1 = up1 p2 =)
@x1 @p1
3 1 1 up1 2 p22 = 2
= cmp 1
1
E = 2up12 p22 =) Given the Marshalian demand x1 =
1 2
E 1 2
1 2
2p1 p2
!
3
1
p1 2 p22 =
1 Ep 2 4 1
(106)
1 1 @E = up1 2 p22 = x1 @p1
(107)
E 2p1
@x1 1 = @E 2p1
(108)
Slutskey decomposition: @x1 @p1
= =
@x1 @p1
Cmp
1 Ep 2 4 1
@E @x1 1 = Ep 2 @p1 @E 4 1 ! 1 1 E 1 p1 2 p22 1 1 2p1 2p12 p22
1
1
up1 2 p22 =
1 2p1
1 Ep 2 4 1
1 E Ep1 2 = 2 4 2p1
(109)
First part is substitution e¤ect and the second part is income e¤ect. If E = 800; p1 = 4 E 800 800 E substitution e¢ ect is - 4p 12:5 and the income e¤ect is also - 4p 2 = 2 42 = 4 4 4 = 2 = 800 4 4 4
1
1
800 2 42
=
= 12:5 . Both reinforce each other and total e¤ect is -25. Blundell R (2014) Income Dynamics and Life-cycle Inequality: Mechanisms and Controversies, Economic Journal, 124, 576, 289–318 25
Jehle G A and P.J. Reny (2005) Advanced Microeconomic Theory, Pearson Education. M. Hoy, J Livernois, C McKenna, R Rees and T. Stengos (2001) Mathematics for Economics, 2nd ed., MIT Press. 2.3.1
Comparative static analyis with matrix
Consider a consumer maximisation problem given below: M ax U (X; Y )
(110)
I = px X + py Y
(111)
X;Y
Subject to
Form a constrained optimisation problem and characterise the demand function X(px ; py ; I) and Y (px ; py ; I). L = U (X; Y ) + [I px X py Y ] (112) @L = I @
px X
@L = Ux @X
py Y = 0
(113)
px = 0
(114)
@L = UY py = 0 (115) @Y Following Henderson and Quandt (1980) take the total di¤erentiation of these FOCS. (Consumer takes prices of commodities as given, x and y are constant values of X and Y): 0:d
In matrix notation 2 4
0 px py
px dX
py dY = xdpx + ydpy
dI
(116)
px d + Uxx dX + Uxy dY = dpx
(117)
py d + Uyx dX + Uyy dY = dpY
(118)
px
py
Uxx Uyx
Uxy Uyy
32
3 2 d xdpx + ydpy 5 4 dX 5 = 4 dpx dY dpY
dI
3 5
(119)
Evaluating the impact of change in shadow price of income d on demand taking all else constant dpx = 0 and dpY = 0 2 32 3 2 3 0 px py d =dI 1 4 px Uxx Uxy 5 4 dX=dI 5 = 4 0 5 (120) py Uyx Uyy dY =dI 0 26
Similarly comparative static when only prices of X change dpx 6= 0 taking everything else is constant dpy = 0 and d = 0 2 32 3 2 3 0 px py d =dpx x 4 px Uxx Uxy 5 4 dX=dpx 5 = 4 5 (121) py Uyx Uyy dY =dpx 0 Similarly comparative static when only prices of Y change dpy 6= 0 taking everything else is constant dpx = 0 and d = 0 2 32 3 2 3 0 px py d =dpy y 4 px Uxx Uxy 5 4 dX=dpy 5 = 4 0 5 (122) py Uyx Uyy dY =dpy Each of these could be solved using the Cramer’s rule. For instance 2
3 2 d =dI 4 dX=dI 5 = 4 dY =dI
0 px py
px Uxx Uyx
py
3
Uxy 5 Uyy
1
2
3 1 4 0 5 0
(123)
Apply Cramer’s rule to …nd how much the shadow price and demand change in response to change in income; solve for d =dI; dX=dI; dY =dI
d =dI =
dX=dI =
dY =dI =
0 px py
1 px Uxx Uyx
0 px py
1 px Uxx Uyx
0 px py
1 px Uxx Uyx
py Uxy Uyy
py Uxy Uyy
py Uxy Uyy
1 0 0
px Uxx Uyx
py Uxy Uyy
=
2 Uxx Uyy + Ux;y 2px py Uxy p2y Uxx p2x Uyy
(124)
0 px py
1 0 0
py Uxy Uyy
=
py Ux;y px Ux;y 2px py Uxy p2y Uxx p2x Uyy
(125)
0 px py
px Uxx Uyx
1 0 0
=
px Ux;y py Ux;x 2px py Uxy p2y Uxx p2x Uyy
(126)
Once more to evaluate the comparative static impact of changes in prices of x, 2
3 2 d =dpx 4 dX=dpx 5 = 4 dY =dpx
0 px py
27
px Uxx Uyx
py
3
Uxy 5 Uyy
1
2 4
x 0
3 5
(127)
d =dpx =
0 px py
1 px Uxx Uyx
x py Uxy Uyy
0
px Uxx Uyx
py Uxy Uyy
=
2 xUxx Uy;y py Uy;x + px Uy;y: xUx;y 2px py Uxy p2y Uxx p2x Uyy
(128) Marshalian Demand Function dX=dpx
=
=
0 px py
1 px Uxx Uyx
xpy Ux;y 2px py Uxy
py Uxy Uyy
0 px py
x 0
py Uxy Uyy
p2y + xpx Uy;y: p2y Uxx p2x Uyy
(129)
(130)
Analysis of signs: Give that px > 0, py > 0, Ux;x: < 0, Uy;y: < 0; dpx > 0, dpy > 0„ the denomenator (determinant) is positive. Whether the numerator is positive depends on the sign of the numerator terms which denotes income and substitution e¤ects of changes in prices of X. Here dX=dpx < 0 because each term in numerator is negative and xpy Ux;y p2y + xpx Uy;y: < 0.
dY =dpx
=
=
0 1 px 0 px py py px Uxx Uxy py Uyx Uyy py px xpx Uyx + xpy Ux;x: 2px py Uxy p2y Uxx p2x Uyy
px Uxx Uyx
x (131) 0
(132)
This is exactly what would be expected, the inverse relation between demand for X and its own price. However the impact of a change in px on the demand for Y is not predictable o¤hand because the …rst term in the numerator, py px xpx U + xpy Ux;x , is positive but the last two terms are negative. Thus dpx would have positive impact only if py px > xpx U + xpy Ux;x . xpy Ux;y
p2 +xpx Uy;y:
Here 2px py Uxy p2yUxx p2 Uyy is the total e¤ect of change in prices. It can be decomposed into y x income and substitution e¤ect by deriving the compensated demand Hicksian demand function (dX=dpx )compensated =
2px py Uxy
xUxx p2y Uxx
p2x Uyy
x = y
1
1
1
1
)
(225)
+ (1
)
+ (1
) py
1
i1
(231)
u
y=
(1
1
)y ]
[ x + (1
(1
y
)y ]
[ x + (1
x y
1
[ x + (1
[ x + (1
@L =U @ px = py (1
U
)y ]
1
h
1
(1
)
1 1
1
px
1
+ (1
) py
1
(232)
i1
u
1
=
1
py
1
h
1
(1
)
Now it is possible to apply the Slutskey equation @x1 @x1 @E @x1 @p1 = @p1 @p1 @E Cmp
37
1 1
px
(1 1
+ (1
) py
1
i1
)
1 1
px py
(233)
1 1
2.4.8
Problem 4: CES Demand
1. Consider a consumer maximisation problem given below: M ax u(x; y) = [ x + (1
)y ]
x;y
1
(234)
Subject to x + py y = m Note that the elasticity of substitution and
(235)
are linked as:
=1
1
(a) Formulate a constrained optimisation problem . (b) Determine the demand functions of x and y. (c) Calibrate the share parameter . (d) Derive the indirect utility function (value function) u(x(p1 ; py ; m); y(p1 ; py ; m)) = V (p1 ; py ; m) (e) What is the meaning of
@V @m
=
? Marginal utility of income?
(f) Explain the meaning of Roy’s identity y(p1 ; py ; m).
@V @px
=
@L @px
=
x(p1 ; py ; m) and
@V @py
=
@L @py
=
2. Consider standard properties of a utility function (a) u(xi = 0) = 0 N (b) It continuous in R++
R+
(c) unbounded above for all prices p
0
(d) Homogenous of degree 1 in xi (e) Strictly increasing in income (f) Decreasing in prices (g) Quasiconvex in (p; m) (h) Ful…lls Roy’s identity Show above properties in the following CES utility maximisation problem max u = (x1 + x2 )
1
x1 ;x2
(236)
Subject to M = p1 x1 + p2 x2 3. Consider standard expenditure function with following properties (a) e = 0 for u(xi = 0) = 0 N (b) It continuos in R++
R+ 38
(237)
(c) unbounded above in u for all prices p
0
(d) Homogenous of degree 1 in p (e) Strictly increasing in income (f) Concave in p (g) ful…ls Shephard’s lemma
@E @pi
=
@L @pi
= xi (p1 ; p2 ; m)
for i = 1; 2
Show above properties in the following CES utility maximisation problem Subject to min E = p1 x1 + p2 x2 x1 ;x2
(238)
Subject to u = (x1 + x2 ) 2.4.9
1
(239)
Extra example on Shephard’s Lemma and Roy’s identity
1. Utility function for a consumer is given by u = x1 x2
(240)
I = p1 x1 + p2 x2
(241)
here budget constraint is
1. What are the Marshallian (uncompensated) demand functions for X and Y? 2. Determine the indirect utility function for this consumer. 3. Solving corresponding duality problem determine the expenditure function for this consumer. 4. Find the compensated (Hicksian) demand curve for X or Y? [hint Slutskey equation]. 5. Prove Shephard’s lemma
@E @pi
@L @pi
= xi (p1 ; p2 ; m) . i @V @L 6. Prove Roy’s identity for this case @p = @pi : i Answer Consumer’s Optimisation
=
h
L = x1 x2 + [m @L = x1 @x1
1
p2 x2 ]
(242)
x2
p1 = 0
(243)
1
p2 = 0
(244)
p 2 x2 = 0
(245)
@L = x1 x2 @x2 @L =m @
p 1 x1
p1 x1
39
Marshallian demand functions x1 = 2.4.10
Indierct utility function m p2
m p1
V (x1 ; x2 ) = 2.4.11
m m ; x2 = p1 p2
(246)
p1 p2
Expenditure function m=
p1 p2 u
@V @L = = @pi @pi
2.4.12
=m
(247)
x1 (p1 ; p2 ; m):
(248)
Duality L = p1 x1 + p1 x2 +
h
1
1
U
x12 x22
1 21 1 12 @L x = p1 x2 = 0 @x1 2 1 @L 1 21 12 1 x x = p2 =0 @x2 2 1 2 1 1 @L = u x12 x22 = 0 @ p1 x2 p2 = > x1 = x2 p2 x1 p1 1
1
u = x12 x22 = x2 =
x2
1 2
p2 p1
1
x22 =
1 2
p1 p2
1
u = up12 p2
p2 p1
i
(249) (250) (251) (252) (253)
1 2
x2
(254)
1 2
(255)
Then x1 = x2
p2 = p1
1 2
p1 p2
u
1 1 p2 = up1 2 p22 p1
(256)
Now the expenditure fucntion 1
1
1
E = p1 x1 + p2 x2 = p1 up1 2 p22 + p2 up12 p2 E
u=
1
1
2p12 p22 40
1 2
1
1
= 2up12 p22
(257) (258)
Slutskey Equation: Total e¤ect of price change = substituion e¤ect and income e¤ect @x1 = @p1
@x1 @p1
@E @x1 @p1 @E
Cmp
(259)
Compensated demand
1 2
1 2
x1 = up1 p2 =)
@x1 @p1
1 3 1 up1 2 p22 = 2
= cmp 1
1
E = 2up12 p22 =) Given the Marshalian demand x1 =
1 2
E 1 2
1 2
2p1 p2
!
3
1
p1 2 p22 =
1 Ep 2 4 1
1 1 @E = up1 2 p22 = x1 @p1
(260)
(261)
E 2p1
1 @x1 = @E 2p1
(262)
Slutskey decomposition: @x1 @p1
= =
@x1 @p1
Cmp
1 Ep 2 4 1
1 @E @x1 = Ep 2 @p1 @E 4 1 ! 1 1 E 1 p1 2 p22 1 1 2p 2 2 1 2p1 p2
1
1
up1 2 p22 =
1 2p1
1 Ep 2 4 1
1 E Ep 2 = 2 4 1 2p1
(263)
First part is substitution e¤ect and the second part is income e¤ect. 2.4.13 @E @pi
=
Shephard’s Lemma
@L @pi
= xi (p1 ; p2 ; m)
u(x1 ; x2 ) = x1 x2 where
+
= 1: L = p 1 x1 + p 2 x2 +
@L @u
h
u
x1 x2
i
(264)
= @L = p1 @x1
x1
@L = p2 @x2
x1 x2
@L =u @ p1 = p2
x2 = 0 1
=0
x1 x2 = 0 1
x2
x1 x2
1
x1
1
41
=
x2 x1
(265) (266) (267) (268)
Shephard’s Lemma p1 x1 = x2 p2 p2 x1 = x2 p1 p2 x2 p1
u = x1 x2 =
x1 =
p1 p2 x2
(272)
p1 p2 u =
1+
1
p1 p12 p1 p12
p1 p12
e=
1
u
p1 p12
e= 1
p1 p12
u
(276)
+1
(277)
+
u u
(274) (275)
u
p1 p12
u +
(273)
p1 p2 u
p1 p12
p2 u +
e=
e=
p1 p2 u
p1 p2 u + p2 p11
e=
(271)
p1 p2 u
e = p1 x1 + p2 x2 = p1
e=
(270)
x2 =
x2 = p2 p1
(269)
(278)
1
(279)
u=
p1 p2 u
(280)
Proof of Shephard’s Lemma @E @L = = xi (p1 ; p2 ; m) @pi @pi e=
p1 p2 u =)
@e = @p1
1
L = p1 x1 + p2 x2 + @L = x1 (p1 ; p2 ; m) = @p1 QED
42
p1 h
u
1
(281)
p2 u =
x1 x2
i
p1 p2 u
p1 p2 u
(282) (283) (284)
2.4.14
Roy’s Identity @V @p1
=
m m p1
= x1
1
p1
1
p2
=
=
m p1
p1 p2
x1 (p1 ; p2 ; m)
(285)
x2 = p 1 :
m p1
1
m p2
p1 1
=
x1 1 x2 = x1 p1
=
p1 p1
This is the proof of Roy’s identity.
43
1
x2 p 1 1 = (286)
2.5
Revealed Preference Theory
Utility function based analysis on derivation of consumer demand is subjective and hence less precise. Neither the utility function nor the preference parameters are observable. Revealed prefernce theory focuses on observed income, price and choices to generalise axioms on consumer behaviour. It makes a number of assumptions: 1) all income is spent (M0 = p0 x0 ) 2) buyers always have a unique x bundlde for given M and p; that means there is a unique bundle for each combination of p and M and preferences are consistent. 1 Weak axiom of revealed preference (WARP) : draw a diagram. 0
1
1
p1 x0 > p1 x1 () p0 x0 > p x1 p x1 < p x0 This will give the substitution and income e¤ect as in the indi¤erence curve analysis. But this does not apply in all circumstances; p1 x0 = M2 = p1 x2 . This leads to p1 p0 x1 x0 < 0 X and in general p1i p0i x1i x0i = p1j p0j x1j x0j < 0. 2. Strong axiom of revealed preference (SARP) 3. Congruent axiom 4. Generalised axiom The WARP and GARP axioms were re…need and generalised by Houthakker (1950), Richter (1966), Afriat (1967) Varian (1982), Bandyopadhyay (1988) and most recently by Echinique (2011). The Economic Journal 2012 has a special section on the recent developments of the revealed preference theory (Varian (2012), Vermeulen (2012)). 2.5.1
Slutsky equation from the Revealed Preference Theory
Following Gravelle and Rees (2004) now derive the Slutskey equation from the Revealed preference theory as follows:
Divide by
x1j
x0j = x2j
x1j
x0j
x0j + x2
x0j
x0j
x0j
pj
pj
=
x2j
pj
+
x2
pj
or x1j
x0j pj
=
xj jM = pj
x2j
x0j pj
xj jpx pj
x0j x0j
x2
x0j M
xj jp M
Thus the utility maximising theory and the revealed preference theory of consumer behavior are equivalent. Revealed preference to Laspeyeres price index: p1 x1 > p1 x0 () M I =
44
p1 x1 p 1 x0 > 0 0 = LP 0 0 p x p x
Revealed preference to Paasche price index: p0 x0 > p0 x1 ()
1 1 p 1 x1 p1 x1 5 () M I = 5 = PP p 0 x0 p0 x1 p 0 x0 p0 x1
Afriat (1967 and 2012) proves correspondence between the revealed preference and the utility function. Readings: Afriat, S. N. (2012), Afriat’s Theorem and the Index Number Problem, Economic Journal, 122: 295–304. Bandyopadhyay TD (1988) Revealed Preference Theory, Ordering and the Axiom of Sequential Path Independence, Review of Economic Studies, 343-351. Richter M. K. (1966) Revealed Preference Theory, Econometrica, 34, 3, 635-645 Samuelson, Paul A. (1938). “A note on the pure theory of consumer’s behavior,”Economica, 5(17), 61–71. — — . (1948). “Consumption theory in terms of revealed preference,” Economica, 15(60), 243–253 Varian, H. R. (2012), Revealed Preference and its Applications, Economic Journal, 122: 332– 338 Vermeulen F. (2012) Foundations ¤ Revealed Preference: Introduction, Economic Journal, 122 (May), 287–294 2.5.2
Further Developments in Consumer Theory Kahneman and Tversky (1979): Prospect theory Bandyopadhyay (1988):Revealed Preference Theory, Ordering and the Axiom of Sequential Path Independence Hey and Orme (1994): Experimental approach to consumer choice Blundell, Pistaferri, and Preston (2008), Blundell, and Preston (2008) Consumption and income inequality and Partial Insurance Balasko (1975) equilibrium manifold.
Emprical analysis of consumer demand Blundell R and I. Preston (1998) Consumption Inequality and Income Uncertainty, Quarterly Journal of Economics, 113,2, 603-640. Blundell R, L.Pistaferri, and I. Preston (2008) Consumption Inequality and Partial Insurance, American Economic Review, 98:5, 1887–1921
45
Carey K. (2000) Hospital Cost Containment and Length of Stay: An Econometric Analysis Southern Economic Journal, 67, 2, 363-380 Deolalikar A. B. and R. E. Evenson (1989) Technology Production and Technology Purchase in Indian Industry: An Econometric Analysis The Review of Economics and Statistics, 71, 4, 687-692 Hey J. D and J. A. Knoll (2011) Strategies in dynamic decision making: An experimental investigation of the rationality of decision behaviour, Journal of Economic Psychology 32,399– 409 Hey J. D and C. Orme (1994) Investigating Generalizations of Expected Utility Theory Using Experimental Data Econometrica, 62, 6, 1291-1326 Hey, J. D., Lotito, G., & Ma¢ oletti, A. (2010). The descriptive and predictive adequacy of theories of decision making under uncertainty/ambiguity. Journal of Risk and Uncertainty, 41(2), 81–111. Experimental lab of John Hey at York http://www.york.ac.uk/economics/ourpeople/sta¤-pro…les/john-hey/ Kahneman, D. and Tversky, A. (1979). Prospect theory: an analysis of decision under risk. Econometrica,67, 263–291. Lee R-S and N. Singh (1994) Patterns in Residential Gas and Electricity Consumption: An Econometric Analysis Journal of Business & Economic Statistics, 12, 2, 233-241 McGuinness T. (1980) Econometric Analysis of Total Demand For Alcoholic Beverages in the U.K., 1956-75 The Journal of Industrial Economics, 29, 1, 85-109 Segal, U. (1987). The Ellsberg Paradox and risk aversion: an anticipated utility approach. International Economic Review, 28, 175–202. Schmeidler, D. (1989). Subjective probability and expected utility without additivity. Econometrica, 57,571–587. Van Soest A and P. Kooreman (1987) A Micro-Econometric Analysis of Vacation Behaviour Journal of Applied Econometrics, 2, 3 , 215-226. 2.5.3
Emprical analysis
Explain whether following two estimations are as expected from the microeconomic theory of consumer demand presented in this section. 1) Consider the cross-regional variation of expenditure on food in the UK. For simplicity, it is assumed that food expenditure (F) depends only on wage and salary income (Y) in each region as: Fi;t =
i
+
1 yi;t
+ ei;t
ei;t
IID 0;
2 e
(287)
This model has been estimated using a pooled time series and cross section data set (with the sample size of T=14 and N=13) available from the web site of the O¢ ce of the National Statistics (food_exp_UK_regional_panel.csv: hhttp://www.statistics.gov.uk). The estimated coe¢ cients, by region, are given in the following table. 2) Determinants of houseprices by regions in UK estimated by 3SLS and data in HousePrice_regional.csv. 46
Table 6: Food expenditure on income : Stacking Data for SURE)l Coe¢ cient t-value t_Prob Emp_income 0.511 69.4 0.000 Constant -252.126 -2.18 0.031 NW -28.2405 -0.178 0.859 YH -362.599 -2.29 0.023 EM 359.178 2.27 0.025 WM 2034.03 12.8 0.000 EA 1715.26 10.3 0.000 GL 753.455 4.77 0.000 SE -700.345 -4.36 0.000 SE_R -326.693 -2.02 0.045 SW 412.537 2.61 0.010 WL 710.626 4.49 0.000 SCT 580.688 3.65 0.000 NI 2374.79 4.66 0.000 R2 = 0.99; N =182; T = 14; Chi2 =4815. [0.000] **
3
L3: Production: Supply
Production is a process to modify or transform inputs into outputs. Goods and services are produced by millions of …rms in the economy. Production functions show how much output is obtained for given combinations of inputs. Production technologies di¤er by production sectors and perhaps by the sizes of production …rms that can be from being small to midium to large or to multinational company. For instance consider types of soft drinks: Coke, Pepsi, Fanta, Tango, Sprite, 7 Up, Dr. Pepper or cars such as BMW, Voxhaul, Poeguet, Chrisler, Ford, GM, Toyota, Nissan, Hyundai, Fiat. There are varieties of daily use products supplied by many …rms. Many big companies are listed in stock markets such as FTSE 5000 or Nikkie or Dow Jones but there are many …rms operating in smaller scales. Most often these productions occur in sectors as recorded in the UNIDO databases. The agriculture sector,- that includes farms crops, livestock, forestry, and …sheries usually is operated by small …rms and requires more land. The mining sector is usually more capital intensive and dominated by larger …rms. Firms in manufacturing sector are mostly very big and require more physical and human capital. These vary a lot by the types of manufacturing that according to the Iput Output Table of UK from the ONS these include meat processing, …sh and fruit processing, oils and fats, dairy products, grain milling and starch and animal feed, bread, biscuits, etc, sugar, confectionery, other food products, alcoholic beverages, soft drinks and mineral waters, tobacco products, textile …bres, textile weaving, textile …nishing, made-up textiles, carpets and rugs, other textiles, knitted goods, wearing apparel and fur products, leather goods, footwear, wood and wood products, pulp, paper and paperboard, paper and paperboard products, printing and publishing, coke ovens, re…ned petroleum & nuclear fuel, industrial gases and dyes, inorganic chemicals, organic chemicals, fertilisers, plastics & synthetic resins etc, pesticides, paints, varnishes, printing ink etc, pharmaceuticals, soap and toilet preparations, other chemical products, man-made …bres, rubber products, plastic products, glass and glass products ceramic goods, structural clay products, cement, lime and plaster, articles of concrete, stone etc , iron and steel, non-ferrous metals, metal
47
Table 7: Determinants of houseprice in UK: SURE (3SLS) estimation Coe¢ cient t-value t_Prob Rincome 4.64 45.2 0.000 Pop 1.25 0.55 0.054 MRT_RT -11.51 -0.022 0.982 M/H_Ratio -237240 -19.9 0.000 CRNTDP 1.94 5.85 0.000 SVDEP 1.10 3.72 0.000 NE 22845.3 2.04 0.042 NW 8064.9 2.85 0.005 YH 14615.8 2.37 0.018 SW 9939.4 1.50 0.134 EN -148092 -1.66 0.097 EM 12868.1 1.61 0.108 WM 12404.2 2.20 0.029 EE 16599.7 2.84 0.005 GL 5454.8 2.00 0.046 Constant 101298.1 5.31 0.000 F(90, 373) = 7.09 (0.00); N =480; Chi2 (2)=59.2. [0.000] **
castings, structural metal products, metal boilers and radiators, metal forging, pressing, etc; cutlery, tools etc, other metal products, mechanical power equipment, general purpose machinery, agricultural machinery, machine tools, special purpose machinery, weapons and ammunition, domestic appliances, o¢ ce machinery & computers, electric motors and generators etc, insulated wire and cable, electrical equipment, electronic components, transmitters for TV, radio and phone receivers for TV and radio, medical and precision instruments, motor vehicles, shipbuilding and repair, other transport equipment, aircraft and spacecraft, furniture, jewelry and related products Sports goods and toys, miscellaneous manufacturing & recycling that rely very much on fossil fuels. The production and distribution of electricity and gas is vital in order to run all these industries. Firms in construction sector contribute both to the supply of residential and non-residential properties but also to big infrastructure in the form of transport, communication and service networks and logistics and supply chain management. The distribution sector consists of motor vehicle distribution and repair, automotive fuel retail, wholesale distribution, retail distribution, hotels, catering, pubs etc. The business service sector represents banking and …nance , insurance and pension funds , auxiliary …nancial services, owning and dealing in real estate, letting of dwellings, estate agent activities, renting of machinery etc, computer services, research and development , legal activities, accountancy services, market research, management consultancy, architectural activities and technical consultancy, advertising and other business services. The other services sector includes public administration and defence, education, health and veterinary services, social work activities, membership organisations, recreational services, other service activities, private households with employed persons and sewage and sanitary services. Datastream provides basic timeseries information on public companies regarding their production, sales, revenue, costs, pro…ts, price of stocks and their leverages. Megazines such as Forbes regularly
48
Top 10 Firms in the World in May 2014 (Forb's List): http://www.forbes.com/global2000/list/ Rank
Company Country
1
ICBC
2 3
China Construction Bank Agricultural Bank of China
4
JPMorgan Chase
5
Berkshire Hathaway
6
Exxon Mobil
7
General Electric
8
Wells Fargo
9
Bank of China
10
PetroChina
Sales
Profits
Assets
Market Value
China
$1 48.7 B
$42.7 B
$3,1 24.9 B
$21 5.6 B
China
$1 21 .3 B
$34.2 B
$2,449.5 B
$1 7 4.4 B
China
$1 36.4 B
$27 B
$2,405.4 B
$1 41 .1 B
United States
$1 05.7 B
$1 7 .3 B
$2,435.3 B
$229.7 B
United States
$1 7 8.8 B
$1 9.5 B
$493.4 B
$309.1 B
United States
$394 B
$32.6 B
$346.8 B
$422.3 B
United States
$1 43.3 B
$1 4.8 B
$656.6 B
$259.6 B
United States
$88.7 B
$21 .9 B
$1 ,543 B
$261 .4 B
China
$1 05.1 B
$25.5 B
$2,291 .8 B
$1 24.2 B
China
$328.5 B
$21 .1 B
$386.9 B
$202 B
update top 100 companies by thier turn over. For instance Industrial & Commercial Bank of China Ltd. had assets of 3.1 trillion dollars in 2014 with pro…ts over 148 billion dollars. It is also possible to …lter prominent …rms in each of above sectors operating in the global economy using such databases. A production technology in each of above sector shows how inputs are transferred into outputs. Usually labour, the human toils and trouble in process of production; capital, the man made means of production, as re‡ected in building, structures including highways, communication networks and education, health and environmental system; natural resources including clear air, water, and mineral and energy products represent such inputs. In addition there are intermediate inputs as presented in the input output table of an economy. There are linear and non-linear production functions. Returns to scale vary and possibility of substitution vary among them . Intensity of use of these factors in a speci…c industry or a …rm is re‡ected in these production function. These are important in process of substitution of more expensive by less expensive inputs. The CES categories of these functions being the most commonly used ones in the economic literature as they capture the cross price elasticity more e¢ ciently than any other linear or Cobb-Douglas production functions [see articles such as (Pigou (1934) , Meade (1934) Lancaster and Chesher (1983), Dolton and Makepeace (1990), Harmatuck (1991), Basu and Fernald (1997), Barmby , Ercolani and Treble (2002), Costinot, Vogel and Wang (2013)) discussion about production; Coase (1937) The Nature of the Firm, Economica, 386-405]. 3.0.4
Popular production functions
Popular production functions where output ( y) is expressed as functions of inputs (xi ): Cobb-Douglas: y = x1 x12 CES: y = (x1 + x2 )
1
Nested: x4 = (x1 + x2 )
1
and then y = x4 x13 49
generalised Leontief: Y =
n P n P
p aij xi xj ;
aij = aji
i=1j=1
Translog: ln Y = a0 +
n P
ai ln xi +
i=1
n P n P
aij ln xi ln xj ;
aij = aji
i=1j=1
A tanslog production function adds squares and product terms to the regual production function as: ln Y = a0 +
n X
ai ln xi +
i=1
n n X X
aij ln xi ln xj ;
aij = aji
i=1 j=1
This function is popular as it allows a large number of substitution posibilities among inputs. n n P P Prove that this function becomes a constant return to scale when ai = 1 and aij = 0: i=1
j=1
Generalised Leontief function:
Y =
n n X X
p aij xi xj ;
aij = aji
i=1 j=1
Nested production function shows how composite inputs are used with other inputs (very popular in the CGE and macro modelling): Let V be the CES composite of labour and capital V = [ L + (1
)K ]
1
(288)
then let E be energy input in production. Then Y is prouced using V and E as: Y = V E1 This is one level nest. There can be many levels of nests in the production process. Questions: What are the elasticities of output to input x1 in above production functions?
3.1
Supply function: an example
1. Let us consider a production function for a fruit …rm operating in the competitive market is given by p y=2 l (289) where y is output and l is labour input. Product price is p and input price is w. What is the cost function for this …rm? What is its pro…t function? What is its supply function? What is the demand function for labour? What are the properties of the these production, pro…t and cost functions? Since this is a one input production funtion the cost function can derived direcly from the production technology as: l=
50
y2 4
(290)
Producer pay wage to supply this commodity: y2 (291) 4 The pro…t is the di¤erence between the revenue and cost of the …rm as given by the pro…t function: c = wl = w
y2 (292) 4 The supply function for commodity y is derived using the …rst orcer condition of the pro…t function as: = py
@ =p @y
c = py
w
w
y 2p = 0 =) y = 2 w
(293)
Supply is positively related to prices and negatively to the input cost, in this case the wage rate. Demand for labour: l= 3.1.1
1 2 1 y = 4 4
2p w
2
(294)
Properties of a pro…t function
1. Increasing in p 2. decreasing in w 3. homogenous of degree one in p and w 4. concave in y and convex w These properites satisfy in this example: This supply function is homegeous of degree zero in price and wage, y = 2p w as there is no change in level of output when price and wage increase by the same amount. It is increasing in p and decreasing in w. 2 Pro…t function is concave as its second derivative wrt to output is negative, @@y2 = w2 < 0; 1
2
@ y 2 =) Production function is also concave. @y @l = l @l2 = 2 y @c @ c w Cost function is convex: @y = w 2 =) @y2 = 2 > 0.
1 2l
3 2
< 0:
Demand function for labour is also homegeous of degree zero in price and wage as l =
1 4
2p 2 . w
Hotelling’s lemma Derivative of pro…t function wrt price gives the supply function; derivative pro…t function wrt input prices gives demand function for inputs: @ (p;y) @ (p;w) = y (p; w) = xi (p; w) @p @wi Properties of output supply and input demand functions 1. Homogeniety of degree zero y (tp; tw) = y (p; w) for all t > 0 2. xi (tp; tw) = xi (p; w) for all t > 0 See substituion matrix 51
3.1.2
Production function and its scale properties
A production function f : Rn+ =) R is continuous, strictly increasing, strictly quasiconcave function in Rn+ f (0) = 0 Isoquant Q(y) = fx 0 = f (x) = yg is set of inputs giving a …xed output y. Returns to scale 1. Constant return to scale
f (tx) = tf (x) for all t > 0 and all x
2. Increasing returns to scale f (tx) > tf (x) for all t > 0 and all x 3. Decreasing returns to scale f (tx) < tf (x) for all t > 0 and all x Elasticity of scale of the production function at point x Xn f (x) xi d ln (f (tx)) i=1 = (x) = lim t !1 d ln (t) f (x) 3.1.3
(295)
Variable returns to scale y = k(1 + x1 x2 )
1
(296)
1
(x) =
@y x1 = (1 + x1 x2 ) @x1 y
2
(x) =
@y x2 = (1 + x1 x2 ) @x2 y
1
x1 x2
(297)
1
x1 x2
(298)
Elasticity of scale is obtained by adding above two: (x) = ( + ) (1 + x1 x2 )
1
x1 x2
(299)
This varies with x. Variable returns to scale y = k(1 + x1 x2 )
1
; =) x1 x2
=
k y
1
(300)
1
(y) = (1
y ) k
(301)
2
(y) = (1
y ) k
(302)
Elasticity of scale y ) (303) k Returns to each input declines with output here. Increasing return for 0 < y < k; constant return 0 < y = k and decreasing return when y > k > 0 . k is the upper bound of output. (see more in Jehle and Reny (2001). (y) = ( + ) (1
52
3.1.4
Cost function 0 and output levels y 2 Rn+
It is a minimum value function for for all input price w c(w; y) = min w:x
(304)
subject to f (x)
y
(305)
Constrained optimisation L (x; ) = w:x + [y @L (x; ) = w1 @x1
f (x)]
(306)
f 0 (x1 ) = 0
(307)
:: @L (x; ) = wn @xn y
(308) f 0 (xn ) = 0
f (x) = 0
(309) (310)
Marginal Rate of Trasformation M RT Sj;i =
@f (x;) @xj @f (x;) @xi
=
wj wi
(311)
Some studies on cost and production: Barmby T. A., M. G. Ercolani and J. G. Treble ( 2002) Sickness Absence: An International Comparison Economic Journal, 112, 480, F315-F331 Basu S and J. G. Fernald (1997) Returns to Scale in U.S. Production: Estimates and Implications, Journal of Political Economy, 105, 2, 249-283 Benabou R. and Tirole, J. (2010), Individual and Corporate Social Responsibility. Economica, 77: 1–19. Bloom N., R. Sadun and J. van Reenen (2012) The organization of …rms across countries, Quarterly Journal of Economics 127 (4), 1663–1705. Costinot, A. and J. S. Vogel and S. Wang (2013), An Elementary Theory of Global Supply Chains, Review of Economic Studies 80, 109–144 Dolton P.J. and G. H. Makepeace (1990) The Earnings of Economics Graduates The Economic Journal, 100, 399, 237-250
53
Harmatuck D.J. (1991) Economies of Scale and Scope in the Motor Carrier Industry: An Analysis of the Cost Functions for Seventeen Large LTL Common Motor Carriers, Journal of Transport Economics and Policy, 25, 2 , 135-151 Lancaster T and A. Chesher (1983) An Econometric Analysis of Reservation Wages Econometrica, 51, 6,1661-1676 Meade (1934) The Elasticity of Substitution and the Elasticity of Demand for One Factor of Production The Review of Economic Studies, 1, 2 ,152-153 Panagariya A (1981) Variable Returns to Scale in Production and Patterns of Specialization, The American Economic Review, 71, 1, 221-230 Pigou A. C. (1934) The Elasticity of Substitution ,The Economic Journal, 44, 174 , 232-241 Tirole J. (1995) The Theory of Industrial Organisation, MIT Press.
3.2
CES and Cobb-Douglas production functions
Cobb-Douglas Production Function Y = AK L1
(312)
CES Production Function 1
Y =A
K
+ (1
)L
(313)
Prove that the elasticity of substitution is 1 in Cobb-Douglas production function and = 1+1 in the CES production function. Also prove that the Cobb-Douglas is a special case of the CES production function. Proof that = 1 in the Cobb-Douglas production function =
d d
K L w r
= =
K L w r
K L
d
= d
(1
K L (1
=
)AK L AK 1 L1
=
=
)AK L AK 1 L1
d d
K L K L
= =
K L K L
=1
(314)
For the CES production function @Y = @K
1
@Y = @K
1
A
K
1
+ (1
)L
1
(
1
A
K
+ (1 dK = dL
)L YL = YK
YL (1 = YK
)L
1
=
) K
1
=
) (1
1
(
(1
)
54
)
K L
K L
(1
) A1+ A
A1+ A
Y K
Y L
1+
(315)
1+
(316)
1+
0; and diminishes when
Hotelling’s Lemma
Derivative of pro…t function wrt output price gives the supply function; derivative pro…t function wrt input prices gives demand function for inputs: y (p; w) =
@ (p; w) @p
(378)
xi (w; p) =
@ (p; w) @w
(379)
62
Pro…t function = py
rK
wL
(380)
Constrained pro…t optimisation problem: = (x; ) = py
rK
wL +
@= (x; ) =p @y @= (x; ) = @L @= (x; ) = @K
K1 (1
K1
L
y
=0 L
)K
1
L
(381)
(382) w=0
(383)
r=0
(384)
@= (x; ) = K1 L y=0 (385) @ Solving these Jacobians one will …nd input demands as L(r; w; p) and K(r; w; p) and the level of output y(r; w; p). Insert these functions into the pro…t function to get the value function as: = py(r; w; p)
rK(r; w; p)
wL(r; w; p) = V (r; w; p)
(386)
Applying the envelop theorem: @V @= = = y(r; w; p) @p @p @= @V = = @w @w
(387)
L(r; w; p)
(388)
@V @= = = K(r; w; p) (389) @r @r Di¤erentiating the value function with respect to price gives the output supply function and di¤erentiating wrt input prices gives input demand functions of the …rm. This is Hotelling’s Lemma. Repeat the same process to technology y = K 0:4 L0:4 as: = (x; ) = py
rK
wL +
K 0:4 L0:4
y
(390)
Hotelling’s Lemma like the Shephard’s lemma is very useful for solving Duals of consumer and producer optimisation problems.
63
3.3.6
Exercise 7: Minimising the cost with Cobb-Douglas and CES production function
1. Consider cost of production of a …rm: C = wK + rL
(391)
and its production technology constraint y=K L
(392)
1. (a) Write the pro…t function for this form (b) Write a Langrangian of pro…t subject to technology constraint (c) Determine the optimal demand for inputs (d) Derive the pro…t function in terms of optimal inputs , V (p; w; r): (e) Determine the cost function. (f) Prove Hotelling’s lemma K(p; w; r):
@V @P
=
@L @P
= y(p; w; r); @V @w =
@L @w
=
L(p; w; r); @V @r =
@L @r
=
(g) Derive input demand, output supply and pro…t functions when the technology is y = K10:4 L0:4 1 2. Derive the short run pro…t function for a …rm under the perfect competion = PY
wL
rK
(393)
with production technology Y =L K
(394)
1. Here is pro…t, L labour supply, K capital, w wage rate, P price of good, Y ouput . For simplicity assume capital is …xed at K. Technology operates under the constant returns to scale + = 1: (a) Derive supply function of the …rm using Hotelling’s Lemma. (b) What are the price and output of this …rm in the short run. Prove that …rm earns positive pro…t in the short run taking = 0:5; w = 4; r = 1; k = 1: (c) Now assume that the market demand is given by p = 39 0:009q and the pro…t function is given by = p2 2p 399: Find output, market demand and the number of …rms in the long run ( = 0) (d) Why is the number of …rms indeterminate in the perfect competion? 3. A …rm’s objective is to minimise cost C = rK + wL
64
(395)
subject to technology constraint as: Y = [ L + (1
)K ]
1
(396)
1. (a) Determine the demand for labour and capital. (b) Derive the cost function of the …rm. (c) Prove that the elasticity of substituion is
=
1
1:
(d) Discuss propperties of CES cost function. Consider a cost minimisation under the perfect competition: A …rm’s objective is to minimise cost C = rK + wL
(397)
subject to CES technology constraint as: Y = [ L + (1
)K ]
1
(398)
1. (a) Determine the demand for labour and capital. (b) Derive the cost function of the …rm. (c) Prove that the elasticity of substitution is
1
=
1:
(d) Discuss properties of CES cost function. Answer Comparative static: Derivation of the CES cost function. Lagrange: h L = rK + wL + Y ( L + (1
)K )
Taking the …rst order conditions: @L =w @L 1h
@L =r @K
1h
( L + (1
( L + (1
)K ) )K )
@L = Y ( L + (1 @ From the …rst two …rst order conditions: w = r
1
L K
1
;K
1
=
r w
L
(1
1
) K
=0
(400)
1
(401)
=0
1
Now put the value of K in the production function: 65
i
i
(399)
)K ) = 0
(L)
1
1
1
i
1
1
;K =
(402)
r w
1
1
1
1
L
(403)
Y = ( L + (1
Y =
(
Y =
8 < :
Y = demand for labour
L=
L + (1
)
8
0 p1 z1 (p) + p2 z2 (p) + ::: + pm 1 zm 1 (p) + pm zm (p) = 0 Represent equilibrium allocations of production, endowment and consumption in a single diagram.
5.3
Pure exchange general equilibrium model
Two Good Pure Exchange General Equilibrium Model Households, h = A B. Two goods X1 and X2 A B B Endowments of two goods ! A 1 !2 !1 !2
Objective of each is to maximise life time utility subject to budget constraints w r t X1 and X2 Equilibrium relative price determines the optimal allocation; It is Pareto Optimal. Main Features of Applied General Equilibrium Model Three conditions 1. Demand = supply ; n markets, n-1 relative prices 2. Income = expenditure 129
+
p1 2
p i =2 2 =
3. Firms maximise pro…t: zero economic pro…t in competitive markets Relative Prices 1. Preferences and technology parameters determine relative prices in 2. equilibrium. 3. Relative prices are determined by forces of demand and supply. 4. Numeraire or anchor price; normalised to 1. Markets allocations depend on relative prices. 1. Demand for a commodity depends on preferences and income. 2. Income of a household is determined by her endowment and price of that endowment. Exchange or trade of goods is mutually bene…cial. Each consumer/ producer optimises in equilibrium. Problem of representative households For household A M ax
U (X1A ; X2A ) = X1A
A
X2A
1
A
(846)
Subject to the budget constraint: A A P1 X1A + P2 X2A = P1 ! A 1 + P2 ! 2 = I
(847)
For household B M ax
U (X1B ; X2B ) = X1B
B
X2B
1
B
(848)
Subject to the budget constraint: B B P1 X1B + P2 X2B = P1 ! B 1 + P2 ! 2 = I
(849)
Intertemporal budget constraint Lagrangian for constrained optimisation for Household A : LA = X1A
A
X2A
1
A
+
A P1 ! A 1 + P2 ! 2
P1 X1A
P2 X2A
(850)
P1 X1B
P2 X2B
(851)
Lagrangian for constrained optimisation for Household V : LB = X1B
B
X2B
1
B
+
B P1 ! B 1 + P2 ! 2
First order conditions for optimisation For household A and B
130
@LA = @X1A
A
@LA = (1 @X2A
1
X1A
A
A)
X1A
1
X2A A
A
X2A
A
@LA A = P1 ! A 1 + P2 ! 2 @
P1 X1A
@LB = @X1B
X2B
B
@LB = (1 @X2B
1
X1B
B
B)
X1B
B
@LB B = P1 ! B 1 + P2 ! 2 @ Demand and market clearing conditions For household A AI
X1A =
A
P1
1
B
(1
;
X2B =
(1
P1 = 0 B
P1 X1B
X2A =
P2 = 0
P2 X2A = 0
X2B
;
P1 = 0
P2 = 0
P2 X2B = 0
A) I
(852) (853) (854) (855) (856) (857)
A
(858)
P2
For household B BI
X1B =
B
P1
B) I
B
P2
(859)
Market clears each period B X1A + X1B = ! A 1 + !1
(860)
B X2A + X2B = ! A 2 + !2
(861)
Market clearing Prices Obtained from the market clearing conditions AI
A
P1 (1
A) I
P2
BI
+ A
+
B
P1 (1
B = !A 1 + !1 B) I
B
P2
B = !A 2 + !2
(862) (863)
Walrasian numeraire: P1 = 1: with this speci…cation I A = !A 1
I B = P2 ! B 2
Equilibrium Relative Price and Proof of Walras’Law
131
(864)
Table 30: Parameters in Pure Exchange Model Household A Household B A B A Endowments ! 1 ; ! 2 = f100; 0g !1 ; !B = f0; 200g 2 Preference for X1 ( ) 0:4 0:6 Preference for X2 (1 ) 0.6 0.4
AI
P1
A
+
BI
B
P1
0:4 (100) + 0:6 (200) P2 I A = !A 1 = 100
=
AI
A
+
BI
=
A A !1
+
B B P2 ! 2
=
100;
B
= !A 1
P2 = 0:5
I B = P2 ! B 2 = 0:5 (200) = 100
(865)
Table 31: Parameters in Pure Exchange Model Household A Household B B A B Endowments !A ! ; ! = f100; 0g = f0; 200g 1 1 ; !2 2 Prices 1 0.5 Demand for X1 ( ) 40 60 Demand for for X2 (1 ) 120 80 Utility 77.3 67.3 Income 100 100 Theoretical observations Relative prices of goods, income and consumption change when preferences ( alpha, beta) change. Change in the relative income a¤ects the level of utility and welfare of households Household A can make household B worse o¤ by increasing the demand of good 1 that he owns ( or supplying less to the market). Household B can increase his relative income and reduce the relative price of good 1 by increasing the demand for good 2 (reducing its supply). Relative prices and allocations depend on preferences and endowments. Homework Do sensitivity analysis (solve model for various parametric speci…cations A B 1) by changing endowments ! A = f100; 50g and ! B = f200; 150g 1 ; !2 1 ; !2 2) by changing preferences f A , B g = f0:50; 0:50g ; f A , B g = f0:750; 0:30g 3) introduce VAT of 20 percent in commodity 1. Assume that revenue collected is spent entirely by the government and does not add to any utility for the household. GAMS programme: pexchange.gms; proto.gms; tax2sector.gms; tax_3sectr_utils.gms 132
5.3.1
Exercise 11: Ricardian trade model
1. Develop the above model for two countries and …nd the global equilibrium in two country world. 2. Show that relative output of country 1 not only in domestic costs but also the output and cost in the foreign country. 3. Represent the global economy by three representative countries with the following problem max Ui = X1;i X2;i X1;i
(866)
Ii = p1 X1;i + p2 X2;i + p3 X3;i
(867)
subject to
Endowments of each country respectively is f(! 1 ; 0; 0) ; (0; ! 2 ; 0) ; (0; 0; ! 2 )g Solve for prices and allocations at equilibrium. Apply this model to a realistic situation for countries producing food, oil and manufacturing commodities.
5.4
Simplest general equilibrium model Households and …rm optimise subject to their constraints – Utility maximisation by households and pro…t maximisation by …rms System of prices when all markets clear simultaneously (all goods and factor markets) D (p1 p2 p3 ; ::::pn ) = S (p1 p2 p3 ; ::::pn )
(868)
Excess demand is zero in equilibrium. Income of agents equals their expenditure Imports equals exports in an open economy model Saving equals investment in a dynamic economy model Public spending accounts are balanced in model with public sector In general equilibrium is obtained by the price system when economy is in perfect harmony. Consider one of the easiest possible example of a general equilibrium model with production Exercise Prove that a …rm need to pay higher wage rate to its workers and lower the price of commodity while expanding output if it operates under an increasing returns to scale technology such as Y = L2 .
133
5.5
General equilibrium with production Bhattarai K. (2006) Macroeconomic Impacts of Taxes: A General Equilibrium Analysis, Indian Economic Journal, 54:2:95-116.
A simple general equilibrium model represents an economy in which a representative household maximises utility by consuming goods and services supplied by producers and paying for them by income that it receives in return of supply of labour and capital inputs to the producers. Firms optimise pro…t combining inputs with the existing technology in production and rewarding inputs according to its marginal productivity. Tax policies of government in‡uence both production and consumption sides of the economy by a¤ecting prices of inputs and outputs. By distorting the marginal conditions of optimisation, these taxes in‡uence choices of goods and services by households and use of inputs by producers. The incidence and impact of taxes on consumption may di¤er from taxes on labour income depending on the key parameters for share or elasticities of substitution in consumption or in the production sides of the economy. A general equilibrium implies a set of prices that eliminate excess supply or excess demand and where these prices and wage rates are consistent with the preferences and endowments of households and technology of …rms. The perfect match between demand and supply for both goods and services and inputs of production follow from the properties of utility and production functions as given by explicit analytical solutions in the next section. Consider an economy consisting of a representative household and a representative …rm. A representative household tries to maximise utility by consuming goods and services and from enjoying leisure subject to his budget constraints. max U = C l(1
)
(869)
Subject to time and budget constraints: l + hs = 1
(870)
pc = whs +
(871)
c > 0; h > 0;and l > 0; where c is consumption,l is leisure and hs is labour supply, p is the price of the commodity, w is the wage rate is the pro…t from owning the …rm. General equilibrium question: problem of the …rm The producer wants to maximise pro…t by selling goods produced using the labour supplied by the household. Problem of the …rm s
= py
whd
(872)
subject to technology constraint as: y = (hs ) y > 0; hd > 0; where y is the output supplied by the …rm and hd is its demand for labour.
134
(873)
Lagrangians for constrained optimisation For the houshold: (1
(1
)
hs )
L (C; L; ) = C (1
+ [whs +
pC]
(874)
First order conditions here given demand for consumption and leisuer C; L , supply of labour l) and shadow price For the …rm: = p (hs )
whd
(875)
First order condition from this will give the demand for labour as a function of real wage rate. First order conditions for the household: @L (C; L; ) = C @C @L (C; L; ) = (1 @hs
1
(1
(1
hs )
) C (1
hs )
)
p=0
( 1)
@L (C; L; ) = whs + @
w=0
pC = 0
(876) (877) (878)
From above two (1
) C (1
hs )
1
(1 hs )
C
(1
( 1) )
=
w p
(879)
First order conditions for households: C=
(1
)
hs )
(1
w p
(880)
from the budgent constraint and this FOC w s h + =C= p p (1
)
hs )
(1
w p
(881)
Supply labour hs =
w p
(1
)
p
(882)
w p
Demand for leisure L=1
w p
(1 w p
First order conditions for households: Demand for consumption
135
)
p
(883)
C=
(1
)
w h ) = p (1 s
(1
w p
1
)
(1
)
p
w p
!
w p
(884)
Demand for consumption and leisure and supply of labour are thus functions of Real wage rate
w p
Preference for consumption ( )and leisure (1
)
pro…t of the …rms First order conditions for …rms @ = p hd hd h =
p
w d h = p ' 1 1
=
(h )
=
w p
w=0
(885)
1
1w p
d
s
1
1
(886)
1w p
w p
1
1
1
1
1
#
1
1w p
1
(887)
Equilibrium real wage the equilibrium in the labour market is determines the real wage rate h = 1
1 w p
1
=
( wp )
1
(1)
(1)
1
1
1w p
d
1
1
1
(1
) wp
(1
p
= hs =
(888)
w p
)w p
w p
w =h p
1
(1
)
1
1
1
1
1
i
(889)
1
Optimal labour supply and leisure demand the equilibrium in the labour supply is the function of the real wage rate 0
and the leisure
B1 hd = @ h
1
(1
)
1
1
136
1
1 1
i
1
C 1A
1 1
(890)
l=1
0
B1 @ h
h=1
1
1
(1
1
)
1
1
1
1
1 1
C 1A
i
(891)
Optimal labour output and consumption the equilibrium in the labour supply is the function of the real wage rate 0
B1 y = (h ) = @ h s
(1
)
1
1
1
1 1
and optimal consumption
C
=
(1
(1
h 5.5.1
(1
hs )
(1 ) 0
B B1 )@ )
1
1
1
i
(892)
w = p
0
1
B1 @ h (1
1
)
1
1
1
1
C 1A
1
1 1
i
1
1 1
i
1 C
1A
1
1 1
1 C C A (893)
A numerical example for the general equilibrium tax model
A simple numerical example is provided here for the above general equilibrium tax model and used it to measure the relative impacts of consumption and income taxes in the economy. These impacts vary according to the preferences of households in relation to technology of production available to …rms. If households strongly prefer leisure rather than consumption, the costs of income taxes are likely to be higher than those of the consumption taxes. On the other hand if the households have stronger preferences for consumption than for leisure then consumption taxes might be costlier. As stated above these preference and technology factors jointly determine the prices and prices in‡uence allocation of resources in the economy. Numerical values of model parameters are as follows: Time endowments of 68 ours per week is used here as is customary in the literature. Two types of simulations are conducted using this model. The …rst one is the base case scenario constructed using a reasonable set of preference and technology parameters as givenabove . Then there is a no tax scenario where all taxes are eliminated, followed by scenarios with taxes either only on consumption or only on labour income. The impacts of tax policy experiments are determined by comparing the utility and changes in output and employment before and after the change in taxes. In addition sensitivity analyses are conducted to see how the welfare and macroeconomic impacts change in response to a distribution of households by preference and that of …rms by the production technology. The grids of parameters for sensitivity analyses to study the impacts of taxes are measured in terms of levels and changes in utility, output, leisure, labour supply and consumption as: 137
Table 32: Parameters of the model in the base scenario Parameter of the model Numerical value in the base model Utility weight on consumption ( ) 0.6 Utility weight on leisure (1 ) 0.4 Elasticity of output to labour input ( ) 0.6 Value of Endowment L 68 hours Base consumption tax rate (tc ) 0.17 Income tax rate in the base case (tl ) 0.35 Time endowments of 68 ours per week is used here as is customary in the literature. Normalisation of price , w + p = 1 Table 33: Parameters of the model in the base scenario Use of both consumption and income taxes and lump sum transfers (base case) Elimination of all taxes and no transfer Only labour income tax and lump sum transfers Only consumption tax and lump sum transfers The benchmark economy includes consumption tax rate of 17 percent and income tax rate of 35 percent, similar to the one that actually exists in many of the OECD countries including the UK. In all experiments the government returns tax revenue to the household in the form of a lump sum transfer. The model is then used to evaluate the impacts of four di¤erent tax reform experiments: (1) the distortionary cost of both income and consumption taxes; (2) impact of a complete switch to the labour income tax holding the revenue …xed; (3) impact of a complete switch towards the consumption tax; and (4) the test of reliability and robustness of the model by examining the sensitivity to the key parameters of the model. The Hicksian equivalent variations (EV) are presented in terms of the money metric utility in the counter factual scenario in comparison to the benchmark scenario by asking how much the household bene…ts from the tax changes equivalent in terms of the original equilibrium. The corresponding compensating variation is also in terms of the money metric utility measuring the amount of compensation a consumer needs to bring back her to the original level of utility after the changes in tax rates. The overall welfare costs of taxes, as presented in Table 3 above, show that the costs of using both consumption and income taxes are higher than of using only either the consumption or only the labour income tax. The overall distortionary impacts of both consumption and labour income taxes are up to 3.2 percent of the benchmark utility. This compares to results as contained in Bhattarai and Whalley (1999, 2003). If the revenue is returned as a lump sum form to the households, the model results con…rm that the labour income tax has highly distortionary impact in the economy. It may cost up to 6.2 percent of the benchmark utility. Higher rate of labour income tax …rst reduces labour supply and output and consumption consequently. In comparison, sole reliance on only consumption taxes signi…cantly lowers distortions than labour income taxes. Model calculations suggest that the cost of moving towards only consumption taxes is 0.05 percent of the benchmark utility. Thus the overall costs are lower when tax is only on consumption than when taxes are both on consumption and labour income simultaneously. For this hypothetical economy taxes a¤ect the aggregate output, employment, leisure, labour 138
Table 34: Parameters of the model in the base scenario Parameter of the model Numerical value in the base model Utility weight on consumption ( ) 0.25 to 0.6 with steps size of 0.05 Utility weight on leisure (1 ) 0.75 to 0.4 with steps size of 0.05 Elasticity of output to labour input ( ) 0.3 to .65 with steps size of 0.05 Value of Endowment L 68 to 108 hours with steps size of 5 Base consumption tax rate (tc ) 0.17 to 0.67 with steps size of 0.05 Income tax rate in the base case (tl ) 0.40 to 0.85 with steps size of 0.05 Time endowments of 68 ours per week is used here as is customary in the literature. Normalisation of price , w + p = 1 Table 35: Overall Welfare Impacts of Tax Changes in the General Equilibrium Model of Taxes Equivalent variation Compensating variation Elimination of all taxes 3.2% -3.1% Labour tax only -6.2% 6.7% Consumption tax only -0.05% 0.05% supply as well as consumption of the household as shown in Table 4 to Table 7. It is obvious that the adverse macroeconomic impacts of only labour income taxes are a lot higher than those of only consumption taxes. When both labour income and consumption taxes are removed, households lose the amount of transfer income from the government but still it has a very good positive impact on output, consumption and utility level of the representative household. In contrast the labour income tax discourages labour supply relative to both the no tax and consumption tax only cases leading to highly distortionary e¤ects on the economy. This model generates predictable results when subject to sensitivity tests along the various rates of consumption and labour income tax rates. It con…rms that the welfare costs of taxes rise proportionately to the squares of tax rates as suggested by the famous Harberger triangle, a measure of the dead weight loss of taxes. The model also behaves well when subject to changes in the endowments. Households receive higher utility as their endowments rise but at a decreasing rate given the law of diminishing marginal utility. Since the consumer values both consumption and leisure, the increase in utility of increasing only consumption show a diminishing utility as does the increase in the share of labour in production which raises the marginal productivity of labour and reduces the amount of leisure in the utility function. The …rst scenario considers the e¢ ciency impacts of removing all the taxes and transfers in the UK. When a representative household does not pay tax it also does not receive any transfer from the government. This is an extreme scenario in which all public services are provided by the private sector. The second scenario considers switching completely to the labour income tax and eliminating all indirect taxes. The third scenario, on the other hand is switching completely to consumption taxes and removing all taxes on the labour income. The results of the model are very intuitive. E¢ ciency Gains in the UK from elimination of all taxes and transfers 139
Table 36: Macroeconomic Impacts of Alternative Taxes Variables Both taxes No taxes Labour income tax Consumption tax Utility 14.142 14.601 13.689 14.526 Output 6.505 8.032 5.849 7.431 Leisure 45.333 35.789 49.012 39.703 Labour Supply 22.667 32.211 18.988 28.297 Consumption 6.505 8.032 5.849 7.431 Revenue 2.109 1.687 1.687 Wage 0.147 0.13 0.156 0.136 Price 0.853 0.87 0.844 0.864 Pro…t 14.142 14.601 13.689 14.526 Consumption tax 0.17 0.263 Labour income tax 0.35 0.57 Table 37: Impact of Alternative Taxes: Percentage changes compared to the base case Variables Both taxes Labour income tax Consumption tax Utility 3.246 -6.246 -0.511 Output 23.471 -27.174 -7.478 Leisure -21.053 36.946 10.936 Labour Supply 42.105 -41.051 -12.151 Consumption 23.471 -27.174 -7.478 Revenue -100 Wage -11.406 19.868 4.595 Price 1.964 -2.972 -0.687 Pro…t 25.896 -29.339 -8.114 Consumption tax 0.17 0.263 Labour income tax 0.35 0.57 (Measured as a percent of benchmark utility level of a representative household) Equivalent Variation = 3.715 Compensating Variation = -3.582 E¢ ciency Gains from Switching to Labour income Taxes Equivalent Variation = -6.9 Compensating Variation = 7.0 E¢ ciency Gains from Switching to Consumption Taxes Equivalent Variation = 2.967 Compensating Variation = -2.882 Summary of results The key results of this exercise in the general equilibrium impacts of tax reforms are the following: 1. The e¢ ciency gains from switching to only consumption taxes are about 80 percent of the gains of eliminating all the taxes. Optimal consumption tax rate given the revenue constraint set
140
equal to 80 percent of the benchmark revenue level is 2.9 percent. This seems a very sensible result considering the fact that consumers ultimately bear the burden of all taxes. Similarly consumers make a choice whether to consume a certain product or not depending upon its price. If the prices are high because of taxes they can increase utility by not consuming the heavily taxed good and by taking more leisure instead of work. 2. Labour income tax is highly distortionary in this model for various reasons. As before 47 percent tax rate of labour income is optimal to meet the required revenue target. It reduces the labour supply. Both output and consumption becomes smaller after such a tax is imposed. The e¢ ciency losses from switching to this sort of taxes can be up to 6.9 percent of the original utility. 3. The …rst result shows that the net deadweight loss of the current tax and transfer system is about 4 percent of GDP. GAMS programe: A1model.gms and A2model.gms, macrotax.gms Household gets utility from consuming goods and leisure M ax U = C L(1
)
(894)
c;l
Subject to p:C + w:Lh = wL
(895)
Lh + Lf = L
(896)
C > 0; L > 0;and Lf > 0; Firms’Problem: maximise pro…t M ax
= PY
Lf
w Lf
(897)
Y = Lf
(898)
Y > 0; Lf > 0; Household Problem: Maximise Utility Household gets utility from consuming goods and leisure M ax U = C L(1
)
(899)
c;l
Subject to (900)
p:C + w:Lh = wL Lagrangian optimisation: L (C; Lh ; ) = C L(1 141
)
+
wL
p:C
w:Lh
(901)
Optimal demand for goods C:solving the …rst order conditions wL = p
C=
L
(902)
p w
Buy more when goods are cheaper and when they have more income Optimal demand for leisure Lh (1
Lh = if L = 1600 and
) wL
= (1
w
= 0:4 then :Lh = 0:6
)L
(903)
1600 = 960:
Firms’Problem: maximise pro…t = PY
w Lf
(904)
Y = Lf
Lf =
P w
Y =
P w
(905) 1 1
(906) 1
(907)
Let = 0:5 Clearing Goods and Labour Markets: Real Wage Rate Y =C Lf + Lh = L; P w
Y =
C=
L p w
=
(908)
Lf = 1600 1
0:4
960 = 640
= 6400:5 = 25:29 1600 p w
= y = 25:29
p 0:4 1600 = = 25:29 w 25:29 if w = 1 set as numeraire labour market clears as Lf + Lh = 640 + 960 = 1600 = L Parameters and shadow prices 142
(909)
(910)
(911) (912)
(913)
Table 38: Parameters of the General Equilibrium Model Parameters Value 0.4 0.5 L 1600 w (normalised) 1
=
Lh 0:6 C
p
0:4
=
640 0:6 25:29
25:29
= 0:12
(914)
= 0:116
(915)
Shadow price in tax scenario
T
=
Lh 0:6 C
p
=
0:4
480 0:6 21:90
21:90
This is the change in utility associated to unit change in income. Allocations and Prices in Equilibrium Table 39: General Equilibrium Solutions Variable Base No Tax Solution Tax Solution output (Y ) 25.29 21.90 Consumption(C) 25.29 21.90 Leisure(Lh ) 960 720 Labour demand(Lf ) 640 480 Utility(U ) 224.19 178.09 Relative price wp 25.29 21.90 Shadow Price 0.12 0.116 Welfare loss to households from the government = (224:19 178:09) =224:19 = 0:2056 = 20:56%: E¤ective labour tax = 400/1600=0.25= 25%. True if households do not get utility of from public spending. How far this is true depends on the e¢ ciency of the public sector. Exercise Prove that a …rm need to pay higher wage rate to its workers and lower the price of commodity while expanding output if it operates under an increasing returns to scale technology such as Y = L2 .
5.6
Social Welfare Function
Q5. An economy is inhabited by type 1 and type 2 people. The type 1 is more productive than the type 2. Policy makers encourage productive people by assigning a greater weight to the utility of 3 1 more productive people. They aim to maximise the social welfare function: W = U14 U24 where W is the index of the social welfare, U1 represents the utility of type 1 people and U2 is the utility of type 2 people. For simplicity assume that resources of this economy produce a given level of output Y. It is consumed either by 1 or by 2 type people. Market clearing condition implies: Y = Y1 + Y2 143
. Preferences for type 1 are given by U1 = output, Y, was 1000 billion pounds.
p
Y1 and for type 2 by U2 =
p
Y2 . In a given year total
1. What is the distribution of output between type 1 and type 2 that maximises the social welfare index? What is the maximum value of the social welfare index of this economy? 2. What would have been the allocation if policy makers had given equal weight to the utility 1 1 of both types of people in the economy such as W = U12 U22 . By how much does the welfare index change in this case than compared to the social welfare in (1) above? 3. How would the social welfare index change in (1) if a tax rate of 20 percent is imposed in consumption and the tax receipts are not given back to any of these consumers? How much would the value of social welfare index be in this case? 3
1
4. Assume that the policy makers still hold the welfare function to be W = U14 U24 . How would the social welfare index change in (3 ) if all tax receipts are transferred to type 2 people? Answer Here Y = Y1 + Y2 = 1000 3
p
1
L = U14 U24 + [1000
Y1
Y2 ] = 3
p
1 4
Y2
[1000
Y1
Y2 ]
(916)
1
L = Y18 Y28
[1000
@L 3 = Y1 @Y1 8
5 8
@L = 1000 @ 5 8
Y1
Y2 ]
(917)
1
Y28
@L 1 3 = Y18 Y2 @Y2 8
3 Y 8 1
3 4
Y1
1
(918)
=0
(919)
Y2 = 0
(920)
7 8
Y1
Y28 =
=0
1 38 Y Y 8 1 2
7 8
(921)
3Y2 = Y1
(922)
1000 = Y1 + Y2 = 3Y2 + Y2 =) Y2 =
1000 = 250 4
Y1 = 3Y2 = 3 (250) = 750
(923) (924)
Index of social welfare in this economy is 3
1
W = U14 U24 =
p
750
3 4
p
250
144
1 4
3
1
= (750) 8 (250) 8 = 23:9
(925)
Answer (2) For both of them to get same level of utility: 1
p
1
L = U12 U22 + [1000
Y1
Y2 ] = 1
1
L = Y14 Y24
Y1
[1000
@L 1 = Y1 @Y1 4
3 4
[1000
Y1
Y2 ]
Y2 ]
(926) (927)
1
@L = 1000 @ 3 4
1 2
Y2
Y1
Y24
@L 1 1 = Y14 Y2 @Y2 4
1 Y 4 1
p
1 1
=0
(928)
=0
(929)
Y2 = 0
(930)
3 4
Y1
1 14 Y Y 4 1 2
1
Y24 =
3 4
(931)
Y2 = Y1
(932)
1000 = Y1 + Y2 =) Y2 =
1000 = 500 2
(933)
Y1 = Y2 = 500
(934)
Index of social welfare in this economy is 1
p
1
W = U12 U22 =
p
1 1
500
500
1 2
1
1
= 500 4 500 4 = 22:4
(935)
Answer (3) 3
1
L = U14 U24 + [1000
Y1
Y2 ] =
p 3
L = 0:8
0:8Y1
p 0:8Y2
3 4
[1000 3 Y 8 1
@L = 0:8 @Y2
1 38 Y Y 8 1 2
@L = 1000 @ 3 Y 8 1
5 8
[1000
Y1
Y2 ]
(936)
1
Y18 Y28
@L = 0:8 @Y1
0:8
1 4
5 8
Y2 ]
(937)
1
Y28
Y1
1
Y28 = 0:8 145
Y1
7 8
=0
(938)
=0
(939)
Y2 = 0 1 38 Y Y 8 1 2
(940) 7 8
(941)
3Y2 = Y1
(942)
1000 = Y1 + Y2 = 3Y2 + Y2 =) Y2 =
1000 = 250 4
Y1 = 3Y2 = 3 (250) = 750
(943) (944)
Index of social welfare in this economy is 3
1
W = U14 U24 = (0:8
3
1
750) 4 (0:8
3
1
250) 4 = (600) 4 (200) 4 = 455:9
(945)
Answer (3) If all tax is given to person 2. Y1 = 600; Y2 = 400 3
1
W = U14 U24 =
5.7
p
600
3 4
p
400
1 4
(946) 3
1
= 600 8 400 8 = 23:3
(947)
Exercise 12: Social Welfare and General Equilibrium Problem 7: General Equilibrium and Welfare Analysis
1. There are two people living in an economy. For simplicity assume that a …xed amount of output of 200 is produced each year. Entire in the same year. Utility of p output is consumed p individual 1 and 2 is represented by U1 = Y1 and U2 = 12 Y1 . (a) What is the utility received by each individual if the output is divided equally between these two people? What is the output received by each if it is distributed so that each of them gets the same amount of the utility? (b) What is the distribution of output that maximises the total utility for the whole economy? (c) If person 2 needs utility 5 in order to survive how should the output be distributed? 1
1
(d) Suppose that the authorities like to maximise the social welfare function W = U12 U22 , how should the output be distributed between them? 2. (a) An economy is inhabited by type 1 and type 2 people. The type 1 is more productive than the type 2. Policy makers encourage productive people by assigning a greater weight to the utility of more productive people. They aim to maximise the social welfare function: 3 1 W = U14 U24 where W is the index of the social welfare, U1 represents the utility of type 1 people and U2 is the utility of type 2 people. For simplicity assume that resources of this economy produce a given level of output Y. It is consumed either by 1 or by 2 type people. Marketpclearing condition implies: p Y = Y1 + Y2 . Preferences for type 1 are given by U1 = Y1 and for type 2 by U2 = Y2 . In a given year total output, Y, was 1000 billion pounds. (b) What is the distribution of output between type 1 and type 2 that maximises the social welfare index? What is the maximum value of the social welfare index of this economy? 146
(c) What would have been the allocation if policy makers had given equal weight to the 1
1
utility of both types of people in the economy such as W = U12 U22 . By how much does the welfare index change in this case than compared to the social welfare in (a) above? (d) How would the social welfare index change in (a) if a tax rate of 20 percent is imposed in consumption and the tax receipts are not given back to any of these consumers? How much would the value of social welfare index be in this case? 3
1
e. Assume that the policy makers still hold the welfare function to be W = U14 U24 . How would the social welfare index change in (c ) if all tax receipts are transferred to type 2 people? 3. Consider an economy consisting of a representative household and a representative …rm. A representative household tries to maximise utility by consuming goods and services and from enjoying leisure subject to his budget constraints. The producer wants to maximise pro…t by selling goods produced using the labour supplied by the household. The household maximisation problem can be stated as the following: max U = C l(1
)
(948)
Subject to time and budget constraints: l + hs = 1
(949)
pc = whs +
(950)
c > 0; h > 0;and l > 0; where c is consumption,l is leisure and hs is labour supply, p is the price of the commodity, w is the wage rate is the pro…t from owning the …rm. Maximisation problem for the representative …rm can be states as: s
= py
whd
(951)
subject to technology constraint as: y = (hs ) y > 0; h > 0; where y is the output supplied by the …rm and hd is its demand for labour. d
1. (a) Form a Lagrangian maximisation problem for this household. (b) Derive its demand for consumption goods and derive its demand for leisure. (c) Write the Lagrangian function for the …rm’s optimisation problem. (d) Derive …rm’s demand for labour. (e) De…ne a competitive equilibrium for this economy. (f) Compute the real wage that brings goods and labour market in equilibrium.
147
(952)
(g) What is the equilibrium quantity of c or y? (h) What is the equilibrium quantity of l and h? (i) Formulate the problem with sales and income tax. Discuss qualitatively the macroeconomic impacts of (a) switching completely to the sales taxes or (b) to labour income taxes or to (c) capital income tax.
148
5.8
Two sector model of nessecity and luxury goods (income distribtuion)
Workers and capitalists dwell in an economy. Workers consume only necessities and capitalists consume necessities and luxury goods. Workers supply all labour and capitalists save 20 percent of their income, spend 20 percent in necessities and 60 percent in luxury goods. Total labour supply is 50. LS = 50; w1 = w2 = w (953) Production function of sector i is Qi = Ai Ki i Li1
(954)
i
Table 40: Parameters in production of the K Necessity sector 0.5 100 Luxury sector 0.5 144
two sector model A 1 1
Demand for labour and supply function of necessities Table 41: Parameters in consumption of the two sector model Workers Capitalist
1
= P1 Q1
1
2
3
1 0.2
0 0.6
0 0.2
rK1 = P1 A1 K1 1 L11
wL1
@ 1 1 = (1 = P1 1 ) A1 K1 L1 @L1 Thus labour demand in necessity goods sector L0:5 1 =
5P1 ; w
0:5
1
wL1
1000:5
L1 = 25
P1 w
(
P1 w
L
rK1
0:5
(955)
w=0
(956)
2
(957)
Supply of necessity goods Q1 = A1 K1 1 L11
1
= 10L0:5 1 = 10
25
2
)0:5
;
Q1 = 50
wL2
rK2
P1 w
(958)
Demand for labour and supply function of luxuries = P2 Q2 @ = (1 @L2
rK2 = P2 A2 K2 2 L21
wL2
2 ) A2 K2
2
L2
2
= P2 149
0:5
2
1440:5
L
0:5
w=0
(959) (960)
Thus labour demand in luxury goods sector L0:5 2 =
6P2 ; w
L2 =
2
6P2 w
P2 w
= 36
2
(961)
Supply of luxury goods Q2 =
A2 K2 L22 2
2
=
12L0:5 2
(
= 12
36
P2 w
2
)0:5
; Q2 = 72
P2 w
(962)
Income of labour and capitalists Income of workers YL = wL1 + wL2 = 50w
(963)
Income of capitalists (from the production function capitalist get the same as the labour) YK = YL = 50w
(964)
Demand for necessities and luxury goods P1 Qd1 = YL + 0:2YK = 50w + 0:2 (50w) = 60w Qd1 = 60
w P1
(965) (966)
Demand for luxury goods P2 Qd2 = 0:6YK + I = 0:6 (50w) + 0:2 (50w) = 40w Qd2 = 40
w P2
(967) (968)
Market clearing conditions in goods and labour markets Q1 = 50
P1 w = Qd1 = 60 w P1
(969)
P2 w = Qd2 = 40 w P2
(970)
Q2 = 72
L1 + L2 = LS = 25
P1 w
2
+ 36
I=S
P2 w
2
= LS = 50
(971) (972)
Walras’Law: sum of excess demand is zero; when two markets clear third market automatically clears. Market clearing prices Set numeraire P1 = 1: From necessity goods market:
150
50
P1 w 1 w 5 = 60 =) 5 = 6 =) w2 = =) w = 0:913 w P1 w 1 6
(973)
From luxury goods market: 72
P2 w 5 5 = 40 =) P22 = w2 = w P2 9 9
5 25 = = 0:463 =) P2 = 0:680 6 54
(974)
Allocations: Q1 = 50 Q2 = 72
w 1 = 54:8 = Qd1 = 60 = 60 0:913 P1
P2 = 72 w
0:913 = 54:8 1
0:680 w = 53:63 = Qd2 = 40 = 40 0:913 P2
L1 = 25
2
P1 w
L2 = 36
P2 w
2
= 36
0:680 0:913
(976)
2
1 0:913
= 25
0:913 = 53:7 0:680
(975)
= 29:97
(977)
2
= 19:97
(978)
50
(979)
0:913 = 45:65
(980)
L1 + L2 = 29:97 + 19:9 Consumption: workers’demand for necessity good YL = C1;;L ;
50w = 50
Capitalist’s demand for necessity good 0:2YK = C1;K ; 0:2 ( 50w) = 0:2 (50
0:913) = 9:13
(981)
54:8
(982)
Total demand for necessity good C1;L + C1;K = 45:65 + 9:13 workers do not consumer luxury good Capitalist’s demand for luxury good C2;;K =
C2;;L = 0; 27:39 0:6YK = = 40:23 P2 0:681
(983)
Investment demand by capitalist for luxury good I 0:2 (50 0:913) 9:13 = = = 13:43 P2 P2 0:680
(984)
C2;;K + I = 40:23 + 13:43 = 53:7
(985)
GAMS programme: UK10.gms
151
Small Open Economy Trade Model: Expansionary Devaluation Open above model by including exports and imports and for simplicity …rst assume …xed output and inputs X = f K; L = C + E
(986)
Exports depends on exchange rate (e), domestic price (P) and foreign price for domestic goods ( ) and the elasticity of exports ( ) E = E0 e
(987)
P
Imports depends on exchange rate (e), domestic price (P) and price of imported goods (Pm ) and the elasticity of exports ( ) C P = K0 e m M P
(988)
Resource Balance with foreign borrowing B P X + eB = P C + ePm M
(989)
Small Open Economy Trade Model: Expansionary Devaluation Devaluation lowers the foreign price of domestic goods ( ) , it raises supply of exports (E), it reduces the amount of imports (M ) raises the production of import substitute goods. Thus both domestic and foreign demand for home products rise. Thus devaluation is expansionary. Excess of imports over exports need to be …nanced by foreign borrowing. eB = ePm M
PE
(990)
If borrowing rise, imports rise or exports fall, domestic consumption rise. Borrowing may be necessary when import prices rise or domestic prices fall. This is an over-determined system. Many things may happen Parameters of the model E0 ; K0 ; ; ; K; L; X; ; B; Pm
(991)
C; E; M; P; e
(992)
Gains from Devaluation Variables of the model
Draw a diagram of import export trade-o¤ and export supply and demand functions. Whether workers or capitalist gain depends on what kind of goods are exported and imported by external borrowing. Imported goods may contain necessary goods such as food, medicine, agricultural inputs used by workers or luxury goods such as cars, perfumes, entertainment goods for capitalists. Exports may contain necessity goods or luxury goods. Redistribution impacts of devaluation thus depend on the composition exports and imports.
152
5.9
General equilibrium model of Trade: Ricardian Comparative Advantage Theory
Theory of international trade has developed over time in works of Ricardo (1817), Ohlin (1933), Stopler-Samuelson (1947), Bhagawati Helpman (1976), Dixit and Stiglitz (1977), Meade(1978), Krugman (1980), Whalley (1985), Neary (1988),Krugman and Venables (1995), Hine and Wright (1998), Edwards(1993), Eaton and Kortum (2002), Markusen (1995), Taylor (1995), Raut and Ranis (1999), Roe and Mohladi (2001), Greenaway, Morgan and Wright (2002),Melitz (2003), Beaulieu, Benarroch and Gaisford (2004). These theories are applicable in explaining real world problems ( eg.Bhattarai and Whalley (2006), Bhattarai and Mallick (2003)). As the growth rate of global trade is greater than the growth of the global GDP there the space of globalization is rapid and it is a¤ecting lives of millions of people. Trade models, essentially involve application of the general models to answer who gains and who loses from exchange of goods and services across borders. Essentially it is application of basic microeconomic theories at international scale. 5.9.1
Two Country Ricardian Trade Model There are two countries indexed by j, producing two goods, manufacturing and services. Each of them have an option to be self reliant or to trade on the basis of comparative advantage. Under the import substituting industrialisation (ISI) regimes countries favoured to be self reliant and infant industries were protected by tari¤s and non-tari¤ barriers. After numerous rounds of trade negotiations under GATT/WTO over the years, all countries now have realised that the autarky solutions like this are economically ine¢ cient. In contrast globalisation is a norm. trade is mutually bene…cial for trading nations and raises welfare in both countries. Aim of this model is to illustrate on these statements analytically and numerically with a small and transparent example. For this it is assumed that each country j specialises in commodities in which it is more e¢ cient and engages in trade. The exchange rate is determined by the relative prices of two commodities in the global market.
Two Country Ricardian Trade Model Preferences in country 1 are expressed by its utility function in consumption of good 1 and 2 , C1;1 and C1;2 respectively: max
U1 = (C1;1 )
1
(C1;2 )
1
1
(993)
Income of country 1 is obtained from the wage income in sector 1 and sector 2 plus the transfers to country 1 I1 = w1;1 L1;1 + w1;2 L1;2 + T R1 153
(994)
where L1;1 and L1;2 are labour employed in sector 1 and sector 2 w1;1 and w1;2 are corresponding wages respectively and T R1 is the transfer income. Technology constraints in sector 1 in country 1 X1;1 = a1;1 :L1;1
(995)
where a1;1 is the productivity of labour in sector 1 in country 1. Technology constraints in sector 2 in country 1 X1;2 = a1;2 :L1;2
(996)
where a1;2 is the productivity of labour in sector 2 in country 1. Resource constraint in country 1 de…ned by the labour endowment as: L1 = L1;1 + L1;2
(997)
Production possibility frontier of country 1 now can be de…ned as L1 =
1 1 :X1;1 + :X1;2 a1;1 a1;2 X11 = a11 :L11
(998) (999)
Given above preferences the demand for good 1 in country 1 is C1;1 =
1 :I1 P1
(1000)
the demand for good 2 therefore is: C1;2 = 5.9.2
(1
1 ) :I1
P2
(1001)
Autarky or Trade Theoretically two trade arrangements are possible in this model. First one is an autarky equilibrium in which each country is separate and isolated from another. It produces just for its own consumption and no trade take place between these two countries. Such autarky solution is close to the production arrangement when countries were adopting ISI trade strategy.
Analytical solutions of autarky and specialisation Proposition 1 Autarky solution is Pareto dominated by trade equilibrium for reasonable parameters of preferences and technology. This is proven below by analytical and numerical solutions.
154
A Lagrangian function is used to express how each country maximises welfare subject to its production possibility frontier constraint under the autarky equilibrium as: (1
$1 = X1;11 X2;1
1)
+
1 X1;1 a11
L1
First order conditions with respect to X11 and X21 and @$1 = @X1;1
1 X1;1 1
1
(1
1 X2;1 a12
as:
1)
X2;1
(1002)
a11
=0
(1003)
Analytical solutions in autarky @$1 = (1 @X2;1
( 1 1 ) X1;1 X2;1
@$1 = L1 @
1 X1;1 a11
1)
a12
=0
(1004)
1 X2;1 = 0 a12
(1005)
Analytical solutions of in autarky 1
1 1 X1;1
From the …rst two …rst order conditions
(1
X2;1 =
(1
X2;1
1)
( 1 1) 1 )X1;1 X2;1
(1
1) 1
=
1
(1
1)
X2;1 X1;1
=
a12 a11
a12 X1;1 a11
(1006)
optimal value of X1;1 is found now putting this condition in the production possibility frontier constraint. 1 1 1 (1 1 X1;1 + 1 X2;1 = 1 X1;1 + 1 a11 a2 a1 a2
1) 1
1 a12 (1 X1;1 = 1 X1;1 1 + a11 a1
1)
= L1
(1007)
1
Analytical solutions of in autarky 1 1 a1 L1
X1;1 =
(1008)
Similarly the optimal value of X2;1 is found by X2;1 =
(1
1) 1
a12 (1 X1;1 = a11
1) 1
a12 a11
1 1 a1 L1
= (1
1 1 ) a2 L1
(1009)
For each of 1 country amount produced depends on productivity and preferences parameters and the endowment of its labour input. The autarky welfare level is: U 1 = (X1;1 )
1
1
(X2;1 )
1
=
1 1 a1 L1
1
(1
(1 1 1 ) a2 L1
1)
(1010)
Two Country Ricardian Trade Model Preferences in country 2 are expressed by its utility function in consumption of good 1 and 2 , C2;1 and C2;2 respectively:
155
max
U2 = (C2;1 )
2
(C2;2 )
1
2
(1011)
Income of country 2 is obtained from the wage income in sector 1 and sector 2 plus the transfers to country 2 I2 = w2;1 L2;1 + w2;2 L2;2 + T R2
(1012)
where L2;1 and L2;2 are labour employed in sector 1 and sector 2 w2;1 and w2;2 are corresponding wages respectively and T R2 is the transfer income. Technology constraints in sector 1 in country 2 X2;1 = a21 :L2;1
(1013)
where a2;1 is the productivity of labour in sector 1 in country 2. Technology constraints in sector 2 in country 2 X2;2 = a2;2 :L2;2
(1014)
where a2;2 is the productivity of labour in sector 2 in country 2. Resource constraint in country 2 de…ned by the labour endowment as: L2 = L2;1 + L2;2
(1015)
Production possibility frontier of country 2 now can be de…ned as L2 =
1 1 :X2;1 + :X2;2 a2;1 a2;2
(1016)
Given above preferences the demand for good 1 in country 2 is C2;1 =
2 :I2
P1
(1017)
the demand for good 2 therefore is: C2;2 =
(1
2 ) :I2
P2
(1018)
Autarky or Trade Theoretically two trade arrangements are possible in this model. First one is an autarky equilibrium in which each country is separate and isolated from another. It produces just for its own consumption and no trade take place between these two countries. Such autarky solution is close to the production arrangement when countries were adopting ISI trade strategy. Proposition 2 Autarky solution is Pareto dominated by trade equilibrium for reasonable parameters of preferences and technology. This is proven below by analytical and numerical solutions.
156
Analytical solutions in autarky A Lagrangian function is used to express how each country 2 maximises welfare subject to its production possibility frontier constraint under the autarky equilibrium as: (1
$2 = X1;22 X2;2
2)
+
1 X1;2 a2;1
L2
First order conditions with respect to X12 and X22 and @$2 = @X1;2
1
2 X1;2 2
@$2 = (1 @X2;2
(1
X2;2
(1019)
=0
(1020)
as:
2)
a2;1
( 2 2 ) X1;2 X2;2
1 X2;2 a2;2
2)
a2;2
=0
(1021)
Analytical solutions in autarky 1
@$2 = L2 @
a2;1
From the …rst two …rst order conditions X2;2 =
1
X1;2
a2;2
1
2 2 X1;2
(1
X2;2
X2;2 = 0
2)
( 2 2) 2 )X1;2 X2;2
(1
(1
2) 2
(1022)
=
2
(1
2)
X2;2 X1;2
=
a2;2 a2;1
a2;2 X1;2 a2;1
(1023)
optimal value of X1;2 is found now putting this condition in the production possibility frontier constraint. 1 1 1 (1 1 X1;2 + X2;2 = X1;2 + a2;1 a2;2 a2;1 a2;2
2) 2
a2;2 1 (1 X1;2 = X1;2 1 + a2;1 a2;1
2)
= L2 (1024)
2
Analytical solutions in autarky X1;2 =
2 a2;1 L2
(1025)
Similarly the optimal value of X2;2 is found by X2;2 =
(1
2) 2
a2;2 (1 X1;2 = a2;1
2) 2
a2;2 a2;1
2 a2;1 L2
= (1
2 ) a2;2 L2
(1026)
For each of 2 country amount produced depends on productivity and preferences parameters and the endowment of its labour input. The autarky welfare level is: U 2 = (X1;2 )
2
1
(X2;2 )
2
=(
2 a2;1 L2 )
2
((1
(1 2 ) a2;2 L2 )
2)
(1027)
Summary of two country trade model in Autarky Thus the level of welfare in country 1 is determined in terms of its preferences for consumption of good 1 and 2 as re‡ected by 1 and its own production technology as re‡ected in a11 and a12 .
157
Numerical version of this model is applied to country 1 and country 2 taking the population as rough indicator of its resource in production. country 1 has 200 million population and country 2 has 400 million population. country 1 is more productive in producing services goods X1 whereas country 2 has more advantage in producing manufacturing goods X2 . Preferences are similar but technologies are di¤erent. These parameters are set out in Table 1. Table 42: Parameters of the Autarky Model a1 a2 L country 1 country 2
0.4 0.6
5 2
2 5
200 400
Summary of two country trade model in autarky Under the autarky equilibrium these two economies are completely isolated and produce only for domestic consumption. The optimal production and consumption and employment of labour for both sectors, prices of commodities and labour, and utility for the representative hocountry 1ehold are as given in Table 2. In per capita terms citizens of the country 1 and country 2 have welfare of 1.46 and 1.76 respectively.
Table 43: Parameters of the Autarky Model X1 X2 L1 L2 U p2 country 1 country 2
400 480
240 800
80 240
120 160
294.4 588.8
0.6 1.67
Each country produces both goods in no trade equilibrium which as explained here is very ine¢ cient. Welfare can be improved by making these countries trade. Analytical solutions for trade equilibrium under specialisation A representative hocountry 1ehold in each country maximises its welfare subject to its budget constraint. Demand for goods are derived by standard constrained optimisation on supply side for each country j . Under trade equilibrium it is optimal for each country to specialise in goods in which it has comparative advantage. The optimisation problem and the …rst order conditions for constrained optimisation are given as follows: (1
$j = X1;jj X2;j
j)
+ [Ij
P1 X1;j
P2 X2;j ]
(1028)
First order conditions: @$j = @X1;j @$j = (1 @X2;j
j
j X1;j
1
(1
X2;j
j)
( j j ) X1;j X2;j
j)
P1 = 0 P2 = 0
Analytical solutions for trade equilibrium under specialisation
158
(1029) (1030)
@$j = Ij @ j
j X1;j
(1
1
P1 X1;j
(1
P2 X2;j = 0
j)
X2;j
( j ) X1;j X2;j
j)
j
X2;j =
=
(1
j
(1 j)
j
P1 X1;j + P2 X2;j = P1 X1;j + P2
(1
j) j
X1;j =
P1 P2 X1;j
(1031)
X2;j P1 = P2 j ) X1;j
P1 X1;j P2
(1032)
(1033)
= Ij
(1 j Ij j ) Ij ; X2;j = P1 P2
(1034)
Analytical solutions for trade equilibrium under specialisation Global market clearing conditions for goods 1 and 2 are N X
X1;j = X1
(1035)
N X
X2;j = X2
(1036)
j
j
Prices adjcountry 1t until this equilibrium condition holds. Under complete specialisation, country 1 country 1 specialises in services X2 and produces 1825 units of it. country 2 specialises in manufacturing X1 goods and produced 6000 units of it. It is easy to determine country 2’s income if we choose good 1 as numeraire setting P1 = 1. Analytical solutions for trade equilibrium under specialisation I 1 = P1 X1 = 1
1000 = 1000
(1037)
Relative price of good 2, P2 need to be determined to …nd the level of income in the country 1. This can be done using the global market clearing condition 1
2 2 :I 1 :I + = 0:4 (1000 P1 P1
1) + 0:6 (2000
P2 ) = 1000
600 1000 400 = = 0:5 1200 1200 Now it is easy to determine the income of the country 1 as:
(1039)
P2 =
I 2 = P2 X2 = 200
5
P2 = 2000
P2 = 1825
(1038)
0:5 = 1000
(1040)
Analytical solutions for trade equilibrium under specialisation Since income level for both country 2 and the country 1 are determined, it is now easy to determine the level of demand in both countries:
159
X1;1 =
X2;1
1 I1 2 I2 = 0:4 (1000) = 400; X1;2 = = 0:6 (1000) = 600 P1 P1
= =
(1
0:6 (1000) (1 1 ) I1 2 ) I2 = = 1200; X2;2 = P2 0:5 P2 0:4 (1000) = 800 0:5
(1041)
(1042) (1043)
Solutions of both autarky and trade equilibria are given in Table 3 and 4. Given the preferences and technology speci…cations, with complete specialisation both countries gain from trade. Comparative static analysis of trade can be done changing the preference or technology parameters. Analytical solutions for trade equilibrium under specialisation Table 44: Comparing Specialisation and Autarky Regimes Production Autarky Trade country 1 country 2
Consumption Autarky Trade
X1
X2
X1
X2
C1
C2
C1
C2
400 480
240 800
1000 0
0 2000
400 480
240 800
400 600
1200 800
P 1 0.5
Analytical solutions for trade equilibrium under specialisation Table 45: Comparing Employment and Welfere under Specialisation and Autarky Employment Autarky Trade
Uitlity Autarky Trade
L1
L2
L1
L2
U
80 240
120 160
200 0
0 400
294.4 588.8
U 773.3 673.2
Gains from trade may be distributed di¤erently across countries (Bhattarai and Whalley (2006)). Further there are opportunities for bargaining on the share of those gains particularly from dynamic strategic considerations and the basic elements required for such dynamic model is provided in the next section. GAMS programme: trade.gms; trade_2.gms Beaulieu E, M. Benarroch and J. Gaisford (2004) Trade barriers and wage inequality in a North-South model with technology-driven intra-industry, trade, Journal of development Economics, 75:113-136 Bhattarai K. and S. Mallick (2013) Impact of China’s currency valuation and labour cost on the US in a trade and exchange rate model. North American Journal of Economics and Finance, 25, 40-59. Bhattarai K and J Whalley (2006), Division and Size of Gains from Liberalization of Trade in Services, Review of International Economics, 14:3:348-361, August. 160
Dixit A K and J E. Stiglitz (1977) Monopolistic Competition and Optimum Product Diversity, American Economic Review, 67:3:297-308. Eaton J and S. Kortum (2002) Technology, Geography, and Trade, Econometrica, 70: 5:Sep:17411779 Edwards S. (1993) Openness, Trade Liberalization, and Growth in Developing Countries, Journal of Economic Literature, 31: 3 :1358-1393, September. Greenaway D. W. Morgan and P. Wright (2002) Trade Liberalisation and Growth in Developing Countries, Journal of Development Economics, vol. 67 229-244. Helpman E (1976) Macroeconomic Policy in a Model of International Trade with a Wage Restriction, International Economic Review, 17:2:262-277. Hine R.C. and P.W. Wright (1998) Trade with Low Wage Economies, Employment and Productivity in UK Manufacturing, Economic Journal, 108:450:1500-1510. Krugman P. (1980) Scale Economies, Product Di¤erentiation and the Pattern of Trade, American Economic Review, 70:5:950-959. Markusen. J, R. (1995) The boundaries of multinational enterprises and the theory of international trade, Journal of Economic Perspective, 9:2:169-189. Meade, James (1978) The Meaning of Internal Balance, Economic Journal, 88 (351): Sep 423-435. Melitz M. T. (2003) The Impact of Trade on Intra-Industry Reallocations and Aggregate Industry Productivity, Econometrica, 71:6:1695-1725. Neary P.J. (1988) Determinants of the Equilibrium Real Exchange Rate, American Economic Review, 78:1:Mar.: 210-215. Ohlin B (1933) Interregional and international trade, Harvard economic studies ; no.39 Raut L and G Ranis (eds.) Trade, Growth and Development: Essays in Honour of Professor T N Srinivasan, Contribution to Economic Analysis 242, Elsevier, NorthHolland, Amsterdam, 1999 Roe T and H. Mohladi (2001) International Trade and Growth: An Overview Using the New Growth Theory, Review of Agricultural Economics, 23:2:423-440 Taylor M. P. (1995) The Economics of Exchange Rates, Journal of Economic Literature, March, vol 33, No. 1, pp. 13-47. Whalley, J. (1985) Trade Liberalization Among Major World Trading Areas, MIT Press
161
5.10
Exercise 12’: migration and factor mobility Problem 8: Migration or Factor Movement Across Countries
1. Consider trade between two countries. One is abundant in capital and another in labour. For simplicity assume that the production functions of these economies are given by Y1 = K1 1 L1 1
(1044)
Y2 = K2 2 L2 2
(1045)
Table 46: Endowment and Technology Country 1 Country 2
K 500 1000
L 1000 500
0.4 0.6
0.6 0.4
Output ? ?
1. (a) What will be the rental rate of capital and wage rate in each country if both goods and factors are immobile across countries? (b) What will be the rental rate and output if there is a global market for capital and labour? Explain the pattern of migration across countries. (c) Is free trade equivalent to free mobility of factor of production according to HeckscherOhlin-Stopler-Samuelson theorem? (d) Trade is not bene…cial to every one. Discuss how labour in labour abundant and capitalists in capital abundant countries gain from trade on the basis of this model. (e) Show that in a static world like this aggregate global income remains the same but there is a change in the distribution of income. Beaulieu E, M. Benarroch and J. Gaisford (2004) Trade barriers and wage inequality in a North-South model with technology-driven intra-industry, trade, Journal of development Economics, 75:113-136 Bhattarai K and J Whalley (2006), Division and Size of Gains from Liberalization of Trade in Services, Review of International Economics, 14:3:348-361, August. Dixit A K and J E. Stiglitz (1977) Monopolistic Competition and Optimum Product Diversity, American Economic Review, 67:3:297-308. Greenaway D. W. Morgan and P. Wright (2002) Trade Liberalisation and Growth in Developing Countries, Journal of Development Economics, 67 229-244. Helpman E (1976) Macroeconomic Policy in a Model of International Trade with a Wage Restriction, International Economic Review, 17:2:262-277. Hine R.C. and P.W. Wright (1998) Trade with Low Wage Economies, Employment and Productivity in UK Manufacturing, Economic Journal, 108:450:1500-1510. 162
Krugman P. (1980) Scale Economies, Product Di¤erentiation and the Pattern of Trade, American Economic Review, 70:5:950-959. Melitz M. T. (2003) The Impact of Trade on Intra-Industry Reallocations and Aggregate Industry Productivity, Econometrica, 71:6:1695-1725. Roe T and H. Mohladi (2001) International Trade and Growth: An Overview Using the New Growth Theory, Review of Agricultural Economics, 23:2:423-440 Taylor Mark (1995) The Economics of Exchange Rates, Journal of Economic Literature, March, vol 33, No. 1, pp. 13-47.
5.11
General equilibrium with taxes
Optimal Tax and Public Goods max U h = (1
) ln Y h
Th +
ln G
(1046)
subject to P Yh
Th + G = I
h
(1047)
h
where V is the utility of households, Y T is the net of tax income, G the public good and the weight in utility from consumption of public goods. The production side of the economy is represented here by income for simplicity. L Y h ; G;
= (1
) ln Y h
Th +
ln G +
I
P Yh
Th
G
(1048)
Optimal Tax and Public Goods L Y h ; G; @Y h
=
(1 (Y h
) T h)
P =0
(1049)
L Y h ; G; = =0 (1050) @G G This implies optimal public good and optimal tax correspond to preference for public goods (1 (Y h
) G =P T h)
(1051)
G = PTh = PT
(1052)
G = PY
(1053)
Samuelson’s Theory on Optimal Public Spending Sum of the marginal rate of substitution of all citizens should equal marginal cost of providing public goods (see two citizen public good model) Consumers consume private (x) and public goods (G) 163
max
u1 = u1 (x1 ; G)
(1054)
subject to a given level of utility for the second consumer max u2 = u2 (x2 ; G)
(1055)
x1 + x2 + c (G) = w1 + w2
(1056)
and the resource contraint
Constrained optimisation for this is L = u1 (x1 ; G)
[u2
u2 (x2 ; G)]
[x1 + x2 + c (G)
w1
w2 ]
Samuelson’s Theory on Optimal Public Spending @u1 (x1 ;G) (x1 ;G) @L @L = 0 =) = @u1@x ; ; @x = @x1 = @x1 1 1 @L @G
=
@u1 (x1 ;G) @G @u1 (x1 ;G) @G @u1 (x1 ;G) @x1
@u2 (x2 ;G) @G
+
@u2 (x2 ;G) @G @u2 (x2 ;G) @x2
=
@c(G) @G
= 0; =)
@c (G) ; @G
@u2 (x2 ;G) (x2 ;G) = 0 =) @u2@x2 @x2 @u2 (x2 ;G) 1 @u1 (x1 ;G) = @c(G) @G @G @G
M RS1 + M RS2 = M C(G):::Q:E:D:
(1057) =
(1058)
Simplest General Equilibrium Tax Model: Demand Side Households problem max U = C L
(1059)
p (1 + t) C + wL = wL
(1060)
Subject to
Lagrangian for houshold optimisation L (C; L; ) = C L +
p (1 + t) C + wL
wL
(1061)
Household Optimisation L (C; L; ) = L + p (1 + t) = 0 @C
(1062)
L (C; L; ) =C+ w=0 @L
(1063)
L (C; L; ) = p (1 + t) C + wL wL = 0 @ Above three FOC equations (1403) - (1064) can be solved for three variables : M RSCL =
L(C;L; ) @C L(C;L; ) @L
=)
164
L p (1 + t) = C w
(1064)
(1065)
Household Optimisation L=
p (1 + t) C w
(1066)
Putting (1410) into (1064) p (1 + t) c + wL
wL = p (1 + t) C + w
wL = 0
(1067)
1 wL 2 p (1 + t)
(1068)
p (1 + t) p (1 + t) 1 wL 1 C= = L w w 2 p (1 + t) 2
(1069)
C=
L=
p (1 + t) C w
Demand for goods is low with higher taxes and prices, high with higher wage rate and labour endowment; high with the higher share of spending on goods and services.Given these preferences the demand for leisure is half of the labour endowment. Supply Side of the General Equilibrium Model Firms’pro…t maximisatin problem max
= p:Y
w:LS
(1070)
Subject to p
r
1 L (1071) 2 Consumers pay tax not the producers. In no tax case, given this production technology and demand side derivations, labour demand equals labour supply Labour market clearing Y =
LS =
L + LS = L
(1072)
C=Y
(1073)
Goods market clearing
Let total labour endowment L be 200. Then labour supply is 1 1 L = L = 100 2 2 Given this labour supply the level of output will be r p p 1 Y = LS = L = 100 = 10 2 From the zero pro…t condition required equilibrium = p:Y numeraire p = 1 LS = L
L=L
165
(1074)
(1075) w:LS = 0 and setting the
p:Y = w:LS
(1076)
Y 10 1 = = (1077) LS 100 10 Numerical Example of the General Equilibrium Model 1 both goods and labour market clear Given the equilibrium relative wage rate of w = 10 when t = 0, demand for good eqauls supply as: w=
C=
1 wL 1 = 2 p (1 + t) 2
1 (200) = 10 10
(1078)
Similarly the demand for labour and leisure equal total endowment of labour L + LS = 100 + 100 = 200 = L
(1079)
Labour market clears. Therefore this is a general equilibrium; at these prices both goods and labour market clear, household maximise utility and …rms maximise pro…t. When t = 0:2 then 1 wL 1 1 (200) C= = = 8:33 (1080) 2 p (1 + t) 2 10 1:2 Government revenue and spending: R = p:t:C = 1
0:2
8:33 = 1:67 = G
(1081)
Markets clear in this case too C + G = 8:33 + 1:67 = 10 = Y
(1082)
Houswhold’s welfare before tax U = C L = 10 100 = 1000
(1083)
U = C L = 8:33 100 = 833
(1084)
Welfare after tax Thus 20 percent tax has reduced the household welfare by 16.7 percent. Can utility from public spending of compensate for this lost welfare? Household Problem: Maximise Utility Household gets utility from consuming goods and leisure M ax U = C L(1
)
(1085)
c;l
Subject to p:C + w:Lh = wL
(1086)
Lagrangian optimisation: L (C; Lh ; ) = C L(1 166
)
+
wL
p:C
w:Lh
(1087)
Optimal demand for goods C:solving the …rst order conditions wL = p
C=
L
(1088)
p w
Households buy more when goods are cheaper and when they have more income Optimal demand for leisure Lh (1
:Lh = if L = 1600 and
) wL
= (1
w
= 0:4 then :Lh = 0:6
)L
(1089)
1600 = 960:
Firms’Problem: maximise pro…t = PY
w Lf
(1090)
Y = Lf
Lf =
P w
Y =
P w
(1091) 1 1
(1092) 1
(1093)
Let = 0:5 Clearing Goods and Labour Markets: Real Wage Rate Y =C Lf + Lh = L; P w
Y =
C=
L p w
=
(1094)
Lf = 1600 1
0:4
960 = 640
= 6400:5 = 25:29 1600 p w
= y = 25:29
p 0:4 1600 = = 25:29 w 25:29 if w = 1 set as numeraire labour market clears as Lf + Lh = 640 + 960 = 1600 = L Parameters and shadow prices 167
(1095)
(1096)
(1097) (1098)
(1099)
Table 47: Parameters of the General Equilibrium Model Parameters Value 0.4 0.5 L 1600 w (normalised) 1
=
Lh 0:6 C
0:4
=
p
640 0:6 25:29
25:29
= 0:12
(1100)
= 0:116
(1101)
Shadow price in tax scenario
T
=
Lh 0:6 C
p
=
0:4
480 0:6 21:90
21:90
This is the change in utility associated to unit change in income. Allocations and Prices in Equilibrium Table 48: General Equilibrium Solutions Variable Base No Tax Solution Tax Solution output (Y ) 25.29 21.90 Consumption(C) 25.29 21.90 Leisure(Lh ) 960 720 Labour demand(Lf ) 640 480 Utility(U ) 224.19 178.09 Relative price wp 25.29 21.90 Shadow Price 0.12 0.116 Welfare loss to households from the government = (224:19 178:09) =224:19 = 0:2056 = 20:56%: E¤ective labour tax = 400/1600=0.25= 25%. True if households do not get utility of from public spending. How far this is true depends on the e¢ ciency of the public sector.
5.12
Exercise 13: Monopolistic Competition
Problem 9: Monopolistic Competition 1. Using geometric method prove the Heckscher-Ohlin theorem that a country will export the commodity that uses its relatively abundant factor with unique relation between prices of factors and products making the commodity trade complete substitutes for trade in factors. (hint: constant return to scale, free trade in goods but complete immobility of factors of production; use PPP and Edgeworth boxes for Xand Y and K and L,px =py and w and r). 2. Consider a …rm in monopolistically competitive industry Q=A 168
B P
(1102)
Prove that its marginal revenue is given by Q B
MR = P
(1103)
(a) If the cost function is C = F + cQ then prove that the average cost declines because of the economy of scale. (b) Further assume that the output sold by a …rm, number of …rms, its own price and average prices of …rms are given by Q=S
1 N
b P
P
(1104)
show that the average cost rises to number of …rms in the industry when all …rms charge same price. AC = n:F s +c (c) Prove that price charged by a particular …rm declines with the number of …rms P = c + b1n (d) Determine the number of …rms and price in equilibrium. Explain entry exit behavior and prices when number of …rms are below or above this equilibrium point. (e) Collusive and strategic behaviors may limit above conclusions. Discuss. (f) Apply above model to explain international trade and its impact on prices and number of …rms in a particular industry. (g) Use this model to explain interindustry and intra-industry trade. (h) Use monopolistic competition model to analyse consequences of dumping practices in international trade. Problem of a Multinational Corporation 1. Assume that the MNC has home and foreign markets, faces distinct demand curves across two countries and faces di¤erent cost curves. Home market is more lucrative than the foreign market both in terms of prices and cost e¤ectiveness. Despite that the MNC has a global ambition, therefore it aims to extend its business in overseas markets. The main objective of the MNC is to control the market and to maximise pro…t. Demand in home country P1 = 130
Q1
(1105)
and associated cost function is C1 = 10Q1
(1106)
Demand in the foreign (host) country P2 = 90 169
Q2
(1107)
and associated cost function is C2 = 20Q2
(1108)
1. (a) what will be output price and welfare under perfect competition? (b) What will be price, output and welfare in the host country if the monopolist forms a cartel with the …rm in the host country? (c) What will be price, output and welfare if the MNC plays Cournot game with the …rm in the host country? (d) How will above result change if the monopolists acts as a price leader in Stackelberg equilibrium? (e) How will above result change if both …rms play under a Bertrand equilibrium adopting a predatory pricing strategy? Atkinson A. B.and N. H. Stern (1974) Pigou, Taxation and Public Goods The Review of Economic Studies, 41:1:119-128. Bhattarai K (2010) Taxes, public spending and growth in OECD countries, Journal of Perspective and Management, 1/2010. Bhattarai K and J. Whalley (2009) Redistribution E¤ects of Transfers, Economica 76:3:413431 July. Blundell R (2010) Empirical Evidence and Tax Policy Design: Lessons from the Mirrlee’s Review, Institute of Fiscal Studies. Darling A. Chancellor of Exchequer, HM Treasury (2009), Securing the Recovery: Growth and Opportunity, Pre-Budget Report, December , 2009. Feldstein M (1974) Incidence of Capital Income Tax in a Growth Economy with Varying Saving Rates, Review of Economic Studies, 41:4:505-513 Fullerton, D., J. Shoven and J. Whalley (1983) Dynamic General Equilibrium Impacts of Replacing the US Income tax with a Progressive Consumption Tax, Journal of Public Economics 38, 265-96. Meade J (1978) Structure of Direct Taxation, Institute of Fiscal Studies, London. Mirlees, J.A. (1971) An exploration in the theory of optimum income taxation,Review of Economic Studies, 38:175-208. Perroni, C. (1995), Assessing the Dynamic E¢ ciency Gains of Tax Reform When Human Capital is Endogenous, International Economic Review 36, 907-925. Main budget: http://www.hm-treasury.gov.uk/; Green Budget: http://www.ifs.org.uk/
170
6
L6: Game theory: Bargaining in Goods and Factors markets
In many circumstances optimal decisions of an economic agent depends on decisions taken by others. Dominants …rms competing for a market shares, political parties contesting for power and research and scienti…c discoveries aimed for path-breaking innovations are in‡uenced by decision taken by others. In all these circumstances there are situations where collective e¤orts rather than individual ones generate the best outcome for the group as a whole and for each individual members of the group. Concepts of bargaining, coalition and repeated games developed over years by economists such as Cournot (1838), Bertrand (1883), Edgeworth (1925) von Neumann and Morgenstern (1944) and Nash (1950, 1953) is developing very fast in recent years following works of Kuhn (1953), Shapley (1953),Shelten ( 1965) Aumman (1966) Scarf (1967), Shapley and Shubik (1969), Harsanyi(1967), Spence (1974), Hurwicz (1973), Myerson (1986), Maskin and Tirole (1989), Kreps (1990), Fundenberg and Tirole (1991) and Binmore (1992), Rubinstein (1982) Sutton (1986) Cho and Kreps (1987) Sobel (1985) Machina (1987) Riley (1979) McCormick (1990), Ghosal and Morelli (2004) and others. These have generated models that can be applied to analyse the relative gains from coalitions rather than without these coalitions. Outcome of a noncooperative games can be more easily explained by Nash bargaining game that is popularly known as a game of splitting a pie between two parties, right or left. Rule of this game is that the sum of the shares of the pie claimed by players cannot exceed more than 1, otherwise each will get zero. Standard technique to solve this problem is by maximising the Nash Product. It is natural that economic agents play a zero sum and non-cooperative game until they realise the bene…ts of coalition and cooperation. When an agreement is made and cooperation is achieved there is a question on whether such coalition is stable or not. There are always incentives at least for one of the player to cheat others from this cooperative agreement in order to raise its own share of the gain. However, it is unlikely that any player can fool all others at all the times. Others will discover such cheating sooner or later. Therefore a peaceful solution requires credibility and a punishment mechanism by which any party that tries to cheat on the agreement is punished unless it amends its uncooperative behaviours toward others. A coalition of players should ful…l individual rationality, group rationality and coalition rationality. These can be ascertained by the supper-additivity property of coalition where the maximisation of gain requires being a member of the coalition rather than playing alone. The imputations in the core guarantees each member of a coalition the value at least as much as it could be obtained by playing independently. At the core of the game each player gets at least as much from the coalition as from the individual action, there does not exist any blocking coalition. This is equivalent to Pareto optimal allocation in a competitive equilibrium (Scarf (1967)). Some imputations are dominated by others; the core of the game is the strong criteria for dominant imputation. Core satis…es coalition rationality. Ability of a player to in‡uence the outcome of the game depends on the pivotal status enjoyed by that player. In a game with 3 players; power of player i is re‡ected by its Shapley value. Consider six possible ordering of 123 pivotal game. Three players can order themselves in 3!= 6 ways. Each of these number can appear only twice in the middle out of six possible combinations. A player located in the middle is pivotal. If parties realise this fact while bargaining, such bargaining is likely to generate a stable and cooperative solution. When intention cannot be directly revealed or stated players can signal indirectly to other players. These signals can take many forms. Signalling plays important roles in strategic choices of 171
individuals, parties, communities, regions, national and the global community as a whole. Formation of payo¤ discussed above depends on signalling - players do not know the moves of their opponents but based on their interpretation of signal they can however, put some numerical values to the payo¤. A rational player interprets signals correctly and chooses actions that support each other. This brings that player up in the progress ladder. Wrong interpretation of signals results in status quo or even a gradual decline in the standard of that player. Success in the game thus relates very much on ability and dexterity in providing right signals and accurate interpretation of signals coming from other players. Interpreting those signals correctly and translating them into actions more accurately brings success; sending wrong signals or interpreting them incorrectly is a recipe of failure. Status of player i, si is thus a stochastic process that depends on ability of signal extraction. Such ability depends on intuition and information set i . Very few games are plaid only once. Economic agents, political parties, live for a long time and play games repeatedly. Economists have applied Cournot-Nash bargaining game of oligopoly to explain the consequences of cooperative and non-cooperative games on the division of gains from bargaining. It can quantitatively be illustrated using a Nash bargaining oligopoly model. Players often do not have enough information about other players in the game. They have to guess intention of other players looking at their choices. People are principals of a political game, they want better standard of living, peace and prosperity in a country but they do not have enough information about the true intention of the members of political parties act as their agents and should in principle be responsible for their principals - the common people who elect political parties frequently in the parliament. Once elected party with majority forms the government and tries to ful…l its collective interest. Political contracts are as similar as wage contracts in a labour market that are designed to match e¤orts put by a worker to their productivities in the labour maker. Political parties know their type but the people do not.Thus in the presence of information asymmetry , the e¤orts by the good party is at the …rst best level as the bad e¤ort by him is not as attractive as the good e¤ort, it is not pro…table for a good party to pretend to be bad party. Good party is not attracted by the contract for the bad party. Similarly it is costly for the bad party to act as a good party - it is optimal for it to select the contract appropriate for a bad party, that is being out of the o¢ ce. Cournot (1838), Bertrand (1883), Edgeworth (1925) von Neumann and Morgenstern (1944), Nash (1950); Kuhn (1953), Shapley (1953),Shelten (1965) Aumann (1966) Luce and Rai¤a (1957) Scarf (1967), Shapley and Shubic (1969), Harsanyi (1967), Spence (1974), Myerson (1986), Kreps (1990), Fundenberg and Tirole (1991) and Binmore (1992), Roth (2008); Sobel (1985), Hey (1987), Kreps (1990) Mirlees (1971), Maskin and Moore (1999), Maskin and Tirole (1992), Cripps (1997), Perlo¤ (2013)Osborne and Robinstein (1994) Cripps and Thomas (1995), Gardener (2003) Bhaskar and To (2004), Mailath and Samuelson (2006) Texts: Holt (2007), Rasmusen (2007), Dixit, Skeath and Reiley (2009), Varian (2010), Perlo¤ (2014) 172
6.1
Formal de…nitions
Strategic form game is a tuple G = (Si ; ui ) for each player i = 1; :::; N where Si is the strategy available to player i and ui : xN j=1 Sj ! R is payo¤ of play i. It is …nite if the strategy set contains …nitely many elements. For instance strategy set of column player Sj = fR; M; Lg and of the row player Sj = fT; M; Bg R M L T 3,0 0,-5 0,-4 M 1,-1 3,3 -2,4 B 2,4 4,1 -1,8 Strictly dominated strategy ui (b si ; si ) > ui (si ; s i ) for (si ; s i ) 2 S and sbi 6= si . Eliminate the dominated strategies for row and column player R L T 3,0 0,-4 B 2,4 -1,8 Weak dominance ui (b si ; si ) ui (si ; s i ) for (si ; s i ) 2 S and sbi 6= si .
6.1.1
Nash equilibrium N
Given G = (Si ; ui )i=1 strategy sbi is pure strategy Nash equilibrium if for each player i ui (b s) ui (si ; sb i ) for all si 2 S. Mixed strategy Mi is probability distribution over Si Expected utility from Neumann-Morgenstern utility function is: X ui (m) = (m1 (s1 ) ::mN (sN )) ui (si ) (1109) s2S
For given pure strategies s = (s1 ; ::::; sN ) 2 S with probabilities m1 (s1 ) ; ::; mN (sN ) Theorem: Every …nite strategic form game possesses at least one Nash equilibrium. Proof of Nash equilibrium De…nitions: a. m b 2 M is a Nash equilibrium b. For every player i, ui (m b i ) = ui (si ; m b i ) for every si 2 Si given positive weight m b i and ui (m b i ) ui (si ; m b i ) for every si 2 Si given zero weight m b i. c. For every player i, ui (m b i ) ui (si ; m b i ) for every si 2 Si 1. construct a continuous function that maps m into itself. fi;j (m) =
mi;j + max (0; ui (j; m i ) ui (m)) n P 1+ max (0; ui (j 0 ; m i ) ui (m)) b
(1110)
j 0 21
2. Apply Brouwer’s …xed point theorem to …nd a …xed point ; here numerator is a continuous function and the denominator is continuos and bounded by contraction mapping f : M ! M it has a …xed point. 3. demonstrate that the …xed point is Nash equilibrium. LSH = RHS in above function. fi;j (m) = m b i;j then above function is 173
n X
max (0; ui (j 0 ; m i )
j 0 21
Multiply both sides by ui (j; m b i) n X
=
j 0 21
(ui (j 0 ; m b i)
ui (m)) b
n X
max (0; ui (j 0 ; m i )
j 0 21
ui (m)) b max (0; ui (j; m b i)
Here the left hand side n n P P m b i;j ui (j 0 ; m b i) m b i;j (0; ui (j 0 ; m b i ) ui (m)) b = j 0 21
j 0 21
Now the RHS
0=
n X
j 0 21
(ui (j 0 ; m b i)
RHS is zero only if ui (j 0 ; m b i) is the Nash equilibrium. 6.1.2
ui (m))
(1111)
ui (m) b and sum over j
m b i;j (ui (j 0 ; m b i)
j 0 21 n X
ui (m)) b = max (0; ui (j; m i )
ui (m)) b
ui (m) b = ui (m) b
ui (m)) b max (0; ui (j; m b i)
ui (m) b
ui (m)) b
0. That implies ui (m) b
ui (m)) b
(1112)
ui (m) b =0 (1113)
ui (j 0 ; m b i ). Therefore m b
Game of incomplete information: N
G = (p; Ti ; Si ; ui )i=1 where p is probability over Ti is the type of the player ui : S T ! R Associated strategic form of this game is G = (Rj ; vj )j2J where j is set of indices of the form j = (i; ti ) where ti 2 Ti and i = 1; :::; N . 0 Now the player j = (i; ti ) s strategy and payo¤ are de…ned as : Rj = Si
(1114)
expected payo¤ vj (r) = t0
n X
1 2T
p (t
i
j ti ) ui rj ; r(k;tk )
k6=i
; ti ; t
1i
(1115)
o
Bayesian Nash equilibrium is the Nash equilibrium of this associated strategic form game. Theorem: Every …nite game of incomplete information possesses at least one Bayesian Nash equilibrium.
174
Extensive form Game ( )
6.1.3
1. A …nite set of players N. 2. A set of actions, A 3. A set of nodes, histories, X. it includes initial nodes and a complete description of actions that have been taken so far. A (x) fa 2 A j (x; a) 2 Xg 4. Probability distribution
over actions A (x)
A:
5. Set of end nodes E fx 2 X j (x; a) 2 = Xg for all a 2 A: Each end node describes the complete play of the game so far. 6. A function : fX n (E [ fx0 g) j (x) = ig. When the game reaches at node X it tells it is the turn player i next. 7. Information set belonging to player i ; Ii
fI (x) j (x) = i , some x 2 X n (E [ fx0 g)g
8. for each i 2 N ; Neumann-Morgenstern payo¤ functions at the end node ui : E ! R. This is payo¤ for every possible complete play of the game. 9. Extensive form game is summarised then,
= < N; A; X; E; ; ; I; (ui )i2N .
Economic activities of consumers, producers, governments and nations or regions are interdependent. Game theory provides tools to study the strategic interactions among such economic agents where decisions taken by one individual depend on actions taken by others. Each game has a number of players who choose a set of strategies and rules. .Optimal choices available to one depend on choices made by others. Pay-o¤s are clearly de…ned for each player strategy pairs. Strategic modelling like this started with classics such as Cournot (1838), Bertrand (1883), Edgeworth (1925) von Neumann and Morgenstern (1944), Nash (1950). It is developing very fast in recent years following works of Kuhn (1953), Shapley (1953),Shelten ( 1965) Aumann (1966) Scarf (1967), Shapley and Shubic (1969), Harsanyi(1967), Spence (1974), Kreps (1990), Fundenberg and Tirole (1991) and Binmore (1992). Elements of a Game Rational Players Strategies Payo¤ matrix R 1;1
is pay-o¤ to row player if he plays strategy 1 and the column player plays strategy 1. Players like to maximise their own pay-o¤ given opponent’s strategy; B will choose strategy 1 or 2 that maximises his/her payo¤ looking at the choice of player A. Most games have equilibrium from which players do not have any incentive to move away. 175
Table 49: Structure of a Game Player A Strategy 1 Strategy 2 Player B R R C C Strategy 1 ; 1;1 1;2 ; 1;2 1;1; R C R C Strategy 2 2;1 ; 2;1; 2;2 ; 2;2
6.2
Story of GAME made easy
Story 1 In the beginning human beings did not know much on how to produce or organise the society and economy. They believed in a static world and might of their muscles and had very little concern to others. They obtained resources of nature for themselves and would play a zero sum game. They believed that there were …xed amount of goods or commodities that were to be distributed among people; if some one got more another person would get less. Each of them wanted more. This was the reason for outbreak of frequent …ghts and quarrels among them as seen in movies of early inter-continental settlers or in history books. Those who lost the war were forced to move to other less productive or less pleasant places. There were strategic interactions; actions of one depended on others but in a two person zero sum (TPZS) framework. Such game can be given by a matrix such as Table 50: Two Person Zero Sum Game Storng Hand pull push Strong Leg walk ( 10; 10) (10; 10) stand (10; 10) ( 10; 10) a) Explain this TPZS game b) Find a mixed strategy for strong-leg and strong-hand c) What is the value of the game? d) Why this game is not realistic in modern world? Story 2 Gradually people learnt to cultivate and grow their food. Civilisations started. Then they realised how the production and consumption can be organised collectively. Each can gain more from cooperation. Peace of mind would make them more productive and they can reap the bene…t of economies of scale. This brings the game to the next stage. Table 51: Two Person Cooperative Game Singer sing quite Writer write (5; 5) (2; 6) read (6; 2) (3; 3) a) b)
What is the solution for cooperative solution in this game? What makes such solution stable one? 176
c) Why the cooperation is Pareto superior than non-cooperation? d) What is the solution in mixed strategy? Story 3 Over time people are taken over by greed and self interest. They started competing out others for material bene…ts. They played non-cooperative game. Private corporations and …rms emerged to organise people in production process. Property rights and legal provisions for protecting those rights got built up in the economic system. Table 52: Non Cooperative Game Singer sing quite Writer write (5; 5) (2; 6) read (6; 2) (3; 3) a) What is the non-cooperative solution of this game? b) Show gains from bargain in this game in a diagram? c) What is the Nash Product? d) What are the threat points? Story 4 Now people learnt that inherently human being is sel…sh. They see war not the real solution of the problem. Gradually clever ones come up with good ideas. They make rules, regulations and constitutions to build mechanism in order to motivate someone to do good works and punish some who does bad works. Classical economists had developed models for a perfect world where information was complete, types and preferences and abilities of economic agents were known. It was easy to apply rules in such a perfect world based on criteria. Table 53: Incomplete Information Game Singer sing quite Writer write (?; ?) (?; ?) read (?; ?) (?; ?) Story 5 N person games Society consists of a large number of individuals. Like minded people enter into a coalition with speci…c targets and objectives in their mind. They form an alliance that would give them more than they did not. The core solutions make every one happier than stand alone solutions. Solutions at the core are more e¢ cient than outside it. Story 6 World does not have complete information. Market fails to provide certain goods or it disappears completely under the asymmetric information situation. People take advantage of opportunities that would make them better even if that hurts others. There are good and bad intentions but it is very di¢ cult to guess it precisely in the beginning. Insurance companies emerge to make up for losses. Story 7
177
Auctioning and competitive bids are mechanism to assign contracts and reveal some information that would otherwise would not be revealed. Binmore K. (1990) Fun and Games: A text on Game Theory, Lexington, Heath. Bhattarai K. (2013) Coalition for constitution and economic growth in Nepal, International Journal of Global Studies (IJGS), 1:1, Feb, 1-4 Cripps, M.W.(1997) Bargaining and the Timing of Investment, International Economic Review, 38:3 :Aug.:527-546 Dixit A., S. Skeath and D. F. Reiley (2009) Games of Strategy, Norton. Gardener R (2003) Games of Business and Economics, Wiley, Second Edition. Hey J. D. (2003) Intermediate microeconomics, McGraw Hill. Holt Charles (2007) Markets, Games and Strategic Behaviour, Pearson, . Hurwicz L (1973) The design of mechanism for resource allocation, American Economic Review, 63:2:1-30. Kreps D. M. (1990) A Course in Microeconomic Theory, Princeton. Maskin E, J. Moore (1999) Implementation and Renegotiation The Review of Economic Studies, Vol. 66, No. 1, Special Issue: Contracts Jan, pp. 39-56 Maskin E and J Tirole (1992) The principal-agent relationship with an informed principle: common values, Econometrica, 60:1:1-42 Mailath G. J. and L. Samuelson (2006) Repeated Games and Reputations: long run relationship, Oxford. Mirlees, J.A. (1971) 'An exploration in the theory of optimum income taxation.' Review of Economic Studies,38:175-208. Myerson R (1986) Multistage game with communication, Econometrica, 54:323-358. Osborne M.J. and A. Robinstein (1994) A course in game theory, MIT Press. Pathak P and T Sönmez (2013) School Admissions Reform in Chicago and England: Comparing Mechanisms by Their Vulnerability to Manipulation.' American Economic Review, 103(1): 80-106. Perlo¤ J. M. (2013) Microeconomics: Theory and Applications with Calculus, Pearson, 3rd Edition. Rasmusen E(2007) Games and Information, Blackwell. Roth A E. (2008) What have we learned from market design?, Economic Journal, 118 (March), 285–310. Shapley L (1953) A Value for n Person Games, Contributions to the Theory of Games II, 307-317, Princeton. 178
Shapley L and M. Shubik (1969) On Market Games, Journal of Economic Theory, 1:9-25 Varian HR (2010) Intermediate Microeconomics: A Modern Approach, Norton,8th ed..
6.3
Types of games Table 54: Zero Sum Game Player A Strategy 1 Strategy 2 Player B Strategy 1 (10; 10) ( 10; 10) Strategy 2 ( 10; 10) (10; 10)
zero sum game: one’s gain = loss of another ; sports ; market shares two or many players; Chess, football Cooperative Games: Global climate change; bargaining game Non-cooperative Games: two or many players ; Competition and Collusion competition between opposing political parties, countries Single period of multiple period: static and dynamic Full information or incomplete information :Firms and consumers; government and public;Among individuals, clubs, parties; nations Solution of Games by the Dominant Strategy Dominant strategy
Table 55: Advertisement Game Player A Advert Dont Advert Player B Advert (10; 5) (15; 0) Dont Advert (6; 8) (10; 2) Dominant strategy is to advertise for both A and B. With a slight change Dominant strategy is to advertise for A but B has no dominant strategy. Solution of Games by Nash Equilibrium (Prisoner’s Dilemma) Punishment structure for a crime F in d in g N a sh so lu tio n (u n d e rsc o re th e b e st stra te g y to a p laye r i g ive n th e ch o ic e o f th e o p p o n e nt.
Nash Equilibrium: Prisoner’s Dilemma Fact: both players did a crime together. Police suspects and arrest both of them. 179
Table 56: Advertisement Game Player A Advert Dont Advert Player B Advert (10; 5) (15; 0) Dont Advert (6; 8) (20; 2) Table 57: Prisoners’Dilemma Game Player A Confess Dont Confess Player B Confess ( 5; 5) ( 1; 10) Dont Confess ( 10; 1) ( 2; 2) Playing non cooperatively each convicts another. Game results in Nash solution (confess, Confess) = ( 5; 5); Each ends up with 5 years in prison. By confessing, each gives evidence to the police to determine the highest possible punishment. If they had cooperated remaining silent, police would not have enough evidence. Each would have been given only two years of prison ( 2; 2) : This is Pareto optimal outcome, 'where no one could be made better o¤ without making someone worse-o¤'. Cooperation is better but each think that other player will cheat and therefore doesn’t cooperate. Therefore stay longer in jail. There are many example of prisoner’s dilemma game in real world -pricing and output in a cartel, pollution, tax-revenue. Solution by the mixed strategy This game does not have equilibrium in pure strategy. Player B will play H is A plays H but A will play T if B plays H. If A plays T it is optimal to play T for B, then it is optimal for B to play H. Game goes in round in circle again. It can be solved my the mixed strategy. Flip the coin to randomise the chosen strategies. If each played H or T half of the times optimal payo¤ is zero to both players. Probability of playing H or T is 0.5. Solution by mixed strategy B plays Top p times and Bottom (1 p) times if A plays Left . B plays Top p times and Bottom (1 p) times if A plays Right. B likes to be equally well o¤ no matter what A plays. Solution by the mixed strategy Expected pay-o¤ for B if A plays Left E(
B;L )
= 50p + 90(1
p)
(1116)
= 80p + 20(1
p)
(1117)
Expected pay-o¤ for B if A plays Right E(
B;R )
180
Table 58: Prisoners’Dilemma Game Player A Confess Dont Confess Player B Confess ( 5; 5) ( 1; 10) Dont Confess ( 10; 1) ( 2; 2) Table 59: Game of matching penny: mixed strategy Player A Head Tail Player B Head (1; 1) ( 1; 1) Tail ( 1; 1) (1; 1) Making these two payo¤s equal 50p + 90(1
p) = 80p + 20(1
p) =) 100p = 70
p = 0:7
(1118) (1119)
B plays Top 70 % of times and Bottom 30% of times. Subsidy Game Between the Airbus and Boeing If both Boeing and Airbus produce a new aircraft each will lose -10. If Airbus does not produce and only Boeing produces Boeing will make 100 pro…t. If Airbus does not produce Airbus can make 100 but then Boeing will decide to produce even at a loss of 10 so that Airbus does not enter in that market. Subsidy Game Between the Airbus and Boeing EU countries want Airbus to produce, they change this by subsidising 20 to Airbus. Producing new aircraft is dominant strategy for Airbus now, no matter whether Boding produces or not. Entry Deterrence Game In‡ation and unemployment game between public and private sectors Higher payo¤ is good. First element represents payo¤ to the row-player (Government). Second element represents payo¤ to the column-player (private sector). Nash solution is (H; H) = (4; 4) Cooperative solution would have been better with (L; L) = (5; 5). Cost of Cheating Cooperative solution would have been better with (L; L) = (5; 5) but distrusting each other results in (H; H) = (4; 4) . If the game is plaid repeatedly what will be value of the game? It is given by the discounted present value of the game for any discount rate 0 < < 1: P V (cooperate) = 5 + 5 + 5
181
2
+ :::: + 5
n
=
5 1
(1120)
Table 60: Competitive Game Player A Left Right Player B Top (50; 50) (80; 80) Bottom (90; 90) (20; 20) Table 61: Subsidy Game Airbus Produce Don’t produce Boeing Produce ( 10; 10) (100; 0) Don’t produce (0; 100) (0; 0) However, there is an incentive to cheat to get 6 instead of 5. when one player deviates from the cooperative strategy this way another will found out being cheated next period. Then he/she will punish the cheater by playing non-cooperatively next period. So the value of game : P V (cheat) = 6 + 4 + 4
2
+ :::: + 4
n
(1121)
P V (cheat) = 6 + 4 + 4
2
+ :::: + 4
n
(1122)
(1123)
Cost of Cheating
multiply it by P V (cheat) = 6 + 4
2
+ :::: + 4
n+1
2
+ :::: + 4
n+1
taking the di¤erence (1
) P V (cheat) = 6
6 +4
4
=6+4
(1
)
(1124)
Whether a person cheats or not depends on discount factor 5 1
=6+4
(1
)
or5 = 6 (1
)+4
1=
2 ;
=
1 2
(1125)
Extensive form of the game Solution by Backward Induction (Is there any …rst movers advantage?) In‡ation and unemployment game in a diagram In‡ation and unemployment game in a diagram Economic policy game between the …scal and monetary authority Binmore K. (1990) Fun and Games: A text on Game Theory, Lexington, Heath. Cripps, M.W.(1997) Bargaining and the Timing of Investment, International Economic Review, 38:3 :Aug.:527-546 Dixit A., S. Skeath and D. F. Reiley (2009) Games of Strategy, Norton.
182
Table 62: Subsidy Game Airbus Produce Don’t produce Boeing Produce ( 10; 10) (100; 0) Don’t produce (0; 120) (0; 0) Table 63: Subsidy Game Entrant Enter Dont Enter Incumbent Enter ( 10; 10) (100; 0) Dont Enter (0; 100) (0; 0)
Gardener R (2003) Games of Business and Economics, Wiley, Second Edition. Holt Charles (2007) Markets, Games and Strategic Behaviour, Pearson, . Mailath G. J. and L. Samuelson (2006) Repeated Games and Reputations: long run relationship, Oxford. Kreps D. M. (1990) A Course in Microeconomic Theory, Princeton. Osborne M.J. and A. Robinstein (1994) A course in game theory, MIT Press. Perlo¤ J. M. (2008) Microeconomics: Theory and Applications with Calculus, Pearson. Rasmusen E(2007) Games and Information, Blackwell,. Varian HR (2010) Intermediate Microeconomics: A Modern Approach, Norton,8th ed..
Equilibrium concepts: Backward induction; subgame perfect equilibrium, sequential equilibrium, Bayes’s rule.
6.4
Bargaining game The very common example for bargaining game is splitting a pie between two individuals. The sum of the shares of the pie claimed by both cannot exceed more than 1, otherwise each will get zero. If we denote these shares by of the game where each get j =0 .
i i
and j then i + j 1 is required for a meaningful solution 0 and j 0 payo¤. When i + j > 1 then and i = 0 and
Standard technique to solve this problem is to use the concept of Nash Product Nash Product in Bargaining Game max U = (
i
183
0) (
j
0)
(1126)
Table 64: Subsidy Game Entrant Enter Dont Enter Incumbent Enter ( 10; 10) (100; 0) Dont Enter (0; 120) (0; 0) Table 65: In‡ation and unemployment game Private Sector H L Government H (4; 4) (6; 3) L (3; 6) (5; 5) subject to i
or by non-satiation property
i
+
j
+
j
1
(1127)
=1
Using a Lagrangian function L ( i;
j;
)=(
0) (
i
j
0) + [1
i
j]
(1128)
First Order Conditions First order conditions of this maximization problem are L ( i; @ L ( i; @
j;
)
=
j
=0
(1129)
=
i
=0
(1130)
i j;
)
j
L ( i; j ; ) =1 (1131) i j =0 @ From the …rst two …rst order conditions j = i implies j = i and putting this into the third …rst order condition j = i = 12 . This is called focal point. Thus Nash solution of this problem is to divide the pie symmetrically into two equal parts. Any other solution of this not stable. Roy Gardner (2003) and Charles Holt (2007) have a number of interesting examples on bargaining game. Application of Bargaining Game Money to be divided between two players M = u1 + u2 The origin of this bargaining game is the disagreement point d(0; 0), the threat point.
184
(1132)
Here the utility of player one (u1 ) is plotted against the utility of player two u2 and the line u1 u2 is the utility possibility frontier (UPF). Starting of bargaining can be (0; M ) or (M; 0) where one player claims all but other nothing. But this is not stable. O¤ers and counter o¤ers will be made until the game is settled at u = each player gets equal share.
1 1 2 M; 2 M
where
Numerical Example of Bargaining Game Suppose there is 1000 in the table to be split between two players. What is the optimal solution from a symmetric bargaining game if the threat point is given by d(0,0)? Using a Lagrangian function for constrained optimisation L (u1; u2 ; ) = u1 u2 + [1000
u1
u2 ]
(1133)
First order conditions of this maximization problem are L (u1; u2 ; ) = u2 @u1
=0
(1134)
L (u1; u2 ; ) = u1 @u2
=0
(1135)
L (u1; u2 ; ) = 1000 u1 u2 = 0 (1136) @ From the …rst two …rst order conditions u2 = u1 implies u2 = u1 and putting this into the third …rst order condition u2 = u1 = 1000 = 500. This is called focal point. 2 Numerical Example of Bargaining Game The Nash bargaining solution is the values of u1 and u2 that maximise the value of the Nash product u1 u2 subject to the resource allocation constraint,u1 + u2 = 1000. This bargaining solution ful…ls four di¤erent properties: 1) symmetry 2) e¢ ciency 3) linear invariance 4) independence of irrelevant alternatives (IIA). Symmetry implies that equal division between two players and e¢ ciency implies no wastage of resources u1 + u2 = M or maximisation of the Nash product, u1 u2 . Linear invariance refers to the location of threat point as can be shown in a bankruptcy game say dividing 50000. If u is a solution to the bargaining game then u + d is a solution to the bargaining problem with disagreement point d. Numerical Example of Bargaining Game L (u1; u2 ; ) = (u1
d1 ) (u2
d2 ) + [50000
u1
u2 ]
(1137)
Suppose the player 1 has side payment d1 = 15000 L (u1; u2 ; ) = (u1
15000) (u2
d2 ) + [50000
First order conditions of this maximization problem are
185
u1
u2 ]
(1138)
L (u1; u2 ; ) = u2 @u1 L (u1; u2 ; ) = u1 @u2
15000
From the …rst two …rst order conditions u2
(1139) =0
(1140)
u1
u2 = 0
(1141)
= u1
15000
L (u1; u2 ; ) = 50000 @ Numerical Example of Bargaining Game
implies u2 = u1
=0
15000 and
putting this into the third …rst order condition u2 + 15000 = u1 ; u2 =
50000 15000 2
= 17500; u1 = 15000 + u2 = 32500.
Then u1 + u2 = 17500 + 32500 = 50000. Risk and Bargaining A risk averse person loses in bargaining but the risk neutral person gains. Suppose the utility 0:5 functions of risk averse person is given byu2 = (m2 ) but the risk neutral person has a linear utility u1 = m1 . m1 + m2 = M .u1 + u22 = 100 Using a Lagrangian function for constrained optimisation L (u1; u2 ; ) = u1 u2 +
100
u1
u22
(1142)
First order conditions of this maximization problem are L (u1; u2 ; ) = u2 @u1 L (u1; u2 ; ) = u1 @u2
=0 2 u2 = 0
(1143) (1144)
L (u1; u2 ; ) = 100 u1 u22 = 0 (1145) @ Numerical Example of Bargaining Game From the …rst two …rst order conditions uu1;2 = 2 u2 implies u1 2u22 and putting this into the third …rst order condition .3u22 = 100 ; u22 = 100 3 = 33:3 ; u2 = 5:77 2 2 u1 = 2u2 = 2 (5:77) = 66:6 u1 + u22 = 66:6 + 33:3 = 100 Thus the risk-averse player’s utility is 66.7 and risk neutral player’s utility is only 5.7. Morale: do not reveal anyone if you are risk averse, otherwise you will lose in the bargaining. Coalition Possibilities 186
2N -1 rule for possible coalition Consider Four Players A,B,C,D A, B, C, D AB, AC, AD BC, BD, CD ABC, ABD,ACD, BCD, ABCD 16 -1=15 6.4.1
Coalition and Shapley Values of the Game
In many circumstances optimal decisions of an economic agent depends on decisions taken by others. Dominants …rms competing for a market shares, political parties contesting for power and research and scienti…c discoveries aimed for path-breaking innovations are in‡uenced by decision taken by others. In all these circumstances there are situations where collective e¤orts rather than individual ones generate the best outcome for the group as a whole and for each individual members of the group. Concepts of bargaining, coalition and repeated games developed over years by economists such as Cournot (1838), Bertrand (1883), Edgeworth (1925) von Neumann and Morgenstern (1944) and Nash (1950, 1953) is developing very fast in recent years following works of Kuhn (1953), Shapley (1953),Shelten ( 1965) Aumman (1966) Scarf (1967), Shapley and Shubik (1969), Harsanyi(1967), Spence (1974), Hurwicz (1973), Myerson (1986),Gale (1986), Maskin and Tirole (1989), Kreps (1990), Fundenberg and Tirole (1991) and Binmore (1992), Rubinstein (1982) Sutton (1986) Cho and Kreps (1987) Sobel (1985) Machina (1987) Riley (1979) McCormick (1990), Ghosal and Morelli (2004) and others. These have generated models that can be applied to analyse the relative gains from coalitions rather than without these coalitions. It is natural that economic agents play a zero sum and non-cooperative game until they realise the bene…ts of coalition and cooperation. When an agreement is made and cooperation is achieved there is a question on whether such coalition is stable or not. There are always incentives at least for one of the player to cheat others from this cooperative agreement in order to raise its own share of the gain. However, it is unlikely that any player can fool all others at all the times. Others will discover such cheating sooner or later. A coalition of players should ful…l individual rationality, group rationality and coalition rationality. These can be ascertained by the supper-additivity property of coalition where the maximisation of gain requires being a member of the coalition rather than playing alone. This can be explained using standard notations. Let us take three players [ in the current Nepalese context N = (1= CPM and 2 =UML, 3 =NC)]. Superadditivity condition implies that the value of the game in a coalition is greater than the sum of the value of the game of playing alone by those individual members. v (1 [ 2 [ 3)
v1 (1) + v (2) + v (3)
(1146)
Coalitions (parties) playing together generate more value for each of its member than by playing alone. Team spirit generates extra bene…ts. When normalised to 0 and 1 the value of the gains from a coalition are: v (1) = 0; v (N ) = 1 for n = 1; :::; N The fact that payo¤ of the merged coalition is larger than the sum of the payo¤ to the separate coalitions is shown by following imputations, that shows ways on how to value of the game can be distributed among N di¤erent players. The imputations of values characterise these allocations: 187
1
=
v (N ) =
2
+
X
i
3
+
(1147)
N X
=
i
(1148)
i=1
i2N
Group rationality implies that total payo¤ to each players in the coalition equals at least the payo¤ of its independent actions. i
v (fig) ;
i2N
(1149)
In the dynamic context players like to maximise the present value, V, of the gain from in…nite period, with a given discount rate r over years: Z t=1 v (i) e rt dt (1150) V = t=0
The imputations in the core guarantees each member of a coalition the value at least as much as it could be obtained by playing independently. At the core of the game each player gets at least as much from the coalition as from the individual action, there does not exist any blocking coalition. This is equivalent to Pareto optimal allocation in a competitive equilibrium (Scarf (1967)). Some imputations are dominated by others; the core of the game is the strong criteria for dominant imputation. Core satis…es coalition rationality. A unique imputation in the core is obtained by Shapley value. This re‡ects additional payo¤ that each individual can bring by adding an extra player to the existing coalition above the pay-o¤ without this player. This is the power of that player. Consider a game of three players in which the 3rd player always brings more to the coalition than the 1st or the 2nd player. Payo¤ for coalition of empty set: v ( ) = 0 Payo¤ from players acting alone: v (1) = 0; v (2) = 0; v (3) = 0 ; Payo¤ from alternative coalitions: v (1; 2) = 0:1; v (1; 3) = 0:2; v (2; 3) = 0:2; Payo¤ from the grand coalition: v (1; 2; 3) = 1 Power of individual i in the coalitions is measured by the di¤erence that person makes in the value of the game v (S [ fig v (S)) = 1 , where S is the subset of players excluding i, S [ fig is the subset including player i. The expected values of game for i is found by taking account of all possible coalition that person i can enter with N number of players, where is the weighting factor that changes according to the number of people in a particular coalition. This is the probability that a player joins coalition,S 2 N and there are (2N -1) ways of forming in N player game: i
=
X
S2N
n
(S) v (S [ fig
v (S)) ;
n
(S) =
s! (n
s n!
1)
(1151)
v (1) v ( ) = 0 v (1; 2) v (1) = 0:1 v (1; 3) v (1) = 0:2 0 = 0:2 v (1; 2; 3) v (2; 3) = 1 0:2 = 0:8 0
(S) =
s! (n
s n!
1)
=
0! (3
188
0 3!
1)
=
2! 2 = 3! 6
(1152)
1
(S) =
2
(S) =
s! (n
s n!
1)
s! (n
s n!
1)
= =
1! (3
1 3!
1)
2! (3
2 3!
1)
=
1! 1 = 3! 6
(1153)
=
2! 2 = 3! 6
(1154)
Shapley value for player 1 is thus 1
=
X
n
(S) v (S [ fig
v (S)) =
2 1 1 2 19 (0) + (0:1) + (0:2) + (0:8) = 6 6 6 6 60
(1155)
n
(S) v (S [ fig
v (S)) =
2 1 1 2 19 (0) + (0:1) + (0:2) + (0:8) = 6 6 6 6 60
(1156)
2 1 1 2 22 (0) + (0:2) + (0:2) + (0:9) = 6 6 6 6 60
(1157)
S2N
For player 2 2
=
X
S2N
Note as before v (2) v ( ) = 0 v (1; 2) v (1) = 0:1 v (2; 3) v (1) = 0:2 0 = 0:2 v (1; 2; 3) v (1; 3) = 1 0:2 = 0:8 For player 3 3
=
X
S2N
n
(S) v (S [ fig
v (S)) =
v (3) v ( ) = 0 v (1; 3) v (1) = 0:2 0 = 0:2 v (2; 3) v (2) = 0:2 0 = 0:2 v (1; 2; 3) v (1; 2) = 1 0:1 = 0:9 As the player 3 brings more into the coalition its expected payo¤ is higher than of players 1 and 2. Similar con…gurations can be made where players 1 and 2 can bring more in the coalition. In the context of Nepal which of three parties mentioned above are pivotal depends on the value they add to the game. The value of grand coalition is the largest possible value of the game with N players. This fact is shown by the core of the game in Figure 1. Figure 1
189
Solutions towards the core are more stable than those towards the corners which are prone to con‡icts. This is equivalent to …nding a central ground in politics. Ego centric solutions are less likely to bring any stable solution to the game. In the most stable equilibrium all players gain in equal proportions to their supporters.
6.5
Pivotal player in a voting game in Nepal
Ability of a player to in‡uence the outcome of the game depends on the pivotal status enjoyed by that player. In a game with 3 players; power of player i is re‡ected by its Shapley value. Consider six possible ordering of 123 pivotal game. Three players can order themselves in 3!= 6 ways. Each of these number can appear only twice in the middle out of six possible combinations. A player located in the middle is pivotal. If parties realise this fact while bargaining, such bargaining is likely to generate a stable and cooperative solution. In the 123 game given in Table 1 the player 3 is pivotal in game (2) and (4); player 1 in (3) and (5) and player 2 in (1) and (6). The marginal contribution (Shapley value) of each player can be presented then as Therefore each player has 1/3 chance of being pivotal. If 1 is pivotal into the coalition any coalition with 1 will win - player 1 is powerful. Players 2 and 3 are powerless. When a party has majority in a parliament that party is in pivotal position. This outcome is reversed in a hung parliamnet where none is pivotal. There is always a chance that a pivotal player now may have to give up that position for other players later on. Another con…guration is to assume that certain party is pivotal all the times. As shown below, in this situation the Shapley value of player 1 is 1 no matter which position it is in the coalition and it is 0 for players 2 and 3. In the context of Nepal’s Constituent Assembly, it seems that depending on circumstances, players NC, CPM and UML each have equal chance of being a pivotal player. Thus non-pivotal con…gurations are more applicable than pivotal con…gurations. 190
Table 66: No Pivotal Player in a Bargaining Game orderings M(1,S) M(2,S) M(3,S) 1 123 0 1 0 2 132 0 0 1 3 213 1 0 0 4 231 0 0 1 5 312 1 0 0 6 321 0 1 0 Table 67: Pivotal Player in a Bargaining Game orderings M(1,S) M(2,S) M(3,S) 1 123 1 0 0 2 132 1 0 0 3 213 1 0 0 4 231 1 0 0 5 312 1 0 0 6 321 1 0 0 Simple game theoretic models applicable to analyse the current Nepalese situation could be developed taking seminal ideas of Shapley and Shubik (1969), Rubinstein (1982), Myerson (1986), Sutton (1986), Dixit (1987), Maskin and Moore (1999) and Riley (2001). 6.5.1
Model of fruitless bargaining and negotiation
There are N parties in the game indexed by i = 1; ::::; N . Each party i is interested in its own pay-o¤ xi (e.g the number of ministries it should have under its command) which it computes using a payo¤ function Ui that depend on strategies available to players and its information set about the reactions of other players: xi = Ui (S1 ; S2 ; ::::; Sn ; a0 ; a1 ; a2 ; ::::; an )
(1158)
where S1 ; S2 ; ::::; Sn denote the strategies available to players, a0 is common knowledge, and a1 ; a2 ; ::::; an denote the unknown characteristic of player i. Each player knows which strategy is better for it given the strategy space of other players but they have less information about the reactions of other players, aj . They make some subjective estimates about other’s actions while calculating its payo¤ xi . This value gives their reservation or threat point in bargaining. The agreement takes place when actual bargaining and negotiation ends up giving zi and when this value is greater than or equal to what the party i had expected, zi = xi . Negotiation breaks down whenever zi 6 xi . 6.5.2
Model of commitment, credibility and reputation
Parties need to learn from each other to create a more realistic beliefs (bj ) about other players replacing unknown characteristics (a0 ; a1 ; a2 ; ::::; an ) by more accurate representation parameters (b0 ; b1 ; b2 ; ::::; bn )
191
xi = Li (S1 ; S2 ; ::::; Sn ; b0 ; b1 ; b2 ; ::::; bn )
(1159)
Beliefs on these parameters could be formed on the basis of history, principles and values of parties and key personalities of the party and studying their relations to other players. Convergence on beliefs among all parties occurs when they understand and trust each other. This gives credibility to the outcome of the game. Equilibrium in such case is more certain and e¢ cient and generates greater payo¤ for parties and welfare of the country. 6.5.3
Endogenous intervention: change in beliefs
When the …rst CAN dissolved without promulgating a constitution of Nepal on May 28, 2012 people changed their beliefs about the true intention and commitment of the UCPN (Maoists) towards development and their ability to promulgate a constitution of Nepal. They re…ned their beliefs about the Nepali Congress and UML and other Tarai based parties. This change is re‡ected in the structure of the second CAN that was elected on November 19, 2013. Still this was a hung CAN as before but the share of the Nepali Congress increased to 33.9 from 19.3 percents and that of UML increased to 30.4 from 18.0 percents. The share of UCPN (Maoist) reduced to 14 from 38.1 percents. The UCPN (M) is no longer in a position of dreaming a totalitarian system in Nepal. After this verdict parties have committed to promulgate the constitution of Nepal by Jan 22, 2015. Committees in the CAN-II have been able to resolve many disputes but still have not converged in their opinions regarding the structure of governance or that of federal system till the end of September 2014. Table 68: Members of First Constituent Assembly by Political Parties in Nepal Total Maoist NC UML MPRF TMLP Others Total 601 229 116 108 53 21 74 FPTP 240 120 38 33 29 9 11 Proportional 335 100 73 70 22 11 59 Nomination 26 9 5 5 2 1 5 Percentage 100% 38.1% 19.3% 18.0% 8.8% 3.5% 12.3% Source: Constituent Assembly of Nepal (CAN).
Table 69: Members of Second Constituent Assembly by Political Parties in Nepal Total NC UML Maoist MPRF RPP Others Total 601 204 183 84 15 11 103 FPTP 240 105 91 26 4 0 14 Proportional 335 91 84 54 10 10 86 Nomination 26 8 8 (0) 4 (0)1 1 (0)3 Percentage 100% 33.9% 30.4% 14.0% 2.5% 1.8% 17.1% Source: Constituent Assembly of Nepal (CAN). Still leaders of Nepal seem to be confused in understanding the basic fact that the gains from he commitment and cooperation should be much larger than of noncooperation to form coalition
192
or in releasing that the bene…ts of dynamic optimisation are far greater than zero sum game being played at the moment. It is important to rethink about the true and realistic social welfare function such as W (Y; S) where Y denotes the level of aggregate economic activities and its growth rates and S the stability of the system, can be one way to redirect resources wasted in the process of unsuccessful coalition formation to bring more e¢ cient and Pareto optimal solution. Reinvigorate the spirits of April 2006 Revolution. Nepal’s per capita income is one third of India and about 12 percent of China though it had similar per capita income with them till 1980. Political instability in the last two decades has been very costly to Nepal. These could have been decades of spectacular growth but turned into disaster. There cannot be bigger irony than this in the context of Nepal and cooperative strategies of each political party is the only way to sort out this problem. Credibility, respect and commitment only can make this happen. B h a tta ra i K . (2 0 1 3 ) C o a litio n fo r c o n stitu tio n a n d e c o n o m ic g row th in N e p a l, Inte rn a tio n a l J o u rn a l o f G lo b a l S tu d ie s (IJ G S ), 1 :1 , F e b r u a r y, 1 - 4
B h a t t a r a i K . ( 2 0 1 1 ) C o n s t it u t io n a n d E c o n o m ic M o d e ls fo r t h e Fe d e r a l R e p u b lic o f N e p a l, E c o n o m ic J o u r n a l o f N e p a l, Vo l. 3 3 , N o .1 , J a nu a ry _ M a rch , Issu e N o . 1 2 9 , p . 1 -1 5
B h a tta ra i K . (2 0 1 1 ) E m p ty C o re in a C o a litio n : W h y N o C o n s titu tio n in N e p a l? , In d ia n J o u rn a l o f E c o n o m ic s a n d B u s in e s s , 1 0 :1 :1 1 9 1 2 6 ,M a rch 2 0 1 1
B h a tta ra i K . (2 0 0 7 ) M o d e ls o f E c o n o m ic a n d P o litic a l G row th in N e p a l, S e ria ls P u b lic a tio n , N e w D e lh i.
B h a tta ra i K . (2 0 0 6 ) C o n se q u e n c e s o f A p ril 2 0 0 6 R e vo lu tio n a ry C h a n g e s in N e p a l: C o ntinu a tio n N e p a le se D ile m m a , In d ia n J o u rn a l o f E c o n o m ic s a n d B u sin e ss, 5 :2 :3 1 5 -3 2 1 .
C rip p s, M .W .(1 9 9 7 ) B a rg a in in g a n d th e T im in g o f Inve stm e nt, Inte rn a tio n a l E c o n o m ic R e v ie w , 3 8 :3 :A u g .:5 2 7 -5 4 6
D ix it A v in a sh (1 9 8 7 ) S tra te g ic B e h av io u r in C o nte sts, A m e ric a n E c o n o m ic R e v ie w , D e c ., 7 7 :5 :8 9 1 -8 9 8 .
K u h n H . W . ( 1 9 9 7 ) C l a s s i c s i n G a m e T h e o r y, P r i n c e t o n U n i v e r s i t y P r e s s .
M a sk in E , J . M o o re (1 9 9 9 ) Im p le m e nta tio n a n d R e n e g o tia tio n , R e v ie w o f E c o n o m ic S tu d ie s, 6 6 ,1 , 3 9 -5 6
M a ila th G . J . a n d L . S a m u e lso n (2 0 0 6 ) R e p e a te d G a m e s a n d R e p u ta tio n s: lo n g ru n re la tio n sh ip , O x fo rd .
M ye rso n R (1 9 8 6 ) M u ltista g e g a m e w ith c o m m u n ic a tio n , E c o n o m e tric a , 5 4 :3 2 3 -3 5 8 .
P a th a k P a n d T S ö n m e z (2 0 1 3 ) S ch o o l A d m is s io n s R e fo rm in C h ic a g o a n d E n g la n d : C o m p a rin g M e ch a n is m s b y T h e ir Vu ln e ra b ility to M a n ip u la tio n .' A m e ric a n E c o n o m ic R e v ie w , 1 0 3 (1 ): 8 0 -1 0 6 .
R ile y J G (2 0 0 1 ) S ilve r S in g a ls : T w e n ty -F ive Ye a rs o f S c re e n in g a n d S ig n a llin g , J o u rn a l o f E c o n o m ic L ite ra tu re , 3 9 :2 :4 3 2 -4 7 8
R o t h A E . ( 2 0 0 8 ) W h a t h a v e w e l e a r n e d f r o m m a r k e t d e s i g n ? , E c o n o m i c J o u r n a l , 1 1 8 ( M a r c h ) , 2 8 5 –3 1 0 .
R u b in ste in A (1 9 8 2 ) P e rfe c t e q u ilib riu m in a b a rg a in in g m o d e l, E c o n o m e tric a , 5 0 :1 :9 7 -1 0 9 .
S h a p le y L (1 9 5 3 ) A Va lu e fo r n P e rso n G a m e s, C o ntrib u tio n s to th e T h e o ry o f G a m e s I I, 3 0 7 -3 1 7 , P rin c e to n .
S h a p le y L lo y d S . a n d M a r t in S h u b ik ( 1 9 6 9 ) P u r e C o m p e t it io n , C o a lit io n a l P o w e r , a n d Fa ir D iv is io n , In t e r n a t io n a l E c o n o m ic R e v ie w , 10 , 3, 33 7-36 2.
S u tto n J . (1 9 8 6 ) N o n -C o o p e ra tive B a rg a in in g T h e o ry : A n Intro d u c tio n , R e v ie w o f E c o n o m ic S tu d ie s, 5 3 , 5 ., 7 0 9 -7 2 4
193
6.6
Equivalence of Core in Games and Core in a General Equilibrium Model
Both game theory and general equilibrium models analyse optimal choices of consumers and producers faced with resource constraints in which the essential process involves bargaining over the gains from the intra and intertemporal trade on goods, services and …nancial assets. In terms of game theory the core of a bargaining game is given by the payo¤ from a non-blocking coalition. It is a Pareto e¢ cient point. Similarly core of a general equilibrium lies in the contract curve where it is di¢ cult to make one economic agent better o¤ without making another worse o¤. The core of the coalition in the game and core of the equilibrium in a general equilibrium model represent basically the same thing. The optimal allocation of resources to economic agents possible with given endowments con…rm to the …rst and second theorems of welfare economics. Abstract solutions of both models can characterise the optimal allocation of resources after more complex bid and o¤er interactions among economic agents. Debreu and Scarf (1963) have proven the equivalence of a competitive equilibrium to the core of the game for economies with and without production by contradiction when preferences are non-satiable, strictly convex and continuous. Scarf (1967) theorem states that a balanced n person game has a nonempty core. This is best illustrated in terms of a Venn diagram with three players as given in Figure below. Assume a pure exchange economy in which each individual i is endowed with ! i endowments, i =1. . . n. Let the competitive allocations beX xi . Then the competitive equilibrium implies X X X xi = ! i with usual preferences, u xSi u (xi ). In the n person game,T = fSg , the i
i
i
i
collection of coalitions, is called balanced X collection if it is possible to …nd factors to weight value of allocations to each coalition such that i = 1 . Competitive allocations are proven to be in T =fSg S fig
core using these weights as: n X X X xi = i
i T =fSg S fig
S i xi
=
X
X
S T =fSg i2S
xSi
=
X
X
S S2T i2S
Shapley Shubik Core in a Venn Diagram
194
n X X !i = !i i
T =fSg S fig
S
=
X !i i
(1160)
Consider three player game as presented in Figure 3. By 2N 1 rule for possible number of coalitions in N person games, there are seven possible coalitions: f1g,f2g ,f3g ,f1; 2g , f1; 3g,f2; 3g ,f1; 2; 3g . There is some parallel between the value of a property in the central business district as the values of coalition in intersection can be far greater than values under no coalition; in addition in a bargaining game there X can be externality from the bargain as shown by points E around three circles. The condition i = 1 required for the core thus represents summative weight assigned T =fSg S fig
to these individual coalitions. Thus the competitive equilibrium is equivalent to the allocation at the core, “An exchange economy with convex preferences always gives rise to a balanced n person game and such will always have a nonempty core (Scarf (1967)).” Above state principle is generally true under full information. However, it does not work under incomplete information. Competitive …nancial markets are perfect under when all agents that have complete information and are e¢ cient in processing such information. This assumption, however, is not always correct. Financial markets are full of asymmetric information, activities of one set of players depend on actions taken by another set of players and the amount of information they have impacts on the likely choices of others. This requires incentive compatible mechanisms for e¢ cient allocation of …nancial resources. Bejan C and J C Gomez (2009) Core extensions for non-balanced TU-games, International Journal of Game Theory, 38:3-16. Bullard James, Alison Butler (1993) Nonlinearity and Chaos in Economic Models: Implications for Policy Decisions, Economic Journal, 103, 419: 849-867 195
Lipsey R. G. and K. Lancaster (1956 - 1957) The General Theory of Second Best, Review of Economic Studies, 24, 1,11-32 Nash J. (1951) Non-cooperative games, Annals of Mathematics, 54:286-295 Roth, A., Erev, I., 1995. Learning in extensive-form games: Experimental data and simple dynamic models in the intermediate term. Games Econ. Behav. 8, 164–212 Scarf H. (1967) Core of n Person Game, Econometrica, 35:50-69. Shapley L (1953) A Value for n Person Games, Contributions to the Theory of Games II,307317, Princeton. Wooders HM and 52:6:1327-1350.
6.7
WR Zame (1984) Approximate cores of large games, Econometrica,
Labour Market and Search and Matching Model
Producers use labour to produce goods and services. A production function shows how labour complements with other inputs in production and the marginal productivity of labour shows the additional unit of output produced by each additional unit of labour. Thus demand for labour is derived from the demand for output. On the supply side every working age person has 168 hours a week, 720 hours per months or 8760 hours per year of time endowment which can be allocated between work and leisre. How many hours does one work and how much is spent in free time really depends upon the preference between consumption and leisure on one side and the job vacancies on the other. In theory ‡exibility of real wages guarantees equality between demand and supply in the labour in a competitive labour market. However, the labour is far from being a perfectly competitive market. Firms exercise monopoly powers, acting as monopsonists in the labour market or use their market power in order to retain more e¤ective workers. Hiring decisions of …rms also are dependent on the aggregate demand. Firms hire more workers during expansion but are reluctant of recruit any workers during the contraction. A signi…cant number of workers become unemployed as a consequence. Given a production fucntion that related output (Yt ) to capital (Kt ), technology (At ) and labour (Lt ) Yt = Kt (At Lt )
1
0
6.8
Exercise 14: Search Equilibrium Search Equilibrium
1. Consider a bargaining model between …rms and workers Matching function aggregates vacancies and unemployment with job creation as: M =V U
(1171)
where M denote the number of matching of vacancies and job seekers, V is number of vacancies and U the number of unemployed, is the parameter between zero and one.Job seekers and employers bargain over expected earnings by maximising the Nash-product of the bargaining game over the di¤erence between the earnings from work (W) rather than in being unemployed (U) and earnings to …rms from …lled and vacant jobs.
199
(Wi
U ) (Ji
V)
(1172)
(a) Show that the dynamics of unemployment depends on the rate of job destruction, (1 , and the rate of job creation,
u)
q ( ) u. Derive the job creation curve. 1. (a) Optimal job creation or (demand for labour curve) shows how …rms balance the marginal revenue product of labour to wage and hiring and …ring costs. Derive the Beveridge curve.
7
L7: Game theory: Principal Agent and Mechanism Games and Auctions
7.1
Original Ideas
Issues of priciple agent games are disscuss in general terms in articles by Harsanyi (1967) Shapley and Shubik (1969). Hurwicz (1973), Spence (1977) , Riley (1979),Sobel (1985), Myerson (1986) Sutton (1986), Milgrom , Roberts (1986), Dixit Avinash (1987) Cho and Kreps (1987) , Rogerson (1988), Moore (1988), Dawes and Thaler (1988), McCormick (1990) Caminal (1990), Frank, Gilovich, and Regan (1993), Jin (1994), Markusen (1995), Camerer and Thaler (1995), Lundberg and Pollak (1996), Mookherjee and Ray (2001),Fehr, Gächter and Simon (2000), Besley and Ghatak (2001), Acemoglu (2001) and Mailath and Samuelson (2006) (see also Watt (2011) and Snyder and Nicholson (2011)). These explain how the moral hazard and adverse selection -asymmetric information in‡uence in decision making of economic agents. Moral hazard Owners of a …rm principals (P) and workers as agents (A) play a production game in which agent exerts e¤orts (a) in return of income (y) and principal maximises net pro…t. Agent can put high or low e¤orts and P cannot observe it. Utility of agent at work is given by V = u(y)
V0
a
(1173)
0
This must be greater than a reservation utility V that is available from alternative work. The income level that an individual worker requires is then given by y=V
1
V0+a
(1174)
Less informed P can make sure that A exerts good e¤ort by making wage contract as V = v(y ) + (1
) v(y1 ) < V 0
(1175)
Principal’s objective when a is observable is then to maximize pro…t by producing x subject to the participation constraint max zi =
i
(x1
y1 ) + (1
subject to
200
i ) (x2
y2 ) i=h,l
(1176)
i v(y1 )
+ (1
i ) v(y2 )
V0
ai
(1177)
There is uncertainty in production resulting in x1 and x2 levels of production, x1 < x2 . Because of this uncertainty x1 may happen despite A putting high level e¤ort, which P cannot observe. 7.1.1
Full information scenario L=
i
(x1
y1 ) + (1
i ) (x2
y2 ) +
i v(y1 )
+ (1
i ) v(y2 )
ai
V0
(1178)
First order conditions (for high e¤ort case) @zh = @y1
h
+ v 0 (y1 ) = 0
(1179)
@zh = @y2
(1
h)
+ (1
h) v
0
(y2 ) = 0
(1180)
@zh = @
(1
h)
+ (1
h) v
0
(y1 ) = 0
(1181)
V0 =0
(1182)
i v(y1 )
+ (1
i ) v(y2 )
ai
Thus in the full information case 1
v 0 (y1 ) = v 0 (y2 ) =
=) y1 = y2
(1183)
Thus the owners of the company force managers to put the same level of e¤orts. Risk-neutral owers bear all risk. P can design contracts similarly if they like A to put low e¤orts. L= 7.1.2
l
(x1
y1 ) + (1
l ) (x2
y2 ) +
l v(y1 )
+ (1
l ) v(y2 )
V0
al
(1184)
Incomplete information scenario
P cannot observe a of A; therefore they must design a contract which makes A put ah This requires adding an incentive compatibility constraint. h v(y1 )
+ (1
h ) v(y2 )
ah
l v(y1 )
+ (1
l ) v(y2 )
al
(1185)
Then the problem is modi…ed as max zi =
i
(x1
y1 ) + (1
i ) (x2
y2 ) i=h,l
(1186)
V0
(1187)
subject to i v(y1 )
+ (1
i ) v(y2 )
and
201
ai
h v(y1 )
L
=
l
[
(x1
+ (1
y1 ) + (1
h v(y1 )
+ (1
h ) v(y2 )
ah
l ) (x2
y2 ) +
h ) v(y2 )
l v(y1 )
+ (1
l v(y1 )
ah
l v(y1 )
l ) v(y2 )
+ (1 (1
l ) v(y2 ) l ) v(y2 )
al
al
(1188)
V0 +
+ al ]
(1189)
The optimising conditions in this case are given by @zh = @y1 @zh = @y2
(1
h
h)
@zh = @
+ v 0 (y1 ) + (
+ (1
l v(y1 )
@zh = h v(y1 ) + (1 @ From these conditions
h) v
+ (1
(
(y2 ) + (
l ) v(y2 )
h ) v(y2 )
+
0
ah
l)
h
l)
h h
1 < v 0 (y1 )
(1203)
Here ' is a random variable with E' = 0 If the shareholder could observe e¤orts, the optimal contract would be w = w is a …xed wage. Here this from the participation constraint is w = w0 +
Re2 2
(1204)
maximisationof shareholder’s expected pro…t is : E( )=E e+'
Re2 2
w0
=e
w0
Re2 2
(1205)
@E ( ) 1 1 = 1 Re = 0 () e = if w0 @e R 2R This is the optimal solution when shareholders could obseve the e¤ort of managers. Now suppose the e¤orts are not observable. Consider a linear incentive scheme: w( ) = a+b
(1206)
(1207)
What is the expected utility of A with linear scheme: Eu a + be + b'
Re2 2
() b
Re =) e =
b R
(1208)
E¤ort grows with the slope of the incentive scheme. If b = 1; then e = e : The expected utility at this level of e¤orts is Eu a +
b2 + b' R
b2 2R
() Eu a +
b2 + b' 2R
(1209)
Shareholder’s expected pro…t e
= E [e + '
a
be
b'] =
b R
Linear optimal scheme: 204
a
b2 b = (1 R R
b)
a
(1210)
max
e
b (1 R
=
b)
a
(1211)
w0
(1212)
subject to Eu a +
b2 + b' 2R
Substitute a from the participation constraint e
Eu
+
b2 + b' 2R
b R
= w0
(1213)
di¤erentiating wrt b Eu0
1
b R
+ Eu0 ' = 0
(1214)
This gives b = 1: This value of the linear scheme optimises pro…t for the shareholder as the agent puts maximum e¤orts at work. Table 70: Principal Agent Games Principal Agent Action Shareholders CEO Pro…t maximisation Landlord Tentants work e¤ort People Government Political power Manager Workers Work e¤ort Central Banks Banks Quality of credit Patient Doctor Intervention Owner Renter Maintenance Insurnace company Policy holder Careful behaviour
7.1.3
Impacts of Assymetric (incomplete) Information on Markets Equilibrium is ine¢ cient relative to full information case Signalling can improve the e¢ ciency: warranty and guarantee Screening: revealing the risk type of agent Credit history from credit card companies Government can improve the market by setting high standards of business contracts or bailing out troubled ones (Northern Rock, Bear Stearns, Lehman Brothers) Right regulations –Financial Services Authority, Fair trade commissions; O¢ ce of standards; Bank of England 205
Moral hazard (hidden action) Probability of bad event is raised by the action of the person iPeople who have theft insurance are likely to haven low quality locksthat are easy to break (in cars, houses, bicycle (car)) most likely to claim insurances Remedy: deductible amount; to ensure that some customers take care in security. 7.1.4
Adverse Selection (hidden information) Problem Uncertainty about the quality of good or services honest borrowers less likely to borrow at higher interest rates. low quality items crowd out high quality items risky borrowers drive out gentle borrowers in the …nancial market. Theft insurance; health insurance; people from safe area are less likely to buy theft insurance; only unsafe customers end up buying theft insurance healthy people are less likely to buy health insurance
Asymmetric information in Used Car Market -Akerlof’s Model of Asymmetric Information Sellers know exactly quality of cars but buyers do not. Equilibrium is a¤ected when sellers have more information than buyers. Market has plums: good cars and lemons: bad cars Seller knows his quality of cars but buyers do not Market for good cars disappear because of existence of bad cars in the market. Demand for high quality car falls and demand for low quality cars rise. Ultimately only low quality cars remain in the market. signals: warranty and Guarantee Providing warranty less costly for high quality cars as they last long. Warranty is costly for low quality cars as they frequently break down. Buyers can decide whether a car is good or bad looking at the warrantee and pay appropriately. Right signalling can remove ine¢ ciency due to incomplete information. Markets for both types of car can operate e¢ ciently by right signals of warranty and Guarantee 206
Pooling, Separating and Mixed Equilibrium Complete market failure pooling equilibrium (same price for good and bad cars; good cars disappear from the market) Complete market success Separating equilibrium where players act as they should according to the signal (prices according to quality) Partial market success (both good and bad cars are bought, some feel cheated) Near Market failure (mixed strategies) Bayesian updating mechanism at work 7.1.5
Signalling and Incentives
1.Education as a signal of productivity Level of education signals quality of a worker. Given the cost of education it is easier for a high quality worker to complete a degree than for a low quality worker. In an e¢ cient market potential employers take level of education as a signal in hiring and deciding wage rates paid to its employees. Spence (1973) model was among the …rst to illustrate how to analyse principal agent and role of signalling in the job market. Pooling equilibrium Consider a situation where there are N individuals applying to work. In absence of education as the criteria of quality employers cannot see who is a high quality worker and who is a low quality worker. Employers know that proportion of workers is of high quality and (1- ) proportion is of bad quality. Therefore they pay each worker an average wage rage as: w = wh + (1
) wl
(1215)
Every worker gets the average wage rate ; there is no wage premium for higher quality in pooling equilibrium. If more productive worker is worth 40000 and less productive worker is worth 20000 and =0.5 then the average wage rate will be 30000; w = wh + (1 ) wl = 0:5 (40000) + 0:5 (20000) = 30000. Let c denote the cost of education. It is worth for high quality worker to go to school only if the wage di¤erence having and not having education is greater than the cost of education which is given by wh
w = wh
[ wh + (1
) wl ]
(1216)
Simpli…cation of this condition implies a signalling condition c
(1219)
This is possible if the cost of education is 5000; then wage net of education cost for high quality is 35000 which is above the pooling wage rate. This makes sense to signal by choosing higher education. Signalling is optimal in this case; fraction of workers will signal by going to education. Aggregate labour cost will be the same but wages will be paid according to the productivity of workers as re‡ected by the level of education of workers.
208
Excel calculations While making a hiring decision employers take level of education as a signal of quality of workers. Government Policy and Signalling It is important to have optimal amount of signalling –too little or too much signalling generates ine¢ cient result. Empirical …nding on signalling is mixed. Public policy could be designed to generate right amount of signalling as following: 1. It can create separating equilibrium by subsidizing education of more able workers. It can ban on wasteful signalling by banning schools that do not produce good workers. 2. High education provides signals, employers pay according to this signal, this will a¤ect the distribution of wages. 7.1.6
Education Level- A Signal of Productive Worker An employer does not know is more productive and who is less productive It pays the same wage rate to both productive and unproductive workers market is ine¢ cient, it drives out more productive workers. Workers can signal their quality by the level of educational attainment, then market may work well. Less costlier for high quality worker to get education. costlier for low quality worker to get the speci…ed education. 209
so the low quality worker gets no education, but the higher quality worker gets education. Employers pay according to the level of education. Education works as a signalling device and makes the market e¢ cient. Education separates the equilibrium. Education Level- A Signal of Productive Worker Consider a level of education e c1 e
c2 e =) c1
c2
(1220)
Cost of eduction of unproductive worker is much higher c2 e < (a2
a1 ) < c1 e
(1221)
Cost of education relative to productivity of low and high quality workers for education e (a2
a1 ) c1
7.2
k2 More Speci…cally p ut (w; e) = 42 wt
kt e1:5
k1 = 2; k2 = 1 w1 = e; w2 = 2e
(1224)
Level of education chosen by less productive worker In perfect information equilibrium, …rms pay according to the marginal productivity Wage of less productive worker: w1 = e; The type 1 worker’s optimisation problem p Max ut (w; e) = 42 wt e
p kt e1:5 = 42 e
@ut (w; e) : 1 = 42 p @e 2 e
2e1:5
1
3e 2 = 0
(1225)
(1226)
1 1 42 =7 42 p = 3e 2 =) e1 = 6 2 e
(1227)
It is optimal for the less productive worker to takes only seven years of education Level of education chosen by more productive worker Wage of less productive worker: w2 = 2e; The type 1 worker’s optimisation problem p Max ut (w; e) = 42 wt e
p kt e1:5 = 42 2e
@ut (w; e) : 1 = 42 p @e 2 2e
2
e1:5
1
1:5e 2 = 0
1 1 1 42 42 p = 1:5e 2 ; 42 p = e =) e2 = = 19:8 2:121 2e 1:5 2
It is optimal for the more productive worker to takes 19.8 years of education. Government Policy and Signalling 211
(1228)
(1229)
(1230)
It is important to have optimal amount of signalling – too little or too much signalling generates ine¢ cient result. Empirical …nding on signalling is mixed. Public policy could be designed to generate right amount of signalling as following It can create separating equilibrium by subsidizing education of more able workers. It can ban on wasteful signalling by banning schools that do not produce good workers. High education provides signals, employers pay according to this signal, this will a¤ect the distribution of wages.
7.3
Popular Principal Agent Games
Principal Agent Model in Job Market: Incomplete Information and Adverse Selection Principal wants to produce output employing workers with a scheme of wage contract that matches e¤orts put by a worker to produce. Worker knows his type but the principal does not. Principal knows the distribution of quality of workers F(s), where s denotes either good or bad state such as probability of observing good is 0.5 and of bad 0.5. Principal o¤ers the agent a wage contract W(q). Worker accepts or rejects this contract based on self-selection and participation constraints. Objective of Principal and Agents Basically worker evaluates the utility from the wage and disutility from work and decides the amount of work to put in. Output from good workers is q (e; good) = 3e and from bad state is q (e; bad) = e If agent rejects the contract there is no work both worker and principal get zero payo¤. If worker accepts the contract Agent’s utility: UA (e; w; s) = w
e2
(1231)
Principal’s pro…t: . . Vp (q; w) = q
w
(1232)
Optimal level of e¤orts by good and bad workers Good worker maximises M ax UG = wG eG
e2G = 3eG
e2G
The …rst part is wage income and the second part of disutility of work.
212
(1233)
The optimal level of e¤orts by good agent is: 3
2eG = 0 =) eG = 1:5
(1234)
Bad worker’s Objective and Optimal E¤orts e2B = eB
M ax UB = wB eB
1
e2B
(1235)
2eG = 0 =) eG = 0:5
(1236)
The principal does not know what levels of e¤orts are appropriate for good and bad workers. Principal’s Objective Principal maximises expected pro…t M ax
qG ;qB ;wG ;wB
UP = [0:5 (qG
wG ) + 0:5 (qB
wB )]
(1237)
by designing separate contracts for good (qG ; wG ) and bad workers (qB ; wB ) and . Wage for good worker: wG = q (e; good) = 3e or e = q3G Wage for bad worker: wB = q (e; bad) = e or e = qB Incentive Compatibility Constraints for Agents Self selection constraint for good worker 2
qG 3 Self selection constraint for bad worker UG = wG
e2G = wG
UB = w B
e2B = wB
UG = wB
e2B = wB
(qB )
2
UB = w G
2
qB 3
(1238)
e2G
(1239)
Participation constraints for good worker 2
qG 3
UG = wG
0
(1240)
Participation constraint for bad worker UB = wB
2
(qB )
0
(1241)
Binding Constraints Participation constraint of bad worker 2 wB = qB
(1242)
Self selection constraint for good worker qG 2 + wB 3 Principal’s Optimal Solution wG =
qB 3
2
=) wG =
qG 3
2
2 + qB
Principal includes agents’optimal choices into his utility function 213
qB 3
2
(1243)
M ax
qG ;qB ;wG ;wB
UP = [0:5 (qG
Including binding constraints of agents: h qG 2 2 UP = 0:5 qG M ax + qB 3
qG ;qB ;wG ;wB
wG ) + 0:5 (qB
qB 2 3
+ 0:5 qB
wB )]
2 qB
i
Now principal decides how much to produce from each type of worker First order conditions with respect to qG and qB @UP = 0:5 1 @qG @UP @qB qB
=
0:5
=
0:265
2qB 9
2qB
2qG 9
= 0 =) qG = 4:5
+ 0:5 (1
(1244)
2qB ) = 0 =) 34qB = 9 =) (1245)
Incentive Compatible First Best Choices of Good and Bad Worker Now wages can be found from the constraints 2
2 wB = qB = (0:265) = 0:07
wG =
qG 3
2
2 + qB
qB 3
2
=
4:5 3
2
+ (0:265)
2
(1246) 0:265 3
2
= 2:32
(1247)
Thus in the presence of information asymmetry , the e¤orts by the good worker is at the …rst best level as the bad e¤ort by him is not as attractive as the good e¤ort. It is not pro…table for good worker to pretend as a bad worker. Good worker is not attracted by the contract for bad worker. It is very costly for the bad worker to accept the contract of good worker. Bad worker’s …rst best to put low e¤ort. Incentive compatible game on renting a piece of agriculatural land If a worker puts x amount of e¤ort, the land produces y = f (x) Then the land owner pays worker s(y). The land owner wants to maximise pro…t = f (x) s(y) = f (x) s(f (x)) Worker has cost of putting e¤ort c(x) and has a reservation utility, u The participation constraint is given by . s(f (x)) c(x) u Including this constraint maximisation problem becomes max = f (x) s(f (x)) subject to sf (x) c(x) u Solution: marginal productivity equals marginal e¤orts f 0 (x)) c0 (x) 214
Incentive compatible game on rending a piece of agriculatural land (a) renting the land where the workers pays a …xed rent R to the owner and takes the residual amount of output, at equilibrium (1248) f (x ) c(x ) R = u (b) Take it or leave it contract where the owner gives some amount such as B
c(x ) = u
(1249)
(c) hourly contract s(f (x)) = wx + K
(1250)
(d) sharecropping, in which both worker and owner divide the output in a certain way. In (a)-(c) burden of risks due to ‡uctuations in the output falls on the worker but it is shared by both owner and worker in (d). Which of these incentives work best depends on the situation. Acemoglu Daren (2001) A Theory of Political Transitions, the American Economic Review, 91:4:938-963 Bardhan Pranab (2002) Decentralization of Governance and Development, Journal of Economic Perspective, 16:4:185-205. Basu Kaushik (1986) One Kind of Power, Oxford Economic Papers, 38:2:259-282. Besley T and M Ghatak (2001) Government versus Private Ownership of Public Goods, Quarterly Journal of Economics, 116:4:1343-1372. Binmore K (1999) Why Experiment in Economics? The Economic Journal 109, 453, Features Feb. pp. F16-F24 Boyd, J.H. and Prescott, E. C. (1986) Financial Intermediary-Coalitions, Journal of Economic Theory, 38: 211-232 Caminal R. (1990) A Dynamic Duopoly Model with Asymmetric Information, Journal of Industrial Economics 38, 3 , 315-333 Cho I.K. and D.M. Kreps (1987) Signalling games and stable equilibria, the Quarterly Journal of Economics, May179-221. Dixit Avinash (1987) Strategic Behaviour in Contests, American Economic Review, Dec., 77:5:891-898. Fundenberg D and J.Tirole (1995) Game Theory, MIT Press. Gardener R (2003) Games of Business and Economics, Wiley, Second Edition. Ghosal S and M. Morelli (2004) Retrading in market games, Journal of Economic Theory, 115:151-181. Harsanyi J.C. (1967) Games with incomplete information played by Baysian Players, Management Science, 14:3:159-182. 215
Hurwicz L (1973) The design of mechanism for resource allocation, American Economic Review, 63:2:1-30. Jin J. Y. (1994) Information Sharing through Sales Report, Journal of Industrial Economics 42, 3, 323-333 Mailath G. J. (1989),Simultaneous Signaling in an Oligopoly Model Quarterly Journal of Economics 104, 2, 417-427 Maskin E and J Tirole (1990) The principal-agent relationship with an informed principle, Econometrica, 58:379-410. McCormick B. (1990) A Theory of Signalling During Job Search, Employment E¢ ciency, and 'Stigmatised' Jobs Review of Economic Studies 57, 2, 299-313 Mookherjee D and D Ray (2001) Readings in the theory of economic development, Blackwell. Moore J. (1988) Contracting between two parties with private information, Review of Economic Studies, 55: 49-70. Mirrlees James A. (1997) Information and Incentives: The Economics of Carrots and Sticks The Economic Journal , 107, 444,1311-1329 Mailath G. J. and L. Samuelson (2006) Repeated Games and Reputations: long run relationship, Oxford. Milgrom P., J. Roberts (1986) Price and Advertising Signals of Product Quality Journal of Political Economy 94, 4,796-821 Myerson R (1986) Multistage game with communication, Econometrica, 54:323-358. Nash J. (1953) Two person cooperative games, Econometrica, 21:1:128-140. Rasmusen E (2007) Games and Information, Blackwell. Rodrik D. (1989) Promises, Promises: Credible Policy Reform via Signalling Economic Journal 99, 397, 756-772 Rogerson W.P.(1988) Price Advertising and the Deterioration of Product Quality Review of Economic Studies 55, 2 (Apr., 1988), pp. 215-229 Riley J.P. (1979) Noncooperative Equilibrium and Market Signalling American Economic Review 69, 2, 303-307 Rubinstein A (1982) Perfect equilibrium in a bargaining model, Econometrica, 50:1:97-109. Ryan M.J. (1998) Multiple criteria and framing of decisions, Journal of Information and Optimisation Sciences, 19:1:25-42. Shapley L and M. Shubik (1969) On Market Games, Journal of Economic Theory, 1:9-25. Sobel J. (1985) A Theory of Credibility Review of Economic Studies 52, 4, 557-573
216
Sutton J. (1986) Non-Cooperative Bargaining Theory: An Introduction, Review of Economic Studies, 53, 5., 709-724 Spence M. (1977) Consumer Misperceptions, Product Failure and Producer Liability, Review of Economic Studies 44, 3, 561-572
7.4
Exercise 15: Principal Agent Problem Problem 10: Asymmetric information: Principal Agent Problem
1. Project B earns more but is riskier than project A. Probability of success of projects A and B are given by PA and PB respectively. a. Illustrate how the rate of interest rate should be lower in project A than in project B in equilibrium? b. Probability of types A and B agents is given by PA and PB respectively. Prove under the asymmetric information a lender charging a pooling interest rate is unfair to the safe borrower A and more generous to the risky borrower B. c. How can agent signal its worth? How can the lender ascertain the degree of moral hazard in B? 2. Principal wants to produce output by employing workers with a scheme of wage contract that matches e¤orts put by a worker to produce. Worker knows his type but the principal does not but the principal knows the distribution of quality of workers F (s), where s denotes either good or bad state. Probability of observing good is 0.5 and of bad 0.5. Principal o¤ers the agent a wage contract W (q) worker accepts or rejects this contract based on selfselection and participation constraints. Basically worker evaluates the utility from the work and disutility from work and decides the amount of work to put in. Output from good worker is q(e; g) = 3e and from bad state is q(e; b) = e . Both of them are risk neutral. If agent rejects the contract there is no work both worker and principal get zero payo¤ otherwise the e3 and . principal = V (q w) = q w agent = U (e; w; s) = w (a) Determine the level of e¤orts by good and bad workers. (b) Formulate the participation and incentive compatibility constraint for workers. (c) What is the principal’s objective function? (d) What wage rates are paid to good and bad workers?
7.5
Mechanism Design for Price Discrimination: Low Cost Airlines Example
There are three steps in mechanism design. 1. Principal designs a mechanism or contract 2. agents accept or reject mechanism 3. Those who accept the mechanism play the game. Consider a case of monopolist which supplies q with marginal cost c and tari¤ (price) T. Its objective is
217
U1 (q; T; ) = V (q)
T
(1251)
V is a common knowledge, is private information. There are high and low type buyers. = with probability p = with probability p and p+ p = 1 First best solution: If the seller knew its tari¤ would be V (q) = T Its pro…t would be V (q) cq and V 0 (q) = c: Most often it does not know , therefore o¤ers q; T and q; T bundles. Then the expected pro…t will be Eu0 = p q; T + p q; T
(1252)
Sellers’constraints 1. consumers need to be willing to purchase and this requires ful…llment of individual rationality constraint and participation constraint. IR1 :
V q
T
0
(1253)
IR2 :
V (q)
T
0
(1254)
Incentive compatibility requires that consumers consume the bundel intended for them. IC1 : IC2 :
V q
T
V (q)
V (q)
T
V q
T T
Sellers problem is to choose q; T and q; T bundles to maximise her pro…t.
218
(1255) (1256)
Binding constraints are IC1 and IR2 This implies T = V q
T + V (q)
T = V (q) p T
cq + p T
cq = p
(1257)
V q V q
p
(1258) pcq + p V (q)
c
V q = 1
p(
)
cq
(1259) (1260)
p
V (q) = c
(1261)
Allocation is socially optimal Reference Fundenbege and Tirole (1995)), Game Theory, MIT press. Economy and Business Class Ticket Problem for Airlines (Based on Dixit et. al. (2009)) Two types of travellers: economy and business
219
Assume 100 travellers and 70 of them economy type tourists and 30 business type …rst class.
Economy First Class
Cost of the Airlines 100 150
Reservation Price Tourists Business 140 225 175 300
Airline’s Pro…t Tourists Business 40 125 25 150
Economy class tickets cost less than the business class. Business traveller is ready to pay higher price than economy class for both economy and …rst class but the airlines cannot separate them out. Why Mechanism Design for Price Discrimination: Low Cost Airlines Example Economy class tickets cost less than the business class. Business traveller is ready to pay higher price than economy class for both economy and …rst class but the airlines cannot separate them out. Business traveller may well buy economy class ticket rather then business class. Airlines likes to build a mechanism so that business class buy business class tickets and economy class buy economy class ticket. What is the pro…t to the airlines if it knows reservation prices of tourists and business group of travellers? How would this pro…t change in business type buy the economy class ticket? What is the incentive compatible price that the airlines can o¤er to the business group? Incentive Compatible Mechanism What would happen if the split between the business and economy class is 50/50? What will be the optimal reaction of the airlines? Pro…t in an ideal scenario ( perfect price discrimination; if the airlines knew each customer type) 30(300
150) + (140
100) (70)
=
30
=
4500 + 2800 = 7300
150 + 40
70 (1262)
Business travellers have consumer surplus of 225 -140 = 85 in economy class ticket. For this all 30 of may decide to buy economy class ticket. Then the pro…t of the airlines when the airlines fails to screen customers will be (140
100) (100) = 4000
(1263)
Airlines should give consumer surplus of 85 to business traveller and charge them (300-85) = 215. This will alter their pro…t 30(215
150) + (140
100) (70)
=
30
=
1950 + 2800 = 4750
Incentive Compatible and Participation Constraints 220
65 + 40
70 (1264)
Airline initially does not have enough information on types of customers It should design incentive compatible pricing scheme so that business class travellers do not defect to economy class. This requirement is contained in the incentive compatible constraint. If it charges 240 for the business class then the their consumer surplus will be equal (300-240) = 60 from business class travel and (225-165)=60 However 140 is the maximum the tourist class traveller is ready to pay. If the airline raises price to 165 they will lose all tourist travellers. Mechanism requires ful…llment of the participation constraint. Airlines should operate taking account of the participation constraint of tourists and incentive compatible constraint of the business travellers. X < 140 is the participation constraint; incentive compatible constraint is 225 -X < 300-Y or Y < X+75 Mechanism when the composition of travellers change Charging 215 for the business class and 140 for the economy class is the solution to the mechanism design problem. 30(215
150) + (140
100) (70)
=
30
=
1950 + 2800 = 4750
65 + 40
70 (1265)
Suppose the composition of travellers changes to 50% of each. Pro…t with the above price mechanism 50(215
150) + (140
100) (50)
=
50
65 + 40
50
=
3250 + 2000 = 5250
(1266)
It is more pro…table to scrap the tourist class tickets instead and charge the business class its full reservation price 50(300 150) = 50 150 = 7500 (1267) There are relatively few customers but all are willing to pay higher price. There is no problem of screening as the airlines now does not serve to the tourist class at all. 7.5.1
Mechanism for e¢ cient contract for a CEO Owners of a company are concerned about a project that would earn them 600,000 if successful. Probability of success with normal e¤ort from the manager is 60 percent and this can increase up to 80 percent if the manager puts extra e¤orts. The basic salary of the manager is 100,000. He would put extra e¤orts only if he is paid additional amount of at least 50,000. Owners cannot monitor whether the manager is putting high or low e¤orts. 221
a) Is it pro…table to pay extra for the manager? Pro…t without paying extra: 0.6 * 600,000 - 100,000 = 260, 000 Pro…t with extra incentive payment: 0.8 * 600,000 - 150,000 = 330, 000 Extra payment can make up to 70,000 with probability of 0.8. Once extra payment is made how can owners make sure that he puts extra e¤orts? This requires evaluation of incentive compatibility and participation constraints. Mechanism to ensure high e¤orts by a CEO a) Incentive compatibility constraint (s + 0:8b)
(s + 0:6b) > 50; 000
(1268)
0:2b > 50; 000
(1269)
(s + 0:8b) > 150; 000
(1270)
b = 250,000 b) Participation constraint:
s = 150; 000
0:8b;
s = 150; 000
0:8 (250; 000) =
50; 000
(1271)
It is not possible to hire manager with negative salary. At most managers can be conditioned to bonus payment but with zero salary. Mechanism to ensure high e¤orts by a CEO (0 + 0:8b) > 150; 000
(1272)
200; 000 > 150; 000
(1273)
Pay 200,000 and the manager will put maximum e¤ort. c) Is it pro…table to pay extra 200,000 as an incentive payment? Pro…t with incentive payment 0.8 * 600,000 - 200,000 = 280, 000 Pro…t without incentive payment 0.6 * 600,000 - 100,000 = 260, 000 Thus pro…t increases by 20,000 with the incentive payments. 7.5.2
E¢ cient contracts of Land
Proposition 1: Results of …xed fee contract and joint pro…t maximisation are equivalent Proposition 2: Hire contract is incentive incompatible and leads to production ine¢ ciency Proposition 3: Moral hazard problem and production ine¢ ciency exists in revenue sharing contingent contract Proposition 4: Pro…t sharing contract is e¢ cient and free of moral hazard problem Price and cost P = 24
0:5q
222
C = 12q
(1274)
Revenue R = P:q
(1275)
Mechanism design in renting lands Under the joint pro…t maximisation agreement (q) = P:q
C = (24
0:5q) q
0:5q 2
12q = 24q
12q
(1276)
Under the …xed fee contract tenant maximises (q) = P:q
C
F = (24
0:5q) q
12q
F = 24q
0:5q 2
12q
F
(1277)
Under both these arrangements 0
(q) = 24
q
12 = 0
q = 12; p = 18; R = 216; C = 144;
(1278) (q) = 72
(1279)
Mechanism design in renting lands 72 is the total pro…t. It is divided between the tenant and the landlord by their mutually agreed arrangement. Under the …xed fee contract landlord may …x the amount that he needs at 48. Then the residual 24 pro…t goes to the tenant. This arrangement achieves production e¢ ciency, is incentive compatible, ful…ls the participation constraint and motivates to put the optimal e¤ort and solves the moral hazard problem. Hire contract Landowner can hire workers in …xed fee basis, say 12 per unit of output a. This does not motivate tenant to work because his cost per a is also 12 and so does not make any pro…t. Landlord has to raise payment to tenant to say 14 to motivate him to work. Then the pro…t maximisation problem of the landlord will be (q) = P:q
C = (24
0
0:5q) q
(q) = 24
q
q = 10; p = 19; R = 190; C = 120;
14q = 24q
0:5q 2
14 = 0 LL
(q) = 50;
(1280)
(1281) T
(q) = 20
The tenant has incentive to overproduce whenever is paid more than 12. Revenue sharing contract
223
14q
(1282)
Let the landlord enter into a revenue sharing contract whereby she gets 14 th of the revenue and leavening 34 of revenue to the tenant who also bears all production cost. The pro…t function of the tenant is now modi…ed as (q) =
3 P:q 4 0
q
C=
(q) = 6
3 (24 4
8; p = 20; R = 160; C = 96;
=
120;
(q) =
12q
3 q=0 4
=
T
0:5q) q
(1283)
(1284)
LL
(q) =
3 (160) 4
1 (160) = 40 4
(1285)
Pro…t of tenant = 120 - 96 =24 This level of production is not incentive compatible for the land-lord who would be interested in maximising revenue by producing 24. Pro…t sharing contract Now let us assume the landlords and tenants enter into a pro…t sharing deal, say 1/3rd of pro…t goes to the tenant and 2/3rd to the landlord. 1 3
(q) =
1 (P:q 3 0
LL
C) =
(q) = 4
1 24q 3
0:5q 2
1 q=0 3
q
=
12; p = 18; R = 216; C = 144;
(q)
=
48;
T
12q
(1286)
(1287)
(q) = 72;
(q) = 24
(1288)
There are many other situations, including optimal tax designs, optimal price discrimination, fund management, management of theme-park, renting of buildings, collection of taxes or tari¤s, union-management contracts, where these types of models have been applied. 7.5.3
Mechanism for Poverty Alleviation There are three players in the poverty game -poor, rich and government; each has three strategies available to it to play, s, l, and k , cooperation, indi¤erence and non cooperation. The outcome of the game is the strategy contingent income for poor and rich, ytp (s; l; k) and ytR (s; l; k) with the probability of being in particular state like this is given by pt (s; l; k) and R t (s; l; k) respectively and tax and transfer pro…les associated to them. The state-space of the game rises exponentially with the length of time period t. T
224
he objective of these rich and poor households is to maximize the expected utility that is assumed to be concave in income. The government can in‡uence this outcome by choices of taxes and transfers that can be liberal, normal or conservative. Mechanism for Poverty Alleviation: Proposition 1 Proposition 1: The state contingent expected money metric utility of poor is less than that of rich, which can be expressed as: s X l X k X T X
p p p t (s; l; k) t u (yt (s; l; k))
s=1 l=1 k=1 t
'
l X k X T s X X
#
T X TtR (s; l; k)
ytR (s; l; k)
t
T k X l X s X X
Ttp (s; l; k)
p p p t (s; l; k) t u (yt (s; l; k))
+
T X t
Ttp (s; l; k)
#
(1290)
#
p p p t (s; l; k) t u (yt (s; l; k))
(1291)
s=1 l=1 k=1 t
and
s X l X k X T X
R R t (s; l; k) t u
ytR (s; l; k)
s=1 l=1 k=1 t
>
' T s X l X k X X s=1 l=1 k=1
R R t (s; l; k) t u
t
ytR (s; l; k)
T X + TtR (s; l; k) t
225
#
(1292)
Mechanism for Poverty Alleviation:Proposition 4 Proposition 4: Growth requires that income of both poor and rich are rising over time: p p p (s; l; k) < ::::: < Tt+T (s; l; k) (s; l; k) < Tt+1 Ttp (s; l; k) < Tt+1
(1293)
p p p Ytp (s; l; k) < Yt+1 (s; l; k) < Yt+1 (s; l; k) < ::::: < Yt+T (s; l; k)
(1294)
R R R YtR (s; l; k) < Yt+1 (s; l; k) < Yt+1 (s; l; k) < ::::: < Yt+T (s; l; k)
(1295)
Mechanism for Poverty Alleviation:Proposition 5 Proposition 5: Termination of poverty requires that every poor individual has at least the level of income equal to the poverty line determined by the society. When the poverty line is de…ned one half of the average income this can be stated as: ! N 1 1X h p Yt (s; l; k) (1296) Yt (s; l; k) > 2 N h=1
Above …ve propositions comprehensively incorporate all possible scenarios in the poverty game mentioned above. Propositions 2-5 present optimistic scenarios for a chosen horizon T . Mechanism for Poverty Alleviation: Tests Testing above propositions in a real world situation is very challenging exercise. It requires modelling of the entire state space of the economy.
Moreover in real situation consumers and producers are heterogeneous regarding their preferences, endowments and technology and economy is more complicated than depicted in the model above. In essence it requires a general equilibrium set up of an economy where poor and rich households participate freely in economic activities taking their share of income received from supplying labour and capital inputs that are a¤ected by tax and transfer system as illustrated in the next section. Bhattarai K. (2010) Strategic and general equilibrium models of poverty, Romanian Jounral of Economic Forecasting, 13:1:137-150
7.6
Repeated Game
Market demand for a product is P = 130
(q1 + q2 )
(1297)
Cost of production for each of two …rms is . Ci = 10qi If played in…nite number of times two …rms form a cartel and monopolise the market.
226
(1298)
Each will supply only 30, set market price to monopoly level at £ 70 and divide total pro…t £ 3600 equally; each getting £ 1800. This is shown by (1800,1800) point in the diagram. It pays to cooperate in the long run; it is sub-game perfect equilibrium. Cooperative Solution It pays to cooperate in the long run; it is sub-game perfect equilibrium. = (130
Q) Q
@ = 130 @Q
2Q
Q2
10Q = 130Q 10 = 0 =) Q =
10Q
(1299)
120 = 60 2
(1300)
Price: P = (130 Q) = 130 60 = 70; Cost: C = 10Q = 10 60 = 600; Pro…t: = P Q C = 60 70 600 = 3600 Non-Cooperative Nash Equilibrium If any one …rm cheats and tries to supply more in order to get more pro…t; it will be found out by another …rm. Opponent …rm will react to this. Game will be non-cooperative, resulting in a Cournot Nash equilibrium. Each …rm produces 40 units, market price is set at 50 and each gets £ 1600 pro…ts. 1
= (130
(q1 + q2 )) q1
10q1 and
2
= (130
(q1 + q2 )) q2
10q2
with reaction functions 2q1 + q2 = 120 and q1 + 2q2 = 120 Total supply is 80, each supplying 40 and making pro…t of 1600 and market price 50. Trigger Strategy and Perpetual Punishment If …rm 1 plays Cournot game but …rm 2 still plays cartel and supplies just 30. Then from the …rm 1’reaction function . 2q1 + q2 = 120 q1 = 60
1 q2 = 60 2
1 (30) = 45 2
(1301)
If …rm 1 supplies 45, market price will be . P = 130
(q1 + q2 ) = 130
45
30 = 55
This makes pro…t margin of …rm 1 to be 45 and its pro…t . 45 45 = 2025 227
1
= (55q1
(1302) 10q1 ) = 45q1 =
Firm 2 will …nd out that …rm 1 has cheated. If it does not react its pro…t will be down to 1350. It will also produce according to its reaction curve. Thus the Nash equilibrium will result with each …rm producing 40 and earning 1600 pro…t for the rest of the periods and the market price will be 50. For whom is it pro…table to Cheat? Does …rm 1 gain or lose by deviation from the agreement. For this evaluate the in…nite series of pro…ts in deviation and in compliance with agreement. Present value of pro…t in case of cheating 2 + :::: + ::: = 425 + 1600 + 1600 + 1600 2 + :::: + ::: 1 = 2025 + 1600 + 1600 2025 1600 + 1600 + 1600 + 1600 2 + :::: + ::: 1 = (Note just with –and + 1600) Using operator to maintain a constant payo¤ from the game i h 1600 ) + 1600] = 425 425 + 1600 = 2025 425 (1 ) 1 = (1 ) 425 + (1 ) = [425 (1 By comparing pro…ts with and without cheating 225 9 2025 425 < 1800 or ; 425 > 2025 1800; > 425 =) > 17 Whether the …rm 1 will stick to agreement or not depends on whether its discount factor if 9 9 greater than > 17 . For discount factor < 17 it is bene…cial to stick to the agreement, which is very high, about 53 percent. Home Work Show above results in a diagram Illustrate repeated game for multiple periods using brach nodes Workout Bertrand type competition for above game and illustrate 'cut-throat' price competion in a diagram. Home Work: Show above results in a diagram,
7.7
Moral Hazard and Adverse Selection
Moral Hazard: Insurance Game with Symmetric Information Under symmetric information full insurance is optimal; insurance company can charge premium according to level of e¤orts exerted by the agent to prevent accident (see Jehle and Reny (2001, Chapter 8)) Let p be insurance premium. (e) probability of accident with e¤ort e, this diminishes with greater care (higher e). Level of bene…t o¤ered in case of accident is BL speci…c to losses L =1,2,. . . .L . The Moral hazard problem is for insurance company to set the premium according to e¤orts max
e;p;Bo ;::::BL
p
L X l=0
subject to participation constraint:
228
(e) Bl
(1303)
L X
l
(e) u (W
p
l + BL )
u
d (e)
(1304)
l=0
Lagrangian function L= p
L X
'
(e) Bl +
l=0
First order condition
@L = 1 @p
' L X
l
L X
l
(e) u (W
p
l + Bl )
l
l
0
(e) u (W
p
l + BL )
#
u =0
d (e)
0
(e) u (W
(e) u (W
p
l + BL
p
l + Bl )
d (e)
d (e)
l=0
From above
u0 (W
u
d (e)
l=0
l=0
@L = 1 @Bl ' L X @L = @
#
p
u) = 0
(1305)
(1306)
(1307)
#
u =0
(1308)
l + Bl ) = d (e) + u
(1309)
Under full insuranceBl = l this implicitly de…nes the insurance premium for e¤ort level . u0 (W
p) = d (e) + u
(1310)
Since low e¤ort is less costly than more e¤ort for the costumer d (e) d (1) ; the premium under lower e¤ort must be set higher than for the higher e¤ort: p (0) p (1) for pro…t maximisation p
L X
(e) :l
(1311)
l=0
This is the prediction of moral hazard with complete information but uncertainty with consumer’s hidden action. Adverse Selection: Insurance Game with Asymmetric Information Insurance company cannot observe the consumer’s choice of accident prevention e¤orts. But the insurance company continues to seek maximize he expected pro…t. It now need to add incentive compatibility constraint. max
e;p;Bo ;::::BL
p
l=0
l
(e) u (W
p
l + BL )
d (e)
(e) Bl
(1312)
l=0
subject to participation constraint: L X
L X
L X l=0
229
l
(e0 ) u (W
p
l + BL )
d (e0 )
(1313)
L X
(e) u (W
l
p
l + BL )
u
d (e)
(1314)
l=0
Incentive compatibility constraint for low e¤orts again full insurance is the best policy from u0 (W
p) = d (e) + u
however the incentive compatibility requires d (0) costs more for the costumer. For high e¤orts case e = 1 Lagrangian function
L =
p 2
6 6 + 6 6 4 @L = 1 @p @L = @Bl
L X
(e) Bl +
l=0
( L X
(l=0 L X
l
l
'
L X
d (1). Therefore lower insurance avoidance
(e) u (W
l
(1315)
p
l + Bl )
l=0
(e) u (W
p
0
(e ) u (W
l + BL )
p
d (e)
l
(1) +
(
l
(1)
l + BL )
l
3
7 7 ) 7 7 5 d (e0 )
l=0
' L X
)
d (e)
0
(0)) u (W
p
(1) +
' L X @L = [( @
l
(1)
l
(e) +
l
[(
l
(1)
(0))] u (W
l
p
0
(0))] u (W
#
(1317)
l + BL ) = 0
(1318)
#
l + BL ) = 0 p
l + Bl ) + d (0)
d (1) = 0
(1319)
l + Bl ) + d (0)
#
(1320)
l=0
' L X @L = [( @
l
(1)
l
(0))] u (W
p
l=0
u0 (W
1 = p l + Bl )
+
u
(1316)
l=0
l
#
1
l l
(1) (0)
d (1)
0
(1321)
since > 0 the RHS is strictly decreasing, this implies that u0 (W p l + Bl ) must be strictly increasing for this to happen l Bl be must increase with e¤ort levels and losses l = 0; 1; 2; ::::L:Optimal high policy does not provide full insurance but the deductible payment increases size of loss. Problem Consider a moral hazard insurance model with an insurance policy fp; B0 ; B1 ; :::::; BL g where p is insurance premium and B0 ; B1 ; :::::; BL denote the bene…t from the insurance company against 230
loss l. Normally the insurance company can observe the loss but not the level of accident avoidance e¤ort (e) of the consumer. The problem of the insurance company is: max
e;p;B0 ;B1 ;:::::;BL
L X
p
subject to participation constraint L X
l
(e) u (w
p
l
(e) BL
(1322)
l=0
l + BL )
d (e)
u
(1323)
l=0
and incentive constraint L X
l (e) u (w
l=0
p
l + BL )
d (e)
L X
l
(e0 ) u (w
p
l + BL )
d (e0 )
u
(1324)
l=0
1. Show that it is Pareto optimal to do full insurance under symmetric information when the insurance company can observe the level of e¤orts of the consumer. 2. How could the insurance company design an e¢ cient contract to induce e¤orts to minimise cost under the assymetric information? Is full insurance still optimal?
7.8
Auction
Types of Auction First price, sealed-bid: person who bids the highest amount gets the good. Second-price, Sealed-bid: Each submit a bid. Higher bidder wins and pays second-highest bid for the good. Dutch Auction: Seller begins from very high price and reduces it until someone raises a hand. English Auction: Begins with very low price, bigger drops out by raising a hand. Which one of these four mechanism is good for the seller?? Online auctions -ebay (http://www.ebay.com/); car auction (http://www.carandvanauctions.co.uk/); Art auction (http://www.artinfo.com/artandauction/); Online advertisement auctions in Google, Microsoft, Yahoo, Facebook, You Tube Auction: Vickrey-Clark-Grove (VCG) mechanism Honesty is the best policy in Vickery auction; truth telling is the winning strategy. Proof Let there be two bidders bidding b1 and b2 but with true values v1 and v2 . Highest bidder wins the auction at the price of the second-highest bid. English auctions and second-highest sealed-bid auctions are equivalent. 231
Expected value for bidder 1 is then given by prob (b1 > b2 ) (v1
b2 )
(1325)
If (v1 > b1 )it is in the best interest of bidder 1 to raise the probability of winning prob (b1 > b2 ) , this can happen when (v1 = b1 ) Similarly If (v1 < b2 ) then it is in the interest of bidder 1 to make prob (b1 < b2 ) as small as possible. It happens when .(v1 = b1 ) Thus the truth telling is the best interest in such action. Auction: Financing Mechanism for Public Goods Let x be a public good such as streetlight or road; x = 1 if it is provided x = 0 if not. If state knew that how much each person is willing to pay for this it could bill e¢ ciently. Each would pay according to the value they put in such public good. Unfortunately it is impossible to know preferences of individuals. Individuals do not tell true value when asked that how much they are ready to pay for this. Let N individuals be indexed by i. Then the utility from the public good to an individual i is given by Ui (x). There is free rider problem with public goods. Individuals may underreport their utility thinking that others will pay higher for it if they act like this but they will have opportunity of full bene…t. Under Vickrey-Clark-Grove mechanism it is in the best interest of individuals to tell the truth. Auction:Financing Mechanism for Public Goods Under Grove mechanism each individual is asked to report his her utility; which is ri (x). . Then N P the state chooses x* that maximises the sum of reported utilities R = ri (x): Each individual i=1
receives a side-payment Ri =
N P
ri (x):.
j6=1
With side payment the total utility of an individual is Ui (x) +
N X ri (x)
(1326)
i=1
State chooses x to maximise ri (x) +
N X
ri (x)
(1327)
i=1
Therefore it is in the best interest of an individual to tell the truth Ui (x) = ri (x). All agents tell truth like this and this mechanism generates e¢ cient outcome. (page 27 of the new handbook on sum of MRS =MC of public good.) (See Varian HR (2010) Intermediate Microeconomics: A Modern Approach, Norton,8th edition). 232
Binmore K. (1990) Fun and Games: A text on Game Theory, Lexington, Heath. Cripps, M.W.(1997) Bargaining and the Timing of Investment, International Economic Review, 38:3 :Aug.:527-546 Dixit A., S. Skeath and D. F. Reiley (2009) Games of Strategy, Norton. Gardener R (2003) Games of Business and Economics, Wiley, Second Edition. Holt Charles (2007) Markets, Games and Strategic Behaviour, Pearson, . Mailath G. J. and L. Samuelson (2006) Repeated Games and Reputations: long run relationship, Oxford. Moore J (1984) Global Incentive Constraints in Auction Design Econometrica, 52:. 6 pp. 1523-1535 Kreps D. M. (1990) A Course in Microeconomic Theory, Princeton. Osborne M.J. and A. Robinstein (1994) A course in game theory, MIT Press. Perlo¤ J. M. (2008) Microeconomics: Theory and Applications with Calculus, Pearson. Rasmusen E(2007) Games and Information, Blackwell,. Varian HR (2010) Intermediate Microeconomics: A Modern Approach, Norton,8th ed.Chapter 17,35.
7.9
Exercise 16 :Optimal production of a multiproduct …rm
Q1. Consider a principal agent model where the gross pro…t ( of the manager and is subject to random shocks (') as:
g
g)
of a …rm depends on e¤orts (e)
=e+'
(1328)
Cost of e¤orts to the manager is given by C(e): C(e); C 0 (e) > 0 and C 00 (e) > 0 Net pro…t (
n)
for the …rms is gross pro…t ( n
=
g
g)
(1329)
minus manager’s salary (s) as:
s=e+'
s
(1330)
Then the expected net pro…t becomes: E(
n)
= E (e + '
s) = e
E (s)
(1331)
Manager’s utility takes the form: A 2 A 2 C(e) = e C(e) (1332) 2 s 2 e where 2s is the variance of salary and A is a risk aversion parameter. Manager dislikes volatility in salary 2s . Two scenarios could be considered in this game; …rst when owens can observe the e¤orts put in by the manager and another where they cannot observe it. E (u) = E (s)
233
A. What is the …rst best solution if the owners can observe the level of e¤orts by the manager? What will be the variance of salary 2s in the …rst best solution? What is the relation between the marginal cost of e¤orts and marginal bene…ts of the manager in this case? If owners are not able to observe the level e¤orts by the managers, they will o¤er him a performance based salary contract subjecting it to gross pro…t as: s(
g)
=a+b
g
(1333)
Owners set salary by setting parameters a and b …rst where a is …xed component and b is incentive payment. Then the manager decides the level of e¤orts conditional on the contract. If the cost of e¤orts is given by: C(e) =
e2 ; 2
2
= 1; A = 2
(1334)
B. What will the optimal value of incetive coe¢ cient (b) be in the second best solution? What will the optimal level of e¤orts (e ) be ? What will the …xed component of the salary (a) be? What will the expected pro…t for the owners be? Solve this game by backward induction to …nd answers to these questions. Q2 This exerice aims to investigate how di¤erent production technologies adoped by a …rm results in di¤erent level of output, employment and investment in each sector. First exercise is …xed input process under the linear programming framework with limited price based substitution. The second problem contains adoption of the Cobb-Douglas technology with elasticity of substituion equal to 1. Third process involves usign the CES production technology with a constant elasticity of substitution. Overall resource available to this …rm is given in each case. Linear programming problem A multinational …rm produces two products Y1 and Y2 , these products are sold at 6 and 3 respectively. Linear programming problem max R = 6Y1 + 5Y2 (1335) 1. Subject to: 1) Labour constraint (workers) 2Y1 + Y2
1000
(1336)
2) capital constraint (machine hours) 12Y1 + 40Y2 1. where Y1
0 and Y2
0; 234
30000
(1337)
What are the revenue maximising levels of output of Y1 and Y2 . How many workers /machine hours are devided between producing these two products to maximise revenue. If wage rate is 2 and capital cost is 0.05 per hour what would be total pro…t? Cobb-Douglas Technology = P1 Y1 + P1 Y1
C1
C2
(1338)
Subject to (1
)
(1339)
(2
)
(1340)
Y1 = K1 L1 Y2 = K2 L2
C1 = rK1 + wL1
(1341)
C2 = rK2 + wL2
(1342)
CES technology = P1 Y1 + P2 Y2
C1
C2
(1343)
Subject to Y1 = [ 1 K1 + (1
1 ) L1 ]
Y2 = [ 2 K2 + (2
2 ) L2 ]
1
(1344)
2
(1345)
C1 = rK1 + wL1
(1346)
C2 = rK2 + wL2
(1347)
Comparision of results among technologies Y K Firm 1 Linear Programming Cobb-Douglas CES Firm 2 Y K Linear Programming Cobb-Douglas CES
235
L
R
C
Pr
L
R
C
Pr
Impacts of microeconomic policies in …rms output decision. 1) cost of capital doubles to …rms because of credit crisis. How would this a¤ect these variables. 2) Wage dispute between the management and workers caused wage rate to rise by 25 percent. Given no change in capital market conditions how would this a¤ect the production. 3) There is a change in preferences of consumers because of substitutable product in the market. How does these variables. Y K Firm 1 Linear Programming Cobb-Douglas CES Firm 2 Y K Linear Programming Cobb-Douglas CES 7.9.1
L
R
C
Pr
L
R
C
Pr
A Microeconomic Model of FDI
Hymer (1976) and Caves (1982), Batra and Ramachandran (1980) and Batra (1986) Rossel (1985), Hortmann and Markusen (1987) and Markusen (1995). MNCs move to a foreign country for a number of reasons: cost advantages in producing there rather than exporting commodities; ownership (O) of …rm speci…c capital; location (L) based advantages of production; licensing abroad for reasons of natural resources or customer bases; internalisation (I) of bene…ts of technical know how by …rms doing R & D.
where 1 ,P1 ,q1 ; C1 and and foreign markets
2 ,P2
1
= P1 q1
C1
(1348)
2
= P2 q2
C2
(1349)
,q2 ; C2 are pro…t, price, demand and cost of production at home
T C > C2 R+D
C1 = 800
F C < 2M
200 = 600
F C < 2R
FC
(1350) (1351)
where R is the rental income from licensing of a partner foreign …rm, (M is the pro…t from subsidiary is FDI takes the form of subsidiary operation), D is the payments made in case the licensee defects in the second period (this deters the licensee from supplying the market itself after gaining the know-how from the MNC in the …rst period), and FC is the …xed cost of FDI to the MNC. Let us assume that the demand of the monopolist in the home market is given by
236
q1 = 21
0:1P1
(1352)
10q1
(1353)
0:4P2
(1354)
2:5q2
(1355)
and its inverse demand is P1 = 210 If demand abroad is q2 = 50 and the inverse demand abroad is P2 = 125
Cost of production is di¤erent across counties di¤er.It is C1 = 200 + 10q1
(1356)
at home but it is costlier to set up business in foreign country because of higher …xed costs, C2 = 800 + 10q2
(1357)
(marginal costs may also be di¤erent). The optimal condition for pro…t maximisation is given by a point where the marginal revenue equals marginal cost in each market: M R1 = M C1 and M R2 = M C2
(1358)
Given the above information, the total revenue from the home market is R1 = P1 q1 = (210
10q1 ) q1 = 210q1
10q12
(1359)
. Therefore, the marginal revenue from home market sales would be M R1 = 210
20q1
(1360)
.. Similarly, the total revenue from the sales in the foreign market would be R2 = P2 q2 = (125
2:5q2 ) q2 = 125q2
2:5q22
(1361)
and associated marginal revenue would therefore be M R2 = 125
5q2
(1362)
Fixed costs of production are di¤erent across countries, but the marginal costs are assumed to be the same in both countries. M C1 = M C2 = 10
(1363)
Now it is possible to solve the model for the optimal amount of goods supplied at home and abroad by using conditions where the marginal revenue in each market needs to equal its marginal cost.
237
M R1
=
M C1 ) 210
20q1 = 10 ) q1 = 10 =) P1 = 210
10q1
M C2 ) 125
5q2 = 10 ) q2 = 23 =) P2 = 125
2:5q2
=) P1 = 110
M R2
=
=) P2 = 67:5
(1364)
(1365)
Thus, the amount supplied at home is much smaller than amount supplied abroad and prices charged at home are much higher than prices charged abroad. Corresponding revenues are: R1 = P1 q1 = 110
10 = 1100; R2 = P2 q2 = 67:5
23 = 1552:5
(1366)
The cost function is assumed to be known here for simplicity. It must be derived from the cost minimisation principle subject to a production technology constraint. The total cost of production at home and abroad are given by C1 = 200 + 10q1 : = 200 + 10
10 = 300
(1367)
C2 = 800 + 10q2 = 800 + 10
23 = 1030
(1368)
Now it is possible to calculate pro…ts to the MNC from home and foreign markets: 1
2
= P1 q1
= P2 q 2
C1 = 1100 C2 = 1552:5
300 = 800
(1369)
1030 = 525:5
(1370)
Conclusion In this paper, the microeconomic e¤ects of FDI have been illustrated with an example of a multi-plant MNC that faces a di¤erent structure of demand and costs between home and foreign countries with strategic consideration of licensing or subsidiary production in foreign countries. On the macro side, the total FDI aggregated over MNCs accounts for a signi…cant proportion of total investment and has a signi…cant impact on economic growth. This growth e¤ect is shown theoretically using an endogenous growth model with FDI in which foreign capital complements domestic capital and contributes to both investment and growth rate of output. Our model predictions have been tested using panel data growth regressions for 30 OECD countries over 1990 to 2004. Our analysis establishes positive impacts of FDI in‡ows and negative impacts of FDI out‡ows on investment and economic growth. The impacts of time and country speci…c e¤ects are found to be consistent with the stylized facts relating to growth rates of output, investment ratios and in‡ows and out‡ows of FDI. The empirical results illustrated in this paper are comparable to Desai et al. (2000).
238
8
L8: Uncertainty and Insurance
Future is uncertain. Arrow (1963), Harsanyi (1967), Roy (1968), Akerlof (1970), Rothschild and Stiglitz (1976), Kahneman and Tversky (1979), Machina (1987), Hey (1987), Moore (1988) Newbery and Stiglitz (1982) Hirshleifer and Riley (1992) Hey and Orme (1994) Hey, Lotito and Ma¢ oletti (2010) Conte and Hey (2013) has analysed how optimal choices under uncertainty occur in many aspects. Income or expenditure of individuals are uncertain. Return on stocks and portfolios are uncertain. Von Neumann-Morgenstern expected utility theory is applied to analyse choices in uncertain world. For an asset or commodity x the expected utlity is sum of u(x) weighted by their densities, dF . Z U (F ) = u(x)dF (1371) Consider a case where safe asset earns £ 1 per £ 1 invested and an unsafe risky assets ears a random amound z per £ 1 invested, z has a distribution F(z). Z zdF (z) > 1 (1372) Return on risky asset exceed that in the safe asset. Investor has initial wealth w to invest, which could be invested in and proportion in risky and safe assets. Return from the portfolios is z+ and
+
(1373)
= w: Investors problem is to choose and to maximise expected utility. Z Z max u ( z + ) dF (z) = max u (w + (z 1)) dF (z) ;
subject to
+ max
Z
u (w + (z
=w 1)) dF (z); 0 6
(1374)
(1375) 6w
(1376)
Optimal > 0: If the risk is acturially fair, the risk averstor will take small amount of risk. Bernoulli utility functions increasing in x are continueous and concave and ful…ll Jensen’s inequality conditions. Z Z u(x)dF (x) 6 u[ xdF (x)] (1377)
239
1 1 u (1) + u (3) 6 u (2) 2 2 In a general expected return maximisation problem Z U ( 1 ; ::; N ) = u( 1 z1 + :: + N zN )dF (z1 ; ::; zN )
(1378)
(1379)
First and second order stochastic dominence For any distributions F ( ) and G ( ) the F ( ) …rst order stochastic dominates G ( ) if Z Z u(x)dF (x) > u(x)dG (x) (1380) and F (x) > G (x) for every x. F ( ) second order stochastic dominates G ( ) if G ( ) is mean preserving spread of F ( ) :
240
See Mas-Colell et at. (1995) Chapter 6.
8.1
Allais’paradox Table 71: Lotteries in Allais’paradox 25 5 0 A 0 1 0 B 0.1 0.89 0.01 C 0 0.11 0.89 D 0.1 0 0.9
A
B =) u(5)
0:1u(25) + 0:89u(5) + 0:01u(0)
0:11u(5) D
0:1u(25) + 0:01u(0)
(1381) (1382)
C =) 0:1u(25) + 0:9u(0)
0:11u(5) + 0:89u(0)
0:1u(25) + 0:01u(0)
0:11u(5)
(1383) (1384)
Certainly a paradox. Savage principle and Ellsbury paradox resolves Allais paradox. Individuals prefer smaller but certain prizes over bigger but uncertain prizes.
241
Two urns R and H contain 100 white and black balls; R has 49 whites and 51 black but the number of white or black balls in unknown for urn H. Lotterty: draw a white ball from R and win 1000 or draw a white balls from H and earn 1000. People prefer to draw from R because they know probability of 0.49 of drawing white ball rather than from H which may have more than 50 black balls. Decision makers do not like risk; so removes Allais paradox. Expected utility theory is still applicable. Portfolio choice and insurance contract Two …rms two states; N shares v price V value V j = v j N j and proportion of 0 < j < 1 1 1 i
+
2 2 i
N 2 6 v1
1 1 0N
wi =
(1385)
Budget constraint v1
1
2
N 1 + v2 1
v1 N 1
1 0
Avoid short selling and choose optimal Lagrangian for optimisation
j
1 1 2
+
L( ; )
= pu +
(1386)
2 0
60
(1387)
for optimal portfolio.
v1 N
0
2
+ v2 N 2
2 2 0N
+ v2
2 2 2 + 1 1
(1
p) u
1 0
v2 N
1 1 1+ 2 2
2 2 1 2 0
(1388)
FOC: L( ; ) = pu0 @ 2
1 1 2
+
2 2 2
2 2
+ (1
p) u0
1 1 1
+
2 2 1
2 1
v2 N 2 = 0
(1389)
L( ; ) = pu0 @ 1
1 1 2
+
2 2 2
1 2
+ (1
p) u0
1 1 1
+
2 2 1
1 1
v1 N 1 = 0
(1390)
+ v2 N 2
2
1
v1 N 1
1 0
2 0
=0
(1391)
From this =
pu0
1 1 2
+
2 2 2
2 2
+ (1 p) u0 v2 N 2
1 1 1
+
2 2 1
2 1
pu0
1 1 2
+
2 2 2
1 2
+ (1 p) u0 v1 N 1
1 1 1
+
2 2 1
1 1
(1392)
or =
(1393)
Then pu0 pu0
1 1 2 1 1 2
+ +
2 2 2 2 2 2
2 2 1 2
+ (1
p) u0
+ (1
p) u0
Now solve for optimal 242
1 1 1 1 1 1
+ +
2 2 1 2 2 1
2 1 1 1
=
v2 N 2 v1 N 1
(1394)
Watt(2011). Insurance contract: Given initial wealth w0 and the possibility of loss L insurance contract means for individual: (1
L) 6 (1
p) u (w0 ) + pu (w0
with the insurance contract: Insurance company’s income:
k
(x1 ; x2 ; (1
z0 + (1
p) u (w0 + x1 ) + pu (w0 + x2 )
(1395)
p) ; p)
p) (y1
x1 ) + p (y2
L
x2 )
(1396)
Di¤erence made by the contract:
[z0 + (1
B(x)
=
[z0 + (1
p) (y1
p) y1 + py2 ]
=
[(1
p) x1 + px2 ]
x1 ) + p (y2 pL
L
x2 )] (1397)
B(x) > 0 for a viable contract. That means pL > [(1
p) x1 + px2 ]
(1398)
For individual w0 + (1 8.1.1
p) x1 + px2 6 w0
L
Uncertainty of Good Times and Bad Times Future is uncertain; can be good or bad; two states. Contingent consumption in good times Cg and in bad times Cb
243
(1399)
Probability of good times
g
and of bad times
b
Prices of good times pg and of bad times pb Utilities from contingent consumption in good times u (Cg ) and in bad times u (Cb ) Budget constraint .I = Pg Cg + Pb Cb Consumer problem under uncertainty Expected utility theorem: utilities under uncertainty are additively separable (von-NeumannMorgenstern Utility) M ax
EU =
g u (Cg )
+
b u (Cb )
(1400)
Subject to I = Pg Cg + Pb Cb
(1401)
Lagrangian for constrained optimisation L=
g u (Cg )
+
b u (Cb )
+ [I
Pg C g
P b Cb ]
(1402)
First order conditions for optimisation For household A and B @L = @Cg
0 gu
(Cg )
Pg = 0
(1403)
@L = @Cb
0 bu
(Cb )
Pb = 0
(1404)
@L = I Pg C g P b C b = 0 (1405) @ Dividing (1403) by (1404) gives the marginal rate of substitution between good and bad times 0 gu 0 bu
(Cg ) Pg = ; (Cb ) Pb
Pg = Pb
g
(1406)
b
Fair market for contingent goods implies ratio of prices in good and bad states equals ratio of respective probabilities. Utility and allocation in good and bad times u0 (Cg ) =1 u0 (Cb )
(1407)
u0 (Cg ) = u0 (Cb )
(1408)
Since preference are symmetric over the states Cg = Cb
244
(1409)
consumer likely to fully insure against any risk; like to have same consumption in both good and bad states. Represent above result in a diagram with certainty line. budget line and indi¤erence curve u (Cg ; Cb ) : It is possible that individuals like to consume a bit more in good times and a bit less in bad times. 8.1.2
Optimal Demand for Insurance
There is certain wealth (W ), if an event occurs there will be a loss (L). probability of loss is (p) : Owner of the property can insure for amount (q) paying premium (m) Expected utility maximisation problem is maxEU = p:u (W
L
q
mq + q) + (1
p) u (W
mq)
(1410)
mq) m = 0
(1411)
Choose q to maximise EU using the …rst order condition as: @EU = p:u0 (W @q Optimal condition
L
mq + q) (1
m)
p) u0 (W
(1
u0 (W L mq + q) (1 p) m = 0 u (W mq) p (1 m) Pro…t function of the insurance company = (1
p) mq
p (1
m) q
(1412)
(1413)
Assume perfect competition in the insurance business, pro…t is zero p (1
m) q
(1
p) mq = 0
(1414)
The premium rate equals the probability of loss in equilibrium p=m
(1415)
This is actuarially fair insurance. Insert (??) into (1411) p:u0 (W
L
mq + q) (1
u0 (W
L
p)
(1
p) u0 (W
mq + q) = u0 (W
mq)
mq) p = 0
(1416) (1417)
00
For risk averse consumer u (W ) < 0 W
L
mq + q = W q=L
Consumer completely insures (q) against the loss (L). Risk spreading and risk diversi…cation 245
mq
(1418) (1419)
Risk can be spread among individuals. Imagine a society with 1000 individuals each endowed with £ 35000. Each faces a risk of losing £ 10000 with probability of 1 percent. Only 10 person in aggregate face this risk. It is a big loss for each individual as it can happen to each of them. Now they create an insurance market. Each contributes 100 to mitigate this uncertainty. This creates 100,000 insurance fund. This is enough to ensure each for any eventual loss. Every one will be certain (ensured) to have 34,900.: endowment minus insurance contribution. This is an example of risk spreading. Risk is spread (divided) among all. Each pays 100 to ensure against loss of 10000. Risk spreading and risk diversi…cation Risk can be diversi…ed by choosing an appropriate portfolio. Consider an excellent example from Varian (2010) on sunglasses and raincoat. You have 100 to invest. Probability of rain or shine is equally likely. You can invest only in sunglasses or raincoats or split 50/50 in each. Value of sunglass investment will double if it is sunny or down by half if it is rainy. Similarly value of investment will be double if rainy and down by half if sunny. If invested all in one then at the end of the day the expected value is 0.5(50)+0.5(200)=125. There is considerable risk. If case of splitting 50/50 the expected value of investment is [0:5(25) + 0:5(100)]+[0:5(100) + 0:5(25)] =125. Thus 125 is guaranteed no matter rainy or shiny. Diversi…cation has ensured 125. Do not put all your eggs in one basket. Homework A person has wealth worth £ 35000. There is 1 percent probability of loss. If this event occurs there is a loss of 10,000. This individual is risk neutral. 1) What is expected wealth without insurance? 2) This person can buy insurance equal to amount K to cover insurance by paying K insurance premium, where is the premium rate. Write individuals budget in case of accident and in case of no accident. 3) Write the expected utility function of this person. Assume that person receives utility from the wealth that he has. 4) What is expected pro…t of the insurance company? 5) Prove that premium rate equals the probability of the event. 6) Prove that consumption is same in both states with insurance. L) Prove that it is optimal to fully insurance against the loss and that is actuarially fair insurance. 246
Dixit A., S. Skeath and D. F. Reiley (2009) Games of Strategy, Norton. Hirshleifer J and J G Riley (1992) The Analytics of Uncertainty and Information, Cambridge. Perlo¤ J. M. (2008) Microeconomics: Theory and Applications with Calculus, Pearson. Nicholson W. (1985) Microeconomic Theory: Basic Principles and Extensions, Norton. (HoltSaunders). Rasmusen E(2007) Games and Information, Blackwell, ISBN 1-140513666-9. Varian HR (2010) Intermediate Microeconomics: A Modern Approach, Norton,8th edition http://cepa.newschool.edu/het/home.htm.
8.2
Expected utility theory
Expected wealth and utility from expected wealth Future is uncertain; two states - high wealth and low wealth. Contingent wealth in high state is WH and in low state is WL : Probability of high wealth
H
and low wealth
L
:
Utilities from high wealth u (WH ) and low wealth u (WL ) : Expected wealth EW = Expected utility EU =
L WL
+
H WH :
L u (WH )
+
H u (WL ) :
Faced with uncertainty people maximise expected utility (von-Neumann-Morgentern preferences). People are ready to pay some amount to insure themselves against the possible risk. Preferences of risk averse consumer Utility functions of risk averse individual U (W ) = ln(W ) U (W ) =
p
(1420) 1
W =W2
(1421)
Expected utility theorem: utilities under uncertainty are additively separable (von-NeumannMorgenstern Utility) M ax
EU =
H
:u (WH ) +
L:
u (WL )
(1422)
Utility from expected wealth U (EW ) = ln (
H
WH +
Certainty equivalent wealth 247
L
WL ) = ln (EW )
(1423)
CEW = exp (ln (EU ))
(1424)
Maximum insurance against risk and a measure of risk aversion Maximum insurance that person is ready to pay to cover risk: Insurance = EW
CEW
Table 72: Uncertainty of Income High Probability 0.75 Income 5000 Expected Income 3750
(1425)
and Wealth Low 0.25 1000 250
Expected wealth EW = L WL + H WH = 0:75 (5000) + 0:25 (1000) = 4000 Do people maximize expected wealth? No. They maximize expected utility. Maximum insurance against risk and a measure of risk aversion EU
=
H
:u (WH ) +
=
0:75
L:
u (WL ) =
H
: ln (WH ) +
L:
ln (WL )
ln (5000) + 0:25 ln (1000) = 6:388 + 1:727 = 8:115
(1426) (1427)
Certainty equivalent wealth CEW = exp (ln (EU )) = exp(8:115) = 3344:26
(1428)
Maximum insurance that person is ready to pay to cover risk: Insurance = EW
CEW = 4000
3344:26 = 655:74
(1429)
After paying 655.74 for the insurance company, this person can be sure that no matter high or low state 3344:26 is guaranteed. Can sleep well as income will not fall to 1000 even in the bad state! Risk pooling is possible. If 100 people ensure like this revenue of insurance company is 65575; only 25 percent people claim (2444.26 25 = 61106:5). Pro…t to the insurer is 65575 - 61106.5 = 4468.5. 8.2.1
Measure of risk aversion
Arrow-Pratt (1964) measure of risk aversion r(W ) =
U 00 (W ) >0 U 0 (W )
(1430)
r(W ) =
U 00 (W ) 0 W
(1434)
with Cobb-Douglus type preferences 1
(1435)
U (W ) = W 2
r(W ) =
1 2
U 00 (W ) = U 0 (W )
1 2
1 2W 1 2W
1
1 2
=
1 1 >0 2W
Maximum insurance against risk and a measure of risk aversion Risk lovers U (W ) = exp(aW ) r(W ) =
U 00 (W ) = U 0 (W )
a2 W 2 exp(aW ) = aW exp(aW )
(1436)
(1437) aW < 0
(1438)
Risk neutral U (W ) = aW r(W ) = 8.2.2
(1439)
U 00 (W ) 0 = =0 U 0 (W ) a
(1440)
St Petersberg Paradox (Bernoulli Game) and Allais Paradox
People are ready to play a small amount for a lottery but do not want to risk a huge amount in it. People care about utility. How much should one pay to play a game that promises to pay 2n if the head turns up in the nth trial? Answer 1.39. How? Expected payo¤ is in…nite E( )=
1
2+
2
22 +
3
23 + ::: +
1 1 > 2= 2 > 2 2 but the Expected Utility is …nite here 1
=
3
=
249
n
2n = 1 + 1 + 1 + ::: + 1 = 1
1 > :::::: > 23
n
=
1 2n
(1441) (1442)
E (u) =
1
ln (2) +
2
ln 22 +
3
ln 23 + ::: +
n
ln (2n ) < 1
(1443)
People buy lotteries for small amount but not for more (Allais Paradox) St Petersberg Paradox (Bernoulli Game) E (u) =
1 1 1 1 ln (2) + 2 ln 22 + 3 ln 23 + ::: + n ln (2n ) < 1 2 2 2 2
E (u) =
1 X 1 i=1
2i
1 X i i ln (2) = ln (2) = ln (2) 2 = 1:39 i 2 i=1
(1444)
(1445)
People are ready to pay small amount to buy lotteries but do not want to risk large sums (Allais Paradox) 8.2.3
Non-linear pricing Scheme
Let us consider a non-linear pricing problem considered by Snyder and Nicholson (2011) and earlier explained in Rochet and Tirole (2003), Fehr and List(2004), Blundell, Dias and Meghir (2004). Consumers of a …rm are of two types, f H ; L g ; H is the probability of consumer who put high value on the production and L is the proportion of consumer that put very low value in the consumption. Non-linear price scheme is to set tarrifs (T ) and output (q) in such a way that maximises …rms pro…t by designing price scheme appropriate to these consumers. Utility function of consumers: u = V (q)
T
V 0 (q) > 0 and V 00 (q) < 0: Firm’s problem is =T
cq
Participation constraint [ V (q)
T] > 0
First best solution This is when …rm knows the consumer type. It is binding , V (q) = T: Substitute this information on consumer into the …rms objective function. = V (q)
cq
Optimal pro…t then means @ = V 0 (q) @q
c = 0 =) V 0 (q) = c
This leads to the …rst degree price discrimination; high value consumer will be sold more goods at discounted per unit price and low value cumtomer will be sold less but ends up paying more per unit. (do a graph here)
250
Sedond best solution This is when the …rm does not know the type of the consumer. It has a probability belief on type of each type of consumer,0 < < 1 for type high and (1 ) for type low. Now the …rms pro…t becomes: =
(TH
cqH ) + (1
) (TL
cqL )
Subject to participation and incentive constraints for low high type consumers as: [ [
LV
(qL )
TL ] > 0
HV
(qH )
TH ] > 0
[
LV
(qL )
TL ] > [
[
HV
(qH )
TH ] > [
LV
(qH )
HV
TH ]
(qL )
TL ]
Participation constraint of the low type customer and incentive constraint of the high type customers are binding; resulting in LV
TH = [
(qL ) = TL
(V (qH )
H
V (qL )) + TL ]
Now put this into the …rm’s pro…t function = =
[f
H
@ = @qL
[f
H
(V (qH )
(V (qH ) HV
HV
0
(
H
0
V (qL )) + TL g
V (qL )) + (qL ) +
LV
(qL ) +
LV
L) V
LV
0
LV
0
0
0
(qL )g
cqH ] + (1
) (TL
cqH ] + (1
)(
(qL ) + (1
)
(qL ) + (1
(qL ) + (1
(qL ) = c +
) )
[
LV
LV
LV
0
(1 0
(qL )
0
(1
(qL ) = (1
(qL ) = (1
L] V
H
0
0
LV
cqL ) (qL )
cqL )
)c = 0 )c
)c
(qL )
)
Since the last term is positive it implies that L V (qL ) > c Since L V 0 (qL ) = c is the …rst best and qL now should be smaller to have L V 0 (qL ) > c. For the high value type H V 0 (qL ) = c, this means 'no distortions at the top'. An example from Nicholson and Snyder about co¤ee market: p V (q) = 2 q and f H ; L g = f20; 15g c=5, = 21 First best solution when the type of consumer is known. 251
V 0 (q) =
c
1 2
V 0 (q) = q
1 2
then V 0 (q) = q ( q=
2
1
= c; q2 = c; q =
c
20 2 = 16 5 15 2 =9 5
2
c
Tari¤ p p16 = 160 9 = 90
20 2 15 2
T = f V (q)j Expected pro…t of the …rm: =
(TH cqH ) + (1 ) (TL cqL ) 1 1 (160 80) + (90 45) = 40 + 22:5 = 62:5 2 2
=
When types arepunknown high type may buy 9 aunce and pay 90 cents thus with consumer 9 30 = 120 90 = 30 surplus of 20 2 He pays not 160 but 130. Thus the pro…t of the …rms witll be =
(TH cqH ) + (1 ) (TL cqL ) 1 1 (130 5 16) + (90 5 16) = 25 + 22:5 = 47:5 2 2
=
Now the shopper reduces the size of the cup ( Check the incentive compatible conditions): LV
0
(qL ) = c + [ 2
1
q2 =
L
H
c
L] V
H
(qL ) ;
Lq 2
2
; q=
0
L
H
=c+[
2
=
c
1 2
15 5
L] q
H
20
1 2
2
= 22 = 4
Tari¤ for the low customer TL =
LV
p 2 4 = 60
(qL ) = 15
For high type HV
0
(qL ) = c =) 20
q
1 2
1
= 5 =) q 2 = 4 =) q = 16
Now tari¤ from the high type TH = [
H
TH
(V (qH )
= =
[ h
V (qL )) + TL ] = [
(V (qH ) p 20 2 16 H
H
V (qL )) + p 2 4 + 15 252
(V (qH )
V (qL )) +
(qL )] p i 2 4 = 160
LV
LV
20 = 140
(qL )]
The pro…t in the second best solution is: = =
(TH cqH ) + (1 ) (TL cqL ) 1 1 (140 5 16) + (60 5 4) = 30 + 20 = 50 2 2
Now the high value type pays 8.2.4
140 16
= 8:75 and low type pays
60 4
= 15:
Job market applications
Principal may design contract in two ways (Nicholson and Snyder (2013)). The …rst one is where he would like to retain the proportionate share and the second on to raise his share over time, (see in pie charts) Gross pro…t of a …rm depends on e¤orts of the manager and is subject to random shocks. g
=e+'
Cost of e¤orts to the manager is C(e); C 0 (e) > 0 and C 00 (e) > 0 Net pro…t for the …rms is gross pro…t minus manager’s salary n
E(
n)
=
s=e+'
g
= E (e + '
s) = e
s E (s)
Manager’s utility E (u) = E (s) First best solution
A 2
s
C(e) = e
A 2
2 e
C(e)
is to …x salary according to e¤orts E (u) = s
C(e) > 0;
in equilibrium s = C(e)
Expected net pro…t for the owners will be: E(
n)
=e
E(s ) = e
This would gaurantee maximum (full) e¤orts.
253
C(e ); C 0 (e ) = 1
Second best case Owners cannot observe the e¤orts. Therefore they can subject salary to gross pro…t to a linear scheme as: s(
g)
=a+b
g
Owner sets salary by setting a and b …rst; a is …xed component and b is incentive payment. Then the manager decides the level of e¤orts conditional on the contract. Solve the game by backward induction: E (u) = E (s)
A 2
C(e); E (u) = E (a + b
s
A var (a + be + b') 2
E (u) = E (a + be + b') = a + be
A var (a + b 2
g)
A var (a + be + b') 2 Ab2 2
= a + be
2
g)
C(e)
C(e)
C(e)
C(e)
C 0 (e) = b How do they choose b? Manager accepts contratct if the E (u) > 0. That implies: Ab2 2 be 2 owener decides whether to o¤er the contract at the 3rd stage. a > C(e) +
E(
n)
= E (e + '
E(
n)
= e(1
s) = e
b)
@E ( n ) =1 @e
E (s) = e Ab2 2
C(e) + C 0 (e)
A
C 0 (e) = b
=
2
2
a
be = e(1
be = e
C 00 (e) = 0;
C(e)
2 C 0 (e)
1 1+A
2 C 00 (e)
u
(1 Constrained optimisation problem: L=p @L =1 @p
x + [(1 ) U 0 (W0
[(1 @L = @x
@L = @e
) U (W1 ) + U (W2 ) e
p) + U 0 (W0
U 0 (W0
+
e
(1 ) U 0 (W0 e p) + U 0 (W0 e p l + x)
@ x @e
p
+
e
u] p
l + x)] = 0
l + x) = 0
@ @e
U (W0 e p) U (W0 e p l + x)
=0
From the …rst two …rst order conditions 1
=
) U 0 (W0
[(1 0
= U (W0
e
p
e
p) + U 0 (W0
e
p
l + x)]
l + x)
This implies l = x, full insurance is optimal in the …rst best world. From the last …rst order condition @@e x = 1: The marginal social bene…t of precautionary e¤orts equals the marginal social cost of precaution. Partial versus full insurance: an example (from NS) U = ln (W ) Individual can …t an alarm and reduce the probability of theft from 25 percent to 15 percent; with W=100000 and L =20000 EUN A
= =
ln (W L) + (1 ) ln (W ) 0:25 ln (80000) + 0:75 ln (100000) = 11:45714
cost of allarm e = 1750 EUA
= =
ln (W
L) + (1
0:15 ln (80000
) ln (W )
1750) + 0:85 ln (100000
1750) = 11:46113
Utility from putting alarm is higher than from not putting it; EUA > EUN A Premium with alarm in the …rst best world 256
U = ln (100000
1750
p) = 11:46113 =) 98250 e11:46113 = 98250
=) p = 98250
p = e11:46113
94252:3 = 3297:7
Insurance pro…t p
L = 3298
0:15
20000 = 298
Second best solution If insured people may not put alarms; then the pro…t of insurance company will decrease: U
=
ln (100000
p) = 11:46113 =) 100000
=) p = 100000
11:46113
e
p = e11:46113
= 5048
Company pro…t p
L = 5048
0:25
20000 = 48
Insurance company can induce individuals to …t alarm with a contract as following with two equations First if the customer …ts the alarm EUA
= =
ln (W
e
p) + (1
0:85 ln (100000
) ln (W
1750
e
p
L + x)
p) + 0:15 ln (100000
1750
p
20000 + x) = 11:46113
Second when the customer does not …t the alarm EUA
= =
ln (W
p) + (1
0:75 ln (100000
) ln (W
p
L + x)
p) + 0:25 ln (100000
p
20000 + x) = 11:46113
These two equation could be solved numerically to …nd optimal p and x. These solutions are p = 601.8 and x = 3374.4 Company pro…t now p
L = 602
0:15
3374 = 96
Thus the partial insurance is more pro…table than the full insurance when the company cannot observe the precaution. Akerlof George A. (1970) The Market for 'Lemons': Quality Uncertainty and the Market Mechanism The Quarterly Journal of Economics, 84, 3. (Aug., 1970), pp. 488-500. Arrow K. J. (1964) The Role of Securities in the Optimal Allocation of Risk-bearing The Review of Economic Studies, 31, 2 (Apr., 1964), pp. 91-96
257
Arrow Kenneth J. (1963) Uncertainty and the Welfare Economics of Medical Care The American Economic Review, 53, 5 (Dec., 1963), pp. 941-973 Blundell R, M. C. Dias and C. Meghir, (2004) Evaluating the employment impact of a mandatory job search program,Journal of European Economic Association, 2:4:569-606.
Conte A & John D. Hey (2013) Assessing multiple prior models of behaviour under ambiguity,J Risk Uncertain 46:113–132 Cook Philip J. and Daniel A. Graham (1977) The Demand for Insurance and Protection: The Case of Irreplaceable The Quarterly Journal of Economics, 91, 1 pp. 143-156 Dixit A., S. Skeath and D. F. Reiley (2009) Games of Strategy, Norton. Epstein L. G. and S E. Zin (1989) Substitution, risk aversion and the temporal behaviour of consumption and asset returns: A theoretical Framework, Econometrica, 57:4:937-969. Gravelle H and R Rees (2004) Microeconomics, 3rd ed. Prentice Hall Fehr E and J. List,(2004) The hidden costs and returns of incentives— trust and trustworthiness among CEOs, Journal of European Economic Association, 2:5:743-771.
Hey J. D. and J. A. Knoll (2001) Strategies in dynamic decision making: An experimental investigation of the rationality of decision behaviour, Journal of Economic Psychology 32 399–409 Hey, J. D., Lotito, G., & Ma¢ oletti, A. (2010). The descriptive and predictive adequacy of theories of decision making under uncertainty/ambiguity. Journal of Risk and Uncertainty, 41(2), 81–111.Experimental lab of John Hey at York Hey J. D and C. Orme (1994) Investigating Generalizations of Expected Utility Theory Using Experimental Data Econometrica, 62, 6, 1291-1326 Hirshleifer J and J G Riley (1992) The Analytics of Uncertainty and Information, Cambridge. Jehle G A and P.J. Reny (2005) Advanced Microeconomic Theory, Pearson Education. Kahneman, D., & Tversky, A. (1979). Prospect theory: an analysis of decision under risk. Econometrica,67, 263–291. Perlo¤ J. M. (2008) Microeconomics: Theory and Applications with Calculus, Pearson. Machina M. J. (1987) Choice under uncertainty: problems solved and unsolved, Journal of economic perspective, 1:1:121-154. Newbery David M. G. Joseph E. Stiglitz (1982) Risk Aversion, Supply Response, and the Optimality of Random Prices: A Diagrammatic Analysis The Quarterly Journal of Economics, 97, 1 (Feb., 1982), pp. 1-26 Newbery David M. G., Joseph E. Stiglitz (1982) The Choice of Techniques and the Optimality of Market Equilibrium with Rational Expectations, Journal of Political Economy, 90, 2, 223246 258
Ö Özak (2014) Optimal consumption under uncertainty, liquidity constraints, and bounded rationality, Journal of Economic Dynamics and Control, 39, 237–254 Radner Roy (1968) Competitive Equilibrium Under Uncertainty Econometrica, 36, 1 31-58 Rothschild Michael and Joseph Stiglitz (1976) Equilibrium in Competitive Insurance Markets: An Essay on the Economics of Imperfect Information The Quarterly Journal of Economics, 90, 4 Nov., pp. 629-649 Rasmusen E(2007) Games and Information, Blackwell, ISBN 1-140513666-9. Rochet JH and J. Tirole (2003) Platform Competition in Two-Sided Markets' Journal of European Economic Association, 1:4:990-1029.
Snyder C and W. Nicholson (2012) Microeconomic Theory: Basic Principles and Extensions, 10th edition, South Western. Varian HR (2010) Intermediate Microeconomics: A Modern Approach, Norton,8th edition Watt Richard (2011) The microeconomics of risk and information, palgrave macmillan. Zizzo D. (2010) Experimenter Demand E¤ects in Economic Experiments, Experimental Economics 13(1), March, 75-98 Kahneman, D., & Tversky, A. (1979). Prospect theory: an analysis of decision under risk. Econometrica,67, 263–291. Machina M. (1987) Choice under uncertainty: problems solved and unsolved, Journal of Economic Perspective, 1:1:121-154.
9
L9:Class Test
Questions are given in sections A and B. Answer two questions, at least one from each section. Each question is worth 100 marks. Each subquestion within a question is of equal value. Use diagrams to illustrate your answers. Section A Q1. A certain consumer has £ 8 to spend in commodities x and y; prices in the market were px = 1 and py = 4. She evaluates utilities from these products according to a standard utility function; u = xa y b and has a = 0:5 and b = 0:5. (a) Derive the Marshallian demand functions for good x and y. Determine the amount purchased of each. (b) Evaluate the optimal utility obtained by this consumer. Derive the expenditure function from that indirect utility function. (c) What is the compensated demand for good x and y for this consumer? What are the compensated and Marshallian demands for good x when the price of x increases to 4, px = 4? 259
(d) Using Slutsky equation decompose the change in demand into the substitution and income e¤ects. (e) Are the relative size of income and substitution e¤ects sensitive to parameters a and b, i.e. share of income spent on x and y? Q2. Illustrate e¢ ciency conditions in allocations of resources: a. When consumers’utility function is given by U (X; Y ) and the production possibility frontier is T (X; Y ): b. E¢ ciency of production when: X = f1 (K1 ; L1 ) + f2 (K2 ; L2 )
(1446)
K1 + K2 = K
(1447)
L1 + L2 = L
(1448)
c. Prove that e¢ cient provision of public goods requires that the sum of the marginal rate of substitution equals the marginal cost of provision of public good with a two consumer economy in which consumers like to maximise utility by consuming private (x) and public goods (G): max
u1 = u1 (x1 ; G)
(1449)
subject to a given level of utility for the second consumer: max u2 = u2 (x2 ; G)
(1450)
x1 + x2 + c (G) = w1 + w2
(1451)
and the resource constraint:
Q3. Consider price adjustments model with N number of …rms: qD
p =
qS
(1452)
where p is percent change in price (p) and is the adustment coe¢ cient. Demand for the product with parameters a > 0 and b < 0 is q D = a + bp; and its supply with a parameter m > 0 is q S = mN . Then: p =
(a + bp
mN )
>0
(1453)
There is free entry and exit of …rms in this market: N =
(p
c)
>0
(1454)
Find the dynamic time path for prices (pt ) and the number of …rms (Nt ) and examine convergence towards the steady state.
260
Q4. Consider general equilibrium with production (x; y) = (x1; x2; :::::xm ; y1; y2; :::::ym ) for consumer i = 1; ; ; m and producer j =1,.,. n. A possible allocation for consumers and producers satisfying following: Consumption set: xi 2 X Production possibility set: yi 2 Y m m n P P P Resource balance condition: xi = ei + yj i=1
i=1
Wealth of the consumer: wi (p) = p:e1 +
j=1
n P
i;j j
(p)
j=1
Supply correspondence: sj (p) = fyi 2 Yj : y 0 2 Yj =) p:y > py 0 g m n P P Excess demand correspondence: Z (p) = (di (p) e1 ) sj (p) i=1
j=1
Here a competitive equilibrium is a pair of prices, demand and supply (p; (x; y)) with a p vector in Rl and x 2 di (p) for consumers i to m, and yj 2 sj (p) for …rms j to n and where the excess demand is zero in equilibrium. Simplify this model to a Robinson Crusoe economy as: Commodity space: R2 (leisure and food) Consumer characteristic: Xi = R2+ Endowment: ei = (24; 0) Preference relation: i U (L; Fn = LF p o Producer characteristics: Yj ( L; F ) : L 0; F L p where F = L is the production function. Solve this model for price vector, demand vector and the output vector that satisfy conditions of general equilibrium in that economy.
261
Section B Q5. Consider a Dixit-Stiglitz model of monopolistic competition in which consumers maximise utility (u) by consuming varieties of di¤erentiated products qi in addition to a unique numeraire product q0 . Their problem is:
max
u = u q0 ;
subject to budget constraint with income (I): X q0 + pi qi
X
1
(1455)
qi
I
(1456)
Producer i maximises pro…t, setting prices pi given marginal cost c and …xed cost f as: max
= (pi
pi
1. Prove that elasticity of demand is
1 1
;i.e.
c) qi =
f @qi pi qi @pi
(1457) =
1 1
:
2. How much does each …rm produce? How does it relate to the elasticity of demand as well as the …xed cost (f ) and the variable cost (c). 3. How many …rms exist in the market? Explain the role of
in it.
Q6. Consider consumers’problems for comparative static analysis max U = U (X; Y )
(1458)
subject to the budget constraint: I = Px X + Py Y
(1459)
where U is utility, I income, X and Y and Px and Py are are amounts and prices of X and Y commodities respectively. 1. Illustrate the …rst order conditions for consumer optimisation in this model. 2. By total di¤erentiation of the …rst order conditions determine (a) the impact of a change in shadow prices on the demand for X and Y: (b) the impact of a change in price of X on the demand for X and Y: (c) the impact of a change the price of Y on the demand for X and Y: (d) the impact of a change in income on the demand for X and Y and the shadow price. 3. Decompose the total e¤ect of a price change into substitution and income e¤ects. 4. Show the major di¤erences between Hicksian and Marshallian demand functions.
262
Q7. A …rm’s objective is to minimise cost (C) C = rK + wL
(1460)
subject to a CES technology constraint: Y = [ L + (1
)K ]
1
(1461)
Here Y is output, K capital, L labour inputs, r interest rate, w wage rate, 0 < labour the substitution parameter.
< 1 share of
1. Determine the demand for labour and capital inputs. 2. Derive the cost function of the …rm. 3. Prove that the elasticity of substitution is
=
1
1:
4. Discuss the properties of the CES cost function. 5. Prove that the Cobb-Douglas production function is a special case of the CES production function.
10
10.1 10.1.1
L10: Impact of Taxes and Public Goods in E¢ ciency, Growth and Redistribution: A General Equilibrium Analysis First best principles E¢ ciency in consumption
Marginal rate of substitution between two products should equal price ratios for a certain consumer Uy Un Ux = = ::::: = Px Py Pn
(1462)
Allocation is Pareto e¢ cient if it is not possible to make one person better o¤ without making another worse o¤. L = U (X; Y ) + [T (X; Y )]
(1463)
@L @U @T = + =0 @X @X @X
(1464)
@L @U @T = + =0 @Y @Y @Y
(1465)
@L = @
(1466)
[T (X; Y )] = 0
263
@U @X @U @Y
=
@T @X @T @Y
(1467)
@Y = RP TX;Y @X This is the optimal point in the production possibility frontier. See the trade model. M RSX;Y =
10.1.2
(1468)
E¢ ciency in production
If it is not possible to more of one good without reducing the production of another good. X = f1 (K1 ; L1 ) + f2 (K2 ; L2 )
(1469)
K1 + K2 = K
(1470)
L1 + L2 = L
(1471)
X = f1 (K1 ; L1 ) + f2 (K
K1 ; L
@f1 @f2 @f1 @X = + = @K1 @K1 @K2 @K1 @f1 @f2 @f1 @X = + = @L1 @L1 @L2 @L1
L1 )
(1472)
@f2 =0 @K2
(1473)
@f2 =0 @L2
(1474)
Marginal productivity of capital and labour inputs are same across both sectors:
10.1.3
@f1 @f2 = @K1 @K2
(1475)
@f1 @f2 = @L1 @L2
(1476)
E¢ ciency of Trade (Exchange)
If it is possible to increase welfare of one country without harming another country. M RSx;y =
Ux Px = Uy Py
= ::::::: = 1
264
M RSx;y =
Ux Px = Uy Py
(1477) N
10.1.4
A simple numerical example of optimal tax or optimal public spending
Assume consumption (C) equals income (Y ); C = Y and public sector is balanced, tax revenue = public spending, T = G Y = Y0 + aG @Y =a @G
2bG = 0
bG2 =) G =
(1478) a 2b
(1479)
2
a is maximum. If Y0 = 50; From second order condition, @@GY2 = 2b < 0 it is clear that G = 2b 60 a = 60 and b = 2; G = 2 2 = 15. Then Y = Y0 + aG bG2 = 50 + 60 15 2 152 = 500: To prove that G = 15 is the maximum evaluate the above function for G = 14 and G = 16: In both cases Y should be less than 500 with optimal G = 15: Y = Y0 + aG bG2 = 50 + 60 14 2 142 = 498: Y = Y0 + aG bG2 = 50 + 60 16 2 162 = 498: This proves that optimal tax revenue in this case is G = T = 15:
10.1.5
E¢ ciency in public goods
When i = 1; :::N individuals live in a society then, the social marginal utility of public goods is the sum of the utilities from public good for individuals SM UP = SM UP1 + SM UP2 + :::: + SM UPN SM RSP;G =
SM UP SM UP1 SM UP2 SM UPN = + + :::: + i i i i M UG M UG M UG M UG SM RSP;G = RP TP;G
(1480) (1481) (1482)
Rate of transformation of private to public goods should equal the social rate of substitution of private to public goods. 10.1.6
Theory of second best
When the optimal point is not achievable, other points in the e¢ ciency frontier are not necessarily optimal. (draw a diagram to prove). 10.1.7
Externality
Externality has been one important issue of research in microeconomics in recent years (Porter and van der Linde (1995) Hanemann (1994) Dixit (1992) Palmer, Oates, Portney (1995), Stern (2008)). What would happen to public parks if city councils do not maintain? Personal and social bene…ts of a beautiful garden? Why does not produce e¢ cient amount of education and health? Why will market produce excessive amount of water, air or noise pollution? Why many cities in England are introducing congestion charges? 265
Aldy et al. (2010) have discussed how much action in carbon reduction is desirable on cost minimisation and welfare maximising grounds along with the alternative policy artitecture at the international level. Earlier Stern (2008) did a very comprehensive study on economics of climate change. Bohringer and Rutherford (2004) Grubb (2004) Green and Newbery (1992), Manne and Richel (1992), McFarland, Reilly and Herzog (2002) Nordhaus (1979), Perroni and Rutherford (1993), Backus and Crucini (2000), Boyd and Doroodian (2001), Coupal and Holland (2002), Grepperud and Rasmussen (2004), Jansen and Klaassen (2000), Kumbaroglu (2003), Spear( 2003) and Thompson (2000) use partial or general equilibrium models with the electricity sector to examine how pollution arises in process of generating energy required for e¢ cient functioning of economies under investigation. How do the production activities over time generate pollution at the local, national or the global level? How do these a¤ect the climate change? What are their consequences? Who bears the burdens of such adjustments? How is the dividend from the improved environment from emission control at the global level shared by advanced or developing economies under the Montreal or Kyoto protocols? These questions are examined in several studies including those of Aronsson, (1999), Bohringer and Conrad and Loschel (2003), Crettez and Aronsson (1999), Crettez (2004) Dissou, Mac Leod, and Souissi (2002), Faehn and Holmoy (2003) Nordhaus and Yang (1996), Proost and Van Regemorter (1992), Rasmussen (2001), Kumbaroglu (2003), Roson (2003) , Uri and Boyd (1996) and Vennemo (1997). Despite so many global negotiations including Cancun (2001) and Copenhagen (2009) summits, apparently very few studies have measured and demonstrated the level of pollution as a consequence of economic activities in multisectoral dynamic general equilibrium framework for the UK. Positive Externality A classic example of positive externality: bees pollinate apple trees and they get materials for honey from apples. For instance cost of producing apples is Ca = a2 and cost of producing honey Ch = h2 a, Private market solution Firms maximise own pro…t independently: a
by marginal cost pricing rule and
@ a @a
a2
= Pa a
= Pa
(1483)
2a = 0 =) Pa = 2a and hence supply of apples
a=
Pa 2
(1484)
Similarly h
= Ph h
h2 + a
(1485)
Supply of honey by the private market @ a 2h = 0 =) Ph = 2h and @a = Ph Ph : 2 Private market does not consider positive externality. Now consider a social planner that produces both to maximise joint pro…t: h=
= Pa a
a2 + Ph h
Then optimal apple supply is : 266
h2 + a
(1486)
(1487)
@ a @a
= Pa
2a + 1 = 0 =) Pa = 2a
1 .and a=
1 Pa + 2 2
(1488)
Ph 2
(1489)
Optimal honey supply is @ h 2h = 0 =) Ph = 2h .and @h = Ph h=
It is optimal to produce more apples taking account of its positive externality. Negative Externality Negative externality production of electricity and pollution and food production Electricity production using coal generates electricity as well as pollution. This pollution raises production cost in the food industry. Cost of electricity production when the environment is not taken into account 2 Ce = e2 (x 3) and its pro…t function is: e = Pe e e2 (x 3) the cost of food production Cf = f 2 + 2x and its pro…t f = Pf f f 2 2x. Pollution adds extra cost in food production. Private market solution 2 e2 (x 3) e = Pe e @ e 2e = 0 =) Pe = 2e and hence supply of electricity @e = Pe e= = Pf f @ f @f = Pf f
Pe 2
(1490)
f 2 2x 2f = 0 =) Pf = 2f and hence supply of food f=
Pf 2
(1491)
Here pollution is produced more than optimal. @ e = 2 (x @x
3) = 0 =) x = 3:
(1492)
Socially optimal solution : = @ e @e
= Pe
e
+
f
= Pe e
e2
(x
= Pf
f2
2x
(1493)
2e = 0 =) Pe = 2e and hence supply of electricity e=
@ f @f
2
3) + Pf f
Pe 2
(1494)
2f = 0 =) Pf = 2f and hence supply of food f=
Pf 2
267
(1495)
@ = 2 (x 3) + 2 = 0 =) x = 2: @x Social solution generates less pollution than the market solution. 10.1.8
(1496)
Samuelson and Nash on Sharing Public Good
Consider a case where two friends share a public good x = x1 + x2 but consume private good yi . 1
1
max u1 = (x1 + x2 ) 2 y12
(1497)
10x1 + y1 = 300
(1498)
subject to
1
1
L1 = (x1 + x2 ) 2 y12
[300
1 @L = (x1 + x2 ) @x1 2
1 2
y1 ]
(1499)
1
10 = 0
y12
1 @L 1 = (x1 + x2 ) 2 y1 @y1 2
@L = 300 @
10x1
1 2
10x1
(1500)
=0
(1501)
y1 = 0
(1502)
From the …rst two FOC y1 = 10 =) y1 = 10(x1 + x2 ) (x1 + x2 )
(1503)
Putting this y1 back in the budget constraint 300 10x1 y1 = 300 10x1 10(x1 + x2 ) = 0 x2 2 As the problem is symmetric dimilar proces for individual 2 we get
(1504)
x1 = 15
1
1
L2 = (x1 + x2 ) 2 y22
[300
x2 = 15 2
1 15 2
y2 ]
(1505)
x1 2
x2 = 15 x1 = 15
10x2
x1 2
(1506)
=) x1 =
41 32
x1 = x2 = 10 =) x1 + x2 = 20 y1 = 10(x1 + x2 ) = 10 268
20 = 200
15 = 10
(1507) (1508) (1509)
Utility level in non-cooperative Nash scenario is: 1
1
1
1
u1 = (x1 + x2 ) 2 y12 = (20) 2 200 2 = 63:2 = u2
(1510)
Under the Samuelsonian rule @L @x1 @L @y1 1 2 (x1 1 2 (x1
1 2
+ x2 ) 1 2
+
1
y12
+ x2 ) y1
1 2
+
@L @x2 @L @y2 1 2 (x1 1 2 (x1
= M RS1 + M RS2 = M RT 1 2
+ x2 )
(1511)
1
y22
1 2
+ x2 ) y2
y2 px 10 y1 + = = = 10 x x py 1
=
1 2
y1 + y2 = 10x
(1512) (1513)
Combined budget constraint of both persons: 10x + y1 + y2 = 600
(1514)
10x + 10x = 600 =) x = 30
(1515)
y1 + y2 = 10x = 10
30 = 300
(1516)
If the private good is equally devided each gets 150. 1
1
1
1
u1 = (x1 + x2 ) 2 y12 = (30) 2 150 2 = 67:1 = u2
(1517)
Table 73: Ine¢ cienty of competitive equilibrium in case of positive externality Nash (CE) Optimal (Samuelson) x 20 30 y1 200 150 y2 200 150 u 63.2 67.1 Key questions: Pollution controls are less important in developing countries such as China and India. How does it a¤ect the global environment? Readings: VAR 35 10.1.9
Sameulson’s Theorem on Public Good
Provision of public goods: two consumers, and public and private goods, but not clear how they should pay for it; valuation of each person is di¤erent. Proposition: Pareto optimality requires that sum of the marginal rate of substitution between private and public goods by two individuals should equal the marginal cost of provision of public goods (see two citizen public good model). Consumers consume private (x) and public goods (G)
269
max
u1 = u1 (x1 ; G)
(1518)
subject to a given level of utility for the second consumer max u2 = u2 (x2 ; G)
(1519)
x1 + x2 + c (G) = w1 + w2
(1520)
and the resource contraint
Constrained optimisation for this is L = u1 (x1 ; G) @L @x1 @L @G
(x1 ;G) = @u1@x 1 (x1 ;G) = @u1@G @u1 (x1 ;G) @G @u1 (x1 ;G) @x1
10.1.10
[u2
= 0 =) @u2 (x2 ;G) @G
+
@u2 (x2 ;G) @G @u2 (x2 ;G) @x2
u2 (x2 ; G)]
[x1 + x2 + c (G)
w1
w2 ]
(x1 ;G) @u2 (x2 ;G) (x2 ;G) @L = @u1@x ; ; @x = = 0 =) @u2@x2 @x2 1 1 @c(G) @u2 (x2 ;G) 1 @u1 (x1 ;G) = @c(G) @G = 0; =) @G @G @G
=
@c (G) ; @G
M RS1 + M RS2 = M C(G):::Q:E:D:
(1521) =
(1522)
Negative externality in production U = U (xc ; yc )
(1523)
xo = f (yi )
(1524)
yo = g(xi ; xo )
(1525)
xc + xi = xo + x
(1526)
yc + yi = yo + y
(1527)
See: Snyder C. and W. Nicholson (2012) Microeconomic Theory: Basic Principles and Extensions, 11th ed.. South Western. Competitive equilibrium is ine¢ cient; There is over production of x. L
= U (xc ; yc ) + +
3
1
[xc + xi
[f (yi ) xo
xo ] + x ]+
3
2
[g(xi ; xo )
[yc + yi
yo
yo ] y ]
(1528)
FOC: i:
@L = U1 + @xc
3
=0
(1529)
@L = U2 + @yc
4
=0
(1530)
ii:
270
iii:
@L = @xi
2 g1
+
3
=0
(1531)
iv:
@L = @yi
1 fy
+
4
=0
(1532)
+
2 g2
@L = @xo
v:
vi:
1
@L = @yo
2
3
4
=0
(1533)
=0
(1534)
3
(1535)
From i and ii U1 = M RSx;y = U2
4
From iii and vi 3
M RSx;y =
=
2 g1
4
= g1
(1536)
2
from iv 1
1 fy
=
4
(1537)
From v M RSx;y =
3
=
1
+
4
2 g2
=
4
1
2 g2
+
4
=
4
1 fy
2
g2 =
2
1 fy
g2
(1538)
Competitive equilibrium condition M RSx;y =
dy py 1 = = dx px fy
(1539)
but the optimal allocation after taking of externality is M RSx;y = here
10.2
1 fy
g2
(1540)
g2 is the measure of negative externality; x is over produced.
Negative externality and Pigouvian tax
Pigouvian tax to correct negative externality in case of upstream-downstream …rms 1
x = 2000Lx2
(1541)
1
y=f
2000Ly2 (x
x0 )
1 2000Ly2
x 6 x0
271
x > x0
(1542)
If = 0 production of x has no e¤ect on production of y. If increase. Assume px = 1; w=50; = 0 no externality w = p:
@x =) 50 = 1:2000 @Lx
1 2
< 0 output of y decrease as x
1
Lx 2 =) Lx = 202 = 400
(1543)
1
x = 2000Lx2 = 40000
(1544)
Downstream …rms also produces same product: 1
y = 2000Ly2 = 40000 Suppose
=
(1545) 1 2
0:1; x0 = 38000; x still produce x = 2000Lx = 40000: 1
y = 2000Ly2 (40000
w
=
p:
0:1
38000)
@y =) 50 = 1:2000 @Ly
1 2
Ly
(1546)
1 2
(2000)
0:1
1
=) Ly2 = 9:35; Ly = 9:352 = 87:5 1
0:1
y = 2000Ly2 (2000)
(1547)
1
= 2000 (87:5) 2 (2000)
0:1
= 8753
(1548)
Pigouvian tax forces x to produce just x0 = 38000 1
x = 2000Lx2 = 38000 =) Lx = 192 = 361
(1
t) w
=
(1
t) p:
=) 50 = (1 =
10.3
@x =) (1 @Lx t) 1:2000
50 =) t = 0:05:
t) 50 = (1 1 2
(362)
t) 1:2000
(1549)
1 2
1
Lx 2
1 2
(1550)
Carbon Emmission in the UK
Warned by these vulnerabilities and complying to the international climate agreements, the UK government now is committed to cut such emissions by 34% by 2020 and 80% by 2050 (Carbon Plan, DECC (2010)).
272
Table 74: Sources of Green-House Gas (GHG) Emission in UK Percentage Electricity Generation 35 Household Consumption 14 Industry and Business 17 Transport 20 Waste 3 Agriculture 8 Public Sector 3 Total 100 Source: DECC Carbon Plan, London, (2010).
Figure 1 Economists together with natural scientists have attempted to assess the amount of damage of such encroachment into the environment using a dynamic general equilibrium models of one or multiple economies with proper appreciation of interactions among them (Nordhaus (DICE-1993); Perroni and Rutherford (1993), Nordhaus and Yang (Rice -1976); Hope and Newberry (2008) 273
,Grubb, Jamasb and Pollit (2008)). The amount of pollution generated in this manner a¤ects not only the national economy but has global consequences (Stern (2007, 2008)). Scienti…cally pollution - carbon dioxide, methane, nitrogen dioxide or nitric oxide, chloro‡uorocarabon (CFC)) in solid, liquid or gaseous form or the explosive, oxidizing, irritant, toxic, carcinogenic, corrosive, infectious, teratogenic, mutagenic hazardous solid wastes - is detrimental to human, animal and plant lives. It not only contaminates and adulterates the natural environment and ecological balances locally but also has global consequences resulting in the rise of global temperature, acid rains, a large Arctic ozone hole ultimately generating a process called the “greenhouse e¤ect”. Despite that it is hard to quantify the overall damage caused by a particular industry or a nation. Containing global warming requires cooperation of all nations, putting energy and environmental taxes might not be a prudent way of controlling such pollution. Using applied general equilibrium models Whalley and Wigle (1991) had estimated consequence of 50 percent reduction in CO2 gases to cause up to 19 percent reduction in GDP, Vennemo (1997) had shown carbon taxes caused a fall in the wage rate of up to 5 percent, Kombaroglu (2003) reported them to dampen the growth rate by up to 6 percent, Bohringer, Conrad and Loscel (2003) found negative impacts of such taxes on output, employment and the wage rate, Perroni and Rutherford (1993) had shown how pollution permits would a¤ect the structure of trade among economies. Imposition of extra environmental and energy taxes to reduce the pollution a¤ects the behaviour of households and …rms. Taxes reduce the pro…tability of …rms. They invest less, have less capital stock and can produce less. Taxes depress the real income of households and their levels of utility despite working more. These a¤ect sectoral and national growth rates and allocations of inputs and distribution of income among households. There is still a great deal of uncertainty about the optimal rate of carbon tax even after several years of intense research activity on carbon taxes and global warming (Poterba (1993), Stern (2008)). Growth of capital stock, output and investment in the agriculture sector,- that includes farms crops, livestock, forestry, and …sheries- is lower when taxes are imposed in the use of inputs. Scienti…cally it is true that the malpractices in agriculture can generate biomass, organic and inorganic wastes that cause environmental problems and hazards to human and animal health which may result from animal manure and other dejections, animal corpses, residues of plastic, rubber and other petrochemicals, pesticides, pharmaceuticals, papers and wood, mineral fertilizer, scrap tools and agricultural machines. Nitrous oxides generated by these processes can bring respiratory infections, burning of eyes, headache, chest tightening, ground water pollution. Inadequate measures taken to control the spread of crop or animal diseases can produce biological hazards. It is questionable, whether extra tax for controlling such pollution in this manner is reasonable as most of agricultural wastes can be valuable resources if properly recycled with adoption of better agricultural recycling practices; more taxes in input merely deter farmers from spending on better environmentally friendly production technology. Extra taxes reduce the growth rate of output, investment and capital stock in the mining sector. It is well understood that pollution emerging from physical and chemical processing of minerals in metal ores extraction as well as other mining and quarrying sector may generate acids. Drilling mud, dangerous substances, land deformation can be minimised by designing dumping sites for sul…dic waste speci…c materials with proper consideration of climate, hydrogeological conditions to prevent air penetration and water in…ltration rather through higher rates of input taxes. Accumulation of capital stock, output, and investment is a¤ected by extra taxes in the manufacturing sector relative to the benchmark. At the current state of technology manufacturing is not possible without burning fossil fuels directly from machines operating from burning such fuels or
274
indirectly through the use of electricity that is generated through CO2 releasing fossil fuels. This is evident from a cursory look at the composition of 45 di¤erent industries within the manufacturing sector [see Appendix A for a complete list of subsectors]. Despite continuous e¤orts for adopting more e¢ cient and environmental friendly production technologies over years, production plants of these industries are known for generating pollutants such as CO2 , S2 or other hazardous gases as by-products in the production process, ever since the time of industrial revolution. Environmental or energy taxes can only raise the cost of production and lower their motivation to search for the better technology. Growth of capital, output and investment in the energy sector - that includes production and distribution of electricity and gas - are a¤ected negatively by extra input taxes. Electricity is generated from coal, oil, gas, wind turbines and nuclear sources. Coal and oil plant generate larger amount of CO2 in atmosphere and the nuclear sources are di¢ cult to build in the beginning and leave plenty of hazardous wastes at the end. If one looks at the current industrial structure of the energy sector, environmentally friendly renewable sources can not ful…l even a fraction of energy demand and this industry is in needs of support for better technology such as carbon tapping, development of hydrogen and other sources of green energy, extra taxes can only cause a setback. The growth of capital, output and investment in the construction sector is relatively higher than in other sectors mainly because of higher taxes in the use of input in this sector in the benchmark. The distribution sector here consists of motor vehicle distribution and repair, automotive fuel retail, wholesale distribution, retail distribution, hotels, catering, pubs etc. Improper scrapping of old vehicles generates solid waste, cold-storages and refrigeration generates CFC and improper treatment of residues at the retail level causes pollution. Again extra taxes slightly lower the growth of output, capital and investment compared to the steady state. Transport and communication sector that comprises of railway transport, other land transport, water transport, air transport, ancillary transport services, postal and courier services and telecommunications, generates air, water, noise and land pollutions. Extra environmental taxes raise cost of operating their businesses and depress the growth of capital, output and investment in this sector. Better technology rather than taxes can promote the growth of this sector. The business service sector represents banking and …nance, insurance and pension funds, auxiliary …nancial services, owning and dealing in real estate, letting of dwellings, estate agent activities, renting of machinery etc, computer services, research and development, legal activities, accountancy services, market research, management consultancy, architectural activities and technical consultancy, advertising and other business services. Negative externality in this sector may be less visible; intense competition for market often generates rivalry, spam, fraud and unhealthy practices that can put extra costs of providing services. It is di¢ cult for any tax system to prevent such malpractices. Higher rates of taxes reduce its growth compared to the benchmark. The other services sector includes public administration and defence, education, health and veterinary services, social work activities, membership organisations, recreational services, other service activities, private households with employed persons and sewage and sanitary services. Malpractices in social services sector appear in the form of corruption, sleaze, unfair treatment and breach of fundamental liberties, trust and social values which may cause anxiety and create psychological burden and create an unhealthy environment for workers as well as entrepreneurs in the economy. It requires more creative thinking and putting extra taxes creates disincentives and cannot contribute to higher productivity required for growth prospect of this sector. Aldy J.E, A. J. Krupnick, R. G. Newell, I. W. H. Parry and W. A. Pizer (2010) Designing Climate Mitigation Policy, Journal of Economic Literature, 48:4, 903–934 . 275
Aronsson, T. (1999) On Cost Bene…t Rules for Green Taxes, Environmental and Resource Economics, January, v. 13, iss. 1, pp. 31-43 . Backus, D. K. and M. J. Crucini (2000) Oil Prices and the Terms of Trade, Journal of International Economics, February, v. 50, iss. 1, pp. 185-213 . Bhattarai K. (2007) Capital Accumulation, Growth and Redistribution in UK: Multisectoral Impacts of Energy and Pollution Taxes, Research Memorandum, 64, Business School, University of Hull, UK. (see 123 sector dynamic general equilibrium model of the UK economy). Barker P.,R. Blundell and J. Micklewright(1989) Modelling household energy expenditure using micro data, Economic Journal 99:397:720-738. Bohringer, C. and T. F. Rutherford ( 2004) Who Should Pay How Much? Compensation for International Spillovers from Carbon Abatement Policies to Developing Countries–A Global CGE Assessment, Computational Economics, February, v. 23, iss. 1, pp. 71-103 Bohringer, C., K. Conrad, K. and,A. Loschel (2003) Carbon Taxes and Joint Implementation: An Applied General Equilibrium Analysis for Germany and India, Environmental and Resource Economics, January, 24:1:49-76 Boyd, R. G. and Doroodian, K. (2001) A Computable General Equilibrium Treatment of Oil Shocks and U.S. Economic Growth,Journal of Energy and Development, Autumn, 27:1: 43-68. Crettez, B. (2004) Tradeable Emissions Permits, Emissions Taxes and Growth, Manchester School, July, 72:4:443-62. Coupal, R. H and Holland, D. (2002) Economic Impact of Electric Power Industry Deregulation on the State of Washington: A General Equilibrium Analysis, Journal of Agricultural and Resource Economics, July, 27:1:244-60. Dissou, Y., Mac Leod, C. and Souissi, Mokhtar (2002) Compliance Costs to the Kyoto Protocol and Market Structure in Canada: A Dynamic General Equilibrium Analysis, Journal of Policy Modeling, November:24: 7-8: 751-79. Faehn, T., and E. Holmoy (2003) Trade Liberalisation and E¤ects on Pollutive Emissions to Air and Deposits of Solid Waste: A General Equilibrium Assessment for Norway; Economic Modelling, July:20: 4: 703-27. Grepperud, S. and I. Rasmussen (2004) A General Equilibrium Assessment of Rebound Effects, Energy Economics, March, 26:2: 261-82. Grubb, M. (2004): Kyoto and the future of International Climate Change Responses: From here to Where? International Review for Environmental Strategies, Summer 5:1:15-38. Green R. J. and D. M. Newbery(1992): Competition in the British Spot market, Journal of Political Economy, 100:5: 929-953. Grubb M, T. Jamasb and M. Pollitt (2010) Delivering a Low Carbon Electricity System: Technologies, Economics and Policy, Cambridge University Press. 276
Jansen, H. and G. Klaassen(2000) Economic Impacts of the 1997 EU Energy Tax: Simulations with Three EU-Wide Models, Environmental and Resource Economics, February, 15:2:179-97. Kumbaroglu, G. S. (2003) Environmental Taxation and Economic E¤ects: A Computable General Equilibrium Analysis for Turkey, Journal of Policy Modeling, November, 25:8:795810. Manne A and R. Richel (1992) Buying Greenhouse Insurance: The Economic Costs of Carbon Dioxide Emission Limits, MIT Press. Cambridge, MA. McFarland J.R. J, Reilly J.M. and Herzog H.J.( 2002) Representing Energy Technologies in Top-Down Economics Models Using Buttom-up Information, Report 89, CEEPR, MIT. Nordhaus, W.D. and Yang, Z( 1996) A Regional Dynamic General-Equilibrium Model of Alternative Climate-Change Strategies, American Economic Review, September, 86:4:741-65 Nordhaus W. D. (1979) The E¢ cient Use of Energy Resources, New Haven: Yale University Press. Perroni, C. and T. F. Rutherford (1993) International Trade in Carbon Emission Rights and Basic Materials: General Equilibrium Calculations for 2020, Scandinavian Journal of Economics, 95:3:257-78 Poterba J. M. (1993) Global Warming Policy: A Public Finance Perspective, Journal of Economic Perspectives, 7:4:47-63 Proost, S. and Van Regemorter, D. (1992) Economic E¤ects of a Carbon Tax: With a General Equilibrium Illustration for Belgium, Energy Economics, April, 14: 2:136-49. Rasmussen, T. N. (2001) CO2 Abatement Policy with Learning-by-Doing in Renewable Energy, Resource and Energy Economics, October, 23: 4: 297-325. Roson, R. (2003) Climate Change Policies and Tax Recycling Schemes: Simulations with a Dynamic General Equilibrium Model of the Italian Economy, Review of Urban and Regional Development Studies, March, 15:1:26-44 . Stern N. (2007) Stern Review on the Economics of Climate Change, Cambridge University Press. http://www.hm-treasury.gov.uk./documents/international_issues/int_globalchallenges_index.cfm Stern N. (2008) The Economics of Climate Change, American Economic Review, 98:2:1-37. Spear, S. E.( 2003) The Electricity Market Game, Journal of Economic Theory, April 109:2:300323. Thompson, H. (2000) Energy Taxes and Wages in a General Equilibrium Model of Production, OPEC Review, September, 24:3:185-93. Uri, N. D. and Boyd, R. (1996) An Assessment of the Economic Impacts of the Energy Price Increase in Mexico, Journal of Economic Development, December 21: 2: 31-60. Vennemo, H. (1997) A Dynamic Applied General Equilibrium Model with Environmental Feedbacks, Economic Modelling, January, v. 14, iss. 1, pp. 99-154 . 277
Yago M. , J. Atkins, K. Bhattarai, R.Green and S. Trotter (2008) Modelling the Economic Impact of Low-Carbon Electricity in in Grubb, Jamasb and Pollitt (eds.) Delivering a Low Carbon Electricity System: Technologies, Economics and Policy, Cambridge University Press, 2008, pp. 394-413 Weyant J. P. (1993) Costs of Reducing Global Carbon Emission, Journal of Economic Perspectives, 7:4:27-46 Whalley J. and R. Wigle (1991) The International Incidence of Carbon Taxes, in Dornbusch R and Poterba eds., Global Warming: Economic Policy Responses, Cambridge, MIT Press 71-97. 10.3.1
A model of growth, …scal policy and welfare
Traditional Macromodel for Fiscal Policy and Growth (Bruce and Turnovsky(2007)) framework with U=
Z1
1
t
(CGc ) e
dt
0
Production function with public (Gc ) and private capital (K) Y = GP K 1 H=
1
(CGc ) +
K +B
;
0
1
) rB + GP K 1
(1
(1
!) C
T
Standard macromodel economicy growth, …scal policy and welfare: Optimisation @H = (CGc ) @C @H = @K
(1
1
Gc
(1
) GP (1
@H = @B
(1
!) = 0
)K
=0
)r = 0
Firms optimal conditions: Y =
(1
)
GP K
K=
r 1
K
Solving this equilibrium results in: (1
) r = (1
) (1
278
)
GP K
=
Transversality conditions: t
Lim Be t!1
t
= Lim Ke t!1
=0
Steady State equilibrium Y = C + Gc + GP + K (
(
1) ln C + n ln Gc = ln + ln (1
1)
C Gc + n = C Gc
=
!)
r (1
)
Balanced growth: C K B Gc GP = = = = = C K B Gc GP Steady state growth (
1) + n = =
r (1 1
r (1
)
) n
Consumption to capital ratio:
Y K GP
C Gc GP K + + + ; Gc = gc Y and K K K K = gP Y ; 0 < gc < 1; 0 < gP < 1;
=
C Y = K K Impact of tax on consumption ratios C Y = K K
Gc K
GP K
Gc K
K = K =r
1
GP K
= gC 1
K = K
r 1
gC
r 1
r 1
gP
Increase in ince tax ( ) reduces growth rate but raises the private consumption ratio with no e¤ect in the interest rate. Consumption tax (!) does not a¤ect growth rate, . 279
C K
Increase in government consumption (gc ) has no e¤ect on growth rate or interest rate but crowds out private consumption. Spending on infrascture (GP ) raises growth rate. [Following is bases on a paper presented to the AEA (Jan 2012) and ESEM/EEA (August 2012)] Lucas, R. E. (2009), Ideas and Growth. Economica, 76: 1–19.
10.4
Fiscal Policy, Growth and Income Distribution in the UK
The annual average growth rate of GDP in the UK was 0.2 percent on average between AD 1 and 1830 and 2.05 percent between the years 1830 and 20083 . Recessionary episodes were frequent but dynamic forces of growth were stronger to pull the economy into its long term growth path (Fig. 1 and 2). Kuznets (1955) had found widening of income inequality in England in the early phases of industrialisation between 1780 and 1850, when the transition from the mercantilist state to the industrial civilisation was very rapid. The process of urbanisation, lower death rate and higher birth rate, rising rate of saving, investment, capital accumulation and pro…t contributed to such inequality that remained high till 1875. It gradually narrowed down after that time as the UK parliament started enacting several laws (Finance Acts) to move towards a more egalitarian tax and transfer system (Bowley 1914) creating a tax-wedge between the original and post-tax income.
Fig. 1 3 They
were 0.08 percent and 1.5 percent in per capita term. These estimates are based on time series data provided by Maddision (1991) at http://www.ggdc.net/maddison/. See also Hansen and Prescott (2002) and Parente and Prescott (2002).
280
Fig. 2 The public …nance in UK, until 1815, was limited to heavy borrowing to …nance military and naval expenses during the wars and redeeming such debts using revenues from rents, royalties and indirect taxes in the peaceful years (O’Brien (1988), Fig. 4). Equity issues were ignored in the traditional feudalistic or mercantilist mind-set of ‘the rich man in his castle, the poor man at his gate, God made them high and lowly and ordered their estate’even after the Magna Carta (1215) and the Glorious Revolution (1688) that had transferred political power to the people and the parliament. According to economic historians the unprecedented economic growth brought by the industrial revolution and development of trade, commerce and capitalism not only made UK a global economic leader from 1750 to up to 1900 but also created wide gaps in the distribution of income and wealth between the rich and the poor (Williamson (1980), Ward(1994), Weaver (1950)). In the celebrated four cannons of taxation Smith (1776) preached for equality, certainty, convenience and economy while taxing rents, wages and pro…ts. He was worried more on e¢ ciency rather than on redistribution. Frustrated by the plight and worsening living conditions of workers, socialist reformers and radical thinkers including Wilberforce, Owen, Marx and Engels supported unions to organise and agitate for more equal rights and better working conditions of workers. This movement raised the number of MPs representing workers such as Snowden (1907), who eventually were able to promulgate a series of entitlements by enshrining them into the laws such as the Income Tax Act (1853) or Finance Act (1909). Clauses to mobilise additional revenues from the direct and indirect taxes to provide for social services including education and health in these Acts truely initiated an egalitarian tax and transfer system raising the size of state in the economy upto 12 percent of the national income (Fig 3).
281
Fig. 3
Fig. 4 A massive expansion in the public sector relative to the GDP that occurred through the public debt during the World Wars I and II, as suggested by Keynes (1940), left a legacy of a large public sector that have become a permanent feature of the UK economy since then (Hicks, 1954). Acts aimed at relieving the war devastated economy resulted in the public commitment to the social security system as proposed in the Beveridge report in 1942 that brought the share of public sector to around 40 percent of GDP as shown in Table 1. While it seems obvious that the public tax and transfer system has eliminated absolute poverty among the bottom income group, inequality of income has widened further after the another wave of reforms of the public …nance that started in early 1980s. It is shown by increase in Gini coe¢ cients of both the original and post tax income from 28.6 in 1970s to 38.3 in 2000s in Table 1. High in‡ation has further made distribution of income more unequal as the burden of higher in‡ation are born mostly by the low income households (Keynes 1940, Sargent 1987). Such an upward trend in the inequality in the last two decades despite a continuos reform of the tax and bene…t are discussed in greater details in Dutta,Sefton and Weale (2001), Johnson and Webb (1993) and Clark and Leicester (2004) for the UK and Aghion et al. (1999) for other countries. Despite above U-shaped trend in inequality one still …nds a signi…cant degree of redistribution taking place in the UK under the existing tax-bene…t system. Net of tax income of the top income
282
Table 75: Fiscal policy, growth and inequality in the UK: Recent Trends 1950s 1960s 1970s 1980s 1990s 2000s Revenue/GDP 41.1 40.0 41.1 42.5 37.1 37.3 Spending/GDP 39.0 40.2 44.4 44.7 40.4 40.7 De…cit/GDP 2.0 -0.2 -3.3 -2.2 -3.3 -3.4 Debt GDP ratio 145.0 89.6 49.9 40.6 34.6 34.2 Growth rate 2.5 3.1 2.4 2.5 2.2 1.7 Gini of original income 41.3 32.1 43.3 48.8 52.4 51.7 Gini of post tax income 35.4 25.1 28.6 33.8 38.6 38.3 In‡ation 4.2 3.6 13.6 7.6 3.6 2.5 Data source: ONS, OBR, IFS, and http://www.ukpublicspending.co.uk/index.php Gini for 1950 and 1960 rely on Stark (1972), Barna (1945), Nicholson (1964).
households are trimmed down and that of bottom income group is raised substantially by it. For instance in 2009, as shown in Table 2, the net tax payment by an average top 20 percent income earner raises enough revenue to …nance bene…ts received by an average bottom 40 percent household, who get net amount around ten thousand pounds each, twice as much as their contribution to the Treasury (Fig. 5). The extent of redistribution is less serious in the middle income group where four thousand pounds received by the third quintile almost matches the net tax payment by the fourth quintile (Table 2). In real income terms, the impacts of redistribution are more pronounced for households in the top and bottom income groups. The absolute amount received in bene…ts or paid in taxes grow with the growth of the economy. Table 76: Net E¤ects of Tax and Transfer to an Average Household by Quintile in 2009 Bene…ts Taxes Net Cash In Kind Total Direct Indirect Total Gain or Loss Bottom 6883 7555 14,438 -1195 -2965 -4,160 10,278 2nd 8280 7252 15,535 -2200 -3466 -5,666 9,866 3rd 6139 7088 13,227 -4850 -4459 -9,309 3,918 4th 3949 6162 10,111 -8403 -5386 -13,789 -3,678 Top 1992 5123 7,115 -19500 -7441 -26,941 -19,826 Average 5448 6636 12,084 -7230 -4743 -11,973 111 Data source: O¢ ce of the National Statistics; in £ .
283
Fig. 5 Time series on a sets of income measures including the original, gross and post-tax income, available from the O¢ ce of the National Statistics (http://www.ons.gov.uk/ons/statbase/) are helpful in estimating the impacts of tax and transfer in income of a household by the decile or quintile they belong to. The original income contains the sum of wages and salaries, interest and pro…t, annuities and pension, investment and other income. The gross income is obtained by adding cash bene…ts, total of contributory and non-contributory types, to the original income. Direct taxes - income tax, national insurance and council tax, are deducted from the gross income to calculate the disposable income. Further deductions of indirect taxes from it results in the post tax income (PTI) which truly measures the real economic position of a household as in Table 3 for each decile. In kind bene…ts including education, health and housing subsidies are then added to get the amount of …nal income. Thus comparing inequality in the original and PTI provides a rough indication of the di¤erence made by the tax and transfer system in the distribution of income among households in UK (Blundell 2001, Bhattarai, and Whalley 2009). As expected the post-tax income is less unequal than the original income; compare 4.0 versus 69.6 thousands of PTI to 1.9 versus 101.5 thousands of original income in Table 3 for the bottom and top income household respectively. Table 77: Pattern Bottom Households (in mln) 2.6 Original income 1.9 Cash bene…ts 4.9 Gross income 6.8 Direct taxes 0.8 Disposable income 6.0 Indirect taxes 2.0 Post-tax income 4.0 Inkind bene…ts 3.5 Final income 7.6 S o u rc e : C o n s tru c te d fro m d a ta ava ila b le a t:
of Income Distribution in 2nd 3rd 4th 5th 2.6 2.6 2.6 2.6 4.2 7.3 11.7 18.0 7.0 7.6 7.5 6.2 11.2 15.0 19.2 24.2 1.1 1.8 2.7 4.0 10.1 13.1 16.5 20.2 2.2 2.7 3.3 4.2 7.9 10.4 13.2 16.1 4.0 4.8 5.7 5.7 11.8 15.2 18.9 21.8
2009 (in ’000 6th 7th 2.6 2.6 24. 33.5 5.2 4.2 30.1 37.7 5.4 7.6 24.7 30.1 4.6 5.4 20.1 24.8 5.7 6.4 25.8 31.2
Pounds) 8th 9th 2.6 2.6 43.3 58.5 3.2 2.3 46.6 60.8 9.9 14.0 36.7 46.8 6.3 7.2 30.4 39.6 6.6 6.1 37.1 45.7
10th 2.6 101.5 2.4 103.9 24.7 79.2 9.6 69.6 6.7 76.3
http :/ / w w w .sta tistic s.g ov .u k / S TAT B A S E / P ro d u c t.a sp ? v ln k = 1 0 3 3 6 ; Ta b le 1 4 A .
284
All 26 30.5 5.0 35.3 7.2 28.4 4.7 23.6 5.5 29.1
A summary of redistribution by taxes and transfers by quintile is given in Table 4. While the average share of the bottom quintile was about 2.5 percent of original income in comparison to 50.7 percent of the top quintile, the operation of the tax and transfer system lifts the share of the bottom quintile up to 6.8 percent in the PTI and drops the share of top quintile down to 43.8 percent of it. These shares have fallen for the bottom groups and risen for the top groups in the last two decades as is clear from the smaller area under the Lorenz curve in 1983 compared to that in 2009 in Fig. 6.
Bottom 2nd 3rd 4th Top
Table 78: Share of origina and post tax income by quintile in UK, 2009 Original income share Post-tax income share Impacts of tax and transfers, % 2.46 6.75 4.29 6.92 11.33 4.41 15.04 15.92 0.88 24.92 22.25 -2.67 50.71 43.75 -6.96 Data source: O¢ ce of the National Statistics
Fig. 6
285
Fig. 7 The ratio of disposable income of 90th to 10th percentile was around 4.5 twice as much to the ratio of the 3rd to the …rst quartile. 10.4.1
Middle income hypothesis
It is a common perception in the UK that workers in the middle income group drive the economy, they generate the value that is distributed to idle rich and needy poor households. In this so called middle income-group hypothesis, the growth rate of the economy depends on the relative share of this group. Support for this hypothesis is found in the data as indicated by signi…cant coe¢ cients on the share of post tax income of the third quintile (3rd_PTI) and that of the fourth quintile (4th_PTI) in the growth equation. In constrast a higher growth rate does not raise inequality as coe¢ cient on growth is not statistically signi…cant in the inequality equation. This implies that income inequality (Gini coe¢ cient) falls only by raising the share of bottom income group (see also Beaudry et al. 2009). Table 79: Growth Inequality Relations: Testing Middle Income Share Hypothesis Change in growth rate on quintile shares Inequality (Gini-all) on growth rate Variables Coe¢ cient t-value t-prob Variables Coe¢ cient t-value t-prob Intercept 0.54** 21.0 0.00 Intercept 44.6** 16.1 0.00 3rd_PTI 1.81* 2.3 0.03 Growth -0.29 -1.7 0.10 4th_PTI -1.32 -2.6 0.02 Bottom_PTI -1.48** -3.6 0.00 R2 = 0.41, F(1,21) = 7.04 [0.00]** 2 =8.77(0.02) DW=1.11, N=24
R2 = 0.45, F(1,21) = 8.9 [0.02]* 2 =1.5(0.47) DW=1.94, N=24
While the fairness of tax system was at the heart of Meade (1951, 1978) as the optimality of taxes were in Mirrlees (1971 and 2011) the reversion of the Kuznets process as indicated by above in the UK in recent years (Gemmell 1985, Atkinson and Voitchovsky 2011) clearly show that those ideas have not been translated into actions adequately. Behavioural responses to welfares system 286
are shaped fundamentally by the structural features of the economy including the preferences of households, technologies and composition of …rms or the trading arrangements with the global economy. Proper evaluation of full impacts of tax transfer policies therefore requires an applied dynamic general equilibrium model that takes account of the decentralised structure of the UK economy. Even though the general equilibrium model have been built for the UK to study intersectoral and multi-household allocation issues since the pioneering work of Whalley (1975, 1977) and then in Piggott and Whalley (1985) only limited e¤orts have been made to measure the dynamic impacts of taxes in e¢ ciency and growth simultaneously (Bhattarai 1999, 2007). In the current context of rising inequality and declincing growth rate, what will happen to these in the next century is an issue of immense interest which we aim to analyse in this paper. 10.4.2
Current Fiscal Policy Context
As the recovery from the 2008-09 recession towards the steady state has been very slow the current …scal policy of UK aims at achieving the macroeconomic stability, supporting the pro-business and low carbon growth, achieving fairness and providing opportunities for all and in protecting the public services. The programmes and activities that the government can implement to achieve these are limited, however, by its intertempoal budget constraints. A careful analyis of the ratios of revenues, spending and de…cit to the GDP in Table 6 (and Fig. 8) shed some lights on this. Current forecasts of spending targets, revenues and public de…cit are set in the context of slow recovery after the recession that lasted for seven quarters from the second quarter of 2008 to the 4th quarter of 2009. Expansionary …scal and monetary policies taken by the government and the Bank of England have taken economy out of the slump but these are projected to raise the debt ratio to 78 percent of GDP by 2015 and exerting an in‡ationary (stag‡ationary) pressure in the economy (OBR and HM-Treasury, 2011). While these short run policy measures were taken to stimulate the economy so that it could return to its long run equilibrium path, what will happen in the next 80 years from such short run policy measures are determined more by the broad parameters that guide choices of households, …rms and traders in the economy. While there is a pressure on the government to stick to the Smith’s cannons of taxations as stated above it faces further challenges in incorporating ability to pay and bene…ts to tax payers from public spending principles that Mirrlees (1971, 2011) and Meade (1978) have proposed for the UK in recent years. While these studies provide hints for the computations or estimations of the excess burden of taxes in the context of current economic climate, proper quanti…cation of the economic e¤ects of policies on equity, e¢ ciency in allocation, growth and sectoral composition of output and employment over time is a task that can be done only with a more elaborate dynamic general equilibrium model of the UK economy. Table 80: Ratios of Revenue, Speding 2016 2015 Revenue/GDP 37.8 37.7 Spending/GDP 39.0 40.5 De…ct/GDP 1.2 2.9 Debt/GDP 75.8 77.7
287
and De…cit to GDP (OBR) 2011 2010 2009 37.8 37.3 36.5 46.2 46.6 47.7 8.4 9.3 11.1 67.5 60.5 52.8
Fig. 8 The UK government has set its activities within the constraints set by the structure of revenue and spending that have evolved over years by striking a balance between the direct taxes (income tax, national insurance, corporate tax and council tax) that bring about 60 percent of total revenue and the indirect taxes (VAT/Excise and Business Rates) for the remaining 40 percent to minimise the burden of taxes (Table 7). This requires assessment of the more complicated economy-wide income and substitution e¤ects which depend on the ‡exibility of markets as re‡ected in the elasticities of demand and supplies of goods and factors of production over time (Whalley (1975), Bhattarai and Whalley (1999)). While the right blending of progressive income and corporation taxes with regressive national insurance contribution, council taxes and VAT, petrol and fuel duties, business and other taxes is necessary to minimise the burden of taxes, the actual post tax distribution is determined not only by the net of tax income but also by the allocation of public provision of various items of public services and accessibility of households to them. As Table 8 shows around 60 percent of the public spending takes the form of transfer of resources mainly from high income to low income individuals or families for social protection, for personal social services, for health, education and transport services and the remaining 40 percent provides for the basic public goods including defence, public order and safety and servicing of debt required for the smooth functioning of the economy. Thus it is important to consider both the revenue and spending sides simultaneously to assess impacts of …scal policy on growth and redistribution.
288
Fig. 9
Table 81: Source of Revenue in UK (GBP Billion) 2011 2010 2009 Sources of Revenue Revenue Percent Revenue Percent Revenue Percent Income tax 158 0.27 150 0.27 146 0.27 National insurance 101 0.17 99 0.18 97 0.18 Corporation tax 48 0.08 43 0.08 42 0.08 Excise tax 46 0.08 46 0.08 46 0.09 VAT 100 0.17 81 0.15 78 0.14 Business tax 25 0.04 25 0.05 25 0.05 Council tax 26 0.04 25 0.05 26 0.05 Other 85 0.14 79 0.14 81 0.15 Total 589 1.00 548 1.00 541 1.00 Source: Budget Report (March 2011) HM Treasury, http://www.hm-treasury.gov. Note: In 2010/11 income tax is paid for any income above £ 7475 at the basic rate of 20% up to income of £ 35,000, at 40% rate on additional income up to £ 150,000 and at 50% for income above this. National insurance contribution rate is 12% for every employee. Council tax rate vary by the value of property in A to H bands, A paying two third and H paying twice of band D which is liable for council tax amount of £ 1332 for each year. VAT is 20 % and corporation tax is 28 % of corporate pro…t, excise and business tax-subsidy rates vary by product, going up to 95 percent.
289
Fig. 10
Table 82: Elements of Public Expenditure in UK (GBP Billion) 2011 2010 2009 Expenditure Items Spending Percent Spending Percent Spending Percent Social protection 200 0.28 194 0.28 190 0.28 Personal social services 32 0.05 32 0.04 29 0.04 Health 126 0.18 122 0.18 119 0.18 Education 89 0.13 89 0.13 88 0.13 Transport 23 0.03 22 0.03 23 0.03 Defence 40 0.06 40 0.06 38 0.05 Industry, Agr, Employment 20 0.03 20 0.03 21 0.03 Housing and Environment 24 0.03 27 0.04 30 0.04 Public order and safety 33 0.05 35 0.04 36 0.05 Debt and interest 50 0.07 44 0.11 43 0.06 Others 74 0.10 73 0.10 74 0.11 Total 711 1.00 696 1.00 704 1.00 Source: Budget Report (March 2011) HM Treasury, http://www.hm-treasury.gov. Above objectives and constraints faced by the UK economy can be successfully studied here with an applied dynamic general equilibrium model bench-marked to the micro-consistent data constructed from the latest input-output table for the decentralised market of the UK. Long run impacts of current policies on capital accumulation, investment, output and distribution among households is evaluated using results of this model for the 21st century (see Hansen and Prescott (2002) applied Malthus model to 1275 to 1800 and Solow model to 1800-1989 for the UK).A multisectoral dynamic general equilibrium model calibrated to the micro-structural features provided by the input-output table and social accounting matrix of the economy is the most appropriate tool to assess long run impacts of …scal policy in the economy.
290
10.5
Features of Dynamic Tax Model of UK
A general equilibrium model in spirit of Walras (1874), Hicks (1939), Arrow and Debreu (1954), Scarf (1973) and Whalley (1975) is a complete speci…cation of the price system in which prices and quantities are determined for each year by the interactions of demand and supply sides of goods and factor markets. In the dynamic version the relative price for every good for each year depends on the intertemporal preferences of households regarding labour-leisure and consumption and of …rms for capital and labout inuts similar to that in Ramsey (1927, 1928), Solow (1956) or Lucas (1988). Government in‡uences market outcome by distorting the prices by means of taxes and transfers which impact on income, savings, investments and the growth rate of the economy and its production sectors. As a regular macro model, households, …rms and traders optimise (Samuelson 1947, Sargent 1987, Prescott 2002) and choose optimal levels of labour supply, employment, consumption, production and trade. Intertemporal optimisation results in the optimal growth rate of output, capital and investment as in Holland and Scott (1998) or Jensen and Rutherford (2002). How can a set of policies be more e¢ cient in terms of welfare to one household rather than to another is evaluated with a social welfare function. Model is good for analysing available alternatives for long run growth prospects from the accumulation of physical and human capital or for evaluating the e¢ ciency gains from inter-temporally balanced budget or from the tax-transfer system or welfare reforms or from the low-carbon growth strategy. Short run ‡uctuations often studied in the Keynesian or the new Keynesian type economy could be introduced incorporating stochastic shocks to the production or the consumption sides of the economy (see Stern 1992 for desirable properties of this type of model). Dynamics of the applied general equilibrium model of UK with tax and transfer system contained here is an advancement on the comparative static frameworks available in the pioneering work of Whalley (1975), Piggott and Whalley (1985) and Bhattarai and Whalley (2000). This model is better suited to study growth and inequality and shows the evolution of the whole economy based on intertemporal optimsation problems of households, …rms and the government for the 21st century. 10.5.1
Preferences
Model adopts a standard Ramsey (1928) type time separable constant elasticity of substitution (CES) utility function to measure the welfare of households in each period. They engage in the intra-period and inter-temporal substitution between consumption and leisure on relative prices, interest rate, wage rate, tax rates and spending allocations in the economy. It contains AIDS demand similar to that in Deaton and Muellbauer (1980) and has multiple nests. The …rst stage h of it is the aggregation at the level of goods and services Ci;t , next stage of the nest is the choice between that composite goods and leisure Cth ; lth and …nally choice is over consumption-saving decisions across various periods based on Euler conditions. Thus the problem of household h is: max U0h =
1 X
t;h
Uth Cth ; lth
(1551)
t=0
Subject to an intertemporal budget constraint of the form: '
1 X h + wj;t 1 Pi;t 1 + tchi Ci;t t=0
twih
h li;t
#
'1 X t=0
291
h wi;t
1
twih
h Li;t
+ rj;t (1
h tki ) Ki;t
#
(1552)
here tax rates on consumption and income tchi ; twih ; tki are set by the policy makers who aim for optimality and revenue neutrality in process of tax reform. 10.5.2
Production Technology
Each …rm in the model has a unit pro…t function ( i;t ) which is the di¤erence between aggregate composite market price - the composite of prices of domestic sales (P Di;t ) and exports (P Ei;t ), and prices of primary inputs (P Yi;t ) and intermediate inputs (Pi;t ). Thus the problem of a …rm i is: 1
y
max
i;t
=
(1
i ) P Di;t
y
1
y
+
1 y 1
y
i P Ei;t
i P Yi;t
d i
1 X
ai;t Pi;t
(1553)
t=0
Subject to production technology: 1
p
Yj;t = (1
i ) Ki;t
p
1
p
+
i Li;t
p
1 p 1 p
(1554)
Sector speci…c capital (Ki;t ) accumulation: Ii;t = Ki;t
(1
) Ki;t
(1555)
1
Here i and i are share parameters, y and p are elasticities of substitution in trade and production, ai;t are the input-output coe¢ cients giving the economy wide forward and backward linkages. The real returns (rj;t ) from investments across sectors are determined by the marginal productivity of capital that adjust until the net of business tax returns are equal across sectors. The nominal interest rates set by the central bank should converge to these real rates in the long run. Wage rate of household h; wth , equals its marginal productivity (Becker et al. 1990, Meyer and Rosenbaum 2001). 10.5.3
Trade arrangements
Economy is open for the trade. Domestic …rms supply products di¤erentiated from corresponding foreign goods. Traders decide on how much to buy (Di;t ) in the domestic markets and how much to import (Mi;t ) while supplying goods (Ai;t ) to the economy. Choice of consumers between imports and domestic consumption depend on the elasticity of substitution ( m ) between domestic supplies and imported commodities in line of Krugman (1980) and Armington (1969). UK exports products that she produces at lower cost and imports products in which she has no comparative advantage. m
Ai;t = 1 X t=0
d m i Di;t
1
+
P Ei;t Ei;t =
m 1 m
m i Mi;t
1 X P Mi;t Mi;t
m y 1
(1556)
(1557)
t=0
UK economy, being one of the most liberal economies in the world, has almost no tax on exports and has very minimal tari¤s and non-tari¤ barriers on imports.
292
10.5.4
Government sector
Government receives revenues from direct and indirect taxes and tari¤s. These taxes are distortionary and a¤ect the marginal conditions of allocation in consumption, production and trade causing widespread shifts in the demand and supply functions of commodities.Which ones of these tax instruments are optimal sources of revenue and which ones are the most ine¢ cient for it and in generating growth process of the economy is a very important question but could be set following the logic of micro level incentive compatible mechanism of Mirrlees (1971, 2011) or in DiamondMirrlees (1971). It can adopt a balanced budget or a de…cit budget or a cyclically balanced budget or inter temporally balanced budget or it may simply peg de…cit to a …xed debt/GDP ratio. Which one of these strategies is adopted may depend on circumstances of the economy, policy debates and rules based on conventions and international commitments made in the treaties or agreements (i.e. EU or G20). Rt =
H X N X
h tchi Pi;t Ci;t +
h=1 i=1
H X N N X X h h twih wj;t LSi;t + tki ri Ki;t
h=1 i=1
Gt
(1558)
i=1
Ideally people’s preference for public good should decide the degree of freedom the government is given in determining the size public sector relative to the aggregate economic activities (Devereux and Love 1995, Barro (1990), Jensen and Rutherford (2002)). 10.5.5
General Equilibrium in a Growing Economy
General equilibrium is a point of rest, where the opposing forces of demand and supply balance across all markets in each period and over the entire model horizon. It is given by the system of prices of commodities and services, wage rate and interest rate in which demand and supply balance for each period (Hicks 1939). When a model is properly calibrated to the benchmark micro-consistent data set, such prices re‡ect the scarcity for those goods in the economy. Cost bene…t analysis or economic decisions can be based on real level of welfare for a set of alternatives available to the households, …rms and the government. Theoretically there has been much work, since the time of Walras, in …nding whether such equilibrium exists, or is unique or is stable (Scarf 1973, Feenberg and Poterba 2000, Feldstein 1985, Friedman 1962, Lee and Gordon 2005, Hines and Summers 2009, Naito 2006, Lockwood and Manning 1993, Bovenberg and Sørensen 2009). Uniqueness is guaranteed by the properties of preferences, technology and trade, such as continuity, concavity or convexity or twice di¤erentiability of functions. Explicit analytical solutions are possible only for very small scale models that are instructive but hardly representative of the economy (Heckman, Lochner and Taber 1998,García-Peñalosa and Turnovsky 2007). It is common to apply numerical methods to …nd the solutions of these models for a realistic policy analysis. Yi;t =
H X
h Ci;t + Ii;t + Ei;t + gi;t
(1559)
h=1 h
h
Lt = L0 en
h
Gt =
;t
= LSth + lth
N X gi;t i=1
293
(1560)
(1561)
Markets for goods clear but the economy may not always be in equilibrium. Imperfections either in goods or input markets are common giving rise to monopolistic or oligopolist situations. Such imperfections in the markets are often represented by appropriately designed mark-up schemes (Dixit and Stiglitz 1977). These mark ups may be sensitive to strategic interactions between consumers and producers, …rms and government or between the national economy and the Rest of the World. With widening gap between number of vacancies and unemployed workers it is possible to incorporate the equilibrium unemployment features of Mortensen and Pissarides (1994) in the model. 10.5.6
Procedure for Calibration
Computation and calibration of dynamic models like this are discussed in greater details in the literature (Blanchard and Kahn 1980, Sims 1980, Rutherford 1995, Smet and Wouters 2003, den Haan and Marcet 1990, Sims 1980, Kehoe 1985, Taylor and Uhlig 1990, Harrison and Vinod 1992). This model is calibrated to the reference path of the economy using the arbitrage condition in the capital market: k t Pi;t = Ri;t
t Ri;t = (r +
k Pi;t+1 k Pi;t
i ) Pi;t
=
k i ) Pi;t+1
(1
= (r +
1 1 + ri
(1
(1562)
k i ) Pi;t+1
i)
(1563)
(1564)
This helps to calibrate the capital stock and the level of investment in equilibrium path: V i;t = (r +
k i ) Pi;t+1 Ki;t ;
Ii;t =
Ki;t = gi + ri +
i
V i;t ; ri + i
V i;t
k Pi;t = Pi;t+1
(1565) (1566)
i
Even a small reform in the public policy of a sector can have a large impact on the welfare and growth over time if such policy has larger knock on e¤ects in the wider economy and removes the root source of the distortions that can have a detrimental impact on output, employment and investment levels in the economy. 10.5.7
Data for the Benchmark Economy
In their seminal works Stone (1942-43; 1961) and Meade and Stone (1941) had developed methods to construct national account and input-output table of the UK economy. The latest versions of these tables available from the O¢ ce of the National Statistics (as presented in the appendix) are used to construct the micro-consistent data for this model. Demand and supply sides for each production sector, income and expenditure for each category of household are balanced in it. The distribution of income and expenditure for di¤erent categories of households are taken from the Department of Work and Pension that is in process of unifying numerous bene…ts going to low income households 294
(http://www.dwp.gov.uk/research-and-statistics/). Share parameters from these tables are used to decompose the labour and capital income as well as consumption across households. It assumes inter temporal balance of budget by the government during the model horizon allowing occasional de…cits, like the current one, in the short run. It uses existing rates of direct and the indirect taxes that in‡uence the stream of income and consumptions of households and input choices of …rms. Detailed discussion of the microconsistent data set and algorithm and GAMS/MPSGE programmes are skiped here for space reasons. Table 83: Key Parameters of the Model values Elasticity of substitution 1.55 Steady state growth rate 0.02 Benchmark interest rate 0.05 Intertemporal substitution 0.98 Rate of depreciation 0.03 Elasticity of transformation 2.00 Capital labour substitution 1.5 Armington substitution 3.0 VAT rate 0.20 Income tax rates: 0, 0.32 0.4 and 0.5 Model uses literature based values of elasticities of substitution among inputs in production for each sector and the demand for consumption of various goods or between consumption goods and leisure for each household. Intertemporal elasticity of substitution provides trade-o¤ between the current and future choices. Thus the income redistribution e¤ect in the model occurs not only through the di¤erentiation in endowments but also by variations in tax rates on labour and capital income as well as full or reduced rate of VAT on consumption of goods and services and di¤erentiated rates of subsidies and transfers according to criteria set for the households and …rms in the economy. The optimal design of the tax system occurs by considering which one of these tax instruments is cost e¤ective in raising a given amount of revenue and has the least distortions in choices of households and …rms. 10.5.8
Results on Redistribution
Model solutions for the benchmark and counterfactual scenarios provide basis for the evaluation of the current tax and transfer system on both the functional and the size distribution of income for the next century which then could be compared to the historical accounts presented in section one. While the distribution of income between capital and labour are broadly determined by their marginal productivities as well as the amount of each factor used in production in line of standard neoclassical principles of …rms and rates of taxes on the use of these inputs, the size distribution on the income of the households depend on socio-economic structure of the economy. It is the post tax income or the level of utility from composite of consumption and leisure that households care the most. In the model these are ultimately determined by inter and intra- temporal preferences of ten categories of households and technological choices available to the producers in all eleven production sectors and the design of the tax-transfer system as proposed in Mirrlees (2011). These model solutions could …t to the available theories of distribution that emphasize on 295
ability or stochastic factors or individual choice or on human capital or on inheritance or educational inequality or life cycle or public choices for redistribution or justice as presented in Sahota (1978) or Aghion et al. (1999). The dynamic general equilibrium theory thus is the the most comprehensive theory of income distribution (Sen 1974, Auerbach, Kotliko¤ and Skinner 1983, Huggett et al. 2011). Model solutions are used to compute the Gini coe¢ cient to measure impacts of reform on distribution taking note of related literature such as Persson and Tabellini (1994), Mookherjee and Shorrocks (1982), Perotti (1993), King (1983) and Cowell and Flachaire (2012). We adopt Z 1 L(u)du = Dorfman (1979) approach to compute the area under the Lorez curve for L(u) A = 0 Z y 2 1 (1 F (y)) dy from the solution of the model for each year and apply Gini (G) coe¢ cient as 2 0
G = AeAe A to measure the inequality of distribution (Fig. 12). Since the level of utility is the most relevant indicator of the welfare of households we focus on growth and inequality in this variable that are caused by changes in the tax and transfer system. It is observed that more equality not necessarily brings the highest possible welfare to all households. While the welfare of every household can rise if the growth rate is higher but more equality with lower growth rate can reduce the level of the lifetime utility of households as is clear from the solution of the model (Fig. 11). This brings us to more di¢ cult question of choosing an appropriate social welfare function based on comparison of all types households in the model. Given utilities of individual households, U (C1 ; l1 ), U (C2 ; l2 )...U (C10 ; l10 ) it is possible to compute the social welfare function W = W fU1 ; U2 ; :::U10 g which has desirable properties (Dasgupta et al. 1973). Philosophical controversy is in whether to use maxmin criterion of Rawls (1971) which requires …nding the welfare level of the lowest income household to base the improvement in the social welfare or to adopt a weak equity axiom of Sen (1978) to justify Gini computed from the model solutions for ranking policies on the ground of distributional objective. In Atkinson’s measure of inequality (I) with income density function 1 1 ' P yi 1 f (yi ) with mean income , I = 1 f (yi ) ; transfer to lower income is weightier i
in the social welfare function as the parameter rises (Rawlsonian case when ! 1) in the measure of inequality. By constraining revenue neutrality or social welfare neutrality of taxes and spending policies, the model presented here can generate optimal numerical values of tax rates that are consistent to the principles set in Mirrlees (1971) or in Diamond-Mirrlees (1971). When tax rates are properly designed in this manner these can not only minimise the risks due to income uncertainty for low income households but also ensure that the economy moves along its long run steady state mitigating impacts of disturbances as seen in the current recession.
296
Fig.11
Fig. 12
297
Fig. 13
Fig. 14 Model results also help us to evaluate the impacts of current …scal policies in the industrial composition over the long run which could be important in evaluating emerging features of UK economy in a very competitive global economy. 10.5.9
Conclusion
UK economy that grew annually at 2.05 percent in the last two centuries had experienced a rise in income inequality during the peak phase of the industrial revolution around 1850. Thus UK became the economic leader in the global economy in 19th century creating inequality in income distribution. 298
Greater concerns towards the plight of ordinary workers in the 19th century led to actions by trade unions, politicians and philanthropists that resulted in the promulgation of a series redistributive tax and transfers measures changing the focus of public …nance from debt …nancing of wars to an egalitarian modern welfare state from the beginning of 20th century. Disappearance of the Kunznets curve phenomenon on inequality in both the original and the post tax income in the last …ve decades has caused quite a lot of discomfort and tension among people and policy makers particularly when the contribution of recent reforms to growth has become controversial. An attempt is made here to provide an evidence based on a dynamic multisectoral, multi-household general equilibrium model with tax and transfer, calibrated to the structural features of microconsistent input-output table, preferences and technological features of the UK economy. Model results are used to study the evolution of the whole economy in the 21st century. These show how the tax- transfer policies could be designed to prevent income inequality rising further and to ensure that growth rates of all sectors converge towards the steady state by the model horizon. Whether the growth enhancing and inequality reducing objectives could be achieved in the long run depend on the degree of cooperative choices from low as well as high income households in response to the public policies aimed at realising the long run vision of the UK economy. Achieving greater equality by increasing the level of utility of all households would be a sensible policy and is possible from higher rate of economic growth. It is not easy to …nd such solutions if the compensation principles are not clear in setting up a social welfare function as the greater equality in income does not automatically guarantee greater welfare for everyone when the economy is not growing at a desirable space.
References [1] Aghion Philippe, Eve Caroli and Cecilia García-Peñalosa.1999. “Inequality and Economic Growth: The Perspective of the New Growth Theories.” Journal of Economic Literature, 37(4): 1615-1660. [2] Altig David, Alan J. Auerbach, Laurence J. Kotliko¤, Kent A. Smetters and Jan Walliser. 2001. 'Simulating Fundamental Tax Reform in the United States.' American Economic Review, 91(3): 574-595. [3] Arrow, Kenneth J. and Gerard Debreu.1954 “Existence of an Equilibrium for a Competitive Economy.” Econometrica 22, 265-90. [4] Atkinson, A.B.1970. On the measurement of inequality, Journal of Economic Theory 2(3):244263. [5] Atkinson, A. B. and Voitchovsky, S. 2011. 'The Distribution of Top Earnings in the UK since the Second World War.' Economica, 78(311): 440-459. [6] Auerbach Alan J., Laurence J. Kotliko¤ and Jonathan Skinner. 1983. 'The E¢ ciency Gains from Dynamic Tax Reform.' International Economic Review, 24(1): 81-100. [7] Bandyopadyay D. and P. Basu. 2005. 'What drives the cross country growth and inequality correlations?' Canadian Journal of Economics, 38(4):1272-1297. [8] Barro Robert J. 1990. 'Government Spending in a Simple Model of Endogeneous Growth.' Journal of Political Economy, 98(5) Part 2: S103-S125. 299
[9] Beaudry P. , C. Blackorby and D. SZalay. 2009. 'Taxes and employment subsidies in Optimal Redistribution Program.' American Economic Review, 99(1):216-241. [10] Becker, Gary S.; Murphy, Kevin M. and Tamura, Robert.1990. 'Human Capital, Fertility, and Economic Growth.' Journal of Political Economy, Pt. 2, 98(5), pp. S12-37.. [11] Blanchard, O. and C.M. Kahn (1980), ‘The Solution of Linear Di¤erence Models under Rational Expectations’, Econometrica, 48, 5, July, 1305-1313. [12] Bowley A. L. 1914. 'The British Super-Tax and the Distribution of Income.' Quarterly Journal of Economics, 28(2): 255-268. [13] Bhattarai Keshab 2007. 'Welfare Impacts of Equal-Yield Tax Experiment in the UK Economy.' Applied Economics, 39(10-12): 1545-1563. [14] Bhattarai Keshab, Jonathan Haughton and David G. Tuerck (2011) The Economic E¤ects of the Fair Tax: Analysis of Results of a Dynamic CGE Model of the US Economy, memio, University Hull Business School. [15] Bhattarai Keshab and John Whalley.1999. 'Role of labour demand elasticities in tax incidence analysis with heterogeneity of labour.' Empirical Economics, 24(4).599-620. [16] Bhattarai, Keshab and J. Whalley. 2000. 'General Equilibrium Modelling of UK Tax Policy in UK' in Holly Sean and Martin Weale (eds.) Econometric Modelling: Technique and Applications, Cambridge University Press. [17] Bhattarai K and J. Whalley (2009) 'Redistribution E¤ects of Transfers.', Economica 76(3):413-431. [18] Blundell, Richard. 2001. 'Welfare reform for low income workers.', Oxford Economic Papers, 53(2):189–214. [19] Bovenberg Lans and Peter B. Sørensen. 2009. 'Optimal Social Insurance with Linear Income Taxation.' Scandinavian Journal of Economics, 111(2): 251-275. [20] Clark Tom and Andrew Leicester. 2004. 'Inequality and Two Decades of British Tax and Bene…t Reforms.' Fiscal Studies 25(2): 129–158. [21] Cowell F. and E. Flachaire (2012) Inequality with Ordinal Data, Paper for the EEA, Malaga, 2012. [22] Dasgupta, Partha, Amartya Sen, and David Starrett.1973. 'Notes on the Measurement of Inequality,' Journal of Economic Theory 6 (2): 180-187. [23] Deaton Angus and John Muellbauer. 1980. An Almost Ideal Demand System, American Economic Review, 70(3):312-326. [24] den Haan W.J. and A Marcet (1990) Solving the stochastic growth model by parameterising expectations, Jounral of Business and Economic Statistics, 8:1:31-34. [25] Devereux Michael B. and David R. F. Love.1995. 'The Dynamic E¤ects of Government Spending Policies in a Two-Sector Endogenous Growth.' Journal of Money Credit and Banking, 27(1): 232-256. 300
[26] Diamond P. A. and J. A. Mirrlees .1971. 'Optimal taxation and public producton I: Production E¢ ciency.', American Economic Review, 61:1:8-27. [27] Diamond P. A. and J. A. Mirrlees .1971. 'Optimal taxation and public producton II: Tax Rules.', American Economic Review, 61(3-1):261-278. [28] Dixit Avinash K and Joseph E. Stiglitz. 1977.Monopolistic Competition and Optimum Product Diversity, American Economic Review, 67(3):297-308. [29] Dorfman Robert.1979. 'A Formula for the Gini Coe¢ cient.', Review of Economics and Statistics, 61(1):146-149. [30] Dutta Jayasri, J. A. Sefton and M. R. Weale (2001) Income Distribution and Income Dynamics in the United Kingdom, Journal of Applied Econometrics, 16(5): 599-617. [31] Feenberg Daniel R. and James M. Poterba.2000. 'The Income and Tax Share of Very HighIncome Households, 1960-1995.' American Economic Review, 90(2): 264-270: Papers and Proceedings. [32] Feldstein Martin. 1985. “Debt and Taxes in the Theory of Public Finance.”, Journal of Public Economics, 28: 233-245. [33] Friedman Milton. 1962. Capitalism and Freedom, Chicago: Chicago University Press. [34] García-Peñalosa Cecilia and Stephen J. Turnovsky.2007. 'Growth, Income Inequality, and Fiscal Policy: What Are the Relevant Trade-o¤s?', Journal of Money, Credit and Banking, 39 (2/3): 369-394. [35] Gemmell Norman.1985. 'The Incidence of Government Expenditure and Redistribution in the United Kingdom.' Economica, New Series, 52(207): 335-344. [36] Hansen Gary D.and Edward C. Prescott.2002. 'Malthus to Solow.' American Economic Review, 92(4): 1205-1217 [37] Harrison Glenn W. and H.D. Vinod. 1992. “The sensitivity analysis of applied general equilibrium models: completely randomised factorial sampling designs.”Review of Economics and Statistics, 74(2): 357-362 [38] Heckman James J., Lance Lochner and Christopher Taber .1998. 'Tax Policy and HumanCapital Formation.' American Economic Review, 88(2): 293-297, Papers and Proceedings. [39] Hicks J. R. 1939. Value and Capital: An inquiry into some fundamental principles of economic theory, Oxford: Oxford University Press. [40] Hicks Ursula K. (1954) British Public Finances: Their Structure and Development, 1880-1952, London: Oxford University Press. [41] Hines James R. Jr. and Lawrence H. Summers .2009. 'How Globalization A¤ects Tax Design,Tax Policy and the Economy.' Tax Policy and Economy, 23 (1): 123-158 [42] Holland Allison and Andrew Scott. 1998. 'The Determinants of UK Business Cycles.', Economic Journal, 108(449): 1067-1092. 301
[43] Huggett Mark, Gustavo Ventura, and Amir Yaron. 2011., Sources of Lifetime Inequality, American Economic Review 101 (7): 2923–2954. [44] Jensen S.E. H. and T. F. Rutherford .2002. 'Distributional E¤ects of Fiscal Consolidation.' Scandinavian Journal of Economics, 104(3):471-493. [45] Johnson Paul and Steven Webb.1993. 'Explaining the Growth in UK Income Inequality: 1979-1988.' Economic Journal, 103(417): 429-435. [46] Keynes John M. 1940. How to Pay for the War, London: Macmillan. [47] Kehoe, Timothey J.1985. 'The Comparative Static Properties of Tax Models.' Canadian Journal of Economics,18 (2): 314-34. [48] King, M. A. 1983. Welfare analysis of tax reforms using household data.' Journal of Public Econonmics 2 (1): 183-214. [49] Krugman, Paul. (1980) “Scale Economies, Product Di¤erentiation, and the Pattern of Trade.” American Economic Review, 70(5): 950–59. [50] Kuznets Simon.1955. Economic Growth and Income Inequality.' American Economic Review, 45(1): 1-28. [51] Lee Young and Roger H. Gordon.2005. 'Tax structure and economic growth.' Journal of Public Economics, 89 (5-6): 1027-1043. [52] Lockwood, B. and Manning, A. .1993. 'Wage setting and the tax system. Theory and evidence for the United Kingdom.' Journal of Public Economics 52, 1-29. [53] Lucas Robert E. 1988. 'On the Mechanics of Economic Development.', Journal of Monetary Economics, 22, 3-42. [54] Maddison Angus.1991. Dynamic of Capital Accumulation and Economic Growth, Oxford: Oxford University Press. [55] Marshall Alfred.1952. Principles of Economics, London: Macmillan. [56] Meade J. E. and Richard Stone (1941) The Construction of Tables of National Income, Expenditure, Savings and Investment Economic Journal, 51(202/203):216-233. [57] Meade J. E., and Ironside, Jones, Bell, Flemming, Kay, King, Macdonald, Sandford and Whittington, Willis.1978. The Structure and Reform of Direct Taxation. IFS, London: George Allen and Unwin. [58] Meyer Bruce D. and Dan T. Rosenbaum .2001. 'Welfare, the Earned Income Tax Credit, and the Labor Supply of Single Mothers.' Quarterly Journal of Economics, 116(3): 1063-1114 [59] Mirrlees J., and S. Adam, T. Besley, R. Blundell, S. Bond, R. Chote, M. Gammie, P. Johnson, G. Myles, J. Poterba.2010. Dimensions of tax design: the Mirrlees review, Oxford: Oxford University Press. [60] Mirlees, J.A. 1971. 'An exploration in the theory of optimum income taxation.' Review of Economic Studies,38:175-208. 302
[61] Naito Takumi. 2006. 'Growth, revenue, and welfare e¤ects of tari¤ and tax reform: Win–win– win strategies.' Journal of Public Economics, 90(6-7):1263-1280 [62] O’Brien Patrick K. .1988. 'The Political Economy of British Taxation, 1660-1815.' Economic History Review, New Series, 41(1):1-32 [63] Mo¢ tt Robert A. .2003. 'The Negative Income Tax and the Evolution of U.S. Welfare Policy.' Journal of Economic Perspectives, 17(3): 119-140 [64] Mookherjee Dilip and Anthony Shorrocks .1982. 'A Decomposition Analysis of the Trend in UK Income Inequality.' Economic Journal, 92(368): 886-902 [65] Mortensen Dale T and Christopher A. Pissarides. 1994. Job Creation and Job Destruction in the Theory of Unemployment, Review of Economic Studies, 61(3):397-415. [66] Parente, S.L. and E.C. Prescott. 2002. Barriers to Riches. MIT Press, Cambridge. [67] Persson Torsten and Guido Tabellini. 1994. “Is inequality harmful for growth?”, American Economic Review, 84(3): 600-621. [68] Perotti Roberto.1993. 'Political Equilibrium, Income Distribution, and Growth.' Review of Economic Studies, 60(4): 755-776 [69] Prescott Edward C. 2002. Prosperity and Depression, American Economic Review, 92(2): 1-15 [70] Piggott, J. and J.Whalley .1985. UK Tax Policy and Applied General Equilibrium Analysis, Cambridge University Press. [71] Ramsey, Frank P. (1928) A Mathematical Theory of Saving, Economic Journal 38, 543-559. [72] Ramsey, Frank P. (1927) A Contribution to the Theory of Taxation, Economic Journal 37(145): 47-61. [73] Rowntree B. S. 1902. Poverty of Town Life, London:MacMillan. [74] Rebelo, Sergio T.1991. 'Long-run Policy Analysis and Long-run Growth.' Journal of Political Economy, 99(3), 500-21. [75] Rutherford, Thomas F. 1995 “Extension of GAMS for Complementary Problems Arising in applied Economic Analysis.” Journal of Economic Dynamics and Control 19 1299-1324. [76] Sahota Gian Singh.1978. Theories of Personal Income Distribution: A Survey Journal of Economic Literature, 16(1): 1-55. [77] Samuelson P. 1947. Foundation of Economic Analysis, Harvard University Press. [78] Sargent Thomas J. 1987. Dynamic Macroeconomic Theory, London:Harvard University Press. [79] Scarf, Herbert. 1973.The computation of economic equilibria, New Haven, Conn. : Yale University Press. [80] Sen Amartya. 1974. 'Informational bases of alternative welfare approaches: Aggregation and income distribution.', Journal of Public Economics, 3(4): 387-403. 303
[81] Snowden Phillip.1907. The Socialist Budget, London: George Allen. [82] Shoup Carl S. 1957. Some distinguishing characteristics of the British, French and United States Public Finance Systems, American Economic Review, 47(2): 187-197. [83] Shoven John B.and John Whalley .1973. 'General Equilibrium with Taxes: A Computational Procedure and an Existence, Review of Economic Studies, 40(4): 475-489 [84] Shoven, J.B. and J.Whalley .1984. 'Applied General-Equilibrium Models of Taxation and International Trade: An Introduction and Survey.' Journal of Economic Literature, 22 (Sept):10071051. [85] Sims Christopher A. 1980. Macroeconomics and Reality, Econometrica, 48(1): 1-48. [86] Smet Frank and Raf Wouters. 2003. An estimated dynamic stochastic general equilibrium model of the Euro Area, Journal of European Economic Association, 1(5):1123-1175. [87] Smith Adam.1776. In Inquiry into the Nature and Cause of Wealth of Nations, vol I and II, Liberty Fund, Indianapolis, Indiana. [88] Solow Robert M. 1956. 'A Contribution to the Theory of Economic Growth.', Quarterly Journal of Economics, 70(1): 65-94. [89] Stern Nicholas.1992. 'From the static to the dynamic: some problems in the theory of taxation.' Journal of Public Economics, 47(2): 273-297 [90] Stiglitz Joseph E. 1978. 'Notes on Estate Taxes, Redistribution, and the Concept of Balanced Growth Path Incidence Notes on Estate.', Journal of Political Economy, 86(2: 2): S137-S150. [91] Stone Richard.1942-43. National Income in the United Kingdom and the United States of America, American Economic Review, 10(1): 1-27. [92] Stone Richard. 1961. Input-output and National Accounts, Paris:OECD. [93] Taylor John B. and Harald Uhlig. 1990. Solving Nonlinear Stochastic Growth Models: A Comparison of Alternative Solution Methods, Journal of Business & Economic Statistics, 8(1): 1-17 [94] Walras, Leon.1874. Elements of Pure Economics, London (1954): Allen and Unwin. [95] Ward J. R. 1994. 'The Industrial Revolution and British Imperialism, 1750-1850.', Economic History Review, 47(1): 44-65. [96] Weaver Findley. 1950. 'Taxation and Redistribution in the United Kingdom.' Review of Economics and Statistics, 32 ( 3): 201-213 [97] Whalley John 1977. 'The United Kingdom Tax System 1968-1970: Some Fixed Point Indications of Its Economic Impact.' Econometrica, 45(8):1837-1858. [98] Whalley John. 1975. 'A General Equilibrium Assessment of the 1973 United Kingdom tax reform.' Economica, 42(166): 139-161. [99] Williamson Je¤rey G. 1980. “Earnings Inequality in Nineteenth-Century Britain.”, Journal of Economic History, 40(3):. 457-475. 304
11
L11: Dynamic Computable General Equilibrium Model: Recent Developments
One sector growth models are analytically tractable but practically they are not designed to answer questions relating to sectoral structure of production, issue of structural transformation and distribution of income as an outcome of the general equilibrium process in the economy. This requires a full dynamic computable general equilibrium (DCGE) model for a decentralised economy. DCGE models contain the relative price system and intertemporal choices of …rms and households as key factors determining the growth of various sectors of the economy and distribution of income among households while studying the long run cycles of model economies (Bhattarai 2010, 2014). The main equations for a typical DCGE model are as follows: h 1) Demand side: welfare of households U0h given by consumption Ci;t and leisure Lht : U0h =
M ax
1 X
t h h Ut ;
0
334
12.2
Dynamic CGE model of the energy and emmission
Households solve the inter-temporal allocation problem by maximising the lifetime utility subject to their lifetime budget constraint as: max U0h =
1 X
t;h
Uth Cth ; lth
t=0
Subject to 1 X
Rt
1
1 X
EMth
(1650)
t=0
h Pi;t 1 + tchi Ci;t + wth 1
twh lth + P Pt EMth
t=0
=
1 X
wth 1
twh LSth + rt (1
tk) Kth + T Rth
(1651)
t=0
h where U0h is lifetime utility of the household h, Ci;t , lth and LSth are respectively composite consumption, leisure and labour supplies of household h in period t, P Pt is the price of pollution
abatement,
EMth is the amount pollution burden in household h, Rt
1
t 1
=
1 s=1 1+rs
is an objective
discount factor whereas is the subjective discount factor of consumer for future consumption relative to current consumption; rs represents the real interest rate on assets at time s; tchi is value added tax on consumption, twh is labour income tax rate, and Kth the capital endowment of household h, Pt is the price of composite consumption (which is based on goods’prices), i.e Pt N
='
i=1
h h i pi;t ,
goods, Cth =
and Cth is the composite consumption, which is composed of sectoral consumption N i=1
h h i Ci;t .
Industries of the economy are represented by …rms that combine both capital and labour input in production and supply goods and services to the market to maximise their pro…ts: 1
y
max
j;t
=
(1
i ) P Dj;t
i P Yj;t
d i
y
+
i P Ej;t
1 X ai;j Pi;t t=0
1
y
1 y 1
y
m i
1 X am i;j Pi;t
(1652)
t=0
where: j;t is the unit pro…t of activity in sector j; P Ej;t is the export price of good j; P Dj;t is the domestic price of good j; P Yj;t is the price of value added per unit of output in activity j; y is a transformation elasticity parameter ; Pi;t is the price of …nal goods used as intermediate goods; i is the share parameter for exports in total production; i is the share of costs paid to labour and capital; di is the cost share of domestic intermediate inputs,( m i for imported intermediate inputs); am are input-output coe¢ cients for domestic supply of intermediate goods. i;j This is an open economy model in which goods produced at home and foreign countries are considered close substitutes by Armington assumption, popular in the applied general equilibrium literature. The production, trade and supply processes by sectors is easy to comprehend with four level nests of functions for each as in Figure 1. 335
Figure 1:
Figure 2:
336
Households pay taxes to the government, which it returns either as transfers to low income households or spends to pay for public consumption. The government revenue is generates from taxes on consumption, income and trade as:
REVt
=
N X H X
h tki;t rt Ki;t +
i=1 h=1
N X H X
h tchi Pi;t Ci;t +
i=1 h=1
N X tgv i Pi;t Gi;t i=1
N N N X X X p vk + ti Pi;t Ii;t + ti Pi;t GYi;t + tm i P Mi;t GYi;t i=1
+
H X
i=1
rt tk Kth +
h=1
H X
i=1
wth twh LSth
(1653)
h=1
where REVt is total government revenue and is a composite tax rate on capital income from sector i, tchi is the ad valorem tax rate on …nal consumption by households, tgv i is that on public h consumption and tvk i is the ad valorem tax rate on investment, tw is the tax rate on labour income of the household, tk is the tax on production, and tm i is the tari¤ on imports. The steady state equilibrium growth path of the economy is determined by relative prices of goods and factors such as the rental rate and the interest rate, that ultimately depend on parameters of the model such as subjective and objective discount factors, elasticities of substitution and many other shift and share parameters. By Walras’ law these prices eliminate the excess demand for goods and factors. These conditions emerge from the resource balance and zero pro…t conditions for the economy and for each household and for the government and for the rest of the world sectors in each period as well as over the entire model horizon. Government tax and transfers policies can alter this equilibrium. Income of each type of household evolves over time as a function of the relative prices of goods and share of households in total endowment of capital and labour. The production process releases emissions that manifests itself in the forms of air, water, land and noise pollutions. Pollution is a by product of production process. This is included by an emission function in the model as following: EM ISt =
N X
i Yi;t
(1654)
i=1
where EM ISt represents the total amount of emission and i is the pollution coe¢ cient for industry i the rate of pollution generated in producing output Yi;t . It is assumed to remain constant for this model. Higher rate of pollution is harmful for growth and hence for the welfare of households. While the consumption of goods generates utility to the households and such pollution generates negative externality. Their utility level falls with the increased amount of emission as it e¤ectively reduces households’life time income. It also raises cost of production as they have spend more on anti-pollution measures.Economists however have paid little attention to the form of social pollution that a¤ects mainly service industries. Corruption, sleaze, malpractices, breach of fundamental human rights and social values create tensions, anxieties, social con‡icts and reduce the creativity and productivity of workers and utility of households though it is very hard to quantify impacts of these externalities. Question 2: Study the dynamic model of UK economy in Bhattarai and Dixon (2014) contained in UK011_HH.GMS and analyse dynamics growth and redistributions impacts of energy sector 337
policies in UKwhen carbon taxes are cut by 20%, 40% and 60% respectively. Write your explanation for the results.
13
Regulation Theory and Practice
13.0.1
Theory of Regulation
Good understanding of microeconomic theories will lead to better policies and regulations for the e¢ cient functioning of the market economy. These policies particularly focus on competition, adoption of better technology, governance and information, correcting externality and good environment, social insurance, more equal distribution of income and identi…cation of cases for government intervention. For recent policies see relevant web page of the government such as in the Department for Business Innovation & Skills https://www.gov.uk/government/organisations/competitionand-markets-authority. Literature on Regulation
Tirole (2014) Market Failure and Public Policy, Nobel Prize Lecture. http://www.nobelprize.org/nobel_prizes/econ sciences/laureates/2014/tirole-lecture.html Fundenberg (191), Markin and Tirole (1990), La¤ront (1997), Jaskow (1996), Rochet and Tirole (1997) Rey (1998), Lerner (1934) Hotz and Mo Xiao (2011) Bundorf and Kosali (2006),Calzolari (2004),Bhattacharya , Goldman, Sood (2004), Buch (2003) Knittel C R V. Stango (2003),Pargal and Mani (2000), Saal and Parker (2000), Cowling (1990) Newbery (1999),Unnevehr, Gómez, Garcia (1998) ,Viscusi (1996), Wheelock and Wilson (1995),Dewatripont and Tirole (1994), Olsen and Torsvik (1993) Berg and Tschirhart (1988), Cowling and Waterson (1976) Ja¤e and Mandelker (1975), Stigler (1971), Bain (1951), Mayson (1939), Lerner (1934),Marshall (1890) Introduction to regulation Tirole (2014) in his Nobel lecture states producer deliver goods to costumers but policy makers should be aware that …rms may provide low quality goods at higher prices. This must be checked by developing a business model speci…c to …rms based empirical analysis or laboratory experiments. Markets fail to provide quality goods in unchecked. Theory of industrial regulations starts from Cournot and Du Point in 19th century; Sherman Act 1890 and Structure Conduct Performance (SCP) hypothesis.
338
Chicago school led by Stigler, Demeltz and Posner in general favoured the competitive market without any speci…c theoretical doctrine for regulation. Collective e¤orts by Fundenberg (1991), Markin (1994), La¤ront (1997), Jaskow (1996), Rochet and Tirole (1997) Rey (1998), Lerner (1934) combined game theory and information economics in designing optimal regulations. Regulatory authorities, for electricity, telecommunication, railways, airlines and road transports, postal o¢ ces, …nancial institutions and ports sprang up in Europe as well as in America. The regulators paid attention to the cost of …rms, prices they charge and the rate of return analysis in regulating these industries. Particularly they compared trade-o¤s between the lower prices and rate of return. Anti-trust laws were designed to prevent horizontal and vertical mergers and to protect patents and innovations. Industries controlling the bottleneck inputs such as railway tracks or postal services were allowed to integrate their downstream services in order provide cheaper commodities to the …nal producers based on cost plus or …xed price contracts to avoid adverse selection and moral hazard problems in the research and development and innovations. Authorities also could auction monopoly rights. Regulators can design incentive compatible mechanism so the it is not in the interest of the …rms with market power to their full extent following Ramsey Boiteux pricing strategy. Such incentive contracts can generate superior outcome as …rms most often have more information about their customers than the regulators particularly in two sided markets. Regulators should not intervene in a¤ecting the price structure and should practice fair reasonable and non-discriminating rates (FRAND) in anti-trust regulation for e¢ cient to make a better world. Better understanding of the cost and demand sides of industries is essential for better regulations. 13.0.2
Measures of concentration and performance
Structure Conduct and Performance (SCP) paradigm Number of …rms (n), buyers and sellers, nature of products and entry barriers Concentration curves, concentration ratios (cumulative market share: CRx =
x P
Si ), He…ndahl-
i=1
Hirschman Index (HHI) : HHI = , Entropy Index: E = 1 N
n P
n P
i=1
Si log
i=1 2
n P
1 Si
Si2
, Hannah and Kay Index (1977): HK =
, Variance of the logarithms of …rm size: V =
(log Si ) , Gini Coe¢ cient
i=1
339
n P
1 1
Si i=1 n P 2 1 (log Si ) N i=1
Welfare measure: Harberger’s welfare loss Dwl =
1 1 q p= 2 2
q
e p
p
p *e=
q
p p
q
Why research need to be subsidized? Consider an economy with production function Y = 10(L labour, w the wage rate.
F ), where F is …xed labour, L is
Then the cost of production is C = wL and the cost function by substituting L from the Y +F : production function: C = w 10 Under the marginal cost pricing rule:
@C @Y
Average cost declines with production:
=
C Y
w 10
=
but the producers experience negative pro…t:
= P.
w 10
+
wF Y
=R
C=
w 10 Y
w
Y 10
+F =
wF < 0
They will not undertake this project on their own. Government need to subsidise to produce optimal amount of research. This example was based on Jones (2002) Introduction to Economic Growth.
Mark-up: basis for regulation T Ri = Pi Yi ;
M Ri
13.0.3
1
(1655)
@ (T Ri = Pi Yi ) @Pi = Pi + Yi @Yi @Yi 1 @Pi Yi = Pi 1 = Pi 1 + Pi @Yi
=
M Ri = M Ci =) Pi = Here
@Yi Pi @Pi Yi
=
M Ci 1
=
1
(1656)
M Ci
is the measure of the mark up.
Regulation for solve the moral hazard problems in the …nancial markets
Asymmetric information Problem: Project B earns more but is riskier than project A. Probability of success of projects A and B are given by a and b respectively. Proportions of types A and B agents is given by pa and pb respectively Interest rate should be lower in project A than in project B in equilibrium. Under the asymmetric information a lender charging a pooling interest rate is unfair to the safe borrower A and more generous to the risky borrower B. 340
Proper signaling (incentive compatible insurance schemes) can remove such ine¢ ciency due to moral hazard in the …nancial markets. Solution of moral hazard problem Here arbitrage condition for the lender implies same expected return from both projects: a rA
=
b rB
(1657)
Since rB > rA then a > b to get this identity. Probability of types of borrower sums to 1: pa + pb = 1. Pooling interest rate: r = pa rA + pb rB
(1658)
Probability of success of projects A is higher than that of project B but the rate of return is lower in it. It need to be proved that r > rA and r < rB . r = (1
pb ) rA + pb rB = rA + pb (rB
rA ) > rA
(1659)
rB ) < rB
(1660)
Similarly r = pa rA + (1 13.0.4
pa ) rB = rB + pa (rA
Regulation by mechanism design by banks
Mechanism Design by banks Consider a bank that has two potential borrowers with amount borrowed B1 and B2 and returns
R1 and R2 respectively. Borrower type 1 has a high yielding project than the borrower type 2. Banker compare returns from the type 1, R1 = 3B1 to that from the type 2. R2 = B2 . Banker is not clear to this bank that which one of the two borrowers is more productive.
Principal’s objective function: UP = [
1
(R1
(B1 )) +
2
(R2
(B2 ))]
(1661)
Here Ri measures the returns to the bank from borrower i and Bi the amount let to the investor i from the bank.
13.0.5
Participation and incentive compatible constraints
Mechanism Design by banks Participation constraints type 1:
U1 = B1
2
(R1 )
Participation constraints type 2:
341
0
U2 = B2
2
(R2 )
0
Let the incentive compatible (the self selection constraints) be given by:
U1 = B1 U 2 = B2 13.0.6
R1 3
2
(R2 )
2
U 2 = B2
R2 3
U 2 = B2
(R1 )
2
0
2
(1662)
0
(1663)
(B2 ))]
(1664)
Solving the mechanism design problem of a bank
Solving the mechanism design problem of a bank For simplicity assume that f 1 ; 2 g = f0:5; 0:5g : Principal’s objective function: UP = [0:5 (R1
(B1 )) + 0:5 (R2
Here Ri is measures the returns to principal from borrower i and Bi the bene…t to the investor i from that. 2 Utility function of agent 1: U1 = B1 (R1 ) given that or R1 = 3B1 simply B1 = R31 . 2 Utility function of agent 2: U2 = B2 (R2 ) given that orR2 = B2 . 2 R1 2 Participation constraint in this game is given by U1 = B1 and U2 = B2 (R2 ) . At the 3 same time the agents need to ful…l the self selection constraint as: U1 = B1 U 2 = B2
R1 3
2
(R2 )
2
U 2 = B2
R2 3
U 2 = B2
(R1 )
Solving the mechanism design problem of a bank 2 R2 2 0 B1 = R31 + B2 3 ' 2 R1 R2 UP = 0:5 R1 + B2 3 3 ' 2 R1 R2 UP = 0:5 R1 + R22 + 3 3 two relevant optimal …rst order conditions 0:5 (1 2R2 ) = 0.
@UP @R1
2
2
! !
= 0:5 1
2
2
0 0
+ 0:5 R2
+ 0:5 R2 2R1 9
(1665)
and
(1666)
2
#
(1667)
2
#
(1668)
(B2 )
(R2 ) @UP @R2
= 0:5
2R2
2R1 3
+
Net bene…t for more productive investor B1 is 2.25 and bene…t for less e¢ cient investor B2 is 0.07. Conclusion of a mechanism design problem
342
Design contracts in this manner to di¤erentiate customers according their potentials the lender ensures proper appropriation of funds according to the productivity of the project This solves the problem of missing market or the credit rationing. It re…nes e¢ cient equilibrium in the presence of asymmetric information in the …nancial markets. 13.0.7
IO Approach to pricing and industrial concentration (HHI)
IO Approach to pricing and industrial concentration (HHI) Start with a Cournot Duopoly model: P =a
bQ
Q = q1 + q2
= P q1
1
dP @ 1 q1 =P + @q1 @Q
cq1
c1 = 0 =) a
2bq1
bq2 = 0
Reaction functions q1 =
a
1 q2 2
c 2b
q1 =
a
c 3b
and q2 = and q2 =
a
c
1 q1 2
2b a
c 3b
Extension to many …rms: = P qi
i
@ i dP qi =P + @qi @Q P
1+
P Multiply both sides by
P
1+
ci = 0
and
@Q =1 @qi
ci = 0
and
@Q =1 @qi
dP Q qi @Q P Q
by de…nition the elasticity is e =
cqi
P @Q Q dP
Si e
ci = 0
and
P
ci P
=
Si P
P
P
Si2 HHI = P e e Mark up relates inversely to the Herphindahl-Hirchman Index P
Si
Si ci
P
cm P
=
=
343
HHI e
Si e
13.0.8
Why regulation? Welfare e¤ects of monopoly
Why regulation? Welfare e¤ects of monopoly TR = PQ ; e =
MR
@Q P @P Q
(1669)
@ (T R = P Q) @P =P + Q @Q @Q @P Q 1 P 1+ =P 1+ P @Q e
= =
M R = M C =) P
1+
1 e
= M C =)
M R = M C =) P
1+
1 e
= M C =) e =
M R = M C =) P
1+
1 e
= M C =) e =
Q=
P
(1670)
MC = P
1 e
P
P = MC
P P
P
P = MC
P P
P Q Qe =) =1 P Q
Pro…t of the …rm: = (P
c )Q =
PQ
Welfare of price changes (a la Harberger): 1 1 P Q= PQ = 2 2 2 Thus welfare cost of monopoly is half of its pro…t. W =
13.0.9
Optimal advertising
What is the optimal intensity of advertising: = PQ
cQ
A and Q = f (P; A)
@ dQ =Q+P @P @P @ dQ =P @A @A Dividing the …rst FOC by
dC dQ =0 @Q @P
dC dQ @Q @A
1=0
dQ @P Q.
@ Q dP = +1 @Q P @Q
dC P @Q dC = 0 =) = P @Q P @Q 344
Q dP P @Q
dC @Q
P
Q dP = P @Q
=
P
1 e
The second FOC: P
dC @Q
dQ @A
1 = 0 =)
dQ 1 = dC @A P @Q
Using above results P
dQ P = dC @A P @Q
=
e
This results in Dorfman-Steiner condition for the optimal advertisement intensity for a period: P
dQ A = @A Q
e
A A ea =) = Q PQ e
Overtime these are discounted by r and the depreciation rate ( ) ea A = PQ e (r + ) 13.0.10
Marginal productivity theory and tax credit
Marginal productivity theory and tax credit =
F (K) (1 + r)
P1K K +
F 0 (K) @ = @K (1 + r)
P1K +
M P K = (1 + r) P1K M P K = (1 + r)
) P2K K (1 + r)
(1
(1671)
(1 ) P2K =0 (1 + r)
(1672)
) P2K = 0
(1673)
(1
(1
k
MPK ' r +
k
) 1+
P1K
P1K
(1674) (1675)
Marginal productivity theory and capital income tax =
(1
) F (K) (1 + r)
@ (1 ) F 0 (K) = @K (1 + r) (1
P1K K + P1K +
) M P K = (1 + r) P1K 345
(1
) P2K K (1 + r)
(1676)
(1 ) P2K =0 (1 + r)
(1677)
) P2K = 0
(1678)
(1
(1
) M P K = (1 + r) (1
13.0.11
(1
k
) 1+ k
) MPK ' r +
P1K
P1K
(1679) (1680)
Capital stock with and without capital income tax
Capital stock without capital income tax Y = K and = 0:75 and = 0:2 What is the optimal capital stock for this manufacturer? (hint ). 1
M RP K = P: K 8000: (0:75) K 0:5
1
k
P1K
(1681)
0:03] 2000
(1682)
' r+
' [0:06 + 0:03
solve for K 6000:K
K=
3 0:06
0:25
' [0:06] 2000
(1683)
4
= 504 = 6; 250; 000
(1684)
Capital stock without capital income tax Y = K and = 0:75 and = 0:2 What is the optimal capital stock for this manufacturer? (hint ). (1 (1
) M RP K = P: K
0:2) 8000: (0:75) K 0:5
1
1
' r+
' [0:06 + 0:03
k
P1K
0:03] 2000
(1685) (1686)
solve for K 4800:K
K= 13.0.12
2:4 0:06
0:25
' [0:06] 2000
(1687)
4
= 404 = 2; 560; 000
(1688)
Technological development, human capital and tax rules
Human Capital and Output in the Lucas Model Production with human capital: 1
Y = K ( hL) h = human capital per worker = fraction of time spent on working (1 ) = fraction of time spent on studies 346
(1689)
L = labour supply –(assume this as given) Example : If K =100, L=100 h =3 =0.8, =0.3 1
= 1000:3 (0:8
Y = K ( hL)
Y = K (L)
1
1 0:3
3
100)
= 1000:3 (100)
1 0:3
= 185
(1690)
= 100
(1691)
Stock of Human Capital without tax Stock of human capital (ht )depends on initial human capital: h0 fraction of time spent on studies: (1
)
the rate of human capital created by per unit of time spent on studying: ht = h0 e
(1
:
)t
(1692)
)
(1693)
growth rate of human capital: gh = if h0 =1,
= 0.4, (1
(1
) = 0.2 at after 20 years (t = 20) the human capital stock becomes 4.95. (1
)t
1
= 1000:3 (0:8
ht = h0 e
e0:4
=1
0:2 20
= e1:6 = 4:95
(1694)
output rises to 262: Y = K ( hL)
4:95
1 0:3
100)
= 262
(1695)
Stock of Human Capital with tax Stock of human capital (ht )depends now on also on the tax: initial human capital: h0 fraction of time spent on studies: (1
) (1
)
the rate of human capital created by per unit of time spent on studying: ht = h0 e
(1
)(1
)t
: (1696)
growth rate of human capital: gh = if h0 =1, = 0.4, (1 becomes 4.95.
) = 0.2 (1
ht = h0 e
(1
)(1
(1
) (1
)
(1697)
) = 0:8 at after 20 years (t = 20) the human capital stock )t
=1
e0:4
0:8 0:2 20
= e1:28 = 3:597
(1698)
output rises to 262: 1
Y = K ( hL)
= 1000:3 (0:8 347
3:597
100)
1 0:3
= 209:6
(1699)
13.0.13
Dixit-Stiglitz Model of Monopolistic Competition
Monopolistic competition Examples Monopolistic competition - (Chamberlain): Brand loyalty Product di¤erentiation characterised the main form of the monopolistic competition. Examples includes: ipod, CD, DVD, diskettes Soft drinks: Coke, Pepsi, Fanta, Tango, Sprite, 7 Up, Dr. Pepper, Cars: BMW, Voxhaul, Poeguet, Chrisler, Ford, GM, Toyota, Nissan, Hyundai, Fiat. Cosmetics , Shoes ,Watches, Camera, PC Computers, Fast food,Yoghurt, Aspirins,Pens, Books in microeconomics or macroeconomics. If a …rm reduces its own price rival …rms will reduce it, when it raises its own price none of the others will raise their prices. Kink in demand - Sweezy model of price and quantity rigidity Two questions are important 1) how much does each …rm produce? 2) How many …rms exist in the market? Consumer likes to consume varieties of products: max
u = u q0 ;
subject to: q0 + First order conditions qi u1 pi = u2
X
X
pi qi
X
1
(1700)
qi
I
1
(1701)
1
qi
qi
1
(1702)
1
qi = k:pi
1
; k>0
(1703)
Demand elasticity =
1 @qi pi = qi @pi 1
(1704)
Producer’s problem max
= (pi
pi
c) qi
f
(1705)
For optimisation apply M R = M C condition. pi 1
1
= c;
Less substitutable the product higher the price. All …rms produce the same quantity qi = q ; c
348
pi =
c
c q=f
(1706)
q=
f c1
(1707)
How many …rms exist in the market? Put this solution in the consumers’optimality condition. u1
c
= u2
X
1
qi
1
1
qi
= u2 (nq )
1
1
q
1
(1708)
Now n can be determined by solving this equation. 13.0.14
Market under imperfect competition and average cost pricing
Market under imperfect competition 1. Consider a …rm in monopolistically competitive industry Q=A
B P
(1709)
Q B
(1710)
Prove that its marginal revenue is given by MR = P
1. (a) If the cost function is C = F + cQ then prove that the average cost declines because of the economy of scale. (b) Further assume that the output sold by a …rm, number of …rms, its own price and average prices of …rms are given by Q=S
1 N
b P
P
(1711)
show that the average cost rises to number of …rms in the industry when all …rms charge same price. AC = n:F s +c Prove that price charged by a particular …rm declines with the number of …rms P =c+
1 b n
(1712)
1. (a) Determine the number of …rms and price in equilibrium. Explain entry exit behavior prices when number of …rms are below or above this equilibrium point. (b) Collusive and strategic behaviors may limit above conclusions. Discuss. (c) Apply above model to explain international trade and its impact on prices and number of …rms in a particular industry. (d) Use this model to explain interindustry and intra-industry trade.
349
(e) Use monopolistic competition model to analyse consequences of dumping practices in international trade. Inverse demand function Inverse demand function A B
P =
A B
A B
(1713) Q B
Q
(1714)
2Q =P B
Q B
(1715)
R = PQ = MR =
Q B
a. For scale economy devide both sides of C = F + cQ by Q. @AC = @Q
AC =
F Q
+c
F 0 and V (q) < 0: Firm’s problem and the participation constraint is =T
cq;
[ V (q)
T] > 0
(1723)
Participation constraint First best solution Participation constraint [ V (q)
T] > 0
(1724)
This is when …rm knows the consumer type. It is binding , V (q) = T: Substitute this information on consumer into the …rms objective function. 352
= V (q)
cq
(1725)
Optimal pro…t then means @ = V 0 (q) @q
c = 0 =) V 0 (q) = c
(1726)
This leads to the …rst degree price discrimination; high value consumer will be sold more goods at discounted per unit price and low value customer will be sold less but ends up paying more per unit. (do a graph here)
Second best solution This is when the …rm does not know the type of the consumer. It has a probability belief on type of each type of consumer,0 < < 1 for type high and (1 ) for type low. Now the …rms pro…t becomes: =
(TH
cqH ) + (1
) (TL
cqL )
(1727)
Subject to participation and incentive constraints for low high type consumers as: [ [
LV
(qL )
TL ] > 0
(1728)
HV
(qH )
TH ] > 0
(1729)
[
LV
(qL )
TL ] > [
[
HV
(qH )
TH ] > [
LV
(qH )
HV
(qL )
TH ]
(1730)
TL ]
(1731)
Binding constraints Participation constraint of the low type customer and incentive constraint of the high type customers are binding; resulting in LV
TH = [
H
(qL ) = TL
(V (qH )
(1732)
V (qL )) + TL ]
(1733)
Expected pro…t maximisation = =
[f
H
H
(V (qH )
V (qL )) + TL g
(V (qH ) V (qL )) + L V (qL )
cqH ] + (1
cqH + (1
353
) (TL )(
LV
(qL )
cqL ) cqL )
(1734) (1735)
@ = @qL
HV
HV
0
(
H
0
(qL ) +
LV
(qL ) +
LV
L) V
0
0
0
(qL ) + (1
)
(qL ) + (1
(qL ) + (1
) )
LV
LV
0
LV
0
0
(qL )
(1
(qL ) = (1
(qL ) = (1
)c = 0 )c
)c
(1736) (1737) (1738)
Quantity implications of non-linear pricing LV
0
[
(qL ) = c +
L] V
H
(1
0
(qL )
)
(1739)
Since the last term is positive it implies that L V 0 (qL ) > c Since L V 0 (qL ) = c is the …rst best and qL now should be smaller to have L V 0 (qL ) > c. For the high value type H V 0 (qH ) = c, this means 'no distortions at the top'. An example from Nicholson and Snyder about co¤ee market: p V (q) = 2 q and f
H ; Lg
= f20; 15g c=5,
=
1 2
First best solution when the type of consumer is known. V 0 (q) = q
1 2
then V 0 (q) = q
1 2
2
q=
2
1
= c; q2 = c; q = ( 2 c
c
20 = 16 5 15 2 =9 5
(1740)
Tari¤ 20 2 q = f V (q)j 15 2
p p16 = 160 9 = 90
(1741)
Expected pro…t of the …rm: = =
(TH
cqH ) + (1 ) (TL cqL ) 1 1 (160 80) + (90 45) = 40 + 22:5 = 62:5 2 2
(1742)
When types arepunknown high type may buy 9 ounce and pay 90 cents thus with consumer surplus of 20 2 9 30 = 120 90 = 30 He pays not 160 but 130. Thus the pro…t of the …rms will be = =
(TH cqH ) + (1 ) (TL cqL ) 1 1 (130 5 16) + (90 5 16) = 25 + 22:5 = 47:5 2 2
Now the shopper reduces the size of the cup: 354
(1743)
LV
0
(qL ) = c + [ 2
1
q2 =
L
H
c
L] V
H
(qL ) ;
Lq 2
2
; q=
0
L
H
=c+[
2
=
c
1 2
15 5
L] q
H
20
1 2
(1744)
2
= 22 = 4
(1745)
Tari¤ for the low customer TL =
LV
p 2 4 = 60
(qL ) = 15
(1746)
For high type HV
0
(qL ) = c =) 20
q
1 2
1
= 5 =) q 2 = 4 =) q = 16
(1747)
Non-linear pricing: co¤ee market Now tari¤ from the high type TH = [
H
TH
(V (qH )
= =
[ h
V (qL )) + TL ] = [
(V (qH ) p 20 2 16 H
V (qL )) + p 2 4 + 15
The pro…t in the second best solution is: = =
H
(V (qH )
V (qL )) +
(qL )] p i 2 4 = 160
LV
LV
140 16
= 8:75 and low type pays
60 4
(1748)
(1749) 20 = 140
(TH cqH ) + (1 ) (TL cqL ) 1 1 (140 5 16) + (60 5 4) = 30 + 20 = 50 2 2
Now the high value type pays
(qL )]
(1750)
= 15:
References [1] Berg, Sanford; John Tschirhart (1988). Natural Monopoly Regulation: Principles and Practices. Cambridge University Press. [2] Bhattacharya J, D. Goldman, N Sood (2004) Price Regulation in Secondary Insurance Markets, The Journal of Risk and Insurance, Vol. 71, No. 4 (Dec., 2004), pp. 643-675 [3] Buch C. M (2003) Information or Regulation: What Drives the International Activities of Commercial Banks? Journal of Money, Credit and Banking, 35, . 6, 851-869 [4] Bundorf K. and Kosali I. Simon (2006),The E¤ects of Rate Regulation on Demand for Supplemental Health Insurance American Economic Review, Vol. 96, No. 2 (May, 2006), pp. 67-71 [5] Calzolari G. (2004) Incentive Regulation of Multinational Enterprises International Economic Review, Vol. 45, No. 1 (Feb., 2004), pp. 257-282 355
[6] Cohen S. I. (2001) Microeconomic Policy, Routledge. [7] Cowling K. and D. Mueller (1978) Social cost of monopoly, Economic Journal 88: 727-48. [8] Cowling K. and M. Waterson (1976) Price-cost margins and market structure, Economica 43: 267-74. [9] Dag Morten Dalen, Steinar Strøm, Tonje Haabeth(2006) Price Regulation and Generic Competition in the Pharmaceutical MarketThe European Journal of Health Economics, Vol. 7, No. 3 (Sep., 2006), pp. 208-214 [10] Daughety A. F (1984) Regulation and Industrial Organization.Journal of Political Economy, 92, 5,. 932-953 [11] Daughety A. F and R. Forsythe (1987) The E¤ects of Industry-Wide Price Regulation on Industrial Organization, Journal of Law, Economics, & Organization, 3, 2 , 397-434 [12] Dewatripont M. and Jean Tirole (1994) The prudential regulation of banks, Cambridge Mass.: MIT Press. [13] Fang L and R. Rogerson (2011) Product Market Regulation and Market Work: A Benchmark Analysis, American Economic Journal: Macroeconomics 3 (April 2011): 163–188 [14] Ferguson P R and G L Ferguson (1994) Industrial Economics: Issues and Perspectives, London: McMillan. [15] Green R (2007) EU Regulation and Competition Policy among the Energy Utilities, Institute for Energy Research and Policy University of Birmingham [16] Hotz V. J. and Mo Xiao (2011) The Impact of Regulations on the Supply and Quality of Care in Child Care Markets, American Economic Review 101 (August 2011): 1775–1805 [17] Ja¤e J. F. and G. Mandelker (1975) The Value of the Firm Under Regulation, The Journal of Finance, Vol. 31, No. 2, December 28-30, 701-713 [18] Knittel C R V. Stango (2003) Price Ceilings as Focal Points for Tacit Collusion: Evidence from Credit Cards, American Economic Review, December 2003 [19] Marshall A. (1890) Principles of Economcis, McMillan. [20] Newbery, D. M. G. (1999) Privatization, restructuring, and regulation of network utilities, Cambridge, Mass. : MIT Press. [21] Olsen T. E and Gaute Torsvik (1993) The Ratchet E¤ect in Common Agency: Implications for Regulation and Privatization, Journal of Law, Economics, & Organization, 9, 1 136-158 [22] Pargal S and M. Mani (2000) Citizen Activism, Environmental Regulation, and the Location of Industrial Plants: Evidence from India, Economic Development and Cultural Change, 48, 829-846 [23] Saal D. S and David Parker (2000) The Impact of Privatization and Regulation on the Water and Sewerage Industry in England and Wales: A Translog Cost Function Model, Managerial and Decision Economics, 21, 6 , 253-268 356
[24] Stigler, G. J. (1971) 'The Theory of Economic Regulation.' Bell J. Econ. and Management Sci. 2: 3-21. [25] Tirole, Jean (2006) The theory of corporate …nance, Princeton, N.J. ; Oxford : Princeton University Press [26] Tirole J. (1995) The Theory of Industrial Organisation, MIT Press. [27] Unnevehr L.J, M. I. Gómez, P. Garcia (1998) The Incidence of Producer Welfare Losses from Food Safety Regulation in the Meat Industry, Review of Agricultural Economics, 20, 1,186-201 [28] Viscusi W. K. (1996) Economic Foundations of the Current Regulatory Reform E¤orts,Journal of Economic Perspectives, 10, 3,1996, 119–134 [29] Wheelock D. C. and P. W. Wilson (1995) Explaining Bank Failures: Deposit Insurance, Regulation, and E¢ ciency, The Review of Economics and Statistics, 77, 4 689-700
13.1
Articles and Texts
13.1.1
Best twenty articles in 100 years in the American Economic Review
Arrow, Kenneth J., B. Douglas Bernheim, Martin S. Feldstein, Daniel L. McFadden, James M. Poterba, and Robert M. Solow. 2011. '100 Years of the American Economic Review: The Top 20 Articles.' American Economic Review, 101(1): 1–8.
1. Alchian, Armen A., and Harold Demsetz. 1972. “Production, Information Costs, and Economic Organization.”American Economic Review, 62(5): 777–95.
2. Arrow, Kenneth J. 1963. “Uncertainty and the Welfare Economics of Medical Care.” American Economic Review, 53(5): 941–73.
3. Cobb, Charles W., and Paul H. Douglas. 1928. “A Theory of Production.” American Economic Review, 18(1): 139–65.
4. Deaton, Angus S., and John Muellbauer. 1980. “An Almost Ideal Demand System.” American Economic Review, 70(3): 312–26.
5. Diamond, Peter A. 1965. “National Debt in a Neoclassical Growth Model.” American Economic Review, 55(5): 1126–50.
6. Diamond, Peter A., and James A. Mirrlees. 1971. “Optimal Taxation and Public Production I: Production E¢ ciency.” American Economic Review, 61(1): 8–27.
7. Diamond, Peter A., and James A. Mirrlees. 1971. “Optimal Taxation and Public Production II: Tax Rules.” American Economic Review, 61(3): 261–78.
8. Dixit, Avinash K., and Joseph E. Stiglitz. 1977. “Monopolistic Competition and Optimum Product Diversity.” American Economic Review, 67(3): 297–308.
9. Friedman, Milton. 1968. “The Role of Monetary Policy.” American Economic Review, 58(1): 1–17.
357
10. Grossman, Sanford J., and Joseph E. Stiglitz. 1980. “On the Impossibility of Informationally E¢ cient Markets.” American Economic Review, 70(3): 393–408.
11. Harris, John R., and Michael P. Todaro. 1970. “Migration, Unemployment and Development: A Two- Sector Analysis.” American Economic Review, 60(1): 126–42.
12. Hayek, F. A. 1945. “The Use of Knowledge in Society.” American Economic Review, 35(4): 519–30. 13. Jorgenson, Dale W. 1963. “Capital Theory and Investment Behavior.” American Economic Review,53(2): 247–59.
14. Krueger, Anne O. 1974. “The Political Economy of the Rent-Seeking Society.” American Economic Review, 64(3): 291–303.
15. Krugman, Paul. 1980. “Scale Economies, Product Di¤erentiation, and the Pattern of Trade.” American Economic Review, 70(5): 950–59.
16. Kuznets, Simon. 1955. “Economic Growth and Income Inequality.”American Economic Review,45(1): 1–28.
17. Lucas, Robert E., Jr. 1973. “Some International Evidence on Output-In‡ation Tradeo¤s.”American Economic Review, 63(3): 326–34.
18. Modigliani, Franco, and Merton H. Miller. 1958. “The Cost of Capital, Corporation Finance and the Theory of Investment.” American Economic Review, 48(3): 261–97.
19. Mundell, Robert A. 1961. “A Theory of Optimum Currency Areas.” American Economic Review,51(4): 657–65.
20. Ross, Stephen A. 1973. “The Economic Theory of Agency: The Principal’s Problem.” American Economic Review, 63(2): 134–39.
21. Shiller, Robert J. 1981. “Do Stock Prices Move Too Much to Be Justi…ed by Subsequent Changes in Dividends?” American Economic Review, 71(3): 421–36. 13.1.2
Ten Best articles in the Journal of European Economic Association
1. Frank Smets and Raf Wouters (2003) An Estimated Dynamic Stochastic General Equilibrium Model of the Euro Area', Journal of European Economic Association, 1:5:1123-1175.
2. Jean-Charles Rochet and Jean Tirole (2003) Platform Competition in Two-Sided Markets' Journal of European Economic Association, 1:4:990-1029.
3. Daron Acemoglu, Philippe Aghion and Fabrizio Zilibotti (2006) Distance to Frontier and Economic Growth',Journal of European Economic Association, 4:1:37-74.
4. Alberto Alesina, Filipe R. Campante and Guido Tabellini (2008) Why is …scal policy often procyclical?Journal of European Economic Association, 6:5:1006-1036.
5. Richard Blundell, Monica Costa Dias and Costas Meghir, (2004) Evaluating the employment impact of a mandatory job search program,Journal of European Economic Association, 2:4:569-606.
358
6. Ernst Fehr and John List,(2004) The hidden costs and returns of incentives— trust and trustworthiness among CEOs, Journal of European Economic Association, 2:5:743-771.
7. Jordi Galí, J. David López-Salido and Javier Vallés (2007) Understanding the e¤ects of government spending on consumption, Journal of European Economic Association, 5:1:277-270.
8. Thomas Laubach New Evidence on the Interest Rate E¤ects of Budget De…cits and Debt, Journal of European Economic Association, 7:4:858-885.
9. James H. Stock and Mark W. Watson (2005) Understanding changes in international business cycle dynamics,Journal of European Economic Association, 3:5:968-1006.
10. Guido Tabellini (2010) Culture and institutions: economic development in the regions of Europe, Journal of European Economic Association, 8:4:677-716.
13.1.3
Best 40 articles in the Journal of Economic Perspectives
David Autor (2012) The Journal of Economic Perspectives at 100, Journal of Economic Perspectives, 26, 2,Spring, 3–18
1. Porter, Michael E.;van der Linde,Claas 1995 Toward a New Conception of the Environment-Competitiveness Relationship 9(4) 657
2. Kahneman, Daniel; Knetsch, Jack L.; Thaler, Richard H. 1991 Anomalies: The Endowment E¤ect, Loss Aversion, and Status Quo Bias 5(1) 572
3. Diamond, Peter A.; Hausman, Jerry A. 1994 Contingent Valuation: Is Some Number Better than No Number? 8(4) 524
4. Fehr, Ernst; Gächter,Simon (2000) Fairness and Retaliation: The Economics of Reciprocity 2000 14(3) 490
5. Katz, Michael L.; Shapiro, Carl 1994 Systems Competition and Network E¤ects 8(2) 448 6. North, Douglass C. 1991 Institutions 5(1) 395 7. Koenker, Roger; Hallock, Kevin F. 2001 Quantile Regression 15(4) 375 8. Markusen, James R. 1995 The Boundaries of Multinational Enterprises and the Theory of International Trade 9(2) 375
9. Bernanke, Ben S.; Gertler, Mark 1995 Inside the Black Box: The Credit Channel of Monetary Policy Transmission 9(4) 365
10. Romer, Paul M. 1994 The Origins of Endogenous Growth 8(1) 365 11. Brynjolfsson, Erik; Hitt, Lorin M. 2000 Beyond Computation: Information Technology, Organizational Transformation and Business Performance14(4) 350
12. Nickell, Stephen 1997 Unemployment and Labor Market Rigidities: Europe versus North America 11(3) 344
359
13. Machina, Mark J. 1987 Choice under Uncertainty: Problems Solved and Unsolved 1(1) 338 14. Hanemann, W. Michael 1994 Valuing the Environment through Contingent Valuation 8(4) 332 15. Camerer, Colin; Thaler, Richard H. 1995 Anomalies: Ultimatums, Dictators, and Manners 9(2) 316 16. Ostrom, Elinor 2000 Collective Action and the Evolution of Social Norms 14(3) 313 17. Smith, James P. 1999 Healthy Bodies and Thick Wallets: The Dual Relation between Health and Economic Status 13(2) 311
18. Jarrell, Gregg A.; Brickley, James A.; Netter, Je¤ry M. 1988 The Market for Corporate Control: The Empirical Evidence since 1980 2(1) 295
19. Andrade, Gregor; Mitchell, Mark; Sta¤ord, Erik 2001 New Evidence and Perspectives on Mergers 15(2) 290
20. Scotchmer, Suzanne 1991Standing on the Shoulders of Giants: Cumulative Research and the Patent Law 5(1) 280
21. Simon, Herbert A. 1991 Organizations and Markets 5(2) 278 22. Bikhchandani, Sushil; Hirshleifer,David; and Welch, Ivo 1998 Learning from the Behavior of Others: Conformity, Fads, and Informational Cascades 12(3) 273
23. Elster, Jon 1989 Social Norms and Economic Theory 3(4) 272 24. Feenstra, Robert C. 1998 Integration of Trade and Disintegration of Production in the Global Economy 12(4) 272
25. Frank, Robert H.; Gilovich, Thomas; Regan, Dennis T. 1993 Does Studying Economics Inhibit Cooperation? 7(2) 272
26. Kirman, Alan P. 1992 Whom or What Does the Representative Individual Represent? 6(2) 272 27. Jensen, Michael C. 1988 Takeovers: Their Causes and Consequences 2(1) 268 28. Przeworski, Adam; Limongi, Fernando 1993 Political Regimes and Economic Growth 7(3) 268 29. Newhouse, Joseph P. 1992 Medical Care Costs: How Much Welfare Loss? 6(3) 265 30. Dixit, Avinash 1992 Investment and Hysteresis 6(1) 259 31. Oliner, Stephen D.; Sichel, Daniel E.2000 The Resurgence of Growth in the Late 1990s: Is Information Technology the Story? 14(4) 257
32. Cutler, David M; Glaeser, Edward L.; Shapiro, Jesse M. 2003 Why Have Americans Become More Obsese? 17(3) 250
33. Milgrom, Paul 1989 Auctions and Bidding: A Primer 3(3) 242 34. Portney, Paul R. 1994 The Contingent Valuation Debate: Why Economists Should Care 8(4) 239
360
35. Babcock, Linda; Loewenstein,George 1997 Explaining Bargaining Impasse: The Role of Self-Serving Biases 11(1) 231
36. Grossman, Gene M.; Helpman, Elhanan 1994 Endogenous Innovation in the Theory of Growth 8(1) 225
37. Palmer, Karen; Oates, Wallace E.; Portney, Paul R. 1995 Tightening Environmental Standards: The Bene… t-Cost or the No-Cost Paradigm 9(4) 222
38. Angrist, Joshua D.; Krueger, Alan B. 2001 Instrumental Variables and the Search for Identi… cation: From Supply and Demand to Natural Experiments 15(4) 221
39. Pritchett, Lant 1997 Divergence, Big Time 11(3) 209 40. Dawes, Robyn M.; Thaler, Richard H. 1988 Anomalies: Cooperation 2(3) 206 41. Lundberg, Shelly; Pollak, Robert A. 1996 Bargaining and Distribution in Marriage10(4) 206 IMF Lists 25 Brightest Young Economists, August 27, 2014 (source IMF.org) 1. Nicholas Bloom, Stanford, Uncertainty 2. Amy Finkelstein, MIT , healthcare 3. Raj Chetty, Harvard, tax policy 4. Melissa Dell, Harvard Poverty 5. Kristin Forbes, BOE and MIT International macro 6. Roland Fryer, Harvard, Randomised experiment 7. Xavier Gabaix, New York, finance and macro 8. Gita Gopinath, Harvard, exchange rate 9. Esther Duflo, MIT microeconomics issues in developing countries 10. Matthew Gentzkow, Chicago, empirical micro and media 11. Emmanuel Farhi, Harvard, Macro 12. Oleg Itskhoki, Princeton, globalisation and inequality
1. Hélène Rey, LBS, international macro 2. Emmanuel Saez, California, income inequality 3. Jonathan Levin, Stanford, market design 4. Atif Mian, Princeton, Debt 5. Emi Nakamura, Columbia, business cycle 6. Nathan Nunn, Harvard, economic development 7. Parag Pathak, MIT, market design 8. Thomas Philippon, NYU, risk and financial intermediation 9. Amit Seru, Chicago, regulation and financial intermediation 10. Amir Sufi, Chicago, house price 11. Iván Werning, MIT, macro prudential policy 12. Justin Wolfers, Peterson Institute, political economy
13.
Thomas Piketty, Paris, income inequality
References [1] Aghion P. and J. Tirole (1994) Opening the blackbox of innovation, Oxford : Nu¢ eld College. [2] Alchian, Armen A., and Harold Demsetz. (1972). Production, Information Costs, and Economic Organization. American Economic Review,62(5): 777–95. [3] Allingham M. (1975) General Equilibrium, McMillan. [4] Aoki M. (1984) The cooperative game theory of …rm, Oxford: Oxford University Press. [5] Arrow K. J. (1964) The Role of Securities in the Optimal Allocation of Risk-bearing, Review of Economic Studies, 31, 2, 91-96 [6] Arrow, Kenneth J. (1963). “Uncertainty and the Welfare Economics of Medical Care.”American Economic Review, 53(5): 941–73. [7] Arrow, K.J. and F.H. Hahn (1971) General Competitive Analysis, San Franscisco: HoldenDay. [8] Armstrong M., Simon C., and J. Vickers (1994) Regulatory reform : economic analysis and British experience, Cambridge, Mass. : MIT Press. 361
[9] Arrow, K.J. and G. Debreu (1954) “Existence of an Equilibrium for a Competitive Economy” Econometrica 22, 265-90. [10] Atkinson A.B. and J. E. Stiglitz (1976) “The design of the tax structure: direct versus indirect taxation”, Journal of Public Economics, 6:1-2:55-75. [11] Baldani J., J. Brad…eld and R. Turner (2004) Mathematical Economics, the Drydon Press, London [12] Baldev Raj and R. Boadway eds. (2000) Advances on Public Economics, Physica-Verlag. [13] Balasko, Yves (1988) Foundations of the theory of general equilibrium, Boston : Academic Press. [14] Balasko Y , D. Cass and K. Shell (1995) Market Participation and Sunspot Equilibria, Review of Economic Studies, 62, 3 ,Jul., p 491-512 [15] Borglin A. (2004) Economic Dynamics and General Equilibrium: Time and Uncertainty, Springer. [16] Binmore K (1999) Why Experiment in Economics? The Economic Journal 109, 453, Features , Feb. F16-F24. [17] Bhaskar V. and Ted To (2004) Is Perfect Price Discrimination Really E¢ cient? An Analysis of Free Entry, The RAND Journal of Economics, 35, 4, 762-776 [18] Bhattarai K. (2007) Welfare Impacts of Equal-Yield Tax Experiment in the UK Economy, Applied Economics, 39, 10-12, 1545-1563. [19] Bhattarai K. (2008) Economic Theory and Models: Derivations, Computations and Applications for Policy Analyses, Serials Publications, New Delhi. [20] Bhattarai K. and J. Whalley (2003) Discreteness and the Welfare Cost of Labour Supply Tax Distortions, International Economic Review 44:3:1117-1133. [21] Bhattarai K. and J. Whalley (1999) Role of labour demand elasticities in tax incidence analysis with heterogeneity of labour, Empirical Economics, 24:4:.599-620. [22] Bloom N., R. Sadun and J. van Reenen (2012) The organization of …rms across countries, Quarterly Journal of Economics 127 (4), 1663–1705. [23] Bloom N and R. Sadun and J van Reenen (2012)Americans do I.T better: US Multinationals and the Productivity Miracle ,American Economic Review 102 (1),167-201 [24] Caminal R. (1990) A Dynamic Duopoly Model with Asymmetric Information Journal of Industrial Economics 38, 3, Mar., 315-333 [25] Cho I.K. and D.M. Kreps (1987) Signalling games and stable equilibria, the Quarterly Journal of Economics, May 179-221. [26] Cobb, Charles W., and Paul H. Douglas. (1928) “A Theory of Production.”American Economic Review,18(1): 139–65. 362
[27] Coase R. H. (1990) The Firm, the Market an the Law, Chicago: University of Chicago Press. [28] Coase R. H. (1937) The Nature of the Firm, Economica, 386-405. [29] Cohen K. J and R M Cyert (1976) Theory of the …rm: Resource Allocation in a Market Economy, New Delhi:Prentice Hall. [30] Cornes, R (1993) Duality and modern economics, Cambridge : Cambridge University Press [31] Cornes, R and T. Saddler (1993) The theory of externalities, public goods, and club goods, Cambridge : Cambridge University Press [32] Cornwall, R. R. (1984) Introduction to the use of general equilibrium analysis, Amsterdam : North-Holland. [33] Cripps, M.W.(1997) Bargaining and the Timing of Investment, International Economic Review, 38:3 :Aug.:527-546 [34] Cripps M. W. and J. P. Thomas (1995) Reputation and Commitment in Two-Person Repeated Games Without Discounting, Econometrica, . 63, 6, 1401-1419 [35] Deaton, Angus S., and John Muellbauer. 1980. “An Almost Ideal Demand System.”American Economic Review, 70(3): 312–26. [36] Debreu, G. (1954) The Theory of Value, Yale University Press, New Haven. [37] Dewatripont M. and Jean Tirole (1994) The prudential regulation of banks, Cambridge Mass.: MIT Press. [38] Dasgupta P (1992) Economic analysis of markets and games : essays in honor of Frank Hahn, Cambridge University Press. [39] Diamond, Peter A., and James A. Mirrlees. 1971. “Optimal Taxation and Public Production I: Production E¢ ciency.” American Economic Review, 61(1): 8–27. [40] Diamond, Peter A., and James A. Mirrlees. 1971. “Optimal Taxation and Public Production II: Tax Rules.” American Economic Review, 61(3): 261–78. [41] Dixit A.K. and R. S. Pindyck (1994) Investment under uncertainty, Princeton, N.J. : Princeton University Press. [42] Dixit A., S. Skeath and D. F. Reiley (2009) Games of Strategy, Norton. [43] Dixit, Avinash K., and Joseph E. Stiglitz. 1977. “Monopolistic Competition and Optimum Product Diversity.” American Economic Review, 67(3): 297–308. [44] Dixon P.B. and M. T. Rimmer (2002) Dynamic general equilibrium modelling for forecasting and policy, Amsterdam: North-Holland. [45] Gale D (1986) Bargaining and competition, Part I and II, Econometrica, 54:785-818. [46] Gardener R (2003) Games of Business and Economics, Wiley, Second Edition.
363
[47] Ginsburgh V and Keyzer M. (1997) The Structure of Applied General Equilibrium Models, MIT Press. [48] Gravelle H and R Rees (2004) Microeconomics, 3rd ed. Prentice Hall [49] Green, R.J. (2005) Electricity and Markets, Oxford Review of Economic Policy, 21, 1, 67-87. [50] Grossman, Sanford J., and Joseph E. Stiglitz. 1980. “On the Impossibility of Informationally E¢ cient Markets.” American Economic Review, 70(3): 393–408. [51] Grubb M, T. Jamasb and M. Pollitt (2010) Delivering a Low Carbon Electricity System: Technologies, Economics and Policy, Cambridge UK:Cambridge University Press. [52] Harberger A.C. (1962),The Incidence of the Corporation Income Tax, Journal of Political Economy 70, 215-40. [53] Harsanyi J.C. (1967) Games with incomplete information played by Baysian Players, Management Science, 14:3:159-182. [54] Hayek, F. A. 1945. “The Use of Knowledge in Society” American Economic Review, 35(4): 519–30. [55] Henderson J. M. and R. E. Quandt (1980) Microeconomic Theory: A Mathematical Approach, McGraw-Hill, London. [56] Henry, C. (1989) Microeconomics for public policy: helping the invisible hand, Oxford : Clarendon Press. [57] Hershleifer J and J. G. Riley (1992) The Analystics of Uncertainty and Information, Cambridge University Press. [58] Hey John D. , Chris Orme (1994) Investigating Generalizations of Expected Utility Theory Using Experimental Data, Econometrica, 62, 6, 1291-1326. [59] Hicks , J. R (1939) Value and Capital: An inquiry into some fundamental principles of economic theory, English Language Book Society, London. [60] Hiller B. (1997) Economics of Asymmetric Information, London: McMillan Press. [61] Holt Charles (2007) Markets, Games and Strategic Behaviour, Pearson. [62] Hoy M, J Livernois, C McKenna, R Rees and T. Stengos (2001) Mathematics for Economics, 2nd ed., MIT Press. [63] Intriligator, M. D. (1791) Mathematical optimization and economic theory, Prentice-Hall series in mathematical economics. [64] Jehle G A and P.J. Reny (2005) Advanced Microeconomic Theory, Pearson Education. [65] Jin J. Y. (1994) Information Sharing through Sales Report Journal of Industrial Economics 42, 3,Sep., 323-333
364
[66] Jorgenson, Dale W. (1963) “Capital Theory and Investment Behavior.” American Economic Review, 53(2): 247–59. [67] Katzner D. W. (1988) Walrasian Microeconomics, Addison Wesley. [68] Kehoe T, T.N. Srinivasan and J Whalley (2005) Frontiers in Applied General Equilibrium Modelling, Cambridge University Press. [69] King, M.A. and D. Fullerton (1984) The taxation of income from capital:a comparative study of the United States, the United Kingdom, Sweden and West Germany Chicago University Press. [70] Kocherlakota N. R. (1996) New dynamic public …nance, Princeton University Press. [71] Krauss M. B. and H.G. Johnson (1974) General equilibrium analysis: a microeconomic text, George Allen & Urwin. [72] Kreps D. M. (1990) A Course in Microeconomic Theory, Princeton. [73] Krueger, Anne O. 1974. “The Political Economy of the Rent-Seeking Society.”American Economic Review, 64(3): 291–303. [74] Krugman, Paul. (1980) “Scale Economies, Product Di¤erentiation, and the Pattern of Trade.” American Economic Review, 70(5): 950–59. [75] Kuhn H. W. (1997) Classics in Game Theory, Princeton University Press. [76] La¤ront J J and J. Tirole (2000) Competition in Telecommunication, London: MIT Press. [77] Layard R. and S. Glaister (1994) Cost-bene…t analysis, Cambridge : Cambridge University Press. [78] Luce R. D. and H Rai¤a (1957) Games and Decisions, New York: John Wiley. [79] Lockwood B. A. Philippopoulos and A. Snell (1996) Fiscal Policy, Public Debt Stabilisation and Politics: Theory and UK Evidence Economic Journal 106, 437,Jul., 894-911. [80] Luce R. D. and H. Rai¤a (1957) Games and Decisions, New York: John Wiley. [81] Machina M. (1987) Choice under uncertainty: problems solved and unsolved, Journal of Economic Perspective, 1:1:121-154. [82] MasColell A, M.D.Whinston and J.R.Green (1995) Microeconomic Theory, Oxford University Press. [83] Mailath G. J. (1989),Simultaneous Signaling in an Oligopoly Model Quarterly Journal of Economics 104, 2 ,May, 417-427 [84] Mailath G. J. and L. Samuelson (2006) Repeated Games and Reputations: long run relationship, Oxford. [85] McCormick B. (1990) A Theory of Signalling During Job Search, Employment E¢ ciency, and 'Stigmatised' Jobs Review of Economic Studies 57, 2,Apr., 299-313
365
[86] Meade, James E. (1936) An introduction to economic analysis and policy,Oxford : Clarendon Press. [87] Milgrom P., J. Roberts (1986) Price and Advertising Signals of Product Quality Journal of Political Economy 94, 4, Aug., 796-821 [88] Mirrlees J. and editors (2010) Dimensions of tax design : the Mirrlees review, Oxford ; Oxford University Press. [89] Mirlees, J.A. (1971) “An exploration in the theory of optimum income taxation”, Review of Economic Studies, 38:175-208. [90] Modigliani, Franco, and Merton H. Miller. (1958) “The Cost of Capital, Corporation Finance and the Theory of Investment.” American Economic Review, 48(3): 261–97. [91] Mookherjee D. and D. Ray (2001) Readings in the theory of economic development, Malden, Mass. : Blackwell. [92] Moore J. (1988) Contracting between two parties with private information, Review of Economic Studies, 55: 49-70. [93] Motta, M. (2004) Competition policy : theory and practice, Cambridge : Cambridge University Press. [94] Myerson R (1986) Multistage game with communication, Econometrica, 54:323-358. [95] Newbery, D. M. G. (1999) Privatization, restructuring, and regulation of network utilities, Cambridge, Mass. : MIT Press. [96] Nash J. (1953) Two person cooperative games, Econometrica, 21:1:128-140. [97] Oliver H and J Moore (1988) Incomplete Contracts and Renegotiation, Econometrica, 56, 4, 755-785 [98] Osborne M.J. and A. Robinstein (1994) A course in game theory, MIT Press. [99] Obstfeld M. and K. Rogo¤ (1996) Foundation of International Macroeconomics, MIT Press. [100] Ok Efe A. (2007) Real Analysis with Economic Applications, Princeton. [101] Pechman J A (1987) Tax reform: theory and practice, Journal of Economic Perspective, 1:1:11-28. [102] Perroni, C. (1995), Assessing the Dynamic E¢ ciency Gains of Tax Reform When Human Capital is Endogenous, International Economic Review 36, 907-925. [103] Perlo¤ J. M. (2008) Microeconomics: Theory and Applications with Calculus, Pearson. [104] Png I. and D Lehman (2007) Managerial Economics,3rd edition, Oxford: Blackwell. [105] Ranis G. and L.K. Raut ed. (1999) Trade, Growth and Development, North-Holland. [106] Rasmusen E(2007) Games and Information, Blackwell. 366
[107] Rawls, J. (1973) Theory of Justice, Oxford: Oxford University Press. [108] Ray I. (2001) On Games with Identical Equilibrium Payo¤s On Games with Identical Equilibrium Payo¤s Economic Theory, 17, 1, 223-231 [109] Riley J.P. (1979) Noncooperative Equilibrium and Market Signalling American Economic Review 69, 2, Papers and Proceedings May, 303-307 [110] Rodrik D. (1989) Promises, Promises: Credible Policy Reform via Signalling Economic Journal 99, 397, Sep., 756-772. [111] Rogerson W.P.(1988) Price Advertising and the Deterioration of Product Quality Review of Economic Studies 55, 2 , Apr., 215-229 [112] Ross, Stephen A. (1973) “The Economic Theory of Agency: The Principal’s Problem.”American Economic Review, 63(2): 134–39. [113] Roth Alvin E. (2008) What have we learned from market design?, Economic Journal, 118, 285–310. [114] Rubinstein A ed. (1990) A course in game theory, Aldershot : Elgar. [115] Rubinstein A (1982) Perfect equilibrium in a bargaining model, Econometrica, 50:1:97-109. [116] Rutherford, T. F. (1995) Extension of GAMS for Complementary Problems Arising in applied Economic Analysis, Journal of Economic Dynamics and Control 19 1299-1324. [117] Samuelson P. (1949) Foundations of Economic Analysis, Harvard University Press. [118] Sawyer M. (1991) The Economics of Industries and Firms, London: Routledge. [119] Scarf, H. E. (1986) The Computation of Equilibrium Prices, in Scarf H. E and Shoven , John B. ed. Applied General Equilibrium Analysis, Cambridge University Press. [120] Sen, Amartya (1970) Collective choice and social welfare, San Francisco(Cal.) : Holden-Day [121] Sen A. (1976) Poverty: An Ordinal Approach to Measurement, Econometrica, 44:2:219-231. [122] Sen Amartya. 1974. Informational bases of alternative welfare approaches: Aggregation and income distribution , Journal of Public Economics, 3(4): 387-403. [123] Shapley L (1953) A Value for n Person Games, Contributions to the Theory of Games II, 307-317, Princeton. [124] Shapley L and M. Shubik (1969) On Market Games, Journal of Economic Theory, 1:9-25 [125] Shiller, Robert J. (1981) “Do Stock Prices Move Too Much to Be Justi…ed by Subsequent Changes in Dividends?” American Economic Review, 71(3): 421–36. [126] Shoven, J.B. and J.Whalley (1984) “Applied General-Equilibrium Models of Taxation and International Trade: An Introduction and Survey”, Journal of Economic Literature, 22, Sept,10071051.
367
[127] Shoven, J.B. and J.Whalley (1992) Applying General Equilibrium, Cambridge University Press, 1992. [128] Simon C. P. and L. Blume (1994) Mathematics for Economists, Norton. [129] Snyder C. and W. Nicholson (2012) Microeconomic Theory: Basic Principles and Extensions, 11th ed.. South Western. [130] Spence M. (1977) Consumer Misperceptions, Product Failure and Producer Liability Review of Economic Studies 44, 3 (ct., 561-572 [131] Sobel J. (1985) A Theory of Credibility, Review of Economic Studies 52, 4,Oct., 557-573 [132] Starr R M (1997) General Equilibrium Theory: An Introduction, Cambridge. [133] Ste¤en H., W. MuÈller and H-T Normann (2001) Stackelberg beats cournot: on collusion and e¢ ciency in experimental marketsThe Economic Journal, 111 (October), 749-765. [134] Stigler, G. J. (1968) The organization of industry,Homewood, Ill. : Irwin. [135] Stone Richard (1961) Input-output and national accounts, Paris : O.E.E.C. [136] Sutton J. (1991) Sunk costs and market structure: price competition, advertising, and the evolution of concentration, Cambridge Mass.: MIT Press. [137] Sutton J. (1986) Non-Cooperative Bargaining Theory: An Introduction, Review of Economic Studies, 53, 5., 709-724 [138] Takayama, Akira (1974) Mathematical economics, Hinsdale, Ill. : Dryden Press. [139] Tirole J. (2006) The Theory of Corporate Finance, Oxford: Princeton University Press. [140] Tirole J. (1995) The Theory of Industrial Organization, MIT Press. [141] Varian H. R. (1992) Microeconomic Analysis, Norton. [142] Varian HR (2010) Intermediate Microeconomics: A Modern Approach, Norton,8th ed.. [143] Walras, L. (1954) Elements of Pure Economics, Allen and Unwin, London. [144] Watt R. (2012) The Microeconomics of Risk and Information, Palgrave Macmillan. [145] Wise D. A. (1998) Inquiries in the Economics of Aging, Cambridge Mass.: NBER.
14
Real Analysis
Basic Concepts for Review 1.Euclidian distance 15.Lower hemicontinuity 2.Convergence of a sequence 16.Fixed point and contraction mapping 3.Cauchy sequence 17.Brower’s and Kakutani …xed point theorems SETS 4.Boundedness in R 5.Compactness 18.Sets and functions and De Morgan’s Laws 368
6.Contineous functions 19.Indexed sets 7.Maximum theorems 20.Relations 8.Convex set 21.Functions 9.Convex Hulls 22.Direct and inverse images 10.Extreme points of convex set 23.Least upper bound principle 11.Strictly convex sets 24.Sequence of real numbers 12.Separation theorems 25.Upper and lower limits 13.Coorespondence 26.In…mum and supremum of functions 14.Upper hemicontinuity 27.lim inf and lim sup of functions
14.1
Methods for constructing Proofs
Direct method a = b; b = c =) a = c: a = b; c = d =) a:c = b:d For x; y; z 2 R prove that x + z = y + z =) x = y Converse and contrapositive A implies B A =) B if it converse B =) A is true then A () B here A and B are equivalent. If A person lives in Hull (A) then that person lives in Yorkshire (B). A =) B but converse is not true in this case B ; A and A < B Contrapositive implies not A implies not B A =) B Methods for constructing Proofs Equivalence A () B Example Pythagorus theorem: h2 = p2 + b2 ; p2 2 b2 h2 + cos2 = 1 h2 = h2 + h2 =) sin Mathematical induction Example: Sum of the N natural numbers is : P (n) = 1 + 2 + 3 + :::: + n = n(n+1) 2 Check if this works for any integer k P (k) = 1 + 2 + 3 + :::: + k = k(k+1) 2 Add and subtract k + 1 from both sides k(k+1) =) P (k + 1) 1 + 2 + 3 + :::: + k + (k + 1) = 2 + (k + 1) = (k + 1) k2 + 1 = (k+1)(k+2) 2 Thus by mathematical induction P (k) True =) P (k + 1): Euclidian distance Take an Euclidian space Rn with elements x = (x1 ; x2 ; x3 ; ::::xn ) q 2 2 d (x; y) = kx yk = (x1 y1 ) + :::: + (xn yn ) (1751) q Xn 2 d (x; y) (xj yj ) = j xj yj j and d (x; y) j xj yj j j=1
Triangular inequality
d (x; z)
d (x; y) + d (y; z)
369
(1752)
see : Syddeater K,P. Hammond P and A. Seierstad and A Strom (2008) . Open Ball : If a is a point in Rn amd r is a positive number then the set of all points x in n R whose distance from a is less than r is called the open ball around a with radius r. Br (a) = B (a; r) = fx 2 Rn : d (x; a) < rg A set in Rn is closed only if its complement is open or A set in Rn is closed if contains all its boundary points. Three properties of a open set a. The whole space Rn and the empty set ? is open b. arbitrary unions of open sets are open c. the intersection of …nitely many open sets is open. Three properties of a closed set a. The whole space Rn and the empty set ? are both closed b. arbitrary unions of close sets is closed c. the intersection of …nitely many closed sets is closed.
14.1.1
Convergence
Convergence of a sequence Existennce of a limit of a sequence is necessary for convergence of a sequence. If the elements 1 for any of sequence get closer to some limit they are getting close to each other for intance X X > 1: De…ne upper bound, least upper bound or supremum; greatest lower bound or in…ma A set bounded both above and below is bounded. For a monotonically increasing sequence xn 1 xn for all n and monotonically decreasing sequence xn xn 1 for all n. Every bounded monotonic sequence always converge. A seqeunce fxk g in Rn converges to a point x if for each ' > 0 there exists a natural number N such that xk 2 B' (x) for k > N or d (xk ; x) ! 0 as k ! 1: A sequence that is not convergent is divergent; lim xk = x: k!1
Cauchy sequence 1 A sequence fxn gn=1 is a cauchy sequence for any ' > 0 if there is an integer N such that for i; j N the distance between xi ; xj d (xi ; xj ) '. Prove that 1) All cauchy sequence is convergent and bounded. 2) Any convergence sequence in Rm is cauchy. Boundedness, Compactness and Continous Functions 14.1.2
Boundedness
A set S in Rn is bounded if there exists a number M such that kxk M for all x in S. No point of S is at a distance greater than M from the origin. A seuence fxk g in Rn is bounded if the the set fxk : k = 1; 2; ::::g is bounded. Any convergent sequence is bounded. An example of spliting a rectangle in four parts then spliting it again and again to …nd a converging sequence fxk g : Compactness (Bolzano- Weirstrass theorem) A set S in Rn is compact (closed and bounded) if and only if every sequence of points in S has a subsequence that converges to a point in S. Continuous functions
370
A function f with domain S Rn is continuous at a point a in S if for every ' > 0 there exists a > 0 such that jf (x) f (a)j < ' for all x in S with kx ak < . If f is continous at every point of a in a set S, then f is continous at S. Maximum theorems maximize f (x; y) subject to y 2 F (x). De…ne the corresponding value function V (x) = max f (x; y) y2F (x)
Then the general case of maximum theorem is 'Suppose f (x; y) and gi (x; y) i = 1; 2; :::; l are continous functions from X Y in R where X Rn , Y Rn and Y is compact. suppose further that for every x in X the constraint asbove is nonempty and equal to the closure of F 0 (x): = fy 2 Y : gi (x; y) < ai , i = 1; 2; ::lg : Then the value function V (x) is continous over X. If the maximization problem has a unique maximum y = y(x) for each x in X, then y(x) is continous. Convex set A set S in the plane in called convex if each points in S can be joined by the line segments lying entirely within S. A set S in Rn is convex if [x,y] S for all x and y in S, or equivalently , if x + (1 )y 2 S for all x, y in S and all in [0; 1] A function f is call concave (convex) if it is de…ned on a convext set and the line segment joining any two points on the graph is never above (below) the graph (dome). 14.1.3
Convex Hull
Convex hull of set S in Rn is the set of all convex combinations of points from S; it is donted by co(S) Extreme points of convex set An extreme point of a convex set S in Rn is a point in S that does not lie properly in any line segment in S. z is extreme point of S if z 2 S and there are no x and y in S and [0,1] such that x 6= y and z = x + (1 )y. Strictly convex sets S is strictly convex if for each pair of distinct points in x and y in S every point of the open line segment (x,y) = f x + (1 ) y : 0 < < 1g is a relative interior point in S. Separation theorems Two disjoint sets in Rn can be separtated by a hyperplane. In two dimensions, hyperplanes are straightlines. Separation theorems are important in optimisation theory. a is nonzero vector in Rn and is a real number:
n
H = fx : a:x = g
(1753)
If S and T are subsets of R , H separates S and T if S is contained in one of the closed half determined by H and T is contained in the another. a:x a:y for all x in S and for all y in T. In case a:x = = a:y H strictly separates S and T. 14.1.4
Correspondence
A correspondece F from a set A into a set B is a rule that maps each x in A to a subset F (x) of B. 371
F :A
B and x
F (x). Most popular example in economics is the budet set B (p; m) = fx 2 Rn : p:x n+1
m; x
0g
(1754)
n
(p; m) B (p; m) is a correspondence from R into R ; e.g. p1 :x1 + p2 :x2 = m: Upper hemicontinuity A correspondence F : X Rn Rm has a closed graph property at a point x0 in X if whenever fxk g is a seqence in X that converges to x0 and fyk g is a seqence in Rm that satis…es yk 2 F (xk ) k = 1,2,..., converges to y 0 , y 0 2 F x0 . A correspondence F : X Rn Rm is said to be upper hemicontinuous (u.h.c) at a point 0 x in X if every open set U that contains F x0 there exists a neighbourhood N of x0 such that F (X) U for every x in N X i.e. such that F (N X) U: F is upper hemicontinuous (or, u.h.c) in X if it is u.h.c. at every x in X. Lower hemicontinuity A correspondence F : X Rn Rm is said to be lower hemicontinuous (l.h.c) at a point x0 in 0 0 X if whenever y 2 F x and fxk g is a seqence in X that converges to x0 there exists a number 1 k0 and a sequence fyk gk=k0 in Rm that converges to y 0 and satisfy yk 2 F (xk ) for all k k0 : F is lower hemicontinuous (or, l.h.c) in X if it is u.h.c. at every x in X. 14.1.5
Fixed Point Theorems
Fixed point and contraction mapping A set S in Rn and B be the set of all bounded functions from S into Rm .Contraction mapping is T : B ! B There exist a unique function ' in B such that ' = T (' ) Brouwer’s …xed point theorems For K an non-empty compact (closed and bounded) convex set in Rn , let f is continous mapping of K into itself. The f has a …xed point f (x ) = x : Kakutani …xed point theorems For K an non-empty compact (closed and bounded) convex set in Rn , let F a correspondence K K and suppose that a F (x) is a nonempty convex set in K for each x in K b. F is upper hemicontinous Then F has a a …xed point x in K i.e a point x such that x 2 F (x ) :
14.2
SETS
Let A and B be some sets. union: A [ B = fx : x 2 A or x 2 Bg intersection: A B = fx : x 2 A and x 2 Bg complement: A-B or AnB A B = fx : x 2 A and x 2 = Bg De Morgan’s Laws An (B [ C) = (AnB) (AnC) ; An (B C) = (AnB) [ (AnC) A B=B A Indexed sets Let set ai be de…ned for each i 2 I then fai gi2I ; is an index set. [i2I Ai consists of all x belonging to Ai A [ (i2I Bi ) = i2I (A [ Bi ) 372
14.2.1
Relations and functions
Relations Let a and b be certain sets; a relation between them aRb can be (a) re‡exive (b) transitive (c) symmetric (d) anti-symmetric (e) complete (f) partial or linear ordering. Functions A function (mapping, map or transformation) f : X ! Y from a set X to a set you is a rule that assings exactly one element y = f (x) in Y to each x in X. Direct and inverse images Let f : A ! B be a function. The direct immage under f of subset S of A is the set f (S) = fy 2 B : y = f (x) for some x in Ag The inverse image under f of a set T B is f 1 (T ) = fx 2 A : f (x) 2 T g Upper and Lower Bounds Least upper bound principle A set of real numbers S is bounded above if there exists a number b such that b > x for all x in S. Any such b is upper bound of S. A number b is the least upper bound of S if b b for every upper bound b. Sequence of real numbers A sequence is a function k 7 ! x (k) with the set of N = f1; 2; 3; ::::g all positive integer in the 1 domain; it is written o¤en as fxk gk=1 or simply fxk g. It is a. increasing if xk xk+1 for k = 1; 2; :::; b. strictly increasing if xk < xk+1 for k = 1; 2; :::; c. decreasing xk xk+1 for k = 1; 2; :::; d. strictly decreasing if xk > xk+1 for k = 1; 2; :::; Increasing or decreasing sequences are monotone sequence.
14.3
Limits
Upper limits Let fxk g be a sequence of real numbers and b be a …ne real number. Then
lim xk = b is
k!1
the upper limit if and only if the following two conditions are satis…ed a. For each ' > 0 there exists an integer N such that xk < b + ' for all k > N b. For each ' > 0 there exists an integer M ther exists a integer k > M such that xk > b '. Lower limits Let fxk g be a sequence of real numbers and b be a …ne real number. Then limk!1 xk = b is the lowrer limit if and only if the following two conditions are satis…ed a. For each ' > 0 there exists an integer N such that xk > b ' for all k > N b. For each ' > 0 there exists an integer M ther exists a integer k > M such that xk < b + ' . In…mum and supremum of functions In…mum and supremum of functions Let f (x) be de…ned on x in B where B Rn then in…mum and supremum are de…ned as: inf f(x) = infff (x) : x 2 Bg ; sup f(x) = supff (x) : x 2 Bg x2B
x2B
If the range of f (x) is the interval [0; 1] and then inf f(x) = 0 and sup f(x) = 1 x2B
lim inf and lim sup of functions
373
x2B
f (x) = lim lim inf 0
r!0
x!x
lim sup f (x) = lim x!x0
r!0
inf f (x) : x 2 B x0 ; r M; x 6= x0 0
sup f (x) : x 2 B x ; r M; x 6= x
;
0
A function is upper semicontinous at point x0 in M if limx!x0 f (x) f x0 A function is lowerer semicontinous at point x0 in M if limx!x0 f (x) f x0 According to the extreme value theorem.if K Rn is a nonempty and compact set anf if f is upper semicontinous, the f has a maximum point in the set K; if f is lower semicontinous, the f has a minimum point in the set K
15
Computation and software
Microeconomic theories after detailed optimisation procedure express variables in terms of behavioural parameters. Application of these model requires calibration or estimation of these parameters with the real world data and computation of alternative scenarios according to …nd out the impacts of economic policies or changes in behaviour. Solving a simultaneous equations becomes more complicated as number of equations increase in the model. Excel is good for small scale examples. Special shoftware such as General Algebraic Modelling System (GAMS) or MATLAB are used for solving bigger models. GAMS/MPSGE is very e¤ective in solving large scale models. Econometrics often involves with estimation parameters using cross section or time series data; PcGive/Stamp/Giviein, Eviews , STATA, Shazam, Limdep are good software for this. SPSS good for processing large scale survey and statistical analysis.
15.1
GAMS
GAMS is good particularly in solving linear and non-linear system of equations. It has widely been used to solve general equilibrium models with many linear or non-linear equations on continuous or discrete variables. It comes with a number of solvers that are useful for numerical analysis such as CONOPT, DICOPT, MILES, MINOS, DNLP, PATH. It can solve very large scale models using detailed structure of consumption, production and trade arrangements on unilateral, bilateral or multilateral basis in the global economy where the optimal choices of consumers and producers are constrained by resources and production technology or arrangements for trade.It is a user friendly software. Any GAMS programme involves declaration of set, parameters, variables, equations, initialisation of variables and setting their lower or upper bounds and solving the model using Newton or other methods for linear or non-linear optimisation and reporting the results in tables or graphs see examples below. GAMS/MPSGE software is good for large scale standard general equilibrium models. GAMS programme can be downloaded from demo version of GAMS free from www.gams.com/download. Learn GAMS by practicing following examples. First write them using a text editor and save …le *.gms. Then execute the program and study the result and then revise the model as necessary. $Title a simple linear programming problem Variables R, X1, X2; Equations ER, Ex1, Ex2; Er.. R =e= 10*x1 +5*x2;
374
Ex1.. 25*x1 +10*x2 =l= 1000; Ex2.. 20*x1 +50*x2 =l= 1500; Model lp / all/; R.lo=1; X1.lo=1; X2.lo=1; Model lp /ER, Ex1, Ex2/; solve lp maximizing R using lp; $Title cartel model Variables P, Q, q1,q2, C1, C2, Pro…t, prof1, prof2; Equations EP, EQ, EC1, EC2, EPro…t,eprof1, eprof2; EP.. P =e= 300 -(1/2)*Q; EQ.. Q =e= q1+q2; EC1.. C1 =e= 500 +20*q1; EC2.. C2 =e= 1000 +(1/4)*q2*q2; EPro…t.. Pro…t =e= p*Q-c1-c2; eprof1.. prof1 =e= p*q1 -c1; eprof2.. prof2 =e= p*q2 -c2; model cartel /all/; solve cartel maximizing pro…t using nlp; $Title General Equilibrium in a Pure Exchange Global Economy $ontext Global economy produces oil and grains. It includes economies A and B. Economy A owns the oil …eld and produces 100 units of oil and economy B produces 200 units of grain. Both economies like to consume oil and grains.Their consumption preferences by given by Cobb-Douglas Utility functions, household in eocnomy A spends 40 percent of income in oil and 60 percent in grains and household in economy B spends 60 percent in apples and 40 percent in grains. Market structure is competitive. Find the relative price in these economies that is consistent with maximization of utility (satisfaction) by representative households in both countries. Choose price of oil 1 as a numeraire. Find the income of both countries, their demands for both oil and grains. Check whether the conditions for equilibrium are ful…lled. Find their levels of utility at equilibrium. $o¤text Parameters WA, WB, a1, b1; WA = 100; WB = 200; a1 =0.4;
375
b1 =0.6; Free variables UA, UB; Variables X1A, X1B, X2A, X2B, IA, IB, P1, P2; Equations EX1A, EX1B, EX2A, EX2B, EI1, EI2, MKT1, MKT2,EUA, EUB; EI1.. IA =e= P1*WA; EI2.. IB=e= P2*WB; EX1A.. X1A =e= (a1*IA)/P1; EX1B.. X1B =e= (b1*IB)/P1; EX2A.. X2A =e= ((1-a1)*IA)/P2; EX2B.. X2B =e= ((1-b1)*IB)/P2; MKT1.. X1A + X1B =e= WA; MKT2.. X2A + X2B =e= WB; EUA.. UA =e= (X1A**a1)*(X2A**(1-a1)); EUB.. UB =e= (X1B**b1)*(X2B**(1-b1)); IA.lo=1; IB.lo=1; X1A.lo=1; X1B.lo=1; X2A.lo=1; X2B.lo=1; P1.fx =1; *P1.lo= 0.001; P2.lo= 0.001; Model pure /all/; option nlp = conopt2; solve pure maximising UA using nlp; Parameters ep report, Report1; set sc /sc1*sc5/; a1= 0.1; ep(sc) = 0.1; loop(sc, solve pure maximising UA using nlp; report(sc,'UA')=UA.L;
376
report(sc,'UB')=UB.L; report(sc,'X1A')=X1A.L; report(sc,'X2A')=X2A.L; report(sc,'X1B')=X1B.L; report(sc,'X2B')=X2B.L; report(sc,'IA')=IA.L; report(sc,'IB')=IB.L; report(sc,'a1')=a1; report(sc,'b1')=b1; a1= a1+ep(sc); ); a1=0.4; b1 =0.1; loop(sc, solve pure maximising UA using nlp; report1(sc,'UA')=UA.L; report1(sc,'UB')=UB.L; report1(sc,'X1A')=X1A.L; report1(sc,'X2A')=X2A.L; report1(sc,'X1B')=X1B.L; report1(sc,'X2B')=X2B.L; report1(sc,'IA')=IA.L; report1(sc,'IB')=IB.L; report1(sc,'a1')=a1; report1(sc,'b1')=b1; b1= b1+ep(sc); ); display report,report1; $Title Cobb Web Model set t time /t1*t20/ t…rst tlast i sectors /i1*i100/; t…rst(t) = Yes$(ord(t) eq 1); tlast(t) = Yes$(ord(t) eq card(t)); alias (t, tt); Parameters al, gm, dl, bt, A, P0, Pbar; dl = 2; bt = 4; al = 500; gm =200; P0 = 100; pbar = (al+gm)/(bt+dl); display pbar; Variables q(t), P(t), QT; Equations Eq(t), EPP(t), EP(t), EQT;
377
Eq(t).. Q(t) =e= al-bt*P(t); EPP(t)$t…rst(t).. P(t)=e= P0; EP(t+1)$(ord(t) gt 1).. P(t+1)=e= -(dl/bt)*P(t) + (al+gm)/bt + P(t)$tlast(t); EQT.. QT =e= sum(t, Q(t)); Model Cobbweb /all/; q.Lo(t) =0.01; P.Lo(t) =0.01; QT.Lo =0.01; Solve Cobbweb maximising QT using nlp; Parameter report base case solution; report(t, 'Price') = P.L(t); report(t, 'Output') = Q.L(t); Display report;
For GAMS/MPSGE systax see http://www.mpsge.org. The check whether the results are consistent with the economic theory underlying the model such as general equilibrium or the ISLM-ASAD analysis for evaluating the impacts of expansionary …scal and monetary policies. Use knowledge of growth theory to explain results of the Solow growth model from Solow.gms. Consult GAMS and GAMS/MPSGE User Manuals, GAMS Development Corporation, 1217 Potomac Street, Washington D.C or www.gams.com. For other relevant software visit: http://www.feweb.vu.nl/econometriclinks/ or https://www.aeaweb.org/rfe/ Brook, A K., D. Kendtrick, A.Meeraus(1992) GAMS: Users’s Guide, release 2.25, The Scienti…c Press, San Francisco, CA. Dirkse SP and Ferris MC (1995) CCPLIB: A collection of nonlinear mixed complementarity Problems. Optimization Methods and Software 5:319-345. Rutherford, T.F. (1995) Extension of GAMS for Complementary Problems Arising in Applied Economic Analysis, Journal of Economic Dynamics and Control 19:1299-1324.
15.2
MATLAB
MATLAB is widely used for solving models. It has script and function …les used in computations. Both have *.m extensions. Its syntax are case sensivite. It is good for solving a system of linear equations and handling matrices Example 1 Write a programme …le matrix.m like the following and execute it. % now solve a linear equation % 5x1 + 2x2 =20 % 3x2 + 4x2 =15 k =[5 2;3 4]; n = [20 15]; kk = inv(k) x = kk*n’
378
One more example of system of equation and factorisation of matrices A=[1 2 3; 3 3 4; 2 3 3] b=[1; 2; 3] %solve AX=b X = inv(A)*b %eigen value and eigenvectors of A [V,D]=eig(A) %LU decomposition of A [L,U]=lu(A) %orthogonal matrix of A [Q,R]=qr(A) %Cholesky decomposition (matrix must be positive de…nite) %R = chol(A) %Singular value decomposition [U,D,V]=svd(A) %simple Markov Chain a = [0.2 0.8]; %transition matrix b = [0.9 0.1; 0.7 0.3]; %initial state c0 = a*b %subsequent states c1 = c0*b c2 = c1*b c3 = c2*b c4 = c3*b c5 = c4*b c6 = c5*b %stationary state c7 = c6*b %eigen values d = eig(b) [V,D]=eig(b) Example 3 Solving …rst order ordinary di¤erential equation write the function simpode.m which has just two lines function xdot = simpode(t,x); xdot = x+t; simpode(t,x) is MATLAB function. Write a scrit …rst_ode.m tspan =[0,2], x0=0; [t,x]=ode23(’simpode’,tspan,x0); plot(t,x) xlabel(’t’), ylabel(’x’) then execute program in the commandline to get the graph
379
>>…rst_ode Example 3 %solving system of ordinary di¤erential equations type the following and save in regid.m functiion …le function dy = rigid(t,y) dy = zeros(3,1); % a column vector dy(1) = y(2) * y(3); dy(2) = -y(1) * y(3); dy(3) = -0.51 * y(1) * y(2); end Then script …le second_ode.m tspan =[0,20], z0=[1;0]; [t,z]=ode23(’pend’,tspan,z0); x=z(:,1); y=z(:,2); plot(t,x,t,y) xlabel(’t’), ylabel(’Consumption and Income’) …gure(2) plot(x,y) xlabel(’consumption’), ylabel(’income’) title(’consumption and income’) then execute the program typing >>second_ode This will give two …gures from the solution. Example 3 Evaluating an integral de…ne the function function y =erfcousin(x); y = exp(-x^2) end write script integral.m y = quad(’erfcousin’,1/2,3/2) >>integral Example 3 Evaluating a double integral write a script …le %evaluating double integral F = inline(’1-6*x.^2*y’); I = dblquad(F,0,2,-1,1) >>integral_d For more see MATLAB help/ examples and documentation/Mathematics. Try sample programs provided there and have a tiny model to practice. See other programs like intro.m; travel.m; simul.m. Contents.m for list of …les in MATLAB demo. http://www.mathworks.com/products/demos/ ; Some demonstrations on how to use MATLAB also available in http://www.youtube.com/.
380
Pratap Rudra (2002) Getting started with MATLAB : a quick introduction for scientists and engineers. http://www.mathworks.com/academia/student_center/tutorials/ps_solve/player.html See Cleve Moler’s text book such as Numerical Computing with MATLAB or Experiments with MATLAB available at http://www.mathworks.com/moler/index.html. Michael Ferris has developed Interface between GAMS and MATLAB. The details of the new package can be found at: http://www.cs.wisc.edu/math-prog/matlab.html. For this a) install a new version of GAMS (23.4) b) put the system directory of GAMS into your MATLAB path .
15.3
Econometric and Statistical Software Excel OX7-GiveWin/PcGive/GARCH/STAMP Eviews 8 Shazam micro…t RATS GAUSS LIMDEP/NLOGIT STATA12/SPSS20 http://www.feweb.vu.nl/econometriclinks/; https://www.aeaweb.org/rfe/
OX-GiveWin/PcGive/STAMP (www.oxmetrics.net) is a very good econometric software for analysing time series, volatility modelling (arch-garch) and cross section data. This software is available in all labs in the network of the university by sequence of clicks Start/applications/economics/givewin. Students can have their own copy of Oxmetrics7 from the helpdesk. Following steps are required to access this software. a. save the data in a standard excel …le. Better to save in *.csv format. b. start oxmetrics7 and givewin in it at start/applications/economics/givewin and pcgive (click them separately) c. open the data …le using …le/open data…le command. d. choose PcGive module for econometric analysis. e. select the package such as descriptive statistics, econometric modelling or panel data models. d. choose dependent and independent variables as asked by the menu. Choose options for output. e. do the estimation and analyse the results, generate graphs of actual and predicted series. A Batch …le can be written in OX for more complicated calculations using a text editor such as pfe32.exe. Such …le contains instructions for computer to compute several tasks in a given sequence.
381
References [1] Doornik J A and D.F. Hendry ((2013) PC-Give Volume I-III, GiveWin Timberlake Consultants Limited, London 15.3.1
Quality ranking of journals in Economics
Findings of theoretical and applied research are published in journals. Better the quality of a paper, more likelihood that it will be published in highly ranked journals, though this relationship is not always perfect one. It is instructive to look into the Association of Business School (ABS) ranking on quality of journals given below in process of reviewing the literature as well as in writing a paper. ABS 4* Journals American Economic Review; Economic Journal; Econometrica; Journal of Labour Economics; Rand Journal of Economics; Journal of Political Economy; Journal of Monetary Economics; International Economic Review; Quarterly Journal of Economics; Review of Economic Studies; Journal of Econometrics; Journal of Economic Literature; Journal of Economic Perspective; Journal of Economic Theory; Journal of Economic Geography; Journal of Environmental Economics and Management; Journal of Financial Economics; Econometric Theory; . ABS 3* Journals Brookings Economics Papers; Journal of Economic Growth; Economic Letters; European Economic Review; Journal of Development Economics; Canadian Journal of Economics; European Review of Agricultural Economics; Cambridge Journal of Economics; Journal of Applied Econometrics ; Journal of Comparative Economics; Journal of Development Studies;Journal of Economic Dynamics and Control; Journal of Health Economics; Journal of Economic Behaviour and Organisation; Journal of Economics and Management Strategy; Journal of Economics of Law and Organisation; Journal of Evolutionary Economics; Journal of Industrial Economics; Economica; Journal of Public Economics; Journal of European Economic Association; Journal of Urban Economics; Kyklos; Labour Economics; Ecological Economics;IMF Economic Review; Land Economics; Oxford Bulletin of Economics and Statistics; Oxford Economics Papers; Review of Economics and Statistics; Review of International Economics;Social Choice and Welfare; Southern Economic Journal; World Bank Economic Review; Journal of International Economics; Economy and Society. ABS 2* Journals Advances in Econometrics; European Journal of Political Economy; Agricultural Economics; Applied Economics; Annals of Public and Cooperative Economics; Applied Financial Economics; ; Australian Journal of Agricultural and Resource Economics; Bulletin of Economic Research; ; Canadian Journal of Agricultural Economics; Contemporary Economic Policy; Contributions to the Political Economy; Defence and Peace Economics; Econometric Reviews; Economics of Education Review; Economics of Innovation and New Technology; Economics of Planning;Economics of Transition; Economist-Netherlands;Environmental Resource Economics; Fiscal Studies; Global Business and Economic Review; History of Political Economy; Oxford Review of Economic Policy; IMF Sta¤ Papers; Insurance Mathematics and Economics; International Journal of Game Theory;International Journal of Economics of Business; International Review of Economics and Finance; Journal of Agricultural and Resource Economics; World Economy; Journal of Economic Methodology, Journal of Economic Psychology; Journal of Industry, competition and Trade; Macroeconomic Dynamics; Journal of Economics; Employee Relations; Empirical Economics; STATA Journal.
382
ABA 1* Journals Applied Economics Letters; Australian Economic Review; Business Economics; Bulletin of Indonesian Economic Studies; Eastern European Economics; ; International Review of Applied Economics; Information Economics and Policy;International Journal of Social Economics; Journal of interdisciplinary Economics; For the latest version visit: http://www.associationofbusinessschools.org/node/1000257. https://ste¤enroth.…les.wordpress.com/2015/06/abs-2015-ste¤en-roth-ch.pdf Note also that there are many journals which have not been ranked by the ABS.
383
15.4
Core texts in Economic Theory and Equivalent reading
References [1] Aumann R.J. and S. Hart. (1994) Handbook of game theory with economic applications, North Holland, 1992-1994. [2] Allen R.G.D. (1956) Mathematical Economics, MacMillan. [3] Atkinson A.B. and J. E. Stiglitz (1980) Lectures on Public Economics, McGraw Hill. [4] Balasko, Yves (1988) Foundations of the theory of general equilibrium, Boston : Academic Press. [5] Baldani J, J Brad…eld and R Turner (1996) Mathematical Economics, Dryden Press. [6] Basu, Kaushik (1993) Lectures in industrial organization theory, Oxford : Blackwell. [7] Bhagwati J. N. and T.N. Srinivasan (1992)Lectures on International Trade, MIT Press [8] Bhattarai K. (2007) Models of Economic and Political Growth in Nepal, Serials Publications, New Delhi. [9] Binmore K. (1990) Fun and Games: A text on Game Theory, Lexington, Heath. [10] Bridel P. (2011) General equilibrium analysis: a century after Walras, Routledge, London. [11] Cohen, S. I. (2001) Microeconomics; Economic policy, London ; New York : Routledge. [12] Cornwall R R (1984) Introduction to the use of general equilibrium analysis, North-Holland. [13] Debreu, G. (1954) The Theory of Value, Yale University Press, New Haven. [14] Estrin S., D Laidler and M. Dietrich (2008) Microeconomics,Prentice Hall. [15] Gardner R (2003) Games for Business and Economics, Willey. [16] Ginsburgh V.and M. Kayzer (1997) The Structure of Applied General Equilibrium Models, MIT Press. [17] Gravelle H and R Rees (2004) Microeconomics, 3rd ed. Prentice Hall [18] Fundenbeg D and J Tirole (1995) Game Theory, MIT Press. [19] Hershleifer J and J G Riley (1992) The Analytics of Uncertainty and Information, Cambridge [20] Harrison GW ed. (2000) Using dynamic general equilibrium models for policy analysis2000, pp. xi, 411, Contributions to Economic Analysis, 248 North-Holland. [21] Hillman Arye (2005) Public Finance, Cambridge University Press.
384
[22] Hirshleifer J and J G Riley (1992) The Analytics of Uncertainty and Information, Cambridge. [23] Hicks J R (1939) Value and Capital, ELBS, MacMillan. [24] Holt C A (2007) Markets, Games and Strategic Behaviour, Pearson. [25] Jehle G A and P.J. Reny (2005) Advanced Microeconomic Theory, Pearson Education. [26] Katzner D W (1988) Walrasian Microeconomics: An Introduction to the Economic Theory and Market Behaviour, Addison Wesley. [27] Kreps D. M. (1990) A Course in Microeconomic Theory, Princeton. [28] Krugman P R and M Obstfeld (2000) International Economics, Addison Wesley. [29] La¤ont JJ and M. Moreaux (1989) Dynamics, incomplete information and industrial economics, Oxford : Blackwell [30] La¤ont, Jean-Jacques (1989) The economics of uncertainty and information, Cambridge, Mass : MIT Press [31] Luce R. D. and Rai¤a H. (1957) Games and Decisions, John Wiley and Sons, New York. [32] Marshall A. (1890) Principles of Economcis, McMillan. [33] Mailath G. J. and L. Samuelson (2006) Repeated Games and Reputations: long run relationship, Oxford. [34] MasColell A, M.D.Whinston and J.R.Green (1995) Microeocnomic Theory, Oxford University Press. [35] Myles G.D. (1995) Public Economics, Cambridge University Press. [36] Nicholson W (1989) Microeconomic Theory and Extensions, 4th edition, Dryden Press. [37] Neumann John von and Oskar Morgenstern (1944) Theory of Games and Economic Behavior, Princeton University Press. [38] Ok Efe A. (2007) Real Analysis with Economic Applications, Princeton University Press. [39] Osborne M.J. and A. Robinstein (1994) A course in game theory, MIT Press. [40] Pigou, A. C. (1932) The economics of welfare, McMillan. [41] Pascal Bridal (2011) General Equilibrium Analysis: A Century After Walras, Routledge. [42] Rasmusen E (2006) Games and Information, Blackwell. [43] Ricardo D. (1817) Principles of Political Economy and Taxation, London, John Murray. [44] Romer D. (2006) Advanced Macroeconomic Theory, McGraw Hill. [45] Romp (1997) Game Theory: Introduction and Applications, Oxford. [46] Rubinstein Ariel (1990), Game theory in economics ,Aldershot : Elgar. 385
[47] Samuelson P. (1947) Foundation of Economic Analysis, Harvard University Press. [48] Schmalensee R. and R. Willig (1989) Handbook of industrial organization, North Holland, 1992-1994. [49] Shoven, J.B. and J.Whalley (1992) Applying General Equilibrium, Cambridge University Press, 1992. [50] Shone R (2001) Economic Dynamics, Cambridge. [51] Simon C. P. and L. Blume (1994) Mathematics for Economists, Norton. [52] Snyder C and W. Nicholson (2011) Microeconomic Theory: Basic Principles and Extensions, 11th edition, South Western. [53] Starr R M (1997) General Equilibrium Theory: An Introduction, Cambridge. [54] Takayama (1974) Mathematical Economics, Dryden Press. [55] Tirole, Jean (2006) The theory of corporate …nance, Princeton, N.J. ; Oxford : Princeton University Press [56] Tirole J. (1995) The Theory of Industrial Organisation, MIT Press. [57] Thijs ten Raa (2005)The economics of input-output analysis Cambridge : Cambridge University Press. [58] Varian H. R. (1992) Microeconomic Analysis, Norton. [59] Watt R. (2011) The Microeconomics of risk and information, Palgrave.
16
Schedule
1. Axioms; optimisations; linear and nonlinear programmes 2. Consumption 3. Production 4. Markets 5. General equilibrium and welfare 6. Game theory: bargaining and coalition 7. Game theory: principal agent problems 8. Game theory: uncertainty and insurance 9. Game theory: mechanism and auction 10. Taxation and public goods and trade
386
11. Class Test (1 hour) 12. Microeconomics for multinationals 13. Microeconomic policies (and welfare analysis)
387
16.1
Sample class test Section A
Q1. Production function for a fruit …rm operating in the competitive market is given by p y=2 l
(1755)
where y is output and l is labour input. Product price is p and input price is w. 1. Determine the cost function for this …rm. 2. What is its pro…t function? 3. Determine its supply function. 4. What is its demand function for labour? 5. Discuss properties of the production, pro…t and cost functions. . Q2. Utility function for a consumer is given by U = X 0:5 Y 0:5
(1756)
I = px X + py Y
(1757)
here budget constraint is
1. What are the Marshallian (uncompensated) demand functions for X and Y? 2. Determine the indirect utility function for this consumer. 3. Solving corresponding duality problem determine the expenditure function for this consumer. 4. Find the compensated (Hicksian) demand curve for X or Y? [hint Slutskey equation]. 5. Prove Shephard’s lemma
@E @pi
=
@L @pi
= xi (p1 ; p2 ; m) . h i @V @L 6. Prove Roy’s identity for this case @p = @pi : i Q3. Consider a two sector two class model of an economy. Workers supply labour and spend all their income in necessity goods. Capitalists do not work but own all capital and spend 60 percent of their income in luxury products, 20 percent in necessity goods and save and invest 20 percent of the remaining income.
388
Table 87: Parameters in production of the K Necessity sector 0.5 100 Luxury sector 0.5 144
two sector model A 1 1
Table 88: Parameters in consumption of the two sector model Workers Capitalist
1
2
3
1 0.2
0 0.6
0 0.2
Total labour supply is 50 and the wage rates are equal in both sectors of production LS = 50;
w1 = w2 = w
(1758)
Production function of sector i is Qi = Ai Ki i Li1
(1759)
i
Parameters of the model are given in two tables below. Capitalist hires workers and allocates labour to maximise its pro…t i
= Pi Qi
wLi
rKi = Pi Ai Ki i L1i
i
wLi
rKi
(1760)
Income of workers YL = wL1 + wL2 = w (L1 + L2 ) = 50w
(1761)
Income of capitalists (from the production function capitalist gets the same as the labour) YK = YL = 50w
(1762)
1. Determine the demand for labour for both necessity and luxury goods sectors. 2. Derive the supply and demand functions for each product. 3. Find the equilibrium relative prices Pi and w (assume necessity good as a numeraire P1 = 1). 4. What is the demand for necessity and luxury goods by workers and capitalists? 5. How much is invested in this economy? 6. What will happen to the level of income and consumption if the labour endowment increases by 10 percent due to migration or better health of the existing working age population?
389
Section B Q3. Market demand function for a certain product with two …rms (Q = q1 + q2 ) is given by P = 120
Q
(1763)
cost function of each …rm is Ci = 10qi
(1764)
Solve for optimal output, revenue, cost, pro…t, consumer, producer and total welfare under following market conditions 1. Cournot duopoly. 2. Stackelberg leadership when …rm 2 follows …rm 1. 3. Cartel. 4. Bertrand duopoly. 5. Perfect competition. 6. Provide brief explanation on …ndings. Q4. Cost function of a …rm producing a certain product under perfectly competitive market is quadratic as: C = 0:1q 2 + 10q + 50 (1765) This product sells in 20 pounds in the market. 7. What is the optimal output of this …rm? 8. What are its total revenue, total cost and pro…t at that optimal output? 9. Derive the supply function of the …rm. 10. Discuss properties of cost, pro…t and supply functions.
Q5. Consider monopoly and oligopoly models given below. Monopoly model: Pro…t function of a monopolist with taxes = PQ
TC
P =a total cost with marginal cost c and …xed cost f
390
bQ
T
(1766) (1767)
T C = cQ + f
(1768)
T = tQ
(1769)
Tax revenue
Oligopoly model There arei = 1; :::; N …rms in the market Market supply Q=
n X
qi
(1770)
i=1
Market price depends on total sales P =a
bQ = a
n X b qi
(1771)
i=1
total cost including taxes (ignore …xed cost for a while) T Ci = (c + t) qi
(1772)
T = tQ
(1773)
Tax revenue
Pro…t of a particular …rm in oligopoly is:
1
= P q1 =
a
(c + t) q1 = n X b qi i=2
!
a
n X b qi i=1
q1
bq12
!
q1
(c + t) q1
(c + t) q1 (1774)
Prove that revenue maximising tax rate is the same for both monopoly or oligopoly. Market structure does not matter for it.
16.2
Sample …nal exam
Q1. Consider consumers’problems for comparative static analysis max U = U (X; Y )
(1775)
I = Px X + Py Y
(1776)
subject to the budget constraint: where U is utility, I income, X and Y and Px and Py are are amounts and prices of X and Y commodities respectively. 1. Illustrate the …rst order conditions for consumer optimisation in this model. 391
2. By total di¤erentiation of the …rst order conditions determine (a) the impact of a change in shadow prices on the demand for X and Y: (b) the impact of a change in price of X on the demand for X and Y: (c) the impact of a change the price of Y on the demand for X and Y: (d) the impact of a change in income on the demand for X and Y and the shadow price. 3. Decompose the total e¤ect of a price change in substitution and income e¤ects. 4. Show the major di¤erences between Hicksian and Marshallian demand functions. Q2. A …rm’s objective is to minimise cost (C) C = rK + wL
(1777)
subject to a CES technology constraint: Y = [ L + (1
)K ]
1
(1778)
Here Y is outpt, K capital, L labour inputs, r interest rate, w wage rate, 0 < labour the substitution parameter.
< 1 share of
1. Determine the demand for labour and capital inputs. 2. Derive the cost function of the …rm. 3. Prove that the elasticity of substitution is
1
=
1:
4. Discuss the properties of the CES cost function. 5. Prove that the Cobb-Douglas production function is a special case of the CES production function. Q3. Consider a Dixit-Stiglitz model of monopolistic competition in which consumers maximise utility by consuming varieties of di¤erentiated products qi in addition to a unique numeraire product q0 . Their problem is: max
u = u q0 ;
subject to: q0 +
X
pi qi
X
1
qi
(1779)
I
(1780)
Producer i maximises pro…t, setting prices pi given marginal cost c and …xed cost f as: max pi
1. Prove that elasticity of demand is
= (pi 1 1
;i.e.
392
c) qi =
f @qi pi qi @pi
(1781) =
1 1
:
2. How much does each …rm produce? How does it relate to the elasticity of demand as well as the …xed cost (f ) and variable costs (c). 3. How many …rms exist in the market? Explain the role of
in it.
Q4. Consider an economy consisting of a representative household and a representative …rm. The representative household tries to maximise utility by consuming goods and services and enjoying leisure subject to its budget constraints. The producer wants to maximise pro…t by selling goods produced using the labour supplied by the household. The household maximisation problem can be stated as the follows: max U = C l(1
)
(1782)
Subject to time and budget constraints: l + hs = 1
(1783)
pc = whs +
(1784)
c > 0; hs > 0;and l > 0:Here c is consumption,l is leisure and hs is labour supply, p is the price of the commodity, w is the wage rate, is the pro…t from owning the …rm. The maximisation problem for the representative …rm can be stated as: = py
whd
(1785)
subject to the technology constraint: y = (hs )
(1786)
y > 0; hd > 0;where y is the output supplied by the …rm and hd is its demand for labour. 1. Form a Lagrangian for the constrained maximisation problem for this household. 2. Derive its demand for consumption goods and its demand for leisure. 3. Write the Lagrangian function for the …rm’s optimisation problem. 4. Derive the …rm’s demand for labour. 5. De…ne a competitive equilibrium for this economy. 6. Compute the real wage that brings goods and labour market to equilibrium. 7. What are the equilibrium quantities of c and y? 8. What are the equilibrium quantities of l and h? 9. Reformulate the problem with a sales tax and an income tax. Discuss qualitatively the economic impacts of (a) switching completely to the sales taxes or (b) to labour income taxes or to (c) a capital income tax. [56278, Continued...]
393
Q5. An economy is inhabited by 'more productive' type 1 and 'less productive' type 2 people. Policy makers encourage more productive people by assigning a greater weight to the utility of more 1 3 productive people. They aim to maximise the social welfare function: W = U14 U24 where W is the index of the social welfare, U1 represents the utility of type 1 people and U2 is the utility of type 2 people. For simplicity, assume that the resources of this economy produce a given level of output Y . It is consumed either by type 1 or by 2ptype. Market clearing condition p implies: Y = Y1 + Y2 . Preferences for type 1 are given by U1 = Y1 and for type 2 by U2 = Y2 . In a given year total output, Y , was 1000 billion pounds. 1. What is the distribution of output between type 1 and type 2 that maximises the social welfare index? What is the maximum value of the social welfare index of this economy? 2. What would have been the allocation if policy makers had given equal weight to 1 1 the utility of both types of people in the economy such as W = U12 U22 . By how much does the optimal welfare index change in this case when compared to the social welfare in (1) above? 3. How would the social welfare index change in (1) if a tax rate of 20 percent is imposed on consumption and the tax receipts are not given back to any of the consumers? What would the value of social welfare index be in this case? 3
1
4. Assume that the policy makers still hold the welfare function to be W = U14 U24 . How would the social welfare index change in (3) if all tax receipts are transferred to type 2 people? Q6. Consider a mechanism design problem in which the owner of a premium quality piece of land can enter into various arrangements with tenants to share output of the land (q). For simplicity assume that the demand for the proceeds of the land and associated costs are given by P = 30
0:5q
C = 10q
(1787)
Prove the following propositions: 1. Proposition 1: results of …xed fee (F ) contract and joint pro…t maximisation are equivalent. 2. Proposition 2: hire contract (e.g. 14 wage per unit of output) is incentive incompatible and leads to production ine¢ ciency. 3. Proposition 3: a moral hazard problem and production ine¢ ciency exist in a revenue sharing contingent contract (assume the owner gets 14 of the revenue leaving 43 of revenue to the tenant). 4. Proposition 4: a pro…t-sharing contract is e¢ cient and free of a moral hazard problem. (assume 1/3rd of pro…t goes to the tenant and 2/3rd to the landlord). Q7. Consider the cost of production (C) and production technology constraint of a …rm that produces output (y) using capital (K) and labour (L) inputs 394
C = rK + wL
(1788)
y=K L
(1789)
Terms w and r represent wage and interest rate respectively and …rm sells output at price p. 1. Write the pro…t function for this …rm and a Langrangian to maximise pro…t subject to the technology constraint. 2. Determine the optimal demand for inputs. 3. Derive the pro…t function in terms of optimal inputs , V (p; w; r): 4. Determine the cost function. @V @P
5. Prove Hotelling’s lemma K(p; w; r):
=
@L @P
= y(p; w; r); @V @w =
@L @w
=
L(p; w; r); @V @r =
@L @r
=
6. Derive input demand, output supply and pro…t functions when the technology is y = K 0:4 L0:4 Q8. Consider a moral hazard insurance model with an insurance policy fp; B0 ; B1 ; :::::; BL g where p is the insurance premium and B0 ; B1 ; :::::; BL denote the bene…ts provided by the insurance company against loss l. Normally the insurance company can observe the loss but not the level of accident avoidance e¤ort (e) of the customer. The problem of the insurance company is: max
e;p;B0 ;B1 ;:::::;BL
L X
p
subject to the participation constraint: L X
l
(e) u (w
p
l
(e) Bl
(1790)
l=0
l + Bl )
d (e)
u
(1791)
l=0
and incentive constraint for level of accidend avoidance e¤orts e; e0 2 f0; 1g ; e 6= e0 ; L X l=0
l (e) u (w
p
l + Bl )
d (e)
L X
l
(e0 ) u (w
p
l + Bl )
d (e0 )
(1792)
l=0
1. Show that it is Pareto optimal to take/provide full insurance under symmetric information when the insurance company can observe the level of e¤ort of the customer. 2. How could the insurance company design an e¢ cient contract to induce accident avoidance e¤orts by customers that would minimise the cost of the insurance company under asymmetric information? Is full insurance still optimal?
395
17 17.1
Tutorials in Advanced Microeconomics Tutorial 1:Consumers’problem
1. What are the properties of a utility function? Find demands functions for x1 and x2 solving the consumer’s optimisation problem in following: max u = x1 x2
(1793)
subject to budget constraint: 2x1 + 4x2 = a
(1794)
Prove that demand for xi is ratios of partial derivative of indirect utility function (u) to price xi and income in the following problem Show that utility e¤ect of price changes will be higher for the commodity that is heavily weighted in the consumer’s consumption basket. Prove that the indirect utillity function ful…lls following properties. Continuous Homegenous of degree zero in (p; y) Strictly increasing in y Decreasing in p Quasiconvex in p and y. Roy’s identity 3. Derive generic demand functions for consumers and examine their properties in the following problem. Consumer optimisation: max u(x)
(1795)
subject to p:x
17.2
y
(1796)
Tutorial 2: Dual of the consumer problem
Q1. Consider a consumer’s utilitymaximisation problem Cobb-Douglas function given below: M ax U = x1 x2 X;Y
+
=1
(1797)
Subject to m = p1 x1 + p2 x2 396
(1798)
1. Derive the demand functions for both x1 and x2 and associated indirect utility function. 2. Formulate the dual of this problem and derive the expenditure function. 3. Prove the Shephard Lemma that,
@E @pi
=
@L @pi
= xi (p1 ; p2 ; m):
4. Prove Roy’s identity. 5. Decompose the total price e¤ect into the compensated (Hicksian) and uncompensated (Marshallian) demand functions using the Slutskey equation. 6. Prove numerically Shephard Lemma,Roy’s identity and Slutskey equation when 0:5 and M =200.
0:5;
=
Q2. Answer all above questions for a CES utility function as given below M ax u(x; y) = [ x + (1 x;y
)y ]
1
(1799)
Subject to x + py y = m Note that the elasticity of substituion and
are linked as:
397
(1800) =1
1
; px = 1:
17.3
Tutorial 3: Dual of the producer’s problem
Q1. A …rm’s objective is to minimise cost (C) taking prices of inputs (r; w) of K and L as given: C = rK + wL
(1801)
subject to CES technology constraint as: Y = [ L + (1
)K ]
1
(1802)
1. Determine the demand for labour and capital. 2. Derive the cost function of the …rm. 3. Prove that the elasticity of substituion is
=
1
1:
4. Discuss properties of CES cost function. 5. Prove that Cobb-Douglas production function is a special case of the CES production function. Use L’Hopital’s rule. Q2. Consider a problem of producer min w1 :x1 + w2 :x2
(1803)
x1; x2;
subject to (x1 + x2 ) Show that solution is
h
1
c (w; y) = y w1 Q3. Consider pro…t ( ) function of a …rm
= py
1
rK
y
+ w2
(1804)
1
wL
i
1
(1805)
(1806)
Derive supply function and input demand function using Hotelling’s Lemma when technology y = K 0:4 L0:4 @ (p; w) y (p; w) = (1807) @p xi (w; p) =
@ (p; w) @w
398
(1808)
17.4
Tutorial 4: Markets, Price War and Stability Analysis
Q1. Market demand fuction is given by P = 30
q1
q2
(1809)
Cost function Ci = 6qi
(1810)
Pro…t: i
= P qi
Ci
(1811)
Solve this duopoly model for market price P;quantities produced (q1 ; q2 );revenue (R1 ; R2 ); cost (C1 ; C2 ) pro…t ( 1 ; 2 );consuer surplus (CS1 ; CS2 ) and welfare cost, W under following market conditions: 1. Cournot duopoly. 2. Stackelberg leader- follower model 3. Cartel 4. Perfect competition (Bertrand price competition model). Summarise all results in one table. Q2. Two rival …rms are competing for a market by engaging in a price war; for 0 < < 1 and 0< 0
(1816)
There is free entry and exit of …rms N =
(p
c)
>0
(1817)
Find the continuous dynamic time path for two prices p(t) and N (t) and examine convergence towards the steady state. Example based on Hoy et al. (2001) Mathematics for Economics, MIT Press. 399
17.5
Tutorial 5: Ricardian General Equilibrium Trade Model
Consider a two good-two country Ricardian pure exchange economy. Preferences in country 1 are expressed by its utility function in consumption of good 1 and 2 , C11 and C21 respectively: max
U 1 = C11
1
C21
1
1
(1818)
Income of country 1 is obtained from the wage income in sector 1 and sector 2 plus the transfers to country 1 I 1 = w11 L11 + w21 L12 + T R1 L11
L12
where and are labour employed in sector 1 and sector 2 wages respectively and T R1 is the transfer income. Technology constraints in sector 1 in country 1
(1819) w11
and
X11 = a11 :L11
w21
are corresponding
(1820)
Two Country Ricardian Trade Model where a11 is the productivity of labour in sector 1 in country 1. Technology constraints in sector 2 in country 1 X21 = a12 :L12
(1821)
where a12 is the productivity of labour in sector 2 in country 1. Resource constraint in country 1 de…ned by the labour endowment as: L1 = L11 + L12
(1822)
Production possibility frontier of country 1 now can be de…ned as L1 =
1 1 :X11 + 1 :X21 1 a1 a2
(1823)
Two Country Ricardian Trade Model Given above preferences the demand for good 1 in country 1 is C11 =
1
:I 1 P1
(1824)
the demand for good 2 therefore is: C21 =
1
1
:I 1
P2 Parameters of the model are given in the table below.
(1825)
Assuming price in country one as a numeraire p1 = 1 solve this model for demand C11 ; C12 C21 ; C22 of output under complete specialisation (X1 ; X2 ) ;level of employment (L1 ; L2 ) ;level of utility U; relative price of commodity two p2 . Compare these results to an autarky. Put solutions into the tables given below. Analytical solutions for trade equilibrium under specialisation 400
levels
Table 89: Parameters of the Autarky Model a1 a2 L country 1 country 2
0.4 0.6
5 2
2 5
200 400
Table 90: Comparing Specialisation and Autarky Regimes Production Autarky Trade
Consumption Autarky Trade
X1
C1
X2
X1
X2
C2
C1
C2
P
country 1 country 2
17.6
Tutorial 6: General equilibrium with production
Q1. Consider an economy consisting of a representative household and a representative …rm. A representative household tries to maximise utility by consuming goods and services and from enjoying leisure subject to his budget constraints. The producer wants to maximise pro…t by selling goods produced using the labour supplied by the household. The household maximisation problem can be stated as the following: max U = C l(1
)
(1826)
Subject to time and budget constraints: l + hs = 1
(1827)
pc = whs +
(1828)
c > 0; hs > 0;and l > 0; where c is consumption,l is leisure and hs is labour supply, p is the price of the commodity, w is the wage rate is the pro…t from owning the …rm. Maximisation problem for the representative …rm can be states as: = py
whd
(1829)
subject to technology constraint as: y = (hs ) y > 0; h > 0; where y is the output supplied by the …rm and hd is its demand for labour. d
1. Form a Lagrangian for constrained maximisation problem for this household. 2. Derive its demand for consumption goods and derive its demand for leisure. 3. Write the Lagrangian function for the …rm’s optimisation problem. 401
(1830)
Table 91: Comparing Employment and Welfere under Specialisation and Autarky Employment Autarky Trade
L1
L2
L1
Uitlity Autarky Trade
L2
U
U
4. Derive …rm’s demand for labour. 5. De…ne a competitive equilibrium for this economy. 6. Compute the real wage that brings goods and labour market in equilibrium. 7. What is the equilibrium quantity of c or y? 8. What is the equilibrium quantity of l and h? 9. Formulate the problem with sales and income tax. Discuss qualitatively the macroeconomic impacts of (a) switching completely to the sales taxes or (b) to labour income taxes or to (c) capital income tax. Q2. Equilibrium with production (x; y)=(x1; x2; :::::xm ; y1; y2; :::::ym ) for consumer i = 1; ; ; m and producer j =1,.,. n A possible allocation for consumers and producers satisfying following: Consumption set: xi 2 X Production possibility set: yi 2 Y m m n P P P Resource balance condition: xi = ei + yj i=1
i=1
Wealth of the consumer: wi (p) = p:e1 +
j=1
n P
i;j j
(p)
j=1
Supply correspondence: sj (p) = fyi 2 Yj : y 0 2 Yj =) p:y > py 0 g m n P P Excess demand correspondence: Z (p) = (di (p) e1 ) sj (p) i=1
j=1
A competitive equilibrium is a pair of prices, demand and supply (p; (x; y)) with a p vector inRl and x 2 di (p)for consumer i to m, and yj 2 sj (p)for …rms j to n and where the excess demand is zero in equilibrium. Now consider a Robinson Crusoe economy Commodity space: R2 (leisure and food) Consumer characteristic: Xi = R2+ Endowment: ei = (24; 0) Preference relation: i U (L; F n = LF p o Producer characteristics: ; Yj ( L; F ) : L 0; F L p where F = L is the production function. Solve this model for price vector, demand vector and the output vector.
402
17.7
Tutorial 7:Monopoly and monopolistic competition and taxes
1. Consider monopoly and oligopoly models given below. Monopoly model: Pro…t function of a monopolist with taxes = PQ
TC
P =a
T
(1831)
bQ
(1832)
T C = cQ + f
(1833)
T = tQ
(1834)
total cost with marginal cost c and …xed cost f
Tax revenue
Oligopoly model There are i = 1,..., N …rms in the market Market supply Q=
n X
qi
(1835)
i=1
Market price depends on total sales P =a
bQ = a
n X b qi
(1836)
i=1
total cost including taxes (ignore …xed cost for a while) T Ci = (c + t) qi
(1837)
T = tQ
(1838)
Tax revenue
Pro…t of a particular …rm in oligopoly is:
1
=
=
P q1 a
(c + t) q1 = n X b qi i=2
!
a
n X b qi i=1
q1
bq12
!
q1
(c + t) q1
(c + t) q1 (1839)
Prove that revenue maximising tax rate is the same for both monopoly or oligopoly. Market structure does not matter for it.
403
Q2. Consider a tripoly market where only three …rms supply to the market. Conjectural variation one …rm against another matter for pricing and output decisions and have impact on pro…ts. P =a
bQ = a
b (q1 + q2 + q3 )
C i = ci q i
(1840) (1841)
1
= [a
b (q1 + q2 + q3 )] q1
C1
(1842)
2
= [a
b (q1 + q2 + q3 )] q2
C2
(1843)
3
= [a
b (q1 + q2 + q3 )] q3
C3
(1844)
Solve for output and pro…t of each …rms and market price in equilibrium. Q3. Consider a tripoly market with the following demand, cost function and conjectural varia@qi =1 tion for each …rm to be one as @q j P =a
bQ = a
b (q1 + q2 + q3 )
Ci = cqi Solve for output and pro…t of each …rms and market price in equilibrium.
404
(1845) (1846)
17.8
Tutorial 8:Moral Hazard and Insurance
Q1 Honesty is the best policy in Vickery auction; truth telling is a winning strategy. Prove it. Q2 Project B earns more but is riskier than project A. Probability of success of projects A and B are given by a and b respectively. a. Illustrate how the rate of interest rate should be lower in project A than in project B in equilibrium? b. Probability of types A and B agents is given by pa and pb respectively. Prove under the asymmetric information, a lender charging a pooling interest rate is unfair to the safe borrower A and more generous to the risky borrower B. c. How can agent signal its worth? How can the lender ascertain the degree of moral hazard in B? Q3 Consider a moral hazard insurance model with an insurance policy fp; B0 ; B1 ; :::::; BL g where p is insurance premium and B0 ; B1 ; :::::; BL denote the bene…t from the insurance company against loss l. Normally the insurance company can observe the loss but not the level of accident avoidance e¤ort (e) of the consumer. The problem of the insurance company is:
max
e;p;B0 ;B1 ;:::::;BL
L X
p
subject to participation constraint L X
l
(e) u (w
p
l
(e) BL
(1847)
l=0
l + BL )
d (e)
u
(1848)
l=0
and incentive constraint L X l=0
l
(e) u (w
p
l + BL )
d (e)
L X
l
(e0 ) u (w
p
l + BL )
d (e0 )
u
(1849)
l=0
1. Show that it is Pareto optimal to do full insurance under symmetric information when the insurance company can observe the level of e¤orts of the consumer. 2. How could the insurance company design an e¢ cient contract to induce e¤orts to minimise cost under the assymetric information? Is full insurance still optimal? Q4. A monopolistic …rm engages in non-linear pricing scheme with its two types of customers f H ; L g; where H is an index of the valuation that high value customers put in its product and L is that of the low value customers. Non-linear price scheme is to set tarrifs (T ) and output (q) in such a way that maximises …rm’s pro…t by designing price scheme appropriate to these consumers. Utility function (u) of consumers in generic form is:
405
u = V (q) 0
T
(1850)
00
V (q) > 0 and V (q) < 0; T is the tarrif paid by the customer. With marginal cost c …rm’s pro…t ( ) is =T
cq
(1851)
T] > 0
(1852)
Participation constraint [ V (q)
p 1. Specialise function and parameters to V (q) = 2 q and f H ; L g = f20; 15g c = 5. Assuming the …rm knows exactly the type of the customers in this way …nd the equilibrium quantities (q) and tari¤s (T ) under the …rst best solution that high and the low value customers would purchase from this producer. What is the expected pro…t of this …rm in this …rst best solution? 2. Now assume that the …rm does not know the true type of customer but it assumes that probability of each type is 50 percent ( = 21 ). This …rm requires to design contracts considering participation and incentive constraints for low and high type consumers as: LV
(qL )
TL ] > 0
(1853)
HV
(qH )
TH ] > 0
(1854)
[ [ [
LV
(qL )
TL ] > [
[
HV
(qH )
TH ] > [
LV
(qH )
HV
(qL )
Then …rm’s objective is to maximise the expected pro…t ( e
=
(TH
cqH ) + (1
) (TL
TH ]
(1855)
TL ]
(1856)
e
) as: cqL )
(1857)
What would be the expected pro…t if the high value type customer defects to the low value type customer? Show how the …rm could reduce the size of qL to make high value customer to stick to its qH . Prove that price discrimation in this manner favours high value customer more than the low value customer.
406
17.9
Tutorial 9:Coalition, Bargaining, Signalling, Contract, Auction and Mechanism
1 Consider Four Players A,B,C,D. How many coalitions are possible? Empty core. 1. Prove that a risk averse person loses but the risk neutral person gains in the bargaining. 0:5 Suppose the utility functions of risk averse person is given by u2 = (m2 ) but the risk neutral person has a linear utility u1 = m1 . m1 + m2 = M ; u1 + u22 = 100: 2. Consider a three player (1,2,3) game in which the 3rd player always brings more to the coalition than the 1st or the 2nd player. Payo¤ for coalition of empty set: v ( ) = 0 Payo¤ from players acting alone: v (1) = 0; v (2) = 0; v (3) = 0 ; Payo¤ from alternative coalitions: v (1; 2) = 0:1; v (1; 3) = 0:2; v (2; 3) = 0:2; Payo¤ from the grand coalition: v (1; 2; 3) = 1 Power of individual i in the coalitions is measured by the di¤erence that person makes in the value of the game v (S [ fig v (S)) = 1 , where S is the subset of players excluding i, S [ fig is the subset including player i. X Compute the Shapley value of the game for each player. [hint : i = v (S)) ; n (S) v (S [ fig S2N
s!(n s 1) ] n!
4 Prove equivalence of core in games and core in general equilibrium. 5 Given the market demand and cost functions P = 24
0:5q
C = 12q
(1858)
6 Prove following four propositions regarding e¢ cient contract. Proposition 1: Results of …xed fee contract and joint pro…t maximisation are equivalent Proposition 2: Hire contract is incentive incompatible and leads to production ine¢ ciency Proposition 3: Moral hazard problem and production ine¢ ciency exists in revenue sharing contingent contract Proposition 4: Pro…t sharing contract is e¢ cient and free of moral hazard problem 7 Level of education signals quality of a worker. Spence (1973) model was among the …rst to illustrate how to analyse principal agent and role of signalling in the job market.Consider a situation where there are N individuals applying to work. In absence of education as the criteria of quality employers cannot see who is a high quality worker and who is a low quality worker. Employers know that proportion of workers is of high quality and (1- ) proportion is of bad quality. Therefore they pay each worker an average wage rate as: w = wh + (1
) wl
(1859)
more productive worker is worth 70000 and less productive worker is worth 30000 and =0.5 then the average wage rate will be 50000. Prove separtating equilibrium is more e¢ cient than the pooling equilibrium and that it is worth for high quality workers to signal their quality by the standard of their education. 407
n
(S) =
8 Consider a situation of lender that has two potential borrowers. Borrower type 1 has a high yielding project than the borrower type 2. It is however not clear to the lender, the principal, which one of the two borrowers is more productive. Faced with this situation the principal is left with two options. Easy option would be to treat both borrowers in the same way and charge the same rate of interest rate to both of them. Such pooling strategy is not e¢ cient because the type 2 borrower does not have enough incentive to put in extra e¤orts in the project. It creates disincentive to be more productive borrower. Part of the market disappears. The second more e¢ cient option is to design a contract that guarantees a separating equilibrium. For this the lender needs to design a mechanism that ful…ls participation and incentive compatibility constraints. Principal’s objective function: UP = [0:5 (R1
(B1 )) + 0:5 (R2
(B2 ))]
(1860)
Here Ri is measures the returns to principal from borrower i and Bi the bene…t to the investor i from that. 2 Utility function of agent 1: U1 = B1 (R1 ) given that or R1 = 3B1 simply B1 = R31 . 2 Utility function of agent 2: U2 = B2 (R2 ) given that orR2 = B2 . a. Formulate participation constraints for both borrowers b. formulate incentive compatible constraits c. Determine the binding constraints. d. solve the game and determine the level of utility for all players in equilibrium.
408
17.10
Tutorial 10: E¢ ciency and Social Welfare
Q1 Illustrate e¢ ciency conditions in allocations of resources a. When consumers’utility function is given by U (X; Y ) and the production possibility frontier is T (X; Y ): b. E¢ ciency of production when X = f1 (K1 ; L1 ) + f2 (K2 ; L2 )
(1861)
K1 + K2 = K
(1862)
L1 + L2 = L
(1863)
c. Prove that e¢ cient provision of public goods require that the sum of the marginal rate of substitution equals the marginal cost of provision of public good with a two consumer economy in which consumers like to maximise utility by consuming private (x) and public goods (G) max
u1 = u1 (x1 ; G)
(1864)
subject to a given level of utility for the second consumer max u2 = u2 (x2 ; G)
(1865)
x1 + x2 + c (G) = w1 + w2
(1866)
and the resource constraint
Q2 There are two people living in an economy. For simplicity assume that a …xed amount of output of 200 is produced each year. Entire in the same year. Utility of p p output is consumed individual 1 and 2 is represented by U1 = Y1 and U2 = 12 Y2 . (a) What is the utility received by each individual if the output is divided equally between these two people? What is the output received by each if it is distributed so that each of them gets the same amount of the utility? (b) What is the distribution of output that maximises the total utility for the whole economy? (c) If person 2 needs utility 5 in order to survive how should the output be distributed? 1
1
(d) Suppose that the authorities like to maximise the social welfare function W = U12 U22 , how should the output be distributed between them? Q3 An economy is inhabited by type 1 and type 2 people. The type 1 is more productive than the type 2. Policy makers encourage productive people by assigning a greater weight to the utility of more productive people. They aim to maximise the social welfare function: 3 1 W = U14 U24 where W is the index of the social welfare, U1 represents the utility of type 1 people and U2 is the utility of type 2 people. For simplicity assume that resources of this economy produce a given level of output Y. It is consumed either by 1 or by 2 type people. Market clearing condition implies: Y = Y1 + Y2 . Preferences for type 1 are given p p by U1 = Y1 and for type 2 by U2 = Y2 . In a given year total output, Y, was 1000 billion pounds. 409
(a) What is the distribution of output between type 1 and type 2 that maximises the social welfare index? What is the maximum value of the social welfare index of this economy? (b) What would have been the allocation if policy makers had given equal weight to the 1
1
utility of both types of people in the economy such as W = U12 U22 . By how much does the welfare index change in this case than compared to the social welfare in (a) above? (c) How would the social welfare index change in (a) if a tax rate of 20 percent is imposed in consumption and the tax receipts are not given back to any of these consumers? How much would the value of social welfare index be in this case? 3
1
e. Assume that the policy makers still hold the welfare function to be W = U14 U24 . How would the social welfare index change in (c ) if all tax receipts are transferred to type 1 people?
17.11
Basic Calculus
17.11.1
Four rules of di¤erentiation
Power rule
@Y = K @K
Y = K =)
1
Product rule R = P Y =) dR = Y
dP + P
dY
Quotient Rule y=
Y L =) L
dY
Y
dL
=
L2
dY L
dL Y L L
Chain Rule W = Y2+L 17.11.2
2
=)
@W =2 Y2+L @Y
2Y = 4Y Y 2 + L
Unconstrained optimisation: using Hessian determinants
Consider a pro…t maximisation problem with the Cobb-Douglas production function. = P L K1
wL
rK
)
w=0
First order conditions for pro…t maximization. L
K
= PL = (1
1
K (1
)PL K
r=0
Hessian determinants should be positive Second order derivatives: LL
=
(
1) P L
410
2
K (1
)
LK
=
K;K
=
K;L
(1
=
)PL
2
K(
(1
)PL K
(1
)PL K
)
1
Hessian determinant: LL K;L
LK K;K
=
(
1) P L 2 K (1 (1 )PL K H1 = j
H2 =
LL j
)
(1 (1
) P L 2K ( )PL K
) 1
0
for maximum and H1 = j H2 =
LL j
>0
LL
LK
K;L
K;K
='>
Abstract This monograph presents major elements of advanced microeconomic models for systematic thinking about the working of modern markets. Problems of consumers and producers are analysed concisely in partial, general equilibrium and game theoretic frameworks relating them to the micro level decision making processes with due consideration on the structure of markets and pulic policies. Exercises and assignments in workbook anticipate reading of relevant journal articles. JEL Classi…cation: D Keywords: microeconomic models H U 6 7 R X , H u ll,
U K . e m a il: K .R .B h a tta ra [email protected] hu ll.a c .u k
1
Contents 1 L1: 1.1 1.2 1.3
Axioms and optimisation Microeconomic Theory: Milestones . . . . . . . Axioms and Consumption Set . . . . . . . . . . Optimisation . . . . . . . . . . . . . . . . . . . 1.3.1 Consumer optimisation: . . . . . . . . . 1.3.2 Producer optimisation . . . . . . . . . . 1.4 A simple computable general equilibrium model 1.5 Methods for constructing Proofs . . . . . . . . 1.5.1 Direct method . . . . . . . . . . . . . 1.5.2 Converse and contrapositive . . . . 1.5.3 Equivalence . . . . . . . . . . . . . . . 1.5.4 Mathematical induction . . . . . . .
. . . . . . . . . . . . . . . with . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . labour . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . leisure . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . choice . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
2 L2: Consumption: Properties of a Utility Function 2.1 Properties of an indirect utility function . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Price elasticities and the elasticity of substitution . . . . . . . . . . . . . . . . 2.1.2 Consumer optimisation model: a numerical example . . . . . . . . . . . . . . 2.1.3 Econometric issues in estimation of demand functions and elasticities of demand 2.1.4 Restrictions in estimating a demand function: . . . . . . . . . . . . . . . . . . 2.2 Exercise 1: Consumer’s problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Marshallian demand functions from the CES preferences: . . . . . . . . . . . 2.2.2 Compensated and uncompensated demands . . . . . . . . . . . . . . . . . . . 2.2.3 Expenditure functions with the CES utility functions . . . . . . . . . . . . . . 2.3 Slutskey equation ( Decomposition of substituion and income e¤ects): Duality on consumer optimisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Comparative static analyis with matrix . . . . . . . . . . . . . . . . . . . . . 2.4 Exercise on Consumption and Demand . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Indirect utility and expenditure functions: Roy’s Identity . . . . . . . . . . . 2.4.2 Dual of the consumer’s optimisation problem . . . . . . . . . . . . . . . . . . 2.4.3 Shephard’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.4 Calibration of the constant elasticity of substitution (CES) demand function 2.4.5 Exercise 1.2: comparative static analysis of consumer choice . . . . . . . . . . 2.4.6 Exercise 2: Indirect utility function, Shephard’s Lemma and Roy’s Identity . 2.4.7 Duality in consumption and Slutskey decomposition . . . . . . . . . . . . . . 2.4.8 Problem 4: CES Demand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.9 Extra example on Shephard’s Lemma and Roy’s identity . . . . . . . . . . . . 2.4.10 Indierct utility function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.11 Expenditure function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.12 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.13 Shephard’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.14 Roy’s Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Revealed Preference Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Slutsky equation from the Revealed Preference Theory . . . . . . . . . . . . . 2.5.2 Further Developments in Consumer Theory . . . . . . . . . . . . . . . . . . .
2
8 8 9 10 10 11 12 14 14 15 15 15 15 16 17 18 19 19 21 22 23 23 24 26 29 30 30 31 32 35 36 37 38 39 40 40 40 41 43 44 44 45
2.5.3
Emprical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
3 L3: Production: Supply 47 3.0.4 Popular production functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.1 Supply function: an example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.1.1 Properties of a pro…t function . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.1.2 Production function and its scale properties . . . . . . . . . . . . . . . . . . . 52 3.1.3 Variable returns to scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.1.4 Cost function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.2 CES and Cobb-Douglas production functions . . . . . . . . . . . . . . . . . . . . . . 54 3.3 Comparative static: derivation of the CES cost function. . . . . . . . . . . . . . . . . 56 3.3.1 Exercise 5: cost minimisation . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.3.2 Properties of a cost function . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.3.3 Exercise 6 : CES and Cobb-Douglas supply functions . . . . . . . . . . . . . 62 3.3.4 Short run supply function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.3.5 Hotelling’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.3.6 Exercise 7: Minimising the cost with Cobb-Douglas and CES production function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.4 Consumer and producer surplus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.4.1 Pro…t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.4.2 Linear programming problem of a …rm . . . . . . . . . . . . . . . . . . . . . . 71 3.4.3 Duality in Linear Programming . . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.5 Input-Output Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.5.1 Numerical Example of Input Output Model . . . . . . . . . . . . . . . . . . . 76 3.5.2 Impact analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 3.5.3 Exercise 10: Input Output Model . . . . . . . . . . . . . . . . . . . . . . . . 80 4 L4: Markets: Perfect and Imperfect Competition 4.1 De…nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Cournot, Stackelberg and Cartel: which is better of consumer welfare? . . . 4.1.2 Price-Leadership by …rm 1 in Stackelberg equilibrium . . . . . . . . . . . . 4.2 Dixit-Stiglitz Model of Monopolistic Competition . . . . . . . . . . . . . . . . . . . 4.2.1 Monopolistic competition and Trade . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Monopolistic competition in an industry with two …rms . . . . . . . . . . . 4.3 Natural Monopoly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Bertrand Game of price competition . . . . . . . . . . . . . . . . . . . . . . 4.4 Price War: stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Monopolistic competition and trade . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Monoply, Oligopoly and tax . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Tripoly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Multinational Company: Microeconomic Theory of FDI . . . . . . . . . . . . . . . 4.8 Predatory pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.1 FDI under uncertainty: Dixit and Pindyk (1994) approach . . . . . . . . . 4.9 General equilibrium model of a multinational …rm:Batra and Ramachandran(1980) 4.9.1 Empirical evidence on growth e¤ects of FDI . . . . . . . . . . . . . . . . . . 4.9.2 Exercise 9: markets and competition . . . . . . . . . . . . . . . . . . . . . .
3
. . . . . . . . . . . . . . . . . .
83 83 87 88 90 91 92 95 97 98 102 103 106 110 114 114 116 119 119
4.9.3
General equilibrium with production . . . . . . . . . . . . . . . . . . . . . . . 122
5 L5: General Equilibrium Model and Welfare 5.1 What is a general equilibrium? . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Existence, uniqueness and stability of general equilibrium . . . . . . . . 5.2 Two fundamental theorems of welfare economics . . . . . . . . . . . . . . . . . 5.3 Pure exchange general equilibrium model . . . . . . . . . . . . . . . . . . . . . 5.3.1 Exercise 11: Ricardian trade model . . . . . . . . . . . . . . . . . . . . . 5.4 Simplest general equilibrium model . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 General equilibrium with production . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 A numerical example for the general equilibrium tax model . . . . . . . 5.6 Social Welfare Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Exercise 12: Social Welfare and General Equilibrium . . . . . . . . . . . . . . . 5.8 Two sector model of nessecity and luxury goods (income distribtuion) . . . . . 5.9 General equilibrium model of Trade: Ricardian Comparative Advantage Theory 5.9.1 Two Country Ricardian Trade Model . . . . . . . . . . . . . . . . . . . 5.9.2 Autarky or Trade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10 Exercise 12’: migration and factor mobility . . . . . . . . . . . . . . . . . . . . 5.11 General equilibrium with taxes . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.12 Exercise 13: Monopolistic Competition . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . .
122 122 124 127 129 133 133 134 137 143 146 149 153 153 154 162 163 168
6 L6: Game theory: Bargaining in Goods and Factors markets 171 6.1 Formal de…nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 6.1.1 Nash equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 . . . . . . . . . . . . .
. . . . . . . . . . . . .
. . . . . . . . . . . . .
. . . . . . . . . . . . .
174 175 176 179 183 187 190 191 191 192 194 196 199
7 L7: Game theory: Principal Agent and Mechanism Games and Auctions 7.1 Original Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 Full information scenario . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.2 Incomplete information scenario . . . . . . . . . . . . . . . . . . . . . . 7.1.3 Impacts of Assymetric (incomplete) Information on Markets . . . . . . . 7.1.4 Adverse Selection (hidden information) Problem . . . . . . . . . . . . . 7.1.5 Signalling and Incentives . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.6 Education Level- A Signal of Productive Worker . . . . . . . . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
200 200 201 201 205 206 207 209
6.2 6.3 6.4 6.5
6.6 6.7 6.8
6.1.2 Game of incomplete information: . . . . . . . . . . . . . . 6.1.3 Extensive form Game ( ) . . . . . . . . . . . . . . . . . . Story of GAME made easy . . . . . . . . . . . . . . . . . . . . . Types of games . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bargaining game . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Coalition and Shapley Values of the Game . . . . . . . . Pivotal player in a voting game in Nepal . . . . . . . . . . . . . . 6.5.1 Model of fruitless bargaining and negotiation . . . . . . . 6.5.2 Model of commitment, credibility and reputation . . . . . 6.5.3 Endogenous intervention: change in beliefs . . . . . . . . Equivalence of Core in Games and Core in a General Equilibrium Labour Market and Search and Matching Model . . . . . . . . . Exercise 14: Search Equilibrium . . . . . . . . . . . . . . . . . . .
4
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Model . . . . . . . .
. . . . . . . . . . . . .
. . . . . . . . . . . . .
. . . . . . . . . . . . .
7.2 7.3 7.4 7.5
7.6 7.7 7.8 7.9
Spence model of education . . . . . . . . . . . . . . . . . . . . . . . . . . Popular Principal Agent Games . . . . . . . . . . . . . . . . . . . . . . . Exercise 15: Principal Agent Problem . . . . . . . . . . . . . . . . . . . Mechanism Design for Price Discrimination: Low Cost Airlines Example 7.5.1 Mechanism for e¢ cient contract for a CEO . . . . . . . . . . . . 7.5.2 E¢ cient contracts of Land . . . . . . . . . . . . . . . . . . . . . . 7.5.3 Mechanism for Poverty Alleviation . . . . . . . . . . . . . . . . . Repeated Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Moral Hazard and Adverse Selection . . . . . . . . . . . . . . . . . . . . Auction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercise 16 :Optimal production of a multiproduct …rm . . . . . . . . . 7.9.1 A Microeconomic Model of FDI . . . . . . . . . . . . . . . . . . .
8 L8: Uncertainty and Insurance 8.1 Allais’paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1 Uncertainty of Good Times and Bad Times . . . . . . . . . . 8.1.2 Optimal Demand for Insurance . . . . . . . . . . . . . . . . . 8.2 Expected utility theory . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Measure of risk aversion . . . . . . . . . . . . . . . . . . . . 8.2.2 St Petersberg Paradox (Bernoulli Game) and Allais Paradox 8.2.3 Non-linear pricing Scheme . . . . . . . . . . . . . . . . . . . . 8.2.4 Job market applications . . . . . . . . . . . . . . . . . . . . . 8.2.5 Insurance market . . . . . . . . . . . . . . . . . . . . . . . . . 9 L9:Class Test
. . . . . . . . .
. . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
210 212 217 217 221 222 224 226 228 231 233 236
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
239 241 243 245 247 248 249 250 253 255 259
10 L10: Impact of Taxes and Public Goods in E¢ ciency, Growth and Redistribution: A General Equilibrium Analysis 263 10.1 First best principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 10.1.1 E¢ ciency in consumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 10.1.2 E¢ ciency in production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 10.1.3 E¢ ciency of Trade (Exchange) . . . . . . . . . . . . . . . . . . . . . . . . . . 264 10.1.4 A simple numerical example of optimal tax or optimal public spending . . . . 265 10.1.5 E¢ ciency in public goods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 10.1.6 Theory of second best . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 10.1.7 Externality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 10.1.8 Samuelson and Nash on Sharing Public Good . . . . . . . . . . . . . . . . . . 268 10.1.9 Sameulson’s Theorem on Public Good . . . . . . . . . . . . . . . . . . . . . . 269 10.1.10 Negative externality in production . . . . . . . . . . . . . . . . . . . . . . . . 270 10.2 Negative externality and Pigouvian tax . . . . . . . . . . . . . . . . . . . . . . . . . 271 10.3 Carbon Emmission in the UK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 10.3.1 A model of growth, …scal policy and welfare . . . . . . . . . . . . . . . . . . . 278 10.4 Fiscal Policy, Growth and Income Distribution in the UK . . . . . . . . . . . . . . . 280 10.4.1 Middle income hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286 10.4.2 Current Fiscal Policy Context . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 10.5 Features of Dynamic Tax Model of UK . . . . . . . . . . . . . . . . . . . . . . . . . . 291 10.5.1 Preferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 5
10.5.2 10.5.3 10.5.4 10.5.5 10.5.6 10.5.7 10.5.8 10.5.9
Production Technology . . . . . . . . . . . Trade arrangements . . . . . . . . . . . . . Government sector . . . . . . . . . . . . . . General Equilibrium in a Growing Economy Procedure for Calibration . . . . . . . . . . Data for the Benchmark Economy . . . . . Results on Redistribution . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
292 292 293 293 294 294 295 298
11 L11: Dynamic Computable General Equilibrium Model: Recent Developments 305 11.1 Capital market: models and issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310 11.1.1 Risk management in asset markets . . . . . . . . . . . . . . . . . . . . . . . . 313 11.1.2 Industrial regulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 11.1.3 IO Approach to pricing and industrial concentration . . . . . . . . . . . . . . 316 11.1.4 Signalling and Incentive Compatibility in the Financial Markets . . . . . . . 320 11.1.5 Moral hazards in the …nancial market . . . . . . . . . . . . . . . . . . . . . . 321 12 Assignment (optional): One in Four 12.1 General equilibrium and game theoretic analysis of …nancial 12.1.1 CGE Modelling of energy sector policies . . . . . . . 12.1.2 CGE Modelling of tax policies . . . . . . . . . . . . 12.1.3 Comparative Static Model . . . . . . . . . . . . . . . 12.2 Dynamic CGE model of the energy and emmission . . . . .
. . . . .
. . . . .
. . . . .
. . . . .
327 327 327 327 329 335
13 Regulation Theory and Practice 13.0.1 Theory of Regulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.0.2 Measures of concentration and performance . . . . . . . . . . . . . . . . 13.0.3 Regulation for solve the moral hazard problems in the …nancial markets 13.0.4 Regulation by mechanism design by banks . . . . . . . . . . . . . . . . . 13.0.5 Participation and incentive compatible constraints . . . . . . . . . . . . 13.0.6 Solving the mechanism design problem of a bank . . . . . . . . . . . . . 13.0.7 IO Approach to pricing and industrial concentration (HHI) . . . . . . . 13.0.8 Why regulation? Welfare e¤ects of monopoly . . . . . . . . . . . . . . . 13.0.9 Optimal advertising . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.0.10 Marginal productivity theory and tax credit . . . . . . . . . . . . . . . . 13.0.11 Capital stock with and without capital income tax . . . . . . . . . . . . 13.0.12 Technological development, human capital and tax rules . . . . . . . . . 13.0.13 Dixit-Stiglitz Model of Monopolistic Competition . . . . . . . . . . . . . 13.0.14 Market under imperfect competition and average cost pricing . . . . . . 13.0.15 Krugman (1980): Trade and scale economy and regulation . . . . . . . . 13.0.16 Regulation by non-linear pricing Mechanism . . . . . . . . . . . . . . . . 13.1 Articles and Texts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1.1 Best twenty articles in 100 years in the American Economic Review . . 13.1.2 Ten Best articles in the Journal of European Economic Association . . . 13.1.3 Best 40 articles in the Journal of Economic Perspectives . . . . . . . . .
. . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . .
338 338 339 340 341 341 342 343 344 344 345 346 346 348 349 351 352 357 357 358 359
6
sector . . . . . . . . . . . . . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
14 Real Analysis 14.1 Methods for constructing Proofs 14.1.1 Convergence . . . . . . . 14.1.2 Boundedness . . . . . . 14.1.3 Convex Hull . . . . . . 14.1.4 Correspondence . . . . 14.1.5 Fixed Point Theorems . . 14.2 SETS . . . . . . . . . . . . . . . 14.2.1 Relations and functions . 14.3 Limits . . . . . . . . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
368 369 370 370 371 371 372 372 373 373
15 Computation and software 15.1 GAMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2 MATLAB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3 Econometric and Statistical Software . . . . . . . . . . . . . . . 15.3.1 Quality ranking of journals in Economics . . . . . . . . 15.4 Core texts in Economic Theory and Equivalent reading
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
374 374 378 381 382 384
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
16 Schedule 386 16.1 Sample class test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388 16.2 Sample …nal exam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391 17 Tutorials in Advanced Microeconomics 17.1 Tutorial 1:Consumers’problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2 Tutorial 2: Dual of the consumer problem . . . . . . . . . . . . . . . . . . . . . 17.3 Tutorial 3: Dual of the producer’s problem . . . . . . . . . . . . . . . . . . . . 17.4 Tutorial 4: Markets, Price War and Stability Analysis . . . . . . . . . . . . . . 17.5 Tutorial 5: Ricardian General Equilibrium Trade Model . . . . . . . . . . . . . 17.6 Tutorial 6: General equilibrium with production . . . . . . . . . . . . . . . . . 17.7 Tutorial 7:Monopoly and monopolistic competition and taxes . . . . . . . . . . 17.8 Tutorial 8:Moral Hazard and Insurance . . . . . . . . . . . . . . . . . . . . . . . 17.9 Tutorial 9:Coalition, Bargaining, Signalling, Contract, Auction and Mechanism 17.10Tutorial 10: E¢ ciency and Social Welfare . . . . . . . . . . . . . . . . . . . . . 17.11Basic Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.11.1 Four rules of di¤erentiation . . . . . . . . . . . . . . . . . . . . . . . . . . 17.11.2 Unconstrained optimisation: using Hessian determinants . . . . . . . . . . 17.11.3 Constrained optimisation: Bordered Hessian Determinants . . . . . . . . 17.11.4 Linear Programming approach to input-output model . . . . . . . . . . .
7
. . . . . . . . . . . . . . .
. . . . . . . . . . . . . . .
396 396 396 398 399 400 401 403 405 407 409 410 410 410 412 413
1
L1: Axioms and optimisation
Rational economic agents use available resources to achieve their objectives in the best possible way. Microeconomics is about choices of these rational individuals who make decisions regarding the allocation of resources, particularly on how much to consume or invest, how to produce and how to interact in the markets remaining within the limits of resources they possess. It is also concerned about the consumption and saving or current or future consumption at the individual level and about the e¢ ciency welfare consequences of public policies that a¤ect them. Game theories have been applied increasingly in recent yeats to study strategic economic behavior of consumers and producers. Main elements of microeconomic theory thus consists of: 1. Consumer choice and demand for products and supply of labour and capital: demand functions. 2. Producer’s choice of products and demand for inputs: production, cost, pro…t and supply functions. 3. Analysis of prices in perfect and imperfect markets with complete and incomplete information. 4. General equilibrium analysis - determination of price system and optimal allocations. 5. Strategic analysis of decision making my consumers, producers, governments in a competitive global economy. 6. Competition and market power; game theory 7. Innovations and adoption new technologies, research and development. 8. Analysis of e¢ ciency and public policies; social welfare function, market failure, negative and positive externalities Good undestanding of microeconomic theories will lead to better policies and regulations for the e¢ cient functioning of the market economy. These policies particularly focus on competition, adoption of better technology, governance and information, correcting externality and good environment, social insurance, more equal distribution of income and identi…cation of cases for govermen intervention. For recent policies see relevant web page of the government such as in the Department for Business Innovation & Skills https://www.gov.uk/government/organisations/competition-andmarkets-authority. Analysis of all above are based on axioms or generally accepted truth about the behavior of economic agents that include axioms of completeness, transitivity, continuity, monotonicity and convexity.
1.1
Microeconomic Theory: Milestones
The existing knowledge in microeconomis is the result of hard work of many prominent economic thinkers: such as Smith (1776), Ricardo (1817), Cournot (1838), Bertrand (1883), Edgeworth (1925), Pareto (1896), Marshall (1890), Walras (1900), Veblen (1904), Slutskey (1915), Hicks (1939), Samuelson (1947), von Neumann and Morgenstern (1944), Nash (1950), Neumann (1957), Arrow (1953), Debreau (1959), Stigler (1961), Kuhn (1953), Shapley (1953), Robinson (1963),
8
Shelten ( 1965), Aumann (1966) Scarf (1967), Shapley and Shubik (1969), Harsanyi (1967), Spence (1974),Kahneman and Tversky (1979), Kreps (1990), Fundenberg and Tirole (1991) and Binmore (1992), Varian (1992), Osborne and Robinstein (1994), MasColell, Whinston and Green (1995), Starr (1997) Gravelle and Rees (2004), Rasmusen (2006), Snyder and Nicholson (2011). Studies by Cobb and Douglas (1928), Arrow (1963), Jorgenson (1963), Diamond and Mirrlees (1971), Alchian and Demsetz (1972), Ross (1973), Dixit and Stiglitz (1977), Deaton, and Muellbauer (1980), Krugman (1980), Shiller (1981), Grossman and Stiglitz (1980), listed in the best 20 articles published in AER in last 100 years, relate to microeconomic issues. http://www.eea-esem.com/eea-esem/2014/prog/list_sessions.asp http://editorialexpress.com/conference/MMF2014/program/MMF2014.html https://www.aeaweb.org/aea/2015conference/program/preliminary.php http://www.webmeets.com/RES/2013/prog/list_sessions.asp http://nobelprize.org/nobel_prizes/economics/laureates/; http://www.economicsnetwork.ac.uk/ http://cepa.newschool.edu/het/schools/game.htm; http://www.hull.ac.uk/php/ecskrb/Confer/research.html; http://homepage.newschool.edu/het/alphabet.htm http://editorialexpress.com/conference/GAMES2012/program/GAMES2012.html http://www.eea-esem.com/EEA-ESEM/2012/prog/list_sessions.asp
It is becoming sophisticated with the recent development of mathematical techniques and computational abilities. This monograph gradually develops these concepts so that all parts of the modern economies could be integrated into a dynamic general equilibrium model of modern economies towards end of this workbook. Axioms and a general summary is in this section. Then behavior of consumers and produced are analysed in the …rst two chapters followed by partial equilibrium analysis under perfectly and imperfectly competitive markets in section 4. Basic principles of general equilibrium models explained in section 5 followed by a short discussion of strategic models in chapters 6 and 7 and uncertainty and asymmetric information in section 8. Impacts of taxes on public goods and externalities are presented in section 10 followed by details of the dynamic general equilibrium model in section 11. Each section have takes problem solving approach to learning and contains exercises at the end. The last part contains details on the software used for empirical testing of various microeconomic theories with some listing of seminal articles and popular text books.
1.2
Axioms and Consumption Set
Let X = (x1; x2; :::::xn ) be quantities of n commodities in nonnegative orthant of X 2 Rn The consumption set X ful…lls following properties. These concepts date back to Pareto (1896), Marshall (1890), Slutskey (1915), Hicks (1939), Samuelson (1947), Debreau (1959) and others. 1. 0 6= X
Rn
2. X is closed 3. X is convex 4. 0
X
Let B 2 X be a feasible set such that x % x for all x 2 B: Axioms of Consumer Choice 9
Axiom 1: Completeness If x1 and x2; are both in X , x1; x2
X either x1 % x2 or x1 - x2 . Consumer can compare.
Axiom 2: Transitivity For x1; x2 ; x3
X
if x1 % x2 ; x2 % x3 then x1 % x3 . Consumer is consistent.
Axiom 3: Continuity Preference relations % xi
- xi are closed in Rn
Axiom 4: Monotonicity For x0 Rn and for all ' > 0 there exist some x 2 B' (x0 ) Rn such that x > x0 . More is prefered for less. Axiom 5: Convexity If x1 % x0 then tx1 + (1 t) x0 % x0 for all t 2 [0; 1] . Demand and supply of the market system are based on above axioms. Think of a simple problem of households and …rms in a market economy.
1.3
Optimisation
Linear and non-linear programming are applied in order to …nd the optimal solution subject to constraints. Objectives like the utility or pro…t or social welfare can be function of one or several variables. Constraints can be one or multiple. Linear programming is applied where the objective functions and constraints are of the linear form and non-linear optimisation techniques is applied when objectives or the constraints are non-linear. By duality theorem every maximisation problem has a corresponding minimisation problem, such as utility maximisation corresponds to expenditure minimisation to achieve a certain level of utility, pro…t maximisation corresponds of cost minimisation given the type technology of production. 1.3.1
Consumer optimisation:
Assuming above axioms are satis…ed, the major objective of a consumer is to maximise utility by consuming x commodities (u (x)) max u(x)
(1)
subject to the budget constraint assuming prices (p) and income (y) as given: p:x
y
Constrained optimisation (Lagrangian function) with price): L (x; ) = u(x) + [y
10
(2) as a Lagrange multiplier (or shadow
p:x]
(3)
First order conditions wrt each xi : @L (x; ) @u(x) = @x1 @x1
p1 = 0
::
(5)
@L (x; ) @u(x) = @xn @xn y
(4)
pn = 0
p:x = 0
(6) (7)
@u(x) @xi
@u(x) @xi
> 0 and pi > 0 =) = pi > 0. Here there are n + 1 …rst order conditions to solve for demand for x1; x2; :::::xn goods and . The maximum utility is obtained when all these optimal values are substituted in the utility function u (x ) : Thus the marginal rate of substitution between xj and xi should equal their price ratios in equilibrium: M RSj;i =
@L(x; ) @xj @L(x; ) @xi
=
pj M Uj M Ui ; = pi pj pi
(8)
Marginal utility of xi represents gain from consumption xi and pi represents pain to the consumer. Equilibrium psychologically is thus a point where the gain equals pain. 1.3.2
Producer optimisation
The major objective of producers is to maximise pro…t. They take prices of commodities (p) and inputs (w) as given: Pro…t function is a value function for for all input price w 0 and output levels y 2 Rn+ (p; w) = p:y
w:x
(9)
subject to f (x)
y
(10)
Constrained optimisation (Lagrangian) function: L (x; ) = py
w:x + [y
@L (x; ) = w1 @x1
f (x)]
f 0 (x1 ) = 0
:: @L (x; ) = wn @xn 11
(11) (12) (13)
f 0 (xn ) = 0
(14)
y
@f (x;) @xj @f (x;) @xi
f (x) = 0;
=
wj wi
(15)
A complete view of microeconomic process requires thinking about the general equilibrium in p the market. The relative price system pji for i = 1; ::; N and j = 1; ::; N determines the optimal allocation of resources in the economy. Consider the following example for this purpose.
1.4
A simple computable general equilibrium model with labour leisure choice
Consider an economy with two individuals, i = 1; 2 and two commodities x (goods) and y (services). Both households are endowed with given amount of capital stock k1 ; k 2 and time L1 ; L2 , which they spend either working or in the form of leisure. Households and …rms optimise taking prices of commodities (px ; py ) and factors (pL ; pk ) as given. Competition between suppliers and consumers or producers sets the equilibrium price of commodities and income of households (I1 ; I2 ) More speci…cally the problems of households and …rms can be stated as: Household’s problem: _
max U1 = xa1 1 y1b1 L1
g1
LS1
; a1 > 0; b1 > 0; g1 > 0:
(16)
subject to: _
_
I1 = px x1 + py y1 + pL L1
LS1 ;
_
I1 = pL L1 + pk K 1
(17)
_
x1
0; y1
0; L1
LS1
0: _
max U2 = xa2 2 y2b2 L2
LS2
g2
; a2 > 0; b2 > 0; g2 > 0:
(18)
subject to: _
_
I2 = px x2 + py y2 + pL L2
LS2 ;
_
I2 = pL L2 + pk K 2
(19)
_
x2 0; y2 0; L2 Firm’s problem:
LS2
max
0:
x
= px x
pk kx
pL LS1x
pL LS2x
(20)
subject to: x = kxx LS1x1x LS2x2x max
y
= py y
pk ky
pL LS1y
(21) pL LS2y
(22)
subject to: y = kyy LS1y1y LS2y2y 12
(23)
Equilibrium conditions: x = x1 + x2
(24)
y = y1 + y2
(25)
kx + ky = k1 + k 2
(26)
_
L1 + LS1x + LS1y = L1 ;
LS1 = LS1x + LS1y
(27)
_
L2 + LS2x + LS2y = L2 :::LS2 = LS2x + LS2y
(28)
Price normalisation: px + py + pL + pk = 1
(29)
Questions 1) Derive demand for x and y and leisure (or labour supply) by households 1 and 2 i. e. determine x1 ; x2 ; y1 ; y2 ; LS1; LS2. 2) Determine the demand for labour and capital by …rms supplying x and y, i,e, evaluate kx ; ky ; LS1x ; LS1y ; LS2x ; LS2y : 3) Compute the equilibrium relative price system for this economy that are consistent to optimisation problems of households and …rms. 4) What are the optimal allocations of resources in this economy? 5) Evaluate the demand for x and y and leisure by both households and …nd the optimal levels of their welfare. Is this Pareto optimal allocation? 6) Suggest tax and transfer scheme in this economy in order to improve the distribution system. 7) Explain notions of Hicksian equivalent and compensating variations in order to evaluate the welfare consequences of tax and welfare reforms proposed above. 8) Write GAMS code to solve the model and few simulation scenarios for comparative static analysis. 9) Propose reforms in the labour and capital markets for improving the e¢ ciency of allocations in this economy. GAMS programe: ge2by2.gms Bhattarai K. and J. Whalley (2003) Discreteness and the Welfare Cost of Labour Supply Tax Distortions, International Economic Review 44:3:1117-1133, August Bridel (2011) for a non-technical introduction to the general equilibrium modelling. Now solve this CGE (computable general equilibrium) model using GAMS. First assign values for behavioural parameters. Secondly, download gams at www.gams.com and install in your PC or laptop. Thirdly write model equations for numerical optimisation routine in GAMS (see GAMS programme …le ge2by2.gms and its result …le ge2by2.lst)
13
a1 0.5
Table 1: Parameters for the 2 by 2 model with _leisure _ a2 b1 b2 g1 g2 1x 1y L1 L1 x y 0.4 0.3 0.4 0.2 0.2 0.2 0.8 0.4 0.1 24 24 Table 2: E¢ cient allocation in the 2 by I1 I2 x1 x2 y1 y2 3.42 2.01 5.9 2.8 1.7 1.3 px py pk pl kx ky 0.289 0.599 0.047 0.064 35.3 14.7 ls11 6.7
ls12 8.9
ls21 6.7
ls22 8.9
_
_
L1 24
L2 24
k1 40
k2 10
2 model with leisure x y u1 u2 8.7 3.1 4.6 2.4 L1 L2 ls1 ls2 10.7 6.3 13.3 17.7 k1 40
k2 10
Fourth study the solution of the model systematically: _ _ _ _ I1 = pL L1 + pk K 1 (=0.064*24+0.047*40)=3.42; I2 = pL L2 + pk K 2 (= 0:064 24 + 0:047 10) = 2:01: Fifth check the equilibrium conditions; check that all equilibrium conditions are satis…ed: x = x1 + x2 = 5:9 + 2:8 = 8:7
(30)
y = y1 + y2 = 1:7 + 1:3 = 3:1
(31)
kx + ky = 35:3 + 14:7 = k1 + k 2 = 44 + 10 = 50
(32)
_
L1 + LS1x + LS1y = 10:7 + 6:7 + 6:7 = 24 = L1 ;
LS1 = LS1x + LS1y
(33)
_
L2 + LS2x + LS2y = 6:3 + 8:9 + 8:9 = 24 = L2 :::LS2 = LS2x + LS2y
(34)
Sixth, consider tax policy analysis a) introducing VAT in commodities x and y b) introducing taxes in labour and capital inputs c) set a revenue target and do equal yield tax reforms …nding model solution when all taxes are i) raised in VAT or ii) by labour income tax or iii) capital income tax or iv) equally by from these sources. Seventh, compute the optimal tax rates that maximise revenue (hint make tax rates endogenous and solve the maximisation routine). The reader is expected to study the real analysis section in the appendix at this point. It is important to understand these basic concepts in order to follow literature on economic theory or to develop concepts in economic theory.
1.5 1.5.1
Methods for constructing Proofs Direct method
a = b; b = c =) a = c: or a = b; c = d =) a:c = b:d For x; y; z 2 R prove that x + z = y + z =) x = y 14
1.5.2
Converse and contrapositive
A implies B A =) B if it converse B =) A is true then A () B here A and B are equivalent. If A person lives in Hull (A) then that person lives in Yorkshire (B). A =) B but converse is not true in this case B ; A and A < B Contrapositive implies not A implies not B A =) B 1.5.3
Equivalence
A () B Example Pythagorus theorem: h2 = p2 + b2 p2 2 b2 h2 + cos2 = 1 h2 = h2 + h2 () sin Revealed preference theory is equivalent to utility maximisation theory in deriving income and substitution e¤ects. 1.5.4
Mathematical induction
Example: Sum of the N natural numbers is : P (n) = 1 + 2 + 3 + :::: + n = n(n+1) 2 Check if this works for any integer k P (k) = 1 + 2 + 3 + :::: + k = k(k+1) ; now prove that P (k + 1) = (k+1)(k+2) : 2 2 Add and subtract k + 1 from both sides 1 + 2 + 3 + :::: + k + (k + 1) = k(k+1) + (k + 1) = (k + 1) k2 + 1 = (k+1)(k+2) =) P (k + 1) 2 2 Thus by mathematical induction P (k) and P (k + 1) are similar.
2
L2: Consumption: Properties of a Utility Function
Maximising the level of utility (satisfaction) from consumption of goods and services is the ultimate objective of all economic activities. Various speci…cations of utility functions are used to represent the level of welfare of households from consuming goods and services and leisure in an economy. From abstract functions to linear and non-linear utility functions are popular in the literature. The Cobb-Douglas and constant elasticity of substitution (CES) utility functions are very popular in the literature. There are also nested utility function for instance in a general equilibrium models with many goods one can consider of three levels of nests to capture the intra- period and inter temporal substitution between consumption and leisure based on relative prices and wage rates in the economy. The …rst level of nest aggregates the goods and services in composite consumption good, then the second level nest aggregates these composite goods with leisure. Then there is the nest of time separable utility functions to arrive at the life time utility for each household. The consumption shares of various goods are calibrated from the benchmark dataset (Blundell (2014), Deaton, and Muellbauer (1980), Barker, Blundell and Micklewright (1989) for more in depth study on demand side parameters of household demand functions). These utility functions are further modi…ed in order to represent positive or negative externalities in consumption. While recreation facilities in the neighbourhood generates positive externalities but pollutions reduce utility levles of households. In dynamic setting most studies apply the time separable utility functions. Aim of this section is to present popular models used in analysing preferences and demands of households for various commodities. It shows how to evaluate the welfare as well as the price and substitution
15
e¤ects of changes in prices and the elasticities of demand. Basics of revealed preference theory is reviewed at the end. Microeconomic theories are often tested econometrically to ascertain their validities (Houthakker (1950), Richter (1966), Afriat (1967), Kahneman and Tversky (1979),Varian (1982), McGuinness (1980), Bandyopadhyay (1988), Hey and Orme (1994), Lee and Singh (1994), Carey (2000), Deolalikar and Evenson (1989) Van Soest and Kooreman (1987), Blundell, and Preston (2008), Echinique (2011), Varian (2012), Vermeulen (2012), Schmeidler, D. (1989), Blundell (2014).). Where preference relations are complete, transitive, continuous, monotonous and convex then there exists a real valued utility function u : Rn+ =) R and this utility function has following properties: u (x) is strictly increasing in x if and only if % is strictly monotonic. u (x) is quasi-concave if and only if % is convex. u (x) is strictly quasi-concave if and only if % is strictly convex.
2.1 v:
Rn+
Properties of an indirect utility function =) R v (p; y) = max u(x) s.t. p:x
y
X 2 Rn
1. Continuous 2. Homegenous of degree zero in (p; y) 3. Strictly increasing in y 4. Decreasing in p 5. Quasiconvex in p and y. 6. Roy’s identity
0
xi p ; y
0
=
@v (p0 ;y 0 ) @pi @v(p0 ;y 0 ) @y
::::i = 1::m
(35)
Numerical Example: Derive demands for (x1 , x2 ) from ratios of marginal utilities (partial derivative of utility functions) given the prices [(p1 , p2 ) = (2; 4)] and income (a). max u = x1 x2 subject to 2x1 + 4x2 = a L (x1 ; x2 ) = x1 x2 + [a
2x1
4x2 ]
(36) (37)
@L (x1 ; x2 ) = x2 @x1
2 =0
(38)
@L (x1 ; x2 ) = x1 @x2
4 =0
(39)
16
From the
@L (x1 ; x2 ) = a 2x1 4x2 = 0 @ = 2: Then put this into the last FOC to get: x1 =
…rst two FOCs xx21 2 = x1 x2 = a4 a8 = a32 :
(40) a 4;
x2 =
a 8 ;,
a = 16 =) u By an envelop theorem evaluating the indirect utility function @L(x1 ;x2 ) a and Lagrange multiplier at the optimal solution: @u = : QED. If consumer @a = 16 = @a a 200 income a = 200 then x1 = a4 = 200 25 = 1250: 4 = 50; x2 = 8 = 8 = 25: Then u = x1 x2 = 50
2.1.1
Price elasticities and the elasticity of substitution
Now if p1 changes to 4 and p2 to 2 what will be elasticities, cross elasticities and elasticities of substition between x1 and x2 ? a 200 New demands: x1 = a8 = 200 8 = 25; x2 = 4 = 4 = 50: Utility is still 1250. Price elasticity of demand e1 =
dx1 =x1 dx1 p1 = = dp1 =p1 dp1 x1
25 2 = 2 50
0:5
(41)
1
(42)
e2 =
dx2 =x2 dx2 p2 25 4 = = = dp2 =p2 dp2 x2 2 50
e1 =
dx1 =x1 dx1 p2 = = dp2 =p2 dp2 x1
25 4 =1 2 50
(43)
e2 =
dx2 p1 25 1 dx2 =x2 = = = 0:5 dp1 =p1 dp1 x2 2 25
(44)
Cross price elasticity:
Elasticity of substituion bewteen x1 and x2 d =
x1 x2
=
p2 p1
=
x1 x2
=
25 50 2 4
= =
50 25 4 2
=1
(45)
e1;a =
dx1 =x1 dx1 a 1 200 = = =1 da=a da x1 4 50
(46)
e2;a =
dx2 =x2 dx2 a 1 200 = = =1 da=a da x2 8 25
(47)
e1;a =
dx1 =x1 dx1 a 1 200 = = =1 da=a da x1 8 25
(48)
e2;a =
dx2 =x2 dx2 a 1 200 = = =1 da=a da x2 4 50
(49)
d Income elasticities of demand
p2 p1
Before the change in price:
After the change in price:
17
2.1.2
Consumer optimisation model: a numerical example M ax U = X10:4 X20:6
(50)
Subject to p1 :X1 + p2 :X2 = 150
(51)
Lagrangian optimisation: L (X1 ; X2 ; ) = X10:4 X 0:6 + [150
p1 :X1
p2 :X2 ]
(52)
For base equilibrium assume that p1 = 3 and p2 = 2: Optimal demand for goods X1 X1 =
0:4 (150) 60 = = 20; p1 3
X2 = 0:4
U0 = X10:4 X20:6 = (20)
0:6 (150) 90 = = 45 p2 2 0:6
(45)
= 32:53
(53) (54)
Now assume that there is a subsidy in X1 of £ 1 and price reduces from 3 to 2; p1 = 2: Equivalent Variation What is the Hicksian Equivalent and compensating variations of price change? What are the income and substitution e¤ects of this price change? First …nd out how much money is required at new prices to guarantee the original utility by solving U0 = U0 =
0:4 (m0 ) 2
0:4
0:4 (m0 ) 2 0:4
0:6 (m0 ) 2
0:6
0:6 (m0 ) 2 0:6
; m0 =
= 32:53 2 (32:53) = 127:49 0:40:4 0:60:6
(55)
(56)
Equivalent variation (money to be taken away when prices fall) EV = 150
127 = 22:51
(57)
Compensating Variation For compensating variation …rst compute the demand in new prices and utility X1 =
0:4 (150) 60 = = 30; p1 2
X2 = 0:4
U1 = X10:4 X20:6 = (30) 0
U1 = @
0:4 m 3
00
10:4 A
18
0:6 (150) 90 = = 45 p2 2 0:6
(45)
0:6 (m00 ) 2
= 38:26
(58)
(59)
0:6
= 38:26
(60)
0
m0=
(38:26) 30:4 20:6 = 176:39 0:40:4 0:60:6
CV = 150
176:39 =
(61)
26:39
(62)
Summarising the Money Metric Utility Changes Due to Taxes Table 3: Summary Fall in Price Rise + -
EV CV
of Equivalent and Compensating Variation in Price Fall in Price Basis of evaluation 22.51 New Price-Old Utility + -26.39 OLD Price- New Utility
Substitution E¤ect : 2.5 =10-7.6; Income e¤ect:7.6=22.5/3 and total e¤ect: 10. 2.1.3
Econometric issues in estimation of demand functions and elasticities of demand
Measure of elasticity di¤ers by the functional forms used to estimate it. With data on quantiy (Y ) and price (X) ; in brief these can be stated as follows: Elasticity around the mean values of X and Y in a linear regression model , Yi = 1 + 2 Xi +ei is @Yi X @Yi de…ned as e = @X = 2 is it obtained as e = 2 X Then given estimate of the slope @X . In a log i Y i Y @Yi X dependent variable linear regression model of the form ln (Yi ) = 1 + 2 Xi +ei e = @X Y = 2X i Y @Yi 1 because @Xi Yi = 2 : Similarly elasticity in a log explanatory variable linear regression model: @Yi X @Yi 1 1 Yi = 1 + 2 ln (Xi ) + ei is given by e = @X = 2 X1i X Y = 2 Y * @Xi = 2 Xi . Then elasticity i Y @Yi X in a double log linear regression model, ln (Yi ) = 1 + 2 ln (Xi ) + ei is e = @Xi Y = 2 XYi X Y = @Yi 1 1 Elasticity in a regression model linear in reciprocal of an explanatory 2 Xi . 2 * @Xi Yi = @Yi X @Yi 1 variable, Yi = 1 + 2 X1i + ei is given by e = @X = 2 X1i Y1i * @X = 2 X 2 : In a a quadratic i Y i i
@Yi X regression model, Yi = 1 + 2 Xi + 3 Xi2 + ei the elasticity is e = @X = ( 2 + 2 3 Xi ) X Y i Y @Yi 2 = + 2 X : How to decide which one these two choose? First, should depend on the * @X 2 3 i i optimisation functions discussed in this chapter. Secondly choice between linear and log-linear models should be econometric tests such as MacKinnon, White and Davidson test.
2.1.4
Restrictions in estimating a demand function:
Suppose that you are interested in estimating the demand for beer in a country and consider the following multiple regression model: ln (Yi ) =
0
+
1
ln (X1;i ) +
2
ln (X2;i ) +
3
ln (X3;i ) +
4
ln (X4;i ) + 'i
i = 1 :::N
(63)
where Yi is the demand for beer, X1;i is the price of beer, X2;i is the price of other liquor products, X3;i is the price of food and other services, X4;i is consumer income. Coe¢ cients 0 , 1 , 2 , 3 ,and 4 are the set of unknown elasticity coe¢ cients you would like to estimate. Again assume that errors 'i are independently normally distributed, 'i N (0; 2 ). Given non-sample information on the relation between the price and income coe¢ cients as following:
19
1. (a)
i. sum of the elasticities equals zero: 1 + 2 + 3 + 4 = 0: ii. two cross elasticities are equal: 3 = 4 = 0 or 3 - 4 = 0 iii. income elasticity is equal to unity: 5 = 1
F-test can be applied to test the validity of such restrictions as: F =
(Rb
0
r) [Rcov (b) R0 ] J
1
(Rb
r)
(64)
Here J = 3 is the number of restrictions 2
1 R=4 0 0
0 1 0
2 3 2 3 3 b 0 0 1 6 7 0 5 ; b = 4 b2 5 ; r = 4 0 5 b 1 0 3
(65)
Thus empirical test of consumer behaviour whether purchase of x1 and x2 are proportionate to the changes in the level of income or prices are measures by income and price elasticities of demand. These are empirically estimated using the cross section and time series data on quantities and prices. These data can be obtained from various organisations1 Utility e¤ect of price changes will be higher for the commodity that is heavily weighted in the consumer’s consumption basket. In real life households vary by their income and have good varieties in consumption bundles as: Table 4: Consumption of households by sectors in UK, 2008 Deciles agri Prod Constr Dist Infcom Finins Rlest Prfspp Ghlthed H1 435 10520 201 3688 1071 1541 3915 432 1835 H2 671 16224 310 5688 1651 2377 6037 666 2831 H3 854 20663 395 7244 2103 3027 7689 848 3605 H4 1037 25073 479 8790 2552 3673 9330 1029 4375 H5 1223 29583 565 10372 3011 4334 11008 1214 5161 H6 1407 34031 650 11931 3463 4985 12663 1397 5937 H7 1676 40532 775 14210 4125 5938 15083 1663 7072 H8 1977 47829 914 16769 4868 7007 17798 1963 8345 H9 2358 57037 1090 19997 5805 8355 21224 2341 9951 H10 3864 93463 1786 32767 9512 13692 34779 3835 16307 Note: Constructed from the ONS data.
Othrsrv 1448 2233 2844 3451 4071 4683 5578 6582 7850 12863
Rich households with more income can consumer more goods and services and enjoy more utility than poor households. Governments apply commodity taxes (VAT of 20%) and income taxes in order transfer some income from the richer to poorer households. Such transfer may reduce the gap between the income of rich and poor but it is very di¢ cult to imagine a society with perfect equality. 1 Such as Food and Agriculture Organisation: http://faostat.fao.org/ or from the Department for Environment, Food & Rural A¤airsas https://www.gov.uk/government/statistical-data-sets/commodity-prices. In general consult government department web pages to …nd such data at https://www.gov.uk/government/organisations or for many other links in http://www.hull.ac.uk/php/ecskrb/Confer/research.html.
20
Table 5: Benchmark production tax and prices by sectors Deciles Leisure Consumption Income share Consshare Income tax rate H1 2577 38163 0.0281 0.0627 0.0 H2 7451 52401 0.0433 0.0552 0.32 H3 14230 66740 0.0551 0.0624 0.32 H4 21877 80983 0.0669 0.0850 0.32 H5 28269 95550 0.0789 0.0966 0.32 H6 35535 109917 0.0908 0.1067 0.32 H7 41156 130916 0.1081 0.1078 0.32 H8 46294 154484 0.1276 0.1323 0.32 H9 54041 178551 0.1521 0.1409 0.40 H10 73363 292582 0.2493 0.1945 0.50 Note: Constructed from the ONS data. More detailed estimates of price, income and cross elasticities of demand can be estimated from the survey data such as food and expenditure survey, travel and tourism survey, multiple household survey including the understanding society dataset that can be obtained from the data archive or could be constructed from the ONS. McFadden Daniel (1963) Constant Elasticity of Substitution Production Functions, Review of Economic Studies, 30, 2, 73-83 McFadden Daniel and Paul A. Ruud (1994) Estimation by Simulation, Review of Economics and Statistics, 76, 4 , 591-608 Stone R (1954) “Linear Expenditure System and Demand Analysis: An Application to the Pattern of British Demand”, Economic Journal 64:511-527.
2.2
Exercise 1: Consumer’s problem
Q1. Consider a utility maximisation problem of a consumer with the CES utility function on goods x1 and x2 : max u = (x1 + x2 )
1
x1; x2;
(66)
subject to the budget constraint with prices p1 and p2 and income y: p1 :x1 + p2 :x2 = y
(67)
Derive Marshallian demand functions x1 and x2 for and indirect utility function u x1; x2; . Prove that the v(p; y) is homegenous of degree zero in p and y. prove that it is increasing in y and decreasing in p. Prove Roy’s identity for this problem. 21
Q2. Show above properties in the following CES utility maximisation problem: max u = (x1 + x2 )
1
(68)
x1 ;x2
Subject to M = p1 x1 + p2 x2 2.2.1
(69)
Marshallian demand functions from the CES preferences: 1
L (x1 ; x2 ; ) = (x1 + x2 ) + [M @L 1 = (x1 + x2 ) @x1
1
1 @L 1 = (x1 + x2 ) @x2
p1 x1
p2 x2 ]
(70)
1
x1
1
p1 = 0
(71)
1
x2
1
p2 = 0
(72)
@L = M p1 x1 p2 x2 = 0 @ The marginal rate of substitution bewteen x1 and x2 1
x1 x2
=
M = p1 x2
p1 p2
1 1
p1 p2
(74)
1
p1 p2
x1 = x2
(73)
1
(75)
M = p1 x1 + p2 x2 ' # 1 h 1 p1 + p2 x2 = x2 p1 + p 2 = x2 p 1 p2
(76) 1
+ p2
1
i
1
p2
1
(77)
Properties of the CES demand functions 1
x2 = h
Now get the value of
p1
1
+ p2
1
M p2
Value function:
x1 = h p1
20
1
1
+ p2
1
i
1
1
M p2
1
i 1
1
p1 p2
1
=h 1
(78)
0
M p1 p1
1
1
+ p2
1
1
i 1 31
1 1 B M p2 6B M p1 C C 7 v (x1 ; x2 ) = [email protected] h iA + @h iA 5 1 1 1 1 p1 + p2 p1 + p2
22
(79)
(80)
If M and p1 and p2 increase by t it does not change the value function: v (x1 (tM; tp1 ; tp2 ) ; x2 (tM; tp1 ; tp2 )) = v (x1 ; x2 ) : @v(x1 ;x2 ) @v(x1 ;x2 ) @v(x1 ;x2 ) < 0 and < 0. > 0 and @M @p1 @p2 Roy’s Identity: @v (p0 ;M 0 ) @Pi @v(p0 ;y 0 ) @M
xi p0 ; M 0 =
::::i = 1::m
(81)
(Marshallian demand for xi equal negative of the ratio derivative of IUF wrt price and income). Note that the derivative of value function wrt price equals derivative of Lagranging function wrt price and this equals negative of lagrange multiplier times the demand for the product as: @L @V = = @pi @pi 2.2.2
x1 (p1 ; p2 ; m):
(82)
Compensated and uncompensated demands
Consumer’s primal problem is to maximise utility (U ) subject to budget constraints. When optimal demands for xi are substituted in the utility function it becomes indirect utility function (V ). By Roy’s identity Marshallian demand for xi equals the negavite of the ratio of the …rst derivative of V wrt pi to its …rst derivative wrt income (M ). Consumer’s dual problem is to minimise the expenditure (E) wrt a target utility U . When optimal values of xi are substituted in E it becomes an expenditure function. The …rst derivative of the expenditure function wrt pi is equal to the compensated demand function xi : This means given E(p1 ; p2 ; U ) compesated demand is 1 ;p2 ;U ) xci = @E([email protected] : While the compensated demand gives the pure substitution e¤ect of price change i and the Marshallian demand minus the compensated demand equals the income e¤ect of price change. Inverse of indirect utility function is the expenditure function. 2.2.3
Expenditure functions with the CES utility functions min E = p1 x1 + p2 x2
(83)
x1 ;x2
Subject to u = (x1 + x2 ) L (x1 ; x2 ; ) = p1 x1 + p2 x2 + @L = p1 @x1
1
@L = p2 @x2
1
(x1 + x2 ) (x1 + x2 )
1
h
(84) u
(x1 + x2 )
1
i
(85)
1
1
x1
1
=0
(86)
1
1
x2
1
=0
(87)
Expenditure Functions with the CES utility functions @L =u @
1
(x1 + x2 ) = 0
23
(88)
p1 = p2
'
1
x2 = u
'
p1 p2
1
1
+1
#
1
1
1
+ x2 h = u p1
#1
1
1 1
= x2 p1
p1 p2
u = (x1 + x2 ) = x2
(89)
1
p1 p2
x1 = x2
1
x1 x2
1
p2
= x2
1
+ p2
(90)
'
p1 p2
i
1
1
+1
( p2
1
)(
1
#1
(91)
)
Expenditure Functions with the CES utility functions h x2 = u p 1
1
+ p2
1
i
1
1
1
p2
=h p1
Putting x2 in x1 1
x1 = x2 p 1
1 1
1
p2
h x1 = u p 1
2.3
1
h = u p1
+ p2
1
i
1
+ p2
1
1
i
p1
1
+ p2
1
1
p2
1
1 1
p1
(92)
i1 1
1
p2
1
(93)
1
1 1
1
u p2
1
=h p1
u p1 1
1
+ p2
1
i1
(94)
Slutskey equation ( Decomposition of substituion and income effects): Duality on consumer optimisation L = p1 x1 + p1 x2 +
h
1
U
1 21 1 12 @L = p1 x x2 = 0 @x1 2 1 1 21 12 1 @L = p2 x x =0 @x2 2 1 2 1 1 @L = u x12 x22 = 0 @ p1 x2 p2 = > x1 = x2 p2 x1 p1 1 2
1 2
u = x1 x2 =
x2 =
1 2
p2 x2 p1 p1 p2
1 2
x2 =
1 2
1
u = up12 p2 24
1
x12 x22
p2 p1 1 2
i
(95) (96) (97) (98) (99)
1 2
x2
(100)
(101)
Then x1 = x2
p2 = p1
1 2
p1 p2
u
1 1 p2 = up1 2 p22 p1
(102)
Now the expenditure fucntion 1
1
1
E = p1 x1 + p2 x2 = p1 up1 2 p22 + p2 up12 p2
1 2
1
1
= 2up12 p22
(103)
E
u=
1
(104)
1
2p12 p22 Slutskey Equation: Total e¤ect of price change = (substituion e¤ect + income e¤ect) @x1 = @p1
@x1 @p1
Cmp
@E @x1 @p1 @E
(105)
Compensated demand
1 2
1 2
x1 = up1 p2 =)
@x1 @p1
3 1 1 up1 2 p22 = 2
= cmp 1
1
E = 2up12 p22 =) Given the Marshalian demand x1 =
1 2
E 1 2
1 2
2p1 p2
!
3
1
p1 2 p22 =
1 Ep 2 4 1
(106)
1 1 @E = up1 2 p22 = x1 @p1
(107)
E 2p1
@x1 1 = @E 2p1
(108)
Slutskey decomposition: @x1 @p1
= =
@x1 @p1
Cmp
1 Ep 2 4 1
@E @x1 1 = Ep 2 @p1 @E 4 1 ! 1 1 E 1 p1 2 p22 1 1 2p1 2p12 p22
1
1
up1 2 p22 =
1 2p1
1 Ep 2 4 1
1 E Ep1 2 = 2 4 2p1
(109)
First part is substitution e¤ect and the second part is income e¤ect. If E = 800; p1 = 4 E 800 800 E substitution e¢ ect is - 4p 12:5 and the income e¤ect is also - 4p 2 = 2 42 = 4 4 4 = 2 = 800 4 4 4
1
1
800 2 42
=
= 12:5 . Both reinforce each other and total e¤ect is -25. Blundell R (2014) Income Dynamics and Life-cycle Inequality: Mechanisms and Controversies, Economic Journal, 124, 576, 289–318 25
Jehle G A and P.J. Reny (2005) Advanced Microeconomic Theory, Pearson Education. M. Hoy, J Livernois, C McKenna, R Rees and T. Stengos (2001) Mathematics for Economics, 2nd ed., MIT Press. 2.3.1
Comparative static analyis with matrix
Consider a consumer maximisation problem given below: M ax U (X; Y )
(110)
I = px X + py Y
(111)
X;Y
Subject to
Form a constrained optimisation problem and characterise the demand function X(px ; py ; I) and Y (px ; py ; I). L = U (X; Y ) + [I px X py Y ] (112) @L = I @
px X
@L = Ux @X
py Y = 0
(113)
px = 0
(114)
@L = UY py = 0 (115) @Y Following Henderson and Quandt (1980) take the total di¤erentiation of these FOCS. (Consumer takes prices of commodities as given, x and y are constant values of X and Y): 0:d
In matrix notation 2 4
0 px py
px dX
py dY = xdpx + ydpy
dI
(116)
px d + Uxx dX + Uxy dY = dpx
(117)
py d + Uyx dX + Uyy dY = dpY
(118)
px
py
Uxx Uyx
Uxy Uyy
32
3 2 d xdpx + ydpy 5 4 dX 5 = 4 dpx dY dpY
dI
3 5
(119)
Evaluating the impact of change in shadow price of income d on demand taking all else constant dpx = 0 and dpY = 0 2 32 3 2 3 0 px py d =dI 1 4 px Uxx Uxy 5 4 dX=dI 5 = 4 0 5 (120) py Uyx Uyy dY =dI 0 26
Similarly comparative static when only prices of X change dpx 6= 0 taking everything else is constant dpy = 0 and d = 0 2 32 3 2 3 0 px py d =dpx x 4 px Uxx Uxy 5 4 dX=dpx 5 = 4 5 (121) py Uyx Uyy dY =dpx 0 Similarly comparative static when only prices of Y change dpy 6= 0 taking everything else is constant dpx = 0 and d = 0 2 32 3 2 3 0 px py d =dpy y 4 px Uxx Uxy 5 4 dX=dpy 5 = 4 0 5 (122) py Uyx Uyy dY =dpy Each of these could be solved using the Cramer’s rule. For instance 2
3 2 d =dI 4 dX=dI 5 = 4 dY =dI
0 px py
px Uxx Uyx
py
3
Uxy 5 Uyy
1
2
3 1 4 0 5 0
(123)
Apply Cramer’s rule to …nd how much the shadow price and demand change in response to change in income; solve for d =dI; dX=dI; dY =dI
d =dI =
dX=dI =
dY =dI =
0 px py
1 px Uxx Uyx
0 px py
1 px Uxx Uyx
0 px py
1 px Uxx Uyx
py Uxy Uyy
py Uxy Uyy
py Uxy Uyy
1 0 0
px Uxx Uyx
py Uxy Uyy
=
2 Uxx Uyy + Ux;y 2px py Uxy p2y Uxx p2x Uyy
(124)
0 px py
1 0 0
py Uxy Uyy
=
py Ux;y px Ux;y 2px py Uxy p2y Uxx p2x Uyy
(125)
0 px py
px Uxx Uyx
1 0 0
=
px Ux;y py Ux;x 2px py Uxy p2y Uxx p2x Uyy
(126)
Once more to evaluate the comparative static impact of changes in prices of x, 2
3 2 d =dpx 4 dX=dpx 5 = 4 dY =dpx
0 px py
27
px Uxx Uyx
py
3
Uxy 5 Uyy
1
2 4
x 0
3 5
(127)
d =dpx =
0 px py
1 px Uxx Uyx
x py Uxy Uyy
0
px Uxx Uyx
py Uxy Uyy
=
2 xUxx Uy;y py Uy;x + px Uy;y: xUx;y 2px py Uxy p2y Uxx p2x Uyy
(128) Marshalian Demand Function dX=dpx
=
=
0 px py
1 px Uxx Uyx
xpy Ux;y 2px py Uxy
py Uxy Uyy
0 px py
x 0
py Uxy Uyy
p2y + xpx Uy;y: p2y Uxx p2x Uyy
(129)
(130)
Analysis of signs: Give that px > 0, py > 0, Ux;x: < 0, Uy;y: < 0; dpx > 0, dpy > 0„ the denomenator (determinant) is positive. Whether the numerator is positive depends on the sign of the numerator terms which denotes income and substitution e¤ects of changes in prices of X. Here dX=dpx < 0 because each term in numerator is negative and xpy Ux;y p2y + xpx Uy;y: < 0.
dY =dpx
=
=
0 1 px 0 px py py px Uxx Uxy py Uyx Uyy py px xpx Uyx + xpy Ux;x: 2px py Uxy p2y Uxx p2x Uyy
px Uxx Uyx
x (131) 0
(132)
This is exactly what would be expected, the inverse relation between demand for X and its own price. However the impact of a change in px on the demand for Y is not predictable o¤hand because the …rst term in the numerator, py px xpx U + xpy Ux;x , is positive but the last two terms are negative. Thus dpx would have positive impact only if py px > xpx U + xpy Ux;x . xpy Ux;y
p2 +xpx Uy;y:
Here 2px py Uxy p2yUxx p2 Uyy is the total e¤ect of change in prices. It can be decomposed into y x income and substitution e¤ect by deriving the compensated demand Hicksian demand function (dX=dpx )compensated =
2px py Uxy
xUxx p2y Uxx
p2x Uyy
x = y
1
1
1
1
)
(225)
+ (1
)
+ (1
) py
1
i1
(231)
u
y=
(1
1
)y ]
[ x + (1
(1
y
)y ]
[ x + (1
x y
1
[ x + (1
[ x + (1
@L =U @ px = py (1
U
)y ]
1
h
1
(1
)
1 1
1
px
1
+ (1
) py
1
(232)
i1
u
1
=
1
py
1
h
1
(1
)
Now it is possible to apply the Slutskey equation @x1 @x1 @E @x1 @p1 = @p1 @p1 @E Cmp
37
1 1
px
(1 1
+ (1
) py
1
i1
)
1 1
px py
(233)
1 1
2.4.8
Problem 4: CES Demand
1. Consider a consumer maximisation problem given below: M ax u(x; y) = [ x + (1
)y ]
x;y
1
(234)
Subject to x + py y = m Note that the elasticity of substitution and
(235)
are linked as:
=1
1
(a) Formulate a constrained optimisation problem . (b) Determine the demand functions of x and y. (c) Calibrate the share parameter . (d) Derive the indirect utility function (value function) u(x(p1 ; py ; m); y(p1 ; py ; m)) = V (p1 ; py ; m) (e) What is the meaning of
@V @m
=
? Marginal utility of income?
(f) Explain the meaning of Roy’s identity y(p1 ; py ; m).
@V @px
=
@L @px
=
x(p1 ; py ; m) and
@V @py
=
@L @py
=
2. Consider standard properties of a utility function (a) u(xi = 0) = 0 N (b) It continuous in R++
R+
(c) unbounded above for all prices p
0
(d) Homogenous of degree 1 in xi (e) Strictly increasing in income (f) Decreasing in prices (g) Quasiconvex in (p; m) (h) Ful…lls Roy’s identity Show above properties in the following CES utility maximisation problem max u = (x1 + x2 )
1
x1 ;x2
(236)
Subject to M = p1 x1 + p2 x2 3. Consider standard expenditure function with following properties (a) e = 0 for u(xi = 0) = 0 N (b) It continuos in R++
R+ 38
(237)
(c) unbounded above in u for all prices p
0
(d) Homogenous of degree 1 in p (e) Strictly increasing in income (f) Concave in p (g) ful…ls Shephard’s lemma
@E @pi
=
@L @pi
= xi (p1 ; p2 ; m)
for i = 1; 2
Show above properties in the following CES utility maximisation problem Subject to min E = p1 x1 + p2 x2 x1 ;x2
(238)
Subject to u = (x1 + x2 ) 2.4.9
1
(239)
Extra example on Shephard’s Lemma and Roy’s identity
1. Utility function for a consumer is given by u = x1 x2
(240)
I = p1 x1 + p2 x2
(241)
here budget constraint is
1. What are the Marshallian (uncompensated) demand functions for X and Y? 2. Determine the indirect utility function for this consumer. 3. Solving corresponding duality problem determine the expenditure function for this consumer. 4. Find the compensated (Hicksian) demand curve for X or Y? [hint Slutskey equation]. 5. Prove Shephard’s lemma
@E @pi
@L @pi
= xi (p1 ; p2 ; m) . i @V @L 6. Prove Roy’s identity for this case @p = @pi : i Answer Consumer’s Optimisation
=
h
L = x1 x2 + [m @L = x1 @x1
1
p2 x2 ]
(242)
x2
p1 = 0
(243)
1
p2 = 0
(244)
p 2 x2 = 0
(245)
@L = x1 x2 @x2 @L =m @
p 1 x1
p1 x1
39
Marshallian demand functions x1 = 2.4.10
Indierct utility function m p2
m p1
V (x1 ; x2 ) = 2.4.11
m m ; x2 = p1 p2
(246)
p1 p2
Expenditure function m=
p1 p2 u
@V @L = = @pi @pi
2.4.12
=m
(247)
x1 (p1 ; p2 ; m):
(248)
Duality L = p1 x1 + p1 x2 +
h
1
1
U
x12 x22
1 21 1 12 @L x = p1 x2 = 0 @x1 2 1 @L 1 21 12 1 x x = p2 =0 @x2 2 1 2 1 1 @L = u x12 x22 = 0 @ p1 x2 p2 = > x1 = x2 p2 x1 p1 1
1
u = x12 x22 = x2 =
x2
1 2
p2 p1
1
x22 =
1 2
p1 p2
1
u = up12 p2
p2 p1
i
(249) (250) (251) (252) (253)
1 2
x2
(254)
1 2
(255)
Then x1 = x2
p2 = p1
1 2
p1 p2
u
1 1 p2 = up1 2 p22 p1
(256)
Now the expenditure fucntion 1
1
1
E = p1 x1 + p2 x2 = p1 up1 2 p22 + p2 up12 p2 E
u=
1
1
2p12 p22 40
1 2
1
1
= 2up12 p22
(257) (258)
Slutskey Equation: Total e¤ect of price change = substituion e¤ect and income e¤ect @x1 = @p1
@x1 @p1
@E @x1 @p1 @E
Cmp
(259)
Compensated demand
1 2
1 2
x1 = up1 p2 =)
@x1 @p1
1 3 1 up1 2 p22 = 2
= cmp 1
1
E = 2up12 p22 =) Given the Marshalian demand x1 =
1 2
E 1 2
1 2
2p1 p2
!
3
1
p1 2 p22 =
1 Ep 2 4 1
1 1 @E = up1 2 p22 = x1 @p1
(260)
(261)
E 2p1
1 @x1 = @E 2p1
(262)
Slutskey decomposition: @x1 @p1
= =
@x1 @p1
Cmp
1 Ep 2 4 1
1 @E @x1 = Ep 2 @p1 @E 4 1 ! 1 1 E 1 p1 2 p22 1 1 2p 2 2 1 2p1 p2
1
1
up1 2 p22 =
1 2p1
1 Ep 2 4 1
1 E Ep 2 = 2 4 1 2p1
(263)
First part is substitution e¤ect and the second part is income e¤ect. 2.4.13 @E @pi
=
Shephard’s Lemma
@L @pi
= xi (p1 ; p2 ; m)
u(x1 ; x2 ) = x1 x2 where
+
= 1: L = p 1 x1 + p 2 x2 +
@L @u
h
u
x1 x2
i
(264)
= @L = p1 @x1
x1
@L = p2 @x2
x1 x2
@L =u @ p1 = p2
x2 = 0 1
=0
x1 x2 = 0 1
x2
x1 x2
1
x1
1
41
=
x2 x1
(265) (266) (267) (268)
Shephard’s Lemma p1 x1 = x2 p2 p2 x1 = x2 p1 p2 x2 p1
u = x1 x2 =
x1 =
p1 p2 x2
(272)
p1 p2 u =
1+
1
p1 p12 p1 p12
p1 p12
e=
1
u
p1 p12
e= 1
p1 p12
u
(276)
+1
(277)
+
u u
(274) (275)
u
p1 p12
u +
(273)
p1 p2 u
p1 p12
p2 u +
e=
e=
p1 p2 u
p1 p2 u + p2 p11
e=
(271)
p1 p2 u
e = p1 x1 + p2 x2 = p1
e=
(270)
x2 =
x2 = p2 p1
(269)
(278)
1
(279)
u=
p1 p2 u
(280)
Proof of Shephard’s Lemma @E @L = = xi (p1 ; p2 ; m) @pi @pi e=
p1 p2 u =)
@e = @p1
1
L = p1 x1 + p2 x2 + @L = x1 (p1 ; p2 ; m) = @p1 QED
42
p1 h
u
1
(281)
p2 u =
x1 x2
i
p1 p2 u
p1 p2 u
(282) (283) (284)
2.4.14
Roy’s Identity @V @p1
=
m m p1
= x1
1
p1
1
p2
=
=
m p1
p1 p2
x1 (p1 ; p2 ; m)
(285)
x2 = p 1 :
m p1
1
m p2
p1 1
=
x1 1 x2 = x1 p1
=
p1 p1
This is the proof of Roy’s identity.
43
1
x2 p 1 1 = (286)
2.5
Revealed Preference Theory
Utility function based analysis on derivation of consumer demand is subjective and hence less precise. Neither the utility function nor the preference parameters are observable. Revealed prefernce theory focuses on observed income, price and choices to generalise axioms on consumer behaviour. It makes a number of assumptions: 1) all income is spent (M0 = p0 x0 ) 2) buyers always have a unique x bundlde for given M and p; that means there is a unique bundle for each combination of p and M and preferences are consistent. 1 Weak axiom of revealed preference (WARP) : draw a diagram. 0
1
1
p1 x0 > p1 x1 () p0 x0 > p x1 p x1 < p x0 This will give the substitution and income e¤ect as in the indi¤erence curve analysis. But this does not apply in all circumstances; p1 x0 = M2 = p1 x2 . This leads to p1 p0 x1 x0 < 0 X and in general p1i p0i x1i x0i = p1j p0j x1j x0j < 0. 2. Strong axiom of revealed preference (SARP) 3. Congruent axiom 4. Generalised axiom The WARP and GARP axioms were re…need and generalised by Houthakker (1950), Richter (1966), Afriat (1967) Varian (1982), Bandyopadhyay (1988) and most recently by Echinique (2011). The Economic Journal 2012 has a special section on the recent developments of the revealed preference theory (Varian (2012), Vermeulen (2012)). 2.5.1
Slutsky equation from the Revealed Preference Theory
Following Gravelle and Rees (2004) now derive the Slutskey equation from the Revealed preference theory as follows:
Divide by
x1j
x0j = x2j
x1j
x0j
x0j + x2
x0j
x0j
x0j
pj
pj
=
x2j
pj
+
x2
pj
or x1j
x0j pj
=
xj jM = pj
x2j
x0j pj
xj jpx pj
x0j x0j
x2
x0j M
xj jp M
Thus the utility maximising theory and the revealed preference theory of consumer behavior are equivalent. Revealed preference to Laspeyeres price index: p1 x1 > p1 x0 () M I =
44
p1 x1 p 1 x0 > 0 0 = LP 0 0 p x p x
Revealed preference to Paasche price index: p0 x0 > p0 x1 ()
1 1 p 1 x1 p1 x1 5 () M I = 5 = PP p 0 x0 p0 x1 p 0 x0 p0 x1
Afriat (1967 and 2012) proves correspondence between the revealed preference and the utility function. Readings: Afriat, S. N. (2012), Afriat’s Theorem and the Index Number Problem, Economic Journal, 122: 295–304. Bandyopadhyay TD (1988) Revealed Preference Theory, Ordering and the Axiom of Sequential Path Independence, Review of Economic Studies, 343-351. Richter M. K. (1966) Revealed Preference Theory, Econometrica, 34, 3, 635-645 Samuelson, Paul A. (1938). “A note on the pure theory of consumer’s behavior,”Economica, 5(17), 61–71. — — . (1948). “Consumption theory in terms of revealed preference,” Economica, 15(60), 243–253 Varian, H. R. (2012), Revealed Preference and its Applications, Economic Journal, 122: 332– 338 Vermeulen F. (2012) Foundations ¤ Revealed Preference: Introduction, Economic Journal, 122 (May), 287–294 2.5.2
Further Developments in Consumer Theory Kahneman and Tversky (1979): Prospect theory Bandyopadhyay (1988):Revealed Preference Theory, Ordering and the Axiom of Sequential Path Independence Hey and Orme (1994): Experimental approach to consumer choice Blundell, Pistaferri, and Preston (2008), Blundell, and Preston (2008) Consumption and income inequality and Partial Insurance Balasko (1975) equilibrium manifold.
Emprical analysis of consumer demand Blundell R and I. Preston (1998) Consumption Inequality and Income Uncertainty, Quarterly Journal of Economics, 113,2, 603-640. Blundell R, L.Pistaferri, and I. Preston (2008) Consumption Inequality and Partial Insurance, American Economic Review, 98:5, 1887–1921
45
Carey K. (2000) Hospital Cost Containment and Length of Stay: An Econometric Analysis Southern Economic Journal, 67, 2, 363-380 Deolalikar A. B. and R. E. Evenson (1989) Technology Production and Technology Purchase in Indian Industry: An Econometric Analysis The Review of Economics and Statistics, 71, 4, 687-692 Hey J. D and J. A. Knoll (2011) Strategies in dynamic decision making: An experimental investigation of the rationality of decision behaviour, Journal of Economic Psychology 32,399– 409 Hey J. D and C. Orme (1994) Investigating Generalizations of Expected Utility Theory Using Experimental Data Econometrica, 62, 6, 1291-1326 Hey, J. D., Lotito, G., & Ma¢ oletti, A. (2010). The descriptive and predictive adequacy of theories of decision making under uncertainty/ambiguity. Journal of Risk and Uncertainty, 41(2), 81–111. Experimental lab of John Hey at York http://www.york.ac.uk/economics/ourpeople/sta¤-pro…les/john-hey/ Kahneman, D. and Tversky, A. (1979). Prospect theory: an analysis of decision under risk. Econometrica,67, 263–291. Lee R-S and N. Singh (1994) Patterns in Residential Gas and Electricity Consumption: An Econometric Analysis Journal of Business & Economic Statistics, 12, 2, 233-241 McGuinness T. (1980) Econometric Analysis of Total Demand For Alcoholic Beverages in the U.K., 1956-75 The Journal of Industrial Economics, 29, 1, 85-109 Segal, U. (1987). The Ellsberg Paradox and risk aversion: an anticipated utility approach. International Economic Review, 28, 175–202. Schmeidler, D. (1989). Subjective probability and expected utility without additivity. Econometrica, 57,571–587. Van Soest A and P. Kooreman (1987) A Micro-Econometric Analysis of Vacation Behaviour Journal of Applied Econometrics, 2, 3 , 215-226. 2.5.3
Emprical analysis
Explain whether following two estimations are as expected from the microeconomic theory of consumer demand presented in this section. 1) Consider the cross-regional variation of expenditure on food in the UK. For simplicity, it is assumed that food expenditure (F) depends only on wage and salary income (Y) in each region as: Fi;t =
i
+
1 yi;t
+ ei;t
ei;t
IID 0;
2 e
(287)
This model has been estimated using a pooled time series and cross section data set (with the sample size of T=14 and N=13) available from the web site of the O¢ ce of the National Statistics (food_exp_UK_regional_panel.csv: hhttp://www.statistics.gov.uk). The estimated coe¢ cients, by region, are given in the following table. 2) Determinants of houseprices by regions in UK estimated by 3SLS and data in HousePrice_regional.csv. 46
Table 6: Food expenditure on income : Stacking Data for SURE)l Coe¢ cient t-value t_Prob Emp_income 0.511 69.4 0.000 Constant -252.126 -2.18 0.031 NW -28.2405 -0.178 0.859 YH -362.599 -2.29 0.023 EM 359.178 2.27 0.025 WM 2034.03 12.8 0.000 EA 1715.26 10.3 0.000 GL 753.455 4.77 0.000 SE -700.345 -4.36 0.000 SE_R -326.693 -2.02 0.045 SW 412.537 2.61 0.010 WL 710.626 4.49 0.000 SCT 580.688 3.65 0.000 NI 2374.79 4.66 0.000 R2 = 0.99; N =182; T = 14; Chi2 =4815. [0.000] **
3
L3: Production: Supply
Production is a process to modify or transform inputs into outputs. Goods and services are produced by millions of …rms in the economy. Production functions show how much output is obtained for given combinations of inputs. Production technologies di¤er by production sectors and perhaps by the sizes of production …rms that can be from being small to midium to large or to multinational company. For instance consider types of soft drinks: Coke, Pepsi, Fanta, Tango, Sprite, 7 Up, Dr. Pepper or cars such as BMW, Voxhaul, Poeguet, Chrisler, Ford, GM, Toyota, Nissan, Hyundai, Fiat. There are varieties of daily use products supplied by many …rms. Many big companies are listed in stock markets such as FTSE 5000 or Nikkie or Dow Jones but there are many …rms operating in smaller scales. Most often these productions occur in sectors as recorded in the UNIDO databases. The agriculture sector,- that includes farms crops, livestock, forestry, and …sheries usually is operated by small …rms and requires more land. The mining sector is usually more capital intensive and dominated by larger …rms. Firms in manufacturing sector are mostly very big and require more physical and human capital. These vary a lot by the types of manufacturing that according to the Iput Output Table of UK from the ONS these include meat processing, …sh and fruit processing, oils and fats, dairy products, grain milling and starch and animal feed, bread, biscuits, etc, sugar, confectionery, other food products, alcoholic beverages, soft drinks and mineral waters, tobacco products, textile …bres, textile weaving, textile …nishing, made-up textiles, carpets and rugs, other textiles, knitted goods, wearing apparel and fur products, leather goods, footwear, wood and wood products, pulp, paper and paperboard, paper and paperboard products, printing and publishing, coke ovens, re…ned petroleum & nuclear fuel, industrial gases and dyes, inorganic chemicals, organic chemicals, fertilisers, plastics & synthetic resins etc, pesticides, paints, varnishes, printing ink etc, pharmaceuticals, soap and toilet preparations, other chemical products, man-made …bres, rubber products, plastic products, glass and glass products ceramic goods, structural clay products, cement, lime and plaster, articles of concrete, stone etc , iron and steel, non-ferrous metals, metal
47
Table 7: Determinants of houseprice in UK: SURE (3SLS) estimation Coe¢ cient t-value t_Prob Rincome 4.64 45.2 0.000 Pop 1.25 0.55 0.054 MRT_RT -11.51 -0.022 0.982 M/H_Ratio -237240 -19.9 0.000 CRNTDP 1.94 5.85 0.000 SVDEP 1.10 3.72 0.000 NE 22845.3 2.04 0.042 NW 8064.9 2.85 0.005 YH 14615.8 2.37 0.018 SW 9939.4 1.50 0.134 EN -148092 -1.66 0.097 EM 12868.1 1.61 0.108 WM 12404.2 2.20 0.029 EE 16599.7 2.84 0.005 GL 5454.8 2.00 0.046 Constant 101298.1 5.31 0.000 F(90, 373) = 7.09 (0.00); N =480; Chi2 (2)=59.2. [0.000] **
castings, structural metal products, metal boilers and radiators, metal forging, pressing, etc; cutlery, tools etc, other metal products, mechanical power equipment, general purpose machinery, agricultural machinery, machine tools, special purpose machinery, weapons and ammunition, domestic appliances, o¢ ce machinery & computers, electric motors and generators etc, insulated wire and cable, electrical equipment, electronic components, transmitters for TV, radio and phone receivers for TV and radio, medical and precision instruments, motor vehicles, shipbuilding and repair, other transport equipment, aircraft and spacecraft, furniture, jewelry and related products Sports goods and toys, miscellaneous manufacturing & recycling that rely very much on fossil fuels. The production and distribution of electricity and gas is vital in order to run all these industries. Firms in construction sector contribute both to the supply of residential and non-residential properties but also to big infrastructure in the form of transport, communication and service networks and logistics and supply chain management. The distribution sector consists of motor vehicle distribution and repair, automotive fuel retail, wholesale distribution, retail distribution, hotels, catering, pubs etc. The business service sector represents banking and …nance , insurance and pension funds , auxiliary …nancial services, owning and dealing in real estate, letting of dwellings, estate agent activities, renting of machinery etc, computer services, research and development , legal activities, accountancy services, market research, management consultancy, architectural activities and technical consultancy, advertising and other business services. The other services sector includes public administration and defence, education, health and veterinary services, social work activities, membership organisations, recreational services, other service activities, private households with employed persons and sewage and sanitary services. Datastream provides basic timeseries information on public companies regarding their production, sales, revenue, costs, pro…ts, price of stocks and their leverages. Megazines such as Forbes regularly
48
Top 10 Firms in the World in May 2014 (Forb's List): http://www.forbes.com/global2000/list/ Rank
Company Country
1
ICBC
2 3
China Construction Bank Agricultural Bank of China
4
JPMorgan Chase
5
Berkshire Hathaway
6
Exxon Mobil
7
General Electric
8
Wells Fargo
9
Bank of China
10
PetroChina
Sales
Profits
Assets
Market Value
China
$1 48.7 B
$42.7 B
$3,1 24.9 B
$21 5.6 B
China
$1 21 .3 B
$34.2 B
$2,449.5 B
$1 7 4.4 B
China
$1 36.4 B
$27 B
$2,405.4 B
$1 41 .1 B
United States
$1 05.7 B
$1 7 .3 B
$2,435.3 B
$229.7 B
United States
$1 7 8.8 B
$1 9.5 B
$493.4 B
$309.1 B
United States
$394 B
$32.6 B
$346.8 B
$422.3 B
United States
$1 43.3 B
$1 4.8 B
$656.6 B
$259.6 B
United States
$88.7 B
$21 .9 B
$1 ,543 B
$261 .4 B
China
$1 05.1 B
$25.5 B
$2,291 .8 B
$1 24.2 B
China
$328.5 B
$21 .1 B
$386.9 B
$202 B
update top 100 companies by thier turn over. For instance Industrial & Commercial Bank of China Ltd. had assets of 3.1 trillion dollars in 2014 with pro…ts over 148 billion dollars. It is also possible to …lter prominent …rms in each of above sectors operating in the global economy using such databases. A production technology in each of above sector shows how inputs are transferred into outputs. Usually labour, the human toils and trouble in process of production; capital, the man made means of production, as re‡ected in building, structures including highways, communication networks and education, health and environmental system; natural resources including clear air, water, and mineral and energy products represent such inputs. In addition there are intermediate inputs as presented in the input output table of an economy. There are linear and non-linear production functions. Returns to scale vary and possibility of substitution vary among them . Intensity of use of these factors in a speci…c industry or a …rm is re‡ected in these production function. These are important in process of substitution of more expensive by less expensive inputs. The CES categories of these functions being the most commonly used ones in the economic literature as they capture the cross price elasticity more e¢ ciently than any other linear or Cobb-Douglas production functions [see articles such as (Pigou (1934) , Meade (1934) Lancaster and Chesher (1983), Dolton and Makepeace (1990), Harmatuck (1991), Basu and Fernald (1997), Barmby , Ercolani and Treble (2002), Costinot, Vogel and Wang (2013)) discussion about production; Coase (1937) The Nature of the Firm, Economica, 386-405]. 3.0.4
Popular production functions
Popular production functions where output ( y) is expressed as functions of inputs (xi ): Cobb-Douglas: y = x1 x12 CES: y = (x1 + x2 )
1
Nested: x4 = (x1 + x2 )
1
and then y = x4 x13 49
generalised Leontief: Y =
n P n P
p aij xi xj ;
aij = aji
i=1j=1
Translog: ln Y = a0 +
n P
ai ln xi +
i=1
n P n P
aij ln xi ln xj ;
aij = aji
i=1j=1
A tanslog production function adds squares and product terms to the regual production function as: ln Y = a0 +
n X
ai ln xi +
i=1
n n X X
aij ln xi ln xj ;
aij = aji
i=1 j=1
This function is popular as it allows a large number of substitution posibilities among inputs. n n P P Prove that this function becomes a constant return to scale when ai = 1 and aij = 0: i=1
j=1
Generalised Leontief function:
Y =
n n X X
p aij xi xj ;
aij = aji
i=1 j=1
Nested production function shows how composite inputs are used with other inputs (very popular in the CGE and macro modelling): Let V be the CES composite of labour and capital V = [ L + (1
)K ]
1
(288)
then let E be energy input in production. Then Y is prouced using V and E as: Y = V E1 This is one level nest. There can be many levels of nests in the production process. Questions: What are the elasticities of output to input x1 in above production functions?
3.1
Supply function: an example
1. Let us consider a production function for a fruit …rm operating in the competitive market is given by p y=2 l (289) where y is output and l is labour input. Product price is p and input price is w. What is the cost function for this …rm? What is its pro…t function? What is its supply function? What is the demand function for labour? What are the properties of the these production, pro…t and cost functions? Since this is a one input production funtion the cost function can derived direcly from the production technology as: l=
50
y2 4
(290)
Producer pay wage to supply this commodity: y2 (291) 4 The pro…t is the di¤erence between the revenue and cost of the …rm as given by the pro…t function: c = wl = w
y2 (292) 4 The supply function for commodity y is derived using the …rst orcer condition of the pro…t function as: = py
@ =p @y
c = py
w
w
y 2p = 0 =) y = 2 w
(293)
Supply is positively related to prices and negatively to the input cost, in this case the wage rate. Demand for labour: l= 3.1.1
1 2 1 y = 4 4
2p w
2
(294)
Properties of a pro…t function
1. Increasing in p 2. decreasing in w 3. homogenous of degree one in p and w 4. concave in y and convex w These properites satisfy in this example: This supply function is homegeous of degree zero in price and wage, y = 2p w as there is no change in level of output when price and wage increase by the same amount. It is increasing in p and decreasing in w. 2 Pro…t function is concave as its second derivative wrt to output is negative, @@y2 = w2 < 0; 1
2
@ y 2 =) Production function is also concave. @y @l = l @l2 = 2 y @c @ c w Cost function is convex: @y = w 2 =) @y2 = 2 > 0.
1 2l
3 2
< 0:
Demand function for labour is also homegeous of degree zero in price and wage as l =
1 4
2p 2 . w
Hotelling’s lemma Derivative of pro…t function wrt price gives the supply function; derivative pro…t function wrt input prices gives demand function for inputs: @ (p;y) @ (p;w) = y (p; w) = xi (p; w) @p @wi Properties of output supply and input demand functions 1. Homogeniety of degree zero y (tp; tw) = y (p; w) for all t > 0 2. xi (tp; tw) = xi (p; w) for all t > 0 See substituion matrix 51
3.1.2
Production function and its scale properties
A production function f : Rn+ =) R is continuous, strictly increasing, strictly quasiconcave function in Rn+ f (0) = 0 Isoquant Q(y) = fx 0 = f (x) = yg is set of inputs giving a …xed output y. Returns to scale 1. Constant return to scale
f (tx) = tf (x) for all t > 0 and all x
2. Increasing returns to scale f (tx) > tf (x) for all t > 0 and all x 3. Decreasing returns to scale f (tx) < tf (x) for all t > 0 and all x Elasticity of scale of the production function at point x Xn f (x) xi d ln (f (tx)) i=1 = (x) = lim t !1 d ln (t) f (x) 3.1.3
(295)
Variable returns to scale y = k(1 + x1 x2 )
1
(296)
1
(x) =
@y x1 = (1 + x1 x2 ) @x1 y
2
(x) =
@y x2 = (1 + x1 x2 ) @x2 y
1
x1 x2
(297)
1
x1 x2
(298)
Elasticity of scale is obtained by adding above two: (x) = ( + ) (1 + x1 x2 )
1
x1 x2
(299)
This varies with x. Variable returns to scale y = k(1 + x1 x2 )
1
; =) x1 x2
=
k y
1
(300)
1
(y) = (1
y ) k
(301)
2
(y) = (1
y ) k
(302)
Elasticity of scale y ) (303) k Returns to each input declines with output here. Increasing return for 0 < y < k; constant return 0 < y = k and decreasing return when y > k > 0 . k is the upper bound of output. (see more in Jehle and Reny (2001). (y) = ( + ) (1
52
3.1.4
Cost function 0 and output levels y 2 Rn+
It is a minimum value function for for all input price w c(w; y) = min w:x
(304)
subject to f (x)
y
(305)
Constrained optimisation L (x; ) = w:x + [y @L (x; ) = w1 @x1
f (x)]
(306)
f 0 (x1 ) = 0
(307)
:: @L (x; ) = wn @xn y
(308) f 0 (xn ) = 0
f (x) = 0
(309) (310)
Marginal Rate of Trasformation M RT Sj;i =
@f (x;) @xj @f (x;) @xi
=
wj wi
(311)
Some studies on cost and production: Barmby T. A., M. G. Ercolani and J. G. Treble ( 2002) Sickness Absence: An International Comparison Economic Journal, 112, 480, F315-F331 Basu S and J. G. Fernald (1997) Returns to Scale in U.S. Production: Estimates and Implications, Journal of Political Economy, 105, 2, 249-283 Benabou R. and Tirole, J. (2010), Individual and Corporate Social Responsibility. Economica, 77: 1–19. Bloom N., R. Sadun and J. van Reenen (2012) The organization of …rms across countries, Quarterly Journal of Economics 127 (4), 1663–1705. Costinot, A. and J. S. Vogel and S. Wang (2013), An Elementary Theory of Global Supply Chains, Review of Economic Studies 80, 109–144 Dolton P.J. and G. H. Makepeace (1990) The Earnings of Economics Graduates The Economic Journal, 100, 399, 237-250
53
Harmatuck D.J. (1991) Economies of Scale and Scope in the Motor Carrier Industry: An Analysis of the Cost Functions for Seventeen Large LTL Common Motor Carriers, Journal of Transport Economics and Policy, 25, 2 , 135-151 Lancaster T and A. Chesher (1983) An Econometric Analysis of Reservation Wages Econometrica, 51, 6,1661-1676 Meade (1934) The Elasticity of Substitution and the Elasticity of Demand for One Factor of Production The Review of Economic Studies, 1, 2 ,152-153 Panagariya A (1981) Variable Returns to Scale in Production and Patterns of Specialization, The American Economic Review, 71, 1, 221-230 Pigou A. C. (1934) The Elasticity of Substitution ,The Economic Journal, 44, 174 , 232-241 Tirole J. (1995) The Theory of Industrial Organisation, MIT Press.
3.2
CES and Cobb-Douglas production functions
Cobb-Douglas Production Function Y = AK L1
(312)
CES Production Function 1
Y =A
K
+ (1
)L
(313)
Prove that the elasticity of substitution is 1 in Cobb-Douglas production function and = 1+1 in the CES production function. Also prove that the Cobb-Douglas is a special case of the CES production function. Proof that = 1 in the Cobb-Douglas production function =
d d
K L w r
= =
K L w r
K L
d
= d
(1
K L (1
=
)AK L AK 1 L1
=
=
)AK L AK 1 L1
d d
K L K L
= =
K L K L
=1
(314)
For the CES production function @Y = @K
1
@Y = @K
1
A
K
1
+ (1
)L
1
(
1
A
K
+ (1 dK = dL
)L YL = YK
YL (1 = YK
)L
1
=
) K
1
=
) (1
1
(
(1
)
54
)
K L
K L
(1
) A1+ A
A1+ A
Y K
Y L
1+
(315)
1+
(316)
1+
0; and diminishes when
Hotelling’s Lemma
Derivative of pro…t function wrt output price gives the supply function; derivative pro…t function wrt input prices gives demand function for inputs: y (p; w) =
@ (p; w) @p
(378)
xi (w; p) =
@ (p; w) @w
(379)
62
Pro…t function = py
rK
wL
(380)
Constrained pro…t optimisation problem: = (x; ) = py
rK
wL +
@= (x; ) =p @y @= (x; ) = @L @= (x; ) = @K
K1 (1
K1
L
y
=0 L
)K
1
L
(381)
(382) w=0
(383)
r=0
(384)
@= (x; ) = K1 L y=0 (385) @ Solving these Jacobians one will …nd input demands as L(r; w; p) and K(r; w; p) and the level of output y(r; w; p). Insert these functions into the pro…t function to get the value function as: = py(r; w; p)
rK(r; w; p)
wL(r; w; p) = V (r; w; p)
(386)
Applying the envelop theorem: @V @= = = y(r; w; p) @p @p @= @V = = @w @w
(387)
L(r; w; p)
(388)
@V @= = = K(r; w; p) (389) @r @r Di¤erentiating the value function with respect to price gives the output supply function and di¤erentiating wrt input prices gives input demand functions of the …rm. This is Hotelling’s Lemma. Repeat the same process to technology y = K 0:4 L0:4 as: = (x; ) = py
rK
wL +
K 0:4 L0:4
y
(390)
Hotelling’s Lemma like the Shephard’s lemma is very useful for solving Duals of consumer and producer optimisation problems.
63
3.3.6
Exercise 7: Minimising the cost with Cobb-Douglas and CES production function
1. Consider cost of production of a …rm: C = wK + rL
(391)
and its production technology constraint y=K L
(392)
1. (a) Write the pro…t function for this form (b) Write a Langrangian of pro…t subject to technology constraint (c) Determine the optimal demand for inputs (d) Derive the pro…t function in terms of optimal inputs , V (p; w; r): (e) Determine the cost function. (f) Prove Hotelling’s lemma K(p; w; r):
@V @P
=
@L @P
= y(p; w; r); @V @w =
@L @w
=
L(p; w; r); @V @r =
@L @r
=
(g) Derive input demand, output supply and pro…t functions when the technology is y = K10:4 L0:4 1 2. Derive the short run pro…t function for a …rm under the perfect competion = PY
wL
rK
(393)
with production technology Y =L K
(394)
1. Here is pro…t, L labour supply, K capital, w wage rate, P price of good, Y ouput . For simplicity assume capital is …xed at K. Technology operates under the constant returns to scale + = 1: (a) Derive supply function of the …rm using Hotelling’s Lemma. (b) What are the price and output of this …rm in the short run. Prove that …rm earns positive pro…t in the short run taking = 0:5; w = 4; r = 1; k = 1: (c) Now assume that the market demand is given by p = 39 0:009q and the pro…t function is given by = p2 2p 399: Find output, market demand and the number of …rms in the long run ( = 0) (d) Why is the number of …rms indeterminate in the perfect competion? 3. A …rm’s objective is to minimise cost C = rK + wL
64
(395)
subject to technology constraint as: Y = [ L + (1
)K ]
1
(396)
1. (a) Determine the demand for labour and capital. (b) Derive the cost function of the …rm. (c) Prove that the elasticity of substituion is
=
1
1:
(d) Discuss propperties of CES cost function. Consider a cost minimisation under the perfect competition: A …rm’s objective is to minimise cost C = rK + wL
(397)
subject to CES technology constraint as: Y = [ L + (1
)K ]
1
(398)
1. (a) Determine the demand for labour and capital. (b) Derive the cost function of the …rm. (c) Prove that the elasticity of substitution is
1
=
1:
(d) Discuss properties of CES cost function. Answer Comparative static: Derivation of the CES cost function. Lagrange: h L = rK + wL + Y ( L + (1
)K )
Taking the …rst order conditions: @L =w @L 1h
@L =r @K
1h
( L + (1
( L + (1
)K ) )K )
@L = Y ( L + (1 @ From the …rst two …rst order conditions: w = r
1
L K
1
;K
1
=
r w
L
(1
1
) K
=0
(400)
1
(401)
=0
1
Now put the value of K in the production function: 65
i
i
(399)
)K ) = 0
(L)
1
1
1
i
1
1
;K =
(402)
r w
1
1
1
1
L
(403)
Y = ( L + (1
Y =
(
Y =
8 < :
Y = demand for labour
L=
L + (1
)
8
0 p1 z1 (p) + p2 z2 (p) + ::: + pm 1 zm 1 (p) + pm zm (p) = 0 Represent equilibrium allocations of production, endowment and consumption in a single diagram.
5.3
Pure exchange general equilibrium model
Two Good Pure Exchange General Equilibrium Model Households, h = A B. Two goods X1 and X2 A B B Endowments of two goods ! A 1 !2 !1 !2
Objective of each is to maximise life time utility subject to budget constraints w r t X1 and X2 Equilibrium relative price determines the optimal allocation; It is Pareto Optimal. Main Features of Applied General Equilibrium Model Three conditions 1. Demand = supply ; n markets, n-1 relative prices 2. Income = expenditure 129
+
p1 2
p i =2 2 =
3. Firms maximise pro…t: zero economic pro…t in competitive markets Relative Prices 1. Preferences and technology parameters determine relative prices in 2. equilibrium. 3. Relative prices are determined by forces of demand and supply. 4. Numeraire or anchor price; normalised to 1. Markets allocations depend on relative prices. 1. Demand for a commodity depends on preferences and income. 2. Income of a household is determined by her endowment and price of that endowment. Exchange or trade of goods is mutually bene…cial. Each consumer/ producer optimises in equilibrium. Problem of representative households For household A M ax
U (X1A ; X2A ) = X1A
A
X2A
1
A
(846)
Subject to the budget constraint: A A P1 X1A + P2 X2A = P1 ! A 1 + P2 ! 2 = I
(847)
For household B M ax
U (X1B ; X2B ) = X1B
B
X2B
1
B
(848)
Subject to the budget constraint: B B P1 X1B + P2 X2B = P1 ! B 1 + P2 ! 2 = I
(849)
Intertemporal budget constraint Lagrangian for constrained optimisation for Household A : LA = X1A
A
X2A
1
A
+
A P1 ! A 1 + P2 ! 2
P1 X1A
P2 X2A
(850)
P1 X1B
P2 X2B
(851)
Lagrangian for constrained optimisation for Household V : LB = X1B
B
X2B
1
B
+
B P1 ! B 1 + P2 ! 2
First order conditions for optimisation For household A and B
130
@LA = @X1A
A
@LA = (1 @X2A
1
X1A
A
A)
X1A
1
X2A A
A
X2A
A
@LA A = P1 ! A 1 + P2 ! 2 @
P1 X1A
@LB = @X1B
X2B
B
@LB = (1 @X2B
1
X1B
B
B)
X1B
B
@LB B = P1 ! B 1 + P2 ! 2 @ Demand and market clearing conditions For household A AI
X1A =
A
P1
1
B
(1
;
X2B =
(1
P1 = 0 B
P1 X1B
X2A =
P2 = 0
P2 X2A = 0
X2B
;
P1 = 0
P2 = 0
P2 X2B = 0
A) I
(852) (853) (854) (855) (856) (857)
A
(858)
P2
For household B BI
X1B =
B
P1
B) I
B
P2
(859)
Market clears each period B X1A + X1B = ! A 1 + !1
(860)
B X2A + X2B = ! A 2 + !2
(861)
Market clearing Prices Obtained from the market clearing conditions AI
A
P1 (1
A) I
P2
BI
+ A
+
B
P1 (1
B = !A 1 + !1 B) I
B
P2
B = !A 2 + !2
(862) (863)
Walrasian numeraire: P1 = 1: with this speci…cation I A = !A 1
I B = P2 ! B 2
Equilibrium Relative Price and Proof of Walras’Law
131
(864)
Table 30: Parameters in Pure Exchange Model Household A Household B A B A Endowments ! 1 ; ! 2 = f100; 0g !1 ; !B = f0; 200g 2 Preference for X1 ( ) 0:4 0:6 Preference for X2 (1 ) 0.6 0.4
AI
P1
A
+
BI
B
P1
0:4 (100) + 0:6 (200) P2 I A = !A 1 = 100
=
AI
A
+
BI
=
A A !1
+
B B P2 ! 2
=
100;
B
= !A 1
P2 = 0:5
I B = P2 ! B 2 = 0:5 (200) = 100
(865)
Table 31: Parameters in Pure Exchange Model Household A Household B B A B Endowments !A ! ; ! = f100; 0g = f0; 200g 1 1 ; !2 2 Prices 1 0.5 Demand for X1 ( ) 40 60 Demand for for X2 (1 ) 120 80 Utility 77.3 67.3 Income 100 100 Theoretical observations Relative prices of goods, income and consumption change when preferences ( alpha, beta) change. Change in the relative income a¤ects the level of utility and welfare of households Household A can make household B worse o¤ by increasing the demand of good 1 that he owns ( or supplying less to the market). Household B can increase his relative income and reduce the relative price of good 1 by increasing the demand for good 2 (reducing its supply). Relative prices and allocations depend on preferences and endowments. Homework Do sensitivity analysis (solve model for various parametric speci…cations A B 1) by changing endowments ! A = f100; 50g and ! B = f200; 150g 1 ; !2 1 ; !2 2) by changing preferences f A , B g = f0:50; 0:50g ; f A , B g = f0:750; 0:30g 3) introduce VAT of 20 percent in commodity 1. Assume that revenue collected is spent entirely by the government and does not add to any utility for the household. GAMS programme: pexchange.gms; proto.gms; tax2sector.gms; tax_3sectr_utils.gms 132
5.3.1
Exercise 11: Ricardian trade model
1. Develop the above model for two countries and …nd the global equilibrium in two country world. 2. Show that relative output of country 1 not only in domestic costs but also the output and cost in the foreign country. 3. Represent the global economy by three representative countries with the following problem max Ui = X1;i X2;i X1;i
(866)
Ii = p1 X1;i + p2 X2;i + p3 X3;i
(867)
subject to
Endowments of each country respectively is f(! 1 ; 0; 0) ; (0; ! 2 ; 0) ; (0; 0; ! 2 )g Solve for prices and allocations at equilibrium. Apply this model to a realistic situation for countries producing food, oil and manufacturing commodities.
5.4
Simplest general equilibrium model Households and …rm optimise subject to their constraints – Utility maximisation by households and pro…t maximisation by …rms System of prices when all markets clear simultaneously (all goods and factor markets) D (p1 p2 p3 ; ::::pn ) = S (p1 p2 p3 ; ::::pn )
(868)
Excess demand is zero in equilibrium. Income of agents equals their expenditure Imports equals exports in an open economy model Saving equals investment in a dynamic economy model Public spending accounts are balanced in model with public sector In general equilibrium is obtained by the price system when economy is in perfect harmony. Consider one of the easiest possible example of a general equilibrium model with production Exercise Prove that a …rm need to pay higher wage rate to its workers and lower the price of commodity while expanding output if it operates under an increasing returns to scale technology such as Y = L2 .
133
5.5
General equilibrium with production Bhattarai K. (2006) Macroeconomic Impacts of Taxes: A General Equilibrium Analysis, Indian Economic Journal, 54:2:95-116.
A simple general equilibrium model represents an economy in which a representative household maximises utility by consuming goods and services supplied by producers and paying for them by income that it receives in return of supply of labour and capital inputs to the producers. Firms optimise pro…t combining inputs with the existing technology in production and rewarding inputs according to its marginal productivity. Tax policies of government in‡uence both production and consumption sides of the economy by a¤ecting prices of inputs and outputs. By distorting the marginal conditions of optimisation, these taxes in‡uence choices of goods and services by households and use of inputs by producers. The incidence and impact of taxes on consumption may di¤er from taxes on labour income depending on the key parameters for share or elasticities of substitution in consumption or in the production sides of the economy. A general equilibrium implies a set of prices that eliminate excess supply or excess demand and where these prices and wage rates are consistent with the preferences and endowments of households and technology of …rms. The perfect match between demand and supply for both goods and services and inputs of production follow from the properties of utility and production functions as given by explicit analytical solutions in the next section. Consider an economy consisting of a representative household and a representative …rm. A representative household tries to maximise utility by consuming goods and services and from enjoying leisure subject to his budget constraints. max U = C l(1
)
(869)
Subject to time and budget constraints: l + hs = 1
(870)
pc = whs +
(871)
c > 0; h > 0;and l > 0; where c is consumption,l is leisure and hs is labour supply, p is the price of the commodity, w is the wage rate is the pro…t from owning the …rm. General equilibrium question: problem of the …rm The producer wants to maximise pro…t by selling goods produced using the labour supplied by the household. Problem of the …rm s
= py
whd
(872)
subject to technology constraint as: y = (hs ) y > 0; hd > 0; where y is the output supplied by the …rm and hd is its demand for labour.
134
(873)
Lagrangians for constrained optimisation For the houshold: (1
(1
)
hs )
L (C; L; ) = C (1
+ [whs +
pC]
(874)
First order conditions here given demand for consumption and leisuer C; L , supply of labour l) and shadow price For the …rm: = p (hs )
whd
(875)
First order condition from this will give the demand for labour as a function of real wage rate. First order conditions for the household: @L (C; L; ) = C @C @L (C; L; ) = (1 @hs
1
(1
(1
hs )
) C (1
hs )
)
p=0
( 1)
@L (C; L; ) = whs + @
w=0
pC = 0
(876) (877) (878)
From above two (1
) C (1
hs )
1
(1 hs )
C
(1
( 1) )
=
w p
(879)
First order conditions for households: C=
(1
)
hs )
(1
w p
(880)
from the budgent constraint and this FOC w s h + =C= p p (1
)
hs )
(1
w p
(881)
Supply labour hs =
w p
(1
)
p
(882)
w p
Demand for leisure L=1
w p
(1 w p
First order conditions for households: Demand for consumption
135
)
p
(883)
C=
(1
)
w h ) = p (1 s
(1
w p
1
)
(1
)
p
w p
!
w p
(884)
Demand for consumption and leisure and supply of labour are thus functions of Real wage rate
w p
Preference for consumption ( )and leisure (1
)
pro…t of the …rms First order conditions for …rms @ = p hd hd h =
p
w d h = p ' 1 1
=
(h )
=
w p
w=0
(885)
1
1w p
d
s
1
1
(886)
1w p
w p
1
1
1
1
1
#
1
1w p
1
(887)
Equilibrium real wage the equilibrium in the labour market is determines the real wage rate h = 1
1 w p
1
=
( wp )
1
(1)
(1)
1
1
1w p
d
1
1
1
(1
) wp
(1
p
= hs =
(888)
w p
)w p
w p
w =h p
1
(1
)
1
1
1
1
1
i
(889)
1
Optimal labour supply and leisure demand the equilibrium in the labour supply is the function of the real wage rate 0
and the leisure
B1 hd = @ h
1
(1
)
1
1
136
1
1 1
i
1
C 1A
1 1
(890)
l=1
0
B1 @ h
h=1
1
1
(1
1
)
1
1
1
1
1 1
C 1A
i
(891)
Optimal labour output and consumption the equilibrium in the labour supply is the function of the real wage rate 0
B1 y = (h ) = @ h s
(1
)
1
1
1
1 1
and optimal consumption
C
=
(1
(1
h 5.5.1
(1
hs )
(1 ) 0
B B1 )@ )
1
1
1
i
(892)
w = p
0
1
B1 @ h (1
1
)
1
1
1
1
C 1A
1
1 1
i
1
1 1
i
1 C
1A
1
1 1
1 C C A (893)
A numerical example for the general equilibrium tax model
A simple numerical example is provided here for the above general equilibrium tax model and used it to measure the relative impacts of consumption and income taxes in the economy. These impacts vary according to the preferences of households in relation to technology of production available to …rms. If households strongly prefer leisure rather than consumption, the costs of income taxes are likely to be higher than those of the consumption taxes. On the other hand if the households have stronger preferences for consumption than for leisure then consumption taxes might be costlier. As stated above these preference and technology factors jointly determine the prices and prices in‡uence allocation of resources in the economy. Numerical values of model parameters are as follows: Time endowments of 68 ours per week is used here as is customary in the literature. Two types of simulations are conducted using this model. The …rst one is the base case scenario constructed using a reasonable set of preference and technology parameters as givenabove . Then there is a no tax scenario where all taxes are eliminated, followed by scenarios with taxes either only on consumption or only on labour income. The impacts of tax policy experiments are determined by comparing the utility and changes in output and employment before and after the change in taxes. In addition sensitivity analyses are conducted to see how the welfare and macroeconomic impacts change in response to a distribution of households by preference and that of …rms by the production technology. The grids of parameters for sensitivity analyses to study the impacts of taxes are measured in terms of levels and changes in utility, output, leisure, labour supply and consumption as: 137
Table 32: Parameters of the model in the base scenario Parameter of the model Numerical value in the base model Utility weight on consumption ( ) 0.6 Utility weight on leisure (1 ) 0.4 Elasticity of output to labour input ( ) 0.6 Value of Endowment L 68 hours Base consumption tax rate (tc ) 0.17 Income tax rate in the base case (tl ) 0.35 Time endowments of 68 ours per week is used here as is customary in the literature. Normalisation of price , w + p = 1 Table 33: Parameters of the model in the base scenario Use of both consumption and income taxes and lump sum transfers (base case) Elimination of all taxes and no transfer Only labour income tax and lump sum transfers Only consumption tax and lump sum transfers The benchmark economy includes consumption tax rate of 17 percent and income tax rate of 35 percent, similar to the one that actually exists in many of the OECD countries including the UK. In all experiments the government returns tax revenue to the household in the form of a lump sum transfer. The model is then used to evaluate the impacts of four di¤erent tax reform experiments: (1) the distortionary cost of both income and consumption taxes; (2) impact of a complete switch to the labour income tax holding the revenue …xed; (3) impact of a complete switch towards the consumption tax; and (4) the test of reliability and robustness of the model by examining the sensitivity to the key parameters of the model. The Hicksian equivalent variations (EV) are presented in terms of the money metric utility in the counter factual scenario in comparison to the benchmark scenario by asking how much the household bene…ts from the tax changes equivalent in terms of the original equilibrium. The corresponding compensating variation is also in terms of the money metric utility measuring the amount of compensation a consumer needs to bring back her to the original level of utility after the changes in tax rates. The overall welfare costs of taxes, as presented in Table 3 above, show that the costs of using both consumption and income taxes are higher than of using only either the consumption or only the labour income tax. The overall distortionary impacts of both consumption and labour income taxes are up to 3.2 percent of the benchmark utility. This compares to results as contained in Bhattarai and Whalley (1999, 2003). If the revenue is returned as a lump sum form to the households, the model results con…rm that the labour income tax has highly distortionary impact in the economy. It may cost up to 6.2 percent of the benchmark utility. Higher rate of labour income tax …rst reduces labour supply and output and consumption consequently. In comparison, sole reliance on only consumption taxes signi…cantly lowers distortions than labour income taxes. Model calculations suggest that the cost of moving towards only consumption taxes is 0.05 percent of the benchmark utility. Thus the overall costs are lower when tax is only on consumption than when taxes are both on consumption and labour income simultaneously. For this hypothetical economy taxes a¤ect the aggregate output, employment, leisure, labour 138
Table 34: Parameters of the model in the base scenario Parameter of the model Numerical value in the base model Utility weight on consumption ( ) 0.25 to 0.6 with steps size of 0.05 Utility weight on leisure (1 ) 0.75 to 0.4 with steps size of 0.05 Elasticity of output to labour input ( ) 0.3 to .65 with steps size of 0.05 Value of Endowment L 68 to 108 hours with steps size of 5 Base consumption tax rate (tc ) 0.17 to 0.67 with steps size of 0.05 Income tax rate in the base case (tl ) 0.40 to 0.85 with steps size of 0.05 Time endowments of 68 ours per week is used here as is customary in the literature. Normalisation of price , w + p = 1 Table 35: Overall Welfare Impacts of Tax Changes in the General Equilibrium Model of Taxes Equivalent variation Compensating variation Elimination of all taxes 3.2% -3.1% Labour tax only -6.2% 6.7% Consumption tax only -0.05% 0.05% supply as well as consumption of the household as shown in Table 4 to Table 7. It is obvious that the adverse macroeconomic impacts of only labour income taxes are a lot higher than those of only consumption taxes. When both labour income and consumption taxes are removed, households lose the amount of transfer income from the government but still it has a very good positive impact on output, consumption and utility level of the representative household. In contrast the labour income tax discourages labour supply relative to both the no tax and consumption tax only cases leading to highly distortionary e¤ects on the economy. This model generates predictable results when subject to sensitivity tests along the various rates of consumption and labour income tax rates. It con…rms that the welfare costs of taxes rise proportionately to the squares of tax rates as suggested by the famous Harberger triangle, a measure of the dead weight loss of taxes. The model also behaves well when subject to changes in the endowments. Households receive higher utility as their endowments rise but at a decreasing rate given the law of diminishing marginal utility. Since the consumer values both consumption and leisure, the increase in utility of increasing only consumption show a diminishing utility as does the increase in the share of labour in production which raises the marginal productivity of labour and reduces the amount of leisure in the utility function. The …rst scenario considers the e¢ ciency impacts of removing all the taxes and transfers in the UK. When a representative household does not pay tax it also does not receive any transfer from the government. This is an extreme scenario in which all public services are provided by the private sector. The second scenario considers switching completely to the labour income tax and eliminating all indirect taxes. The third scenario, on the other hand is switching completely to consumption taxes and removing all taxes on the labour income. The results of the model are very intuitive. E¢ ciency Gains in the UK from elimination of all taxes and transfers 139
Table 36: Macroeconomic Impacts of Alternative Taxes Variables Both taxes No taxes Labour income tax Consumption tax Utility 14.142 14.601 13.689 14.526 Output 6.505 8.032 5.849 7.431 Leisure 45.333 35.789 49.012 39.703 Labour Supply 22.667 32.211 18.988 28.297 Consumption 6.505 8.032 5.849 7.431 Revenue 2.109 1.687 1.687 Wage 0.147 0.13 0.156 0.136 Price 0.853 0.87 0.844 0.864 Pro…t 14.142 14.601 13.689 14.526 Consumption tax 0.17 0.263 Labour income tax 0.35 0.57 Table 37: Impact of Alternative Taxes: Percentage changes compared to the base case Variables Both taxes Labour income tax Consumption tax Utility 3.246 -6.246 -0.511 Output 23.471 -27.174 -7.478 Leisure -21.053 36.946 10.936 Labour Supply 42.105 -41.051 -12.151 Consumption 23.471 -27.174 -7.478 Revenue -100 Wage -11.406 19.868 4.595 Price 1.964 -2.972 -0.687 Pro…t 25.896 -29.339 -8.114 Consumption tax 0.17 0.263 Labour income tax 0.35 0.57 (Measured as a percent of benchmark utility level of a representative household) Equivalent Variation = 3.715 Compensating Variation = -3.582 E¢ ciency Gains from Switching to Labour income Taxes Equivalent Variation = -6.9 Compensating Variation = 7.0 E¢ ciency Gains from Switching to Consumption Taxes Equivalent Variation = 2.967 Compensating Variation = -2.882 Summary of results The key results of this exercise in the general equilibrium impacts of tax reforms are the following: 1. The e¢ ciency gains from switching to only consumption taxes are about 80 percent of the gains of eliminating all the taxes. Optimal consumption tax rate given the revenue constraint set
140
equal to 80 percent of the benchmark revenue level is 2.9 percent. This seems a very sensible result considering the fact that consumers ultimately bear the burden of all taxes. Similarly consumers make a choice whether to consume a certain product or not depending upon its price. If the prices are high because of taxes they can increase utility by not consuming the heavily taxed good and by taking more leisure instead of work. 2. Labour income tax is highly distortionary in this model for various reasons. As before 47 percent tax rate of labour income is optimal to meet the required revenue target. It reduces the labour supply. Both output and consumption becomes smaller after such a tax is imposed. The e¢ ciency losses from switching to this sort of taxes can be up to 6.9 percent of the original utility. 3. The …rst result shows that the net deadweight loss of the current tax and transfer system is about 4 percent of GDP. GAMS programe: A1model.gms and A2model.gms, macrotax.gms Household gets utility from consuming goods and leisure M ax U = C L(1
)
(894)
c;l
Subject to p:C + w:Lh = wL
(895)
Lh + Lf = L
(896)
C > 0; L > 0;and Lf > 0; Firms’Problem: maximise pro…t M ax
= PY
Lf
w Lf
(897)
Y = Lf
(898)
Y > 0; Lf > 0; Household Problem: Maximise Utility Household gets utility from consuming goods and leisure M ax U = C L(1
)
(899)
c;l
Subject to (900)
p:C + w:Lh = wL Lagrangian optimisation: L (C; Lh ; ) = C L(1 141
)
+
wL
p:C
w:Lh
(901)
Optimal demand for goods C:solving the …rst order conditions wL = p
C=
L
(902)
p w
Buy more when goods are cheaper and when they have more income Optimal demand for leisure Lh (1
Lh = if L = 1600 and
) wL
= (1
w
= 0:4 then :Lh = 0:6
)L
(903)
1600 = 960:
Firms’Problem: maximise pro…t = PY
w Lf
(904)
Y = Lf
Lf =
P w
Y =
P w
(905) 1 1
(906) 1
(907)
Let = 0:5 Clearing Goods and Labour Markets: Real Wage Rate Y =C Lf + Lh = L; P w
Y =
C=
L p w
=
(908)
Lf = 1600 1
0:4
960 = 640
= 6400:5 = 25:29 1600 p w
= y = 25:29
p 0:4 1600 = = 25:29 w 25:29 if w = 1 set as numeraire labour market clears as Lf + Lh = 640 + 960 = 1600 = L Parameters and shadow prices 142
(909)
(910)
(911) (912)
(913)
Table 38: Parameters of the General Equilibrium Model Parameters Value 0.4 0.5 L 1600 w (normalised) 1
=
Lh 0:6 C
p
0:4
=
640 0:6 25:29
25:29
= 0:12
(914)
= 0:116
(915)
Shadow price in tax scenario
T
=
Lh 0:6 C
p
=
0:4
480 0:6 21:90
21:90
This is the change in utility associated to unit change in income. Allocations and Prices in Equilibrium Table 39: General Equilibrium Solutions Variable Base No Tax Solution Tax Solution output (Y ) 25.29 21.90 Consumption(C) 25.29 21.90 Leisure(Lh ) 960 720 Labour demand(Lf ) 640 480 Utility(U ) 224.19 178.09 Relative price wp 25.29 21.90 Shadow Price 0.12 0.116 Welfare loss to households from the government = (224:19 178:09) =224:19 = 0:2056 = 20:56%: E¤ective labour tax = 400/1600=0.25= 25%. True if households do not get utility of from public spending. How far this is true depends on the e¢ ciency of the public sector. Exercise Prove that a …rm need to pay higher wage rate to its workers and lower the price of commodity while expanding output if it operates under an increasing returns to scale technology such as Y = L2 .
5.6
Social Welfare Function
Q5. An economy is inhabited by type 1 and type 2 people. The type 1 is more productive than the type 2. Policy makers encourage productive people by assigning a greater weight to the utility of 3 1 more productive people. They aim to maximise the social welfare function: W = U14 U24 where W is the index of the social welfare, U1 represents the utility of type 1 people and U2 is the utility of type 2 people. For simplicity assume that resources of this economy produce a given level of output Y. It is consumed either by 1 or by 2 type people. Market clearing condition implies: Y = Y1 + Y2 143
. Preferences for type 1 are given by U1 = output, Y, was 1000 billion pounds.
p
Y1 and for type 2 by U2 =
p
Y2 . In a given year total
1. What is the distribution of output between type 1 and type 2 that maximises the social welfare index? What is the maximum value of the social welfare index of this economy? 2. What would have been the allocation if policy makers had given equal weight to the utility 1 1 of both types of people in the economy such as W = U12 U22 . By how much does the welfare index change in this case than compared to the social welfare in (1) above? 3. How would the social welfare index change in (1) if a tax rate of 20 percent is imposed in consumption and the tax receipts are not given back to any of these consumers? How much would the value of social welfare index be in this case? 3
1
4. Assume that the policy makers still hold the welfare function to be W = U14 U24 . How would the social welfare index change in (3 ) if all tax receipts are transferred to type 2 people? Answer Here Y = Y1 + Y2 = 1000 3
p
1
L = U14 U24 + [1000
Y1
Y2 ] = 3
p
1 4
Y2
[1000
Y1
Y2 ]
(916)
1
L = Y18 Y28
[1000
@L 3 = Y1 @Y1 8
5 8
@L = 1000 @ 5 8
Y1
Y2 ]
(917)
1
Y28
@L 1 3 = Y18 Y2 @Y2 8
3 Y 8 1
3 4
Y1
1
(918)
=0
(919)
Y2 = 0
(920)
7 8
Y1
Y28 =
=0
1 38 Y Y 8 1 2
7 8
(921)
3Y2 = Y1
(922)
1000 = Y1 + Y2 = 3Y2 + Y2 =) Y2 =
1000 = 250 4
Y1 = 3Y2 = 3 (250) = 750
(923) (924)
Index of social welfare in this economy is 3
1
W = U14 U24 =
p
750
3 4
p
250
144
1 4
3
1
= (750) 8 (250) 8 = 23:9
(925)
Answer (2) For both of them to get same level of utility: 1
p
1
L = U12 U22 + [1000
Y1
Y2 ] = 1
1
L = Y14 Y24
Y1
[1000
@L 1 = Y1 @Y1 4
3 4
[1000
Y1
Y2 ]
Y2 ]
(926) (927)
1
@L = 1000 @ 3 4
1 2
Y2
Y1
Y24
@L 1 1 = Y14 Y2 @Y2 4
1 Y 4 1
p
1 1
=0
(928)
=0
(929)
Y2 = 0
(930)
3 4
Y1
1 14 Y Y 4 1 2
1
Y24 =
3 4
(931)
Y2 = Y1
(932)
1000 = Y1 + Y2 =) Y2 =
1000 = 500 2
(933)
Y1 = Y2 = 500
(934)
Index of social welfare in this economy is 1
p
1
W = U12 U22 =
p
1 1
500
500
1 2
1
1
= 500 4 500 4 = 22:4
(935)
Answer (3) 3
1
L = U14 U24 + [1000
Y1
Y2 ] =
p 3
L = 0:8
0:8Y1
p 0:8Y2
3 4
[1000 3 Y 8 1
@L = 0:8 @Y2
1 38 Y Y 8 1 2
@L = 1000 @ 3 Y 8 1
5 8
[1000
Y1
Y2 ]
(936)
1
Y18 Y28
@L = 0:8 @Y1
0:8
1 4
5 8
Y2 ]
(937)
1
Y28
Y1
1
Y28 = 0:8 145
Y1
7 8
=0
(938)
=0
(939)
Y2 = 0 1 38 Y Y 8 1 2
(940) 7 8
(941)
3Y2 = Y1
(942)
1000 = Y1 + Y2 = 3Y2 + Y2 =) Y2 =
1000 = 250 4
Y1 = 3Y2 = 3 (250) = 750
(943) (944)
Index of social welfare in this economy is 3
1
W = U14 U24 = (0:8
3
1
750) 4 (0:8
3
1
250) 4 = (600) 4 (200) 4 = 455:9
(945)
Answer (3) If all tax is given to person 2. Y1 = 600; Y2 = 400 3
1
W = U14 U24 =
5.7
p
600
3 4
p
400
1 4
(946) 3
1
= 600 8 400 8 = 23:3
(947)
Exercise 12: Social Welfare and General Equilibrium Problem 7: General Equilibrium and Welfare Analysis
1. There are two people living in an economy. For simplicity assume that a …xed amount of output of 200 is produced each year. Entire in the same year. Utility of p output is consumed p individual 1 and 2 is represented by U1 = Y1 and U2 = 12 Y1 . (a) What is the utility received by each individual if the output is divided equally between these two people? What is the output received by each if it is distributed so that each of them gets the same amount of the utility? (b) What is the distribution of output that maximises the total utility for the whole economy? (c) If person 2 needs utility 5 in order to survive how should the output be distributed? 1
1
(d) Suppose that the authorities like to maximise the social welfare function W = U12 U22 , how should the output be distributed between them? 2. (a) An economy is inhabited by type 1 and type 2 people. The type 1 is more productive than the type 2. Policy makers encourage productive people by assigning a greater weight to the utility of more productive people. They aim to maximise the social welfare function: 3 1 W = U14 U24 where W is the index of the social welfare, U1 represents the utility of type 1 people and U2 is the utility of type 2 people. For simplicity assume that resources of this economy produce a given level of output Y. It is consumed either by 1 or by 2 type people. Marketpclearing condition implies: p Y = Y1 + Y2 . Preferences for type 1 are given by U1 = Y1 and for type 2 by U2 = Y2 . In a given year total output, Y, was 1000 billion pounds. (b) What is the distribution of output between type 1 and type 2 that maximises the social welfare index? What is the maximum value of the social welfare index of this economy? 146
(c) What would have been the allocation if policy makers had given equal weight to the 1
1
utility of both types of people in the economy such as W = U12 U22 . By how much does the welfare index change in this case than compared to the social welfare in (a) above? (d) How would the social welfare index change in (a) if a tax rate of 20 percent is imposed in consumption and the tax receipts are not given back to any of these consumers? How much would the value of social welfare index be in this case? 3
1
e. Assume that the policy makers still hold the welfare function to be W = U14 U24 . How would the social welfare index change in (c ) if all tax receipts are transferred to type 2 people? 3. Consider an economy consisting of a representative household and a representative …rm. A representative household tries to maximise utility by consuming goods and services and from enjoying leisure subject to his budget constraints. The producer wants to maximise pro…t by selling goods produced using the labour supplied by the household. The household maximisation problem can be stated as the following: max U = C l(1
)
(948)
Subject to time and budget constraints: l + hs = 1
(949)
pc = whs +
(950)
c > 0; h > 0;and l > 0; where c is consumption,l is leisure and hs is labour supply, p is the price of the commodity, w is the wage rate is the pro…t from owning the …rm. Maximisation problem for the representative …rm can be states as: s
= py
whd
(951)
subject to technology constraint as: y = (hs ) y > 0; h > 0; where y is the output supplied by the …rm and hd is its demand for labour. d
1. (a) Form a Lagrangian maximisation problem for this household. (b) Derive its demand for consumption goods and derive its demand for leisure. (c) Write the Lagrangian function for the …rm’s optimisation problem. (d) Derive …rm’s demand for labour. (e) De…ne a competitive equilibrium for this economy. (f) Compute the real wage that brings goods and labour market in equilibrium.
147
(952)
(g) What is the equilibrium quantity of c or y? (h) What is the equilibrium quantity of l and h? (i) Formulate the problem with sales and income tax. Discuss qualitatively the macroeconomic impacts of (a) switching completely to the sales taxes or (b) to labour income taxes or to (c) capital income tax.
148
5.8
Two sector model of nessecity and luxury goods (income distribtuion)
Workers and capitalists dwell in an economy. Workers consume only necessities and capitalists consume necessities and luxury goods. Workers supply all labour and capitalists save 20 percent of their income, spend 20 percent in necessities and 60 percent in luxury goods. Total labour supply is 50. LS = 50; w1 = w2 = w (953) Production function of sector i is Qi = Ai Ki i Li1
(954)
i
Table 40: Parameters in production of the K Necessity sector 0.5 100 Luxury sector 0.5 144
two sector model A 1 1
Demand for labour and supply function of necessities Table 41: Parameters in consumption of the two sector model Workers Capitalist
1
= P1 Q1
1
2
3
1 0.2
0 0.6
0 0.2
rK1 = P1 A1 K1 1 L11
wL1
@ 1 1 = (1 = P1 1 ) A1 K1 L1 @L1 Thus labour demand in necessity goods sector L0:5 1 =
5P1 ; w
0:5
1
wL1
1000:5
L1 = 25
P1 w
(
P1 w
L
rK1
0:5
(955)
w=0
(956)
2
(957)
Supply of necessity goods Q1 = A1 K1 1 L11
1
= 10L0:5 1 = 10
25
2
)0:5
;
Q1 = 50
wL2
rK2
P1 w
(958)
Demand for labour and supply function of luxuries = P2 Q2 @ = (1 @L2
rK2 = P2 A2 K2 2 L21
wL2
2 ) A2 K2
2
L2
2
= P2 149
0:5
2
1440:5
L
0:5
w=0
(959) (960)
Thus labour demand in luxury goods sector L0:5 2 =
6P2 ; w
L2 =
2
6P2 w
P2 w
= 36
2
(961)
Supply of luxury goods Q2 =
A2 K2 L22 2
2
=
12L0:5 2
(
= 12
36
P2 w
2
)0:5
; Q2 = 72
P2 w
(962)
Income of labour and capitalists Income of workers YL = wL1 + wL2 = 50w
(963)
Income of capitalists (from the production function capitalist get the same as the labour) YK = YL = 50w
(964)
Demand for necessities and luxury goods P1 Qd1 = YL + 0:2YK = 50w + 0:2 (50w) = 60w Qd1 = 60
w P1
(965) (966)
Demand for luxury goods P2 Qd2 = 0:6YK + I = 0:6 (50w) + 0:2 (50w) = 40w Qd2 = 40
w P2
(967) (968)
Market clearing conditions in goods and labour markets Q1 = 50
P1 w = Qd1 = 60 w P1
(969)
P2 w = Qd2 = 40 w P2
(970)
Q2 = 72
L1 + L2 = LS = 25
P1 w
2
+ 36
I=S
P2 w
2
= LS = 50
(971) (972)
Walras’Law: sum of excess demand is zero; when two markets clear third market automatically clears. Market clearing prices Set numeraire P1 = 1: From necessity goods market:
150
50
P1 w 1 w 5 = 60 =) 5 = 6 =) w2 = =) w = 0:913 w P1 w 1 6
(973)
From luxury goods market: 72
P2 w 5 5 = 40 =) P22 = w2 = w P2 9 9
5 25 = = 0:463 =) P2 = 0:680 6 54
(974)
Allocations: Q1 = 50 Q2 = 72
w 1 = 54:8 = Qd1 = 60 = 60 0:913 P1
P2 = 72 w
0:913 = 54:8 1
0:680 w = 53:63 = Qd2 = 40 = 40 0:913 P2
L1 = 25
2
P1 w
L2 = 36
P2 w
2
= 36
0:680 0:913
(976)
2
1 0:913
= 25
0:913 = 53:7 0:680
(975)
= 29:97
(977)
2
= 19:97
(978)
50
(979)
0:913 = 45:65
(980)
L1 + L2 = 29:97 + 19:9 Consumption: workers’demand for necessity good YL = C1;;L ;
50w = 50
Capitalist’s demand for necessity good 0:2YK = C1;K ; 0:2 ( 50w) = 0:2 (50
0:913) = 9:13
(981)
54:8
(982)
Total demand for necessity good C1;L + C1;K = 45:65 + 9:13 workers do not consumer luxury good Capitalist’s demand for luxury good C2;;K =
C2;;L = 0; 27:39 0:6YK = = 40:23 P2 0:681
(983)
Investment demand by capitalist for luxury good I 0:2 (50 0:913) 9:13 = = = 13:43 P2 P2 0:680
(984)
C2;;K + I = 40:23 + 13:43 = 53:7
(985)
GAMS programme: UK10.gms
151
Small Open Economy Trade Model: Expansionary Devaluation Open above model by including exports and imports and for simplicity …rst assume …xed output and inputs X = f K; L = C + E
(986)
Exports depends on exchange rate (e), domestic price (P) and foreign price for domestic goods ( ) and the elasticity of exports ( ) E = E0 e
(987)
P
Imports depends on exchange rate (e), domestic price (P) and price of imported goods (Pm ) and the elasticity of exports ( ) C P = K0 e m M P
(988)
Resource Balance with foreign borrowing B P X + eB = P C + ePm M
(989)
Small Open Economy Trade Model: Expansionary Devaluation Devaluation lowers the foreign price of domestic goods ( ) , it raises supply of exports (E), it reduces the amount of imports (M ) raises the production of import substitute goods. Thus both domestic and foreign demand for home products rise. Thus devaluation is expansionary. Excess of imports over exports need to be …nanced by foreign borrowing. eB = ePm M
PE
(990)
If borrowing rise, imports rise or exports fall, domestic consumption rise. Borrowing may be necessary when import prices rise or domestic prices fall. This is an over-determined system. Many things may happen Parameters of the model E0 ; K0 ; ; ; K; L; X; ; B; Pm
(991)
C; E; M; P; e
(992)
Gains from Devaluation Variables of the model
Draw a diagram of import export trade-o¤ and export supply and demand functions. Whether workers or capitalist gain depends on what kind of goods are exported and imported by external borrowing. Imported goods may contain necessary goods such as food, medicine, agricultural inputs used by workers or luxury goods such as cars, perfumes, entertainment goods for capitalists. Exports may contain necessity goods or luxury goods. Redistribution impacts of devaluation thus depend on the composition exports and imports.
152
5.9
General equilibrium model of Trade: Ricardian Comparative Advantage Theory
Theory of international trade has developed over time in works of Ricardo (1817), Ohlin (1933), Stopler-Samuelson (1947), Bhagawati Helpman (1976), Dixit and Stiglitz (1977), Meade(1978), Krugman (1980), Whalley (1985), Neary (1988),Krugman and Venables (1995), Hine and Wright (1998), Edwards(1993), Eaton and Kortum (2002), Markusen (1995), Taylor (1995), Raut and Ranis (1999), Roe and Mohladi (2001), Greenaway, Morgan and Wright (2002),Melitz (2003), Beaulieu, Benarroch and Gaisford (2004). These theories are applicable in explaining real world problems ( eg.Bhattarai and Whalley (2006), Bhattarai and Mallick (2003)). As the growth rate of global trade is greater than the growth of the global GDP there the space of globalization is rapid and it is a¤ecting lives of millions of people. Trade models, essentially involve application of the general models to answer who gains and who loses from exchange of goods and services across borders. Essentially it is application of basic microeconomic theories at international scale. 5.9.1
Two Country Ricardian Trade Model There are two countries indexed by j, producing two goods, manufacturing and services. Each of them have an option to be self reliant or to trade on the basis of comparative advantage. Under the import substituting industrialisation (ISI) regimes countries favoured to be self reliant and infant industries were protected by tari¤s and non-tari¤ barriers. After numerous rounds of trade negotiations under GATT/WTO over the years, all countries now have realised that the autarky solutions like this are economically ine¢ cient. In contrast globalisation is a norm. trade is mutually bene…cial for trading nations and raises welfare in both countries. Aim of this model is to illustrate on these statements analytically and numerically with a small and transparent example. For this it is assumed that each country j specialises in commodities in which it is more e¢ cient and engages in trade. The exchange rate is determined by the relative prices of two commodities in the global market.
Two Country Ricardian Trade Model Preferences in country 1 are expressed by its utility function in consumption of good 1 and 2 , C1;1 and C1;2 respectively: max
U1 = (C1;1 )
1
(C1;2 )
1
1
(993)
Income of country 1 is obtained from the wage income in sector 1 and sector 2 plus the transfers to country 1 I1 = w1;1 L1;1 + w1;2 L1;2 + T R1 153
(994)
where L1;1 and L1;2 are labour employed in sector 1 and sector 2 w1;1 and w1;2 are corresponding wages respectively and T R1 is the transfer income. Technology constraints in sector 1 in country 1 X1;1 = a1;1 :L1;1
(995)
where a1;1 is the productivity of labour in sector 1 in country 1. Technology constraints in sector 2 in country 1 X1;2 = a1;2 :L1;2
(996)
where a1;2 is the productivity of labour in sector 2 in country 1. Resource constraint in country 1 de…ned by the labour endowment as: L1 = L1;1 + L1;2
(997)
Production possibility frontier of country 1 now can be de…ned as L1 =
1 1 :X1;1 + :X1;2 a1;1 a1;2 X11 = a11 :L11
(998) (999)
Given above preferences the demand for good 1 in country 1 is C1;1 =
1 :I1 P1
(1000)
the demand for good 2 therefore is: C1;2 = 5.9.2
(1
1 ) :I1
P2
(1001)
Autarky or Trade Theoretically two trade arrangements are possible in this model. First one is an autarky equilibrium in which each country is separate and isolated from another. It produces just for its own consumption and no trade take place between these two countries. Such autarky solution is close to the production arrangement when countries were adopting ISI trade strategy.
Analytical solutions of autarky and specialisation Proposition 1 Autarky solution is Pareto dominated by trade equilibrium for reasonable parameters of preferences and technology. This is proven below by analytical and numerical solutions.
154
A Lagrangian function is used to express how each country maximises welfare subject to its production possibility frontier constraint under the autarky equilibrium as: (1
$1 = X1;11 X2;1
1)
+
1 X1;1 a11
L1
First order conditions with respect to X11 and X21 and @$1 = @X1;1
1 X1;1 1
1
(1
1 X2;1 a12
as:
1)
X2;1
(1002)
a11
=0
(1003)
Analytical solutions in autarky @$1 = (1 @X2;1
( 1 1 ) X1;1 X2;1
@$1 = L1 @
1 X1;1 a11
1)
a12
=0
(1004)
1 X2;1 = 0 a12
(1005)
Analytical solutions of in autarky 1
1 1 X1;1
From the …rst two …rst order conditions
(1
X2;1 =
(1
X2;1
1)
( 1 1) 1 )X1;1 X2;1
(1
1) 1
=
1
(1
1)
X2;1 X1;1
=
a12 a11
a12 X1;1 a11
(1006)
optimal value of X1;1 is found now putting this condition in the production possibility frontier constraint. 1 1 1 (1 1 X1;1 + 1 X2;1 = 1 X1;1 + 1 a11 a2 a1 a2
1) 1
1 a12 (1 X1;1 = 1 X1;1 1 + a11 a1
1)
= L1
(1007)
1
Analytical solutions of in autarky 1 1 a1 L1
X1;1 =
(1008)
Similarly the optimal value of X2;1 is found by X2;1 =
(1
1) 1
a12 (1 X1;1 = a11
1) 1
a12 a11
1 1 a1 L1
= (1
1 1 ) a2 L1
(1009)
For each of 1 country amount produced depends on productivity and preferences parameters and the endowment of its labour input. The autarky welfare level is: U 1 = (X1;1 )
1
1
(X2;1 )
1
=
1 1 a1 L1
1
(1
(1 1 1 ) a2 L1
1)
(1010)
Two Country Ricardian Trade Model Preferences in country 2 are expressed by its utility function in consumption of good 1 and 2 , C2;1 and C2;2 respectively:
155
max
U2 = (C2;1 )
2
(C2;2 )
1
2
(1011)
Income of country 2 is obtained from the wage income in sector 1 and sector 2 plus the transfers to country 2 I2 = w2;1 L2;1 + w2;2 L2;2 + T R2
(1012)
where L2;1 and L2;2 are labour employed in sector 1 and sector 2 w2;1 and w2;2 are corresponding wages respectively and T R2 is the transfer income. Technology constraints in sector 1 in country 2 X2;1 = a21 :L2;1
(1013)
where a2;1 is the productivity of labour in sector 1 in country 2. Technology constraints in sector 2 in country 2 X2;2 = a2;2 :L2;2
(1014)
where a2;2 is the productivity of labour in sector 2 in country 2. Resource constraint in country 2 de…ned by the labour endowment as: L2 = L2;1 + L2;2
(1015)
Production possibility frontier of country 2 now can be de…ned as L2 =
1 1 :X2;1 + :X2;2 a2;1 a2;2
(1016)
Given above preferences the demand for good 1 in country 2 is C2;1 =
2 :I2
P1
(1017)
the demand for good 2 therefore is: C2;2 =
(1
2 ) :I2
P2
(1018)
Autarky or Trade Theoretically two trade arrangements are possible in this model. First one is an autarky equilibrium in which each country is separate and isolated from another. It produces just for its own consumption and no trade take place between these two countries. Such autarky solution is close to the production arrangement when countries were adopting ISI trade strategy. Proposition 2 Autarky solution is Pareto dominated by trade equilibrium for reasonable parameters of preferences and technology. This is proven below by analytical and numerical solutions.
156
Analytical solutions in autarky A Lagrangian function is used to express how each country 2 maximises welfare subject to its production possibility frontier constraint under the autarky equilibrium as: (1
$2 = X1;22 X2;2
2)
+
1 X1;2 a2;1
L2
First order conditions with respect to X12 and X22 and @$2 = @X1;2
1
2 X1;2 2
@$2 = (1 @X2;2
(1
X2;2
(1019)
=0
(1020)
as:
2)
a2;1
( 2 2 ) X1;2 X2;2
1 X2;2 a2;2
2)
a2;2
=0
(1021)
Analytical solutions in autarky 1
@$2 = L2 @
a2;1
From the …rst two …rst order conditions X2;2 =
1
X1;2
a2;2
1
2 2 X1;2
(1
X2;2
X2;2 = 0
2)
( 2 2) 2 )X1;2 X2;2
(1
(1
2) 2
(1022)
=
2
(1
2)
X2;2 X1;2
=
a2;2 a2;1
a2;2 X1;2 a2;1
(1023)
optimal value of X1;2 is found now putting this condition in the production possibility frontier constraint. 1 1 1 (1 1 X1;2 + X2;2 = X1;2 + a2;1 a2;2 a2;1 a2;2
2) 2
a2;2 1 (1 X1;2 = X1;2 1 + a2;1 a2;1
2)
= L2 (1024)
2
Analytical solutions in autarky X1;2 =
2 a2;1 L2
(1025)
Similarly the optimal value of X2;2 is found by X2;2 =
(1
2) 2
a2;2 (1 X1;2 = a2;1
2) 2
a2;2 a2;1
2 a2;1 L2
= (1
2 ) a2;2 L2
(1026)
For each of 2 country amount produced depends on productivity and preferences parameters and the endowment of its labour input. The autarky welfare level is: U 2 = (X1;2 )
2
1
(X2;2 )
2
=(
2 a2;1 L2 )
2
((1
(1 2 ) a2;2 L2 )
2)
(1027)
Summary of two country trade model in Autarky Thus the level of welfare in country 1 is determined in terms of its preferences for consumption of good 1 and 2 as re‡ected by 1 and its own production technology as re‡ected in a11 and a12 .
157
Numerical version of this model is applied to country 1 and country 2 taking the population as rough indicator of its resource in production. country 1 has 200 million population and country 2 has 400 million population. country 1 is more productive in producing services goods X1 whereas country 2 has more advantage in producing manufacturing goods X2 . Preferences are similar but technologies are di¤erent. These parameters are set out in Table 1. Table 42: Parameters of the Autarky Model a1 a2 L country 1 country 2
0.4 0.6
5 2
2 5
200 400
Summary of two country trade model in autarky Under the autarky equilibrium these two economies are completely isolated and produce only for domestic consumption. The optimal production and consumption and employment of labour for both sectors, prices of commodities and labour, and utility for the representative hocountry 1ehold are as given in Table 2. In per capita terms citizens of the country 1 and country 2 have welfare of 1.46 and 1.76 respectively.
Table 43: Parameters of the Autarky Model X1 X2 L1 L2 U p2 country 1 country 2
400 480
240 800
80 240
120 160
294.4 588.8
0.6 1.67
Each country produces both goods in no trade equilibrium which as explained here is very ine¢ cient. Welfare can be improved by making these countries trade. Analytical solutions for trade equilibrium under specialisation A representative hocountry 1ehold in each country maximises its welfare subject to its budget constraint. Demand for goods are derived by standard constrained optimisation on supply side for each country j . Under trade equilibrium it is optimal for each country to specialise in goods in which it has comparative advantage. The optimisation problem and the …rst order conditions for constrained optimisation are given as follows: (1
$j = X1;jj X2;j
j)
+ [Ij
P1 X1;j
P2 X2;j ]
(1028)
First order conditions: @$j = @X1;j @$j = (1 @X2;j
j
j X1;j
1
(1
X2;j
j)
( j j ) X1;j X2;j
j)
P1 = 0 P2 = 0
Analytical solutions for trade equilibrium under specialisation
158
(1029) (1030)
@$j = Ij @ j
j X1;j
(1
1
P1 X1;j
(1
P2 X2;j = 0
j)
X2;j
( j ) X1;j X2;j
j)
j
X2;j =
=
(1
j
(1 j)
j
P1 X1;j + P2 X2;j = P1 X1;j + P2
(1
j) j
X1;j =
P1 P2 X1;j
(1031)
X2;j P1 = P2 j ) X1;j
P1 X1;j P2
(1032)
(1033)
= Ij
(1 j Ij j ) Ij ; X2;j = P1 P2
(1034)
Analytical solutions for trade equilibrium under specialisation Global market clearing conditions for goods 1 and 2 are N X
X1;j = X1
(1035)
N X
X2;j = X2
(1036)
j
j
Prices adjcountry 1t until this equilibrium condition holds. Under complete specialisation, country 1 country 1 specialises in services X2 and produces 1825 units of it. country 2 specialises in manufacturing X1 goods and produced 6000 units of it. It is easy to determine country 2’s income if we choose good 1 as numeraire setting P1 = 1. Analytical solutions for trade equilibrium under specialisation I 1 = P1 X1 = 1
1000 = 1000
(1037)
Relative price of good 2, P2 need to be determined to …nd the level of income in the country 1. This can be done using the global market clearing condition 1
2 2 :I 1 :I + = 0:4 (1000 P1 P1
1) + 0:6 (2000
P2 ) = 1000
600 1000 400 = = 0:5 1200 1200 Now it is easy to determine the income of the country 1 as:
(1039)
P2 =
I 2 = P2 X2 = 200
5
P2 = 2000
P2 = 1825
(1038)
0:5 = 1000
(1040)
Analytical solutions for trade equilibrium under specialisation Since income level for both country 2 and the country 1 are determined, it is now easy to determine the level of demand in both countries:
159
X1;1 =
X2;1
1 I1 2 I2 = 0:4 (1000) = 400; X1;2 = = 0:6 (1000) = 600 P1 P1
= =
(1
0:6 (1000) (1 1 ) I1 2 ) I2 = = 1200; X2;2 = P2 0:5 P2 0:4 (1000) = 800 0:5
(1041)
(1042) (1043)
Solutions of both autarky and trade equilibria are given in Table 3 and 4. Given the preferences and technology speci…cations, with complete specialisation both countries gain from trade. Comparative static analysis of trade can be done changing the preference or technology parameters. Analytical solutions for trade equilibrium under specialisation Table 44: Comparing Specialisation and Autarky Regimes Production Autarky Trade country 1 country 2
Consumption Autarky Trade
X1
X2
X1
X2
C1
C2
C1
C2
400 480
240 800
1000 0
0 2000
400 480
240 800
400 600
1200 800
P 1 0.5
Analytical solutions for trade equilibrium under specialisation Table 45: Comparing Employment and Welfere under Specialisation and Autarky Employment Autarky Trade
Uitlity Autarky Trade
L1
L2
L1
L2
U
80 240
120 160
200 0
0 400
294.4 588.8
U 773.3 673.2
Gains from trade may be distributed di¤erently across countries (Bhattarai and Whalley (2006)). Further there are opportunities for bargaining on the share of those gains particularly from dynamic strategic considerations and the basic elements required for such dynamic model is provided in the next section. GAMS programme: trade.gms; trade_2.gms Beaulieu E, M. Benarroch and J. Gaisford (2004) Trade barriers and wage inequality in a North-South model with technology-driven intra-industry, trade, Journal of development Economics, 75:113-136 Bhattarai K. and S. Mallick (2013) Impact of China’s currency valuation and labour cost on the US in a trade and exchange rate model. North American Journal of Economics and Finance, 25, 40-59. Bhattarai K and J Whalley (2006), Division and Size of Gains from Liberalization of Trade in Services, Review of International Economics, 14:3:348-361, August. 160
Dixit A K and J E. Stiglitz (1977) Monopolistic Competition and Optimum Product Diversity, American Economic Review, 67:3:297-308. Eaton J and S. Kortum (2002) Technology, Geography, and Trade, Econometrica, 70: 5:Sep:17411779 Edwards S. (1993) Openness, Trade Liberalization, and Growth in Developing Countries, Journal of Economic Literature, 31: 3 :1358-1393, September. Greenaway D. W. Morgan and P. Wright (2002) Trade Liberalisation and Growth in Developing Countries, Journal of Development Economics, vol. 67 229-244. Helpman E (1976) Macroeconomic Policy in a Model of International Trade with a Wage Restriction, International Economic Review, 17:2:262-277. Hine R.C. and P.W. Wright (1998) Trade with Low Wage Economies, Employment and Productivity in UK Manufacturing, Economic Journal, 108:450:1500-1510. Krugman P. (1980) Scale Economies, Product Di¤erentiation and the Pattern of Trade, American Economic Review, 70:5:950-959. Markusen. J, R. (1995) The boundaries of multinational enterprises and the theory of international trade, Journal of Economic Perspective, 9:2:169-189. Meade, James (1978) The Meaning of Internal Balance, Economic Journal, 88 (351): Sep 423-435. Melitz M. T. (2003) The Impact of Trade on Intra-Industry Reallocations and Aggregate Industry Productivity, Econometrica, 71:6:1695-1725. Neary P.J. (1988) Determinants of the Equilibrium Real Exchange Rate, American Economic Review, 78:1:Mar.: 210-215. Ohlin B (1933) Interregional and international trade, Harvard economic studies ; no.39 Raut L and G Ranis (eds.) Trade, Growth and Development: Essays in Honour of Professor T N Srinivasan, Contribution to Economic Analysis 242, Elsevier, NorthHolland, Amsterdam, 1999 Roe T and H. Mohladi (2001) International Trade and Growth: An Overview Using the New Growth Theory, Review of Agricultural Economics, 23:2:423-440 Taylor M. P. (1995) The Economics of Exchange Rates, Journal of Economic Literature, March, vol 33, No. 1, pp. 13-47. Whalley, J. (1985) Trade Liberalization Among Major World Trading Areas, MIT Press
161
5.10
Exercise 12’: migration and factor mobility Problem 8: Migration or Factor Movement Across Countries
1. Consider trade between two countries. One is abundant in capital and another in labour. For simplicity assume that the production functions of these economies are given by Y1 = K1 1 L1 1
(1044)
Y2 = K2 2 L2 2
(1045)
Table 46: Endowment and Technology Country 1 Country 2
K 500 1000
L 1000 500
0.4 0.6
0.6 0.4
Output ? ?
1. (a) What will be the rental rate of capital and wage rate in each country if both goods and factors are immobile across countries? (b) What will be the rental rate and output if there is a global market for capital and labour? Explain the pattern of migration across countries. (c) Is free trade equivalent to free mobility of factor of production according to HeckscherOhlin-Stopler-Samuelson theorem? (d) Trade is not bene…cial to every one. Discuss how labour in labour abundant and capitalists in capital abundant countries gain from trade on the basis of this model. (e) Show that in a static world like this aggregate global income remains the same but there is a change in the distribution of income. Beaulieu E, M. Benarroch and J. Gaisford (2004) Trade barriers and wage inequality in a North-South model with technology-driven intra-industry, trade, Journal of development Economics, 75:113-136 Bhattarai K and J Whalley (2006), Division and Size of Gains from Liberalization of Trade in Services, Review of International Economics, 14:3:348-361, August. Dixit A K and J E. Stiglitz (1977) Monopolistic Competition and Optimum Product Diversity, American Economic Review, 67:3:297-308. Greenaway D. W. Morgan and P. Wright (2002) Trade Liberalisation and Growth in Developing Countries, Journal of Development Economics, 67 229-244. Helpman E (1976) Macroeconomic Policy in a Model of International Trade with a Wage Restriction, International Economic Review, 17:2:262-277. Hine R.C. and P.W. Wright (1998) Trade with Low Wage Economies, Employment and Productivity in UK Manufacturing, Economic Journal, 108:450:1500-1510. 162
Krugman P. (1980) Scale Economies, Product Di¤erentiation and the Pattern of Trade, American Economic Review, 70:5:950-959. Melitz M. T. (2003) The Impact of Trade on Intra-Industry Reallocations and Aggregate Industry Productivity, Econometrica, 71:6:1695-1725. Roe T and H. Mohladi (2001) International Trade and Growth: An Overview Using the New Growth Theory, Review of Agricultural Economics, 23:2:423-440 Taylor Mark (1995) The Economics of Exchange Rates, Journal of Economic Literature, March, vol 33, No. 1, pp. 13-47.
5.11
General equilibrium with taxes
Optimal Tax and Public Goods max U h = (1
) ln Y h
Th +
ln G
(1046)
subject to P Yh
Th + G = I
h
(1047)
h
where V is the utility of households, Y T is the net of tax income, G the public good and the weight in utility from consumption of public goods. The production side of the economy is represented here by income for simplicity. L Y h ; G;
= (1
) ln Y h
Th +
ln G +
I
P Yh
Th
G
(1048)
Optimal Tax and Public Goods L Y h ; G; @Y h
=
(1 (Y h
) T h)
P =0
(1049)
L Y h ; G; = =0 (1050) @G G This implies optimal public good and optimal tax correspond to preference for public goods (1 (Y h
) G =P T h)
(1051)
G = PTh = PT
(1052)
G = PY
(1053)
Samuelson’s Theory on Optimal Public Spending Sum of the marginal rate of substitution of all citizens should equal marginal cost of providing public goods (see two citizen public good model) Consumers consume private (x) and public goods (G) 163
max
u1 = u1 (x1 ; G)
(1054)
subject to a given level of utility for the second consumer max u2 = u2 (x2 ; G)
(1055)
x1 + x2 + c (G) = w1 + w2
(1056)
and the resource contraint
Constrained optimisation for this is L = u1 (x1 ; G)
[u2
u2 (x2 ; G)]
[x1 + x2 + c (G)
w1
w2 ]
Samuelson’s Theory on Optimal Public Spending @u1 (x1 ;G) (x1 ;G) @L @L = 0 =) = @u1@x ; ; @x = @x1 = @x1 1 1 @L @G
=
@u1 (x1 ;G) @G @u1 (x1 ;G) @G @u1 (x1 ;G) @x1
@u2 (x2 ;G) @G
+
@u2 (x2 ;G) @G @u2 (x2 ;G) @x2
=
@c(G) @G
= 0; =)
@c (G) ; @G
@u2 (x2 ;G) (x2 ;G) = 0 =) @u2@x2 @x2 @u2 (x2 ;G) 1 @u1 (x1 ;G) = @c(G) @G @G @G
M RS1 + M RS2 = M C(G):::Q:E:D:
(1057) =
(1058)
Simplest General Equilibrium Tax Model: Demand Side Households problem max U = C L
(1059)
p (1 + t) C + wL = wL
(1060)
Subject to
Lagrangian for houshold optimisation L (C; L; ) = C L +
p (1 + t) C + wL
wL
(1061)
Household Optimisation L (C; L; ) = L + p (1 + t) = 0 @C
(1062)
L (C; L; ) =C+ w=0 @L
(1063)
L (C; L; ) = p (1 + t) C + wL wL = 0 @ Above three FOC equations (1403) - (1064) can be solved for three variables : M RSCL =
L(C;L; ) @C L(C;L; ) @L
=)
164
L p (1 + t) = C w
(1064)
(1065)
Household Optimisation L=
p (1 + t) C w
(1066)
Putting (1410) into (1064) p (1 + t) c + wL
wL = p (1 + t) C + w
wL = 0
(1067)
1 wL 2 p (1 + t)
(1068)
p (1 + t) p (1 + t) 1 wL 1 C= = L w w 2 p (1 + t) 2
(1069)
C=
L=
p (1 + t) C w
Demand for goods is low with higher taxes and prices, high with higher wage rate and labour endowment; high with the higher share of spending on goods and services.Given these preferences the demand for leisure is half of the labour endowment. Supply Side of the General Equilibrium Model Firms’pro…t maximisatin problem max
= p:Y
w:LS
(1070)
Subject to p
r
1 L (1071) 2 Consumers pay tax not the producers. In no tax case, given this production technology and demand side derivations, labour demand equals labour supply Labour market clearing Y =
LS =
L + LS = L
(1072)
C=Y
(1073)
Goods market clearing
Let total labour endowment L be 200. Then labour supply is 1 1 L = L = 100 2 2 Given this labour supply the level of output will be r p p 1 Y = LS = L = 100 = 10 2 From the zero pro…t condition required equilibrium = p:Y numeraire p = 1 LS = L
L=L
165
(1074)
(1075) w:LS = 0 and setting the
p:Y = w:LS
(1076)
Y 10 1 = = (1077) LS 100 10 Numerical Example of the General Equilibrium Model 1 both goods and labour market clear Given the equilibrium relative wage rate of w = 10 when t = 0, demand for good eqauls supply as: w=
C=
1 wL 1 = 2 p (1 + t) 2
1 (200) = 10 10
(1078)
Similarly the demand for labour and leisure equal total endowment of labour L + LS = 100 + 100 = 200 = L
(1079)
Labour market clears. Therefore this is a general equilibrium; at these prices both goods and labour market clear, household maximise utility and …rms maximise pro…t. When t = 0:2 then 1 wL 1 1 (200) C= = = 8:33 (1080) 2 p (1 + t) 2 10 1:2 Government revenue and spending: R = p:t:C = 1
0:2
8:33 = 1:67 = G
(1081)
Markets clear in this case too C + G = 8:33 + 1:67 = 10 = Y
(1082)
Houswhold’s welfare before tax U = C L = 10 100 = 1000
(1083)
U = C L = 8:33 100 = 833
(1084)
Welfare after tax Thus 20 percent tax has reduced the household welfare by 16.7 percent. Can utility from public spending of compensate for this lost welfare? Household Problem: Maximise Utility Household gets utility from consuming goods and leisure M ax U = C L(1
)
(1085)
c;l
Subject to p:C + w:Lh = wL
(1086)
Lagrangian optimisation: L (C; Lh ; ) = C L(1 166
)
+
wL
p:C
w:Lh
(1087)
Optimal demand for goods C:solving the …rst order conditions wL = p
C=
L
(1088)
p w
Households buy more when goods are cheaper and when they have more income Optimal demand for leisure Lh (1
:Lh = if L = 1600 and
) wL
= (1
w
= 0:4 then :Lh = 0:6
)L
(1089)
1600 = 960:
Firms’Problem: maximise pro…t = PY
w Lf
(1090)
Y = Lf
Lf =
P w
Y =
P w
(1091) 1 1
(1092) 1
(1093)
Let = 0:5 Clearing Goods and Labour Markets: Real Wage Rate Y =C Lf + Lh = L; P w
Y =
C=
L p w
=
(1094)
Lf = 1600 1
0:4
960 = 640
= 6400:5 = 25:29 1600 p w
= y = 25:29
p 0:4 1600 = = 25:29 w 25:29 if w = 1 set as numeraire labour market clears as Lf + Lh = 640 + 960 = 1600 = L Parameters and shadow prices 167
(1095)
(1096)
(1097) (1098)
(1099)
Table 47: Parameters of the General Equilibrium Model Parameters Value 0.4 0.5 L 1600 w (normalised) 1
=
Lh 0:6 C
0:4
=
p
640 0:6 25:29
25:29
= 0:12
(1100)
= 0:116
(1101)
Shadow price in tax scenario
T
=
Lh 0:6 C
p
=
0:4
480 0:6 21:90
21:90
This is the change in utility associated to unit change in income. Allocations and Prices in Equilibrium Table 48: General Equilibrium Solutions Variable Base No Tax Solution Tax Solution output (Y ) 25.29 21.90 Consumption(C) 25.29 21.90 Leisure(Lh ) 960 720 Labour demand(Lf ) 640 480 Utility(U ) 224.19 178.09 Relative price wp 25.29 21.90 Shadow Price 0.12 0.116 Welfare loss to households from the government = (224:19 178:09) =224:19 = 0:2056 = 20:56%: E¤ective labour tax = 400/1600=0.25= 25%. True if households do not get utility of from public spending. How far this is true depends on the e¢ ciency of the public sector.
5.12
Exercise 13: Monopolistic Competition
Problem 9: Monopolistic Competition 1. Using geometric method prove the Heckscher-Ohlin theorem that a country will export the commodity that uses its relatively abundant factor with unique relation between prices of factors and products making the commodity trade complete substitutes for trade in factors. (hint: constant return to scale, free trade in goods but complete immobility of factors of production; use PPP and Edgeworth boxes for Xand Y and K and L,px =py and w and r). 2. Consider a …rm in monopolistically competitive industry Q=A 168
B P
(1102)
Prove that its marginal revenue is given by Q B
MR = P
(1103)
(a) If the cost function is C = F + cQ then prove that the average cost declines because of the economy of scale. (b) Further assume that the output sold by a …rm, number of …rms, its own price and average prices of …rms are given by Q=S
1 N
b P
P
(1104)
show that the average cost rises to number of …rms in the industry when all …rms charge same price. AC = n:F s +c (c) Prove that price charged by a particular …rm declines with the number of …rms P = c + b1n (d) Determine the number of …rms and price in equilibrium. Explain entry exit behavior and prices when number of …rms are below or above this equilibrium point. (e) Collusive and strategic behaviors may limit above conclusions. Discuss. (f) Apply above model to explain international trade and its impact on prices and number of …rms in a particular industry. (g) Use this model to explain interindustry and intra-industry trade. (h) Use monopolistic competition model to analyse consequences of dumping practices in international trade. Problem of a Multinational Corporation 1. Assume that the MNC has home and foreign markets, faces distinct demand curves across two countries and faces di¤erent cost curves. Home market is more lucrative than the foreign market both in terms of prices and cost e¤ectiveness. Despite that the MNC has a global ambition, therefore it aims to extend its business in overseas markets. The main objective of the MNC is to control the market and to maximise pro…t. Demand in home country P1 = 130
Q1
(1105)
and associated cost function is C1 = 10Q1
(1106)
Demand in the foreign (host) country P2 = 90 169
Q2
(1107)
and associated cost function is C2 = 20Q2
(1108)
1. (a) what will be output price and welfare under perfect competition? (b) What will be price, output and welfare in the host country if the monopolist forms a cartel with the …rm in the host country? (c) What will be price, output and welfare if the MNC plays Cournot game with the …rm in the host country? (d) How will above result change if the monopolists acts as a price leader in Stackelberg equilibrium? (e) How will above result change if both …rms play under a Bertrand equilibrium adopting a predatory pricing strategy? Atkinson A. B.and N. H. Stern (1974) Pigou, Taxation and Public Goods The Review of Economic Studies, 41:1:119-128. Bhattarai K (2010) Taxes, public spending and growth in OECD countries, Journal of Perspective and Management, 1/2010. Bhattarai K and J. Whalley (2009) Redistribution E¤ects of Transfers, Economica 76:3:413431 July. Blundell R (2010) Empirical Evidence and Tax Policy Design: Lessons from the Mirrlee’s Review, Institute of Fiscal Studies. Darling A. Chancellor of Exchequer, HM Treasury (2009), Securing the Recovery: Growth and Opportunity, Pre-Budget Report, December , 2009. Feldstein M (1974) Incidence of Capital Income Tax in a Growth Economy with Varying Saving Rates, Review of Economic Studies, 41:4:505-513 Fullerton, D., J. Shoven and J. Whalley (1983) Dynamic General Equilibrium Impacts of Replacing the US Income tax with a Progressive Consumption Tax, Journal of Public Economics 38, 265-96. Meade J (1978) Structure of Direct Taxation, Institute of Fiscal Studies, London. Mirlees, J.A. (1971) An exploration in the theory of optimum income taxation,Review of Economic Studies, 38:175-208. Perroni, C. (1995), Assessing the Dynamic E¢ ciency Gains of Tax Reform When Human Capital is Endogenous, International Economic Review 36, 907-925. Main budget: http://www.hm-treasury.gov.uk/; Green Budget: http://www.ifs.org.uk/
170
6
L6: Game theory: Bargaining in Goods and Factors markets
In many circumstances optimal decisions of an economic agent depends on decisions taken by others. Dominants …rms competing for a market shares, political parties contesting for power and research and scienti…c discoveries aimed for path-breaking innovations are in‡uenced by decision taken by others. In all these circumstances there are situations where collective e¤orts rather than individual ones generate the best outcome for the group as a whole and for each individual members of the group. Concepts of bargaining, coalition and repeated games developed over years by economists such as Cournot (1838), Bertrand (1883), Edgeworth (1925) von Neumann and Morgenstern (1944) and Nash (1950, 1953) is developing very fast in recent years following works of Kuhn (1953), Shapley (1953),Shelten ( 1965) Aumman (1966) Scarf (1967), Shapley and Shubik (1969), Harsanyi(1967), Spence (1974), Hurwicz (1973), Myerson (1986), Maskin and Tirole (1989), Kreps (1990), Fundenberg and Tirole (1991) and Binmore (1992), Rubinstein (1982) Sutton (1986) Cho and Kreps (1987) Sobel (1985) Machina (1987) Riley (1979) McCormick (1990), Ghosal and Morelli (2004) and others. These have generated models that can be applied to analyse the relative gains from coalitions rather than without these coalitions. Outcome of a noncooperative games can be more easily explained by Nash bargaining game that is popularly known as a game of splitting a pie between two parties, right or left. Rule of this game is that the sum of the shares of the pie claimed by players cannot exceed more than 1, otherwise each will get zero. Standard technique to solve this problem is by maximising the Nash Product. It is natural that economic agents play a zero sum and non-cooperative game until they realise the bene…ts of coalition and cooperation. When an agreement is made and cooperation is achieved there is a question on whether such coalition is stable or not. There are always incentives at least for one of the player to cheat others from this cooperative agreement in order to raise its own share of the gain. However, it is unlikely that any player can fool all others at all the times. Others will discover such cheating sooner or later. Therefore a peaceful solution requires credibility and a punishment mechanism by which any party that tries to cheat on the agreement is punished unless it amends its uncooperative behaviours toward others. A coalition of players should ful…l individual rationality, group rationality and coalition rationality. These can be ascertained by the supper-additivity property of coalition where the maximisation of gain requires being a member of the coalition rather than playing alone. The imputations in the core guarantees each member of a coalition the value at least as much as it could be obtained by playing independently. At the core of the game each player gets at least as much from the coalition as from the individual action, there does not exist any blocking coalition. This is equivalent to Pareto optimal allocation in a competitive equilibrium (Scarf (1967)). Some imputations are dominated by others; the core of the game is the strong criteria for dominant imputation. Core satis…es coalition rationality. Ability of a player to in‡uence the outcome of the game depends on the pivotal status enjoyed by that player. In a game with 3 players; power of player i is re‡ected by its Shapley value. Consider six possible ordering of 123 pivotal game. Three players can order themselves in 3!= 6 ways. Each of these number can appear only twice in the middle out of six possible combinations. A player located in the middle is pivotal. If parties realise this fact while bargaining, such bargaining is likely to generate a stable and cooperative solution. When intention cannot be directly revealed or stated players can signal indirectly to other players. These signals can take many forms. Signalling plays important roles in strategic choices of 171
individuals, parties, communities, regions, national and the global community as a whole. Formation of payo¤ discussed above depends on signalling - players do not know the moves of their opponents but based on their interpretation of signal they can however, put some numerical values to the payo¤. A rational player interprets signals correctly and chooses actions that support each other. This brings that player up in the progress ladder. Wrong interpretation of signals results in status quo or even a gradual decline in the standard of that player. Success in the game thus relates very much on ability and dexterity in providing right signals and accurate interpretation of signals coming from other players. Interpreting those signals correctly and translating them into actions more accurately brings success; sending wrong signals or interpreting them incorrectly is a recipe of failure. Status of player i, si is thus a stochastic process that depends on ability of signal extraction. Such ability depends on intuition and information set i . Very few games are plaid only once. Economic agents, political parties, live for a long time and play games repeatedly. Economists have applied Cournot-Nash bargaining game of oligopoly to explain the consequences of cooperative and non-cooperative games on the division of gains from bargaining. It can quantitatively be illustrated using a Nash bargaining oligopoly model. Players often do not have enough information about other players in the game. They have to guess intention of other players looking at their choices. People are principals of a political game, they want better standard of living, peace and prosperity in a country but they do not have enough information about the true intention of the members of political parties act as their agents and should in principle be responsible for their principals - the common people who elect political parties frequently in the parliament. Once elected party with majority forms the government and tries to ful…l its collective interest. Political contracts are as similar as wage contracts in a labour market that are designed to match e¤orts put by a worker to their productivities in the labour maker. Political parties know their type but the people do not.Thus in the presence of information asymmetry , the e¤orts by the good party is at the …rst best level as the bad e¤ort by him is not as attractive as the good e¤ort, it is not pro…table for a good party to pretend to be bad party. Good party is not attracted by the contract for the bad party. Similarly it is costly for the bad party to act as a good party - it is optimal for it to select the contract appropriate for a bad party, that is being out of the o¢ ce. Cournot (1838), Bertrand (1883), Edgeworth (1925) von Neumann and Morgenstern (1944), Nash (1950); Kuhn (1953), Shapley (1953),Shelten (1965) Aumann (1966) Luce and Rai¤a (1957) Scarf (1967), Shapley and Shubic (1969), Harsanyi (1967), Spence (1974), Myerson (1986), Kreps (1990), Fundenberg and Tirole (1991) and Binmore (1992), Roth (2008); Sobel (1985), Hey (1987), Kreps (1990) Mirlees (1971), Maskin and Moore (1999), Maskin and Tirole (1992), Cripps (1997), Perlo¤ (2013)Osborne and Robinstein (1994) Cripps and Thomas (1995), Gardener (2003) Bhaskar and To (2004), Mailath and Samuelson (2006) Texts: Holt (2007), Rasmusen (2007), Dixit, Skeath and Reiley (2009), Varian (2010), Perlo¤ (2014) 172
6.1
Formal de…nitions
Strategic form game is a tuple G = (Si ; ui ) for each player i = 1; :::; N where Si is the strategy available to player i and ui : xN j=1 Sj ! R is payo¤ of play i. It is …nite if the strategy set contains …nitely many elements. For instance strategy set of column player Sj = fR; M; Lg and of the row player Sj = fT; M; Bg R M L T 3,0 0,-5 0,-4 M 1,-1 3,3 -2,4 B 2,4 4,1 -1,8 Strictly dominated strategy ui (b si ; si ) > ui (si ; s i ) for (si ; s i ) 2 S and sbi 6= si . Eliminate the dominated strategies for row and column player R L T 3,0 0,-4 B 2,4 -1,8 Weak dominance ui (b si ; si ) ui (si ; s i ) for (si ; s i ) 2 S and sbi 6= si .
6.1.1
Nash equilibrium N
Given G = (Si ; ui )i=1 strategy sbi is pure strategy Nash equilibrium if for each player i ui (b s) ui (si ; sb i ) for all si 2 S. Mixed strategy Mi is probability distribution over Si Expected utility from Neumann-Morgenstern utility function is: X ui (m) = (m1 (s1 ) ::mN (sN )) ui (si ) (1109) s2S
For given pure strategies s = (s1 ; ::::; sN ) 2 S with probabilities m1 (s1 ) ; ::; mN (sN ) Theorem: Every …nite strategic form game possesses at least one Nash equilibrium. Proof of Nash equilibrium De…nitions: a. m b 2 M is a Nash equilibrium b. For every player i, ui (m b i ) = ui (si ; m b i ) for every si 2 Si given positive weight m b i and ui (m b i ) ui (si ; m b i ) for every si 2 Si given zero weight m b i. c. For every player i, ui (m b i ) ui (si ; m b i ) for every si 2 Si 1. construct a continuous function that maps m into itself. fi;j (m) =
mi;j + max (0; ui (j; m i ) ui (m)) n P 1+ max (0; ui (j 0 ; m i ) ui (m)) b
(1110)
j 0 21
2. Apply Brouwer’s …xed point theorem to …nd a …xed point ; here numerator is a continuous function and the denominator is continuos and bounded by contraction mapping f : M ! M it has a …xed point. 3. demonstrate that the …xed point is Nash equilibrium. LSH = RHS in above function. fi;j (m) = m b i;j then above function is 173
n X
max (0; ui (j 0 ; m i )
j 0 21
Multiply both sides by ui (j; m b i) n X
=
j 0 21
(ui (j 0 ; m b i)
ui (m)) b
n X
max (0; ui (j 0 ; m i )
j 0 21
ui (m)) b max (0; ui (j; m b i)
Here the left hand side n n P P m b i;j ui (j 0 ; m b i) m b i;j (0; ui (j 0 ; m b i ) ui (m)) b = j 0 21
j 0 21
Now the RHS
0=
n X
j 0 21
(ui (j 0 ; m b i)
RHS is zero only if ui (j 0 ; m b i) is the Nash equilibrium. 6.1.2
ui (m))
(1111)
ui (m) b and sum over j
m b i;j (ui (j 0 ; m b i)
j 0 21 n X
ui (m)) b = max (0; ui (j; m i )
ui (m)) b
ui (m) b = ui (m) b
ui (m)) b max (0; ui (j; m b i)
ui (m) b
ui (m)) b
0. That implies ui (m) b
ui (m)) b
(1112)
ui (m) b =0 (1113)
ui (j 0 ; m b i ). Therefore m b
Game of incomplete information: N
G = (p; Ti ; Si ; ui )i=1 where p is probability over Ti is the type of the player ui : S T ! R Associated strategic form of this game is G = (Rj ; vj )j2J where j is set of indices of the form j = (i; ti ) where ti 2 Ti and i = 1; :::; N . 0 Now the player j = (i; ti ) s strategy and payo¤ are de…ned as : Rj = Si
(1114)
expected payo¤ vj (r) = t0
n X
1 2T
p (t
i
j ti ) ui rj ; r(k;tk )
k6=i
; ti ; t
1i
(1115)
o
Bayesian Nash equilibrium is the Nash equilibrium of this associated strategic form game. Theorem: Every …nite game of incomplete information possesses at least one Bayesian Nash equilibrium.
174
Extensive form Game ( )
6.1.3
1. A …nite set of players N. 2. A set of actions, A 3. A set of nodes, histories, X. it includes initial nodes and a complete description of actions that have been taken so far. A (x) fa 2 A j (x; a) 2 Xg 4. Probability distribution
over actions A (x)
A:
5. Set of end nodes E fx 2 X j (x; a) 2 = Xg for all a 2 A: Each end node describes the complete play of the game so far. 6. A function : fX n (E [ fx0 g) j (x) = ig. When the game reaches at node X it tells it is the turn player i next. 7. Information set belonging to player i ; Ii
fI (x) j (x) = i , some x 2 X n (E [ fx0 g)g
8. for each i 2 N ; Neumann-Morgenstern payo¤ functions at the end node ui : E ! R. This is payo¤ for every possible complete play of the game. 9. Extensive form game is summarised then,
= < N; A; X; E; ; ; I; (ui )i2N .
Economic activities of consumers, producers, governments and nations or regions are interdependent. Game theory provides tools to study the strategic interactions among such economic agents where decisions taken by one individual depend on actions taken by others. Each game has a number of players who choose a set of strategies and rules. .Optimal choices available to one depend on choices made by others. Pay-o¤s are clearly de…ned for each player strategy pairs. Strategic modelling like this started with classics such as Cournot (1838), Bertrand (1883), Edgeworth (1925) von Neumann and Morgenstern (1944), Nash (1950). It is developing very fast in recent years following works of Kuhn (1953), Shapley (1953),Shelten ( 1965) Aumann (1966) Scarf (1967), Shapley and Shubic (1969), Harsanyi(1967), Spence (1974), Kreps (1990), Fundenberg and Tirole (1991) and Binmore (1992). Elements of a Game Rational Players Strategies Payo¤ matrix R 1;1
is pay-o¤ to row player if he plays strategy 1 and the column player plays strategy 1. Players like to maximise their own pay-o¤ given opponent’s strategy; B will choose strategy 1 or 2 that maximises his/her payo¤ looking at the choice of player A. Most games have equilibrium from which players do not have any incentive to move away. 175
Table 49: Structure of a Game Player A Strategy 1 Strategy 2 Player B R R C C Strategy 1 ; 1;1 1;2 ; 1;2 1;1; R C R C Strategy 2 2;1 ; 2;1; 2;2 ; 2;2
6.2
Story of GAME made easy
Story 1 In the beginning human beings did not know much on how to produce or organise the society and economy. They believed in a static world and might of their muscles and had very little concern to others. They obtained resources of nature for themselves and would play a zero sum game. They believed that there were …xed amount of goods or commodities that were to be distributed among people; if some one got more another person would get less. Each of them wanted more. This was the reason for outbreak of frequent …ghts and quarrels among them as seen in movies of early inter-continental settlers or in history books. Those who lost the war were forced to move to other less productive or less pleasant places. There were strategic interactions; actions of one depended on others but in a two person zero sum (TPZS) framework. Such game can be given by a matrix such as Table 50: Two Person Zero Sum Game Storng Hand pull push Strong Leg walk ( 10; 10) (10; 10) stand (10; 10) ( 10; 10) a) Explain this TPZS game b) Find a mixed strategy for strong-leg and strong-hand c) What is the value of the game? d) Why this game is not realistic in modern world? Story 2 Gradually people learnt to cultivate and grow their food. Civilisations started. Then they realised how the production and consumption can be organised collectively. Each can gain more from cooperation. Peace of mind would make them more productive and they can reap the bene…t of economies of scale. This brings the game to the next stage. Table 51: Two Person Cooperative Game Singer sing quite Writer write (5; 5) (2; 6) read (6; 2) (3; 3) a) b)
What is the solution for cooperative solution in this game? What makes such solution stable one? 176
c) Why the cooperation is Pareto superior than non-cooperation? d) What is the solution in mixed strategy? Story 3 Over time people are taken over by greed and self interest. They started competing out others for material bene…ts. They played non-cooperative game. Private corporations and …rms emerged to organise people in production process. Property rights and legal provisions for protecting those rights got built up in the economic system. Table 52: Non Cooperative Game Singer sing quite Writer write (5; 5) (2; 6) read (6; 2) (3; 3) a) What is the non-cooperative solution of this game? b) Show gains from bargain in this game in a diagram? c) What is the Nash Product? d) What are the threat points? Story 4 Now people learnt that inherently human being is sel…sh. They see war not the real solution of the problem. Gradually clever ones come up with good ideas. They make rules, regulations and constitutions to build mechanism in order to motivate someone to do good works and punish some who does bad works. Classical economists had developed models for a perfect world where information was complete, types and preferences and abilities of economic agents were known. It was easy to apply rules in such a perfect world based on criteria. Table 53: Incomplete Information Game Singer sing quite Writer write (?; ?) (?; ?) read (?; ?) (?; ?) Story 5 N person games Society consists of a large number of individuals. Like minded people enter into a coalition with speci…c targets and objectives in their mind. They form an alliance that would give them more than they did not. The core solutions make every one happier than stand alone solutions. Solutions at the core are more e¢ cient than outside it. Story 6 World does not have complete information. Market fails to provide certain goods or it disappears completely under the asymmetric information situation. People take advantage of opportunities that would make them better even if that hurts others. There are good and bad intentions but it is very di¢ cult to guess it precisely in the beginning. Insurance companies emerge to make up for losses. Story 7
177
Auctioning and competitive bids are mechanism to assign contracts and reveal some information that would otherwise would not be revealed. Binmore K. (1990) Fun and Games: A text on Game Theory, Lexington, Heath. Bhattarai K. (2013) Coalition for constitution and economic growth in Nepal, International Journal of Global Studies (IJGS), 1:1, Feb, 1-4 Cripps, M.W.(1997) Bargaining and the Timing of Investment, International Economic Review, 38:3 :Aug.:527-546 Dixit A., S. Skeath and D. F. Reiley (2009) Games of Strategy, Norton. Gardener R (2003) Games of Business and Economics, Wiley, Second Edition. Hey J. D. (2003) Intermediate microeconomics, McGraw Hill. Holt Charles (2007) Markets, Games and Strategic Behaviour, Pearson, . Hurwicz L (1973) The design of mechanism for resource allocation, American Economic Review, 63:2:1-30. Kreps D. M. (1990) A Course in Microeconomic Theory, Princeton. Maskin E, J. Moore (1999) Implementation and Renegotiation The Review of Economic Studies, Vol. 66, No. 1, Special Issue: Contracts Jan, pp. 39-56 Maskin E and J Tirole (1992) The principal-agent relationship with an informed principle: common values, Econometrica, 60:1:1-42 Mailath G. J. and L. Samuelson (2006) Repeated Games and Reputations: long run relationship, Oxford. Mirlees, J.A. (1971) 'An exploration in the theory of optimum income taxation.' Review of Economic Studies,38:175-208. Myerson R (1986) Multistage game with communication, Econometrica, 54:323-358. Osborne M.J. and A. Robinstein (1994) A course in game theory, MIT Press. Pathak P and T Sönmez (2013) School Admissions Reform in Chicago and England: Comparing Mechanisms by Their Vulnerability to Manipulation.' American Economic Review, 103(1): 80-106. Perlo¤ J. M. (2013) Microeconomics: Theory and Applications with Calculus, Pearson, 3rd Edition. Rasmusen E(2007) Games and Information, Blackwell. Roth A E. (2008) What have we learned from market design?, Economic Journal, 118 (March), 285–310. Shapley L (1953) A Value for n Person Games, Contributions to the Theory of Games II, 307-317, Princeton. 178
Shapley L and M. Shubik (1969) On Market Games, Journal of Economic Theory, 1:9-25 Varian HR (2010) Intermediate Microeconomics: A Modern Approach, Norton,8th ed..
6.3
Types of games Table 54: Zero Sum Game Player A Strategy 1 Strategy 2 Player B Strategy 1 (10; 10) ( 10; 10) Strategy 2 ( 10; 10) (10; 10)
zero sum game: one’s gain = loss of another ; sports ; market shares two or many players; Chess, football Cooperative Games: Global climate change; bargaining game Non-cooperative Games: two or many players ; Competition and Collusion competition between opposing political parties, countries Single period of multiple period: static and dynamic Full information or incomplete information :Firms and consumers; government and public;Among individuals, clubs, parties; nations Solution of Games by the Dominant Strategy Dominant strategy
Table 55: Advertisement Game Player A Advert Dont Advert Player B Advert (10; 5) (15; 0) Dont Advert (6; 8) (10; 2) Dominant strategy is to advertise for both A and B. With a slight change Dominant strategy is to advertise for A but B has no dominant strategy. Solution of Games by Nash Equilibrium (Prisoner’s Dilemma) Punishment structure for a crime F in d in g N a sh so lu tio n (u n d e rsc o re th e b e st stra te g y to a p laye r i g ive n th e ch o ic e o f th e o p p o n e nt.
Nash Equilibrium: Prisoner’s Dilemma Fact: both players did a crime together. Police suspects and arrest both of them. 179
Table 56: Advertisement Game Player A Advert Dont Advert Player B Advert (10; 5) (15; 0) Dont Advert (6; 8) (20; 2) Table 57: Prisoners’Dilemma Game Player A Confess Dont Confess Player B Confess ( 5; 5) ( 1; 10) Dont Confess ( 10; 1) ( 2; 2) Playing non cooperatively each convicts another. Game results in Nash solution (confess, Confess) = ( 5; 5); Each ends up with 5 years in prison. By confessing, each gives evidence to the police to determine the highest possible punishment. If they had cooperated remaining silent, police would not have enough evidence. Each would have been given only two years of prison ( 2; 2) : This is Pareto optimal outcome, 'where no one could be made better o¤ without making someone worse-o¤'. Cooperation is better but each think that other player will cheat and therefore doesn’t cooperate. Therefore stay longer in jail. There are many example of prisoner’s dilemma game in real world -pricing and output in a cartel, pollution, tax-revenue. Solution by the mixed strategy This game does not have equilibrium in pure strategy. Player B will play H is A plays H but A will play T if B plays H. If A plays T it is optimal to play T for B, then it is optimal for B to play H. Game goes in round in circle again. It can be solved my the mixed strategy. Flip the coin to randomise the chosen strategies. If each played H or T half of the times optimal payo¤ is zero to both players. Probability of playing H or T is 0.5. Solution by mixed strategy B plays Top p times and Bottom (1 p) times if A plays Left . B plays Top p times and Bottom (1 p) times if A plays Right. B likes to be equally well o¤ no matter what A plays. Solution by the mixed strategy Expected pay-o¤ for B if A plays Left E(
B;L )
= 50p + 90(1
p)
(1116)
= 80p + 20(1
p)
(1117)
Expected pay-o¤ for B if A plays Right E(
B;R )
180
Table 58: Prisoners’Dilemma Game Player A Confess Dont Confess Player B Confess ( 5; 5) ( 1; 10) Dont Confess ( 10; 1) ( 2; 2) Table 59: Game of matching penny: mixed strategy Player A Head Tail Player B Head (1; 1) ( 1; 1) Tail ( 1; 1) (1; 1) Making these two payo¤s equal 50p + 90(1
p) = 80p + 20(1
p) =) 100p = 70
p = 0:7
(1118) (1119)
B plays Top 70 % of times and Bottom 30% of times. Subsidy Game Between the Airbus and Boeing If both Boeing and Airbus produce a new aircraft each will lose -10. If Airbus does not produce and only Boeing produces Boeing will make 100 pro…t. If Airbus does not produce Airbus can make 100 but then Boeing will decide to produce even at a loss of 10 so that Airbus does not enter in that market. Subsidy Game Between the Airbus and Boeing EU countries want Airbus to produce, they change this by subsidising 20 to Airbus. Producing new aircraft is dominant strategy for Airbus now, no matter whether Boding produces or not. Entry Deterrence Game In‡ation and unemployment game between public and private sectors Higher payo¤ is good. First element represents payo¤ to the row-player (Government). Second element represents payo¤ to the column-player (private sector). Nash solution is (H; H) = (4; 4) Cooperative solution would have been better with (L; L) = (5; 5). Cost of Cheating Cooperative solution would have been better with (L; L) = (5; 5) but distrusting each other results in (H; H) = (4; 4) . If the game is plaid repeatedly what will be value of the game? It is given by the discounted present value of the game for any discount rate 0 < < 1: P V (cooperate) = 5 + 5 + 5
181
2
+ :::: + 5
n
=
5 1
(1120)
Table 60: Competitive Game Player A Left Right Player B Top (50; 50) (80; 80) Bottom (90; 90) (20; 20) Table 61: Subsidy Game Airbus Produce Don’t produce Boeing Produce ( 10; 10) (100; 0) Don’t produce (0; 100) (0; 0) However, there is an incentive to cheat to get 6 instead of 5. when one player deviates from the cooperative strategy this way another will found out being cheated next period. Then he/she will punish the cheater by playing non-cooperatively next period. So the value of game : P V (cheat) = 6 + 4 + 4
2
+ :::: + 4
n
(1121)
P V (cheat) = 6 + 4 + 4
2
+ :::: + 4
n
(1122)
(1123)
Cost of Cheating
multiply it by P V (cheat) = 6 + 4
2
+ :::: + 4
n+1
2
+ :::: + 4
n+1
taking the di¤erence (1
) P V (cheat) = 6
6 +4
4
=6+4
(1
)
(1124)
Whether a person cheats or not depends on discount factor 5 1
=6+4
(1
)
or5 = 6 (1
)+4
1=
2 ;
=
1 2
(1125)
Extensive form of the game Solution by Backward Induction (Is there any …rst movers advantage?) In‡ation and unemployment game in a diagram In‡ation and unemployment game in a diagram Economic policy game between the …scal and monetary authority Binmore K. (1990) Fun and Games: A text on Game Theory, Lexington, Heath. Cripps, M.W.(1997) Bargaining and the Timing of Investment, International Economic Review, 38:3 :Aug.:527-546 Dixit A., S. Skeath and D. F. Reiley (2009) Games of Strategy, Norton.
182
Table 62: Subsidy Game Airbus Produce Don’t produce Boeing Produce ( 10; 10) (100; 0) Don’t produce (0; 120) (0; 0) Table 63: Subsidy Game Entrant Enter Dont Enter Incumbent Enter ( 10; 10) (100; 0) Dont Enter (0; 100) (0; 0)
Gardener R (2003) Games of Business and Economics, Wiley, Second Edition. Holt Charles (2007) Markets, Games and Strategic Behaviour, Pearson, . Mailath G. J. and L. Samuelson (2006) Repeated Games and Reputations: long run relationship, Oxford. Kreps D. M. (1990) A Course in Microeconomic Theory, Princeton. Osborne M.J. and A. Robinstein (1994) A course in game theory, MIT Press. Perlo¤ J. M. (2008) Microeconomics: Theory and Applications with Calculus, Pearson. Rasmusen E(2007) Games and Information, Blackwell,. Varian HR (2010) Intermediate Microeconomics: A Modern Approach, Norton,8th ed..
Equilibrium concepts: Backward induction; subgame perfect equilibrium, sequential equilibrium, Bayes’s rule.
6.4
Bargaining game The very common example for bargaining game is splitting a pie between two individuals. The sum of the shares of the pie claimed by both cannot exceed more than 1, otherwise each will get zero. If we denote these shares by of the game where each get j =0 .
i i
and j then i + j 1 is required for a meaningful solution 0 and j 0 payo¤. When i + j > 1 then and i = 0 and
Standard technique to solve this problem is to use the concept of Nash Product Nash Product in Bargaining Game max U = (
i
183
0) (
j
0)
(1126)
Table 64: Subsidy Game Entrant Enter Dont Enter Incumbent Enter ( 10; 10) (100; 0) Dont Enter (0; 120) (0; 0) Table 65: In‡ation and unemployment game Private Sector H L Government H (4; 4) (6; 3) L (3; 6) (5; 5) subject to i
or by non-satiation property
i
+
j
+
j
1
(1127)
=1
Using a Lagrangian function L ( i;
j;
)=(
0) (
i
j
0) + [1
i
j]
(1128)
First Order Conditions First order conditions of this maximization problem are L ( i; @ L ( i; @
j;
)
=
j
=0
(1129)
=
i
=0
(1130)
i j;
)
j
L ( i; j ; ) =1 (1131) i j =0 @ From the …rst two …rst order conditions j = i implies j = i and putting this into the third …rst order condition j = i = 12 . This is called focal point. Thus Nash solution of this problem is to divide the pie symmetrically into two equal parts. Any other solution of this not stable. Roy Gardner (2003) and Charles Holt (2007) have a number of interesting examples on bargaining game. Application of Bargaining Game Money to be divided between two players M = u1 + u2 The origin of this bargaining game is the disagreement point d(0; 0), the threat point.
184
(1132)
Here the utility of player one (u1 ) is plotted against the utility of player two u2 and the line u1 u2 is the utility possibility frontier (UPF). Starting of bargaining can be (0; M ) or (M; 0) where one player claims all but other nothing. But this is not stable. O¤ers and counter o¤ers will be made until the game is settled at u = each player gets equal share.
1 1 2 M; 2 M
where
Numerical Example of Bargaining Game Suppose there is 1000 in the table to be split between two players. What is the optimal solution from a symmetric bargaining game if the threat point is given by d(0,0)? Using a Lagrangian function for constrained optimisation L (u1; u2 ; ) = u1 u2 + [1000
u1
u2 ]
(1133)
First order conditions of this maximization problem are L (u1; u2 ; ) = u2 @u1
=0
(1134)
L (u1; u2 ; ) = u1 @u2
=0
(1135)
L (u1; u2 ; ) = 1000 u1 u2 = 0 (1136) @ From the …rst two …rst order conditions u2 = u1 implies u2 = u1 and putting this into the third …rst order condition u2 = u1 = 1000 = 500. This is called focal point. 2 Numerical Example of Bargaining Game The Nash bargaining solution is the values of u1 and u2 that maximise the value of the Nash product u1 u2 subject to the resource allocation constraint,u1 + u2 = 1000. This bargaining solution ful…ls four di¤erent properties: 1) symmetry 2) e¢ ciency 3) linear invariance 4) independence of irrelevant alternatives (IIA). Symmetry implies that equal division between two players and e¢ ciency implies no wastage of resources u1 + u2 = M or maximisation of the Nash product, u1 u2 . Linear invariance refers to the location of threat point as can be shown in a bankruptcy game say dividing 50000. If u is a solution to the bargaining game then u + d is a solution to the bargaining problem with disagreement point d. Numerical Example of Bargaining Game L (u1; u2 ; ) = (u1
d1 ) (u2
d2 ) + [50000
u1
u2 ]
(1137)
Suppose the player 1 has side payment d1 = 15000 L (u1; u2 ; ) = (u1
15000) (u2
d2 ) + [50000
First order conditions of this maximization problem are
185
u1
u2 ]
(1138)
L (u1; u2 ; ) = u2 @u1 L (u1; u2 ; ) = u1 @u2
15000
From the …rst two …rst order conditions u2
(1139) =0
(1140)
u1
u2 = 0
(1141)
= u1
15000
L (u1; u2 ; ) = 50000 @ Numerical Example of Bargaining Game
implies u2 = u1
=0
15000 and
putting this into the third …rst order condition u2 + 15000 = u1 ; u2 =
50000 15000 2
= 17500; u1 = 15000 + u2 = 32500.
Then u1 + u2 = 17500 + 32500 = 50000. Risk and Bargaining A risk averse person loses in bargaining but the risk neutral person gains. Suppose the utility 0:5 functions of risk averse person is given byu2 = (m2 ) but the risk neutral person has a linear utility u1 = m1 . m1 + m2 = M .u1 + u22 = 100 Using a Lagrangian function for constrained optimisation L (u1; u2 ; ) = u1 u2 +
100
u1
u22
(1142)
First order conditions of this maximization problem are L (u1; u2 ; ) = u2 @u1 L (u1; u2 ; ) = u1 @u2
=0 2 u2 = 0
(1143) (1144)
L (u1; u2 ; ) = 100 u1 u22 = 0 (1145) @ Numerical Example of Bargaining Game From the …rst two …rst order conditions uu1;2 = 2 u2 implies u1 2u22 and putting this into the third …rst order condition .3u22 = 100 ; u22 = 100 3 = 33:3 ; u2 = 5:77 2 2 u1 = 2u2 = 2 (5:77) = 66:6 u1 + u22 = 66:6 + 33:3 = 100 Thus the risk-averse player’s utility is 66.7 and risk neutral player’s utility is only 5.7. Morale: do not reveal anyone if you are risk averse, otherwise you will lose in the bargaining. Coalition Possibilities 186
2N -1 rule for possible coalition Consider Four Players A,B,C,D A, B, C, D AB, AC, AD BC, BD, CD ABC, ABD,ACD, BCD, ABCD 16 -1=15 6.4.1
Coalition and Shapley Values of the Game
In many circumstances optimal decisions of an economic agent depends on decisions taken by others. Dominants …rms competing for a market shares, political parties contesting for power and research and scienti…c discoveries aimed for path-breaking innovations are in‡uenced by decision taken by others. In all these circumstances there are situations where collective e¤orts rather than individual ones generate the best outcome for the group as a whole and for each individual members of the group. Concepts of bargaining, coalition and repeated games developed over years by economists such as Cournot (1838), Bertrand (1883), Edgeworth (1925) von Neumann and Morgenstern (1944) and Nash (1950, 1953) is developing very fast in recent years following works of Kuhn (1953), Shapley (1953),Shelten ( 1965) Aumman (1966) Scarf (1967), Shapley and Shubik (1969), Harsanyi(1967), Spence (1974), Hurwicz (1973), Myerson (1986),Gale (1986), Maskin and Tirole (1989), Kreps (1990), Fundenberg and Tirole (1991) and Binmore (1992), Rubinstein (1982) Sutton (1986) Cho and Kreps (1987) Sobel (1985) Machina (1987) Riley (1979) McCormick (1990), Ghosal and Morelli (2004) and others. These have generated models that can be applied to analyse the relative gains from coalitions rather than without these coalitions. It is natural that economic agents play a zero sum and non-cooperative game until they realise the bene…ts of coalition and cooperation. When an agreement is made and cooperation is achieved there is a question on whether such coalition is stable or not. There are always incentives at least for one of the player to cheat others from this cooperative agreement in order to raise its own share of the gain. However, it is unlikely that any player can fool all others at all the times. Others will discover such cheating sooner or later. A coalition of players should ful…l individual rationality, group rationality and coalition rationality. These can be ascertained by the supper-additivity property of coalition where the maximisation of gain requires being a member of the coalition rather than playing alone. This can be explained using standard notations. Let us take three players [ in the current Nepalese context N = (1= CPM and 2 =UML, 3 =NC)]. Superadditivity condition implies that the value of the game in a coalition is greater than the sum of the value of the game of playing alone by those individual members. v (1 [ 2 [ 3)
v1 (1) + v (2) + v (3)
(1146)
Coalitions (parties) playing together generate more value for each of its member than by playing alone. Team spirit generates extra bene…ts. When normalised to 0 and 1 the value of the gains from a coalition are: v (1) = 0; v (N ) = 1 for n = 1; :::; N The fact that payo¤ of the merged coalition is larger than the sum of the payo¤ to the separate coalitions is shown by following imputations, that shows ways on how to value of the game can be distributed among N di¤erent players. The imputations of values characterise these allocations: 187
1
=
v (N ) =
2
+
X
i
3
+
(1147)
N X
=
i
(1148)
i=1
i2N
Group rationality implies that total payo¤ to each players in the coalition equals at least the payo¤ of its independent actions. i
v (fig) ;
i2N
(1149)
In the dynamic context players like to maximise the present value, V, of the gain from in…nite period, with a given discount rate r over years: Z t=1 v (i) e rt dt (1150) V = t=0
The imputations in the core guarantees each member of a coalition the value at least as much as it could be obtained by playing independently. At the core of the game each player gets at least as much from the coalition as from the individual action, there does not exist any blocking coalition. This is equivalent to Pareto optimal allocation in a competitive equilibrium (Scarf (1967)). Some imputations are dominated by others; the core of the game is the strong criteria for dominant imputation. Core satis…es coalition rationality. A unique imputation in the core is obtained by Shapley value. This re‡ects additional payo¤ that each individual can bring by adding an extra player to the existing coalition above the pay-o¤ without this player. This is the power of that player. Consider a game of three players in which the 3rd player always brings more to the coalition than the 1st or the 2nd player. Payo¤ for coalition of empty set: v ( ) = 0 Payo¤ from players acting alone: v (1) = 0; v (2) = 0; v (3) = 0 ; Payo¤ from alternative coalitions: v (1; 2) = 0:1; v (1; 3) = 0:2; v (2; 3) = 0:2; Payo¤ from the grand coalition: v (1; 2; 3) = 1 Power of individual i in the coalitions is measured by the di¤erence that person makes in the value of the game v (S [ fig v (S)) = 1 , where S is the subset of players excluding i, S [ fig is the subset including player i. The expected values of game for i is found by taking account of all possible coalition that person i can enter with N number of players, where is the weighting factor that changes according to the number of people in a particular coalition. This is the probability that a player joins coalition,S 2 N and there are (2N -1) ways of forming in N player game: i
=
X
S2N
n
(S) v (S [ fig
v (S)) ;
n
(S) =
s! (n
s n!
1)
(1151)
v (1) v ( ) = 0 v (1; 2) v (1) = 0:1 v (1; 3) v (1) = 0:2 0 = 0:2 v (1; 2; 3) v (2; 3) = 1 0:2 = 0:8 0
(S) =
s! (n
s n!
1)
=
0! (3
188
0 3!
1)
=
2! 2 = 3! 6
(1152)
1
(S) =
2
(S) =
s! (n
s n!
1)
s! (n
s n!
1)
= =
1! (3
1 3!
1)
2! (3
2 3!
1)
=
1! 1 = 3! 6
(1153)
=
2! 2 = 3! 6
(1154)
Shapley value for player 1 is thus 1
=
X
n
(S) v (S [ fig
v (S)) =
2 1 1 2 19 (0) + (0:1) + (0:2) + (0:8) = 6 6 6 6 60
(1155)
n
(S) v (S [ fig
v (S)) =
2 1 1 2 19 (0) + (0:1) + (0:2) + (0:8) = 6 6 6 6 60
(1156)
2 1 1 2 22 (0) + (0:2) + (0:2) + (0:9) = 6 6 6 6 60
(1157)
S2N
For player 2 2
=
X
S2N
Note as before v (2) v ( ) = 0 v (1; 2) v (1) = 0:1 v (2; 3) v (1) = 0:2 0 = 0:2 v (1; 2; 3) v (1; 3) = 1 0:2 = 0:8 For player 3 3
=
X
S2N
n
(S) v (S [ fig
v (S)) =
v (3) v ( ) = 0 v (1; 3) v (1) = 0:2 0 = 0:2 v (2; 3) v (2) = 0:2 0 = 0:2 v (1; 2; 3) v (1; 2) = 1 0:1 = 0:9 As the player 3 brings more into the coalition its expected payo¤ is higher than of players 1 and 2. Similar con…gurations can be made where players 1 and 2 can bring more in the coalition. In the context of Nepal which of three parties mentioned above are pivotal depends on the value they add to the game. The value of grand coalition is the largest possible value of the game with N players. This fact is shown by the core of the game in Figure 1. Figure 1
189
Solutions towards the core are more stable than those towards the corners which are prone to con‡icts. This is equivalent to …nding a central ground in politics. Ego centric solutions are less likely to bring any stable solution to the game. In the most stable equilibrium all players gain in equal proportions to their supporters.
6.5
Pivotal player in a voting game in Nepal
Ability of a player to in‡uence the outcome of the game depends on the pivotal status enjoyed by that player. In a game with 3 players; power of player i is re‡ected by its Shapley value. Consider six possible ordering of 123 pivotal game. Three players can order themselves in 3!= 6 ways. Each of these number can appear only twice in the middle out of six possible combinations. A player located in the middle is pivotal. If parties realise this fact while bargaining, such bargaining is likely to generate a stable and cooperative solution. In the 123 game given in Table 1 the player 3 is pivotal in game (2) and (4); player 1 in (3) and (5) and player 2 in (1) and (6). The marginal contribution (Shapley value) of each player can be presented then as Therefore each player has 1/3 chance of being pivotal. If 1 is pivotal into the coalition any coalition with 1 will win - player 1 is powerful. Players 2 and 3 are powerless. When a party has majority in a parliament that party is in pivotal position. This outcome is reversed in a hung parliamnet where none is pivotal. There is always a chance that a pivotal player now may have to give up that position for other players later on. Another con…guration is to assume that certain party is pivotal all the times. As shown below, in this situation the Shapley value of player 1 is 1 no matter which position it is in the coalition and it is 0 for players 2 and 3. In the context of Nepal’s Constituent Assembly, it seems that depending on circumstances, players NC, CPM and UML each have equal chance of being a pivotal player. Thus non-pivotal con…gurations are more applicable than pivotal con…gurations. 190
Table 66: No Pivotal Player in a Bargaining Game orderings M(1,S) M(2,S) M(3,S) 1 123 0 1 0 2 132 0 0 1 3 213 1 0 0 4 231 0 0 1 5 312 1 0 0 6 321 0 1 0 Table 67: Pivotal Player in a Bargaining Game orderings M(1,S) M(2,S) M(3,S) 1 123 1 0 0 2 132 1 0 0 3 213 1 0 0 4 231 1 0 0 5 312 1 0 0 6 321 1 0 0 Simple game theoretic models applicable to analyse the current Nepalese situation could be developed taking seminal ideas of Shapley and Shubik (1969), Rubinstein (1982), Myerson (1986), Sutton (1986), Dixit (1987), Maskin and Moore (1999) and Riley (2001). 6.5.1
Model of fruitless bargaining and negotiation
There are N parties in the game indexed by i = 1; ::::; N . Each party i is interested in its own pay-o¤ xi (e.g the number of ministries it should have under its command) which it computes using a payo¤ function Ui that depend on strategies available to players and its information set about the reactions of other players: xi = Ui (S1 ; S2 ; ::::; Sn ; a0 ; a1 ; a2 ; ::::; an )
(1158)
where S1 ; S2 ; ::::; Sn denote the strategies available to players, a0 is common knowledge, and a1 ; a2 ; ::::; an denote the unknown characteristic of player i. Each player knows which strategy is better for it given the strategy space of other players but they have less information about the reactions of other players, aj . They make some subjective estimates about other’s actions while calculating its payo¤ xi . This value gives their reservation or threat point in bargaining. The agreement takes place when actual bargaining and negotiation ends up giving zi and when this value is greater than or equal to what the party i had expected, zi = xi . Negotiation breaks down whenever zi 6 xi . 6.5.2
Model of commitment, credibility and reputation
Parties need to learn from each other to create a more realistic beliefs (bj ) about other players replacing unknown characteristics (a0 ; a1 ; a2 ; ::::; an ) by more accurate representation parameters (b0 ; b1 ; b2 ; ::::; bn )
191
xi = Li (S1 ; S2 ; ::::; Sn ; b0 ; b1 ; b2 ; ::::; bn )
(1159)
Beliefs on these parameters could be formed on the basis of history, principles and values of parties and key personalities of the party and studying their relations to other players. Convergence on beliefs among all parties occurs when they understand and trust each other. This gives credibility to the outcome of the game. Equilibrium in such case is more certain and e¢ cient and generates greater payo¤ for parties and welfare of the country. 6.5.3
Endogenous intervention: change in beliefs
When the …rst CAN dissolved without promulgating a constitution of Nepal on May 28, 2012 people changed their beliefs about the true intention and commitment of the UCPN (Maoists) towards development and their ability to promulgate a constitution of Nepal. They re…ned their beliefs about the Nepali Congress and UML and other Tarai based parties. This change is re‡ected in the structure of the second CAN that was elected on November 19, 2013. Still this was a hung CAN as before but the share of the Nepali Congress increased to 33.9 from 19.3 percents and that of UML increased to 30.4 from 18.0 percents. The share of UCPN (Maoist) reduced to 14 from 38.1 percents. The UCPN (M) is no longer in a position of dreaming a totalitarian system in Nepal. After this verdict parties have committed to promulgate the constitution of Nepal by Jan 22, 2015. Committees in the CAN-II have been able to resolve many disputes but still have not converged in their opinions regarding the structure of governance or that of federal system till the end of September 2014. Table 68: Members of First Constituent Assembly by Political Parties in Nepal Total Maoist NC UML MPRF TMLP Others Total 601 229 116 108 53 21 74 FPTP 240 120 38 33 29 9 11 Proportional 335 100 73 70 22 11 59 Nomination 26 9 5 5 2 1 5 Percentage 100% 38.1% 19.3% 18.0% 8.8% 3.5% 12.3% Source: Constituent Assembly of Nepal (CAN).
Table 69: Members of Second Constituent Assembly by Political Parties in Nepal Total NC UML Maoist MPRF RPP Others Total 601 204 183 84 15 11 103 FPTP 240 105 91 26 4 0 14 Proportional 335 91 84 54 10 10 86 Nomination 26 8 8 (0) 4 (0)1 1 (0)3 Percentage 100% 33.9% 30.4% 14.0% 2.5% 1.8% 17.1% Source: Constituent Assembly of Nepal (CAN). Still leaders of Nepal seem to be confused in understanding the basic fact that the gains from he commitment and cooperation should be much larger than of noncooperation to form coalition
192
or in releasing that the bene…ts of dynamic optimisation are far greater than zero sum game being played at the moment. It is important to rethink about the true and realistic social welfare function such as W (Y; S) where Y denotes the level of aggregate economic activities and its growth rates and S the stability of the system, can be one way to redirect resources wasted in the process of unsuccessful coalition formation to bring more e¢ cient and Pareto optimal solution. Reinvigorate the spirits of April 2006 Revolution. Nepal’s per capita income is one third of India and about 12 percent of China though it had similar per capita income with them till 1980. Political instability in the last two decades has been very costly to Nepal. These could have been decades of spectacular growth but turned into disaster. There cannot be bigger irony than this in the context of Nepal and cooperative strategies of each political party is the only way to sort out this problem. Credibility, respect and commitment only can make this happen. B h a tta ra i K . (2 0 1 3 ) C o a litio n fo r c o n stitu tio n a n d e c o n o m ic g row th in N e p a l, Inte rn a tio n a l J o u rn a l o f G lo b a l S tu d ie s (IJ G S ), 1 :1 , F e b r u a r y, 1 - 4
B h a t t a r a i K . ( 2 0 1 1 ) C o n s t it u t io n a n d E c o n o m ic M o d e ls fo r t h e Fe d e r a l R e p u b lic o f N e p a l, E c o n o m ic J o u r n a l o f N e p a l, Vo l. 3 3 , N o .1 , J a nu a ry _ M a rch , Issu e N o . 1 2 9 , p . 1 -1 5
B h a tta ra i K . (2 0 1 1 ) E m p ty C o re in a C o a litio n : W h y N o C o n s titu tio n in N e p a l? , In d ia n J o u rn a l o f E c o n o m ic s a n d B u s in e s s , 1 0 :1 :1 1 9 1 2 6 ,M a rch 2 0 1 1
B h a tta ra i K . (2 0 0 7 ) M o d e ls o f E c o n o m ic a n d P o litic a l G row th in N e p a l, S e ria ls P u b lic a tio n , N e w D e lh i.
B h a tta ra i K . (2 0 0 6 ) C o n se q u e n c e s o f A p ril 2 0 0 6 R e vo lu tio n a ry C h a n g e s in N e p a l: C o ntinu a tio n N e p a le se D ile m m a , In d ia n J o u rn a l o f E c o n o m ic s a n d B u sin e ss, 5 :2 :3 1 5 -3 2 1 .
C rip p s, M .W .(1 9 9 7 ) B a rg a in in g a n d th e T im in g o f Inve stm e nt, Inte rn a tio n a l E c o n o m ic R e v ie w , 3 8 :3 :A u g .:5 2 7 -5 4 6
D ix it A v in a sh (1 9 8 7 ) S tra te g ic B e h av io u r in C o nte sts, A m e ric a n E c o n o m ic R e v ie w , D e c ., 7 7 :5 :8 9 1 -8 9 8 .
K u h n H . W . ( 1 9 9 7 ) C l a s s i c s i n G a m e T h e o r y, P r i n c e t o n U n i v e r s i t y P r e s s .
M a sk in E , J . M o o re (1 9 9 9 ) Im p le m e nta tio n a n d R e n e g o tia tio n , R e v ie w o f E c o n o m ic S tu d ie s, 6 6 ,1 , 3 9 -5 6
M a ila th G . J . a n d L . S a m u e lso n (2 0 0 6 ) R e p e a te d G a m e s a n d R e p u ta tio n s: lo n g ru n re la tio n sh ip , O x fo rd .
M ye rso n R (1 9 8 6 ) M u ltista g e g a m e w ith c o m m u n ic a tio n , E c o n o m e tric a , 5 4 :3 2 3 -3 5 8 .
P a th a k P a n d T S ö n m e z (2 0 1 3 ) S ch o o l A d m is s io n s R e fo rm in C h ic a g o a n d E n g la n d : C o m p a rin g M e ch a n is m s b y T h e ir Vu ln e ra b ility to M a n ip u la tio n .' A m e ric a n E c o n o m ic R e v ie w , 1 0 3 (1 ): 8 0 -1 0 6 .
R ile y J G (2 0 0 1 ) S ilve r S in g a ls : T w e n ty -F ive Ye a rs o f S c re e n in g a n d S ig n a llin g , J o u rn a l o f E c o n o m ic L ite ra tu re , 3 9 :2 :4 3 2 -4 7 8
R o t h A E . ( 2 0 0 8 ) W h a t h a v e w e l e a r n e d f r o m m a r k e t d e s i g n ? , E c o n o m i c J o u r n a l , 1 1 8 ( M a r c h ) , 2 8 5 –3 1 0 .
R u b in ste in A (1 9 8 2 ) P e rfe c t e q u ilib riu m in a b a rg a in in g m o d e l, E c o n o m e tric a , 5 0 :1 :9 7 -1 0 9 .
S h a p le y L (1 9 5 3 ) A Va lu e fo r n P e rso n G a m e s, C o ntrib u tio n s to th e T h e o ry o f G a m e s I I, 3 0 7 -3 1 7 , P rin c e to n .
S h a p le y L lo y d S . a n d M a r t in S h u b ik ( 1 9 6 9 ) P u r e C o m p e t it io n , C o a lit io n a l P o w e r , a n d Fa ir D iv is io n , In t e r n a t io n a l E c o n o m ic R e v ie w , 10 , 3, 33 7-36 2.
S u tto n J . (1 9 8 6 ) N o n -C o o p e ra tive B a rg a in in g T h e o ry : A n Intro d u c tio n , R e v ie w o f E c o n o m ic S tu d ie s, 5 3 , 5 ., 7 0 9 -7 2 4
193
6.6
Equivalence of Core in Games and Core in a General Equilibrium Model
Both game theory and general equilibrium models analyse optimal choices of consumers and producers faced with resource constraints in which the essential process involves bargaining over the gains from the intra and intertemporal trade on goods, services and …nancial assets. In terms of game theory the core of a bargaining game is given by the payo¤ from a non-blocking coalition. It is a Pareto e¢ cient point. Similarly core of a general equilibrium lies in the contract curve where it is di¢ cult to make one economic agent better o¤ without making another worse o¤. The core of the coalition in the game and core of the equilibrium in a general equilibrium model represent basically the same thing. The optimal allocation of resources to economic agents possible with given endowments con…rm to the …rst and second theorems of welfare economics. Abstract solutions of both models can characterise the optimal allocation of resources after more complex bid and o¤er interactions among economic agents. Debreu and Scarf (1963) have proven the equivalence of a competitive equilibrium to the core of the game for economies with and without production by contradiction when preferences are non-satiable, strictly convex and continuous. Scarf (1967) theorem states that a balanced n person game has a nonempty core. This is best illustrated in terms of a Venn diagram with three players as given in Figure below. Assume a pure exchange economy in which each individual i is endowed with ! i endowments, i =1. . . n. Let the competitive allocations beX xi . Then the competitive equilibrium implies X X X xi = ! i with usual preferences, u xSi u (xi ). In the n person game,T = fSg , the i
i
i
i
collection of coalitions, is called balanced X collection if it is possible to …nd factors to weight value of allocations to each coalition such that i = 1 . Competitive allocations are proven to be in T =fSg S fig
core using these weights as: n X X X xi = i
i T =fSg S fig
S i xi
=
X
X
S T =fSg i2S
xSi
=
X
X
S S2T i2S
Shapley Shubik Core in a Venn Diagram
194
n X X !i = !i i
T =fSg S fig
S
=
X !i i
(1160)
Consider three player game as presented in Figure 3. By 2N 1 rule for possible number of coalitions in N person games, there are seven possible coalitions: f1g,f2g ,f3g ,f1; 2g , f1; 3g,f2; 3g ,f1; 2; 3g . There is some parallel between the value of a property in the central business district as the values of coalition in intersection can be far greater than values under no coalition; in addition in a bargaining game there X can be externality from the bargain as shown by points E around three circles. The condition i = 1 required for the core thus represents summative weight assigned T =fSg S fig
to these individual coalitions. Thus the competitive equilibrium is equivalent to the allocation at the core, “An exchange economy with convex preferences always gives rise to a balanced n person game and such will always have a nonempty core (Scarf (1967)).” Above state principle is generally true under full information. However, it does not work under incomplete information. Competitive …nancial markets are perfect under when all agents that have complete information and are e¢ cient in processing such information. This assumption, however, is not always correct. Financial markets are full of asymmetric information, activities of one set of players depend on actions taken by another set of players and the amount of information they have impacts on the likely choices of others. This requires incentive compatible mechanisms for e¢ cient allocation of …nancial resources. Bejan C and J C Gomez (2009) Core extensions for non-balanced TU-games, International Journal of Game Theory, 38:3-16. Bullard James, Alison Butler (1993) Nonlinearity and Chaos in Economic Models: Implications for Policy Decisions, Economic Journal, 103, 419: 849-867 195
Lipsey R. G. and K. Lancaster (1956 - 1957) The General Theory of Second Best, Review of Economic Studies, 24, 1,11-32 Nash J. (1951) Non-cooperative games, Annals of Mathematics, 54:286-295 Roth, A., Erev, I., 1995. Learning in extensive-form games: Experimental data and simple dynamic models in the intermediate term. Games Econ. Behav. 8, 164–212 Scarf H. (1967) Core of n Person Game, Econometrica, 35:50-69. Shapley L (1953) A Value for n Person Games, Contributions to the Theory of Games II,307317, Princeton. Wooders HM and 52:6:1327-1350.
6.7
WR Zame (1984) Approximate cores of large games, Econometrica,
Labour Market and Search and Matching Model
Producers use labour to produce goods and services. A production function shows how labour complements with other inputs in production and the marginal productivity of labour shows the additional unit of output produced by each additional unit of labour. Thus demand for labour is derived from the demand for output. On the supply side every working age person has 168 hours a week, 720 hours per months or 8760 hours per year of time endowment which can be allocated between work and leisre. How many hours does one work and how much is spent in free time really depends upon the preference between consumption and leisure on one side and the job vacancies on the other. In theory ‡exibility of real wages guarantees equality between demand and supply in the labour in a competitive labour market. However, the labour is far from being a perfectly competitive market. Firms exercise monopoly powers, acting as monopsonists in the labour market or use their market power in order to retain more e¤ective workers. Hiring decisions of …rms also are dependent on the aggregate demand. Firms hire more workers during expansion but are reluctant of recruit any workers during the contraction. A signi…cant number of workers become unemployed as a consequence. Given a production fucntion that related output (Yt ) to capital (Kt ), technology (At ) and labour (Lt ) Yt = Kt (At Lt )
1
0
6.8
Exercise 14: Search Equilibrium Search Equilibrium
1. Consider a bargaining model between …rms and workers Matching function aggregates vacancies and unemployment with job creation as: M =V U
(1171)
where M denote the number of matching of vacancies and job seekers, V is number of vacancies and U the number of unemployed, is the parameter between zero and one.Job seekers and employers bargain over expected earnings by maximising the Nash-product of the bargaining game over the di¤erence between the earnings from work (W) rather than in being unemployed (U) and earnings to …rms from …lled and vacant jobs.
199
(Wi
U ) (Ji
V)
(1172)
(a) Show that the dynamics of unemployment depends on the rate of job destruction, (1 , and the rate of job creation,
u)
q ( ) u. Derive the job creation curve. 1. (a) Optimal job creation or (demand for labour curve) shows how …rms balance the marginal revenue product of labour to wage and hiring and …ring costs. Derive the Beveridge curve.
7
L7: Game theory: Principal Agent and Mechanism Games and Auctions
7.1
Original Ideas
Issues of priciple agent games are disscuss in general terms in articles by Harsanyi (1967) Shapley and Shubik (1969). Hurwicz (1973), Spence (1977) , Riley (1979),Sobel (1985), Myerson (1986) Sutton (1986), Milgrom , Roberts (1986), Dixit Avinash (1987) Cho and Kreps (1987) , Rogerson (1988), Moore (1988), Dawes and Thaler (1988), McCormick (1990) Caminal (1990), Frank, Gilovich, and Regan (1993), Jin (1994), Markusen (1995), Camerer and Thaler (1995), Lundberg and Pollak (1996), Mookherjee and Ray (2001),Fehr, Gächter and Simon (2000), Besley and Ghatak (2001), Acemoglu (2001) and Mailath and Samuelson (2006) (see also Watt (2011) and Snyder and Nicholson (2011)). These explain how the moral hazard and adverse selection -asymmetric information in‡uence in decision making of economic agents. Moral hazard Owners of a …rm principals (P) and workers as agents (A) play a production game in which agent exerts e¤orts (a) in return of income (y) and principal maximises net pro…t. Agent can put high or low e¤orts and P cannot observe it. Utility of agent at work is given by V = u(y)
V0
a
(1173)
0
This must be greater than a reservation utility V that is available from alternative work. The income level that an individual worker requires is then given by y=V
1
V0+a
(1174)
Less informed P can make sure that A exerts good e¤ort by making wage contract as V = v(y ) + (1
) v(y1 ) < V 0
(1175)
Principal’s objective when a is observable is then to maximize pro…t by producing x subject to the participation constraint max zi =
i
(x1
y1 ) + (1
subject to
200
i ) (x2
y2 ) i=h,l
(1176)
i v(y1 )
+ (1
i ) v(y2 )
V0
ai
(1177)
There is uncertainty in production resulting in x1 and x2 levels of production, x1 < x2 . Because of this uncertainty x1 may happen despite A putting high level e¤ort, which P cannot observe. 7.1.1
Full information scenario L=
i
(x1
y1 ) + (1
i ) (x2
y2 ) +
i v(y1 )
+ (1
i ) v(y2 )
ai
V0
(1178)
First order conditions (for high e¤ort case) @zh = @y1
h
+ v 0 (y1 ) = 0
(1179)
@zh = @y2
(1
h)
+ (1
h) v
0
(y2 ) = 0
(1180)
@zh = @
(1
h)
+ (1
h) v
0
(y1 ) = 0
(1181)
V0 =0
(1182)
i v(y1 )
+ (1
i ) v(y2 )
ai
Thus in the full information case 1
v 0 (y1 ) = v 0 (y2 ) =
=) y1 = y2
(1183)
Thus the owners of the company force managers to put the same level of e¤orts. Risk-neutral owers bear all risk. P can design contracts similarly if they like A to put low e¤orts. L= 7.1.2
l
(x1
y1 ) + (1
l ) (x2
y2 ) +
l v(y1 )
+ (1
l ) v(y2 )
V0
al
(1184)
Incomplete information scenario
P cannot observe a of A; therefore they must design a contract which makes A put ah This requires adding an incentive compatibility constraint. h v(y1 )
+ (1
h ) v(y2 )
ah
l v(y1 )
+ (1
l ) v(y2 )
al
(1185)
Then the problem is modi…ed as max zi =
i
(x1
y1 ) + (1
i ) (x2
y2 ) i=h,l
(1186)
V0
(1187)
subject to i v(y1 )
+ (1
i ) v(y2 )
and
201
ai
h v(y1 )
L
=
l
[
(x1
+ (1
y1 ) + (1
h v(y1 )
+ (1
h ) v(y2 )
ah
l ) (x2
y2 ) +
h ) v(y2 )
l v(y1 )
+ (1
l v(y1 )
ah
l v(y1 )
l ) v(y2 )
+ (1 (1
l ) v(y2 ) l ) v(y2 )
al
al
(1188)
V0 +
+ al ]
(1189)
The optimising conditions in this case are given by @zh = @y1 @zh = @y2
(1
h
h)
@zh = @
+ v 0 (y1 ) + (
+ (1
l v(y1 )
@zh = h v(y1 ) + (1 @ From these conditions
h) v
+ (1
(
(y2 ) + (
l ) v(y2 )
h ) v(y2 )
+
0
ah
l)
h
l)
h h
1 < v 0 (y1 )
(1203)
Here ' is a random variable with E' = 0 If the shareholder could observe e¤orts, the optimal contract would be w = w is a …xed wage. Here this from the participation constraint is w = w0 +
Re2 2
(1204)
maximisationof shareholder’s expected pro…t is : E( )=E e+'
Re2 2
w0
=e
w0
Re2 2
(1205)
@E ( ) 1 1 = 1 Re = 0 () e = if w0 @e R 2R This is the optimal solution when shareholders could obseve the e¤ort of managers. Now suppose the e¤orts are not observable. Consider a linear incentive scheme: w( ) = a+b
(1206)
(1207)
What is the expected utility of A with linear scheme: Eu a + be + b'
Re2 2
() b
Re =) e =
b R
(1208)
E¤ort grows with the slope of the incentive scheme. If b = 1; then e = e : The expected utility at this level of e¤orts is Eu a +
b2 + b' R
b2 2R
() Eu a +
b2 + b' 2R
(1209)
Shareholder’s expected pro…t e
= E [e + '
a
be
b'] =
b R
Linear optimal scheme: 204
a
b2 b = (1 R R
b)
a
(1210)
max
e
b (1 R
=
b)
a
(1211)
w0
(1212)
subject to Eu a +
b2 + b' 2R
Substitute a from the participation constraint e
Eu
+
b2 + b' 2R
b R
= w0
(1213)
di¤erentiating wrt b Eu0
1
b R
+ Eu0 ' = 0
(1214)
This gives b = 1: This value of the linear scheme optimises pro…t for the shareholder as the agent puts maximum e¤orts at work. Table 70: Principal Agent Games Principal Agent Action Shareholders CEO Pro…t maximisation Landlord Tentants work e¤ort People Government Political power Manager Workers Work e¤ort Central Banks Banks Quality of credit Patient Doctor Intervention Owner Renter Maintenance Insurnace company Policy holder Careful behaviour
7.1.3
Impacts of Assymetric (incomplete) Information on Markets Equilibrium is ine¢ cient relative to full information case Signalling can improve the e¢ ciency: warranty and guarantee Screening: revealing the risk type of agent Credit history from credit card companies Government can improve the market by setting high standards of business contracts or bailing out troubled ones (Northern Rock, Bear Stearns, Lehman Brothers) Right regulations –Financial Services Authority, Fair trade commissions; O¢ ce of standards; Bank of England 205
Moral hazard (hidden action) Probability of bad event is raised by the action of the person iPeople who have theft insurance are likely to haven low quality locksthat are easy to break (in cars, houses, bicycle (car)) most likely to claim insurances Remedy: deductible amount; to ensure that some customers take care in security. 7.1.4
Adverse Selection (hidden information) Problem Uncertainty about the quality of good or services honest borrowers less likely to borrow at higher interest rates. low quality items crowd out high quality items risky borrowers drive out gentle borrowers in the …nancial market. Theft insurance; health insurance; people from safe area are less likely to buy theft insurance; only unsafe customers end up buying theft insurance healthy people are less likely to buy health insurance
Asymmetric information in Used Car Market -Akerlof’s Model of Asymmetric Information Sellers know exactly quality of cars but buyers do not. Equilibrium is a¤ected when sellers have more information than buyers. Market has plums: good cars and lemons: bad cars Seller knows his quality of cars but buyers do not Market for good cars disappear because of existence of bad cars in the market. Demand for high quality car falls and demand for low quality cars rise. Ultimately only low quality cars remain in the market. signals: warranty and Guarantee Providing warranty less costly for high quality cars as they last long. Warranty is costly for low quality cars as they frequently break down. Buyers can decide whether a car is good or bad looking at the warrantee and pay appropriately. Right signalling can remove ine¢ ciency due to incomplete information. Markets for both types of car can operate e¢ ciently by right signals of warranty and Guarantee 206
Pooling, Separating and Mixed Equilibrium Complete market failure pooling equilibrium (same price for good and bad cars; good cars disappear from the market) Complete market success Separating equilibrium where players act as they should according to the signal (prices according to quality) Partial market success (both good and bad cars are bought, some feel cheated) Near Market failure (mixed strategies) Bayesian updating mechanism at work 7.1.5
Signalling and Incentives
1.Education as a signal of productivity Level of education signals quality of a worker. Given the cost of education it is easier for a high quality worker to complete a degree than for a low quality worker. In an e¢ cient market potential employers take level of education as a signal in hiring and deciding wage rates paid to its employees. Spence (1973) model was among the …rst to illustrate how to analyse principal agent and role of signalling in the job market. Pooling equilibrium Consider a situation where there are N individuals applying to work. In absence of education as the criteria of quality employers cannot see who is a high quality worker and who is a low quality worker. Employers know that proportion of workers is of high quality and (1- ) proportion is of bad quality. Therefore they pay each worker an average wage rage as: w = wh + (1
) wl
(1215)
Every worker gets the average wage rate ; there is no wage premium for higher quality in pooling equilibrium. If more productive worker is worth 40000 and less productive worker is worth 20000 and =0.5 then the average wage rate will be 30000; w = wh + (1 ) wl = 0:5 (40000) + 0:5 (20000) = 30000. Let c denote the cost of education. It is worth for high quality worker to go to school only if the wage di¤erence having and not having education is greater than the cost of education which is given by wh
w = wh
[ wh + (1
) wl ]
(1216)
Simpli…cation of this condition implies a signalling condition c
(1219)
This is possible if the cost of education is 5000; then wage net of education cost for high quality is 35000 which is above the pooling wage rate. This makes sense to signal by choosing higher education. Signalling is optimal in this case; fraction of workers will signal by going to education. Aggregate labour cost will be the same but wages will be paid according to the productivity of workers as re‡ected by the level of education of workers.
208
Excel calculations While making a hiring decision employers take level of education as a signal of quality of workers. Government Policy and Signalling It is important to have optimal amount of signalling –too little or too much signalling generates ine¢ cient result. Empirical …nding on signalling is mixed. Public policy could be designed to generate right amount of signalling as following: 1. It can create separating equilibrium by subsidizing education of more able workers. It can ban on wasteful signalling by banning schools that do not produce good workers. 2. High education provides signals, employers pay according to this signal, this will a¤ect the distribution of wages. 7.1.6
Education Level- A Signal of Productive Worker An employer does not know is more productive and who is less productive It pays the same wage rate to both productive and unproductive workers market is ine¢ cient, it drives out more productive workers. Workers can signal their quality by the level of educational attainment, then market may work well. Less costlier for high quality worker to get education. costlier for low quality worker to get the speci…ed education. 209
so the low quality worker gets no education, but the higher quality worker gets education. Employers pay according to the level of education. Education works as a signalling device and makes the market e¢ cient. Education separates the equilibrium. Education Level- A Signal of Productive Worker Consider a level of education e c1 e
c2 e =) c1
c2
(1220)
Cost of eduction of unproductive worker is much higher c2 e < (a2
a1 ) < c1 e
(1221)
Cost of education relative to productivity of low and high quality workers for education e (a2
a1 ) c1
7.2
k2 More Speci…cally p ut (w; e) = 42 wt
kt e1:5
k1 = 2; k2 = 1 w1 = e; w2 = 2e
(1224)
Level of education chosen by less productive worker In perfect information equilibrium, …rms pay according to the marginal productivity Wage of less productive worker: w1 = e; The type 1 worker’s optimisation problem p Max ut (w; e) = 42 wt e
p kt e1:5 = 42 e
@ut (w; e) : 1 = 42 p @e 2 e
2e1:5
1
3e 2 = 0
(1225)
(1226)
1 1 42 =7 42 p = 3e 2 =) e1 = 6 2 e
(1227)
It is optimal for the less productive worker to takes only seven years of education Level of education chosen by more productive worker Wage of less productive worker: w2 = 2e; The type 1 worker’s optimisation problem p Max ut (w; e) = 42 wt e
p kt e1:5 = 42 2e
@ut (w; e) : 1 = 42 p @e 2 2e
2
e1:5
1
1:5e 2 = 0
1 1 1 42 42 p = 1:5e 2 ; 42 p = e =) e2 = = 19:8 2:121 2e 1:5 2
It is optimal for the more productive worker to takes 19.8 years of education. Government Policy and Signalling 211
(1228)
(1229)
(1230)
It is important to have optimal amount of signalling – too little or too much signalling generates ine¢ cient result. Empirical …nding on signalling is mixed. Public policy could be designed to generate right amount of signalling as following It can create separating equilibrium by subsidizing education of more able workers. It can ban on wasteful signalling by banning schools that do not produce good workers. High education provides signals, employers pay according to this signal, this will a¤ect the distribution of wages.
7.3
Popular Principal Agent Games
Principal Agent Model in Job Market: Incomplete Information and Adverse Selection Principal wants to produce output employing workers with a scheme of wage contract that matches e¤orts put by a worker to produce. Worker knows his type but the principal does not. Principal knows the distribution of quality of workers F(s), where s denotes either good or bad state such as probability of observing good is 0.5 and of bad 0.5. Principal o¤ers the agent a wage contract W(q). Worker accepts or rejects this contract based on self-selection and participation constraints. Objective of Principal and Agents Basically worker evaluates the utility from the wage and disutility from work and decides the amount of work to put in. Output from good workers is q (e; good) = 3e and from bad state is q (e; bad) = e If agent rejects the contract there is no work both worker and principal get zero payo¤. If worker accepts the contract Agent’s utility: UA (e; w; s) = w
e2
(1231)
Principal’s pro…t: . . Vp (q; w) = q
w
(1232)
Optimal level of e¤orts by good and bad workers Good worker maximises M ax UG = wG eG
e2G = 3eG
e2G
The …rst part is wage income and the second part of disutility of work.
212
(1233)
The optimal level of e¤orts by good agent is: 3
2eG = 0 =) eG = 1:5
(1234)
Bad worker’s Objective and Optimal E¤orts e2B = eB
M ax UB = wB eB
1
e2B
(1235)
2eG = 0 =) eG = 0:5
(1236)
The principal does not know what levels of e¤orts are appropriate for good and bad workers. Principal’s Objective Principal maximises expected pro…t M ax
qG ;qB ;wG ;wB
UP = [0:5 (qG
wG ) + 0:5 (qB
wB )]
(1237)
by designing separate contracts for good (qG ; wG ) and bad workers (qB ; wB ) and . Wage for good worker: wG = q (e; good) = 3e or e = q3G Wage for bad worker: wB = q (e; bad) = e or e = qB Incentive Compatibility Constraints for Agents Self selection constraint for good worker 2
qG 3 Self selection constraint for bad worker UG = wG
e2G = wG
UB = w B
e2B = wB
UG = wB
e2B = wB
(qB )
2
UB = w G
2
qB 3
(1238)
e2G
(1239)
Participation constraints for good worker 2
qG 3
UG = wG
0
(1240)
Participation constraint for bad worker UB = wB
2
(qB )
0
(1241)
Binding Constraints Participation constraint of bad worker 2 wB = qB
(1242)
Self selection constraint for good worker qG 2 + wB 3 Principal’s Optimal Solution wG =
qB 3
2
=) wG =
qG 3
2
2 + qB
Principal includes agents’optimal choices into his utility function 213
qB 3
2
(1243)
M ax
qG ;qB ;wG ;wB
UP = [0:5 (qG
Including binding constraints of agents: h qG 2 2 UP = 0:5 qG M ax + qB 3
qG ;qB ;wG ;wB
wG ) + 0:5 (qB
qB 2 3
+ 0:5 qB
wB )]
2 qB
i
Now principal decides how much to produce from each type of worker First order conditions with respect to qG and qB @UP = 0:5 1 @qG @UP @qB qB
=
0:5
=
0:265
2qB 9
2qB
2qG 9
= 0 =) qG = 4:5
+ 0:5 (1
(1244)
2qB ) = 0 =) 34qB = 9 =) (1245)
Incentive Compatible First Best Choices of Good and Bad Worker Now wages can be found from the constraints 2
2 wB = qB = (0:265) = 0:07
wG =
qG 3
2
2 + qB
qB 3
2
=
4:5 3
2
+ (0:265)
2
(1246) 0:265 3
2
= 2:32
(1247)
Thus in the presence of information asymmetry , the e¤orts by the good worker is at the …rst best level as the bad e¤ort by him is not as attractive as the good e¤ort. It is not pro…table for good worker to pretend as a bad worker. Good worker is not attracted by the contract for bad worker. It is very costly for the bad worker to accept the contract of good worker. Bad worker’s …rst best to put low e¤ort. Incentive compatible game on renting a piece of agriculatural land If a worker puts x amount of e¤ort, the land produces y = f (x) Then the land owner pays worker s(y). The land owner wants to maximise pro…t = f (x) s(y) = f (x) s(f (x)) Worker has cost of putting e¤ort c(x) and has a reservation utility, u The participation constraint is given by . s(f (x)) c(x) u Including this constraint maximisation problem becomes max = f (x) s(f (x)) subject to sf (x) c(x) u Solution: marginal productivity equals marginal e¤orts f 0 (x)) c0 (x) 214
Incentive compatible game on rending a piece of agriculatural land (a) renting the land where the workers pays a …xed rent R to the owner and takes the residual amount of output, at equilibrium (1248) f (x ) c(x ) R = u (b) Take it or leave it contract where the owner gives some amount such as B
c(x ) = u
(1249)
(c) hourly contract s(f (x)) = wx + K
(1250)
(d) sharecropping, in which both worker and owner divide the output in a certain way. In (a)-(c) burden of risks due to ‡uctuations in the output falls on the worker but it is shared by both owner and worker in (d). Which of these incentives work best depends on the situation. Acemoglu Daren (2001) A Theory of Political Transitions, the American Economic Review, 91:4:938-963 Bardhan Pranab (2002) Decentralization of Governance and Development, Journal of Economic Perspective, 16:4:185-205. Basu Kaushik (1986) One Kind of Power, Oxford Economic Papers, 38:2:259-282. Besley T and M Ghatak (2001) Government versus Private Ownership of Public Goods, Quarterly Journal of Economics, 116:4:1343-1372. Binmore K (1999) Why Experiment in Economics? The Economic Journal 109, 453, Features Feb. pp. F16-F24 Boyd, J.H. and Prescott, E. C. (1986) Financial Intermediary-Coalitions, Journal of Economic Theory, 38: 211-232 Caminal R. (1990) A Dynamic Duopoly Model with Asymmetric Information, Journal of Industrial Economics 38, 3 , 315-333 Cho I.K. and D.M. Kreps (1987) Signalling games and stable equilibria, the Quarterly Journal of Economics, May179-221. Dixit Avinash (1987) Strategic Behaviour in Contests, American Economic Review, Dec., 77:5:891-898. Fundenberg D and J.Tirole (1995) Game Theory, MIT Press. Gardener R (2003) Games of Business and Economics, Wiley, Second Edition. Ghosal S and M. Morelli (2004) Retrading in market games, Journal of Economic Theory, 115:151-181. Harsanyi J.C. (1967) Games with incomplete information played by Baysian Players, Management Science, 14:3:159-182. 215
Hurwicz L (1973) The design of mechanism for resource allocation, American Economic Review, 63:2:1-30. Jin J. Y. (1994) Information Sharing through Sales Report, Journal of Industrial Economics 42, 3, 323-333 Mailath G. J. (1989),Simultaneous Signaling in an Oligopoly Model Quarterly Journal of Economics 104, 2, 417-427 Maskin E and J Tirole (1990) The principal-agent relationship with an informed principle, Econometrica, 58:379-410. McCormick B. (1990) A Theory of Signalling During Job Search, Employment E¢ ciency, and 'Stigmatised' Jobs Review of Economic Studies 57, 2, 299-313 Mookherjee D and D Ray (2001) Readings in the theory of economic development, Blackwell. Moore J. (1988) Contracting between two parties with private information, Review of Economic Studies, 55: 49-70. Mirrlees James A. (1997) Information and Incentives: The Economics of Carrots and Sticks The Economic Journal , 107, 444,1311-1329 Mailath G. J. and L. Samuelson (2006) Repeated Games and Reputations: long run relationship, Oxford. Milgrom P., J. Roberts (1986) Price and Advertising Signals of Product Quality Journal of Political Economy 94, 4,796-821 Myerson R (1986) Multistage game with communication, Econometrica, 54:323-358. Nash J. (1953) Two person cooperative games, Econometrica, 21:1:128-140. Rasmusen E (2007) Games and Information, Blackwell. Rodrik D. (1989) Promises, Promises: Credible Policy Reform via Signalling Economic Journal 99, 397, 756-772 Rogerson W.P.(1988) Price Advertising and the Deterioration of Product Quality Review of Economic Studies 55, 2 (Apr., 1988), pp. 215-229 Riley J.P. (1979) Noncooperative Equilibrium and Market Signalling American Economic Review 69, 2, 303-307 Rubinstein A (1982) Perfect equilibrium in a bargaining model, Econometrica, 50:1:97-109. Ryan M.J. (1998) Multiple criteria and framing of decisions, Journal of Information and Optimisation Sciences, 19:1:25-42. Shapley L and M. Shubik (1969) On Market Games, Journal of Economic Theory, 1:9-25. Sobel J. (1985) A Theory of Credibility Review of Economic Studies 52, 4, 557-573
216
Sutton J. (1986) Non-Cooperative Bargaining Theory: An Introduction, Review of Economic Studies, 53, 5., 709-724 Spence M. (1977) Consumer Misperceptions, Product Failure and Producer Liability, Review of Economic Studies 44, 3, 561-572
7.4
Exercise 15: Principal Agent Problem Problem 10: Asymmetric information: Principal Agent Problem
1. Project B earns more but is riskier than project A. Probability of success of projects A and B are given by PA and PB respectively. a. Illustrate how the rate of interest rate should be lower in project A than in project B in equilibrium? b. Probability of types A and B agents is given by PA and PB respectively. Prove under the asymmetric information a lender charging a pooling interest rate is unfair to the safe borrower A and more generous to the risky borrower B. c. How can agent signal its worth? How can the lender ascertain the degree of moral hazard in B? 2. Principal wants to produce output by employing workers with a scheme of wage contract that matches e¤orts put by a worker to produce. Worker knows his type but the principal does not but the principal knows the distribution of quality of workers F (s), where s denotes either good or bad state. Probability of observing good is 0.5 and of bad 0.5. Principal o¤ers the agent a wage contract W (q) worker accepts or rejects this contract based on selfselection and participation constraints. Basically worker evaluates the utility from the work and disutility from work and decides the amount of work to put in. Output from good worker is q(e; g) = 3e and from bad state is q(e; b) = e . Both of them are risk neutral. If agent rejects the contract there is no work both worker and principal get zero payo¤ otherwise the e3 and . principal = V (q w) = q w agent = U (e; w; s) = w (a) Determine the level of e¤orts by good and bad workers. (b) Formulate the participation and incentive compatibility constraint for workers. (c) What is the principal’s objective function? (d) What wage rates are paid to good and bad workers?
7.5
Mechanism Design for Price Discrimination: Low Cost Airlines Example
There are three steps in mechanism design. 1. Principal designs a mechanism or contract 2. agents accept or reject mechanism 3. Those who accept the mechanism play the game. Consider a case of monopolist which supplies q with marginal cost c and tari¤ (price) T. Its objective is
217
U1 (q; T; ) = V (q)
T
(1251)
V is a common knowledge, is private information. There are high and low type buyers. = with probability p = with probability p and p+ p = 1 First best solution: If the seller knew its tari¤ would be V (q) = T Its pro…t would be V (q) cq and V 0 (q) = c: Most often it does not know , therefore o¤ers q; T and q; T bundles. Then the expected pro…t will be Eu0 = p q; T + p q; T
(1252)
Sellers’constraints 1. consumers need to be willing to purchase and this requires ful…llment of individual rationality constraint and participation constraint. IR1 :
V q
T
0
(1253)
IR2 :
V (q)
T
0
(1254)
Incentive compatibility requires that consumers consume the bundel intended for them. IC1 : IC2 :
V q
T
V (q)
V (q)
T
V q
T T
Sellers problem is to choose q; T and q; T bundles to maximise her pro…t.
218
(1255) (1256)
Binding constraints are IC1 and IR2 This implies T = V q
T + V (q)
T = V (q) p T
cq + p T
cq = p
(1257)
V q V q
p
(1258) pcq + p V (q)
c
V q = 1
p(
)
cq
(1259) (1260)
p
V (q) = c
(1261)
Allocation is socially optimal Reference Fundenbege and Tirole (1995)), Game Theory, MIT press. Economy and Business Class Ticket Problem for Airlines (Based on Dixit et. al. (2009)) Two types of travellers: economy and business
219
Assume 100 travellers and 70 of them economy type tourists and 30 business type …rst class.
Economy First Class
Cost of the Airlines 100 150
Reservation Price Tourists Business 140 225 175 300
Airline’s Pro…t Tourists Business 40 125 25 150
Economy class tickets cost less than the business class. Business traveller is ready to pay higher price than economy class for both economy and …rst class but the airlines cannot separate them out. Why Mechanism Design for Price Discrimination: Low Cost Airlines Example Economy class tickets cost less than the business class. Business traveller is ready to pay higher price than economy class for both economy and …rst class but the airlines cannot separate them out. Business traveller may well buy economy class ticket rather then business class. Airlines likes to build a mechanism so that business class buy business class tickets and economy class buy economy class ticket. What is the pro…t to the airlines if it knows reservation prices of tourists and business group of travellers? How would this pro…t change in business type buy the economy class ticket? What is the incentive compatible price that the airlines can o¤er to the business group? Incentive Compatible Mechanism What would happen if the split between the business and economy class is 50/50? What will be the optimal reaction of the airlines? Pro…t in an ideal scenario ( perfect price discrimination; if the airlines knew each customer type) 30(300
150) + (140
100) (70)
=
30
=
4500 + 2800 = 7300
150 + 40
70 (1262)
Business travellers have consumer surplus of 225 -140 = 85 in economy class ticket. For this all 30 of may decide to buy economy class ticket. Then the pro…t of the airlines when the airlines fails to screen customers will be (140
100) (100) = 4000
(1263)
Airlines should give consumer surplus of 85 to business traveller and charge them (300-85) = 215. This will alter their pro…t 30(215
150) + (140
100) (70)
=
30
=
1950 + 2800 = 4750
Incentive Compatible and Participation Constraints 220
65 + 40
70 (1264)
Airline initially does not have enough information on types of customers It should design incentive compatible pricing scheme so that business class travellers do not defect to economy class. This requirement is contained in the incentive compatible constraint. If it charges 240 for the business class then the their consumer surplus will be equal (300-240) = 60 from business class travel and (225-165)=60 However 140 is the maximum the tourist class traveller is ready to pay. If the airline raises price to 165 they will lose all tourist travellers. Mechanism requires ful…llment of the participation constraint. Airlines should operate taking account of the participation constraint of tourists and incentive compatible constraint of the business travellers. X < 140 is the participation constraint; incentive compatible constraint is 225 -X < 300-Y or Y < X+75 Mechanism when the composition of travellers change Charging 215 for the business class and 140 for the economy class is the solution to the mechanism design problem. 30(215
150) + (140
100) (70)
=
30
=
1950 + 2800 = 4750
65 + 40
70 (1265)
Suppose the composition of travellers changes to 50% of each. Pro…t with the above price mechanism 50(215
150) + (140
100) (50)
=
50
65 + 40
50
=
3250 + 2000 = 5250
(1266)
It is more pro…table to scrap the tourist class tickets instead and charge the business class its full reservation price 50(300 150) = 50 150 = 7500 (1267) There are relatively few customers but all are willing to pay higher price. There is no problem of screening as the airlines now does not serve to the tourist class at all. 7.5.1
Mechanism for e¢ cient contract for a CEO Owners of a company are concerned about a project that would earn them 600,000 if successful. Probability of success with normal e¤ort from the manager is 60 percent and this can increase up to 80 percent if the manager puts extra e¤orts. The basic salary of the manager is 100,000. He would put extra e¤orts only if he is paid additional amount of at least 50,000. Owners cannot monitor whether the manager is putting high or low e¤orts. 221
a) Is it pro…table to pay extra for the manager? Pro…t without paying extra: 0.6 * 600,000 - 100,000 = 260, 000 Pro…t with extra incentive payment: 0.8 * 600,000 - 150,000 = 330, 000 Extra payment can make up to 70,000 with probability of 0.8. Once extra payment is made how can owners make sure that he puts extra e¤orts? This requires evaluation of incentive compatibility and participation constraints. Mechanism to ensure high e¤orts by a CEO a) Incentive compatibility constraint (s + 0:8b)
(s + 0:6b) > 50; 000
(1268)
0:2b > 50; 000
(1269)
(s + 0:8b) > 150; 000
(1270)
b = 250,000 b) Participation constraint:
s = 150; 000
0:8b;
s = 150; 000
0:8 (250; 000) =
50; 000
(1271)
It is not possible to hire manager with negative salary. At most managers can be conditioned to bonus payment but with zero salary. Mechanism to ensure high e¤orts by a CEO (0 + 0:8b) > 150; 000
(1272)
200; 000 > 150; 000
(1273)
Pay 200,000 and the manager will put maximum e¤ort. c) Is it pro…table to pay extra 200,000 as an incentive payment? Pro…t with incentive payment 0.8 * 600,000 - 200,000 = 280, 000 Pro…t without incentive payment 0.6 * 600,000 - 100,000 = 260, 000 Thus pro…t increases by 20,000 with the incentive payments. 7.5.2
E¢ cient contracts of Land
Proposition 1: Results of …xed fee contract and joint pro…t maximisation are equivalent Proposition 2: Hire contract is incentive incompatible and leads to production ine¢ ciency Proposition 3: Moral hazard problem and production ine¢ ciency exists in revenue sharing contingent contract Proposition 4: Pro…t sharing contract is e¢ cient and free of moral hazard problem Price and cost P = 24
0:5q
222
C = 12q
(1274)
Revenue R = P:q
(1275)
Mechanism design in renting lands Under the joint pro…t maximisation agreement (q) = P:q
C = (24
0:5q) q
0:5q 2
12q = 24q
12q
(1276)
Under the …xed fee contract tenant maximises (q) = P:q
C
F = (24
0:5q) q
12q
F = 24q
0:5q 2
12q
F
(1277)
Under both these arrangements 0
(q) = 24
q
12 = 0
q = 12; p = 18; R = 216; C = 144;
(1278) (q) = 72
(1279)
Mechanism design in renting lands 72 is the total pro…t. It is divided between the tenant and the landlord by their mutually agreed arrangement. Under the …xed fee contract landlord may …x the amount that he needs at 48. Then the residual 24 pro…t goes to the tenant. This arrangement achieves production e¢ ciency, is incentive compatible, ful…ls the participation constraint and motivates to put the optimal e¤ort and solves the moral hazard problem. Hire contract Landowner can hire workers in …xed fee basis, say 12 per unit of output a. This does not motivate tenant to work because his cost per a is also 12 and so does not make any pro…t. Landlord has to raise payment to tenant to say 14 to motivate him to work. Then the pro…t maximisation problem of the landlord will be (q) = P:q
C = (24
0
0:5q) q
(q) = 24
q
q = 10; p = 19; R = 190; C = 120;
14q = 24q
0:5q 2
14 = 0 LL
(q) = 50;
(1280)
(1281) T
(q) = 20
The tenant has incentive to overproduce whenever is paid more than 12. Revenue sharing contract
223
14q
(1282)
Let the landlord enter into a revenue sharing contract whereby she gets 14 th of the revenue and leavening 34 of revenue to the tenant who also bears all production cost. The pro…t function of the tenant is now modi…ed as (q) =
3 P:q 4 0
q
C=
(q) = 6
3 (24 4
8; p = 20; R = 160; C = 96;
=
120;
(q) =
12q
3 q=0 4
=
T
0:5q) q
(1283)
(1284)
LL
(q) =
3 (160) 4
1 (160) = 40 4
(1285)
Pro…t of tenant = 120 - 96 =24 This level of production is not incentive compatible for the land-lord who would be interested in maximising revenue by producing 24. Pro…t sharing contract Now let us assume the landlords and tenants enter into a pro…t sharing deal, say 1/3rd of pro…t goes to the tenant and 2/3rd to the landlord. 1 3
(q) =
1 (P:q 3 0
LL
C) =
(q) = 4
1 24q 3
0:5q 2
1 q=0 3
q
=
12; p = 18; R = 216; C = 144;
(q)
=
48;
T
12q
(1286)
(1287)
(q) = 72;
(q) = 24
(1288)
There are many other situations, including optimal tax designs, optimal price discrimination, fund management, management of theme-park, renting of buildings, collection of taxes or tari¤s, union-management contracts, where these types of models have been applied. 7.5.3
Mechanism for Poverty Alleviation There are three players in the poverty game -poor, rich and government; each has three strategies available to it to play, s, l, and k , cooperation, indi¤erence and non cooperation. The outcome of the game is the strategy contingent income for poor and rich, ytp (s; l; k) and ytR (s; l; k) with the probability of being in particular state like this is given by pt (s; l; k) and R t (s; l; k) respectively and tax and transfer pro…les associated to them. The state-space of the game rises exponentially with the length of time period t. T
224
he objective of these rich and poor households is to maximize the expected utility that is assumed to be concave in income. The government can in‡uence this outcome by choices of taxes and transfers that can be liberal, normal or conservative. Mechanism for Poverty Alleviation: Proposition 1 Proposition 1: The state contingent expected money metric utility of poor is less than that of rich, which can be expressed as: s X l X k X T X
p p p t (s; l; k) t u (yt (s; l; k))
s=1 l=1 k=1 t
'
l X k X T s X X
#
T X TtR (s; l; k)
ytR (s; l; k)
t
T k X l X s X X
Ttp (s; l; k)
p p p t (s; l; k) t u (yt (s; l; k))
+
T X t
Ttp (s; l; k)
#
(1290)
#
p p p t (s; l; k) t u (yt (s; l; k))
(1291)
s=1 l=1 k=1 t
and
s X l X k X T X
R R t (s; l; k) t u
ytR (s; l; k)
s=1 l=1 k=1 t
>
' T s X l X k X X s=1 l=1 k=1
R R t (s; l; k) t u
t
ytR (s; l; k)
T X + TtR (s; l; k) t
225
#
(1292)
Mechanism for Poverty Alleviation:Proposition 4 Proposition 4: Growth requires that income of both poor and rich are rising over time: p p p (s; l; k) < ::::: < Tt+T (s; l; k) (s; l; k) < Tt+1 Ttp (s; l; k) < Tt+1
(1293)
p p p Ytp (s; l; k) < Yt+1 (s; l; k) < Yt+1 (s; l; k) < ::::: < Yt+T (s; l; k)
(1294)
R R R YtR (s; l; k) < Yt+1 (s; l; k) < Yt+1 (s; l; k) < ::::: < Yt+T (s; l; k)
(1295)
Mechanism for Poverty Alleviation:Proposition 5 Proposition 5: Termination of poverty requires that every poor individual has at least the level of income equal to the poverty line determined by the society. When the poverty line is de…ned one half of the average income this can be stated as: ! N 1 1X h p Yt (s; l; k) (1296) Yt (s; l; k) > 2 N h=1
Above …ve propositions comprehensively incorporate all possible scenarios in the poverty game mentioned above. Propositions 2-5 present optimistic scenarios for a chosen horizon T . Mechanism for Poverty Alleviation: Tests Testing above propositions in a real world situation is very challenging exercise. It requires modelling of the entire state space of the economy.
Moreover in real situation consumers and producers are heterogeneous regarding their preferences, endowments and technology and economy is more complicated than depicted in the model above. In essence it requires a general equilibrium set up of an economy where poor and rich households participate freely in economic activities taking their share of income received from supplying labour and capital inputs that are a¤ected by tax and transfer system as illustrated in the next section. Bhattarai K. (2010) Strategic and general equilibrium models of poverty, Romanian Jounral of Economic Forecasting, 13:1:137-150
7.6
Repeated Game
Market demand for a product is P = 130
(q1 + q2 )
(1297)
Cost of production for each of two …rms is . Ci = 10qi If played in…nite number of times two …rms form a cartel and monopolise the market.
226
(1298)
Each will supply only 30, set market price to monopoly level at £ 70 and divide total pro…t £ 3600 equally; each getting £ 1800. This is shown by (1800,1800) point in the diagram. It pays to cooperate in the long run; it is sub-game perfect equilibrium. Cooperative Solution It pays to cooperate in the long run; it is sub-game perfect equilibrium. = (130
Q) Q
@ = 130 @Q
2Q
Q2
10Q = 130Q 10 = 0 =) Q =
10Q
(1299)
120 = 60 2
(1300)
Price: P = (130 Q) = 130 60 = 70; Cost: C = 10Q = 10 60 = 600; Pro…t: = P Q C = 60 70 600 = 3600 Non-Cooperative Nash Equilibrium If any one …rm cheats and tries to supply more in order to get more pro…t; it will be found out by another …rm. Opponent …rm will react to this. Game will be non-cooperative, resulting in a Cournot Nash equilibrium. Each …rm produces 40 units, market price is set at 50 and each gets £ 1600 pro…ts. 1
= (130
(q1 + q2 )) q1
10q1 and
2
= (130
(q1 + q2 )) q2
10q2
with reaction functions 2q1 + q2 = 120 and q1 + 2q2 = 120 Total supply is 80, each supplying 40 and making pro…t of 1600 and market price 50. Trigger Strategy and Perpetual Punishment If …rm 1 plays Cournot game but …rm 2 still plays cartel and supplies just 30. Then from the …rm 1’reaction function . 2q1 + q2 = 120 q1 = 60
1 q2 = 60 2
1 (30) = 45 2
(1301)
If …rm 1 supplies 45, market price will be . P = 130
(q1 + q2 ) = 130
45
30 = 55
This makes pro…t margin of …rm 1 to be 45 and its pro…t . 45 45 = 2025 227
1
= (55q1
(1302) 10q1 ) = 45q1 =
Firm 2 will …nd out that …rm 1 has cheated. If it does not react its pro…t will be down to 1350. It will also produce according to its reaction curve. Thus the Nash equilibrium will result with each …rm producing 40 and earning 1600 pro…t for the rest of the periods and the market price will be 50. For whom is it pro…table to Cheat? Does …rm 1 gain or lose by deviation from the agreement. For this evaluate the in…nite series of pro…ts in deviation and in compliance with agreement. Present value of pro…t in case of cheating 2 + :::: + ::: = 425 + 1600 + 1600 + 1600 2 + :::: + ::: 1 = 2025 + 1600 + 1600 2025 1600 + 1600 + 1600 + 1600 2 + :::: + ::: 1 = (Note just with –and + 1600) Using operator to maintain a constant payo¤ from the game i h 1600 ) + 1600] = 425 425 + 1600 = 2025 425 (1 ) 1 = (1 ) 425 + (1 ) = [425 (1 By comparing pro…ts with and without cheating 225 9 2025 425 < 1800 or ; 425 > 2025 1800; > 425 =) > 17 Whether the …rm 1 will stick to agreement or not depends on whether its discount factor if 9 9 greater than > 17 . For discount factor < 17 it is bene…cial to stick to the agreement, which is very high, about 53 percent. Home Work Show above results in a diagram Illustrate repeated game for multiple periods using brach nodes Workout Bertrand type competition for above game and illustrate 'cut-throat' price competion in a diagram. Home Work: Show above results in a diagram,
7.7
Moral Hazard and Adverse Selection
Moral Hazard: Insurance Game with Symmetric Information Under symmetric information full insurance is optimal; insurance company can charge premium according to level of e¤orts exerted by the agent to prevent accident (see Jehle and Reny (2001, Chapter 8)) Let p be insurance premium. (e) probability of accident with e¤ort e, this diminishes with greater care (higher e). Level of bene…t o¤ered in case of accident is BL speci…c to losses L =1,2,. . . .L . The Moral hazard problem is for insurance company to set the premium according to e¤orts max
e;p;Bo ;::::BL
p
L X l=0
subject to participation constraint:
228
(e) Bl
(1303)
L X
l
(e) u (W
p
l + BL )
u
d (e)
(1304)
l=0
Lagrangian function L= p
L X
'
(e) Bl +
l=0
First order condition
@L = 1 @p
' L X
l
L X
l
(e) u (W
p
l + Bl )
l
l
0
(e) u (W
p
l + BL )
#
u =0
d (e)
0
(e) u (W
(e) u (W
p
l + BL
p
l + Bl )
d (e)
d (e)
l=0
From above
u0 (W
u
d (e)
l=0
l=0
@L = 1 @Bl ' L X @L = @
#
p
u) = 0
(1305)
(1306)
(1307)
#
u =0
(1308)
l + Bl ) = d (e) + u
(1309)
Under full insuranceBl = l this implicitly de…nes the insurance premium for e¤ort level . u0 (W
p) = d (e) + u
(1310)
Since low e¤ort is less costly than more e¤ort for the costumer d (e) d (1) ; the premium under lower e¤ort must be set higher than for the higher e¤ort: p (0) p (1) for pro…t maximisation p
L X
(e) :l
(1311)
l=0
This is the prediction of moral hazard with complete information but uncertainty with consumer’s hidden action. Adverse Selection: Insurance Game with Asymmetric Information Insurance company cannot observe the consumer’s choice of accident prevention e¤orts. But the insurance company continues to seek maximize he expected pro…t. It now need to add incentive compatibility constraint. max
e;p;Bo ;::::BL
p
l=0
l
(e) u (W
p
l + BL )
d (e)
(e) Bl
(1312)
l=0
subject to participation constraint: L X
L X
L X l=0
229
l
(e0 ) u (W
p
l + BL )
d (e0 )
(1313)
L X
(e) u (W
l
p
l + BL )
u
d (e)
(1314)
l=0
Incentive compatibility constraint for low e¤orts again full insurance is the best policy from u0 (W
p) = d (e) + u
however the incentive compatibility requires d (0) costs more for the costumer. For high e¤orts case e = 1 Lagrangian function
L =
p 2
6 6 + 6 6 4 @L = 1 @p @L = @Bl
L X
(e) Bl +
l=0
( L X
(l=0 L X
l
l
'
L X
d (1). Therefore lower insurance avoidance
(e) u (W
l
(1315)
p
l + Bl )
l=0
(e) u (W
p
0
(e ) u (W
l + BL )
p
d (e)
l
(1) +
(
l
(1)
l + BL )
l
3
7 7 ) 7 7 5 d (e0 )
l=0
' L X
)
d (e)
0
(0)) u (W
p
(1) +
' L X @L = [( @
l
(1)
l
(e) +
l
[(
l
(1)
(0))] u (W
l
p
0
(0))] u (W
#
(1317)
l + BL ) = 0
(1318)
#
l + BL ) = 0 p
l + Bl ) + d (0)
d (1) = 0
(1319)
l + Bl ) + d (0)
#
(1320)
l=0
' L X @L = [( @
l
(1)
l
(0))] u (W
p
l=0
u0 (W
1 = p l + Bl )
+
u
(1316)
l=0
l
#
1
l l
(1) (0)
d (1)
0
(1321)
since > 0 the RHS is strictly decreasing, this implies that u0 (W p l + Bl ) must be strictly increasing for this to happen l Bl be must increase with e¤ort levels and losses l = 0; 1; 2; ::::L:Optimal high policy does not provide full insurance but the deductible payment increases size of loss. Problem Consider a moral hazard insurance model with an insurance policy fp; B0 ; B1 ; :::::; BL g where p is insurance premium and B0 ; B1 ; :::::; BL denote the bene…t from the insurance company against 230
loss l. Normally the insurance company can observe the loss but not the level of accident avoidance e¤ort (e) of the consumer. The problem of the insurance company is: max
e;p;B0 ;B1 ;:::::;BL
L X
p
subject to participation constraint L X
l
(e) u (w
p
l
(e) BL
(1322)
l=0
l + BL )
d (e)
u
(1323)
l=0
and incentive constraint L X
l (e) u (w
l=0
p
l + BL )
d (e)
L X
l
(e0 ) u (w
p
l + BL )
d (e0 )
u
(1324)
l=0
1. Show that it is Pareto optimal to do full insurance under symmetric information when the insurance company can observe the level of e¤orts of the consumer. 2. How could the insurance company design an e¢ cient contract to induce e¤orts to minimise cost under the assymetric information? Is full insurance still optimal?
7.8
Auction
Types of Auction First price, sealed-bid: person who bids the highest amount gets the good. Second-price, Sealed-bid: Each submit a bid. Higher bidder wins and pays second-highest bid for the good. Dutch Auction: Seller begins from very high price and reduces it until someone raises a hand. English Auction: Begins with very low price, bigger drops out by raising a hand. Which one of these four mechanism is good for the seller?? Online auctions -ebay (http://www.ebay.com/); car auction (http://www.carandvanauctions.co.uk/); Art auction (http://www.artinfo.com/artandauction/); Online advertisement auctions in Google, Microsoft, Yahoo, Facebook, You Tube Auction: Vickrey-Clark-Grove (VCG) mechanism Honesty is the best policy in Vickery auction; truth telling is the winning strategy. Proof Let there be two bidders bidding b1 and b2 but with true values v1 and v2 . Highest bidder wins the auction at the price of the second-highest bid. English auctions and second-highest sealed-bid auctions are equivalent. 231
Expected value for bidder 1 is then given by prob (b1 > b2 ) (v1
b2 )
(1325)
If (v1 > b1 )it is in the best interest of bidder 1 to raise the probability of winning prob (b1 > b2 ) , this can happen when (v1 = b1 ) Similarly If (v1 < b2 ) then it is in the interest of bidder 1 to make prob (b1 < b2 ) as small as possible. It happens when .(v1 = b1 ) Thus the truth telling is the best interest in such action. Auction: Financing Mechanism for Public Goods Let x be a public good such as streetlight or road; x = 1 if it is provided x = 0 if not. If state knew that how much each person is willing to pay for this it could bill e¢ ciently. Each would pay according to the value they put in such public good. Unfortunately it is impossible to know preferences of individuals. Individuals do not tell true value when asked that how much they are ready to pay for this. Let N individuals be indexed by i. Then the utility from the public good to an individual i is given by Ui (x). There is free rider problem with public goods. Individuals may underreport their utility thinking that others will pay higher for it if they act like this but they will have opportunity of full bene…t. Under Vickrey-Clark-Grove mechanism it is in the best interest of individuals to tell the truth. Auction:Financing Mechanism for Public Goods Under Grove mechanism each individual is asked to report his her utility; which is ri (x). . Then N P the state chooses x* that maximises the sum of reported utilities R = ri (x): Each individual i=1
receives a side-payment Ri =
N P
ri (x):.
j6=1
With side payment the total utility of an individual is Ui (x) +
N X ri (x)
(1326)
i=1
State chooses x to maximise ri (x) +
N X
ri (x)
(1327)
i=1
Therefore it is in the best interest of an individual to tell the truth Ui (x) = ri (x). All agents tell truth like this and this mechanism generates e¢ cient outcome. (page 27 of the new handbook on sum of MRS =MC of public good.) (See Varian HR (2010) Intermediate Microeconomics: A Modern Approach, Norton,8th edition). 232
Binmore K. (1990) Fun and Games: A text on Game Theory, Lexington, Heath. Cripps, M.W.(1997) Bargaining and the Timing of Investment, International Economic Review, 38:3 :Aug.:527-546 Dixit A., S. Skeath and D. F. Reiley (2009) Games of Strategy, Norton. Gardener R (2003) Games of Business and Economics, Wiley, Second Edition. Holt Charles (2007) Markets, Games and Strategic Behaviour, Pearson, . Mailath G. J. and L. Samuelson (2006) Repeated Games and Reputations: long run relationship, Oxford. Moore J (1984) Global Incentive Constraints in Auction Design Econometrica, 52:. 6 pp. 1523-1535 Kreps D. M. (1990) A Course in Microeconomic Theory, Princeton. Osborne M.J. and A. Robinstein (1994) A course in game theory, MIT Press. Perlo¤ J. M. (2008) Microeconomics: Theory and Applications with Calculus, Pearson. Rasmusen E(2007) Games and Information, Blackwell,. Varian HR (2010) Intermediate Microeconomics: A Modern Approach, Norton,8th ed.Chapter 17,35.
7.9
Exercise 16 :Optimal production of a multiproduct …rm
Q1. Consider a principal agent model where the gross pro…t ( of the manager and is subject to random shocks (') as:
g
g)
of a …rm depends on e¤orts (e)
=e+'
(1328)
Cost of e¤orts to the manager is given by C(e): C(e); C 0 (e) > 0 and C 00 (e) > 0 Net pro…t (
n)
for the …rms is gross pro…t ( n
=
g
g)
(1329)
minus manager’s salary (s) as:
s=e+'
s
(1330)
Then the expected net pro…t becomes: E(
n)
= E (e + '
s) = e
E (s)
(1331)
Manager’s utility takes the form: A 2 A 2 C(e) = e C(e) (1332) 2 s 2 e where 2s is the variance of salary and A is a risk aversion parameter. Manager dislikes volatility in salary 2s . Two scenarios could be considered in this game; …rst when owens can observe the e¤orts put in by the manager and another where they cannot observe it. E (u) = E (s)
233
A. What is the …rst best solution if the owners can observe the level of e¤orts by the manager? What will be the variance of salary 2s in the …rst best solution? What is the relation between the marginal cost of e¤orts and marginal bene…ts of the manager in this case? If owners are not able to observe the level e¤orts by the managers, they will o¤er him a performance based salary contract subjecting it to gross pro…t as: s(
g)
=a+b
g
(1333)
Owners set salary by setting parameters a and b …rst where a is …xed component and b is incentive payment. Then the manager decides the level of e¤orts conditional on the contract. If the cost of e¤orts is given by: C(e) =
e2 ; 2
2
= 1; A = 2
(1334)
B. What will the optimal value of incetive coe¢ cient (b) be in the second best solution? What will the optimal level of e¤orts (e ) be ? What will the …xed component of the salary (a) be? What will the expected pro…t for the owners be? Solve this game by backward induction to …nd answers to these questions. Q2 This exerice aims to investigate how di¤erent production technologies adoped by a …rm results in di¤erent level of output, employment and investment in each sector. First exercise is …xed input process under the linear programming framework with limited price based substitution. The second problem contains adoption of the Cobb-Douglas technology with elasticity of substituion equal to 1. Third process involves usign the CES production technology with a constant elasticity of substitution. Overall resource available to this …rm is given in each case. Linear programming problem A multinational …rm produces two products Y1 and Y2 , these products are sold at 6 and 3 respectively. Linear programming problem max R = 6Y1 + 5Y2 (1335) 1. Subject to: 1) Labour constraint (workers) 2Y1 + Y2
1000
(1336)
2) capital constraint (machine hours) 12Y1 + 40Y2 1. where Y1
0 and Y2
0; 234
30000
(1337)
What are the revenue maximising levels of output of Y1 and Y2 . How many workers /machine hours are devided between producing these two products to maximise revenue. If wage rate is 2 and capital cost is 0.05 per hour what would be total pro…t? Cobb-Douglas Technology = P1 Y1 + P1 Y1
C1
C2
(1338)
Subject to (1
)
(1339)
(2
)
(1340)
Y1 = K1 L1 Y2 = K2 L2
C1 = rK1 + wL1
(1341)
C2 = rK2 + wL2
(1342)
CES technology = P1 Y1 + P2 Y2
C1
C2
(1343)
Subject to Y1 = [ 1 K1 + (1
1 ) L1 ]
Y2 = [ 2 K2 + (2
2 ) L2 ]
1
(1344)
2
(1345)
C1 = rK1 + wL1
(1346)
C2 = rK2 + wL2
(1347)
Comparision of results among technologies Y K Firm 1 Linear Programming Cobb-Douglas CES Firm 2 Y K Linear Programming Cobb-Douglas CES
235
L
R
C
Pr
L
R
C
Pr
Impacts of microeconomic policies in …rms output decision. 1) cost of capital doubles to …rms because of credit crisis. How would this a¤ect these variables. 2) Wage dispute between the management and workers caused wage rate to rise by 25 percent. Given no change in capital market conditions how would this a¤ect the production. 3) There is a change in preferences of consumers because of substitutable product in the market. How does these variables. Y K Firm 1 Linear Programming Cobb-Douglas CES Firm 2 Y K Linear Programming Cobb-Douglas CES 7.9.1
L
R
C
Pr
L
R
C
Pr
A Microeconomic Model of FDI
Hymer (1976) and Caves (1982), Batra and Ramachandran (1980) and Batra (1986) Rossel (1985), Hortmann and Markusen (1987) and Markusen (1995). MNCs move to a foreign country for a number of reasons: cost advantages in producing there rather than exporting commodities; ownership (O) of …rm speci…c capital; location (L) based advantages of production; licensing abroad for reasons of natural resources or customer bases; internalisation (I) of bene…ts of technical know how by …rms doing R & D.
where 1 ,P1 ,q1 ; C1 and and foreign markets
2 ,P2
1
= P1 q1
C1
(1348)
2
= P2 q2
C2
(1349)
,q2 ; C2 are pro…t, price, demand and cost of production at home
T C > C2 R+D
C1 = 800
F C < 2M
200 = 600
F C < 2R
FC
(1350) (1351)
where R is the rental income from licensing of a partner foreign …rm, (M is the pro…t from subsidiary is FDI takes the form of subsidiary operation), D is the payments made in case the licensee defects in the second period (this deters the licensee from supplying the market itself after gaining the know-how from the MNC in the …rst period), and FC is the …xed cost of FDI to the MNC. Let us assume that the demand of the monopolist in the home market is given by
236
q1 = 21
0:1P1
(1352)
10q1
(1353)
0:4P2
(1354)
2:5q2
(1355)
and its inverse demand is P1 = 210 If demand abroad is q2 = 50 and the inverse demand abroad is P2 = 125
Cost of production is di¤erent across counties di¤er.It is C1 = 200 + 10q1
(1356)
at home but it is costlier to set up business in foreign country because of higher …xed costs, C2 = 800 + 10q2
(1357)
(marginal costs may also be di¤erent). The optimal condition for pro…t maximisation is given by a point where the marginal revenue equals marginal cost in each market: M R1 = M C1 and M R2 = M C2
(1358)
Given the above information, the total revenue from the home market is R1 = P1 q1 = (210
10q1 ) q1 = 210q1
10q12
(1359)
. Therefore, the marginal revenue from home market sales would be M R1 = 210
20q1
(1360)
.. Similarly, the total revenue from the sales in the foreign market would be R2 = P2 q2 = (125
2:5q2 ) q2 = 125q2
2:5q22
(1361)
and associated marginal revenue would therefore be M R2 = 125
5q2
(1362)
Fixed costs of production are di¤erent across countries, but the marginal costs are assumed to be the same in both countries. M C1 = M C2 = 10
(1363)
Now it is possible to solve the model for the optimal amount of goods supplied at home and abroad by using conditions where the marginal revenue in each market needs to equal its marginal cost.
237
M R1
=
M C1 ) 210
20q1 = 10 ) q1 = 10 =) P1 = 210
10q1
M C2 ) 125
5q2 = 10 ) q2 = 23 =) P2 = 125
2:5q2
=) P1 = 110
M R2
=
=) P2 = 67:5
(1364)
(1365)
Thus, the amount supplied at home is much smaller than amount supplied abroad and prices charged at home are much higher than prices charged abroad. Corresponding revenues are: R1 = P1 q1 = 110
10 = 1100; R2 = P2 q2 = 67:5
23 = 1552:5
(1366)
The cost function is assumed to be known here for simplicity. It must be derived from the cost minimisation principle subject to a production technology constraint. The total cost of production at home and abroad are given by C1 = 200 + 10q1 : = 200 + 10
10 = 300
(1367)
C2 = 800 + 10q2 = 800 + 10
23 = 1030
(1368)
Now it is possible to calculate pro…ts to the MNC from home and foreign markets: 1
2
= P1 q1
= P2 q 2
C1 = 1100 C2 = 1552:5
300 = 800
(1369)
1030 = 525:5
(1370)
Conclusion In this paper, the microeconomic e¤ects of FDI have been illustrated with an example of a multi-plant MNC that faces a di¤erent structure of demand and costs between home and foreign countries with strategic consideration of licensing or subsidiary production in foreign countries. On the macro side, the total FDI aggregated over MNCs accounts for a signi…cant proportion of total investment and has a signi…cant impact on economic growth. This growth e¤ect is shown theoretically using an endogenous growth model with FDI in which foreign capital complements domestic capital and contributes to both investment and growth rate of output. Our model predictions have been tested using panel data growth regressions for 30 OECD countries over 1990 to 2004. Our analysis establishes positive impacts of FDI in‡ows and negative impacts of FDI out‡ows on investment and economic growth. The impacts of time and country speci…c e¤ects are found to be consistent with the stylized facts relating to growth rates of output, investment ratios and in‡ows and out‡ows of FDI. The empirical results illustrated in this paper are comparable to Desai et al. (2000).
238
8
L8: Uncertainty and Insurance
Future is uncertain. Arrow (1963), Harsanyi (1967), Roy (1968), Akerlof (1970), Rothschild and Stiglitz (1976), Kahneman and Tversky (1979), Machina (1987), Hey (1987), Moore (1988) Newbery and Stiglitz (1982) Hirshleifer and Riley (1992) Hey and Orme (1994) Hey, Lotito and Ma¢ oletti (2010) Conte and Hey (2013) has analysed how optimal choices under uncertainty occur in many aspects. Income or expenditure of individuals are uncertain. Return on stocks and portfolios are uncertain. Von Neumann-Morgenstern expected utility theory is applied to analyse choices in uncertain world. For an asset or commodity x the expected utlity is sum of u(x) weighted by their densities, dF . Z U (F ) = u(x)dF (1371) Consider a case where safe asset earns £ 1 per £ 1 invested and an unsafe risky assets ears a random amound z per £ 1 invested, z has a distribution F(z). Z zdF (z) > 1 (1372) Return on risky asset exceed that in the safe asset. Investor has initial wealth w to invest, which could be invested in and proportion in risky and safe assets. Return from the portfolios is z+ and
+
(1373)
= w: Investors problem is to choose and to maximise expected utility. Z Z max u ( z + ) dF (z) = max u (w + (z 1)) dF (z) ;
subject to
+ max
Z
u (w + (z
=w 1)) dF (z); 0 6
(1374)
(1375) 6w
(1376)
Optimal > 0: If the risk is acturially fair, the risk averstor will take small amount of risk. Bernoulli utility functions increasing in x are continueous and concave and ful…ll Jensen’s inequality conditions. Z Z u(x)dF (x) 6 u[ xdF (x)] (1377)
239
1 1 u (1) + u (3) 6 u (2) 2 2 In a general expected return maximisation problem Z U ( 1 ; ::; N ) = u( 1 z1 + :: + N zN )dF (z1 ; ::; zN )
(1378)
(1379)
First and second order stochastic dominence For any distributions F ( ) and G ( ) the F ( ) …rst order stochastic dominates G ( ) if Z Z u(x)dF (x) > u(x)dG (x) (1380) and F (x) > G (x) for every x. F ( ) second order stochastic dominates G ( ) if G ( ) is mean preserving spread of F ( ) :
240
See Mas-Colell et at. (1995) Chapter 6.
8.1
Allais’paradox Table 71: Lotteries in Allais’paradox 25 5 0 A 0 1 0 B 0.1 0.89 0.01 C 0 0.11 0.89 D 0.1 0 0.9
A
B =) u(5)
0:1u(25) + 0:89u(5) + 0:01u(0)
0:11u(5) D
0:1u(25) + 0:01u(0)
(1381) (1382)
C =) 0:1u(25) + 0:9u(0)
0:11u(5) + 0:89u(0)
0:1u(25) + 0:01u(0)
0:11u(5)
(1383) (1384)
Certainly a paradox. Savage principle and Ellsbury paradox resolves Allais paradox. Individuals prefer smaller but certain prizes over bigger but uncertain prizes.
241
Two urns R and H contain 100 white and black balls; R has 49 whites and 51 black but the number of white or black balls in unknown for urn H. Lotterty: draw a white ball from R and win 1000 or draw a white balls from H and earn 1000. People prefer to draw from R because they know probability of 0.49 of drawing white ball rather than from H which may have more than 50 black balls. Decision makers do not like risk; so removes Allais paradox. Expected utility theory is still applicable. Portfolio choice and insurance contract Two …rms two states; N shares v price V value V j = v j N j and proportion of 0 < j < 1 1 1 i
+
2 2 i
N 2 6 v1
1 1 0N
wi =
(1385)
Budget constraint v1
1
2
N 1 + v2 1
v1 N 1
1 0
Avoid short selling and choose optimal Lagrangian for optimisation
j
1 1 2
+
L( ; )
= pu +
(1386)
2 0
60
(1387)
for optimal portfolio.
v1 N
0
2
+ v2 N 2
2 2 0N
+ v2
2 2 2 + 1 1
(1
p) u
1 0
v2 N
1 1 1+ 2 2
2 2 1 2 0
(1388)
FOC: L( ; ) = pu0 @ 2
1 1 2
+
2 2 2
2 2
+ (1
p) u0
1 1 1
+
2 2 1
2 1
v2 N 2 = 0
(1389)
L( ; ) = pu0 @ 1
1 1 2
+
2 2 2
1 2
+ (1
p) u0
1 1 1
+
2 2 1
1 1
v1 N 1 = 0
(1390)
+ v2 N 2
2
1
v1 N 1
1 0
2 0
=0
(1391)
From this =
pu0
1 1 2
+
2 2 2
2 2
+ (1 p) u0 v2 N 2
1 1 1
+
2 2 1
2 1
pu0
1 1 2
+
2 2 2
1 2
+ (1 p) u0 v1 N 1
1 1 1
+
2 2 1
1 1
(1392)
or =
(1393)
Then pu0 pu0
1 1 2 1 1 2
+ +
2 2 2 2 2 2
2 2 1 2
+ (1
p) u0
+ (1
p) u0
Now solve for optimal 242
1 1 1 1 1 1
+ +
2 2 1 2 2 1
2 1 1 1
=
v2 N 2 v1 N 1
(1394)
Watt(2011). Insurance contract: Given initial wealth w0 and the possibility of loss L insurance contract means for individual: (1
L) 6 (1
p) u (w0 ) + pu (w0
with the insurance contract: Insurance company’s income:
k
(x1 ; x2 ; (1
z0 + (1
p) u (w0 + x1 ) + pu (w0 + x2 )
(1395)
p) ; p)
p) (y1
x1 ) + p (y2
L
x2 )
(1396)
Di¤erence made by the contract:
[z0 + (1
B(x)
=
[z0 + (1
p) (y1
p) y1 + py2 ]
=
[(1
p) x1 + px2 ]
x1 ) + p (y2 pL
L
x2 )] (1397)
B(x) > 0 for a viable contract. That means pL > [(1
p) x1 + px2 ]
(1398)
For individual w0 + (1 8.1.1
p) x1 + px2 6 w0
L
Uncertainty of Good Times and Bad Times Future is uncertain; can be good or bad; two states. Contingent consumption in good times Cg and in bad times Cb
243
(1399)
Probability of good times
g
and of bad times
b
Prices of good times pg and of bad times pb Utilities from contingent consumption in good times u (Cg ) and in bad times u (Cb ) Budget constraint .I = Pg Cg + Pb Cb Consumer problem under uncertainty Expected utility theorem: utilities under uncertainty are additively separable (von-NeumannMorgenstern Utility) M ax
EU =
g u (Cg )
+
b u (Cb )
(1400)
Subject to I = Pg Cg + Pb Cb
(1401)
Lagrangian for constrained optimisation L=
g u (Cg )
+
b u (Cb )
+ [I
Pg C g
P b Cb ]
(1402)
First order conditions for optimisation For household A and B @L = @Cg
0 gu
(Cg )
Pg = 0
(1403)
@L = @Cb
0 bu
(Cb )
Pb = 0
(1404)
@L = I Pg C g P b C b = 0 (1405) @ Dividing (1403) by (1404) gives the marginal rate of substitution between good and bad times 0 gu 0 bu
(Cg ) Pg = ; (Cb ) Pb
Pg = Pb
g
(1406)
b
Fair market for contingent goods implies ratio of prices in good and bad states equals ratio of respective probabilities. Utility and allocation in good and bad times u0 (Cg ) =1 u0 (Cb )
(1407)
u0 (Cg ) = u0 (Cb )
(1408)
Since preference are symmetric over the states Cg = Cb
244
(1409)
consumer likely to fully insure against any risk; like to have same consumption in both good and bad states. Represent above result in a diagram with certainty line. budget line and indi¤erence curve u (Cg ; Cb ) : It is possible that individuals like to consume a bit more in good times and a bit less in bad times. 8.1.2
Optimal Demand for Insurance
There is certain wealth (W ), if an event occurs there will be a loss (L). probability of loss is (p) : Owner of the property can insure for amount (q) paying premium (m) Expected utility maximisation problem is maxEU = p:u (W
L
q
mq + q) + (1
p) u (W
mq)
(1410)
mq) m = 0
(1411)
Choose q to maximise EU using the …rst order condition as: @EU = p:u0 (W @q Optimal condition
L
mq + q) (1
m)
p) u0 (W
(1
u0 (W L mq + q) (1 p) m = 0 u (W mq) p (1 m) Pro…t function of the insurance company = (1
p) mq
p (1
m) q
(1412)
(1413)
Assume perfect competition in the insurance business, pro…t is zero p (1
m) q
(1
p) mq = 0
(1414)
The premium rate equals the probability of loss in equilibrium p=m
(1415)
This is actuarially fair insurance. Insert (??) into (1411) p:u0 (W
L
mq + q) (1
u0 (W
L
p)
(1
p) u0 (W
mq + q) = u0 (W
mq)
mq) p = 0
(1416) (1417)
00
For risk averse consumer u (W ) < 0 W
L
mq + q = W q=L
Consumer completely insures (q) against the loss (L). Risk spreading and risk diversi…cation 245
mq
(1418) (1419)
Risk can be spread among individuals. Imagine a society with 1000 individuals each endowed with £ 35000. Each faces a risk of losing £ 10000 with probability of 1 percent. Only 10 person in aggregate face this risk. It is a big loss for each individual as it can happen to each of them. Now they create an insurance market. Each contributes 100 to mitigate this uncertainty. This creates 100,000 insurance fund. This is enough to ensure each for any eventual loss. Every one will be certain (ensured) to have 34,900.: endowment minus insurance contribution. This is an example of risk spreading. Risk is spread (divided) among all. Each pays 100 to ensure against loss of 10000. Risk spreading and risk diversi…cation Risk can be diversi…ed by choosing an appropriate portfolio. Consider an excellent example from Varian (2010) on sunglasses and raincoat. You have 100 to invest. Probability of rain or shine is equally likely. You can invest only in sunglasses or raincoats or split 50/50 in each. Value of sunglass investment will double if it is sunny or down by half if it is rainy. Similarly value of investment will be double if rainy and down by half if sunny. If invested all in one then at the end of the day the expected value is 0.5(50)+0.5(200)=125. There is considerable risk. If case of splitting 50/50 the expected value of investment is [0:5(25) + 0:5(100)]+[0:5(100) + 0:5(25)] =125. Thus 125 is guaranteed no matter rainy or shiny. Diversi…cation has ensured 125. Do not put all your eggs in one basket. Homework A person has wealth worth £ 35000. There is 1 percent probability of loss. If this event occurs there is a loss of 10,000. This individual is risk neutral. 1) What is expected wealth without insurance? 2) This person can buy insurance equal to amount K to cover insurance by paying K insurance premium, where is the premium rate. Write individuals budget in case of accident and in case of no accident. 3) Write the expected utility function of this person. Assume that person receives utility from the wealth that he has. 4) What is expected pro…t of the insurance company? 5) Prove that premium rate equals the probability of the event. 6) Prove that consumption is same in both states with insurance. L) Prove that it is optimal to fully insurance against the loss and that is actuarially fair insurance. 246
Dixit A., S. Skeath and D. F. Reiley (2009) Games of Strategy, Norton. Hirshleifer J and J G Riley (1992) The Analytics of Uncertainty and Information, Cambridge. Perlo¤ J. M. (2008) Microeconomics: Theory and Applications with Calculus, Pearson. Nicholson W. (1985) Microeconomic Theory: Basic Principles and Extensions, Norton. (HoltSaunders). Rasmusen E(2007) Games and Information, Blackwell, ISBN 1-140513666-9. Varian HR (2010) Intermediate Microeconomics: A Modern Approach, Norton,8th edition http://cepa.newschool.edu/het/home.htm.
8.2
Expected utility theory
Expected wealth and utility from expected wealth Future is uncertain; two states - high wealth and low wealth. Contingent wealth in high state is WH and in low state is WL : Probability of high wealth
H
and low wealth
L
:
Utilities from high wealth u (WH ) and low wealth u (WL ) : Expected wealth EW = Expected utility EU =
L WL
+
H WH :
L u (WH )
+
H u (WL ) :
Faced with uncertainty people maximise expected utility (von-Neumann-Morgentern preferences). People are ready to pay some amount to insure themselves against the possible risk. Preferences of risk averse consumer Utility functions of risk averse individual U (W ) = ln(W ) U (W ) =
p
(1420) 1
W =W2
(1421)
Expected utility theorem: utilities under uncertainty are additively separable (von-NeumannMorgenstern Utility) M ax
EU =
H
:u (WH ) +
L:
u (WL )
(1422)
Utility from expected wealth U (EW ) = ln (
H
WH +
Certainty equivalent wealth 247
L
WL ) = ln (EW )
(1423)
CEW = exp (ln (EU ))
(1424)
Maximum insurance against risk and a measure of risk aversion Maximum insurance that person is ready to pay to cover risk: Insurance = EW
CEW
Table 72: Uncertainty of Income High Probability 0.75 Income 5000 Expected Income 3750
(1425)
and Wealth Low 0.25 1000 250
Expected wealth EW = L WL + H WH = 0:75 (5000) + 0:25 (1000) = 4000 Do people maximize expected wealth? No. They maximize expected utility. Maximum insurance against risk and a measure of risk aversion EU
=
H
:u (WH ) +
=
0:75
L:
u (WL ) =
H
: ln (WH ) +
L:
ln (WL )
ln (5000) + 0:25 ln (1000) = 6:388 + 1:727 = 8:115
(1426) (1427)
Certainty equivalent wealth CEW = exp (ln (EU )) = exp(8:115) = 3344:26
(1428)
Maximum insurance that person is ready to pay to cover risk: Insurance = EW
CEW = 4000
3344:26 = 655:74
(1429)
After paying 655.74 for the insurance company, this person can be sure that no matter high or low state 3344:26 is guaranteed. Can sleep well as income will not fall to 1000 even in the bad state! Risk pooling is possible. If 100 people ensure like this revenue of insurance company is 65575; only 25 percent people claim (2444.26 25 = 61106:5). Pro…t to the insurer is 65575 - 61106.5 = 4468.5. 8.2.1
Measure of risk aversion
Arrow-Pratt (1964) measure of risk aversion r(W ) =
U 00 (W ) >0 U 0 (W )
(1430)
r(W ) =
U 00 (W ) 0 W
(1434)
with Cobb-Douglus type preferences 1
(1435)
U (W ) = W 2
r(W ) =
1 2
U 00 (W ) = U 0 (W )
1 2
1 2W 1 2W
1
1 2
=
1 1 >0 2W
Maximum insurance against risk and a measure of risk aversion Risk lovers U (W ) = exp(aW ) r(W ) =
U 00 (W ) = U 0 (W )
a2 W 2 exp(aW ) = aW exp(aW )
(1436)
(1437) aW < 0
(1438)
Risk neutral U (W ) = aW r(W ) = 8.2.2
(1439)
U 00 (W ) 0 = =0 U 0 (W ) a
(1440)
St Petersberg Paradox (Bernoulli Game) and Allais Paradox
People are ready to play a small amount for a lottery but do not want to risk a huge amount in it. People care about utility. How much should one pay to play a game that promises to pay 2n if the head turns up in the nth trial? Answer 1.39. How? Expected payo¤ is in…nite E( )=
1
2+
2
22 +
3
23 + ::: +
1 1 > 2= 2 > 2 2 but the Expected Utility is …nite here 1
=
3
=
249
n
2n = 1 + 1 + 1 + ::: + 1 = 1
1 > :::::: > 23
n
=
1 2n
(1441) (1442)
E (u) =
1
ln (2) +
2
ln 22 +
3
ln 23 + ::: +
n
ln (2n ) < 1
(1443)
People buy lotteries for small amount but not for more (Allais Paradox) St Petersberg Paradox (Bernoulli Game) E (u) =
1 1 1 1 ln (2) + 2 ln 22 + 3 ln 23 + ::: + n ln (2n ) < 1 2 2 2 2
E (u) =
1 X 1 i=1
2i
1 X i i ln (2) = ln (2) = ln (2) 2 = 1:39 i 2 i=1
(1444)
(1445)
People are ready to pay small amount to buy lotteries but do not want to risk large sums (Allais Paradox) 8.2.3
Non-linear pricing Scheme
Let us consider a non-linear pricing problem considered by Snyder and Nicholson (2011) and earlier explained in Rochet and Tirole (2003), Fehr and List(2004), Blundell, Dias and Meghir (2004). Consumers of a …rm are of two types, f H ; L g ; H is the probability of consumer who put high value on the production and L is the proportion of consumer that put very low value in the consumption. Non-linear price scheme is to set tarrifs (T ) and output (q) in such a way that maximises …rms pro…t by designing price scheme appropriate to these consumers. Utility function of consumers: u = V (q)
T
V 0 (q) > 0 and V 00 (q) < 0: Firm’s problem is =T
cq
Participation constraint [ V (q)
T] > 0
First best solution This is when …rm knows the consumer type. It is binding , V (q) = T: Substitute this information on consumer into the …rms objective function. = V (q)
cq
Optimal pro…t then means @ = V 0 (q) @q
c = 0 =) V 0 (q) = c
This leads to the …rst degree price discrimination; high value consumer will be sold more goods at discounted per unit price and low value cumtomer will be sold less but ends up paying more per unit. (do a graph here)
250
Sedond best solution This is when the …rm does not know the type of the consumer. It has a probability belief on type of each type of consumer,0 < < 1 for type high and (1 ) for type low. Now the …rms pro…t becomes: =
(TH
cqH ) + (1
) (TL
cqL )
Subject to participation and incentive constraints for low high type consumers as: [ [
LV
(qL )
TL ] > 0
HV
(qH )
TH ] > 0
[
LV
(qL )
TL ] > [
[
HV
(qH )
TH ] > [
LV
(qH )
HV
TH ]
(qL )
TL ]
Participation constraint of the low type customer and incentive constraint of the high type customers are binding; resulting in LV
TH = [
(qL ) = TL
(V (qH )
H
V (qL )) + TL ]
Now put this into the …rm’s pro…t function = =
[f
H
@ = @qL
[f
H
(V (qH )
(V (qH ) HV
HV
0
(
H
0
V (qL )) + TL g
V (qL )) + (qL ) +
LV
(qL ) +
LV
L) V
LV
0
LV
0
0
0
(qL )g
cqH ] + (1
) (TL
cqH ] + (1
)(
(qL ) + (1
)
(qL ) + (1
(qL ) + (1
(qL ) = c +
) )
[
LV
LV
LV
0
(1 0
(qL )
0
(1
(qL ) = (1
(qL ) = (1
L] V
H
0
0
LV
cqL ) (qL )
cqL )
)c = 0 )c
)c
(qL )
)
Since the last term is positive it implies that L V (qL ) > c Since L V 0 (qL ) = c is the …rst best and qL now should be smaller to have L V 0 (qL ) > c. For the high value type H V 0 (qL ) = c, this means 'no distortions at the top'. An example from Nicholson and Snyder about co¤ee market: p V (q) = 2 q and f H ; L g = f20; 15g c=5, = 21 First best solution when the type of consumer is known. 251
V 0 (q) =
c
1 2
V 0 (q) = q
1 2
then V 0 (q) = q ( q=
2
1
= c; q2 = c; q =
c
20 2 = 16 5 15 2 =9 5
2
c
Tari¤ p p16 = 160 9 = 90
20 2 15 2
T = f V (q)j Expected pro…t of the …rm: =
(TH cqH ) + (1 ) (TL cqL ) 1 1 (160 80) + (90 45) = 40 + 22:5 = 62:5 2 2
=
When types arepunknown high type may buy 9 aunce and pay 90 cents thus with consumer 9 30 = 120 90 = 30 surplus of 20 2 He pays not 160 but 130. Thus the pro…t of the …rms witll be =
(TH cqH ) + (1 ) (TL cqL ) 1 1 (130 5 16) + (90 5 16) = 25 + 22:5 = 47:5 2 2
=
Now the shopper reduces the size of the cup ( Check the incentive compatible conditions): LV
0
(qL ) = c + [ 2
1
q2 =
L
H
c
L] V
H
(qL ) ;
Lq 2
2
; q=
0
L
H
=c+[
2
=
c
1 2
15 5
L] q
H
20
1 2
2
= 22 = 4
Tari¤ for the low customer TL =
LV
p 2 4 = 60
(qL ) = 15
For high type HV
0
(qL ) = c =) 20
q
1 2
1
= 5 =) q 2 = 4 =) q = 16
Now tari¤ from the high type TH = [
H
TH
(V (qH )
= =
[ h
V (qL )) + TL ] = [
(V (qH ) p 20 2 16 H
H
V (qL )) + p 2 4 + 15 252
(V (qH )
V (qL )) +
(qL )] p i 2 4 = 160
LV
LV
20 = 140
(qL )]
The pro…t in the second best solution is: = =
(TH cqH ) + (1 ) (TL cqL ) 1 1 (140 5 16) + (60 5 4) = 30 + 20 = 50 2 2
Now the high value type pays 8.2.4
140 16
= 8:75 and low type pays
60 4
= 15:
Job market applications
Principal may design contract in two ways (Nicholson and Snyder (2013)). The …rst one is where he would like to retain the proportionate share and the second on to raise his share over time, (see in pie charts) Gross pro…t of a …rm depends on e¤orts of the manager and is subject to random shocks. g
=e+'
Cost of e¤orts to the manager is C(e); C 0 (e) > 0 and C 00 (e) > 0 Net pro…t for the …rms is gross pro…t minus manager’s salary n
E(
n)
=
s=e+'
g
= E (e + '
s) = e
s E (s)
Manager’s utility E (u) = E (s) First best solution
A 2
s
C(e) = e
A 2
2 e
C(e)
is to …x salary according to e¤orts E (u) = s
C(e) > 0;
in equilibrium s = C(e)
Expected net pro…t for the owners will be: E(
n)
=e
E(s ) = e
This would gaurantee maximum (full) e¤orts.
253
C(e ); C 0 (e ) = 1
Second best case Owners cannot observe the e¤orts. Therefore they can subject salary to gross pro…t to a linear scheme as: s(
g)
=a+b
g
Owner sets salary by setting a and b …rst; a is …xed component and b is incentive payment. Then the manager decides the level of e¤orts conditional on the contract. Solve the game by backward induction: E (u) = E (s)
A 2
C(e); E (u) = E (a + b
s
A var (a + be + b') 2
E (u) = E (a + be + b') = a + be
A var (a + b 2
g)
A var (a + be + b') 2 Ab2 2
= a + be
2
g)
C(e)
C(e)
C(e)
C(e)
C 0 (e) = b How do they choose b? Manager accepts contratct if the E (u) > 0. That implies: Ab2 2 be 2 owener decides whether to o¤er the contract at the 3rd stage. a > C(e) +
E(
n)
= E (e + '
E(
n)
= e(1
s) = e
b)
@E ( n ) =1 @e
E (s) = e Ab2 2
C(e) + C 0 (e)
A
C 0 (e) = b
=
2
2
a
be = e(1
be = e
C 00 (e) = 0;
C(e)
2 C 0 (e)
1 1+A
2 C 00 (e)
u
(1 Constrained optimisation problem: L=p @L =1 @p
x + [(1 ) U 0 (W0
[(1 @L = @x
@L = @e
) U (W1 ) + U (W2 ) e
p) + U 0 (W0
U 0 (W0
+
e
(1 ) U 0 (W0 e p) + U 0 (W0 e p l + x)
@ x @e
p
+
e
u] p
l + x)] = 0
l + x) = 0
@ @e
U (W0 e p) U (W0 e p l + x)
=0
From the …rst two …rst order conditions 1
=
) U 0 (W0
[(1 0
= U (W0
e
p
e
p) + U 0 (W0
e
p
l + x)]
l + x)
This implies l = x, full insurance is optimal in the …rst best world. From the last …rst order condition @@e x = 1: The marginal social bene…t of precautionary e¤orts equals the marginal social cost of precaution. Partial versus full insurance: an example (from NS) U = ln (W ) Individual can …t an alarm and reduce the probability of theft from 25 percent to 15 percent; with W=100000 and L =20000 EUN A
= =
ln (W L) + (1 ) ln (W ) 0:25 ln (80000) + 0:75 ln (100000) = 11:45714
cost of allarm e = 1750 EUA
= =
ln (W
L) + (1
0:15 ln (80000
) ln (W )
1750) + 0:85 ln (100000
1750) = 11:46113
Utility from putting alarm is higher than from not putting it; EUA > EUN A Premium with alarm in the …rst best world 256
U = ln (100000
1750
p) = 11:46113 =) 98250 e11:46113 = 98250
=) p = 98250
p = e11:46113
94252:3 = 3297:7
Insurance pro…t p
L = 3298
0:15
20000 = 298
Second best solution If insured people may not put alarms; then the pro…t of insurance company will decrease: U
=
ln (100000
p) = 11:46113 =) 100000
=) p = 100000
11:46113
e
p = e11:46113
= 5048
Company pro…t p
L = 5048
0:25
20000 = 48
Insurance company can induce individuals to …t alarm with a contract as following with two equations First if the customer …ts the alarm EUA
= =
ln (W
e
p) + (1
0:85 ln (100000
) ln (W
1750
e
p
L + x)
p) + 0:15 ln (100000
1750
p
20000 + x) = 11:46113
Second when the customer does not …t the alarm EUA
= =
ln (W
p) + (1
0:75 ln (100000
) ln (W
p
L + x)
p) + 0:25 ln (100000
p
20000 + x) = 11:46113
These two equation could be solved numerically to …nd optimal p and x. These solutions are p = 601.8 and x = 3374.4 Company pro…t now p
L = 602
0:15
3374 = 96
Thus the partial insurance is more pro…table than the full insurance when the company cannot observe the precaution. Akerlof George A. (1970) The Market for 'Lemons': Quality Uncertainty and the Market Mechanism The Quarterly Journal of Economics, 84, 3. (Aug., 1970), pp. 488-500. Arrow K. J. (1964) The Role of Securities in the Optimal Allocation of Risk-bearing The Review of Economic Studies, 31, 2 (Apr., 1964), pp. 91-96
257
Arrow Kenneth J. (1963) Uncertainty and the Welfare Economics of Medical Care The American Economic Review, 53, 5 (Dec., 1963), pp. 941-973 Blundell R, M. C. Dias and C. Meghir, (2004) Evaluating the employment impact of a mandatory job search program,Journal of European Economic Association, 2:4:569-606.
Conte A & John D. Hey (2013) Assessing multiple prior models of behaviour under ambiguity,J Risk Uncertain 46:113–132 Cook Philip J. and Daniel A. Graham (1977) The Demand for Insurance and Protection: The Case of Irreplaceable The Quarterly Journal of Economics, 91, 1 pp. 143-156 Dixit A., S. Skeath and D. F. Reiley (2009) Games of Strategy, Norton. Epstein L. G. and S E. Zin (1989) Substitution, risk aversion and the temporal behaviour of consumption and asset returns: A theoretical Framework, Econometrica, 57:4:937-969. Gravelle H and R Rees (2004) Microeconomics, 3rd ed. Prentice Hall Fehr E and J. List,(2004) The hidden costs and returns of incentives— trust and trustworthiness among CEOs, Journal of European Economic Association, 2:5:743-771.
Hey J. D. and J. A. Knoll (2001) Strategies in dynamic decision making: An experimental investigation of the rationality of decision behaviour, Journal of Economic Psychology 32 399–409 Hey, J. D., Lotito, G., & Ma¢ oletti, A. (2010). The descriptive and predictive adequacy of theories of decision making under uncertainty/ambiguity. Journal of Risk and Uncertainty, 41(2), 81–111.Experimental lab of John Hey at York Hey J. D and C. Orme (1994) Investigating Generalizations of Expected Utility Theory Using Experimental Data Econometrica, 62, 6, 1291-1326 Hirshleifer J and J G Riley (1992) The Analytics of Uncertainty and Information, Cambridge. Jehle G A and P.J. Reny (2005) Advanced Microeconomic Theory, Pearson Education. Kahneman, D., & Tversky, A. (1979). Prospect theory: an analysis of decision under risk. Econometrica,67, 263–291. Perlo¤ J. M. (2008) Microeconomics: Theory and Applications with Calculus, Pearson. Machina M. J. (1987) Choice under uncertainty: problems solved and unsolved, Journal of economic perspective, 1:1:121-154. Newbery David M. G. Joseph E. Stiglitz (1982) Risk Aversion, Supply Response, and the Optimality of Random Prices: A Diagrammatic Analysis The Quarterly Journal of Economics, 97, 1 (Feb., 1982), pp. 1-26 Newbery David M. G., Joseph E. Stiglitz (1982) The Choice of Techniques and the Optimality of Market Equilibrium with Rational Expectations, Journal of Political Economy, 90, 2, 223246 258
Ö Özak (2014) Optimal consumption under uncertainty, liquidity constraints, and bounded rationality, Journal of Economic Dynamics and Control, 39, 237–254 Radner Roy (1968) Competitive Equilibrium Under Uncertainty Econometrica, 36, 1 31-58 Rothschild Michael and Joseph Stiglitz (1976) Equilibrium in Competitive Insurance Markets: An Essay on the Economics of Imperfect Information The Quarterly Journal of Economics, 90, 4 Nov., pp. 629-649 Rasmusen E(2007) Games and Information, Blackwell, ISBN 1-140513666-9. Rochet JH and J. Tirole (2003) Platform Competition in Two-Sided Markets' Journal of European Economic Association, 1:4:990-1029.
Snyder C and W. Nicholson (2012) Microeconomic Theory: Basic Principles and Extensions, 10th edition, South Western. Varian HR (2010) Intermediate Microeconomics: A Modern Approach, Norton,8th edition Watt Richard (2011) The microeconomics of risk and information, palgrave macmillan. Zizzo D. (2010) Experimenter Demand E¤ects in Economic Experiments, Experimental Economics 13(1), March, 75-98 Kahneman, D., & Tversky, A. (1979). Prospect theory: an analysis of decision under risk. Econometrica,67, 263–291. Machina M. (1987) Choice under uncertainty: problems solved and unsolved, Journal of Economic Perspective, 1:1:121-154.
9
L9:Class Test
Questions are given in sections A and B. Answer two questions, at least one from each section. Each question is worth 100 marks. Each subquestion within a question is of equal value. Use diagrams to illustrate your answers. Section A Q1. A certain consumer has £ 8 to spend in commodities x and y; prices in the market were px = 1 and py = 4. She evaluates utilities from these products according to a standard utility function; u = xa y b and has a = 0:5 and b = 0:5. (a) Derive the Marshallian demand functions for good x and y. Determine the amount purchased of each. (b) Evaluate the optimal utility obtained by this consumer. Derive the expenditure function from that indirect utility function. (c) What is the compensated demand for good x and y for this consumer? What are the compensated and Marshallian demands for good x when the price of x increases to 4, px = 4? 259
(d) Using Slutsky equation decompose the change in demand into the substitution and income e¤ects. (e) Are the relative size of income and substitution e¤ects sensitive to parameters a and b, i.e. share of income spent on x and y? Q2. Illustrate e¢ ciency conditions in allocations of resources: a. When consumers’utility function is given by U (X; Y ) and the production possibility frontier is T (X; Y ): b. E¢ ciency of production when: X = f1 (K1 ; L1 ) + f2 (K2 ; L2 )
(1446)
K1 + K2 = K
(1447)
L1 + L2 = L
(1448)
c. Prove that e¢ cient provision of public goods requires that the sum of the marginal rate of substitution equals the marginal cost of provision of public good with a two consumer economy in which consumers like to maximise utility by consuming private (x) and public goods (G): max
u1 = u1 (x1 ; G)
(1449)
subject to a given level of utility for the second consumer: max u2 = u2 (x2 ; G)
(1450)
x1 + x2 + c (G) = w1 + w2
(1451)
and the resource constraint:
Q3. Consider price adjustments model with N number of …rms: qD
p =
qS
(1452)
where p is percent change in price (p) and is the adustment coe¢ cient. Demand for the product with parameters a > 0 and b < 0 is q D = a + bp; and its supply with a parameter m > 0 is q S = mN . Then: p =
(a + bp
mN )
>0
(1453)
There is free entry and exit of …rms in this market: N =
(p
c)
>0
(1454)
Find the dynamic time path for prices (pt ) and the number of …rms (Nt ) and examine convergence towards the steady state.
260
Q4. Consider general equilibrium with production (x; y) = (x1; x2; :::::xm ; y1; y2; :::::ym ) for consumer i = 1; ; ; m and producer j =1,.,. n. A possible allocation for consumers and producers satisfying following: Consumption set: xi 2 X Production possibility set: yi 2 Y m m n P P P Resource balance condition: xi = ei + yj i=1
i=1
Wealth of the consumer: wi (p) = p:e1 +
j=1
n P
i;j j
(p)
j=1
Supply correspondence: sj (p) = fyi 2 Yj : y 0 2 Yj =) p:y > py 0 g m n P P Excess demand correspondence: Z (p) = (di (p) e1 ) sj (p) i=1
j=1
Here a competitive equilibrium is a pair of prices, demand and supply (p; (x; y)) with a p vector in Rl and x 2 di (p) for consumers i to m, and yj 2 sj (p) for …rms j to n and where the excess demand is zero in equilibrium. Simplify this model to a Robinson Crusoe economy as: Commodity space: R2 (leisure and food) Consumer characteristic: Xi = R2+ Endowment: ei = (24; 0) Preference relation: i U (L; Fn = LF p o Producer characteristics: Yj ( L; F ) : L 0; F L p where F = L is the production function. Solve this model for price vector, demand vector and the output vector that satisfy conditions of general equilibrium in that economy.
261
Section B Q5. Consider a Dixit-Stiglitz model of monopolistic competition in which consumers maximise utility (u) by consuming varieties of di¤erentiated products qi in addition to a unique numeraire product q0 . Their problem is:
max
u = u q0 ;
subject to budget constraint with income (I): X q0 + pi qi
X
1
(1455)
qi
I
(1456)
Producer i maximises pro…t, setting prices pi given marginal cost c and …xed cost f as: max
= (pi
pi
1. Prove that elasticity of demand is
1 1
;i.e.
c) qi =
f @qi pi qi @pi
(1457) =
1 1
:
2. How much does each …rm produce? How does it relate to the elasticity of demand as well as the …xed cost (f ) and the variable cost (c). 3. How many …rms exist in the market? Explain the role of
in it.
Q6. Consider consumers’problems for comparative static analysis max U = U (X; Y )
(1458)
subject to the budget constraint: I = Px X + Py Y
(1459)
where U is utility, I income, X and Y and Px and Py are are amounts and prices of X and Y commodities respectively. 1. Illustrate the …rst order conditions for consumer optimisation in this model. 2. By total di¤erentiation of the …rst order conditions determine (a) the impact of a change in shadow prices on the demand for X and Y: (b) the impact of a change in price of X on the demand for X and Y: (c) the impact of a change the price of Y on the demand for X and Y: (d) the impact of a change in income on the demand for X and Y and the shadow price. 3. Decompose the total e¤ect of a price change into substitution and income e¤ects. 4. Show the major di¤erences between Hicksian and Marshallian demand functions.
262
Q7. A …rm’s objective is to minimise cost (C) C = rK + wL
(1460)
subject to a CES technology constraint: Y = [ L + (1
)K ]
1
(1461)
Here Y is output, K capital, L labour inputs, r interest rate, w wage rate, 0 < labour the substitution parameter.
< 1 share of
1. Determine the demand for labour and capital inputs. 2. Derive the cost function of the …rm. 3. Prove that the elasticity of substitution is
=
1
1:
4. Discuss the properties of the CES cost function. 5. Prove that the Cobb-Douglas production function is a special case of the CES production function.
10
10.1 10.1.1
L10: Impact of Taxes and Public Goods in E¢ ciency, Growth and Redistribution: A General Equilibrium Analysis First best principles E¢ ciency in consumption
Marginal rate of substitution between two products should equal price ratios for a certain consumer Uy Un Ux = = ::::: = Px Py Pn
(1462)
Allocation is Pareto e¢ cient if it is not possible to make one person better o¤ without making another worse o¤. L = U (X; Y ) + [T (X; Y )]
(1463)
@L @U @T = + =0 @X @X @X
(1464)
@L @U @T = + =0 @Y @Y @Y
(1465)
@L = @
(1466)
[T (X; Y )] = 0
263
@U @X @U @Y
=
@T @X @T @Y
(1467)
@Y = RP TX;Y @X This is the optimal point in the production possibility frontier. See the trade model. M RSX;Y =
10.1.2
(1468)
E¢ ciency in production
If it is not possible to more of one good without reducing the production of another good. X = f1 (K1 ; L1 ) + f2 (K2 ; L2 )
(1469)
K1 + K2 = K
(1470)
L1 + L2 = L
(1471)
X = f1 (K1 ; L1 ) + f2 (K
K1 ; L
@f1 @f2 @f1 @X = + = @K1 @K1 @K2 @K1 @f1 @f2 @f1 @X = + = @L1 @L1 @L2 @L1
L1 )
(1472)
@f2 =0 @K2
(1473)
@f2 =0 @L2
(1474)
Marginal productivity of capital and labour inputs are same across both sectors:
10.1.3
@f1 @f2 = @K1 @K2
(1475)
@f1 @f2 = @L1 @L2
(1476)
E¢ ciency of Trade (Exchange)
If it is possible to increase welfare of one country without harming another country. M RSx;y =
Ux Px = Uy Py
= ::::::: = 1
264
M RSx;y =
Ux Px = Uy Py
(1477) N
10.1.4
A simple numerical example of optimal tax or optimal public spending
Assume consumption (C) equals income (Y ); C = Y and public sector is balanced, tax revenue = public spending, T = G Y = Y0 + aG @Y =a @G
2bG = 0
bG2 =) G =
(1478) a 2b
(1479)
2
a is maximum. If Y0 = 50; From second order condition, @@GY2 = 2b < 0 it is clear that G = 2b 60 a = 60 and b = 2; G = 2 2 = 15. Then Y = Y0 + aG bG2 = 50 + 60 15 2 152 = 500: To prove that G = 15 is the maximum evaluate the above function for G = 14 and G = 16: In both cases Y should be less than 500 with optimal G = 15: Y = Y0 + aG bG2 = 50 + 60 14 2 142 = 498: Y = Y0 + aG bG2 = 50 + 60 16 2 162 = 498: This proves that optimal tax revenue in this case is G = T = 15:
10.1.5
E¢ ciency in public goods
When i = 1; :::N individuals live in a society then, the social marginal utility of public goods is the sum of the utilities from public good for individuals SM UP = SM UP1 + SM UP2 + :::: + SM UPN SM RSP;G =
SM UP SM UP1 SM UP2 SM UPN = + + :::: + i i i i M UG M UG M UG M UG SM RSP;G = RP TP;G
(1480) (1481) (1482)
Rate of transformation of private to public goods should equal the social rate of substitution of private to public goods. 10.1.6
Theory of second best
When the optimal point is not achievable, other points in the e¢ ciency frontier are not necessarily optimal. (draw a diagram to prove). 10.1.7
Externality
Externality has been one important issue of research in microeconomics in recent years (Porter and van der Linde (1995) Hanemann (1994) Dixit (1992) Palmer, Oates, Portney (1995), Stern (2008)). What would happen to public parks if city councils do not maintain? Personal and social bene…ts of a beautiful garden? Why does not produce e¢ cient amount of education and health? Why will market produce excessive amount of water, air or noise pollution? Why many cities in England are introducing congestion charges? 265
Aldy et al. (2010) have discussed how much action in carbon reduction is desirable on cost minimisation and welfare maximising grounds along with the alternative policy artitecture at the international level. Earlier Stern (2008) did a very comprehensive study on economics of climate change. Bohringer and Rutherford (2004) Grubb (2004) Green and Newbery (1992), Manne and Richel (1992), McFarland, Reilly and Herzog (2002) Nordhaus (1979), Perroni and Rutherford (1993), Backus and Crucini (2000), Boyd and Doroodian (2001), Coupal and Holland (2002), Grepperud and Rasmussen (2004), Jansen and Klaassen (2000), Kumbaroglu (2003), Spear( 2003) and Thompson (2000) use partial or general equilibrium models with the electricity sector to examine how pollution arises in process of generating energy required for e¢ cient functioning of economies under investigation. How do the production activities over time generate pollution at the local, national or the global level? How do these a¤ect the climate change? What are their consequences? Who bears the burdens of such adjustments? How is the dividend from the improved environment from emission control at the global level shared by advanced or developing economies under the Montreal or Kyoto protocols? These questions are examined in several studies including those of Aronsson, (1999), Bohringer and Conrad and Loschel (2003), Crettez and Aronsson (1999), Crettez (2004) Dissou, Mac Leod, and Souissi (2002), Faehn and Holmoy (2003) Nordhaus and Yang (1996), Proost and Van Regemorter (1992), Rasmussen (2001), Kumbaroglu (2003), Roson (2003) , Uri and Boyd (1996) and Vennemo (1997). Despite so many global negotiations including Cancun (2001) and Copenhagen (2009) summits, apparently very few studies have measured and demonstrated the level of pollution as a consequence of economic activities in multisectoral dynamic general equilibrium framework for the UK. Positive Externality A classic example of positive externality: bees pollinate apple trees and they get materials for honey from apples. For instance cost of producing apples is Ca = a2 and cost of producing honey Ch = h2 a, Private market solution Firms maximise own pro…t independently: a
by marginal cost pricing rule and
@ a @a
a2
= Pa a
= Pa
(1483)
2a = 0 =) Pa = 2a and hence supply of apples
a=
Pa 2
(1484)
Similarly h
= Ph h
h2 + a
(1485)
Supply of honey by the private market @ a 2h = 0 =) Ph = 2h and @a = Ph Ph : 2 Private market does not consider positive externality. Now consider a social planner that produces both to maximise joint pro…t: h=
= Pa a
a2 + Ph h
Then optimal apple supply is : 266
h2 + a
(1486)
(1487)
@ a @a
= Pa
2a + 1 = 0 =) Pa = 2a
1 .and a=
1 Pa + 2 2
(1488)
Ph 2
(1489)
Optimal honey supply is @ h 2h = 0 =) Ph = 2h .and @h = Ph h=
It is optimal to produce more apples taking account of its positive externality. Negative Externality Negative externality production of electricity and pollution and food production Electricity production using coal generates electricity as well as pollution. This pollution raises production cost in the food industry. Cost of electricity production when the environment is not taken into account 2 Ce = e2 (x 3) and its pro…t function is: e = Pe e e2 (x 3) the cost of food production Cf = f 2 + 2x and its pro…t f = Pf f f 2 2x. Pollution adds extra cost in food production. Private market solution 2 e2 (x 3) e = Pe e @ e 2e = 0 =) Pe = 2e and hence supply of electricity @e = Pe e= = Pf f @ f @f = Pf f
Pe 2
(1490)
f 2 2x 2f = 0 =) Pf = 2f and hence supply of food f=
Pf 2
(1491)
Here pollution is produced more than optimal. @ e = 2 (x @x
3) = 0 =) x = 3:
(1492)
Socially optimal solution : = @ e @e
= Pe
e
+
f
= Pe e
e2
(x
= Pf
f2
2x
(1493)
2e = 0 =) Pe = 2e and hence supply of electricity e=
@ f @f
2
3) + Pf f
Pe 2
(1494)
2f = 0 =) Pf = 2f and hence supply of food f=
Pf 2
267
(1495)
@ = 2 (x 3) + 2 = 0 =) x = 2: @x Social solution generates less pollution than the market solution. 10.1.8
(1496)
Samuelson and Nash on Sharing Public Good
Consider a case where two friends share a public good x = x1 + x2 but consume private good yi . 1
1
max u1 = (x1 + x2 ) 2 y12
(1497)
10x1 + y1 = 300
(1498)
subject to
1
1
L1 = (x1 + x2 ) 2 y12
[300
1 @L = (x1 + x2 ) @x1 2
1 2
y1 ]
(1499)
1
10 = 0
y12
1 @L 1 = (x1 + x2 ) 2 y1 @y1 2
@L = 300 @
10x1
1 2
10x1
(1500)
=0
(1501)
y1 = 0
(1502)
From the …rst two FOC y1 = 10 =) y1 = 10(x1 + x2 ) (x1 + x2 )
(1503)
Putting this y1 back in the budget constraint 300 10x1 y1 = 300 10x1 10(x1 + x2 ) = 0 x2 2 As the problem is symmetric dimilar proces for individual 2 we get
(1504)
x1 = 15
1
1
L2 = (x1 + x2 ) 2 y22
[300
x2 = 15 2
1 15 2
y2 ]
(1505)
x1 2
x2 = 15 x1 = 15
10x2
x1 2
(1506)
=) x1 =
41 32
x1 = x2 = 10 =) x1 + x2 = 20 y1 = 10(x1 + x2 ) = 10 268
20 = 200
15 = 10
(1507) (1508) (1509)
Utility level in non-cooperative Nash scenario is: 1
1
1
1
u1 = (x1 + x2 ) 2 y12 = (20) 2 200 2 = 63:2 = u2
(1510)
Under the Samuelsonian rule @L @x1 @L @y1 1 2 (x1 1 2 (x1
1 2
+ x2 ) 1 2
+
1
y12
+ x2 ) y1
1 2
+
@L @x2 @L @y2 1 2 (x1 1 2 (x1
= M RS1 + M RS2 = M RT 1 2
+ x2 )
(1511)
1
y22
1 2
+ x2 ) y2
y2 px 10 y1 + = = = 10 x x py 1
=
1 2
y1 + y2 = 10x
(1512) (1513)
Combined budget constraint of both persons: 10x + y1 + y2 = 600
(1514)
10x + 10x = 600 =) x = 30
(1515)
y1 + y2 = 10x = 10
30 = 300
(1516)
If the private good is equally devided each gets 150. 1
1
1
1
u1 = (x1 + x2 ) 2 y12 = (30) 2 150 2 = 67:1 = u2
(1517)
Table 73: Ine¢ cienty of competitive equilibrium in case of positive externality Nash (CE) Optimal (Samuelson) x 20 30 y1 200 150 y2 200 150 u 63.2 67.1 Key questions: Pollution controls are less important in developing countries such as China and India. How does it a¤ect the global environment? Readings: VAR 35 10.1.9
Sameulson’s Theorem on Public Good
Provision of public goods: two consumers, and public and private goods, but not clear how they should pay for it; valuation of each person is di¤erent. Proposition: Pareto optimality requires that sum of the marginal rate of substitution between private and public goods by two individuals should equal the marginal cost of provision of public goods (see two citizen public good model). Consumers consume private (x) and public goods (G)
269
max
u1 = u1 (x1 ; G)
(1518)
subject to a given level of utility for the second consumer max u2 = u2 (x2 ; G)
(1519)
x1 + x2 + c (G) = w1 + w2
(1520)
and the resource contraint
Constrained optimisation for this is L = u1 (x1 ; G) @L @x1 @L @G
(x1 ;G) = @u1@x 1 (x1 ;G) = @u1@G @u1 (x1 ;G) @G @u1 (x1 ;G) @x1
10.1.10
[u2
= 0 =) @u2 (x2 ;G) @G
+
@u2 (x2 ;G) @G @u2 (x2 ;G) @x2
u2 (x2 ; G)]
[x1 + x2 + c (G)
w1
w2 ]
(x1 ;G) @u2 (x2 ;G) (x2 ;G) @L = @u1@x ; ; @x = = 0 =) @u2@x2 @x2 1 1 @c(G) @u2 (x2 ;G) 1 @u1 (x1 ;G) = @c(G) @G = 0; =) @G @G @G
=
@c (G) ; @G
M RS1 + M RS2 = M C(G):::Q:E:D:
(1521) =
(1522)
Negative externality in production U = U (xc ; yc )
(1523)
xo = f (yi )
(1524)
yo = g(xi ; xo )
(1525)
xc + xi = xo + x
(1526)
yc + yi = yo + y
(1527)
See: Snyder C. and W. Nicholson (2012) Microeconomic Theory: Basic Principles and Extensions, 11th ed.. South Western. Competitive equilibrium is ine¢ cient; There is over production of x. L
= U (xc ; yc ) + +
3
1
[xc + xi
[f (yi ) xo
xo ] + x ]+
3
2
[g(xi ; xo )
[yc + yi
yo
yo ] y ]
(1528)
FOC: i:
@L = U1 + @xc
3
=0
(1529)
@L = U2 + @yc
4
=0
(1530)
ii:
270
iii:
@L = @xi
2 g1
+
3
=0
(1531)
iv:
@L = @yi
1 fy
+
4
=0
(1532)
+
2 g2
@L = @xo
v:
vi:
1
@L = @yo
2
3
4
=0
(1533)
=0
(1534)
3
(1535)
From i and ii U1 = M RSx;y = U2
4
From iii and vi 3
M RSx;y =
=
2 g1
4
= g1
(1536)
2
from iv 1
1 fy
=
4
(1537)
From v M RSx;y =
3
=
1
+
4
2 g2
=
4
1
2 g2
+
4
=
4
1 fy
2
g2 =
2
1 fy
g2
(1538)
Competitive equilibrium condition M RSx;y =
dy py 1 = = dx px fy
(1539)
but the optimal allocation after taking of externality is M RSx;y = here
10.2
1 fy
g2
(1540)
g2 is the measure of negative externality; x is over produced.
Negative externality and Pigouvian tax
Pigouvian tax to correct negative externality in case of upstream-downstream …rms 1
x = 2000Lx2
(1541)
1
y=f
2000Ly2 (x
x0 )
1 2000Ly2
x 6 x0
271
x > x0
(1542)
If = 0 production of x has no e¤ect on production of y. If increase. Assume px = 1; w=50; = 0 no externality w = p:
@x =) 50 = 1:2000 @Lx
1 2
< 0 output of y decrease as x
1
Lx 2 =) Lx = 202 = 400
(1543)
1
x = 2000Lx2 = 40000
(1544)
Downstream …rms also produces same product: 1
y = 2000Ly2 = 40000 Suppose
=
(1545) 1 2
0:1; x0 = 38000; x still produce x = 2000Lx = 40000: 1
y = 2000Ly2 (40000
w
=
p:
0:1
38000)
@y =) 50 = 1:2000 @Ly
1 2
Ly
(1546)
1 2
(2000)
0:1
1
=) Ly2 = 9:35; Ly = 9:352 = 87:5 1
0:1
y = 2000Ly2 (2000)
(1547)
1
= 2000 (87:5) 2 (2000)
0:1
= 8753
(1548)
Pigouvian tax forces x to produce just x0 = 38000 1
x = 2000Lx2 = 38000 =) Lx = 192 = 361
(1
t) w
=
(1
t) p:
=) 50 = (1 =
10.3
@x =) (1 @Lx t) 1:2000
50 =) t = 0:05:
t) 50 = (1 1 2
(362)
t) 1:2000
(1549)
1 2
1
Lx 2
1 2
(1550)
Carbon Emmission in the UK
Warned by these vulnerabilities and complying to the international climate agreements, the UK government now is committed to cut such emissions by 34% by 2020 and 80% by 2050 (Carbon Plan, DECC (2010)).
272
Table 74: Sources of Green-House Gas (GHG) Emission in UK Percentage Electricity Generation 35 Household Consumption 14 Industry and Business 17 Transport 20 Waste 3 Agriculture 8 Public Sector 3 Total 100 Source: DECC Carbon Plan, London, (2010).
Figure 1 Economists together with natural scientists have attempted to assess the amount of damage of such encroachment into the environment using a dynamic general equilibrium models of one or multiple economies with proper appreciation of interactions among them (Nordhaus (DICE-1993); Perroni and Rutherford (1993), Nordhaus and Yang (Rice -1976); Hope and Newberry (2008) 273
,Grubb, Jamasb and Pollit (2008)). The amount of pollution generated in this manner a¤ects not only the national economy but has global consequences (Stern (2007, 2008)). Scienti…cally pollution - carbon dioxide, methane, nitrogen dioxide or nitric oxide, chloro‡uorocarabon (CFC)) in solid, liquid or gaseous form or the explosive, oxidizing, irritant, toxic, carcinogenic, corrosive, infectious, teratogenic, mutagenic hazardous solid wastes - is detrimental to human, animal and plant lives. It not only contaminates and adulterates the natural environment and ecological balances locally but also has global consequences resulting in the rise of global temperature, acid rains, a large Arctic ozone hole ultimately generating a process called the “greenhouse e¤ect”. Despite that it is hard to quantify the overall damage caused by a particular industry or a nation. Containing global warming requires cooperation of all nations, putting energy and environmental taxes might not be a prudent way of controlling such pollution. Using applied general equilibrium models Whalley and Wigle (1991) had estimated consequence of 50 percent reduction in CO2 gases to cause up to 19 percent reduction in GDP, Vennemo (1997) had shown carbon taxes caused a fall in the wage rate of up to 5 percent, Kombaroglu (2003) reported them to dampen the growth rate by up to 6 percent, Bohringer, Conrad and Loscel (2003) found negative impacts of such taxes on output, employment and the wage rate, Perroni and Rutherford (1993) had shown how pollution permits would a¤ect the structure of trade among economies. Imposition of extra environmental and energy taxes to reduce the pollution a¤ects the behaviour of households and …rms. Taxes reduce the pro…tability of …rms. They invest less, have less capital stock and can produce less. Taxes depress the real income of households and their levels of utility despite working more. These a¤ect sectoral and national growth rates and allocations of inputs and distribution of income among households. There is still a great deal of uncertainty about the optimal rate of carbon tax even after several years of intense research activity on carbon taxes and global warming (Poterba (1993), Stern (2008)). Growth of capital stock, output and investment in the agriculture sector,- that includes farms crops, livestock, forestry, and …sheries- is lower when taxes are imposed in the use of inputs. Scienti…cally it is true that the malpractices in agriculture can generate biomass, organic and inorganic wastes that cause environmental problems and hazards to human and animal health which may result from animal manure and other dejections, animal corpses, residues of plastic, rubber and other petrochemicals, pesticides, pharmaceuticals, papers and wood, mineral fertilizer, scrap tools and agricultural machines. Nitrous oxides generated by these processes can bring respiratory infections, burning of eyes, headache, chest tightening, ground water pollution. Inadequate measures taken to control the spread of crop or animal diseases can produce biological hazards. It is questionable, whether extra tax for controlling such pollution in this manner is reasonable as most of agricultural wastes can be valuable resources if properly recycled with adoption of better agricultural recycling practices; more taxes in input merely deter farmers from spending on better environmentally friendly production technology. Extra taxes reduce the growth rate of output, investment and capital stock in the mining sector. It is well understood that pollution emerging from physical and chemical processing of minerals in metal ores extraction as well as other mining and quarrying sector may generate acids. Drilling mud, dangerous substances, land deformation can be minimised by designing dumping sites for sul…dic waste speci…c materials with proper consideration of climate, hydrogeological conditions to prevent air penetration and water in…ltration rather through higher rates of input taxes. Accumulation of capital stock, output, and investment is a¤ected by extra taxes in the manufacturing sector relative to the benchmark. At the current state of technology manufacturing is not possible without burning fossil fuels directly from machines operating from burning such fuels or
274
indirectly through the use of electricity that is generated through CO2 releasing fossil fuels. This is evident from a cursory look at the composition of 45 di¤erent industries within the manufacturing sector [see Appendix A for a complete list of subsectors]. Despite continuous e¤orts for adopting more e¢ cient and environmental friendly production technologies over years, production plants of these industries are known for generating pollutants such as CO2 , S2 or other hazardous gases as by-products in the production process, ever since the time of industrial revolution. Environmental or energy taxes can only raise the cost of production and lower their motivation to search for the better technology. Growth of capital, output and investment in the energy sector - that includes production and distribution of electricity and gas - are a¤ected negatively by extra input taxes. Electricity is generated from coal, oil, gas, wind turbines and nuclear sources. Coal and oil plant generate larger amount of CO2 in atmosphere and the nuclear sources are di¢ cult to build in the beginning and leave plenty of hazardous wastes at the end. If one looks at the current industrial structure of the energy sector, environmentally friendly renewable sources can not ful…l even a fraction of energy demand and this industry is in needs of support for better technology such as carbon tapping, development of hydrogen and other sources of green energy, extra taxes can only cause a setback. The growth of capital, output and investment in the construction sector is relatively higher than in other sectors mainly because of higher taxes in the use of input in this sector in the benchmark. The distribution sector here consists of motor vehicle distribution and repair, automotive fuel retail, wholesale distribution, retail distribution, hotels, catering, pubs etc. Improper scrapping of old vehicles generates solid waste, cold-storages and refrigeration generates CFC and improper treatment of residues at the retail level causes pollution. Again extra taxes slightly lower the growth of output, capital and investment compared to the steady state. Transport and communication sector that comprises of railway transport, other land transport, water transport, air transport, ancillary transport services, postal and courier services and telecommunications, generates air, water, noise and land pollutions. Extra environmental taxes raise cost of operating their businesses and depress the growth of capital, output and investment in this sector. Better technology rather than taxes can promote the growth of this sector. The business service sector represents banking and …nance, insurance and pension funds, auxiliary …nancial services, owning and dealing in real estate, letting of dwellings, estate agent activities, renting of machinery etc, computer services, research and development, legal activities, accountancy services, market research, management consultancy, architectural activities and technical consultancy, advertising and other business services. Negative externality in this sector may be less visible; intense competition for market often generates rivalry, spam, fraud and unhealthy practices that can put extra costs of providing services. It is di¢ cult for any tax system to prevent such malpractices. Higher rates of taxes reduce its growth compared to the benchmark. The other services sector includes public administration and defence, education, health and veterinary services, social work activities, membership organisations, recreational services, other service activities, private households with employed persons and sewage and sanitary services. Malpractices in social services sector appear in the form of corruption, sleaze, unfair treatment and breach of fundamental liberties, trust and social values which may cause anxiety and create psychological burden and create an unhealthy environment for workers as well as entrepreneurs in the economy. It requires more creative thinking and putting extra taxes creates disincentives and cannot contribute to higher productivity required for growth prospect of this sector. Aldy J.E, A. J. Krupnick, R. G. Newell, I. W. H. Parry and W. A. Pizer (2010) Designing Climate Mitigation Policy, Journal of Economic Literature, 48:4, 903–934 . 275
Aronsson, T. (1999) On Cost Bene…t Rules for Green Taxes, Environmental and Resource Economics, January, v. 13, iss. 1, pp. 31-43 . Backus, D. K. and M. J. Crucini (2000) Oil Prices and the Terms of Trade, Journal of International Economics, February, v. 50, iss. 1, pp. 185-213 . Bhattarai K. (2007) Capital Accumulation, Growth and Redistribution in UK: Multisectoral Impacts of Energy and Pollution Taxes, Research Memorandum, 64, Business School, University of Hull, UK. (see 123 sector dynamic general equilibrium model of the UK economy). Barker P.,R. Blundell and J. Micklewright(1989) Modelling household energy expenditure using micro data, Economic Journal 99:397:720-738. Bohringer, C. and T. F. Rutherford ( 2004) Who Should Pay How Much? Compensation for International Spillovers from Carbon Abatement Policies to Developing Countries–A Global CGE Assessment, Computational Economics, February, v. 23, iss. 1, pp. 71-103 Bohringer, C., K. Conrad, K. and,A. Loschel (2003) Carbon Taxes and Joint Implementation: An Applied General Equilibrium Analysis for Germany and India, Environmental and Resource Economics, January, 24:1:49-76 Boyd, R. G. and Doroodian, K. (2001) A Computable General Equilibrium Treatment of Oil Shocks and U.S. Economic Growth,Journal of Energy and Development, Autumn, 27:1: 43-68. Crettez, B. (2004) Tradeable Emissions Permits, Emissions Taxes and Growth, Manchester School, July, 72:4:443-62. Coupal, R. H and Holland, D. (2002) Economic Impact of Electric Power Industry Deregulation on the State of Washington: A General Equilibrium Analysis, Journal of Agricultural and Resource Economics, July, 27:1:244-60. Dissou, Y., Mac Leod, C. and Souissi, Mokhtar (2002) Compliance Costs to the Kyoto Protocol and Market Structure in Canada: A Dynamic General Equilibrium Analysis, Journal of Policy Modeling, November:24: 7-8: 751-79. Faehn, T., and E. Holmoy (2003) Trade Liberalisation and E¤ects on Pollutive Emissions to Air and Deposits of Solid Waste: A General Equilibrium Assessment for Norway; Economic Modelling, July:20: 4: 703-27. Grepperud, S. and I. Rasmussen (2004) A General Equilibrium Assessment of Rebound Effects, Energy Economics, March, 26:2: 261-82. Grubb, M. (2004): Kyoto and the future of International Climate Change Responses: From here to Where? International Review for Environmental Strategies, Summer 5:1:15-38. Green R. J. and D. M. Newbery(1992): Competition in the British Spot market, Journal of Political Economy, 100:5: 929-953. Grubb M, T. Jamasb and M. Pollitt (2010) Delivering a Low Carbon Electricity System: Technologies, Economics and Policy, Cambridge University Press. 276
Jansen, H. and G. Klaassen(2000) Economic Impacts of the 1997 EU Energy Tax: Simulations with Three EU-Wide Models, Environmental and Resource Economics, February, 15:2:179-97. Kumbaroglu, G. S. (2003) Environmental Taxation and Economic E¤ects: A Computable General Equilibrium Analysis for Turkey, Journal of Policy Modeling, November, 25:8:795810. Manne A and R. Richel (1992) Buying Greenhouse Insurance: The Economic Costs of Carbon Dioxide Emission Limits, MIT Press. Cambridge, MA. McFarland J.R. J, Reilly J.M. and Herzog H.J.( 2002) Representing Energy Technologies in Top-Down Economics Models Using Buttom-up Information, Report 89, CEEPR, MIT. Nordhaus, W.D. and Yang, Z( 1996) A Regional Dynamic General-Equilibrium Model of Alternative Climate-Change Strategies, American Economic Review, September, 86:4:741-65 Nordhaus W. D. (1979) The E¢ cient Use of Energy Resources, New Haven: Yale University Press. Perroni, C. and T. F. Rutherford (1993) International Trade in Carbon Emission Rights and Basic Materials: General Equilibrium Calculations for 2020, Scandinavian Journal of Economics, 95:3:257-78 Poterba J. M. (1993) Global Warming Policy: A Public Finance Perspective, Journal of Economic Perspectives, 7:4:47-63 Proost, S. and Van Regemorter, D. (1992) Economic E¤ects of a Carbon Tax: With a General Equilibrium Illustration for Belgium, Energy Economics, April, 14: 2:136-49. Rasmussen, T. N. (2001) CO2 Abatement Policy with Learning-by-Doing in Renewable Energy, Resource and Energy Economics, October, 23: 4: 297-325. Roson, R. (2003) Climate Change Policies and Tax Recycling Schemes: Simulations with a Dynamic General Equilibrium Model of the Italian Economy, Review of Urban and Regional Development Studies, March, 15:1:26-44 . Stern N. (2007) Stern Review on the Economics of Climate Change, Cambridge University Press. http://www.hm-treasury.gov.uk./documents/international_issues/int_globalchallenges_index.cfm Stern N. (2008) The Economics of Climate Change, American Economic Review, 98:2:1-37. Spear, S. E.( 2003) The Electricity Market Game, Journal of Economic Theory, April 109:2:300323. Thompson, H. (2000) Energy Taxes and Wages in a General Equilibrium Model of Production, OPEC Review, September, 24:3:185-93. Uri, N. D. and Boyd, R. (1996) An Assessment of the Economic Impacts of the Energy Price Increase in Mexico, Journal of Economic Development, December 21: 2: 31-60. Vennemo, H. (1997) A Dynamic Applied General Equilibrium Model with Environmental Feedbacks, Economic Modelling, January, v. 14, iss. 1, pp. 99-154 . 277
Yago M. , J. Atkins, K. Bhattarai, R.Green and S. Trotter (2008) Modelling the Economic Impact of Low-Carbon Electricity in in Grubb, Jamasb and Pollitt (eds.) Delivering a Low Carbon Electricity System: Technologies, Economics and Policy, Cambridge University Press, 2008, pp. 394-413 Weyant J. P. (1993) Costs of Reducing Global Carbon Emission, Journal of Economic Perspectives, 7:4:27-46 Whalley J. and R. Wigle (1991) The International Incidence of Carbon Taxes, in Dornbusch R and Poterba eds., Global Warming: Economic Policy Responses, Cambridge, MIT Press 71-97. 10.3.1
A model of growth, …scal policy and welfare
Traditional Macromodel for Fiscal Policy and Growth (Bruce and Turnovsky(2007)) framework with U=
Z1
1
t
(CGc ) e
dt
0
Production function with public (Gc ) and private capital (K) Y = GP K 1 H=
1
(CGc ) +
K +B
;
0
1
) rB + GP K 1
(1
(1
!) C
T
Standard macromodel economicy growth, …scal policy and welfare: Optimisation @H = (CGc ) @C @H = @K
(1
1
Gc
(1
) GP (1
@H = @B
(1
!) = 0
)K
=0
)r = 0
Firms optimal conditions: Y =
(1
)
GP K
K=
r 1
K
Solving this equilibrium results in: (1
) r = (1
) (1
278
)
GP K
=
Transversality conditions: t
Lim Be t!1
t
= Lim Ke t!1
=0
Steady State equilibrium Y = C + Gc + GP + K (
(
1) ln C + n ln Gc = ln + ln (1
1)
C Gc + n = C Gc
=
!)
r (1
)
Balanced growth: C K B Gc GP = = = = = C K B Gc GP Steady state growth (
1) + n = =
r (1 1
r (1
)
) n
Consumption to capital ratio:
Y K GP
C Gc GP K + + + ; Gc = gc Y and K K K K = gP Y ; 0 < gc < 1; 0 < gP < 1;
=
C Y = K K Impact of tax on consumption ratios C Y = K K
Gc K
GP K
Gc K
K = K =r
1
GP K
= gC 1
K = K
r 1
gC
r 1
r 1
gP
Increase in ince tax ( ) reduces growth rate but raises the private consumption ratio with no e¤ect in the interest rate. Consumption tax (!) does not a¤ect growth rate, . 279
C K
Increase in government consumption (gc ) has no e¤ect on growth rate or interest rate but crowds out private consumption. Spending on infrascture (GP ) raises growth rate. [Following is bases on a paper presented to the AEA (Jan 2012) and ESEM/EEA (August 2012)] Lucas, R. E. (2009), Ideas and Growth. Economica, 76: 1–19.
10.4
Fiscal Policy, Growth and Income Distribution in the UK
The annual average growth rate of GDP in the UK was 0.2 percent on average between AD 1 and 1830 and 2.05 percent between the years 1830 and 20083 . Recessionary episodes were frequent but dynamic forces of growth were stronger to pull the economy into its long term growth path (Fig. 1 and 2). Kuznets (1955) had found widening of income inequality in England in the early phases of industrialisation between 1780 and 1850, when the transition from the mercantilist state to the industrial civilisation was very rapid. The process of urbanisation, lower death rate and higher birth rate, rising rate of saving, investment, capital accumulation and pro…t contributed to such inequality that remained high till 1875. It gradually narrowed down after that time as the UK parliament started enacting several laws (Finance Acts) to move towards a more egalitarian tax and transfer system (Bowley 1914) creating a tax-wedge between the original and post-tax income.
Fig. 1 3 They
were 0.08 percent and 1.5 percent in per capita term. These estimates are based on time series data provided by Maddision (1991) at http://www.ggdc.net/maddison/. See also Hansen and Prescott (2002) and Parente and Prescott (2002).
280
Fig. 2 The public …nance in UK, until 1815, was limited to heavy borrowing to …nance military and naval expenses during the wars and redeeming such debts using revenues from rents, royalties and indirect taxes in the peaceful years (O’Brien (1988), Fig. 4). Equity issues were ignored in the traditional feudalistic or mercantilist mind-set of ‘the rich man in his castle, the poor man at his gate, God made them high and lowly and ordered their estate’even after the Magna Carta (1215) and the Glorious Revolution (1688) that had transferred political power to the people and the parliament. According to economic historians the unprecedented economic growth brought by the industrial revolution and development of trade, commerce and capitalism not only made UK a global economic leader from 1750 to up to 1900 but also created wide gaps in the distribution of income and wealth between the rich and the poor (Williamson (1980), Ward(1994), Weaver (1950)). In the celebrated four cannons of taxation Smith (1776) preached for equality, certainty, convenience and economy while taxing rents, wages and pro…ts. He was worried more on e¢ ciency rather than on redistribution. Frustrated by the plight and worsening living conditions of workers, socialist reformers and radical thinkers including Wilberforce, Owen, Marx and Engels supported unions to organise and agitate for more equal rights and better working conditions of workers. This movement raised the number of MPs representing workers such as Snowden (1907), who eventually were able to promulgate a series of entitlements by enshrining them into the laws such as the Income Tax Act (1853) or Finance Act (1909). Clauses to mobilise additional revenues from the direct and indirect taxes to provide for social services including education and health in these Acts truely initiated an egalitarian tax and transfer system raising the size of state in the economy upto 12 percent of the national income (Fig 3).
281
Fig. 3
Fig. 4 A massive expansion in the public sector relative to the GDP that occurred through the public debt during the World Wars I and II, as suggested by Keynes (1940), left a legacy of a large public sector that have become a permanent feature of the UK economy since then (Hicks, 1954). Acts aimed at relieving the war devastated economy resulted in the public commitment to the social security system as proposed in the Beveridge report in 1942 that brought the share of public sector to around 40 percent of GDP as shown in Table 1. While it seems obvious that the public tax and transfer system has eliminated absolute poverty among the bottom income group, inequality of income has widened further after the another wave of reforms of the public …nance that started in early 1980s. It is shown by increase in Gini coe¢ cients of both the original and post tax income from 28.6 in 1970s to 38.3 in 2000s in Table 1. High in‡ation has further made distribution of income more unequal as the burden of higher in‡ation are born mostly by the low income households (Keynes 1940, Sargent 1987). Such an upward trend in the inequality in the last two decades despite a continuos reform of the tax and bene…t are discussed in greater details in Dutta,Sefton and Weale (2001), Johnson and Webb (1993) and Clark and Leicester (2004) for the UK and Aghion et al. (1999) for other countries. Despite above U-shaped trend in inequality one still …nds a signi…cant degree of redistribution taking place in the UK under the existing tax-bene…t system. Net of tax income of the top income
282
Table 75: Fiscal policy, growth and inequality in the UK: Recent Trends 1950s 1960s 1970s 1980s 1990s 2000s Revenue/GDP 41.1 40.0 41.1 42.5 37.1 37.3 Spending/GDP 39.0 40.2 44.4 44.7 40.4 40.7 De…cit/GDP 2.0 -0.2 -3.3 -2.2 -3.3 -3.4 Debt GDP ratio 145.0 89.6 49.9 40.6 34.6 34.2 Growth rate 2.5 3.1 2.4 2.5 2.2 1.7 Gini of original income 41.3 32.1 43.3 48.8 52.4 51.7 Gini of post tax income 35.4 25.1 28.6 33.8 38.6 38.3 In‡ation 4.2 3.6 13.6 7.6 3.6 2.5 Data source: ONS, OBR, IFS, and http://www.ukpublicspending.co.uk/index.php Gini for 1950 and 1960 rely on Stark (1972), Barna (1945), Nicholson (1964).
households are trimmed down and that of bottom income group is raised substantially by it. For instance in 2009, as shown in Table 2, the net tax payment by an average top 20 percent income earner raises enough revenue to …nance bene…ts received by an average bottom 40 percent household, who get net amount around ten thousand pounds each, twice as much as their contribution to the Treasury (Fig. 5). The extent of redistribution is less serious in the middle income group where four thousand pounds received by the third quintile almost matches the net tax payment by the fourth quintile (Table 2). In real income terms, the impacts of redistribution are more pronounced for households in the top and bottom income groups. The absolute amount received in bene…ts or paid in taxes grow with the growth of the economy. Table 76: Net E¤ects of Tax and Transfer to an Average Household by Quintile in 2009 Bene…ts Taxes Net Cash In Kind Total Direct Indirect Total Gain or Loss Bottom 6883 7555 14,438 -1195 -2965 -4,160 10,278 2nd 8280 7252 15,535 -2200 -3466 -5,666 9,866 3rd 6139 7088 13,227 -4850 -4459 -9,309 3,918 4th 3949 6162 10,111 -8403 -5386 -13,789 -3,678 Top 1992 5123 7,115 -19500 -7441 -26,941 -19,826 Average 5448 6636 12,084 -7230 -4743 -11,973 111 Data source: O¢ ce of the National Statistics; in £ .
283
Fig. 5 Time series on a sets of income measures including the original, gross and post-tax income, available from the O¢ ce of the National Statistics (http://www.ons.gov.uk/ons/statbase/) are helpful in estimating the impacts of tax and transfer in income of a household by the decile or quintile they belong to. The original income contains the sum of wages and salaries, interest and pro…t, annuities and pension, investment and other income. The gross income is obtained by adding cash bene…ts, total of contributory and non-contributory types, to the original income. Direct taxes - income tax, national insurance and council tax, are deducted from the gross income to calculate the disposable income. Further deductions of indirect taxes from it results in the post tax income (PTI) which truly measures the real economic position of a household as in Table 3 for each decile. In kind bene…ts including education, health and housing subsidies are then added to get the amount of …nal income. Thus comparing inequality in the original and PTI provides a rough indication of the di¤erence made by the tax and transfer system in the distribution of income among households in UK (Blundell 2001, Bhattarai, and Whalley 2009). As expected the post-tax income is less unequal than the original income; compare 4.0 versus 69.6 thousands of PTI to 1.9 versus 101.5 thousands of original income in Table 3 for the bottom and top income household respectively. Table 77: Pattern Bottom Households (in mln) 2.6 Original income 1.9 Cash bene…ts 4.9 Gross income 6.8 Direct taxes 0.8 Disposable income 6.0 Indirect taxes 2.0 Post-tax income 4.0 Inkind bene…ts 3.5 Final income 7.6 S o u rc e : C o n s tru c te d fro m d a ta ava ila b le a t:
of Income Distribution in 2nd 3rd 4th 5th 2.6 2.6 2.6 2.6 4.2 7.3 11.7 18.0 7.0 7.6 7.5 6.2 11.2 15.0 19.2 24.2 1.1 1.8 2.7 4.0 10.1 13.1 16.5 20.2 2.2 2.7 3.3 4.2 7.9 10.4 13.2 16.1 4.0 4.8 5.7 5.7 11.8 15.2 18.9 21.8
2009 (in ’000 6th 7th 2.6 2.6 24. 33.5 5.2 4.2 30.1 37.7 5.4 7.6 24.7 30.1 4.6 5.4 20.1 24.8 5.7 6.4 25.8 31.2
Pounds) 8th 9th 2.6 2.6 43.3 58.5 3.2 2.3 46.6 60.8 9.9 14.0 36.7 46.8 6.3 7.2 30.4 39.6 6.6 6.1 37.1 45.7
10th 2.6 101.5 2.4 103.9 24.7 79.2 9.6 69.6 6.7 76.3
http :/ / w w w .sta tistic s.g ov .u k / S TAT B A S E / P ro d u c t.a sp ? v ln k = 1 0 3 3 6 ; Ta b le 1 4 A .
284
All 26 30.5 5.0 35.3 7.2 28.4 4.7 23.6 5.5 29.1
A summary of redistribution by taxes and transfers by quintile is given in Table 4. While the average share of the bottom quintile was about 2.5 percent of original income in comparison to 50.7 percent of the top quintile, the operation of the tax and transfer system lifts the share of the bottom quintile up to 6.8 percent in the PTI and drops the share of top quintile down to 43.8 percent of it. These shares have fallen for the bottom groups and risen for the top groups in the last two decades as is clear from the smaller area under the Lorenz curve in 1983 compared to that in 2009 in Fig. 6.
Bottom 2nd 3rd 4th Top
Table 78: Share of origina and post tax income by quintile in UK, 2009 Original income share Post-tax income share Impacts of tax and transfers, % 2.46 6.75 4.29 6.92 11.33 4.41 15.04 15.92 0.88 24.92 22.25 -2.67 50.71 43.75 -6.96 Data source: O¢ ce of the National Statistics
Fig. 6
285
Fig. 7 The ratio of disposable income of 90th to 10th percentile was around 4.5 twice as much to the ratio of the 3rd to the …rst quartile. 10.4.1
Middle income hypothesis
It is a common perception in the UK that workers in the middle income group drive the economy, they generate the value that is distributed to idle rich and needy poor households. In this so called middle income-group hypothesis, the growth rate of the economy depends on the relative share of this group. Support for this hypothesis is found in the data as indicated by signi…cant coe¢ cients on the share of post tax income of the third quintile (3rd_PTI) and that of the fourth quintile (4th_PTI) in the growth equation. In constrast a higher growth rate does not raise inequality as coe¢ cient on growth is not statistically signi…cant in the inequality equation. This implies that income inequality (Gini coe¢ cient) falls only by raising the share of bottom income group (see also Beaudry et al. 2009). Table 79: Growth Inequality Relations: Testing Middle Income Share Hypothesis Change in growth rate on quintile shares Inequality (Gini-all) on growth rate Variables Coe¢ cient t-value t-prob Variables Coe¢ cient t-value t-prob Intercept 0.54** 21.0 0.00 Intercept 44.6** 16.1 0.00 3rd_PTI 1.81* 2.3 0.03 Growth -0.29 -1.7 0.10 4th_PTI -1.32 -2.6 0.02 Bottom_PTI -1.48** -3.6 0.00 R2 = 0.41, F(1,21) = 7.04 [0.00]** 2 =8.77(0.02) DW=1.11, N=24
R2 = 0.45, F(1,21) = 8.9 [0.02]* 2 =1.5(0.47) DW=1.94, N=24
While the fairness of tax system was at the heart of Meade (1951, 1978) as the optimality of taxes were in Mirrlees (1971 and 2011) the reversion of the Kuznets process as indicated by above in the UK in recent years (Gemmell 1985, Atkinson and Voitchovsky 2011) clearly show that those ideas have not been translated into actions adequately. Behavioural responses to welfares system 286
are shaped fundamentally by the structural features of the economy including the preferences of households, technologies and composition of …rms or the trading arrangements with the global economy. Proper evaluation of full impacts of tax transfer policies therefore requires an applied dynamic general equilibrium model that takes account of the decentralised structure of the UK economy. Even though the general equilibrium model have been built for the UK to study intersectoral and multi-household allocation issues since the pioneering work of Whalley (1975, 1977) and then in Piggott and Whalley (1985) only limited e¤orts have been made to measure the dynamic impacts of taxes in e¢ ciency and growth simultaneously (Bhattarai 1999, 2007). In the current context of rising inequality and declincing growth rate, what will happen to these in the next century is an issue of immense interest which we aim to analyse in this paper. 10.4.2
Current Fiscal Policy Context
As the recovery from the 2008-09 recession towards the steady state has been very slow the current …scal policy of UK aims at achieving the macroeconomic stability, supporting the pro-business and low carbon growth, achieving fairness and providing opportunities for all and in protecting the public services. The programmes and activities that the government can implement to achieve these are limited, however, by its intertempoal budget constraints. A careful analyis of the ratios of revenues, spending and de…cit to the GDP in Table 6 (and Fig. 8) shed some lights on this. Current forecasts of spending targets, revenues and public de…cit are set in the context of slow recovery after the recession that lasted for seven quarters from the second quarter of 2008 to the 4th quarter of 2009. Expansionary …scal and monetary policies taken by the government and the Bank of England have taken economy out of the slump but these are projected to raise the debt ratio to 78 percent of GDP by 2015 and exerting an in‡ationary (stag‡ationary) pressure in the economy (OBR and HM-Treasury, 2011). While these short run policy measures were taken to stimulate the economy so that it could return to its long run equilibrium path, what will happen in the next 80 years from such short run policy measures are determined more by the broad parameters that guide choices of households, …rms and traders in the economy. While there is a pressure on the government to stick to the Smith’s cannons of taxations as stated above it faces further challenges in incorporating ability to pay and bene…ts to tax payers from public spending principles that Mirrlees (1971, 2011) and Meade (1978) have proposed for the UK in recent years. While these studies provide hints for the computations or estimations of the excess burden of taxes in the context of current economic climate, proper quanti…cation of the economic e¤ects of policies on equity, e¢ ciency in allocation, growth and sectoral composition of output and employment over time is a task that can be done only with a more elaborate dynamic general equilibrium model of the UK economy. Table 80: Ratios of Revenue, Speding 2016 2015 Revenue/GDP 37.8 37.7 Spending/GDP 39.0 40.5 De…ct/GDP 1.2 2.9 Debt/GDP 75.8 77.7
287
and De…cit to GDP (OBR) 2011 2010 2009 37.8 37.3 36.5 46.2 46.6 47.7 8.4 9.3 11.1 67.5 60.5 52.8
Fig. 8 The UK government has set its activities within the constraints set by the structure of revenue and spending that have evolved over years by striking a balance between the direct taxes (income tax, national insurance, corporate tax and council tax) that bring about 60 percent of total revenue and the indirect taxes (VAT/Excise and Business Rates) for the remaining 40 percent to minimise the burden of taxes (Table 7). This requires assessment of the more complicated economy-wide income and substitution e¤ects which depend on the ‡exibility of markets as re‡ected in the elasticities of demand and supplies of goods and factors of production over time (Whalley (1975), Bhattarai and Whalley (1999)). While the right blending of progressive income and corporation taxes with regressive national insurance contribution, council taxes and VAT, petrol and fuel duties, business and other taxes is necessary to minimise the burden of taxes, the actual post tax distribution is determined not only by the net of tax income but also by the allocation of public provision of various items of public services and accessibility of households to them. As Table 8 shows around 60 percent of the public spending takes the form of transfer of resources mainly from high income to low income individuals or families for social protection, for personal social services, for health, education and transport services and the remaining 40 percent provides for the basic public goods including defence, public order and safety and servicing of debt required for the smooth functioning of the economy. Thus it is important to consider both the revenue and spending sides simultaneously to assess impacts of …scal policy on growth and redistribution.
288
Fig. 9
Table 81: Source of Revenue in UK (GBP Billion) 2011 2010 2009 Sources of Revenue Revenue Percent Revenue Percent Revenue Percent Income tax 158 0.27 150 0.27 146 0.27 National insurance 101 0.17 99 0.18 97 0.18 Corporation tax 48 0.08 43 0.08 42 0.08 Excise tax 46 0.08 46 0.08 46 0.09 VAT 100 0.17 81 0.15 78 0.14 Business tax 25 0.04 25 0.05 25 0.05 Council tax 26 0.04 25 0.05 26 0.05 Other 85 0.14 79 0.14 81 0.15 Total 589 1.00 548 1.00 541 1.00 Source: Budget Report (March 2011) HM Treasury, http://www.hm-treasury.gov. Note: In 2010/11 income tax is paid for any income above £ 7475 at the basic rate of 20% up to income of £ 35,000, at 40% rate on additional income up to £ 150,000 and at 50% for income above this. National insurance contribution rate is 12% for every employee. Council tax rate vary by the value of property in A to H bands, A paying two third and H paying twice of band D which is liable for council tax amount of £ 1332 for each year. VAT is 20 % and corporation tax is 28 % of corporate pro…t, excise and business tax-subsidy rates vary by product, going up to 95 percent.
289
Fig. 10
Table 82: Elements of Public Expenditure in UK (GBP Billion) 2011 2010 2009 Expenditure Items Spending Percent Spending Percent Spending Percent Social protection 200 0.28 194 0.28 190 0.28 Personal social services 32 0.05 32 0.04 29 0.04 Health 126 0.18 122 0.18 119 0.18 Education 89 0.13 89 0.13 88 0.13 Transport 23 0.03 22 0.03 23 0.03 Defence 40 0.06 40 0.06 38 0.05 Industry, Agr, Employment 20 0.03 20 0.03 21 0.03 Housing and Environment 24 0.03 27 0.04 30 0.04 Public order and safety 33 0.05 35 0.04 36 0.05 Debt and interest 50 0.07 44 0.11 43 0.06 Others 74 0.10 73 0.10 74 0.11 Total 711 1.00 696 1.00 704 1.00 Source: Budget Report (March 2011) HM Treasury, http://www.hm-treasury.gov. Above objectives and constraints faced by the UK economy can be successfully studied here with an applied dynamic general equilibrium model bench-marked to the micro-consistent data constructed from the latest input-output table for the decentralised market of the UK. Long run impacts of current policies on capital accumulation, investment, output and distribution among households is evaluated using results of this model for the 21st century (see Hansen and Prescott (2002) applied Malthus model to 1275 to 1800 and Solow model to 1800-1989 for the UK).A multisectoral dynamic general equilibrium model calibrated to the micro-structural features provided by the input-output table and social accounting matrix of the economy is the most appropriate tool to assess long run impacts of …scal policy in the economy.
290
10.5
Features of Dynamic Tax Model of UK
A general equilibrium model in spirit of Walras (1874), Hicks (1939), Arrow and Debreu (1954), Scarf (1973) and Whalley (1975) is a complete speci…cation of the price system in which prices and quantities are determined for each year by the interactions of demand and supply sides of goods and factor markets. In the dynamic version the relative price for every good for each year depends on the intertemporal preferences of households regarding labour-leisure and consumption and of …rms for capital and labout inuts similar to that in Ramsey (1927, 1928), Solow (1956) or Lucas (1988). Government in‡uences market outcome by distorting the prices by means of taxes and transfers which impact on income, savings, investments and the growth rate of the economy and its production sectors. As a regular macro model, households, …rms and traders optimise (Samuelson 1947, Sargent 1987, Prescott 2002) and choose optimal levels of labour supply, employment, consumption, production and trade. Intertemporal optimisation results in the optimal growth rate of output, capital and investment as in Holland and Scott (1998) or Jensen and Rutherford (2002). How can a set of policies be more e¢ cient in terms of welfare to one household rather than to another is evaluated with a social welfare function. Model is good for analysing available alternatives for long run growth prospects from the accumulation of physical and human capital or for evaluating the e¢ ciency gains from inter-temporally balanced budget or from the tax-transfer system or welfare reforms or from the low-carbon growth strategy. Short run ‡uctuations often studied in the Keynesian or the new Keynesian type economy could be introduced incorporating stochastic shocks to the production or the consumption sides of the economy (see Stern 1992 for desirable properties of this type of model). Dynamics of the applied general equilibrium model of UK with tax and transfer system contained here is an advancement on the comparative static frameworks available in the pioneering work of Whalley (1975), Piggott and Whalley (1985) and Bhattarai and Whalley (2000). This model is better suited to study growth and inequality and shows the evolution of the whole economy based on intertemporal optimsation problems of households, …rms and the government for the 21st century. 10.5.1
Preferences
Model adopts a standard Ramsey (1928) type time separable constant elasticity of substitution (CES) utility function to measure the welfare of households in each period. They engage in the intra-period and inter-temporal substitution between consumption and leisure on relative prices, interest rate, wage rate, tax rates and spending allocations in the economy. It contains AIDS demand similar to that in Deaton and Muellbauer (1980) and has multiple nests. The …rst stage h of it is the aggregation at the level of goods and services Ci;t , next stage of the nest is the choice between that composite goods and leisure Cth ; lth and …nally choice is over consumption-saving decisions across various periods based on Euler conditions. Thus the problem of household h is: max U0h =
1 X
t;h
Uth Cth ; lth
(1551)
t=0
Subject to an intertemporal budget constraint of the form: '
1 X h + wj;t 1 Pi;t 1 + tchi Ci;t t=0
twih
h li;t
#
'1 X t=0
291
h wi;t
1
twih
h Li;t
+ rj;t (1
h tki ) Ki;t
#
(1552)
here tax rates on consumption and income tchi ; twih ; tki are set by the policy makers who aim for optimality and revenue neutrality in process of tax reform. 10.5.2
Production Technology
Each …rm in the model has a unit pro…t function ( i;t ) which is the di¤erence between aggregate composite market price - the composite of prices of domestic sales (P Di;t ) and exports (P Ei;t ), and prices of primary inputs (P Yi;t ) and intermediate inputs (Pi;t ). Thus the problem of a …rm i is: 1
y
max
i;t
=
(1
i ) P Di;t
y
1
y
+
1 y 1
y
i P Ei;t
i P Yi;t
d i
1 X
ai;t Pi;t
(1553)
t=0
Subject to production technology: 1
p
Yj;t = (1
i ) Ki;t
p
1
p
+
i Li;t
p
1 p 1 p
(1554)
Sector speci…c capital (Ki;t ) accumulation: Ii;t = Ki;t
(1
) Ki;t
(1555)
1
Here i and i are share parameters, y and p are elasticities of substitution in trade and production, ai;t are the input-output coe¢ cients giving the economy wide forward and backward linkages. The real returns (rj;t ) from investments across sectors are determined by the marginal productivity of capital that adjust until the net of business tax returns are equal across sectors. The nominal interest rates set by the central bank should converge to these real rates in the long run. Wage rate of household h; wth , equals its marginal productivity (Becker et al. 1990, Meyer and Rosenbaum 2001). 10.5.3
Trade arrangements
Economy is open for the trade. Domestic …rms supply products di¤erentiated from corresponding foreign goods. Traders decide on how much to buy (Di;t ) in the domestic markets and how much to import (Mi;t ) while supplying goods (Ai;t ) to the economy. Choice of consumers between imports and domestic consumption depend on the elasticity of substitution ( m ) between domestic supplies and imported commodities in line of Krugman (1980) and Armington (1969). UK exports products that she produces at lower cost and imports products in which she has no comparative advantage. m
Ai;t = 1 X t=0
d m i Di;t
1
+
P Ei;t Ei;t =
m 1 m
m i Mi;t
1 X P Mi;t Mi;t
m y 1
(1556)
(1557)
t=0
UK economy, being one of the most liberal economies in the world, has almost no tax on exports and has very minimal tari¤s and non-tari¤ barriers on imports.
292
10.5.4
Government sector
Government receives revenues from direct and indirect taxes and tari¤s. These taxes are distortionary and a¤ect the marginal conditions of allocation in consumption, production and trade causing widespread shifts in the demand and supply functions of commodities.Which ones of these tax instruments are optimal sources of revenue and which ones are the most ine¢ cient for it and in generating growth process of the economy is a very important question but could be set following the logic of micro level incentive compatible mechanism of Mirrlees (1971, 2011) or in DiamondMirrlees (1971). It can adopt a balanced budget or a de…cit budget or a cyclically balanced budget or inter temporally balanced budget or it may simply peg de…cit to a …xed debt/GDP ratio. Which one of these strategies is adopted may depend on circumstances of the economy, policy debates and rules based on conventions and international commitments made in the treaties or agreements (i.e. EU or G20). Rt =
H X N X
h tchi Pi;t Ci;t +
h=1 i=1
H X N N X X h h twih wj;t LSi;t + tki ri Ki;t
h=1 i=1
Gt
(1558)
i=1
Ideally people’s preference for public good should decide the degree of freedom the government is given in determining the size public sector relative to the aggregate economic activities (Devereux and Love 1995, Barro (1990), Jensen and Rutherford (2002)). 10.5.5
General Equilibrium in a Growing Economy
General equilibrium is a point of rest, where the opposing forces of demand and supply balance across all markets in each period and over the entire model horizon. It is given by the system of prices of commodities and services, wage rate and interest rate in which demand and supply balance for each period (Hicks 1939). When a model is properly calibrated to the benchmark micro-consistent data set, such prices re‡ect the scarcity for those goods in the economy. Cost bene…t analysis or economic decisions can be based on real level of welfare for a set of alternatives available to the households, …rms and the government. Theoretically there has been much work, since the time of Walras, in …nding whether such equilibrium exists, or is unique or is stable (Scarf 1973, Feenberg and Poterba 2000, Feldstein 1985, Friedman 1962, Lee and Gordon 2005, Hines and Summers 2009, Naito 2006, Lockwood and Manning 1993, Bovenberg and Sørensen 2009). Uniqueness is guaranteed by the properties of preferences, technology and trade, such as continuity, concavity or convexity or twice di¤erentiability of functions. Explicit analytical solutions are possible only for very small scale models that are instructive but hardly representative of the economy (Heckman, Lochner and Taber 1998,García-Peñalosa and Turnovsky 2007). It is common to apply numerical methods to …nd the solutions of these models for a realistic policy analysis. Yi;t =
H X
h Ci;t + Ii;t + Ei;t + gi;t
(1559)
h=1 h
h
Lt = L0 en
h
Gt =
;t
= LSth + lth
N X gi;t i=1
293
(1560)
(1561)
Markets for goods clear but the economy may not always be in equilibrium. Imperfections either in goods or input markets are common giving rise to monopolistic or oligopolist situations. Such imperfections in the markets are often represented by appropriately designed mark-up schemes (Dixit and Stiglitz 1977). These mark ups may be sensitive to strategic interactions between consumers and producers, …rms and government or between the national economy and the Rest of the World. With widening gap between number of vacancies and unemployed workers it is possible to incorporate the equilibrium unemployment features of Mortensen and Pissarides (1994) in the model. 10.5.6
Procedure for Calibration
Computation and calibration of dynamic models like this are discussed in greater details in the literature (Blanchard and Kahn 1980, Sims 1980, Rutherford 1995, Smet and Wouters 2003, den Haan and Marcet 1990, Sims 1980, Kehoe 1985, Taylor and Uhlig 1990, Harrison and Vinod 1992). This model is calibrated to the reference path of the economy using the arbitrage condition in the capital market: k t Pi;t = Ri;t
t Ri;t = (r +
k Pi;t+1 k Pi;t
i ) Pi;t
=
k i ) Pi;t+1
(1
= (r +
1 1 + ri
(1
(1562)
k i ) Pi;t+1
i)
(1563)
(1564)
This helps to calibrate the capital stock and the level of investment in equilibrium path: V i;t = (r +
k i ) Pi;t+1 Ki;t ;
Ii;t =
Ki;t = gi + ri +
i
V i;t ; ri + i
V i;t
k Pi;t = Pi;t+1
(1565) (1566)
i
Even a small reform in the public policy of a sector can have a large impact on the welfare and growth over time if such policy has larger knock on e¤ects in the wider economy and removes the root source of the distortions that can have a detrimental impact on output, employment and investment levels in the economy. 10.5.7
Data for the Benchmark Economy
In their seminal works Stone (1942-43; 1961) and Meade and Stone (1941) had developed methods to construct national account and input-output table of the UK economy. The latest versions of these tables available from the O¢ ce of the National Statistics (as presented in the appendix) are used to construct the micro-consistent data for this model. Demand and supply sides for each production sector, income and expenditure for each category of household are balanced in it. The distribution of income and expenditure for di¤erent categories of households are taken from the Department of Work and Pension that is in process of unifying numerous bene…ts going to low income households 294
(http://www.dwp.gov.uk/research-and-statistics/). Share parameters from these tables are used to decompose the labour and capital income as well as consumption across households. It assumes inter temporal balance of budget by the government during the model horizon allowing occasional de…cits, like the current one, in the short run. It uses existing rates of direct and the indirect taxes that in‡uence the stream of income and consumptions of households and input choices of …rms. Detailed discussion of the microconsistent data set and algorithm and GAMS/MPSGE programmes are skiped here for space reasons. Table 83: Key Parameters of the Model values Elasticity of substitution 1.55 Steady state growth rate 0.02 Benchmark interest rate 0.05 Intertemporal substitution 0.98 Rate of depreciation 0.03 Elasticity of transformation 2.00 Capital labour substitution 1.5 Armington substitution 3.0 VAT rate 0.20 Income tax rates: 0, 0.32 0.4 and 0.5 Model uses literature based values of elasticities of substitution among inputs in production for each sector and the demand for consumption of various goods or between consumption goods and leisure for each household. Intertemporal elasticity of substitution provides trade-o¤ between the current and future choices. Thus the income redistribution e¤ect in the model occurs not only through the di¤erentiation in endowments but also by variations in tax rates on labour and capital income as well as full or reduced rate of VAT on consumption of goods and services and di¤erentiated rates of subsidies and transfers according to criteria set for the households and …rms in the economy. The optimal design of the tax system occurs by considering which one of these tax instruments is cost e¤ective in raising a given amount of revenue and has the least distortions in choices of households and …rms. 10.5.8
Results on Redistribution
Model solutions for the benchmark and counterfactual scenarios provide basis for the evaluation of the current tax and transfer system on both the functional and the size distribution of income for the next century which then could be compared to the historical accounts presented in section one. While the distribution of income between capital and labour are broadly determined by their marginal productivities as well as the amount of each factor used in production in line of standard neoclassical principles of …rms and rates of taxes on the use of these inputs, the size distribution on the income of the households depend on socio-economic structure of the economy. It is the post tax income or the level of utility from composite of consumption and leisure that households care the most. In the model these are ultimately determined by inter and intra- temporal preferences of ten categories of households and technological choices available to the producers in all eleven production sectors and the design of the tax-transfer system as proposed in Mirrlees (2011). These model solutions could …t to the available theories of distribution that emphasize on 295
ability or stochastic factors or individual choice or on human capital or on inheritance or educational inequality or life cycle or public choices for redistribution or justice as presented in Sahota (1978) or Aghion et al. (1999). The dynamic general equilibrium theory thus is the the most comprehensive theory of income distribution (Sen 1974, Auerbach, Kotliko¤ and Skinner 1983, Huggett et al. 2011). Model solutions are used to compute the Gini coe¢ cient to measure impacts of reform on distribution taking note of related literature such as Persson and Tabellini (1994), Mookherjee and Shorrocks (1982), Perotti (1993), King (1983) and Cowell and Flachaire (2012). We adopt Z 1 L(u)du = Dorfman (1979) approach to compute the area under the Lorez curve for L(u) A = 0 Z y 2 1 (1 F (y)) dy from the solution of the model for each year and apply Gini (G) coe¢ cient as 2 0
G = AeAe A to measure the inequality of distribution (Fig. 12). Since the level of utility is the most relevant indicator of the welfare of households we focus on growth and inequality in this variable that are caused by changes in the tax and transfer system. It is observed that more equality not necessarily brings the highest possible welfare to all households. While the welfare of every household can rise if the growth rate is higher but more equality with lower growth rate can reduce the level of the lifetime utility of households as is clear from the solution of the model (Fig. 11). This brings us to more di¢ cult question of choosing an appropriate social welfare function based on comparison of all types households in the model. Given utilities of individual households, U (C1 ; l1 ), U (C2 ; l2 )...U (C10 ; l10 ) it is possible to compute the social welfare function W = W fU1 ; U2 ; :::U10 g which has desirable properties (Dasgupta et al. 1973). Philosophical controversy is in whether to use maxmin criterion of Rawls (1971) which requires …nding the welfare level of the lowest income household to base the improvement in the social welfare or to adopt a weak equity axiom of Sen (1978) to justify Gini computed from the model solutions for ranking policies on the ground of distributional objective. In Atkinson’s measure of inequality (I) with income density function 1 1 ' P yi 1 f (yi ) with mean income , I = 1 f (yi ) ; transfer to lower income is weightier i
in the social welfare function as the parameter rises (Rawlsonian case when ! 1) in the measure of inequality. By constraining revenue neutrality or social welfare neutrality of taxes and spending policies, the model presented here can generate optimal numerical values of tax rates that are consistent to the principles set in Mirrlees (1971) or in Diamond-Mirrlees (1971). When tax rates are properly designed in this manner these can not only minimise the risks due to income uncertainty for low income households but also ensure that the economy moves along its long run steady state mitigating impacts of disturbances as seen in the current recession.
296
Fig.11
Fig. 12
297
Fig. 13
Fig. 14 Model results also help us to evaluate the impacts of current …scal policies in the industrial composition over the long run which could be important in evaluating emerging features of UK economy in a very competitive global economy. 10.5.9
Conclusion
UK economy that grew annually at 2.05 percent in the last two centuries had experienced a rise in income inequality during the peak phase of the industrial revolution around 1850. Thus UK became the economic leader in the global economy in 19th century creating inequality in income distribution. 298
Greater concerns towards the plight of ordinary workers in the 19th century led to actions by trade unions, politicians and philanthropists that resulted in the promulgation of a series redistributive tax and transfers measures changing the focus of public …nance from debt …nancing of wars to an egalitarian modern welfare state from the beginning of 20th century. Disappearance of the Kunznets curve phenomenon on inequality in both the original and the post tax income in the last …ve decades has caused quite a lot of discomfort and tension among people and policy makers particularly when the contribution of recent reforms to growth has become controversial. An attempt is made here to provide an evidence based on a dynamic multisectoral, multi-household general equilibrium model with tax and transfer, calibrated to the structural features of microconsistent input-output table, preferences and technological features of the UK economy. Model results are used to study the evolution of the whole economy in the 21st century. These show how the tax- transfer policies could be designed to prevent income inequality rising further and to ensure that growth rates of all sectors converge towards the steady state by the model horizon. Whether the growth enhancing and inequality reducing objectives could be achieved in the long run depend on the degree of cooperative choices from low as well as high income households in response to the public policies aimed at realising the long run vision of the UK economy. Achieving greater equality by increasing the level of utility of all households would be a sensible policy and is possible from higher rate of economic growth. It is not easy to …nd such solutions if the compensation principles are not clear in setting up a social welfare function as the greater equality in income does not automatically guarantee greater welfare for everyone when the economy is not growing at a desirable space.
References [1] Aghion Philippe, Eve Caroli and Cecilia García-Peñalosa.1999. “Inequality and Economic Growth: The Perspective of the New Growth Theories.” Journal of Economic Literature, 37(4): 1615-1660. [2] Altig David, Alan J. Auerbach, Laurence J. Kotliko¤, Kent A. Smetters and Jan Walliser. 2001. 'Simulating Fundamental Tax Reform in the United States.' American Economic Review, 91(3): 574-595. [3] Arrow, Kenneth J. and Gerard Debreu.1954 “Existence of an Equilibrium for a Competitive Economy.” Econometrica 22, 265-90. [4] Atkinson, A.B.1970. On the measurement of inequality, Journal of Economic Theory 2(3):244263. [5] Atkinson, A. B. and Voitchovsky, S. 2011. 'The Distribution of Top Earnings in the UK since the Second World War.' Economica, 78(311): 440-459. [6] Auerbach Alan J., Laurence J. Kotliko¤ and Jonathan Skinner. 1983. 'The E¢ ciency Gains from Dynamic Tax Reform.' International Economic Review, 24(1): 81-100. [7] Bandyopadyay D. and P. Basu. 2005. 'What drives the cross country growth and inequality correlations?' Canadian Journal of Economics, 38(4):1272-1297. [8] Barro Robert J. 1990. 'Government Spending in a Simple Model of Endogeneous Growth.' Journal of Political Economy, 98(5) Part 2: S103-S125. 299
[9] Beaudry P. , C. Blackorby and D. SZalay. 2009. 'Taxes and employment subsidies in Optimal Redistribution Program.' American Economic Review, 99(1):216-241. [10] Becker, Gary S.; Murphy, Kevin M. and Tamura, Robert.1990. 'Human Capital, Fertility, and Economic Growth.' Journal of Political Economy, Pt. 2, 98(5), pp. S12-37.. [11] Blanchard, O. and C.M. Kahn (1980), ‘The Solution of Linear Di¤erence Models under Rational Expectations’, Econometrica, 48, 5, July, 1305-1313. [12] Bowley A. L. 1914. 'The British Super-Tax and the Distribution of Income.' Quarterly Journal of Economics, 28(2): 255-268. [13] Bhattarai Keshab 2007. 'Welfare Impacts of Equal-Yield Tax Experiment in the UK Economy.' Applied Economics, 39(10-12): 1545-1563. [14] Bhattarai Keshab, Jonathan Haughton and David G. Tuerck (2011) The Economic E¤ects of the Fair Tax: Analysis of Results of a Dynamic CGE Model of the US Economy, memio, University Hull Business School. [15] Bhattarai Keshab and John Whalley.1999. 'Role of labour demand elasticities in tax incidence analysis with heterogeneity of labour.' Empirical Economics, 24(4).599-620. [16] Bhattarai, Keshab and J. Whalley. 2000. 'General Equilibrium Modelling of UK Tax Policy in UK' in Holly Sean and Martin Weale (eds.) Econometric Modelling: Technique and Applications, Cambridge University Press. [17] Bhattarai K and J. Whalley (2009) 'Redistribution E¤ects of Transfers.', Economica 76(3):413-431. [18] Blundell, Richard. 2001. 'Welfare reform for low income workers.', Oxford Economic Papers, 53(2):189–214. [19] Bovenberg Lans and Peter B. Sørensen. 2009. 'Optimal Social Insurance with Linear Income Taxation.' Scandinavian Journal of Economics, 111(2): 251-275. [20] Clark Tom and Andrew Leicester. 2004. 'Inequality and Two Decades of British Tax and Bene…t Reforms.' Fiscal Studies 25(2): 129–158. [21] Cowell F. and E. Flachaire (2012) Inequality with Ordinal Data, Paper for the EEA, Malaga, 2012. [22] Dasgupta, Partha, Amartya Sen, and David Starrett.1973. 'Notes on the Measurement of Inequality,' Journal of Economic Theory 6 (2): 180-187. [23] Deaton Angus and John Muellbauer. 1980. An Almost Ideal Demand System, American Economic Review, 70(3):312-326. [24] den Haan W.J. and A Marcet (1990) Solving the stochastic growth model by parameterising expectations, Jounral of Business and Economic Statistics, 8:1:31-34. [25] Devereux Michael B. and David R. F. Love.1995. 'The Dynamic E¤ects of Government Spending Policies in a Two-Sector Endogenous Growth.' Journal of Money Credit and Banking, 27(1): 232-256. 300
[26] Diamond P. A. and J. A. Mirrlees .1971. 'Optimal taxation and public producton I: Production E¢ ciency.', American Economic Review, 61:1:8-27. [27] Diamond P. A. and J. A. Mirrlees .1971. 'Optimal taxation and public producton II: Tax Rules.', American Economic Review, 61(3-1):261-278. [28] Dixit Avinash K and Joseph E. Stiglitz. 1977.Monopolistic Competition and Optimum Product Diversity, American Economic Review, 67(3):297-308. [29] Dorfman Robert.1979. 'A Formula for the Gini Coe¢ cient.', Review of Economics and Statistics, 61(1):146-149. [30] Dutta Jayasri, J. A. Sefton and M. R. Weale (2001) Income Distribution and Income Dynamics in the United Kingdom, Journal of Applied Econometrics, 16(5): 599-617. [31] Feenberg Daniel R. and James M. Poterba.2000. 'The Income and Tax Share of Very HighIncome Households, 1960-1995.' American Economic Review, 90(2): 264-270: Papers and Proceedings. [32] Feldstein Martin. 1985. “Debt and Taxes in the Theory of Public Finance.”, Journal of Public Economics, 28: 233-245. [33] Friedman Milton. 1962. Capitalism and Freedom, Chicago: Chicago University Press. [34] García-Peñalosa Cecilia and Stephen J. Turnovsky.2007. 'Growth, Income Inequality, and Fiscal Policy: What Are the Relevant Trade-o¤s?', Journal of Money, Credit and Banking, 39 (2/3): 369-394. [35] Gemmell Norman.1985. 'The Incidence of Government Expenditure and Redistribution in the United Kingdom.' Economica, New Series, 52(207): 335-344. [36] Hansen Gary D.and Edward C. Prescott.2002. 'Malthus to Solow.' American Economic Review, 92(4): 1205-1217 [37] Harrison Glenn W. and H.D. Vinod. 1992. “The sensitivity analysis of applied general equilibrium models: completely randomised factorial sampling designs.”Review of Economics and Statistics, 74(2): 357-362 [38] Heckman James J., Lance Lochner and Christopher Taber .1998. 'Tax Policy and HumanCapital Formation.' American Economic Review, 88(2): 293-297, Papers and Proceedings. [39] Hicks J. R. 1939. Value and Capital: An inquiry into some fundamental principles of economic theory, Oxford: Oxford University Press. [40] Hicks Ursula K. (1954) British Public Finances: Their Structure and Development, 1880-1952, London: Oxford University Press. [41] Hines James R. Jr. and Lawrence H. Summers .2009. 'How Globalization A¤ects Tax Design,Tax Policy and the Economy.' Tax Policy and Economy, 23 (1): 123-158 [42] Holland Allison and Andrew Scott. 1998. 'The Determinants of UK Business Cycles.', Economic Journal, 108(449): 1067-1092. 301
[43] Huggett Mark, Gustavo Ventura, and Amir Yaron. 2011., Sources of Lifetime Inequality, American Economic Review 101 (7): 2923–2954. [44] Jensen S.E. H. and T. F. Rutherford .2002. 'Distributional E¤ects of Fiscal Consolidation.' Scandinavian Journal of Economics, 104(3):471-493. [45] Johnson Paul and Steven Webb.1993. 'Explaining the Growth in UK Income Inequality: 1979-1988.' Economic Journal, 103(417): 429-435. [46] Keynes John M. 1940. How to Pay for the War, London: Macmillan. [47] Kehoe, Timothey J.1985. 'The Comparative Static Properties of Tax Models.' Canadian Journal of Economics,18 (2): 314-34. [48] King, M. A. 1983. Welfare analysis of tax reforms using household data.' Journal of Public Econonmics 2 (1): 183-214. [49] Krugman, Paul. (1980) “Scale Economies, Product Di¤erentiation, and the Pattern of Trade.” American Economic Review, 70(5): 950–59. [50] Kuznets Simon.1955. Economic Growth and Income Inequality.' American Economic Review, 45(1): 1-28. [51] Lee Young and Roger H. Gordon.2005. 'Tax structure and economic growth.' Journal of Public Economics, 89 (5-6): 1027-1043. [52] Lockwood, B. and Manning, A. .1993. 'Wage setting and the tax system. Theory and evidence for the United Kingdom.' Journal of Public Economics 52, 1-29. [53] Lucas Robert E. 1988. 'On the Mechanics of Economic Development.', Journal of Monetary Economics, 22, 3-42. [54] Maddison Angus.1991. Dynamic of Capital Accumulation and Economic Growth, Oxford: Oxford University Press. [55] Marshall Alfred.1952. Principles of Economics, London: Macmillan. [56] Meade J. E. and Richard Stone (1941) The Construction of Tables of National Income, Expenditure, Savings and Investment Economic Journal, 51(202/203):216-233. [57] Meade J. E., and Ironside, Jones, Bell, Flemming, Kay, King, Macdonald, Sandford and Whittington, Willis.1978. The Structure and Reform of Direct Taxation. IFS, London: George Allen and Unwin. [58] Meyer Bruce D. and Dan T. Rosenbaum .2001. 'Welfare, the Earned Income Tax Credit, and the Labor Supply of Single Mothers.' Quarterly Journal of Economics, 116(3): 1063-1114 [59] Mirrlees J., and S. Adam, T. Besley, R. Blundell, S. Bond, R. Chote, M. Gammie, P. Johnson, G. Myles, J. Poterba.2010. Dimensions of tax design: the Mirrlees review, Oxford: Oxford University Press. [60] Mirlees, J.A. 1971. 'An exploration in the theory of optimum income taxation.' Review of Economic Studies,38:175-208. 302
[61] Naito Takumi. 2006. 'Growth, revenue, and welfare e¤ects of tari¤ and tax reform: Win–win– win strategies.' Journal of Public Economics, 90(6-7):1263-1280 [62] O’Brien Patrick K. .1988. 'The Political Economy of British Taxation, 1660-1815.' Economic History Review, New Series, 41(1):1-32 [63] Mo¢ tt Robert A. .2003. 'The Negative Income Tax and the Evolution of U.S. Welfare Policy.' Journal of Economic Perspectives, 17(3): 119-140 [64] Mookherjee Dilip and Anthony Shorrocks .1982. 'A Decomposition Analysis of the Trend in UK Income Inequality.' Economic Journal, 92(368): 886-902 [65] Mortensen Dale T and Christopher A. Pissarides. 1994. Job Creation and Job Destruction in the Theory of Unemployment, Review of Economic Studies, 61(3):397-415. [66] Parente, S.L. and E.C. Prescott. 2002. Barriers to Riches. MIT Press, Cambridge. [67] Persson Torsten and Guido Tabellini. 1994. “Is inequality harmful for growth?”, American Economic Review, 84(3): 600-621. [68] Perotti Roberto.1993. 'Political Equilibrium, Income Distribution, and Growth.' Review of Economic Studies, 60(4): 755-776 [69] Prescott Edward C. 2002. Prosperity and Depression, American Economic Review, 92(2): 1-15 [70] Piggott, J. and J.Whalley .1985. UK Tax Policy and Applied General Equilibrium Analysis, Cambridge University Press. [71] Ramsey, Frank P. (1928) A Mathematical Theory of Saving, Economic Journal 38, 543-559. [72] Ramsey, Frank P. (1927) A Contribution to the Theory of Taxation, Economic Journal 37(145): 47-61. [73] Rowntree B. S. 1902. Poverty of Town Life, London:MacMillan. [74] Rebelo, Sergio T.1991. 'Long-run Policy Analysis and Long-run Growth.' Journal of Political Economy, 99(3), 500-21. [75] Rutherford, Thomas F. 1995 “Extension of GAMS for Complementary Problems Arising in applied Economic Analysis.” Journal of Economic Dynamics and Control 19 1299-1324. [76] Sahota Gian Singh.1978. Theories of Personal Income Distribution: A Survey Journal of Economic Literature, 16(1): 1-55. [77] Samuelson P. 1947. Foundation of Economic Analysis, Harvard University Press. [78] Sargent Thomas J. 1987. Dynamic Macroeconomic Theory, London:Harvard University Press. [79] Scarf, Herbert. 1973.The computation of economic equilibria, New Haven, Conn. : Yale University Press. [80] Sen Amartya. 1974. 'Informational bases of alternative welfare approaches: Aggregation and income distribution.', Journal of Public Economics, 3(4): 387-403. 303
[81] Snowden Phillip.1907. The Socialist Budget, London: George Allen. [82] Shoup Carl S. 1957. Some distinguishing characteristics of the British, French and United States Public Finance Systems, American Economic Review, 47(2): 187-197. [83] Shoven John B.and John Whalley .1973. 'General Equilibrium with Taxes: A Computational Procedure and an Existence, Review of Economic Studies, 40(4): 475-489 [84] Shoven, J.B. and J.Whalley .1984. 'Applied General-Equilibrium Models of Taxation and International Trade: An Introduction and Survey.' Journal of Economic Literature, 22 (Sept):10071051. [85] Sims Christopher A. 1980. Macroeconomics and Reality, Econometrica, 48(1): 1-48. [86] Smet Frank and Raf Wouters. 2003. An estimated dynamic stochastic general equilibrium model of the Euro Area, Journal of European Economic Association, 1(5):1123-1175. [87] Smith Adam.1776. In Inquiry into the Nature and Cause of Wealth of Nations, vol I and II, Liberty Fund, Indianapolis, Indiana. [88] Solow Robert M. 1956. 'A Contribution to the Theory of Economic Growth.', Quarterly Journal of Economics, 70(1): 65-94. [89] Stern Nicholas.1992. 'From the static to the dynamic: some problems in the theory of taxation.' Journal of Public Economics, 47(2): 273-297 [90] Stiglitz Joseph E. 1978. 'Notes on Estate Taxes, Redistribution, and the Concept of Balanced Growth Path Incidence Notes on Estate.', Journal of Political Economy, 86(2: 2): S137-S150. [91] Stone Richard.1942-43. National Income in the United Kingdom and the United States of America, American Economic Review, 10(1): 1-27. [92] Stone Richard. 1961. Input-output and National Accounts, Paris:OECD. [93] Taylor John B. and Harald Uhlig. 1990. Solving Nonlinear Stochastic Growth Models: A Comparison of Alternative Solution Methods, Journal of Business & Economic Statistics, 8(1): 1-17 [94] Walras, Leon.1874. Elements of Pure Economics, London (1954): Allen and Unwin. [95] Ward J. R. 1994. 'The Industrial Revolution and British Imperialism, 1750-1850.', Economic History Review, 47(1): 44-65. [96] Weaver Findley. 1950. 'Taxation and Redistribution in the United Kingdom.' Review of Economics and Statistics, 32 ( 3): 201-213 [97] Whalley John 1977. 'The United Kingdom Tax System 1968-1970: Some Fixed Point Indications of Its Economic Impact.' Econometrica, 45(8):1837-1858. [98] Whalley John. 1975. 'A General Equilibrium Assessment of the 1973 United Kingdom tax reform.' Economica, 42(166): 139-161. [99] Williamson Je¤rey G. 1980. “Earnings Inequality in Nineteenth-Century Britain.”, Journal of Economic History, 40(3):. 457-475. 304
11
L11: Dynamic Computable General Equilibrium Model: Recent Developments
One sector growth models are analytically tractable but practically they are not designed to answer questions relating to sectoral structure of production, issue of structural transformation and distribution of income as an outcome of the general equilibrium process in the economy. This requires a full dynamic computable general equilibrium (DCGE) model for a decentralised economy. DCGE models contain the relative price system and intertemporal choices of …rms and households as key factors determining the growth of various sectors of the economy and distribution of income among households while studying the long run cycles of model economies (Bhattarai 2010, 2014). The main equations for a typical DCGE model are as follows: h 1) Demand side: welfare of households U0h given by consumption Ci;t and leisure Lht : U0h =
M ax
1 X
t h h Ut ;
0
334
12.2
Dynamic CGE model of the energy and emmission
Households solve the inter-temporal allocation problem by maximising the lifetime utility subject to their lifetime budget constraint as: max U0h =
1 X
t;h
Uth Cth ; lth
t=0
Subject to 1 X
Rt
1
1 X
EMth
(1650)
t=0
h Pi;t 1 + tchi Ci;t + wth 1
twh lth + P Pt EMth
t=0
=
1 X
wth 1
twh LSth + rt (1
tk) Kth + T Rth
(1651)
t=0
h where U0h is lifetime utility of the household h, Ci;t , lth and LSth are respectively composite consumption, leisure and labour supplies of household h in period t, P Pt is the price of pollution
abatement,
EMth is the amount pollution burden in household h, Rt
1
t 1
=
1 s=1 1+rs
is an objective
discount factor whereas is the subjective discount factor of consumer for future consumption relative to current consumption; rs represents the real interest rate on assets at time s; tchi is value added tax on consumption, twh is labour income tax rate, and Kth the capital endowment of household h, Pt is the price of composite consumption (which is based on goods’prices), i.e Pt N
='
i=1
h h i pi;t ,
goods, Cth =
and Cth is the composite consumption, which is composed of sectoral consumption N i=1
h h i Ci;t .
Industries of the economy are represented by …rms that combine both capital and labour input in production and supply goods and services to the market to maximise their pro…ts: 1
y
max
j;t
=
(1
i ) P Dj;t
i P Yj;t
d i
y
+
i P Ej;t
1 X ai;j Pi;t t=0
1
y
1 y 1
y
m i
1 X am i;j Pi;t
(1652)
t=0
where: j;t is the unit pro…t of activity in sector j; P Ej;t is the export price of good j; P Dj;t is the domestic price of good j; P Yj;t is the price of value added per unit of output in activity j; y is a transformation elasticity parameter ; Pi;t is the price of …nal goods used as intermediate goods; i is the share parameter for exports in total production; i is the share of costs paid to labour and capital; di is the cost share of domestic intermediate inputs,( m i for imported intermediate inputs); am are input-output coe¢ cients for domestic supply of intermediate goods. i;j This is an open economy model in which goods produced at home and foreign countries are considered close substitutes by Armington assumption, popular in the applied general equilibrium literature. The production, trade and supply processes by sectors is easy to comprehend with four level nests of functions for each as in Figure 1. 335
Figure 1:
Figure 2:
336
Households pay taxes to the government, which it returns either as transfers to low income households or spends to pay for public consumption. The government revenue is generates from taxes on consumption, income and trade as:
REVt
=
N X H X
h tki;t rt Ki;t +
i=1 h=1
N X H X
h tchi Pi;t Ci;t +
i=1 h=1
N X tgv i Pi;t Gi;t i=1
N N N X X X p vk + ti Pi;t Ii;t + ti Pi;t GYi;t + tm i P Mi;t GYi;t i=1
+
H X
i=1
rt tk Kth +
h=1
H X
i=1
wth twh LSth
(1653)
h=1
where REVt is total government revenue and is a composite tax rate on capital income from sector i, tchi is the ad valorem tax rate on …nal consumption by households, tgv i is that on public h consumption and tvk i is the ad valorem tax rate on investment, tw is the tax rate on labour income of the household, tk is the tax on production, and tm i is the tari¤ on imports. The steady state equilibrium growth path of the economy is determined by relative prices of goods and factors such as the rental rate and the interest rate, that ultimately depend on parameters of the model such as subjective and objective discount factors, elasticities of substitution and many other shift and share parameters. By Walras’ law these prices eliminate the excess demand for goods and factors. These conditions emerge from the resource balance and zero pro…t conditions for the economy and for each household and for the government and for the rest of the world sectors in each period as well as over the entire model horizon. Government tax and transfers policies can alter this equilibrium. Income of each type of household evolves over time as a function of the relative prices of goods and share of households in total endowment of capital and labour. The production process releases emissions that manifests itself in the forms of air, water, land and noise pollutions. Pollution is a by product of production process. This is included by an emission function in the model as following: EM ISt =
N X
i Yi;t
(1654)
i=1
where EM ISt represents the total amount of emission and i is the pollution coe¢ cient for industry i the rate of pollution generated in producing output Yi;t . It is assumed to remain constant for this model. Higher rate of pollution is harmful for growth and hence for the welfare of households. While the consumption of goods generates utility to the households and such pollution generates negative externality. Their utility level falls with the increased amount of emission as it e¤ectively reduces households’life time income. It also raises cost of production as they have spend more on anti-pollution measures.Economists however have paid little attention to the form of social pollution that a¤ects mainly service industries. Corruption, sleaze, malpractices, breach of fundamental human rights and social values create tensions, anxieties, social con‡icts and reduce the creativity and productivity of workers and utility of households though it is very hard to quantify impacts of these externalities. Question 2: Study the dynamic model of UK economy in Bhattarai and Dixon (2014) contained in UK011_HH.GMS and analyse dynamics growth and redistributions impacts of energy sector 337
policies in UKwhen carbon taxes are cut by 20%, 40% and 60% respectively. Write your explanation for the results.
13
Regulation Theory and Practice
13.0.1
Theory of Regulation
Good understanding of microeconomic theories will lead to better policies and regulations for the e¢ cient functioning of the market economy. These policies particularly focus on competition, adoption of better technology, governance and information, correcting externality and good environment, social insurance, more equal distribution of income and identi…cation of cases for government intervention. For recent policies see relevant web page of the government such as in the Department for Business Innovation & Skills https://www.gov.uk/government/organisations/competitionand-markets-authority. Literature on Regulation
Tirole (2014) Market Failure and Public Policy, Nobel Prize Lecture. http://www.nobelprize.org/nobel_prizes/econ sciences/laureates/2014/tirole-lecture.html Fundenberg (191), Markin and Tirole (1990), La¤ront (1997), Jaskow (1996), Rochet and Tirole (1997) Rey (1998), Lerner (1934) Hotz and Mo Xiao (2011) Bundorf and Kosali (2006),Calzolari (2004),Bhattacharya , Goldman, Sood (2004), Buch (2003) Knittel C R V. Stango (2003),Pargal and Mani (2000), Saal and Parker (2000), Cowling (1990) Newbery (1999),Unnevehr, Gómez, Garcia (1998) ,Viscusi (1996), Wheelock and Wilson (1995),Dewatripont and Tirole (1994), Olsen and Torsvik (1993) Berg and Tschirhart (1988), Cowling and Waterson (1976) Ja¤e and Mandelker (1975), Stigler (1971), Bain (1951), Mayson (1939), Lerner (1934),Marshall (1890) Introduction to regulation Tirole (2014) in his Nobel lecture states producer deliver goods to costumers but policy makers should be aware that …rms may provide low quality goods at higher prices. This must be checked by developing a business model speci…c to …rms based empirical analysis or laboratory experiments. Markets fail to provide quality goods in unchecked. Theory of industrial regulations starts from Cournot and Du Point in 19th century; Sherman Act 1890 and Structure Conduct Performance (SCP) hypothesis.
338
Chicago school led by Stigler, Demeltz and Posner in general favoured the competitive market without any speci…c theoretical doctrine for regulation. Collective e¤orts by Fundenberg (1991), Markin (1994), La¤ront (1997), Jaskow (1996), Rochet and Tirole (1997) Rey (1998), Lerner (1934) combined game theory and information economics in designing optimal regulations. Regulatory authorities, for electricity, telecommunication, railways, airlines and road transports, postal o¢ ces, …nancial institutions and ports sprang up in Europe as well as in America. The regulators paid attention to the cost of …rms, prices they charge and the rate of return analysis in regulating these industries. Particularly they compared trade-o¤s between the lower prices and rate of return. Anti-trust laws were designed to prevent horizontal and vertical mergers and to protect patents and innovations. Industries controlling the bottleneck inputs such as railway tracks or postal services were allowed to integrate their downstream services in order provide cheaper commodities to the …nal producers based on cost plus or …xed price contracts to avoid adverse selection and moral hazard problems in the research and development and innovations. Authorities also could auction monopoly rights. Regulators can design incentive compatible mechanism so the it is not in the interest of the …rms with market power to their full extent following Ramsey Boiteux pricing strategy. Such incentive contracts can generate superior outcome as …rms most often have more information about their customers than the regulators particularly in two sided markets. Regulators should not intervene in a¤ecting the price structure and should practice fair reasonable and non-discriminating rates (FRAND) in anti-trust regulation for e¢ cient to make a better world. Better understanding of the cost and demand sides of industries is essential for better regulations. 13.0.2
Measures of concentration and performance
Structure Conduct and Performance (SCP) paradigm Number of …rms (n), buyers and sellers, nature of products and entry barriers Concentration curves, concentration ratios (cumulative market share: CRx =
x P
Si ), He…ndahl-
i=1
Hirschman Index (HHI) : HHI = , Entropy Index: E = 1 N
n P
n P
i=1
Si log
i=1 2
n P
1 Si
Si2
, Hannah and Kay Index (1977): HK =
, Variance of the logarithms of …rm size: V =
(log Si ) , Gini Coe¢ cient
i=1
339
n P
1 1
Si i=1 n P 2 1 (log Si ) N i=1
Welfare measure: Harberger’s welfare loss Dwl =
1 1 q p= 2 2
q
e p
p
p *e=
q
p p
q
Why research need to be subsidized? Consider an economy with production function Y = 10(L labour, w the wage rate.
F ), where F is …xed labour, L is
Then the cost of production is C = wL and the cost function by substituting L from the Y +F : production function: C = w 10 Under the marginal cost pricing rule:
@C @Y
Average cost declines with production:
=
C Y
w 10
=
but the producers experience negative pro…t:
= P.
w 10
+
wF Y
=R
C=
w 10 Y
w
Y 10
+F =
wF < 0
They will not undertake this project on their own. Government need to subsidise to produce optimal amount of research. This example was based on Jones (2002) Introduction to Economic Growth.
Mark-up: basis for regulation T Ri = Pi Yi ;
M Ri
13.0.3
1
(1655)
@ (T Ri = Pi Yi ) @Pi = Pi + Yi @Yi @Yi 1 @Pi Yi = Pi 1 = Pi 1 + Pi @Yi
=
M Ri = M Ci =) Pi = Here
@Yi Pi @Pi Yi
=
M Ci 1
=
1
(1656)
M Ci
is the measure of the mark up.
Regulation for solve the moral hazard problems in the …nancial markets
Asymmetric information Problem: Project B earns more but is riskier than project A. Probability of success of projects A and B are given by a and b respectively. Proportions of types A and B agents is given by pa and pb respectively Interest rate should be lower in project A than in project B in equilibrium. Under the asymmetric information a lender charging a pooling interest rate is unfair to the safe borrower A and more generous to the risky borrower B. 340
Proper signaling (incentive compatible insurance schemes) can remove such ine¢ ciency due to moral hazard in the …nancial markets. Solution of moral hazard problem Here arbitrage condition for the lender implies same expected return from both projects: a rA
=
b rB
(1657)
Since rB > rA then a > b to get this identity. Probability of types of borrower sums to 1: pa + pb = 1. Pooling interest rate: r = pa rA + pb rB
(1658)
Probability of success of projects A is higher than that of project B but the rate of return is lower in it. It need to be proved that r > rA and r < rB . r = (1
pb ) rA + pb rB = rA + pb (rB
rA ) > rA
(1659)
rB ) < rB
(1660)
Similarly r = pa rA + (1 13.0.4
pa ) rB = rB + pa (rA
Regulation by mechanism design by banks
Mechanism Design by banks Consider a bank that has two potential borrowers with amount borrowed B1 and B2 and returns
R1 and R2 respectively. Borrower type 1 has a high yielding project than the borrower type 2. Banker compare returns from the type 1, R1 = 3B1 to that from the type 2. R2 = B2 . Banker is not clear to this bank that which one of the two borrowers is more productive.
Principal’s objective function: UP = [
1
(R1
(B1 )) +
2
(R2
(B2 ))]
(1661)
Here Ri measures the returns to the bank from borrower i and Bi the amount let to the investor i from the bank.
13.0.5
Participation and incentive compatible constraints
Mechanism Design by banks Participation constraints type 1:
U1 = B1
2
(R1 )
Participation constraints type 2:
341
0
U2 = B2
2
(R2 )
0
Let the incentive compatible (the self selection constraints) be given by:
U1 = B1 U 2 = B2 13.0.6
R1 3
2
(R2 )
2
U 2 = B2
R2 3
U 2 = B2
(R1 )
2
0
2
(1662)
0
(1663)
(B2 ))]
(1664)
Solving the mechanism design problem of a bank
Solving the mechanism design problem of a bank For simplicity assume that f 1 ; 2 g = f0:5; 0:5g : Principal’s objective function: UP = [0:5 (R1
(B1 )) + 0:5 (R2
Here Ri is measures the returns to principal from borrower i and Bi the bene…t to the investor i from that. 2 Utility function of agent 1: U1 = B1 (R1 ) given that or R1 = 3B1 simply B1 = R31 . 2 Utility function of agent 2: U2 = B2 (R2 ) given that orR2 = B2 . 2 R1 2 Participation constraint in this game is given by U1 = B1 and U2 = B2 (R2 ) . At the 3 same time the agents need to ful…l the self selection constraint as: U1 = B1 U 2 = B2
R1 3
2
(R2 )
2
U 2 = B2
R2 3
U 2 = B2
(R1 )
Solving the mechanism design problem of a bank 2 R2 2 0 B1 = R31 + B2 3 ' 2 R1 R2 UP = 0:5 R1 + B2 3 3 ' 2 R1 R2 UP = 0:5 R1 + R22 + 3 3 two relevant optimal …rst order conditions 0:5 (1 2R2 ) = 0.
@UP @R1
2
2
! !
= 0:5 1
2
2
0 0
+ 0:5 R2
+ 0:5 R2 2R1 9
(1665)
and
(1666)
2
#
(1667)
2
#
(1668)
(B2 )
(R2 ) @UP @R2
= 0:5
2R2
2R1 3
+
Net bene…t for more productive investor B1 is 2.25 and bene…t for less e¢ cient investor B2 is 0.07. Conclusion of a mechanism design problem
342
Design contracts in this manner to di¤erentiate customers according their potentials the lender ensures proper appropriation of funds according to the productivity of the project This solves the problem of missing market or the credit rationing. It re…nes e¢ cient equilibrium in the presence of asymmetric information in the …nancial markets. 13.0.7
IO Approach to pricing and industrial concentration (HHI)
IO Approach to pricing and industrial concentration (HHI) Start with a Cournot Duopoly model: P =a
bQ
Q = q1 + q2
= P q1
1
dP @ 1 q1 =P + @q1 @Q
cq1
c1 = 0 =) a
2bq1
bq2 = 0
Reaction functions q1 =
a
1 q2 2
c 2b
q1 =
a
c 3b
and q2 = and q2 =
a
c
1 q1 2
2b a
c 3b
Extension to many …rms: = P qi
i
@ i dP qi =P + @qi @Q P
1+
P Multiply both sides by
P
1+
ci = 0
and
@Q =1 @qi
ci = 0
and
@Q =1 @qi
dP Q qi @Q P Q
by de…nition the elasticity is e =
cqi
P @Q Q dP
Si e
ci = 0
and
P
ci P
=
Si P
P
P
Si2 HHI = P e e Mark up relates inversely to the Herphindahl-Hirchman Index P
Si
Si ci
P
cm P
=
=
343
HHI e
Si e
13.0.8
Why regulation? Welfare e¤ects of monopoly
Why regulation? Welfare e¤ects of monopoly TR = PQ ; e =
MR
@Q P @P Q
(1669)
@ (T R = P Q) @P =P + Q @Q @Q @P Q 1 P 1+ =P 1+ P @Q e
= =
M R = M C =) P
1+
1 e
= M C =)
M R = M C =) P
1+
1 e
= M C =) e =
M R = M C =) P
1+
1 e
= M C =) e =
Q=
P
(1670)
MC = P
1 e
P
P = MC
P P
P
P = MC
P P
P Q Qe =) =1 P Q
Pro…t of the …rm: = (P
c )Q =
PQ
Welfare of price changes (a la Harberger): 1 1 P Q= PQ = 2 2 2 Thus welfare cost of monopoly is half of its pro…t. W =
13.0.9
Optimal advertising
What is the optimal intensity of advertising: = PQ
cQ
A and Q = f (P; A)
@ dQ =Q+P @P @P @ dQ =P @A @A Dividing the …rst FOC by
dC dQ =0 @Q @P
dC dQ @Q @A
1=0
dQ @P Q.
@ Q dP = +1 @Q P @Q
dC P @Q dC = 0 =) = P @Q P @Q 344
Q dP P @Q
dC @Q
P
Q dP = P @Q
=
P
1 e
The second FOC: P
dC @Q
dQ @A
1 = 0 =)
dQ 1 = dC @A P @Q
Using above results P
dQ P = dC @A P @Q
=
e
This results in Dorfman-Steiner condition for the optimal advertisement intensity for a period: P
dQ A = @A Q
e
A A ea =) = Q PQ e
Overtime these are discounted by r and the depreciation rate ( ) ea A = PQ e (r + ) 13.0.10
Marginal productivity theory and tax credit
Marginal productivity theory and tax credit =
F (K) (1 + r)
P1K K +
F 0 (K) @ = @K (1 + r)
P1K +
M P K = (1 + r) P1K M P K = (1 + r)
) P2K K (1 + r)
(1
(1671)
(1 ) P2K =0 (1 + r)
(1672)
) P2K = 0
(1673)
(1
(1
k
MPK ' r +
k
) 1+
P1K
P1K
(1674) (1675)
Marginal productivity theory and capital income tax =
(1
) F (K) (1 + r)
@ (1 ) F 0 (K) = @K (1 + r) (1
P1K K + P1K +
) M P K = (1 + r) P1K 345
(1
) P2K K (1 + r)
(1676)
(1 ) P2K =0 (1 + r)
(1677)
) P2K = 0
(1678)
(1
(1
) M P K = (1 + r) (1
13.0.11
(1
k
) 1+ k
) MPK ' r +
P1K
P1K
(1679) (1680)
Capital stock with and without capital income tax
Capital stock without capital income tax Y = K and = 0:75 and = 0:2 What is the optimal capital stock for this manufacturer? (hint ). 1
M RP K = P: K 8000: (0:75) K 0:5
1
k
P1K
(1681)
0:03] 2000
(1682)
' r+
' [0:06 + 0:03
solve for K 6000:K
K=
3 0:06
0:25
' [0:06] 2000
(1683)
4
= 504 = 6; 250; 000
(1684)
Capital stock without capital income tax Y = K and = 0:75 and = 0:2 What is the optimal capital stock for this manufacturer? (hint ). (1 (1
) M RP K = P: K
0:2) 8000: (0:75) K 0:5
1
1
' r+
' [0:06 + 0:03
k
P1K
0:03] 2000
(1685) (1686)
solve for K 4800:K
K= 13.0.12
2:4 0:06
0:25
' [0:06] 2000
(1687)
4
= 404 = 2; 560; 000
(1688)
Technological development, human capital and tax rules
Human Capital and Output in the Lucas Model Production with human capital: 1
Y = K ( hL) h = human capital per worker = fraction of time spent on working (1 ) = fraction of time spent on studies 346
(1689)
L = labour supply –(assume this as given) Example : If K =100, L=100 h =3 =0.8, =0.3 1
= 1000:3 (0:8
Y = K ( hL)
Y = K (L)
1
1 0:3
3
100)
= 1000:3 (100)
1 0:3
= 185
(1690)
= 100
(1691)
Stock of Human Capital without tax Stock of human capital (ht )depends on initial human capital: h0 fraction of time spent on studies: (1
)
the rate of human capital created by per unit of time spent on studying: ht = h0 e
(1
:
)t
(1692)
)
(1693)
growth rate of human capital: gh = if h0 =1,
= 0.4, (1
(1
) = 0.2 at after 20 years (t = 20) the human capital stock becomes 4.95. (1
)t
1
= 1000:3 (0:8
ht = h0 e
e0:4
=1
0:2 20
= e1:6 = 4:95
(1694)
output rises to 262: Y = K ( hL)
4:95
1 0:3
100)
= 262
(1695)
Stock of Human Capital with tax Stock of human capital (ht )depends now on also on the tax: initial human capital: h0 fraction of time spent on studies: (1
) (1
)
the rate of human capital created by per unit of time spent on studying: ht = h0 e
(1
)(1
)t
: (1696)
growth rate of human capital: gh = if h0 =1, = 0.4, (1 becomes 4.95.
) = 0.2 (1
ht = h0 e
(1
)(1
(1
) (1
)
(1697)
) = 0:8 at after 20 years (t = 20) the human capital stock )t
=1
e0:4
0:8 0:2 20
= e1:28 = 3:597
(1698)
output rises to 262: 1
Y = K ( hL)
= 1000:3 (0:8 347
3:597
100)
1 0:3
= 209:6
(1699)
13.0.13
Dixit-Stiglitz Model of Monopolistic Competition
Monopolistic competition Examples Monopolistic competition - (Chamberlain): Brand loyalty Product di¤erentiation characterised the main form of the monopolistic competition. Examples includes: ipod, CD, DVD, diskettes Soft drinks: Coke, Pepsi, Fanta, Tango, Sprite, 7 Up, Dr. Pepper, Cars: BMW, Voxhaul, Poeguet, Chrisler, Ford, GM, Toyota, Nissan, Hyundai, Fiat. Cosmetics , Shoes ,Watches, Camera, PC Computers, Fast food,Yoghurt, Aspirins,Pens, Books in microeconomics or macroeconomics. If a …rm reduces its own price rival …rms will reduce it, when it raises its own price none of the others will raise their prices. Kink in demand - Sweezy model of price and quantity rigidity Two questions are important 1) how much does each …rm produce? 2) How many …rms exist in the market? Consumer likes to consume varieties of products: max
u = u q0 ;
subject to: q0 + First order conditions qi u1 pi = u2
X
X
pi qi
X
1
(1700)
qi
I
1
(1701)
1
qi
qi
1
(1702)
1
qi = k:pi
1
; k>0
(1703)
Demand elasticity =
1 @qi pi = qi @pi 1
(1704)
Producer’s problem max
= (pi
pi
c) qi
f
(1705)
For optimisation apply M R = M C condition. pi 1
1
= c;
Less substitutable the product higher the price. All …rms produce the same quantity qi = q ; c
348
pi =
c
c q=f
(1706)
q=
f c1
(1707)
How many …rms exist in the market? Put this solution in the consumers’optimality condition. u1
c
= u2
X
1
qi
1
1
qi
= u2 (nq )
1
1
q
1
(1708)
Now n can be determined by solving this equation. 13.0.14
Market under imperfect competition and average cost pricing
Market under imperfect competition 1. Consider a …rm in monopolistically competitive industry Q=A
B P
(1709)
Q B
(1710)
Prove that its marginal revenue is given by MR = P
1. (a) If the cost function is C = F + cQ then prove that the average cost declines because of the economy of scale. (b) Further assume that the output sold by a …rm, number of …rms, its own price and average prices of …rms are given by Q=S
1 N
b P
P
(1711)
show that the average cost rises to number of …rms in the industry when all …rms charge same price. AC = n:F s +c Prove that price charged by a particular …rm declines with the number of …rms P =c+
1 b n
(1712)
1. (a) Determine the number of …rms and price in equilibrium. Explain entry exit behavior prices when number of …rms are below or above this equilibrium point. (b) Collusive and strategic behaviors may limit above conclusions. Discuss. (c) Apply above model to explain international trade and its impact on prices and number of …rms in a particular industry. (d) Use this model to explain interindustry and intra-industry trade.
349
(e) Use monopolistic competition model to analyse consequences of dumping practices in international trade. Inverse demand function Inverse demand function A B
P =
A B
A B
(1713) Q B
Q
(1714)
2Q =P B
Q B
(1715)
R = PQ = MR =
Q B
a. For scale economy devide both sides of C = F + cQ by Q. @AC = @Q
AC =
F Q
+c
F 0 and V (q) < 0: Firm’s problem and the participation constraint is =T
cq;
[ V (q)
T] > 0
(1723)
Participation constraint First best solution Participation constraint [ V (q)
T] > 0
(1724)
This is when …rm knows the consumer type. It is binding , V (q) = T: Substitute this information on consumer into the …rms objective function. 352
= V (q)
cq
(1725)
Optimal pro…t then means @ = V 0 (q) @q
c = 0 =) V 0 (q) = c
(1726)
This leads to the …rst degree price discrimination; high value consumer will be sold more goods at discounted per unit price and low value customer will be sold less but ends up paying more per unit. (do a graph here)
Second best solution This is when the …rm does not know the type of the consumer. It has a probability belief on type of each type of consumer,0 < < 1 for type high and (1 ) for type low. Now the …rms pro…t becomes: =
(TH
cqH ) + (1
) (TL
cqL )
(1727)
Subject to participation and incentive constraints for low high type consumers as: [ [
LV
(qL )
TL ] > 0
(1728)
HV
(qH )
TH ] > 0
(1729)
[
LV
(qL )
TL ] > [
[
HV
(qH )
TH ] > [
LV
(qH )
HV
(qL )
TH ]
(1730)
TL ]
(1731)
Binding constraints Participation constraint of the low type customer and incentive constraint of the high type customers are binding; resulting in LV
TH = [
H
(qL ) = TL
(V (qH )
(1732)
V (qL )) + TL ]
(1733)
Expected pro…t maximisation = =
[f
H
H
(V (qH )
V (qL )) + TL g
(V (qH ) V (qL )) + L V (qL )
cqH ] + (1
cqH + (1
353
) (TL )(
LV
(qL )
cqL ) cqL )
(1734) (1735)
@ = @qL
HV
HV
0
(
H
0
(qL ) +
LV
(qL ) +
LV
L) V
0
0
0
(qL ) + (1
)
(qL ) + (1
(qL ) + (1
) )
LV
LV
0
LV
0
0
(qL )
(1
(qL ) = (1
(qL ) = (1
)c = 0 )c
)c
(1736) (1737) (1738)
Quantity implications of non-linear pricing LV
0
[
(qL ) = c +
L] V
H
(1
0
(qL )
)
(1739)
Since the last term is positive it implies that L V 0 (qL ) > c Since L V 0 (qL ) = c is the …rst best and qL now should be smaller to have L V 0 (qL ) > c. For the high value type H V 0 (qH ) = c, this means 'no distortions at the top'. An example from Nicholson and Snyder about co¤ee market: p V (q) = 2 q and f
H ; Lg
= f20; 15g c=5,
=
1 2
First best solution when the type of consumer is known. V 0 (q) = q
1 2
then V 0 (q) = q
1 2
2
q=
2
1
= c; q2 = c; q = ( 2 c
c
20 = 16 5 15 2 =9 5
(1740)
Tari¤ 20 2 q = f V (q)j 15 2
p p16 = 160 9 = 90
(1741)
Expected pro…t of the …rm: = =
(TH
cqH ) + (1 ) (TL cqL ) 1 1 (160 80) + (90 45) = 40 + 22:5 = 62:5 2 2
(1742)
When types arepunknown high type may buy 9 ounce and pay 90 cents thus with consumer surplus of 20 2 9 30 = 120 90 = 30 He pays not 160 but 130. Thus the pro…t of the …rms will be = =
(TH cqH ) + (1 ) (TL cqL ) 1 1 (130 5 16) + (90 5 16) = 25 + 22:5 = 47:5 2 2
Now the shopper reduces the size of the cup: 354
(1743)
LV
0
(qL ) = c + [ 2
1
q2 =
L
H
c
L] V
H
(qL ) ;
Lq 2
2
; q=
0
L
H
=c+[
2
=
c
1 2
15 5
L] q
H
20
1 2
(1744)
2
= 22 = 4
(1745)
Tari¤ for the low customer TL =
LV
p 2 4 = 60
(qL ) = 15
(1746)
For high type HV
0
(qL ) = c =) 20
q
1 2
1
= 5 =) q 2 = 4 =) q = 16
(1747)
Non-linear pricing: co¤ee market Now tari¤ from the high type TH = [
H
TH
(V (qH )
= =
[ h
V (qL )) + TL ] = [
(V (qH ) p 20 2 16 H
V (qL )) + p 2 4 + 15
The pro…t in the second best solution is: = =
H
(V (qH )
V (qL )) +
(qL )] p i 2 4 = 160
LV
LV
140 16
= 8:75 and low type pays
60 4
(1748)
(1749) 20 = 140
(TH cqH ) + (1 ) (TL cqL ) 1 1 (140 5 16) + (60 5 4) = 30 + 20 = 50 2 2
Now the high value type pays
(qL )]
(1750)
= 15:
References [1] Berg, Sanford; John Tschirhart (1988). Natural Monopoly Regulation: Principles and Practices. Cambridge University Press. [2] Bhattacharya J, D. Goldman, N Sood (2004) Price Regulation in Secondary Insurance Markets, The Journal of Risk and Insurance, Vol. 71, No. 4 (Dec., 2004), pp. 643-675 [3] Buch C. M (2003) Information or Regulation: What Drives the International Activities of Commercial Banks? Journal of Money, Credit and Banking, 35, . 6, 851-869 [4] Bundorf K. and Kosali I. Simon (2006),The E¤ects of Rate Regulation on Demand for Supplemental Health Insurance American Economic Review, Vol. 96, No. 2 (May, 2006), pp. 67-71 [5] Calzolari G. (2004) Incentive Regulation of Multinational Enterprises International Economic Review, Vol. 45, No. 1 (Feb., 2004), pp. 257-282 355
[6] Cohen S. I. (2001) Microeconomic Policy, Routledge. [7] Cowling K. and D. Mueller (1978) Social cost of monopoly, Economic Journal 88: 727-48. [8] Cowling K. and M. Waterson (1976) Price-cost margins and market structure, Economica 43: 267-74. [9] Dag Morten Dalen, Steinar Strøm, Tonje Haabeth(2006) Price Regulation and Generic Competition in the Pharmaceutical MarketThe European Journal of Health Economics, Vol. 7, No. 3 (Sep., 2006), pp. 208-214 [10] Daughety A. F (1984) Regulation and Industrial Organization.Journal of Political Economy, 92, 5,. 932-953 [11] Daughety A. F and R. Forsythe (1987) The E¤ects of Industry-Wide Price Regulation on Industrial Organization, Journal of Law, Economics, & Organization, 3, 2 , 397-434 [12] Dewatripont M. and Jean Tirole (1994) The prudential regulation of banks, Cambridge Mass.: MIT Press. [13] Fang L and R. Rogerson (2011) Product Market Regulation and Market Work: A Benchmark Analysis, American Economic Journal: Macroeconomics 3 (April 2011): 163–188 [14] Ferguson P R and G L Ferguson (1994) Industrial Economics: Issues and Perspectives, London: McMillan. [15] Green R (2007) EU Regulation and Competition Policy among the Energy Utilities, Institute for Energy Research and Policy University of Birmingham [16] Hotz V. J. and Mo Xiao (2011) The Impact of Regulations on the Supply and Quality of Care in Child Care Markets, American Economic Review 101 (August 2011): 1775–1805 [17] Ja¤e J. F. and G. Mandelker (1975) The Value of the Firm Under Regulation, The Journal of Finance, Vol. 31, No. 2, December 28-30, 701-713 [18] Knittel C R V. Stango (2003) Price Ceilings as Focal Points for Tacit Collusion: Evidence from Credit Cards, American Economic Review, December 2003 [19] Marshall A. (1890) Principles of Economcis, McMillan. [20] Newbery, D. M. G. (1999) Privatization, restructuring, and regulation of network utilities, Cambridge, Mass. : MIT Press. [21] Olsen T. E and Gaute Torsvik (1993) The Ratchet E¤ect in Common Agency: Implications for Regulation and Privatization, Journal of Law, Economics, & Organization, 9, 1 136-158 [22] Pargal S and M. Mani (2000) Citizen Activism, Environmental Regulation, and the Location of Industrial Plants: Evidence from India, Economic Development and Cultural Change, 48, 829-846 [23] Saal D. S and David Parker (2000) The Impact of Privatization and Regulation on the Water and Sewerage Industry in England and Wales: A Translog Cost Function Model, Managerial and Decision Economics, 21, 6 , 253-268 356
[24] Stigler, G. J. (1971) 'The Theory of Economic Regulation.' Bell J. Econ. and Management Sci. 2: 3-21. [25] Tirole, Jean (2006) The theory of corporate …nance, Princeton, N.J. ; Oxford : Princeton University Press [26] Tirole J. (1995) The Theory of Industrial Organisation, MIT Press. [27] Unnevehr L.J, M. I. Gómez, P. Garcia (1998) The Incidence of Producer Welfare Losses from Food Safety Regulation in the Meat Industry, Review of Agricultural Economics, 20, 1,186-201 [28] Viscusi W. K. (1996) Economic Foundations of the Current Regulatory Reform E¤orts,Journal of Economic Perspectives, 10, 3,1996, 119–134 [29] Wheelock D. C. and P. W. Wilson (1995) Explaining Bank Failures: Deposit Insurance, Regulation, and E¢ ciency, The Review of Economics and Statistics, 77, 4 689-700
13.1
Articles and Texts
13.1.1
Best twenty articles in 100 years in the American Economic Review
Arrow, Kenneth J., B. Douglas Bernheim, Martin S. Feldstein, Daniel L. McFadden, James M. Poterba, and Robert M. Solow. 2011. '100 Years of the American Economic Review: The Top 20 Articles.' American Economic Review, 101(1): 1–8.
1. Alchian, Armen A., and Harold Demsetz. 1972. “Production, Information Costs, and Economic Organization.”American Economic Review, 62(5): 777–95.
2. Arrow, Kenneth J. 1963. “Uncertainty and the Welfare Economics of Medical Care.” American Economic Review, 53(5): 941–73.
3. Cobb, Charles W., and Paul H. Douglas. 1928. “A Theory of Production.” American Economic Review, 18(1): 139–65.
4. Deaton, Angus S., and John Muellbauer. 1980. “An Almost Ideal Demand System.” American Economic Review, 70(3): 312–26.
5. Diamond, Peter A. 1965. “National Debt in a Neoclassical Growth Model.” American Economic Review, 55(5): 1126–50.
6. Diamond, Peter A., and James A. Mirrlees. 1971. “Optimal Taxation and Public Production I: Production E¢ ciency.” American Economic Review, 61(1): 8–27.
7. Diamond, Peter A., and James A. Mirrlees. 1971. “Optimal Taxation and Public Production II: Tax Rules.” American Economic Review, 61(3): 261–78.
8. Dixit, Avinash K., and Joseph E. Stiglitz. 1977. “Monopolistic Competition and Optimum Product Diversity.” American Economic Review, 67(3): 297–308.
9. Friedman, Milton. 1968. “The Role of Monetary Policy.” American Economic Review, 58(1): 1–17.
357
10. Grossman, Sanford J., and Joseph E. Stiglitz. 1980. “On the Impossibility of Informationally E¢ cient Markets.” American Economic Review, 70(3): 393–408.
11. Harris, John R., and Michael P. Todaro. 1970. “Migration, Unemployment and Development: A Two- Sector Analysis.” American Economic Review, 60(1): 126–42.
12. Hayek, F. A. 1945. “The Use of Knowledge in Society.” American Economic Review, 35(4): 519–30. 13. Jorgenson, Dale W. 1963. “Capital Theory and Investment Behavior.” American Economic Review,53(2): 247–59.
14. Krueger, Anne O. 1974. “The Political Economy of the Rent-Seeking Society.” American Economic Review, 64(3): 291–303.
15. Krugman, Paul. 1980. “Scale Economies, Product Di¤erentiation, and the Pattern of Trade.” American Economic Review, 70(5): 950–59.
16. Kuznets, Simon. 1955. “Economic Growth and Income Inequality.”American Economic Review,45(1): 1–28.
17. Lucas, Robert E., Jr. 1973. “Some International Evidence on Output-In‡ation Tradeo¤s.”American Economic Review, 63(3): 326–34.
18. Modigliani, Franco, and Merton H. Miller. 1958. “The Cost of Capital, Corporation Finance and the Theory of Investment.” American Economic Review, 48(3): 261–97.
19. Mundell, Robert A. 1961. “A Theory of Optimum Currency Areas.” American Economic Review,51(4): 657–65.
20. Ross, Stephen A. 1973. “The Economic Theory of Agency: The Principal’s Problem.” American Economic Review, 63(2): 134–39.
21. Shiller, Robert J. 1981. “Do Stock Prices Move Too Much to Be Justi…ed by Subsequent Changes in Dividends?” American Economic Review, 71(3): 421–36. 13.1.2
Ten Best articles in the Journal of European Economic Association
1. Frank Smets and Raf Wouters (2003) An Estimated Dynamic Stochastic General Equilibrium Model of the Euro Area', Journal of European Economic Association, 1:5:1123-1175.
2. Jean-Charles Rochet and Jean Tirole (2003) Platform Competition in Two-Sided Markets' Journal of European Economic Association, 1:4:990-1029.
3. Daron Acemoglu, Philippe Aghion and Fabrizio Zilibotti (2006) Distance to Frontier and Economic Growth',Journal of European Economic Association, 4:1:37-74.
4. Alberto Alesina, Filipe R. Campante and Guido Tabellini (2008) Why is …scal policy often procyclical?Journal of European Economic Association, 6:5:1006-1036.
5. Richard Blundell, Monica Costa Dias and Costas Meghir, (2004) Evaluating the employment impact of a mandatory job search program,Journal of European Economic Association, 2:4:569-606.
358
6. Ernst Fehr and John List,(2004) The hidden costs and returns of incentives— trust and trustworthiness among CEOs, Journal of European Economic Association, 2:5:743-771.
7. Jordi Galí, J. David López-Salido and Javier Vallés (2007) Understanding the e¤ects of government spending on consumption, Journal of European Economic Association, 5:1:277-270.
8. Thomas Laubach New Evidence on the Interest Rate E¤ects of Budget De…cits and Debt, Journal of European Economic Association, 7:4:858-885.
9. James H. Stock and Mark W. Watson (2005) Understanding changes in international business cycle dynamics,Journal of European Economic Association, 3:5:968-1006.
10. Guido Tabellini (2010) Culture and institutions: economic development in the regions of Europe, Journal of European Economic Association, 8:4:677-716.
13.1.3
Best 40 articles in the Journal of Economic Perspectives
David Autor (2012) The Journal of Economic Perspectives at 100, Journal of Economic Perspectives, 26, 2,Spring, 3–18
1. Porter, Michael E.;van der Linde,Claas 1995 Toward a New Conception of the Environment-Competitiveness Relationship 9(4) 657
2. Kahneman, Daniel; Knetsch, Jack L.; Thaler, Richard H. 1991 Anomalies: The Endowment E¤ect, Loss Aversion, and Status Quo Bias 5(1) 572
3. Diamond, Peter A.; Hausman, Jerry A. 1994 Contingent Valuation: Is Some Number Better than No Number? 8(4) 524
4. Fehr, Ernst; Gächter,Simon (2000) Fairness and Retaliation: The Economics of Reciprocity 2000 14(3) 490
5. Katz, Michael L.; Shapiro, Carl 1994 Systems Competition and Network E¤ects 8(2) 448 6. North, Douglass C. 1991 Institutions 5(1) 395 7. Koenker, Roger; Hallock, Kevin F. 2001 Quantile Regression 15(4) 375 8. Markusen, James R. 1995 The Boundaries of Multinational Enterprises and the Theory of International Trade 9(2) 375
9. Bernanke, Ben S.; Gertler, Mark 1995 Inside the Black Box: The Credit Channel of Monetary Policy Transmission 9(4) 365
10. Romer, Paul M. 1994 The Origins of Endogenous Growth 8(1) 365 11. Brynjolfsson, Erik; Hitt, Lorin M. 2000 Beyond Computation: Information Technology, Organizational Transformation and Business Performance14(4) 350
12. Nickell, Stephen 1997 Unemployment and Labor Market Rigidities: Europe versus North America 11(3) 344
359
13. Machina, Mark J. 1987 Choice under Uncertainty: Problems Solved and Unsolved 1(1) 338 14. Hanemann, W. Michael 1994 Valuing the Environment through Contingent Valuation 8(4) 332 15. Camerer, Colin; Thaler, Richard H. 1995 Anomalies: Ultimatums, Dictators, and Manners 9(2) 316 16. Ostrom, Elinor 2000 Collective Action and the Evolution of Social Norms 14(3) 313 17. Smith, James P. 1999 Healthy Bodies and Thick Wallets: The Dual Relation between Health and Economic Status 13(2) 311
18. Jarrell, Gregg A.; Brickley, James A.; Netter, Je¤ry M. 1988 The Market for Corporate Control: The Empirical Evidence since 1980 2(1) 295
19. Andrade, Gregor; Mitchell, Mark; Sta¤ord, Erik 2001 New Evidence and Perspectives on Mergers 15(2) 290
20. Scotchmer, Suzanne 1991Standing on the Shoulders of Giants: Cumulative Research and the Patent Law 5(1) 280
21. Simon, Herbert A. 1991 Organizations and Markets 5(2) 278 22. Bikhchandani, Sushil; Hirshleifer,David; and Welch, Ivo 1998 Learning from the Behavior of Others: Conformity, Fads, and Informational Cascades 12(3) 273
23. Elster, Jon 1989 Social Norms and Economic Theory 3(4) 272 24. Feenstra, Robert C. 1998 Integration of Trade and Disintegration of Production in the Global Economy 12(4) 272
25. Frank, Robert H.; Gilovich, Thomas; Regan, Dennis T. 1993 Does Studying Economics Inhibit Cooperation? 7(2) 272
26. Kirman, Alan P. 1992 Whom or What Does the Representative Individual Represent? 6(2) 272 27. Jensen, Michael C. 1988 Takeovers: Their Causes and Consequences 2(1) 268 28. Przeworski, Adam; Limongi, Fernando 1993 Political Regimes and Economic Growth 7(3) 268 29. Newhouse, Joseph P. 1992 Medical Care Costs: How Much Welfare Loss? 6(3) 265 30. Dixit, Avinash 1992 Investment and Hysteresis 6(1) 259 31. Oliner, Stephen D.; Sichel, Daniel E.2000 The Resurgence of Growth in the Late 1990s: Is Information Technology the Story? 14(4) 257
32. Cutler, David M; Glaeser, Edward L.; Shapiro, Jesse M. 2003 Why Have Americans Become More Obsese? 17(3) 250
33. Milgrom, Paul 1989 Auctions and Bidding: A Primer 3(3) 242 34. Portney, Paul R. 1994 The Contingent Valuation Debate: Why Economists Should Care 8(4) 239
360
35. Babcock, Linda; Loewenstein,George 1997 Explaining Bargaining Impasse: The Role of Self-Serving Biases 11(1) 231
36. Grossman, Gene M.; Helpman, Elhanan 1994 Endogenous Innovation in the Theory of Growth 8(1) 225
37. Palmer, Karen; Oates, Wallace E.; Portney, Paul R. 1995 Tightening Environmental Standards: The Bene… t-Cost or the No-Cost Paradigm 9(4) 222
38. Angrist, Joshua D.; Krueger, Alan B. 2001 Instrumental Variables and the Search for Identi… cation: From Supply and Demand to Natural Experiments 15(4) 221
39. Pritchett, Lant 1997 Divergence, Big Time 11(3) 209 40. Dawes, Robyn M.; Thaler, Richard H. 1988 Anomalies: Cooperation 2(3) 206 41. Lundberg, Shelly; Pollak, Robert A. 1996 Bargaining and Distribution in Marriage10(4) 206 IMF Lists 25 Brightest Young Economists, August 27, 2014 (source IMF.org) 1. Nicholas Bloom, Stanford, Uncertainty 2. Amy Finkelstein, MIT , healthcare 3. Raj Chetty, Harvard, tax policy 4. Melissa Dell, Harvard Poverty 5. Kristin Forbes, BOE and MIT International macro 6. Roland Fryer, Harvard, Randomised experiment 7. Xavier Gabaix, New York, finance and macro 8. Gita Gopinath, Harvard, exchange rate 9. Esther Duflo, MIT microeconomics issues in developing countries 10. Matthew Gentzkow, Chicago, empirical micro and media 11. Emmanuel Farhi, Harvard, Macro 12. Oleg Itskhoki, Princeton, globalisation and inequality
1. Hélène Rey, LBS, international macro 2. Emmanuel Saez, California, income inequality 3. Jonathan Levin, Stanford, market design 4. Atif Mian, Princeton, Debt 5. Emi Nakamura, Columbia, business cycle 6. Nathan Nunn, Harvard, economic development 7. Parag Pathak, MIT, market design 8. Thomas Philippon, NYU, risk and financial intermediation 9. Amit Seru, Chicago, regulation and financial intermediation 10. Amir Sufi, Chicago, house price 11. Iván Werning, MIT, macro prudential policy 12. Justin Wolfers, Peterson Institute, political economy
13.
Thomas Piketty, Paris, income inequality
References [1] Aghion P. and J. Tirole (1994) Opening the blackbox of innovation, Oxford : Nu¢ eld College. [2] Alchian, Armen A., and Harold Demsetz. (1972). Production, Information Costs, and Economic Organization. American Economic Review,62(5): 777–95. [3] Allingham M. (1975) General Equilibrium, McMillan. [4] Aoki M. (1984) The cooperative game theory of …rm, Oxford: Oxford University Press. [5] Arrow K. J. (1964) The Role of Securities in the Optimal Allocation of Risk-bearing, Review of Economic Studies, 31, 2, 91-96 [6] Arrow, Kenneth J. (1963). “Uncertainty and the Welfare Economics of Medical Care.”American Economic Review, 53(5): 941–73. [7] Arrow, K.J. and F.H. Hahn (1971) General Competitive Analysis, San Franscisco: HoldenDay. [8] Armstrong M., Simon C., and J. Vickers (1994) Regulatory reform : economic analysis and British experience, Cambridge, Mass. : MIT Press. 361
[9] Arrow, K.J. and G. Debreu (1954) “Existence of an Equilibrium for a Competitive Economy” Econometrica 22, 265-90. [10] Atkinson A.B. and J. E. Stiglitz (1976) “The design of the tax structure: direct versus indirect taxation”, Journal of Public Economics, 6:1-2:55-75. [11] Baldani J., J. Brad…eld and R. Turner (2004) Mathematical Economics, the Drydon Press, London [12] Baldev Raj and R. Boadway eds. (2000) Advances on Public Economics, Physica-Verlag. [13] Balasko, Yves (1988) Foundations of the theory of general equilibrium, Boston : Academic Press. [14] Balasko Y , D. Cass and K. Shell (1995) Market Participation and Sunspot Equilibria, Review of Economic Studies, 62, 3 ,Jul., p 491-512 [15] Borglin A. (2004) Economic Dynamics and General Equilibrium: Time and Uncertainty, Springer. [16] Binmore K (1999) Why Experiment in Economics? The Economic Journal 109, 453, Features , Feb. F16-F24. [17] Bhaskar V. and Ted To (2004) Is Perfect Price Discrimination Really E¢ cient? An Analysis of Free Entry, The RAND Journal of Economics, 35, 4, 762-776 [18] Bhattarai K. (2007) Welfare Impacts of Equal-Yield Tax Experiment in the UK Economy, Applied Economics, 39, 10-12, 1545-1563. [19] Bhattarai K. (2008) Economic Theory and Models: Derivations, Computations and Applications for Policy Analyses, Serials Publications, New Delhi. [20] Bhattarai K. and J. Whalley (2003) Discreteness and the Welfare Cost of Labour Supply Tax Distortions, International Economic Review 44:3:1117-1133. [21] Bhattarai K. and J. Whalley (1999) Role of labour demand elasticities in tax incidence analysis with heterogeneity of labour, Empirical Economics, 24:4:.599-620. [22] Bloom N., R. Sadun and J. van Reenen (2012) The organization of …rms across countries, Quarterly Journal of Economics 127 (4), 1663–1705. [23] Bloom N and R. Sadun and J van Reenen (2012)Americans do I.T better: US Multinationals and the Productivity Miracle ,American Economic Review 102 (1),167-201 [24] Caminal R. (1990) A Dynamic Duopoly Model with Asymmetric Information Journal of Industrial Economics 38, 3, Mar., 315-333 [25] Cho I.K. and D.M. Kreps (1987) Signalling games and stable equilibria, the Quarterly Journal of Economics, May 179-221. [26] Cobb, Charles W., and Paul H. Douglas. (1928) “A Theory of Production.”American Economic Review,18(1): 139–65. 362
[27] Coase R. H. (1990) The Firm, the Market an the Law, Chicago: University of Chicago Press. [28] Coase R. H. (1937) The Nature of the Firm, Economica, 386-405. [29] Cohen K. J and R M Cyert (1976) Theory of the …rm: Resource Allocation in a Market Economy, New Delhi:Prentice Hall. [30] Cornes, R (1993) Duality and modern economics, Cambridge : Cambridge University Press [31] Cornes, R and T. Saddler (1993) The theory of externalities, public goods, and club goods, Cambridge : Cambridge University Press [32] Cornwall, R. R. (1984) Introduction to the use of general equilibrium analysis, Amsterdam : North-Holland. [33] Cripps, M.W.(1997) Bargaining and the Timing of Investment, International Economic Review, 38:3 :Aug.:527-546 [34] Cripps M. W. and J. P. Thomas (1995) Reputation and Commitment in Two-Person Repeated Games Without Discounting, Econometrica, . 63, 6, 1401-1419 [35] Deaton, Angus S., and John Muellbauer. 1980. “An Almost Ideal Demand System.”American Economic Review, 70(3): 312–26. [36] Debreu, G. (1954) The Theory of Value, Yale University Press, New Haven. [37] Dewatripont M. and Jean Tirole (1994) The prudential regulation of banks, Cambridge Mass.: MIT Press. [38] Dasgupta P (1992) Economic analysis of markets and games : essays in honor of Frank Hahn, Cambridge University Press. [39] Diamond, Peter A., and James A. Mirrlees. 1971. “Optimal Taxation and Public Production I: Production E¢ ciency.” American Economic Review, 61(1): 8–27. [40] Diamond, Peter A., and James A. Mirrlees. 1971. “Optimal Taxation and Public Production II: Tax Rules.” American Economic Review, 61(3): 261–78. [41] Dixit A.K. and R. S. Pindyck (1994) Investment under uncertainty, Princeton, N.J. : Princeton University Press. [42] Dixit A., S. Skeath and D. F. Reiley (2009) Games of Strategy, Norton. [43] Dixit, Avinash K., and Joseph E. Stiglitz. 1977. “Monopolistic Competition and Optimum Product Diversity.” American Economic Review, 67(3): 297–308. [44] Dixon P.B. and M. T. Rimmer (2002) Dynamic general equilibrium modelling for forecasting and policy, Amsterdam: North-Holland. [45] Gale D (1986) Bargaining and competition, Part I and II, Econometrica, 54:785-818. [46] Gardener R (2003) Games of Business and Economics, Wiley, Second Edition.
363
[47] Ginsburgh V and Keyzer M. (1997) The Structure of Applied General Equilibrium Models, MIT Press. [48] Gravelle H and R Rees (2004) Microeconomics, 3rd ed. Prentice Hall [49] Green, R.J. (2005) Electricity and Markets, Oxford Review of Economic Policy, 21, 1, 67-87. [50] Grossman, Sanford J., and Joseph E. Stiglitz. 1980. “On the Impossibility of Informationally E¢ cient Markets.” American Economic Review, 70(3): 393–408. [51] Grubb M, T. Jamasb and M. Pollitt (2010) Delivering a Low Carbon Electricity System: Technologies, Economics and Policy, Cambridge UK:Cambridge University Press. [52] Harberger A.C. (1962),The Incidence of the Corporation Income Tax, Journal of Political Economy 70, 215-40. [53] Harsanyi J.C. (1967) Games with incomplete information played by Baysian Players, Management Science, 14:3:159-182. [54] Hayek, F. A. 1945. “The Use of Knowledge in Society” American Economic Review, 35(4): 519–30. [55] Henderson J. M. and R. E. Quandt (1980) Microeconomic Theory: A Mathematical Approach, McGraw-Hill, London. [56] Henry, C. (1989) Microeconomics for public policy: helping the invisible hand, Oxford : Clarendon Press. [57] Hershleifer J and J. G. Riley (1992) The Analystics of Uncertainty and Information, Cambridge University Press. [58] Hey John D. , Chris Orme (1994) Investigating Generalizations of Expected Utility Theory Using Experimental Data, Econometrica, 62, 6, 1291-1326. [59] Hicks , J. R (1939) Value and Capital: An inquiry into some fundamental principles of economic theory, English Language Book Society, London. [60] Hiller B. (1997) Economics of Asymmetric Information, London: McMillan Press. [61] Holt Charles (2007) Markets, Games and Strategic Behaviour, Pearson. [62] Hoy M, J Livernois, C McKenna, R Rees and T. Stengos (2001) Mathematics for Economics, 2nd ed., MIT Press. [63] Intriligator, M. D. (1791) Mathematical optimization and economic theory, Prentice-Hall series in mathematical economics. [64] Jehle G A and P.J. Reny (2005) Advanced Microeconomic Theory, Pearson Education. [65] Jin J. Y. (1994) Information Sharing through Sales Report Journal of Industrial Economics 42, 3,Sep., 323-333
364
[66] Jorgenson, Dale W. (1963) “Capital Theory and Investment Behavior.” American Economic Review, 53(2): 247–59. [67] Katzner D. W. (1988) Walrasian Microeconomics, Addison Wesley. [68] Kehoe T, T.N. Srinivasan and J Whalley (2005) Frontiers in Applied General Equilibrium Modelling, Cambridge University Press. [69] King, M.A. and D. Fullerton (1984) The taxation of income from capital:a comparative study of the United States, the United Kingdom, Sweden and West Germany Chicago University Press. [70] Kocherlakota N. R. (1996) New dynamic public …nance, Princeton University Press. [71] Krauss M. B. and H.G. Johnson (1974) General equilibrium analysis: a microeconomic text, George Allen & Urwin. [72] Kreps D. M. (1990) A Course in Microeconomic Theory, Princeton. [73] Krueger, Anne O. 1974. “The Political Economy of the Rent-Seeking Society.”American Economic Review, 64(3): 291–303. [74] Krugman, Paul. (1980) “Scale Economies, Product Di¤erentiation, and the Pattern of Trade.” American Economic Review, 70(5): 950–59. [75] Kuhn H. W. (1997) Classics in Game Theory, Princeton University Press. [76] La¤ront J J and J. Tirole (2000) Competition in Telecommunication, London: MIT Press. [77] Layard R. and S. Glaister (1994) Cost-bene…t analysis, Cambridge : Cambridge University Press. [78] Luce R. D. and H Rai¤a (1957) Games and Decisions, New York: John Wiley. [79] Lockwood B. A. Philippopoulos and A. Snell (1996) Fiscal Policy, Public Debt Stabilisation and Politics: Theory and UK Evidence Economic Journal 106, 437,Jul., 894-911. [80] Luce R. D. and H. Rai¤a (1957) Games and Decisions, New York: John Wiley. [81] Machina M. (1987) Choice under uncertainty: problems solved and unsolved, Journal of Economic Perspective, 1:1:121-154. [82] MasColell A, M.D.Whinston and J.R.Green (1995) Microeconomic Theory, Oxford University Press. [83] Mailath G. J. (1989),Simultaneous Signaling in an Oligopoly Model Quarterly Journal of Economics 104, 2 ,May, 417-427 [84] Mailath G. J. and L. Samuelson (2006) Repeated Games and Reputations: long run relationship, Oxford. [85] McCormick B. (1990) A Theory of Signalling During Job Search, Employment E¢ ciency, and 'Stigmatised' Jobs Review of Economic Studies 57, 2,Apr., 299-313
365
[86] Meade, James E. (1936) An introduction to economic analysis and policy,Oxford : Clarendon Press. [87] Milgrom P., J. Roberts (1986) Price and Advertising Signals of Product Quality Journal of Political Economy 94, 4, Aug., 796-821 [88] Mirrlees J. and editors (2010) Dimensions of tax design : the Mirrlees review, Oxford ; Oxford University Press. [89] Mirlees, J.A. (1971) “An exploration in the theory of optimum income taxation”, Review of Economic Studies, 38:175-208. [90] Modigliani, Franco, and Merton H. Miller. (1958) “The Cost of Capital, Corporation Finance and the Theory of Investment.” American Economic Review, 48(3): 261–97. [91] Mookherjee D. and D. Ray (2001) Readings in the theory of economic development, Malden, Mass. : Blackwell. [92] Moore J. (1988) Contracting between two parties with private information, Review of Economic Studies, 55: 49-70. [93] Motta, M. (2004) Competition policy : theory and practice, Cambridge : Cambridge University Press. [94] Myerson R (1986) Multistage game with communication, Econometrica, 54:323-358. [95] Newbery, D. M. G. (1999) Privatization, restructuring, and regulation of network utilities, Cambridge, Mass. : MIT Press. [96] Nash J. (1953) Two person cooperative games, Econometrica, 21:1:128-140. [97] Oliver H and J Moore (1988) Incomplete Contracts and Renegotiation, Econometrica, 56, 4, 755-785 [98] Osborne M.J. and A. Robinstein (1994) A course in game theory, MIT Press. [99] Obstfeld M. and K. Rogo¤ (1996) Foundation of International Macroeconomics, MIT Press. [100] Ok Efe A. (2007) Real Analysis with Economic Applications, Princeton. [101] Pechman J A (1987) Tax reform: theory and practice, Journal of Economic Perspective, 1:1:11-28. [102] Perroni, C. (1995), Assessing the Dynamic E¢ ciency Gains of Tax Reform When Human Capital is Endogenous, International Economic Review 36, 907-925. [103] Perlo¤ J. M. (2008) Microeconomics: Theory and Applications with Calculus, Pearson. [104] Png I. and D Lehman (2007) Managerial Economics,3rd edition, Oxford: Blackwell. [105] Ranis G. and L.K. Raut ed. (1999) Trade, Growth and Development, North-Holland. [106] Rasmusen E(2007) Games and Information, Blackwell. 366
[107] Rawls, J. (1973) Theory of Justice, Oxford: Oxford University Press. [108] Ray I. (2001) On Games with Identical Equilibrium Payo¤s On Games with Identical Equilibrium Payo¤s Economic Theory, 17, 1, 223-231 [109] Riley J.P. (1979) Noncooperative Equilibrium and Market Signalling American Economic Review 69, 2, Papers and Proceedings May, 303-307 [110] Rodrik D. (1989) Promises, Promises: Credible Policy Reform via Signalling Economic Journal 99, 397, Sep., 756-772. [111] Rogerson W.P.(1988) Price Advertising and the Deterioration of Product Quality Review of Economic Studies 55, 2 , Apr., 215-229 [112] Ross, Stephen A. (1973) “The Economic Theory of Agency: The Principal’s Problem.”American Economic Review, 63(2): 134–39. [113] Roth Alvin E. (2008) What have we learned from market design?, Economic Journal, 118, 285–310. [114] Rubinstein A ed. (1990) A course in game theory, Aldershot : Elgar. [115] Rubinstein A (1982) Perfect equilibrium in a bargaining model, Econometrica, 50:1:97-109. [116] Rutherford, T. F. (1995) Extension of GAMS for Complementary Problems Arising in applied Economic Analysis, Journal of Economic Dynamics and Control 19 1299-1324. [117] Samuelson P. (1949) Foundations of Economic Analysis, Harvard University Press. [118] Sawyer M. (1991) The Economics of Industries and Firms, London: Routledge. [119] Scarf, H. E. (1986) The Computation of Equilibrium Prices, in Scarf H. E and Shoven , John B. ed. Applied General Equilibrium Analysis, Cambridge University Press. [120] Sen, Amartya (1970) Collective choice and social welfare, San Francisco(Cal.) : Holden-Day [121] Sen A. (1976) Poverty: An Ordinal Approach to Measurement, Econometrica, 44:2:219-231. [122] Sen Amartya. 1974. Informational bases of alternative welfare approaches: Aggregation and income distribution , Journal of Public Economics, 3(4): 387-403. [123] Shapley L (1953) A Value for n Person Games, Contributions to the Theory of Games II, 307-317, Princeton. [124] Shapley L and M. Shubik (1969) On Market Games, Journal of Economic Theory, 1:9-25 [125] Shiller, Robert J. (1981) “Do Stock Prices Move Too Much to Be Justi…ed by Subsequent Changes in Dividends?” American Economic Review, 71(3): 421–36. [126] Shoven, J.B. and J.Whalley (1984) “Applied General-Equilibrium Models of Taxation and International Trade: An Introduction and Survey”, Journal of Economic Literature, 22, Sept,10071051.
367
[127] Shoven, J.B. and J.Whalley (1992) Applying General Equilibrium, Cambridge University Press, 1992. [128] Simon C. P. and L. Blume (1994) Mathematics for Economists, Norton. [129] Snyder C. and W. Nicholson (2012) Microeconomic Theory: Basic Principles and Extensions, 11th ed.. South Western. [130] Spence M. (1977) Consumer Misperceptions, Product Failure and Producer Liability Review of Economic Studies 44, 3 (ct., 561-572 [131] Sobel J. (1985) A Theory of Credibility, Review of Economic Studies 52, 4,Oct., 557-573 [132] Starr R M (1997) General Equilibrium Theory: An Introduction, Cambridge. [133] Ste¤en H., W. MuÈller and H-T Normann (2001) Stackelberg beats cournot: on collusion and e¢ ciency in experimental marketsThe Economic Journal, 111 (October), 749-765. [134] Stigler, G. J. (1968) The organization of industry,Homewood, Ill. : Irwin. [135] Stone Richard (1961) Input-output and national accounts, Paris : O.E.E.C. [136] Sutton J. (1991) Sunk costs and market structure: price competition, advertising, and the evolution of concentration, Cambridge Mass.: MIT Press. [137] Sutton J. (1986) Non-Cooperative Bargaining Theory: An Introduction, Review of Economic Studies, 53, 5., 709-724 [138] Takayama, Akira (1974) Mathematical economics, Hinsdale, Ill. : Dryden Press. [139] Tirole J. (2006) The Theory of Corporate Finance, Oxford: Princeton University Press. [140] Tirole J. (1995) The Theory of Industrial Organization, MIT Press. [141] Varian H. R. (1992) Microeconomic Analysis, Norton. [142] Varian HR (2010) Intermediate Microeconomics: A Modern Approach, Norton,8th ed.. [143] Walras, L. (1954) Elements of Pure Economics, Allen and Unwin, London. [144] Watt R. (2012) The Microeconomics of Risk and Information, Palgrave Macmillan. [145] Wise D. A. (1998) Inquiries in the Economics of Aging, Cambridge Mass.: NBER.
14
Real Analysis
Basic Concepts for Review 1.Euclidian distance 15.Lower hemicontinuity 2.Convergence of a sequence 16.Fixed point and contraction mapping 3.Cauchy sequence 17.Brower’s and Kakutani …xed point theorems SETS 4.Boundedness in R 5.Compactness 18.Sets and functions and De Morgan’s Laws 368
6.Contineous functions 19.Indexed sets 7.Maximum theorems 20.Relations 8.Convex set 21.Functions 9.Convex Hulls 22.Direct and inverse images 10.Extreme points of convex set 23.Least upper bound principle 11.Strictly convex sets 24.Sequence of real numbers 12.Separation theorems 25.Upper and lower limits 13.Coorespondence 26.In…mum and supremum of functions 14.Upper hemicontinuity 27.lim inf and lim sup of functions
14.1
Methods for constructing Proofs
Direct method a = b; b = c =) a = c: a = b; c = d =) a:c = b:d For x; y; z 2 R prove that x + z = y + z =) x = y Converse and contrapositive A implies B A =) B if it converse B =) A is true then A () B here A and B are equivalent. If A person lives in Hull (A) then that person lives in Yorkshire (B). A =) B but converse is not true in this case B ; A and A < B Contrapositive implies not A implies not B A =) B Methods for constructing Proofs Equivalence A () B Example Pythagorus theorem: h2 = p2 + b2 ; p2 2 b2 h2 + cos2 = 1 h2 = h2 + h2 =) sin Mathematical induction Example: Sum of the N natural numbers is : P (n) = 1 + 2 + 3 + :::: + n = n(n+1) 2 Check if this works for any integer k P (k) = 1 + 2 + 3 + :::: + k = k(k+1) 2 Add and subtract k + 1 from both sides k(k+1) =) P (k + 1) 1 + 2 + 3 + :::: + k + (k + 1) = 2 + (k + 1) = (k + 1) k2 + 1 = (k+1)(k+2) 2 Thus by mathematical induction P (k) True =) P (k + 1): Euclidian distance Take an Euclidian space Rn with elements x = (x1 ; x2 ; x3 ; ::::xn ) q 2 2 d (x; y) = kx yk = (x1 y1 ) + :::: + (xn yn ) (1751) q Xn 2 d (x; y) (xj yj ) = j xj yj j and d (x; y) j xj yj j j=1
Triangular inequality
d (x; z)
d (x; y) + d (y; z)
369
(1752)
see : Syddeater K,P. Hammond P and A. Seierstad and A Strom (2008) . Open Ball : If a is a point in Rn amd r is a positive number then the set of all points x in n R whose distance from a is less than r is called the open ball around a with radius r. Br (a) = B (a; r) = fx 2 Rn : d (x; a) < rg A set in Rn is closed only if its complement is open or A set in Rn is closed if contains all its boundary points. Three properties of a open set a. The whole space Rn and the empty set ? is open b. arbitrary unions of open sets are open c. the intersection of …nitely many open sets is open. Three properties of a closed set a. The whole space Rn and the empty set ? are both closed b. arbitrary unions of close sets is closed c. the intersection of …nitely many closed sets is closed.
14.1.1
Convergence
Convergence of a sequence Existennce of a limit of a sequence is necessary for convergence of a sequence. If the elements 1 for any of sequence get closer to some limit they are getting close to each other for intance X X > 1: De…ne upper bound, least upper bound or supremum; greatest lower bound or in…ma A set bounded both above and below is bounded. For a monotonically increasing sequence xn 1 xn for all n and monotonically decreasing sequence xn xn 1 for all n. Every bounded monotonic sequence always converge. A seqeunce fxk g in Rn converges to a point x if for each ' > 0 there exists a natural number N such that xk 2 B' (x) for k > N or d (xk ; x) ! 0 as k ! 1: A sequence that is not convergent is divergent; lim xk = x: k!1
Cauchy sequence 1 A sequence fxn gn=1 is a cauchy sequence for any ' > 0 if there is an integer N such that for i; j N the distance between xi ; xj d (xi ; xj ) '. Prove that 1) All cauchy sequence is convergent and bounded. 2) Any convergence sequence in Rm is cauchy. Boundedness, Compactness and Continous Functions 14.1.2
Boundedness
A set S in Rn is bounded if there exists a number M such that kxk M for all x in S. No point of S is at a distance greater than M from the origin. A seuence fxk g in Rn is bounded if the the set fxk : k = 1; 2; ::::g is bounded. Any convergent sequence is bounded. An example of spliting a rectangle in four parts then spliting it again and again to …nd a converging sequence fxk g : Compactness (Bolzano- Weirstrass theorem) A set S in Rn is compact (closed and bounded) if and only if every sequence of points in S has a subsequence that converges to a point in S. Continuous functions
370
A function f with domain S Rn is continuous at a point a in S if for every ' > 0 there exists a > 0 such that jf (x) f (a)j < ' for all x in S with kx ak < . If f is continous at every point of a in a set S, then f is continous at S. Maximum theorems maximize f (x; y) subject to y 2 F (x). De…ne the corresponding value function V (x) = max f (x; y) y2F (x)
Then the general case of maximum theorem is 'Suppose f (x; y) and gi (x; y) i = 1; 2; :::; l are continous functions from X Y in R where X Rn , Y Rn and Y is compact. suppose further that for every x in X the constraint asbove is nonempty and equal to the closure of F 0 (x): = fy 2 Y : gi (x; y) < ai , i = 1; 2; ::lg : Then the value function V (x) is continous over X. If the maximization problem has a unique maximum y = y(x) for each x in X, then y(x) is continous. Convex set A set S in the plane in called convex if each points in S can be joined by the line segments lying entirely within S. A set S in Rn is convex if [x,y] S for all x and y in S, or equivalently , if x + (1 )y 2 S for all x, y in S and all in [0; 1] A function f is call concave (convex) if it is de…ned on a convext set and the line segment joining any two points on the graph is never above (below) the graph (dome). 14.1.3
Convex Hull
Convex hull of set S in Rn is the set of all convex combinations of points from S; it is donted by co(S) Extreme points of convex set An extreme point of a convex set S in Rn is a point in S that does not lie properly in any line segment in S. z is extreme point of S if z 2 S and there are no x and y in S and [0,1] such that x 6= y and z = x + (1 )y. Strictly convex sets S is strictly convex if for each pair of distinct points in x and y in S every point of the open line segment (x,y) = f x + (1 ) y : 0 < < 1g is a relative interior point in S. Separation theorems Two disjoint sets in Rn can be separtated by a hyperplane. In two dimensions, hyperplanes are straightlines. Separation theorems are important in optimisation theory. a is nonzero vector in Rn and is a real number:
n
H = fx : a:x = g
(1753)
If S and T are subsets of R , H separates S and T if S is contained in one of the closed half determined by H and T is contained in the another. a:x a:y for all x in S and for all y in T. In case a:x = = a:y H strictly separates S and T. 14.1.4
Correspondence
A correspondece F from a set A into a set B is a rule that maps each x in A to a subset F (x) of B. 371
F :A
B and x
F (x). Most popular example in economics is the budet set B (p; m) = fx 2 Rn : p:x n+1
m; x
0g
(1754)
n
(p; m) B (p; m) is a correspondence from R into R ; e.g. p1 :x1 + p2 :x2 = m: Upper hemicontinuity A correspondence F : X Rn Rm has a closed graph property at a point x0 in X if whenever fxk g is a seqence in X that converges to x0 and fyk g is a seqence in Rm that satis…es yk 2 F (xk ) k = 1,2,..., converges to y 0 , y 0 2 F x0 . A correspondence F : X Rn Rm is said to be upper hemicontinuous (u.h.c) at a point 0 x in X if every open set U that contains F x0 there exists a neighbourhood N of x0 such that F (X) U for every x in N X i.e. such that F (N X) U: F is upper hemicontinuous (or, u.h.c) in X if it is u.h.c. at every x in X. Lower hemicontinuity A correspondence F : X Rn Rm is said to be lower hemicontinuous (l.h.c) at a point x0 in 0 0 X if whenever y 2 F x and fxk g is a seqence in X that converges to x0 there exists a number 1 k0 and a sequence fyk gk=k0 in Rm that converges to y 0 and satisfy yk 2 F (xk ) for all k k0 : F is lower hemicontinuous (or, l.h.c) in X if it is u.h.c. at every x in X. 14.1.5
Fixed Point Theorems
Fixed point and contraction mapping A set S in Rn and B be the set of all bounded functions from S into Rm .Contraction mapping is T : B ! B There exist a unique function ' in B such that ' = T (' ) Brouwer’s …xed point theorems For K an non-empty compact (closed and bounded) convex set in Rn , let f is continous mapping of K into itself. The f has a …xed point f (x ) = x : Kakutani …xed point theorems For K an non-empty compact (closed and bounded) convex set in Rn , let F a correspondence K K and suppose that a F (x) is a nonempty convex set in K for each x in K b. F is upper hemicontinous Then F has a a …xed point x in K i.e a point x such that x 2 F (x ) :
14.2
SETS
Let A and B be some sets. union: A [ B = fx : x 2 A or x 2 Bg intersection: A B = fx : x 2 A and x 2 Bg complement: A-B or AnB A B = fx : x 2 A and x 2 = Bg De Morgan’s Laws An (B [ C) = (AnB) (AnC) ; An (B C) = (AnB) [ (AnC) A B=B A Indexed sets Let set ai be de…ned for each i 2 I then fai gi2I ; is an index set. [i2I Ai consists of all x belonging to Ai A [ (i2I Bi ) = i2I (A [ Bi ) 372
14.2.1
Relations and functions
Relations Let a and b be certain sets; a relation between them aRb can be (a) re‡exive (b) transitive (c) symmetric (d) anti-symmetric (e) complete (f) partial or linear ordering. Functions A function (mapping, map or transformation) f : X ! Y from a set X to a set you is a rule that assings exactly one element y = f (x) in Y to each x in X. Direct and inverse images Let f : A ! B be a function. The direct immage under f of subset S of A is the set f (S) = fy 2 B : y = f (x) for some x in Ag The inverse image under f of a set T B is f 1 (T ) = fx 2 A : f (x) 2 T g Upper and Lower Bounds Least upper bound principle A set of real numbers S is bounded above if there exists a number b such that b > x for all x in S. Any such b is upper bound of S. A number b is the least upper bound of S if b b for every upper bound b. Sequence of real numbers A sequence is a function k 7 ! x (k) with the set of N = f1; 2; 3; ::::g all positive integer in the 1 domain; it is written o¤en as fxk gk=1 or simply fxk g. It is a. increasing if xk xk+1 for k = 1; 2; :::; b. strictly increasing if xk < xk+1 for k = 1; 2; :::; c. decreasing xk xk+1 for k = 1; 2; :::; d. strictly decreasing if xk > xk+1 for k = 1; 2; :::; Increasing or decreasing sequences are monotone sequence.
14.3
Limits
Upper limits Let fxk g be a sequence of real numbers and b be a …ne real number. Then
lim xk = b is
k!1
the upper limit if and only if the following two conditions are satis…ed a. For each ' > 0 there exists an integer N such that xk < b + ' for all k > N b. For each ' > 0 there exists an integer M ther exists a integer k > M such that xk > b '. Lower limits Let fxk g be a sequence of real numbers and b be a …ne real number. Then limk!1 xk = b is the lowrer limit if and only if the following two conditions are satis…ed a. For each ' > 0 there exists an integer N such that xk > b ' for all k > N b. For each ' > 0 there exists an integer M ther exists a integer k > M such that xk < b + ' . In…mum and supremum of functions In…mum and supremum of functions Let f (x) be de…ned on x in B where B Rn then in…mum and supremum are de…ned as: inf f(x) = infff (x) : x 2 Bg ; sup f(x) = supff (x) : x 2 Bg x2B
x2B
If the range of f (x) is the interval [0; 1] and then inf f(x) = 0 and sup f(x) = 1 x2B
lim inf and lim sup of functions
373
x2B
f (x) = lim lim inf 0
r!0
x!x
lim sup f (x) = lim x!x0
r!0
inf f (x) : x 2 B x0 ; r M; x 6= x0 0
sup f (x) : x 2 B x ; r M; x 6= x
;
0
A function is upper semicontinous at point x0 in M if limx!x0 f (x) f x0 A function is lowerer semicontinous at point x0 in M if limx!x0 f (x) f x0 According to the extreme value theorem.if K Rn is a nonempty and compact set anf if f is upper semicontinous, the f has a maximum point in the set K; if f is lower semicontinous, the f has a minimum point in the set K
15
Computation and software
Microeconomic theories after detailed optimisation procedure express variables in terms of behavioural parameters. Application of these model requires calibration or estimation of these parameters with the real world data and computation of alternative scenarios according to …nd out the impacts of economic policies or changes in behaviour. Solving a simultaneous equations becomes more complicated as number of equations increase in the model. Excel is good for small scale examples. Special shoftware such as General Algebraic Modelling System (GAMS) or MATLAB are used for solving bigger models. GAMS/MPSGE is very e¤ective in solving large scale models. Econometrics often involves with estimation parameters using cross section or time series data; PcGive/Stamp/Giviein, Eviews , STATA, Shazam, Limdep are good software for this. SPSS good for processing large scale survey and statistical analysis.
15.1
GAMS
GAMS is good particularly in solving linear and non-linear system of equations. It has widely been used to solve general equilibrium models with many linear or non-linear equations on continuous or discrete variables. It comes with a number of solvers that are useful for numerical analysis such as CONOPT, DICOPT, MILES, MINOS, DNLP, PATH. It can solve very large scale models using detailed structure of consumption, production and trade arrangements on unilateral, bilateral or multilateral basis in the global economy where the optimal choices of consumers and producers are constrained by resources and production technology or arrangements for trade.It is a user friendly software. Any GAMS programme involves declaration of set, parameters, variables, equations, initialisation of variables and setting their lower or upper bounds and solving the model using Newton or other methods for linear or non-linear optimisation and reporting the results in tables or graphs see examples below. GAMS/MPSGE software is good for large scale standard general equilibrium models. GAMS programme can be downloaded from demo version of GAMS free from www.gams.com/download. Learn GAMS by practicing following examples. First write them using a text editor and save …le *.gms. Then execute the program and study the result and then revise the model as necessary. $Title a simple linear programming problem Variables R, X1, X2; Equations ER, Ex1, Ex2; Er.. R =e= 10*x1 +5*x2;
374
Ex1.. 25*x1 +10*x2 =l= 1000; Ex2.. 20*x1 +50*x2 =l= 1500; Model lp / all/; R.lo=1; X1.lo=1; X2.lo=1; Model lp /ER, Ex1, Ex2/; solve lp maximizing R using lp; $Title cartel model Variables P, Q, q1,q2, C1, C2, Pro…t, prof1, prof2; Equations EP, EQ, EC1, EC2, EPro…t,eprof1, eprof2; EP.. P =e= 300 -(1/2)*Q; EQ.. Q =e= q1+q2; EC1.. C1 =e= 500 +20*q1; EC2.. C2 =e= 1000 +(1/4)*q2*q2; EPro…t.. Pro…t =e= p*Q-c1-c2; eprof1.. prof1 =e= p*q1 -c1; eprof2.. prof2 =e= p*q2 -c2; model cartel /all/; solve cartel maximizing pro…t using nlp; $Title General Equilibrium in a Pure Exchange Global Economy $ontext Global economy produces oil and grains. It includes economies A and B. Economy A owns the oil …eld and produces 100 units of oil and economy B produces 200 units of grain. Both economies like to consume oil and grains.Their consumption preferences by given by Cobb-Douglas Utility functions, household in eocnomy A spends 40 percent of income in oil and 60 percent in grains and household in economy B spends 60 percent in apples and 40 percent in grains. Market structure is competitive. Find the relative price in these economies that is consistent with maximization of utility (satisfaction) by representative households in both countries. Choose price of oil 1 as a numeraire. Find the income of both countries, their demands for both oil and grains. Check whether the conditions for equilibrium are ful…lled. Find their levels of utility at equilibrium. $o¤text Parameters WA, WB, a1, b1; WA = 100; WB = 200; a1 =0.4;
375
b1 =0.6; Free variables UA, UB; Variables X1A, X1B, X2A, X2B, IA, IB, P1, P2; Equations EX1A, EX1B, EX2A, EX2B, EI1, EI2, MKT1, MKT2,EUA, EUB; EI1.. IA =e= P1*WA; EI2.. IB=e= P2*WB; EX1A.. X1A =e= (a1*IA)/P1; EX1B.. X1B =e= (b1*IB)/P1; EX2A.. X2A =e= ((1-a1)*IA)/P2; EX2B.. X2B =e= ((1-b1)*IB)/P2; MKT1.. X1A + X1B =e= WA; MKT2.. X2A + X2B =e= WB; EUA.. UA =e= (X1A**a1)*(X2A**(1-a1)); EUB.. UB =e= (X1B**b1)*(X2B**(1-b1)); IA.lo=1; IB.lo=1; X1A.lo=1; X1B.lo=1; X2A.lo=1; X2B.lo=1; P1.fx =1; *P1.lo= 0.001; P2.lo= 0.001; Model pure /all/; option nlp = conopt2; solve pure maximising UA using nlp; Parameters ep report, Report1; set sc /sc1*sc5/; a1= 0.1; ep(sc) = 0.1; loop(sc, solve pure maximising UA using nlp; report(sc,'UA')=UA.L;
376
report(sc,'UB')=UB.L; report(sc,'X1A')=X1A.L; report(sc,'X2A')=X2A.L; report(sc,'X1B')=X1B.L; report(sc,'X2B')=X2B.L; report(sc,'IA')=IA.L; report(sc,'IB')=IB.L; report(sc,'a1')=a1; report(sc,'b1')=b1; a1= a1+ep(sc); ); a1=0.4; b1 =0.1; loop(sc, solve pure maximising UA using nlp; report1(sc,'UA')=UA.L; report1(sc,'UB')=UB.L; report1(sc,'X1A')=X1A.L; report1(sc,'X2A')=X2A.L; report1(sc,'X1B')=X1B.L; report1(sc,'X2B')=X2B.L; report1(sc,'IA')=IA.L; report1(sc,'IB')=IB.L; report1(sc,'a1')=a1; report1(sc,'b1')=b1; b1= b1+ep(sc); ); display report,report1; $Title Cobb Web Model set t time /t1*t20/ t…rst tlast i sectors /i1*i100/; t…rst(t) = Yes$(ord(t) eq 1); tlast(t) = Yes$(ord(t) eq card(t)); alias (t, tt); Parameters al, gm, dl, bt, A, P0, Pbar; dl = 2; bt = 4; al = 500; gm =200; P0 = 100; pbar = (al+gm)/(bt+dl); display pbar; Variables q(t), P(t), QT; Equations Eq(t), EPP(t), EP(t), EQT;
377
Eq(t).. Q(t) =e= al-bt*P(t); EPP(t)$t…rst(t).. P(t)=e= P0; EP(t+1)$(ord(t) gt 1).. P(t+1)=e= -(dl/bt)*P(t) + (al+gm)/bt + P(t)$tlast(t); EQT.. QT =e= sum(t, Q(t)); Model Cobbweb /all/; q.Lo(t) =0.01; P.Lo(t) =0.01; QT.Lo =0.01; Solve Cobbweb maximising QT using nlp; Parameter report base case solution; report(t, 'Price') = P.L(t); report(t, 'Output') = Q.L(t); Display report;
For GAMS/MPSGE systax see http://www.mpsge.org. The check whether the results are consistent with the economic theory underlying the model such as general equilibrium or the ISLM-ASAD analysis for evaluating the impacts of expansionary …scal and monetary policies. Use knowledge of growth theory to explain results of the Solow growth model from Solow.gms. Consult GAMS and GAMS/MPSGE User Manuals, GAMS Development Corporation, 1217 Potomac Street, Washington D.C or www.gams.com. For other relevant software visit: http://www.feweb.vu.nl/econometriclinks/ or https://www.aeaweb.org/rfe/ Brook, A K., D. Kendtrick, A.Meeraus(1992) GAMS: Users’s Guide, release 2.25, The Scienti…c Press, San Francisco, CA. Dirkse SP and Ferris MC (1995) CCPLIB: A collection of nonlinear mixed complementarity Problems. Optimization Methods and Software 5:319-345. Rutherford, T.F. (1995) Extension of GAMS for Complementary Problems Arising in Applied Economic Analysis, Journal of Economic Dynamics and Control 19:1299-1324.
15.2
MATLAB
MATLAB is widely used for solving models. It has script and function …les used in computations. Both have *.m extensions. Its syntax are case sensivite. It is good for solving a system of linear equations and handling matrices Example 1 Write a programme …le matrix.m like the following and execute it. % now solve a linear equation % 5x1 + 2x2 =20 % 3x2 + 4x2 =15 k =[5 2;3 4]; n = [20 15]; kk = inv(k) x = kk*n’
378
One more example of system of equation and factorisation of matrices A=[1 2 3; 3 3 4; 2 3 3] b=[1; 2; 3] %solve AX=b X = inv(A)*b %eigen value and eigenvectors of A [V,D]=eig(A) %LU decomposition of A [L,U]=lu(A) %orthogonal matrix of A [Q,R]=qr(A) %Cholesky decomposition (matrix must be positive de…nite) %R = chol(A) %Singular value decomposition [U,D,V]=svd(A) %simple Markov Chain a = [0.2 0.8]; %transition matrix b = [0.9 0.1; 0.7 0.3]; %initial state c0 = a*b %subsequent states c1 = c0*b c2 = c1*b c3 = c2*b c4 = c3*b c5 = c4*b c6 = c5*b %stationary state c7 = c6*b %eigen values d = eig(b) [V,D]=eig(b) Example 3 Solving …rst order ordinary di¤erential equation write the function simpode.m which has just two lines function xdot = simpode(t,x); xdot = x+t; simpode(t,x) is MATLAB function. Write a scrit …rst_ode.m tspan =[0,2], x0=0; [t,x]=ode23(’simpode’,tspan,x0); plot(t,x) xlabel(’t’), ylabel(’x’) then execute program in the commandline to get the graph
379
>>…rst_ode Example 3 %solving system of ordinary di¤erential equations type the following and save in regid.m functiion …le function dy = rigid(t,y) dy = zeros(3,1); % a column vector dy(1) = y(2) * y(3); dy(2) = -y(1) * y(3); dy(3) = -0.51 * y(1) * y(2); end Then script …le second_ode.m tspan =[0,20], z0=[1;0]; [t,z]=ode23(’pend’,tspan,z0); x=z(:,1); y=z(:,2); plot(t,x,t,y) xlabel(’t’), ylabel(’Consumption and Income’) …gure(2) plot(x,y) xlabel(’consumption’), ylabel(’income’) title(’consumption and income’) then execute the program typing >>second_ode This will give two …gures from the solution. Example 3 Evaluating an integral de…ne the function function y =erfcousin(x); y = exp(-x^2) end write script integral.m y = quad(’erfcousin’,1/2,3/2) >>integral Example 3 Evaluating a double integral write a script …le %evaluating double integral F = inline(’1-6*x.^2*y’); I = dblquad(F,0,2,-1,1) >>integral_d For more see MATLAB help/ examples and documentation/Mathematics. Try sample programs provided there and have a tiny model to practice. See other programs like intro.m; travel.m; simul.m. Contents.m for list of …les in MATLAB demo. http://www.mathworks.com/products/demos/ ; Some demonstrations on how to use MATLAB also available in http://www.youtube.com/.
380
Pratap Rudra (2002) Getting started with MATLAB : a quick introduction for scientists and engineers. http://www.mathworks.com/academia/student_center/tutorials/ps_solve/player.html See Cleve Moler’s text book such as Numerical Computing with MATLAB or Experiments with MATLAB available at http://www.mathworks.com/moler/index.html. Michael Ferris has developed Interface between GAMS and MATLAB. The details of the new package can be found at: http://www.cs.wisc.edu/math-prog/matlab.html. For this a) install a new version of GAMS (23.4) b) put the system directory of GAMS into your MATLAB path .
15.3
Econometric and Statistical Software Excel OX7-GiveWin/PcGive/GARCH/STAMP Eviews 8 Shazam micro…t RATS GAUSS LIMDEP/NLOGIT STATA12/SPSS20 http://www.feweb.vu.nl/econometriclinks/; https://www.aeaweb.org/rfe/
OX-GiveWin/PcGive/STAMP (www.oxmetrics.net) is a very good econometric software for analysing time series, volatility modelling (arch-garch) and cross section data. This software is available in all labs in the network of the university by sequence of clicks Start/applications/economics/givewin. Students can have their own copy of Oxmetrics7 from the helpdesk. Following steps are required to access this software. a. save the data in a standard excel …le. Better to save in *.csv format. b. start oxmetrics7 and givewin in it at start/applications/economics/givewin and pcgive (click them separately) c. open the data …le using …le/open data…le command. d. choose PcGive module for econometric analysis. e. select the package such as descriptive statistics, econometric modelling or panel data models. d. choose dependent and independent variables as asked by the menu. Choose options for output. e. do the estimation and analyse the results, generate graphs of actual and predicted series. A Batch …le can be written in OX for more complicated calculations using a text editor such as pfe32.exe. Such …le contains instructions for computer to compute several tasks in a given sequence.
381
References [1] Doornik J A and D.F. Hendry ((2013) PC-Give Volume I-III, GiveWin Timberlake Consultants Limited, London 15.3.1
Quality ranking of journals in Economics
Findings of theoretical and applied research are published in journals. Better the quality of a paper, more likelihood that it will be published in highly ranked journals, though this relationship is not always perfect one. It is instructive to look into the Association of Business School (ABS) ranking on quality of journals given below in process of reviewing the literature as well as in writing a paper. ABS 4* Journals American Economic Review; Economic Journal; Econometrica; Journal of Labour Economics; Rand Journal of Economics; Journal of Political Economy; Journal of Monetary Economics; International Economic Review; Quarterly Journal of Economics; Review of Economic Studies; Journal of Econometrics; Journal of Economic Literature; Journal of Economic Perspective; Journal of Economic Theory; Journal of Economic Geography; Journal of Environmental Economics and Management; Journal of Financial Economics; Econometric Theory; . ABS 3* Journals Brookings Economics Papers; Journal of Economic Growth; Economic Letters; European Economic Review; Journal of Development Economics; Canadian Journal of Economics; European Review of Agricultural Economics; Cambridge Journal of Economics; Journal of Applied Econometrics ; Journal of Comparative Economics; Journal of Development Studies;Journal of Economic Dynamics and Control; Journal of Health Economics; Journal of Economic Behaviour and Organisation; Journal of Economics and Management Strategy; Journal of Economics of Law and Organisation; Journal of Evolutionary Economics; Journal of Industrial Economics; Economica; Journal of Public Economics; Journal of European Economic Association; Journal of Urban Economics; Kyklos; Labour Economics; Ecological Economics;IMF Economic Review; Land Economics; Oxford Bulletin of Economics and Statistics; Oxford Economics Papers; Review of Economics and Statistics; Review of International Economics;Social Choice and Welfare; Southern Economic Journal; World Bank Economic Review; Journal of International Economics; Economy and Society. ABS 2* Journals Advances in Econometrics; European Journal of Political Economy; Agricultural Economics; Applied Economics; Annals of Public and Cooperative Economics; Applied Financial Economics; ; Australian Journal of Agricultural and Resource Economics; Bulletin of Economic Research; ; Canadian Journal of Agricultural Economics; Contemporary Economic Policy; Contributions to the Political Economy; Defence and Peace Economics; Econometric Reviews; Economics of Education Review; Economics of Innovation and New Technology; Economics of Planning;Economics of Transition; Economist-Netherlands;Environmental Resource Economics; Fiscal Studies; Global Business and Economic Review; History of Political Economy; Oxford Review of Economic Policy; IMF Sta¤ Papers; Insurance Mathematics and Economics; International Journal of Game Theory;International Journal of Economics of Business; International Review of Economics and Finance; Journal of Agricultural and Resource Economics; World Economy; Journal of Economic Methodology, Journal of Economic Psychology; Journal of Industry, competition and Trade; Macroeconomic Dynamics; Journal of Economics; Employee Relations; Empirical Economics; STATA Journal.
382
ABA 1* Journals Applied Economics Letters; Australian Economic Review; Business Economics; Bulletin of Indonesian Economic Studies; Eastern European Economics; ; International Review of Applied Economics; Information Economics and Policy;International Journal of Social Economics; Journal of interdisciplinary Economics; For the latest version visit: http://www.associationofbusinessschools.org/node/1000257. https://ste¤enroth.…les.wordpress.com/2015/06/abs-2015-ste¤en-roth-ch.pdf Note also that there are many journals which have not been ranked by the ABS.
383
15.4
Core texts in Economic Theory and Equivalent reading
References [1] Aumann R.J. and S. Hart. (1994) Handbook of game theory with economic applications, North Holland, 1992-1994. [2] Allen R.G.D. (1956) Mathematical Economics, MacMillan. [3] Atkinson A.B. and J. E. Stiglitz (1980) Lectures on Public Economics, McGraw Hill. [4] Balasko, Yves (1988) Foundations of the theory of general equilibrium, Boston : Academic Press. [5] Baldani J, J Brad…eld and R Turner (1996) Mathematical Economics, Dryden Press. [6] Basu, Kaushik (1993) Lectures in industrial organization theory, Oxford : Blackwell. [7] Bhagwati J. N. and T.N. Srinivasan (1992)Lectures on International Trade, MIT Press [8] Bhattarai K. (2007) Models of Economic and Political Growth in Nepal, Serials Publications, New Delhi. [9] Binmore K. (1990) Fun and Games: A text on Game Theory, Lexington, Heath. [10] Bridel P. (2011) General equilibrium analysis: a century after Walras, Routledge, London. [11] Cohen, S. I. (2001) Microeconomics; Economic policy, London ; New York : Routledge. [12] Cornwall R R (1984) Introduction to the use of general equilibrium analysis, North-Holland. [13] Debreu, G. (1954) The Theory of Value, Yale University Press, New Haven. [14] Estrin S., D Laidler and M. Dietrich (2008) Microeconomics,Prentice Hall. [15] Gardner R (2003) Games for Business and Economics, Willey. [16] Ginsburgh V.and M. Kayzer (1997) The Structure of Applied General Equilibrium Models, MIT Press. [17] Gravelle H and R Rees (2004) Microeconomics, 3rd ed. Prentice Hall [18] Fundenbeg D and J Tirole (1995) Game Theory, MIT Press. [19] Hershleifer J and J G Riley (1992) The Analytics of Uncertainty and Information, Cambridge [20] Harrison GW ed. (2000) Using dynamic general equilibrium models for policy analysis2000, pp. xi, 411, Contributions to Economic Analysis, 248 North-Holland. [21] Hillman Arye (2005) Public Finance, Cambridge University Press.
384
[22] Hirshleifer J and J G Riley (1992) The Analytics of Uncertainty and Information, Cambridge. [23] Hicks J R (1939) Value and Capital, ELBS, MacMillan. [24] Holt C A (2007) Markets, Games and Strategic Behaviour, Pearson. [25] Jehle G A and P.J. Reny (2005) Advanced Microeconomic Theory, Pearson Education. [26] Katzner D W (1988) Walrasian Microeconomics: An Introduction to the Economic Theory and Market Behaviour, Addison Wesley. [27] Kreps D. M. (1990) A Course in Microeconomic Theory, Princeton. [28] Krugman P R and M Obstfeld (2000) International Economics, Addison Wesley. [29] La¤ont JJ and M. Moreaux (1989) Dynamics, incomplete information and industrial economics, Oxford : Blackwell [30] La¤ont, Jean-Jacques (1989) The economics of uncertainty and information, Cambridge, Mass : MIT Press [31] Luce R. D. and Rai¤a H. (1957) Games and Decisions, John Wiley and Sons, New York. [32] Marshall A. (1890) Principles of Economcis, McMillan. [33] Mailath G. J. and L. Samuelson (2006) Repeated Games and Reputations: long run relationship, Oxford. [34] MasColell A, M.D.Whinston and J.R.Green (1995) Microeocnomic Theory, Oxford University Press. [35] Myles G.D. (1995) Public Economics, Cambridge University Press. [36] Nicholson W (1989) Microeconomic Theory and Extensions, 4th edition, Dryden Press. [37] Neumann John von and Oskar Morgenstern (1944) Theory of Games and Economic Behavior, Princeton University Press. [38] Ok Efe A. (2007) Real Analysis with Economic Applications, Princeton University Press. [39] Osborne M.J. and A. Robinstein (1994) A course in game theory, MIT Press. [40] Pigou, A. C. (1932) The economics of welfare, McMillan. [41] Pascal Bridal (2011) General Equilibrium Analysis: A Century After Walras, Routledge. [42] Rasmusen E (2006) Games and Information, Blackwell. [43] Ricardo D. (1817) Principles of Political Economy and Taxation, London, John Murray. [44] Romer D. (2006) Advanced Macroeconomic Theory, McGraw Hill. [45] Romp (1997) Game Theory: Introduction and Applications, Oxford. [46] Rubinstein Ariel (1990), Game theory in economics ,Aldershot : Elgar. 385
[47] Samuelson P. (1947) Foundation of Economic Analysis, Harvard University Press. [48] Schmalensee R. and R. Willig (1989) Handbook of industrial organization, North Holland, 1992-1994. [49] Shoven, J.B. and J.Whalley (1992) Applying General Equilibrium, Cambridge University Press, 1992. [50] Shone R (2001) Economic Dynamics, Cambridge. [51] Simon C. P. and L. Blume (1994) Mathematics for Economists, Norton. [52] Snyder C and W. Nicholson (2011) Microeconomic Theory: Basic Principles and Extensions, 11th edition, South Western. [53] Starr R M (1997) General Equilibrium Theory: An Introduction, Cambridge. [54] Takayama (1974) Mathematical Economics, Dryden Press. [55] Tirole, Jean (2006) The theory of corporate …nance, Princeton, N.J. ; Oxford : Princeton University Press [56] Tirole J. (1995) The Theory of Industrial Organisation, MIT Press. [57] Thijs ten Raa (2005)The economics of input-output analysis Cambridge : Cambridge University Press. [58] Varian H. R. (1992) Microeconomic Analysis, Norton. [59] Watt R. (2011) The Microeconomics of risk and information, Palgrave.
16
Schedule
1. Axioms; optimisations; linear and nonlinear programmes 2. Consumption 3. Production 4. Markets 5. General equilibrium and welfare 6. Game theory: bargaining and coalition 7. Game theory: principal agent problems 8. Game theory: uncertainty and insurance 9. Game theory: mechanism and auction 10. Taxation and public goods and trade
386
11. Class Test (1 hour) 12. Microeconomics for multinationals 13. Microeconomic policies (and welfare analysis)
387
16.1
Sample class test Section A
Q1. Production function for a fruit …rm operating in the competitive market is given by p y=2 l
(1755)
where y is output and l is labour input. Product price is p and input price is w. 1. Determine the cost function for this …rm. 2. What is its pro…t function? 3. Determine its supply function. 4. What is its demand function for labour? 5. Discuss properties of the production, pro…t and cost functions. . Q2. Utility function for a consumer is given by U = X 0:5 Y 0:5
(1756)
I = px X + py Y
(1757)
here budget constraint is
1. What are the Marshallian (uncompensated) demand functions for X and Y? 2. Determine the indirect utility function for this consumer. 3. Solving corresponding duality problem determine the expenditure function for this consumer. 4. Find the compensated (Hicksian) demand curve for X or Y? [hint Slutskey equation]. 5. Prove Shephard’s lemma
@E @pi
=
@L @pi
= xi (p1 ; p2 ; m) . h i @V @L 6. Prove Roy’s identity for this case @p = @pi : i Q3. Consider a two sector two class model of an economy. Workers supply labour and spend all their income in necessity goods. Capitalists do not work but own all capital and spend 60 percent of their income in luxury products, 20 percent in necessity goods and save and invest 20 percent of the remaining income.
388
Table 87: Parameters in production of the K Necessity sector 0.5 100 Luxury sector 0.5 144
two sector model A 1 1
Table 88: Parameters in consumption of the two sector model Workers Capitalist
1
2
3
1 0.2
0 0.6
0 0.2
Total labour supply is 50 and the wage rates are equal in both sectors of production LS = 50;
w1 = w2 = w
(1758)
Production function of sector i is Qi = Ai Ki i Li1
(1759)
i
Parameters of the model are given in two tables below. Capitalist hires workers and allocates labour to maximise its pro…t i
= Pi Qi
wLi
rKi = Pi Ai Ki i L1i
i
wLi
rKi
(1760)
Income of workers YL = wL1 + wL2 = w (L1 + L2 ) = 50w
(1761)
Income of capitalists (from the production function capitalist gets the same as the labour) YK = YL = 50w
(1762)
1. Determine the demand for labour for both necessity and luxury goods sectors. 2. Derive the supply and demand functions for each product. 3. Find the equilibrium relative prices Pi and w (assume necessity good as a numeraire P1 = 1). 4. What is the demand for necessity and luxury goods by workers and capitalists? 5. How much is invested in this economy? 6. What will happen to the level of income and consumption if the labour endowment increases by 10 percent due to migration or better health of the existing working age population?
389
Section B Q3. Market demand function for a certain product with two …rms (Q = q1 + q2 ) is given by P = 120
Q
(1763)
cost function of each …rm is Ci = 10qi
(1764)
Solve for optimal output, revenue, cost, pro…t, consumer, producer and total welfare under following market conditions 1. Cournot duopoly. 2. Stackelberg leadership when …rm 2 follows …rm 1. 3. Cartel. 4. Bertrand duopoly. 5. Perfect competition. 6. Provide brief explanation on …ndings. Q4. Cost function of a …rm producing a certain product under perfectly competitive market is quadratic as: C = 0:1q 2 + 10q + 50 (1765) This product sells in 20 pounds in the market. 7. What is the optimal output of this …rm? 8. What are its total revenue, total cost and pro…t at that optimal output? 9. Derive the supply function of the …rm. 10. Discuss properties of cost, pro…t and supply functions.
Q5. Consider monopoly and oligopoly models given below. Monopoly model: Pro…t function of a monopolist with taxes = PQ
TC
P =a total cost with marginal cost c and …xed cost f
390
bQ
T
(1766) (1767)
T C = cQ + f
(1768)
T = tQ
(1769)
Tax revenue
Oligopoly model There arei = 1; :::; N …rms in the market Market supply Q=
n X
qi
(1770)
i=1
Market price depends on total sales P =a
bQ = a
n X b qi
(1771)
i=1
total cost including taxes (ignore …xed cost for a while) T Ci = (c + t) qi
(1772)
T = tQ
(1773)
Tax revenue
Pro…t of a particular …rm in oligopoly is:
1
= P q1 =
a
(c + t) q1 = n X b qi i=2
!
a
n X b qi i=1
q1
bq12
!
q1
(c + t) q1
(c + t) q1 (1774)
Prove that revenue maximising tax rate is the same for both monopoly or oligopoly. Market structure does not matter for it.
16.2
Sample …nal exam
Q1. Consider consumers’problems for comparative static analysis max U = U (X; Y )
(1775)
I = Px X + Py Y
(1776)
subject to the budget constraint: where U is utility, I income, X and Y and Px and Py are are amounts and prices of X and Y commodities respectively. 1. Illustrate the …rst order conditions for consumer optimisation in this model. 391
2. By total di¤erentiation of the …rst order conditions determine (a) the impact of a change in shadow prices on the demand for X and Y: (b) the impact of a change in price of X on the demand for X and Y: (c) the impact of a change the price of Y on the demand for X and Y: (d) the impact of a change in income on the demand for X and Y and the shadow price. 3. Decompose the total e¤ect of a price change in substitution and income e¤ects. 4. Show the major di¤erences between Hicksian and Marshallian demand functions. Q2. A …rm’s objective is to minimise cost (C) C = rK + wL
(1777)
subject to a CES technology constraint: Y = [ L + (1
)K ]
1
(1778)
Here Y is outpt, K capital, L labour inputs, r interest rate, w wage rate, 0 < labour the substitution parameter.
< 1 share of
1. Determine the demand for labour and capital inputs. 2. Derive the cost function of the …rm. 3. Prove that the elasticity of substitution is
1
=
1:
4. Discuss the properties of the CES cost function. 5. Prove that the Cobb-Douglas production function is a special case of the CES production function. Q3. Consider a Dixit-Stiglitz model of monopolistic competition in which consumers maximise utility by consuming varieties of di¤erentiated products qi in addition to a unique numeraire product q0 . Their problem is: max
u = u q0 ;
subject to: q0 +
X
pi qi
X
1
qi
(1779)
I
(1780)
Producer i maximises pro…t, setting prices pi given marginal cost c and …xed cost f as: max pi
1. Prove that elasticity of demand is
= (pi 1 1
;i.e.
392
c) qi =
f @qi pi qi @pi
(1781) =
1 1
:
2. How much does each …rm produce? How does it relate to the elasticity of demand as well as the …xed cost (f ) and variable costs (c). 3. How many …rms exist in the market? Explain the role of
in it.
Q4. Consider an economy consisting of a representative household and a representative …rm. The representative household tries to maximise utility by consuming goods and services and enjoying leisure subject to its budget constraints. The producer wants to maximise pro…t by selling goods produced using the labour supplied by the household. The household maximisation problem can be stated as the follows: max U = C l(1
)
(1782)
Subject to time and budget constraints: l + hs = 1
(1783)
pc = whs +
(1784)
c > 0; hs > 0;and l > 0:Here c is consumption,l is leisure and hs is labour supply, p is the price of the commodity, w is the wage rate, is the pro…t from owning the …rm. The maximisation problem for the representative …rm can be stated as: = py
whd
(1785)
subject to the technology constraint: y = (hs )
(1786)
y > 0; hd > 0;where y is the output supplied by the …rm and hd is its demand for labour. 1. Form a Lagrangian for the constrained maximisation problem for this household. 2. Derive its demand for consumption goods and its demand for leisure. 3. Write the Lagrangian function for the …rm’s optimisation problem. 4. Derive the …rm’s demand for labour. 5. De…ne a competitive equilibrium for this economy. 6. Compute the real wage that brings goods and labour market to equilibrium. 7. What are the equilibrium quantities of c and y? 8. What are the equilibrium quantities of l and h? 9. Reformulate the problem with a sales tax and an income tax. Discuss qualitatively the economic impacts of (a) switching completely to the sales taxes or (b) to labour income taxes or to (c) a capital income tax. [56278, Continued...]
393
Q5. An economy is inhabited by 'more productive' type 1 and 'less productive' type 2 people. Policy makers encourage more productive people by assigning a greater weight to the utility of more 1 3 productive people. They aim to maximise the social welfare function: W = U14 U24 where W is the index of the social welfare, U1 represents the utility of type 1 people and U2 is the utility of type 2 people. For simplicity, assume that the resources of this economy produce a given level of output Y . It is consumed either by type 1 or by 2ptype. Market clearing condition p implies: Y = Y1 + Y2 . Preferences for type 1 are given by U1 = Y1 and for type 2 by U2 = Y2 . In a given year total output, Y , was 1000 billion pounds. 1. What is the distribution of output between type 1 and type 2 that maximises the social welfare index? What is the maximum value of the social welfare index of this economy? 2. What would have been the allocation if policy makers had given equal weight to 1 1 the utility of both types of people in the economy such as W = U12 U22 . By how much does the optimal welfare index change in this case when compared to the social welfare in (1) above? 3. How would the social welfare index change in (1) if a tax rate of 20 percent is imposed on consumption and the tax receipts are not given back to any of the consumers? What would the value of social welfare index be in this case? 3
1
4. Assume that the policy makers still hold the welfare function to be W = U14 U24 . How would the social welfare index change in (3) if all tax receipts are transferred to type 2 people? Q6. Consider a mechanism design problem in which the owner of a premium quality piece of land can enter into various arrangements with tenants to share output of the land (q). For simplicity assume that the demand for the proceeds of the land and associated costs are given by P = 30
0:5q
C = 10q
(1787)
Prove the following propositions: 1. Proposition 1: results of …xed fee (F ) contract and joint pro…t maximisation are equivalent. 2. Proposition 2: hire contract (e.g. 14 wage per unit of output) is incentive incompatible and leads to production ine¢ ciency. 3. Proposition 3: a moral hazard problem and production ine¢ ciency exist in a revenue sharing contingent contract (assume the owner gets 14 of the revenue leaving 43 of revenue to the tenant). 4. Proposition 4: a pro…t-sharing contract is e¢ cient and free of a moral hazard problem. (assume 1/3rd of pro…t goes to the tenant and 2/3rd to the landlord). Q7. Consider the cost of production (C) and production technology constraint of a …rm that produces output (y) using capital (K) and labour (L) inputs 394
C = rK + wL
(1788)
y=K L
(1789)
Terms w and r represent wage and interest rate respectively and …rm sells output at price p. 1. Write the pro…t function for this …rm and a Langrangian to maximise pro…t subject to the technology constraint. 2. Determine the optimal demand for inputs. 3. Derive the pro…t function in terms of optimal inputs , V (p; w; r): 4. Determine the cost function. @V @P
5. Prove Hotelling’s lemma K(p; w; r):
=
@L @P
= y(p; w; r); @V @w =
@L @w
=
L(p; w; r); @V @r =
@L @r
=
6. Derive input demand, output supply and pro…t functions when the technology is y = K 0:4 L0:4 Q8. Consider a moral hazard insurance model with an insurance policy fp; B0 ; B1 ; :::::; BL g where p is the insurance premium and B0 ; B1 ; :::::; BL denote the bene…ts provided by the insurance company against loss l. Normally the insurance company can observe the loss but not the level of accident avoidance e¤ort (e) of the customer. The problem of the insurance company is: max
e;p;B0 ;B1 ;:::::;BL
L X
p
subject to the participation constraint: L X
l
(e) u (w
p
l
(e) Bl
(1790)
l=0
l + Bl )
d (e)
u
(1791)
l=0
and incentive constraint for level of accidend avoidance e¤orts e; e0 2 f0; 1g ; e 6= e0 ; L X l=0
l (e) u (w
p
l + Bl )
d (e)
L X
l
(e0 ) u (w
p
l + Bl )
d (e0 )
(1792)
l=0
1. Show that it is Pareto optimal to take/provide full insurance under symmetric information when the insurance company can observe the level of e¤ort of the customer. 2. How could the insurance company design an e¢ cient contract to induce accident avoidance e¤orts by customers that would minimise the cost of the insurance company under asymmetric information? Is full insurance still optimal?
395
17 17.1
Tutorials in Advanced Microeconomics Tutorial 1:Consumers’problem
1. What are the properties of a utility function? Find demands functions for x1 and x2 solving the consumer’s optimisation problem in following: max u = x1 x2
(1793)
subject to budget constraint: 2x1 + 4x2 = a
(1794)
Prove that demand for xi is ratios of partial derivative of indirect utility function (u) to price xi and income in the following problem Show that utility e¤ect of price changes will be higher for the commodity that is heavily weighted in the consumer’s consumption basket. Prove that the indirect utillity function ful…lls following properties. Continuous Homegenous of degree zero in (p; y) Strictly increasing in y Decreasing in p Quasiconvex in p and y. Roy’s identity 3. Derive generic demand functions for consumers and examine their properties in the following problem. Consumer optimisation: max u(x)
(1795)
subject to p:x
17.2
y
(1796)
Tutorial 2: Dual of the consumer problem
Q1. Consider a consumer’s utilitymaximisation problem Cobb-Douglas function given below: M ax U = x1 x2 X;Y
+
=1
(1797)
Subject to m = p1 x1 + p2 x2 396
(1798)
1. Derive the demand functions for both x1 and x2 and associated indirect utility function. 2. Formulate the dual of this problem and derive the expenditure function. 3. Prove the Shephard Lemma that,
@E @pi
=
@L @pi
= xi (p1 ; p2 ; m):
4. Prove Roy’s identity. 5. Decompose the total price e¤ect into the compensated (Hicksian) and uncompensated (Marshallian) demand functions using the Slutskey equation. 6. Prove numerically Shephard Lemma,Roy’s identity and Slutskey equation when 0:5 and M =200.
0:5;
=
Q2. Answer all above questions for a CES utility function as given below M ax u(x; y) = [ x + (1 x;y
)y ]
1
(1799)
Subject to x + py y = m Note that the elasticity of substituion and
are linked as:
397
(1800) =1
1
; px = 1:
17.3
Tutorial 3: Dual of the producer’s problem
Q1. A …rm’s objective is to minimise cost (C) taking prices of inputs (r; w) of K and L as given: C = rK + wL
(1801)
subject to CES technology constraint as: Y = [ L + (1
)K ]
1
(1802)
1. Determine the demand for labour and capital. 2. Derive the cost function of the …rm. 3. Prove that the elasticity of substituion is
=
1
1:
4. Discuss properties of CES cost function. 5. Prove that Cobb-Douglas production function is a special case of the CES production function. Use L’Hopital’s rule. Q2. Consider a problem of producer min w1 :x1 + w2 :x2
(1803)
x1; x2;
subject to (x1 + x2 ) Show that solution is
h
1
c (w; y) = y w1 Q3. Consider pro…t ( ) function of a …rm
= py
1
rK
y
+ w2
(1804)
1
wL
i
1
(1805)
(1806)
Derive supply function and input demand function using Hotelling’s Lemma when technology y = K 0:4 L0:4 @ (p; w) y (p; w) = (1807) @p xi (w; p) =
@ (p; w) @w
398
(1808)
17.4
Tutorial 4: Markets, Price War and Stability Analysis
Q1. Market demand fuction is given by P = 30
q1
q2
(1809)
Cost function Ci = 6qi
(1810)
Pro…t: i
= P qi
Ci
(1811)
Solve this duopoly model for market price P;quantities produced (q1 ; q2 );revenue (R1 ; R2 ); cost (C1 ; C2 ) pro…t ( 1 ; 2 );consuer surplus (CS1 ; CS2 ) and welfare cost, W under following market conditions: 1. Cournot duopoly. 2. Stackelberg leader- follower model 3. Cartel 4. Perfect competition (Bertrand price competition model). Summarise all results in one table. Q2. Two rival …rms are competing for a market by engaging in a price war; for 0 < < 1 and 0< 0
(1816)
There is free entry and exit of …rms N =
(p
c)
>0
(1817)
Find the continuous dynamic time path for two prices p(t) and N (t) and examine convergence towards the steady state. Example based on Hoy et al. (2001) Mathematics for Economics, MIT Press. 399
17.5
Tutorial 5: Ricardian General Equilibrium Trade Model
Consider a two good-two country Ricardian pure exchange economy. Preferences in country 1 are expressed by its utility function in consumption of good 1 and 2 , C11 and C21 respectively: max
U 1 = C11
1
C21
1
1
(1818)
Income of country 1 is obtained from the wage income in sector 1 and sector 2 plus the transfers to country 1 I 1 = w11 L11 + w21 L12 + T R1 L11
L12
where and are labour employed in sector 1 and sector 2 wages respectively and T R1 is the transfer income. Technology constraints in sector 1 in country 1
(1819) w11
and
X11 = a11 :L11
w21
are corresponding
(1820)
Two Country Ricardian Trade Model where a11 is the productivity of labour in sector 1 in country 1. Technology constraints in sector 2 in country 1 X21 = a12 :L12
(1821)
where a12 is the productivity of labour in sector 2 in country 1. Resource constraint in country 1 de…ned by the labour endowment as: L1 = L11 + L12
(1822)
Production possibility frontier of country 1 now can be de…ned as L1 =
1 1 :X11 + 1 :X21 1 a1 a2
(1823)
Two Country Ricardian Trade Model Given above preferences the demand for good 1 in country 1 is C11 =
1
:I 1 P1
(1824)
the demand for good 2 therefore is: C21 =
1
1
:I 1
P2 Parameters of the model are given in the table below.
(1825)
Assuming price in country one as a numeraire p1 = 1 solve this model for demand C11 ; C12 C21 ; C22 of output under complete specialisation (X1 ; X2 ) ;level of employment (L1 ; L2 ) ;level of utility U; relative price of commodity two p2 . Compare these results to an autarky. Put solutions into the tables given below. Analytical solutions for trade equilibrium under specialisation 400
levels
Table 89: Parameters of the Autarky Model a1 a2 L country 1 country 2
0.4 0.6
5 2
2 5
200 400
Table 90: Comparing Specialisation and Autarky Regimes Production Autarky Trade
Consumption Autarky Trade
X1
C1
X2
X1
X2
C2
C1
C2
P
country 1 country 2
17.6
Tutorial 6: General equilibrium with production
Q1. Consider an economy consisting of a representative household and a representative …rm. A representative household tries to maximise utility by consuming goods and services and from enjoying leisure subject to his budget constraints. The producer wants to maximise pro…t by selling goods produced using the labour supplied by the household. The household maximisation problem can be stated as the following: max U = C l(1
)
(1826)
Subject to time and budget constraints: l + hs = 1
(1827)
pc = whs +
(1828)
c > 0; hs > 0;and l > 0; where c is consumption,l is leisure and hs is labour supply, p is the price of the commodity, w is the wage rate is the pro…t from owning the …rm. Maximisation problem for the representative …rm can be states as: = py
whd
(1829)
subject to technology constraint as: y = (hs ) y > 0; h > 0; where y is the output supplied by the …rm and hd is its demand for labour. d
1. Form a Lagrangian for constrained maximisation problem for this household. 2. Derive its demand for consumption goods and derive its demand for leisure. 3. Write the Lagrangian function for the …rm’s optimisation problem. 401
(1830)
Table 91: Comparing Employment and Welfere under Specialisation and Autarky Employment Autarky Trade
L1
L2
L1
Uitlity Autarky Trade
L2
U
U
4. Derive …rm’s demand for labour. 5. De…ne a competitive equilibrium for this economy. 6. Compute the real wage that brings goods and labour market in equilibrium. 7. What is the equilibrium quantity of c or y? 8. What is the equilibrium quantity of l and h? 9. Formulate the problem with sales and income tax. Discuss qualitatively the macroeconomic impacts of (a) switching completely to the sales taxes or (b) to labour income taxes or to (c) capital income tax. Q2. Equilibrium with production (x; y)=(x1; x2; :::::xm ; y1; y2; :::::ym ) for consumer i = 1; ; ; m and producer j =1,.,. n A possible allocation for consumers and producers satisfying following: Consumption set: xi 2 X Production possibility set: yi 2 Y m m n P P P Resource balance condition: xi = ei + yj i=1
i=1
Wealth of the consumer: wi (p) = p:e1 +
j=1
n P
i;j j
(p)
j=1
Supply correspondence: sj (p) = fyi 2 Yj : y 0 2 Yj =) p:y > py 0 g m n P P Excess demand correspondence: Z (p) = (di (p) e1 ) sj (p) i=1
j=1
A competitive equilibrium is a pair of prices, demand and supply (p; (x; y)) with a p vector inRl and x 2 di (p)for consumer i to m, and yj 2 sj (p)for …rms j to n and where the excess demand is zero in equilibrium. Now consider a Robinson Crusoe economy Commodity space: R2 (leisure and food) Consumer characteristic: Xi = R2+ Endowment: ei = (24; 0) Preference relation: i U (L; F n = LF p o Producer characteristics: ; Yj ( L; F ) : L 0; F L p where F = L is the production function. Solve this model for price vector, demand vector and the output vector.
402
17.7
Tutorial 7:Monopoly and monopolistic competition and taxes
1. Consider monopoly and oligopoly models given below. Monopoly model: Pro…t function of a monopolist with taxes = PQ
TC
P =a
T
(1831)
bQ
(1832)
T C = cQ + f
(1833)
T = tQ
(1834)
total cost with marginal cost c and …xed cost f
Tax revenue
Oligopoly model There are i = 1,..., N …rms in the market Market supply Q=
n X
qi
(1835)
i=1
Market price depends on total sales P =a
bQ = a
n X b qi
(1836)
i=1
total cost including taxes (ignore …xed cost for a while) T Ci = (c + t) qi
(1837)
T = tQ
(1838)
Tax revenue
Pro…t of a particular …rm in oligopoly is:
1
=
=
P q1 a
(c + t) q1 = n X b qi i=2
!
a
n X b qi i=1
q1
bq12
!
q1
(c + t) q1
(c + t) q1 (1839)
Prove that revenue maximising tax rate is the same for both monopoly or oligopoly. Market structure does not matter for it.
403
Q2. Consider a tripoly market where only three …rms supply to the market. Conjectural variation one …rm against another matter for pricing and output decisions and have impact on pro…ts. P =a
bQ = a
b (q1 + q2 + q3 )
C i = ci q i
(1840) (1841)
1
= [a
b (q1 + q2 + q3 )] q1
C1
(1842)
2
= [a
b (q1 + q2 + q3 )] q2
C2
(1843)
3
= [a
b (q1 + q2 + q3 )] q3
C3
(1844)
Solve for output and pro…t of each …rms and market price in equilibrium. Q3. Consider a tripoly market with the following demand, cost function and conjectural varia@qi =1 tion for each …rm to be one as @q j P =a
bQ = a
b (q1 + q2 + q3 )
Ci = cqi Solve for output and pro…t of each …rms and market price in equilibrium.
404
(1845) (1846)
17.8
Tutorial 8:Moral Hazard and Insurance
Q1 Honesty is the best policy in Vickery auction; truth telling is a winning strategy. Prove it. Q2 Project B earns more but is riskier than project A. Probability of success of projects A and B are given by a and b respectively. a. Illustrate how the rate of interest rate should be lower in project A than in project B in equilibrium? b. Probability of types A and B agents is given by pa and pb respectively. Prove under the asymmetric information, a lender charging a pooling interest rate is unfair to the safe borrower A and more generous to the risky borrower B. c. How can agent signal its worth? How can the lender ascertain the degree of moral hazard in B? Q3 Consider a moral hazard insurance model with an insurance policy fp; B0 ; B1 ; :::::; BL g where p is insurance premium and B0 ; B1 ; :::::; BL denote the bene…t from the insurance company against loss l. Normally the insurance company can observe the loss but not the level of accident avoidance e¤ort (e) of the consumer. The problem of the insurance company is:
max
e;p;B0 ;B1 ;:::::;BL
L X
p
subject to participation constraint L X
l
(e) u (w
p
l
(e) BL
(1847)
l=0
l + BL )
d (e)
u
(1848)
l=0
and incentive constraint L X l=0
l
(e) u (w
p
l + BL )
d (e)
L X
l
(e0 ) u (w
p
l + BL )
d (e0 )
u
(1849)
l=0
1. Show that it is Pareto optimal to do full insurance under symmetric information when the insurance company can observe the level of e¤orts of the consumer. 2. How could the insurance company design an e¢ cient contract to induce e¤orts to minimise cost under the assymetric information? Is full insurance still optimal? Q4. A monopolistic …rm engages in non-linear pricing scheme with its two types of customers f H ; L g; where H is an index of the valuation that high value customers put in its product and L is that of the low value customers. Non-linear price scheme is to set tarrifs (T ) and output (q) in such a way that maximises …rm’s pro…t by designing price scheme appropriate to these consumers. Utility function (u) of consumers in generic form is:
405
u = V (q) 0
T
(1850)
00
V (q) > 0 and V (q) < 0; T is the tarrif paid by the customer. With marginal cost c …rm’s pro…t ( ) is =T
cq
(1851)
T] > 0
(1852)
Participation constraint [ V (q)
p 1. Specialise function and parameters to V (q) = 2 q and f H ; L g = f20; 15g c = 5. Assuming the …rm knows exactly the type of the customers in this way …nd the equilibrium quantities (q) and tari¤s (T ) under the …rst best solution that high and the low value customers would purchase from this producer. What is the expected pro…t of this …rm in this …rst best solution? 2. Now assume that the …rm does not know the true type of customer but it assumes that probability of each type is 50 percent ( = 21 ). This …rm requires to design contracts considering participation and incentive constraints for low and high type consumers as: LV
(qL )
TL ] > 0
(1853)
HV
(qH )
TH ] > 0
(1854)
[ [ [
LV
(qL )
TL ] > [
[
HV
(qH )
TH ] > [
LV
(qH )
HV
(qL )
Then …rm’s objective is to maximise the expected pro…t ( e
=
(TH
cqH ) + (1
) (TL
TH ]
(1855)
TL ]
(1856)
e
) as: cqL )
(1857)
What would be the expected pro…t if the high value type customer defects to the low value type customer? Show how the …rm could reduce the size of qL to make high value customer to stick to its qH . Prove that price discrimation in this manner favours high value customer more than the low value customer.
406
17.9
Tutorial 9:Coalition, Bargaining, Signalling, Contract, Auction and Mechanism
1 Consider Four Players A,B,C,D. How many coalitions are possible? Empty core. 1. Prove that a risk averse person loses but the risk neutral person gains in the bargaining. 0:5 Suppose the utility functions of risk averse person is given by u2 = (m2 ) but the risk neutral person has a linear utility u1 = m1 . m1 + m2 = M ; u1 + u22 = 100: 2. Consider a three player (1,2,3) game in which the 3rd player always brings more to the coalition than the 1st or the 2nd player. Payo¤ for coalition of empty set: v ( ) = 0 Payo¤ from players acting alone: v (1) = 0; v (2) = 0; v (3) = 0 ; Payo¤ from alternative coalitions: v (1; 2) = 0:1; v (1; 3) = 0:2; v (2; 3) = 0:2; Payo¤ from the grand coalition: v (1; 2; 3) = 1 Power of individual i in the coalitions is measured by the di¤erence that person makes in the value of the game v (S [ fig v (S)) = 1 , where S is the subset of players excluding i, S [ fig is the subset including player i. X Compute the Shapley value of the game for each player. [hint : i = v (S)) ; n (S) v (S [ fig S2N
s!(n s 1) ] n!
4 Prove equivalence of core in games and core in general equilibrium. 5 Given the market demand and cost functions P = 24
0:5q
C = 12q
(1858)
6 Prove following four propositions regarding e¢ cient contract. Proposition 1: Results of …xed fee contract and joint pro…t maximisation are equivalent Proposition 2: Hire contract is incentive incompatible and leads to production ine¢ ciency Proposition 3: Moral hazard problem and production ine¢ ciency exists in revenue sharing contingent contract Proposition 4: Pro…t sharing contract is e¢ cient and free of moral hazard problem 7 Level of education signals quality of a worker. Spence (1973) model was among the …rst to illustrate how to analyse principal agent and role of signalling in the job market.Consider a situation where there are N individuals applying to work. In absence of education as the criteria of quality employers cannot see who is a high quality worker and who is a low quality worker. Employers know that proportion of workers is of high quality and (1- ) proportion is of bad quality. Therefore they pay each worker an average wage rate as: w = wh + (1
) wl
(1859)
more productive worker is worth 70000 and less productive worker is worth 30000 and =0.5 then the average wage rate will be 50000. Prove separtating equilibrium is more e¢ cient than the pooling equilibrium and that it is worth for high quality workers to signal their quality by the standard of their education. 407
n
(S) =
8 Consider a situation of lender that has two potential borrowers. Borrower type 1 has a high yielding project than the borrower type 2. It is however not clear to the lender, the principal, which one of the two borrowers is more productive. Faced with this situation the principal is left with two options. Easy option would be to treat both borrowers in the same way and charge the same rate of interest rate to both of them. Such pooling strategy is not e¢ cient because the type 2 borrower does not have enough incentive to put in extra e¤orts in the project. It creates disincentive to be more productive borrower. Part of the market disappears. The second more e¢ cient option is to design a contract that guarantees a separating equilibrium. For this the lender needs to design a mechanism that ful…ls participation and incentive compatibility constraints. Principal’s objective function: UP = [0:5 (R1
(B1 )) + 0:5 (R2
(B2 ))]
(1860)
Here Ri is measures the returns to principal from borrower i and Bi the bene…t to the investor i from that. 2 Utility function of agent 1: U1 = B1 (R1 ) given that or R1 = 3B1 simply B1 = R31 . 2 Utility function of agent 2: U2 = B2 (R2 ) given that orR2 = B2 . a. Formulate participation constraints for both borrowers b. formulate incentive compatible constraits c. Determine the binding constraints. d. solve the game and determine the level of utility for all players in equilibrium.
408
17.10
Tutorial 10: E¢ ciency and Social Welfare
Q1 Illustrate e¢ ciency conditions in allocations of resources a. When consumers’utility function is given by U (X; Y ) and the production possibility frontier is T (X; Y ): b. E¢ ciency of production when X = f1 (K1 ; L1 ) + f2 (K2 ; L2 )
(1861)
K1 + K2 = K
(1862)
L1 + L2 = L
(1863)
c. Prove that e¢ cient provision of public goods require that the sum of the marginal rate of substitution equals the marginal cost of provision of public good with a two consumer economy in which consumers like to maximise utility by consuming private (x) and public goods (G) max
u1 = u1 (x1 ; G)
(1864)
subject to a given level of utility for the second consumer max u2 = u2 (x2 ; G)
(1865)
x1 + x2 + c (G) = w1 + w2
(1866)
and the resource constraint
Q2 There are two people living in an economy. For simplicity assume that a …xed amount of output of 200 is produced each year. Entire in the same year. Utility of p p output is consumed individual 1 and 2 is represented by U1 = Y1 and U2 = 12 Y2 . (a) What is the utility received by each individual if the output is divided equally between these two people? What is the output received by each if it is distributed so that each of them gets the same amount of the utility? (b) What is the distribution of output that maximises the total utility for the whole economy? (c) If person 2 needs utility 5 in order to survive how should the output be distributed? 1
1
(d) Suppose that the authorities like to maximise the social welfare function W = U12 U22 , how should the output be distributed between them? Q3 An economy is inhabited by type 1 and type 2 people. The type 1 is more productive than the type 2. Policy makers encourage productive people by assigning a greater weight to the utility of more productive people. They aim to maximise the social welfare function: 3 1 W = U14 U24 where W is the index of the social welfare, U1 represents the utility of type 1 people and U2 is the utility of type 2 people. For simplicity assume that resources of this economy produce a given level of output Y. It is consumed either by 1 or by 2 type people. Market clearing condition implies: Y = Y1 + Y2 . Preferences for type 1 are given p p by U1 = Y1 and for type 2 by U2 = Y2 . In a given year total output, Y, was 1000 billion pounds. 409
(a) What is the distribution of output between type 1 and type 2 that maximises the social welfare index? What is the maximum value of the social welfare index of this economy? (b) What would have been the allocation if policy makers had given equal weight to the 1
1
utility of both types of people in the economy such as W = U12 U22 . By how much does the welfare index change in this case than compared to the social welfare in (a) above? (c) How would the social welfare index change in (a) if a tax rate of 20 percent is imposed in consumption and the tax receipts are not given back to any of these consumers? How much would the value of social welfare index be in this case? 3
1
e. Assume that the policy makers still hold the welfare function to be W = U14 U24 . How would the social welfare index change in (c ) if all tax receipts are transferred to type 1 people?
17.11
Basic Calculus
17.11.1
Four rules of di¤erentiation
Power rule
@Y = K @K
Y = K =)
1
Product rule R = P Y =) dR = Y
dP + P
dY
Quotient Rule y=
Y L =) L
dY
Y
dL
=
L2
dY L
dL Y L L
Chain Rule W = Y2+L 17.11.2
2
=)
@W =2 Y2+L @Y
2Y = 4Y Y 2 + L
Unconstrained optimisation: using Hessian determinants
Consider a pro…t maximisation problem with the Cobb-Douglas production function. = P L K1
wL
rK
)
w=0
First order conditions for pro…t maximization. L
K
= PL = (1
1
K (1
)PL K
r=0
Hessian determinants should be positive Second order derivatives: LL
=
(
1) P L
410
2
K (1
)
LK
=
K;K
=
K;L
(1
=
)PL
2
K(
(1
)PL K
(1
)PL K
)
1
Hessian determinant: LL K;L
LK K;K
=
(
1) P L 2 K (1 (1 )PL K H1 = j
H2 =
LL j
)
(1 (1
) P L 2K ( )PL K
) 1
0
for maximum and H1 = j H2 =
LL j
>0
LL
LK
K;L
K;K
='>
by B. Douglas Bernheim First published September 1st 2012
Rubinfeld, Microeconomics, Pearson Education, 2008. Analysis and creation of dream meaning. Edition 36 holt elements of literature. Microeconomics: Bernheim. 6th Edition, Addison-Wesley Publishers, 2008, ISBN-13. Required Readings Bernheim & Whinston Chapters 4-5 and 7-8. Microeconomics-Bernheim -Whinston. Microeconomics-Bernheim -Whinston.
Published December 1st 2007 by Irwin/McGraw-Hill
007290027X (ISBN13: 9780072900279)
English
3.88 (17 ratings)
Rate this book
Published January 1st 2013 by Business And Economics
B00DC90HJK
English
3.86 (7 ratings)
Rate this book
Published January 1st 2008 by MCGRAW-HILL Higher Education
0070080569 (ISBN13: 9780070080560)
0.0 (0 ratings)
Rate this book
Published 2011 by McGraw-Hill Ryerson
,
0070969272 (ISBN13: 9780070969278)
English
2.00 (1 rating)
Rate this book
Published March 5th 2013 by McGraw-Hill Education
0073375853 (ISBN13: 9780073375854)
English
0.0 (0 ratings)
Rate this book
Published February 1st 2013 by McGraw-Hill Higher Education
0071314628 (ISBN13: 9780071314626)
0.0 (0 ratings)
Rate this book
Published January 1st 2008 by Irwin/McGraw-Hill
0071287612 (ISBN13: 9780071287616)
English
0.0 (0 ratings)
Rate this book
Published by Mcgraw Hill Higher Education
0071277552
English
0.0 (0 ratings)
Rate this book
Published February 15th 2011 by McGraw-Hill Ryerson Higher Education
0071091343
English
0.0 (0 ratings)
Rate this book
Published September 28th 2012 by Irwin/McGraw-Hill
0077716310 (ISBN13: 9780077716318)
0.0 (0 ratings)
Rate this book
Published September 28th 2012 by McGraw-Hill Education
0077716329 (ISBN13: 9780077716325)
0.0 (0 ratings)
Rate this book
Published March 7th 2013 by McGraw-Hill Education
Manual For Courts Martial 2008 Edition
0077491866 (ISBN13: 9780077491864)
0.0 (0 ratings)
Rate this book
Add a new edition