Frank A. Cowell STICERD and Department of Economics London School of Economics December 2004 Contents iii List of Tables xiii List of Figures xv Preface xxiii 1 Introduction 1 1.1 The rÙle of microeconomic principles 1 1.2 Microeconomic models 2 1.2.1 Purpose 2 1.2.2 The economic actors . 2 1.2.3 Motivation 2 1.2.4 The economic environment . 3 1.2.5 Assumptions and axioms 4 1.2.6 ìTestingîa model 4 1.3 Equilibrium analysis . 5 1.3.1 Equilibrium and economic context . 5 1.3.2 The comparative statics method 5 1.3.3 Dynamics and stability . 6 1.4 Background to this book . 6 1.4.1 Economics . 6 1.4.2 Mathematics . 6 1.5 Using the book 7 1.5.1 A route map . 7 1.5.2 Some tips . 7 The Firm 9 2.1 Basic setting . 9 2.1.1 The Örm: basic ingredients . 10 2.1.2 Properties of the production function . 12 2.2 The optimisation problem 19 2.2.1 Optimisation stage 1: cost minimisation 20 2.2.2 The cost function . 23 2.2.3 Optimisation stage 2: choosing output . 25 2.2.4 Assembling the solution . 26 2.3 The Örm as a ìblack boxî 27 2.3.1 Demand and supply functions of the Örm . 29 2.3.2 Comparative statics: the general case . 31 2.4 The short run . 35 2.5 The multiproduct Örm 39 2.6 Summary . 44 2.7 Reading notes . 44 2.8 Exercises . 44 3 The Firm and the Market 49 3.1 Introduction 49 3.2 The market supply curve 50 3.3 Large numbers and the supply curve 53 3.4 Interaction amongst Örms 55 3.5 The size of the industry . 56 3.6 Price-setting 58 3.6.1 Simple monopoly . 58 3.6.2 Discriminating monopolist 60 3.6.3 Entry fee . 61 3.7 Product variety 62 3.8 Summary . 64 3.9 Reading notes . 64 3.10 Exercises . 64 4 The Consumer 69 4.1 Introduction 69 4.2 The consumerís environment 70 4.3 Revealed preference . 72 4.4 Preferences: axiomatic approach 75 4.5 Consumer optimisation: Öxed income 80 4.5.1 Cost-minimisation 82 4.5.2 Utility-maximisation . 83 4.6 Welfare 87 4.6.1 An application: price indices 92 4.7 Summary . 93 4.8 Reading notes . 93 4.9 Exercises . 94 5 The Consumer and the Market 99 5.1 Introduction 99 5.2 The market and incomes . 100 5.3 Supply by households 100 5.3.1 Labour supply 103 5.4 Household production 107 5.5 Aggregation over goods . 112 5.6 Aggregation of consumers 112 5.7 Summary . 116 5.8 Reading notes . 117 5.9 Exercises . 117 6 A Simple Economy 121 6.1 Introduction 121 6.2 Another look at production . 122 6.2.1 Processes and net outputs 122 6.2.2 The technology 124 6.2.3 The production function again . 128 6.2.4 Externalities and aggregation 128 6.3 The Robinson Crusoe economy . 129 6.4 Decentralisation and trade 132 6.5 Summary . 139 6.6 Reading notes . 139 6.7 Exercises . 139 7 General Equilibrium 143 7.1 Introduction 143 7.2 A more interesting economy . 144 7.2.1 Allocations 145 7.2.2 Incomes 147 7.2.3 An illustration: the exchange economy . 147 7.3 The logic of price-taking . 149 7.3.1 The core of the exchange economy . 151 7.3.2 Competitive equilibrium and the core: small economy 152 7.3.3 Competitive equilibrium and the core: large economy 152 7.4 The excess-demand approach 156 7.4.1 Properties of the excess demand function . 157 7.4.2 Existence . 160 7.4.3 Uniqueness 162 7.4.4 Stability 163 7.5 The rÙle of prices . 166 7.5.1 The equilibrium allocation 167 7.5.2 Decentralisation again 167 7.6 Summary . 171 7.7 Reading notes . 172 7.8 Exercises . 172 8 Uncertainty and Risk 177 8.1 Introduction 177 8.2 Consumption and uncertainty 177 8.2.1 The nature of choice . 179 8.2.2 State-space diagram . 180 8.3 A model of preferences 184 8.3.1 Key axioms 185 8.3.2 Von-Neumann-Morgenstern utility . 188 8.3.3 The ìfelicityîfunction 189 8.4 Risk aversion . 190 8.4.1 Risk premium . 191 8.4.2 Indices of risk aversion 192 8.4.3 Special cases . 195 8.5 Lotteries and preferences . 197 8.5.1 The probability space 198 8.5.2 Axiomatic approach . 199 8.6 Trade . 202 8.6.1 Contingent goods: competitive equilibrium 203 8.6.2 Financial assets 203 8.7 Individual optimisation . 205 8.7.1 The attainable set 205 8.7.2 Components of the optimum 209 8.7.3 The portfolio problem 211 8.7.4 Insurance . 217 8.8 Summary . 219 8.9 Reading notes . 220 8.10 Exercises . 220 9 Welfare 227 9.1 Introduction 227 9.2 The constitution . 228 9.3 Principles for social judgments: e¢ ciency . 234 9.3.1 Private goods and the market 237 9.3.2 Departures from e¢ ciency 242 9.3.3 Externalities . 246 9.3.4 Public goods . 250 9.3.5 Uncertainty 251 9.3.6 Extending the e¢ ciency idea 254 9.4 Principles for social judgments: equity . 257 9.4.1 Fairness 257 9.4.2 Concern for inequality 258 9.5 The social-welfare function . 258 9.5.1 Welfare, national income and expenditure . 259 9.5.2 Inequality and welfare loss 261 9.6 Summary . 264 9.7 Reading notes . 264 9.8 Exercises . 265 10 Strategic Behaviour 271 10.1 Introduction 271 10.2 Games ñbasic concepts . 272 10.2.1 Players, rules and payo§s 272 10.2.2 Information and Beliefs . 273 10.2.3 Strategy 274 10.2.4 Representing a game . 274 10.3 Equilibrium 276 10.3.1 Multiple equilibria 278 10.3.2 E¢ ciency . 279 10.3.3 Existence . 281 10.4 Application: duopoly . 285 10.4.1 Competition in quantities 286 10.4.2 Competition in prices 291 10.5 Time 293 10.5.1 Games and subgames 295 10.5.2 Equilibrium: more on concept and method 297 10.5.3 Repeated interactions 300 10.6 Application: market structure 305 10.6.1 Market leadership 305 10.6.2 Market entry . 306 10.6.3 Another look at duopoly 309 10.7 Uncertainty 310 10.7.1 A basic model 311 10.7.2 An application: entry again . 314 10.7.3 Mixed strategies again 317 10.7.4 A ìdynamicîapproach 317 10.8 Summary . 318 10.9 Reading notes . 319 10.10Exercises . 320 11 Information 327 11.1 Introduction 327 11.2 Hidden characteristics: adverse selection 328 11.2.1 Information and monopoly power 329 11.2.2 One customer type 329 11.2.3 Multiple types: Full information 333 11.2.4 Imperfect information 336 11.2.5 Adverse selection: Competition . 345 11.2.6 Application: Insurance 346 11.3 Hidden characteristics: Signalling 352 11.3.1 Costly signals . 352 11.3.2 Costless signals 360 11.4 Hidden actions 361 11.4.1 The issue . 361 11.4.2 Outline of the problem 362 11.4.3 A simpliÖed model 362 11.4.4 Principal-and-Agent: a richer model 367 11.5 Summary . 373 11.6 Reading notes . 374 11.7 Exercises . 374 2 Design 381 12.1 Introduction 381 12.2 Social choice . 382 12.3 Markets and manipulation 385 12.3.1 Markets: another look 385 12.3.2 Simple trading 386 12.3.3 Manipulation: power and misrepresentation 387 12.3.4 A design issue? 388 12.4 Mechanisms 388 12.4.1 Implementation 390 12.4.2 Direct mechanisms 391 12.4.3 The revelation principle . 392 12.5 The design problem . 393 12.6 Design: applications . 395 12.6.1 Auctions 395 12.6.2 A public project . 403 12.6.3 Contracting again 409 12.6.4 Taxation 415 12.7 Summary . 422 12.8 Reading notes . 424 12.9 Exercises . 424 13 Government and the Individual 431 13.1 Introduction 431 13.2 Market failure? 432 13.3 Nonconvexities 433 13.3.1 Large numbers and convexity 435 13.3.2 Interactions and convexity 435 13.3.3 The infrastructure problem . 436 13.3.4 Regulation 438 13.4 Externalities . 442 13.4.1 Production externalities: the e¢ ciency problem 443 13.4.2 Corrective taxes . 443 13.4.3 Production externalities: Private solutions . 444 13.4.4 Consumption externalities 447 13.4.5 Externalities: assessment 448 13.5 Public consumption . 448 13.5.1 Nonrivalness and e¢ ciency conditions . 449 13.5.2 Club goods 449 13.6 Public goods . 451 13.6.1 The issue . 451 13.6.2 Voluntary provision . 452 13.6.3 Personalised prices? . 454 13.6.4 Public goods: market failure and the design problem . 457 13.6.5 Public goods: alternative mechanisms . 458 13.7 Optimal allocations? . 460 13.7.1 Optimum with lump-sum transfers . 461 13.7.2 Second-best approaches . 464 13.8 Conclusion: Economic Prescriptions 467 13.9 Reading notes . 468 13.10Exercises . 468 Bibliography 473 A Mathematics Background 485 A.1 Introduction 485 A.2 Sets 486 A.2.1 Sets in Rn . 486 A.3 Functions . 487 A.3.1 Linear and a¢ ne functions . 487 A.3.2 Continuity . 488 A.3.3 Homogeneous functions . 490 A.3.4 Homothetic functions 491 A.4 Di§erentiation 492 A.4.1 Function of one variable . 492 A.4.2 Function of several variables . 493 A.4.3 Function-of-a-Function Rule . 493 A.4.4 The Jacobian derivative . 494 A.4.5 The Taylor expansion 494 A.4.6 Elasticities 494 A.5 Mappings and systems of equations . 496 A.5.1 Fixed-point results 496 A.5.2 Implicit functions . 497 A.6 Convexity and Concavity 499 A.6.1 Convex sets 499 A.6.2 Hyperplanes 500 A.6.3 Separation results 500 A.6.4 Convex and concave functions 503 A.6.5 quasiconcave functions 505 A.6.6 The Hessian property 507 A.7 Maximisation . 507 A.7.1 The basic technique . 508 A.7.2 Constrained maximisation 510 A.7.3 More on constrained maximisation . 511 A.7.4 Envelope theorem 514 A.7.5 A point on notation . 515 A.8 Probability 515 A.8.1 Statistics . 516 A.8.2 Bayesírule 518 A.8.3 Probability distributions: examples . 518 A.9 Reading notes . 519 B Answers to Footnote Questions 521 B.1 Introduction 521 B.2 The Örm 521 B.3 The Örm and the market . 528 B.4 The consumer . 529 B.5 The consumer and the market 535 B.6 A simple economy 539 B.7 General equilibrium . 542 B.8 Uncertainty and risk . 548 B.9 Welfare 556 B.10 Strategic behaviour 561 B.11 Information 570 B.12 Design . 583 B.13 Government and individual . 598 C Selected Proofs 607 C.1 The Örm 607 C.1.1 Marginal cost and the Lagrange multiplier . 607 C.1.2 Properties of the cost function (Theorem 2.2) . 608 C.1.3 Firmís demand and supply functions (Theorem 2.4) . 611 C.1.4 Firmís demand and supply functions (continued) . 612 C.1.5 Properties of proÖt function (Theorem 2.7) 613 C.2 The consumer . 614 C.2.1 The representation theorem (Theorem 4.1) 614 C.2.2 Existence of ordinary demand functions (Theorem 4.5) 615 C.2.3 Quasiconvexity of the indirect utility function . 615 C.3 The consumer and the market 616 C.3.1 Composite commodity (Theorem 5.1): . 616 C.3.2 The representative consumer (Theorem 5.2): . 616 C.4 A simple economy 617 C.4.1 Decentralisation (Theorem 6.2) . 617 C.5 General equilibrium . 617 C.5.1 Competitive equilibrium and the core (Theorem 7.1) . 617 C.5.2 Existence of competitive equilibrium (Theorem 7.4) . 618 C.5.3 Uniqueness of competitive equilibrium (Theorem 7.5) 619 C.5.4 Valuation in general equilibrium (Theorem 7.6) 619 C.6 Uncertainty and risk . 620 C.6.1 Risk-taking and wealth (Theorem 8.7) . 620 C.7 Welfare 620 C.7.1 Arrowís theorem (Theorem 9.1) . 620 C.7.2 Blackís theorem (Theorem 9.2) . 622 C.7.3 The support theorem (Theorem 9.5) 623 C.7.4 Potential superiority (Theorem 9.10) 625 C.8 Strategic behaviour 626 C.8.1 Nash equilibrium in pure strategies with inÖnite strategy sets (Theorem 10.2) . 626 C.8.2 Existence of Nash equilibrium (Theorem 10.1) 627 C.8.3 The Folk theorem 627 C.9 Design . 628 C.9.1 Revenue equivalence (Theorem 12.6) 628 C.9.2 The Clark-Groves mechanism (Theorem 12.7) 630 Index 632
