Lecture Notes for Microeconomics Theory ECON 630 :
Contents
1 Preliminaries on Modern Economics and Mathematics
1.1
1
Nature of Modern Economics . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.1.1 Modern E
...
Lecture Notes for Microeconomics Theory ECON 630 :
Contents
1 Preliminaries on Modern Economics and Mathematics
1.1
1
Nature of Modern Economics . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.1.1 Modern Economics and Economic Theory . . . . . . . . . . . . . .
1
1.1.2 Modern Economics and Modern Market System Governance . . . .
4
1.1.3 Modern Economics and Ancient Chinese Economic Thought . . . . 13
1.1.4 The Most Basic Assumption in Modern Economics . . . . . . . . . 17
1.1.5 Other Assumptions Usually adopted in Economics . . . . . . . . . . 19
1.1.6 A Proper Understanding of Modern Economics . . . . . . . . . . . 27
1.1.7 Basic Analytical Framework of Modern Economics . . . . . . . . . . 28
1.1.8 Basic Research Methodologies in Modern Economics . . . . . . . . 37
1.1.9 Basic Requirements for Understanding Modern Economic Theory . 43
1.1.10 Roles of Modern Economic Theory . . . . . . . . . . . . . . . . . . 44
1.1.11 Some Remarks on Modern Economic Theory . . . . . . . . . . . . . 45
1.1.12 Distinguishing Sufficient and Necessary Conditions . . . . . . . . . 50
1.1.13 The Role of Mathematics in Modern Economics . . . . . . . . . . . 50
1.1.14 Conversion between Economic and Mathematical Language . . . . . 53
1.2
Language and Methods of Mathematics . . . . . . . . . . . . . . . . . . . . 54
1.2.1 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
1.2.2 Separating Hyperplane Theorem . . . . . . . . . . . . . . . . . . . . 56
1.2.3 Concave and Convex Functions . . . . . . . . . . . . . . . . . . . . 57
1.2.4 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
1.2.5 The Envelope Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 62
1.2.6 Point-to-Set Mappings . . . . . . . . . . . . . . . . . . . . . . . . . 63
1.2.7 Continuity of a Maximum . . . . . . . . . . . . . . . . . . . . . . . 68
i1.2.8
I
Fixed Point Theorems . . . . . . . . . . . . . . . . . . . . . . . . . 68
Individual Decision Making
75
2 Consumer Theory
77
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
2.2 Consumption Set and Budget Constraint . . . . . . . . . . . . . . . . . . . 78
2.3
2.4
2.5
2.2.1 Consumption Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
2.2.2 Budget Constraint . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
Preferences and Utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
2.3.1 Preferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
2.3.2 The Utility Function . . . . . . . . . . . . . . . . . . . . . . . . . . 85
Utility Maximization and Optimal Choice . . . . . . . . . . . . . . . . . . 90
2.4.1 Consumer Behavior: Utility Maximization . . . . . . . . . . . . . . 90
2.4.2 Consumer’s Optimal Choice . . . . . . . . . . . . . . . . . . . . . . 90
2.4.3 Consumer’s First Order-Conditions . . . . . . . . . . . . . . . . . . 91
2.4.4 Sufficiency of Consumer’s First-Order Conditions . . . . . . . . . . 94
Indirect Utility, and Expenditure, and Money Metric Utility Functions . . 98
2.5.1 The Indirect Utility Function . . . . . . . . . . . . . . . . . . . . . 98
2.5.2 The Expenditure Function and Hicksian Demand . . . . . . . . . . 100
2.5.3 The Money Metric Utility Functions . . . . . . . . . . . . . . . . . 103
2.5.4 Some Important Identities . . . . . . . . . . . . . . . . . . . . . . . 105
2.6 Duality Between Direct and Indirect Utility . . . . . . . . . . . . . . . . . 109
2.7 Properties of Consumer Demand
. . . . . . . . . . . . . . . . . . . . . . . 111
2.7.1 Income Changes and Consumption Choice . . . . . . . . . . . . . . 111
2.7.2 Price Changes and Consumption Choice . . . . . . . . . . . . . . . 112
2.7.3 Income-Substitution Effect: The Slutsky Equation . . . . . . . . . . 113
2.7.4 Continuity and Differentiability of Demand Functions . . . . . . . . 116
2.7.5 Inverse Demand Functions . . . . . . . . . . . . . . . . . . . . . . . 117
2.8 The Integrability Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
2.9 Revealed Preference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
2.9.1
Axioms of Revealed Preferences . . . . . . . . . . . . . . . . . . . . 121
ii2.9.2
Characterization of Revealed Preference Maximization . . . . . . . 123
2.10 Recoverability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
2.11 Topics in Demand Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . 128
2.11.1 Endowments in the Budget Constraint . . . . . . . . . . . . . . . . 128
2.11.2 Income-Leisure Choice Model . . . . . . . . . . . . . . . . . . . . . 129
2.11.3 Homothetic Utility Functions . . . . . . . . . . . . . . . . . . . . . 129
2.11.4 Aggregating Across Goods . . . . . . . . . . . . . . . . . . . . . . . 130
2.11.5 Aggregating Across Consumers . . . . . . . . . . . . . . . . . . . . 135
3 Production Theory
142
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
3.2 Production Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
3.3
3.4
3.5
3.2.1 Measurement of Inputs and Outputs . . . . . . . . . . . . . . . . . 143
3.2.2 Specification of Technology . . . . . . . . . . . . . . . . . . . . . . . 143
3.2.3 Common Properties of Production Sets . . . . . . . . . . . . . . . . 147
3.2.4 Returns to Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
3.2.5 The Marginal Rate of Technical Substitution . . . . . . . . . . . . . 150
3.2.6 The Elasticity of Substitution . . . . . . . . . . . . . . . . . . . . . 151
Profit Maximization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
3.3.1 Producer Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
3.3.2 Producer’s Optimal Choice . . . . . . . . . . . . . . . . . . . . . . . 154
3.3.3 Producer’s First-Order Conditions . . . . . . . . . . . . . . . . . . . 155
3.3.4 Sufficiency of Producer’s First-Order Condition . . . . . . . . . . . 156
3.3.5 Properties of Net Supply Functions . . . . . . . . . . . . . . . . . . 158
3.3.6 Weak Axiom of Profit Maximization . . . . . . . . . . . . . . . . . 159
3.3.7 Recoverability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
Profit Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
3.4.1 Properties of the Profit Function . . . . . . . . . . . . . . . . . . . 163
3.4.2 Deriving Net Supply Functions from Profit Function
. . . . . . . . 164
Cost Minimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
3.5.1 First-Order Conditions of Cost Minimization . . . . . . . . . . . . . 166
3.5.2 Sufficiency of First-Order Conditions for Cost Minimization . . . . 167
iii3.6
3.7
Cost Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
3.6.1 Properties of Cost Functions . . . . . . . . . . . . . . . . . . . . . . 171
3.6.2 Properties of Conditional Input Demand . . . . . . . . . . . . . . . 172
3.6.3 Average and Marginal Costs . . . . . . . . . . . . . . . . . . . . . . 173
3.6.4 The Geometry of Costs . . . . . . . . . . . . . . . . . . . . . . . . . 175
3.6.5 Long-Run and Short-Run Cost Curves . . . . . . . . . . . . . . . . 176
Duality in Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
3.7.1 Recovering a Production Set from a Cost Function . . . . . . . . . 178
3.7.2 Characterization of Cost Functions . . . . . . . . . . . . . . . . . . 182
3.7.3 The Integrability for Cost Functions
4 Choice Under Uncertainty
186
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
4.2 Expected Utility Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
4.3
II
. . . . . . . . . . . . . . . . . 183
4.2.1 Lotteries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
4.2.2 Expected Utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
4.2.3 Uniqueness of the Expected Utility Function . . . . . . . . . . . . . 191
4.2.4 Other Notations for Expected Utility . . . . . . . . . . . . . . . . . 192
Risk aversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
4.3.1 Absolute Risk Aversion . . . . . . . . . . . . . . . . . . . . . . . . . 192
4.3.2 Global Risk Aversion . . . . . . . . . . . . . . . . . . . . . . . . . . 195
4.3.3 Relative Risk Aversion . . . . . . . . . . . . . . . . . . . . . . . . . 199
4.4 State Dependent Utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
4.5 Subjective Probability Theory . . . . . . . . . . . . . . . . . . . . . . . . . 201
Strategic Behavior and Markets
5 Game Theory
206
208
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
5.2 Description of a game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
5.2.1
5.3
Strategic Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
Solution Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
iv5.3.1 Mixed Strategies and Pure Strategies . . . . . . . . . . . . . . . . . 213
5.3.2 Nash equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
5.3.3 Dominant strategies . . . . . . . . . . . . . . . . . . . . . . . . . . 218
5.4 Repeated games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
5.5 Refinements of Nash equilibrium . . . . . . . . . . . . . . . . . . . . . . . . 221
5.6
5.5.1 Elimination of dominated strategies . . . . . . . . . . . . . . . . . . 221
5.5.2 Sequential Games and Subgame Perfect Equilibrium . . . . . . . . . 222
5.5.3 Repeated games and subgame perfection . . . . . . . . . . . . . . . 227
Games with incomplete information . . . . . . . . . . . . . . . . . . . . . . 228
5.6.1 Bayes-Nash Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . 228
5.6.2 Discussion of Bayesian-Nash equilibrium . . . . . . . . . . . . . . . 231
6 Theory of the Market
233
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
6.2 The Role of Prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
6.3 Perfect Competition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
6.4
6.3.1 Assumptions on Competitive Market . . . . . . . . . . . . . . . . . 235
6.3.2 The Competitive Firm . . . . . . . . . . . . . . . . . . . . . . . . . 235
6.3.3 The Competitive Firm’s Short-Run Supply Function . . . . . . . . 235
6.3.4 Partial Market Equilibrium . . . . . . . . . . . . . . . . . . . . . . 237
6.3.5 Competitive in the Long Run . . . . . . . . . . . . . . . . . . . . . 239
Pure Monopoly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
6.4.1 Profit Maximization Problem of Monopolist . . . . . . . . . . . . . 240
6.4.2 Inefficiency of Monopoly . . . . . . . . . . . . . . . . . . . . . . . . 242
6.4.3 Monopoly in the Long Run
. . . . . . . . . . . . . . . . . . . . . . 243
6.5 Monopolistic Competition . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
6.6 Oligopoly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
6.7
6.6.1 Cournot Oligopoly . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
6.6.2 Stackelberg Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
6.6.3 Bertrand Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
6.6.4 Collusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
Monopsony . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
vIII
General Equilibrium Theory and Social Welfare
254
7 Positive Theory of Equilibrium: Existence, Uniqueness, and Stability 256
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
7.2 The Structure of General Equilibrium Model . . . . . . . . . . . . . . . . . 258
7.3
7.4
7.2.1 Economic Environments . . . . . . . . . . . . . . . . . . . . . . . . 258
7.2.2 Institutional Arrangement: Private Market Mechanism . . . . . . . 260
7.2.3 Individual Behavior Assumptions: . . . . . . . . . . . . . . . . . . . 261
7.2.4 Competitive Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . 261
Some Examples of GE Models: Graphical Treatment . . . . . . . . . . . . 263
7.3.1 Pure Exchange Economies . . . . . . . . . . . . . . . . . . . . . . . 263
7.3.2 One-Consumer and One Producer Economy . . . . . . . . . . . . . 269
Existence of Competitive Equilibrium . . . . . . . . . . . . . . . . . . . . . 272
7.4.1 Existence of CE for Aggregate Excess Demand Functions . . . . . . 273
7.4.2 Existence of CE for Aggregate Excess Demand Correspondences . . 292
7.4.3 Existence of CE for General Production Economies . . . . . . . . . 294
7.5 Uniqueness of Competitive Equilibria . . . . . . . . . . . . . . . . . . . . . 295
7.6 Stability of Competitive Equilibrium . . . . . . . . . . . . . . . . . . . . . 299
7.7 Abstract Economy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
7.7.1 Equilibrium in Abstract Economy . . . . . . . . . . . . . . . . . . . 306
7.7.2 Existence of Equilibrium for General Preferences . . . . . . . . . . . 307
8 Normative Theory of Equilibrium: Its Welfare Properties
313
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313
8.2 Pareto Efficiency of Allocation . . . . . . . . . . . . . . . . . . . . . . . . . 314
8.3 The First Fundamental Theorem of Welfare Economics . . . . . . . . . . . 320
8.4 Calculations of Pareto Optimum by First-Order Conditions . . . . . . . . . 322
8.4.1 Exchange Economies . . . . . . . . . . . . . . . . . . . . . . . . . . 323
8.4.2 Production Economies . . . . . . . . . . . . . . . . . . . . . . . . . 324
8.5 The Second Fundamental Theorem of Welfare Economics . . . . . . . . . . 325
8.6 Non-Convex Production Technologies and Marginal Cost Pricing . . . . . . 331
8.7 Pareto Optimality and Social Welfare Maximization . . . . . . . . . . . . . 334
vi8.8
8.7.1 Social Welfare Maximization for Exchange Economies . . . . . . . . 335
8.7.2 Welfare Maximization in Production Economy . . . . . . . . . . . . 336
Political Overtones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338
9 Economic Core, Fair Allocations, and Social Choice Theory
341
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341
9.2 The Core of Exchange Economies . . . . . . . . . . . . . . . . . . . . . . . 342
9.3 Fairness of Allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
9.4 Social Choice Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
9.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
9.4.2 Basic Settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
9.4.3 Arrow’s Impossibility Theorem . . . . . . . . . . . . . . . . . . . . 355
9.4.4 Some Positive Result: Restricted Domain . . . . . . . . . . . . . . . 358
9.4.5 Gibbard-Satterthwaite Impossibility Theorem . . . . . . . . . . . . 361
10 General Equilibrium Under Uncertainty
367
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367
10.2 A Market Economy with Contingent Commodities . . . . . . . . . . . . . . 368
10.3 Arrow-Debreu Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . 372
10.4 Sequential Trade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374
10.5 Incomplete Markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380
IV
Externalities and Public Goods
11 Externalities
385
388
11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388
11.2 Consumption Externalities . . . . . . . . . . . . . . . . . . . . . . . . . . . 389
11.3 Production Externality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396
11.4 Solutions to Externalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398
11.4.1 Pigovian Tax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399
11.4.2 Coase’ Voluntary Negotiation Property Rights Approach . . . . . . 400
11.4.3 Missing Market . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410
11.4.4 The Compensation Mechanism . . . . . . . . . . . . . . . . . . . . 411
vii12 Public Goods
416
12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416
12.2 Notations and Basic Settings . . . . . . . . . . . . . . . . . . . . . . . . . . 416
12.3 Discrete Public Goods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418
12.3.1 Efficient Provision of Public Goods . . . . . . . . . . . . . . . . . . 418
12.3.2 Free-Rider Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 419
12.3.3 Voting for a Discrete Public Good . . . . . . . . . . . . . . . . . . . 420
12.4 Continuous Public Goods
. . . . . . . . . . . . . . . . . . . . . . . . . . . 421
12.4.1 Efficient Provision of Public Goods . . . . . . . . . . . . . . . . . . 421
12.4.2 Lindahl Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . 423
12.4.3 Free-Rider Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 427
V
Information, Incentives, Mechanism Design, and Contract
Theory
431
13 Optimal Mechanism Design: Contracts with One-Agent and Hidden
Information
436
13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436
13.2 Basic Settings of Principal-Agent Model with Adverse Selection . . . . . . 438
13.2.1 Economic Environment (Technology, Preferences, and Information) 438
13.2.2 Contracting Variables: Outcomes . . . . . . . . . . . . . . . . . . . 438
13.2.3 Timing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438
13.3 The Complete Information Optimal Contract(Benchmark Case) . . . . . . 439
13.3.1 First-Best Production Levels . . . . . . . . . . . . . . . . . . . . . . 439
13.3.2 Implementation of the First-Best . . . . . . . . . . . . . . . . . . . 440
13.3.3 A Graphical Representation of the Complete Information Optimal
Contract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441
13.4 Incentive Feasible Contracts . . . . . . . . . . . . . . . . . . . . . . . . . . 442
13.4.1 Incentive Compatibility and Participation . . . . . . . . . . . . . . 442
13.4.2 Special Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443
13.4.3 Monotonicity Constraints . . . . . . . . . . . . . . . . . . . . . . . 444
13.5 Information Rents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444
viii13.6 The Optimization Program of the Principal . . . . . . . . . . . . . . . . . 445
13.7 The Rent Extraction-Efficiency Trade-Off . . . . . . . . . . . . . . . . . . . 446
13.7.1 The Optimal Contract Under Asymmetric Information . . . . . . . 446
13.7.2 A Graphical Representation of the Second-Best Outcome . . . . . . 449
13.7.3 Shutdown Policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449
13.8 The Theory of the Firm Under Asymmetric Information . . . . . . . . . . 451
13.9 Asymmetric Information and Marginal Cost Pricing . . . . . . . . . . . . . 451
13.10The Revelation Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452
13.11A More General Utility Function for the Agent . . . . . . . . . . . . . . . . 454
13.11.1 The Optimal Contract . . . . . . . . . . . . . . . . . . . . . . . . . 454
13.11.2 More than One Good . . . . . . . . . . . . . . . . . . . . . . . . . . 456
13.12Ex Ante versus Ex Post Participation Constraints . . . . . . . . . . . . . . 457
13.12.1 Risk Neutrality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457
13.12.2 Risk Aversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459
13.13Commitment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464
13.13.1 Renegotiating a Contract . . . . . . . . . . . . . . . . . . . . . . . . 464
13.13.2 Reneging on a Contract . . . . . . . . . . . . . . . . . . . . . . . . 465
13.14Informative Signals to Improve Contracting . . . . . . . . . . . . . . . . . 465
13.14.1 Ex Post Verifiable Signal . . . . . . . . . . . . . . . . . . . . . . . . 465
13.14.2 Ex Ante Nonverifiable Signal . . . . . . . . . . . . . . . . . . . . . . 467
13.15Contract Theory at Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 467
13.15.1 Regulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468
13.15.2 Nonlinear Pricing by a Monopoly . . . . . . . . . . . . . . . . . . . 468
13.15.3 Quality and Price Discrimination . . . . . . . . . . . . . . . . . . . 469
13.15.4 Financial Contracts . . . . . . . . . . . . . . . . . . . . . . . . . . . 470
13.15.5 Labor Contracts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471
13.16Challenges to “No Distortion at the Top” Rule . . . . . . . . . . . . . . . . 472
13.16.1 Network Externalities . . . . . . . . . . . . . . . . . . . . . . . . . . 472
13.16.2 Countervailing Incentives . . . . . . . . . . . . . . . . . . . . . . . . 476
13.17The Optimal Contract with a Continuum of Types . . . . . . . . . . . . . 488
13.18Further Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492
ix14 Optimal Mechanism Design: Contracts with One-Agent and Hidden
Action
496
14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496
14.2 Basic Settings of Principal-Agent Model with Moral Hazard . . . . . . . . 497
14.2.1 Effort and Production . . . . . . . . . . . . . . . . . . . . . . . . . 497
14.2.2 Incentive Feasible Contracts . . . . . . . . . . . . . . . . . . . . . . 498
14.2.3 The Complete Information Optimal Contract . . . . . . . . . . . . 499
14.3 Risk Neutrality and First-Best Implementation . . . . . . . . . . . . . . . . 500
14.4 The Trade-Off Between Limited Liability Rent Extraction and Efficiency . 502
14.5 The Trade-Off Between Insurance and Efficiency . . . . . . . . . . . . . . . 504
14.5.1 Optimal Transfers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504
14.5.2 The Optimal Second-Best Effort . . . . . . . . . . . . . . . . . . . . 506
14.6 More than Two Levels of Performance . . . . . . . . . . . . . . . . . . . . 507
14.6.1 Limited Liability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507
14.6.2 Risk Aversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509
14.7 Contract Theory at Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 510
14.7.1 Efficiency Wage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510
14.7.2 Sharecropping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511
14.7.3 Wholesale Contracts . . . . . . . . . . . . . . . . . . . . . . . . . . 513
14.7.4 Financial Contracts . . . . . . . . . . . . . . . . . . . . . . . . . . . 513
14.8 A Continuum of Performances . . . . . . . . . . . . . . . . . . . . . . . . . 515
14.9 A Mixed Model of Moral Hazard and Adverse Selection . . . . . . . . . . . 516
14.9.1 Optimal Wage Contract with Unobservable Efforts Only . . . . . . 517
14.9.2 Optimal Wage Contract with Unobservable Efforts and Risk Aversion520
14.9.3 Optimal Wage Contract with Unobservable Efforts and Cost . . . . 525
14.10Further Extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528
15 General Mechanism Design: Contracts with Multi-Agents
531
15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531
15.2 Basic Settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533
15.2.1 Economic Environments . . . . . . . . . . . . . . . . . . . . . . . . 533
15.2.2 Social Goal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534
x15.2.3 Economic Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . 535
15.2.4 Solution Concept of Self-Interested Behavior . . . . . . . . . . . . . 536
15.2.5 Implementation and Incentive Compatibility . . . . . . . . . . . . . 537
15.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539
15.4 Dominant Strategy and Truthful Revelation Mechanisms . . . . . . . . . . 540
15.5 Gibbard-Satterthwaite Impossibility Theorem . . . . . . . . . . . . . . . . 544
15.6 Hurwicz Impossibility Theorem . . . . . . . . . . . . . . . . . . . . . . . . 545
15.7 Vickrey-Clark-Groves Mechanisms . . . . . . . . . . . . . . . . . . . . . . . 548
15.7.1 Vickrey-Clark-Groves Mechanisms for Discrete Public Good . . . . 549
15.7.2 Vickrey-Clark-Groves Mechanisms with Continuous Public Goods . 552
15.7.3 Uniqueness of VCG for Efficient Decision . . . . . . . . . . . . . . . 557
15.7.4 Balanced VCG Mechanisms . . . . . . . . . . . . . . . . . . . . . . 558
15.8 Nash Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 560
15.8.1 Nash Equilibrium and General Mechanism Design . . . . . . . . . . 560
15.8.2 Characterization of Nash Implementation . . . . . . . . . . . . . . . 561
15.9 Better Mechanism Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 568
15.9.1 Groves-Ledyard Mechanism . . . . . . . . . . . . . . . . . . . . . . 568
15.9.2 Walker’s Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . 570
15.9.3 Tian’s Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572
15.10Incomplete Information and Bayesian-Nash Implementation . . . . . . . . 575
15.10.1 Bayesian-Nash Implementation Problem . . . . . . . . . . . . . . . 575
15.10.2 Ex-Post Efficient Implementation . . . . . . . . . . . . . . . . . . . 578
15.10.3 Participation Constraints . . . . . . . . . . . . . . . . . . . . . . . . 583
15.10.4 The Revenue Equivalence Theorem in Auctions . . . . . . . . . . . 588
15.10.5 Correlated Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592
15.10.6 Ex Post Implementation . . . . . . . . . . . . . . . . . . . . . . . . 595
15.10.7 Ex post Implementation with Common Values . . . . . . . . . . . . 596
15.11Informational Efficiency in Mechanism Design . . . . . . . . . . . . . . . . 596
15.11.1 Framework of Informational Mechanism . . . . . . . . . . . . . . . 598
15.11.2 Informational Efficiency and Uniqueness of Competitive Market
Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 600
xi16 Dynamic Mechanism Design
615
16.1 Dynamic Contracts with Commitment . . . . . . . . . . . . . . . . . . . . 615
16.1.1 Dynamic P-A Model with Constant Type . . . . . . . . . . . . . . . 616
16.1.2 Dynamic P-A Model with Markov Types . . . . . . . . . . . . . . . 617
16.2 Dynamic Contracts without Full Commitment . . . . . . . . . . . . . . . . 619
16.2.1 Basic Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 619
16.2.2 Full Analysis of A Simple Model . . . . . . . . . . . . . . . . . . . . 622
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