Advanced Microeconomic Theory1 Guoqiang TIAN Department of Economics Texas A&M University College Station, Texas 77843 ([email protected]) August, 2002/Revised: February 2021 1This book draft is for my teaching and convenience of my students in class. Please not distribute it. ii Contents Preface I Preliminary Knowledge and Methods 3 1 Nature of Modern Economics 7 1.1 Economics and Modern Economics . 7 1.1.1 What is economics about? . 7 1.1.2 Four Basic Questions in Economics . 8 1.1.3 What is Modern Economics? . 9 1.1.4 Economics vs. Natural Science . 11 1.2 Two Categories of Economic Theory . 12 1.2.1 Benchmark Theory and Relatively Realistic Theory . 13 1.2.2 The Domain and Scientific Rigor of Economics . 18 1.2.3 The Roles of Economic Theory . 21 1.2.4 Microeconomic Theory . 22 1.3 Economics and Market System . 23 1.3.1 Market and Market Mechanism . 23 1.3.2 Three Functions of Price . 26 1.3.3 The Superiority of the Market System . 29 1.4 Government, Market, and Society . 35 1.4.1 Three Elements of State Governance and Development 35 1.4.2 Good Market Economies vs. Bad Market Economies 37 1.4.3 The Boundaries of Government-Market-Society . 40 1.5 Comprehensive Governance by the Three Arrangements . 42 1.5.1 Governance on Rules . 43 iii iv CONTENTS 1.5.2 Incentive Mechanism . 44 1.5.3 Social Norms . 45 1.5.4 The Hierarchical Structure of the Three Arrangements 47 1.6 Ancient Chinese Thoughts on the Market . 49 1.7 A Cornerstone Assumption in Economics . 55 1.7.1 Self-love, Selfishness, and Self-interest . 55 1.7.2 Practical Rationality of Self-interested Behavior . 56 1.7.3 Boundaries of Self-interest and Altruism . 57 1.8 Key Points in Economics . 59 1.8.1 Scarcity of Resources . 60 1.8.2 Information Asymmetries and Decentralization . 60 1.8.3 Economic Freedom and Voluntary Exchange . 62 1.8.4 Acting under Constraints . 63 1.8.5 Incentives and Incentive Compatibility . 65 1.8.6 Property Rights as an Incentive Scheme . 66 1.8.7 Equality of Opportunity and Equity in Outcome . 67 1.8.8 Efficient Allocation of Resources . 68 1.9 Understanding Economics Properly . 69 1.9.1 On the Scientification of Economics . 70 1.9.2 On the Mathematical Feature of Economics . 72 1.9.3 Misunderstandings on Economic Theory . 72 1.9.4 On Experiments in Economics . 75 1.10 Basic Analytical Framework of Modern Economics . 80 1.10.1 Specifying Economic Environment . 82 1.10.2 Making Behavioral Assumptions . 85 1.10.3 Setting-up Institutional Arrangements . 87 1.10.4 Determining Equilibrium . 89 1.10.5 Making Evaluations . 90 1.11 Basic Research Methodologies in Economics . 92 1.11.1 Setting up a Benchmark . 93 1.11.2 Establishing a Reference System . 93 1.11.3 Building Studying Platforms . 95 1.11.4 Developing Analytical Tools . 97 1.11.5 Constructing Rigorous Analytical Models . 97 CONTENTS v 1.11.6 Making Positive and Normative Analysis . 98 1.12 Practical Role of the Analytical Framework and Methodologies 99 1.13 Requirements for Learning Economics . 102 1.14 Distinguishing Sufficient and Necessary Conditions . 103 1.15 The Role of Mathematics and Statistics in Economics . 104 1.16 Conversion between Economic and Mathematical Languages 108 1.17 Biographies . 109 1.17.1 Adam Smith . 109 1.17.2 David Ricardo . 111 1.18 Exercises . 112 1.19 References . 118 2 Preliminary Knowledge and Methods of Mathematics 125 2.1 Basic Set Theory . 125 2.1.1 Set . 125 2.1.2 Mapping . 126 2.2 Basic Linear Algebra . 128 2.2.1 Matrix and Vector . 128 2.2.2 Matrix Operations . 129 2.2.3 Linear Dependence of Vectors . 131 2.2.4 Transpose and Inverse of Matrix . 132 2.2.5 Solving a Linear System . 133 2.2.6 Quadratic Form and Matrix . 137 2.2.7 Eigenvalues, Eigenvectors, and Traces . 139 2.3 Basic Topology . 141 2.3.1 Topological Space . 141 2.3.2 Metric Space . 142 2.3.3 Open Sets, Closed Sets, and Compact Sets . 143 2.3.4 Connectedness of Sets . 146 2.3.5 Sequences and Convergence . 147 2.3.6 Convex Set and Convexity . 148 2.4 Single-Valued Function . 149 2.4.1 Continuity of functions . 149 2.4.2 Upper Semi-continuity and Lower Semi-continuity . 151 vi CONTENTS 2.4.3 Transfer Upper and Lower Continuity . 152 2.4.4 Differentiation and Partial Differentiation of Functions 152 2.4.5 Mean Value Theorem and Taylor Expansion . 154 2.4.6 Homogeneous Functions and Euler’s Theorem . 156 2.4.7 Implicit Function Theorem . 156 2.4.8 Concave and Convex Function . 158 2.4.9 Quasi-concave and Quasi-convex Function . 159 2.4.10 Separating and Supporting Hyperplane Theorems . 163 2.5 Multi-Valued Function . 164 2.5.1 Point-to-Set Mappings . 164 2.5.2 Upper Hemi-continuous and Lower Hemi-continuous Correspondence . 166 2.5.3 Open and Closed Graphs of Correspondence . 169 2.5.4 Transfer Closed-valued Correspondence . 170 2.6 Static Optimization . 173 2.6.1 Unconstrained Optimization . 173 2.6.2 Optimization with Equality Constraints . 180 2.6.3 Optimization with Inequality Constraints . 184 2.6.4 The Envelope Theorem . 186 2.6.5 Maximum Theorems . 188 2.6.6 Continuous Selection Theorems . 190 2.6.7 Fixed Point Theorems . 191 2.6.8 Variation Inequality . 194 2.6.9 FKKM Theorems . 195 2.7 Dynamic Optimization . 197 2.7.1 Calculus of Variation . 197 2.7.2 Optimal Control . 201 2.7.3 Dynamic Programming . 204 2.8 Differential Equations . 208 2.8.1 Existence and Uniqueness Theorem of Solutions for Ordinary Differential Equations . 210 2.8.2 Some Common Ordinary Differential Equations with Explicit Solutions . 211 CONTENTS vii 2.8.3 Higher Order Linear Equations with Constant Coef- ficients . 214 2.8.4 System of Ordinary Differential Equations . 218 2.8.5 Stability of Simultaneous Differential Equations . 222 2.8.6 The Global Stability of Dynamical System . 224 2.9 Difference Equations . 225 2.9.1 First-order Difference Equations . 227 2.9.2 Second-order Difference Equation . 230 2.9.3 Difference Equations of Order n . 231 2.9.4 Stability of nth-Order Difference Equations . 232 2.9.5 Difference Equations with Constant Coefficients . 233 2.10 Basic Probability . 234 2.10.1 Probability and Conditional Probability . 234 2.10.2 Mathematical Expectation and Variance . 235 2.10.3 Continuous Distributions . 236 2.10.4 Common Probability Distributions . 237 2.11 Stochastic Dominance and Affiliation . 239 2.11.1 Order Stochastic Dominance . 239 2.11.2 Hazard Rate Dominance . 243 2.11.3 Reverse Hazard Rate Dominance . 244 2.11.4 Likelihood Ratio Dominance . 245 2.11.5 Order Statistics . 246 2.11.6 Affiliation . 247 2.12 Biographies . 250 2.12.1 Friedrich August Hayek . 250 2.12.2 Joseph Alois Schumpeter . 252 2.13 Exercises . 254 2.14 References . 267 III Game Theory and Market Theory 271 6 Non-Cooperative Game Theory 277 6.1 Introduction . 277 6.2 Basic Concepts . 277 viii CONTENTS 6.2.1 Strategic Form Representation of Games . 278 6.2.2 Extensive Form Representation of Games . 281 6.2.3 Mixed Strategies and Behavior Strategies . 287 6.3 Static Games with Complete Information . 289 6.3.1 Dominant and Dominated Strategies . 289 6.3.2 Best Response and Rationalizability . 302 6.3.3 Nash Equilibrium . 304 6.3.4 Refinements of Nash Equilibrium . 312 6.4 Dynamic Games of Complete Information . 316 6.4.1 Subgame . 318 6.4.2 Backward Induction and Subgame Perfect Nash E- quilibrium . 319 6.5 Static Games of Incomplete Information . 331 6.5.1 Bayesian Game . 332 6.5.2 Bayesian-Nash Equilibrium . 336 6.6 Dynamic Games of Incomplete Information . 344 6.6.1 Beliefs, Sequential Rationality and Bayes’ Rule . 346 6.6.2 Weak Perfect Bayesian Equilibrium . 351 6.6.3 Sequential Equilibrium . 356 6.6.4 Forward Induction . 362 6.6.5 Signaling Game . 365 6.6.6 Reasonable-Beliefs Refinements in Signaling Games . 369 6.7 Existence of Nash Equilibrium . 375 6.7.1 Existence of Nash Equilibrium in Continuous Games 375 6.7.2 Existence of Nash Equilibrium in Discontinuous Games376 6.8 Biographies . 382 6.8.1 John Forbes Nash Jr. 382 6.8.2 John C. Harsanyi . 384 6.9 Exercises . 386 6.10 References . 408 7 Repeated Games 413 7.1 Introduction . 413 7.2 Examples of Repeated Games . 416 CONTENTS ix 7.3.