Mathematical Methods and Economic Theory Anjan Mukherji Subrata Guha C 263944 OXTORD UNIVERSITY PRESS
Contents Preface SECTION I 1 Introduction 3 1.1 The Objective 3 1.2 The Tools for Section I 4 2 Basic Mathematical Logic 5 2.1 Introduction 5 2.2 Sentential Logic 7 2.2.1 Sentences, truth values, and notations 7 2.2.2 Truth rules and truth tables 11 2.2.3 Tautologies, contradictions, and contingent sentences 13 2.2.4 Logical consequence and the validity of arguments 15 2.2.5 Logical consistency and independence 19 2.3 Predicate Logic 20 2.3.1 Universe of discourse, universal and existential sentences 20 2.3.2 Individual constants, variables, quantifiers, and predicates 21 2.3.3 Well-formed formulas, scope of a quantifier, bound and free variables 23 2.3.4 Truth rules of predicate logic 24 2.3.5 Using multiple quantifiers 25 3 Set Theory 31 3.1 Operations with Sets 31 3.2 Binary Relations 32
vl CONTENTS 3.3 Even and Odd Integers 33 3.4 Real Numbers 33 3.5 Infimum and Supremum 35 3.6 Functions 35 3.7 Countable Sets 36 3.8 Open and Closed Sets 38 3.9 Compactness 41 4 Functions of a Single Variable 46 4.1 Limits 46 4.2 Continuity 47 4.2.1 Uniform continuity 49 4.2.2 Existence of extrema 50 4.3 Differentiability 51 4.3.1 Approximations 53 4.4 Integration 55 4.4.1 Introduction 56 4.4.2 Functions of bounded variation 56 4.4.3 Basic properties of the integral 57 4.4.4 Integration by parts 58 4.4.5 The Riemann-Stieltjes integral as a Riemann integral 59 4.4.6 The Riemann-Stieltjes integral as a finite Sum 60 4.4.7 The integral as a function 63 4.4.8 Improper integrals 66 5 Economic Applications I: Choice, Utility, and Aggregation 69 5.1 Introduction 69 5.2 Possibility of Choosing the 'Best' 69 5.3 The Construction of a Continuous Utility Indicator Function 71 5.4 Arrow's Theorem 73 5.4.1 Notation and definitions 74 5.4.2 A lemma 75 5.4.3 The theorem 75 Further Readings for Section I 76
CONTENTS vii SECTION II 6 Introduction 79 6.1 The Objective of Section II 79 7 Real Linear Algebra 80 7.1 Preliminaries: Vector Spaces, Sub-spaces, Linear Dependence, Rank of a Sub-space, Matrices 80 7.2 Solution to Equations and Inequalities 85 7.3 Determinants 89 7.4 Characteristic Roots and Vectors 91 7.5 Quadratic Forms 92 7.6 Dominant Diagonal Matrices 94 7.6.1 Non-negative square matrices 97 7.6.2 Stable matrices. 100 8 Functions of Several Variables 103 8.1 Differentiability 103 8.2 Some Special Functions 107 8.3 Maps and Fixed Points 108 8.4 Separation Theorems 109 9 Static Optimization * 112 9.1 Unconstrained Optimization 112 9.2 Constrained Optimization 113 9.3 Equality Constraints 113 9.4 Inequality Constraints 115 9.5 A Duality Theorem 117 10 Economic Applications II: Demand and Supply 122 10.1 Static Optimization I 122 10.2 The Hicks-Allen Theory 127 10.3 Producer Behaviour 130 10.3.1 Cost and profit functions 131 10.4 Market Equilibria 137 10.4.1 The excess demand function 137 10.4.2 The Existence Theorem and the Fixed Point Theorem 139
viii CONTENTS 10.5 Non-competitive Market Equilibria 141 10.6 Perfect Competition 142 10.7 Monopoly and Monopsony 142 10.8 Bilateral Monopoly 142 10.9 Social Welfare Maximization 144 10.10 Efficiency and Competitive Equilibria 144 11 Decision-making under Alternative Scenarios 149 11.1 Introduction, 149 11.2 Decision-making under Uncertainty 149 11.2.1 Lotteries ' 149 11.2.2 Ranking over lotteries 150 11.2.3 The expected utility function 151 11.3 Risk Aversion 154 11.3.1 Preliminaries 154 11.3.2 Measures of risk aversion 154 11.3.3 Risk aversion and choice of risky assets 155 11.3.4 Global measures of risk aversion 160 11.3.5 Portfolio choice with more than one risky asset 162 11.4 Interactive Decision-making 163 11.4.1 Introduction 163 11.4.2 Games in normal form 164 11.4.3 Refinements of Nash equilibria 169 11.4.4 Bayesian-Nash equilibria 173 11.4.5 Repeated games 174 Further Readings for Section II 176 SECTION in 12 Introduction 181 13 Dynamical Systems 182 13.1 Continuous Time Processes 182 13.1.1 Introduction 182 13.1.2 Solutions to some standard forms 183 13.1.3 Definitions and propositions 187 13.1.4 The linear case 190
CONTENTS ix 13.1.5 Motion on the plane 192 13.1.6 Lotka-Volterra system of equations 197 13.2 Discrete Processes 200 13.2.1 Preliminary definitions 200 13.3 Stability of Periodic Points 201 13.3.1 The logistic map 203 «14 Dynamic Optimization 207 14.1 Introduction to the Optimal Control Theory 207 14.2 A Basic Optimal Control Problem 209 14.3 Necessary Conditions 211 14.3.1 Some special assumptions 211 14.3.2 A maximization condition 212 14.3.3 A differential equation 216 14.3.4 The backward value function 219 14.4 The Maximum Principle for the Basic Problem 221 14.4.1 The maximum principle for problem (A) 221 14.5 Sufficient Conditions for an Optimal Control 223 14.5.1 The Mangasarian sufficiency conditions for problem (A) 223 14.5.2 The Arrow sufficiency conditions for problem (A) 224 14.6 Variants of the Basic Problem 225 14.6.1 Alternative conditions on the terminal state 225 14.6.2 Addition of a salvage value function 227 14.6.3 An important note 229 14.6.4 Variable terminal time 229 14.6.5 Inequality constraints with control variables 232 14.7 Infinite Horizon Problems 237 14.7.1 Definition of an optimal control 238 14.7.2 Necessary conditions for optimality 240 14.7.3 Sufficient conditions for optimality 242 14.8 Infinite Horizon Problem: An Alternative Aproach 243 14.8.1 The value function and the Bellman equation 245 14.8.2 The existence of the value function 250 14.8.3 Some properties of bounded continuous functions on X 250 14.8.4 Restrictions on T and application of the Contraction Mapping Theorem 253
X CONTENTS 14.8.5 Differentiability of the value function and the Euler equation 256 15 Economic Applications III: Economic Dynamics 260 15.1 Introduction. 260 15.2 The Stability of Competitive Equilibrium 260 15.2.1 Gross substitutes and the weak axiom of revealed preference 263 15.2.2 Scarf example 267 15.2.3 Discrete price adjustment 271 15.2.4 Bifurcation and complex dynamics in a discrete tatonnement * 273 15.3 Optimal Economic Growth 277 15.3.1 The outlines of the model 277 15.3.2 Solution to the optimal control problem for the household 282 15.3.3 The solution using Arrow-type sufficiency conditions ' 285 15.3.4 Aggregate dynamics in the model 287 15.4 The Social Planner's Problem 292 Further Readings for Section III 295 References 297 Index 302