Mathematical Methods and Economic Theory

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1 Mathematical Methods and Economic Theory Anjan Mukherji Subrata Guha C OXTORD UNIVERSITY PRESS

2 Contents Preface SECTION I 1 Introduction The Objective The Tools for Section I 4 2 Basic Mathematical Logic Introduction Sentential Logic Sentences, truth values, and notations Truth rules and truth tables Tautologies, contradictions, and contingent sentences Logical consequence and the validity of arguments Logical consistency and independence Predicate Logic Universe of discourse, universal and existential sentences Individual constants, variables, quantifiers, and predicates Well-formed formulas, scope of a quantifier, bound and free variables Truth rules of predicate logic Using multiple quantifiers 25 3 Set Theory Operations with Sets Binary Relations 32

3 vl CONTENTS 3.3 Even and Odd Integers Real Numbers Infimum and Supremum Functions Countable Sets Open and Closed Sets Compactness 41 4 Functions of a Single Variable Limits Continuity Uniform continuity Existence of extrema Differentiability Approximations Integration Introduction Functions of bounded variation Basic properties of the integral Integration by parts The Riemann-Stieltjes integral as a Riemann integral The Riemann-Stieltjes integral as a finite Sum The integral as a function Improper integrals 66 5 Economic Applications I: Choice, Utility, and Aggregation Introduction Possibility of Choosing the 'Best' The Construction of a Continuous Utility Indicator Function Arrow's Theorem Notation and definitions A lemma The theorem 75 Further Readings for Section I 76

4 CONTENTS vii SECTION II 6 Introduction The Objective of Section II 79 7 Real Linear Algebra Preliminaries: Vector Spaces, Sub-spaces, Linear Dependence, Rank of a Sub-space, Matrices Solution to Equations and Inequalities Determinants Characteristic Roots and Vectors Quadratic Forms Dominant Diagonal Matrices Non-negative square matrices Stable matrices Functions of Several Variables Differentiability Some Special Functions Maps and Fixed Points Separation Theorems Static Optimization * Unconstrained Optimization Constrained Optimization Equality Constraints Inequality Constraints A Duality Theorem Economic Applications II: Demand and Supply Static Optimization I The Hicks-Allen Theory Producer Behaviour Cost and profit functions Market Equilibria The excess demand function The Existence Theorem and the Fixed Point Theorem 139

5 viii CONTENTS 10.5 Non-competitive Market Equilibria Perfect Competition Monopoly and Monopsony Bilateral Monopoly Social Welfare Maximization Efficiency and Competitive Equilibria Decision-making under Alternative Scenarios Introduction, Decision-making under Uncertainty Lotteries ' Ranking over lotteries The expected utility function Risk Aversion Preliminaries Measures of risk aversion Risk aversion and choice of risky assets Global measures of risk aversion Portfolio choice with more than one risky asset Interactive Decision-making Introduction Games in normal form Refinements of Nash equilibria Bayesian-Nash equilibria Repeated games 174 Further Readings for Section II 176 SECTION in 12 Introduction Dynamical Systems Continuous Time Processes Introduction Solutions to some standard forms Definitions and propositions The linear case 190

6 CONTENTS ix Motion on the plane Lotka-Volterra system of equations Discrete Processes Preliminary definitions Stability of Periodic Points The logistic map 203 «14 Dynamic Optimization Introduction to the Optimal Control Theory A Basic Optimal Control Problem Necessary Conditions Some special assumptions A maximization condition A differential equation The backward value function The Maximum Principle for the Basic Problem The maximum principle for problem (A) Sufficient Conditions for an Optimal Control The Mangasarian sufficiency conditions for problem (A) The Arrow sufficiency conditions for problem (A) Variants of the Basic Problem Alternative conditions on the terminal state Addition of a salvage value function An important note Variable terminal time Inequality constraints with control variables Infinite Horizon Problems Definition of an optimal control Necessary conditions for optimality Sufficient conditions for optimality Infinite Horizon Problem: An Alternative Aproach The value function and the Bellman equation The existence of the value function Some properties of bounded continuous functions on X Restrictions on T and application of the Contraction Mapping Theorem 253

7 X CONTENTS Differentiability of the value function and the Euler equation Economic Applications III: Economic Dynamics Introduction The Stability of Competitive Equilibrium Gross substitutes and the weak axiom of revealed preference Scarf example Discrete price adjustment Bifurcation and complex dynamics in a discrete tatonnement * Optimal Economic Growth The outlines of the model Solution to the optimal control problem for the household The solution using Arrow-type sufficiency conditions ' Aggregate dynamics in the model The Social Planner's Problem 292 Further Readings for Section III 295 References 297 Index 302

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