Advanced Mathematical Economics

This concise textbook presents students with all they need for advancing in mathematical economics. Higher level undergrads as well as postgrad students in mathematical economics will find this book to be extremely useful in the course of their development as economists.

Preface

viii

Things to know

1

(12)

Sets

1

(1)

The space we work in

1

(1)

Facts from real analysis

2

(4)

Facts from linear algebra

6

(3)

Facts from graph theory

9

(4)

Feasibility

13

(20)

Fundamental theorem of linear algebra

13

(2)

Linear inequalities

15

(1)

Non-negative solutions

15

(4)

The general case

19

(1)

Application: arbitrage

20

(4)

Application: co-operative games

24

(1)

Application: auctions

25

(8)

Convex sets

33

(20)

Separating hyperplane theorem

34

(6)

Polyhedrons and polytopes

40

(6)

Dimension of a set

46

(1)

Properties of convex sets

47

(2)

Application: linear production model

49

(4)

Linear programming

53

(34)

Basic solutions

56

(4)

Duality

60

(4)

Writing down the dual

64

(1)

Interpreting the dual

64

(4)

Marginal value theorem

68

(1)

Application: zero-sum games

69

(3)

Application: Afriat's theorem

72

(3)

Integer programming

75

(3)

Application: efficient assignment

78

(2)

Application: Arrow's theorem

80

(7)

Non-linear programming

87

(30)

Necessary conditions for local optimality

89

(6)

Sufficient conditions for optimality

95

(5)

Envelope theorem

100

(3)

An aside on utility functions

103

(2)

Application: market games

105

(4)

Application: principal--agent problem

109

(8)

Fixed points

117

(25)

Banach fixed point theorem

117

(1)

Brouwer fixed point theorem

118

(7)

Application: Nash equilibrium

125

(2)

Application: equilibrium in exchange economies

127

(6)

Application: Hex

133

(3)

Kakutani's fixed point theorem

136

(6)

Lattices and supermodularity

142

(21)

Abstract lattices

149

(2)

Application: supermodular games

151

(1)

Application: transportation problem

152

(2)

Application: efficient assignment and the core

154

(5)

Application: stable matchings

159

(4)

Matroids

163

(28)

Introduction

163

(2)

Matroid optimization

165

(1)

Rank functions

166

(3)

Deletion and contraction

169

(1)

Matroid intersection and partitioning

170

(5)

Polymatroids

175

(4)

Application: efficient allocation with indivisibilities

179

(7)

Application: Shannon switching game

186

(5)

Index

191

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Advanced Mathematical Economics

Summary

Table of Contents

Supplemental Materials