ISBN-10:
9812837183
ISBN-13:
9789812837189
Pub. Date:
05/21/2010
Publisher:
World Scientific Publishing Company, Incorporated
Fixed Points And Economic Equilibria

Fixed Points And Economic Equilibria

by Ken UraiKen Urai

Hardcover

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Overview

This book presents a systematic approach to problems in economic equilibrium based on fixed-point arguments and rigorous set-theoretical (axiomatic) methods. It describes the highest-level research on the classical theme, fixed points and economic equilibria, in the theory of mathematical economics, and also presents basic results in this area, especially in the general equilibrium theory and non-co-operative game theory. The arguments also contain distinguishable developments of the main theme in the homology theory for general topological spaces, in the model theory and mathematical logic, and in the methodology and philosophy of social sciences. It can thus serve as a graduate-level textbook on mathematical economics as well as an advanced monograph for students and researchers who are concerned about rigorous mathematical treatment in the social sciences.

Product Details

ISBN-13: 9789812837189
Publisher: World Scientific Publishing Company, Incorporated
Publication date: 05/21/2010
Series: Series On Mathematical Economics And Game Theory , #5
Pages: 312
Product dimensions: 6.00(w) x 9.00(h) x 0.75(d)

Table of Contents

Preface vii

1 Introduction 1

1.1 Mathematics is Language 1

1.2 Notes on Some Mathematical Tools in This Book 2

1.3 Basic Mathematical Concepts and Definitions 12

2 Fixed-Point Theorems 27

2.1 Classical Results and Basic Extensions 27

2.2 Convexity and Duality for General Spaces 39

2.3 Extension of Classical Results to General Spaces 46

3 Nash Equilibrium and Abstract Economy 57

3.1 Multi-Agent Product Settings for Games 57

3.2 Nash Equilibrium 67

3.3 Abstract Economy 75

4 Gale-Nikaido-Debreu's Theorem 87

4.1 Gale-Nikaido-Debreu's Theorem 87

4.2 Market Equilibria in General Vector Spaces 91

4.3 Demand-Supply Coincidence in General Spaces 99

5 General Economic Equilibrium 101

5.1 General Preferences and Basic Existence Theorems 101

5.2 Pareto Optimal Allocations 109

5.3 Existence of General Equilibrium 111

6 The Cech Type Homology Theory and Fixed Points 125

6.1 Basic Concepts in Algebraic Topology 125

6.2 Vietoris-Begle Mapping and Local Connectedness 142

6.3 Nikaido's Analogue of Sperner's Lemma 150

6.4 eilenberg-Montgomery's Theorem 164

7 Convex Structure and Fixed-Point Index 167

7.1 Lefschetz's Fixed-Point Theorem and Its Extensions 167

7.2 Cohomology Theory for General Spaces 175

7.3 Dual-System Structure and Differentiability 177

7.4 Linear Approximation for Isolated Fixed Points 177

7.5 Indices for Compact Set of Fixed Points 180

8 Applications to Related Topics 183

8.1 KKM, KKMS, and Core Existence 183

8.2 Eaves' Theorem 187

8.3 Fan-Browder's Coincidence Theorem 189

8.4 L-majorized Mappings 190

8.5 Variational Inequality Problem 191

8.6 Equilibrium with Cooperative Concepts 192

8.7 System of Inequalities and Affine Transformations 195

9 Mathematics and Social Science 201

9.1 Basic Concepts in Axiomatic Set Theory 201

9.2 Individuals and Rationality 218

9.3 Society and Values 232

10 Concluding Discussions 243

10.1 Fixed Points and Economic Equilibria 243

10.2 Rationality and Fixed-Point Views of the World 250

Mathematical Appendix I 257

Mathematical Appendix II 263

Mathematical Appendix III 273

References 277

Index 285

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