Mathematical Economics

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ISBN: 9780521314985 | 0521314984
Cover: Paperback
Copyright: 8/30/1985

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This book provides a systematic exposition of mathematical economics, presenting and surveying existing theories and showing ways in which they can be extended. One of its strongest features is that it emphasises the unifying structure of economic theory in such a way as to provide the reader with the technical tools and methodological approaches necessary for undertaking original research. The author offers explanations and discussion at an accessible and intuitive level providing illustrative examples. He begins the work at an elementary level and progessively takes the reader to the frontier of current research. This second edition brings the reader fully up to date with recent research in the field.

Preface To Second Edition

iii

Preface To First Edition

Introduction

Scope of the Book

Outline of the Book

xviii

SOME FREQUENTLY USED NOTATIONS

(718)

Preliminaries

(54)

Mathematical Preliminaries

(34)

Some Basic Concepts and Notations

(4)

Rn and Linear Space

(5)

Basis and Linear Functions

(6)

Convex Sets

(3)

A Little Topology

(16)

Separation Theorems

(10)

Activity Analysis and the General Production Set

(10)

Developments of Nonlinear Programming

(110)

Introduction

(7)

Concave Programming--Saddle-Point Characterization

(13)

Differentiation and the Unconstrained Maximum Problem

(11)

Differentiation

(7)

Unconstrained Maximum

(4)

The Quasi-Saddle-Point Characterization

(22)

Appendix to Section D: A Further Note on the Arrow-Hurwicz-Uzawa Theorem

106

(6)

Some Extensions

112

(21)

Quasi-Concave Programming

113

(3)

The Vector Maximum Problem

116

(5)

Quadratic Forms, Hessians, and Second-Order Conditions

121

(12)

Applications, Envelope Theorem, Duality, and Related Topics

133

(22)

Some Applications

133

(4)

The Envelope Theorem

137

(4)

Elements of Micro Theory

141

(3)

Elasticities of Factor Substitution, Duality, and Translog Estimation

144

(11)

Linear Programming and Classical Optimization

155

(14)

Linear Programming

156

(3)

The Classical Theory of Optimization

159

(2)

Comparative Statics

161

(1)

The Second-Order Conditions and Comparative Statics

162

(1)

An Example: Hicks-Slutsky Equation

163

(6)

The Theory of Competitive Markets

169

(126)

Introduction

169

(6)

Consumption Set and Preference Ordering

175

(10)

Consumption Set

175

(1)

Quasi-Ordering and Preference Ordering

176

(3)

Utility Function

179

(2)

The Convexity of Preference Ordering

181

(4)

The Two Classical Propositions of Welfare Economics

185

(49)

Appendix to Section C: Introduction to the Theory of the Core

204

(1)

Introduction

204

(3)

Some Basic Concepts

207

(6)

Theorems of Debreu and Scarf

213

(5)

Some Illustrations

218

(6)

Some Remarks

224

(10)

Demand Theory

234

(21)

Appendix to Section D: Various Concepts of Semicontinuity and the Maximum Theorem

249

(1)

Various Concepts of Semicontinuity

249

(4)

The Maximum Theorem

253

(2)

The Existence of Competitive Equilibrium

255

(30)

Historical Background

255

(10)

McKenzie's Proof

265

(15)

Appendix to Section E: On the Uniqueness of Competitive Equilibrium

280

(5)

Programming, Pareto Optimum, and the Existence of Competitive Equilibria

285

(10)

The Stability of Competitive Equilibrium

295

(64)

Introduction

295

(7)

Elements of the Theory of Differential Equations

302

(11)

The Stability of Competitive Equilibrium---The Historical Background

313

(8)

A Proof of Global Stability for the Three-Commodity Case (with Gross Substitutability)---An Illustration of the Phase Diagram Technique

321

(4)

A Proof of Global Stability with Gross Substitutability---The n-commodity Case

325

(6)

Some Remarks

331

(8)

An Example of Gross Substitutability

331

(2)

Scarf's Counterexample

333

(2)

Consistency of Various Assumptions

335

(1)

Nonnegative Prices

336

(3)

The Tatonnement and the Non-Tatonnement Processes

339

(8)

The Behavioral Background and the Tatonnement Process

340

(1)

The Tatonnement and the Non-Tatonnement Processes

341

(6)

Liapunov's Second Method

347

(12)

Frobenius Theorems, Dominant Diagonal Matrices, and Applications

359

(51)

Introduction

359

(8)

Frobenius Theorems

367

(13)

Dominant Diagonal Matrices

380

(11)

Some Applications

391

(19)

Summary of Results

391

(3)

Input-Output Analysis

394

(2)

The Expenditure Lag Input-Output Analysis

396

(1)

Multicountry Income Flows

397

(1)

A Simple Dynamic Leontief Model

398

(1)

Stability of Competitive Equilibrium

399

(4)

Comparative Statics

403

(7)

The Calculus of Variations and the Optimal Growth of an Aggregate Economy

410

(76)

Elements of the Calculus of Variations and its Applications

410

(9)

Statement of the Problem

410

(3)

Euler's Equation

413

(2)

Solutions of Illustrative Problems

415

(4)

Spaces of Functions and the Calculus of Variations

419

(13)

Introduction

419

(2)

Spaces of Functions and Optimization

421

(5)

Euler's Condition and a Sufficiency Theorem

426

(6)

A Digression: The Neo-Classical Aggregate Growth Model

432

(12)

The Structure of the Optimal Growth Problem for an Aggregate Economy

444

(42)

Introduction

444

(6)

The Case of a Constant Capital: Output Ratio

450

(9)

Nonlinear Production Function with Infinite Time Horizon

459

(9)

Appendix to Section D: A Discrete Time Model of One-Sector Optimal Growth and Sensitivity Analysis

468

(1)

Introduction

468

(2)

Model

470

(4)

The Optimal Attainable Paths

474

(6)

Sensitivity Analysis: Brock's Theorem

480

(6)

Multisector Models of Economic Growth

486

(73)

The von Neumann Model

486

(17)

Introduction

486

(5)

Major Theorems

491

(6)

Two Remarks

497

(6)

The Dynamic Leontief Model

503

(56)

Introduction

503

(4)

The Output System

507

(10)

The Price System

517

(5)

Inequalities and Optimization Model (Solow)

522

(5)

Morishima's Model of the Dynamic Leontief System

527

(14)

Appendix to Section B: Some Problems in the Dynamic Leontief Model---The One-Industry Illustration

541

(18)

Multisector Optimal Growth Models

559

(41)

Turnpike Theorems

559

(16)

Introduction

559

(2)

The Basic Model and Optimality

561

(2)

Free Disposability and the Condition for Optimality

563

(4)

The Radner Turnpike Theorem

567

(8)

Multisector Optimal Growth with Consumption

575

(25)

Introduction

575

(2)

The Model

577

(3)

Finite Horizon: Optimality and Competitiveness

580

(3)

Optimal Stationary Program

583

(4)

O. S. P. and Eligibility

587

(7)

Optimal Program for an Infinite Horizon Problem

594

(6)

Developments of Optimal Control Theory and its Applications

600

(121)

Pontryagin's Maximum Principle

600

(27)

Optimal Control: A Simple Problem and the Maximum Principle

600

(6)

The Proof of a Simple Case

606

(3)

Various Cases

609

(8)

An Illustrative Problem: The Optimal Growth Problem

617

(10)

Some Applications

627

(19)

Regional Allocation of Investment

627

(11)

Optimal Growth with a Linear Objective Function

638

(8)

Further Developments in Optimal Control Theory

646

(21)

Constraint

646

(5)

Hestenes' Theorem

651

(9)

A Sufficiency Theorem

660

(7)

Two Illustrations: The Constraint and the Use of the Control Parameter

667

(18)

Optimal Growth Once Again

667

(4)

Two Peak-Load Problems

671

(14)

The Neo-Classical Theory of Investment and Adjustment Costs---An Application of Optimal Control Theory

685

(36)

Introduction

685

(3)

The Case of No Adjustment Costs

688

(9)

The Case with Adjustment Costs

697

(6)

Some Remarks

703

(18)

Indexes

721

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Mathematical Economics

Summary

Table of Contents

Supplemental Materials