Mathematical Optimization and Economic Theory

by Intriligator, Michael D.

• ISBN13: 9780898715118
• ISBN10: 0898715113
• Format: Paperback
• Copyright: 2002-03-01
• Publisher: Society for Industrial & Applied

Summary

Mathematical Optimization and Economic Theory provides a self-contained introduction to and survey of mathematical programming and control techniques and their applications to static and dynamic problems in economics, respectively. It is distinctive in showing the unity of the various approaches to solving problems of constrained optimization that all stem back directly or indirectly to the method of Lagrange multipliers. In the 30 years since its initial publication, there have been many more applications of these mathematical techniques in economics, as well as some advances in the mathematics of programming and control. Nevertheless, the basic techniques remain the same today as when the book was originally published. Thus, it continues to be useful not only to its original audience of advanced undergraduate and graduate students in economics, but also to mathematicians and other researchers who are interested in learning about the applications of the mathematics of optimization to economics.

Author Biography

Michael D. Intriligator is Professor of Economics at the University of California, Los Angeles (UCLA), where he is also Professor of Political Science, Professor of Policy Studies, Director of the Burkle Center for International Relations, and Co-Director of the Jacob Marschak Interdisciplinary Colloquium on Mathematics in the Behavioral Sciences

Table of Contents

Preface to the Classics Edition	xv
Preface	xvii
PART ONE INTRODUCTION	1	(6)
Economizing and the Economy	2	(5)
The Economizing Problem	2	(1)
Institutions of the Economy	3
Economics	1	(6)
PART TWO STATIC OPTIMIZATION	7	(134)
The Mathematical Programming Problem	8	(12)
Formal Statement of the Problem	8	(4)
Types of Maxima, the Weierstrass Theorem, and the Local-Global Theorem	12	(4)
Geometry of the Problem	16	(4)
Classical Programming	20	(24)
The Unconstrained Case	22	(6)
The Method of Lagrange Multipliers	28	(8)
The Interpretation of the Lagrange Multipliers	36	(8)
Problems	38	(6)
Nonlinear Programming	44	(28)
The Case of No Inequality Constraints	46	(3)
The Kuhn-Tucker Conditions	49	(7)
The Kuhn-Tucker Theorem	56	(4)
The Interpretation of the Lagrange Multipliers	60	(2)
Solution Algorithms	62	(10)
Problems	64	(8)
Linear Programming	72	(34)
The Dual Problems of Linear Programming	77	(2)
The Lagrangian Approach Existence, Duality and Complementary Slackness Theorems	79	(7)
The Interpretation of the Dual	86	(3)
The Simplex Algorithm	89	(17)
Problems	96	(10)
Game Theory	106	(35)
Classification and Description of Games	107	(3)
Two-person, Zero-sum Games	110	(10)
Two-person Nonzero-sum Games	120	(3)
Cooperative Games	123	(7)
Games With Infinitely Many Players	130	(11)
Problems	131	(10)
PART THREE APPLICATIONS OF STATIC OPTIMIZATION	141	(150)
Theory of the Household	142	(36)
Commodity Space	142	(1)
The Preference Relation	143	(5)
The Neoclassical Problem of the Household	148	(6)
Comparative Statics of the Household	154	(9)
Revealed Preference	163	(3)
von Neumann-Morgenstern Utility	166	(12)
Problems	169	(9)
Theory of the Firm	178	(42)
The Production Function	178	(11)
The Neoclassical Theory of the Firm	189	(7)
Comparative Statics of the Firm	196	(5)
Imperfect Competition: Monopoly and Monopsony	201	(4)
Competition Among the Few: Oligopoly and Oligopsony	205	(15)
Problems	213	(7)
General Equilibrium	220	(38)
The Classical Approach: Counting Equations and Unknowns	221	(6)
The Input-Output Linear Programming Approach	227	(11)
The Neoclassical Excess Demand Approach	238	(3)
Stability of Equilibrium	241	(5)
The von Neumann Model of an Expanding Economy	246	(12)
Problems	249	(9)
Welfare Economics	258	(33)
The Geometry of the Problem in the 2 X T X T Case	259	(10)
Competitive Equilibrium and Pareto Optimality	269	(9)
Market Failure	278	(1)
Optimality Over Time	279	(12)
Problems	282	(9)
PART FOUR DYNAMIC OPTIMIZATION	291	(106)
The Control Problem	292	(14)
Formal Statement of the Problem	293	(5)
Some Special Cases	298	(1)
Types of Control	299	(3)
The Control Problem as One of Programming in an Infinite Dimensional Space the Generalized Weierstrass Theorem	302	(4)
Calculus of Variations	306	(20)
Euler Equation	308	(4)
Necessary Conditions	312	(3)
Transversality Condition	315	(2)
Constraints	317	(9)
Problems	320	(6)
Dynamic Programming	326	(18)
The Principle of Optimality and Bellman's Equation	327	(3)
Dynamic Programming and the Calculus of Variations	330	(3)
Dynamic Programming Solution of Multistage Optimization Problems	333	(11)
Problems	338	(6)
Maximum Principle	344	(26)
Costate Variables, the Hamillonion, and the Maximum Principle	345	(6)
The Interpretation of the Calculus of Variables	351	(2)
The Maximum Principle and the Calculus of Variations	353	(2)
The Maximum Principle and Dynamic Programming	355	(2)
Examples	357	(13)
Problems	362	(8)
Differential Games	370	(27)
Two-Person Deterministic Continuous Differential Games	371	(2)
Two-Person Zero-Sum Differential Games	373	(4)
Pursuit Games	377	(6)
Coordination Differential Games	383	(4)
Noncooperative Differential Games	387	(10)
Problems	388	(9)
PART FIVE APPLICATIONS OF DYNAMIC OPTIMIZATION	397	(52)
Optimal Economic Growth	398	(51)
The Neoclassical Growth Model	399	(6)
Neoclassical Optimal Economic Growth	405	(11)
The Two Sector Growth Model	416	(14)
Heterogeneous Capitol Goods	430	(19)
Problems	435	(14)
APPENDICES	449	(52)
Appendix A Analysis	450	(26)
A.1 Sets	450	(2)
A.2 Relations and Functions	452	(2)
A.3 Metric Spaces	454	(3)
A.4 Vector Spans	457	(3)
A.5 Convex Sets and Functions	460	(5)
A.6 Differential Calculus	465	(2)
A.7 Differential Equations	467	(9)
Appendix B Matrices	476	(25)
B.1 Basic Definitions and Examples	476	(2)
B.2 Some Special Matrices	478	(1)
B.3 Matrix Relations and Operations	479	(5)
B.4 Scalar Valued Functions Defined on Matrices	484	(3)
B.5 Inverse Matrix	487	(1)
B.6 Linear Equations and Linear Inequalities	488	(5)
B.7 Linear Transformations; Characteristic Roots and Vectors	493	(2)
B.8 Quadratic Forms	495	(2)
B.9 Matrix Derivatives	497	(4)
Index	501