Mathematical Economics

758 Pages·1985·33.14 MB·English

by  Akira Takayama

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AKITRAAK AYAMA
This book provides a systematic exposition of mathematical economic .... pre­
v.
senting and surveying existing theories and showing way.., in hich the� can
be extended. A strong feature is its emphasis on the unifying structure of eco­
nomic theory in a way that provides the reader with the technical tool.., and
methodological approaches necessary for undertaking original research. The
author offers explanations and discussions at an accessible and intuitive le\el,
providing illustrative examples. He begins the work at an elementary level and
progressively takes the reader to the frontier of current research.
In this second edition, Professor Takayama has made major revisiom to the
presentation of nonlinear programming and worked through the book to improve
the presentation of the central mathematical tools of the economist.
Professor Akira Takayama is currently professor of economics at Kyoto Uni­
versity and Vandeveer professor of economics at Southern Illinois University
at Carbondale. He received his PhD from the University of Rochester in 1982
and his Doctorate of Economics from Hitotsubashi University in Tokyo in 196-l.
International Trade: An Approach to the Theory
He is the author of ( 1972).
Praise for the first edition:
"An excellent book, which is capable of filling these gaps in knowledge in an
Mathematical Eco110111ic.L
exceptional way, is Takayama's monumental work
This is not only by far the most comprehensive book currently in the field. It
is also one of the most lucid and clear-sighted, as well as one of the most en­
joyable to read."
K\'J..lo.\
"This author has undertaken a difficult and useful task and has succeeded in
covering a broad range of topics in depth. He has brought together in a unified
manner a large number of contributions scattered in the literature. pointing out
common features of the topics discussed and drawing attention to earlier con­
.
tributions in economic analysis and in the relevant mathematical techniques ..
/:'co11omin1
MATHEMATICAL
ECONOMICS
2ndE dition
AKIRTAA KAYAMA
Southern Illinois University and Kyoto University
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Cambridge University Press
Cambridge
New York New Rochelle Melbourne Sydney
Tom yl atpea rents
Published by the Press Syndicate of the University of Cambridge
The Pitt Building, Trumpington Street, Cambridge CB2 I RP
32 East 57th Street, New York, NY 10022, USA
10 Stamford Road, Oakleigh, Melbourne 3 I 66, Australia
© Cambridge University Press 1985
First published 1985
Reprinted 1986 (twice), 1987
Printed in the United States of America
Library of Congress Cataloging in Publication Data
Takayama, Akira, 1932-
Mathematical economics.
Includes bibliographies.
1. Economics, Mathematical. I. Title.
HB135.T34 1985 330'.01'51 84-23766
ISBN O 521 25707 7 hard covers
ISBN O 521 31498 4 paperback
The first edition of Mathematical Economics was published
by The Dryden Press in 197 4.
Prefatcoe
thes econedd ition
In the first edition of this book, I set out for myself the task of presenting a
systematic treatment of mathematical economics which would emphasize the uni­
fying structure of economic theory and the mathematical methods involved in
modern economic theory. It was my hope that students in economics would
recognize the importance of the analytical approach to economics and become
familiar with basic and powerful (and yet accessible to students with minimal pre­
requisites) mathematical tools. Economics is the amalgam of "poetry" and pre­
cise logic (or analytics) as well as a wide knowledge of facts. Although these arc
all important, as I wrote at the outset of the first edition, "this book chooses to
discuss the analytical and mathematical approach."
I have been pleased with the positive reception of the first edition. Numer­
ous letters and verbal communications as well as various book reviews have been
Kyklm,
most gratifying and useful in shaping the present edition. The comments and cri­
Economica, Economic Studi£'s
ticisms contained in the book reviews by Professors C. Dcisscnbcrg (in
Quarterly,
1975), D. Glycopantis (in 1976), and Y. Murata (in
Bulletin of the American Mathematical Society,
1977) were particularly heartening. A remark on the boo!- by Profc,­
sor L. J. Billera (in 1980), though
passing, was encouraging. It was also gratifying, as a native Japanese, that the
first edition received the 1975 Best Books of the Year Award in Fconomics and
Management ("Nikkei-sho") from the Japan Economic Research Center.
I am grateful for this opportunity to prepare the new edition. Among other
things, it enables me to correct some typographical errors and to irnprcn c the
exposition in a number of places. Furthermore, the new cdi1io11 gi,·es me an
opportunity to write some new material towards the end of Chapl<;r 1, which
intends to provide an updated, streamlined exposition 011 the \\ay in ,,hich 1hc
constrained maximization technique is to be applied to economic problerm, �cn­
sitivity analysis, the envelope theorem, duality, and translog estimation. It i,
iii
iv PREFACE TO THE SECOND EDITION
hoped that the reader can, in this relatively brief section, understand the power of
constrained maximization in economics in a proper way, and realize that the
analytics of basic micro theory are really simple and straightforward.
Recent developments in mathematical economics are important and enor­
mous, and some of these require a considerable amount of mathematical prepara­
tion by the interested economist. In this book, I have decided not to attempt to
exposit all such developments, since there are some other sources that the reader
Hand­
can be referred to. For example, a (rather massive) three volume series,
book of Mathematical Economics,
edited by K. J. Arrow and M. D. Intriligator
(North-Holland), contains many excellent chapters on such recent developments.
On the other hand, I also believe that the present book should provide the core
of the analytical approach to economics which is essential to most modern-day
economists.
I have received many valuable comments and suggestions on the first edition
from my friends, colleagues, and students, which were useful in preparing the
present edition. In particular, I would like to mention John Z. Drabicki, Yasuhiro
Sakai, and Richard K. Anderson.
A.T.
November, 1984
Kyoto, Japan
Prefatcoe
thef iresdti tion
This book is intended to provide a systematic treatment of mathe­
matical economics, a field that has progressed enormously in recent decades.
It discusses existing theories in the field and attempts to extend them. The
coverage herein is much broader than in any other book currently used in
the field.
The literature on mathematical economics is enormous. The tradi­
tional method of education in economics-that of assigning many books
and articles to be read by the student-is clearly inappropriate for the study
of mathematical economics. This is both because of the size and complexity
of the field and because the traditional method fails to make the student
aware of the importance of the analytical character of economic theory.
Here an attempt is made to provide all of the material usually obtained from
a multitude of different sources but within a single framework, using con­
sistent terminology, and requiring a minimum of outside reading.
More than a mere survey of the literature, this book strongly empha­
sizes both the unifying structure of economic theory and the mathematical
methods involved in modern economic theory with the intention of provid­
ing the reader both the technical tools and the methodological approach
necessary for doing original research in the field.
Furthermore, the book is not an exposition of elementary calculus and
matrix theory with applications to economic problems: rather it is a
book on economic problems using mathematical tools to aid in the analysis.
Nor is it an introduction to a higher level text. It begins at a rather elementary
level and brings the reader right to the frontiers of current research. C!rc is
also taken so that each chapter can be read more or less independently (that
is, each chapter can be read without careful reading of other chapters).
Needless to say, economics is concerned with real world problems,
V
viii PREFACE
P. Manes, and Jay W. Wiley of the Krannert School of Industrial Admin­
istration of Purdue University, who have provided me with generous en­
couragement as well as unusually favorable research conditions. Finally,
my wife, Machiko, greatly helped me in preparing the indexes of the book,
as well as providing me with encouragement.
A. T.
December, 1973
West Lafayette, Indiana
Contents
PREFACE TO SECOND EDITION
iii
PREFACE TO FIRST EDITION V
INTRODUCTION
A. Scope of the Book xv
B. Outline of the Book xviii
SOME FREQUENTLY USED NOTATIONS 3
CHAPTER O PRELIMINARIES 5
A. Mathematical Preliminaries 5
a. Some Basic Concepts and Notations 5
b. Rn and Linear Space 9
c. Basis and Linear Functions 14
d. Convex Sets 20
e. A Little Topology 23
B. Separation Theorems 39
C. Activity Analysis and the General Production Set 49
CHAPTER 1 DEVELOPMENTS OF NONLINEAR PROGRAMMING 59
A. Introduction 59
B. Concave Programming-Saddle-Point Characterization 66
C. Differentiation and the Unconstrained Maximum Problem 79
a. Differentiation 79
b. Unconstrained Maximum 86
D. The Quasi-Saddle-Point Characterization 90
Appendix to Section D: A Further Note on the Arrow-Hurwicz-Uzawa
Theorem 106
ix
CO�TENTS
X
E. Some Extensions 112
a. Quasi-Concave Programming 113
b. The Vector Maximum Problem 116
c. Quadratic Forms, Hessians, and Second-Order Conditions 121
F. Applications, Envelope Theorem, Duality, and Related Topics 133
a. Some Applications 133
b. The Envelope Theorem 137
c. Elements of Micro Theory 141
d. Elasticities of Factor Substitution, Duality, and
Translog Estimation 144
G. Linear Programming and Classical Optimization 155
a. Linear Programming 156
b. The Classical Theory of Optimization 159
c. Comparative Statics 161
d. The Second-Order Conditions and Comparative Statics 162
e. An Example: Hicks-Slutsky Equation 163
CHAPTER 2 THE THEORY OF COMPETITIVE MARKETS 169
A. Introduction 169
B. Consumption Set and Preference Ordering 175
a. Consumption Set 175
b. Quasi-Ordering and Preference Ordering 176
c. Utility Function 179
d. The Convexity of Preference Ordering 181
C. The Two Classical Propositions of Welfare Economics 185
Appendix to Section C: Introduction to the Theory of the Core 204
a. Introduction 204
b. Some Basic Concepts 207
c. Theorems of Debreu and Scarf 213
d. Some Illustrations 218
e. Some Remarks 224
D. Demand Theory 234
Appendix to Section D: Various Concepts of Semicontinuity and the
Maximum Theorem 249
a. Various Concepts of Semicontinuity 249
b. The Maximum Theorem 253
E. The Existence of Competitive Equilibrium 255
a. Historical Background 255
b. McKenzie's Proof 265
Appendix to Section E: On the Uniqueness of Competitive
Equilibrium 280