Table Of Content•.........----------.
introduction to
1
sets and
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\ mappings in
modern
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economil!ll cs
north-holland /american elsevier
Introduction to Sets and Mappings
in Modern Economics
Introduction to Sets
and
Mappin~s
in Modern Economics
Hukukane Nikaido
Hitotsubashi University, Tokyo, Japan
translated by
Kazuo Sato
United Nations, New York, USA
NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM• OXFORD
AMERICAN ELSEVIER PUBLISHING COMPANY, INC. - NEW YORK
Copyright© 1970 by Hukukane Nikaido
All Rights Reserved. No part of this book may be reproduced, stored in a retrieval system,
or transmitted, in any form or by any means, electronic, mechanical, photocopying,
recording or otherwise without written permission.
This book was originally published in the Japanese language as:
GENDAI KEIZAIGAKU NO SUGAKlITEKI HOHO
By Hukukane Nikaido
Published by Iwanami Shoten, Publishers, Tokyo, 1960
Copyright© 1960 by Hukukane Nikaido
Published by:
North-Holland Publishing Company - Amsterdam
North-Holland Publishing Company, Ltd. - Oxford
Sole distributors for the U.S.A. and-Canada:
American Elsevier Publishing Company, Inc.
52 Vanderbilt Avenue
New York, N.Y. 10017
ISBN North-Holland 7204 3043 7
ISBN American Elsevier 0 444 10038 5
Library of Congress Catalog Card Number 69-18376
First edition 1970
Second printing 1972
Third printing 1975
Printed in The Netherlands
PREFACE
This book is written as an introduction to the modern mathematical
approach that is gaining popularity in modern economics, specially directed
to students who wish to become familiar with the approach. At the same
time, it may also serve as an introduction to elementary topology and its
applications. Topology is a field occupying a central position in modern
mathematics; it has come to be recognized as a very efficient tool for solving
various problems that are posed in economics.
The only mathematical prerequisite for the reader of this book is the
college junior's level of differential and integral calculus and the basic theory
of determinants. Though complex numbers appear in a few places in Chap
ter 3 (viz. § 16, 17 and 21 ), they do not require more than a very elementary
level of mathematics. As for economic theory, no special prior knowledge is
presupposed; whenever necessary, an exposition is given in a form under
standable even to the reader specialized in sciences or engineering.
The book proceeds from a mathematical introduction to such problems
that can be handled with this preparation. This is repeated by giving another
mathematical introduction followed by more advanced problems. Thus, the
exposition of basic mathematics and the economic analysis of relevant
problems are closely intertwined; this process will gradually elevate the
academic standard.
Needless to say, it is impossible in a small volume like this to attempt an
exhaustive coverage of problems in modern mathematical economics. There
fore, the book highlights a small number of basic problems and treats them in
a detailed manner. The author believes that this is a more effective way of
revealing the essentials of the mathematical approach.
The book is arranged as follows: in the first part under the heading of
vi PREFACE
modern static analysis, it discusses the inter-industry analysis, linear pro
gramming, the modern analysis of maximum and minimum problems, the
activity analysis of production, game theory and saddle point problems, the
von Neumann model, and the analysis of consumer behavior. In the second
part, it analyzes the existence problem of an equilibrium solution in a
Walrasian model. Sprinkled in between, three chapters are devoted to
mathematical expositions.
The author has tried to keep expositions elementary. The book, however,
does not intend to expound economic doctrines as such. Rather, it aims to
show, in a form as palatable as possible to the reader, how economic
problems are mathematically studied. In this connection, a special effort has
been made to keep the expositions easy and at the same time not to lose
mathematical rigor.
The author will be more than happy if this book turns out to be a good
study companion not only to economists but also to those students in other
disciplines like humanities and sciences who are interested in learning
mathematical methods in social sciences.
It should be acknowledged that the second part of this book is based on an
earlier booklet written jointly with Professor Yasui in the Iwanami series of
Modern Applied Mathematics. The author is indebted to Professor Yasui for
his kind permission to incorporate its substance in the present book.
Finally, the author wishes to express his hearty gratitude to Professors
Shokichi Iyanaga and Takuma Yasui for their generous assistance and
encouragement in many respects from suggesting the publication to reading
the manuscript.
Winter, 1959 Hukukane Nikaido
PREF ACE TO THE ENGLISH EDITION
This book takes a principal aim at introducing the reader to concepts and
theorems pertaining to sets and mappings and illustrating, in terms of a few
important typical topics, how fruitfully they are applied to the solution of
economic problems. In order to achieve this objective, emphasis is placed on a
detailed exposition of a very limited number of selected subject matters at
the cost of abandoning a broader coverage of material.
Fortunately, since its original publication almost a decade ago, this book
seems to have been of some use as a study companion to Japanese students
wishing to get acquainted with modern mathematical methods in economic
theory. I think an introduction of this kind to modern mathematical methods
to be equally helpful to non-Japanese-speaking students and make it worth
while removing the language barrier lying between them and the original
edition of this book. To my satisfaction, this undertaking is now accom
plished by the present translation by Dr. Sato. I hope that English speaking
readers will find this book useful as their study companion.
Although I myself find many flaws in this book by reviewing it now some
years after the original writing, the book as a whole is almost a living
organism to me, which would lose vigor if a drastic revision is made. I
therefore desired that the original text remain essentially intact in the present
English edition. The translation is based on a slightly revised version of the
original edition, prepared by myself by re-examining its entire pages,
correcting slips and rewording certain passages on a few pages, to the extent
which leaves the essential content and characteristic of the original edition
intact. I am extremely happy to note that Dr. Sato's translation transmits the
spirit of the original book vividly. I wish to express my most cordial thanks to
Dr. Sato for his excellent translation and invaluable co-operation.
vii
viii PREFACE TO THE ENGLISH EDITION
The publication of this English edition has only been made possible by the
courtesy of the publisher of the original Japanese edition, lwanami Shoten,
Publishers, Tokyo, to whom many thanks are due. I am also very grateful to
the North-Holland Publishing Company for undertaking the publication of
this English edition of my book.
January, 1970 Hukukane Nikaido
