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•.........----------. introduction to 1 sets and l ! \ mappings in modern ! 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