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MATHEMATICAL ECONOMICS

Allen

Allen

Atiyah

Blaug

Some Other ELBS Low-priced Editions

MACRO-EcONOMIC THEORY

MATHEMATICAL ANALYSIS FOR ECONOMISTS

THE SALE OF GOODS

ECONOMICS OF EDUCATION, VOL. I

Macmillan Macmillan

Bridger and FAMINE IN RETREAT?

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Pitman

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Drover and SHELDON'S PRACTICE AND LAW OF BANKING

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Hanson

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Hicks

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MANAGERIAL ECONOMICS Longman

A DICTIONARY OF ECONOMICS AND COMMERCE Macdonald &1

MONETARY THEORY AND PRACTICE

A TEXTBOOK OF ECONOMICS

Evans Macdonald &1 Evans M acdonald &1 Evans

VALUE AND CAPITAL Oxford University Press

AN INTRODUCTION TO POSITIVE ECONOMICS

PRINCIPLES OF ECONOMICS

THE THEORY OF INTERNATIONAL ECONOMIC

POLICY, VOLS. lAND II

BENHAM'S ECONOMICS

PUBLIC FINANCE IN DEVELOPING COUNTRIES

PUBLIC FINANCE IN THEORY AND PRACTICE

Weidenfield &1 Nicolson

Macmillan

Oxford University Press

Pitman

Weidenfield &f Nicolson

Weidenfield &1 Nicolson

THE ACCUMULATION OF CAPITAL

MATHEMATICS FOR ECONOMISTS

WORKBOOK TO ACCOMPANY 'AN

INTRODUCTION TO POSITIVE ECONOMICS'

Macmillan

Macdonald &1 Evans

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Stonier and A TEXTBOOK OF ECONOMIC THEORY

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MATHEMATICAL

ECONOMICS

R.G.D.ALLEN

Second Edition

CfY0 @\E

ENGLISH LANGUAGE BOOK SOCIETY and

M MACMILLAN

R. G. D. Allen 1956, 1959

All rights reserved. No part of this publication may be reproduced or transmitted, in any form or by any

means, without permission.

First Edition 1956 Reprinted 1957

Second Edition 1959 Reprinted 1960

Reprinted (with alterations) 1963 Reprinted 1964, 1965, 1966, 1970, 1972, 1973, 1976

ELBS edition first published 1973 Reprinted 1976

Puhlished by THE MACMILLAN PRESS LTD

London and Basingstoke Associated companies in New York Dublin

Melbourne Johannesburg and Madras

Also by R. G. D. Allen

BASIC MATHEMATICS

MACRO-ECONOMIC THEORY

MATHEMATICAL ANALYSIS FOR ECONOMISTS

Trus book is sold subject to the standard conditions of the Net Book Agreement.

The paperback edition of trus book is sold subject to the condition that it shall not, by way of trade or otherwise, be lent, re-sold, hired out, or other-wise circulated without the publisher's prior consent in any form of binding or cover other than that in wruch it is published and without a similar con-dition including trus condition being imposed on the subsequent purchaser.

ISBN 978-1-349-81549-4 ISBN 978-1-349-81547-0 (eBook) DOI 10.1007/978-1-349-81547-0

PREFACE

THIS book has evolved from three strands of thought. When I was first interested in mathematical economics in the early 1930's, it seemed to me that the main need was for a grounding in calculus and I wrote my text-book, Mathematical Analysis/oT Economists (1938), for that purpose. I did not venture into the higher reaches of algebra and I made no use of the complex variable. lieft such matters to be written up by those with specialist applications to economics in mind. Subsequently the uses of matrix algebra, of vectors and complex variables, of operational processes and of other such mathematical devices have been greatly developed in many parts of mathematical economics. There are in my view no mathe-matical texts on higher algebra and on operational methods which are really suitable for economists.

A second development of the last twenty years is the growth of econo-metrics. This has been so rapid that I think there is some risk that the necessary development of economic theory, formulated in a way which makes econometric sense to a statistician, will lag behind rather seriously. Such formulations of economic theory must be in mathematical terms but simplified as far as possible.

Finally, the change in the direction of economic thought over the past twenty years has involved a considerable upheaval in the structure of economic theory. This is partly, though by no means entirely, the result of the work of Keynes. I believe that there is now areal need for some synthesis of the " new " economics, for some calm survey of the form and scope of econornic theory.

With these things in mind, I feel that the best contribution I can make is not to extend my 1938 book but rather to write a completely different one, a text on economic theory written in mathematical terms. The present work is not mathematics for the economist, nor is it econometrics. It aims at a fairly systematic treatment of some of the more important and simpler parts of mathematical economics. My main problem is one of selection from a vast range of economic topics, of making the book reasonably up-to-date and of keeping it down to reasonable dimen-sions. Even if I include only what is of particular interest to me personally, I would still find I have too much. To those who look for other topics or different methods of treatment, I can only make the familiar excuses of the anthologist.

VI PREFACE

I am sure that the book is very different from what it would have been if I had written it five years ago. How long it will remain even approximately up-to-date I cannot pretend to guess. I feel, however, that many of the techniques I attempt to describe will be relevant for some time to come ; they are now relatively new and they need to be consolidated and absorbed into the general body of economic theory. Moreover, my guiding principle is to start with, and to remain as elose as possible to, economic problems of the real world, simplified as an economist might simplify them, and translated into mathematics of no more than moderate difficulty. I hope that this approach will appeal to many economists now and in the future.

I could not write a book of this kind without assistance on a scale far beyond my powers to acknowledge. I am grateful for the constant encouragement and advice of my colleagues, particularly Professor Lionel Robbins, Professor James Meade, Mr. David Knox, and Mr. Ralph Turvey. Mr. W. M. Gorman and Dr. F. H. Hahn of the University of Birmingham have kindly read through the manuscript; they see the need for a book on the present lines, they are ideally qualified to write it, and yet they stand aside and allow me to monopolise the market. Above all, I am in debt to Dr. Helen Makower, Dr! G. Morton and Dr. A. W. Phillips; this is clear to anyone who reads the middle and later chapters as now written.

R. G.D.ALLEN LoNDON SCHOOL OF EcONOMICS

PREFACE TO SECOND EDITION

I am glad to have this opportunity of making some changes in the text. Firstly, I have corrected errors and misprints which escaped me, all too frequently, during my original proof-reading. Secondly, I have made numerous revisions in the wording and exposition throughout the text. I hope that these will make the argument more clear and precise.

Thirdly, I have included some additional references, mainly to books and articles published since 1956. It happens that half-a-dozen books of outstanding merit have appeared in the two years 1957 and 1958. They are Goldberg : IntroductWn to Difference Equations (1958); Murdoch : Linear Algebra JOT Undergraduates (1957); Kemeny, Snell and Thompson : Introduction toFinite Mathematics (1957); Thrall and Tomheim : VectOT Spaces and Matrices (1957); Luce and Raiffa: Games and Decisions (1957); and Dorfman, Samuelson and Solow: Linear Programming and Economic Analysis (1958). None of these can be described, mathe-matically, as very simple; they vary in level from the moderately difficult to the quite advanced. But, equally, none of them can be ignored in the education of the mathematical economist.

Finally, I have complete1y re-written some critical sections: 1.9 and 5.8 on time lags in dynamic models; 2.3, 10.3, 17.1 and 17.9 on general economic equilibrium; and 16.2 on the dual problem in linear programm-ing. Substantial parts of the chapters on vectors and matrices have been drastically re-cast. I have extended Appendix A which deals with the " practical " mathematics of operators and linear systems. I have added a complete1y new Appendix B on "modem" algebra to serve as an introduction to the rigorous and postulational development of algebra. In asense, this is a complement to Appendix A. The strict, axiomatic approach of the modem algebraist should be of considerable interest in itself to the economic theorist concemed with model-building. In any case, I hope that the new Appendix provides an under-pinning for the algebra of the text, where a compromise is adopted in the interests of simplicity.

My thanks are due to a number of correspondents. I am particularly grateful for the suggestions made by Sven Dan" of the University of Copenhagen; by Lucien Foldes of the London School of lconomics;

viii PREFACE

by Maurice McManus of the U niversity of Birmingham and the School of Business Administration, Minnesota; by Peter Newrnan of the University College of the West Indies; and by Ciro Tognetti of Centro per la Ricorca Economica ed Econometrica, Genoa.

UNIVERSITY OF CALIFORNIA, BERKELEY

]ANUARY, 1959

R. G. p. ALLEN

CHAPTI!R

PREFACE

CONTENTS

PREFACE TO SECOND EDITION

INTRODUCTION

1. THE COBWEB AND OTHER SIMPLE DYNAMIC