THEORY OF COST AND PRODUCTION FUNCTIONS PRINCETON STUDIES IN MATHEMATICAL ECONOMICS Edited by David Gale and Harold W. Kuhn 1. Spectral Analysis of Economic Time Series, by C. W. J. Granger and M. Hatanaka 2. The Economics of Uncertainty, by Karl Henrik Borch 3. Production Theory and Indivisible Commodities, by Charles Frank, Jr. 4. Theory of Cost and Production Functions, by Ronald W. Shephard THEORY OF COST AND PRODUCTION FUNCTIONS BY RONALD W. SHEPHARD PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY 1970 Copyright© 1970 by Princeton University Press ALL RIGHTS RESERVED LCC 75-120762 ISBN 0-691-04198-9 This book is composed in Fotosetter Times Roman Printed in the United States of America by Princeton University Press To Hilda Maloy Shephard PREFACE Fifteen years have passed since my original monograph on cost and production functionst was published by Princeton University Press. Until recently there has been little if any reference to the work. The monograph has long been out of print and for some time I have been aware that individuals were seeking copies of the apple green booklet. Some of the ideas and conceptions of the monograph seem to have per colated to the surface of theoretical and econometric studies, renewing my interest in the subject. About three years ago, my friend Oskar Morgenstern, whose interest in my early work on cost and production functions was largely respon sible for the publication of the first monograph, began urging me to re write the booklet, and I set myself the task of doing so. It soon became evident that considerable modernization and extension of the subject matter was desirable, and in this book I have tried to develop the theory of cost and production functions in a more complete and systematic way. The subject matter is essentially mathematical and, although there is a predilection in mathematical economics for the use of symbolism in the place of words, I have not hesitated to use words when the precision of the discussion is not lost. The mathematical arguments are simple and direct, although perhaps inelegant, but minimally invoking theorems which disconnect the reasoning. One may ask: why devote a book to the theory of cost and produc tion functions? In a narrow sense, the mathematical economic theory of production is a theory of cost and production functions, with the central topic being an understanding of the possibilities of substitution between the factors of production to achieve a given output. Optimiza tion in production planning is yet another topic now largely being pursued in Operations Research, where the models reflect the peculiari ties of the individual firm and the difficulties are mainly computational and algorithmic. Econometric studies of capital expansion, returns to scale and factor substitution lean heavily upon a clear understanding of cost and production functions, complicated by problems of aggregation which are still unsolved. Realistically, one may hope to advance the economic theory of production by concentrating upon the core of this subject, i.e., cost and production functions. Discussions of this subject are at best confusing. I have not tried to reference comprehensively the work of others, this being a distracting t Ronald W. Shephard: Cost and Production Functions, Princeton University Press 1953. ( vii ) PREFACE chore. The references used in this connection have been chosen at my convenience to contrast viewpoints. The material for this book has been developed in a series of prelimi nary reports issued at the Operations Research Center, College of Engineering, University of California, Berkeley. These reports have not been referenced because of their limited distribution. I take this opportunity to express my gratitude to Oskar Morgenstern for his supporting interest in the research from which this book has evolved. I also wish to express my indebtedness to Dr. Stephen Jacobsen for his reading of my manuscript as it developed and the many helpful suggestions which he has made. I gratefully acknowledge the financial support of Professor Morgen stem's Econometric Research Project at Princeton University, supported by the Office of Naval Research, and the support of the Office of Nav al Research and the National Science Foundation research grants to the Operations Research Center at the University of California, Berkeley, both of which assisted the research which has led to the publication of this book. Also, I take this means of expressing my appreciation to Mrs. Linda Betters for typing the manuscript. June 1969 RONALD w. SHEPHARD Berkeley, California ( vm )