Application of Linear Programming to the Theory of the Firm [Reprint 2020 ed.] 9780520339446

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PUBLICATIONS OF THE BUREAU OF BUSINESS AND ECONOMIC RESEARCH Previously published in this series: A TREATISE O N W A R

INFLATION

by William Fellner (1942)

BREAD AND D E M O C R A C Y IN G E R M A N Y

THE E C O N O M I C S OF THE PACIFIC COAST PETROLEUM INDUSTRY. PART 3: PUBLIC POLICY T O W A R D COMPETITION AND P R I C I N G by Joe S. Bain (1947)

by Alexander Gerschenltron (1943)

THE E C O N O M I C S O F THE PACIFIC COAST PETROLEUM INDUSTRY.

THE STRUCTURE OF TRANSCONTINENTAL RAILROAD RATES by Stuart Daggett and John F. Carter ( 1947)

PART I: MARKET STRUCTURE by Joe S. Bain (1944)

THE C H A N G I N G COMPETITIVE STRUCTURE OF THE W H O L E S A L E G R O C E R Y TRADE:

LAND TENURE PROBLEMS IN THE SANTA FE RAILROAD GRANT AREA

A CASE STUDY OF THE LOS A N G E L E S MARKET by Ralph Cassady, Jr., and Wylie L. Jones (1949)

by Sanford A. Moslt (1944) NATIONAL POWER AND THE STRUCTURE

T A X I N G M U N I C I P A L BOND I N C O M E by Lyle C. Fitch (1950)

OF FOREIGN TRADE by Albert O. Hirschmann (1945) THE E C O N O M I C S O F THE PACIFIC COAST PETROLEUM INDUSTRY. PART 2: PRICE

CRISIS IN BRITAIN: PLANS AND ACHIEVEMENTS OF THE LABOUR GOVERNMENT by Robert A. Brady (1950)

BEHAVIOR AND COMPETITION by Joe S. Bain (1945)

SCIENTIFIC METHOD FOR AUDITING. APPLICATIONS OF STATISTICAL S A M P L I N G

C A L I F O R N I A BUSINESS CYCLES

THEORY TO AUDITING PROCEDURE

by Frank L. Kidner ( 1946)

by Lawrence L. Vance

MONETARY POLICIES AND

A COMPREHENSIVE CLASSIFIED MARKETING

FULL EMPLOYMENT

BIBLIOGRAPHY. PARTS I AND II

by William Fellner (1946, 1947)

by David A. Revzan ( 1951)

application of linear programming to the theory of the firm

A P P L I C A T I O N OF L I N E A R PROGRAMMING TO THE T H E O R Y OF T H E FIRM INCLUDING AN ANALYSIS OF MONOPOLISTIC FIRMS BY NON-LINEAR PROGRAMMING

A PUBLICATION OF T H E BUREAU OF BUSINESS AND ECONOMIC RESEARCH, UNIVERSITY OF CALIFORNIA PUBLISHED BY THE UNIVERSITY OF CALIFORNIA PRESS BERKELEY AND LOS ANGELES NINETEEN FIFTY ONE

University of California Press Berkeley and Los Angeles, California C a m b r i d g e University Press, London, England

C o p y r i g h t 1951 By the Regents of the University of California Printed in the United States of America

BUREAU OF BUSINESS AND E C O N O M I C R E S E A R C H William Crum, Chairman Robert A . Brady

Charles A . Gulick

Roy W . Jastram

Maurice Moonitz

Sanford A . Mösle, Acting Director

The opinions expressed in this study are those of the author. The functions of the Bureau of Business and Economic Research are confined to facilitating the prosecution of independent scholarly research by members of the faculty.

Preface

This monograph, except for the last part of chapter iii.was prepared as a doctoral dissertation in the Department of Economics, University of California. The committee in charge of the research was headed successively by Professors William Fellner and R. A. Gordon. Chapter iv, especially, bears the imprint of Professor Gordon's strict and constructive criticisms. Another member of the committee, Professor G. C. Evans of the Department of Mathematics, served as the guardian of the author's mathematical conscience. Whatever nathematical elegance the treatment contains must be credited to the high standards which Professor Evans imposed and insisted on. All work in the field of linear programming must be deeply indebted to Dr. George B. Dantzig of the Department of the Air Force, who opened up the field and has been one of its most productive investigators. But Dr. Dantzig's contribution to the present monograph goes far beyond that. He gave much of his time to instructing the author in the concepts and methods of linear programming and after the research was under way allowed his patience to be imposed on many times when difficulties beyond the mathematical competence of the author threatened to bring the project to an ignominious halt. Professor A. W. Tucker, Princeton University, and Professor E. W. Barankin, University of California, were also consulted about mathematical difficulties, and the author is gratified to note that each of them subsequently undertook mathematical studies of first importance in problems suggested by this thesis. Professor Tucker, especially, by his pioneering study of " Non-Linear Prog r a m i n g , " cleared the way for the treatment of quadratic programming, which appears in chapter iii. If this volume were dedicated to anyone, it would be to the officers and officials who staff the headquarters of the United States Air Force. These far-sighted men, faced with the difficulties of administering one of the largest integrated organizations in the world, have had the vision required to turn away from traditional methods of administration and to encourage an ambitious

vii

program of research into the theory of progranming. The research whose results are reported in this monograph was originally undertaken as part of that research program and was' inspired to a great extent by the needs of the Air Force. It is the author's earnest wish that these results may assist in some small measure in easing and improving the work of Air Force headquarters. But, of course, the author also feels that the theories here developed invite far wider application.

I wish to thank the various publishers who have given permission for the use of quotations from their publications, cluding:

in-

Harper & Brothers for permission to quote from Kenneth E. Boulding, Economic Analysis, 1941. Harvard University Press for permission to quote from Paul A. Samuelson, Foundations of Economic Analysis, 1948. Houghton Mifflin Company for permission to quote from Frank H. Knight, Risk, Uncertainty and Profit, 1921. The Macmillan Company for permission to quote from Joan Robinson, The Economics of Imperfect Competition, 1948. Oxford University Press, Inc., for permission to quote from J. R. Hicks, Value and Capital, 1941. University of California Berkeley, California April 13, 1951

R. D.

viii

Contents I.

Two Approaches to the Theory of the Firm 1. 2. 3.

II.

III.

IV.

Historical Perspective The Marginal Analysis Basic Concepts of Linear

1 1 5 12

Programming

The Competitive Firm Using Fixed Factors

23

1. 2. 3. 4. 5. 6. 7.

23 27 31 35 39 45 51

Formulation of the Problem The Basic Theorem Computational Procedure The Three Assumptions Firm with Two Fixed Factors Price Imputation for Fixed Factors The Concept of the " Dual"

Production Scheduling for Monopolized Products

53

1. 2. 3. 4.

53 56 65 67

The Monopolist's Problem The Optimum Program Application to Monopsony Quadratic Programming

Assumptions, Limitations, and Possibilities 1. 2. 3. 4. 5.

General Postulates of Maximization The Specific Postulates of Linear The Problem of Time Variations of Linear Programming What is Linear Programming

References

Programming

79

....

79 80 85 88 93 95

ix

CHAPTER I

Two Approaches to the Theory of the Firm

1.

Historical

Perspective.

Hie modern theory of the firm as expounded by Chamberlin (3)*, Hicks (16), Robinson (31), Samuelson (32), Viner (38), et al., traces its lineage back to Ricardo's discussion of the problems of English agriculture a century and a half ago. Ricardo and his contemporary, Malthus, share credit for the development of the concept of " deceasing returns," the progenitor of the modern Ushaped cost curve. The classicists also used the idea of incremental variation of factors of production and thus set the stage for the modern " marginal" analysis. The fact that an analytic apparatus which was inspired by the predicament of early nineteenth-century English agriculture is so deeply embedded in the analysis of modern industrial problems calls for a reconsideration of the appropriateness of the model. Ricardo's theory (30) was based on a highly abstracted model of a single type of farming: English wheat cultivation. He grouped all things necessary for the cultivation of wheat into three factors of production: land, labor, and capital. He assumed that these factors of production could be combined in, essentially, any proportions desired. The technical process involved was considered from a broad point of view which ignored the existence of subsidiary steps and intermediate products; the three necessary factors were applied and after a proper interval the final product emerged. The entire process resulted in a single, homogeneous comnodity—wheat. * Numbers in parentheses refer to the references at the end of the volume.

1

2

LINEAR

PROGRAMMING

AND

THEORY

OF

THE

FIRM

Ricardo applied this model, essentially, to calculate the distribution of product between the landowner, who supplied the land, and the farmer, who provided the capital. He assumed that the total quantity of land was fixed and this total could be subdivided into the amounts of land of first, second, third, etc., qualities, which were also fixed. It is now that the technique of incremental variation and the postulate of decreasing returns enter. As new capital seeks to enter agricultural production, it may be applied to new,and less productive, land or to the more intensive cultivation of land already in use. In either case, it will give rise to a smaller physical product per unit than capital previously applied. An increased application of capital to a fixed supply of land will therefore tend to raise rents, lower interest, and increase the real cost of production. It is not necessary for our purposes to describe Ricardo's theory further; the foregoing is sufficient to show that it contained the essentials of the modern analysis of the valuation of factors of production and the allocation of resources. This model answered rather well to the problem which interested Ricardo. Almost as soon as it was laid down, however, economists began to apply it to a broader range of questions and to make the necessary emendations. All that was necessary, really, was to combine the Ricardian technique of analysis with Say's concept of firm (34) which purchased the services of the three factors of production in a competitive market. Within a generation Nassau Senior adapted the Ricardian mode of analysis to the study of industrial production by treating the three factors of production more synmietrically than Ricardo had (35). Contemporaneously with Senior, Cournot made the pioneer application of the differential calculus to the theory of the firm (5), but this did not bear fruit until the 1870's when Jevons (17), Menger (27), and their foil owers introduced the formal methods of marginal analysis. As applied to the theory of production, the marginal analysis was an improvement more in form than in content, however. The marginalist contribution was essentially a more symmetrical, compact, and formal technique for handling the received Ricardian model. The marginalist analysis of the behavior of the firm and the allocation of productive resources has enjoyed practically complete acceptance for the past seventy-five years, so that it is fair to say that in spite of more than a century of refinement, the analysis

TWO APPROACHES T O THE THEORY OF THE

FIRM

3

of production to t h i s very day i s based upon t h e o r i g i n a l R i c a r dian-Malthusian concepts. Hie c l a s s i c a l Ricardian a n a l y s i s was a study of long-run economic e q u i l i b r i u m , of " t h e high theme of economic p r o g r e s s . " But, as our quick survey has j u s t i n d i c a t e d , as i t passed through succ e s s i v e hands i t evolved i n t o a theory of e n t r e p r e n e u r i a l behavior, t h a t i s i n t o an e x p l a n a t i o n of t h e f a c t o r s which govern an e n t r e p r e n e u r ' s d e c i s i o n s . In the 1 9 3 0 ' s t h i s a p p l i c a t i o n of the theory underwent r a p i d r e f i n e m e n t , e s p e c i a l l y a t the hands of Chanberlin ( 3 ) , Robinson (31), Viner (38), and t h e i r f o l l o w e r s . A r e a c t i o n a g a i n s t t h i s r e f i n e d a p p l i c a t i o n of the marginal approach i s now i n p r o g r e s s . Hall and Hitch (14) and L e s t e r (25) have made f i e l d s t u d i e s of b u s i n e s s decision-making which show t h a t the c o n s i d e r a t i o n s which businessmen t a k e i n t o account are q u i t e d i f f e r e n t from the concepts of the marginal a n a l y s i s and perhaps i n c o n s i s t e n t with i t . Gordon (13) and Eiteman (11) have r e i n f o r c e d t h e s e doubts by showi n g t h a t i t i s h a r d l y conceivable t h a t businessmen can o b t a i n the i n f o r m a t i o n which the marginal a n a l y s i s assumes they have a t t h e i r d i s p o s a l . Dean (10) and L e s t e r (25) have made e m p i r i c a l s t u d i e s which r a i s e q u e s t i o n s as to the v a l i d i t y of the shape of the s h o r t run production curve assumed by the marginal a n a l y s i s . I t deserves to be r e p e a t e d , of course, t h a t t h i s a r r a y of c r i t i cism of t h e marginal a n a l y s i s b e a r s most p a r t i c u l a r l y on i t s a p p l i c a t i o n t o e n t r e p r e n e u r i a l behavior in the r e l a t i v e l y s h o r t run, b u t , s i n c e a l l d e c i s i o n s of i n d i v i d u a l s and f i r m s are s h o r t - r u n dec i s i o n s , these c r i t i c i s m s concern the e n t i r e theory of t h e f i r m . The c r i t i c s have a l s o a t t e n p t e d the c o n s t r u c t i v e t a s k of answering the question: I f t h e marginal a n a l y s i s does not explain e n t r e p r e n e u r i a l behavior, what does? The s t r o n g i n f l u e n c e of custom and the concept of a " f a i r p r o f i t " have been emphasized by Hall and Hitch and L e s t e r on the b a s i s of t h e i r empirical s t u d i e s . Eiteman has sketched a theory which i s based on the concept of m a i n t a i n i n g normal i n v e n t o r i e s and a normal r a t e of turnover of working c a p i t a l . A very ambitious attempt was made by von Neumann and Morgens t e r n (40) t o explain e n t r e p r e n e u r i a l and market behavior i n terms of a theory of competitive s t r a t e g y . U n f o r t u n a t e l y , t h e empirical t h e o r i e s a l l r e s t on t h e c o n t e n t i o n t h a t businessmen attempt t o maintain sane s o r t of " n o n n a l " r e l a t i o n s h i p s i n the p r i c e s t r u c t u r e s of t h e i r i n d u s t r i e s and the f i n a n c i a l s t r u c t u r e s of t h e i r

4

LINEAR PROGRAMMING AND THEORY OF THE FIRM

firms. Hie process by which these norms come to be established or modified is still largely conjectural. In essence these theories assert that any equilibrium position, once attained, acquires a sort of social inertia which helps to perpetuate it. It is perhaps this superficial level of explanation, this abandonment of the maximization principle in favor of the non-rational following of sociological norms, which has deterred economists by and large from accepting the conclusions of this school as a satisfactory explanation of entrepreneurial behavior. The von Neumann and Morgenstern theory attempts to supplant the marginal analysis from another direction. It is ultra-rational. It assumes that all businessmen are oligopolists engaged in an unremitting strategic struggle, each to maximize his own profits at the expense of his competitors and the world at large. Unfortunately for this theory, as von Neumann and Morgenstern themselves show, the rational conclusion of such a struggle is the formation of monopolistic cartels; the assumption of oligopoly is itself inconsistent with the assunption of complete rationality. Von Neumann and Morgenstern attempt to salvage the theory by introducing an irrational element, the so-called "standards of behavior" (40, p.40) but these remain ill-defined and do not lead to stable responses to market situations. Von Neumann and Morgenstern's theory is open to other criticisms, too, but this is not the place to review their monumental work. In recent years interest has been growing in a rather different approach to economic and managerial problems, the "input-output" or "linear progranming" approach. Even at this writing the content and methodology of linear progranming are in so formative a state that it would be premature to attempt to delimit them. It represents a confluence of several diverse streams of development: mathematical studies of the geometry of higher spaces dating back, probably to Laplace and certainly to Weyl (42), a simple dynamic economic model due to von Neumann (39), Leontief's matrix description of the interrelations of American industry (23), and Wood and Dantzig's research into the managerial problems of the Air Force (9, 36, 37, 43, 44, 45). Research into the theory of games by von Neumann and others certainly helped pave the way for linear progranming which mathematically, though not conceptually, is a closely related problem. In spite of the long existence of these varied anticipatory developments, the first integration of them into the viewpoint and the tool which is now called linear progranming was probably achieved by Dantzig in 1947.

TWO

APPROACHES

TO THE

THEORY

OF THE

FIRM

5

Hie o r i g i n a l appearance of l i n e a r progranming, then, was in the f i e l d of s c i e n t i f i c management r a t h e r than i n t h a t of economics (and t h i s may account f o r the f a c t t h a t l i n e a r progranming l a r g e l y n e g l e c t s t h e demand s i d e of economic problems). Hie i n t e r e s t of economists i n the new technique awakened quickly, however, and Koopmans (20 and 22) took the lead i n s t u d y i n g the a p p l i c a t i o n of l i n e a r progranming t o problems of economic w e l f a r e . Hie p r e s e n t essay appears t o be the f i r s t attempt t o apply l i n e a r progranming to the s h o r t - r u n behavior of the i n d i v i d u a l f i r m or e n t r e p r e n e u r and t o a s s e s s i t s value f o r d e a l i n g with t h i s problem. Linear programming w i l l be d e s c r i b e d in t h i s book i n terms of i t s r e l a t i o n s h i p to the t h e o r y of t h e f i r m . Hie b e s t s t a r t i n g p o i n t w i l l be an o u t l i n e of t h e formal s t r u c t u r e of the marginali s t approach to t h i s same theory.

2.

The Marginal

Analysis

Hie marginal a n a l y s i s ( l i k e l i n e a r programming) i s concerned with the d e c i s i o n s of an economic u n i t c a l l e d a firm. This firm, i t i s assumed, has j u s t one motive: to maximize i t s p r o f i t s , or the excess of i t s revenues over i t s expenses. Hie d e c i s i o n s which the f i r m has to make are of t h r e e s o r t s : technical decisions, which determine the methods of production to be enployed, q u a n t i t y d e c i s i o n s , which concern the amount of goods t o be o f f e r e d and s o l d , and marketing d e c i s i o n s which deal with procedures f o r a l t e r i n g , i f p o s s i b l e , the demand curves of the f i r m ' s customers. I t should be noted t h a t a firm in pure competition cannot a f f e c t the p r i c e s a t which i t buys and s e l l s , while a monopolist can determine e i t h e r the p r i c e s a t which he buys and s e l l s or the q u a n t i t i e s which he buys and s e l l s but not both. Hie marketing d e c i s i o n s w i l l n o t be discussed f u r t h e r in t h i s paper. With the exception of Chamberlin (3, chapter v i i ) , t h e r e has been l i t t l e e f f o r t to i n t e g r a t e them with the body of economi c d o c t r i n e , presumably because the c e n t r a l i s s u e s of economics concern t h e response to the w a n t - p a t t e r n of s o c i e t y r a t h e r than manipulation of i t . Technical d e c i s i o n s are u s u a l l y banned from the economic f i e l d of d i s c o u r s e and r e l e g a t e d to t h a t of engineering by the assumpt i o n t h a t t h e r e i s some most e f f i c i e n t way t o combine given

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PROGRAMMING

AND

THEORY

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THE

FIRM

quantities o f raw materials to produce a desired b i l l of products, and that t h i s i s the technical procedure which will be adopted. Hie solution to the technical problems of the firm i s regarded as summarized by a "production function." Samuelson presents a typical formulation: We assume as given by technical considerations the maximum amount of output, x, which can be produced from any given set of inputs (Vy v n ) , Hiis catalogue of p o s s i b i l i t i e s i s the production function and may be written * =

f(vv

In general, there will be a maximum output for each set of inputs, and so t h i s function i s single-valued, and will be assumed i n i t i a l l y to have continuous p a r t i a l derivatives of desired order. (32, pp. 57, 58). The formal essence of the marginal analysis of the firm can now be stated very b r i e f l y and in somewhat more general terms than Samuelson's. We begin with the firm, which i s assumed to be a r a t i o n a l , independent, decision-making unit whose whole objective i s to maximize p r o f i t s . I t s f i e l d of decision making and the measure of merit of any decisions are defined in terms of production and price functions. Assume that the firm produces m salable commodities by consuming n inputs. Let x. denote the quantity produced o f the ith salable comnodity and Vj the quantity consumed o f the j t h input. We assume also that these quantities are connected by the production or transformation function 0.1

f(xv

...,

xm; v v . . . . vn) = 0

A l i t t l e care i s needed in the definition o f t h i s function. For any given set of inputs, Vj, . . . , v n , there will generally be several technically possible sets of outputs, i j , . . . . x . Assign arbitrary values to m-1 of these outputs and determine the large s t value of the remaining output which i s consistent with equation 0 . 1 . In t h i s manner we are assured that the transformation function will be single-valued, that i s , i f a l l the inputs and a l l but one of the outputs are assigned, then the remaining output i s fully determined. I t i s assumed also that the transformation function i s defined over the domain of non-negative inputs and outputs and that within the domain of definition i t has continuous partial derivatives o f f i r s t and second order.

TWO A P P R O A C H E S

TO

THE

THEORY

OF

THE

FIRM

7

We assume t h a t a p r i c e f u n c t i o n i s a s s o c i a t e d with each s a l a b l e commodity and t h a t i t s value i s dependent on the q u a n t i t i e s produced of a l l s a l a b l e conmodities. We denote these f u n c t i o n s by: 0 2

P P

i- i

Then we d e f i n e g r o s s revenue to be: m - 2 UÎ

0.3

P

i*i-

I t i s assumed t h a t a p r i c e f u n c t i o n i s a s s o c i a t e d a l s o with each i n p u t , and t h a t t h e s e f u n c t i o n s may be w r i t t e n

0.4