Theory of Cost and Production Functions 9781400871087

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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 functionsf 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 axe 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.

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 Umited 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 Morgenstern's Econometric Research Project at Princeton University, supported by the Office of Naval Research, and the support of the Office of Naval 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 Berkeley, California

RONALD W. SHEPHARD

< vin >

TABLE OF CONTENTS vii

PREFACE CHAPTER 1. INTRODUCTION

3

CHAPTER 2. THE PRODUCTON FUNCTON

2.1 2.2 2.3 2.4 2.5 2.6 2.7

13

DefinitionofaTechnology Definition and Properties of the Production Function Transforms of Production Functions Homothetic Production Functions A Classification of the Factors of Production The Production Function of a Limited Unit Law of Diminishing Returns

CHAPTER 3. THE DISTANCE FUNCTION OF A PRODUCTION STRUCTURE

3.1 Definition of the Distance Function I (UjX) 3.2 Properties of the Distance Function 3.3 Expression of the Production Function Φ(χ) in Terms of the Distance Fimction Ψ(ιι,χ) 3.4 The Distance Function of Homothetic Production Structures 1 r

CHAPTER 4. THE FACTOR MINIMAL COST FUNCTION

4.1 4.2 4.3 4.4

Definition of the Cost Function Q(u,p) Geometric Interpretation of the Cost Function Properties of the Cost Function The Cost Function of Homothetic Production Structures

CHAPTER 5. THE COST STRUCTURE

5.1 Definition of the Cost Structure £Q(U), U ε [Ο,οο) 5.2 Efficient Price Vectors of the Cost Structure 5.3 The Cost Structure of Homothetic Production Structures 5.4 Cost Limited Maximal Output Function Γ(ρ) 5.5 Cost Limited Output Function for Homothetic Cost Structures

13 20 23 30 36 39 42

64

64 67 74 76 79

79 81 83 92 96

96 100 103 105 111

TABLE OF CONTENTS

CHAPTER 6. THE AGGREGATION PROBLEM FOR COST AND PRODUCTION FUNCTIONS

6.1 CriteriaforAggregates 6.2 Gross Aggregation of Homothetic Production, Cost and Cost Limited Output Functions 6.3 Aggregation of Cobb-Douglas Production, Cost and Cost Limited Output Functions 6.4 Aggregation of ACMSU Production, Cost and Cost Limited Output Functions 6.5 Aggregation of a Class of Homothetic Cost, Production and Cost Limited Output Fimctions CHAPTER 7. THE PRICE MINIMAL COST FUNCTION

7.1 Definition of the Price Minimal Cost Function ^*(u,x) 7.2 Properties of the Price Minimal Cost Function 7.3 The Production Structure L*(u) Defined By the Price Minimal Cost Function 7.4 Equivalence of the Production Structures L*(u), L41(U) and their Distance Functions ^*(u,x), ^(u,x) CHAPTER 8. DUALITY OF COST AND PRODUCTION STRUCTURES AND RELATED FUNCTIONS

8.1 DuaHty of the Cost and Production Structures

114

114 119 123 131 139 147

147 148 153 157

159

159 161 163 167 169 178

178 192 199 206

TABLE OF CONTENTS

9.5 The Joint Production Function 9.6 Distance Functions for Homothetic Production Correspondences

212 220

CHAPTER 10. COST AND BENEFIT (REVENUE) FUNCTIONS FOR PRODUCTION CORRESPONDENCES, AND THE RELATED COST, BENEFIT (REVENUE), COST-LIMITED-OUTPUT AND BENEFIT (REVENUE)-AFFORDED-INPUT CORRESPONDENCES 223

10.1 Definition and Properties of the Cost and Benefit (Revenue) Functions 10.2 CostandBenefit(Revenue)Correspondences 10.3 Cost-Limited-Output and Benefit (Revenue)-AffordedInput Correspondences 10.4 Special Forms for Homotheticity of Input and Output Structure of a Production Correspondence 10.5 Returns to Scale for Production Correspondences CHAPTER 11. DUALITIES FOR PRODUCTION CORRESPONDENCES

11.1 Duality Between the Cost Fimction Q(u,p) and the Distance Function ^(u,x) for the Input Sets L(u) of Ρ: X -» U 11.2 DuaHty Between the Benefit (Revenue) Fimction B(x,r) and the Distance Function fl(x,u) for the Output Sets P(x) ofP:X->U 11.3 Two Theorems Concerning the Cost and Benefit (Revenue) Functions 11.4 DualitiesforAccounting(Shadow)Prices 11.5 Implications for Linear Production Models

APPENDIX 2. APPENDIX 3. INDEX

243 250 255 261

261

266 272 275 283 292

REFERENCES APPENDIX 1.

223 231

MATHEMATICAL CONCEPTS AND THEOREMS FOR SEMI-CONTINUITY AND QUASI-CONCAVITY (CONVEXITY)

295

MATHEMATICAL CONCEPTS AND PROPOSITIONS FOR CORRESPONDENCES

298

UTILITY FUNCTIONS

301 306

THEORY OF COST AND PRODUCTION FUNCTIONS

CHAPTER 1 INTRODUCTIONf In economic theory the production function is a mathematical state­ ment relating quantitatively the purely technological relationship be­ tween the output of a process and the inputs of the factors of produc­ tion, the chief purpose of which is to display the possibilities of substitution between the factors of production to achieve a given output. The distinct kinds of goods and services which are usable in a production technology are referred to as the factors of production of that technology and, for any set of inputs of these factors, the production function is interpreted to define the maximal output realizable therefrom. The more or less traditional treatments of the production function exclude free goods as inputs and require that the function express the variable, substitutional and Umitational character or other qualifica­ tions of the factors of production peculiar to some hypothetical produc­ tion unit. Sune Carlson [5] states: "As regards the productive services which constitute the input to the technical unit during the period we shall only consider the services which are limited in supply." He makes a distinction between fixed and variable productive factors and asserts: "The production function, it must be remembered, is defined relative to a given plant; that is certain fixed services." Erich Schneider [25] dis­ tinguishes between "substitutional" and "limitational" factors and ex­ presses the production function in terms of substitutional inputs with side equations between output and each Umitational factor. Samuelson [24] explains that: "The production function must be associated with a particular institution (accounting, decision making, etc.), and must be drawn up as of any unique circumstances pertaining to this unit." AU of these qualifications are made in the context of a general theory of production! Sometimes the productive factor inputs are classified as to whether they are flow or stock quantities, the former referring to labor services, raw materials, energy, etc. and the latter designating real capital goods such as plant, machinery and equipment. KreUe [17] makes this distinc­ tion and introduces stock variables in the production function along with flow variables for the inputs of consumable factors. In order to exhibit the structure of investment planning, V. L. Smith [27] puts forth the notion of a "stock-flow production function" in which capital stock in­ puts are freely variable along with current input flows for treating f The preliminary paragraphs of this chapter have appeared in UnternehmensJorschung, Band 11, Heft 4,1967, and are used here with the permission of Physica Verlag, Wurzburg.

THEORY OF COST AND PRODUCTION FUNCTIONS

hypothetical alternative production plans, but once the physical con­ figuration has been chosen the real capital inputs can no longer be varied like current inputs. Also, it is common in the economic theory of production to distin­ guish between "short run" and "long run" production functions, the form of the function being essentially different in the two cases. But the pro­ duction function is ideally a statement of purely technological alter­ natives, without regard to their execution, and one need not define different production functions for these two situations. In doing so, institutional conditions of specific economic planning are brought into the definition, confusing the purely technological (engineering) charac­ ter of the production function. The significance of the short run is that there are constraints on the amounts and kinds of factor inputs and these qualifications are best kept in this form, leaving the production function as a statement of unconstrained technological alternatives relative to some horizon of planning, encompassing arrangements which have not yet been realized as well as those which have been put into operation. The viewpoint taken in this study is that neither the exclusion of free goods nor the requirement that the production function express the variable, substitutional, consumable character or the limitational, fixed stock character of the productive factors, as qualifications peculiar to a particular production unit, are logically necessary for the definition of the production function. The production function is regarded here as a mathematical construc­ tion for some well defined production technology. This technology consists of a family of conceivable and feasible engineering arrange­ ments, not restricted necessarily to particular realizations found in practice and possibly spanning historical changes in the application of the technology. Once defined, the technology implies a certain set of factors of production and no limitations will be put upon the inputs of these factors both as to type and amount available. Thus the production function will be taken to describe the unconstrained technical possibili­ ties of a technology without limitation to any existing or realized production units. The productive factors are not restricted to economic goods and services, i.e., those with a positive market price, because this implies some particular resource availabilities relative to demand in an exchange economy, which is irrelevant to the technical alternatives defined by the production function. However, the situations of interest in economics are those for which not all factors of production are free. No limitations will be put upon the available amounts of the factors of production, because this implies reference to some particular produc-

INTRODUCTION

tion unit which confounds the notion of a production function with some implicit economic decisions or production plan, the variety of which is unlimited, preventing a clear, unambiguous and generally applicable definition of the production function. Both the input and output variables will be defined as time rates. The unconstrained service flows from real capital (plant, machinery and equipment) imply freely variable physical counterparts, in whatever units and capacity they arise in the technology, and unutilized capacity of a physical item is merely excess input flow of the related capital service which does not hinder output. If the production function is to define purely technological possibilities, the available means of a firm or other production unit are not relevant. Such limitations merely prescribe a particular realization of the tech­ nology which may be considered by imposing constraints on the input flows which restrict the analysis to a particular subset of the factor in­ put space. For example, if the production unit uses only certain ma­ chinery and equipment, then the input flows of these factors can be bounded by the positive capacities involved, while the input rates of other real capital conceived for the technology but not available to the production unit may be bounded by zero. In such circumstances the substitution possibilities of one factor for another are likely to be limited, i.e., specific realizations of the technology will have a high degree of factor complementarity, whereas the substitutability of fac­ tors sought in economics will arise for a broadly defined technology not constrained to particular realizations, which is the kind of produc­ tion structure most interesting for economic planning. These matters will become clear when we consider such constraints as defining subsets of input vectors available to the firm. In Chapter 2, the foregoing conception of the production function is developed in some detail. From an engineering viewpoint, the structure of production may be conceived as a family of production possibility sets, specifying for each nonnegative output rate the set of input vec­ tors which yield at least the given output rate. On this structure the pro­ duction function may be defined as the maximum output rate obtainable for any given nonnegative input vector, giving to it the traditional meaning in economic theory. Conversely, one may postulate the exist­ ence of a production function with certain properties and determine the production possibility sets as the level sets of this function, and the uniqueness of the production function is a question of some interest. These ideas are developed at length in Chapter 2. Of particular significance to the theory of production for study of returns to scale is the discovery of the circumstances under which cost data may be "deflated" to real terms by an index function of the

THEORY OF COST AND PRODUCTION FUNCTIONS

prices of the factors of production. To pursue this matter, a class of production functions was defined in the first Princeton monograph [26] and named homothetic. There it was shown that the cost function fac­ tors into a function of output rate and a linear homogeneous function of the prices of the factors of production (an index function of prices), if and only if the production function is homothetic. Interestingly, the ACMS production function [2] and Uzawa's extension of this function [29], the Cobb-Douglas production function and its modifications, used for the study of returns to scale, are all very special cases of homothetic production functions. One might speculate that these endeavors could profit from a conscious use of the general definition of a homothetic production function, taking some special mathematical form for the Unear homogeneous function of the prices into which the cost function factors, but not forcing any special form for the other term (i.e., the function of output rate), the inverse function of which defines the returns to scale. In Chapter 2, a slightly more general definition of homotheticityf will be given, with a discussion of the properties of the corresponding pro­ duction possibility sets. Also, a brief discussion of a classification of the factors of production is presented which seems to be more useful than the traditional notion of complementarity, and a discussion of the pro­ duction function of a limited unit or firm is given. The chapter is closed with a discussion of the law of diminishing returns which provides a proof of a form of the law without assumptions on the fine structure of production, and the implications for commonly used production functions like the Cobb-Douglas and CES are developed. In Chapter 3 the distance function of a production structure is intro­ duced as an alternative to the production function. The properties of this function are determined and the special form of the distance func­ tion for homothetic production structures is deduced. At first it may seem strange that the distance function is considered. But, as will be seen in the subsequent chapters, the minimum cost function is a distance function of a pricw-output cost structure and the duality between cost and production function is naturally formulated in terms of these dis­ tance functions. When production correspondences are considered in Chapter 9, the significance of the distance function will become further apparent, because it affords a means of investigating the possibilities for a joint production function. Chapter 4 is devoted to the factor minimal cost function, i.e., the traditional cost function defining the minimum total cost rate for any

t To avoid an unnecessary assumption of continuity.

< 6 >

INTRODUCTION

output rate u and vector ρ of the prices of the factors of production, with the input rates of the factors of production adjusted to yield mini­ mum total cost. This function is called the "factor minimal" cost func­ tion in order to distinguish it from another cost function introduced in Chapter 7 for discussion of the duality between cost and production function. The properties of the factor minimal cost function are stated and proved in Chapter 4, after which the special form and properties of this function for homothetic production structures are developed. In Chapter 5 it is observed that the factor minimal cost function has the properties of a distance function and it is shown that indeed it is a distance function for a certain cost structure consisting of a family of subsets of price vectors for the factors of production. The properties of the price sets in this cost structure are determined, and the special properties of the cost structure corresponding to homothetic production structures are developed. Then as a dual (to be shown later) to the production function (factor maximal output function) a cost limited output functionf is defined on the cost structure, which provides for any nonnegative price vector of the factors of production the supremal out­ put which can be obtained for any positive cost rate. When differentiable, this function enables a calculation of the marginal productivity of money capital to supply the cost of production, and, in the case of the cost structure for a homothetic production structure, simple formulas are given. Chapter 6 is addressed to the aggregation problem for the theory of cost and production functions. Certain criteria are set fortn for aggre­ gating the input variables and prices of the factors of production. An aggregation of homothetic production, cost and cost limited output functions in terms of one variable for inputs and one variable for prices is determined and shown to satisfy the criteria. The usual aggregation for Cobb-Douglas production and cost functions is then shown to satisfy the criteria, and an aggregation of the ACMS production and cost function is given which yields the same aggregate form as that for the Cobb-Douglas function. Then it is shown that for a certain class of homothetic production and cost functions the aggregate form is a Cobb-Douglas production and cost function. The chapter is closed with a demonstration by construction that a generalization of homothetic cost and production functions can be aggregated to satisfy the criteria. As preparation for the duality between cost and production functions, a price minimal cost function defined on the cost structure is defined in Chapter 7. It is shown that this function has the same properties as the distance function of the production structure from which the cost f Also known as the Indirect Production Function.

THEORY OF COST AND PRODUCTION FUNCTIONS

structure was derived, and, when treated as a distance function in the factor input space, it defines a production structure identical to the parent production structure. Thus in Chapter 8, where the duality between cost and production functions is discussed, the production possibility sets of the production structure and the price sets of the cost structure are shown to be duals, derivable from each other by dual cost minimizations which determine the factor minimal and price minimal cost functions as dual distance functions. A duality between the production function and the cost limited output function is then developed by showing that they may be determined in terms of each other by dual maximum problems. After which, the elegant geometric relationship between the dual cost and production structures is demonstrated, and a theorem is proved estab­ lishing homotheticity as an if and only if property for the factorization of the cost function into a function of output rate and a linear homo­ geneous function of the prices of the factors of production. The chapter is closed with a discussion of dual expansion paths in the cost and production structures. All of the previous considerations apply to production structures with a single output, but they are extendable to technologies with multiple or joint outputs. In Chapter 9, the concept of a production correspond­ ence P is introduced for joint outputs by treating the production rela­ tionship as a mapping of each input vector into a subset of output vectors which can be realized with the given input vector. Certain well defined properties of this mapping are assumed, which are a natural extension of those assumed in Chapter 2 for a production technology with single output. The inverse correspondence L of P is a mapping of each output vector into a subset of input vectors which yield at least the given output vector, analogous to the level sets of the production function. Thus, for the production relationship of a technology with multiple outputs, we have point to set mappings defining outputs realizable with each input vector and, inversely, point to set mappings defining for each output vector a subset of input vectors yielding at least the given output vector. The assumptions made for the mapping P imply certain properties for the inverse mapping L which are analogous to the properties of the level sets of the production function for a technology with single output. Except for modifications to permit nondisposable outputs, the prop­ erties taken for the production correspondence P follow those used by my erstwhile student Dr. Stephen Jacobsen in his doctoral thesis [15], where he extended my duality between cost function and distance func­ tion for the level sets of the production function to the cost function and distance function of the map sets of the inverse correspondence L,

INTRODUCTION

applying my notions of distance function and cost structure induced by the cost function and giving a somewhat more general definition of homothetidty of input structure for the production correspondence. These formulations of the production relationship for joint outputs include those of the production function for a single output as a special case. Indeed, the production correspondence with single output defines a production function with the properties assumed in Chapter 2, and conversely the production function induces a suitable production corre­ spondence with one component output vectors. After developing these notions for the production correspondence, the property of homothetidty is considered. Both the input structure and the output structure of the production correspondence may independ­ ently have a homothetic property. My definition of homothetic input structure is equivalent to that used by Jacobsen, although stated in diiferent terms. Symmetrically, the output structure may also have a homothetic property and a definition of his property is given. If both the input structure and the output structure of the production corre­ spondence are homothetic, the production relationship is homogeneous of degree one.f This discussion of homothetidty of structure contains certain propositions concerning the representation of the input sets, and also the output sets of the production correspondence when they are homothetic. The next topic of Chapter 9 is the definition of two distance functions, one for the input sets of the inverse correspondence L, like that used by Jacobsen, and another for the output sets of the correspondence P. The properties of these two distance functions are developed in some detail, and demonstration is given that they may be used to define the mappings P and L of the production correspondence. In terms of these two distance functions, the question of the existence of the frequently used joint production function is examined. It is found that the joint production function does exist if outputs are disposable and the pro­ duction correspondence is continuous. However, the two distance func­ tions can always be used separately to define the boundary substitutable output vectors for a given input vector (i.e., the output isoquants) and the boundary substitutable inputs to yield a given output vector (i.e., the input isoquants). Chapter 9 is concluded with a discussion of the special forms which the two distance functions take when the related structure is homothetic. Chapter 10 is initiated by consideration of the cost function defined on the input sets of the production correspondence as the minimal cost t The extended definition of homothetidty, given in Section 10.5 and used following, avoids this result.

THEORY OF COST AND PRODUCTION FUNCTIONS

of attaining an output vector u with a price vector ρ for the factors of production, showing that it has properties analogous to those set forth in Chapter 4 for the cost function related to a production function with single output. Then a formulation is given for the revenue (or benefit) function defined on the output sets of the correspondence P as the maximal revenue (or benefit) obtainable with an input vector χ and prices (or unit values) r for the outputs, depending upon whether the outputs are disposable (or nondisposable). The term benefit is used to indicate coverage of situations where the outputs do not have market prices but may have unit values (positive or negative) in accordance with some social weighting system, the negative values applying to un­ desirable outputs. Just as in the case of the cost function for a produc­ tion relationship with single output, the cost function for the production correspondence induces a cost structure. This cost structure is definable by a correspondence which maps an output vector into a subset of price vectors for the inputs which yield at least unit minimal cost in attaining the output vector; or any level of cost for that matter, since the cost function is homogeneous of degree one in the price vectors for the factors of production. Similarly, the revenue (or benefit) function induces a revenue (or benefit) structure which may be taken as a correspondence mapping an input vector into a subset of output price (or unit value) vectors which yield at most unit revenue (or benefit) for the maximal revenue (or benefit) which may be obtained by using the input vector; and the unit level is not restrictive for defining this structure, since the revenue (or benefit) function is homogeneous of degree one in the price (unit value) vector for the outputs. The properties of these two corre­ spondences are considered in some detail, and it is shown that the cost function and the revenue (or benefit) function are distance functions for the map sets of the correspondence which they induce. By way of a digression in order to show the connection with the cost limited maximal output function introduced in Chapter 5, two additional correspondences are defined. One is a cost limited output correspond­ ence mapping a price vector for the factors of production into a subset of output vectors for which the minimal cost of attaining the output vector with the given price vector for the inputs is less than unit value. Similarly, the other correspondence is a mapping of price (or unit value) vectors for outputs into a subset of input vectors for which the maxi­ mal revenue (or benefit) obtainable with the input vector is greater than unit value. The first correspondence enables one to determine for any input price vector the output vectors which can be attained at cost less than any given level of cost, and the second provides a determina­ tion for any output price (or unit value) vector the input vectors which will yield more than any given level of maximal revenue (or benefit). < io >

INTRODUCTION

After showing the special forms taken by the cost and benefit func­ tions if the input and output structures of a production correspondence are homothetic, the final section of Chapter 10 is devoted to a consider­ ation of returns to scale and for this purpose an extended definition of homothetic structure is given. With this extended definition, the altera­ tions of previous propositions for homothetic structures are straight­ forward. The final chapter of the book takes up various dualities, starting with the extension for production correspondences of the duality between cost function and distance function for the input sets of a production rela­ tionship with single output (developed in Chapter 8). This extension was made by Jacobsen [15] in his doctoral thesis. Another duality between the revenue (or benefit) function and the distance function for the out­ put sets of the production correspondence is also developed. The first duality consists of dual cost minimization problems which enable one to determine the cost function and distance function for the input sets of the production correspondence in terms of each other, and the sec­ ond duality provides a determination of the revenue (or benefit) func­ tion and the distance function for the output sets of the production correspondence in terms of each other by dual revenue (or benefit) maximization problems. As a consequence of these two dualities, it is shown that the cost function and the revenue (or benefit) function fac­ tor in a certain way if and only if the input sets and the output sets of the production correspondence respectively have homothetic structure. For the cost function, the factorization is into a product of a function of the output vector and a function of the price vector for inputs, and the benefit function factors into a product of a function of the input vector and a function of the price (unit value) vector for outputs. These factorizations provide simple forms for cost and revenue (or benefit) function which enable one to express price deflated minimal costs as a function of output vector alone and price deflated maximal revenue (or benefit) as a function of input vector alone. Correspondingly, the distance functions for the input sets and the output sets of the produc­ tion correspondence also take simple factored forms if and only if these two structures are homothetic. The next three dualities considered are directed to the determination of accounting (shadow) prices for input vectors (given prices (or unit values) for outputs), for output vectors (given prices for inputs) and simultaneously for both output and input vectors. These dualities are obtained by recombination of the two dual cost minimization problems and the two dual revenue (or benefit) maximization problems described above. The connection between these three dual problems Eind the duality arising in mathematical programming is only incidental, because < Π >

THEORY OF COST AND PRODUCTION FUNCTIONS

the dual program in mathematical programming is merely one aspect of a saddle point problem for the Lagrangian of the primal problem, a device which is not of immediate economic significance except in the case of the simple linear model of production. In the final section of Chapter 11, the constant coefficient model of production is discussed as a Unear production correspondence. There it is shown that, although diiferently motivated, the dual problems in linear programming, one for imputing input prices and another for imputing output prices, are special forms of the first two of the three dualities introduced for determination of shadow prices with more restrictive constraints for the prices imputed. A discussion of the aggregation problem for production correspond­ ences is not given, because, due to the symmetry of definition for homotheticity of input structure and output structure, the arguments of Chapter 6 on the problem of aggregation may be applied for either the input structure or the output structure of the production correspondence. Throughout the exposition to follow no strict attempt has been made to avoid repetition of argument in different contexts or restatement of properties, for the purpose of enabling the reader to follow the discourse without frequent referral to the text of previous sections. Propositions are numbered consecutively as statements which summarize arguments and they are used when applicable as parts of subsequent arguments. Some frequently used mathematical notions are included in Appendix 1 and Appendix 2. Since the results of the first eight chapters carry over for utihty functions, they are summarized in Appendix 3.

CHAPTER 2 THE PRODUCTION FUNCTION^ 2.1 Definition of a Technology A production technology consists of certain alternative means, ar­ rangements of these means and uses of materials and services by which goods or services may be produced. The distinct goods and services which may be used as inputs to the technology are called factors of production. Free goods or services are not excluded as factors of pro­ duction, since market prices have no bearing upon the technical roles of these inputs. The technology exists independently of the political and social structure in which it may operate and also of the scarcity of inputs, i.e., it is a blueprint for production. At this point of our study, it is assumed that a single good or service is obtainable as an output of the technology.f f Let u ε [0, + oo) denote the output rate and take χ = (xi,x2,... ,Xn) to denote the input rates of the factors of production. The input vector χ ranges over the nonnegative domain D of a Euclidian space Rn, i.e., χ ε D, where D = {χ I χ Ξϊ 0,xeRn}.

(1)

It is not assumed that χ must be strictly positive for u to be positive, i.e., some of the factors of production may be substituted completely for others. Definition: A production input set L(u) of a technology is the set of all input vectors χ yielding at least the output rate u, for u ε [0,+ oo). Obviously not all input vectors χ belonging to an input set L(u) are technologically efficient. The efficient subset E(u) of a production input set L(u) is given by the following definition: Definition: E(u) = (χ | χ ε L(u),y < χ => y f L(u)}.ftt A production technology is defined as follows: Definition: A production technology is a family of input sets T: L(u), u ε [0, + oo) satisfying:

f The contents of this chapter, except for Section 2.7, have appeared in Unternehmensforschung, Band II, Heft 4, 1967, and are used here in modified form with the permission of Physica Verlag, Wfirzburg.. ft See Chapter 9 for extension to multiple outputs. t t t y ^ x = > y i ^ x i , i = 1, 2 , . . . , n . y < X = > y , g X b i = 1, 2 , . . . , n , y Φ χ .

THEORY OF COST AND PRODUCTION FUNCTIONS

P.l

L(O) = D, 0 iL(u) for u > 0.

P.2 χ ε L(u) and x' ^ χ imply χ' ε L(u). P.3 If (a) χ > 0, or (b) χ > 0 and (λ · χ) ε L(u) for some X > 0 and ΰ > 0, the ray {λχ | λ ΞΪ 0} intersects L(u) for all u ε [0, +oo). P.4 u2 ^ U ^ 0 implies L(U ) C L(U ). 1

2

i

Π L(u) = L(U ) for Uo > 0. P.6 Π L(u) is empty. ιιε[0, + οο) P.5

O^u^uo

0

P.7

L(u) is closed for all u ε [0, + oo).

P.8 P.9

L(u) is convex for all u ε [0, + oo). E(u) is bounded for all u ε [0, + oo).

The Properties P.l,. .., P.9 are taken as valid for any technology. Property P. 1 states merely that any nonnegative input vector yields at least zero output (a truism), and positive output cannot be obtained from a null input vector. Property P.2 implies disposability of inputs. For example, if chemical fertilizer is used as an input with land to pro­ duce a crop and excessive amounts of fertilizer have been provided, one merely disposes of the surplus. Fortuitous events, such as floods supply­ ing excess water, are not encompassed. Excess capacity of machinery and equipment imply merely that the services of such capital are fore­ gone. Thus, the technology is regarded as a rational, controllable arrangement. Property P.3 states first that any output rate u ε [0, + oo) can be realized by scalar magnification of a positive input vector x, although not necessarily in an efficient way, and second that, if a positive output rate can be obtained by scalar magnification of a semi-positive input vector x, any null inputs of χ are not required for production and the same attainability of all output rates holds by scalar magnification of the semi-positive input vector x. Divisibility of output rate is not implied. Property P.4 is clearly appropriate, since an input vector yielding at least an output rate U Si ui also yields at least ui, and Property P.6 is merely a precise way of stating that an unbounded output rate cannot be attained by a bounded input vector. Properties P.5 and P.7 have only mathematical significance. Property P.5 is imposed in order to guarantee the existence of the production function Φ(χ) as the maximum output rate attainable with x. Property P.7 is imposed in order to be able to define the production isoquant for an output rate u as a subset of the boundary of the input set L(u) relative to Rn. 2

THE PRODUCTION FUNCTION

Property P.8 is valid for time divisibly-operable technologies. For ex­ ample, if χ ε L(u), y ε L(u) and θ ε [0,1], the input vector [(1 — θ)χ + θγ\ may be interpreted as an operation of the technology a fraction (1—0) of some vuiit time interval with the input vector χ and a fraction θ with y, assuring at least the output rate u.f Nothing is implied about the efficiency of such an operation. Property P.9 is imposed as an obvious physical fact that no output rate is attained efficiently (in a technological sense) by an unbounded input vector. In the foregoing definition of a technology, nothing is assumed which is peculiar to any particular physical system of production. Substitutions between the factors of production are permitted, both as alternative and complementary means of production. The family of sets L(u) defines the input unconstrained technical possibilities. From the definition of the efficient subset E(u) of a production input set L(u), it is clear that the technologically efficient input vectors belong to the boundary of L(u). Because, suppose χ ε interior L(u). Then there would exist a spherical neighborhood S8(x), centered at x, composed entirely of points of L(u), implying y ε L(u), y < x, a contradiction. The set of efficient points E(u) need not be closed, because there is a counterexample to closure. See [1]. Even so, it is sufficient for our pur­ poses to use the closure E(u) of E(u). Note that E(u) C L(u), since L(u) is closed. It is necessary to verify that the efficient subsets E(u) are not empty for all u ε [0, + oo). Clearly, the null input vector is efficient for u = 0. Hence, consider u > 0 and let BR(O) = {x I ||x|| Si R,x ε Rn}, R > 0

be a closed ball centered at χ = 0 with radius R. One may choose R large enough so that BR(O) Π L(u) is a nonempty, closed and bounded convex subset of L(u). Let IIx0II = Min {||x|| I χ ε BR(O) Π L(u)}. The vector x° exists, since it minimizes a continuous function

over a nonempty, closed and bounded set, and x° ε L(u). Moreover x° ε E(u), because, if y < x°, then y # BR(O) D L(U) since y < X0 implies Ilyll < ||x°||. Thus, the following proposition holds: f Indeed the input vector [(1 — θ ) χ + 6 y ] may have no meaning unless so interpreted.

THEORY OF COST AND PRODUCTION FUNCTIONS

Proposition 1: The efficient subset E(u) of a production input set L(u) is nonempty for all u ε [0,+ oo). Each production input set L(u) may be partitioned into the sum of the efficient subset E(u) and the set D = (χ | χ Ξϊ Ο,χ ε Rn}, and the fol­ lowing proposition holds: Proposition 2: L(u) = E(u) + D = E(u) + D. The operator symbol + is used to denote the usual addition of sets, i.e., E(u) + D is the set of all input vectors of the form (x + y) where χ ε E(u) and y ε D. First, we show that (E(u) + D) C L(u). Since E(u) is nonempty, let χ ε E(u) and y ε D. E(u) C L(u) implies χ ε L(u), and (x + y) ε L(u) due to Property P.2 since (x + y) ^ x. Thus, any input vector belonging to (E(u) + D) also belongs to L(u). Next, we show that L(u) C (E(u) + D). Let y ε L(u) be arbitrarily chosen. The vector y belongs to the closed ball Bn yn (0). Define Dy = (x I χ 2: 0,x ^ y} and let K(u) = {λχ I χ ε E(u),A > 0}. The intersection L(u) D Dy is a bounded, closed subset of L(u). There are two cases to consider: (a) y ε K(u), (b) y f K(u) (see Figures 1 and x,

2

L(u)

x.

FIGURE 1:

ytK(u)

THE PRODUCTION FUNCTION

K(u)

FIGURE 2:

y f K(u)

2). If y ε K(u), the ray {fly | 0 2:0} intersects E(u) at a point x, and y = χ + (y — x) with (y — χ) ε D since y Ξ5 x. Hence, for Case (a), y ε (E(u) + D). In Case (b), consider}·

Minjy1i Zi I ζ rg y,ζ ε K(u) Π Dy Π L(u)j The set K(u) Π Dy Π L(u) is not empty and the minimum exists. Let χ denote the vector yielding this minimum. Then χ ε E(u) and y = χ + (y — χ) with y ^ χ, so that y ε (E(u) + D). Consequently L(u) C (E(u) + D) and L(u) = E(u) + D. _ The equality between L(u) and (E(u) + D) is verified simply, as fol­ lows: (E(u) + D) C L(u), because, if ζ ε (E(u) + D), ζ = χ + y with χ ε E(u), y ^ 0, and χ ε L(u) since E(u) C L(u), whence ζ ε L(u) due to Property P.2. Conversely, L(u) C (E(u) + D), because (E(u) + D) C (E(u) + D), and L(u) = (E(u) + D) C (E(u) + D). Another subset of the boundary of a production input set L(u), called the production isoquant, is of use in the theory of production. Definition: The production isoquant corresponding to an output rate u > 0 is a subset of the boundary of the input set L(u) defined by {x I χ > Ο,χ ε L(u),X-x # L(u) for λ ε [0,1)} t This proof suggested by K. Arrow (see Math. Reviews, 5460, 1969) is simpler than the original proof given by the author in Unternehmensforschung, Heft 4, 1967.

< Π >

THEORY OF COST AND PRODUCTION FUNCTIONS

O FIGURE 3: SINGLE FACTOR MIX ALTERNATIVE

The isoquant for u = 0 is {0}. The production isoquant for u ε [0,+ oo) is a closed subset of L(u), and the definition applies whether or not the output rate u exists as a Max{u I χ ε L(u)} for some χ ε D since the sets L(u) are defined for all u ε [0,+ oo). Various production isoquants are illustrated in Figures 3, 4, 5, 6 for two factors of production. In Figure 3, there is illustrated a technology for which the two factors of production can be used efficiently only in a fixed proportion, typical of the Leontief model of production. In Fig­ ures 4, 5, and 6, the efficient subsets of the isoquants are indicated by darkened lines. Figure 4 illustrates a technology with 4 alternatives of mixing two factors of production. Figure 5 illustrates that a factor of production need not be essential. Figure 6 shows the usual neoclassical continuity of substituting one factor for another, with both factors of production nonessential. Notice that in all four figures the efficient sub­ sets are bounded, but the isoquants need not be bounded. For each figure the production input set L(O) is bounded by the positive axes with the null input vector the single efficient point. The production in­ put sets are convex and closed, and for Ui < U2 < u3 these sets are nonincreasing with each contained in its predecessor. Also if an input vector χ belongs to one of the input sets any input vector at least as large as χ also belongs to that input set. Further if χ > 0 and the ray (λχ I λ ^ 0} intersects an input set for positive output, the ray inter-

THE PRODUCTION FUNCTION

FIGURE 4: SEVERAL FACTOR MIX ALTERNATIVES

FIGURE 5: LINEAR SUBSTITUTION WITH ONE FACTOR NONESSENTIAL

THEORY OF COST AND PRODUCTION FUNCTIONS

FIGURE 6:

SMOOTH SUBSTITUTION WITH BOTH FACTORS NONESSENnAL

sects all input sets. Properties P.5 and P.6 cannot be illustrated in such figures. Observe that for each figure L(u) = E(u) + D, and a similar decomposition holds in terms of the production isoquant. 2.2 Definition and Properties of the Production Function

The production function is a mathematical form defined on the pro­ duction input sets of a technology, with properties following from those of the family of sets L(u), u ε [0,+ oo) which can be best understood this way instead of making assumptions ab initio on a mathematical func­ tion. For any input vector χ ε D, consider a function Φ(χ) defined on the sets L(u) by Φ(χ) = Max{u I χ ε L(u),u ε [0,+ oo)}, χ ε D

(2)

giving to the production function Φ(χ) the traditional meaning as the largest output rate obtainable with x. It is not obvious that Max exists for all χ ε D, and this fact needs to be proved first. Let χ ε D be chosen arbitrarily. The input vector χ belongs to L(O), see Property P.l, and there exists a finite value ΰ > 0 such that χ ^L(H), due to Properties P.4 and P.6. Hence, Sup{u | χ ε L(u), u ε [0, + oo)} = U0 is finite. Then, it follows that χ ε L(u) for u ε [0,Uo) and Property P.5

THE PRODUCTION FUNCTION

implies that χ ε L(U0). Thus, χ ε L(u) for u belonging to the closed inter­ val [0,uo]. Consequently, Φ(χ) exists, it is finite for bounded χ and Φ(0) = 0, solely due to Properties P.l, P.4, P.5 and P.6. The significance of Property P.5 is evident. It guarantees the existence of the production function and was introduced for this purpose. One may then define a production function using only these four properties of a technology, but such generality omits the substance of a technology which differen­ tiates it as a production structure. From Property P.2 it follows directly that Φ(χ') 2: Φ(χ) for χ' 2: χ, since {u I χ' ε L(u)} D {u I χ ε L(u)} is implied because χ' ε L(u) if χ ε L(u). The ray Property P.3 implies an interesting and useful property for the production function. If χ > 0, or χ > 0 and Φ(λχ) > 0 for some positive scalar λ, the value of Φ(λχ) may be made to exceed any output rate u ε [0, + oo) by choosing a sufficiently large scalar λ, i.e., Φ(λχ) —» + oo as λ —» -)- oo. The closure Property P.7 implies that the production function Φ(χ) is upper semi-continuous. In order to see this, we need to consider the level sets of the function Φ(χ) defined by L4(U) = {x I Φ(χ) 5; u}, u e (— oo, + oo) For u ^ 0, L41(U) = D since any input vector χ ε D yields at least zero output and any value of u less than zero. For u > 0, L4(U) — (x I Max (v I χ ε L(v)} Sr u).

If x° ε LI(U), Max {ν | x° ε L(u)} =S u, implying x° ε L(u) due to Property P.4, and L41(U) C L(u). Conversely, if x° ε L(u), Φ(χ°) Ξϊ u implying X0 ε LI(U) and L(u) C L41(U). Thus, for u Sr 0, L41(U) = L(u), i.e., the level sets of the production function Φ(χ) for u ε [0,+ oo) are identical to the production input sets L(u). Since the input sets L(u) are closed for u ^0, see Property P.7, it then follows that the level sets L4(u) of the production function Φ(χ) are closed for all u ε ( — oo, + oo) which is equivalent to the property of upper semi-continuity for Φ(χ). See Ap­ pendix 1 for the definition of upper semi-continuity and this equivalence. Finally, the convex Property P.8 of the input sets L(u) implies that the production function Φ(χ) is quasi-concave, i.e., for χ ε D, y ε D and θ ε [0,1], Φ((1 - θ)χ + θγ) ^ Μΐη[Φ(χ),Φφ)]. The convexity of the level sets L®(u) of the production function for all u ε (— oo, + oo) is synonymous with this property. See Appendix 1.

THEORY OF COST AND PRODUCTION FUNCTIONS

The properties of the production function Φ(χ) implied by those of the input sets L(u) of the technology on which it is defined are sum­ marized in the following proposition. Proposition 3: The production function Φ(χ) = Max{u | χ e L(u),u ε [0, + oo) }, χ ε D, defined on the input sets L(u) of a technology with the Properties P. 1,..., P.9, has the following properties: A.l

Φ(0) = 0.

A.2 Φ(χ) is finite for all χ ε D. A.3 Φ(χ') = Φ(χ) if χ' = XΑ.4 For any χ > 0, or χ > 0 such that Φ(λχ) > 0 for some scalar λ > 0, Φ(λχ) -» + oo as λ ^ + oo. A.5 Φ(χ) is upper semi-continuous on D. A.6 Φ(χ) is quasi-concave on D. The implications of Property P.9, i.e., the boundedness of the efficient subsets E(u) of the input sets L(u), have not been stated. However, the connection between the level sets of the production function Φ(χ) and the input sets of the technology, on which Φ(χ) is defined, is given in the proposition: Proposition 3.1: The level set Lt(u) = {χ | Φ(χ) ^ u} of a production function Φ(χ) defined on a technology T: L(u), u ε [0,+ oo) is iden­ tical to the input set L(u) for each u 0. and Property P.9 implies that the efficient subset of the level set L41(U) is bounded for all u ε [0,+ oo). Thus, it is clear that any function Φ(χ) satisfying A.1,..., A.6, with bounded efficient subsets for its level sets, defines a technology by the sets L (U), u ε [0,+ oo). From the definition of the production function Φ(χ), it is clear that this function is unique for any technology T on which it is defined. Hence, the structure of production may be defined either in terms of a suitable production function Φ(χ) or by a family of production input sets L(u), since one is uniquely determined from the other. In either case, the Properties A.1,... ,A.6 with bounded efficient subsets of L„(u), u ε [ 0 , + o o ) f o r t h e p r o d u c t i o n f u n c t i o n Φ ( χ ) , o r t h e P r o p e r t i e s P .1 , . . . , P.9 for the input sets L(u) of the technology T, seem natural for the un­ constrained technical possibilities of production. These notions have application for an economic theory of the firm, where given facilities and limited availability of the factors of produc­ tion are involved, merely by restricting consideration to an appropriate subset of the domain D of the input vectors x. il

THE PRODUCTION FUNCTION Φ(Xxu)

u.

'3

2

u

u.

1

O

u

0 FIGURE 7: PROFILE OF THE PRODUCTION FUNCTION

Qualitatively, one may comprehend the generality of the production function by considering profiles of the function Φ(χ). Let x° be an input vector such that Φ(Χχ°) > 0 for some positive scalar λ, and consider the profile Φ(λ · x°) for λ ε [0, + oo) as illustrated in Figure 7. The profile is nondecreasing (A.3) and may have discontinuities as indicated at A0, A2, λ3. At each point of discontinuity the profile is continuous from the right, (A.3) and (A.5), and all values of u ε [0,+ oo) are realized by dis­ posal of output when needed, as for u ε (ui,u2). The profile may be zero for inputs less than some value like (λ0 * x°) and positive but constant over some intervals like [Aix0jA2X0).

23 Transforms of Production Functions Let Φ(χ) be a production function with the Properties A.1,..., A.6. The transform of Φ(χ) is defined by: Definition: A function Ρ(Φ(χ)), where F( ·) is a finite, nonnegative, upper semi-continuous and nondecreasing function of Φ(χ) with F(O) = 0, is a transform of the production function Φ(χ).

THEORY OF COST AND PRODUCTION FUNCTIONS

If ν ε [Ο,οο) is an index of the values of Φ(χ) and u is an index of the values of F(v), then u ranges over a set J C [Ο,οο) as ν ranges over [Ο,οο) and the graph of F(v) may be unbounded, or bounded and tend­ ing toward an asymptote, asv-> oo. (See Figures 8 and 9.) Also, F(v) may be concave or convex for various intervals of v. For transforms of production functions, the following proposition holds: Proposition 4: If F(v) —» oo as ν -» oo, then as a function of χ the transform F((x)) is a production function for an unconstrained technology with the Properties A.1,. .., A.6. The transform F( χ may introduce positive inputs of some factors which are zero in x, involving the application of new technical apparatus and facilities the significance of which for output is a jump in the level of input. Here the scalar measure Φ(χ) measures the input χ according to its significance in this respect. Last, Property A.6 is certainly a necessary property for interpreting Φ(χ) as a scalar measure of input for time divisible technologies, because if the technology is operated a fraction of the time with an input χ and the remaining fraction of the time for some arbitrary time interval with the input y, the scalar measure of input for the entire interval of time must be at least as large as the smaller of the two scalar measures Φ(χ) and Φ(γ). Thus, at least, the Properties A.1,..., A.6 are ncessary for interpret­ ing Φ(χ) as a scalar measure of input in a production function of the form Ρ(Φ(χ)) where returns to scale are then expressed by the function F(v).

THEORY OF COST AND PRODUCTION FUNCTIONS

Now ordinarily a property of linear homogeneity is demanded for a scalar measure of input, i.e., Φ(χ) would be taken homogeneous of degree one in χ so that Φ(λχ) = λΦ(χ), implying that a multiple of an input combination χ has scalar measure which is the same multiple of the scalar measure of x. This assumption leads us to a special and more restricted class of production functions which we shall designate as homothetic [26] and consider in the next section. As a final comment, the Properties A.1,..., A.6 are appropriate for the interpretation of the function Φ(χ) as the utility function of an indi­ vidual, when satiation is not assumed. Whereas, if for any consumption bundle χ the utility Φ(λχ) is bounded as λ —» oo, A.4 may be dropped or retained for Φ(χ) in a utility function Ρ(Φ(χ)) where the function F(v) is bounded as ν —> oo. This kind of correspondence between production and utility function is not surprising, because both have the properties of a nonnegative, nondecreasing transfer function with free disposal. 2.4 Homothetic Production Functions

Here we consider, in some detail, transforms Ρ(Φ(χ)) of production functions Φ(χ) which are homogeneous of degree one, allowing Φ(χ) to be interpreted as a conventional index number of the level of input of the factors of production. Definition: A homothetic production function is one of the form Ρ(Φ(χ)), where Φ(χ) is positively homogeneous of degree one in χ in addition to having the Properties A.1,..., A.6 and Ρ(Φ(χ)) is a trans­ form of Φ(χ) with F(v) —» oo as ν oo.f The restriction of the homogeneity of Φ(χ) to be linear is no loss of generality in the definition of this class of transforms, because suppose Φ(χ) is homogeneous of degree k > 0. Then v(x) = Φ(χ)1/1{ is homo­ geneous of degree one and we may define a function G( ·) by and

G(v(x)) ΕΞ Ρ(Φ(χ)) G(v(x)) ΞΞ F(v(x)*).

The function G( ·) is a transform of the positively homogeneous func­ tion v(x) of degree one, because: (i) it is nonnegative, (ii) it is finite for finite v(x), since F is a finite function of its argu­ ment and v(x)k is finite, fThe definition of homotheticity originally introduced in [26], see Section 7, p. 41, treated F( ·) as a continuous, strictly increasing function.

THE PRODUCTION FUNCTION

(iii) it is nondecreasing, since F( ·) is a nondecreasing function of its argument and v(x)k is nondecreasing, (iv) G(O) = F(O) = 0. (v) it is upper semi-continuous in v(x), since F( ·) is upper semicontinuous and v(x)k is nondecreasing and continuous in v(x). (See the proof of semi-continuity for Proposition 4). Thus, one may always express the transform of a homogeneous func­ tion of degree k > 0 as a transform of a homogeneous function of degree one. The linear homogeneity property of the production function Φ(χ) is a strong one and considerably strengthens the Properties A.3, A.5 and A.6. Proposition 7: If a production function Φ(χ) with Properties A.1,..., A.6 is homogeneous of degree one in x, it is a super-additive, con­ tinuous and concave function of χ ε D. Consider any two nonnegative vectors χ and y such that Φ(χ) > 0 and Φ(γ) > 0. Then, due to the homogeneity of Φ( ·)

for all u ε (Ο,οο), and the input combinations χ · f(u)/^(x), y · f(u)/0(y) belong to the level set L«(f(u)) of the production function Ρ(Φ(χ)) (see Proposition 6.) Since L4(f(u)) is a convex set, because we have assumed that Φ(χ) is a quasi-concave function of x, it follows that

for all scalars θ satisfying 0 5Ξ 0 :S 1. Take θ= and

H

Φ(χ) + Φφ)

®wW(x

+ y))a,(u)·

Then due to the homogeneity of Φ( ·) it follows that Φ(χ + y) ^ Φ(χ) + Φφ). If either Φ(χ) or Φφ) or both equal zero, the same inequality will hold due to the nondecreasing and nonnegative properties of Φ(·)· Thus Φ(χ) is super-additive on D. Next, using the superadditivity and homogeneity of Φ( ·), it follows that Φ((1 - 0)Χ + 0y) ^ (1 - 0)Φ(Χ) + 0Φφ)

THEORY

OF COST AND PRODUCTION

FUNCTIONS

for any two vectors x and y of D and for all scalars 0 satisfying 0 ^ 0 ^ 1. Thus, is also a concave function on D. For the continuity of observe first that 3»(x) is continuous on the open convex subset of interior points of D due to a well known mathematical theorem for convex functions defined on convex sets [3] stating that a convex function of x e R" is continuous on any open convex subset of R° in the domain of definition, whence and thus is continuous on Di. It remains to show that $(x) is continuous on the boundary of D. It will be shown that is lower semi-continuous for all points on the boundary of D, and then due to the upper semi-continuity assumed on D it follows that is continuous on the boundary of D, whence is continuous for all D. Theorem: A function F(x), which is convex and defined for all points in a closed convex region R of R" formed by the intersection of halfspaces, is upper semi-continuous on the boundary of R. Let

be a boundary point of R, i.e.,

belongs to some hyperplane n

and suppose x e R implies 2 ^iXi ^ 1 (the opposite inequahty being merely a matter of the sign of r). Since R*^ is a linear space of dimension n, there are (n -)- 1) points of R° defining an n-dimensional simplex S containing x^ as an interior point. Let be an open spherical neighborhood of

contained in S. Then

is an open neighborhood of relative to R. Now the set S n R is a closed convex polyhedron, each point of which is representable in the form

where and

points of function on R,

are vertices of for aU indices

Likewise the

are so representable. Then, since F(x) is a convex

f o r T h e n , letting it follows that all and F(x) is bounded in the neighborhood

for Simi-

t In seeking to establish thistheorem for the continuity of the production function on the boundary of D when is homogeneous, I am indebted to my colleague, David Gale, who suggested that it might hold when the region R is an intersection of half spaces. For a more general formulation see [12].

< 32 >

THE PRODUCTION

FUNCTION

larly there is an such that a, i.e., f(x) is positively bounded, for . For any e > 0, restrict x so that Translate the origin of to Then and implies Hence, for

Consequently, for any positive number take , and there exists a neighborhood such that for all , Thus F(x) is upper semi-continuous on the boundary of R since x" was arbitrarily chosen. Now, we may apply this theorem directly to show that is lower semi-continuous on the boundary of D. Simply define and let x® belong to the boundary of D. Then F(x) is convex on a closed convex region D formed by the intersection of half-spaces, and or for any positive provided where and a is a positive bound on F(x) in the neighborhood of Thus is lower semi-continuous on the boundary of D. Homothetic production structures have a special geometric property, expressable in terms of the isoquants (boundaries) of the production possibility sets. Consider the level set for unit output rate. This set may, or may not, intersect the boundary of the nonnegative domain D of input vectors, as illustrated in Figure 12 for two factors of production. For any point x > 0 belonging to the boundary of , let y denote the point of intersection of the ray with the boundary of any other production set as illustrated in Figure 12. The coordinates of x and y are related by where Since is continuous (see Proposition 7), the points x and y satisfy and respectively. Then, it follows from the homogeneity of that

and hence for any output rate

< 33 >

THEORY OF COST AND PRODUCTION FUNCTIONS

χ 1

χ

O FIGURE 12:

ISOQUANTS FOR HOMOTHETIC PRODUCTION FUNCTION

independently of the point χ > 0 on the boundary of the production set Lt(f(l)). If the isoquant of L4^f(I)) intersects the boundary of the domain D = {χ I χ 0}, as illustrated for the coordinate Xi in Figure 12, then on the boundary of D we consider only those points χ such that χ = Min {λζ I ζ > Ο,λζ ε L4(f(l))}. For any other output rate u, the ray λ

{λχ I λ Sr 0} also intersects the boundary of L„(f(u)), since Φ(χ) = f(l) due to continuity of Φ(χ) and it is possible to find a scalar τ > 0 such that Φ(τχ) = τΦ(χ) = rf(l) = f(u) and the pointy = τχ where τ = f(u)/f(l) Ues on the isoquant of Lt(f(u)). Also y = Min {τχ | χ > Ο,τχ ε L„(f(u))}. λ

Thus the following proposition holds. Proposition 8: The isoquant for any output rate u 0 of a homothetic production function (structure) may be obtained from that for unit output rate by radial expansion from the origin in a fixed ratio f(u)/f(l).

THE PRODUCTION FUNCTION

When u = 0 only the efficient point χ = 0 will be generated since f(0) = 0, but the isoquant of L®(f(0)) consists only of this point. We might have defined the isoquant of a production set L»(u) = (χ I Φ(χ) ^ u} for a production function Φ(χ) as the set E(u) of the effi­ cient points of L41(U). Then, the isoquants for any u ^ 0 of a homothetic production function are generated by radial expansion of the efficient points of the set L„(f(l)). This alternative is probably appealing, since the isoquant is intended to represent the set of inputs which yield exactly a given output rate, displaying the substitution between inputs of the factors of production to obtain this output. Even so, the efficient points of a production set L41(U) do not necessarily yield exactly the out­ put rate u unless the production function is continuous, and we have chosen to define the isoquant of a production set L®(u) as a subset of the boundary of this set which excludes only the nonefficient points co­ incident with the boundary of D. Various particular forms of a homothetic production function have been used in theoretical and statistical economic studies. The familiar Cobb-Douglas production function Φ(χ) = ΑΠXi«i, A > 0, Σ B1 = 1 (4) 1 1 is the simplest of these forms, exhibiting constant returns to scale since F(0(x)) = Φ(χ) and Φ(χ) is linear homogeneous. Nerlov [22] has used η this form with Σ ai = r not constrained to be unity for a study of re­ turns to scale in electricity supply. The most general form of the CobbDouglas production function is F(®(x)), with Φ(χ) = A ·Π Xia', A > O1Jal = 1, (4.1) ι ι where F( ·) is any finite, nonnegative, nondecreasing, upper semi-continuous function with F(O) = 0 and F(v) oo as ν -» oo. The special form of (4.1) used by Nerlov is one where Ρ(Φ(χ)) = [Φ(χ)]Γ. Arrow, Chenery, Minhas and Solow [2] use a homothetic production function of the form

Φ(χ)

+ a2x2_b]"1/b f(u) Ξ u, ai > 0, a2 > 0, b > -1 = [aixrb

for a discussion of capital labor substitution and economic efficiency. This function, which has come to be known as the ACMS production function, is linear homogeneous and requires constant returns to scale. Uzawa [29] has generalized this production function to accommodate η factors of production grouped in S categories, giving it the form

THEORY OF COST AND PRODUCTION FUNCTIONS

= π 1

{{ς^ Ύ "' 1

f '

S

(6)

f(u) = u, aj > 0, bj > -1, C j > 0,^ C j = 1 ι for a production function with category constant elasticities of substitu­ tion. Again, the production function is linear homogeneous, demanding constant returns to scale. A more general form of the Uzawa-ACMS homothetic production function can be obtained by taking f(u) as a nonnegative, nondecreasing lower semi-continuous function with f(u) = 0 and f(u) -» oo as u —» oo. The special forms used for the linear homogeneous function Φ(χ) have no obvious justification. Why demand constant elasticities of substitution? However, the Cobb-Douglas form of Φ(χ) does lend itself to a simple interpretation of the parameters involved (see Chapter 6, Section 6.3) and provides a convenient form for aggregation of the fac­ tors of production. But is should be remarked that any nonnegative, nondecreasing, superadditive and linear homogeneous function of the input vector χ with Φ(0) = 0 will do for a homothetic production func­ tion. Moreover, not all factors of production need be essential, as they are in the Cobb-Douglas form, but not so in the Uzawa-ACMS form of Φ(χ), and the isoquants need not be smooth curves, i.e., Φ(χ) need not be differentiable. Also, the efficient set E(u) of a production set L (f(u)) need not be unbounded. In fact, it is essential on technological grounds to require bounded efficient sets. Homothetic production functions are useful for the study of returns to scale, since they provide via the function f( ·) a means of convenient formulation for such investigations, and, as we shall see in Chapter 6, Section 6.2, cost data may be used to estimate the function F( ·). What is meant by scale needs to be clarified. It should refer to scale of input, i.e., a scalar measure of the level of input, and, for any vector of inputs χ which yields positive output, replication of this input by a positive real number λ results in various scales (levels) of input. The variation of the function F( ·) is independent of the particular input vector χ being rep­ licated. This is the special property of homothetic production functions (see Proposition 8). Whether the returns to scale are in fact so inde­ pendent is another matter. It is clear that the measurement or deter­ mination of returns to scale is a complex problem. Perhaps one should seek tests for homotheticity in econometrics. 2.5 A Classification of the Factors of Production

Notions of "complementary" factors of production and "complemen­ tarity" of groups of factors have been used in the economic theory of

THE PRODUCTION

FUNCTION

production (see Dano [6], pp. 16, 103) to distinguish various kinds of limitational character of factor inputs. These ideas aim at describing how efficient inputs of individual factors or groups of factors may be limited by some strict functional relationship to output rate, the extreme case being where all factor inputs are required to be in fixed proportions to output rate. These matters are precisely determined by the structiu-e of the efficient point sets for the family of production possibihty sets L(u) which describe the technology, and the production function 0(x) = Max expresses these relationships without requiring constraint equations on the inputs x. Whether a factor of production or groups of factors are essential to output is still another matter which can be investigated in the following way. The boundary points of the nonnegative domain of factor inputs represent an input combination for which one or more factors of production have zero input. We classify these boundary points by the following definitions. Definition: The set of points D2 consists of all the boundary points of D except and the boundary of D equals Definition: The set of points is the subset of iti' factor of production is zero.

for which the input of the

Definition:

The set of points inputs of the factors

is the subset of are zero.

for which the

Definition: Interior of The interior of the set consists of the input vectors for which the input of the factor of production is zero and the inputs of all other factors are positive. Definition: Interior of for j The interior of consists of the input vectors for which the inputs of the group of factors are zero and the inputs of all other factors are positive.

< 37 >

THEORY

It is clear that interior of some set

OF COST AND PRODUCTION

FUNCTIONS

and each point of D2 belongs to the where

Proposition 9: If Interior of and then for aU and the essential by itself.

for all factor of production is

The proof of Proposition 9 follows directly from the nondecreasing property of the production function Let x be a point belonging to the interior of for which for all , Then, for any vector y belonging to D2(i) there exists a positive scalar such that and whence since . Thus, if the production function is zero along a ray belonging to the interior of it is zero for all inputs of D2(i) and no positive output can be obtained with zero input of the i''' factor of production. Proposition 10: If x e Interior of and for some then for all input vectors y belonging to the interior of D2(i) there exists a scalar such that and by itself the i'^ factor of production is nonessential. Now suppose x belongs to the interior of D2(i) and

_

for some

Then, for any vector y belonging to the interior of there exists a scalar jx such that and Thus the itii factor of production by itself, i.e., only with is nonessential, because for any positive inputs of the other factors it is always possible to apply them in sufficient scale to obtain a positive output. Groups of factors may be essential, that is, with zero inputs for a subset of the factors of production no positive output may be obtained, even though they are not essential individually. Proposition 11: If x e Interior n, and 0 for all then for aU and the group of the factors of production is essential. The proof of this proposition is exactly analogous to that given for Proposition 9. Proposition 12: If Interior and 0 for some then for all input vectors y belonging to the inte rior of there exists a scalar such that 0, and the group of the factors of production is non essential. The proof of this proposition is similar to that given for Proposition 10.

< 38 >

THE PRODUCTION FUNCTION

2.6 The Production Function of a Limited Unit The large part of the usual analyses of production functions is con­ cerned with "short run" production functions related to some firm or other unit with limited resources. These limitations may take a variety of forms. If certain capital services are not available to the physical apparatus of the unit, then the inputs of certain factors of production are constrained to be zero. Those capital services available may be Hmited in capacity and the inputs per unit time of certain factors are then bounded by certain positive constants. Financial restrictions or other institutional constraints such as labor contracts may require cer­ tain linear constraints on the factor inputs. For example, if a labor agreement requires at least a certain proportion of labor services per unit service of some equipment, then the constraint may take the form Xi ^ axj.

All of these constraints can take the form Ax ^ b where A is an (m χ n) matrix of real coefficients, χ is an n-component column vector and b is an m-component column vector of nonnegative coefficients. The inequalities Ax ^ b define a closed convex subset of Rn containing the origin 0 and the domain of feasible input vectors is D = D η {x I Ax^ b}

(7)

Then, the production function of a Umited unit is merely stated by Φ(χ); χ ε t)

(8)

where Φ(χ) has the Properties A.1,..., A.6 given in Section 2.2. For this formulation all the inputs which are relevant to the technology should be included in the Ust of the factors of production, and by "the tech­ nology" is meant the totality of aU feasible engineering arrangements which may yield the given output. We assume, of course, that the set £) is not empty and contains at least one point χ > 0 such that Φ(χ) > 0, otherwise the constraints on the factor inputs preclude positive output. Thus, the production function of a limited unit is the production func­ tion of the technology in which it operates, with the inputs of the factors of production constrained to a convex subset of the domain D of nonneg­ ative input vectors which contains the vector χ = 0. The dimension of the set D, i.e., the dimension of the smaUest Unear space containing D, will generally be less than n, and we assume that none of the factors with zero constrained inputs or any grouping of such factors is essential, otherwise positive output is impossible. FinaUy D may be bounded. The production possibiUty sets of the limited unit are subsets of the family of sets L (U), u ^ 0 for the production function of the technoli

THEORY OF COST AND PRODUCTION FUNCTIONS

ogy, formed by the intersections of D with the sets Lw(u). Denote them by Lw(u), and (9) The properties of the constrained production sets LoI>(u) are given in the following proposition.

Proposition 13: The properties of the production possibility sets Lw(u) of a limited unit operating in a technology with production function and with inputs constrained to a convex subset D, are

> O.

P.l

Lw(O) = D and x = 0 does not belong to Lw(u) for any u

P.2

If xED, x' E D, x' > x and x E Lw(u), then x' E Lw(u).

P.3

If x 0, or x ~ 0 and (.\x) E Lw(u) for some .\ 0 and u 0, the ray {Ax /.\ > O} may not intersect all nonempty sets Lw(u), u>O.

P.4

Lw(uz)

P.5 P.6

>

>

c

Lw(UI) if Uz > UI.

n

Lw(u)

n

Lw(u) is empty.

O~U:S:Uo

uqO,oo)

>

= Lw(uo),

P.7

Lw(u) is closed for any u E[0,00).

P.8

Lw(u) is convex for any u E [0,00).

These properties more or less evidently follow from those of the family Liu). Since Lw(O) D and D C D, it follows that LoI>(O) D. Also, since o¢ Lw(u) for any u 0, 0 ¢ Lw(u) for any u O. Thus Property P.l holds. If x' > x E Lw(u), x E Lw(u), x' E Lw(u) and x' E Lw(u) since x' E D, and therefore Property P.2 holds. Regarding P.3, a strictly positive ineut vector x mayor may not belong to D and hence not to any subset Lw(u), depending upon whether or not the constraint set D implies one of the factor inputs is zero. In any event, the validity of Property P.3 for x ~ 0 is evidenced by the example of Figure l3. For Property P.4, we note that Lw(UI) mayor may not be empty. If Lw(UI) is empty, then LoI>(uz) is empty since Lw(uz) C Lw(UI), and Lw(uz) C Lw(UI)' If LiuI) is non empty, then Lw(uz) mayor may not be empty. If Lw(uz) is empty it is certainly contained in Lw(UI) because each point of Lw(uz) (there are none) belongs to Lw(Ul). If Lw(uz) is nonempty, x E Lw('!z) implies x E Lw(uz) C L.iuI) and xED, so that x ELw(UI), and Lw(uz) C Lw(UI). The proof of Property P.5 follows that of P.4. First, Lw(uo) mayor may not be empty. If Lw(uo) is empty, then Lw(uo) C Lw(u), be-

=

>

>

=

n

Uf[O,UO)

cause all points of Lw(uo) (there are none) certainly belong to each set

( 40

>

THE PRODUCTION FUNCTION

~~

____

~

____

~

__

~~

__

~

____

~

____________-+ xl

o

xl FIGURE 13:

COUNTER EXAMPLE FOR THE RAY PROPERTY; f> = {XIXl~xf,x2~X~X;:;;0} L.(u) = {x IIIXt + bX2 ;:;; u,x ;:;; O}

t.(u) for u € [O,uo) and hence to their intersection whether or not the latter be empty. If t.(uo) is nonempty, then

n

U.[O,I1o)

t.(u) is nonempty

because, by Property P.4, t.(uo) is a subset of all t.(u) for u € [O,\lv) and t.(uo) is contained in their intersection. Thus t.(uo) c t.(u). It remains to show that t.(\lv)::J •

n

u.[O,Uo)

n

.

ue[O,Uo)

t.(u). Suppose that x €

n

Ue{O,I1o)

t.(u)

< \lv. Consequently, there exists an output rate U > °such that (x) < u < \lv. But, since x n t.(u), U.[O,I1o)

and x , t.(uo). Then x € f> and (x)

€

(x) > u for all u € [O,uo), a contradiction. If

n

U.[O,I1o)

t.(u) is empty, it

certainly belongs to t.(uo). Since

n

u.[O,oo)

L.(u) is empty, a fortiori

n

ue[O,oo)

t.(u) is empty, and Prop-

erty P.6 holds. Finally, since f> is a closed convex set and the sets L.(u) are closed and convex, it follows that for any u > t.(u) is either a closed, convex nonempty subset or t.(u) is empty in which case it is also closed and convex. Hence Properties P.7 and P.S hold. Thus, with the exception of the ray property, i.e., P.3, the production

°

< 41

)

THEORY OF COST AND PRODUCTION FUNCTIONS

oossibility sets for a limited unit have the same properties as those for the unconstrained technology. If one were to start with the production sets L (U) as defining the struc­ ture of production for the limited unit, the corresponding production function is 41

Φ(χ) = Max {u I χ ε L (U)) = Max {u I χ e D Π L„(u)} 4

from which it is evident that, if the input vector χ is restricted to the subset D, Φ(χ) = Max {u I χ ε Lij(U)} = Φ(χ), the existence of which has already been established in Section 2.2. Thus, the production function of a Umited unit is in no way different than the production function of the technology in which it operates, the differ­ ence being merely a restriction of the input vectors to some closed, con­ vex subset £) of the domain of all possible nonnegative input vectors. 2.7 Law of Diminishing Returnsf *

The previous sections of this chapter have been addressed to a math­ ematical model of production in general terms, without assumptions concerning the fine structure of production. A question of some interest is whether this model implies a law of diminishing returns for the phys­ ical output of production, stating in some fashion that the output will suffer decreasing increments or decreasing average return if the inputs of some factors of production are fixed and the others are increased in­ definitely by some equal increments. For more than 200 years since such a law was first expressed for ag­ riculture, with land as a fixed factor, by the physiocrat Turgot (1767),f various arguments have been put forth to justify a law of diminishing returns. The classical arguments contended that if the law did not hold, the output of every piece of land could be unbounded and the agricul­ tural products needed could be met by using a small area with suffi­ ciently large outlay of other factors. Boehm-Bawerk's argumentff re­ duces to the contention that, if output is strictly increasing with inputs and the production function is sub-homogeneous, average return for t* The contents of this section have appeared in the Zeitschrift fur Nationalokonomie, No. 1-2,1970, and are used here in modified form with the permission of Springer-VerlagWien/New York. f Anne Robert Jaques Turgot: "Observations sur Ie Memoire de M. Saint-Peravy" republished in Oeuvres de Turgot, Ed. Daire, Vol. 1, pp. 418-433, Paris (1844). See also J. A. Schumpeter: History of Economic Analysis, pp. 259-260, Oxford University Press, N.Y. (1954). ft Boehm-Bawerk, Gesammelte Schriften, Vol. 1, pg. 198.

THE PRODUCTION FUNCTION

scalar extension of an input vector is strictly decreasing over the whole range of the scalar extension. Wicksell's prooff implies three assump­ tions: (a) that the production function is sub-homogeneous, (b) the pro­ duction function is super-additive and (c) that output is positive for land as the single positive input. For a penetrating, albeit entertaining, discussion of such arguments see the two papers of K. Mengerff (1936), subsequently reissued in Eng­ lish in Economic Activity Analysis (1954), edited by Oskar Morgenstern, as "The Logic of the Laws of Return—A Study in Meta-Economics." Menger shows that there has been considerable confusion in the state­ ments of the law and the arguments adduced for it. With the advent of the notion of a production function (circa 1910), the law has been implied by the mathematical properties assumed for the production function. Most recently, Eichhornfff (1968), prompted by the issues raised by Menger, has deduced an over the whole range (of inputs) law of diminishing product increments and average return, from assumptions that the production function is homogeneousof degree one and also homogeneous in any combination of (n — 1) factors of production. Divorced of its reference solely to agriculture, a law of diminishing returns is taken as a fundamental proposition for technology to support economic theories of equilibrium and price determination. In this sec­ tion a proof of a form of the law is given as a deduction from the basic properties given in Section 2.1 for a production technology, particularly by Property P.9 that the efficient subsets E(u) of the production sets L(u) are bounded for all u ε [0, -(- oo). Also, a discussion is given of the implications for commonly used production functions such as the Cobb-Douglas and CES functions. To begin with, since the law is a statement concerning technology, the following definition of a technology is used.f * Definition: A production technology is a family of production input ( p ossibility) sets T: L(u), u ε [0, + oo) with the Properties P.l,..., P.8 and the Property P.9 that the efficient subsets E(u) are bounded for all u ε [0, + oo). t Thuenen Archiv 2, pg. 354, (1909). ft K. Menger: "Bemerkungen zu den Ertragsgesetzen," Zeitschrift fttr NationalOkonomie, Vol. VII, pp. 25-26, (1936), and "Weitere Bemerkungen zu den Ertragsgesetzen," ibid, pp. 388-397. ftt W.Eichhorn: "Deduktion der Ertragsgesetzen aus Pramissen," Zeitschrift fur Nationalokonomie, Vol. 28, pp. 191-205, (1968). t*This definition is restricted to the case involving a single output, since the law classically refers to this situation. See Chapter 9, Section 9.1, for the more general defini­ tion as a production correspondence.

THEORY

OF COST AND PRODUCTION

FUNCTIONS

The Property P.9 asserts that technologically efficient production of an output rate u is not made with an input vector which has infinitely large apphcation of any factor of production. A further property seems reasonable, although it is not needed for the arguments to follow: SH For X ^ 1, u e [0, + oo), AL(u) C L(Xu) and -iL(u) D stating that the technology is super-homogeneous. Property SH may be justified for the following reasons. If an input vector x realizes at least the output rate u, i.e., x e L(u), and X ^ 1, then (A • x) may realize at least the output rate (X • u), merely by time-divisiblef replication of the arrangement with x producing u, but X • L(u) may be a proper subset of L(X • u), since, if x is an efficient input vector for u, i.e., x e E(u), then (X • x) may not be efficient for the output rate (X • u). For the same reasons, if X e L(u/X) and X ^ 1, then (X • x) will yield at least the output rate u, implying x e (l/X)L(u) and L(u/X) C (l/X)L(u). A direct consequence of Property SH is A.8

0(Xx) ^ X$(x) and

< 1 $(x) for X ^ 1 and x e D,

i.e., the production function is super-homogeneous. Further, Property A.8, taken with the convexity Property P.8 of the input sets L(u), implies A.9

$(x + y) ^ a»(x) -t- «>(y) for x, y e D,

i.e., the production function is super-additive. Thus, it would appear that the properties of sub-homogeneity and super-additivity implied by Wicksell's argument are contrived to obtain the result sought. In order to verify the Properties A.8 and A.9, following from SH, note first that if u = 4>(x), whence x e L(u), SH implies (X • x) e L(Xu) for X ^ 1, and $(Xx) ^ Xu = X$(x), while, if (u/X) = $(x/X) and (x/X) e L(u/X) C (l/X)L(u), then x e L(u) and $(x) ^ u = X$(x/X), whence ^•(x/X) ^ (1/X)4)(x). Next, let $(x) > 0, $(y) > 0, and define u = Max [$(x),$(y)]. Then, since A.8 holds,

implying that the input vectors (u/$(x)) • x and (u/0(y)) • y belong to L(u). The convexity of the production sets L(u), i.e., Property P.8, then implies for 0 ^ 0 ^ 1, that t Time-divisibility is also involved in Property P.8, i.e., the convexity of the input sets L(u).

< 44 >

THE PRODUCTION

FUNCTION

Take

and use the super-homogeneity Property A.8 to obtain

whence If either or _ or both are zero, this same inequality holds by virtue of Property A. 3 for the production fimction. The definition of a technology given above permits substitutions between the factors of production, both as alternative and complementary means, to attain efficiently any given output rate u and it is not assumed that a positive input of any particular factor of production, or positive inputs for any combination of the factors (except aU), are required for positive output, nor that a positive bound upon the inputs of a factor or combination of factors Umits the output which may be realized imder increasing applications of the other factors. In a word, we have been concerned wiA a general model for the unconstrained alternatives of a technology. Further, nothing has been assumed about the fine structure of the technology, and no premises which are contrived to obtain a law of diminishing returns have been made. For an investigation of a law of diminishing returns we must turn our attention to the possible limitational character of the factors of production. The classification of the factors of production and propositions related thereto, given in Section 2.5, suit our purposes in this respect. Definition: The combination n factors of production is essential if and only if L(u) is empty for all

of the

Propositions 9, 10, 11, and 12 of Section 2.5 show that either a combination of the factors of production is essential by or for all y e Interior there exists a scalar such that positive output may be obtained with the input vector . If a combination is essential, clearly any combination is likewise essential for (n — 1). The combination consisting of all factors of production is obviously essential by virtue of Property P.l or A.1, but this does not imply that any factor or lesser combination of factors is essential. In fact, the Properties P . l , . . . , P.8, P.9 used for the definition of a tech-

< 45 >

THEORY

OF COST AND PRODUCTION

FUNCTIONS

nology do not imply the existence of an essential combination for Now suppose for a combination of the factors of production that a positive bound is imposed on the inputs of these factors. How may this bound impose a limitation (if at aU) upon the outputs which may be obtained under unrestricted application of the other factors? For the investigation of this question two definitions are introduced: Definition: A combination of the factors of production is Weak Limitational if there exists a positive bound such that is bounded for and

Definition: A combination of the factors of production is Strong Limitational if, for all positive subvectors is bounded for and Clearly, if a combination is strong limitational it is weak limitational. Also, it would seem that if a combination is not essential it would not be limitational in either sense. To pursue this issue, suppose that a combination is not essential. Then, it follows from Propositions 10 and 12, Section 2.5 that for all input vectors y e interior of there exists a positive scalar such that and it follows from Property A.4 of the pro duction function that, for any input vector y e interior of is unbounded for Consequently, the combination is not weak limitational, and the following proposition holds: Proposition 12.1: A combination of the factors of pro duction is limitational (weak or strong) only if it is essential. Next suppose that a combination is essential. Then for all For any positive output rate u, due to Proposition 2, Section 2.1. Since is boimded (Property P.9) and closed, there exists a hyperplane which strictly separates E(u) and the closed set of Because all points of L(u) may be expressed as (x -F y) where and there exists a strictly separating hyperplane

t The so-called strict separation theorem, see [3].

< 46 >

THE PRODUCTION

FUNCTION

for L(u) and

then

and if

t

h

e

n

A

n

y

point

belongs to the hyperplane H, and hence there exists a bound such that

is bounded for

because for Hence, the following proposition holds:

implying

Proposition 12.2: A combination weak limitational if and only if the combination is essential.

is

However, essentiahty by itself does not imply that a combination is strong hmitational, a fact which is easily seen from the counter example of Figure 13.1. There, for u ranging over [0,+oo), the efficient subset is the closed line segment where p (0,u) and The family of sets so generated clearly satisfies Properties P . l , . . . , P.8, P.9 for a technology. Further, Property SH also holds. To see this, we need only consider the efficient subset E(u), the points of which are given by:

In order that that or that

for all

for

and and compute

Since arbitrary

for Now,

and

. Clearly,

it is sufficient to show

for

and

Hence, take

is strictly convex in and

< 47 >

for

THEORY OF COST AND PRODUCTION FUNCTIONS

u

1

-

~------------------~~--------____ ~~________----

---

Q(u,1 - e

-u

)

/' -1 / ' Q(1,1 - e )

/

/ o

1 FIGURE 13.1:

= PQ, L(u) = E(u) + {x I x ~ O}

COUNTER EXAMPLE: E(u)

F'(1)

= 1_

>0

1+ u eU

>

for arbitrary u O. Hence, F(A) > 0 for A > 1 and u > O. Similarly, in order that x e (l/A)L(u) for x e E(U/A) and A > 1, it is sufficient to show that G(A)

= A(1

- e- u /)..)

for all U > 0 and A > 1. Since G(A) u 0 arbitrarily and

>

G'(A)

(1 - e-u ) > 0

-

=0

for u

=1-

e-U /)..

=1_

1 + U/A

_

eU /)..

< 48 >

E. e-U /).. A

>0

=0

and A > 1, take

THE PRODUCTION FUNCTION

for all A > 1. Since G(1) = 0 for u > 0 and G(A) is strictly increasing in A for A > I and u > 0, it follows that G(A) > 0 for all A > 1 and u>O. Hence, the example of Figure 13.1 satisfies the properties required of a technology and also the super-homogeneous property. In this example, the factor of production with input denoted by Xl is not essential, while the factor with input X2 is essential because any input vector of the form (Xl,O) does not belong to any input set L(u) for u > 0 and O. For any positive bound xg such that xg I, 0 and 0 0 and 0 0 for j ; {l,2, ... ,k},

THEORY OF COST AND PRODUCTION

for

FUNCTIONS

and

If the second condition of Proposition 12.3 does not hold, it is implied, for some bounded positive inputs for an essential combination of the factors of production, that output is unbounded for unrestricted inputs of the other factors. The previous considerations lead one to consider, more specifically than given in Section 2.6, the production possibility sets of a technology when the inputs of some (but not all) of the factors of production are limited by some positive bounds. Let be a combination of the factors of production and suppose that the inputs of the factors of this combination are bounded by Let

denote subvectors of the input vector x, and take order of the inputs is not important. Define

since the

The sets LO(u) are the production possibility sets for the Hmited operation of the technology when It is rather straightforward to verify that the sets satisfy analogues of the Properties P.4,.. ., P.8 for the production sets of an unrestricted technology, with P.l and P.2 replaced by P.P P.20

If

and

then

Regarding Property P.3, suppose first that the combination is nonessential. Then Propositions 9, 10,11, 12, Section 2.5, imply that if there exists a scalar for any u > 0 such that Then by Property A.3 it follows, for any for which , that

or for any such that P.30(a)

there exists a scalar (depending also perhaps on Thus, the following property holds: If the combination

is nonessential:

< 50 >

THE PRODUCTION

FUNCTION

(i) L''(u) is nonempty for all (ii) For and the ray {xk,Xyk) | X ^ 0} emanating from the point (xk,0) intersects all sets LO(u) for u e [0, + oo). Assume now that the combination is essential. Two situations arise. Either the combination is weak limitational or it is strong hmitational. If it is strong hmitational, is bounded for and not all sets LO(u) are nonempty. If it is weak hmitational, it may happen that is unbounded for Then for and for any u > 0 there exists a scalar such that and the sets | are nonempty for Hence, Property P.3 takes the following second form when the combination is essential: P.3°(b)

If the combination is unboimded for

(i) (ii)

is nonempty for aU the ray intersects all sets

is weak limitationalt and

emanating from the point for u e [0, + oo).

It remains to consider Properties P.9 and SH. The efficient subset of a nonempty hmited production possibUity set is defined by

An efficient subset E(Xk,~)

+ hk)
(Xk'Y(Xk'Y N(e), ~Xk'Y(~) + h0 - «l>(Xk'Y(Xk.Yk + h k) - «l>(Xk,Yk), such that for Yk > Yk, «l>(Xk,Yk

+ hk) -

«l>(Xk,Yk) < «l>(Xk,yk

< 55 >

+ hk) -

«l>(Xk,yk).

THEORY

OF COST AND PRODUCTION

FUNCTIONS

Statement (b) is merely a statement of the possibility illustrated in Figure 13.2. There, for any product increment the product differences are all equal to zero. Weak Law of Diminishing Average Product: If the combination is weak limitational there exists a bound such that is boimded for Then for every such that is finite and, if then

is nonempty,

and

This law is merely a restatement of Property for the production possibility sets L®(u) of a limited technology. See Proposition 13.1. It has the form described by Menger as an "assertion intersecting" a "proposition of diminishing average product," the latter implying that beyond some input i.e., for the average return is strictly decreasing, while the "intersecting assertion" impHes merely that for any there exists a value such that for the average return is less than the positive output associated with when Xk does not exceed the bound The existence of the bound follows from Proposition 12.2. Strictly decreasing average returns for is a property of the fine structure of the technology and cannot be deduced without assumptions contrived to obtain this result. Strong Law of Diminishing Product Increments: For every combination related to a minimal essential combination of the factors of production which is strong hmitational, there exists a subvector yk such that

if depends upon

and

: where

and hk.

In this strong law no restriction is put upon the vector other than because for any fixed input the output . is bounded for and the proof follows exactly that given for the corresponding weak law. The statement (b) is omitted because due to Property A.4 and the property of strong limitationality there exists a vector such that and Strong Law of Diminishing Average Product: If the combination is strong limitational, then, for every nonempty,

and

is finite and, if then

< 56 >

is

THE PRODUCTION FUNCTION

For this strong law, no restriction is put on Xk because 0(xk,yk) is bounded for yk = 0 and xk fixed, since the combination is strong limitational. The foregoing laws are precise laws of diminishing returns for any technology T: L(u), u ε [0, + oo), and provable for such structures with­ out assumptions on the fine structure of T. Nothing is said about any particular physical production system. It is presumed, however, that the ideal structure T describes microscopically all actual production sys­ tems. Only in this sense does a law of diminishing returns have mean­ ing. If an actual physical production system can be found which violates the laws, excepting situations where the output u for an input vector χ does not correspond to Φ(χ) = Max (u | χ ε L(u)}, i.e., inefficient sys­ tems, then the properties defining the technology T must be modified in some way to encompass this critical observation and new forms of the laws sought which are not contradicted. It is useful to look at some functions which are commonly used in econometric studies, i.e., the Cobb-Douglas and CES functions. The Cobb-Douglas functionf may be represented by

η with ai > 0, x^ > 0 (i = 1,... ,n) and ^ εα = 1. The quantities X are ι some positive inputs at a reference point of the set D, taken to give an expression which is independent of the diverse physical units of the fac­ tors of production. This function does not satisfy Property P.9 for the implied input sets L(u), i.e., the efficient subsets E(u) are not bounded, and hence it is not a valid production function over the entire domain D of the input vectors x. It has the further restrictive property that each factor of production is essential, that is, no factor may be completely substituted for another. Similarly, the CES function [2] fails to be a valid production function over the entire domain D of the input vectors for significant parameter values. This function, presented for two aggregate factors of production (capital and labor) by the expression 0 i

Φ(χ) = [aixr" + a2x2^ri//3, with ai > 0, a2 > 0, β > — 1 is offered as a "new class of production functions," but the efficient subsets of the implied production possi­ bility sets are not bounded (i.e., P.9 fails to hold) when β > 0, the case t C . W . C o b b a n d P. H. Douglas: " A T h e o r y o f Production," American Economic Review, Papers and Proceedings, Vol. 18, pp. 139-105, (1926).

THEORY OF COST AND PRODUCTION FUNCTIONS

described by the authors as most interesting empirically. If (— 1 < β < 0), neither factor of production is essential and the function is unbounded for indefinite increase of either factor with the other held fixed, but the efficient subsets of the implied production input sets are bounded. A law of diminishing returns holds over the whole range of inputs of one fac­ tor with the other fixed, because the function has the property of being strictly concave in each factor when β > — 1. When β > 0, both factors of production are essential. For both the Cobb-Douglas and CES function with β > 0, output is unbounded if a positive bound is put on any factor of production and the others are increased indefinitely, that is the factors are individually essential but not even weak Umitational! From three premises concerning the production function: (a) increase in output for an appropriate increase of an input Xj, (b) positive homo­ geneity of degree one, and (c) homogeneity in (n — 1) factors for fixed input of the remaining factor, all of which are satisfied by the CobbDouglas function (but not boundedness of the efficient subsets), Eichhornf has deduced that the production function satisfies an over the whole range strictly decreasing product increments. For the class of production structures which (in addition to P.1,..., P.8,P.9) are positively homogeneous of degree one, the following propo­ sition holds. Proposition 7.1: If the technology satisfies L(Xu) = AL(u) for all λ > 0, the production function Φ(χ) is positively homogeneous of degree one and a concave and continuous function of χ on D.

For λ > 0, Φ(λχ) = Max (u I (λχ) ε L(u),u Ξϊ 0}

= λΦ(χ). From the homogeneity of Φ(χ) it follows that the production function is a concave and continuous function of χ for all χ e D, see Proposition 7, Section 2.4. Thus, a restricted law of nonincreasing product increments holds: Restricted Law of Nonincreasing Returns: If the technology is posit W. Eichhorn, "Deduktion der Ertragsgesetzen aus Pramissen," Zeitschrift fur Nationalokonomie, Vol. 28, pp. 191-205, (1968).

THE PRODUCTION FUNCTION

tively homogeneous of degree one, then for any input vector XOsuch that IP(XO) > 0 and arbitrary increment h (h1 ,h2 , .•. ,hn ) subject to h p; = 0 for ie {l,2, ... ,k} and hpj> 0 for j, {1,2, ... ,k}, 1 0 for N > 2, then

=

IP(XO+ N· h) - IP(xo + (N - 1)· h) < IP(XO+ (N - 1)· h) - IP(XO + (N - 2)· h) for all integers N > 2. For any () e [0,1], consider x ()·x

+ (1

= XO+ (N -

2)h, N > 2, and

+ 2h) = x + 2(1

- ()(x

- O)h.

Then, from the concavity of cP(x), IP«()· x and for () =

+ (1

- ()(x

+ 2h»

> ()IP(x)

+ (1

- O)IP(x

+ 2h),

f, 2IP(x

whence IP(x

+ h) >

+ 2h) -

IP(x

+ IP(x + 2h),

IP(x)

+ h)
2. Further, if the Property P.9 for the input sets L(u) is deleted foru > 0, i.e., the efficient subsets E(u) are not held to be bounded, and the sets L(u) are assumed to be strictly convex for u > 0 (which is the case for the Cobb-Douglas and CES functions), the following restricted law for strictly decreasing returns holds: Restricted Law of Decreasing Returns: If the technology is positively homogeneous of degree one, Property P.9 is deleted for u > 0 and the input sets L(u) are strictly convex for u > 0, then for any input vector XOsuch that IP(XO) >0 and arbitrary increment h = (h1,h2, ... ,hn ) with h pi = 0 for i e {l,2, ... ,k} and hpj> 0 for j , {l,2, ... ,k}, for 1 0 for N > 2, then


2.

< 59

)

THEORY OF COST AND PRODUCTION FUNCTIONS

The strict convexity of the input sets and homogeneity of the tech0, cP(y) then cP(x + y) cP(x) + cP(Y), nology imply that if cP(x) i.e., for input vectors x and y yielding positive output the production function is strictly super additive. To see this, note that for any u

>

>

°

>

>°

due to the homogeneity of cP( • ). Hence, the input vectors u

u

cP(x) • x, cP(y) • Y belong to the boundary of L(u), because if not, say for the first, then

~. = tcP(x + 2h) > t 0, cP(x + h) = cP(tx + !(x + 2h» > tcP(x) + t 0 and Φ(γ) > 0, and θ ε (0,1), the production function is strictly concave, since it is strictly super additive and homogeneous. Thus, it is seen that, if boundedness of the efficient subsets E(u) for u > 0 is discarded and one assumes that the technology is positively homogeneous of degree one with strictly convex input sets L(u) for u > 0, the technology will obey a law of strictly decreasing product in­ crements over the whole range of inputs, no matter which inputs are fixed and which are incremented, explaining Eichhorn's result and the properties of pseudo production functions like the Cobb-Douglas and CES. The reasons why boundedness of the efficient subsets E(u) is deleted and each factor of production is essential, when the production possi­ bility sets L(u) are strictly convex for u > 0, are explained by the fol­ lowing proposition. Proposition 12.4: A production possibility set L(u) is strictly convex for u > 0 if and only if each factor of production is essential and the efficient subset E(u) is unbounded. If a factor, say the first, is nonessential then there exists an input vector (0,X2,... ,xn) such that (0,X2,... ,xn) ε L(u). This vector belongs to the boundary of L(u) and likewise all input vectors (0,λχ2,... ,λχη) for λ > 1 belong to the boundary of L(u), implying that L(u) is not strictly convex. If E(u) is bounded, then for any factor of production, say the first, Min (X11 χ ε L(u)} = Min (χι | χ ε (E(u) + D)} = Min (xi I χ ε E(u)} exists, where E(u) is the closure of E(u), since E(u) is a bounded and closed set. Let x* yield this Min. Then χ* ε Boundary L(u), since E(u) C L(u) because L(u) is a closed set, and there does not exist any e > 0 such that (χ | ||x — x*|| < e) C L(u) because χ ?L(u) if Xi < x*, and all input vectors (χ^,λχ^,... ,λχ£) for λ > 1 belong to the boundary of L(u), implying that L(u) is not strictly convex. Now suppose that L(u) is strictly convex. Then clearly, each factor of production is essential. Also, if η 2 PiXi = α, P = (pi,... ,pn) > 0, a > 0 1 is a supporting hyperplane of L(u), it contacts the set L(u) at a unique point x*(p) of L(u). For pi > 0 and p2 = p3 = · · · = pn = 0, Inf {ρ · χ I χ ε L(u)} = Inf{p · χ | χ ε (E(u) + D)} = Inf {ρ · χ | χ ε E(u)}

THEORY OF COST AND PRODUCTION FUNCTIONS

does not occur for a bounded X* e B(u), because then all points (xV'x;, ... ,Ax~) likewise belong to L(u) and moreover these points belong to the boundary ofL(u), implying that L(u) is not strictly convex. Hence, the set L(u) = B(u) + D) is not supported by a hyperplane Xl = IX at a finite point x*(p) for p = (1,0,0, ... ,0), and the efficient subset E(u) is not bounded. Eichhorn's assumptions reduce to: (a) «P(AX) = A«P(X), A > 0, xeD (b) «P(Axl, ... ,AXi-l,Xi,Axi+l, ... ,AXn) = Ari . «p(x), ie{I,2, ... ,n},A>0,0 0, ie {I,2, ... ,n} and xe D,

and, for A ~ +00, it follows that

= 0, i e {I,2, ... ,n}

«P(Xh ... ,Xi-l,O,Xi+l, ... ,xn)

since the production function is a continuous, concave function for xeD when it is homogeneous of degree one. Thus, each factor of production is implied to be essential. Moreover, for x 0,

>

«p(Xl,X2, ... ,Xn) Xl

= «P(1,X2, ... ,xn) = (Xl;1= (Xl)1-r 1(X2)1-r . «p(x) 1

2

~

• «p(x)

'

and, continuing in this fashion, one obtains

Due to Property (a) it follows that: n

«p(x) = «Po' II Xi", 1

n

2: Vi = 1

1,

°< Vi < 1, i e {l, ... ,n},

< 62 >

THE PRODUCTION FUNCTION

where Φο = ¢(1,1,. · · ,1) and V = (1 — Ti). Thus, his assumptions im­ ply that the production function is a Cobb-Douglas production function with strictly convex level sets (production possibility sets), which is a special case of a positively homogeneous technology (degree one) with strictly convex production possibility sets, just as is the CES production function for β > 0, both of which violate an essential property of a technology, i.e., boundedness of the efficient subset for any positive output rate. The proposition described above as the restricted law of diminishing returns encompasses all cases of this kind. When — 1 < )8 < 0 for the CES function, the case where the implied efficient sub­ sets E(u) are bounded, the property of this function to show a law of diminishing returns over the entire range of each input, with neither input being essential, is not significant, being merely due to the fact that the function is strictly concave in each factor of production. If neither is es­ sential, i.e., they are complete substitutes for each other, there is no reason for the technology to exhibit diminishing returns when the input of either factor is put equal to zero. Of course as a statistical approximation there is no objection to using the CES function, but one should be careful about describing it as a "new class of production functions." 1

CHAPTER 3 THE DISTANCE FUNCTION OF A PRODUCTION STRUCTURE 3.1 Definition of the Distance Function Ϋ(ι^χ)

In economics the production function Φ(χ) is primarily intended to define the alternatives of substitution between inputs of the factors of production to achieve a given output rate u, but as we have seen in Chapter 2 these alternatives are not generally definable in terms of a simple equation Φ(χ) = (u). Only if the production function is con­ tinuous and strictly increasing in x, will this equation allow calculation of the locus of input vectors which yields exactly the output rate u. An upper semi-continuous production function may not take certain out­ put rates u for all nonnegative input vectors x. Neither are the vectors χ which yield Φ(χ) = u necessarily boundary points of the production possibility set L (U), since Φ(χ) is only nondecreasing in x, i.e., the solutions of the equation Φ(χ) = u are not necessarily points on the isoquant corresponding to the output rate u. Also, it is common in the theory of production to write the produc­ tion relation as an equation F(u,x) = 0, particularly when u is a vector of joint outputs. Again, the existence of such an equation to determine the substitutions between inputs to achieve a mix of output rates is not obvious. For these reasons, and others which will become apparent later, it is useful to seek a function defined on the production sets L4(u) which permits definition of substitution alternatives by a simple equation on u and x. As a preliminary for the definition of such a function, we par­ tition the nonnegative domain D of input vectors into mutually exclusive and exhaustive subsets, as follows: 4

(a) The origin: {0}. (b) The interior points of D: Di = {x|x>0}. (c) The boundary points of D excluding the origin:

Clearly D = {0} U Di U D2. Further, we partition the subset D2 into Dg = (x I χ ε D2,(λχ) ε L»(u) for some u > Ο,λ > 0} D'2' = (x I χ ε D2j(Ax) $ Li(U) for all u > Ο,λ > 0}.

THE DISTANCE

FUNCTION

This partitioning of the boundary points D2 is proper, because if x e Dg then by the Property P.3 of the production sets (see Chapter 2, Section 2.1) the ray intersects all sets for . Consequently and these partitions of D are mutually exclusive. The fimction we seek is given by the following definition: Definition: A nonnegative distance function for the production possibiHty sets

is defined on D by

(11) where

and

If and it follows from the ray Property P.3 of the production sets that the ray intersects all production sets and is the point of intersection of the ray 0} with the boimdary of , see Figure 14 illustrating for two factors ofproduction. Note that if the ray intersects the set only on the boundary, however, the definition (11) uses the intersection with smallest norm. When the ray is either not defined for X = 0 or fails to intersect each production set for u > 0. But the value zero is a natural value to take for the distance function in both cases, and this definition will serve our purposes. Consider the perturbed points , and (0 A) where and take the limits of and as thru points of Di U D^. Then

and, as

because if the

were finite the Umit point belongs to contrary to the definition of the set

and, as since When

due to the closure of For the perturbed point

while is bounded away from zero due to the Property P.l. and, for any and we

< 65 >

THEORY OF COST AND PRODUCTION FUNCTIONS

(u)

Isoquant

Mu)

( 0 + Δ)

{0} U .** x'

FIGURE 14:

INTERSECTIONS OF A LEVEL SET L,(u) BY RAYS FROM THE ORIGIN

take Ψ(0,χ) = + oo by adjoining + oo to the real line. If χ = 0 and u = 0, the distance ratio ||x|]/||£|| is not defined, however, we take Ψ(0,0) =+oo because this value will serve our purposes. The foregoing definition of the distance function for each production set L (U), u Sr 0, is an adaptation for the production sets L„,(u) of the Minkowski distance function for convex bodies [21], A convex body is a closed bounded convex set in Rn and Minkowski took the point 0 as an interior point of the convex body for definition of the distance func­ tion. In our situation the production sets are closed, convex and un­ bounded with the origin 0 exterior to the set, and the properties of the function ^(u,x) are different than those of the Minkowski metric for a finite dimensional linear space. In fact, Sf(U1X) is not a metric, because it does not satisfy the triangle inequality (see Property D.4, Section 3.2). Yet the function ^(u,x) will serve exactly our purposes to define the isoquants of the production sets L41(U). 0

THE DISTANCE

FUNCTION

Proposition 14: For any u

Clearly, if

for u > 0 then and and then , and When

if

. Also, since the point x

does not belong to and . Thus for if and only if , For and for all D so that The isoquants of the production sets for are defined by a simple equation in terms of the distance function Proposition 15: For any the isoquant of a production set consists of those input vectors x > 0 such that Recall that the definition we use for the isoquant of a production set excludes those boxmdary points of common to the boimdary of D which are not at minimal distance from the origin along the ray from the origin passing through the boundary point. TTius, when X belongs to the isoquant of if and only if When the point x likewise belongs to the isoquant of if and only if . The closure of the production sets is essential. The isoquant of the production set consists merely of The efficient points E(u) of a production set are clearly contained in because if the point x is either an interior point of or a point on the boundary of common with the boundary of D such that , But not all points of this set are necessarily efficient (see the lower portion of the isoquant illustrated in Figure 14). Proposition 15.1:

(12) Expression (12) defines exactly the technologically efiicient substitution alternatives to achieve at least an output rate 3.2 Properties of the Distance Function The properties of the distance function for a production structure D are given by the following proposition: Proposition 16: The distance function

for the production pos-

t First used in [26], p. 5 as an alternative for defining the isoquants.

< 67 >

THEORY

sibility sets properties: D.l D.l

OF COST AND PRODUCTION

FUNCTIONS

of a production function for all

For aU

D.3

and is finite for finite

for all

has the following for all

when

and positive for

but

for

u = 0. D.4 D.5 D . D.7 D.8 D.9 D.IO

for all and for all if 6 i s a concave function of x on D for all is a continuous function of x on D for aU For a n y D , For any D and For any D and is possibly finite.

D . l l For a n y D , u for all [0,oo).

is an upper semi-continuous function of

The remainder of this section will be devoted to proofs of these properties. Property D.l is merely a restatement of the definition of the function given in Section 3.1. Regarding Property D.l,if and then the intersection I has and is finite for finite x. When and is finite. Finally, if and is positive since Property D.3 holds when and u > 0 because

If for

If u = 0 and

for all and the equality stiU holds, but not for

For verification of Property D.4 note first that, for and and by Property D.3

Then, it follows from Proposition 14 that the points

< 68 >

THE DISTANCE FUNCTION

belong to the production set L.(u) and, since L.(u) is convex by virtue of Property P.8, the point [(1 - fJ)

i'(~,x) + fJ i'lu,y)]

belongs to L.(u) for any scalar fJ E [O,IJ. Consequently, due to Proposition 14,

'1'(u,(1 - fJ) i'(~,X) + fJ i'(~,y») >

1

for any scalar fJ E [O,lJ. Take fJ

==

i'(u,y) i'(u,x) + 'It(u,y)

and use Property D.3 to obtain 'It(u,x

+ y) >

'It(u,x)

+ 'I'(u,y).

> 0 and both x and y belong to {O} U D~, 'I'(u,x) == 'I'(u,y) == 0 and the inequality still holds since i'(u,x + y) is nonnegative. If one point (say y) belongs to {O} U D~ and the other (say x) belongs to Dl U D~, then 'I'(u,y) == 0 by D.l and i'(u,x) > 0 by D.2. The point x/i'(u,x) belongs to the production set L.(u) and by the Property P.2 for L.(u) it follows that (x + y)/i'(u,x) belongs to L.(u). Hence, by Proposition 14

If u

'1'(u, ;(!1;) >

1

and due to the homogeneity of i'(u,x) it follows that 'I'(u,x

+ y) >

'I'(u,x) == i'(u,x)

+ 'I'(u,y).

Titus the super-additivity of the distance function holds for all u > 0 and XED. It remains to consider u == 0, and here the super-additivity trivially holds since i'(O,x) == + 00 for all XED. Therefore, Property D.4 is proved. Property D.5 is a simple consequence of D.4 and the nonnegativity of the distance function. Let x' > x be written x' == x + Ax where Ax == (x' - x) > 0 and 'I'(U,X/) > 'I'(u,x) + 'I'(u,Ax) > 'l'(u,x). The concavity of'l'(u,x) on D for u > 0 follows directly from the super-additivity and homogeneity properties merely by taking x == (1 - fJ)z, y == Ow for any fJ E [0,1 J and any z, w belonging to D to obtain 'It(u,(l - fJ)z

+ Ow) >

(1 - fJ)'I'(u,z)

( 69 )

+ fJ'I'(u,w).

THEORY OF COST AND PRODUCTION

FUNCTIONS

For u = 0, the inequality holds trivially since

We consider next the continuity in x of on D. If u = 0, for all and we need concern ourselves only with u > 0. The function is continuous in the interior of D, i.e., for because there is a well-known theorem that a convex function defined on a convex open subset of R° is continuous in this open set (see [3], p. 193) and is convex on Di. Now, regarding the boundary of D, i.e., for is lower semi-continuous in (see theorem, Section 2.4). But the distance function is also upper semi-continuous in . To show the upper semi-continuity we need only demonstrate for u > 0 that the set is closed for aU v e R^ (see Appendix 1). If this set is closed since it is given by

since

is nonnegative, and D is closed. If

due to the homogeneity of the distance function, and by Proposition 14 it follows that

a closed set. Thus , is both lower and upper semi-continuous in and hence it is continuous on the boundaryof D. Regarding Property D.8, , ^ for all u > 0, if and for all Thus we need only consider Since implies and letting and denote the intersections of the ray with the boundaries of and such that , it follows that and hence (See the definition of the distance function.) When ui = 0 the inequality is trivially satisfied since for aU For Property D.9, let be an infinitive sequence of output rates. Then Property D.9 is trivially satisfied if since for all u > 0 (see Property D. 1). Hence, we confine our attention to For any such point x, let denote the intersection of the ray with the boundary of the production set Corresponding to the sequence there is a sequence

< 70 >

THE DISTANCE

If

FUNCTION

the sequence

must be boimded, imply-

ing that there exists a finite input vector on the ray which belongs to all production sets , contrary to the Property P.6 of the production sets. Thus Sup Next, regarding Property D.IO, consider first and let be an arbitrary sequence of output rates tending to zero. Then, since for all u > 0 when Sup (finite). Further, for the sequence of intersections with the boundaries of the production sets L,f(Un) may be bounded uniformly away from the null vector 0, a property not excluded by the properties of the production sets Then (finite) for aU n and lim Sup is finite. See Figure 14.1. n—>00 In general, there may exist a neighborhood

of the null input vector 0 such that for any u > 0 when implying that the technology may require certain minimal inputs of the factors to obtain a positive output. Finally, we turn to a proof of the upper semi-continuity of in , First, it follows from Property D.IO that for any sequence and any that Sup Hence the distance function is upper semi-continuous in u at u = 0 for all Thus we need only consider u > 0. Now for any and and the distance function is continuous in u > 0 for any Thus we need consider further only u > 0 and . By counter example it may be seen that the distance function is not always lower semi-continuous. Consider the following example: is a nondecreasing step function, where x is a vector of dimension one, i.e., X e Ri, as illustrated in Figure 15. This production function is upper semi-continuous and satisfies aU of the Properties A.1, ...,A.6. The corresponding distance fimction is only upper semi-continuous as shown in Figure 16. For any where Hence the corresponding intersection is and

The function

is clearly not lower semi-continuous because

< 71 >

THEORY OF COST AND PRODUCTION FUNCTIONS

Hx)

/ :,/

-u o

/

/

/

./

-'"

1 x

x

o

FIGURE 14.1: EXAMPLE FOR BOUNDED DISTANCE FUNCTION AS {Un) -> 0, SINGLE FACTOR OF PRODUCTION L.(un) = [~.. oo) for un> il. L.(u.) [x",oo) for Un ;2i ii.

=

>

let u = 2, for example. Then, for any u 2, no matter how close to u = 2, (l/x)i'(u,x) (l/x)i'(2,x) - aori'(u,l) i'(2,1) - aforO< a < I and u > 2. However, the distance function is upper semi-continuous in u > 0 for all x £ Dl U D~, and we proceed to verify this fact. Let x be any point belonging to Dl U D~, and consider an arbitrary value of u £ (0,00), say 110. Corresponding to uo,




< 72 >




THEORY OF COST AND PRODUCTION FUNCTIONS

'l'(uo,x) + a if u:> Uo. Hence, to show the upper semi-continuity of 'l'(u,x) we need concern ourselves only with values u < uo. Now for all scalars A and a > such that

°

we have

and ~

l+Aoa that

°

'l'(u,x)

= ::~II

~

it is clear

i ;'I, (1 + "Aoa) = ii~l, + a = 'l'(Uo,x) + a.


uo, 'l'(u,x) < 'l'(uo,x) + a for any a> 0, and the distance function is upper semicontinuous. 3.3 Expression of the Production Function cp(x) in Terms of the Distance Function 'l'(u,x)

The output rate corresponding to an input vector Xo cannot be determined simply as a solution of the equation 'l'(u,XO) = l. To see this, consider the counter example of Figure 17, where for a single factor of production the production function cp(x) has a finite discontinuity at xO. Since cp(x) is upper semi-continuous it takes the value Uo at xO. The production sets L~(u) for u f (Ul,UO] are L~(u)

= [XO, + (0).

The ray {Ax 1 A :> O} for any x e (0, + (0) intersects these sets on their boundaries at ~o = XO and 'l'(u,XO) = 1 for all u f (Ul,UO]. Hence, this equation does not determine a unique value of output rate. This simple example suggests the following proposition: Proposition 17:

cp(x)

= Max {u

1

'l'(u,x) :> I}, xeD.

Proposition 17 follows immediately from Proposition 14, since cp(x)

= Max {u x f < 74 ) 1

L~(u)}

THE DISTANCE FUNCTION Φ(χ)

U

O

for all

u.

1

FIGURE 17: COUNTER EXAMPLE FOR *(u,x°) = 1 NOT DEFINING A UNIQUE OUTPUT RATE

and χ ε L (U) if and only if ^(u,x) ^ 1. See also Properties D.8 and D.11. From a technological viewpoint the structure of production is given by a family of production possibility sets L(u) which define for each output rate u ε [0,+ oo) the set of input vectors χ which yield at least the output rate u. In fact, we may define a technology with single output by: lt

Definition: A technology with single output u is a family of produc­ tion sets T: L(u), u ε [0,+ oo) with the Properties P.1,..., P.9, defin­ ing for each nonnegative output rate u the set of input vectors which yield at least u. For this structure the distance function Ψ(υ,χ) completely characterizes the production possibilities in a general way, since L(u)= {χ | ^(u,x)^1}, u ε [Ο,οο). The properties of the function ^(u,x) follow from the Prop­ erties P.1,..., P.9 assumed for production technology, the latter being

THEORY OF COST AND PRODUCTION FUNCTIONS

justified on technological grounds. The isoquant (as defined in Sec­ tion 2.1) for any positive output rate u is given in terms of the distance function ^(u,x) as the solution set of the equation Ψ^,χ) = 1. Not all of these solutions are necessarily efficient. However, efficiency of a boundary point χ of a set L(u) is a local property requiring the test that ¥(u,y) u}]-\ χ ^ 0, u ^ 0. The validity of this calculation for u > 0 and χ e Di U is evident from the definition (11) of the distance function, since (λχ) ε L(u) if and only if Φ(λχ) > u. For u = 0 and χ ε D, Φ(λχ) > 0 for λ ε [Ο,οο) and calculation yields Ψ(υ,χ) = + oo. For u > 0 and χ ε {0} U D'2', (λ · χ) $ L(u) for all λ ε [Ο,οο), implying the set (λ | Φ(λ · χ) ^ u} is empty, whence Min {λ I Φ(λ · χ) ^ u} = + oo and Ψ(υ,χ) = 0. Thus, by a maximum problem and a minimum problem the produc­ tion function Φ(χ) and the distance function ^(u,x) are determinable from each other. 3.4 The Distaoce Function of Homotfaetic Production Structures As defined in Chapter 2, Section 2.4, a homothetic production struc­ ture is one with a production function of the form Ρ(Φ(χ)), where Φ(χ) is a linear homogeneous function satisfying A.1,..., A.6 and F( ·) is any finite, nonnegative, upper semi-continuous nondecreasing function with F(O) = 0 and F(v) oo as ν oo. The production sets corres­ ponding to such production functions have a special structure in that the family may be generated from the production set L®(1).

THE DISTANCE

FUNCTION

Proposition i&'f

To prove this proposition we note from Proposition 6 that But

The homogeneity of the function makes possible this conversion. Thus, the production sets for a homothetic production structure are derived by scalar magnification f r o m t h e production set of the linear homogeneous production function corresponding to unit output rate. The fact that the production sets of a homothetic production structure may be expressed as (see Proposition 14)

suggests that the distance function of a homothetic production structure is given by (13) Consider first an input vector x belonging to the subset of the boundary of the nonnegative domain D of input vectors. Then, by the definition of the set Dg and the Property A. I of it follows that Also, by the property (ii) of the inverse function f(u) (see Section 2.3, Proposition 5), f(u) > 0 since u > 0. Hence and the expression (13) is valid when (see (11)). If and let denote the point on the ray such that . Now, since is continuous (Proposition 7, Section 2.4) and strictly mcreasing along the ray (homogeneity), it follows that Hence and, since t Shown by Stephen Jacobsen, see [15].

< 77 >

THEORY OF COST AND PRODUCTION FUNCTIONS

Therefore, the expression (13) is a proper formula for the distance func­ tion of a homothetic production structure when the output rate u is positive. We are then led to the following proposition: Proposition 19: The distance function of a homothetic production

structure with production function F( 0 by ^(u,x) = (x)/f(u), where f(u) is the inverse function of F( · )· The Properties D.l,..., D.9 of a distance function are clearly satis­ fied by the expression (13). Property D.10 holds, since, for {un} —» 0,

because f(u) is lower semi-continuous and Um Inf f(un) =5 f(0) = 0. The OO

upper semi-continuity of (x)/f(u) in u, i.e., Property D.ll, is implied by the lower semi-continuity of f(u). In general, no simple explicit formula like (13) relates the production function Φ(χ) and the distance function ^(u,x) when the production structure is not homothetic. However, for any χ e Di U Ό'2, the point x/^(u,x) lies on the boundary of Li(U), because

since the distance function is homogeneous of degree one in x, and, if the production function is continuous and strictly increasing in x, (14) Equation (14) provides an implicit relationship between the distance function and the production function under these circumstances.

CHAPTER 4 THE FACTOR MINIMAL COST FUNCTION 4.1 Definition of the Cost Fimction Q(u,p) Here we are concerned with the traditional cost function of economic theory. The title of this chapter describes the cost function as factor minimal in order to distinguish it from another cost function to be dis­ cussed in Chapter 7. Denote the prices per unit of the factors of production by a vector ρ = (pi,p2,... ,pn)· TTie price vector ρ defines a point in the nonnegative domain D of Rn and the coordinate systems of input vectors χ and price vectors ρ are superimposed in Rn. The cost per unit time of an input vector χ is denoted by the inner product ρ · χ of the two vectors ρ and x. Some convention is required for the costing of mixed input vectors. If x, y ε D are two input vectors, the input vector [(1 — θ)χ + 0y], where θ ε [0,1], may be interpreted either as a single input vector ζ or that the input χ is used a fraction (1 — θ) of the time interval and the input y is used the remaining frac­ tion Θ. In either case, the cost per unit time of the input [(1 — ff)x + θγ] is calculated by the inner product ρ · [(1 — ff)x + 0y]. As a combined input, the interpretation of this cost is straightforward. The capital ser­ vice components imply certain amounts of these services are inputed per unit time, and the corresponding components of the price vector ρ denote the costs per unit of these services calculated by whatever prac­ tices may be used for amortization of investments involved. If the input [(1 — ff)x + θγ] is regarded as time fractional applications of two distinct input vectors, the costing of the capital service components of each input vector is time prorated. The price vector need not be positive, i.e., each component positive, and free goods are allowed as inputs. When ρ = 0, all of the factors of production are free goods or services, a situation of trivial interest in economics but not excluded for the definition of the cost function. For all price vectors ρ ε D and output rates u ε [0,oo), the cost func­ tion Q(u,p) denotes the smallest total cost per unit time attainable for input vectors which yield at least the output rate u, i.e., Q(u,p) = Min (ρ · χ I χ ε L4(u)}, ρ ε D, u ε [Ο,οο). X

(15)

It is permissible for us to regard the greatest lower bound of cost as at­ tainable, because we assume that the efficient point sets E(u) of the pro­ duction sets L4(U) are bounded, and since L41(U) = E(u) + D (see Prop­ osition 2, Section 2.1), the cost function may be calculated by

THEORY OF COST AND PRODUCTION FUNCTIONS

Q(u,p) = Min (ρ · χ I χ ε E(u)}, ρ ε D, u e [Ο,οο).

(15.1)

X

The cost function Q(u,p) is described as factor minimal, because the inner product is minimized with respect to the inputs of the factors of production. Evidently Q(u,0) = 0 for all u ε [Ο,οο), and Q(0,p) = 0 for all ρ ε D since E(O) = {0}. Neither of these two situations are of particular in­ terest for economic theory, but they are included for completeness. In the minimum problem (15), the components of the price vector ρ and the output rate u are arbitrary parameters, and the cost function Q(u,p) gives the minimum total cost per unit time for all nonnegative output rates and nonnegative prices of the factors of production. In effect, it is assumed that the prices pi(i = 1,2,... ,n) do not depend upon the amounts xj(i = 1,2,... ,n) of the inputs of the factors of production. But this is apparently not a serious loss of generality for the economic theory of production, because, if the price of a factor varies stepwise with the amount demanded, each quantity range may be considered qualitatively as a different factor of production with inputs of multiples of this level treated as replications of use. The most economic of these so considered qualitatively different levels will be chosen for any output rate sought. The dependence of price of a factor upon the amount used has primary significance for the analysis of the total economy. We do not consider stocks explicitly. Let D

i = {p I ρ > °} = {χ I χ > 0}

D2 = {ρ I ρ > 0 ,JJpi = 0j = |x I χ > 0,]¾¾ = oj, as in Chapter 3. If ρ ε D2 and u > 0, the cost minimizing vector x*(u,p) may have zero components for factors with positive prices and positive components for factors which are free goods. Hence, it is useful to pursue further a classification of the boundary points D2 for price vectors. Let £>2 = {p I P e D2,Q(u,p) > 0 for all u >0}, SD2 = (p I ρ ε D2,Q(u,p) = 0 for all u > 0}. Proposition 20: SD2 Π SD2 = 0 (the empty set) and D2 = SD2 U SD2. The sets SD2 and SD2 are exclusive by their definition and SD2 HSD2 is an empty set. Next suppose that there exists a price vector ρ ε D2 such that Q(u,p) = 0 and Q(u,p) > 0 for some u > 0 and u > 0, i.e., D2 φ SD2 U SD2. Then the cost minimizing input x*(u,p) is such that ρ·χ*(ϋ,ρ) = 0, which implies for each index i that ρι·χ*(ϋ,ρ) = 0 since ρ is semi-positive and x*(u,p) is also semi-positive. Hence, when

THE FACTOR

MINIMAL

COST

FUNCTION

and . Moreover, the ray intersects all production sets for since for (see Property P.3, Section 2.1). Hence, for any u > 0 there exists a positive scalar X^ such that and the inner product p• because by hypothesis. Thus, for all u > 0 if for some and since no price can yield both a positive and zero cost Q(u,p) for two distinct positive output rates. Recall, from Section 3.1, Chapter 3 preceding the definition of the distance function ^(u,x), the classification of the boundary points given by for some for all Regarding whether Proposition 21:

and and

is nonempty, the following proposition holds: is nonempty if and only if

is nonempty.

Assume is nonempty and let Then there is a price point such that . Moreover, the ray intersects all sets for u > 0, by virtue of Property P.3, and for any there is a point with Hence and is nonempty. If Dg is empty, no input vector of D2 yields positive output and hence for all p e D2 any cost minimizing output for u > 0. Consequently if and which implies is empty. Thus, in order to exclude zero minimum total cost for positive output we must require that positive output is possible only with positive input for all factors of production, i.e., none of the boundaries of the production possibiUty sets coincide with the boundary of D for u > 0. This restriction is too strong, since some of the factors may be alternatives for others, i.e., not aU the factors are essential by themselves. (See Section 2.5.) We need not require the set D2 to be empty, since the subset of prices Dg is merely a formal possibility, which may or may not be reaUzed in practice. On the other hand, we do not wish to require the price vector p to be positive. Some factors may be free goods, and it is not correct technologically to exclude them from the input vectors. What is free depends upon the exchange economy which may vary from place to place and from time to time. 4.2 Geometric Interpretation of the Cost Function Consider the hyperplane p • x = Q(u,p). Since Q(u,p) is the minimum of p • x for all points it follows that

< 81 >

THEORY OF COST AND PRODUCTION FUNCTIONS {O} U

Di

L4>(u) , u > 0

~

__________________________________________-+ {O}U D

Z

o FIGURE 18:

RELATION OF HYPERPLANE p' x = Q(u,p) TO L.(u)

L~(u)

c

{X I p' X >Q(u,p)} for

all p =1= 0,

and this hyperplane is a supporting hyperplane of the production possibility set L.(u) (see [31], Part II, Section B) and the cost function Q(u,p) is a support functional of L.(u) (see [31], Part V) for any u > O. The relation of the hyperplane p' X = Q(u,p) to the production possibility set L.(u) is depicted in Figure 18, where x*(u,p) denotes an input at which the minimum of p . x is attained for x e L..(u). Note that the contact point x*(u,p) is not necessarily unique. Let r denote the intersection of the ray {Op I () > O} from the origin normal to the hyperplane p' x = Q(u,p), for p =1= O. For some value of (), say (Jo, r = ()o' p, and, since r lies in the hyperplane p . x = Q(u,p), p·r

= (JoJjpll2 = Q(u,p)

and (J

Q(u,p)

-I-

o=~,P-r-

0

.

Consequently -1-0 - Q(u,p) rp, p -r- ,

liPii2'

< 82 >

THE FACTOR

MINIMAL

COST

FUNCTION

and

Thus

(16) If the price vector p is normalized so that , the minimum total cost Q(u,p) is merely the normal distance of the supporting hyperplane p • X = Q(u,p) from the point 0. The closure of the efficient point set E(u) of in Figure 18 is the boundary of the set comprised between the points Pi and P2. The total minimum cost occurs for an input vector and x*(u,p) will be understood always to be such a point, unless otherwise specified. 43

P r i ^ r t i e s of tbe Cost Fimctkm

The properties of the factor minimal cost function Q(u,p) are given in the following proposition: Proposition 22: If the production structure P . l , . . . , P.9, then

has the Properties

Q.l

for aU

Q.2 Q.3 Q.4 Q.5 Q . Q.7 Q.8 Q.9

for aU

For aU

and

and

is finite and positive for all for all and for aU

6 i s

a concave function of p on D for all is a continuous function of p on D for all For any For and and possibly less than Q.IO For p e D and is possibly greater than zero. Q . l l For any is a lower semi-continuous function of u for Q.12 Q(u,p) is convex in u e for if the graph G® of the production structure is convex.

Property Q.l is merely a recapitulation of the discussion in Section 4.1.

< 83 >

THEORY OF COST AND PRODUCTION FUNCTIONS

Regarding Property Q.2, note first from (15.1) that Q(u,p) is finite for finite p since x belongs to the bounded set of efficient points E(u) ofL40(u). When p e 0 1 , Q(u,p) for u since the minimizing vector is (Property P.l). Ifp e:D~, x*(u,p) ~ 0, because 0, L40(U) for any u it follows from the definition of the set:D~ that Q(u,p) for all u 0. Property Q.3 is valid, because

>°

>°

>°

>°

>

Q(U,Ap) = Min {(Ap)· x I xe L.(u)}

= AMin {p. x I x e L.(u)} = A· Q(u,p)

>°

=° =°

=

and p -=1= 0. When A or p 0, or both are zero, for any u > 0, A AQ(U,p) for all u > 0. it is obvious that Q(U,Ap) For the proof of Property Q.4, note first that the inequality holds for u = 0, since Q(o,p

=

+ q) = Q(O,p) = Q(O,q) =

°

for all p and q. Thus, we need concern ourselves further only with and suppose p -=1= 0, q -=1= 0. Denote by positive output rates. Let u x*(u,p + q), x*(u,p), x*(u,q) the input vectors minimizing (p + q) . x, p. x and q. x respectively. Then

>°

Q(u,p

+ q) = p . x*(u,p + q) + q. x*(u,p + q).

But, clearly p. x*(u,p q . x*(u,p

+ q) > + q) >

p. x*(u,p) q. x*(u,q)

Q(u,p

+ q) >

Q(u,p)

= Q(u,p) = Q(u,q)

and If p

+ Q(u,q).

= q = 0, this inequality holds since

°°

Q(u,p

+ q) = Q(u,p) = Q(u,q) = 0.

°

Also, ifp -=1= and q = 0, or p = and q -=1= 0, the inequality holds since Q(u,O) = for all u > 0. Property Q.5 follows directly from the nonnegativity and superadditivity (Property Q.4) of Q(u,p), since p' = p + Ap where Ap > and

°

Q(u,p') > Q(u,p)

+ Q(u,Ap) >

Q(u,p).

The concavity of Q(u,p) on 0 is a simple consequence of the homogeneity and super-additivity of Q(u,p), since for any p, qeD, (l - O)p and O· q belong to 0 for any 0 e [0,1] and Q(u,(1 - O)p

+ Oq) >

(1 - O)Q(u,p)

< 84

)

+ OQ(u,q).

THE FACTOR

MINIMAL

COST

FUNCTION

The continuity of in p on D may be established as follows: First, for any the function is continuous on the interior of D, i.e., for since is a convex function and a convex function defined on a convex open set in R» is continuous on this open set (see [3], p. 193). Second, regarding the boundary of D, i.e., for is lower semi-continuous (see Theorem, Section 2.4). But is also upper semi-continuous for In order to show this last statement, we extend the definition of the cost fimction to all points in the following way: Let

where

The set

is bounded and closed. Then let

for any and The extended cost function is a concave function defined on R ^ since the arguments used for apply alsoto Q(u,p/R), and the function is continuous in p for all Now suppose and let be a nondecreasing sequence of values of R tending to The sequence is nonincreasing, since

for any n, because for any n. Also, since the sequence is uniformly bounded below by zero and exists. The greatest lower boimd of the sequence is

since

monotonically as . In fact,

i and for R suffl

ciently large. Thus,

Let a be positive. Then there is a positive integer N such that

Further, since such that, for aU

i is continuous at p" for n = N, there is a

< 85 >

THEORY

OF COST AND PRODUCTION

FUNCTIONS

Thus,

for all

and therefore also for all Then, since for all it follows that

for aU and is upper semi-continuous on the boxmdary of D. Therefore, is continuous for all We turn our attention now to Property Q.8. Since if (Property P.4, Section 2.1), it follows for all ' and any that

Hence Q(u,p) is nondecreasing in u for any Regarding Property Q.9, for

for any

because, suppose that this Umit is finite and denote by the cost minimizing vector belonging to Then for some subsequence

and the sequence is bounded since for aU Consequently, there exists a bounded input vector for aU contrary to Property P.6 of the production sets Thus, if Inf However, if then ~ for all permitting Inf , as it is for the point p" illustrated in Figure 19. Further, when it is possible for to be positive and finite for all u e (0,oo), as shown by the example in Figure 19 where and for aU , and again Inf For a sequence pUcated. If neighborhood when

the possibilities are somewhat more com(i.e., p > 0) and there exists for an open such that for any then

since p > 0 and the cost minimizing vectors are semi-positive for aU n. If the neighborhood does not exist, then we may have

< 86 >

THE FACTOR MINIMAL COST FUNCTION

0':i)~----~------~--~--~~--~---------------------pox = Q(u,p)

o~--------------------~~-----------------------+

p"

FIGURE 19:

EXAMPLE OF BOUNDED Q(u,p) FOR p. ~

= O. For most technologies, the neighborhood Na(O) is

lim InfQ(un,p)

n .... oo

likely to exist. When p £:D~ and the neighborhood Na(O) exists, it is possible for lim InfQ(un,p) to be zero as illustrated in Figure 20 where n->oo

the boundaries of the production sets L.(un) converge to the axes as Un ~ O. Moreover, when p £:0 2 it is also possible for Q(u,p) = Qo 0 for all u £ (0,00) (see Figure 19) so that lim Inf Q(un,p) O.

>

When p £

n .... oo

:D~,

Q(u,p) = 0 for all u £[0,00) and lim Sup Q(un,p) = lim InfQ(un,p) == O. n~ao

D-+OO

< 87 >

>

THEORY OF COST AND PRODUCTION FUNCTIONS

FIGURE 20:

EXAMPLE OF l~~ InfQ(un,p)

= 0 FOR 4>(x) = 0 IF

xeN,(O)

Turning now to Property Q.ll, i.e., the lower semi-continuity of Q(u,p) in u € [0,00), we note first that the cost function is generally not upper semi-continuous. The counterexample shown in Figures 21 and 22 illustrates this fact. The step function of Figure 21 satisfies the and Properties A.1, ... , A.6 (see Section 2.2). At u 0, Q(O,p) (l/p)Q(O,p) = 0. For any u € (i,i + 1], clearly the cost minimizing vector x*(u,p) = (i + 1), p . x*(u,p) = P(i + 1) and (l/p)Q(u,p) = (i + 1). The cost function is evidently not upper semi-continuous in u, because supQ(3,1) + 0' for 0 0' 1 pose u = 3. Then (l/p)Q(u,p) = Q(u,l) 3. and u For the lower semi-continuity of Q(u,p) in u when p 0 it is convenient to consider the graph of the production structure defined by Definition: The graph G~ of the production structure determined by the production sets L~(u), u > 0 is a subset of ~+1 = R~ X Rlgiven by

=°

=

<


>

>

G~

= {(x,u) I x >O,u > < 88 >

O,X €

L~(u)}

THE FACTOR MINIMAL COST FUNCTION 4>(x)

r~I

7 6

~

5

~I

4

r--l~II

3 2 1 "

__

0

~

I I I __- L __- L __ __ ~

2

1

FIGURE 21:

p1 Q(u,p)

':

3

4

:

I

I

I

~

__

5

I

~~

6

__

L-________ 7

UPPER SEMI-CONTINUOUS PRODUCTION STEP FUNCTION

• p > 0

7

r--T

rl r-1 I

6

5

~ r--jl rl I I

4

3 2

1

0

1

FIGURE 22:

I I

I

I

I I

I

(

I

I I I I

I

I

I

I I

I I I

2

3

4

5

6

I

I I

r

7

COST FUNCTION FOR THE PRODUCTION STEP FUNCTION

< 89 >

x

THEORY

OF COST AND PRODUCTION

FUNCTIONS

FIGURE 22.1: ILLUSTRATION OF THE GRAPH OF A PRODUCTION STRUCTURE WITH SINGLE INPUT

or equivalently by

Figure 22.1 illustrates the graph of a production structure for input vectors with a single component. For any input the points for which belong to the graph. Likewise, for any the points for which belong to the graph. The upper semi-continuity of the production function , . or equivalently the closure of the sets for all (see Appendix 1), is equivalent to the closure of the graph . Let be an arbitrary sequence of points belonging to i.e., with for all n, and suppose that this sequence converges to the hmit point Since for all n,

< 90 >

THE FACTOR MINIMAL COST FUNCTION

lim Sup cll(xn) > lim Sup Un

n--->oo

= no,

while the upper semi-continuity of the production function cll(x) implies (see Appendix I) lim Sup cll(xn) < «P(XO), n--->oo

whence cll(xO) > no and the limit point (XO,uo) belongs to the graph G~. Now suppose that the graph G~ is closed. If for any U e Rl the subset S(U) ! {x I cll(x) > U,X e~} of G~ is closed, the production function cll(x) is upper semi-continuous. When U < 0, S(U) D, a closed set. For U > 0, let {xn} ~ XO with xn e S(U) for all n, i.e., cll(xn) > Ufor all n. The sequence {(xn,U)} ~ (XO,U) belongs to G~ and the closure ofG~ implies cll(xO) > U, whence XO e S(U) and the set S(U) is closed for all U e Rl. Thus, the production function cll(x) is upper semi-continuous if and only if the graph G~ is closed. Returning to the argument for the lower semi-continuity in u of the 0, let {Un} ~ Uo (arbitrary) and consider cost function Q(u,p) for p the sequence

=

>

where x*(un,p) is the cost minimizing input vector belonging to the bounded efficient subset of L~(un). Each term p x*(un,p) is bounded. If {p x*(un,p)} is not a bounded set of real numbers, 0

0

lim InfQ(un,p) > Q(no,p), n--->oo

since Q(uo,p) is finite (Properties Q.I,Q.2) and Q(u,p) is lower semicontinuous at no. Thus, take {p x*(un,p)} as a bounded set, and, since 0, there is a subsequence {unk } ~ Uo such that p 0

>

lim InfQ(un,p) D-+OO

= k-+oo lim Q(Unk>p) = poXO(p)

where {x*(unk,p)} ~ XO(p). But, due to the closure of the graph G~, following since cll(x) is upper semi-continuous (Property A.S), XO(p) e L~(uo). Hence, po XO > Q(UO,p), and lim Inf Q(un,p) > Q(uo,p). n--->oo

Since {Un} ~ UO is arbitrary, the cost function Q(u,p) is lower semi0. continuous in u for p When the graph G~isconvex, L«(1-0)u + Ov) ~ [(I -O)L(u) +OL(v»). To see this let x e [(1 - O)L(u) + OL(v»), i.e., x = (I - O)y + Ow where ye L(u) and we L(v). Since G~ is convex, [(1 - O)(u,y) + O(v,w») e G~,

>

< 91 >

THEORY OF COST AND PRODUCTION

FUNCTIONS

and Property Q.12 holds. If the super-homogeneity property (SH), see Section 2.7, is used for the production technology, a further interesting property follows for the cost structure. For u > 0 and a scalar

since

and

i.e., the cost function is subhomogeneous in output rate. Then, for any and u > 0,

stating, for any price vector p of the factors of production, that average cost of output is nonincreasing in scale of output for all positive levels of output. 4.4 The Cost Function of Homothetic Production Structures We consider now the special form of the cost function Q(u,p) when the production structure has a production function of the form where is a homogeneous function of degree one satisfying A . 1 , . . . , A.6, and F( • ) is any nonnegative, finite, upper semi-continuous and nondecreasing function with F(0) = 0 and The production possibility sets of this homothetic structure may be written (see Proposition 6, Section 2.3)

< 92 >

THE FACTOR MINIMAL COST FUNCTION

=

where f(u) Min {v I F(v) > u} for u > 0 and has the properties stated in Proposition 5, Section 2.3. Then, for u > 0, L.(f(u»

= f(u)· L.(I)

(see Proposition 18, Section 3.4), and the cost function Q(u,p) is given by Q(u,p) = Min {p. x I x E f(u)· L.(1)} for u

x

> 0 and

Q(u,p) = f(u)· ~in (p. ftu)

I f(:)

E

L.(I)}.

whence

= f(u)· Min {p. xI x L.(1)} for u > 0, where x= x/f(u). Therefore, for u > 0, Q(u,p) = f(u)· Pep), Q(u,p)

E

x

where the function P(p) is independent of u and homogeneous of degree one in the price vector p (Property Q.3). Thus, we are led to the following proposition:

Proposition 23: The cost function for a homothetic production structure has for u > 0 the form Q(u,p) = f(u)· pep),

(17)

in which feu) = Min {v I F(v) > u} with the properties stated in Proposition 5, and pep) is a homogeneous function of degree one in the price vector p.

= 0, clearly L.(f(u» = D, since feu) = 0 and 0 for all XED. Then Q(O,p) = 0 for all p E D. The properties of the homogeneous factor price function pep) follow from those for the cost function given above. First, if p E {O} U :D~, then by Q.I we have Pep) = 0, and by Q.2 it follows that pep) 0 for p E Dl U :D~. Thus,

If u

>

pep) is { =0 'V P E {O} U :D~ >0 'V P E Dl U :D~. It is interesting to note that if the price vector p belongs to :D~, the scalar measure Pep) of this price vector is zero. Property Q.3 is consistent with the homogeneity of the factor price function pep) and adds nothing new, while the Properties Q.4, Q.5,

( 93 )

THEORY OF COST AND PRODUCTION FUNCTIONS

Q.6, and Q.7 imply that the price function P(p) is super-additive, nondecreasing, concave and continuous for p e D. Thus, the following proposition holds for the price function P(p) of the cost function of homothetic production structures:

Proposition 24: The cost function of a homothetic production structure has the form Q(u,p) = f(u)· P(p) where the price function P(p) has the following properties: HQ.l HQ.2 HQ.3 HQ.4 HQ.5 HQ.6 HQ.7

P(p) = 0 for all p e {O} U ~~, P(p) is finite for finite p e D and P(p) 0 for all p e Dl U :D 2, P(Ap) = AP(p) for A >0 and all p e D. P(p + q) > P(p) + P(q) for all p,q eD, P(p') > P(p) ifp' > P e D, P(p) is a concave function ofp on D, P(p) is a continuous function ofp on D.

>

The Property Q.8 of the cost function Q(u,p) is consistent with the non decreasing character of f(u). But Property Q.9 is strengthened to 0 and lim InfQ(un,p) = + 00 for all p e Dl U :D~, since P(p)

>

n->oo

f(u n ) ~ + 00 as Un ~ 00 because F(v) ~ + 00 as v ~ + 00. The example of Figure 19 does not apply since the production sets of the homothetic production structure are formed by radical magnification of L(1).

For {un}

~

0 and p eD, lim InfQ(un,p) > 0 for p e Dl U n->oo

lim Inf f(un) > 0 because f(O)

n->oo

~~,

since

= 0 and f(u) is lower semi-continuous.

Thus, Property Q.ll holds. Hence, regarding the properties of the cost function Q(u,p) in respect to output rate for homothetic production structures the following proposition holds:

Proposition 25: If the production structure is homothetic, the cost function Q(u,p) satisfies: HQ.8 For any p e D, Q(U2,P) > Q(Ul,P) ifu2 > Ul > O. HQ.9 For any p e Dl U :D2 and {Un} -~ + 00, lim Inf Q(un,p) = n->oo

HQ.lO For any p e D and {un}

~

+ 00.

0, lim InfQ(un,p) is possibly n->oo

greater than zero. HQ.ll For anyp e D, Q(u,p)is a lower semi-continuous function of u for all u e [0,00).

< 94 >

THE FACTOR MINIMAL COST FUNCTION

The special form (17) of the cost function is of some interest for the study of changing returns to scale, because for any ρ ε Dx U 3¾ it implies

and, if cost data reflects minimum cost operation for the output rates and factor prices encountered, then f(u) and hence F( ·) may be investi­ gated by studying the relation between output rate and factor price de­ flated costs. TTie function F( ·) has direct meaning for changing returns to scale, since the homogeneous function Φ(χ) has the properties of a scalar measure or index of input. It will be shown later that homotheticity of production structure is an if and only if condition for the factori­ zation of the cost function given by Equation (17), and thus for the use of factor price deflated costs to estimate changing returns to scale. See Section 6.2 for further discussion of this topic. t For any U0^O the inverse function F( ·) of f( ·) is defined by F(f(u°)) = Max {u I f(u) £ f(u°)}.

CHAPTER 5 THE COST STRUCTURE 5.1 Definition of the Cost Structure

JBQ(U), U

ε [Ο,οο)

We note that the cost function Q(u,p) has properties which are similar to those of the distance function ^(u,x) (compare Propositions 16 and 22). Those properties of the function ^(u,x) which essentially character­ ize it as a distance function are its homogeneity, super-additivity and concavity in χ and these same properties are possessed by the cost func­ tion Q(u,p) for the price vector p. Thus, one is led to regard the cost function as a distance function for a family of subsets of the price vec­ tor ρ in the nonnegative domain D. Hence, in order to proceed carefully along these lines, a cost struc­ ture is defined by: Definition: The cost structure is a family of subsets of D = (p j ρ = 0} given by £Q(U)

= {p I Q(u,p) ^ Ι,ρ ε D}, u ε [Ο,οο).

The set JBQ(U) of the cost structure for any nonnegative output rate is the subset of price vectors which yield a minimum total cost equal to or greater than unity. Corresponding to u = 0, the subset JBQ(O) = {p I Q(0,p) ^ Ι,ρ ε D} is empty since Q(0,p) = 0 for all ρ ε D (Property Q.1 of the cost function Q(u,p)). Before demonstrating that the cost function Q(u,p) is a distance func­ tion for the sets JBQ(U), u > 0, consider first the properties of the price sets of the cost structure which are summarized in the following proposition: Proposition 26: The price sets of the cost structure £Q(U), U ε [Ο,οο) corresponding to a cost function Q(u,p) of a production structure have the following properties: 17.1 JE (O) is empty and 0 # JEQ(U) for any u >0. OT.2 If ρ ε JBQ(U) and p' ^ p, then ρ' ε £Q(U). 7Γ.3 If ρ > 0, or ρ > 0 and (ffp) ε 0 and θ > 0, then the ray {θ · ρ | θ ^ 0} intersects all price sets £Q(U) for u > 0. Q

17.4 17.5

JEQ(U2) D JBQ(UI)

Π u>u

if

U2 Ξ2 UI ^

JBQ(U) = JBQ(U0) 0

Uo for all Ρ ε

Π

u>u 0

0. for any U > 0, if Q(u,p) is continuous at

JEQ(U).

0

THE COST

STRUCTURE

may not be empty. Closure of is closed for all is convex for all u We shall verify these properties in turn. First, since O(0,p) = 0 for all it is evident that is empty; and the price vector 0 does not belong to any because for all Property follows directly from Property Q.5 of the cost function, smce implies implying and Regarding Property note that if then for all see Property Q.2 oi the cost function). Hence, it then for any 0 there is a positive scalar such that and the ray for u ; intersects all price sets the other hand, if and for some and then p see Proposition 2U) and, by Property Q.2 oi the cost function for u intersects all price sets Property follows directly from the Property Q.8 of the cost function. If p ther we have and for and When the set is empty and all points of (there are none) belong to . Regarding Property due to Property but m general the reverse is not true. Suppose

From

the discussion of Property Q . l l of the cost function it is clear thai is lower semi-continuous at u but need not be continuous, i.e., also upper semi-continuous, and the graph of the cost function may have at the form indicated in Figure 23 where it is only continuous from the left. Then but Howfor all u ever, if it follows is continuous at then for because if there exists an output rate for all u such that contradicting For the first part excluding u = 0, since is empty, we is not necessarily empty due to the possibility lliusnote that that

trated in Figure 19, Section 4.3. The verification of the second part of D and the Property is made as follows: I f f then D, since

D for all

P interior of D, i.e., p < 97 >

Contrariwise, if

THEORY OF COST AND PRODUCTION

FIGURE 23:

FUNCTIONS

DISCONTINUITY OF THE COST FUNCTION

(see Property Q.9 of the cost function). Accordingly, if p there exists a positive output rate u such that and Thus,

with and hence closure ol the cost function Q(u,p) is Property holds, because for any a continuous function of p on D and therefore upper semi-contmuous on D which implies that is closed for all numbers Qo e R 1 , since this property is an if and only if condition for the upper semi-continuity of Q(u,p) in p on D. (See Appendix 1.) Hence, for the set < 98 >

THE COST

is closed for any Finally, Property the cost function, i.e., and any scalar

STRUCTURE

For u is empty and therefore closed. follows directly from the concavity in p on D of then for Property 6. Let p any u

1 it implies Hence the point empty and therefore convex. Now, in order to verify that Q(u,p) is a distance function for the price sets £Q(U) of the cost structure, we must show that and since p follows that

implies

min the vector being p and where with the boundary of the set the intersection of the ray when and (See Figure 24.) Property Q.l of the cost function Q(u,p) implies imIfp moreover, the ray does not mediately that for since < intersect any of the price sets then by Property Q.2 we for all Ifp the cost structure, it follows have and, by Property . Then intersects all price sets i for that the ray we may define for any u

are closed (Property since the price sets and the distance ratio is given by

p when

But

and by the continuity of the since belongs to the boundary of 1. W h e n u 0 for all cost function in p we have

< 99 >

THEORY OF COST AND PRODUCTION

FIGURE 24:

FUNCTIONS

INTERSECTIONS OF PRICE RAYS WITH

Proposition 26.1: The cost function Q(u,p) is a distance function for he cost structun 5.2 Efficient Price Vectors of the Cost Structure Analogous to the definition of the efficient points of a production set, we use the following definition of an efficient price vector of the set Definition: A price vector if and only if Q(u,q

is efficient relative to the price set for all price vectors

Hence, for any positive output rate u a price vector is efficient if and only if the minimum total cost is less than unity for all price vectors which are equal to or less than but not identically equal to the given price vector.

< ioo >

THE COST

Definition: The efficient subset structure is defined by

STRUCTURE

of a price set

of the cost

From a cost factor-price standpoint, the efficient price vectors are those which for the given output rate cannot be decreased without making the minimum total cost less than unity. Now, in all essential respects so far as efficiency is concerned, the price sets have the same properties in regards to the price vectors p as the production possibility sets have m terms of the input vectors x—compare with P.2, P.3, P.4, P.7, and P.8. In particular, the argument given in Section 2.1 to show that E(u) i nonempty may be used here to verify that: is non-

Proposition 27: The efficient point set £(u) of a price set empty for all positive output rates.

The counterexample referred to in Section 2.1, Chapter 2, shows that need not be closed. However, for our purposes it will be sufficient and < since is to work with the closure closed. For reasons explained in Section 2.1, Chapter 2, it is suitable to assume that the efficient point set E(u) of a production possibility set is bounded. But the question remains whether boundedness of is bounded. implies that < A simple counterexample will suffice to show that E(u) bounded does not imply S(u) bounded. Consider the production possibility set defined by

and illustrated in Figure 25. The efficient point set of

is

which is evidently bounded. Now, calculate Q(u,p) by

where p = is a price vector for x. Necessary conditions for this minimization are:

and

so that

Hence, < 101 >

THEORY OF COST AND PRODUCTION

FIGURE 25:

FUNCTIONS

COUNTEREXAMPLE FOR BOUNDEDNESS OF

Thus, and the price set £Q(U) is defined by

The boundary of

THE COST

STRUCTURE

or in polar coordinates:

implying and

Moreover,

and

Thus, the boundary of is a strictly convex locus with negative slope foi unbounded over this range and the efficient set is unbounded when the corresponding efficient set E(u) of the production structure is bounded. This counterexample leads to the conclusion. Proposition 28: Boundedness of E(u) does not imply

bounded.

Finally, by a proof which parallels that given in Section 2.1 for the one may verify the following properties: property

53

The Cost Structure of Homothetic Production Structures

Recall that for a homothetic production structure the cost function has the special form where is a homogeneous (see function of degree one having the Properties Proposition 24, Section 4.4), and f(u) is a finite, nonnegative, nonand decreasing and lower semi-continuous function witl (See Proposition 5, Section 2.3.) Also of the cost structure are The corresponding price sets homothetic, i.e., the isoquantf for any positive output rate may be obtained from that for unit output rate by radial contraction from the origin in a fixed ratio. To see this, let denote a ray from (Property the origin in the price domain D. If p Such rays do not intersect any of the price sets. The remaining rays in D, i.e., those for which Defined analogous to the definition of a production isoquant, given in Section 2.1

< 103 >

THEORY OF COST AND PRODUCTION

FIGURE 26:

FUNCTIONS

INTERSECTIONS OF PRICE RAYS WITH PRICE SETS A HOMOTHETIC PRODUCTION STRUCTURE

FOR

intersect all of the price sets for u due to Property Thus, we need only consider rays where p, and denote the intersections of the ray with the isoquants of the price sets and respectively. Then p are price vectors such that 1, as illustrated in Figure 26, since P(p) is continuous in p. Also, is homogeneous of degree one (Property HQ.3), and

so that

Hence,

< 104 >

THE COST

STRUCTURE

and the price point on the isoquant of the price set is obtained from the point for by radial contraction with a scalar independently of the price direction p. The price sets of the cost structure for a homothetic production structure have a special structure in that they may be generated from the price set where is the homogeneous function of the price vector p in the form i of the cost function. Proposition 29: For any u

But, due to the homogeneity of the function P(p),

Compare Proposition 29 with Proposition 18, Section 3.4, Chapter 3. The cost structure for a homothetic production structure is thus homothetic, since the price sets can be generated by radial contraction of the price se 5.4 Cost Limited Maximal Output Function T(p) (discussed in Chapter 2) is deThe classical production function fined on a production structure with production possibility sets L(u), as the maximal output rate corresponding to the input vec) for this structure tor x, and in terms of the distance function

Similarly, for a cost structure with price sets a cost limited maximal output function 1 (p) may be denned as the supremum output rate, corresponding to an input price vector p, such that the minimum cost is less than unity. Definition: Now, although the function T(p) gives the supremum output rate, for a < 105 >

THEORY OF COST AND PRODUCTION

FUNCTIONS

price vector p, the minimum cost of which is less than unity, it may be interpreted for any positive cost rate c. Write, for c > 0,

and, since the cost function Q(u,p) is homogeneous of degree one in p (Property Q.3),

Thus, by normalizing the price vector p in units of total minimum cost per unit time, the function T(p/c) defines the supremum output rate attainable at a cost rate less than c dollars per unit time, for any nonnegative factor price vector p. Consider next the properties of the function T(p): Proposition 30: The cost limited maximal output function I\p), defined for a cost function Q(u,p) with the Properties Q . l , . . . , Q.l 1, has the following properties:

and finite if p

and with

Sup

for some

when

is positive and finite for some scalar r(p) is lower semi-continuous on D. T(p) is quasi-convex on D. Regarding Property a.l, if (Property Q.l) and! since the set it is possible that and hence

for all u with

for all 1 for

For Property a.l, note that if p for all (see the discussion of Property Q.10 for this possibility), then for all svitl and Sup for some

when p

is obviously positive and finite. < 106 >

THE COST

STRUCTURE

Property a.3 follows simply from the fact that, due to Property Q.5 of the cost function,

For verification of Property a.4 note that, when p ; for all u > 0 (Property Q.2). Further, when p and then and finite for some

is positive

and, by Proposition 20, Section with u' bounded, implying for all u with 4.1, it follows that Q(u,p) and Consider next Then there exists a monofor some 1 such that for all k and tone subsequence

since is bounded below by zero for all n. Suppose implying there exists a positive output rate u less than 'with

for all k. But, since that

and Hence, there exists an integer N such contradiction. Thus, and

In order to show that a.5 holds, we need only verify that the level sets of the function F(p) defined by are closed for all v e R 1 (see Appendix . First for we show that is empty and hence closed.

the set

Sup whence

and

and then

Contrariwise, if p and

Hence the equality (20)

implying < 107 >

THEORY OF COST AND PRODUCTION

holds. Consequently, since the sets erty it follows the

FUNCTIONS

are closed for all u is closed and (20) implies

(Propis

closed for all Thus, the cost limited maximal output function T(p) is lower semi-continuous for p e D. We note that the set may be empty or nonempty. (See the discussion of Property and Figure 19.) Finally for Property note that is empty and hence convex equals the intersection of for all while, for for the sets convex sets (Property and is, therefore, convex. Accordingly, the level sets of the function IYp) are convex for al implying that is quasi-convex for p e D (see Appendix 1). The interchanges (substitutions) of prices of the factors of production for a cost limited maximal output not exceeding v and cost rate less than c > 0 are given by the isoquantsf of the sets (21) These isoquants define for any cost rate the substitutions between pnee vectors p such that the supremum output rate does not exceed any positive output rate v. In analogy with Utility Theory the function represents the indirect production function, and, in the cost structure, the cost limited supremal output plays a dual role to that of the production function It is to be noted that the assumptions made for the production structure do not imply that the level sets of the function are identical to the price sets of the cost structure on which cost limited supremal output is defined. then p for all (Property of the cost If 1 structure) and p due to (20). Hence However, the reverse mclusion does not necessarily hold. Since the cost function Q(u,p) is only lower semi-continuous in u for and for some output rate it may happen that for all while 1 for all u so that implying by (20) that P Property

But this does not imply

(See Figure 27 and

5 of the cost structure.)

Defined as in Section 2.1 for the production sets See H. S. Houthakker, "Additive Preferences," Econometrica, Vol. 28, No. 2, (April 1960).

< 108 >

THE COST

FIGURE 27:

STRUCTURE

DISCONTINUITY IN

AT

'OR GIVEN PRICE SECTOR

will hold if Equivalence of the sets and for the cost function Q(u,p) is continuous in u for all This anomaly between the level sets of the cost limited maximal output function T(p) and the price sets of the cost structure on which it i are is defined may be avoided if the sets taken as the cost structure on which the cost limited maximal output function is defined by

(22)

defined by (22) is identical to the function Moreover, the function T(p) defined by (18). In order to see this, the following two lemmas are established. Lemma (a): The level sets for all u ol Let

Let

of IYp) are identical to the level sets

be chosen arbitrarily. The level set

for all

implying, by

is given by

that

THEORY OF COST AND PRODUCTION

for all

Hence,

FUNCTIONS

implying JBr(u)

and

then

Contrariwise,

Thus.

Lemma (b): Let p be an arbitrary price vector of D. The price vector p belongs to Thus, it follows that if and only if

Then, by Lemma (b), Now, suppose for some and there exists an output rate u such that

implying while But then to Lemma (a). Thus, the following proposition holds: Proposition 31:

, contrary

for all

Hence, by using the family of sets for the cost structure on which the cost limited maximal output function is defined by (22), the level sets of the function T(p) are identical to the price sets of the cost structure. However, for the development of the dualities of Chapter 8, we shall use the price sets of the cost structure because the production input sets dual to the cost structure price sets are not necessarily completely determined by the since the relation price sets

does not generally hold. The cost limited output function T(p) may be used to calculate the marginal productivity of a positive cost rate c to supply the cost of production. Proposition 32: For the marginal productivity of the cost rate c to supply the cost of production is given by the left-hand derivative when and

for some u

when p

Under the conditions of Proposition 32, IYp) is positive and finite. Let ) and consider the profile F(p/c) as c ranges over the open < no >

THE COST STRUCTURE

given)

O

c1

c2

FIGURE 27.1: PROFILE OF THE COST LIMITED OUTPUT FUNCTION

interval (0, + oo) for some price vector p. The vector (p/c) ranges the ray {(1/c) ·ρ | c > 0} toward the null price vector as c increases. The cost limited output function T(p/c) is nondecreasing in c, it may be constant over some intervals like [ci,C2] and discontinuous at some points c3, as illustrated in Figure 27.1. Since the profile T(p/c) is lower semi-continu­ ous and nondecreasing as c increases, the profile is continuous from the left. Hence, the left-hand derivative exists since T(p/c) is nondecreasing and it is noimegative, implying that the marginal productivity of the cost rate c to supply the cost of production is nonnegative. 5.5 Cost Limited Output Function for Homoflietic Cost Structures In Section 5.3 above it was found that the cost structure of a homothetic production structure is likewise homothetic, and it is to be ex­ pected that the cost limited output function defined on a homothetic cost structure will have special form. In fact, using Proposition 23, Γ(ρ) = Sup {u I f(u)P(p) < l,u ε [0,+ oo)}, ρ ε D. < πι >

THEORY OF COST AND PRODUCTION

0 (Property HQ.l) and output rates may be obtained at zero cost. But, positive and finite (Property HQ.2) and If

FUNCTIONS

since all is

But, by the definition of the inverse function F( •) of f( • ) (see Section 2.3)

and (23) Equation (23) has particular significance for the statistical study of returns to scale, assuming that the production structure is homothetic. Let c denote a positive cost rate, and (24) since the function P(p) is homogeneous of degree one. Thus, assuming that the technology is operated at minimum cost for each output rate realized, one may relate various output rates with cost rates deflated by and index of the level of the prices of the factors of production to investigate the form of the function F( • ) which defines the returns to scale (see Section 6.2, Chapter 6, for the properties of P(p) as an index number function). A word of caution regarding such statistical studies is in order. Our theory applies strictly only to operations of the technology which are not restricted as to the inputs of the factors of production, i.e., Equation (23) is rigorously correct only for unconstrained homothetic production technologies, and if the output rates are realized optimally on the boundary of the constraint set D of a limited unit (see Section 2.6, Chapter 2), Equation (23) may not strictly apply. Homotheticity is also a limiting assumption. The marginal productivity of money capital to supply the cost of production also takes a special form. For any given price vector p of the factors of production, it is given by the left-hand derivative

(25) Here Sup

THE COST

STRUCTURE

of the function T(p/c) and thus equals the marginal returns of a price deflated cost rate input c/P(p), deflated by an index of the level of the prices of the factors of production, varying inversely with the level of the prices of the factors of production. If the homothetic production function is homogeneous of degree one,

(24.1) i.e., the cost limited output function T(p/c) equals the cost rate deflated by an index of the level of the prices of the factors of production, and, for any given price vector p

(25.1) so that the marginal productivity of money capital to supply the cost of production is simply the reciprocal of the level of the prices of the factors of production. We close this discussion with some remarks concerning the properties of the cost limited output function T(p) for homothetic cost structures. Proposition 33: If the cost structure is homothetic: for p is finite for p For any p r(p) is lower semi-continuous on D. r(p) is quasi-convex on D. Thus, the Properties a.l, a.l, and a.4 are sharpened for homothetic cost structures. One need only use Equation (23) and Proposition 30 to verify these alterations.

< 113 >

CHAPTER 6 THE AGGREGATION PROBLEM FOR COST AND PRODUCTION FUNCTIONS 6.1 Criteria for Aggregates A large number of variables in a mathematical economic model has certain disadvantages for economic theory. One reasons intuitively in terms of collections of these variables which appear to have a similar role, and concepts such as capital and labor for the factors of produc­ tion, and producers goods and consumers goods for the outputs of pro­ duction processes, are common in economic theory. The heuristic justification for the use of single quantities (scalar measures or index functions) for vectors of economic variables is the conviction that the individual variables of an aggregate are not important in economic re­ lationships and that the latter can be significantly expressed in terms of mathematical equations between index functions of suitably chosen aggregates. The tradition in economic theory for reasoning in terms of aggregates is long standing, but seldom rigorously justified. Dresch [8] has shown that Divisia index functionals of the microeconomic variables for general economic equilibrium can be used to define aggregate variables which satisfy the same equations for equilib­ rium as those in Evan's simplified economic system of the total econ­ omy [9], thereby justifying Evan's model. May [20] performed a similar aggregation for a one industry model which underlies the theories of J. M. Keynes. Klein [16] suggested two criteria for the aggregation problem of the general economic equilibrium: (a) If there exist produc­ tion functions relating output to input for the individual firm, there should also exist functional relations that connect aggregate output and aggregate input for the economy as a whole, (b) If profits are maximized by individual firms according to certain marginal productivity equa­ tions, then the aggregate variables should satisfy analogous equations. The first criterion of Klein requires that if the aggregate variables are to be treated technologically as inputs and outputs of an economy wide process, there should be derivable from the individual production func­ tions an economy production function of the aggregate variables. This requirement is independent of the satisfaction of any equilibrium con­ ditions for maximum profits, since technology alone is involved. The second criterion of Klein requires that at equilibrium for the economy the aggregate variables should satisfy the classical marginal productivity equations, thereby justifying the practice of treating aggre­ gate variables in this way in economic theory.

THE AGGREGATION PROBLEM

Pu [23] questioned the criteria of Klein, arguing that the first criterion implies that aggregate output must be independent of the distribution of the various inputs, and that the second criterion is unnecessary and arbitrary because, if a unique macroeconomic production function exists, the conditions for maximization of profits can be represented just as well by some other form of equations. He suggested instead that the aggregation be based upon fixed patterns of distribution of the values of the microvariables in each aggregate, after the fashion of Leontief [19] in his input-output models, relying therefore upon the existence of such fixed patterns. Neither the formulation of the criteria by Klein nor Pu's objections are clear. If we adhere to the classical definition of the production func­ tion as peculiar to the special circumstances of the firm, i.e., the given plant, equipment and other resources, the production relationships to be aggregated reflect some optimization (equilibrium) decisions and they describe a limited arrangement of inputs to outputs relative to those available in the technology. Then, perhaps there are some fixed patterns of distribution of the microvariables in the aggregates. But then the aggregate production function does not give a proper state­ ment of the alternatives available in the technology. Ipso facto, it is merely a statement of the net effect of individual optimizing decisions (rational or not) for some given set of prices and other circumstances when the decisions were made, and as these determining conditions change so will the decisions change and with them the so-called fixed patterns of the distribution of the microvariables. Thus, one may pre­ sume that Klein intended the production functions of the individual technologies to reflect the full range of the alternatives available, realized or not by the firms at a given time, after the fashion of the defi­ nition of the production function given in Chapter 2 above. Then, if there be aggregate variables for the inputs of an economy wide process which can be properly related to some defined aggregate outputs, the aggregate production function should accurately relate the technological alternatives between aggregate inputs and outputs, independently of the distributions of the values of the microvariables in each aggregate vari­ able, to which Pu specifically objects. Only in this way can the aggre­ gate production functions serve the purpose of prediction and explana­ tion for which they are intended. The issue of "short rim" versus "long run" analysis is most clearly put by the constraints imposed on the variation of the input variables, and the aggregate production function should be accompanied by proper constraints on the aggregate input variables. In the very short run, one may expect to find the fixed distri­ butions of the microvariables which Pu seeks. One cannot treat this issue in an off-hand manner without confusion. For the period in mind, < Π5 >

THEORY OF COST AND PRODUCTION

FUNCTIONS

the constraints on the aggregate variables should accurately reflect physical reality, and, when adjoined to a production function expressing the unconstrained technological alternatives, the combination will express the alternatives available to the limited aggregate production unit. For the theory of cost and production functions the aggregation problem is more specific. The criteria of aggregation may be formulated to serve the theory. We are dealing with a given technology having certain microeconomic factors of production. These factors are to be aggregated into one or more sets of input vectors, say and with corresponding price vectors p and as for an aggregate model with two aggregate input variables and two corresponding aggregate price variables, such as Capital and Labor inputs and prices. For the discussion to follow, the criteria will be stated for two vector pairs (x,p), (z,w), since the extension to an arbitrary number of input and price vector pairs is straightforward. The aggregation to aggregate variables X, P and Z, W is to be obtained for any nonnegative output rate u by finite mappings

which satisfy the following criteria:

For a scalar For any price vectors p and w, the values of the aggregate variables X, Z corresponding to the cost minimizing vectors x and z satisfy

The aggregate production and cost functions have the Properties A . 1 , . . . , A.6 and Q . l , . . . , Q.ll respectively, possessed by their microcounterparts 0(x,z) and Q(u,p,w), and satisfy < 116 >

THE AGGREGATION

PROBLEM

for all nonnegative vectors x, z, p, w such that X(x) = X, For any nonnegative aggregate variables P, W and nonnegative output rate u,

For all vectors x, z, p, w such that W, if the partial derivatives i exist (and the functions are differentiable) they satisfy

where

C.8

The aggregate cost limited output function f(P,W) has the Properties a . l , . . . , a.6 of the microfunction T(p,w) and, for all nonnegative vectors p and w such that P(p) = P, W(w) = W,

t (P,W) = r(p,w), and for any nonnegative aggregate variables P, W

< H7 >

THEORY OF COST AND PRODUCTION FUNCTIONS

If the functions Γ and Γ are differentiable,

0 the corresponding cost minimal input vector χ belongs to the efficient set E(u) and 1Ir(UjX) = 1, whence Φ(χ) = f(u) and § piXi = Q(u,p) = Φ(χ) · P(p) = P-X. 1 Thus the criterion C.4 is satisfied for u > 0 and ρ > 0. If u = 0, the cost minimizing vector χ equals the null vector and C.4 is trivially satisfied since Φ(0) = 0. If u ^ 0 and ρ = 0, C.4 is likewise trivially satisfied since P(O) = 0. Thus with the definitions (26) and (27) of the aggregate variables X and P, the first four criteria are satisfied. The aggregate production and cost functions are Φ(Χ) = F(X) Q(u,P) = f(u) · P.

(28) (29)

The function F(X) has the Properties A.1,..., A.6 of a production function (see Proposition 4, Section 2.3), and F(X) = Ρ(Φ(χ)) for all vectors χ Ξϊ 0 such that Φ(χ) = X. The function f(u) · P clearly has the Properties Q.l,..., Q.ll of the micro cost function Q(u,p) = f(u) · P(p) (see Sections 4.3 and 4.4 of Chapter 4) and Q(u,p) = f(u) · P(p) = Q(u,P) for all price vectors ρ Ξϊ 0 such that P(p) = P. Hence the Criterion C.5 is satisfied. For any P g; 0, Min {Ρ · X I F(X) ^ u,X ^ 0} = f(u) · P = Q(u,P), since F(X) is nondecreasing and upper semi-continuous and the mini-

THE AGGREGATION

PROBLEM

miring X, say satisfiies Fi and so that P f(u) • P. Therefore, the Criterion C.6 is satisfied. The partial derivatives of the aggregate production and cost function are (when they exist)

By simple computation

and

are differentiable,)

and hence, for vectors (x,p) such that

Thus, Criterion C.7 is satisfied. The aggregate cost limited output function is computed for

by

But, since f(u) is nondecreasing and lower semi-continuous with f(0) = 0, the set is a closed interval [0,F(1/P)] and f(P) satisfies

Hence

for P > 0. Also, P = P(p) = 0 if p (see Proposition 24, Seetion 4.4) and when P = 0. Thus, the macro cost limited output function is given by < 121 >

THEORY OF COST AND PRODUCTION

FUNCTIONS

(30) The micro cost limited output function for a homothetic cost strucure is (see Equation (23), Section 5.5)

such that Consequently, for all price vectors has the same properties = P. Moreover, the aggregate function (See Proposition 30, Section 5.4 for these as the microfunction properties.) First, fYP) is nonincreasing in P, since F( •) is nondecreasing. Second since Third, f (P) is lower semi-continuous in P, since F( •) is upper semicontinuous. Finally, for any

and, since F( •) is nondecreasing

for all is quasi-convex in P (see Apand the aggregate function pendix 1). It is more or less immediate that for all price vectors p such that P(p) = P, the marginal productivity of money capital to supply the cost of production is given by (when the derivative exists)

Thus the aggregation defined by Equations (26), (27), (28), (29) satisfies all criteria C. 1 , . . . , C.8. The returns to scale are given by the aggregate production function F(X) for any magnitude (scale) X of the inputs. Note that the measure of scale, i.e., the aggregate variable X, < 122 >

THE AGGREGATION PROBLEM

and the corresponding measure P of the level of factor prices has the well defined properties C.1,..., C.4 and equality between the aggregate and micro production function, cost function and cost limited output function is preserved. If cost and price data are to be used, instead of the aggregate vari­ able X, Equation (30) may be written for any P > 0 and cost rate C > 0 as (30.1) and the supremum output rate for any positive cost rate C and positive level P of the prices of the factors of production is given by the righthand side of Equation (30.1). Hence, factor price deflated cost data may be related to output rate to investigate the character of the function F( ·). For either input or cost-price data, some index function Φ(χ) will have to be assumed, and the corresponding index function P(p) can be determined by minimizing cost. In the summary it has been shown that Proposition 34: The homothetic production function F(0(x)) may be aggregated to satisfy the criteria C.l,..., C.8 by aggregate variables X = Φ(χ), P = P(p), where P(p) is the homogeneous function of factor prices in the cost function Q(u,p) = f(u) · P(p), and the aggregate production, cost and cost limited output functions are Φ(Χ) = F(X), X^O 0(u,P) = f(u) · P, u ^ 0, P ^ 0 m= F(l),P>0.

63 Aggregation of Cobb-Dougias Production, Cost and Cost Limited Output Functionst In the previous section the homogeneous function Φ(χ) was not given explicitly. The Cobb-Douglas production function is a homothetic production function Ρ(Φ(χ)) where Φ(χ) is a weighted geometric mean of the factor input rates χ,. To give this function specific form, with no­ tation for aggregation into more than one aggregate input variable, write t The aggregation for Cobb-Douglas cost and production function was given in Section 8, [26].

THEORY OF COST AND PRODUCTION

FUNCTIONS

(31)

The quantities are input rates at some reference level of the function and the expression (31) is a weighted geometric mean ot input relatives. The function with defined by (31), is the general form of a Cobb-Douglas production function with segregation of the input rates of the factors of production mto two vectors x rhe function F( •) is a transform (see Section 2.3, Chapter 2) of a production function with F(v) It can be verified that 0 are not bounded. denote price vectors corresponding to the input vectors x and z respectively. The aggregate input rates corresponding to the vectors x and z are taken as

The functions X(x), Z(z) defining the aggregate variables X and Z satisfy the Properties A . 1 , . . . , A.6 for a production function and they are nonnegatively homogeneous of degree one. Hence, they satisfy the criteria C.l, C.2, C.3. Now, by Proposition 23, Section 4.4, the cost function of the produc< 124 >

THE AGGREGATION

PROBLEM

tion function where f(u) is the inverse function ol r ( •) and r(p,w) is a nonnegativeiy homogeneous function of degree one with the properties stated in Proposition 24, Section 4.4. In order to determine the function P(p,w) we minimize the subject to the input vector (x,z) belonging to the production set

where

is the distance function of the production function given by (see Section 3.4, Chapter 3)

If u > 0, the efficient point set E(u) of the production set for the CobbDouglas production function is unbounded and, for any price vector (p,w) with at least one component zero, Min 0. As a convendoes not exist, but Inf ient convention we shall take 0 if any price component is zero. If u = 0, the efficient set E(0) consists of the pomt {0} and Q(0,p,w) = 0. Thus for the determination of the function P(p,w) we confine ourselves to u > 0, p > 0, w > 0. Then E(u), and we formulate the cost minimum problem as the Lagrangian problem

The necessary conditions upon x, z and A are

Solve the first two sets of these equations for Xj, Zk in terms of and substitute into the last to obtain

and calculation of

THEORY OF COST AND PRODUCTION

FUNCTIONS

(34) denote positive prices of the factors of producNow, by letting and output rate with positive inputs tion at the reference level and

Also, it follows from the necessary conditions for minimum cost and Equation (34) that

and the cost function is given by

Thus, the function P(p,w) is given by

and it has the same form as $(x,z) in terms of price relatives of the factors of production. Equation (35) gives the cost function for the Cobb-Douglas homothetic production structure, and the homogeneous function P(p,w) is a weighted geometric mean of price relatives. The aggregate prices corresponding to the vectors p and w are taken as

The functions P(p) and W(w) defining the aggregate variables P and W clearly satisfy the criteria C.l, C.2, C.3. The criterion C.4 is satisfied by proper choice of the constants Note first that from the necessary conditions tor minimum cost that < 126 >

THE AGGREGATION

PROBLEM

(39)

i.e., at positive minimum cost the value of the input of each factor is a constant fraction of the total cost. Then letting (40) it follows for any output rate u and price vectors

that (41)

Thus we choose the reference values

so that (42)

Then, by a straight forward calculation, using the necessary conditions for minimum cost,

Thus, by choice of the proper initial conditions the criterion C.4 w k equals zero, is satisfied for If the price Q(u,p,w) = 0 and the optimal inputs are zero for positive prices so that C.4 holds also. Thus the criterion C.4 is satisfied for all p if we take the cost function as an infinum when at least one price pi or w k is zero. The optimal inputs for factors of production with zero prices are unbounded and not actually realized. Turning now to the criterion C.5, the aggregate production and cost functions are (43) (44)

These conditions have to be satisfied by reason of Equations (39).

< 127 >

THEORY OF COST AND PRODUCTION

FUNCTIONS

and the aggregate forms of the homogeneous functions P(p,w) are

and (43.1) (44.1)

Clearly, for all nonnegaand tive vectors (x,z) and (p,w) such that X(x) W(w) = W, and the criterion C.5 is satisfied. Likewise and P(P,W) = P(p,w). Continuing with the criterion C.6, for any P > 0, W>0

is obtained by

since the distance function 1 on the efficient point set where the Mm occurs. Thus, we formulate this minimum problem as the Lagrangian problem

the necessary conditions for which are

The solution of these equations yields (note that

as the minimum value of P • X + W *Z. Hence criterion C.6 is satisfied If P = 0 or W = 0, the minimum of P • X + W • Z does not exist, but, taking Q(u,P,W) as the infimum of P • X + W • Z, Q(u,P,W) = 0 for P = 0 or W = 0 and criterion C.6 is then satisfied for Consider now the criterion C.7. Aggregate marginal productivities < 128 >

THE AGGREGATION

PROBLEM

and aggregate marginal costs computed from Equations (43), (43.1) and (44), (44.1) as partial derivatives are (when they exist)

By a straightforward calculation (when differentiable),

and for all vectors (x,z) such that ]

Similarly,

and, for all vectors (x,z) such that

Similarly,

< 129 >

Z we obtain

THEORY OF COST AND PRODUCTION

FUNCTIONS

and, for all vectors (p,w) such that

Finally, for all vectors (p,w) such that P(p) = P, W(w) = W,

Thus, the criterion C.7 is satisfied. For criterion C.8, we compute Sup I f P = Oor W = 0, Sup and, in this case, tor all vectors (p,w) such that P(p) = P, W(w) = W there exists a zero price among pi or w k so that Q(u,p,w) = 0 and the micro cost limited output function If P > 0 and W > 0

and, since f(u) is nondecreasing and lower semi-continuous with f(0) = 0, the set is a closed interval and Thu; satisfies (45) and for all vectors (p,w) such that P(p) = P, W(w) = W,

Moreover, it is direct to verify that the function f(P,W) has the same properties as T(p,w) (see Proposition (30), Section 5.4 for these properties). f(P,W) is nondecreasing in the vector i is lower semi-continuous in (P,W) since F( • ) is upper semi-continuous, and it is quasi-convex in (P,W) due to the concavity of P(P,W). < 130 >

THE AGGREGATION

PROBLEM

In summary, it has been shown that Proposition 35: The Cobb-Douglas production function may be aggregated, to satisfy the criteria C . l , . . . , C.8, by weighted geometric means of input and price relatives. The macrovariables are

with the reference values ! chosen so that and the aggregate production, cost and cost limited output functions are

where v 6.4 Aggregation of ACMSU Production, Cost and Cost Limited Output Functions In [28] a positively homogeneous function of degree one of the form

was used to illustrate a discussion of economic growth, and subsequently it was shown in [2] that the elasticity of substitution is constant for two factors of production if and only if the production function has this form. More recently, Uzawa [29] has shown the same result for more than two factors of production. We shall, for the following discussion, consider Uzawa's extension of this production function for two groupings of the factors of production, because the results to follow are easily seen to apply to any number of groupings of the factors. Let the factor input variables be grouped into two input vectors and denote by p = ( p i , . . . ,PN), < 131 >

THEORY OF COST AND PRODUCTION

FUNCTIONS

w = ( w i , . . . ,WL) the corresponding price vectors. Then the ACMSU production function has the form

The two factor of production form of this function has come to be known as the ACMS production function, and we shall dub Uzawa's extension of it as the ACMSU production function. This production function is another special case of a homothetic production function, and one need not restrict to homogeneity. Hence, we define the ACMSU production function as one of the forn where F( •) is a transform such that F(0) = 0 and and is a homogeneous function of degree one taking the form (46) for two groupings of the factors of production. The input rates of the function are inputs corresponding to some reference level Before proceeding further, we shall verify that the function (46) has the Properties A . 1 , . . . , A.6 of a production function so that indeed is a homothetic production function.-)" For this purpose, define

The parameters a and /5 range over the sets

for any input and and hence equals zero. Thus, rate

However, the efficient subsets

if any input

for the implied input sets L(u) are not bounded

< 132 >

THE AGGREGATION

PROBLEM

and

When this same property holds for The functions and are clearly finite for finite x and z, and continuous for Thus, the Properties A.1, A.2, A.4, A.5 hold for the function $(x,z) given by Equation (46). l o verity that the Properties A.3 and A.6 also apply, observe that g(x) is strictly concave (convex) for if and only if g(x + At) is strictly concave (convex) in the scalar X for all Section 11]. Compute

and t for x' nent is positive. If

this derivative is positive for all and is increasing in x since the expothe derivative is negative for all Then again is increasing in since the exponent of g(x) is negative. In the same way it may be shown is increasing in z. Next compute that h(z)

and, using the Cauchy inequality on the numerator of the first term above,

whence

for any and all is a concave function of x for any similarly, it may be shown that

Thus, and and 1 is a concave function of z for

Now, let

where

and the function

THEORY OF COST AND PRODUCTION

Then for any vectors of the inputs of the factors and scalar

FUNCTIONS

and

and

Since

and

for any the Property A.3. Further,

it follows that the function $(x,z) satisfies

and since the functions X(x), Z(z) are concave, it follows that 0 and hence is a concave function and the Property A.6 holds. It is interesting to observe that when ( the factors of production related to each of the two vectors x and z arc not essential by themselves, nor is any proper subgroup of them essential, and within each group the factors may be pure alternatives. But eact group as a whole is essential, since for all rhus, for negative exponents a and fi, the ACMSU production function permits the factors of production within each group to substitute completely for each other, whereas each factor of production is essential for the Cobb-Douglas production function. If the parameters a and B are positive, each factor of production is essential for the ACMSU production function. Thus, the production function given by Equation (46) spans both technologies where the with factors of production in each group are essential and where they are nonessential. < 134 >

THE AGGREGATION

PROBLEM

Having established that (x,z) given by (46) is a production function, homogeneous ot degree one, we turn now to the aggregation ot the homothetic production function in terms of two aggregate variables for the input vectors x and z. Define the aggregate variables X and Z by (47) (48) The functions X(x), Z(z) are finite, nonnegative homogeneous functions of degree one, and nondecreasing in their arguments with X(0) Z(0) = 0. Hence, these definitions of aggregate variables satisfy the criteria C.l, C.2, C.3. We note that, il and the aggregate variables X and Z are weighted arithmetic means of input relatives taken to a common positive power and a root taken corresponding to this power. While, if a X and Z are and weighted harmonic means of input relatives taken to a common positive power and a root taken corresponding to this power. Now, by Proposition 23, Section 4.4, the cost function of the production function ] where f(u) i has the form is the inverse function of the transform F( •) and P(p,w) is a homogeneous function of degree one with the properties stated in Proposition 24, Section 4.4. If the efficient set for any u > 0 is bounded, otherwise it is unbounded and we shall use the convention in this case that Q(u,p,w) is the infimum cost. As done for the Cobb-Douglas production function, we determine the function P(p,w) by the minimum problem

and the necessary conditions upon x, z, A for this minimization are

(49)

< 135 >

THEORY OF COST AND PRODUCTION

FUNCTIONS

From these equations it is clear that

and

Hence, if the are adjusted to minimum cost for any output rate u and price vector (p,w) > 0 (50) i

Further, at the reference inputs Equations (49) that

and prices

it follows from

and we may write (49) as

Then, multiplying the two sets of equations in (49.1) by spectively and summing, we obtain

and substitution into

and

yields

Q(u,p,w) = f(u)

as the cost function, where geneous function P(p,w) of the price vectors (p,w) is

and the homo-

(52) The function P(p,w) has the Properties A . 1 , . . . , A.6 (see Proposition 24, Section 4.4) and from Proposition 7, Section 2.4 it follows that the function P(p,w) is continuous on the boundary of the domain Hence, we may use Equations (51) and (52) as the cost function and price function for all taking the infimum cost for p and w on the boundary of this domain when < 136 >

THE AGGREGATION PROM EM Note that when either the cost function is zero if either the price vector p or the price vector w belongs to the boundary of the domains because the factor inputs in the vectors x and z with positive price will be substituted to zero, and as a group of factors either x or z may substitute out the other due to the Cobb-Douglas-like form of the function 0(x,z) in terms of these two vectors. But, when a > 0 and ft > 0, the cost function is positive for all price vectors p > 0, w > 0, because each factor of production in the vectors x and z is essential. The aggregate price variables P and W corresponding to the price vectors p and w are defined by (53) (54) By an argument paralleling that used above for the corresponding terms in the function (46), it is easy to show that the functions P(p), W(w) are nondecreasing in their arguments for p 0 with P(0) = W(0) = 0, as well as being finite and nonnegative homogeneous functions of degree one. Hence, these definitions satisfy the criteria C.1, C.2 and C.3. The criterion C.4 is satisfied by proper choice of the reference values Referring to Equations (50), we take them to satisfy (55) Then, by a straightforward calculation, using the necessary conditions for minimum cost (i.e., Equations (49.2)) and (55), it follows from (50) for inputs minimizing cost that

and the criterion C.4 is satisfied. The aggregate production and cost functions are (56) (57) where (58) (59) < 137 >

THEORY OF COST AND PRODUCTION FUNCTIONS

Notice that these aggregate functions have exactly the same form as the corresponding aggregate functions for the Cobb-Douglas produc­ tion structure (see Equations (43) and (44), Section 6.3). However, the aggregate variables X, Z, P, W are defined by different index functions, i.e., for each structure there is a special form of these index functions peculiar to the mathematical form of the production function. For both structures the aggregate factors of production are essential and the ag­ gregate cost function Q(u,P,W) is zero if in aggregate the factors of either group are free goods. The satisfaction of the criterion C.5 is immediate for all vectors x, z, p, w such that X(x) = X, Z(z) = Z, P(p) = P, W(w) = W. When the functions involved are differentiable, a straightforward calculation yields verification that the criterion C.7 is satisfied. Finally, for the criterion C.8 a calculation like that made for the CobbDouglas production structure yields f(P,W) = F(^—); P > 0, W > 0 v 7 \P(P,W)/

(60)

for the aggregate cost limited output function and the verification of C.8 is similar. But here the function P(P5W) is defined by Equations (59), (53) and (54). The micro and aggregate calculation of the marginal productivity of money capital to supply the cost of production are equivalent for all vectors p, w such that P(p) = P, W(w) = W, if the functions involved are differentiable. For positive C, P and W

(61)

p/ P(P5W)

Formula (61) obviously holds also for the Cobb-Douglas structure, using the definitions (37) and (38) for the aggregate variables P and W. To summarize, it has been shown that Proposition 36: The aggregate forms of the ACMSU cost, produc­ tion and cost limited output functions are the same as the corre­ sponding functions for the Cobb-Douglas structure, with the aggregate variables defined by the index number functions

=^srr =4i^n „[i^ri

Wk\~0~|~1//s

χ

r

o w

"w=w

α+β)/β

THE AGGREGATION

PROBLEM

and the aggregate initial values chosen so that

All criteria C . l , . . . , C.8 are satisfied. Strangely, the aggregate form of the ACMSU production function is not one with constant aggregate elasticities of substitution—nor do we require this property to hold, since it is not of much importance for the theory of cost and production functions. 6.5 Aggregation of a Class of Homothetic Cost, Production and Cost Limited Output Functions It is not surprising that the Cobb-Douglas and ACMSU cost and production structures have the same aggregate form for the production function, cost function and cost limited output functions. To see this, consider the class of homothetic production functions defined by

(62) where X(x) and Z(z) are finite, positive, nondecreasing, concave, continuous and difierentiablef homogeneous functions of degree one of the vectors x 0, and F( •) is a continuous and differentiable transformf of the production function with That i and F(v) is a production function, one may readily see by comparison of its properties with those ( A . 1 , . . . ,A.6) of a production function. Clearly, the Cobb-Douglas and ACMSU production functions are members of this class. The number of groupings of the factors of production in the form (62) is a trivial consideration, i.e., the arguments given in Sections 6.3 and 6.4, as well as here, hold for any number of groups of the factors in the function a as long as the exponents sum to unity. Define aggregate variables for inputs by (63) and these index number functions obviously satisfy the criteria C.l, C.2, denote price vectors for the input vectors x and z. The cost function for (62) is given by the minimum problem

Assumed for the existence of the marginal productivities.

< 139 >

THEORY OF COST AND PRODUCTION

FUNCTIONS

and the equations on x, z, X for this minimization are

(64)

Multiply the first set by Xj and the second set by z k and sum to find

and, using Euler's theorem on homogeneous functions, we obtain

(65)

since

(66) Thus, the Lagrangian multiplier is the minimum cost function Q(u,p,w). Solve Equations (64) in terms of the variables

and let (67) denote the solutions. The functions degree minus one m the variables positive scalar T and arbitrary price vector Equations (64) that

< 140 >

are homogeneous of because, tor any it follows from

THE AGGREGATION

PROBLEM

and

for arbitrary

and

implying

Similarly, the functions minus one in the variables tions (67) may be written

are homogeneous of degree Hence, the solutions of Equa-

(68) and the solutions x z j are homogeneous functions of degree zero in the variables p and w respectively, since the cost function is homogeneous of degree plus one in p and w. Substitute these solutions in the last of Equations (64), using the homogeneity of the function a(x,z), to obtain (69) where (70)

as the cost function for the production function (62). Since the function • W(w)" has the Properties A . 1 , . . . , A.6 (see Proposition (24), Section 4.4) and is homogeneous of degree one, it follows from Proposition 7, Section 2.4 that the cost function is continuous on Hence, Equation (69) the boundary of the price domain p and w holds for i For the aggregation of the cost function we define the aggregate variables P and W by (71) Since and are homogeneous of degree minus one and X(x) and Z(z) are homogeneous of degree plus one, it follows that P(p) and W(w) are homogeneous of degree one in the variables p and w respectively, and the criterion C.3 for aggregation is satisfied. In order to see what happens to P(p) for p = 0, replace p by T • p (where T IS a nonnegative scalar) in Equations (70) and, assuming p

< 141 >

THEORY OF COST AND PRODUCTION

FUNCTIONS

and Similarly, W(0) = 0 and the criterion C. 1 is satisfied. rhe criterion C.2 is satisfied, because for given u > 0 and follows from Equation (69) and the property of the cost function that (see Proposition 22, Section 4.3), that and similarly > To verify that the criterion C.4 is satisfied, note from Equations (70) and (68) that

and it follows from Equations (65) that

The aggregate forms of the production function and cost function are (72) for all vectors x and z such that X(x) = X Obviously, and Z(z) = Z, and likewise for all vectors p and w such that P(p) = P and W(w) = W. Moreover, these two aggregate functions have the same properties as their microvariable counterparts, i.e., A . 1 , . . . , A.6 and Q . l , . . . , Q.l 1, since they are merely the CobbDouglas production and cost function for two factors of production. The satisfaction of criterion C.6 follows by the argument given above in Section 6.3 for the fulfillment of this criterion by the aggregate form of the Cobb-Douglas production function. Likewise, the satisfaction of the criterion C.8 may be verified. Hence, it remains for us to verify the criterion C.7. But a simple calculation yields

and these ratios are clearly equal to the corresponding partial derivatives of for all vectors x, z such that X(x) = X and Z(z) = Z. Similarly < 142 >

THE AGGREGATION

PROBLEM

and these ratios are clearly equal to the corresponding partial derivatives of the function Q(u,P,W) f o r ail vectors p, w such that P(p) = P and W(w) = W. Thus the following proposition has been shown, f Proposition 37: The production function, cost function and cost limited output function for production functions of the form

aggregate to satisfy the criteria C . l , . . . , C.8 in the form of the corresponding functions for a Cobb-Douglas production structure with two factors of production. Thus, if a technology has homothetic structure of this form, one can expect that, by suitable definition of the index number functions, the aggregate data will follow the Cobb-Douglas production function, cost function and cost limited output function for two or more factors of production, depending upon how many groups of factors there are in the function a. We consider next a more general class of production structures. For as defined above, take the functions ] (73) be any nonnegative, finite, nondecreasing, quasi-concave, continuous and differentiate function of X = X(x), Z = Z(z) such that F(0,Z) = as and Then, as a for any vectors x, z such thai function of (x,z) as well as (X,Z), F satisfies the Properties A . 1 , . . . , A.6 for a production function. Moreover, as a group the factors x and z are essential. The continuity and differentiability of F in X and Z is assumed merely as a convenience for the existence of marginal productivities. The expression (73) is a natural generalization of a homothetic producSee [26], Section 9.

< 143 >

THEORY OF COST AND PRODUCTION FUNCTIONS

tion structvire for two groups of factors of production. In particular, we may have F(X(x),Z(z)) ΞΞ G(a(X(x),Z(z))), (73.1) where σ(Χ(χ),Ζ(ζ)) is a homogeneous function of degree one in X and Z with the properties stated for F(X,Z) and G( ·) is an arbitrary finite, nonnegative, nondecreasing, continuous and differentiable function of a with G(O) = 0 and G(a) + oo as σ —> + oo. Define aggregate variables for inputs by X = X(x), Z = Z(z). Then as defined above for the production function (62), these index number functions satisfy the criteria C.l, C.2 and C.3. Let ρ = pi,... ,pN), w = (wi,... ,wL) denote price vectors for χ and z. Then the cost function for the production function (73) is defined by Q(u,p,w) = Min JJp1Xi + 2 WkZk ~ MF(x(x),z(z)) - u)| Because of the separability of the variables χ and z, the nonnegativity of p, x, w, ζ and the nondecreasing property of the functions X(x), Z(z) and F(X,Z), this minimum problem may be formulated as Q(u,p,w) = Min {X · P(p) + Z · W(w) | F(X,Z) = u)

(74)

X1Z

where X · P(p) = Min if PiXi I X(x) = x) x ^ i J

a S )

Z · W(x) -- Min f ^ wkzk | Z(z) - Z Z I χ It is clear from the form of the sub-minimum problems that the related cost functions have the forms given, in which P(p) and W(w) have the Properties Q.l,... ,Q.7 of a cost function (see Proposition 24, Section 4.3), and in particular they are homogeneous functions of degree one. Thus the cost function of the production function (73) takes the form Q(u,p,w) = Q(u,P,(p),W(w))

(76)

in which the variables p, w are separable. Hence, we take as the aggre­ gate variables for the prices ρ and w, P = P(p), W = W(w). The index number functions P(p), P(w) clearly satisfy the criteria C.l, C.2, C.3 (see Proposition 24, Section 4.3). Regarding the criterion C.4, the aggregate variables X, Z, P, W satisfy

THE AGGREGATION

PROBLEM

for inputs w, z which minimize cost, see (75). The aggregate production and cost functions

obviously have the properties of a production function and cost function respectively and

for all vectors^ x, z, p, w such that X(x) = X, Z(z) = Z, P(p) = P and W(w) = W. Hence, the criterion C.4 is satisfied. Moreover, it is clear from the calculation of Q(u,P(p),W(w)) that the criterion C.6 holds. Concerning C.l, the satisfaction of this criterion is verifiable in a straightforward manner. The micro cost limited output function F(p,w) is defined by

and is a closed interval due to the lower semi-continuity of the cost function in u. Hence, the function IYp,w) exists and since the constraint 1 depends upon P(p), W(w) it follows that the micro cost limited output function has the form

The corresponding macro cost limited output function evidently has the form T(P,W) and

for all vectors p and w such that P(p) = P and W(w) = W. Thus, the following proposition holds Proposition 38: A production function F(X(x),Z(z)) where (a) X(x), Z(z) are finite, positive, nondecreasing, concave, continuous and differentiable homogeneous functions of degree one of 0 and the vectors x (b) F(X,Z) is a nonnegative, finite, nondecreasing, quasi-concave, contmuous and diiferentiable function of X = X(x), Z = Z(z) such that for any vectors (x,z) such that < 145 >

THEORY OF COST AND PRODUCTION FUNCTIONS

may be aggregated to satisfy the criteria C.l,..., C.8 by the aggregate variables X = X(x), Z = Z(z) ι

In particular, if F(X(x),Z(z)) has the form (73.1), it is a homothetic production function of general type for separability of the factor inputs into two groups of factors, and the aggregation problem is solved. The extension of these results (i.e., propositions 37 and 38) for sepa­ rability of the factor inputs into an arbitrary number of groups of factors is trivial.

CHAPTER 7 THE PRICE MINIMAL COST

FUNCTION

7.1 Definition of the Price Minimal Cost Function To review the framework in which the price minimal cost function is to be defined, we consider a production structure of production poswith production function sibility sets It is assumed that the subsets of the nonnegative domain D of a Euclidian space have the Properties P . l , . . . , P.9 (see Definition, Section 2.1) and the efficient subsets E(u) ol are bounded. Independently of boundedness oi the efficient subsets E(u), it was shown in Section 2.1, Chapter 2, that The distance function of the production structure is (see Section 3.1)

where

for some and the production possibility sets of the production structure are defined in terms of this distance function by (see Proposition 14, Section 3.1) For this production structure, there is a unique production function defined alternatively by

and this function has the Properties A . 1 , . . . , A.6 (see Section 2.2). It has been shown in Section 5.1 that there is a cost structure defined in terms of the factor minimal cost function Q(u,p) by

Used in [26] as a dual problem to that defining the cost function, see pp. 18, 19.

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THEORY OF COST AND PRODUCTION FUNCTIONS

The cost structure consists of subsets of price vectors in the nonnegative domain D of Rn for which the factor minimal cost is equal to or greater than unity corresponding to output rates u Ξϊ 0. The subset JEQ(O) is empty, since Q(0,p) = 0 for all Ρ ^ 0. Now because the cost function Q(u,p) is a distance function for the sets £Q(U) of the cost structure (see Proposition 26.1, Section 5.1), it appears that the cost function Q(u,p) is a dual of the production struc­ ture distance function ^(u,x). It is in this setting that we consider here the minimization of cost with respect to the factor price vector ρ for any output rate u ^ 0 and input vector χ ε D. This minimization is a dual operation to that defining the cost function Q(u,p). Precisely, the price minimal cost function Ψ*(υ,χ) is defined by

J (U5X) = Inf {ρ · χ I Ρ e £ Q(U),P ^0) , u ^ 0 , x e D .

1 FT

ρ

(77)

Clearly, ^*(u,0) = 0 for u > 0. If u = 0, the set JCQ(O) is empty and Ψ*(0,χ) = + oo for any χ ε D, because a bounded value implies the ex­ istence of a bounded price vector Ρ ε £Q(0), a contradiction since £Q(0) is empty. The operation Inf has been used instead of Min, because, although the price sets 0 there is a price point of the form such that Thus, if for some 0 for all 0. Hence, there does not it follows that 0 for some u' with exist points of D2 such that for another value u > 0. Consequently The possibility of obtaining zero price minimal cost for all positive being not empty, is a formal result which need output rates, i.e., not be realizable lor any economic situation, since the cost function merely expresses mathematical combinations of price and input vectors. We may pursue this issue by investigating when is an empty set. For this purpose, classify the boundary points D 2 relative to factor prices by for some

< 149 >

THEORY OF COST AND PRODUCTION

FUNCTIONS

of the price sets It follows from the ray Property of the cost structure (see Proposition 26) that the ray intersects all if and if p then p Hence, the sets sets and are exclusive and This classification of the boundary points of is identical to that provided by the sets denned in Section 4.1, Chapter 4, i.e., A proof of this fact follows: If then for any u > 0 we have Q(u,p) > 0 and, due to the homogeneity of the cost function, there exists a scalar 6 > 0 such that 1. Hence, for some u > 0 and and implies ] Conversely, if there is for some u 0 a scalar l such that and, by virtue of the ray Property 77.3 of the cost structure, intersects all price sets the ray of the cost structure tor u > U. Ihus, tor any u > 0 there is a scalar 0 > 0 such that and consequently for any u > 0 we have Hence, p implies p Therefore, and, since where the mtersections are empty, it follows also that Now, regarding whether is nonempty, assume is nonempty and let I hen there is an input vector such that and the ray mtersects all price sets by virtue of the Property of the cost structure. Hence for any u > 0, there will be a price poinl such that 0, implying 0, and there is an input vector Thus, if I is nonempty, the set is nonempty. Contrariwise if is empty, all points of do not belong to any of the price sets 0 and, for all i for 1 since and (semi-positive). Thus, if is empty, the set is empty. The foregoing arguments regarding the sets may be summarized in terms of the following propositions: Proposition i:

is empty.

Proposition ii: Proposition iii:

empty; and is nonempty, if and only if

is nonempty.

With these preliminaries, we state and verify the following proposition concermng the properties of the price minimal cost function Proposition 39: If the cost structure has the Properties ( (see Proposition 26, Section 5.1), the price minimal cost function has the following properties: < 150 >

THE PRICE MINIMAL COST FUNCTION

and is finite for finite

and positive for but

for

is a concave function of x on D for all is a continuous function of x on D for all and and finite. For any of u for all

Sup

is possibly

is an upper semi-continuous function

These properties are clearly the same as those of the distance function for the production possibility sets (see Section 3.2, Chapter 3) and, treating the price minimal cost function as a distance function in the factor input domain D, a family of sets is defined by

In subsequent sections, it will be shown that the family of sets is a production structure, that is a distance function for this for all structure, that anc for all and any We turn now to the verification of the properties stated in Proposition 39. From the definition (77) of Section 7.1 above for all u anu for all , Further, by definition of the set fhus, the Property D.l* holds. Regarding Property D. suppose Since x > 0 (semi-positive) and x is finite, this implies that there does not exist a bounded pnce vector a contradiction, since is nonempty for u > 0. Further, by definition of the set When clearly since for any Thus, Property D.2* holds. The Properties may be proved for u > 0, D by arguments which are exact analogous of those given in Section < 151 >

THEORY

OF COST

AND PRODUCTION

FUNCTIONS

4.3, Chapter 4, to prove the Properties Q.3, Q.4, Q.5, Q.6, Q.7 for the factor minimal cost function Q(u,p). When u = 0, they follow from the fact that for all Thus, these properties may be estabhshed for the price minimal cost function Property is immediate, since implies by virtue of the Property of the cost structure (Proposition 26, Section 5.1), and 0, since then obviously holds when The inequality of for any D. Hence, the price minimal cost function is nonincreasing in u for any : For verification of Property The sequence is nonincreasing and bounded below by zero. Suppose f h e n there exists a nondecreasing subsequence The sequence is nonincreasing such that (Property

Now, since p • x is

and

for p e closure of

continuous in p, it follows that p • (see Property D and Regarding

the cost structure), a contradiction. Thus for any

and

is a nonempty chosen subset of D. Then

finite for all 0 on an open subset ) be nonempty. Then

to be nonempty, it is necessary that of D containing the origin. Let for all

Suppose

and

where is the cost minimizing input vector corresponding to p. for all u ~ which implies that there Thus for exists an open subset of D contaimng the origin such that: all u on this subset. As a final remark, certainly for all is finite if since then Finally, we turn to the proof of the upper semi-continuity of lor any in u on the interval be an arbitrary seWe need to quence of output rates tending to an output rate show that

< 152 >

THE PRICE MINIMAL COST FUNCTION

Now, for any scalar there exists a price vector p£ belonging to the interior of such that and The lower semi-continuity of the cost function for p > 0 implies

the last inequality holding because p£ is an interior point of and Q(u,p) is a distance function for the cost structure . Then there exists an integer N such that for all n 1 and pf belongs to for all n > N. Therefore,

Sup

When

+ oo and clearly this last inequality holds. Thus, ^*(u,x) is upper semicontinuous in u on [0,oo) for any x e D. 73

The Production Structure L*(u) Defined by the Price Minimal Cost Function

Consider now the family of sets of input vectors indexed by the output rate u and defined by It has been shown in Section 3.2, Chapter 3, that the distance function of a production structure : has the Properties D.ll*. However, the converse, that is, a function with these properties does indeed define a production structure for which it is a distance function, has not been demonstrated. We proceed to verify that the family , is a production structure, i.e., it has the Properties First, since by the Property D.l* implying

, and

implying 1 family If and the Property

for any for

Thus, Property P.l holds for the it follows that

because by

and Property P.2 holds. , Property P.9, i.e., the boundedness of the efficient subsets, will follow by showing that , see Section 7.4.

< 153 >

THEORY OF COST AND PRODUCTION

FUNCTIONS

For the verification of Property P.3, we need concern ourselves only with because if then from Property and the homogeneity Property it follows that for all and the ray does not intersect any of the sets for When for (see Property and clearly the ray intersects all sets of the family since can be made at least as large as unity for by choice of the scalar due to the homogeneityProperty D.3*. Property P.4 follows directly from the Property because if and ii, then and Property P.5 isestablished as follows: If then bv Property P.4 we have for all u e [0,u0) and hence "ontrariwise, if

then

because

i then

and due to the upper semi-continuity of the function (Property it follows that there exists an and a ich that By choice of a small enough, for positive contradicting the hypothesis that Property P.6 holds, because suppose there exists a finite input vectoi Let

and

is not empty. Then,

such that

Sup

for all

, contradicting Property

Property P.7 holds, because for any the function is continuous in x on D and therefore upper semi-continuous on D which implies for all positive numbers and any that the set

is closed. (See Appendix 1.) In particular, for

we have

closed for all Finally, regarding the Property P. 8, the sets are convex for all because let be any two input vectors belonging to a set L*(u) for any Then, due to Property concavity of vl/*(u,x) in x on D, it follows for any scalar

But whence

implies

and

< 154 >

implies

, i.e., the hat

THE PRICE MINIMAL COST FUNCTION

Hence, if and the points belong to for and the set is convex. In summary, the following proposition has been established: Proposition 40: The family of sets , where has the Properties duction structure with the Properties P . l , . . . , P.8.

is a pro-

We show, next, that for the production structure the function is a distance function, see Section 3.1, Chapter 3, i.e.,

where and section of the ray First, if

. The vector is the interwith the boundary of the set for (See Figure 28.) the ray does not intersect any of the sets

FIGURE 28: INTERSECTIONS OF INPUT RAYS WITH L*(u)

< 155 >

THEORY

OF COST AND PRODUCTION

FUNCTIONS

for and by Property we have for and Second, if the ray intersects all sets for (see Property P.3) and, for any positive output rate, let

since is closed (P.7). The intersection of the the boundary of and

since in

1 for Thus,

boundary

because

r a y w i t h

is continuous

for any and Finally, if u = • for all (Property , and the function is a distance function for the production structure L*(u). Thus, the following proposition has been shown: Proposition 41: The price minimal cost function function for the production structure

is a distance

One thing remains to be done in this discussion,namely to eliminate the asymmetry of the definition o f t h e sets ' relative to that given in Section 3.1 for the sets to define the distance function of the production structure . If then for any we have positive and finite for x finite, and there exists for each positive output rate u a scalar such that due to the homogeneity of the distance function Hence, for some and and for sonu

ant

Conversely, if foi and some then the ray intersects all production sets by virtue o f t h e Property P.3 holding for the sets Consequently, for all there is some such that , and 0 for all u > 0. Hence, x e D£ implies x e and for some Similarly, it may be shown that

< 156 >

THE PRICE MINIMAL COST FUNCTION

In fact, since is empty and and tion holds:

it follows that

is empty,

. Thus, the following proposi-

Proposition 42: 7.4 Equivalence of the Production Structures and Their Distance Functions In order to establish a duality between the distance function ^(u,x) of the production structure L»(u) and the factor minimal cost function Q(u,p), we now prove the following two propositions: Proposition 43: Proposition 44:

for any x e D and all

We consider first the Proposition 43. If u = 0, clearly , since theProperty P.l is commonly held by Next for we show first that Then, since

it follows that

Since

Inf

it follows that for any scalar

p

there exists a price vector pe belonging to

Consider the ray • pe and therefore

such that

be a point on this ray. Then

since pe can always be chosen so that p£ belongs to the efficient set S(u) of Consequently,

and

t Shown in [26], pp. 20-22, under more restrictive conditions.

< 157 >

THEORY OF COST AND PRODUCTION FUNCTIONS

>

>

Hence, for any £ 0, 'Y*(u,XO) + £ 1 and 'Y*(u,XO) > 1, implying that XO belongs to L*(u). Next, in order to show L",(u) :::> L*(u), assume XO belongs to L*(u) and that XO does not belong to L",(u). Then, by the strict separation theorem for convex sets (see [3], p. 163), there exists a semi-positive price vector po such that po . XO

< Q(u,pO),

po being semi-positive because of Property P.2 of a production structure. Hence, p0 • XO

< 1, where p0 = Q(u,pO) po

=

But, Q(u,pO) 1, due to the homogeneity of the cost function, implying that p0 E £Q(u). Consequently, 'Y*(u,XO) < 1'0' XO

O.

Hence, the production structure L*(u) is identical to the production structure L",(u), because L*(u) = L",(u) for all u > O. Proposition 43 is therefore proved. Having shown that the production sets L*(u) and L",(u) are identical for any u E[0,00), it follows that the distance function 'Y*(u,x) and 'Y(u,x) are identical for any XED and any u E [0,00). To see this, note first that by their definitions

'Y*(O,x) 'Y*(u,O)

= 'Y(O,x) = + 00, 'v'x ED. = 'Y(u,O) = 0, 'v'u > O.

N ext, the sets D 2, D:f and Dz, Dz* satisfy (see Proposition 42) D2 Thus, 'Y*(u,x)

= Dz, D:f = Dz*.

= 'Y(u,x) = 0, 'v'u > 0

and

x EDz*

= D:f,

and it remains for us to consider the equivalence of these two distance functions only for x E Dl U Dl; = Dl U D 2. In this case, the identity of the sets L*(u) and L",(u) implies that the intersection ~ of the ray {Ax I A >O} with L*(u) and L",(u) is the same point, whence by the definitions of the functions 'Y*(u,x) and 'Y(u,x) they have the same value 0 and x E Dl U Dz Dl U D 2. Therefore, Proposition 44 holds. for u

>

=

< 158 >

CHAPTER 8 DUALITY OF COST AND PRODUCTION STRUCTURES AND RELATED FUNCTIONS 8.1 Duality of the Cost and Production Structures and Their Related Distance Functions Q(u,p), For any production structure with production possibility sets L*(u), having the Properties P . l , . . . , P.9, there is a distance function defined on the sets The production structure is specified in terms of the distance function ^(u,x) by

The cost structure dual to the production structure is a family of sets in the nonnegative price domain defined in terms of the factor minimal cost function Q(u,p) by

and the cost function Q(u,p) is a distance function for this structure. The two functions Q(u,p) and ^(u,p) are dualistically determined from each other by:

Min is used for the calculation of Q(u,p) because the efficient subsets of the production structure are assumed to be bounded; and, if the efficient subsets of the cost structure , are bounded for all , Inf may be replaced by Min in the calculation of This duality provides algorithms for calculation of each function and Q(u,p) in terms of the other by dual cost minimization processes, and the dual production and cost structures are determined from each other in terms of these two functions. Mathematically, then, the price minimal cost function , taken with the factor minimal cost function Q(u,p) is a more convenient function for definition of a production structure than the classical production function. One need only specify the technology either by a function with the Properties D.l, . . . , D.ll (see Section 3.2) or a function Q(u,p) with the Properties Q.l, . . . , Q.l 1 (see Section 4.3), since the other is determined by the dual cost minimization process. The set defined production and t This duality was shown in [26] under more restrictive conditions, see Section 4, p. 17.

< 159 >

THEORY OF COST AND PRODUCTION FUNCTIONS

cost structures L41(U) and J3Q(U) are then easily determined as defined above in terms of these two functions. From an engineering study of the technology (economic analysis of cost) one might specify the production structure sets L41(U) (price sets £Q(U)). Then the function ^(u,X) (Q(u,p)) is merely determined as a distance function on the structure LCT(U) ( 0. Now, using the homo­ geneity of the function P(p), {p I f(u) · P(p) ^ 1} = -X- {(f(u) · p) I P(f(u)-p)^ 1), since f(u) > 0 for u > 0 (see Proposition 5, Section 2.3). Hence, the minimal problem defining ^(u,x) may be expressed as *(u'x) =~L·· < ¥U «Ρ · f(u» 'x I p^u)" P) = 1), u > 0, χ ε D. I(U) (ρ · f(u)) f Shown in [26], see pp. 43-47, under more restrictive conditions.

DUALITY OF COST AND PRODUCTION STRUCTURES

Then, for any positive output rate u, let ρ = f(u) ·ρ and

ik)' Ύ '

*(u,x)=

=1}'χε D

{p x 1 Ρφ)

From this it follows that *(η,χ) = ^·Φ(χ), since the solution of the minimal problem is independent of the output rate u. Moreover, the function Φ(χ) is homogeneous of degree one in χ and otherwise has the Properties A.1,..., A.6 of a production function. (See Proposition 39, Section 7.2.) Consequently, the production sets Lp(u) corresponding to a cost structure with cost function Q(u,p) = f(u) · P(p) are

= {x I F(O(X) ) ^ u ) , with F( ·) having the required properties for homotheticity. Thus, if the factor minimal cost function has the form Q(u,p) = f(u) · P(p), the pro­ duction structure is homothetic, which coupled with Proposition 23 establishes the theorem. We may therefore use factor price deflated costs to study the returns to scale, either by f(u) = Q(u,p)/P(p) or F(Q(u,p)/P(p)), if and only if the production structure is homothetic. Corollary: 1Jr(UjX) = (x)/f(u) if and only if the production structure is homothetic.f Corollary: T(p/c) = F(c/P(p)) for c > 0 and ρ ε Di U if the production structure is homothetic.

if and only

8.5 Dual Expansion Paths The notion of expansion paths in the theory of production is usually introduced for production structures with production possibility sets which have single contact point supporting hyperplanes, i.e., regular support planes. Under these circumstances then there is a unique path (locus) of least cost input vectors as a function of output rate for any given vector ρ of factor prices. If we assume that the production sets L4(u), u ε [Ο,οο) have only regu­ lar support planes, the cost function Q(u,p) has continuous partial f Stated in [26], p. 41.

THEORY OF COST AND PRODUCTION

FUNCTIONS

derivatives of the first order in the components of the price vectors p (see [4], p. 26). Further, assume Q(u,p) is continuous and differentiable in u for all Then letting denote the least cost input vector corresponding to any price vector p e D and output rate u it follows that

and

But, for any

the identity

holds and

Also, since occurs on the boundary of the s e t , minimum problem may be expressed as

where

is a positive multiplier, and the point

the cost

satisfies

whence

Thus,

and the least cost input rate of each factor of production equals the "marginal factor minimum cost" with respect to the price of the factor along the expansion path, a familiar economic proposition. We shall define the dual expansion (contraction) path in the price structure to be the locus of least cost factor-price vectors as a function of output rate for any input vector x, and the distance function of the production structure (price minimal cost function) is given by (see Chapter 7)

< 170 >

DUALITY

OF COST AND PRODUCTION

By a similar argument using the identity

STRUCTURES

it follows that

and along the expansion path the price of each factor equals the "marginal price minimum cost" with respect to the input of that factor. Thus, using the duality between the distance function of the production structure (factor price minimal cost function) and the distance function of the cost structure (factor input minimal cost function) (see Section 8.1), the expansion paths relate the prices and inputs of the factors by the following elegant dual equations :f

The first set of equations is in the customary form in which they appear in the classical literature of the theory of production, and the second set appeared first in [26], The duality between the cost function Q(u,p) and the distance function may be expressed by an involution (transformation) defined by the equations

(80.1)

We note that the functions tions as defined by Fenchel.

and Q(u,p) are not conjugate funcsince

Fenchel's notion of conjugate functions is associated with polarity with respect to a parabolid of revolution, while the transformation (80.1) is Given in [26] by Equations and (6.1), pp. 13 and 19. See [10], Chapter III, Section 5.

< 171 >

THEORY OF COST AND PRODUCTION

FUNCTIONS

a polarity with respect to the unit sphere in centered at the origin (see Section 8.3), i.e., the point coordinates of the surface Q(u,p) = 1 for arbitrary are the poles of the tangent planes of the surface with respect to the unit sphere and vice versa. However, the transformation (80.1) is a contact transformation. For the transformation of the first set of equations into another form, we note that (assuming the production function to be continuous, differentiable and strictly increasing) the distance function and the production function are related for any input x and output rate u by the identity

because due to the homogeneity of ^(u,x)

implying that the point lies on the isoquant of the production set , and with this input vector the production function yields the output rate u. Then, by differentiating this identity one finds

But

is a contact point of the production set Hence,

and

(a)

(b)

where the superscript * denotes evaluation at the input vector < 172 >

DUAliTY OF COST AND PRODUCTION STRUCTURES

which minimizes cost with respect to x for the given price vector p and output rate u. The cost function Q(u,p) is given in terms of x "'(u,p) by n

Q(u,p)

= 2.: Pixi(u,p) 1

and

oQ _ ~ . oxtcu,p) ou -

Since 'I'(u,x"'(u,p»

1

PI

ou

(c)

.

= 1, for any p and u,

- . ox*(u,p) = O. (-0'ou1')* + 2.: (0'1')* OU n

(d)

j

j=l

OXj

Further, for the minimum cost Q(u,p), the minimum problem may be expressed as Q(u,p) = Min {p. x I 'I'(u,x) = I}, P e D, u x

> 0,

and letting ~(x I p,u)

n

=2.: PiXi + A(u,p)[1

- 'I'(u,x)]

1

denote a Lagrange function for this problem; necessary conditions to be satisfied by x*(u,p) are Pi

= A(U,P)( o'l'(U,X»)* OXi

(i

= 1,2, ... ,n)

(e)

and, since 'I'(u,x) is homogeneous of degree one in x, it follows that n

2.: Pixi(u,p) = Q(u,p) = A(u,p) 1

for all u

> 0, p e D. Then, using Equations (b), (c), (d), (e) and (f)

(0'1')*

aQ -_ L,., ~ Pi oxi(u,p) -_ - Q(u ,) p · au i=1 ou OU _ Q(u,p)

-

-

~ ~(u,p)(afl>(x»)*' ':l

L,., Xl j=1

uXj

and, using Equations (e), (f) and (a)

Pi = Q(u,p)

n

ofl>(X»)* ( aXi * (ofl>(X»)*

j~Xj(U,P) ~

< 173 >

(i

= 1,2, ... ,n).

(f)

THEORY OF COST AND PRODUCTION FUNCTIONS

By combining these last two equations, one obtains:

>-(#)•

m

*-*·•·*

and, along the expansion path in the production structure the ratios of the marginal productivities of the factors to their prices have a common value equal to the reciprocal of the marginal cost with respect to output rate, i.e., the marginal productivity of money capital as an input. Similarly, for the expansion path in the cost structure, the cost func­ tion Q(u,p) and the cost limited output function Γ(ρ) are connected for any price vector ρ and output rate u by the identity

since, due to the homogeneity of Q(u,p),

4-0(¾) =1 and the price point p/Q(u,p) lies on the isoquant of the price set £r(u)f. Then by differentiating this identity

(i = 1,2,... ,n),

But, since p*(u,x) is a contact point of the price set and Q(u,p*(u,x)) = 1, it follows that

(i = 1,2,... ,n)

(a')

(b')

f Due to continuity of the cost function in u.

DUALITY OF COST AND PRODUCTION

STRUCTURES

where the superscript * denotes evaluation at the price which yields minimum cost for given input vector x and output rate u. The price minimal cost function is given in terms of by

and (C) But, since

for any p and u, (dO

Further, for the price minimal cost function ^(u,x), the minimum problem defining this function is

and, letting

be a Lagrangian for this minimum problem; necessary conditions to be satisfied by are (eO and, since the cost function Q(u,p) is homogeneous of degree one in p, it follows that (fO for all and (f')

Then, as before, using Equations

and using Equations

< 175 >

THEORY OF COST AND PRODUCTION FUNCTIONS

By combining these last two sets of equations, one obtains Xi

= (oi'). (Or(p»)* ou

(i

OPi

= 1,2, ... ,n)

and along the expansion path in the cost structure, for any input vector x, the ratio to the input of a factor of its "marginal productivity with respect to the factor price" has a common value equal to the reciprocal of the marginal value of the price-minimal cost function with respect to output rate. Thus along the expansion paths in the production and cost structures, one may express the relationship between.the prices and inputs of the factors by the dual equations

(i

= 1,2, ... ,n).

(81)

ou Moreover, if the price vector p and input vector x related to the expansion paths are taken as dual correspondents, i.e., as p*(u,x*), x*(u,p*) (see Section 8.3), these equations simplify to o(X»)* OXi --p-:--~(

_(or(p»)* 0Pi +1 = --x~~"--- - (~~

r

(i

= 1,2, ... ,n),

(82)

where the superscript * indicates that price and input vectors are taken as dual correspondents. One need only take the expressions given above for (oi' /ou)* or (oQ/ou)* and substitute from the last set of dual equations to obtain ( oi')* au

=

( n

L

oQ)* au , pixi

i=l

and observe that (see Section 8.3) n

L

pixi

= 1.

i=l

Hence, along what might be called the Dual Expansion Paths the mar( 176

>

DUALITY OF COST AND PRODUCTION STRUCTURES

ginal productivities with respect to factor inputs and factor prices (taken as dual correspondents) bear a common ratio to price and input of the factor respectively which are the negative of each other, and the numer­ ical value of this common ratio equals the marginal productivity of money capital as an input into the process.

CHAPTER 9 PRODUCTION CORRESPONDENCES 9.1 The Definition of a Production Correspondence Here we are concerned with technologies which yield several differ­ ent joint products for a given input vector of the factors of production. For the most general treatment, all of these products need not be desirablef or have a positive economic or social value. In particular, waste products, which lead to pollution of air, stream and land and cost society for their control, may be explicitly treated as part of the joint outputs of the technology. The classical example of wool and mutton, as joint products of a livestock technology, is an example of products occurring more or less in fixed proportions. In other technologies, such as chemical and petroleum refining, it may be possible to obtain vari­ ous mixes of several outputs for a given set of inputs. Even disjoint technologies may be combined and operated coherently to produce jointly the outputs derivable from each, particularly when there is a subset of the factors of production encompassing all of the disjoint technologies which is commonly applicable. All of these possibilities are allowed in our notion of a production correspondence. As before, let χ = (xi,x2,... ,xn) denote a vector of input rates of the factors of production related to the technology, and the input vectors χ are restricted to the nonnegative domain {χ | χ 0} of an n-dimensional Euclidian space Rn. We assume that there are m distinct kinds of goods and services which may be jointly produced by the technology, using the input vectors x. Let u = (ui,u2,.. . ,um) denote a vector of output rates for the technology. The output vectors u are restricted to the nonnegative domain (u | u ^ 0} = U of an m-dimensional Euclidian space Rm. For the vectors u we use the same convention regarding inequality signs as that used for the input vectors x, i.e., if u and ν are output vectors u > v = > U i > V i V i e {1 , 2 , . . . , m } u ^ ν =* Uj 2: V V i ε {1,2,... ,m} u > ν => U 2 ; V but u ^ v . i

i

i

For an input vector χ the possible outputs of the technology are gen­ erally not a single vector u, but a set of nonnegative vectors u denoted by P(x). f Throughout the text to follow "desirable" will be used synonymously with "disposable."

PRODUCTION CORRESPONDENCES

Definition: P(x) C {u | u ^ 0} = Rf denotes the set of alternative output vectors u which may be obtained by use of the input vector χ ε {χ I χ ^ 0}. Thus for a technology with joint outputs, we can no longer define the production relationship by a production function Φ(χ) mapping χ onto the nonnegative real line. In order to distinguish this situation, we shall refer to the technical relationship between the inputs and outputs as a Production Correspondence, f For the definition of a production correspondence, let X = {χ | χ ^ 0} = R+ and U = {u | u ^ 0} = Rif1 denote the sets of nonnegative input and output vectors of the technology. Definition: Ρ: X U denotes a production correspondence mapping X into U with output sets P(x) C U corresponding to χ ε X. Thus the production correspondence Ρ: X -» U is a function mapping points of X into subsets of U. Analogous to the role of the production function (single output), the production correspondence Ρ: X U defines output substitution alter­ natives. For any input vector χεΧ, the map set P(x) in the output space constitutes the set of output vectors which may be realized and substituted for one another with the given inputs χ of the factors of production. Inversely, the substitutions between input vectors χ (alternative inputs) to obtain a given output vector u is a subset of the total input set X defined by L(u) = (x I u ε P(x),x ε X}. The subsets L(u) for u ε U define a correspondence inverse to the production correspondence Ρ: X —» U.ff Definition: The inverse correspondence L: U X is a mapping of U into X such that for any u ε U the map set is L(u) = {χ | u ε P(x), χ ε X}. The subset L(u) of X defines the set of input vectors χ which yield at least the output vector u. As in the case of the production function for a technology with a single output, we might speak of the sets L(u), u ε U as production possibility sets or simply production sets. But, in order to distinguish the sets L(u), u ε U from the sets P(x), χ ε X, we shall refer to a set L(u) as an input possibility set and a set P(x) as an output possibility set, or just simply as input and output sets respectively. f The terms correspondence, function, mapping are all synonymous. ft Sometimes called the lower inverse, see [3], Section 3.

THEORY OF COST AND PRODUCTION FUNCTIONS

The significant substitution alternatives for outputs and inputs are not given by the entire sets P(x) and L(u) respectively. Mainly, we are concerned with the boundaries of these sets relative to U and X respec­ tively, and more particularly with the subsets of these boundaries which are efficient. For the definition of efficiency, let u = (u(D), u(D)), where u(D) is the subvector of u representing nondesirable outputs and u(D) is the subvector of desirable or intended outputs. Definition:f The efficient subset of an output set P(x) is u ε P(x); Max {θ \ θ · u ε P(x), θ e [Ο,οο)} = 1; Ep(X) =

ν = (v(D),v(D)) # Ρ(χ) if u(D) is not void and (i) v(D) > u(D), v(D) ^ u(D) (ii) v(D) ^ u(D), v(D) < u(D).

Definition: The efficient subset of an input set L(u) is EL(U)

= {x I

Χ

ε L(u),y ¢ L(u) if y < x}, u ε U.

Clearly, for efficiency of an output vector, neither decrease in any com­ ponent of u(D) with u(D) not decreasing, nor an increase in any com­ ponent of u(D) with u(D) not increasing, should be possible. Also, we exclude proportionate decreases along the boundary of P(x) from the efficient subset. If u(D) is null, i.e., all outputs are desirable, the defini­ tion of Ep(x) reduces to the conventional one. Thus, the production correspondence Ρ: X -» U and its inverse cor­ respondence L: U ^ X provide a complete statement of the production relationship, characterizing the substitution alternatives for both out­ puts and inputs. We might have started with the correspondence L to define the pro­ duction correspondence, i.e., considered L: U —> X where the map set of an output vector u ε U is the subset L(u) of input vectors χ belong­ ing to X which yield at least the output vector u. Then the correspond­ ence Ρ: X U is obtained as the inverse correspondence of L: U ^ X, where the map set of an input vector χ ε X is defined by P(x) = (u I χ ε L(u),u ε U}. It is clear that u ε P(x) is equivalent to χ ε L(u), i.e., u ε P(x) if and only if χ ε L(u). Another way of expressing the production relationship is in terms of the graph of the production correspondence P: U —> X or the graph of the inverse correspondence L: X —» U. For this purpose, let (u,x) and (x,u) denote ordered pairs of vectors in the product spaces Rm χ Rn and Rn X Rm respectively. Then (u,x) is restricted to the nonnegative f The definition of EP(X) as given is possible if P(x) is closed for all χ ε X, and similarly for an input set L(u).

PRODUCTION CORRESPONDENCES

domain of Rm χ Rn, i.e., (u,x) ε U χ X, and (x,u) is restricted so that (x,u) ε X χ U. The graph of the correspondence Ρ: X U is defined by Definition: The graph of the correspondence Ρ: X

U is

{(x,u) I χ ε X,u ε P(x)}. Similarly, the graph of the correspondence L(u) is defined by Definition: The graph of the correspondence L: U -» X is {(u,x) I u ε U,x ε L(u)}. Clearly, these graphs represent the same subset of the product spaces Rm χ Rn, the difference being merely the ordering of the two vectors u and x. The graphs of the production correspondences Ρ: X —> U and L: U -» X are sometimes referred to as Technologies. The properties of closure and convexity for the input sets L(u), u ε U and the output sets P(x), χ ε X are important for our purposes. Hence, we introduce the following definitions: Ru χ Rm and

Definition: The production correspondence Ρ: X -* U has closed structure if the graph of P is closed. Definition: The inverse production correspondence L: U -» X has closed structure if the graph of L is closed. By these two definitions, the production correspondence Ρ: X -> U has closed structure if and only if the inverse correspondence L: U —* X has closed structure. Definition: The production correspondence Ρ: X —> U has convex structure if the input sets L(u) of the inverse correspondence are convex for all u ε U. Definition: The inverse correspondence L: U —» X has convex struc­ ture if the output sets P(x) of the correspondence Ρ: X -» U inverse to it are convex for all χ ε X. Note that the correspondence P and its inverse L separately may have convex structures. Convex structures for the correspondence P and its inverse L do not imply that the graphs of these correspondences are convex sets. Closure of the structures of the production correspondence Ρ: X —> U and its inverse L: U -» X is synonymous with the property that P and L are upper semi-continuous on X and U respectively. (See Appendix 2 for the definition of the upper semi-continuity of a production corre­ spondence, and Proposition 1, Appendix 2.)

THEORY OF COST AND PRODUCTION

FUNCTIONS

Properties of quasi-concavity and quasi-convexity may be defined for correspondences as follows: Definition: A correspondence for all where R(x) is the map set of x in

is quasi-concave on

if

Definition: A correspondence for all and where is the map set of x in

is quasi-convex on R n if,

These definitions are set theoretic extensions for correspondences of the notions of quasi-concave and quasi-convex functions mapping points of into points of If F(x) is a numerical function defined for x e R n , then F(x) is quasi-concave (quasi-convex) if min Two propositions are relevant to an understanding of these definitions. Proposition 45: The map sets of the inverse correspondence of a correspondence are convex if and only if R is quasi-concave. First, suppose the correspondence is quasi-concave. Let x, y denote vectors of R n and u denote a vector of R m . Suppose are twoarbitrary points of the map set of the inverse correspondence for arbitrary For any scalar consider the point is quasi-concave R((l — and, since it follows that Thus, implying that R _1 (u) is convex. Conversely, suppose there exist such that denote the complement of a set relative to the space containing the subset. Then, there exists and

implying that the map sets of the inverse correspondence are not convex for some Proposition 46: A correspondence is quasi-concave if and only if the c o m p l e m e n t s o f the map sets of R are a quasiconvex correspondence. Suppose is quasi-concave. Then for any and and

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Conversely, if the c o m p l e m e n t s o f the map s e t s o f R are quasi-convex, i.e.,

for any

then

Now, for the function a natural correspondence defined by this function is where the map set of a point is defined as , as indicated in Figure 30 for There , where u denotes a point in R 1 on which x is mapped. If the correspondence R is quasi-concave, i.e., for then and the function F(x) is quasi-concave (see Figure 30). If the correspond-

FIGURE 30: GRAPH OF A QUASI-CONCAVE CORRESPONDENCE

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THEORY OF COST AND PRODUCTION FUNCTIONS

F(y)

FIGURE 31:

GRAPH OF A QUASI-CONVEX CORRESPONDENCE R: Ri -> Ri

ence R is quasi-convex, i.e., for XeR1JeR1 and λ ε [0,1], [—oo, F((l - λ)χ + Ay)] C [ — oo,F(x)] U [-oo,F(y)],thenF((l - λ)χ + Ay) ΞΞ max [F(x),F(y)] and the function F(x) is quasi-convex (see Figure 31). Conversely, the correspondence R(x) is quasi-concave or quasi-convex according as the function F(x) is quasi-concave or quasi-convex. Fur­ ther, the level sets R_1(u) = {χ | F(x) ^ u,x e R1} of the function F(x) are the map sets of the inverse correspondence, and they are convex if and only if the correspondence R(x) is quasi-concave (see Figure 30). The property of quasi-convexity for the correspondence R(x) implies that the closed complements of its level sets, i.e., R_1(u)c = (χ | F(x) ^ u, χ ε R1}, are convex and conversely (see Figure 31). From these simple examples, the role of the property of quasi-concavity (quasi-convexity) for production correspondences and its relation­ ship to quasi-concavity (quasi-convexity) of numerical functions defined on Rn may be understood. Due to Proposition 45, the production correspondence Ρ: X -» U and its inverse L: X —> U have convex struc­ ture if and only if the correspondences P and L are respectively quasiconcave. Quasi-concavity of the production correspondence Ρ: X U and its inverse L: U X is thus synonymous with convex structures

PRODUCTION

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for the correspondences P and L respectively. But convexity of the graph of the correspondence P (inverse correspondence L) is not guaranteed by the property of quasi-concavity for the correspondences P and L (see the counter example in Section 9.2). For convexity of the graph of P (graph of L) we need the stronger property that

for all

and

or

for all and (see Appendix 2). That this condition is stronger is apparent, since if then One may designate this stronger property as concavity for the correspondence P (see Appendix 2). In general, one cannot expect a production correspondence to have a convex graph since this property implies nonincreasing returns to scale throughout (see Section 9.2, Figure 30). Consequently, we do not assume the Technology (i.e., the graph of the correspondence P and its inverse L) to be convex. The convexity of the input sets L(u) (i.e., quasi-concavity of the correspondence does not imply convexity of the output sets P(x) and conversely. See Figure 31, where is convex, but the sets are obviously not convex. It is convenient here to state all of the properties which we shall assume for the production correspondence Definition: A mapping of input vectors x into subsets P(x) of output vectors is a production correspondence if:

is bounded for all implies and for some scalar , then for any scalar there exists a scalar such that or/and and for some i and then for any for some scalar P is upper semi-continuous on X, implying closed for all P is quasi-concave on X. The output sets of P are convex for all < 185 >

THEORY OF COST AND PRODUCTION FUNCTIONS

A.8 (a) u ε P(x) implies {0u | θ ε [0,1]} C P(x).f or/and

(b) u ε P(x) and 0 ^ u' 5Ξ u implies u' ε P(x). Property Al asserts that the null output vector is the only possible result from a zero input vector. It is a generalization of Property A.1 for the production function Φ(χ), that nothing comes from nothing. Property A.2 requires that only bounded output vectors can be real­ ized from finite input vectors, like A.2 for the production function Φ(χ). Property A.3 implies that excess inputs for some factors of production when others are limitational does not hinder output, just as the Property A.3 does for the production function Φ(χ). Property A.4 is intended to specify the unconstrained technological alternatives, like A.4, but two alternatives are given to correspond to two situations: (a) one where not all conceivable output vectors u ε U are attainable, such as might arise when some of the outputs are by-products, particularly when they are not desirable, like waste products, which cannot be controlled beyond certain minimum levels depending upon the rates of output of the other products, (b) another where all output vectors are attainable by sufficiently large input vectors. Not all of the factors of production need be essential to produce a positive output vector. Hence, we allow for the possibility that a semi-positive vector x, with zero input for one or more but not all factors of production, may yield a strictly positive output vector u. In situation (a), the unconstrained technological alternatives are such that if χ > 0 yields u > 0, then any scalar magnification of u is attainable by a suitable scalar magnification of the input vector x, and the set of all possibly attainable output vec­ tors is a cone with vertex at u = 0 which may be a proper subset of U = Rg> In situation (b), a positive input vector χ can, by suitable magnification, yield any output vector u ε U, and a semi-positive input vector, which by some magnification yields a positive output vector, can, by suitable magnification, yield any output vector u ε U. These two possibilities, i.e., (a) or/and (b), correspond respectively to what we may call weak and strong attainability of outputs under no limitations on the input vectors x, i.e., for an unconstrained technology. The conjunc­ tion "or/and" is used to indicate that for strong attainability of outputs both (a) and (b) apply. Property A.5 implies that the graph of the correspondence P is closed. The semi-continuity of P and the closure of the graph of P are equiva­ lent (see Proposition 1, Appendix 2). In particular, the input sets L(u), u ε U of the correspondence inverse to P are closed, and in this respect f Property A.8(a) is a consequence of A.1, A.3 and A.7.

PRODUCTION CORRESPONDENCES

A.5 is synonymous to A.5 for the production function Φ(χ). Beyond this, the output sets P(x) of the correspondence P are likewise closed. Assumption A.5 is a mathematical convenience, and it imposes Uttle or no restriction on the generality of the correspondence P. It enables us to define the input and output isoquants as subsets of the boundaries of the input sets L(u) and the output sets P(x), respectively, relative to Rn and Rm respectively. On these isoquants the efficient subsets EL(U) and Ep(x) constitute the technologically efficient substitutions between input vectors χ to attain a given output vector u and between output vectors realizable with an input vector x. Property A.6 is synonymous with convexity of the input sets L(u) of the inverse correspondence L of P (see Proposition 45), and it is analo­ gous to A.6 for the production function Φ(χ). Property A.7 has a trivial counterpart for the production function Φ(χ), since the latter maps the point χ into a single output rate u ε R1. However, A.6 and A.7 are suitable for time divisible technologies. If u and ν belong to P(x), then the output rate [(1 — ff)u + 0v], θ ε [0,1] may be obtained for any time interval by operating the technology a fraction 9 of the time interval to obtain ν and the remaining fraction (1 — Θ) to obtain u. Similarly, if χ and y belong to L(u), the input vector [(1 — λ)χ + Ay] will certainly yield at least the output rate u. The graph of the cor­ respondence P is not assumed to be convex in order to allow for increas­ ing returns to scale in the technology. Property A. 8 has two alternative forms, one for weak disposability and the other for strong disposability of outputs. If none of the outputs are undesirable, i.e., all components of u have positive value, economic or social, Property A.8 (b) applies, since any output can be disposed. Moreover, a strong disposal of outputs together with weak attainability of outputs implies a strong attainability of outputs, because, if χ > 0, or χ > 0 and ϋ ε P(Xx) for some λ > 0 where ΰ > 0, then for any vector u ε U there exists a scalar θ such that u ^ θ · ΰ and a scalar X such that (θ · ΰ) ε Ρ(λ„· χ), whence, by A.8 (b), u ε Ρ(λ9 · χ). Thus, Property A.8 (b) should be taken with both Properties A.4 (a), (b). The properties of weak attainability of outputs, i.e., A.4 (a), and weakly disposable outputs, i.e., A.8 (a), go together to characterize a production correspondence where not all outputs have positive value and cannot be disposed of indis­ criminately if the structure is to be analyzed without disregard of the undesirable outputs. It is useful to reiterate at this point that the foregoing assumptions for the production correspondence do not exclude the technology being composed of separable processes (or sub-technologies) which are to be jointly planned, as well as situations where joint outputs are inherently involved. e

THEORY OF COST AND PRODUCTION FUNCTIONS

The output sets P(x) are illustrated for u ε in Figures 32 (a) and 32 (b), corresponding to weak and strong disposal of outputs. For two input vectors χ' χ note that P(x) and P(x') are bounded, closed con­ vex sets containing the origin with P(x) C P(x')· If an output vector u ε P(x), the entire line segment {#u | θ e [0,1]} belongs to P(x) for weak disposal, whereas the set (ν | 0 ^ ν ^ u} belongs to P(x) for strong disposal. The darkened portions of the boundaries of P(x) and P(x') are efficient subsets. It may happen (as illustrated in Figure 32 (b)) that for an input vector χ with one component zero the output set P(x) is con­ tained in the boundary of U. Note that if x" ^ χ and χ" φ χ, P(x") may not be contained in P(x) and vice versa. It may also happen that the outputs Ui and u2 are naturally obtained in a fixed proportion as illustrated in Figures 32 (a) and 32 (b) for P(x), with a single efficient point u. The properties of the inputs sets L(u), u ε U of the inverse correspond­ ence L: U -» X follow from those of Ρ: X -» U.

FIGURE 32 (a): OUTPUT SETS FOR A PRODUCTION CORRESPONDENCE WITH WEAK DISPOSAL (U1 NOT DESIRABLE), x' g χ

PRODUCTION

CORRESPONDENCES

FIGURE 32 (b): OUTPUT SETS FOR A PRODUCTION CORRESPONDENCE WITH STRONG DISPOSAL,

Proposition 47: I f t h e production correspondence has the Properties the input sets L(u) of the inverse correspondence have the following properties: and for all and then and for some the ray intersects all input sets or/and (b) If and for some the ray intersects all input sets L(u) for or/and

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THEORY OF COST AND PRODUCTION FUNCTIONS

P.S

n

(a)

0.[0,1)

(b) P.6

n

L(O· U O)

or/and L(u)

n

O~u:S;uo

= L(uO).

= L(uO).

L(u) is empty.

UEU

P.7

The correspondence L is upper semi-continuous, implying L(u) closed for all u E U.

P.8 P.9

The input sets L(u) are convex for all u E U. The correspondence L is quasi-concave on U.

It is apparent that, except for P.9, these properties are analogous to those of the input sets L",(u) of the production function (x). Property P.l holds, since L(O) {x I E P(x)} X because Al and A3 imply E P(O) C P(x) for all x E X. Property P.2 follows from A3, since x' :> x E L(u) implies u E P(x) C P(x /), and x' E L(u). Properties P.3 (a), (b) follow respectively from A4 (a), (b). If x 2: and (Xx) E L(U) for X> 0, U 2: 0, then U E P(Xx) and by A4 (a) there exists a scalar Au > such that (orr) E P(Ao • x) for any 0, or (Au· x) E L(O· U) for any 0 > 0. For 0 = 0, L(O . U) = L(O) = X and the ray {Ax I A :> O} intersects this set. Property P.3 (b) follows from A4 (b), since the existence of X > such that (X • x) E L(U) for U > implies U E P(X . x) and for all u E U, U E P(Au • x) for some Au > 0, or (Au· x) E L(u) for all u E U. Property P.4 (a) follows from A8 (a), since, if x E L(Ou) for a scalar o:> 1, then (0· u) E P(x) and 0:> 1 implies u E P(x) or x E L(u). By a similar argument, Property P.4 (b) follows from the strong disposal Property A8 (b). The Property P.S (a) may be established by an argument identical to that given for the Property P.5 of the level sets Liu) of the production function (x) (see Section 2.2, Chapter 2). Regarding Property P.5 (b), it follows from P.4 (b) that L(uO) C L(u). Let XO belong to o::o;u

°

°

n

n

O~u:S;uO

°

Take {un} ~ U Owith un ::; UOfor all n. Let {xn} be an infinite sequence with xn = XO for all n. Then {xn} ~ XO, un E p(xn) for all nand {un} ~ U O. Consequently, since P is upper semi-continuous (A5), it follows that UO E P(XO) (see the definition of upper semi-continuity in Appendix 2) and XO E L(uO). Thus L(u) C L(uO) and P.5 (b) is

n

O~u:S;uO

established. Property P.6 follows from A2, because suppose there exists a finite

< 190

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vector such that for all Then for all contradicting the boundedness of Property follows from For any and then and by the upper semi-continuity of P, whence and L is upper semi-continuous. Since the graph of P is closed, L(u) is closed for all Property follows from since the quasi-concavity of P implies that the input sets are convex (see proposition 45), and Property follows from since the quasi-concavity of L is an if and only if property for the output sets to be convex. We note that there does not appear to be a property like for the output sets of the correspondence P, otherwise they are similar. However, a property like does hold for the sets Note that follows from and Similarly, the following property for follows from and

First, belongs to

due to

Second, let

and u°

for all

implying for all For any with for all n, and take as an infinite sequence with for all n. Then with rnd, due to the upper semi-continuity of P, u° e P(x°). Hence Thus Property A.9 holds. 1

We might have defined the production correspondence in termsof the input sets rather than starting with the output sets . Technologically, one can define the production correspondence either in terms of its input sets or its output sets. The question arises whether a definition in terms of the input sets leads to output sets by the correspondence inverse to L, i.e., , which have the Properties This question is resolved by the following proposition: Proposition 48: If the correspondence its inverse correspondence has the properties correspondence is unique. Clearly the inverse correspondence

has the Properties defined by and this inverse is unique, since

Alternatively, the graph of the correspondence is closed if and only if the corresDondences P and L are upper semi-continuous. Then P.7 is a direct consequence ol See Appendix 2.

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Property Al follows from P.I, since P(O) = {u 10 e L(u)} = {O} because 0 ¢L(u) for u ~ O. Property A2 follows from P.6, since, if P(x) is unbounded for some XO e X, then there exists a finite vector XO such that XO e L(u) contrary UfU to P.6. Property A3 follows from P.2, because, if x e L(u) and x' > x, then P.2 implies x' e L(u), i.e., if u e P(x) and x' > x then u e P(x' ) and P(x) C P(x'). Property A4 (a) follows from P.3 (a), because, if x ~ 0, u ~ 0 and u e P(X· x), then x ~ 0, u ~ 0, (X· x) e L(u) and P.3 (a) implies that the ray {Ax I A >O} intersects all input sets L(OU) for 0 >0, or for any 0 such that (0· u) e P(A o • x). Similarly, scalar 0 0 there exists Ao A4 (b) follows from P.3 (b). Property AS is synonymous with P.7, since the closure of the graph of L: U -? X implies that the graph of P: X -? U is closed which is an if and only if condition 'for the upper semi-continuity of P (see Proposition 1, Appendix 2). Property A6 follows from P.8, because the quasi-concavity of the correspondence P: X -? U is an if and only if condition for the convexity of the sets L(u), u e U of the correspondence L: U -? X inverse to P. Property A7 is likewise a direct consequence of P.9. Property A8 (a), (b) follows from P.4 (a), (b). First if A e [0,1] and u e P(x), then x e L(u) C L(Au) and (AU) e P(x). Second, if 0 < u' < u and u e P(x), then x e L(u) C L(u') and u' e P(x). Finally, Property P.9 follows from A3 and AS as shown above. Thus, the production correspondence P: X -? U and its inverse L: U -? X define the production structure by output sets and input sets, respectively, which uniquely characterize the possibilities of production with properties which imply those of the other. The map sets of the production correspondence P: X -? U and its inverse L: U -? X are a coherent generalization of the production sets of the production function 0 to observe that the Properties P.I, ... , P.8 imply P.I, ... ,P.8 for this case. Examples of the input sets for the correspondence P: X -? U are shown in Figure 33 (a) and 33 (b) corresponding to the weak and strong disposal property for outputs.

n

>

>

9.2 Relationship Between Production Correspondences and Production Functions Consider a production correspondence P: X -? U with a one dimensional output vector u, i.e., u e U where U = R4- is the nonnegative portion of the real line and X = R~.

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FIGURE 33 (a): ILLUSTRATION OF INPUT SETS FOR A PRODUCTION CORRESPONDENCE WITH WEAK DISPOSAL OF OUTPUTS

The output sets the form

of this correspondence are closed intervals of

where (83) since is a closed, convex and bounded interval in R | containing the origin (see We note that the correspondence so defined has a strong disposal property, assuming that the single output is desirable. Proposition 49: The function definedby (83) in terms of the correspondence exists for all and it is a production function with the Properties < 193 >

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OF COST AND PRODUCTION

FUNCTIONS

FIGURE 33 (b): ILLUSTRATION OF THE INPUT SETS L(u) FOR A PRODUCTION CORRESPONDENCE WITH STRONG DISPOSAL OF OUTPUTS

Clearly is single valued, nonnegative and exists for all It remains to show that has the Properties (see Section 2.2, Chapter 2). PropertyA.1 holds because = 0. A.2 holds since the sets P(x) are bounded for all (see Regarding A.3, let so that

For the verification of A.4, let be such that contains as a proper subsetfor some for some positive scalar then by it follows that for each there exists a scalar < 194 >

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such that and Consequently, The same holds for The upper semi-continuity of (i.e., A.5) follows from that of correspondenceP (i.e., For any point we seek to show sup for all sequences Consider a sequence as

and suppose and

;up

Then

and, by the upper semi-continuity of a contradiction. Hence, up

The function is quasi-concave (i.e., A.6 holds), since the correspondence P is quasi-concave on X and for any and

or and hence Thus the function

induced from the correspondence is a production function relating a single commodity output to n factors of production. The production possibility sets are clearly identical to the input sets L(u) of the correspondence P, because

Therefore, the notion of a production correspondence as defined in Section 9.1 above is entirely consistent with the notion of the production function given in Chapter 2. Moreover, if the outputs of a correspondence can occur only in fixed proportions, i.e., they are of the form then the induced production function is (83.1) Conversely, we may ask whether a production function with the Properties A.1,..., A.6 defines a production correspondence The c o r r e s p o n d e n c e i n d u c e d by the production function is defined by (84) and the following proposition holds. Proposition 50: If is a production function with Properties A . 1 , . . . , A.6, the correspondence defined by P(x) = is a production correspondence with strong dis< 195 >

THEORY OF COST AND PRODUCTION FUNCTIONS

posal form of the Properties A.1,..., A.8 and input sets L(u) = L„(u) — (x I Φ(χ) ^ u,x ε R£}. Clearly, L(u) = (x I u ε [0,Φ(χ)],χ ε R$} = {χ I Φ(χ) Ξϊ u,x ER»} = L„(u), and the input sets L(u) have the Properties P. 1,..., P.3 (b), P.4 (b), P.5 (b),..., P.9. Compare Properties P.l,. .., P.8, Section2.1, Chapter 2, and the Properties P.l,... ,P.8. Property P.9 follows directly from the quasi-concavity of the production function Φ(χ) (i.e., A.6), the definition (84) of P(x) and Proposition 45. Then due to Proposition 48, P(x) is a p r o d u c t i o n c o r r e s p o n d e n c e w i t h t h e P r o p e r t i e s A .1 , . . . , A . 4 ( b ) , . . . , X8(b). It was stated above (Section 9.1) that quasi-concavity of the produc­ tion correspondence Ρ: X U and its inverse L: U —» X did not guar­ antee convexity of the graph of P, although the input sets L(u), u ε U and the output sets P(x), χ ε X were convex. We consider now the counter example validating this statement,f which at the same time geometrically illustrates the production function Φ(χ) induced by a cor­ respondence P: R| -* Rf and the correspondence induced by a produc­ tion function Φ(χ), χ ε Rj. Let X = Rf and U = R|, and consider the graph of the correspond­ ence P: Rf -» Rf illustrated in Figure 34. The graph of P is the set of points in the plane bounded above by the curve drawn and below by the X-axis, while the graph of the inverse correspondence L: Rf —» is the set of points in the plane bounded below by the curve and to the right by the X-axis (they are the same). For any χ ε Rf, the output set P(x) of the correspondence P is the interval [Ο,Φ(χ)] where Φ(χ) = Max {u I (u,x) ε Graph of P} is the classical production function. In particular, at xo the value of Φ(χο) is the upper point as illustrated in Figure 34. The function Φ(χ) is discontinuous at X but upper semicontinuous. For any u ε Rf, the input set L(u) of the inverse correspond­ ence L: U Xis the interval [F(u), + oo) where F(u) = Min (χ | (u,x) ε Graph of P}. The function F(u) may be taken naturally in the example as the inverse function of the production function Φ(χ) by 0

F(u) = Min {χ I Φ(χ) 2: u,x ε Rf}, and it has the lower value Xi coresponding to Ui as illustrated in Figure 34. F(u) is discontinuous at Ui, but lower semi-continuous (See Proposition 5, Section 2.3). t Since u ^ 0, we use [Ο,Φ(χ)] instead of [ — οο,Φ(χ)].

< 196 >

PRODUCTION CORRESPONDENCES

U

Graph Boundary

=Φ(χ )

FIGURE 34: COUNTER EXAMPLE FOR CONVEXITY OF THE GRAPH OF P:X -» U

The sets P(x) = [Ο,Φ(χ)], χ ε Rf and L(u) = [F(u), + oo), u ε Rf are obviously convex, and the correspondence P: Rf -» Rf and its inverse L: Rf —> Rf are quasi-concave (see Proposition 45), but obviously the graph of these correspondences is not convex. Thus, the property of a convex graph is not implied by quasi-concavity for a correspondence and its inverse. Convexity of the graph is unnecessarily restrictive, im­ plying nonincreasing returns to scale for the related production function. It is interesting to consider at this point a production relationship suggested by the example of Figure 34 but not heretofore considered in detail. If the technology involves a single factor of production, say χ ε Rf, with a vector of outputs, say u ε U = Rij?, then the production relationship may be summarized in terms of an inverse production function defined on the family of output sets P(x), χ ε Rf of the produc­ tion correspondence by _1(u) = Min {x I u ε P(x),x ε Rf}.

(2)-1

THEORY OF COST AND PRODUCTION FUNCTIONS

See Equation (2), Section 2.2, for comparison of this definition with that of the production function (x) in terms of the input sets L(u) for u e Rt- and x e X = R~. The production function -l(U) determines the smallest input x e Rtsuch that the output vector u is producible. Clearly, -l(U) is single valued and nonnegative, and it exists for all u e Rt-. Note that the sets P(x) are monotone nondecreasing in x e Ri. The properties of the inverse production function -l(U) are summarized in the following proposition. Proposition 51: The function -l(U) = Min {x I u e P(x),x e Ri} defined on a production correspondence P: X ~ U, U = R~, X = Ri with the Properties AI, ... , A8 has the following properties:

= O.

(Al)-l

-1(0)

(A2)-1 (A3)-1

-l(U) is finite for finite u e U. (a) -l(AU):> -l(U) for A :> 1. or, and (b) -l(U') :> -l(U) if u' :> U.

(A4)-1

For {lIunll}

~

+00, lim Inf-l(Un) n--.oo

= +00.

(A5)-1

-l(U) is lower semi-continuous on U.

(A6)-1

-l(U) is quasi-convex on U.

Thus, the properties of the inverse production function -l(U) are similar but not identical to those of the production function (x). (AI )-1 holds because due to Al and A3, 0 e P(x) for all x e Rt-, and (A2)-1 holds because by AA there exists Au for x 0 such that u e P(Au • x) and P(AuX) is bounded due to A2. (A3)-1 (a) holds since A· u :> u for A :> I and (AU) e P(x) implies u e P(x) due to A8 (a), and

>

{x I (AU) e P(x),x e Ri} C {x I u e P(x),x e Ri}, whence -l(AU) :> -l(U). Similarly, (A3)-1 (b) holds. Property (A4)-1 holds, because otherwise there exists an infinite monotone subsequence {liunkll} ~ +00 with lim -l(U nk) = lim x*(u nk) = Xo +00, and k--.oo

k--.oo


x*(unk) for all k, whence Xo e L(unk) for all k, contradicting P.6. The lower semi-continuity of -l(U) may be established as follows: The upper semi-continuity of the correspondence P: X ~ U implies that for any Xo e Ri, Uo e P(xo) if {xn} ~ XO, un e P(xn) for all nand {un} ~ uo. Consider a sequence {un} ~ Uo and let Xn -l(Un) for all n, so that n un e P(xn) for all n. Suppose -l(U ) is not lower semi-continuous. Then lim Inf -l(Un) = lim Inf Xn -l(UO). Consequently, {xn} ~ X

=




THEORY OF COST AND PRODUCTION

FUNCTIONS

on X, mapping RJJ into Rj, with Properties A . 1 , . . . , A.6 stated in Section 2.2 for the production function Then we introduce the following two definitions: Definition: The production correspondence has homothetic output structure if the output sets ! are representable by with f(u) a homogeneous function of degree one for Definition: The production correspondence has homothetic input structure if the input sets ^ of the inverse correspondence are representable by with O(x) a homogeneous function of degree one for The suitability of these two definitions requires verification by showing that the sets and have the Properties respectively, and that these sets have a ray property like that of Proposition 8, Section 2.4. Proposition 52: The sets with a homogeneous function of degree one for u e U, have the Properties Property

holds because by A. 1 and since f(u) is nonnegative and f(u) = 0 only for u = 0 due to the properties (a) and (b). Property holds since is finite forx finite by A.2 and is a finite function of For Property suppose Then by A.3, and u e P(x) implies so that . Hence, ] Property , holds because if and and some scalar then, by the property (b) for and, by Property A.4 for so that for any u e U there exists a scalar such that since f(u) is a finite function, and for all there exists a scalar such that For Property Then

. let implies

with

the lower semi-continuity of f(u) implies f See the extended definition given in Section 10.5.

< 200 >

for all n.

, and the

PRODUCTION CORRESPONDENCES

upper semi-continuity of Φ(χ) implies Φ(χ°) 5; Iim Sup Φ(χη), whence n—>00 f(u°) ^ Φ(χ°) and u0 e P(x°). Thus, the correspondence P is upper semi-continuous. For Property A.6, note that, due to the quasi-concavity of Φ(χ), the sets L(u) = (χ I Φ(χ) ^ f(u),x ε X}, u ε U of the inverse correspondence L: U X are convex, and by Proposition 45 it follows that the corre­ spondence Ρ: X -» U is a quasi-concave. Property A. 7 holds because the quasi-convexity of the function f(u) implies that the sets P(x), χ ε X are convex (see Appendix 1). Finally, Property A.8 (b) holds because if u ε P(x) then f(u) S Φ(χ) and u' u implies f(u') ^ f(u) ^ Φ(χ), due to the property (c) for the function f(u) and u' ε P(x). Thus, a production correspondence with homothetic output structure has strong disposability of outputs. By entirely analogous arguments, it may be shown that: Proposition 53: The sets L(u) = {χ | Φ(χ) ^ f(u),x ε X}, u ε U, with Φ(χ) a homogeneous function of degree one for χ ε X, have the Prop­ erties?!, ... ,ί\3 (b), PA (b), P3 (b),..., R9. Note that by Proposition 48, homothetic input structure implies strong disposability of outputs. (See details of proof.) Thus, the two definitions for homotheticity of output and input struc­ ture are consistent with P: U -» X and L: X —> U being production correspondences. It is also clear that Ρ: X U may have homothetic output structure with the inverse correspondence L: U -» X not hav­ ing homothetic input structure, and vice versa, since homotheticity of each results from homogeneity of the corresponding function f(u) or Φ(χ); the other properties of these two functions being required in order that the sets P(x), χ ε X and L(u), u ε U be output and input sets of a production correspondence and its inverse. Turning now to the geometric structure of homothetic production correspondences, consider first a homothetic output structure. Let χ,χ'εΧ be two distinct input vectors such that Φ(χ) > 0 and Φ(χ') > 0, and {0u I θ ^ 0, u > 0} be a ray from the origin. Now, by arguments exactly analogous to those for Proposition 7, Section 2.4, it follows that f(u) is a continuous and convex function of u ε U with f(0) = 0. Also, f(u) > 0 for u > 0 (see property (b) of the function f(u)). Hence, the sets P(x) and P(x') span the cone R? = U, and for any u > 0 the ray {ffu j θ^ 0 } intersects the boundaries of the sets P(x), P(x'). Let ξ and η be the intersections of the ray (0u | 0 0} with the boundary of P(x) and P(x') respectively, with maximal Θ, and η = θ· ξ for some scalar θ > 0. Then, due to the continuity of the function f(u), ΐ(θ · ξ) =

THEORY OF COST AND PRODUCTION FUNCTIONS

Φ(χ'), f(£) = Φ(χ), and, since f(u) is homogeneous of degree one, f(0· ξ) = θΐ(ξ), whence θ — Φ(χ')/Φ(χ) and η = Φ(χ')/Φ(χ) · £ independently of the direction of the ray (flu | θ Ξϊ 0, u > 0}. Thus, the following propo­ sition holds. Proposition 54: If the output structure of the correspondence Ρ: X U is homothetic, then, for any two input vectors Χ,Χ'ΕΧ such that Φ(χ) > 0, Φ(χ') > 0, the isoquant of the set P(x') relative to U is generated by radial extension of the isoquant of P(x) relative to U in a fixed ratio Φ(χ')/Φ(χ). The definition of homotheticity of output structure is thereby justified. Compare Proposition 54 with Proposition 8, Section 2.4, and see Figure 35 illustrating for U = 1¾. Note that the sets P(x) span the cone U = R|. For homotheticity of input structure, we observe that the function Φ(χ) is homogeneous of degree one, with the Properties A.1,.. ., A.6, and by Proposition 7, Section 2.4, it is a continuous and concave func­ tion of χ ε X. Letting u > 0 and u' > 0 be two distinct output vectors, then f(u) > 0, f(u') > 0 due to property (b) of the function f(u). Then

Φ(χ') Φ(χ)

P(x')

P (x)

FIGURE 35: HOMOTHETIC OUTPUT STRUCTURE, U = R|, FOR A PRODUCTION CORRESPONDENCE

PRODUCTION CORRESPONDENCES

~ f (u)

(u)

O

1

FIGURE 36: HOMOTHETIC INPUT STRUCTURE, X = RJ, FOR A PRODUCTION CORRESPONDENCE

for x > 0 such that (λχ) ε L(u) for some λ > 0, it follows that the ray {λχ I λ ^ 0,x > 0} intersects the set L(u'), since Φ(λχ) = λ · Φ(χ) > 0 and there exists Xu' such that Au< · Φ(χ) = Φ(λ„< · χ) 5 f(u'). Let £ and η =z θ' ξ denote the intersections of the ray {λχ | λ | λ ^ 0} with mini­ mal λ for the sets L(u) and L(u') respectively. Since Φ(χ) is homogeneous of degree one and continuous, Φ(0 · £) = 0Φ(ξ) = f'(u) and Φ(£) = f(u), so that θ = fi(u')/f(u) independently of the direction of the ray {λχ | λ ^ 0,x > 0}. Thus, the following proposition holds.

Proposition 55: If the input structure of a production correspondence Ρ: X —> U is homothetic, then, for u > 0, u' > 0, the isoquant of the set L(u') relative to X is generated by radial extension of the isoquant of the set L(u) relative to X in a fixed ratio f(u') f(u).f Consequently, the definition of homotheticity of input structure is justified. Compare Proposition 55 with Proposition 8, Section 2.4 and see Figure 36 illustrating for X = R^. Another way of expressing the property of homotheticity for output or input structure is given by the following two propositions. t See definition of isoquant given in Section 2.1.

THEORY OF COST AND PRODUCTION

FUNCTIONS

Proposition 56: If the production correspondence has homothetic output structure, for all where

Proposition 57: If the production correspondence has homothetic input structure, For x such that

and when homogeneity of

use the homogeneity of f(u) to obtain

. Similarly, if

use the

to obtain

and when Homotheticity for input or output structure, but not both, does not imply that the graph of the correspondence is convex, since in either case the opposite function i is only quasi-concave or quasi-convex respectively. However, if is concave or f(u) is convex when the output structure or input structure respectively is homothetic, the graph of the correspondence is convex. Proposition 58: If the output structure of the production correspondence is homothetic and is concave for the graph of P is convex. Proposition 59: If the input structure of the production correspondence is homothetic and f(u) is convex for u e U, the graph of P is convex. Let (x,u) and (y,v) belong to the graph of P. Then, if the output structure of P is homothetic it follows from Proposition 56 that and for all

Used by Jacobsen [15] to define homothetic input structure.

< 204 >

PRODUCTION CORRESPONDENCES

«(1 - A) 0, of a produc­ tion correspondence is a subset of the boundary of L(u) defined by (x I χ > Ο,χ ε L(u),(Xx) # L(u) for λ ε [0,1)}.

PRODUCTION CORRESPONDENCES

The isoquant of L(O) is {0}. Definition: The output isoquant of a production correspondence cor­ responding to an input vector χ such that P(x) φ {0} is a subset of the boundary of P(x) defined by {u I u ε P(x),(0 · u) ¢ P(x) for θ ε (1,+oo)}. The isoquant of P(x) = {0} is {0}. The term "boundary" may need clarification. Due to Property P.2, the boundary of an input set L(u) is taken relative to R$. However, for the output sets P(x), a semi-positive input vector χ may preclude certain outputs and the boundary of an output set P(x) is taken relative to the nonnegative domain of the linear subspace of Rm of smallest dimension which contains P(x). The input and output isoquants of the production correspondence P are interpreted as being sets of "minimal" input vectors and "maximal" output vectors, respectively, in a global sense, but they are not neces­ sarily efficient in a local sense. A local test is needed to determine the efficient subsets Ep(x) and EL(U) of the output and input isoquants re­ spectively (see the definitions in Section 9.1 and Figure 37). The classical definition of production isoquants presupposes the ex­ istence of a joint production function F(x,u), with the property that the equation F(x°,u) = 0 defines the output isoquant corresponding to an input vector x° and the equation F(x,u°) = 0 defines the input isoquant corresponding to an output vector u0, defined for a production corre­ spondence with disposable outputs and usually also for a situation where all of the factors of production are essential. For these special circumstances, the isoquant definitions given above define the isoquants as the entire boundaries of the input sets L(u) and output sets P(x). Since we are concerned with the existence of the joint production function F(x,u), the input and output isoquants have to be defined beforehand as subsets of theinput and output sets of the production correspondence. To make the issue of the existence of a Joint Production Function precise, it is assumed in the economic theory of production that there exists a function F(x,u) such that: (a) For given output vector u > 0, F(x,u) = 0 if and only if χ belongs to the input isoquant corresponding to the output vector u. (b) For given input vector χ > 0 such that P(x) φ {0}, F(x,u) = 0 if and only if u belongs to the output isoquant corresponding to the input vector x. The existence of a joint production function has not been demon­ strated, i.e., derived from the properties of the input and output sets of the production correspondence to which it refers. Before proceeding

THEORY OF COST AND PRODUCTION FUNCTIONS

•fr input isoquant

efficient points

FIGURE 37:

ILLUSTRATION OF THE SOLUTIONS OF *(u,x) = 1 FOR u > 0 WHEN u IS PRODUCIBLE WITHOUT X3

with this issue, the following two propositions characterize the roles of the distance functions fi(x,u) and Φ^,χ) for determination of the output and input isoquants respectively of a production correspondence: Proposition 64: An output vector u belongs to the output isoquant for an input vector χ such that P(x) φ {0} if and only if B(x,u) = 1. Proposition 65: An input vector χ belongs to the input isoquant for an output vector u > 0 if and only if ^(u,x) = 1. These two propositions follow directly from the definitions of the iso­ quants and the distance functions fi(x,u) and xKu,x). The following proposition is verified by counterexample: Proposition 66: Satisfaction of the Properties A.1,..., A.8 by the pro­ duction correspondence Ρ: X -» U does not imply the existence of the joint production function F(x,u).

PRODUCTION

CORRESPONDENCES

Consider the production correspondence illustrated in Figure 34, which satisfies with strong attainability and strong disposability of outputs. If the joint production function F(x,u) exists, if and only i f f o r given if and only if for given implying from (a) that contradiction. Similarly,

and from (b) that

if and only i f f o r given if and only if for given u', implying from (c) that and from (d) that contradiction. This counterexample suggests that, if outputs are strongly attainable and disposable and the correspondence P and its inverse L are continuous, a joint production function will exist. For this purpose, consider the following two propositions: Proposition 61 Bis: If the production correspondence P is continuous with strong attainability and disposability of outputs, i.e., with and holding, the distance function is continuous in for all u e U, along rays Proposition 60 Bis: If the inverse correspondence L of the production correspondence P is continuous, the distance function is continuous in u, along rays Observe from Figure 34 that continuity of the correspondence P does not imply continuity of the inverse correspondence L, because continuity of the correspondence P at making P continuous for all , does not imply continuity of the inverse correspondence L, since with L for all n and does not imply the existence of a sequence with for all n and required for the lower semi-continuity of the inverse correspondence L (see Appendix 2 for the definition of lower semi-continuity). A correspondence is defined to be continuous if and only if it is both upper and lower semi-continuous. Consider first the proof of Proposition 61 Bis and let the correspondence P be continuous. For for all (Property and hence is continuous in x for x e X. Therefore, consider and let be arbitrary. If and clearly for any infinite sequence

< 215 >

THEORY OF COST AND PRODUCTION

FUNCTIONS

implying that the distance function is upper semi-continuous at as well as being lower semi-continuous (Property and hence is continuous at for Thus, we need consider further only the case where and is positive and finite. Then, assuming that the distance function is not continuous at along the ray there exists an infinite sequence such that

or, what is the same thing since the distance function geneous of degree one in u and

is homo-

where is a positive scalar such that Then, in view of Property there exists a nondecreasing infinite subsequence such that

Further, since Property holds for the correspondence P,the ray intersects the boundary of at a point i with where

Now for all k and because otherwise where and, by the strict separation theorem for convex sets, , there exists a hyperplane separating the sets from for all k, implying that there does not exist a sequence with for all k, when with contradicting the lower semi-continuity of P. Consequently,

contradicting the supposition that is not continuous at along the ray Consider now the proof of Proposition 60 Bis. Let with be arbitrary. Then, by Property of the correspondence L, the ray intersects for all Hence, by Property it follows that is not continuous at u° along the ray there < 216 >

PRODUCTION

CORRESPONDENCES

exists an infinite sequence

such that

or

where is a positive scalar such that , since the function is homogeneous of degree one in x for and (see Property . Then, in view of Property there exists a nonincreasing infinite subsequence such that

The ray intersects the boundary of at a point with since for all k due to . Now I for all k and because otherwise with and, by the strict separation theorem for convex sets there exists a hyperplane separating the bounded, closed, convex sets I, from for all k, implying that there does not exist a sequence i with for all k, when with contradicting the lower semi-continuity of L. Consequently

contradicting the supposition that is not continuous at along the ray In Proposition 61 Bis, a strong disposability was assumed for the correspondence P, i.e., holds, because continuity of the distance function in x along rays is not implied by continuity of the correspondence P under weak disposability, i.e.. See the counterexample of Figure 32 Bis, where outputs are weakly disposable. Forlhis counterexample to apply we must show that the correspondence illustrated is continuous, i.e., both upper and lower semi-continuous. The upper semi-continuity is apparent, because if i for all n and then , since the inequalities defining the set are obviously satisfied. For the lower semi-continuity, it is required to show that, if and , there exists an infinite sequence with for all n and . Three cases are considered. First, if implying take Then the inequalities defining the sets of the correspondence are obviously < 217 >

THEORY

OF COST AND PRODUCTION

FUNCTIONS

FIGURE 32 Bis: COUNTER EXAMPLE FOR CONTINUITY IN P CONTINUOUS AND OUTPUTS WEAKLY DISPOSABLE

satisfied and

Then with

WITH

take

implies

and as defined above for

, and clearly for all n. Moreover, when Finally, if we may take | with given by

< 218 >

PRODUCTION

CORRESPONDENCES

with Hence, the correspondence illustrated in Figure 32 Bis is continuous. But, the distance function is not upper semi-continuous at , because, consider with for all n. Then, for all n and

implying that the distance function is not continuous at

and

Now returning to the question of the existence of a joint production function F(x,u), the following proposition holds as a constructive statement: Proposition 67: If the production correspondence P is continuous with strong disposal of outputs (i.e., holds) and the inverse correspondence L of P is continuous, the function is a joint production iuncuon. Let

be some arbitrarily given output vector and consider to be any point belonging to the isoquant of Then (Proposition 65), and , implying (Proposition 63). We seek to show that 0, i.e., any point x on the isoquant of yields Suppose ,implying Since and there exists with implying and . By Proposition 61 Bis, there exists such that Then (Proposition 64) and implying that there exists an input vector belonging toi contradicting the supposition that x belongs to the isoquant of (see the definition of the input isoquant corresponding to Next let be arbitrary for and consider to be any point belonging to the isoquant of Then (Proposition 64). and , implying (Proposition 62). We seek to show that , i.e., any point u on the isoquant of P(x°) yieldsF(x°,u) = 0. Suppose implying Since there exists such that implying ) and By Proposition 60 Bis, there exists a bounded scalar such thai . Then (Proposition 62) and ,implying that there exists an output vector belonging to contradicting the supposition that u belongs to the isoquant of (see the definition of the output isoquant corresponding to < 219 >

THEORY OF COST AND PRODUCTION FUNCTIONS

Conversely, suppose F(x°,u°) = 0, i.e., Q(x°,u°) = xP(U05X0)j for some arbitrary pair (x°,u°). Then, either u0 ε P(x°) or u0 ¢ P(x°). If u0 ε P(x°) then x° ε L(u°) and by Propositions 62 and 63 it follows that Ω(χ°,u0) 1 and ^(u°,x°) S 1, whence B(x°,u°) = ^(u°,x°) = 1, implying u0 belongs to the isoquant of P(x°) and X0 belongs to the isoquant of L(u°). If u0 f P(x°) then X0 £ L(u°), and S(x°,u°) > 1 with ^(u°,x0) < 1, con­ tradicting F(x°,u°) = 0. Thus a joint production function exists if the correspondence P is continuous, outputs are strongly disposable and the inverse correspond­ ence L is continuous. The function F(x,u) = ±(*(u,x) - Ω(χ,υ))

(91)

is not a unique joint production function, since G(x,u) = (*(u,x) - I)2 + (S(x,u) - I)2

(92)

is Ukewise a joint production function. Nothing new to the information obtainable directly from the distance functions fl(x,u) and ^(u,x) is added by the joint production function. The example of Figure 32 Bis shows that a joint production function may not exist when outputs are not strongly disposable. Consider the output vector ΰ = ^(l,x°) belonging to the output set P(x°). Clearly, x° belongs to the input isoquant for this output vector. Suppose the joint production function F(x,u) exists. Then F(x°,u) = 0. If X0 is taken as a given input vector then the joint production function will be zero for an output vector ΰ not belonging to the output isoquant, as we have defined it, for the input vector x°. Why not include the ray portions of the boundary of an output set P(x) as part of the isoquant? We have not done so, because except for the end point which is part of the iso­ quant, the points of a ray portion of the boundary of a set P(x) are globally determined as being inefficient because for such boundary points the distance function fl(x,u) < 1. 9.6 Distance Functions for Homothetic Production Correspondences Recall from Section 9.3 above that either the output structure or the input structure, or both, may be homothetic. If the production correspondence has homothetic output structure, the output sets P(x) are given by P(x) = (u I f(u) Ξ= (x),u ε U}, χ ε X, where f(u) is a continuous, convex and homogeneous function of degree one in u, with f(0) = 0, f(u) > 0 for u > 0, f(u') Ξ=: f(u) for u' > u, and f(un) -*• + oo for {||un||} -» + oo. See Proposition 7, Section 2.4, for the

PRODUCTION

CORRESPONDENCES

continuity and convexity of

Then,

and the ray intersects the set | in an output vector such that due to the continuity of f(u), and s i n c c i t follows from the homogeneity of f(u) that

and Consequently, proposition holds.

i, and the following

Proposition 68: If the production correspondence has homothetic output structure, the distance function of the output sets is given by (93) When

it has been shown that If and Moreover, since it follows that for all Thus, the formula (93) is consistent with the Properties and for the distance function The Properties follow directly from those of the function f(u), while and follow from the Properties of the function Similarly, if the production correspondence has homothetic input structure, the input sets are given by

where f(u) has the Properties (see Section 9.3), and is a continuous, concave and homogeneous function of degree one with Then for some for all If vector since

the ray intersects the set I in an input such that , due to the continuity of and it follows from the homogeneity of $(x) that

and Consequently. proposition holds: < 221 >

and the following

THEORY OF COST AND PRODUCTION FUNCTIONS

Proposition 69: If the production correspondence Ρ: X -» U has homothetic input structure, the distance function of the input sets L(u) = {x I Φ(χ) ^ f(u),x ε X}, u ε U is given by

*(u,x) = i§r

(94)

The verification of this proposition exactly parallels that given for Proposition 68. Now, if the production correspondence has both homothetic input and output structure, the structure of production is homogeneous of degree one, i.e., Ρ(λχ) = λΡ(χ) and L(Au) = AL(u) for χ ε X, u ε U and λ ε [0,1]. In this case, the distance functions fi(x,u) and ^(u,x) satisfy

=¾)=If·

(95 >

and the joint production function may be taken as F(x,u) = ±(Φ(χ) - f(u)).f

(96)

Note that outputs are strongly disposable for homothetic output structure. Without repeating the arguments of Chapter 6 we observe here that, if the production structure has homothetic input structure, aggregation of inputs may be made for the Cobb-Douglas or ACMSU forms of the function Φ(χ), and, if the correspondence has homothetic output struc­ ture, an aggregation may be performed for similar representations of the function f(u). t This is not the most general form of the joint production function for homothetic correspondences. See Section 10.5 and Corollary F, Section 11.3 below.

CHAPTER 10 COST AND BENEFIT (REVENUE) FUNCTIONS FOR PRODUCTION CORRESPONDENCES, AND THE RELATED COST, BENEFIT (REVENUE), COSTLIMITED-OUTPUT AND BENEFIT (REVENUE)-AFFORDED-INPUT CORRESPONDENCES 10.1 Definition and Propwties of the Cost and Benefit (Revenue) Functions, By proofs which exactly parallel those given in Chapter 2, Section 2.1, it may be verified that the efficient subsets EL(U) = {Χ | Χ ε L(u),y $ L(u) if y < x), of the input sets of a production correspondence Ρ: X -» U are nonempty for all u ε U and L(u) may be partitioned as a sum of EL(U) and X. Proposition 70: EL(U) is nonempty for all u ε U and L(u) = EL(u) + X = EL(U) + X, where EL(u) is the closure of EL(u). Again, we assume as a technological constraint that EL(u) is bounded for all u ε U. Since we wish to encompass situations for the benefit function where not all of the outputs are desirable and consequently the sets P(x) may have the weak disposal Property A.8(a), the following definition is used for decomposition of an output vector into desirable and nondesirable subvectors. Definition: u = (u(D),u(D)) where u(D) is the subvector of compo­ nents of u which are not disposable. Then the efficient subset of an output set P(x) is defined by Definition: The efficient subset of an ouput set P(x) is

E (X) p

u ε P(x); Max [θ | θ · u ε Ρ(χ),0 ε [Ο,οο)} = 1; ν = (v(D),v(D)) $ Ρ(χ) if u(D) is not empty and = {u ^ ^ u(D^ y(D) ^ U(D) (ii) v(D) ^ u(D), v(D) < u(D).

THEORY OF COST AND PRODUCTION FUNCTIONS

Note that if the subvector u(D) is void, this definition provides the cus­ tomary statement for efficient output vectors. If u(D) is void, the strong disposal Property A.8(b) holds, otherwise the weak disposal Property A.8(a) may apply, see Figure 38(b), while the strong disposal Property A.8(b) holds for desirable outputs, see Figure 38(a). Proposition 71: Ep(X) is nonempty and bounded. The proposition holds trivially for χ ε X such that P(x) = {0}. Hence, consider χ > 0 such that P(x) — {0} is not empty. Since P(x) is bounded, Property A.2, the subset Ep(x) is bounded. Consider first the case where u(D) is void and let B = Max {||u|| | u ε P(x)). The quantity B is posi­ tive and exists for all χ such that P{x} — {0} is not empty, since it is the maximum of a continuous function Hull =

ι

on a closed and bounded set P(x). Let u0 denote a point of P(x) at which ||u°|| = B. See Figure 38(a). The point u0 is evidently efficient, because ν > u0 and ν ε P(x) implies ||v|j > ||u°||. Next, suppose the vector u(D)

ο FIGURE 38 (a): EFFICIENT SUBSET E (X) FOR DISPOSABLE OUTPUT SETS p

COST AND BENEFIT

FUNCTIONS

FOR PRODUCTION

CORRESPONDENCES

FIGURE 38 (b): EFFICIENT SUBSET FOR OUTPUT SETS WITH SOME NONDISPOSABLE OUTPUTS

is not void, and let B = Max . We assume that for some otherwise the production correspondence does not yield positive desirable outputs. For for all . Otherwise, the quantity B exists and is positive, and in this case consider Min and let denote a point of at which , See Figure 38(b). The point u° exists, since is closed and bounded, being the intersection of a closed, bounded set ! by a closed set.Then •ecause a point with and or with cannot belong to since then either Thus, the efficient subset is nonempty forall Let denote the prices per unit of the factors of production, and we consider , i.e., the price vectors of interest are such that superimposing the factor input and price spaces. Since the inputs of real capital, i.e., plant and equipment, are treated as the services of these items, some convention regarding the measurement of such service inputs per unit time and the correspond< 225 >

THEORY OF COST AND PRODUCTION FUNCTIONS

ing prices per unit is required. One possibility, frequently used in eco­ nomic theory, is to measure the service input per unit time of an item of real capital by the acquisition cost K of the item, distinguishing still the qualitatively different items of real capital so that the quantity X1 = K is regarded as having dimension [J]/[t], where [J] denotes the kind of item and [t] denotes time. Then assuming straight line deprecia­ tion recovery of the money capital outlay, the price per unit pi of the real capital input is (1 · $)/T where T is the working life of the item or the legal period of depreciation, and pi · Xi = K/T, the cost per unit time. In this way, the indivisibility of real capital service inputs is avoided by treating the cost K as a continuously varying quantity, the values of which denote rational number levels of inputing the services of the physical item. Values of K leading to a fractional number of physical items are then interpreted as inputing the entire item for a correspond­ ing fraction of the unit time interval with costing on a time fractional basis. One might also distinguish each item of equipment as a distinct machine input with a corresponding designated dimension [J], If the in­ put Xj of this item per unit time is 3.5, for example, this quantity denotes the services of three machines inputed during the unit time interval plus the services of a fourth machine inputed for one-half the time interval. Here the price pi denotes the dollars per unit time charged as deprecia­ tion (or cost recovery) for the item and the time fractional inputs of machine services are costed on a time fractional basis. Prices are institutional and there is no absolute basis for calculating cost. If χ and y are two input vectors of the factors of production, the input vector [(1 — θ)χ + 0y], θ ε [0,1] may or may not be regarded as time fractional applications of χ and y depending upon whether the compo­ nents are integers or nonintegers. In either case, the cost per unit time of this input is computed by ρ · [(1 — θ)χ + 0y], These conventions are probably suitable as long as the capital inputs of each kind are not small where the decision to be made is essentially integral in character. The factor minimal cost function is defined by Q(u,p) = Min (ρ · χ I χ ε L(u)}, ρ ε X, u ε U. X

(97)

This cost function is essentially the same as the cost function defined by Equation (15), Chapter 4, the difference being that u is a multi­ dimensional vector for production correspondences. Therefore, the properties of the cost function (97) are similar to those of the cost function (15). As in the case of the cost function for a production function, we may

COST AND BENEFIT FUNCTIONS

FOR PRODUCTION

CORRESPONDENCES

define certain subsets of X, by

It is clear that, since for all . Also for all for all s i n c e f o r If we consider the two subsets of

then certainly their intersection is empty. But it cannot be said that these two sets exhaust the set because even under strong attainability of outputs, i.e., where both » and (b)apply, if a ray intersecting for all may not intersect for all . Hence, we define instead the following two subsets in the product space

Then the following proposition may be stated. Proposition 72: The properties of the factor minimal cost function for a production correspondence satisfying A.1,..., are for all for all for all For all is positive and finite f o positive and finite for for all and for all for all is a concave function of p on X for all is a continuous function of p on X for all For any for and or/and (b) For any ond then

r

i

For any is lower semi-continuous in u on U. i is convex in u on U, if the graph of P is convex. < 227 >

s

THEORY OF COST AND PRODUCTION

Properties „ and

FUNCTIONS

are obvious. Proofs of the Properties

will not be given here, since theycorrespond exactly to those previously given for the Properties in Proposition 22, Section 4.3, Chapter 4. Property has two forms depending upon whether the correspondence has weak or strong disposal of outputs, i.e., or applies. For weak disposal, the sets L(u) have Property i.e., Then X

X

Under strong disposal,

For Property

and

suppose

and let

denote the minimizing input vector corresponding to taken on the bounded set Then the sequence is bounded since and there exists an infinite subsequence with and a limit point of with for all k. Then for all k, contradicting The lower semi-continuity of the cost function Q(u,p) in u is verified by an argument analogous to that given for Property in Section 4.3, using the upper semi-continuity of the correspondence P for closure of the graph of the correspondence. Let be arbitrary and consider the sequence

where is the cost minimizing input vector belonging to the bounded efficient subset of Each term is bounded, and if is not a bounded set of real numbers,

since is finite (Properties semi-continuous in u at Thus, take Since there is a subsequence

where

, and is lower as a bounded set. such that

But, since the graph of P and thus L is closed, Hence and the cost function is lower semi-continuous in u for Finally consider Property . If the graph of P is convex, < 228 >

COST AND BENEFIT FUNCTIONS FOR PRODUCTION

CORRESPONDENCES

For the definition of a Benefit Function, let r = (ri,r 2 ,... ,r m ) be a vector with real components. The quantity ri is assigned to the output Ui as a "price" or relative value for the i th output. If all of the outputs are desirable and marketable, the vector r 0 and may be interpreted as a market exchange price vector, and the inner product r • u denotes revenue from the sale of the output vector u. On the other hand, some of the outputs u may be undesirable products from a social viewpoint, if all outputs of the production correspondence P are covered by the vector u, and there are no market prices for such products. Also, many technologies are being developed and operated for public service with joint outputs which have no prices determined by an exchange economy, and the vector r specifies the relative benefits in a social accounting system, however determined. Thus, in order to define a benefit function which may apply to all of these situations, we consider the vector r to range over the entire Euclidian space R m . The output maximal Benefit Function is defined by (98) for a production correspondence as the maximal benefit obtainable with an input vector and output unit value vector Clearly, the function B(x,r) exists for all and since the set P(x) is bounded and closed for each In the revenue function case, replace Proposition 73: The properties of the output maximal benefit (revenue) function B(x,r), for a production correspondence satisfying are B.l B.2 B.3 < 229 >

THEORY OF COST AND PRODUCTION FUNCTIONS

BA B.5 B.6 B.7 B.8 B.9

B(x,r + s) < B(x,r) + B(x,s) for all x E X; r,s E Rm(R~). B(x,r') < B(x,r) for all x E X, if r' < r. B(x,r) is a convex function of r on Rm(~) for all x E X. B(x,r) is a continuous function of r on Rm(~) for all x E X. B(y,r) > B(x,r) for all r E Rm(R~) ify > XEX. For {llxnll} ~ 0, lim Sup B(xn,r) = for any r E Rm(R~).

°

n-->(x,

B.IO For r E Rm(~), B(x,r) is upper semi-continuous in x EX. B.ll If the graph ofP: X ~ U is convex, B(x,r) is concave in x for all re Rm(R~). The Properties B.l and B.2 are evident and somewhat trivial statements. For Property B.3, let A > and B(X,M) = Max {M' u I u E P(x)}

°

u

= A Max {r' u I u E P(x)} = AB(x,r). u

The verification of Property B.4 is direct by letting u* denote the maximizing output vector for (r + s) and noting that B(x,r + s) = (r + s) • u* = r • u* + s . u* < B(x,r) + B(x,s). The Property B.S follows directly in the benefit function case from BA and B.l, since r' < r implies r' = r + ~ where dr < 0, and B(x,r')

= B(x,r + dr)
u

Max {r' u I u E P(x)} u

< 230 >

= B(x,r).

COST AND BENEFIT FUNCTIONS

FOR PRODUCTION

CORRESPONDENCES

To show that Property B. 10 holds, let be arbitrary and consider the sequence Then, there is a subsequence such that sup Denote by the maximizing output vector for yielding The sequence is bounded for any , and there exists a convergent subsequence with a limit u°.The upper semi-continuity of the correspondence P implies that since ^ for all k and (see Appendix 2). Hence,

and the benefit function B(x,r) is upper semi-continuous in x for all Then Property B.9 is a simple consequence of B.10. Let and Sup The proof of B. 10 does not use B.9. For Property B.l 1, the graph of P is convex if and only if

for any

(see Appendix 2), and

Note that we have omitted for the benefit function B(x,r) a property analogous to for the cost function, i.e., if because can occur by unbounded increase of the input rate of one factor of production with bounded input rate of an essential factor of production implying P(xn) is bounded. On the other hand, due to Property

10.2 Cost and Benefit (Revenue) Correspondences Analogous to the cost structure for a production function mapping the cost structure for the production correspondence is a correspondence is < 231 >

THEORY

OF COST AND PRODUCTION

the set of nonnegative price vectors production. This correspondence is defined by:

FUNCTIONS

of the factors of

Definition: The cost structure correspondence is a mapping for which the map set of an output vector u is the set (99) The price vectors p e

THEORY

OF COST AND PRODUCTION

is concave in x for all and

FUNCTIONS

(see Property B.l 1). If then

and Thus, the sets are convex for all An empty set is convex. Note that the input sets of the correspondence are unbounded. The efficient subsets of the closures of , defined as for the input sets of the correspondence P, specify the "infimal" input vectors yielding a maximal benefit rate greater than B, and similarly for the sets No analogue has been given for the benefit (revenue)-afforded-input correspondence, like that of the cost limited output function T(p) for the correspondence G. However, for the correspondence with and on which the inverse production function _1(u) was defined in Section 9.2, such an analogue may be defined. Let and define a benefit (revenue) afforded input function by (104) The properties of the Indirect Inverse Production Function given in the following proposition:

are

Proposition 81: The properties of the benefit (revenue)-affordedinput function for a correspondence with Properties

F.5 F(r) is upper semi-continuous on F.6 F(r) is quasi-concave on Property F.l holds, because the set

is empty (see Property B.l, Proposition 73), whence Since x represents the input of a single factor of production, a positive input x must yield a bounded positive output vector, otherwise the correspondence always has null output for one component of u and this component may be omitted from consideration. Thus, due to Property of the correspondence P, F(r) is positive and finite for Hence, Property F.2 holds. < 248 >

COST AND BENEFIT FUNCTIONS FOR PRODUCTION CORRESPONDENCES

Property F.3 is an immediate consequence of Property B.5 of the function B(x,r) which implies {xl B(x,r)

> 1,xe[0,+00)} C {x I B(x,r') > 1,xe[0,+00)}.

Property F.4 is a consequence of the continuity of the function B(x,r) in r e RID(~), for all x e X. Since B(x,O) = for all x e X, it follows that lim B(x,rn) = 0. Now suppose that lim InfF(rn) = Fo + 00. Then ~oo

n~oo

° °




> Fo

n-->oo

In order to prove Properties F.5 and F.6, we show that the level sets CBF(y) = {r I F(r) > y,r e R~)}, ye Rl are closed and convex (see Appendix 1). Ify 0.

Now r e CBF(y) implies F(r) > y or Inf {x I B(x,r) which implies B(x,r) CBF(y)

c

n

u[O,y)

l,x e [0,+ oo)} >

y,

1 for x e [O,y), and r e n

xe[O,y)

CB(x). Hence,

CB(x). Conversely, yen CB(x) implies B(x,r) u[O,y)

l,x e [0,+ oo)} >

implying r e CBF(y). Thus, CBF(y)::J n

xe[O,y)

y,

CB(x), and CBF(y) equals an

intersection of closed and convex sets and is, therefore, closed and convex for y 0. Therefore, Properties F.5 and F.6 hold. The economic meaning of the function F(r) is: for any output price vector r, F(r) gives th\! infimal value of the single input x e Rl to yield a revenue (benefit) greater than unity. Again, letting B (or R) denote any positive benefit (revenue) rate,

>

F(~) = Inf {x I B(x,r) > B,x e [O,+oo)}, r e Rm,

F(~) =

Inf {x I B(x,r)

> R,x e [0,+ oo)}, r e Rtf

define the infimal input x of a single factor of production to yield a maximal benefit (revenue) greater than B (or R) at prices r for the output vector u e ~.

< 249 >

THEORY

OF COST AND PRODUCTION

FUNCTIONS

The function F(r) induces a benefit (revenue)-afforded-input correspondence with map sets

and the efficient subset of the closure of the sets defines the infimal input yielding a maximal benefit (revenue) rate greater than B at prices r for the outputs. 10.4 Special Forms for Homotheticity of Input and Output Structure of a Production Correspondence Recall that the production correspondence defined (see Section 9.3) to have homothetic input structure if the input sets L(u), u e U are representable by where is a homogeneous function of degree one and otherwise O.

The definitions of the cost and benefit (revenue) correspondence "c(u), u e R'f, (B(x)(CR(x», x e R~ take the following simple forms, when the correspondence P has homothetic input and output structure respectively, "c(u) (B(x)(CR(x»

= {p I feu) . 'IT(p) >

=

l,p eX}, u e U = R'f {r I (x)' f3(r) < l,r e Rm(R'f)}, x e X,

(99H) (lOOH)

and the strong disposal Properties 'IT.5 (b), 'IT.6 (b) apply for the cost structure. Note that the cost function defining "c(u) is lower semicontinuous in u. Propositions.72H and 73H are merely partial restatements of Propositions 72 and 73, but they serve to remind that the functions 'IT(p) and f3(r) have suitable properties for interpreting them as index number functions. Similarly, the cost limited output sets G(p) of the cost-limited-output correspondence G: R~ ~ R'f and the benefit (revenue)-afforded-input sets «(B-l(r)Y of the correspondence «(B-l)c: Rm(R'f) ~ R~ take the explicit forms G(p) (CB-l(r»c

= {u I f(u)' 'IT(p) < l,u e R'f}, p e R~ = {x I (x)' f3(r) > l,x e R~}, r e Rm(R'f)

(l02H) (103H)

when the input and output structures of the correspondence Pare respectively homothetic. It is noted that in this case the correspondence G defines by the output sets G(p/C), C 0, the set of output vectors u such that feu) is less than the cost rate C deflated by the "level" 'IT(p) of

>

< 252 >

COST AND BENEFIT FUNCTIONS FOR PRODUCTION CORRESPONDENCES

the prices of the factors of production. Also, the correspondence «B-l)c defines by the sets ( 0, the set of input vectors x such that (x) is greater than the benefit (revenue) rate B deflated by the "level" f3(r) of the unit output value (prices) of the outputs of production. In Section 10.2 it was shown (Propositions 77 and 78) that the cost function Q(u,p) and the benefit (revenue) function B(x,r) are distance functions respectively for the sets £(u), u e R~ and O} intersects the set G(p) for o > O. Homotheticity of both input and output structures for P implies Q(u,p) = f(u)· '1T(p) where both f(u) and '1T(p) are homogeneous functions of degree one in u and p respectively. Then Oo(u,p)

= Sup {O I f(O· u) . '1T(P) < I} = Sup { 0 0 < f(u) ~ '1T(p) } 1

1

1

- f(u)· '1T(p) - Q(u,p) . Note that u Z 0 implies f(u) > 0 and the boundedness of G(p) implies '1T(p) > O. Next suppose u = 0, or u Z 0 and p Z 0 with '1T(p) = 0, i.e., the set G(p) = ~. Then Q(u,p) = 0, and Oo(u,p) = + 00 implying l/Oo(u,p) equals zero. Hence, in this case also Q(u,p) = l/Oo(u,p). Thus, the cost function Q(u,p) is a distance function for the correspondence G. t See modification in Section 10.5 for the extended definition of homotheticity.

< 253 >

THEORY OF COST AND PRODUCTION FUNCTIONS

Consider next the correspondence (B-l)c. Here we seek to show that B(x,r)

= II~~;,~)II = Ao(~,r)'

where ~(x,r) = Ao(x,r)· x is the infimal intersection of the ray {A· x I A>O} with (B-l(r»c, and Ao(x,r)

°

= Inf {x I (A· x)

°

E

(B-l(r»c}.

°

Suppose that x ~ with clI(x) > and r ~ with f3(r) > 0. Then B(x,r) = clI(x) • f3(r) where both clI(x) and f3(r) are homogeneous functions of degree one in x and u respectively, and the ray {Ax I A > o} intersects the set (B-l(r»c. Then Ao(x,r)

= Inf {A I clI(Ax) . f3(r) > I} = Inf { A I A > clI(x) 1. f3(r)} 1

1

= clI(x)f3(r) = B(x,r) .

°

°

= 0, or x ~ with clI(x) = 0, or r = 0, or r ~ with f3(r) = 0, the set (B-l(r»c is empty with B(x,r) = and Ao(x,r) = + 00, and in these cases we also have B(x,r) = l/Ao(x,r). Thus, the benefit (revenue) function B(x,r) is a distance function for the correspondence (B-l)c. When the output vector u has a single component, i.e., G: ~ ~ Rl, and 'IT(p) 0, the map sets of the correspondence G take the form

If x

°

>

G(p)

= {u I u < F(~p») J,

and (see Section 5.4) the cost limited maximal output function (indirect production function) f(p) becomes f(p)

= Sup { u I u < F('IT~»)'u Rl} _F(_l) t E

-

'IT(p)'

Thus, the correspondence G is a consistent generalization of the indirect production function. Further, when the input vector x has a single component, i.e., (B-l)c: RIn{~) ~ Rl, and f3(r) > 0, the map sets ofthe correspondence (B-l)c take the form (B-l(r»c t As defined,

= {x I x > H( f3~r») J,

F( . ) is a transform with F(v) -->

+ 00

< 254 >

as v -->

00.

COST AND BENEFIT FUNCTIONS FOR PRODUCTION CORRESPONDENCES

where H( . ) is the inverse function of 4>(x) defined by H(v)

= Min {x I cp(x) >

v,x E Rl}, v E Rl,

and (see Section 10.3) the Benefit (Revenue)-Afforded-Input function (indirect inverse production function) F(r) becomes F(r) = Inf{ x I x > H( f3~r»)'x E Rl} =

H(f3~rJt,

the applicable form of the function F(r) when the output structure is homothetic. 10.5 Returns to Scale for Production Correspondences For the study of returns to scale it is convenient to extend the definition of homothetic input structure given in Section 9.3 as follows: Extended Definition: The production correspondence P has homothetic input structure if the input sets are given by

L(u)

= {x I F(cp(x»

> f(u),x E ~},

where F(v) is a transformtt with F(v) ~ is homogeneous of degree one.

U E~

+ 00 as v ~ + 00 and 4>(x)

In this definition the functions cp(x) and f(u) are otherwise assumed to have the Properties AI, ... , A6 and (a) ... (f) respectively listed in Section 9.3. With this extended definition, the analysis given in Section 9.3 is largely unaltered. To see this, let F-l(W) = Min {v I F(v) > w,v E Rl} be taken as the definition of the inverse function of F(v) = w, which is proper since F(v) is upper semi-continuous and nondecreasing in v E Rl, and the set {v I F(v) > w, V E Rl} is a closed interval [F-l(W), + 00). Then the input and output sets of the correspondence P become L(u) = {x I cp(x) > F-l(f(u»,x E ~}, u E R~ P(x) = {u I f(u) < F(cp(x»,u E ~}, x E~,

(105) (106)

the representation for P(x) holding since P is a correspondence inverse to L. All we need to show is that F-l(f(u» has the Properties (a) ... (f) (listed in Section 9.3) as a function ofu, since Proposition 4 (Section 2.3) states that F(cp(x» has the Properties AI, ... , A6. tHere F(r) is not a transform as used for the function f(p).

tt See Section 2.3 for the definition of transforms.

< 255 >

THEORY

OF COST AND PRODUCTION

FUNCTIONS

By using Proposition 5 (Section 2.3) it is trivial to verifythat has the Properties (a). . .(d). In order to show that is lower semi-continuous in u , Property (e) holds, we need only show that the set

is closed for all is empty and therefore closed, since and are nonnegative functions. For note that if and only if and if and only if whence closed set. Therefore, we need only consider further the case where Suppose is not closed for Then there exists a sequence with

implying thatf,

and

a contradiction. Hence, is closed for all and is lowersemi-continuous in Property (f), i.e., the quasi-convexity followsfrom the quasi-convexity of f(u) and the nondecreasing property of Thus, when is homogeneous of degree one, statement (105) is equivalent to that given for homotheticity of input structure in Section 9.3. Similarly, when f(u) is homogeneous of degree one statement (106) is equivalent to thatgiven for homotheticity of output structure with replaced by With this extended definition of homothetic input structure and the implied extension of the definition of homothetic output structure (i.e., Equation (106) with f(u) homogeneous of degree one), the Propositions 52, . . . , 59 of Section 9.3 hold with replaced by when is not homogeneous and f(u) replaced by when f(u) is not homogeneous. However, when the production structure has both homothetic input and homothetic output structure, the correspondence P and its inverse L are not homogeneous. In Section 9.6, Propositions 68 and 69 hold with replaced by and respectively. In Section 10.4, Propositions 82 and 83 hold with f(u) replaced by The inverse function of if we define it as Max Note that is lower semi-continuous by Proposition 5, Section 2.3.

< 256 >

COST AND BENEFIT FUNCTIONS FOR PRODUCTION CORRESPONDENCES

F_1(f(u)) and Φ(χ) by F( Q(u,p), and, in particular, if p e £(u), p' XO > Q(u,p) > 1. Then, since i'*(U,XO) there exists for any

t:

= Inf {p . XO I p e £(u)}, p

> °a p. e £(u) such that i'*(u,XO) + e > p •. XO > 1.

Consequently, i'*(u,XO) > 1 and XO e L*(u), so that L(u) C L*(u). For the converse, define ~ P P = Q(u,p)' A

=

and, since Q(u,p) is homogeneous of degree one in p, Q(u,p/Q(u,p» 1 and f> e £(u) for any u ~ 0. Now, assume XO e L*(u) and that XO ¢ L(u). By the strict separation theorem for convex sets, see [3], p. 163, there exists p ~ such that

°

p . XO

< Min {p . x I x e L(u)} = Q(u,p) x

or

f>' XO < 1 with f> e £(u). Consequently, i'*(u,XO)< f> •XO < 1

and XO ¢L*(u), a contradiction. Hence, XO e L(u), and L*(u) C L(u). L(u). Thus, L*(u) Since L is the inverse correspondence of P, the following proposition holds,

=

Proposition 87: The correspondence L *: U ~ X is the inverse correspondence of a production correspondence P with the Properties A.l, ... , A.8.

= i'(u,x) for all x e X, u e U. First, i'(O,x) = i'*(O,x) = + 00 for all x e X, and both functions are PropOSition 88: ¥(u,x)

finite for u ~ 0, x e X. Confining our attention to u ~ 0, let x e X and suppose i'(u,x) =1= i'*(u,x) for some XO e X. Four situations are of interest

< 264

)

DUAliTIES FOR PRODUCTION CORRESPONDENCES

(a) (b) (c) (d)

< <
>

>

In Case (a), i'*(U,XO) 0 and taking A = (i'*(u,XO»-l it follows, due to the homogeneity ofi'*(u,xO) in x, i.e., Property d.3*, that i'*(u,Ax°) = I, implying that (Ax°) e L*(u) L(u) for u 2:: O. Then

=

i'(U,Ax°) = Ai'(U,XO) ::> I

>

and i'(u,x0) o. Thus, Case (a) is impossible. Next, for Case (b), let d i'*(u,XO) - i'(u,XO)

=

and take A i'(u,x) in x,

= l/(i'(u,xO) + d/2). i'(U,Ax°)

> 0,

Then, due to the homogeneity of

= Ai'(u,XO) = i'(U~~~:)d/2 < I,

implying (Ax0) ~ L(u). Also, i'*(u,xO)

i'(u,XO)

+d

= i'(u,XO) + d/2 = i'(u,XO) + d/2 > 1, But since L*(u) = L(u), Case (b) is impossible.

i'*(u,Ax°)

and (Ax°) e L*(u). By symmetry, Cases (c) and (d), are likewise shown to be impossible.t Corollary 1: The price minimal cost function i'*(u,x) is a distance function for the sets L*(u) of the correspondence L* = L. Corollary 2: The function i'*(u,x) is a quasi-concave function of ueU. Thus, the distance i'(u,x) for the input sets L(u) of a production correspondence P: X ~ U is given by the price minimal cost function i'*(u,x) defined on the cost structure correspondence .c: U ~ X induced by the cost function Q(u,p) defined on the inverse correspondence L: U ~ X ofP. The two cost functions Q(u,p) and i'(u,x) are dualistically determined from each other by the following dual problems: Dual Problems for Q(u,p) and i'(u,x): (a) Q(u,p) = Min {p' x I i'(u,x) ::> I,x eX}, u e U, p e X. x

(b) i'(u,x)

= Inf {p' x I Q(u,p) ::> l,p e X},u e U, x e X. p

tThis proof, used by Jacobsen [15], is more convenient than my original argument for -t*(u,x) defined on £Q(u) for the production function, since it avoids detailed classification of the boundary points of X.

< 265 >

THEORY OF COST AND PRODUCTION FUNCTIONS

This duality provides cost minimum problems for calculation of Q(u,p) and ^(u,x) in terms of each other, and the dual production and cost structures L: U —> X and £: U —» X are determined from each other in terms of these two functions. The distance function ^(u,x) provides a basis for defining a produc­ tion input correspondence L: U X by the sets L(u) = {χ | ^(u,x) Ξϊ 1, χ ε X}, u ε U. The correspondence L: U —> X defines the input sets of the production structure, and the correspondence Ρ: X —» U inverse to L: U —» X, i.e., the family of sets P(x) = (u | χ ε L(u),u ευ} = {u I ^(u,x) ^ l,u ε U}, for χ ε X, defines the output sets of the produc­ tion structure. Similarly, the distance function Q(u,p) provides a basis for defin­ ing a cost structure correspondence £: U -» X by the sets £(u) = {p I Q(u,p) Ι,ρ ε X}, u ε U. If this cost structure is known, the pro­ duction structure is then determined by the price minimal cost function Φ(ϋ,χ) given by the Problem (b). Dually, if the production structure is known by the input structure correspondence L: U —> X, the cost struc­ ture is determined by the factor minimal cost function Q(u,p) given by the dual Problem (a). The inverse correspondence £-1: X U, by the complements of the sets £_1(p) = {u I Q(u,p) ^ l,u ε U}, ρεΧ, also plays an economic role (see Section 10.3 above). 11.2 Duality Between the Benefit (Revenue) Function B(x,r) and the Distance Function fi(x,u) for the Output Sets P(x) of Ρ: X —> U On the benefit (or revenue) correspondence (B(or (R) a unit value (price) maximal benefit (revenue) function may be defined by Definition: S*(x,u) = Sup {r · u | r ε (B(x)((R(x))}, u ε U, χ ε Χ. r

The sets (B(x) (or (R(x)) are not necessarily bounded for all χ ε X. For example, let χ ε R^ and suppose P(x) = [0,uj]. Then B(x,r) = Max {r · u | u ε P(x)} = ri · uj U

and the sets

are unbounded as illustrated in Figures 42 (a), (b). Consequently, unless the vector u has the form (ui,0), the inner product r · u has no finite upper bound for r ε ®(x) or r ε (R(x). In this case, the supremum of

DUALITIES FOR PRODUCTION CORRESPONDENCES

FIGURE 42 (a):

UNBOUNDED SET x. V.l* V.2* V.3* V.4* V.5*

>

In Proposition S9 the set V* is defined by

= {(x,u) I x ~ O,u ~ O,O*(x,u) < + oo}

V*

and (V"')c is the complement ofV* with respect to (X X U) - (X X {O}). For x = 0, c:B(0) = Rm( g"'(x,u'). Property V.6'" follows directly from V.4'" by taking u = (1 - fJ)u, v = (Jv for" e [0,1] and using V.3*. Then, it follows from the theorem of Section 2.4 that U*(x,u) is upper semi-continuous in u on the boundary of Rlf as well as being continuous in u in the interior of Rlf. Hence, for Property V.7* we need only show that g"'(x,u) is lower semi-continuous on the boundary of R!f, which may be established by an argument analogous to that given in Section 4.3 for the upper semi-continuity in p of the cost function Q(u,p) on the boundary of X ~, merely by extending the definition of g*(x,u) to all r e Rm in the revenue case by

=

g*(x,u/R)

= Sup {r' u 1 r e ffi.(x/R)}, u e Rm, x e X, r

where SR(O) ffi.(x/R)

= {r Illrll < R,r e Rm}, = ffi.(x) n SR(O).

For Property V.8"', note that, by the Property

l,p e X} l,p e X}

= f(~) {f(u)· pi 'IT(f(u)· p) > l,f(u)· p e X} and, from the duality between the functions 'l'(u,x) and Q(u,p), 'l'(u,x)

= Inf {p . x I p e .c(u),p eX} p =

I~f {p. x I p e f(~) {f(u)· pi 'IT(f(u)· p) >

=

f(~) I~f {p. x I 'IT(p) >

l,f(u)· p eX}}

l,p eX}

~(x)

= f(u)' where ~(x)

!

Inf {p. x I 'IT(p) > l,p e X} p

is homogeneous of degree one in the input vector x and otherwise has the Properties AI, ... , A6 of a production function ~(x). (See Propositions 85 and 88.) Thus, due to Proposition 62 L(u) = {x I 'l'(u,x) > l,x eX}, u e U, and

= {x I ~(x) > f(u),x eX}, u ~ o. f(O) = 0 for all x e X and L(O) = X. Consequently,

L(u)

For u = 0, ~(x) > the production structure is homothetic (see the definition Section 9.3).

Corollary 'l':t The distance function (price minimal cost function) 'l'(u,x) takes the separable variable form 'l'(u,x) = ~(x)/f(u), if and only if the input structure of the production correspondence P is homothetic. See Proposition 69. Corollary G: The isoquants of the cost-limited-output correspondence G are given by Equations (110) and (110.1), if and only if the input structure of the correspondence P has both homothetic input and output structure, using the extended definition of homothetic input structure. t Under extended definition of homothetic input structure i'(u,x) =

< 273 >

(X)/F-l(f(U)).

THEORY OF COST AND PRODUCTION FUNCTIONS

Corollary QE: The factor minimal cost function Q(u,p) factors into the form Q(u,p) = P-l(f(u»· 7/"(p) if and only if the correspondence P has extended homothetic input structure. Corollary '1' E: The distance function (price minimal cost function) '1'(u,x) takes the separable variable form CP(x)/P-l(f(u», if and only if the production correspondence P has extended homothetic input structure.

Concerning the output maximal benefit (revenue) function B(x,r), it was shown by Propositions 83 and 73H that, if the production correspondence P has homothetic output structure, the function B(x,r) takes the form B(x,r) = cp(x) . f3(r) where f3(r) is a homogeneous function of degree one in the unit value (price) vector r with certain properties and cp(x) has the Properties A.I, ... , A.6 of a production function. Theorem B:t The output maximal benefit (revenue) function B(x,r) factors into the form B(x,r) = cp(x)· f3(r) if and only if the correspondence P has homothetic output structure. If the production correspondence P has homothetic output structure,

the function B(x,r) = cp(x)· per) by virtue ofthe Proposition 83 and 73H. Conversely, suppose B(x,r) = cp(x) . f3(r) where cp(x) and f3(r) have the properties indicated. Then, for x 2:: 0, and cp(x) 0,

>

- z· A for some z >- 0 such that z· B >- AU}

= A{'!'/'!' >- !:.. • A for some !:.. >- 0 such that ~ • B >- u} A A-A AA-

= AL(u). Taking p and r as column vectors of prices for the factors of production and the outputs respectively, the factor minimal cost function and the output maximal benefit function. are: Q(u,p) = Min {x'p I x >- z'A for some x z >- 0 such that z· B >- u}, u E~, P E ~ B(x,r) Max {u • riO < u < Z • B for some u Z >- 0 such that z· A - I,p E~}, p

U E~,

XII ~

(ID) Q(u,p) = Min {x, p I x >- z· A for some z >- 0 such that x

z· B >- u}, u E:RJf, p E ~ (2)

O(x,u) = Sup {u' r I B(x,r) < l,r E ~}, XE~,

U E~

r

(2D) B(x,r) = Max {u· riO < u
- 0 such that

u

z·A o. Then for output vectors u such that

< 287

)

THEORY

OF COST AND PRODUCTION

B for some (2D) become

such t h a t P r o b l e m s (1) and

B for some Letting lem

FUNCTIONS

such that z •

denote an output vector yielding the solution of ProbProblem may be specialized to

and Problems determine dual vectors with as outlined in the previous section. Considering the usual duality of linear programming, Problem (2D)° may be taken as a primal problem, which may be expressed conventionally by: Maximize with

Letting p and w be "price" vectors for x and deficits between u and z • B respectively, the corresponding dual problem in linear programming is Minimize with

This dual problem determines a "price" vector such that

for the input vector

i.e., such that the imputed minimal cost of input for each activity at unit level is not exceeded by the value of output produced by that activity at unit level, whereas the dual Problem (l) 00 determines an imputed minimal total cost of the input vector x° which is not exceeded by the maximal total output value obtainable with x° at prices r°. The < 288 >

DUALITIES FOR PRODUCTION CORRESPONDENCES

special form of the dual problem arising in linear programming is pos­ sible, because the matrices A and B impute cost and output value for each activity. But, in a general production correspondence such imputa­ tion by activities cannot be sorted out, and Problems (2D)° and (I)00 are suitable problems (with weaker and more appropriate constraint conditions than those in linear programming) for determination of shadow prices of a given input vector x°. It is clear that the price im­ putation of the duality of linear programming is not generalizable for production correspondences, being merely a result of the special form of the Unear production correspondence. Moreover, for a production correspondence with truly joint outputs, an imputation by disjoint activities is artificial. One need only require that the total imputed minimal cost of x° be not exceeded by the maximal total value of all outputs which can be obtained with x°. Assuming that the given vectors x° and r0 are such that a solution exists for the Unear programming imputation of a shadow price vector p* for the input vector x°, the DuaUty Theorem for linear programming implies that x° · p* = u* · r0, where u* is an optimal output vector for the linear programming Problem (2D)°. It is to be expected that the solution of Problem (I)00 yields an imputed price vector p*(x°,u*) such that x° · p*(x°,u*) = u* · r0, but this fact needs proving. First, note that Problem (2D)° is B(x°,r°) = Max {u · r0 | u ε P(x0)}, U

and, since outputs are strongly disposable for the general linear model of production, u* belongs to the isoquant of P(x°), i.e., for a scalar λ > 1 there does not exist a nonnegative activity vector ζ such that ζ · A :£ x° and Xu* ^ ζ · B, or, what is the same (letting θ = l/λ and w = ζ/θ), there does not exist a vector w 3:0 such that w · B 2: u* and θ' x° 5; w · A for θ < 1. Thus, x° belongs to the isoquant of L(u*), im­ plying that *(u*,x°) = 1 (see Proposition 65). Hence, since the cost function is a distance function and the solution p*(x°,u*) occurs on the boundary of the price set £(u*), the solution of Problem (I)00 yields x° · p*(x°,u*) = B(x°,r°) = u* · r0, and, Q(u*,p*(x°,u*)) = B(x°,r°) = X0 · p*(x°,u*), implying that x° yields a solution to Problem (ID) with u = u* and ρ = p*(x°,u*) (see Proposi­ tion 93, Section 11.4). Therefore, the duaUty of linear programming for imputing shadow prices for an input vector x° is merely a special form of the dual Problems (2D)°, (I)00 arising for the particular structure of linear production correspondences. Similarly Problems (ID) and (2) determine dual vectors x*(u°,p°), r*(u°,x*), for given output vector u0 and input price vector p0, such that u0 · r*(u°,x*) g x*(u°,p°) · p0, when expressed in the form

THEORY OF COST AND PRODUCTION

FUNCTIONS

and

with denoting the input vector yielding the solution of Problem The price vector is an accounting (shadow) price vector for outputs imputing a price maximal value of not exceeding the minimal cost of obtaining at prices for the factors of production. If is taken as a primal problem for linear programming, expressed conventionally as Minimize with

the corresponding dual problem is Maximize with

This dual problem determines a price vector r* for outputs such that

i.e., such that the imputed value of output for each activity at unit level does not exceed the cost of operating that activity at unit level with prices as compared to Problem which determines a price vector for outputs such that the imputed total valueof the output vector u° does not exceed the minimal cost of obtaining with a price vector p°. Again, the special form of the dual problem for prices r in linear programming is only possible if the production structure is linear. It involves a more restrictive constraint set for the price variables r than does Problem i , and this restricted constraint set is not applicable for more general production correspondences. For coherent operation of a technology, there does not seem to be a compelling need to restrict the feasible price vectors r so that the output value of each activity at unit level does not exceed the cost of operating the activity at unit level, even when the activities have physical counterparts. Again, assuming that the given vectors and yield a solution for < 290 >

DUALITIES FOR PRODUCTION CORRESPONDENCES

the linear programming imputation of a shadow price vector r*, u0 · r* = x* · p0 where x* is an optimal input vector for Problem (ID)0, and also the solution r*(u°,x*) of Problem (2)00 satisfies u0 · r*(u°,x*) = x* · p0, showing that the determination of shadow prices for an output vector by linear programming duality is a special form of the dual Problems (ID)0 and (2)00 arising for the particular structure of linear production correspondences. Just as in the previous pair of dual prob­ lems, x* belongs to the isoquant of L(u°) implying for the linear produc­ tion correspondence that u0 belongs to the isoquant of P(x*) and fi(x*,u°) = 1 by virtue of proposition 64, so that Problem (2)00 yields u0 · r*(u°,x*) = Q(u°,p°) = x* · p0, with B(x*,r*(u°,x*)) = Q(u°,p°) = u0 · r*(u°,x*) implying that u0 yields a solution to Problem (2D) with χ = x* and r = r*(u°,x*) (see Proposi­ tion 94, Section 11.4). The duality of Unear progamming is a mathematical relationship with Uttle or no original purpose for imputing economic-theoretic prices, arising from the structure of linear inequaUty systems and subsequently generalized by interpreting the dual (price) variables as Lagrange multipUers in a saddle point problem. The problem pairs (I)00, (2D)° and (2)00, (ID)0 are described in the previous section as duals because they reciprocaUy define each other, i.e., the data to formulate one problem also determine the structure for expression of the other problem. Outside of the framework of duality in linear programming, we may consider any feasible vector pair (u°,x°) (i.e., one for which there exists an activity vector ζ such that ζ · A 5Ξ x°,u° ^z-B) and use Problems (1) and (2) to impute shadow prices for both the output vector u0 and the input vector x° by *(u°,x°) = Inf {x° · ρ I Q(u°,p) Ι,ρ ε 1¾} ρ S2(x°,u°) = Sup (u0 · r I B(x°,r) g l,r ε Rg?}. r

(I)0 (2)0

An analogous calculation does not arise in Unear programming duaUty, but may be so formulated for the Unear production correspondence in terms of the functions Q(u,p) and B(x,r). As a final topic, it is useful to consider the interpretation of Problems (l)o and (2)o in the context of a total economy. Let Ρ: X -» U be a production correspondence, linear or not, with the Properties A.1,..., A.8, and take it to describe the technological re­ lationship between net outputs u0 and inputs x° of primary resources for a total economy by regarding att intermediate products as transfers within the technology which need not be treated expUcitly as variables. The primary resources may include certain capacities of plant and

THEORY OF COST AND PRODUCTION FUNCTIONS

equipment. We suppose further that u0 and x° are feasibly related, i.e., u0 ε P(x°) and x° ε L(u°), and that the price sets £(u°) and ®(x0)((R(x0)) have bounded efficient subsets. Then, for u0 > 0 and a resource vector X0 ε L(u°), the problems *(u°,x°) = Min {x° · ρ I Q(u°,p) g 1}

(l)o

i2(x°,u°) = Max (u0 · r I B(x°,r)ί 1}

(2)0

ρ

r

determine net output prices r*(x°,u°) and primary resource prices p*(u°,x°) such that u0 · r*(x°,u°) ^ x° · p*(u°,x°), i.e., the imputed value of net output does not exceed the imputed value of primary resources. If the resource vector x° and the net output vec­ tor u0 > 0 are related so that the given resource vector x° yields the minimal cost of attaining u0 at prices p*(u°,x°) for resource vectors be­ longing to L(u°) and the net output vector u0 yields the maximal bene­ fit (revenue) obtainable with x° at prices r*(x°,u°) for net output vectors belonging to P(x°), then (see Proposition 97) u0 · r*(x°,u°) = x° · p*(u°,x°), and the imputed prices p*(u°,x°), r*(x°,u°) are balanced prices in the sense that total value of primary resources equals total value of outputs. Hence, the following proposition holds. Proposition 99: If a net output vector u0 > 0 and a primary resource vector x° ε L(u°) are given for a production correspondence P of a total economy with bounded efficient subsets for the induced corre­ spondences

APPENDICES

Theorem: is upper semi-continuous at

if and only if

for all sequences Suppose is upper semi-continuous at Let be any sequence converging to Then for any there exists an integer Ne such that for Consequently, from definition (a) it follows that Sup . Conversely, suppose is not upper semi-continuous at Then there exists a positive real number such that forall neighborhoods there exists such that which implies that there is a sequence with for all n and Theorem: is lower semi-continuous at x° if and only if for all sequences The proof of this theorem exactly parallels that given above for upper semi-continuity. Theorem: is upper semi-continuous for if and only if the sets are closed for all Suppose is upper semi-continuous for Let be an arbitrary cluster point (limit point) of arbitrary L(u) for Then there exists a sequence such that with u for all n. The upper semi-continuity of on implies

and the limit point x° belongs to L(u). Conversely, suppose is not upper semi-continuous at some point Then there exists a sequence such that Sup . Then, for some value

there is a subsequence

converging to x° with

Consequently, there is an integer Ksuch that for all k _ which implies for all while since and L(u) is not closed for some u e K1.

But

,

Theorem: , ) is lower semi-continuous foi are closed for all < 296 >

if and only if the sets S(u) =

APPENDICES

The proof of this theorem exactly parallels that given for the preceding theorem. 2. Quasi-Concavity (Convexity) Definition: A numerical function Φ(χ) defined on a convex subset D C Rn is quasi-concave on D if for all points χ and y of D, Φ((1 — ff)x + 0y) =2; Min [Φ(χ),Φ(γ)] for all β ε [0,1]. Theorem: The sets L(u) = {χ | Φ(χ) ^ u,x ε D} are convex for all u ε R1 if and only if Φ(χ) is quasi-concave on D. Suppose L(u) is convex for all u ε R1. Let χ and y be any two points of D, and take τ = Min [Φ(χ),Φφ)]. Then χ ε L(T), y ε L(T) and, since L(r) is convex, ((1 — ff)x + 6y) ε L(T) for all θ ε [0,1], which implies Φ((1 - ff)x + 0y) ^ τ = Min [Φ(χ),Φ^)] for all θ ε [0,1] and Φ(χ) is quasi-concave on D. Next, suppose Φ(χ) is quasi-concave on D. Then for any u ε R1, let χ and y be any two points of L(u) and Φ(χ) S u, Φφ) ^ u. It follows from the quasi-concavity of Φ(χ) on D that Φ((1 — 0)x + 0y) ^ Min [Φ(χ),Φφ)]^ u for all θ ε [0,1], implying that [(1 — ff)x + 0y] belongs to L(u) for all θ ε {0,1]. Hence, L(u) is convex for all u ε R1. Definition: A numerical function Φ(χ) defined on a convex subset D C Rn is quasi-convex on D if for all points χ and y of D, Φ((1 — θ)χ + 0y) 5Ξ Max [Φ(χ),Φ^)] for all θ ε [0,1 ]. Theorem: The sets S(u) = {χ | Φ(χ) ^ u,x ε D} are convex for all u ε R1 if and only if Φ(χ) is quasi-concave on D. The proof of this theorem exactly parallels that given for the preceding theorem.

APPENDIX 2 MATHEMATICAL CONCEPTS AND PROPOSITIONS FOR CORRESPONDENCES The discussion and arguments of Chapter 9 on cost and production correspondences require certain mathematical concepts and proposi­ tions which will be briefly reviewed in this appendix. Chiefly, we are concerned with closed graph correspondences and convex structures of correspondences. 1. Closed Graph Correspondences Let X and U denote subsets of two Euclidian spaces Rn and Rm respectively. For the discussion of Chapter 9, X and U are the nonnegative domains of Rn and Rm respectively. Definition of a Correspondence:

A function P mapping the points of X into subsets of U is called a correspondence of X into U, denoted by Ρ: X -> U. The terms corre­ spondence, function, mapping, transformation are all synonymous. We use the term correspondence as opposed to function in order to serve a distinction between multi-dimensional output production technologies and those where the output is a single good or service for which the term production function has been used. As a consistent notation, we shall use P(x) to denote the subset of U into which an element χ ε X is mapped. The graph of a correspondence Ρ: X -» U is defined by Definition of the Graph of Ρ: X

U:

The graph of the correspondence P is {(x,u) | χ e X,u ε P(x)}, as the set of ordered pairs (x,u) belonging to the product space Rn χ Rm such that u ε P(x) for χ ε X. The inverse correspondence of P will enable us to define the "level sets" of a correspondence P. Hence, we shall use the following definition suitable for our purposes. Definition of the Inverse Correspondence L: U —» X:

The inverse of a correspondence Ρ: X —» U is a correspondence L: U ^ X such that for any u ε U the mapping of u is the subset L(u) of X defined by L(u) = {χ | u ε P(x)}. This definition is that used for the lower inverse of a correspondence (see [3], Section 3, Chapter 2). The graph of the correspondence L: U —» X is defined by

APPENDICES

Definition of the Graph of L: U

X:

The graph of the correspondence L is {(u,x) | u ε U,x ε L(u)}, as the set of ordered pairs (u,x) belonging to the product space Rm χ Rn such that χ ε L(u) for u ε U, and, except for the ordering of the vectors u and x, the graphs of L and P are the same. The notions of upper and lower semi-continuity are extended for correspondences by the following two definitions: Definition of an Upper Semi-Continuous Correspondence Ρ: X -» U: The correspondence P is upper semi-continuous at x° ε X, if {xn} -» x°, for all η and {un} —> u0, imply u0 ε P(x°).

un ε P(xn)

Definition of a Lower Semi-Continuous Correspondence Ρ: X —> U: The correspondence P is lower semi-continuous at x°, if {xn} x° and u0 ε P(x°) implies that there exists a sequence {un} -» u0 with un ε P(xn) for all n. With these definitions, the following proposition holds. Proposition 1: The correspondences Ρ: X -» U and L: U -» X are upper semi-continuous on X and U respectively, if and only if the graphs of P and L are closed in Rn χ Rm and Rm χ Rn respectively. Definition of Continuity of a Correspondence Ρ: X -* U (or L: U -> X) at a Point X (or u0) Respectively: 0

The correspondence P (or L) is continuous at x° (or u0) if it is upper and lower semi-continuous at x° (or u0). 2. Convex Structures of Correspondences Let X and U be convex subsets of Rn and Rm respectively. The cor­ respondence Ρ: X U is said to have convex structure if the sets L(u) of the inverse correspondence L are convex for all u ε U. The inverse correspondence L: U X of P is said to have convex structure if the sets P(x) are convex for all χ ε X. The following two propositions (see [15]) characterize the convexity of the structures of the correspondence P (or L) and the graph of P (or L). Proposition 2: The correspondence Ρ: X U, where X is convex, has convex structure if and only if, for all u,v ε U and λ ε [0,1], L(Au + (1 — λ)ν) D L(u) Π L(v). Proposition 2.1: The inverse correspondence L: U -* X of Ρ: X —> U, where U is convex, has convex structure if and only if, for all x,y ε X and λ ε [0,1], Ρ(λχ + (1- A)y) D P(x) Π P(y).

APPENDICES

Proposition 3: The correspondence Ρ: X —> U, where X and U are convex cones, has a convex graph if and only if, for all x,y ε X and λ ε [0,1], P((l - λ)χ + Xy) D ((1 - λ)Ρ(χ) + AP(y)). Proposition 3.1: The inverse correspondence L: U —» X of P, where U and X are convex cones, has a convex graph if and only if, for all u,v ε U and λ ε [0,1], L((l — X)u + λν) D ((1 — A)L(u) + XL(v)).

APPENDIX 3 UTILITY FUNCTIONS The theory developed in the chapters for the production function is readily adaptable to utility functions. Let χ ε {χ | χ 0,χ ε Rn} = 1¾ denote a vector of goods and services for consumption. The components of χ are assumed to cover all goods and services available for consumption. The preferences of the consumer for the vectors χ of R$ are assumed to be completely preordered, i.e., for any two vectors χ and y one of the three following possibilities holds: (a) χ is preferred to y, (b) y is preferred to x, or (c) χ is indiffer­ ent to y. Under these circumstances there exists a correspondence be­ tween the points u of the half real line R+ and the consumption vectors χ ε R£, such that u = 0 corresponds to χ = 0 and, if χ is preferred to y, the value u corresponding to χ is greater than the value ν corresponding to y, and if indifferent u = ν (see [7], 4.6). Hence, letting u ε [0,+ oo), we may suppose that the completely preordered consumer preferences exists with a nonnegative real number assigned to each consumption vector, in such a way that u = 0 for χ = 0, u > u' for χ preferred to x' and u = u' for χ indifferent to x'.f Then let L(u) C be the subset of consumption vectors χ such that the associated nonnegative real number is at least as large as u, i.e., the utility of χ is at least u. Certain assumptions are made for the consump­ tion sets L(u), u ε [Ο,οο). In fact, we may assume that the Properties P.l, ..., P.8 (stated in Section 2.2 for the production input sets) hold. Prop­ erty P.l implies for consumer preferences that all possible consumption vectors (including the null vector) yield at least zero utility. Negative utility has no meaning in this context. Property P.2 is a disposal assump­ tion that, if χ yields at least u and x' ^ x, then x' yields at least u. Con­ cerning Property P.3 (a), if χ is a positive consumption vector, then all utility (satisfaction) levels may be obtained by scalar magnification of x. When χ is semi-positive with zero input for a commodity and positive utility can be obtained for some scalar magnification of x, the zero inputed commodity is not essential and all levels of satisfaction may be obtained by scalar magnification of x, implying indifference with respect to a positive input vector.ff Property P.4 states merely that the set of consumption vectors yield­ ing at least a utility level U2 ^ ui must also yield at least ui. Regarding t This measurement of satisfaction (utility) is not unique. Any strictly increasing trans­ formation of u will serve. tt This assumption is not essential for the arguments to follow.

APPENDICES

Property P.5, clearly L(u0)

C

erty P.4. But also L(uo) •

Π

O^=IKUo

Π

0^u< Uo

L(u) for any u0 > 0, due to Prop-

L(u), because otherwise there exists

a consumption vector χ ε

Π L(u) with χ ¢ L(u χ is indifferent to y Φ(χ) < Φ(γ) => y is preferred to x. Note that the indifference class of a consumption vector χ may have interior points. Also, since only upper semi-continuity is implied for the utihty function Φ(χ), the consumption sets L(u), L(v) for two distinct utility levels u and ν may be equal, i.e., all utility levels need not be as­ signed to a consumption vector. Continuity of the utility function is not essential. With these interpretations the previous results of Chapters 1,...,8 carry over for utility functions. In particular for M > 0, the function (Μ )

Γ

= Sup

< M)' P ε R+

t In [7], Section 4.6, Debreu states that the utility "function would be of little interest if it were not continuous."

APPENDICES

is the Indirect Utility Function, being the supremum utility obtainable with an expenditure less than M at prices p for the commodities of x. If the utility function is homothetic (see Section 2.4), then

where F( •) is a transform with (see Section 2.3) and P(p) is a positively homogeneous function of degree one, nondecreasing, continuous and concave. (See Proposition 24, Section 4.4.) In the simple case where i.e., the utility function is homogeneous of degree one, the indirect utility function

is merely a commodity price deflated expenditure budget. Considering the utility function for some "average family," and two points of time indexed by 0 and 1, an index function of the "real standard of living" is defined by

taking into account both price and income changes. When the utility function is homothetic with i.e., it is homogeneous of degree one,

and, if the price level

at the reference period 0 is taken as 100,

a particularly simple form of the index, and homotheticity of the utility function with is an if and only if condition for this simple form (see the second corollary, Section 8.4). For interpretation of this index function with a fixed level of income, one merely sets and the index function becomes

< 303 >

APPENDICES

in the general homothetic case, with τ

P(Pq) 01

~ P(P1) '

i.e., merely the reciprocal of a price index, when F(