# Theory of Cost and Production Functions

RONALD W. SHEPHARD
Copyright Date: 1970
Pages: 322
https://www.jstor.org/stable/j.ctt13x11vf

## Table of Contents

1. Front Matter
(pp. i-vi)
2. PREFACE
(pp. vii-viii)
Ronald W. Shephard
3. Table of Contents
(pp. ix-2)
4. CHAPTER 1 INTRODUCTION
(pp. 3-12)

In economic theory the production function is a mathematical statement relating quantitatively the purely technological relationship between the output of a process and the inputs of the factors of production, 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...

5. CHAPTER 2 THE PRODUCTION FUNCTION
(pp. 13-63)

A production technology consists of certain alternative means, arrangements 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 production, 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...

6. CHAPTER 3 THE DISTANCE FUNCTION OF A PRODUCTION STRUCTURE
(pp. 64-78)

In economics the production function Φ(X) 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(X) = (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 output rates u for all nonnegative input vectors...

7. CHAPTER 4 THE FACTOR MINIMAL COST FUNCTION
(pp. 79-95)

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 p = (p₁, p₂, ...,pn)· The price vector p defines a point in the nonnegative domain D of Rnand the coordinate systems of input vectors X and price vectors p are superimposed in Rn.

The cost per unit time of an input vector X is denoted...

8. CHAPTER 5 THE COST STRUCTURE
(pp. 96-113)

We note that the cost function Q(u,p) has properties which are similar to those of the distance function Ψ(u,x) (compare Propositions 16 and 22). Those properties of the function Ψ(u,x) which essentially characterize it as a distance function are its homogeneity, super-additivity and concavity in x and these same properties are possessed by the cost function 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 vector p in the nonnegative domain D.

Hence, in order to proceed carefully along these lines, a...

9. CHAPTER 6 THE AGGREGATION PROBLEM FOR COST AND PRODUCTION FUNCTIONS
(pp. 114-146)

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 production, and producers goods and consumers goods for the outputs of production 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 relationships and that the...

10. CHAPTER 7 THE PRICE MINIMAL COST FUNCTION
(pp. 147-158)

To review the framework in which the price minimal cost function is to be defined, we consider a production structure of production possibility sets LΦ(u) with production function Φ(x). It is assumed that the subsets${L_\phi }\left( u \right),u\underline{\underline > } 0$, of the nonnegative domain D of a Euclidian space${R^n}$, have the Properties P.1, . . . , P.9 (see Definition, Section 2.1) and the efficient subsets EΦ(u) are bounded. Independently of boundedness of the efficient subsets E(u), it was shown in Section 2.1, Chapter 2, that${L_\phi }\left( u \right) = \overline E \left( u \right) + D$for all$u\underline{\underline > } 0$. The distance function of the production structure is (see Section 3.1)

a a...

11. CHAPTER 8 DUALITY OF COST AND PRODUCTION STRUCTURES AND RELATED FUNCTIONS
(pp. 159-177)

For any production structure with production possibility sets${L_\phi }\left( u \right),u\varepsilon \left[ {0,} \right.\left. \infty \right)$, having the Properties P.1, . . . , P.9, there is a distance function Ψ(u, x) defined on the sets Lϕ(u). The production structure is specified in terms of the distance function Ψ(u, x) by

${L_\phi }\left( u \right) = \left\{ {X\left| {\psi \left( {u,x} \right)\underline{\underline > } 1,x\underline{\underline > } } \right.0} \right\},u\varepsilon \left[ {0,\left. \infty \right)} \right..$

The cost structure dual to the production structure is a family of sets${L_Q}\left( u \right),u\varepsilon \left[ {0,\left. \infty \right)} \right.$, in the nonnegative price domain defined in terms of the factor minimal cost function Q(u,p) by

${L_Q}\left( u \right) = \left\{ {P\left| {Q\left( {u,p} \right)} \right.\underline{\underline > } 1,p\underline{\underline > } 0} \right\},u\varepsilon \left[ {0,\left. \infty \right)} \right.$

and the cost function Q(u,p) is a distance function for this structure.

The two functions Q(u,p) and Ψ(u,p) are dualistically determined from each other...

12. CHAPTER 9 PRODUCTION CORRESPONDENCES
(pp. 178-222)

Here we are concerned with technologies which yield several different 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...

13. CHAPTER 10 COST AND BENEFIT (REVENUE) FUNCTIONS FOR PRODUCTION CORRESPONDENCES, AND THE RELATED COST, BENEFIT (REVENUE), COSTLIMITED-OUTPUT AND BENEFIT (REVENUE)-AFFORDED-INPUT CORRESPONDENCES
(pp. 223-260)

By proofs which exactly parallel those given in Chapter 2, Section 2.1, it may be verified that the efficient subsets${E_L}(u) = \left\{ {X\left| X \right.} \right.\varepsilon L\left( u \right),y\varepsilon L(u)$if$y \le \left. x \right\}$, of the input sets of a production correspondence$P:X \to U$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${L_{\left( u \right)}} = {E_L}\left( u \right) + X = \overline {{E_L}\left( u \right)} + X$, where$\overline {{E_L}\left( u \right)}$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 mot all of...

14. CHAPTER 11 DUALITIES FOR PRODUCTION CORRESPONDENCES
(pp. 261-292)

Analogous to the price minimal cost function Ψ*(u,x) defined on the cost structure LQ(u) of the production function Φ(x), we define the following function.

Definition:${\psi ^*}\left( {u,x} \right) = \mathop {\inf }\limits_P \left\{ P \right.x\left| {P\varepsilon L\left. {(u)} \right\}} \right.,u\varepsilon U,x\varepsilon X.$

Infimum rather than minimum is used, because the efficient subsets of the price sets L(u) of the cost structure correspondence$R_ + ^m \to X = R_ + ^n$are not necessarily bounded, even though we assume the efficient subsets of the input sets L(u) of$P:X \to U$are bounded for all$u\varepsilon U = R_ + ^m.$

The properties of the price minimal cost function Ψ*(u, x) are given by the following proposition.

Proposition 85:The price minimal cost function Ψ*(u,x) for the cost structure$L:U \to X$of...

15. References
(pp. 292-294)
16. APPENDIX 1. MATHEMATICAL CONCEPTS AND THEOREMS FOR SEMI-CONTINUITY AND QUASI-CONCAVITY (CONVEXITY)
(pp. 295-297)
17. APPENDIX 2. MATHEMATICAL CONCEPTS AND PROPOSITIONS FOR CORRESPONDENCES
(pp. 298-300)
18. APPENDIX 3. UTILITY FUNCTIONS
(pp. 301-305)
19. INDEX
(pp. 306-308)