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Theory of Cost and Production Functions
By Ronald William Shepherd PRINCETON UNIVERSITY PRESS
Copyright © 1970 Princeton University Press
All rights reserved.
ISBN: 978-0-691-04198-8
CHAPTER 1
INTRODUCTION†
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 or less traditional treatments of the production function exclude free goods as inputs and require that the function express the variable, substitutional and limitational character or other qualifications of the factors of production peculiar to some hypothetical production unit. Sune Carlson 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 distinguishes between "substitutional" and "limitational" factors and expresses the production function in terms of substitutional inputs with side equations between output and each limitational factor. Samuelson 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." All 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. Krelle makes this distinction 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 puts forth the notion of a "stock-flow production function" in which capital stock inputs are freely variable along with current input flows for treating hypothetical alternative production plans, but once the physical configuration 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 distinguish between "short run" and "long run" production functions, the form of the function being essentially different in the two cases. But the production function is ideally a statement of purely technological alternatives, 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) character 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 construction for some well defined production technology. This technology consists of a family of conceivable and feasible engineering arrangements, 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 possibilities 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 production 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 technology which may be considered by imposing constraints on the input flows which restrict the analysis to a particular subset of the factor input space. For example, if the production unit uses only certain machinery 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 factors sought in economics will arise for a broadly defined technology not constrained to particular realizations, which is the kind of production 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 vectors which yield at least the given output rate. On this structure the production 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 existence 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 prices of the factors of production. To pursue this matter, a class of production functions was defined in the first Princeton monograph and named homothetic. There it was shown that the cost function factors 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 and Uzawa's extension of this function, 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 linear 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 homotheticity will be given, with a discussion of the properties of the corresponding production 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 production 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 introduced as an alternative to the production function. The properties of this function are determined and the special form of the distance function 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 price-output cost structure and the duality between cost and production function is naturally formulated in terms of these distance 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 output rate u and vector p of the prices of the factors of production, with the input rates of the factors of production adjusted to yield minimum total cost. This function is called the "factor minimal" cost function 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 function is defined on the cost structure, which provides for any nonnegative price vector of the factors of production the supremal output 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 forth for aggregating 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 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 establishing homotheticity as an if and only if property for the factorization of the cost function into a function of output rate and a linear homogeneous 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 correspondence P is introduced for joint outputs by treating the production relationship 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.
(Continues...)
Excerpted from Theory of Cost and Production Functions by Ronald William Shepherd. Copyright © 1970 Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS.
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