The Correspondence of Efficiency Frontier as a Generalization of the Cost Function

Abstract

As THE TITLE SUGGESTS, this paper generalizes the concept of a cost function, first developed by Shephard [5], refined and extended by Uzawa [10], McFadden f 3], Diewart [1] and Hanoch [2]. The nature of the generalization is introduced -below in a somewhat general context. Empirical analyses of behavioral functions, such as product supply or factor ,demand, based on cross-section data often lead to the conclusion that prices are relatively unimportant explanatory variables. Such a result is simply an indication of the fact that there is a wide spread in the output and input composition of different firms operating under similar economic conditions. This spread can be attributed to unobserved random variables which, among other things, indicate that producers fail to locate their optimal points. Such an explanation is unsatisfactory and leaves room for a broader framework which can account for the empirical evidence. An attempt in this direction is undertaken in this paper. The starting point is the assumption that the utility function of the firm is a function of several arguments, rather than of profit alone (cf. [4]). From this assumption it immediately follows that differences among firms may result from ,differences in the components that enter the utility functions and the weights given to them. Technically, the problem can be stated as follows: Let u(c) be a quasi concave utility function monotonic in the components of the m-vector c. Let x be a k-vector of outputs and inputs that come from a production set X possessing some desirable properties. Let P be a m X k matrix (m < k) of coefficients to be referred to as prices. The problem of maximization is: