Testing Efficiency Hypotheses in Joint Production: A Parametric Approach

Abstract

N recent years a great deal of research has been directed to the modelling and measurement of technical and allocative efficiency in production. With few exceptions this research has been restricted to single-product firms.' However, recent developments in duality theory have facilitated the extension of this research to multi-product firms. The main purpose of this paper is to develop a model of the multiproduct firm in which the possibilities of both technical and allocative inefficiency are incorporated in an econometrically useful way. The first model we develop includes a nonneutral2 type of technical inefficiency and three distinguishable types of allocative inefficiency-output mix, input mix, and scale. Each type of inefficiency is costly to the firm, in the sense that each causes a reduction in profit beneath the maximum value attainable under full efficiency. The cost of each type of inefficiency depends on the magnitude of the inefficiency and the structure of the underlying production technology. In the second model we develop, technical inefficiency remains nonneutral, but allocative inefficiency is not generally decomposable into output mix, input mix and scale components. However, both technical and allocative inefficiency remain costly to the firm, the cost of each type of inefficiency depending on its magnitude and the structure of the underlying production technology. We model the technology of a competitive profit maximizing multi-product firm with the dual profit function. This enables us to use Hotelling's Lemma to generate a system of profit maximizing output supply and input demand equations. These equations are then modified to allow for the possibility of technical and three types of allocative inefficiency. A virtue of using the profit function to represent production technology is that it permits a straightforward comparison of maximum profit under full efficiency with actual profit, and with the profit that would result from any combination of the four types of inefficiency. This enables us to allocate the cost of inefficiency to each of four components. Our model of inefficiency is parametric, and is embedded in a Generalized Leontief profit function, although any flexible specification of the profit function can be used. The model is developed in sections II-IV. Estimation of the model is considered in section V. An empirical example designed to illustrate the workings of the model is discussed in section VI. Section VII concludes.