Profit Functions of Technologies with Multiple Inputs and Outputs

PpTHE application of duality in economic analysis was initiated by Hotelling (1932) and Roy (1942) in the area of consumer demand. Subsequently, Shephard (1953) in his pioneering work on cost and production functions extended the duality concepts to the theory of production and derived many of the fundamental results, including the basic duality theorems and Shephard's (1953) Lemma, which asserts that the gradient of the dual function is equal to the supply and demand correspondences. More recently McFadden (1972) generalized the duality concepts in production theory to include profit and revenue functions. The profit function is a function of the output and input prices which gives the value of the maximized profit of a profit-maximizing and pricetaking firm endowed with a given technology. Because of Shephard's (1953) Lemma, the partial derivatives of the profit function with respect to the output and input prices give the supply and demand functions. Thus, the econometric analysis of the behavior and technology of the profit-maximizing and price-taking firms is greatly facilitated.' In this paper we are concerned with the properties of the profit functions of technologies with multiple inputs and outputs. Several theorems which relate the properties of the transformation functions to the properties of the profit functions and vice versa are proved. The specific properties considered are: (1) homogeneity, (2) separability, and (3) nonjointness. The properties of the profit function have been studied by McFadden (1972), Diewert (1969), and Jorgenson, Christensen and Lau (1970), among others. In addition, Hall (1972) has approached the problem of multi-output, multi-input firms from the point of view of joint cost functions.