On the Existence and Equivalence of Three Joint Production Functions

Abstract

Production functions play an important role in the study of production technologies. In the case of a single output technology, the production function is defined as the maximal output obtainable from a given input vector; cf. Shephard (1970). No simple maximal output exists for a multi-output technology, so that the definition and existence of a multioutput production function has become an interesting issue. Two different multi-output production functions or, as they are also called, joint production functions, can be found in the literature. First, Shephard (1970) defined an isoquant joint production function in the manner that for a given pair of input and output vectors, the input vector belongs to the isoquant of the input correspondence and the output vector belongs to the isoquant of the output correspondence. Second, Hanoch (1970) introduced an efficient joint production function for which input and output vectors are simultaneously input and output efficient. A third joint production function which relates weakly efficient input vectors to weakly efficient output vectors is introduced in this paper. This third class of joint production functions is intermediate to the two mentioned above. In general, none of the above three classes of joint production functions exist. Thus one of the purposes of this paper is to show the conditions under which they do exist. A second purpose is to show the conditions on the technology under which these joint production functions are equivalent.