On the existence of joint production functions

Abstract

Within a general framework of production correspondences satisfying a set of weak axioms necessary and sufficient conditions for the existence of a joint production function are given. Without enforcing the strong disposability of inputs or outputs it is shown that a joint production function exists if and only if both input and output correspondences are strictly increasing along rays.

Revisiting Transformation and Directional Technology Distance Functions

“…This definition entails arbitrarily choosing an output with respect to which maximization is performed given a vector of inputs and of the remaining outputs (Diewert, 1973, p. 286). Shephard (1970), Bol and Moeschlin (1975), Al-Ayat and Färe (1979), and Färe et al (1985) study the concept of an isoquant joint production function (JPF) and investigate conditions sufficient for its existence. This function assumes the value zero at the output-input bundles belonging to the isoquant of the production possibilities set whenever this function exists; see Section 2.1 and Färe et al (1985, pp.…”

On Inverse Homotheticity

“…In the multi-output case these two concepts are distinct. Equivalent to linear (input) expansion paths is ray homotheticity of the input correspondence (Fare and Shephard, 1977), and equivalent to cost-function separability is input homotheticity (Fare and Mitchell, 1993). Moreover in the multi-output case one may define linear output expansion paths (Fare and Shephard, 1977) and separability of the revenue function (Shephard, 1970).…”

“…These functions are complete characterizations of technology (Shephard, 1970;Fare, 1988), and here we assume that they meet properties that suffice for the following dualities' to hold (Fare and Primont, 199 5 ):…”