Measuring the Technical Efficiency of Multiple Output Production Technologies

Abstract

In this paper we consider a production unit employing many inputs to produce many outputs, subject to the constraints imposed by given technology. Our purpose is to present and analyze various measures of the effectiveness with which inputs are transformed into outputs. Early efforts in this direction were made by Koopmans [1951] and Debreu [1951]. Koopmans defined a feasible input-output vector to be efficient if it is technologically impossible to increase any output and/or to reduce any input without simultaneously reducing other outputs and/or increasing other inputs. Using this definition, he was able to prove that a vector is efficient if, and only if, it possesses a positive normal to the production possibilities set. While Koopmans offered a definition and a characterization of efficiency, Debreu provided a measure of efficiency with his “coefficient of resource utilization.” This coefficient is computed as one minus the maximum equiproportionate reduction in all inputs consistent with continued production of existing outputs, and from it Debreu obtained a measure of the cost of inefficiency. More recently Vincze [1960] and Eichhorn [1972, 1978a, 1978b] have considered a production process involving many variable inputs, one fixed input, many outputs and a time dimension. In this context he defined the price-dependent notions of technical and economic “effectiveness” in terms of families of functional equations, and derived closed-form parametric indexes of technical and economic effectiveness as solutions to the respective families of functional equations.

We are indebted to W. Erwin Diewert and Wolfgang Eichhorn for their helpful comments.