Theory and application of directional distance functions

In 1957 Farrell demonstrated how cost inefficiency could be decomposed into two mutually exclusive and exhaustive components: technical and allocative inefficiency. This result is consequence of the fact that-as shown by Shep hard-the cost function and the input distance function (the reciprocal of F arrell's technical efficiency measure) are 'dual' to each other. Similarly, the revenue function and the output distance function are dual providing t he basis for the decomposition of revenue inefficiency into technical and a llocative components (see for example, Fare, Grosskopf and Lovell (1994)). Here we extend those results to include the directional distance function a nd its dual, the profit function. This provides the basis for defining and decomposing profit efficiency. As we show, the output and input distance fu nctions (reciprocals of Farrell efficiency measures) are special cases of t he directional distance function. We also show how to use the directional d istance function as a tool for measuring capacity utilization using DEA typ e techniques.