Revisiting the Decomposition of Cost Efficiency for Non-homothetic Technologies: A Directional Distance Function Approach

In the early 1980’s Kopp and Diewert proposed the decomposition of cost efficiency into allocative and technical efficiency for parametric functional forms based on the radial approach initiated by Farrell. We show that such approach is only valid for homothetic technologies where the radial distance function can be correctly interpreted as a technical efficiency measure since allocative efficiency is independent of the output level and radial input reductions leave it unchanged. For the general case of non-homothetic technologies where optimal input demands depend on the output targeted by the firm, as does the inequality between marginal rates of substitution and market prices allocative inefficiency, we show that the correct definition of technical efficiency corresponds to the directional distance function. Indeed, its flexibility ensures that allocative efficiency is kept unchanged through movements in the input production possibility set when solving technical inefficiency, and therefore the associated cost reductions can be solely and rightly ascribed to technical engineering improvements. The new methodology to consistently decompose cost inefficiency, which generalizes the approach introduced by Kopp and Diewert as well as subsequent refinements, is illustrated resorting to simple examples of both homothetic and non-homothetic production functions.