Dual Representations of Non-Parametric Technologies and Measurement of Technical Efficiency

This paper extends the recent work by Frei and Harker on projections onto efficient frontiers (1999) in three ways. First, we provide a formal definition of the production set as the intersection of a finite number of closed halfspaces. We emphasize the necessity of a complete enumeration of the supporting hyperplanes to define the production set properly. We focus on the problem of exhaustive enumeration of the supporting hyperplanes to characterize the production set. Second, we consider the problem of an arbitrary-norm projection on the boundary of the production set. We use the concept of the Hölder distance function and we derive the necessary mathematics to calculate distances and projections of inefficient DMUs onto the efficient frontier. Third, we introduce a relevant weighting scheme for inputs and outputs so that the Hölder distance function respects the commensurability axiom defined by Russell (1988). Finally, we present an illustration using the same data set as Frei and Harker (1999) to highlight some of the extensions proposed in the paper.