The Generalized Distance Function (GDF): Profitability Efficiency Decomposition

This chapter is concerned with the measurement of profitability efficiency, defined as the ratio of revenue to cost, and its multiplicative decomposition into a productive efficiency measure—including technical and scale efficiencies, corresponding to the generalized distance function introduced by Chavas and Cox (1999), and allocative efficiency. The generalized distance function, GDF, received such name by these authors because it generalizes Shepard’s radial distance functions and the graph (hyperbolic) efficiency measure introduced by Färe et al. (1985:110–111). Building upon this measure, which can be reinterpreted in terms of a distance function, these authors extended the input- and output-oriented measures to a graph representation of the technology including both dimensions of the production technology. In contrast to the partial dimensions represented by input and output orientations, the hyperbolic technical efficiency measure, presented in Sect 2.1.3 of Chap. 2 , is a scalar value function that projects the firm under evaluation to the production frontier by simultaneously reducing its inputs and increasing its outputs. As we show below, Chavas and Cox (1999) qualified this definition by making these changes dependent on an exponent that weights the outputs and inputs differently. Therefore, setting the value of such bearing (or directional) parameter to a specific value, it is possible to recover, among others, the hyperbolic efficiency measure as well as Farrell’s input and output radial counterparts. Since the latter corresponds to Shepard’s input and output distance functions, as shown in Chap. 3 , the generalized distance function represents an improvement over the previous definitions, by adding flexibility to the orientation and as we show below providing a dual counterpart to the profitability function.