The Reverse Directional Distance Function (RDDF): Economic Inefficiency Decompositions

The reverse directional distance function, shortly RDDF, is a relatively recent concept introduced by Pastor et al. (2016). It is an apparently simple idea and, at the same time, a fruitful one. Let us start considering an efficiency measure (EM) and a finite sample of firms to be analyzed, F_J. As we have shown throughout most of the previous chapters, by solving a mathematical program for each firm of the sample, we obtain two relevant outcomes: the maximum of its objective function, which usually corresponds to the technical (in)efficiency associated with the mentioned firm, and a point that belongs to the technological frontier and corresponds to the technical-efficient projection of the firm. Consequently, the efficiency measure (EM) we are going to consider assigns to each firm being rated an inefficiency score as well as a single projection. These are the two conditions required for introducing the RDDF. Let us denote as EM^S any efficiency measure that generates a single inefficiency projection together with its inefficiency score for each inefficient firm—denoted by superscript S. This condition is satisfied by all the additive measures presented in Part II of the book, except for the modified DDF-related efficiency measures introduced by Aparicio et al. (2013a, b). The modified DDF assigns two different scores to the input and output dimensions: βx∗ and βy∗, respectively—see expression (11.5 ) of the previous chapter. Given a directional vector g = (g_x, g_y), observed outputs are increased by the amount βy∗gy, while observed inputs are reduced by the amount βx∗gx. Moreover, several multiplicative measures presented in Part I fail to satisfy this condition, calculating instead two inefficiency scores, one related to inputs and the other one to outputs. The most prominent example is the hyperbolic (graph) measure (Färe et al., 1985), which calculates an efficiency score φ^∗ measuring the proportion by which inputs can be reduced and outputs increased—the latter is numerically the inverse of the former assuming CRS and represents a particular case of the generalized distance function introduced by Chavas and Cox (1999) (see expression (4.4 ) in Chap. 4 ).