The Directional Distance Function (DDF): Economic Inefficiency Decompositions

The birth of the directional distance function as an inefficiency measure was linked to the consumer theory work developed by Luenberger in the early 1900s. Luenberger (1992a) introduced the concept of the benefit function in consumer theory in order to develop group welfare relations and, particularly, considered the Shephard’s input distance function, claiming that it would be useful in developing relations between individuals. Chambers et al. (1996) redefined Luenberger’s benefit function as an inefficiency measure and called it the directional input distance function. Although mathematically related, the last two distance functions are conceptual opposites: while Shephard’s distance function is multiplicative by nature, the directional distance function is additive by nature. The same happens with the Shephard’s output distance function and the directional output distance function, first studied by Chung (1996) in his Ph.D. dissertation, where he also considered undesirable outputs, which have experienced since then a great diffusion, boosting the analysis of environmental issues. Luenberger (1992b), transposing the benefit function into a production context, defined the so-called shortage function, which basically measures the distance, in the direction of a vector g, from a production plan toward the boundary of the production possibility set. However, Luenberger interprets the distance as a shortage of the production plan to reach the frontier of T, while Chambers et al.’s (1998) interpretation was that of an inefficiency measure, giving rise to the general definition of a directional distance function, DDF, and analyzing its properties., In the aforementioned paper, they considered three specific DDFs: the two mentioned oriented distance functions and the graph inefficiency measure of Briec (1997) presently known as the proportional DDF (see Boussemart et al., 2003).