Directional distance functions: Optimal endogenous directions

A substantial literature has dealt with the problem of estimating multiple-input and multiple-output production functions, where inputs and outputs can be good and bad. Numerous studies can be found in the areas of productivity analysis, industrial organization, labor economics, and health economics. While many papers have estimated the more restrictive output- and input-oriented distance functions, here we estimate a more general directional distance function. A seminal paper on directional distance functions by Chambers (1998) as well as papers by Fare et al. (1997), Chambers et al. (1998), Fare and Grosskopf (2000), Grosskopf (2003), Fare et al. (2005), and Hudgins and Primont (2007) do not address the issue of how to choose an optimal direction set. Typically the direction is arbitrarily selected to be 1 for good outputs and −1 for inputs and bad outputs. By estimating the directional distance function together with the first-order conditions for cost minimization and profit maximization using Bayesian methods, we are able to estimate optimal firm-specific directions for each input and output which are consistent with allocative and technical efficiency. We apply these methods to an electric-utility panel data set, which contains firm-specific prices and quantities of good inputs and outputs as well as the quantities of bad inputs and outputs. Estimated firm-specific directions for each input and output are quite different from those normally assumed in the literature. The computed firm-specific technical efficiency, technical change, and productivity change based on estimated optimal directions are substantially higher than those calculated using fixed directions.