Directional Distance Functions Revisited: Selective Overview and Update

Suggested Citation
R. Färe, S. Grosskopf and D. Margaritis (2015), “Directional Distance Functions Revisited: Selective Overview and Update”, Data Envelopment Analysis Journal: Vol. 1: No. 2, pp 57-79.
Publication Date: 30 Jul 2015
© 2015 R. Färe, S. Grosskopf and D. Margaritis
Theory: Empirical methods of evaluation,  Nonparametric methods,  Optimization,  Operations Research
C61, D24, D92, O33
Directional distance functions, Profit efficiency, Shadow pricing, Productivity indicators, Endogenous directions


In this article:

1. Introduction
2. Directional Distance Functions and their Duals
3. Efficiency Measurement
4. A Parametric Specification
5. A Price Space Application
6. Shadow Pricing with Non-Marketed Goods
7. Productivity and Undesirable Outputs
8. Bennet-Bowley Productivity Indicators
9. Further Issues and Summary


Since its introduction in the performance measurement literature in 1996 by Chambers and coauthors, the directional distance function has proved to be useful and popular: Google Scholar provides over 22,000 citations since 1996. In this paper, we include a selective overview and update of both theoretical underpinnings and various applications, including implementation using DEA, robust estimators as well as parametric forms. We begin with its important dual association with profit, revenue and cost support functions as well as its relationship as a generalization of the familiar Shephard (1970) distance functions. We include the derivation of a profit efficiency measure using the directional distance function. We then turn to applications of directional distance functions in price space and show how the price space technology can be given a functional representation as directional distance functions. These can then be used to assign prices to nonmarket outputs (or bads). A discussion of their role in productivity measurement is included and current and future issues of concern to practitioners such as endogenizing the direction vector concludes.

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