Journal of Productivity Analysis

, Volume 40, Issue 3, pp 257–266 | Cite as

The directional profit efficiency measure: on why profit inefficiency is either technical or allocative

  • Jose L. Zofio
  • Jesus T. Pastor
  • Juan Aparicio


The directional distance function encompasses Shephard’s input and output distance functions and also allows nonradial projections of the assessed firm onto the frontier of the technology in a preassigned direction. However, the criteria underlying the choice of its associated directional vector are numerous. When market prices are observed and firms have a profit maximizing behavior, it seems natural to choose as the directional vector that projecting inefficient firms towards profit maximizing benchmarks. Based on that choice of directional vector, we introduce the directional profit efficiency measure and show that, in this general setting, profit inefficiency can be categorized as either technical, for firms situated within the interior of the technology, or allocative, for firms lying on the frontier. We implement and illustrate the analytical model by way of Data Envelopment Analysis techniques, and introduce the necessary optimization programs for profit inefficiency measurement.


Directional distance function Profit efficiency Technical efficiency Allocative efficiency 

JEL Classification

C61 D21 D24 



We are grateful to the participants and discussants at the International Workshop on Efficiency and Productivity in honor of Prof. Knox Lovell (October 4–5, 2010, Elche, Spain). We acknowledge financial support from the Ministerio de Ciencia e Innovacion, Spain, and the Conselleria de Educacion, Generalitat Valenciana, for supporting this research with grants MTM2009-10479 and ACOMP/2011/115, respectively. Finally, we are also indebted to two referees for their comments and suggestions that have contributed to improving this paper.


  1. Aparicio J, Ruiz JL, Sirvent I (2007) Closest targets and minimum distance to the Pareto-efficient frontier in DEA. J Prod Anal 28:209–218CrossRefGoogle Scholar
  2. Asmild M, Paradi JC, Reese DN, Tam F (2007) Measuring overall efficiency and effectiveness using DEA. Eur J Oper Res 178:305–321CrossRefGoogle Scholar
  3. Ball E, Färe R, Grosskopf S, Nehring R (2001) Productivity of the U.S. agricultural sector: the case of undesirable outputs. In: Hulten CR, Dean ER, Harper MJ (eds) New developments in productivity analysis. University of Chicago Press, Chicago, pp 541–586CrossRefGoogle Scholar
  4. Banker RD, Maindiratta A (1988) Nonparametric analysis of technical and allocative efficiencies in production. Econometrica 56(6):1315–1332CrossRefGoogle Scholar
  5. Bogetoft P, Färe R, Obel B (2006) Allocative efficiency of technically inefficient production units. Eur J Oper Res 168:450–462CrossRefGoogle Scholar
  6. Chambers R (1998) Input and output indicators. In: Färe R, Grosskopf S, Russell RR (eds) Index numbers: essays in honour of Sten Malmquist. Kluwer Academic Publishers, Boston, pp 241–271CrossRefGoogle Scholar
  7. Chambers R, Chung Y, Färe R (1996) Benefit and distance functions. J Econom Theory 70:407–419CrossRefGoogle Scholar
  8. Chambers R, Chung Y, Färe R (1998) Profit, directional distance functions and Nerlovian efficiency. J Optim Theory Appl 95(2):351–364CrossRefGoogle Scholar
  9. Charnes A, Cooper WW, Golany B, Seiford L (1985) Foundations of data envelopment analysis for Pareto-Koopmans efficient empirical production functions. J Econom 30:91–107CrossRefGoogle Scholar
  10. Chavas J-P, Cox TM (1999) A generalized distance function and the analysis of production efficiency. South Econ J 66(2):295–318CrossRefGoogle Scholar
  11. Cooper WW, Park KS, Pastor JT (1999) RAM: a range adjusted measure of inefficiency for use with additive models, and relations to other models and measures in DEA. J Prod Anal 11:5–42CrossRefGoogle Scholar
  12. Cooper WW, Pastor JT, Aparicio J, Borras F (2011) Decomposing profit inefficiency in DEA through the weighted additive model. Eur J Oper Res 212:411–416CrossRefGoogle Scholar
  13. Cyert RM, March JG (1963) A behavioral theory of the firm. Prentice Hall, Englewood Cliffs, NJGoogle Scholar
  14. Färe R, Grosskopf S (2000a) Notes on some inequalities in economics. Econ Theor 15(1):227–233CrossRefGoogle Scholar
  15. Färe R, Grosskopf S (2000b) Theory and applications of directional distance functions. J Prod Anal 13(2):93–103CrossRefGoogle Scholar
  16. Färe R, Primont D (1995) Multi-output production and duality: theory and applications. Kluwer Academic Publishers, DordrechtCrossRefGoogle Scholar
  17. Färe R, Grosskopf S, Whittaker G (2007) Network DEA. In: Zhu J, Cook WD (eds) Modeling data irregularities and structural complexities in data envelopment analysis, chapter 12. Springer, BerlinGoogle Scholar
  18. Farrell M (1957) The measurement of productive efficiency. J R Stat Soc Ser A 120(3):253–281CrossRefGoogle Scholar
  19. Leibenstein H (1966) Allocative efficiency vs. X-efficiency. Am Econ Rev 56:392–415Google Scholar
  20. Luenberger DG (1992) New optimality principles for economic efficiency and equilibrium. J Optim Theory Appl 75:221–264CrossRefGoogle Scholar
  21. Portela M, Thanassoulis E (2005) Profitability of a sample of Portuguese bank branches and its decomposition into technical and allocative components. Eur J Oper Res 162(3):850–866CrossRefGoogle Scholar
  22. Portela M, Thanassoulis E (2007) Developing a decomposable measure of profit efficiency using DEA. J Oper Res Soc 58:481–490CrossRefGoogle Scholar
  23. Portela M, Borges PC, Thanassoulis E (2003) Finding closest targets in non-oriented DEA models: the case of convex and non-convex technologies. J Prod Anal 19:251–269CrossRefGoogle Scholar
  24. Ruiz JL, Sirvent I (2011) A DEA approach to derive individual lower and upper bounds for the technical and allocative components of the overall profit efficiency. J Oper Res Soc 62(11):1907–1916CrossRefGoogle Scholar
  25. Shephard R (1970) Theory of cost and production functions. Princeton University Press, New JerseyGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Jose L. Zofio
    • 1
  • Jesus T. Pastor
    • 2
  • Juan Aparicio
    • 2
  1. 1.Departamento de Analisis Economico: Teoria Economica e Historia EconomicaUniversidad Autonoma de MadridMadridSpain
  2. 2.Center of Operations Research (CIO)Universidad Miguel Hernandez de ElcheElcheSpain

Personalised recommendations