Journal of Productivity Analysis

, Volume 38, Issue 2, pp 109–120 | Cite as

Families of linear efficiency programs based on Debreu’s loss function

  • Jesus T. Pastor
  • C. A. Knox Lovell
  • Juan Aparicio


Gerard Debreu introduced a well known radial efficiency measure which he called a “coefficient of resource utilization.” He derived this scalar from a much less well known “dead loss” function that characterizes the monetary value sacrificed to inefficiency, and which is to be minimized subject to a normalization condition. We use Debreu’s loss function, together with a variety of normalization conditions, to generate several popular families of linear efficiency programs. Our methodology also can be employed to generate entirely new families of linear efficiency programs.


Loss function Linear efficiency programs DEA 

JEL Classification

C51 C61 



We thank two anonymous referees for providing constructive comments and help in improving the contents and the presentation of this paper. Also, we are grateful to Professor Prasada Rao, Director of the CEPA at the University of Queensland, for his hospitality and to Ministerio de Ciencia e Innovacion, Spain for supporting this research with grant MTM2009-10479.


  1. Ali AI, Seiford LM (1993) The mathematical programming approach to efficiency analysis, chapter 3. In: Fried HO, Lovell CAK, Schmidt SS (eds) The measurement of productive efficiency. Oxford University Press, New YorkGoogle Scholar
  2. Asmild M, Pastor JT (2010) Slack free MEA and RDM with comprehensive efficiency measures. OMEGA 38(6):475–483CrossRefGoogle Scholar
  3. Banker RD, Cooper WW (1994) Validation and generalization of DEA and its uses. TOP 2:249–314CrossRefGoogle Scholar
  4. Banker RD, Charnes A, Cooper WW (1984) Some models for estimating technical and scale inefficiencies in data envelopment analysis. Manage Sci 30(9):1078–1092CrossRefGoogle Scholar
  5. Bardhan I, Bowlin WF, Cooper WW, Sueyoshi T (1996) Models and measures for efficiency dominance in DEA, parts I and II. J Oper Res Soc Jpn 39(3):322–344Google Scholar
  6. Bogetoft P, Hougaard JL (1999) Efficiency evaluations based on potential (non-proportional) improvements. J Prod Anal 12(3):233–247CrossRefGoogle Scholar
  7. Briec W (1997) A graph-type extension of Farrell technical efficiency measure. J Prod Anal 8(1):95–110CrossRefGoogle Scholar
  8. Briec W (1999) Hölder distance function and measurement of technical efficiency. J Prod Anal 11(2):111–131CrossRefGoogle Scholar
  9. Chambers RG, Chung Y, Färe R (1996) Benefit and distance functions. J Econ Theory 70(2):407–419CrossRefGoogle Scholar
  10. Chambers RG, Chung Y, Färe R (1998) Profit, directional distance functions, and Nerlovian efficiency. J Optim Theory Appl 98(2):351–364CrossRefGoogle Scholar
  11. Charnes A, Cooper WW, Rhodes E (1978) Measuring the efficiency of decision-making units. Eur J Oper Res 2(6):429–444CrossRefGoogle Scholar
  12. Charnes A, Cooper WW, Golany B, Seiford L, Stutz J (1985) Foundations of data envelopment analysis for Pareto-Koopmans efficient empirical production functions. J Econom 30(1–2):91–107CrossRefGoogle Scholar
  13. Charnes A, Cooper WW, Rousseau J, Semple J (1987) Data envelopment analysis and axiomatic notions of efficiency and reference sets. Research Report CCS558, Center for Cybernetic Studies, University of Texas, Austin TX, USAGoogle Scholar
  14. 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(1):5–42CrossRefGoogle Scholar
  15. Cooper WW, Pastor JT, Borras F, Aparicio J, Pastor D (2011) BAM: a bounded adjusted measure of efficiency for use with bounded additive models. J Prod Anal 35:85–94CrossRefGoogle Scholar
  16. Debreu G (1951) The coefficient of resource utilization. Econometrica 19(3):273–292CrossRefGoogle Scholar
  17. Diewert WE (1983) The measurement of waste within the production sector of an open economy. Scand J Econ 85(2):159–179CrossRefGoogle Scholar
  18. Färe R, Lovell CAK (1978) Measuring the technical efficiency of production. J Econ Theory 19(1):150–162CrossRefGoogle Scholar
  19. Färe R, Primont D (1995) Multi-output production and duality: theory and applications. Kluwer Academic Publishers, BostonCrossRefGoogle Scholar
  20. Färe R, Grosskopf S, Lovell CAK (1985) The measurement of efficiency of production. Kluwer-Nijhoff Publishing, BostonGoogle Scholar
  21. Farrell MJ (1957) The measurement of productive efficiency. J R Stat Soc Ser A General 120(3):253–282CrossRefGoogle Scholar
  22. Koopmans TC (1951) Analysis of production as an efficient combination of activities. In: Koopmans TC (ed) Activity analysis of production and allocation. Cowles Commission for Research in Economics Monograph No. 13. Wiley, New YorkGoogle Scholar
  23. Lovell CAK, Pastor JT (1995) Units invariant and translation invariant DEA models. Oper Res Lett 18(3):147–151CrossRefGoogle Scholar
  24. Luenberger DG (1992a) Benefit functions and duality. J Math Econ 21(5):461–481CrossRefGoogle Scholar
  25. Luenberger DG (1992b) New optimality principles for economic efficiency and equilibrium. J Optim Theory Appl 75(2):221–264CrossRefGoogle Scholar
  26. Pastor JT, Aparicio J (2010) Distance functions and efficiency measurement. Indian Econ Rev (forthcoming)Google Scholar
  27. Pastor JT, Ruiz JL, Sirvent I (1999) An enhanced DEA Russell-graph efficiency measure. Eur J Oper Res 115(3):187–198Google Scholar
  28. Portela MCAS, Thanassoulis E (2006) Zero weights and non-zero slacks: different solutions to the same problem. Ann Oper Res 145:129–147CrossRefGoogle Scholar
  29. Ray SC (2007) Shadow profit maximization and a measure of overall inefficiency. J Prod Anal 27(3):231–236CrossRefGoogle Scholar
  30. Silva Portela MCA, Thanassoulis E, Simpson G (2004) Negative data in DEA: a directional distance approach applied to bank branches. J Oper Res Soc 55(10):1111–1121CrossRefGoogle Scholar
  31. Ten Raa T (2008) Debreu’s coefficient of resource utilization, the solow residual, and FTP: the connection by Leontief preferences. J Prod Anal 30(3):191–199CrossRefGoogle Scholar
  32. Thompson RG, Singleton F, Thrall R, Smith B (1986) Comparative site evaluations for locating a high-energy physics lab in Texas. Interfaces 16:35–49CrossRefGoogle Scholar
  33. Tone K (2001) A slacks-based measure of efficiency in data envelopment analysis. Eur J Oper Res 130(3):498–509CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Jesus T. Pastor
    • 1
  • C. A. Knox Lovell
    • 2
    • 3
  • Juan Aparicio
    • 1
  1. 1.Center of Operations Research (CIO) at University Miguel Hernandez of ElcheElcheSpain
  2. 2.School of EconomicsUniversity of QueenslandBrisbaneAustralia
  3. 3.Department of Economics, Terry College of BusinessUniversity of GeorgiaAthensUSA

Personalised recommendations