IDEAS home Printed from https://ideas.repec.org/a/inm/oropre/v58y2010i6p1743-1754.html
   My bibliography  Save this article

Differential Characteristics of Efficient Frontiers in Data Envelopment Analysis

Author

Listed:
  • Victor V. Podinovski

    (Warwick Business School, University of Warwick, Coventry CV4 7AL, United Kingdom)

  • Finn R. Førsund

    (Department of Economics, University of Oslo, 0317 Blindern, Oslo, Norway)

Abstract

The implicit definition and nondifferentiability of efficient frontiers used in data envelopment analysis are two major obstacles to obtaining their differential characteristics, including various elasticity measures and marginal rates of substitution. In this paper we invoke the theorem of the directional derivative of the optimal value function and show how this can be used to define and calculate the required elasticities without any simplifying assumptions. This approach allows us to extend the known elasticity measures and introduce new ones to the entire efficient frontier, including all its extreme points, in one single development. We also construct linear programs that are required for the calculation of elasticity measures. Our main development is undertaken in the variable returns-to-scale technology but extends to other polyhedral technologies of efficiency analysis.

Suggested Citation

  • Victor V. Podinovski & Finn R. Førsund, 2010. "Differential Characteristics of Efficient Frontiers in Data Envelopment Analysis," Operations Research, INFORMS, vol. 58(6), pages 1743-1754, December.
    • Handle: RePEc:inm:oropre:v:58:y:2010:i:6:p:1743-1754
    DOI: 10.1287/opre.1090.0794
    as

    Download full text from publisher

    File URL: http://dx.doi.org/10.1287/opre.1090.0794
    Download Restriction: no
    File URL: https://libkey.io/10.1287/opre.1090.0794?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    References listed on IDEAS

    as
    1. Banker, Rajiv D. & Thrall, R. M., 1992. "Estimation of returns to scale using data envelopment analysis," European Journal of Operational Research, Elsevier, vol. 62(1), pages 74-84, October.
    2. Blackorby, Charles & Russell, R Robert, 1989. "Will the Real Elasticity of Substitution Please Stand Up? (A Comparison of the Allen/Uzawa and Morishima Elasticities)," American Economic Review, American Economic Association, vol. 79(4), pages 882-888, September.
    3. Charnes, A. & Cooper, W. W. & Rhodes, E., 1978. "Measuring the efficiency of decision making units," European Journal of Operational Research, Elsevier, vol. 2(6), pages 429-444, November.
    4. F R Førsund & L Hjalmarsson, 2004. "Calculating scale elasticity in DEA models," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 55(10), pages 1023-1038, October.
    5. Hanoch, Giora, 1970. "Homotheticity in joint production," Journal of Economic Theory, Elsevier, vol. 2(4), pages 423-426, December.
    6. John C. Panzar & Robert D. Willig, 1977. "Economies of Scale in Multi-Output Production," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 91(3), pages 481-493.
    7. Hadjicostas, Petros & Soteriou, Andreas C., 2006. "One-sided elasticities and technical efficiency in multi-output production: A theoretical framework," European Journal of Operational Research, Elsevier, vol. 168(2), pages 425-449, January.
    8. Podinovski, Victor V. & Førsund, Finn R. & Krivonozhko, Vladimir E., 2009. "A simple derivation of scale elasticity in data envelopment analysis," European Journal of Operational Research, Elsevier, vol. 197(1), pages 149-153, August.
    9. Timo Kuosmanen, 2005. "Weak Disposability in Nonparametric Production Analysis with Undesirable Outputs," American Journal of Agricultural Economics, Agricultural and Applied Economics Association, vol. 87(4), pages 1077-1082.
    10. J Ruggiero, 2000. "Nonparametric estimation of returns to scale in the public sector with an application to the provision of educational services," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 51(8), pages 906-912, August.
    11. Charnes, A. & Cooper, W. W. & Golany, B. & Seiford, L. & Stutz, J., 1985. "Foundations of data envelopment analysis for Pareto-Koopmans efficient empirical production functions," Journal of Econometrics, Elsevier, vol. 30(1-2), pages 91-107.
    12. W. Cooper & Dr. Park & Professor Ciurana, 2000. "Marginal Rates and Elasticities of Substitution with Additive Models in DEA," Journal of Productivity Analysis, Springer, vol. 13(2), pages 105-123, March.
    13. Bougnol, M.-L. & Dulá, J.H., 2009. "Anchor points in DEA," European Journal of Operational Research, Elsevier, vol. 192(2), pages 668-676, January.
    14. Rajiv D. Banker & Ajay Maindiratta, 1986. "Erratum to: "Piecewise Loglinear Estimation of Efficient Production Surfaces"," Management Science, INFORMS, vol. 32(3), pages 385-385, March.
    15. Dan Rosen & Claire Schaffnit & Joseph Paradi, 1998. "Marginal Rates and Two-dimensional Level Curves in DEA," Journal of Productivity Analysis, Springer, vol. 9(3), pages 205-232, March.
    16. Holger Scheel & Stefan Scholtes, 2003. "Continuity of DEA Efficiency Measures," Operations Research, INFORMS, vol. 51(1), pages 149-159, February.
    17. V E Krivonozhko & O B Utkin & A V Volodin & I A Sablin & M Patrin, 2004. "Constructions of economic functions and calculations of marginal rates in DEA using parametric optimization methods," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 55(10), pages 1049-1058, October.
    18. F R Førsund & S A C Kittelsen & V E Krivonozhko, 2009. "Farrell revisited–Visualizing properties of DEA production frontiers," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 60(11), pages 1535-1545, November.
    19. Asmild, Mette & Paradi, Joseph C. & Reese, David N., 2006. "Theoretical perspectives of trade-off analysis using DEA," Omega, Elsevier, vol. 34(4), pages 337-343, August.
    20. Finn Førsund & Lennart Hjalmarsson & Vladimir Krivonozhko & Oleg Utkin, 2007. "Calculation of scale elasticities in DEA models: direct and indirect approaches," Journal of Productivity Analysis, Springer, vol. 28(1), pages 45-56, October.
    21. Starrett, David A, 1977. "Measuring Returns to Scale in the Aggregate, and the Scale Effect of Public Goods," Econometrica, Econometric Society, vol. 45(6), pages 1439-1455, September.
    22. O. B. Olesen & N. C. Petersen, 1996. "Indicators of Ill-Conditioned Data Sets and Model Misspecification in Data Envelopment Analysis: An Extended Facet Approach," Management Science, INFORMS, vol. 42(2), pages 205-219, February.
    23. R. D. Banker & A. Charnes & W. W. Cooper, 1984. "Some Models for Estimating Technical and Scale Inefficiencies in Data Envelopment Analysis," Management Science, INFORMS, vol. 30(9), pages 1078-1092, September.
    24. Rajiv D. Banker & Ajay Maindiratta, 1986. "Piecewise Loglinear Estimation of Efficient Production Surfaces," Management Science, INFORMS, vol. 32(1), pages 126-135, January.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Victor V. Podinovski & Robert G. Chambers & Kazim Baris Atici & Iryna D. Deineko, 2016. "Marginal Values and Returns to Scale for Nonparametric Production Frontiers," Operations Research, INFORMS, vol. 64(1), pages 236-250, February.
    2. Chia-Yen Lee, 2017. "Directional marginal productivity: a foundation of meta-data envelopment analysis," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 68(5), pages 544-555, May.
    3. Podinovski, Victor V., 2017. "Returns to scale in convex production technologies," European Journal of Operational Research, Elsevier, vol. 258(3), pages 970-982.
    4. Vladimir Krivonozhko & Finn Førsund & Andrey Lychev, 2012. "Returns-to-scale properties in DEA models: the fundamental role of interior points," Journal of Productivity Analysis, Springer, vol. 38(2), pages 121-130, October.
    5. Sahoo, Biresh K & Khoveyni, Mohammad & Eslami, Robabeh & Chaudhury, Pradipta, 2016. "Returns to scale and most productive scale size in DEA with negative data," European Journal of Operational Research, Elsevier, vol. 255(2), pages 545-558.
    6. Atici, Kazim Baris & Podinovski, Victor V., 2012. "Mixed partial elasticities in constant returns-to-scale production technologies," European Journal of Operational Research, Elsevier, vol. 220(1), pages 262-269.
    7. Asmild, Mette & Paradi, Joseph C. & Reese, David N., 2006. "Theoretical perspectives of trade-off analysis using DEA," Omega, Elsevier, vol. 34(4), pages 337-343, August.
    8. Michael Zschille, 2014. "Nonparametric measures of returns to scale: an application to German water supply," Empirical Economics, Springer, vol. 47(3), pages 1029-1053, November.
    9. Zelenyuk, Valentin, 2015. "Aggregation of scale efficiency," European Journal of Operational Research, Elsevier, vol. 240(1), pages 269-277.
    10. Zarepisheh, M. & Soleimani-damaneh, M., 2008. "Global variation of outputs with respect to the variation of inputs in performance analysis; generalized RTS," European Journal of Operational Research, Elsevier, vol. 186(2), pages 786-800, April.
    11. Zelenyuk, Valentin, 2013. "A scale elasticity measure for directional distance function and its dual: Theory and DEA estimation," European Journal of Operational Research, Elsevier, vol. 228(3), pages 592-600.
    12. Podinovski, Victor V., 2019. "Direct estimation of marginal characteristics of nonparametric production frontiers in the presence of undesirable outputs," European Journal of Operational Research, Elsevier, vol. 279(1), pages 258-276.
    13. Andreas Dellnitz & Andreas Kleine & Wilhelm Rödder, 2018. "CCR or BCC: what if we are in the wrong model?," Journal of Business Economics, Springer, vol. 88(7), pages 831-850, September.
    14. Walheer, Barnabé, 2018. "Scale efficiency for multi-output cost minimizing producers: The case of the US electricity plants," Energy Economics, Elsevier, vol. 70(C), pages 26-36.
    15. Mehdiloo, Mahmood & Podinovski, Victor V., 2021. "Strong, weak and Farrell efficient frontiers of technologies satisfying different production assumptions," European Journal of Operational Research, Elsevier, vol. 294(1), pages 295-311.
    16. Tone, Kaoru & Sahoo, Biresh K., 2005. "Evaluating cost efficiency and returns to scale in the Life Insurance Corporation of India using data envelopment analysis," Socio-Economic Planning Sciences, Elsevier, vol. 39(4), pages 261-285, December.
    17. Petros Hadjicostas & Andreas Soteriou, 2010. "Different orders of one-sided scale elasticities in multi-output production," Journal of Productivity Analysis, Springer, vol. 33(2), pages 147-167, April.
    18. Podinovski, Victor V. & Førsund, Finn R. & Krivonozhko, Vladimir E., 2009. "A simple derivation of scale elasticity in data envelopment analysis," European Journal of Operational Research, Elsevier, vol. 197(1), pages 149-153, August.
    19. Olesen, Ole Bent & Petersen, Niels Christian, 2013. "Imposing the Regular Ultra Passum law in DEA models," Omega, Elsevier, vol. 41(1), pages 16-27.
    20. K. Tone & M. Tsutsui, 2015. "How to Deal with Non-Convex Frontiers in Data Envelopment Analysis," Journal of Optimization Theory and Applications, Springer, vol. 166(3), pages 1002-1028, September.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:inm:oropre:v:58:y:2010:i:6:p:1743-1754. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Chris Asher (email available below). General contact details of provider: https://edirc.repec.org/data/inforea.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.