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Duality of Shephard’s weakly disposable technology under a directional output distance function

  • Hervé Leleu

    ()

    (CNRS-LEM and IESEG School of Management)

In a recent paper Kuosmanen and Matin (2011) have developed a dual formulation of the weakly disposable Kuosmanen’s technology (Kuosmanen, 2005). Their work sheds new light on the economic interpretation of weak disposability and allows estimation of shadow prices for undesirable/bad outputs. More precisely, they have derived the multiplier formulation for the Kuosmanen weakly disposable technology and have found a ‘limited liability condition’ as the economic interpretation of weak disposability. Considering further the traditional Shephard technology, they argued that: “Obviously, the duality theory of linear programming does not apply to the classic Shephard technology because it is generally non-convex. This explains why the dual formulations of the weakly disposable technology are currently unavailable” (p.505). We prove here that by considering a directional output distance function there exists a duality for the Shephard weakly disposable technology. Moreover it turns out with a clear, intuitive and relevant economic interpretation of the weak disposability axiom. Thanks to our dual framework, we finally provide an original proof of the existing relationship between weak disposability and returns to scale.

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Paper provided by IESEG School of Management in its series Working Papers with number 2013-ECO-03.

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Length: 13 pages
Date of creation: Mar 2013
Date of revision:
Handle: RePEc:ies:wpaper:e201303
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  1. Kuosmanen, Timo & Post, Thierry, 2001. "Measuring economic efficiency with incomplete price information: With an application to European commercial banks," European Journal of Operational Research, Elsevier, vol. 134(1), pages 43-58, October.
  2. Jens Kjærsgaard & Niels Vestergaard & Kristiaan Kerstens, 2009. "Ecological Benchmarking to Explore Alternative Fishing Schemes to Protect Endangered Species by Substitution: The Danish Demersal Fishery in the North Sea," Environmental & Resource Economics, European Association of Environmental and Resource Economists, vol. 43(4), pages 573-590, August.
  3. Byung M. Jeon & Robin C. Sickles, 2004. "The role of environmental factors in growth accounting," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 19(5), pages 567-591.
  4. William L. Weber & Bruce Domazlicky, 2001. "Productivity Growth and Pollution in State Manufacturing," The Review of Economics and Statistics, MIT Press, vol. 83(1), pages 195-199, February.
  5. M. Murty & Surender Kumar & Kishore Dhavala, 2007. "Measuring environmental efficiency of industry: a case study of thermal power generation in India," Environmental & Resource Economics, European Association of Environmental and Resource Economists, vol. 38(1), pages 31-50, September.
  6. Fare, Rolf, et al, 1989. "Multilateral Productivity Comparisons When Some Outputs Are Undesirable: A Nonparametric Approach," The Review of Economics and Statistics, MIT Press, vol. 71(1), pages 90-98, February.
  7. Isabelle Piot-lepetit & Monique Le Moing, 2007. "Productivity and environmental regulation: the effect of the nitrates directive in the French pig sector," Working Papers 13564, Institut National de la Recherche Agronomique, France.
  8. Atakelty Hailu & Terrence S. Veeman, 2001. "Non-parametric Productivity Analysis with Undesirable Outputs: An Application to the Canadian Pulp and Paper Industry," American Journal of Agricultural Economics, Agricultural and Applied Economics Association, vol. 83(3), pages 605-616.
  9. Podinovski, Victor V. & Kuosmanen, Timo, 2011. "Modelling weak disposability in data envelopment analysis under relaxed convexity assumptions," European Journal of Operational Research, Elsevier, vol. 211(3), pages 577-585, June.
  10. Rolf F�re & Shawna Grosskopf, 2003. "Nonparametric Productivity Analysis with Undesirable Outputs: Comment," American Journal of Agricultural Economics, Agricultural and Applied Economics Association, vol. 85(4), pages 1070-1074.
  11. Fare, Rolf & Grosskopf, Shawna & Noh, Dong-Woon & Weber, William, 2005. "Characteristics of a polluting technology: theory and practice," Journal of Econometrics, Elsevier, vol. 126(2), pages 469-492, June.
  12. Zhou, P. & Ang, B.W. & Poh, K.L., 2008. "Measuring environmental performance under different environmental DEA technologies," Energy Economics, Elsevier, vol. 30(1), pages 1-14, January.
  13. Timo Kuosmanen & Victor Podinovski, 2008. "Weak Disposability in Nonparametric Production Analysis: Reply to F�re and Grosskopf," American Journal of Agricultural Economics, Agricultural and Applied Economics Association, vol. 91(2), pages 539-545.
  14. Rolf F�re & Shawna Grosskopf, 2007. "A Comment on Weak Disposability in Nonparametric Production Analysis," American Journal of Agricultural Economics, Agricultural and Applied Economics Association, vol. 91(2), pages 535-538.
  15. Atakelty Hailu, 2003. "Nonparametric Productivity Analysis with Undesirable Outputs: Reply," American Journal of Agricultural Economics, Agricultural and Applied Economics Association, vol. 85(4), pages 1075-1077.
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