Abstract
In this paper, we intend to establish relations between the way efficiency is measured in the literature on efficiency analysis and the notion of distance in topology. To this effect, we are interested particularly in the Hölder norm concept, providing a duality result based upon the profit function. Along this line, we prove that the Luenberger shortage function and the directional distance function of Chambers, Chung, and Färe appear as special cases of some l p distance (also called Hölder distance), under the assumption that the production set is convex. Under a weaker assumption (convexity of the input correspondence), we derive a duality result based on the cost function, providing several examples in which the functional form of the production set is specified.
Similar content being viewed by others
References
Koopmans, T. C., Analysis of Production as an Efficient Combination of Activities, Activity Analysis of Production and Allocation, Edited by T. C. Koopmans, Vol. 36, pp. 27–56, 1951.
Debreu, G., The Coefficient of Resource Utilization, Econometrica, Vol. 19, pp. 273–292, 1951.
Farell, M. J., The Measurement of Productive Efficiency, Journal of the Royal Statistical Society, Vol. 120, pp. 253–281, 1957.
Luenberger, D. G., Benefit Function and Duality, Journal of Mathematical Economics, Vol. 21, pp. 461–481, 1992.
Luenberger, D. G., Microeconomic Theory, McGraw Hill, Boston, Massachusetts, 1995.
Chambers, R., Chung, Y., and FÄre, R., Benefit and Distance Functions, Journal of Economic Theory, Vol. 70, pp. 407–419, 1996.
Chambers, R., Chung, Y., and FÄre, R., Profit, Directional Distance Functions, and Nerlovian Efficiency, Journal of Optimization Theory and Applications, Vol. 98, pp. 351–364, 1998.
Briec, W., Minimum Distance to the Complement of a Convex Set: Duality Result, Journal of Optimization Theory and Applications, Vol. 95, pp. 301–319, 1997.
Briec, W., A Graph Type Extension of Farrell Technical Efficiency Measure, Journal of Productivity Analysis, Vol. 8, pp. 95–110, 1997.
Dupuit, J., De la Mesure de l'Utilité des Travaux Publics, Annales des Ponts et Chaussées, Vol. 8, pp. 332–375, 1844.
Luenberger, D. G., Optimization by Vector Space Methods, John Wiley and Sons, New York, New York, 1969.
Shephard, R. W., Theory of Cost and Production Functions, Princeton University Press, Princeton, New Jersey, 1970.
Nerlove, M., Estimation and Identification of Cobb-Douglas Production Function, Rand McNally Company, Chicago, Illinois, 1965.
FÄre, R., Grosskopf, S., and Lovell, C. A. K., The Measurement of Efficiency of Production, Kluwer Nijhof Publishers, Boston, Massachusetts, 1985.
Fuss, M., and MacFadden, D., Production Economics: A Dual Approach to Theory and Applications, North-Holland, Amsterdam, Holland, 1978.
Russell, R. R., Continuity of Measures of Technical Efficiency, Journal of Economic Theory, Vol. 51, pp. 255–267, 1990.
Rockafellar, R. T., Convex Analysis, Princeton University Press, Princeton, New Jersey, pp. 349–387, 1970.
Seidel, R., Constructing Higher-Dimensional Convex Hulls at Logarithmic Cost per Face, Proceedings of the 18th Annual ACM Symposium on Theoretical Computing, pp. 404–413, 1986.
Briec, W., A Note on the Benefit Function, Working Paper, 1997.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Briec, W., Lesourd, J.B. Metric Distance Function and Profit: Some Duality Results. Journal of Optimization Theory and Applications 101, 15–33 (1999). https://doi.org/10.1023/A:1021762809393
Issue Date:
DOI: https://doi.org/10.1023/A:1021762809393