Infeasibility and Directional Distance Functions with Application to the Determinateness of the Luenberger Productivity Indicator

  • W. Briec
  • K. Kerstens


The purpose of this contribution is to highlight an underexplored property of the directional distance function, a recently introduced generalization of the Shephard distance function. It diagnoses in detail the economic conditions under which infeasibilities may occur for the case of directional distance functions and explores whether there exist any solutions that remedy the problem in an economically meaningful way. This discussion is linked to determinateness as a property in index theory and is illustrated by analyzing the Luenberger total factor productivity indicator, based upon directional distance functions. This indicator turns out to be impossible to compute under certain weak conditions. A fortiori, the same problems can also occur for less general productivity indicators and indexes.


Directional distance function Shortage function Well-definedness Infeasibility Determinateness 


  1. 1.
    Shephard, R.W.: Theory of Cost and Production Functions. Princeton University Press, Princeton (1970) MATHGoogle Scholar
  2. 2.
    Luenberger, D.G.: Benefit functions and duality. J. Math. Econ. 21(5), 461–481 (1992) MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Luenberger, D.G.: New optimality principles for economic efficiency and equilibrium. J. Optim. Theory Appl. 75(2), 221–264 (1992) MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Luenberger, D.G.: Microeconomic Theory. McGraw Hill, New York (1995) Google Scholar
  5. 5.
    Chambers, R.G., Chung, Y., Färe, R.: Benefit and distance functions. J. Econ. Theory 70(2), 407–419 (1996) MATHCrossRefGoogle Scholar
  6. 6.
    Chambers, R.G., Chung, Y., Färe, R.: Profit, directional distance functions, and Nerlovian efficiency. J. Optim. Theory Appl. 98(2), 351–364 (1998) MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Chambers, R.G., Färe, R., Grosskopf, S.: Productivity growth in APEC countries. Pac. Econ. Rev. 1(3), 181–190 (1996) CrossRefGoogle Scholar
  8. 8.
    Mc Fadden, D.: Cost, revenue, and profit functions. In: Fuss, M., McFadden, D. (eds.) Production Economics: A Dual Approach to Theory and Applications, vol. 1, pp. 3–109. North-Holland, Amsterdam (1978) Google Scholar
  9. 9.
    Fischer, I.: The Making of Index Numbers. Houghton-Mifflin, Boston (1922) Google Scholar
  10. 10.
    Swamy, S.: Consistency of Fisher’s tests. Econometrica 33(3), 619–623 (1965) MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Eichorn, W.: Fisher’s tests revisited. Econometrica 44(2), 247–256 (1976) CrossRefMathSciNetGoogle Scholar
  12. 12.
    Samuelson, P., Swamy, S.: Invariant economic index numbers and canonical duality: survey and synthesis. Am. Econ. Rev. 64(4), 566–593 (1974) Google Scholar
  13. 13.
    Färe, R., Lyon, V.: The determinateness test and economic price indices. Econometrica 49(1), 209–213 (1981) MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Caves, D.W., Christensen, L.R., Diewert, W.E.: The economic theory of index numbers and the measurement of input, output, and productivity. Econometrica 50(6), 1393–1414 (1982) MATHCrossRefGoogle Scholar
  15. 15.
    Färe, R., Grosskopf, S., Lindgren, B., Roos, P.: Productivity developments in Swedish hospitals: a Malmquist output index approach. In: Charnes, A., Cooper, W.W., Lewin, A.Y., Seiford, L.M. (eds.) Data Envelopment Analysis: Theory, Methodology and Applications, pp. 253–272. Kluwer Academic, Dordrecht (1995) Google Scholar
  16. 16.
    Chambers, R.G., Pope, R.D.: Aggregate productivity measures. Am. J. Agric. Econ. 78(5), 1360–1365 (1996) CrossRefGoogle Scholar
  17. 17.
    Chambers, R.G.: Exact nonradial input, output, and productivity measurement. Econ. Theory 20(4), 751–765 (2002) MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Diewert, W.E.: Index number theory using differences rather than ratios. Am. J. Econ. Soc. 64(1), 347–395 (2005) CrossRefGoogle Scholar
  19. 19.
    Färe, R., Grosskopf, S., Lovell, C.A.K.: Production Frontiers. Cambridge University Press, Cambridge (1994) Google Scholar
  20. 20.
    Mukherjee, K., Ray, S.C., Miller, S.M.: Productivity growth in large US commercial banks: the initial post-deregulation experience. J. Bank. Finance 25(5), 913–939 (2001) CrossRefGoogle Scholar
  21. 21.
    Varian, H.: The nonparametric approach to production analysis. Econometrica 52(3), 579–597 (1984) MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Banker, R.D., Maindiratta, A.: Nonparametric analysis of technical and allocative efficiencies in production. Econometrica 56(6), 1315–1332 (1988) MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Atkinson, S.E., Cornwell, C., Honerkamp, O.: Measuring and decomposing productivity change: stochastic distance function estimation versus data envelopment analysis. J. Bus. Econ. Stat. 21(2), 284–294 (2003) CrossRefMathSciNetGoogle Scholar
  24. 24.
    Chavas, J.-P., Cox, T.: A generalized distance function and the analysis of production efficiency. South. Econ. J. 66(2), 294–318 (1999) CrossRefGoogle Scholar
  25. 25.
    Jaenicke, E.C.: Testing for intermediate outputs in dynamic DEA models: accounting for soil capital in rotational crop production and productivity measures. J. Prod. Anal. 14(3), 247–266 (2000) CrossRefGoogle Scholar
  26. 26.
    Briec, W.: A graph type extension of Farrell technical efficiency measure. J. Prod. Anal. 8(1), 95–110 (1997) CrossRefGoogle Scholar
  27. 27.
    Blackorby, C., Donaldson, D.: A theoretical treatment of indices of absolute inequality. Int. Econ. Rev. 21(1), 107–136 (1980) MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Glass, J.C., Mac Killop, D.G.: A post deregulation analysis of the sources of productivity growth in UK building societies. Manch. Sch. 68(3), 360–385 (2000) CrossRefGoogle Scholar
  29. 29.
    Briec, W., Lesourd, J.-B.: Metric distance function and profit: some duality result. J. Optim. Theory Appl. 101(1), 15–33 (1999) MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Ouellette, P., Vierstraete, V.: Technological change and efficiency in the presence of quasi-fixed inputs: a DEA application to the hospital sector. Eur. J. Oper. Res. 154(3), 755–763 (2004) MATHCrossRefGoogle Scholar
  31. 31.
    Balk, B.M.: Industrial Price, Quantity, and Productivity Indices: The Micro-Economic Theory and an Application. Kluwer Academic, Dordrecht (1998) Google Scholar
  32. 32.
    Ray, S.C., Mukherjee, K.: Decompositions of the Fisher ideal index of productivity: a non-parametric dual analysis of US airlines data. Econ. J. 106(439), 1659–1678 (1996) CrossRefGoogle Scholar
  33. 33.
    Grifell-Tatjé, E., Lovell, C.A.K.: Profits and productivity. Manag. Sci. 45(9), 1177–1193 (1999) CrossRefGoogle Scholar
  34. 34.
    Malmquist, S.: Index numbers and indifference surfaces. Trab. Estad. 4(2), 209–242 (1953) MATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Konüs, A.: The problem of the true index of the cost of living. Econometrica 7(1), 10–29 (1939) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Université de PerpignanLAMPSPerpignanFrance
  2. 2.CNRS-LEM (UMR 8179)IESEG School of ManagementLilleFrance

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