Journal of Productivity Analysis

, Volume 24, Issue 1, pp 31–48 | Cite as

Hyperbolic Efficiency and Parametric Distance Functions: With Application to Spanish Savings Banks

  • Rafael A. Cuesta
  • José L. Zofío


Distance functions are gaining relevance as alternative representations of production technologies, with growing numbers of empirical applications being made in the productivity and efficiency field. Distance functions were initially defined on the input or output production possibility sets by Shephard (1953, 1970) and extended to a graph representation of the technology by Färe, Grosskopf and Lovell (1985) through their graph hyperbolic distance function. Since then, different techniques such as non parametric-DEA and parametric-SFA have been used to calculate these distance functions. However, in the latter case we know of no study in which the restriction to input or output orientation has been relaxed. What we propose is to overcome such restrictiveness on dimensionality by defining and estimating a parametric hyperbolic distance function which simultaneously allows for the maximum equiproportionate expansion of outputs and reduction of inputs. In particular, we introduce a translog hyperbolic specification that complies with the conventional properties that the hyperbolic distance function satisfies. Finally, to illustrate its applicability in efficiency analysis we implement it using a data set of Spanish savings banks.


production frontiers parametric distance functions hyperbolic efficiency banking efficiency 


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  1. Aczel, J. 1966“Lectures on Functional Equations and their Applications”Academic PressNew YorkGoogle Scholar
  2. Aigner, D., Lovell, C. A. K., Schmidt, P. 1977“Formulation and Estimation of Stochastic Frontier Production Functions Models”International Economic Review17377396Google Scholar
  3. Bauer, P.W., Berger, A. N., Humphrey, D. B. 1993“Efficiency And Productivity Growth in US Banking”Fried, H.Lovell, C. A. K.Schmidt, S. eds. The Measurement of Productive Efficiency: Techniques and ApplicationsOxford University PressOxfordGoogle Scholar
  4. Ball, V. E., Lovell, C. A. K., Nehring, R., Somwaru, A. 1994“Incorporating Undesirable Outputs into Models of Production: An Application to US Agriculture”Cahiers d’économie et sociologie rurales36074Google Scholar
  5. Berger, A. N., Humphrey, D. B. 1992“Measurement and Efficiency Issues in Commercial Banking”Griliches, Z. eds. Output Measurement in the Service SectorThe University of Chicago PressChicagoNBER Studies in Income and Wealth, 56.Google Scholar
  6. Berger, A. N., Humphrey, D. B. 1997“Efficiency of Financial Institutions: International Survey and Directions for Future Research”European Journal of Operational Research98175212CrossRefGoogle Scholar
  7. Battese, G. E., Coelli, T. J. 1988“Prediction of Firm-level Technical Efficiencies with a Generalized Production Function and Panel Data”Journal of Econometrics38387399CrossRefMathSciNetGoogle Scholar
  8. Chambers, R. 1988“Applied Production Analysis”Cambridge University PressNew YorkGoogle Scholar
  9. Chambers, R., Chung, Y., Färe, R. 1996“Benefit and Distance Functions”Journal of Economic Theory70407419CrossRefGoogle Scholar
  10. Christensen, L., Jorgenson, D., Lau, L. 1971“Conjugate Duality and the Trascendental Logarithmic Production Function”Econometrica39255256Google Scholar
  11. Christensen, L., Jorgenson, D., Lau, L. 1973“Trascendental Logarithmic Production Frontiers”Review of Economics and Statistics552845Google Scholar
  12. Chung, Y. H., Färe, R., Grosskopf, S. 1997“Productivity and Undesirable Outputs: A Directional Distance Function Approach”Journal of Environmental Management51229240CrossRefGoogle Scholar
  13. Coelli, T. J. 2000On the Econometric Estimation of the Distance Function Representation of a Production TechnologyUniversite Catholique de LouvainBelgiumCORE Discussion Paper: 2000/42.Google Scholar
  14. Coelli, T. J., Perelman, S. 1996“Efficiency Measurement, Multiple-output Technologies and Distance Functions: With Application to European Railways”University of LiegeBelgiumCREPP Working Paper, 96/05.Google Scholar
  15. Coelli, T. J., Perelman, S. 2000“Technical Efficiency of European Railways: A Distance Function Approach”Applied Economics3219671997CrossRefGoogle Scholar
  16. Cooper, W. W., Seiford, L. M., Tone, K. 2000“Data Envelopment Analysis, A Comprehensive Text with Models, Applications, References and DEA-Software”Kluwer Academic PublishersBostonGoogle Scholar
  17. Cuesta, R. A., Orea, L. 2002“Mergers and Technical Efficiency in Spanish Saving Banks: A Stochastic Distance Function Approach”Journal of Banking and Finance2622312247CrossRefGoogle Scholar
  18. Cuesta, R. A., Zofío, J. L. 2002A Hyperbolic Distance Functions: With Application to Spanish Savings Banks, 3rd Workshop on Efficiency and ProductivityUniversidad de OviedoOviedo, SpainJuly 2002.Google Scholar
  19. Debreu, G. 1951“The Coefficient of Resource Utilization”Econometrica19273292Google Scholar
  20. English, M., Grosskopf, S., Hayes, K., Yaisawarng, S. 1993“Output Allocative and Technical Efficiency of Banks”Journal of Banking and Finance17349366CrossRefGoogle Scholar
  21. Färe, R. 1988“Fundamentals of Production Theory”Springer VerlagBerlinGoogle Scholar
  22. Färe, R., Grosskopf, S., Lovell, C. A. K. 1985“The Measurement of Efficiency of Production”Kluwer Nijhoff PublishingBostonGoogle Scholar
  23. Färe, R., Grosskopf, S., Lovell, C. A. K. 1994“Production Frontiers”Cambridge University PressNew YorkGoogle Scholar
  24. Färe, R., Grosskopf, S., Lovell, C. A. K., Pasurka, C. 1989“Multilateral Productivity Comparisons When Some Outputs are Undesirable: A Non-parametric Approach”Review of Economics and Statistics759098Google Scholar
  25. Färe, R., Grosskopf, S., Zaim, O. 2002“Hyperbolic Efficiency and Return to Dollar”European Journal of Operational Research136671679CrossRefGoogle Scholar
  26. Farrell, M. 1957“The Measurement of Productive Efficiency”Journal of the Royal Statistical Society Series, A General120253290Google Scholar
  27. Ferrier, G., Lovell, C. A. K. 1990“Measuring Cost Efficiency in Banking: Econometric and Linear Programming Evidence”Journal of Econometrics46229245CrossRefGoogle Scholar
  28. Georgescu-Roegen, N. 1951“The Aggregate Linear Production Function and its Application to von Newman’s Economic Model”Koopmans, T. eds. Activity Analysis of Production and AllocationWileyNew YorkGoogle Scholar
  29. Grifell-Tajté, E., Lovell, C. A. K. 1997“The Sources of Productivity Change in Spanish Banking”European Journal of Operational Research98364380CrossRefGoogle Scholar
  30. Grosskopf, S., Hayes, K., Taylor, L., Weber, W. 1997“Budget Constraint Frontier Measures of Fiscal Equality and Efficiency in Schooling”Review of Economics and Statistics7911624CrossRefGoogle Scholar
  31. Hasan, I., Hunter, W. 1996“Efficiency of Japanese Multinational Banks in the US”Research in Finance14157173Google Scholar
  32. Hughes, , Mester, J. L., Moon, C. 2001“Are Scale Economies in Banking Elusive? Evidence Obtained by Incorporating Capital Structure and Risk-Taking into Models of Bank Production”Journal of Banking and Finance1521692208CrossRefGoogle Scholar
  33. Lau, L. J. 1972“Profit Functions of Technologies with Multiple Inputs and Outputs”Review of Economics and Statistics54281289Google Scholar
  34. Lovell, C. A. K., Pastor, J.T. 1997“Target Setting: An Application to a Branch Bank Network”European Journal of Operational Research98291300Google Scholar
  35. Lovell, C. A. K., Richardson, S., Travers, P., Wood, L. 1994“Resources and Functionings: A New View of Inequality in Australia”Eichhorn, W. eds. Models and Measurement of Welfare and InequalitySpringer-VerlagBerlinGoogle Scholar
  36. Maudos, J. 1996A Comparison of Different Stochastic Frontier Techniques with Panel Data: An Application for Efficiency in Spanish Savings Banks, Working PaperUniversity of ValenciaSpainGoogle Scholar
  37. Mester, L. J. 1997“Measuring Efficiency at US Banks: Accounting for Heterogeneity Is Important”European Journal of Operational Research98230243CrossRefGoogle Scholar
  38. Pitt, M., Lee, L. 1981“The Measurement and Sources of Technical Inefficiency in the Indonesian Weaving Industry”Journal of Development Economics94364CrossRefGoogle Scholar
  39. Sealey, C. W., Lindley, J. T. 1977“Inputs, Outputs, and a Theory of Production and Cost at Depository Financial Institutions”Journal of Finance3212511266Google Scholar
  40. Shephard, R. W. 1953“Cost and Production Functions”Princeton University PressPrincetonGoogle Scholar
  41. Shephard, R. W. 1970“Theory of Cost and Production Functions”Princeton University PressPrincetonGoogle Scholar
  42. Zofío, J. L., Lovell, C. A. K. 2001“Graph Efficiency and Productivity Measures: An Application to US Agriculture”Applied Economics3314331442CrossRefGoogle Scholar
  43. Zofío, J. L., Prieto, A. M. 2001“Environmental Efficiency and Regulatory Standards: The Case of CO2 Emissions from OECD Industries”Resource and Energy Economics236386CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Caja de Ahorros de Asturias and Departamento de EconomíaUniversidad de OviedoOviedoSpain
  2. 2.Departamento de Análisis Económico: Teoría Económica e Historia EconómicaUniversidad Autónoma de MadridCantoblancoSpain

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