Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Published in:
Estimation of Technical Inefficiency in Production Frontier Models Using Cross-Sectional Data Read chapter Read first chapter

2015 | OriginalPaper | Chapter

6. Multiplicative and Additive Distance Functions: Efficiency Measures and Duality

Authors : Jesus T. Pastor, Juan Aparicio

Published in: Benchmarking for Performance Evaluation

Publisher: Springer India

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

In this survey, we present, for the first time, a classification scheme for distance functions, considering two broad groups: the multiplicative and the additive distance functions. Guiding empirical work is one of the objectives of this paper; for this reason, we consider only linear distance functions within a data envelopment analysis (DEA) framework. This also constitutes an easy way of connecting distance functions and efficiency measures. Further, we analyze two classes of distance functions: the ratio-directional distance function and the loss distance function. The former opens the possibility of evaluating productivity change combining directional distance functions, additive in nature, with Malmquist indexes, multiplicative in nature. The latter unifies all the known linear distance functions under a common structure, allowing the numerical evaluation of any linear distance function, as shown by a numerical example. We end up with a revision of duality results so as to highlight the economic relevance of distance functions.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

inform now
previous chapter Efficiency Measures for Industrial Organization
Footnotes
1
There are other functions, such as the “returns to the dollar” (see Färe et al. 2002; Zofío and Prieto 2006), that generate duality results associated with already known distance functions, such as the hyperbolic function (see Färe et al. 1985) or the generalized graph distance function (see Chavas and Cox 1999). We do not revise them because they are beyond the linear scope of this paper, at least for the variable returns to scale case.
 
2
Actually, it is enough to assume that the set of feasible inputs is a compact set, \(\left( {x_{m} ,0} \right) \in T\), and weak disposability of inputs (Färe and Primont 1995).
 
3
If \(\left( {x_{0} ,y_{0} } \right) \in T\), both programs have the same nonnegative finite optimal value. But if \(\left( {x_{0} ,y_{0} } \right) \notin T\), the envelopment form may be non-finite, unless we are dealing with a CRS model. In all the other returns to scale options, the directional distance function may be undefined in certain points outside T. From the point of view of the efficiency measurement of points inside T, there is no problem at all. From the point of view of the productivity change measurement, this can cause infeasibilities. In order to overcome this problem, Chambers et al. (1998) gave a more refined and complete definition of directional distance function, as follows:
$$\begin{aligned} \delta_{0} & = \sup \left\{ {\beta \in R:\left( {x_{0} - \beta g^{ - } ,y_{0} + \beta g^{ + } } \right) \in T} \right\},\;{\text{if}}\;\left( {x_{0} - \beta g^{ - } ,y_{0} + \beta g^{ + } } \right) \in T\;{\text{for}}\;{\text{some}}\;\beta , \\ \delta_{0} & = \inf \left\{ {\mu \in R:y_{0} + \mu g^{ + } \in R_{ + }^{s} } \right\},\;{\text{otherwise}} .\\ \end{aligned}$$
 
4
Boussemart et al. (2003) argue that Malmquist indexes are less reliable than Luenberger indicators because they may overestimate productivity change. We disagree with this interpretation and argue that the size of the directional vector determines the value of the directional distance function, which in turn determines the value of the Luenberger indicator.
 
5
Our definition is closely related to the geometric distance function (GDF) of Portela and Thanassoulis (2005), although (1) we do not cover all types of DEA inefficiencies, as they do; (2) we rely on a directional distance function, while they do not; and (3) we consider the inverse of their expression. Moreover, we consider orientations, while they declare that they are dealing with a non-oriented distance function.
 
6
See also Ray (2007) where the author suggests considering a particular normalization condition on the shadow prices for avoiding the problem of an unbounded objective function.
 
7
The models proposed in Cooper et al. (2011a) and Pastor et al. (2013) are recent examples of new weighted additive models in DEA.
 
8
Other derivatives of the distance functions are the measurement of the productivity change and the measurement and decomposition of the economic efficiency (see Russell 1998).
 
Literature
Ali, A.I., and L.M. Seiford. 1993. The mathematical programming approach to efficiency analysis, In The measurement of productive efficiency. Techniques and applications, eds. H.O. Fried, C.A.K. Lovell and S. Schmidt. New York: Oxford University Press.
Andersen, P., and N.C. Petersen. 1993. A procedure for ranking efficient units in data envelopment analysis. Management Science 39: 1261–1264.CrossRef
Aparicio, J., and Pastor, J.T. 2011. A general input distance function based on opportunity costs. Advances in Decision Sciences. Vol. 2011, Article ID 505241, 11 pp.
Aparicio, J., F. Borras, J.T. Pastor, and F. Vidal. 2013a. Accounting for slacks to measure and decompose revenue efficiency in the Spanish designation of origin wines with DEA. European Journal of Operational Research 231: 443–451.CrossRef
Aparicio, J., J.T. Pastor, and S.C. Ray. 2013b. An overall measure of technical inefficiency at the firm and at the industry level: the ‘lost profit on outlay’. European Journal of Operational Research 226: 154–162.CrossRef
Banker, R.D., A. Charnes, and W.W. Cooper. 1984. Some models for estimating technical and scale inefficiencies in data envelopment analysis. Management Science 30: 1078–1092.CrossRef
Briec, W. 1997. A Graph-Type Extension of Farrell Technical Efficiency Measure. Journal of Productivity Analysis 8: 95–110.CrossRef
Briec, W. 1998. Hölder distance functions and measurement of technical efficiency. Journal of Productivity Analysis 11: 111–132.CrossRef
Briec, W., and J.B. Lesourd. 1999. Metric distance function and profit: some duality results. Journal of Optimization Theory and Applications 101(1): 15–33.CrossRef
Boussemart, J.P., W. Briec, K. Kerstens, and J.C. Poutineau. 2003. Luenberger and Malmquist productivity indices: theoretical comparisons and empirical illustration. Bulletin of Economic Research 55(4): 391–405.CrossRef
Chambers, R.G., Y. Chung, and R. Färe. 1996. Benefit and distance functions. Journal of Economic Theory 70: 407–419.CrossRef
Chambers, R.G., Y. Chung, and R. Färe. 1998. Profit, directional distance functions, and Nerlovian efficiency. Journal of Optimization Theory and Applications 98(2): 351–364.
Charnes, A., W.W. Cooper, and E. Rhodes. 1978. Measuring the efficiency of decision making units. European Journal of Operational Research 2: 429–444.CrossRef
Chavas, J.P., and T.L. Cox. 1999. A generalized distance function and the analysis of production efficiency. Southern Economic Journal 66: 294–318.CrossRef
Cook, W.D., and L.M. Seiford. 2009. Data envelopment analysis (DEA)—thirty years on. European Journal of Operational Research 192(1): 1–17.CrossRef
Cooper, W.W., K.S. Park, and J.T. Pastor. 1999. RAM: a range adjusted measure of inefficiency for use with additive models, and relations to others models and measures in DEA. Journal of Productivity Analysis 11: 5–42.CrossRef
Cooper, W.W., J.T. Pastor, F. Borras, J. Aparicio, and D. Pastor. 2011a. BAM: a bounded adjusted measure of efficiency for use with bounded additive models. Journal of Productivity Analysis 35: 85–94.CrossRef
Cooper, W.W., J.T. Pastor, J. Aparicio, and F. Borras. 2011b. Decomposing profit inefficiency in DEA through the weighted additive model. European Journal of Operational Research 212: 411–416.CrossRef
Debreu, G. 1951. The coefficient of resource utilization. Econometrica 19: 273–292.CrossRef
Färe, R., S. Grosskopf, and C.A.K. Lovell. 1985. The measurement of efficiency of production. Boston: Kluwer Nijhof Publishing.
Färe, R., and D. Primont. 1995. Multi-output production and duality: theory and applications. Boston: Kluwer Academic Publishers.
Färe, R., S. Grosskopf, and Z. Osman. 2002. Hyperbolic efficiency and return to the dollar. European Journal of Operational Research 136(3): 671–679.CrossRef
Farrell, M.J. 1957. The measurement of productive efficiency. Journal of the Royal Statistical Society, Series A: General 120: 253–281.CrossRef
Luenberger, D.G. 1992a. Benefit functions and duality. Journal of Mathematical Economics 21(5): 461–481.CrossRef
Luenberger, D.G. 1992b. New optimality principles for economic efficiency and equilibrium. Journal of Optimization Theory and Applications 75(2): 221–264.
Luenberger, D. 1995. Microeconomic theory. New York: McGraw Hill.
Malmquist, S. 1953. Index numbers and indifference surfaces. Trabajos de Estadistica 4: 209–242.CrossRef
Mangasarian, O.L. 1994. Nonlinear programming. In Classics in applied mathematics, vol. 10. Philadelphia: SIAM.
Pastor, J.T., C.A.K. Lovell, and J. Aparicio. 2012. Families of linear efficiency programs based on Debreu’s loss function. Journal of Productivity Analysis 38: 109–120.CrossRef
Pastor, J.T., J. Aparicio, J.F. Monge, and D. Pastor. 2013. Modeling CRS bounded additive DEA models and characterizing their Pareto-efficient points. Journal of Productivity Analysis 40: 285–292.CrossRef
Portela, M.C.A.S., and E. Thanassoulis. 2005. Profitability of a sample of Portuguese bank branches and its decomposition into technical and allocative components. European Journal of Operational Research 162: 850–866.CrossRef
Portela, M.C.S., and E. Thanassoulis. 2007. Developing a decomposable measure of profit efficiency using DEA. Journal of the Operational Research Society 58: 481–490.CrossRef
Ray, S.C. 2007. Shadow profit maximization and a measure of overall inefficiency. Journal of Productivity Analysis 27: 231–236.CrossRef
Rockafellar, R.T. 1970. Convex analysis. Princeton: Princeton University Press.
Russell, R. 1985. Measures of technical efficiency. Journal of Economic Theory 35: 109–126.CrossRef
Russell, R. 1998. Distance functions in consumer and production theory. In Index numbers: essays in honour of Sten Malmquist, eds. by R. Färe, S. Grosskopf and R.R. Russell. Boston: Kluwer Academic Publishers.
Shephard, R.W. 1953. Cost and production functions. Princeton: Princeton University Press.
Shephard, R.W. 1970. Theory of cost and production functions. Princeton: Princeton University Press.
Varian, H.R. 1992. Microeconomic analysis. 3rd ed. New York: Norton.
Zofio, J.L., and A. Prieto. 2006. Return to dollar, generalized distance function and the fisher productivity index. Spanish Economic Review 8(2): 113–138.CrossRef
Metadata
Title
Multiplicative and Additive Distance Functions: Efficiency Measures and Duality
Authors
Jesus T. Pastor
Juan Aparicio
Copyright Year
2015
Publisher
Springer India
Book
Benchmarking for Performance Evaluation

Print ISBN: 978-81-322-2252-1

Electronic ISBN: 978-81-322-2253-8

Copyright Year: 2015

DOI
https://doi.org/10.1007/978-81-322-2253-8_6

Premium Partner

    Image Credits