Abstract
Overall efficiency measures were introduced in the literature for evaluating the economic performance of firms when reference prices are available. These references are usually observed market prices. Recently, Aparicio and Zofío (Economic cross-efficiency: Theory and DEA methods. ERIM Report Series Research in Management, No. ERS-2019-001-LIS. Erasmus Research Institute of Management (ERIM). Erasmus University Rotterdam, The Netherlands. http://hdl.handle.net/1765/115479, 2019) have shown that the result of applying cross-efficiency methods (Sexton, T. R., Silkman, R. H., & Hogan, A. J. (1986). Data envelopment analysis: Critique and extensions. In R. H. Silkman (Ed.), Measuring efficiency: An assessment of data envelopment analysis, new directions for program evaluation (Vol. 32, pp. 73–105). San Francisco/London: Jossey-Bass), yielding an aggregate multilateral index that compares the technical performance of firms using the shadow prices of competitors, can be precisely reinterpreted as a measure of economic efficiency. They termed the new approach “economic cross-efficiency.” However, these authors restrict their analysis to the basic definitions corresponding to the Farrell (Journal of the Royal Statistical Society, Series A, General 120, 253–281, 1957) and Nerlove (Estimation and identification of Cobb-Douglas production functions. Chicago: Rand McNally, 1965) approaches, i.e., based on the duality between the cost function and the input distance function and between the profit function and the directional distance function, respectively. Here we complete their proposal by introducing new economic cross-efficiency measures related to other popular approaches for measuring economic performance, specifically those based on the duality between the profitability (maximum revenue to cost) and the generalized (hyperbolic) distance function and between the profit function and either the weighted additive or the Hölder distance function. Additionally, we introduce panel data extensions related to the so-called cost-Malmquist index and the profit-Luenberger indicator. Finally, we illustrate the models resorting to data envelopment analysis techniques—from which shadow prices are obtained and considering a banking industry dataset previously used in the cross-efficiency literature.
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Notes
- 1.
This cross-efficiency evaluation with respect to alternative peers results in smaller technical efficiency scores, because DEA searches for the most favorable weights when performing own evaluations.
- 2.
Based on these technological characterizations, in what follows we define several measures that allow the decomposition of economic cross-efficiency into technical and allocative components. As it is now well-established in the literature, we rely on the following terminology: We refer to the different factors in which economic cross-efficiency can be decomposed multiplicatively as efficiency measures (e.g., Farrell cost-efficiency). Numerically, the greater their value, the more efficient observations are. For these measures, one is the upper bound signaling an efficient behavior. Alternatively, we refer to the different terms in which economic cross-inefficiency can be decomposed additively as inefficiency measures (e.g., Nerlovian profit inefficiency). Now the greater their numerical value, the greater the inefficiency, with zero being the lower bound associated to an efficient behavior.
- 3.
For a list of relevant properties, see Aparicio and Zofío (2019).
- 4.
Färe et al. (2002) defined this relationship in terms of the hyperbolic distance function; i.e., \( {D}_c^H\left({X}_k,{Y}_k\right)=\underset{\delta, z}{\mathit{\min}}\left\{\delta :\sum \limits_{j=1}^n{\lambda}_j{X}_j\le \delta {X}_k,\frac{Y_k}{\delta}\le \sum \limits_{j=1}^n{\lambda}_j{Y}_j,{\lambda}_j\ge 0,j=1,\dots, n\right\} \).
- 5.
- 6.
In terms of the hyperbolic distance function, \( {ITE}_c\left({X}_k,{Y}_k\right)={D}_c^H{\left({X}_k,{Y}_k\right)}^2 \).
- 7.
Shadow prices are obtained for DMUk through the linear dual of program (33).
- 8.
These shadow prices come from the optimization model that appears in Proposition 1.
- 9.
The number of technically efficient banks reduces to seven under constant returns to scale, the standard assumption in the cross-efficiency literature.
References
Aczél, J., & Roberts, F. S. (1989). On the possible merging functions. Mathematical Social Sciences, 17, 205–243.
Akbarian, D. (2015). Ranking all DEA-efficient DMUs based on cross efficiency and analytic hierarchy process methods. Journal of Optimization, 2015, 594727, 10 pages.
Ali, A. I., & Seiford, L. M. (1993). The mathematical programming approach to efficiency analysis. In H. O. Fried, C. A. K. Lovell, & S. S. Schmidt (Eds.), The measurement of productive efficiency: Techniques and applications (pp. 120–159). New York, Oxford: Oxford University Press.
Amirteimoori, A., & Kordrostami, S. (2005). Efficient surfaces and an efficiency index in DEA: A constant returns to scale. Applied Mathematics and Computation, 163(2), 683–691.
Aparicio, J., & Zofío, J. L. (2019). Economic cross-efficiency: Theory and DEA methods. ERIM Report Series Research in Management, No. ERS-2019-001-LIS. Erasmus Research Institute of Management (ERIM). Erasmus University Rotterdam, The Netherlands. http://hdl.handle.net/1765/115479
Aparicio, J., Pastor, J. T., & Zofio, J. L. (2015). How to properly decompose economic efficiency using technical and allocative criteria with non-homothetic DEA technologies. European Journal of Operational Research, 240(3), 882–891.
Aparicio, J., Borras, J., Pastor, J. T., & Zofío, J. L. (2016). Loss distance functions and profit function: General duality results. In J. Aparicio, C. A. Knox Lovell, & J. T. Pastor (Eds.), Advances in efficiency and productivity (pp. 76–91). New York: Springer.
Aparicio, J., Pastor, J. T., Sainz-Pardo, J. L., & Vidal, F. (2017a). Estimating and decomposing overall inefficiency by determining the least distance to the strongly efficient frontier in data envelopment analysis. Operational Research, 1–24. https://doi.org/10.1007/s12351-017-0339-0.
Aparicio, J., Borras, F., Ortiz, L., Pastor, J.T., & Vidal, F. (2017b). Luenberger-type indicators based on the weighted additive distance function. Annals of Operations Research, 1–19. https://doi.org/10.1007/s10479-017-2620-2.
Balk, B. M. (2018). Profit-oriented productivity change: A comment. Omega, 78, 176–178.
Balk, B. M., de Koster, M. B. M., Kaps, C., & Zofío, J. L. (2017). An evaluation of cross-efficiency methods, applied to measuring warehouse performance. ERIM Report Series Research in Management No. ERS-2017-015-LIS, Erasmus Research Institute of Management (ERIM), Erasmus University Rotterdam, The Netherlands. http://hdl.handle.net/1765/103185
Banker, R. D., Charnes, A., & Cooper, W. W. (1984). Some models for estimating technical and scale inefficiencies in data envelopment analysis. Management Science, 30(9), 1078–1092.
Briec, W. (1998). Hölder distance function and measurement of technical efficiency. Journal of Productivity Analysis, 11(2), 111–131.
Briec, W., & Lesourd, J. B. (1999). Metric distance function and profit: Some duality results. Journal of Optimization Theory and Applications, 101(1), 15–33.
Chambers, R. G., Chung, Y., & Färe, R. (1998). Profit, directional distance functions and Nerlovian efficiency. Journal of Optimization Theory and Applications, 98(2), 351–364.
Charnes, A., Cooper, W. W., & Rhodes, E. (1978). Measuring the efficiency of decision making units. European Journal of Operational Research, 2(6), 429–444.
Chavas, J.-P., & Cox, T. M. (1999). A generalized distance function and the analysis of production efficiency. Southern Economic Journal, 66(2), 295–318.
Cooper, W. W., Pastor, J. T., Aparicio, J., & Borras, F. (2011). Decomposing profit inefficiency in DEA through the weighted additive model. European Journal of Operational Research, 212(2), 411–416.
Doyle, J. R., & Green, R. H. (1994). Efficiency and cross-efficiency in DEA: Derivations, meanings, and uses. Journal of the Operational Research Society, 45, 567–578.
Färe, R., & Primont, D. (1995). Multi-output production and duality: Theory and applications. Boston: Kluwer Academic.
Färe, R., Grosskopf, S., & Zaim, O. (2002). Hyperbolic efficiency and return to the dollar. European Journal of Operational Research, 136(3), 671–679.
Farrell, M. J. (1957). The measurement of productive efficiency. Journal of the Royal Statistical Society, Series A, General, 120, 253–281.
Førsund, F. R. (2018a). Economic interpretations of DEA. Socio-Economic Planning Sciences, 61, 9–15.
Førsund, F. R. (2018b). Cross-efficiency: A critique. Data Envelopment Analysis Journal, 4, 1–25.
Georgescu-Roegen, N. (1951). The aggregate linear production function and its application to von Newman’s economic model. In T. Koopmans (Ed.), Activity analysis of production and allocation (pp. 98–115). New York: Wiley.
Juo, J. C., Fu, T. T., Yu, M. M., & Lin, Y. H. (2015). Profit-oriented productivity change. Omega, 57, 176–187.
Liang, L., Wu, J., Cook, W. D., & Zhu, J. (2008a). Alternative secondary goals in DEA cross-efficiency evaluation. International Journal of Production Economics, 113(2), 1025–1030.
Liang, L., Wu, J., Cook, W. D., & Zhu, J. (2008b). The DEA game cross-efficiency model and its Nash equilibrium. Operations Research, 56(5), 1278–1288.
Lim, S. (2012). Minimax and maximin formulations of cross-efficiency in DEA. Computers and Industrial Engineering, 62(3), 726–731.
Lim, S., & Zhu, J. (2015). DEA cross-efficiency evaluation under variable returns to scale. Journal of the Operational Research Society, 66(3), 476–487.
Luenberger, D. G. (1992). New optimality principles for economic efficiency and equilibrium. Journal of Optimization Theory and Applications, 75(2), 221–264.
Maniadakis, N., & Thanassoulis, E. (2004). A cost Malmquist productivity index. European Journal of Operational Research, 154(2), 396–409.
Nerlove, M. (1965). Estimation and identification of Cobb-Douglas production functions. Chicago: Rand McNally.
Olesen, O. B. (2018). Cross efficiency analysis and extended facets. Data Envelopment Analysis Journal, 4, 27–64.
Ramón, N., Ruiz, J. L., & Sirvent, I. (2010). On the choice of weights profiles in cross-efficiency evaluations. European Journal of Operations Research, 207, 1564–1572.
Ruiz, J. L. (2013). Cross-efficiency evaluation with directional distance functions. European Journal of Operational Research, 228(1), 181–189.
Sexton, T. R., Silkman, R. H., & Hogan, A. J. (1986). Data envelopment analysis: Critique and extensions. In R. H. Silkman (Ed.), Measuring efficiency: An assessment of data envelopment analysis, new directions for program evaluation (Vol. 32, pp. 73–105). San Francisco/London: Jossey-Bass.
Shephard, R. W. (1953). Cost and production functions. Princeton: Princeton University Press.
Wu, J., Liang, L., & Chen, Y. (2009). DEA game cross-efficiency approach to Olympic rankings. Omega, 37(4), 909–918.
Zofío, J. L., & Prieto, A. M. (2006). Return to dollar, generalized distance function and the Fisher productivity index. Spanish Economic Review, 8(2), 113–138.
Acknowledgments
J. Aparicio and J. L. Zofío thank the financial support from the Spanish Ministry of Economy and Competitiveness (Ministerio de Economía, Industria y Competitividad), the State Research Agency (Agencia Estatal de Investigación), and the European Regional Development Fund (Fondo Europeo de Desarrollo Regional) under grant no. MTM2016-79765-P (AEI/FEDER, UE).
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Aparicio, J., Zofío, J.L. (2020). New Definitions of Economic Cross-efficiency. In: Aparicio, J., Lovell, C., Pastor, J., Zhu, J. (eds) Advances in Efficiency and Productivity II. International Series in Operations Research & Management Science, vol 287. Springer, Cham. https://doi.org/10.1007/978-3-030-41618-8_2
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