Abstract
This chapter is concerned with the measurement of profitability efficiency, defined as the ratio of revenue to cost, and its multiplicative decomposition into a productive efficiency measure—including technical and scale efficiencies, corresponding to the generalized distance function introduced by Chavas and Cox (1999), and allocative efficiency. The generalized distance function, GDF, received such name by these authors because it generalizes Shepard’s radial distance functions and the graph (hyperbolic) efficiency measure introduced by Färe et al. (1985:110–111). Building upon this measure, which can be reinterpreted in terms of a distance function, these authors extended the input- and output-oriented measures to a graph representation of the technology including both dimensions of the production technology. In contrast to the partial dimensions represented by input and output orientations, the hyperbolic technical efficiency measure, presented in Sect 2.1.3 of Chap. 2, is a scalar value function that projects the firm under evaluation to the production frontier by simultaneously reducing its inputs and increasing its outputs. As we show below, Chavas and Cox (1999) qualified this definition by making these changes dependent on an exponent that weights the outputs and inputs differently. Therefore, setting the value of such bearing (or directional) parameter to a specific value, it is possible to recover, among others, the hyperbolic efficiency measure as well as Farrell’s input and output radial counterparts. Since the latter corresponds to Shepard’s input and output distance functions, as shown in Chap. 3, the generalized distance function represents an improvement over the previous definitions, by adding flexibility to the orientation and as we show below providing a dual counterpart to the profitability function.
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Notes
- 1.
They refer to this index as the “Farrell Graph Measure of Technical Efficiency.” The measure inherits its name from the hyperbolic path it follows toward the production frontier.
- 2.
These characteristics are not satisfied by the additive directional distance functions, whose numeric interpretation as measure of productivity change, based on the so-called Luenberger indicator, cannot be directly related to conventional index number theory. Index numbers define as ratios, while productivity indicators define as differences. Therefore, the interpretation of the latter is not straightforward. See Diewert (2005), Balk et al. (2008), and Briec et al. (2012) for a discussion of the differences between the two approaches. A recent application of productivity analysis based on the Luenberger indicator is carried out by Juo et al. (2015), whose data is used in this book to illustrate the different models—see Balk (2018) for relevant qualifications.
- 3.
Cuesta and Zofío (2005) introduce the parametric methods allowing the calculation of the hyperbolic distance function using Stochastic Frontier Analysis. These authors show that the almost homogeneity condition can be easily imposed on a translog specification of the distance function. Subsequently, Cuesta et al. (2009) extend the model to the field of environmental economics including the production of undesirable outputs as by-product.
- 4.
- 5.
Resorting to trigonometry, scale efficiency defines equivalently as the ratio of the slope of the ray vector joining the origin and the efficient projection on the variable returns to scale technology to that of observation A, i.e., in Fig. 2.3b, SEG(x, y; α) = tan β/tan α.
- 6.
This is one of the regularity conditions imposed on the production function or, more generally, the transformation function, so they are well-behaved; i.e., they are characterized by a decreasing (increasing) marginal rates of substitution (transformation) between inputs (outputs) resulting from strictly convex (concave) isoquants.
- 7.
An exception can be found in the field of environmental economics, where the technological trade-off between desirable and undesirable outputs, including the null-jointness axiom, is usually modelled through their weak disposability. Färe et al. (1989) develop the environmental efficiency model based on the hyperbolic distance function where inputs and undesirable outputs are reduced, while desirable outputs are increased. See Zaim and Taskin (2000) for a later application of this model.
- 8.
Again, this issue is not of concern in the parametric approach based on well-behaved production functions satisfying the desirable regularity conditions. The usual functional forms present equal strong, weak, and isoquant efficiency subsets. An example is the Cobb-Douglas function, whose input set corresponds to \( L(y)=\left\{{\prod}_{m=1}^M{x}_m^{\alpha_m}\ge y,\kern0.5em {\alpha}_m>0\right\} \).
- 9.
- 10.
A facet is a subset of a T supporting hyperplane satisfying that each of its points can be expressed as a convex linear combination of a finite subset of strong efficient firms belonging to the facet.
- 11.
Again, it is relevant to remark that the weakly efficient subset ∂W(TS), generated under the DEA assumption of strong disposability of inputs and outputs, must not be confused with the production set under the assumption of weakly disposable inputs and outputs, TW.
- 12.
In passing, we note that a second-stage evaluation of the CRS model, searching for input and output slacks, can be performed to shed light on these sources of inefficiency.
- 13.
The relationship between the hyperbolic and input technical efficiency measures is explored in detail by Halická and Trnovská (2019:415).
- 14.
We also note that there may exist multiple solutions to problem (4.15), which results in different total factor productivity values depending on the specific vector of multipliers that is obtained (μCRS∗, νCRS∗). This lack of uniqueness is especially important in interpreting the multipliers and slacks. Note that the existence of output and input slacks, associated with zero multipliers, implies that the attained productivity would be larger if the aggregate output and aggregate input are larger or smaller, respectively.
- 15.
Ipopt, short for “Interior Point Optimizer,” is a software library for large-scale nonlinear optimization of continuous systems. At the time of writing, the latest stable is 3.14.4, from Sep. 20, 2021. See https://github.com/coin-or/Ipopt. A list of commercial and free solvers compatible with the Julia (JuMP) environment can be found in http://www.juliaopt.org/JuMP.jl/v0.20.0/installation/
- 16.
It is assumed that the optimization problem associated with the calculation of Γ(w, p) always attains its maximum in T.
- 17.
In the standard, single output-multiple input production function, y = f(x), scale elasticity defines as follows: \( \varepsilon \left(x,y\right)={\left.\left(\partial \ln f\left(\psi x\right)/\partial \ln \psi \right)\right|}_{\psi =1}={\sum}_{m=1}^M\left(\partial f(x)/\partial {x}_m\right)\cdot \left({x}_m/y\right)={\sum}_{m=1}^M{\varepsilon}_m \). The definition implies proportional changes in the inputs quantities; i.e., the input mix (relative input quantities) remains unchanged.
- 18.
This notation stresses that if the technology exhibits global constant returns to scale, the benchmark and actual technologies coincide. Despite the fact that the actual technology exhibits variable returns to scale, when using the nonparametric Data Envelopment Analysis (DEA) methods discussed in Sect. 4.2.3, researchers approximate the benchmark technology from observed data both under global CRS and VRS and then recover scale efficiency through (4.6). The global CRS characterization allows identifying the reference hyperplanes (faces) consistent with local CRS, i.e., the outer approximation of TCRS.
- 19.
In this case, it is assumed once again that the profitability function is continuous and twice differentiable.
- 20.
Here, X(x,ν) and Y (y,μ) are aggregator functions that are nonnegative, nondecreasing, and linearly homogeneous, i.e., satisfy constant returns to scale.
- 21.
For the first-level decomposition of profitability efficiency, involving CRS, the value of the bearing parameter α is irrelevant, and the function takes advantage of the known relationship \( {D}_G^{CRS}\left(x,y;\alpha \right) \) = \( {D}_O^{CRS}\left(x,y\right) \), ∀α∈[0, 1], Then, the function internally solves the equivalent model corresponding to the output distance function, whose inverse, in turn, can be easily computed with a linear DEA program; see program (3.25) of the preceding chapter.
- 22.
We refer the reader to Sect. 2.6.1 in Chap. 2 for the installation of the “Benchmarking Economic Efficiency” Julia package. All Jupyter Notebooks implementing the different economic models in this book can be downloaded from the reference site: https://www.benchmarkingeconomicefficiency.com
- 23.
To prevent that the reported set of peers include observations with infinitesimal intensity variables, it is possible to adjust the tolerance of the solver in the “peers” function, for example,
. - 24.
In Table 4.4, firm F has as peers firms D and G, being the last one a weak efficient firm. In case we deleted G, F would have as peer only firm D plus the slack input amount \( {s}_{\mathrm{F}}^{-} \) =0.57, determining the new weak efficient point (12,9.57).
.References
Aczél, J. (1966). Lectures on functional equations and their applications. Academic.
Ali, A. I., & Seiford, L. M. (1993). Computational accuracy and infinitesimals in data envelopment analysis. Infor, 31(4), 290–297.
Balk, B. M. (2003). The residual: On monitoring and benchmarking firms, industries and economies with respect to productivity. Journal of Productivity Analysis, 20, 5–47.
Balk, B. M. (2008). Price and quantity index numbers: Models for measuring aggregate change and difference. Cambridge University Press.
Balk, B. M. (2018). Profit-oriented productivity change: A comment. Omega, 78(C), 176–178.
Balk, B. M., and Zofío, J. L. (2018). The many decompositions of Total factor productivity change (ERIM report series research in management, no. ERS-2018-003-LIS). Erasmus Research Institute of Management, Erasmus University, Rotterdam. Available from http://hdl.handle.net/1765/104721
Balk, B. M., Färe, R., Grosskopf, S., & Margaritis, D. (2008). Exact relations between Luenberger productivity indicators and Malmquist productivity indexes. Economic Theory, 35(1), 187–190.
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. (1997). A graph-type extension of Farrell technical efficiency measure. Journal of Productivity Analysis, 8(1), 95–110.
Briec, W., Kerstens, K., & Peypoch, N. (2012). Exact relations between four definitions of productivity indices a indicators. Bulleting of Economic Research, 64(2), 265–274.
Camanho, A. S., & Dyson, R. G. (2008). A generalisation of the Farrell cost efficiency measure applicable to non-fully competitive settings. Omega, 36, 147–162.
Chambers, R. G., Chung, Y., & Färe, R. (1996). Benefit and distance functions. Journal of Economic Theory, 70, 407–419.
Chambers, R. G., Chung, Y., & Färe, R. (1998). Profit, directional distance functions, and Nerlovian efficiency. Journal of Optimization Theory and Applications, 98, 351–364.
Charnes, A., Cooper, W. W., & Rhodes, E. (1978). Measuring the efficiency of decision making units. European Journal of Operational Research, 2, 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.
Cherchye, L., De Rock, B., & Walheer, B. (2015). Multi-output efficiency with good and bad outputs. European Journal of Operational Research, 240, 872–881.
Cooper, W., Seiford, L., & Tone, K. (2007). Data envelopment analysis. A comprehensive text with models, applications, references and DEA-software (2nd ed.). Kluwer, Boston.
Cuesta, R. A., & Zofío, J. L. (2005). Hyperbolic efficiency and parametric distance functions: With application to Spanish savings banks. Journal of Productivity Analysis, 24, 31–48.
Cuesta, R. A., Lovell, C. A. K., & Zofío, J. L. (2009). Environmental efficiency measurement with translog distance functions. Ecological Economics, 68, 2232–2242.
Dakpo, K. H., Jeanneaux, P., & Latruffe, L. (2016). Modelling pollution-generating technologies in performance benchmarking: Recent developments, limits and future prospects in the nonparametric framework. European Journal of Operational Research, 250(2), 347–359.
Diewert, W. E. (2005). Index number theory using differences rather than ratios. The American Journal of Economics and Sociology, 64(1), 311–360.
Diewert, W. E. (2014). Decompositions of profitability change using cost functions. Journal of Econometrics, 183, 58–66.
Dunning, I., Huchette, J., & Lubin, M. (2017). JuMP: A modeling language for mathematical optimization. SIAM Review, 59(2), 295–320.
Färe, R., & Primont, D. (1995). Multi-output production and duality: Theory and applications. Kluwer Academic Publishers.
Färe, R., Grosskopf, S., & Lovell, C. A. K. (1985). The measurement of efficiency of production. Kluwer-Nijhoff.
Färe, R., Grosskopf, S., Lovell, C. A. K., & Pasurka, C. (1989). Multilateral productivity comparisons when some outputs are undesirable: A nonparametric approach. The Review of Economics and Statistics, 71(1), 90–98.
Färe, R., Grosskopf, S., & Lovell, C. A. K. (1994). Production Frontiers. Cambridge University Press.
Färe, R., Grosskopf, S., & Zaim, T. (2002). Hyperbolic efficiency and return to dollar. European Journal of Operational Research, 136, 671–679.
Färe, R., Margaritis, D., Rouse, P., & Roshdi, I. (2016). Estimating the hyperbolic distance function: A directional distance function approach. European Journal of Operational Research, 254, 312–319.
Farrell, M. J. (1957). The measurement of productive efficiency of production. Journal of the Royal Statistical Society, Series A, 120(III), 253–281.
Georgescu-Roegen, N. (1951). The aggregate linear production function and its application to von Neumann’s economic model. In T. Koopmans (Ed.), Activity analysis of production and allocation (pp. 116–131). Wiley.
Grifell-Tatjé, E., & Lovell, C. K. (2015). Productivity accounting. Cambridge University Press.
Hailu, A., & Veeman, T. S. (2001). Non-parametric productivity analysis with undesirable outputs: An application to the Canadian pulp and paper industry. American Journal of Agricultural Economics, 83, 605–616.
Halická, M., & Trnovská, M. (2019). Duality and profit efficiency for the hyperbolic measure model. European Journal of Operational Research, 278, 410–421.
Jiménez-Sáez, F., Zabala-Iturriagagoitia, J. M., Zofío, J. L., & Castro-Martinez, E. (2011). Evaluating research efficiency within National R&D Programs. Research Policy, 40, 230–241.
Jiménez-Sáez, F., Zabala-Iturriagagoitia, J. M., & Zofío, J. L. (2013). Who leads research productivity growth? Guidelines for R&D policy-makers. Scientometrics, 94, 273–303.
Johnson, A. L., & McGinnis, L. F. (2009). The hyperbolic-oriented efficiency measure as a remedy to infeasibility of super efficiency models. Journal of the Operational Research Society, 60(11), 1511–1517.
Juo, J.-C., Fu, T.-T., Yu, M.-M., & Lin, Y.-H. (2015). Profit-oriented productivity change. Omega, 57, 176–187.
Kendrick, J. W. (1984). Improving company productivity: Handbook with case studies. Johns Hopkins University Press.
Koopmans, T. C. (1951). An analysis of production as an efficient combination of activities. In T. C. Koopmans (Ed.), Activity analysis of production and allocation (Cowles Commission for Research in economics, monograph N° 13). Wiley.
Luenberger, D. G. (1992). Benefit functions and duality. Journal of Mathematical Economics, 21, 461–481.
McFadden, D. (1978). Cost, revenue, and profit functions. In M. Fuss & D. McFadden (Eds.), Production economics: A dual approach to theory and applications. North-Holland.
MirHassani, S. A., & Alirezaee, M. (2005). An efficient approach for computing non-Archimedean ε in DEA based on integrated models. Applied Mathematics and Computation, 166(2), 449–456.
O’Donnell, C. J. (2008). An aggregate quantity-price framework for measuring and decomposing productivity and profitability change (Centre for Efficiency and Productivity Analysis Working Papers WP07/2008). University of Queensland.
O’Donnell, C. J. (2012). An aggregate quantity framework for measuring and decomposing.
Portela, M. C. A. S., & Thanassoulis, E. (2014). Economic efficiency when prices are not fixed: Disentangling quantity and price efficiency. Omega, 47, 36–44.
Seiford, L. M., & Zhu, J. (2002). Modeling undesirable factors in efficiency evaluation. Europena Journal of Operational Research, 142(1), 16–20.
Smith, I. G. (1973). The measurement of productivity: A systems approach in the context of productivity agreements. Gower Press.
Wächter, A., & Biegler, L. T. (2006). On the implementation of a primal-dual interior point filter line search algorithm for large-scale nonlinear programming. Mathematical Programming, 106, 25–57.
Wheelock, D. C., & Wilson, P. W. (2009). Robust nonparametric quantile estimation of efficiency and productivity change in U.S. commercial banking, 1985-2004. Journal of Business and Economic Statistics, 27(3), 354–368.
Yu, M. M., & Lee, B. C. Y. (2009). Efficiency and effectiveness of service business: Evidence from international tourist hotels in Taiwan. Tourism Management, 30(4), 571–580.
Zaim, O., & Taskin, F. (2000). Environmental efficiency in carbon dioxide emissions in the OECD: A non-parametric approach. Journal of Environmental Management, 58(2), 95–107.
Zofío, J. L., & Lovell, C. A. K. (2001). Graph efficiency and productivity measures: An application to US agriculture. Applied Economics, 33(11), 1433–1442.
Zofío, J. L., & Prieto, A. M. (2001). Environmental efficiency and regulatory standards: The case of CO2 emission from OECD industries. Resource and Energy Economics, 23(1), 63–83.
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.
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Pastor, J.T., Aparicio, J., Zofío, J.L. (2022). The Generalized Distance Function (GDF): Profitability Efficiency Decomposition. In: Benchmarking Economic Efficiency. International Series in Operations Research & Management Science, vol 315. Springer, Cham. https://doi.org/10.1007/978-3-030-84397-7_4
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