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Performance analysis: economic foundations and trends. (English) Zbl 1498.91223

Summary: The goal of this monograph is to very concisely outline the economic theory foundations and trends of the field of efficiency and productivity analysis, also sometimes referred to as performance analysis. I start with the profit maximization paradigm of mainstream economics, use it to derive a general profit efficiency measure and then present its special cases: revenue maximization and revenue efficiency, cost minimization and cost efficiency. I then consider various types of technical and allocative efficiencies (directional and Shephard’s distance functions and related Debreu-Farrell measures as well as non-directional measures of technical efficiency), showing how they fit into or decompose the profit maximization paradigm. I then cast the efficiency and productivity concepts in a dynamic perspective that is frequently used to analyze the productivity changes of economic systems (firms, hospitals, banks, countries, etc.) over time. I conclude this monograph with an overview of major results on aggregation in productivity and efficiency analysis, where the aggregate productivity and efficiency measures are theoretically connected to their individual analogues.

MSC:

91B38 Production theory, theory of the firm
91-02 Research exposition (monographs, survey articles) pertaining to game theory, economics, and finance

Software:

DEA
Full Text: DOI

References:

[1] Aigner, D., C. Lovell, and P. Schmidt (1977). “Formulation and estima-tion of stochastic frontier production function models”. Journal of Econometrics. 6(1): 21-37. · Zbl 0366.90026
[2] Aparicio, J., J. T. Pastor, and S. C. Ray (2013). “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(1): 154-162. · Zbl 1292.90136
[3] Badunenko, O., D. J. Henderson, and V. Zelenyuk (2008). “Technological change and transition: Relative contributions to worldwide growth during the 1990s”. Oxford Bulletin of Economics and Statistics. 70(4): 461-492.
[4] Badunenko, O., D. J. Henderson, and V. Zelenyuk (2017). “The produc-tivity of nations”. In: The Oxford Handbook of Productivity. Ed. by E. Grifell-Tatjé, C. A. K. Lovell, and R. C. Sickles. New York, NY: Oxford University Press. Chap. 8.
[5] Balk, B. M. (1998). Industrial Price, Quantity, and Productivity In-dices: The Micro-Economic Theory and an Application. Boston, MA: Kluwer Academic Publishers.
[6] Balk, B. M. (2008). Price and Quantity Index Numbers: Models for Mea-suring Aggregate Change and Difference. New York, NY: Cambridge University Press. · Zbl 1167.91001
[7] Bjurek, H. (1996). “The Malmquist total factor productivity index”. The Scandinavian Journal of Economics. 98(2): 303-313.
[8] Blackorby, C., C. A. K. Lovell, and M. C. Thursby (1976). “Extended Hicks neutral technical change”. The Economic Journal. 86(344): 845-852.
[9] Blackorby, C. and R. R. Russell (1999). “Aggregation of efficiency indices”. Journal of Productivity Analysis. 12(1): 5-20.
[10] Bogetoft, P. and D. Wang (2005). “Estimating the potential gains from mergers”. Journal of Productivity Analysis. 23(2): 145-171.
[11] Boussofiane, A., R. G. Dyson, and E. Thanassoulis (1991). “Applied data envelopment analysis”. European Journal of Operational Research. 52(1): 1-15.
[12] Briec, W., B. Dervaux, and H. Leleu (2003). “Aggregation of directional distance functions and industrial efficiency”. Journal of Economics. 79(3): 237-261. · Zbl 1058.91048
[13] Briec, W. and K. Kerstens (2004). “A Luenberger-Hicks-Moorsteen productivity indicator: Its relation to the Hicks-Moorsteen produc-tivity index and the Luenberger productivity indicator”. Economic Theory. 23(4): 925-939. · Zbl 1100.91070
[14] Briec, W. and K. Kerstens (2011). “The Hicks-Moorsteen productivity index satisfies the determinateness axiom”. The Manchester School. 79(4): 765-775.
[15] Caves, D. W., L. R. Christensen, and W. E. Diewert (1982). “The economic theory of index numbers and the measurement of input, output, and productivity”. Econometrica. 50(6): 1393-1414. · Zbl 0524.90028
[16] Chambers, R., Y. Chung, and R. Färe (1998). “Profit, directional dis-tance functions, and Nerlovian efficiency”. Journal of Optimization Theory and Applications. 98(2): 351-364. · Zbl 0909.90040
[17] Chambers, R. G., Y. Chung, and R. Färe (1996). “Benefit and distance functions”. Journal of Economic Theory. 70(2): 407-419. · Zbl 0866.90027
[18] Chambers, R. G. and R. Färe (1994). “Hicks neutrality and trade biased growth: A taxonomy”. Journal of Economic Theory. 64(2): 554-567. · Zbl 0813.90015
[19] Chambers, R. G. and R. Färe (1998). “Translation homotheticity”. Economic Theory. 11(3): 629-641. · Zbl 0903.90013
[20] Charnes, A., W. Cooper, and E. Rhodes (1978). “Measuring the effi-ciency of decision making units”. European Journal of Operational Research. 2(6): 429-444. · Zbl 0416.90080
[21] Charnes, A., W. W. Cooper, A. Y. Lewin, and L. M. Seiford, eds. (1994). Data Envelopment Analysis: Theory, Methodology, and Applications. Amsterdam, Netherlands: Springer.
[22] Cook, W. and L. Seiford (2009). “Data Envelopment Analysis (DEA) -Thirty years on”. European Journal of Operational Research. 192(1): 1-17. · Zbl 1180.90151
[23] Cooper, W., Z. Huang, S. Li, B. Parker, and J. Pastor (2007b). “Ef-ficiency aggregation with enhanced Russell measures in data en-velopment analysis”. Socio-Economic Planning Sciences. 41(1): 1-21.
[24] Cooper, W., J. T. Pastor, J. Aparicio, and F. Borras (2011b). “De-composing profit inefficiency in DEA through the weighted additive model”. European Journal of Operational Research. 212(2): 411-416. · Zbl 1237.90157
[25] Cooper, W. W., L. M. Seiford, and K. Tone (2007a). Data Envelop-ment Analysis: A Comprehensive Text with Models, Applications, References and DEA-Solver Software. New York, NY: Springer. · Zbl 1111.90001
[26] Cooper, W. W., L. M. Seiford, and J. Zhu, eds. (2011a). Handbook on Data Envelopment Analysis. New York, NY: Springer.
[27] Debreu, G. (1951). “The coefficient of resource utilization”. Economet-rica. 19(3): 273-292. · Zbl 0045.41404
[28] Diewert, W. E. (1971). “An application of the Shephard duality theorem: A generalized Leontief production function”. Journal of Political Economy. 79(3): 481-507.
[29] Diewert, W. E. (1973). “Functional forms for profit and transformation functions”. Journal of Economic Theory. 6(3): 284-316.
[30] Diewert, W. E. (1974). “Applications of duality theory”. In: Frontiers in Quantitative Economics. Ed. by M. D. Intriligator and D. A. Kendrick. Vol. 2. Amsterdam, Netherlands: Elsevier. 106-171. · Zbl 0331.90051
[31] Diewert, W. E. (1982). “Duality approaches to microeconomic the-ory”. In: Handbook of Mathematical Economics. Vol. 2. Amsterdam, Netherlands: Elsevier. 535-599. · Zbl 0522.90007
[32] Diewert, W. E. (1992). “Fisher ideal output, input, and productivity indexes revisited”. Journal of Productivity Analysis. 3(3): 211-248.
[33] Dmitruk, A. V. and G. A. Koshevoy (1991). “On the existence of a technical efficiency criterion”. Journal of Economic Theory. 55(1): 121-144. · Zbl 0754.90004
[34] Emrouznejad, A., B. Parker, and G. Tavares (2008). “Evaluation of research in efficiency and productivity: A survey and analysis of the first 30 years of scholarly literature in DEA”. Journal of Socio-Economics Planning Science. 42(3): 151-157.
[35] Epure, M., K. Kerstens, and D. Prior (2011). “Technology-based to-tal factor productivity and benchmarking: New proposals and an application”. Omega. 39(6): 608-619.
[36] Färe, R., F. R. Førsund, S. Grosskopf, K. Hayes, and A. Heshmati (2001). “A note on decomposing the Malmquist productivity index by means of subvector homotheticity”. Economic Theory. 17(1): 239-245. · Zbl 0970.91057
[37] Färe, R., H. Fukuyama, S. Grosskopf, and V. Zelenyuk (2015). “De-composing profit efficiency using a slack-based directional distance function”. European Journal of Operational Research. 247(1): 335-337. · Zbl 1346.90422
[38] Färe, R. and S. Grosskopf (1983). “Measuring congestion in production”. Zeitschrift für Nationalökonomie/Journal of Economics. 43(3): 257-271. · Zbl 0519.90019
[39] Färe, R. and S. Grosskopf (2009). “A comment on weak disposabil-ity in nonparametric production analysis”. American Journal of Agricultural Economics. 91(2): 535-538.
[40] Färe, R., S. Grosskopf, M. Norris, and Z. Zhang (1994). “Productivity growth, technical progress, and efficiency change in industrialized countries”. American Economic Review. 84(1): 66-83.
[41] Färe, R., S. Grosskopf, and P. Roos (1996). “On two definitions of productivity”. Economics Letters. 53(3): 269-274. · Zbl 0897.90018
[42] Färe, R., S. Grosskopf, and V. Zelenyuk (2004). “Aggregation of cost efficiency: Indicators and indexes across firms”. Academia Economic Papers. 32(3): 395-411.
[43] Färe, R., S. Grosskopf, and V. Zelenyuk (2007). “Finding common ground: Efficiency indices”. In: Aggregation, Efficiency and Mea-surement. Ed. by R. Färe, S. Grosskopf, and D. Primont. Boston, MA: Springer. 83-95. · Zbl 1165.90490
[44] Färe, R., S. Grosskopf, and V. Zelenyuk (2008). “Aggregation of Nerlo-vian profit indicator”. Applied Economics Letters. 15(11): 845-847.
[45] Färe, R., X. He, S. K. Li, and V. Zelenyuk (2019). “A unifying framework for Farrell profit efficiency measurement”. Operations Research. 67(1): 183-197. · Zbl 1455.91132
[46] Färe, R. and G. Karagiannis (2014). “A postscript on aggregate Farrell efficiencies”. European Journal of Operational Research. 233(3): 784-786. · Zbl 1339.90116
[47] Färe, R. and C. K. Lovell (1978). “Measuring the technical efficiency of production”. Journal of Economic Theory. 19(1): 150-162. · Zbl 0398.90012
[48] Färe, R. and T. Mitchell (1993). “Multiple outputs and ”homotheticity’”. Southern Economic Journal. 60(2): 287-296.
[49] Färe, R., H. Mizobuchi, and V. Zelenyuk (2021). “Hicks neutrality and homotheticity in technologies with multiple inputs and multiple outputs”. Omega. 101: 102240.
[50] Färe, R. and D. Primont (1995). Multi-Output Production and Du-ality: Theory and Applications. New York, NY: Kluwer Academic Publishers.
[51] Färe, R. and L. Svensson (1980). “Congestion of production factors”. Econometrica: Journal of the Econometric Society. 48(7): 1745-1753. · Zbl 0457.90011
[52] Färe, R. and V. Zelenyuk (2003). “On aggregate Farrell efficiencies”. European Journal of Operational Research. 146(3): 615-620. · Zbl 1037.90516
[53] Färe, R. and V. Zelenyuk (2005). “On Farrell”s decomposition and aggregation”. International Journal of Business and Economics. 4: 167-171.
[54] Färe, R. and V. Zelenyuk (2007). “Extending Färe and Zelenyuk (2003)”. European Journal of Operational Research. 179(2): 594-595. · Zbl 1131.90380
[55] Färe, R. and V. Zelenyuk (2012). “Aggregation of scale elasticities across firms”. Applied Economics Letters. 19(16): 1593-1597.
[56] Färe, R. and V. Zelenyuk (2019). “On Luenberger input, output and productivity indicators”. Economics Letters. 179: 72-74. · Zbl 1414.91257
[57] Färe, R. and V. Zelenyuk (2020). “Profit efficiency: Generalization, business accounting and the role of convexity”. Economics Letters. 196: 109483. · Zbl 1451.91066
[58] Färe, R. and V. Zelenyuk (2021a). “On aggregation of multi-factor productivity indexes”. Journal of Productivity Analysis. 55(2): 107-133.
[59] Färe, R. and V. Zelenyuk (2021b). “Sequential data envelopment analy-sis”. Annals of Operations Research. 300: 307-312. · Zbl 1476.90145
[60] Farrell, M. J. (1957). “The measurement of productive efficiency”. Jour-nal of the Royal Statistical Society. Series A (General). 120(3): 253-290.
[61] Fisher, I. (1921). “The best form of index number”. Journal of the American Statistical Association. 17(133): 533-537.
[62] Førsund, F. R. and L. Hjalmarsson (1979). “Generalised Farrell measures of efficiency: An application to milk processing in Swedish dairy plants”. The Economic Journal. 89(354): 294-315.
[63] Førsund, F. R. and N. Sarafoglou (2002). “On the origins of data envelopment analysis”. Journal of Productivity Analysis. 17(1): 23-40.
[64] Foster, L., J. Haltiwanger, and C. Syverson (2008). “Reallocation, firm turnover, and efficiency: Selection on productivity or profitability?” American Economic Review. 98(1): 394-425.
[65] Fukuyama, H. and W. Weber (2009). “A directional slacks-based mea-sure of technical inefficiency”. Socio-Economic Planning Sciences. 43(4): 274-287.
[66] Georgescu-Roegen, N. (1951). “The aggregate linear production function and its applications to von Neumann”s economic model”. In: Activity Analysis of Production and Allocation. Ed. by T. Koopmans. New York, NY: Wiley. · Zbl 0045.09601
[67] Greene, W. H. (2005a). “Fixed and random effects in stochastic frontier models”. Journal of Productivity Analysis. 23(1): 7-32.
[68] Greene, W. H. (2005b). “Reconsidering heterogeneity in panel data estimators of the stochastic frontier model”. Journal of Econometrics. 126(2): 269-303. · Zbl 1334.62214
[69] Greene, W. H. (2007). “The econometric approach to efficiency analy-sis”. In: The Measurement of Productive Efficiency: Techniques and Applications. Ed. by H. O. Fried, C. A. K. Lovell, and S. S. Schmidt. New York, NY: Oxford University Press. References Hanoch, G. (1970). “Homotheticity in joint production”. Journal of Economic Theory. 2(4): 423-426.
[70] Henderson, D. J. and R. R. Russell (2005). “Human capital and con-vergence: A production-frontier approach”. International Economic Review. 46(4): 1167-1205.
[71] Henderson, D. J. and V. Zelenyuk (2007). “Testing for (efficiency) catching-up”. Southern Economic Journal. 73(4): 1003-1019.
[72] Hicks, J. R. (1932). The Theory of Wages. London, UK: MacMillan.
[73] Karagiannis, G. (2015). “On structural and average technical efficiency”. Journal of Productivity Analysis. 43(3): 259-267.
[74] Karagiannis, G. and C. A. K. Lovell (2015). “Productivity measurement in radial DEA models with a single constant input”. European Journal of Operational Research. 251(1): 323-328. · Zbl 1346.90591
[75] Konüs, A. A. (1939 [1924]). “The problem of the true index of the cost of living”. Econometrica. 7(1): 10-29. Translated into English and published in 1939. · JFM 65.0631.03
[76] Koopmans, T. (1957). Three Essays on the State of Economic Science. New York, NY: McGraw-Hill.
[77] Krein, M. and V. Smulian (1940). “On regulary convex sets in the space conjugate to a Banach space”. Annals of Mathematics. 41(2): 556-583. · Zbl 0024.41305
[78] Kumar, S. and R. R. Russell (2002). “Technological change, technological catch-up, and capital deepening: Relative contributions to growth and convergence”. American Economic Review. 92(3): 527-548.
[79] Kumbhakar, S. and P. Schmidt (2016). Endogeneity Problems in Econo-metrics, Special Issue of the Journal of Econometrics. Vol. 190. Amsterdam, Netherlands: Elsevier.
[80] Kumbhakar, S. C. (1987). “The specification of technical and allocative inefficiency in stochastic production and profit frontiers”. Journal of Econometrics. 34(3): 335-348. · Zbl 0611.90014
[81] Kumbhakar, S. C., C. F. Parmeter, and V. Zelenyuk (2021). “Stochastic frontier analysis: Foundations and advances I”. In: Handbook of Pro-duction Economics. Ed. by S. Ray, R. Chambers, and S. Kumbhakar. Singapore: Springer Singapore.
[82] Kumbhakar, S. C., C. F. Parmeter, and V. Zelenyuk (2021). “Stochastic frontier analysis: Foundations and advances II”. In: Handbook of Pro-duction Economics. Ed. by S. Ray, R. Chambers, and S. Kumbhakar. Singapore: Springer Singapore.
[83] Kuosmanen, T., M. Kortelainen, T. Sipiläinen, and L. Cherchye (2010). “Firm and industry level profit efficiency analysis using absolute and uniform shadow prices”. European Journal of Operational Research. 202(2): 584-594. · Zbl 1179.90111
[84] Lau, L. J. (1976). “A characterization of the normalized restricted profit function”. Journal of Economic Theory. 12(1): 131-163. · Zbl 0341.90012
[85] Leibenstein, H. (1966). “Allocative efficiency vs. ”X-efficiency’”. Ameri-can Economic Review. 56(3): 392-415.
[86] Leibenstein, H. (1975). “Aspects of the X-efficiency theory of the firm”. The Bell Journal of Economics. 6(2): 580-606.
[87] Leibenstein, H. (1987). Inside the Firm: The Inefficiencies of Hierarchy. Cambridge, MA: Harvard University Press.
[88] Leibenstein, H. and S. Maital (1992). “Empirical estimation and parti-tioning of X-inefficiency: A data-envelopment approach”. American Economic Review. 82(2): 428-433.
[89] Li, S.-K. and Y.-S. Cheng (2007). “Solving the puzzles of structural efficiency”. European Journal of Operational Research. 180(2): 713-722. · Zbl 1123.90325
[90] Li, S.-K. and Y. C. Ng (1995). “Measuring the productive efficiency of a group of firms”. International Advances in Economic Research. 1(4): 377-390.
[91] Malmquist, S. (1953). “Index numbers and indifference surfaces”. Tra-bajos de Estadistica. 4(2): 209-242. · Zbl 0052.15903
[92] Mas-Colell, A., M. D. Whinston, and J. R. Green (1995). Microeconomic Theory. New York: Oxford University Press. · Zbl 1256.91002
[93] Mayer, A. and V. Zelenyuk (2014). “Aggregation of Malmquist pro-ductivity indexes allowing for reallocation of resources”. European Journal of Operational Research. 238(3): 774-785. · Zbl 1338.91085
[94] Mayer, A. and V. Zelenyuk (2019). “Aggregation of individual efficiency measures and productivity indices”. In: The Palgrave Handbook of Economic Performance Analysis. Ed. by T. ten Raa and W. H.
[95] Greene. Cham, Switzerland: Springer International Publishing. 527-557.
[96] McFadden, D. (1978). “Cost, revenue, and profit functions”. In: Produc-tion Economics: A Dual Approach to Theory and Applications. Ed. by D. McFadden and M. A. Fuss. Vol. 1: The Theory of Production. Amsterdam, Netherlands: North-Holland. Chap. 1.
[97] Mizobuchi, H. (2017a). “Productivity indexes under Hicks neutral tech-nical change”. Journal of Productivity Analysis. 48(1): 63-68.
[98] Mizobuchi, H. (2017b). “A superlative index number formula for the Hicks-Moorsteen productivity index”. Journal of Productivity Anal-ysis. 48(2-3): 167-178.
[99] Mizobuchi, H. and V. Zelenyuk (2021). “Quadratic-mean-of-order-r indexes of output, input and productivity”. Journal of Productivity Analysis. Forthcoming.
[100] Mussard, S. and N. Peypoch (2006). “On multi-decomposition of the aggregate Malmquist productivity index”. Economics Letters. 91(3): 436-443.
[101] Nesterenko, V. and V. Zelenyuk (2007). “Measuring potential gains from reallocation of resources”. Journal of Productivity Analysis. 28(1-2): 107-116.
[102] Nguyen, B. H. and V. Zelenyuk (2021a). “Aggregate efficiency of industry and its groups: The case of Queensland public hospitals”. Empirical Economics. 60: 2795-2836.
[103] Nguyen, B. H. and V. Zelenyuk (2021b). “Aggregation of outputs and inputs for DEA analysis of hospital efficiency: Economics, operations research and data science perspectives”. In: Data-Enabled Analyt-ics: DEA for Big Data. Ed. by J. Zhu and V. Charles. Springer. Forthcoming.
[104] Oks, E. and M. Sharir (2006). “Minkowski sums of monotone and general simple polygons”. Discrete and Computational Geometry. 35(2): 223-240. · Zbl 1086.68140
[105] Pachkova, E. V. (2009). “Restricted reallocation of resources”. European Journal of Operational Research. 196(3): 1049-1057. · Zbl 1176.90309
[106] Pham, M. D. and V. Zelenyuk (2018). “Slack-based directional distance function in the presence of bad outputs: Theory and application to Vietnamese banking”. Empirical Economics. 54(1): 153-187.
[107] Pham, M. D. and V. Zelenyuk (2019). “Weak disposability in nonpara-metric production analysis: A new taxonomy of reference technology sets”. European Journal of Operational Research. 274(1): 186-198. · Zbl 1431.91195
[108] Podinovski, V. V. and T. Kuosmanen (2011). “Modelling weak dis-posability in data envelopment analysis under relaxed convexity assumptions”. European Journal of Operational Research. 211(3): 577-585. · Zbl 1237.90158
[109] Raa, T. T. (2011). “Benchmarking and industry performance”. Journal of Productivity Analysis. 36(3): 285-292.
[110] Ray, S. C. (2020). Data Envelopment Analysis: A Nonparametric Method of Production Analysis. Singapore: Springer. 1-62.
[111] Rubinstein, A. (2006). Lecture Notes in Microeconomic Theory. Prince-ton, NJ: Princeton University Press.
[112] Russell, R. R. (1990). “Continuity of measures of technical efficiency”. Journal of Economic Theory. 51(2): 255-267. · Zbl 0715.90024
[113] Schmidt, P. and R. C. Sickles (1984). “Production frontiers and panel data”. Journal of Business & Economic Statistics. 2(4): 367-374.
[114] Shephard, R. W. (1970). Theory of Cost and Production Functions. Princeton, NJ: Princeton University Press. · Zbl 0244.90011
[115] Shephard, R. W. (1953). Cost and Production Functions. Princeton, NJ: Princeton University Press. · Zbl 0052.15901
[116] Shephard, R. W. (1974). “Indirect production functions”. In: Mathemat-ical Systems in Economics. Vol. 10. Meisenheim Am Glan, Germany: Verlag Anton Hain. · Zbl 0278.90036
[117] Sickles, R. and V. Zelenyuk (2019). Measurement of Productivity and Ef-ficiency: Theory and Practice. New York, NY: Cambridge University Press. · Zbl 1409.91006
[118] Sickles, R. C., W. Song, and V. Zelenyuk (2020). “Chapter 8 -Econo-metric analysis of productivity: Theory and implementation in R”. In: Financial, Macro and Micro Econometrics Using R. Ed. by H. D. Vinod and C. Rao. Vol. 42. Handbook of Statistics. Elsevier. 267-297. References · Zbl 1443.62507
[119] Simar, L. and P. W. Wilson (2013). “Estimation and inference in non-parametric frontier models: Recent developments and perspectives”. Foundations and Trends in Econometrics. 5(3-4): 183-337. · Zbl 1281.62243
[120] Simar, L. and V. Zelenyuk (2020). “Improving finite sample approxima-tion by central limit theorems for estimates from data envelopment analysis”. European Journal of Operational Research. 284(3): 1002-1015. · Zbl 1441.90075
[121] Simar, L. and V. Zelenyuk (2007). “Statistical inference for aggregates of Farrell-type efficiencies”. Journal of Applied Econometrics. 22(7): 1367-1394.
[122] Simar, L. and V. Zelenyuk (2018). “Central limit theorems for aggregate efficiency”. Operations Research. 166(1): 139-149. · Zbl 1457.91211
[123] Solow, R. M. (1957). “Technical change and the aggregate production function”. Review of Economics and Statistics. 39(3): 312-320.
[124] Solow, R. M. (1956). “A contribution to the theory of economic growth”. The Quarterly Journal of Economics. 70(1): 65-94.
[125] Starr, R. M. (2008). “Shapley-Folkman theorem”. In: The New Palgrave Dictionary of Economics. Ed. by S. N. Durlauf and L. E. Blume. Basingstoke, UK: Palgrave Macmillan. 317-318.
[126] Thaler, R. H. and C. R. Sunstein (2009). Nudge: Improving Decisions About Health, Wealth, and Happiness. London, UK: Penguin.
[127] Tone, K. (2001). “A slacks-based measure of efficiency in data envelop-ment analysis”. European Journal of Operational Research. 130(3): 498-509. · Zbl 0990.90523
[128] von Neumann, J. (1945). “A model of general equilibrium”. Review of Economic Studies. 13(1): 1-9.
[129] Zelenyuk, V. (2006). “Aggregation of Malmquist productivity indexes”. European Journal of Operational Research. 174(2): 1076-1086. · Zbl 1103.90366
[130] Zelenyuk, V. (2013). “A scale elasticity measure for directional distance function and its dual: Theory and DEA estimation”. European Journal of Operational Research. 228(3): 592-600. · Zbl 1317.90110
[131] Zelenyuk, V. (2014a). “Directional vs. Shephard”s distance functions”. Optimization Letters. 8(8): 2185-2189. · Zbl 1309.90105
[132] Zelenyuk, V. (2014b). “Scale efficiency and homotheticity: Equivalence of primal and dual measures”. Journal of Productivity Analysis. 42(1): 15-24.
[133] Zelenyuk, V. (2015). “Aggregation of scale efficiency”. European Journal of Operational Research. 240(1): 269-277. · Zbl 1339.90178
[134] Zelenyuk, V. (2020a). “Aggregation of efficiency and productivity: From firm to sector and higher levels”. In: Handbook of Production Eco-nomics. Ed. by S. C. Ray, R. Chambers, and S. Kumbhakar. Singa-pore: Springer.
[135] Zelenyuk, V. (2020b). “Aggregation of inputs and outputs prior to Data Envelopment Analysis under big data”. European Journal of Operational Research. 282(1): 172-187. · Zbl 1430.90418
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