Abstract
In the Operations Research/Management Science literature, the nonparametric method of Data Envelopment Analysis (DEA) has gained wide popularity as a valid analytical format for efficiency evaluation. In economics, however, its reception has been far less enthusiastic. Yet, the intellectual roots of DEA go back to the seminal contributions to nonparametric analysis of production by Debreu, Shephard, Farrell, Afriat, and others. Over the past four decades, DEA has matured into a full blown non-parametric methodology for measuring productive efficiency that serves as an alternative to parametric Stochastic Frontier Analysis (SFA). Both grounded into the neoclassical theory of production, DEA and SFA provide the researcher alternative ways to calibrate testable relations between inputs, outputs, costs, revenue, and profit.
Staring from the central concept of the Production Possibility set, this chapter provides a broad overview of the literature on DEA methodology for radial and non-radial measurement of technical efficiency from input and output quantity data under alternative returns to scale assumptions. This is followed by models for performance evaluation in the presence of market prices through cost, revenue, and overall profit efficiency- both in the long run when all inputs are variable and in the short run, when some inputs are fixed. DEA models for physical measures of the capacity output in the short run and economic measures of capacity in the long run are discussed. Alternative ways to incorporate the production of ‘bad’ or undesirable outputs collaterally with the ‘good’ or intended output in DEA models for efficiency measurement are presented. Finally, the role of contextual or environmental variables that affect efficiency is also discussed.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
- 2.
In this chapter, vectors are denoted by superscripts and scalars by subscripts.
- 3.
The more popular alternative is Stochastic frontier analysis (SFA) introduced by Aigner, Lovell, and Schmidt [2] where one uses the maximum likelihood procedure to estimate a parametrically specified frontier production function incorporating a one-sided error term representing inefficiency and another two-sided error term representing random noise. See Kumbhakar and Lovell [46] for a detailed discussion of Stochastic frontier analysis (SFA).
- 4.
If the transformation function is differentiable, weak disposability of inputs and outputs will imply \( \frac{\partial F}{\partial {x}_i}\le 0 \) for each input i and \( \frac{\partial F}{\partial {y}_r}\ge 0 \) for each output r. In a later section in this chapter and in much greater details in the Chapter 12, “Bad Outputs” by Murty and Russell in this volume of the Handbook, free or strong disposability is contrasted with weak disposability where an output cannot be decreased (or an input increased) unilaterally but simultaneous reduction in multiple outputs or increase in multiple inputs may be feasible.
- 5.
In the DEA literature, this is often described as minimum extrapolation. It should be noted that this is a criterion for estimation rather than a property of the technology.
- 6.
When (x0, y0) is not one of the observed bundles, non-negativity of the λs and the outputs will ensure that φ will never be negative even if it is lower than 1. However, if any individual input in the bundle x0 is smaller than the smallest value of the corresponding input across all observations in the data set \( {\mathcal{D}} \), (23) will not have a feasible solution.
- 7.
The true maximum may actually be considerably higher than φ∗y0. But we cannot infer that on the basis of the observed input-output bundles without making additional assumptions about the technology. However, it cannot be any smaller than φ∗y0 if the assumptions (A1)–(A4) hold.
- 8.
The case of k = 0 corresponds to inaction when no input is used and no output is produced.
- 9.
Note the absence of the restriction that the λs add up to unity. Under convexity only, \( \left(\overline{x}=\sum \limits_{j=1}^N{\lambda}_j{x}^j,\overline{y}=\sum \limits_{j=1}^N{\lambda}_j{y}^j\right) \) is feasible so long as \( \sum \limits_{j=1}^N{\lambda}_j=1 \) and no λj is negative. With the added assumption of CRS, \( \left(k\overline{x},k\overline{y}\right) \) is also feasible for any k ≥ 0. CRS implies that for \( {\mu}_j=k{\lambda}_j,k\ge 0,\left(\sum \limits_{j=1}^N{\mu}_j{x}^j,\sum \limits_{j=1}^N{\mu}_j{y}^j\right) \) is feasible. But \( \sum \limits_{j=1}^N{\mu}_j=k \) need not be equal to 1.
- 10.
These shadow prices or multipliers are uniquely designed for the unit under evaluation.
- 11.
If (x0, y0) ∈ T but ∉G, then there will exist either some β < 1 such that (βx0, y0) ∈ T or some α > 1 such that (x0, αy0) ∈ T. In the former case, one gets \( \frac{\alpha }{\beta }>1 \) for α = 1. In the latter case, \( \frac{\alpha }{\beta }>1 \) for β = 1.
- 12.
The more interested reader may see Ray [57] for detailed treatment of all these three approaches.
- 13.
See also the earlier paper by Seiford and Zhu [71].
- 14.
See Färe, Grosskopf, Lovell, and Pasurka [33].
- 15.
Rumor has it that when Färe raised this point at a conference in Austin TX, Charnes was so irritated that he excluded the former from his guest list to a barbecue!
- 16.
Portela and Thanassoulis [54] use the measure \( \frac{{\left(\Pi {\theta}_i\right)}^{\frac{1}{n}}}{{\left(\Pi {\varphi}^r\right)}^{\frac{1}{m}}} \) and called it the geometric distance function.
- 17.
See also the range-adjusted measure (RAM) introduced by Cooper, Park, and Pastor [23].
- 18.
An exception is when there is any slack in the technically efficient input bundle.
- 19.
See the discussion in Ray and Mukherjee [64].
- 20.
See, for an example, Tone [75].
- 21.
This problem assumes homogeneous technology across all locations. RCM also consider the case where the technology varies across locations.
- 22.
- 23.
- 24.
See their Chap. 12 on “Bad Outputs” in this volume of the Handbook.
- 25.
MRL [51] also consider another model including pollution abatement as a separate desired output produced by diverting resources from the production of the desired output g. An example would be treatment of polluted waste water before discharging into the stream.
- 26.
The joint disposability of the bad output and the polluting input is comparable to the two materials balance postulates MB1 and MB2 in Dakpo et al. ([26], p. 352).
- 27.
Ayres and Kneese [4] introduced the question of materials balance in economics. In a number of subsequent papers, it has been extensively discussed in the context of production efficiency by a number of authors including Pethig [53]; Coelli, Lauwers, and Van Huylenbroeck [21]; Chambers and Melkonyan [14]; Hampf [41]; Rodseth [68, 69]; and Førsund [39] among others. See, in particular, Dakpo, Jenneauxe, and Latruffe [26].
- 28.
Chapters 20, “Conceptualization and Measurement of Productivity Growth and Technical Change: A Nonparametric Approach” by Ray and 21, “Modeling Technical Change: Theory and Practice” by Kumbhakar in this volume of the Handbook cover non-parametric DEA and parametric SFA approaches to measuring productivity growth and technical change.
- 29.
For a detailed discussion of non-convexity in general, refer to the Chap. 18, “Nonconvexity in Production and Cost Functions: An Exploratory and Selective Review” by Briec, Kerstens, and Van de Woestyne in this volume of the Handbook.
- 30.
References
Afriat S (1972) Efficiency estimation of production functions. Int Econ Rev 13(3):568–598
Aigner DJ, Lovell CAK, Schmidt P (1977) Formulation and estimation of stochastic frontier production function models. J Econ 6(1):21–37
Aparicio J, Pastor JT, Ray SC (2013) An overall measure of technical inefficiency at the firm and at the industry level: the ‘lost profit on outlay’. Eur J Oper Res 226(1):154–162
Ayres RU, Kneese AV (1969) Production, consumption, and externalities. Am Econ Rev 59:282–297
Banker RD (1984) Estimating the most productive scale size using data envelopment analysis. Eur J Oper Res 17(1):35–44
Banker RD, Maindiratta A (1988) Nonparametric analysis of technical and allocative efficiencies in production. Econometrica 56(5):1315–1332
Banker RD, Morey RC (1986) Efficiency analysis for exogenously fixed inputs and outputs. Oper Res 34(4):513–521
Banker RD, Thrall RM (1992) Estimating most productive scale size using data envelopment analysis. Eur J Oper Res 62:74–84
Banker RD, Charnes A, Cooper WW (1984) Some models for estimating technical and scale inefficiencies in data envelopment analysis. Manage Sci 30(9):1078–1092
Banker RD, Chang H, Natarajan R (2007) Estimating DEA technical and allocative inefficiency using aggregate cost or revenue data. J Prod Anal 27:115–121
Baumol WJ, Panzar JC, Willig RD (1982) Contestable Markets and the Theory of Industry Structure. New York: Harcourt, Brace, Jovanovich.
Baumol WJ, Oates WE (1988) The theory of environmental policy, 2nd edn. Cambridge University Press, Cambridge
Cassell JM (1937) Excess capacity and monopolistic competition. Q J Econ 51(3):426–443
Chambers RG, Melkonyan T (2012) Production technologies, material balance, and the income-environmental quality trade-off. University of Exeter working paper
Chambers RG, Chung Y, Färe R (1996) Benefit and distance functions. J Econ Theory 70:407–419
Chambers RG, Chung Y, Färe R (1998) Profit, directional distance functions, and nerlovian efficiency. J Optim Theory Appl 98:351–364
Charnes A, Cooper WW (1968) Programming with linear fractional functionals. Nav Res Logist Q 15:517–522
Charnes AC, Cooper WW, Mellon B (1952) Blending aviation gasolines – a study in programming interdependent activities in an integrated oil company. Econometrica 20(2):135–159
Charnes A, Cooper WW, Rhodes E (1978) Measuring the efficiency of decision making units. Eur J Oper Res 2(6):429–444
Charnes A, Cooper WW, Rhodes E (1979) Short communication: measuring the efficiency of decision making units. Eur J Oper Res 3(4):339
Coelli T, Lauwers L, Van Huylenbroeck GV (2007) Environmental efficiency measurement and the materials balance condition. J Prod Anal 28:3–12
Cooper WW, Thompson RG, Thrall RM (1996) Introduction: extensions and new developments in DEA. Ann Oper Res 66:3–45
Cooper WW, Park SK, Pastor JT (1999) RAM: a range adjusted measure of inefficiency for use with additive models, and relations to other models and measures in DEA. J Prod Anal 11:5–42
Cooper WW, Seiford L, Tone K (2002) Data envelopment analysis: a comprehensive text with uses, example applications, references and DEA-solver software. Kluwer, Norwell
Cropper ML, Oates WE (1992) Environmental economics: a survey. J Econ Lit 30:675–740
Dakpo KH, Jeanneauxe P, Latruffe L (2016) Modeling pollution generating technologies in performance benchmarking: recent developments, limits, and future prospects in the non-parametric framework. Eur J Oper Res 250:347–359
Dantzig GB (1951) Maximization of a linear function of variables subject to linear inequalities. In: Koopmans TC (ed) Activity analysis of production and allocation. Wiley, New York, pp 339–347
Debreu G (1951) The coefficient of resource utilization. Econometrica 19(3):273–292
Färe R, Grosskopf S (2003) Nonparametric productivity analysis with undesirable outputs: comment. Am J Agric Econ 85:1070–1074
Färe R, Lovell CAK (1978) Measuring the technical efficiency of production. J Econ Theory 19(1):150–162
Färe R, Grosskopf S, Lovell CAK (1985) The measurement of efficiency of production. Kluwer-Nijhoff, Boston
Färe R, Grosskopf S, Kokkelenberg EC (1989) Measuring plant capacity, utilization and technical change: a nonparametric approach. Int Econ Rev 30(3):655–666
Färe R, Grosskopf S, Lovell CAK, Pasurka C (1989) Multilateral productivity comparisons when some outputs are undesirable: a non-parametric approach. Rev Econ Stat 71(1):90–98
Färe R, Grosskopf S, Lovell CAK, Yaisawarng S (1993) Derivation of shadow prices for undesirable outputs: a distance function approach. Rev Econ Stat 75:374–380
Färe R, Grosskopf S, Lovell CAK (1994) Production frontiers. Cambridge University Press, Cambridge
Färe R, Grosskopf S, Noh DW, Weber W (2005) Characteristics of a polluting technology: theory and practice. J Econ 126:469–492
Farrell MJ (1957) The measurement of technical efficiency. J R Stat Soc Ser A Gen 120(Part 3):253–281
Førsund F (2009) Good modelling of bad outputs: pollution and multiple-output production. Int Rev Environ Resour Econ 3(1):1–38
Førsund F (2018) Multi-equation modeling of desirable and undesirable outputs satisfying the material balance. Empir Econ, online 54(1):67–99
Frisch R (1965) Theory of production. Rand McNally and Company, Chicago
Hampf B (2014) Separating environmental efficiency into production and abatement efficiency: a nonparametric model with application to US power plants. J Prod Anal 41:457–473
Hanoch G, Rothschild M (1972) Testing the assumptions of production theory: a nonparametric approach. J Polit Econ 80(2):256–275
Johansen L (1968) Production functions and the concept of capacity. Reprinted in Førsund FR (ed) Collected works of Leif Johansen, vol 1. North Holland, Amsterdam
Koopmans TJ (1951) Analysis of production as an efficient combination of activities. In: Koopmans TJ (ed) Activity analysis of production and allocation. Wiley, New York, pp 33–97
Koopmans TJ (1957) Three essays on the state of economic science. McGraw Hill, New York
Kumbhakar S, Lovell CAK (2000) Stochastic frontier analysis. Cambridge University Press, New York
Leleu H, Briec W (2009) A DEA estimation of a lower bound for firms’ allocative efficiency without information on price data. Int J Prod Econ 121:203–211
Lozano SC (2015) A joint-inputs network DEA approach to production and pollution-generating technologies. Expert Syst Appl 42:7960–7968
Luenberger DG (1992) Benefit functions and duality. J Math Econ 21:115–145
Murty S, Russell RR (2016) Modeling emission-generating technologies: reconciliation of axiomatic and by-production approaches. Empir Econ 54(1):7–30
Murty S, Russell R, Levkoff SB (2012) On modeling pollution-generating technologies. J Environ Econ Manag 64:117–135
Pastor JT, Louis JL, Sirvent I (1999) An enhanced DEA Russell-graph efficiency measure. Eur J Oper Res 115:596–607
Pethig R (2006) Non-linear production, abatement, pollution and materials balance reconsidered. J Environ Econ Manag 51:185–204
Portela MCAS, Thanassoulis E (2005) Profitability of a sample of Portuguese bank branches and its decomposition into technical and allocative components. Eur J Oper Res 162(3):850–866
Ray SC (1988) Data envelopment analysis, non-discretionary inputs and efficiency: an alternative interpretation. Socio Econ Plan Sci 22(4):167–176
Ray SC (1991) Resource-use efficiency in public schools: a study of Connecticut data. Manag Sci 37(12):1620–1628
Ray SC (2004) Data envelopment analysis: theory and techniques for economics and operations research. Cambridge University Press, New York
Ray SC (2007) Shadow profit maximization and a measure of overall inefficiency. J Prod Anal 27:231–236
Ray SC (2009) Are Indian firms too small? A nonparametric analysis of cost efficiency and the optimal organization of industry in Indian manufacturing. Indian Econ Rev XXXXVI(1):49–67
Ray SC (2010) A one-step procedure for returns to scale classification of decision making units in data envelopment analysis. University of Connecticut Economics working paper 2010-07
Ray SC (2015) Nonparametric measures of scale economies and capacity utilization: an application to U.S. manufacturing. Eur J Oper Res 245:602–611
Ray SC, Ghose A (2014) Production efficiency in Indian agriculture: an assessment of the post green revolution years. Omega 44:58–69
Ray SC, Jeon Y (2009) Reputation and efficiency: a non-parametric assessment of America’s top-rated MBA programs. Eur J Oper Res 189(2008):245–268
Ray SC, Mukherjee K (2016) Data envelopment analysis with aggregated inputs and a test of allocative efficiency when input prices vary across firms. Data Envel Anal J 2(2):141–161
Ray SC, Chen L, Mukherjee K (2008) Input price variation across locations and a generalized measure of cost efficiency. Int J Prod Econ 116:208–218
Ray SC, Mukherjee K, Venkatesh A (2018) Nonparametric measures of efficiency in the presence of undesirable outputs: a by-production approach with weak disposability. Empir Econ 54(1):31–65
Ray SC, Walden J, Chen L (2018) Economic Measures of Capacity Utilization: A Nonparametric Cost Function Analysis. Working Paper 2018--02, University of Connecticut, Department of Economics
Rodseth KL (2015) Axioms of a polluting technology: a materials balance approach. Environ Res Econ 67(1):1–22. Online October 2015
Rodseth KL (2016) Environmental efficiency measurement and the materials balance condition reconsidered. Eur J Oper Res 250:342–346
Ruggiero J (1998) Non-discretionary inputs in data envelopment analysis. Eur J Oper Res 111:461–469
Seiford L, Zhu J (1999) An investigation of returns to scale in data envelopment analysis. Omega Int J Manag Sci 27:1–11
Shephard RW (1953) Cost and production functions. Princeton University Press, Princeton
Shephard RW (1970) Theory of cost and production functions. Princeton University Press, Princeton
Tone K (2001) A slacks-based measure of efficiency in data envelopment analysis. Eur J Oper Res 130:498–509
Tone K (2002) A strange case of the cost and allocative efficiencies in DEA. J Oper Res Soc 53:1225–1231
Varian HR (1984) The nonparametric approach to production analysis. Econometrica 52(3):579–597
Zhu J (2003) Quantitative models for performance evaluation and benchmarking: data envelopment analysis with spreadsheets and DEA excel solver. Kluwer Academic, Boston
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2022 Springer Nature Singapore Pte Ltd.
About this entry
Cite this entry
Ray, S.C. (2022). Data Envelopment Analysis: A Nonparametric Method of Production Analysis. In: Ray, S.C., Chambers, R.G., Kumbhakar, S.C. (eds) Handbook of Production Economics. Springer, Singapore. https://doi.org/10.1007/978-981-10-3455-8_24
Download citation
DOI: https://doi.org/10.1007/978-981-10-3455-8_24
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-10-3454-1
Online ISBN: 978-981-10-3455-8
eBook Packages: Economics and FinanceReference Module Humanities and Social SciencesReference Module Business, Economics and Social Sciences