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Data Envelopment Analysis: A Nonparametric Method of Production Analysis

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Handbook of Production Economics

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.

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Notes

  1. 1.

    By contrast, the earlier models in DEA and activity analysis assumed CRS (e.g., CCR [20]) and generalization to the VRS technology was a subsequent development [10].

  2. 2.

    In this chapter, vectors are denoted by superscripts and scalars by subscripts.

  3. 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 [47] for a detailed discussion of Stochastic frontier analysis (SFA).

  4. 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 on “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. 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. 6.

    When (x0,?y0) is not one of the observed bundles, non-negativity of the ?s and the outputs will ensure that f 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 \( \mathbf{\mathcal{D}} \), (23) will not have a feasible solution.

  7. 7.

    The true maximum may actually be considerably higher than f*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 f*y0 if the assumptions (A1)–(A4) hold.

  8. 8.

    The case of k = 0 corresponds to inaction when no input is used and no output is produced.

  9. 9.

    Note the absence of the restriction that the ?s 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 ?j 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. 10.

    These shadow prices or multipliers are uniquely designed for the unit under evaluation.

  11. 11.

    If (x0,?y0) ? T but ?G, then there will exist either some ß < 1 such that (ßx0,?y0) ? T or some a > 1 such that (x0,?ay0) ? T. In the former case, one gets \( \frac{\alpha }{\beta }>1 \) for a = 1. In the latter case, \( \frac{\alpha }{\beta }>1 \) for ß = 1.

  12. 12.

    The more interested reader may see Ray [58] for detailed treatment of all these three approaches.

  13. 13.

    See also the earlier paper by Seiford and Zhu [72].

  14. 14.

    See Färe, Grosskopf, Lovell, and Pasurka [34].

  15. 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. 16.

    Portela and Thanassoulis [55] 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. 17.

    See also the range-adjusted measure (RAM) introduced by Cooper, Park, and Pastor [24].

  18. 18.

    An exception is when there is any slack in the technically efficient input bundle.

  19. 19.

    See the discussion in Ray and Mukherjee [65].

  20. 20.

    See, for an example, Tone [76].

  21. 21.

    This problem assumes homogeneous technology across all locations. RCM also consider the case where the technology varies across locations see Ray [61].

  22. 22.

    Also see Afriat [1] and Hanoch and Rothschild [43]. Banker and Maindiratta [6] further extended Varian [77] to construct upper and lower bounds on technical, allocative efficiency, and overall profit efficiencies.

  23. 23.

    See, for example, Färe, Grosskopf, Lovell, and Pasurka [34]; Färe, Grosskopf, Lovell, and Yaisawarng [35]; and Färe, Grosskopf, Noh, and Weber [37] among many others.

  24. 24.

    See their chapter (24) by Squires and Segerson on “Bad Outputs” in this volume of the Handbook.

  25. 25.

    MRL [52] 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. 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. ([28], p. 352).

  27. 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 [54]; Coelli, Lauwers, and Van Huylenbroeck [22]; Chambers and Melkonyan [15]; Hampf [42]; Rodseth [69, 70]; and Forsund (2016) among others. See, in particular, Dapko, Jenneauxe, and Latruffe [28].

  28. 28.

    Chapters 20 by Ray and 21 by Kumbhakar in this volume of the Handbook cover non-parametric DEA and parametric SFA approaches to measuring productivity growth and technical change.

  29. 29.

    For a detailed discussion of non-convexity in general, refer to the chapter 18 by Briec, Kerstens, and Van de SWoestyne in this volume of the Handbook.

  30. 30.

    For detailed discussion of DEA from an OR/MS perspective, the reader should refer to Zhu [78] and Cooper, Seiford, and Tone (25). Fare, Grosskopf, and Lovell [36] and Ray [58] explain the economic theory behind DEA.

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Authors and Affiliations

  1. Department of Economics, University of Connecticut, Storrs, CT, USA

    Subhash C. Ray

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Editors and Affiliations

  1. Department of Economics, University of Connecticut, Storrs, CT, USA

    Subhash C. Ray

  2. Department of Agricultural and Resource Economics, University of Maryland, College Park, MD, USA

    Robert Chambers

  3. Department of Economics, Binghamton University, Binghamton, NY, USA

    Subal Kumbhakar

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Ray, S.C. (2020). Data Envelopment Analysis: A Nonparametric Method of Production Analysis. In: Ray, S., Chambers, R., Kumbhakar, S. (eds) Handbook of Production Economics. Springer, Singapore. https://doi.org/10.1007/978-981-10-3450-3_24-1

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