Nonparametric measures of efficiency in the presence of undesirable outputs: a by-production approach


In empirical research on productivity measurement adjusted for undesirable outputs on the side, the good and the bad outcomes are treated as joint products of the underlying production process. In the present paper, following Murty, Russell, and Levkoff, we conceptualize the good output as technologically separable from the bad output. Joint disposability is assumed between the bad output and the polluting input, rather than weak disposability and null jointness between the good and bad outputs. Moreover, we set up an integrated DEA optimization problem over the intersection of these two subtechnologies to measure the efficiency of a firm that produces a bad output alongside the good output. In an empirical illustration of our methodology, we use country-level data for an unbalanced panel of 64 countries over the years 1986 through 2011 where per capita GDP is the good and per capita \(\hbox {CO}_{2}\) emission is the bad output. We then utilize our DEA results to compute opportunity costs of a targeted reduction in \(\hbox {CO}_{2}\) emission in terms of required dollar amounts of reduction in per capita GDP for the individual countries in selected years.

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  1. 1.

    Formal definitions of these two concepts are provided in “Methodology” section.

  2. 2.

    See, for example, Färe et al. (1989, 1993, 2005).

  3. 3.

    In fact, approximately 65% of the global greenhouse gas emissions result from the combustion of fossil fuel (Covert et al. 2016).

  4. 4.

    Ray and Mukherjee (2007) describe this as ‘reverse disposability’ of the bad output.

  5. 5.

    Geometrically, in a diagram showing the polluting input along the horizontal and the bad output up the vertical axes, the empirically constructed feasible set will be bounded from below (rather than from above). Algebraically, the signs of the inequalities showing the output and input constraints will be exactly the opposite of what they are in the conventional DEA models.

  6. 6.

    In consumer theory, a public good (like the beach or a toll-free highway) is something that can be consumed simultaneously by multiple consumers.

  7. 7.

    For example, the quantity of fuel used for power generation has to be exactly the same quantity of fuel that causes air pollution.

  8. 8.

    Following the practice in the OR literature, Lozano (2015) describes this as the Slack-based measure.

  9. 9.

    Note that the output- and input-oriented efficiencies can both be obtained as special cases of the directional model by setting \((g^{x}=0,g^{y}=y)\) to get \(\varphi =1+\beta \) for the output-oriented model or by setting \((g^{x}=-x,g^{y}=0)\) to get \(\theta =1-\beta \) for the input-oriented model.

  10. 10.

    \(F(x,y)\le 0\Leftrightarrow D(x,y)\le 1.\)

  11. 11.

    This proof differs somewhat from Färe and Grosskopf (2004, pp 49–51).

  12. 12.

    For an example of \(T^{b}\) with proactive abatement, consider \(T^{b}=\left( {(x_1 ,x_2 ,b):b\ge \frac{0.25x_2^2}{x_1} } \right) .\) In this case, it is possible to reduce the bad output without reducing the polluting input by using the neutral input for abatement. As before, joint disposability between the polluting input and the bad output holds.

  13. 13.

    When CRS holds the constraint \(\mathop {\sum }\nolimits _{j=1}^N {\lambda _j =1} \) is removed. As in (8)–(8a) above we can define \(\mu _j =\alpha \lambda _j .\) Even though \(0\le \alpha \le 1\) , because \(\mathop {\sum }\nolimits _{j=1}^N {\lambda _j} \) is not constrained to be 1, \(\mathop {\sum }\limits _{j=1}^N {\mu _j} \)is only constrained to be nonnegative. Hence, under CRS one can set \(\alpha \) equal to 1 in (13).

  14. 14.

    The output set of a given input bundle \((x^{0})\) consists of all output bundles that can be produced from \(x^{0}.\)

  15. 15.

    In principle, one could simply burn coal and produce smoke without generating electricity.

  16. 16.

    See, for example, Färe et al. (1989) or Ray (2004, p 87).

  17. 17.

    Ray (2004, p. 125 fn 2) proposed an iterated estimation procedure updating the point of approximation. Some other studies (e.g., Pham and Zelenyuk 2016) have directly used a nonlinear method.

  18. 18.

    Several authors have used the dual variables of the DEA LP problem to compute shadow prices or marginal rates of transformation between the good and the bad outputs along the frontier. However, often these multipliers are not unique. Besides they are usually extremely unstable and are not useful for measuring opportunity costs of discrete changes. For this approach, see Färe and Grosskopf (1998), Lee et al. (2002), Färe et al. (2005), Salnykov and Zelenyuk (2005), and Ray and Mukherjee (2007).

  19. 19.

    Hereafter, we consider CRS models only.

  20. 20.

    Here we are measuring partial elasticity. Note that this is different from other elasticities, such as scale elasticity measured by Fukuyama (2003) and Zelenyuk (2013).

  21. 21.

    An increase in the good output even with a corresponding increase in the bad output may not be feasible without any increase in any input.

  22. 22.

    Our sample includes 52 countries in 1986, 53 in 1987, 54 in 1988 and 1989, and 64 in the years 1990 through 2011.

  23. 23.

    A study by the Federation of American Scientists (1999) had predicted that over the following decade Russia will be unable to deal effectively with the formidable challenges posed by decades of Soviet and post-Soviet environmental mismanagement and recurrent economic crises. Especially with respect to air pollution, the study anticipated that increase in emissions from an increased number of vehicles on the road will offset any reductions in industrial air pollution owing to reduced economic activity and greater reliance on natural gas. The study also linked the environmental degradation to the incidence of severe health impacts in the country, reducing labor productivity. Our findings seem to corroborate the predictions of that research.

  24. 24.

    This possibility is acknowledged by Murty et al. (2012) as well. See their footnote 15.


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The authors thank two anonymous referees and Sushama Murty for valuable comments on an earlier version of this paper. The usual disclaimer applies.

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Correspondence to Subhash C. Ray.

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Ray, S.C., Mukherjee, K. & Venkatesh, A. Nonparametric measures of efficiency in the presence of undesirable outputs: a by-production approach. Empir Econ 54, 31–65 (2018).

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  • Bad output
  • Weak disposability
  • Null jointness
  • By-production
  • Joint disposability

JEL Classification

  • C61
  • Q52