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Shadow pricing of undesirable outputs in nonparametric analysis

https://doi.org/10.1016/j.ejor.2013.05.028Get rights and content

Highlights

  • We contribute to a methodological debate in the nonparametric literature.

  • Way of modeling undesirables as inputs or outputs, or by transformation functions.

  • We point out a current error in the modeling of weak disposability under VRS.

  • We introduce a new hybrid model with an economically meaningful interpretation.

  • We introduce a Law of One Price rule for shadow prices of undesirable outputs.

Abstract

For three decades a growing interest in the modeling of desirable and undesirable outputs has led to a theoretical and methodological debate in the nonparametric literature on production technology and efficiency. The first main discussion is about the way of modeling ‘bad/undesirables’ as inputs or outputs, or by transformation functions. The second debate concerns the implications of the weak disposability assumption in the modeling of bad outputs, in particular the possibility of assigning unexpected signs to shadow prices of bad outputs. In addition, we point out a current error in the modeling of weak disposability under a variable returns to scale technology. In this paper we introduce a hybrid model to ensure the economically meaningful jointness of good and bad outputs while constraining shadow prices of bad outputs to their expected sign. We argue that it is a sound compromise to model undesirable outputs with a meaningful primal/dual economic interpretation. Finally we propose an extension to define shadow prices for undesirable outputs following the Law of One Price (LoOP) rule.

Introduction

The amount of literature on bad or undesirable outputs in nonparametric analysis has been extensively increasing since the 1980s. In a recent survey on Data Envelopment Analysis (DEA) in energy and environmental studies, Zhou et al. (2008a) listed a total of 100 studies published from 1983 to 2006 and the tendency has not yet decreased. Zhou et al. (2008a) classified DEA methods for modeling bad outputs in two groups namely those based on data transformation and the use of traditional DEA and those using the original data but relying on the weak disposability assumption. Based on this classification, Sahoo et al. (2011) clearly summarized the approaches in modeling bad outputs.

Among the first group, Lovell et al. (1995) and Athanassopoulos and Thanassoulis (1995) transformed the values to their reciprocals as discussed by Liang et al. (2009). Scheel (2001) and Seiford and Zhu (2002) modeled undesirable outputs as desirable using a linear monotone decreasing transformation. A variant was the approach directly dealing with bad outputs as inputs, such as that introduced by Reinhard et al., 2000, Hailu and Veeman, 2001 or more recently by Mahlberg and Sahoo (2011) and Mahlberg et al. (2011).

The second group is composed of the studies starting from the model introduced by Färe et al. (1989). The modeling of bad outputs is based on the assumption of a joint production between outputs that relies on the weak disposability axiom introduced by Shephard, 1970, Shephard, 1974. Intuitively this axiom states that any decrease in bad outputs necessarily implies a decrease in good outputs for efficient Decision Making Units (DMUs). Therefore the bad outputs are not freely disposable and cannot be reduced without affecting the production of desirable outputs. This approach has been widely used in nonparametric literature. We can cite among others, Färe et al., 1996, Färe et al., 2001, Färe et al., 2004, Chung et al., 1997, Tyteca, 1997, Boyd and McClelland, 1999, Boyd et al., 2002, Zaim, 2004, Arcelus and Arocena, 2005, Picazo-Tadeoa et al. (2005), Mandal, 2010, Zhou et al., 2007, Zhou et al., 2008b, Zhou et al., 2012. Other authors like Kuosmanen (2005) and Kuosmanen and Podinovski, 2009, Kuosmanen and Matin, 2011 proposed alternative ways of modeling undesirable outputs.

An additional question concerns the direction used in order to reach the efficient frontier. Hyperbolic, radial input, radial output or directional distance measures have been proposed in the literature. Moreover, a surprisingly ignored pitfall in this literature is the modeling of weak disposability under various assumptions on returns to scale. Currently, constant returns to scale (CRS) and non-increasing returns to scale (NIRS) models lead to ‘ready to use’ linear programs. However, variable returns to scale (VRS) and non-decreasing returns to scale (NDRS) models are more complicated. Nonlinear in their original form, as defined for example by Färe and Grosskopf (2003), they have led to an incorrect linearization used in many studies. It is only recently that correct linearizations have occurred in the literature. We will discuss this issue while proposing a primal/dual formulation with a relevant economic interpretation of the weak disposability.

The above classification is essentially methodological but recently, a more theoretical discussion has emerged on the ‘good modeling of bad outputs’. Two inspiring papers have been proposed by Førsund (2009) and Murty et al. (2012). In the latter, they argued that: “many commonly used models of pollution-generating technologies, which treat pollution as a freely disposable input or as a weakly disposable and null-joint output, may generate unacceptable implications for the trade-offs among inputs, outputs and pollution”. They also argued that: “the correct trade-offs in production are best captured if pollution-generating technology is modeled as an intersection of an intended-production technology of the firm and nature’s residual-generation set.” Another theoretical question is related to the inclusion of a material balance condition in the modeling. A deeper discussion can be found in Pethig (2006) while its application into a nonparametric framework is discussed by Coelli et al., 2007, Førsund, 2009 and Murty et al. (2012).

While these approaches seem promising, we still confine our work to the ‘reduced form’ for which we propose a hybrid model. It appears to be a good compromise between the necessity to model jointness of good and bad outputs and the expectation of getting shadow prices on bad outputs that define them as a cost. Our approach can be viewed as the nonparametric counterpart of the Färe et al. (1993) approach in which they constrained shadow prices of bad outputs in a parametric translog output distance function.

Finally we propose an extension to define shadow prices for undesirable outputs following the Law of One Price (LoOP) rule. This extension proves useful in the case for example where a bad output is a pollutant and by taking the point of view of Society who wants to define a unique price for the pollutant in order to implement a pollution tax.

Section snippets

Strengths and weaknesses of modeling bads as inputs or outputs

We start with the definition of the weakly disposable Shephard’s technology as given in Färe and Grosskopf (2003). We denote inputs by x=(x1,,xN)R+N, desirable or good outputs by v=(v1,,vM)R+M and undesirable or bad outputs by w=(w1,,wJ)R+J. The technology consisting of all feasible (v, w, x) and the corresponding output set are denoted respectively by:T={(v,w,x):xcan produce(v,w)}P(x)={(v,w):(x,v,w)T}Next we turn to the definition of weak disposability as defined by Shephard, 1970,

A hybrid approach to model undesirable outputs

Recently Leleu (2013) proposed a slightly different but equivalent linearization of the VRS technology under the weak disposability assumption as defined in (P4). This leads to a relevant dual economic interpretation of the weak disposability axiom. The corresponding primal/dual programs are given by:

(P6), (P7) Primal/Dual programs for the weakly disposable VRS technologymaxδ,λ,σδs.t.-k=1Kλkvm,k-vmk+σvmk+δgmv0m=1,,Mk=1Kλkwj,k-wjk-σwjk+δgjw=0j=1,,Jk=1Kλk(xn,k-xnk)0n=1,,Nk=1Kλk+σ=1λk

A ‘Law of One Shadow Price’ (LoOSP) model for bad outputs

Under the traditional DEA framework, dual prices in program (P7), (P8) are specific to each of the evaluated firms. Each firm has its own set of shadow prices for good and bad outputs (and inputs) which are computed in such a way that they make the firm as efficient as possible. It is the benefit of the doubt principle of DEA which corresponds to imperfect competition where a bunch of prices may exist for the same good in the economy. However, following the Law of One Price (LoOP), in an

Conclusion

In this paper we have explored some pitfalls associated with the modeling of bad outputs. We have particularly, discussed the implications of the weak disposability assumption on the sign of shadow prices associated to bad outputs. We have also focused on strengths and weaknesses of modeling bad outputs as weakly disposable outputs or strongly disposable inputs. We have also pointed out a current error in the modeling of bad outputs under variable returns to scale technology and provided a new

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