بازده نسبت به مقیاس در فناوری های تولید محدب Returns to scale in convex production technologies

۱٫ Introduction The notion of returns to scale (RTS) is well-established in data envelopment analysis (DEA) — see, e.g., Cooper, Seiford, and Tone (2007), Ray (2004) and Thanassoulis, Portela, and Despic´ (۲۰۰۸). Extending the earlier results of Banker (1984) and Banker and Thrall (1992), the DEA literature has primarily focused on the definition and evaluation of RTS in the variable returns-to-scale (VRS) production technology, for which several different methods are now available (for a review, see Banker, Cooper, Seiford, and Zhu, 2011, and Sahoo & Tone, 2015). The RTS characterization of decision making units (DMUs) is also related to the notions of scale efficiency and most productive scale size (MPSS) introduced by Banker, Charnes, and Cooper (1984) and Banker (1984). Further connections can be made to the notion of global RTS (GRS) introduced by Podinovski (2004a), Podinovski (2004b). The GRS characterization is global in the sense that its types are indicative of the direction to MPSS and are not defined by the local (marginal) properties of production function. In the VRS technology, some relevant known results describing the relationship between RTS and other scale characteristics can be summarized as follows. 1. A standard procedure for testing if a DMU is at MPSS arises from the definition of MPSS by Banker (1984). It is based on evaluation of input or output radial efficiency of the DMU in the reference constant returns-to-scale (CRS) technology of Charnes, Cooper, and Rhodes (1978), which, from a gen eral perspective, is the cone technology generated by the VRS technology. 2. An alternative way to test for MPSS is to evaluate the type of RTS exhibited by a DMU. Namely, a DMU is at MPSS if and only if it exhibits CRS (Banker & Thrall, 1992). 3. The GRS characterization of DMUs in the VRS technology, while generally different from the conventional local RTS characterization, in the case of VRS technology coincides with the latter. This effectively follows from Proposition 1 proved by Banker (1984). In recent years, a number of new production technologies have been developed and studied in the DEA literature. Most of these technologies are polyhedral (and therefore convex) sets in the input and output dimensions. Podinovski, Chambers, Atici, and Deineko (2016) refer to such technologies as polyhedral technologies.1 The class of polyhedral technologies is very large and includes most of the known convex DEA technologies, such as the CRS and VRS technologies of Charnes et al. (1978) and Banker et al. (1984). Further examples include the VRS and CRS technologies expanded by weight restrictions or production trade-offs (Atici & Podinovski, 2015; Joro & Korhonen, 2015; Podinovski, 2004d; 2007; 2015; 2016; Podinovski & Bouzdine-Chameeva, 2013; 2015), the weakly disposable VRS technology (Kuosmanen, 2005; Kuosmanen & Kazemi Matin, 2011; Kuosmanen & Podinovski, 2009), the hybrid returns-to-scale (HRS) technology (Podinovski, 2004c; Podinovski, Ismail, Bouzdine-Chameeva, & Zhang, 2014), the convex CRS technology with exogenously fixed inputs and outputs (Podinovski & Bouzdine-Chameeva, 2011), some models of technologies with multiple component processes (Cherchye, De Rock, Dierynck, Roodhooft, & Sabbe, 2013; Cherchye, De Rock, & Walheer, 2015; 2016; Cook & Zhu, 2011) and various network DEA models (see, e.g., Kao, 2014; Sahoo, Zhu, Tone, and Klemen, 2014). It is clear that RTS and related scale characterizations such as MPSS are important for all polyhedral technologies. Thus, several authors develop bespoke methodologies for evaluation of RTS in particular technologies (see, e.g., Tone, 2001; Sahoo et al., 2014). Podinovski et al. (2016) develop a universal methodology for the RTS characterization of DMUs in any polyhedral technology. This approach uses linear programming techniques for calculation of one-sided scale elasticities that subsequently define the types of RTS. Although the current DEA literature allows us to define and evaluate the RTS types for any polyhedral technology, and further methods exist for their GRS characterization, the relationship between RTS and GRS types (including MPSS) has so far remained unexplored. An exception here is the equivalence of RTS and GRS characterizations for convex technologies whose boundaries are smooth, established by Podinovski (2004a). This result does not, however, apply to polyhedral technologies. This paper addresses the above gap. Its main contribution is the establishment of equivalence of local and global characterizations of RTS in any polyhedral technology. In particular, this implies that a DMU exhibits CRS if and only if it is at MPSS. In fact, from the theoretical perspective, it is straightforward to generalize and prove this result in a larger class of closed convex technologies, of which polyhedral technologies are a special case. From a practical perspective, the established equivalence of the notions of RTS and GRS gives us a new tool for evaluating the GRS types in any polyhedral (and, more generally, closed and convex) technology, by evaluating the RTS types instead. More precisely, standard methods for the evaluation of MPSS and GRS types require the use of reference technologies (such as the CRS, nonincreasing and non-decreasing RTS technologies, if the underlying true technology is VRS). For many polyhedral technologies, their reference technologies may not be immediately available and would require further development before they could be used. The new theoretical results established in this paper allow us to avoid this and, instead, use the existing methodologies for evaluation of RTS. We illustrate the usefulness of the new results by a numerical example involving the RTS and GRS characterizations of a VRS technology expanded by the specification of weight restrictions. We also discuss the application of new results to a two-stage network DEA model.