Returns to Scale in DEA

Abstract

This chapter discusses returns to scale (RTS) in data envelopment analysis (DEA). The BCC and CCR models described in Chap. 1 of this handbook are treated in input-oriented forms, while the multiplicative model is treated in output-oriented form. (This distinction is not pertinent for the additive model, which simultaneously maximizes outputs and minimizes inputs in the sense of a vector optimization.) Quantitative estimates in the form of scale elasticities are treated in the context of multiplicative models, but the bulk of the discussion is confined to qualitative characterizations such as whether RTS is identified as increasing, decreasing, or constant. This is discussed for each type of model, and relations between the results for the different models are established. The opening section describes and delimits approaches to be examined. The concluding section outlines further opportunities for research and an Appendix discusses other approaches in DEA treatment of RTS.

^*Part of the material in this chapter is adapted from European Journal of Operational Research, Vol 154, Banker, R.D., Cooper, W.W., Seiford, L.M., Thrall, R.M. and Zhu, J, Returns to scale in different DEA models, 345–362, 2004, with permission from Elsevier Science.

Appendix

In this Appendix, we first present the FGL approach. We then present a simple RTS approach without the need for checking the multiple optimal solutions as in Zhu and Shen (1995) and Seiford and Zhu (1999) where only the BCC and CCR models are involved. This approach will substantially reduce the computational burden because it relies on the standard CCR and BCC computational codes (see Zhu (2009) for a detailed discussion).

To start, we add to the BCC and CCR models by the following DEA model whose frontier exhibits nonincreasing returns to scale (NIRS), as in Färe, Grosskopf and Lovell (FGL 1985, 1994)

$$ \begin{array}{lllll} \theta_{\rm{NIRS}}^{*} = { \min }\,\theta_{\rm{NIRS}}, \hfill \\ {\hbox{subject}}\,{\hbox{to}} \hfill \\ {\theta_{\rm{NIRS}}}{x_{io}} = \sum\limits_{j = 1}^n {{x_{ij}}{\lambda_j}} + s_i^{-}, \quad\hskip 1.0pt i = 1,2, \ldots, m, \hfill \\ {y_{ro}} \hskip 21pt= \sum\limits_{j = 1}^n {{y_{rj}}{\lambda_j}} - s_r^{+}, \quad r = 1,2, \ldots, s, \hfill \\ 1 \ge \sum\limits_{j = 1}^n {{\lambda_j}}, \hfill \\ 0 \le {\lambda_j},s_i^{-}, s_r^{+} \quad \forall \,i,\,r,\,j{.}\end{array} $$

(2.34)

The development used by FGL (1985, 1994) rests on the following relation

$$ \theta_{\rm{CCR}}^{*} \le \theta_{\rm{NIRS}}^* \le \theta_{\rm{BCC}}^*, $$

where “*” refers to an optimal value and $\theta_{\rm{NIRS}}^{*}$ is defined in (2.34), while $\theta_{\rm{BCC}}^{*}$ and $\theta_{\rm{CCR}}^{*}$ refer to the BCC and CCR models as developed in Theorems 2.3 and 2.4.

FGL utilize this relation to form ratios that provide measures of RTS. However, we turn to the following tabulation that relates their RTS characterization to Theorems 2.3 and 2.4 (and accompanying discussion). See also Färe and Grosskopf (1994), Banker et al. (1996b), and Seiford and Zhu (1999)

	FGL Model	RTS	CCR Model
Case 1	If $\theta_{\rm{CCR}}^{*} = \theta_{\rm{BCC}}^*$	Constant	$\sum {\lambda_j^*} = 1$
Case 2	If $\theta_{\rm{CCR}}^{*} \ \ <\ \ \theta_{\rm{BCC}}^*$ then
Case 2a	If $\theta_{\rm{CCR}}^{*} = \theta_{\rm{NIRS}}^*$	Increasing	$\sum {\lambda_j^*} < 1$
Case 2b	If $\theta_{\rm{CCR}}^{*}\ \ <\ \ \theta_{\rm{NIRS}}^*$	Decreasing	$\sum {\lambda_j^*} > 1$

It should be noted that the problem of nonuniqueness of results in the presence of alternative optima is not encountered in the FGL approach (unless output-oriented as well as input-oriented models are used), whereas they do need to be coincided, as in Theorem 2.3. However, Zhu and Shen (1995) and Seiford and Zhu (1999) develop an alternative approach that is not troubled by the possibility of such alternative optima.

We here present their results with respect to Theorems 2.3 and 2.4 (and accompanying discussion). See also Zhu (2009).

	Seiford and Zhu (1999)	RTS	CCR Model
Case 1	If $\theta_{\rm{CCR}}^{*} = \theta_{\rm{BCC}}^{*}$	Constant	$\sum {\lambda_j^*} = 1$
Case 2	$\theta_{\rm{CCR}}^{*} \ne \theta_{\rm{BCC}}^{*}$
Case 2a	If $\sum {\lambda_j^*} \ \ <\ \ 1$ in any CCR outcome	Increasing	$\sum {\lambda_j^*}<1$
Case 2b	If $\sum {\lambda_j^*}\ >\ 1$ in any CCR outcome	Decreasing	$\sum {\lambda_j^*}>1$

The significance of Seiford and Zhu’s (1999) approach lies in the fact that the possible alternate optimal $\lambda_j^*$ obtained from the CCR model only affect the estimation of RTS for those DMUs that truly exhibit CRS and have nothing to do with the RTS estimation on those DMUs that truly exhibit IRS or DRS. That is, if a DMU exhibits IRS (or DRS), then $\sum\nolimits_j^n {\lambda_j^*}$ must be less (or greater) than one, no matter whether there exist alternate optima of ${\lambda_j}$, because these DMUs do not lie in the MPSS region. This finding is also true for the $u_o^*$ obtained from the BCC multiplier models.

Thus, in empirical applications, we can explore RTS in two steps. First, select all the DMUs that have the same CCR and BCC efficiency scores regardless of the value of $\sum\nolimits_j^n {\lambda_j^*}$ obtained from model (2.5). These DMUs are CRS. Next, use the value of $\sum\nolimits_j^n {\lambda_j^*}$ (in any CCR model outcome) to determine the RTS for the remaining DMUs. We observe that in this process we can safely ignore possible multiple optimal solutions of ${\lambda_j}$.