Estimating scale economies in non-convex production models

Abstract

The literature on nonparametric frontier technologies lacks a method for the measurement of scale economies in non-convex settings. This paper proposes a general procedure which is based on the minimization of the ray average cost and requires the solution of a single programming problem. Our approach allows for multiple optima to introduce the case of global sub-constant scale economies, and it also permits the estimation of scale economies at a local level. The empirical application investigates the role of replicability and the relationship between global and local indicators. It also points out the managerial implications for companies operating in the Italian public transit industry.

Keywords:

• Free Disposal Hull
• returns to scale
• scale economies

Electronic supplementary material

The online version of this article (doi:10.1057/s41274-016-0162-7) contains supplementary material, which is available to authorized users.

Electronic supplementary material

The online version of this article (doi:10.1057/s41274-016-0162-7) contains supplementary material, which is available to authorized users.

Notes

1 This expression, along with “cost-based returns to scale”, is due to Sueyoshi (1999, p. 1593). In the rest of the article we use returns to scale and scale economies to denote production-based and cost-based returns to scale, respectively.

2 A scale size variable indicates the level at which either inputs or outputs are actually being employed by a unit under evaluation (i.e., current). In the analysis of scale economies, this scale size variable is normally expressed in terms of the outputs (see, e.g., Färe and Grosskopf, 1985, p. 600). We follow this convention (as moreover discussed in Section 3).

3 The calculation of CRS and NIRS scores according to model (2) is shown in Appendix 1. The following discussion exploits the equivalence mentioned later on at point 2. Section 3.1 to illustrate the CRS cost efficiency score in terms of ray average cost.

4 In principle our method allows for a variation of input prices associated to a change in the scale size (e.g. bulk buying of inputs may lower their prices) but this choice implies that an optimal scale size is not necessarily a most productive scale size, i.e. it is not CRS technically efficient. For a discussion see Cesaroni and Giovannola (2015, Sect. 4.3).

5 See Case 2 in Banker and Thrall (1992), p. 81. A numerical illustration is given in Appendix 2.

6 See part 2, p. 41 and following in Isfort et al (2014).

7 We are considering radial-efficient projections of the original observations in the sense of Podinovski (2004, p. 244): see his Definition 5.

8 On the suitability of an upper bound in the replicability assumption, see Tone and Sahoo (2003, pp. 171–172). Mairesse and Vanden Eeckaut (2002) develop a similar argument in an FDH context.