Static efficiency decompositions and capacity utilization: integrating economic and technical capacity notions

Abstract

Starting from existing static decompositions of overall economic efficiency on nonparametric production and cost frontiers, this article proposes more comprehensive decompositions including several cost-based notions of capacity utilization. Furthermore, in case prices are lacking, we develop additional decompositions of overall technical efficiency integrating a technical concept of capacity utilization. These new efficiency decompositions provide a link between short and long run economic analysis and, in empirical work, avoid conflating inefficiency and differences in capacity utilization. An empirical analysis using a monthly panel of Chilean hydro-electric power plants illustrates the potential of these decomposition proposals.

Keywords:

• efficiency
• capacity utilization

JEL Classification:

• D24
• M21

View correction statement:

Static efficiency decompositions and capacity utilization: integrating economic and technical capacity notions

Acknowledgements

The authors are grateful to the referees, W. Briec and G. Hites for most helpful comments.

Notes

¹ By contrast, parametric frontiers utilize parametric, locally if not even globally flexible specifications with a finite number of parameters to estimate the underlying technology.

² In line with tradition, we maintain convexity throughout the analysis. Notice that Tone and Sahoo (2003) stress indivisibilities in selecting among technological options and plea in favour of using nonconvex nonparametric technologies. The latter are systematically developed in Briec et al. (2004). Note that in principle one can dispense with convexity in the analysis developed in this contribution.

³ In the parametric literature various productivity decompositions have been suggested to include measures of capacity utilization (see Hulten (1986), among others). Some productivity decompositions have been recently proposed in the nonparametric frontier literature (see below).

⁴ Mainly dual multiple output concepts are known in the parametric literature (e.g. Squires, 1987), while primal capacity notions are difficult to estimate. Färe (1984) shows that a primal capacity notion cannot be obtained for certain popular parametric specifications of technology (e.g. the Cobb–Douglas).

⁵ This does not preclude an eventual extension of our proposals into a parametric framework.

⁶ For convenience, we stick to the traditional radial input efficiency measure. Recently, more general distance functions have been introduced to measure profit efficiency (Chambers et al., 1998). Apart from the fact that these new measures lead to additive rather than multiplicative decompositions, these are related to the traditional radial efficiency measures employed here.

⁷ The duality relation between input distance function and cost function is

While C(y, w) can be obtained from D_i (x, y) by optimizing with respect to input quantities, D_i (x, y) can be resolved from C(y, w) by minimizing with respect to input prices.

⁸ Other classifications include Banker et al. (1984) and Førsund and Hjalmarsson (1974, 1979).

⁹ To simplify matters, we ignore efficiency analysis in noncompetitive settings, leading to price inefficiencies in addition to inefficiencies in quantities (e.g. Färe et al., 1994; Grifell-Tatjé and Lovell, 2000; Kallio and Kallio, 2002).

¹⁰ In addition, one can obtain qualitative information on scale economies by identifying local returns to scale. When SCE_i (x, y) = 1, then the unit is compatible with CRS. When SCE_i (x, y) < 1, then the unit does not operate with optimal size. But, one cannot know whether it is subject to increasing (IRS) or decreasing (DRS) returns to scale. By computing input efficiency also relative to a SD technology with nonincreasing returns to scale (DF_i (x, y|N, S)) and by exploiting the nestedness of technologies, one discriminates between IRS and DRS (Färe et al., 1983): (i) IRS holds when DF_i (x, y|C, S) = DF_i (x, y|N, S) ≤ DF_i (x, y|V, S) ≤ 1; (ii) DRS holds when DF_i (x, y|C, S) ≤ DF_i (x, y|N, S) = DF_i (x, y|V, S) ≤ 1.

¹¹ Identification of local economies of scale proceeds as follows. When CSCE_i (x, y, w) = 1, then the unit minimizes costs and enjoys CRS. When CSCE_i (x, y, w) < 1, then computing a cost function relative to a nonincreasing returns to scale technology (OE_i (x, y, w|N)) and knowing that OE_i (x, y, w|C) ≤ OE_i (x, y, w|N) ≤ OE_i (x, y,w|V) ≤ 1 (Grosskopf, 1986), the same reasoning as above applies to infer local economies of scale. This procedure applies to any dual formulation.

¹² Just as price-dependent parametric approaches have been popular in the literature, this very similar cost-based scale term has repeatedly appeared in the literature since Seitz (1970). See, for instance, Fukuyama and Weber (1999), Rowland et al. (1998) or Sueyoshi (1999).

¹³ One could introduce the notation AE _i (x, y, w|C) in (DEC1) to distinguish this component from the one in (DEC2).

¹⁴ See also Sueyoshi (1999). Actually, scale efficiency in Färe et al. (1994, pp. 84–7) is defined on technologies based on limited data, i.e. using information on cost data and the output vector solely. They show that scale efficiency under cost and production approaches is identical when: (i) all organizations face identical input prices; and (ii) AE_i (x, y, w|C) = AE_i (x, y, w|V). When input price information is available and cost function estimates are employed, however, the first of these conditions is redundant.

¹⁵ See Färe and Grosskopf (2000): in defense to McDonald (1996) who proposes an alternative order of some components, they justify their position by referring to economic tradition, but without mentioning a time perspective.

¹⁶ Bogetoft et al. (2006) discuss how to measure allocative efficiency while maintaining technical inefficiency, which is relevant when it is easier to introduce reallocations than improvements of technical efficiency.

¹⁷ Briec et al. (2010) show that it is possible to develop dual capacity measures for the case of other objective functions using nonparametric technologies: e.g. profit maximization (following Squires (1987)). The case of revenue maximization (Segerson and Squires, 1995) remains to be developed.

¹⁸ There is little agreement on how to define capacity utilization measures: some define it as a ratio of observed to ‘optimal’ costs, while others define it the reverse way (see, e.g. Segerson and Squires (1990)).

¹⁹ Note that Coelli et al. (2002) define an alternative ray economic capacity measure using nonparametric frontiers that involves short-run profit maximization whereby the output mix is held constant. Though this notion has some appeal, it is rarely applied and we simply note that it does not coincide with any of the traditional capacity notions.

²⁰Though strictly speaking transgressing our framework, multiple divisions within an organization may, for instance, make such output adjustments among units in terms of respective installed capacities and their optimal utilization and eventually shut down temporarily redundant units.

²¹ Johansen (1968) also proposes a synthetic capacity concept as the maximal output with existing plant and equipment while accounting for the restricted availability of variable inputs. This corresponds to technical efficiency. Since the latter notion is already part of current efficiency taxonomies, this synthetic capacity concept is ignored.

²² Unless one would be settling for an input efficiency measure defined on the fixed input dimensions only. But we believe this contrasts too much with the focus on variable inputs in the economic capacity concepts.

²³ On the one hand, the link between both scale efficiency terms is simply the ratio of capacity terms

where the ratio of capacity notions is an adjustment factor that can be smaller, equal or larger than unity. On the other hand, the link between both economic capacity utilization notions is made by the scale terms as follows:

where this ratio of scale terms also forms an adjustment factor that can be smaller, equal or larger than unity.

²⁴ Appendices are available on the web site of the journal.

²⁵ On the one hand, the link between both scale efficiency terms is simply the ratio of capacity terms 

where the ratio of capacity notions forms an adjustment factor that can be smaller, equal or larger than unity. On the other hand, the link between both primal capacity utilization notions is provided by the scale terms as follows:

where also this ratio of scale terms offers an adjustment factor that can be smaller, equal or larger than unity.

²⁶ We maintain all observations rather than opting for a preliminary screening looking for any potential outliers.

²⁷ Atkinson and Dorfman (2009) also found considerable differences in allocative and technical inefficiencies among plants. Their efficiency levels are higher because they allow for productivity change over time as well as flexible returns to scale.

²⁸ The nonparametric Li (1996) test statistic compares two unknown distributions making use of kernel densities. It is rather widely used in the frontier estimation literature. Figures of the densities entering this statistic that turn out to be significantly different between run-of-river plants and reservoir plants are plotted in Appendix D.

²⁹ In this empirical illustration, since there is only a single output, weak and strong output disposability coincide. Therefore, we have specified weak disposability in the inputs for these output-oriented decompositions.

³⁰ Though this may well not be that easy, given that even the precise integration of scale efficiency into the Malmquist productivity index has been the source of considerable controversy (see Balk (2001) for an overview).