Aggregation of scale efficiency

doi:10.1016/j.ejor.2014.06.038

European Journal of Operational Research

Volume 240, Issue 1, 1 January 2015, Pages 269-277

https://doi.org/10.1016/j.ejor.2014.06.038 Get rights and content

Highlights

•
We extend the aggregation theory in efficiency and productivity analysis.
•
We derive solutions for aggregation of individual scale efficiency measures.
•
We provide practical way of estimating scale efficiency of an industry (e.g., from DEA).
•
The new aggregation result is coherent with previous aggregation frameworks.

Abstract

In this article we generalize the aggregation theory in efficiency and productivity analysis by deriving solutions to the problem of aggregation of individual scale efficiency measures, primal and dual, into aggregate primal and dual scale efficiency measures of a group (e.g., industry). The new aggregation result is coherent with aggregation framework and solutions that were earlier derived for other related efficiency measures and can be used in practice for estimation of scale efficiency of an industry or other groups of firms within it.

Introduction

Analysis of economies of scale has been one of the fundamental subjects in economics, operations research and production management, both in theory and especially in practice. Indeed, the issue of attaining optimal scale of operations frequently appears among priority questions on management agendas of various types of companies, whether private or public. In this article we focus on the scale efficiency, which combines both the notion of optimal scale and the notion of (relative) production efficiency. Our particular interest is in how to appropriately measure the scale efficiency of a group—an industry consisting of firms, a firm consisting of plants, a union consisting of countries, etc.

For several decades, many studies in various economics and business literature challenged the issue of proper measurement of scale economies for various contexts and for various estimators. These studies include the seminal works of Hanoch, 1975, Panzar and Willig, 1977, Føsund and Hjalmarsson, 1979, Banker, 1984, Banker et al., 1984, Färe and Grosskopf, 1985, Färe et al., 1986, Banker and Thrall, 1992, Banker et al., 1996, Føsund, 1996, Golany and Yu, 1997, as well as more recent works of Føsund and Hjalmarsson, 2004, Krivonozhko et al., 2004, Hadjicostas and Soteriou, 2006, Hadjicostas and Soteriou, 2010, Podinovski et al., 2009, Zelenyuk, 2013a, Zelenyuk, 2013b, Zelenyuk, 2014, Peyrache, 2013, to mention just a few.

In previous studies, researchers primarily focused on the measurement of scale economies for an individual or disaggregate decision making unit (DMU). In the present work we will focus on the issue of how to appropriately aggregate such individual scale efficiency measures, or their estimates/scores obtained from these measures for individual DMUs, into aggregate measures (or aggregate scores) of scale efficiency of a group. Indeed, after obtaining many individual scale efficiency scores, researchers may truly need a proper way to summarize these many scores into one or few numbers to present to their audience concisely. Clearly, one could simply use a sample average of individual estimates—but which one: the arithmetic or the geometric? More importantly, a problem with a sample average, whether arithmetic or geometric, is that it ignores a relative weight of each DMU in the aggregation. On the other hand, while a weighted average can account for a relative weight of each DMU, a critical question arises along the use of a weighted average: Which set of weights should be used? Indeed, conclusions and related policy implications may heavily depend on the weights chosen for the aggregation. Therefore, our primary focus is on deriving the weights for the aggregation of scale efficiency measures, which has not been done in the literature so far.

Studies on the aggregation problem in efficiency analysis go back to at least the seminal work of Farrell (1957), who coined the term structural efficiency of an industry. This notion was then criticized and elaborated by Føsund and Hjalmarsson (1979), who introduced the notion of efficiency of an average unit, and by Li and Ng (1995) who synthesized these latter works, considering them in the context of aggregation weights based on shadow prices. (See also a related discussion in Ylvinger (2000).) On the pure theoretical front, in their seminal work, Blackorby and Russell (1999) unveiled several impossibility theorems for a general aggregation problem of efficiency measures. One important implication of their work was that any positive result on aggregation in efficiency measurement must involve additional assumptions. This route was taken by Färe and Zelenyuk (2003) who, upon accepting certain assumptions on aggregate technology, optimization behavior and prices, and applying a revenue analogue of the fundamental theorem from Koopmans (1957), derived a solution to the aggregation problem for the output oriented Farrell-type technical efficiency measures. Similar approach was later used for deriving various aggregation results, such as aggregation of input oriented technical efficiency measures (Färe, Grosskopf, & Zelenyuk, 2004), aggregation of productivity indexes (Zelenyuk, 2006), aggregation within and between the sub-groups (Simar & Zelenyuk, 2007), aggregation of economic growth rates (Zelenyuk, 2011), aggregation of scale elasticities (Färe & Zelenyuk, 2012), but these works do not answer how the commonly used scale efficiency measures should be aggregated.

In the present work, we generalize the existing approach of aggregation of efficiency measures to the context of aggregation of scale efficiency measures, such that it is coherent with and encompass previous aggregation results for the related efficiency measures. This is a new theoretical result that can be relatively easily applied in practice for obtaining aggregate scale efficiency measures from suitable estimates of the individual scale efficiency scores.

The paper is structured as following: Section 2 briefly outlines the theory of measuring scale efficiency on individual level. Section 3 briefly outlines useful relationships between various efficiency measures that will be used in derivations of solutions to the aggregation problem. Section 4 proposes a solution to the aggregation problem from the perspective of mathematical (functional equations) approach. Section 5 outlines economic approach to solve the aggregation problem. Section 6 presents some special cases and Section 7 concludes.

Section snippets

Characterization of individual efficiencies

To keep our context general yet as simple as possible, let us consider a group of n decision making units (plants or firms or countries, etc.), hereafter DMUs, indexed by $k = 1, 2, \dots, n$ . Our main focus in the paper will be on aggregate efficiencies-measures that would represent various types of efficiency of a group of DMUs (e.g., a firm consisting of plants, an industry consisting of firms, etc.). We also want such aggregate measures of a group to be related or constructed from the individual

Key decompositions of individual efficiency

In our further derivations we will utilize the following well-known decomposition of individual revenue efficiency measure, that hold for any $k \in {1, 2, \dots, n}$ , ${RE}^{k} (x, y, p) = {TE}^{k} (x, y) \times {AE}^{k} (x, y, p), \forall (x, y) \in T^{k}, \forall p \in R_{+ +}^{M}$ for the actual technology and, analogously for the CRS-hypothetical technology, ${\overset{ˇ}{RE}}^{k} (x, y, p) = {\overset{ˇ}{TE}}^{k} (x, y) \times {\overset{ˇ}{AE}}^{k} (x, y, p), \forall (x, y) \in T^{k}, \forall p \in R_{+ +}^{M},$ where ${AE}^{k} (x, y, p)$ and ${\overset{ˇ}{AE}}^{k} (x, y, p)$ are so-called allocative efficiency measures (output oriented), measuring inefficiency due to non-optimal (w.r.t. revenue

Efficiency aggregation: mathematical approach

One way to describe the problem of aggregation we face here is by formulating a goal to find a sequence of aggregation functions $f_{1}, f_{2}, f_{3}, \dots$ that would relate the aggregate efficiency measures, which we denote here with $\overset{ˇ}{\overline{RE}}$ , $\overline{RE}, \overline{TE}, \overline{AE}$ , $\overline{RSE}, \overline{TSE}$ and $\overline{ASE}$ , to the sets of their individual analogues. That is, we want to find some appropriate functions $f_{1}, f_{2}, f_{3}, f_{4}, f_{5}, f_{6}$ and $f_{7}$ , where $\overset{ˇ}{\overline{RE}} = f_{1} ({\overset{ˇ}{RE}}^{1}, \dots, {\overset{ˇ}{RE}}^{n}),$ $\overline{RE} = f_{2} ({RE}^{1}, \dots, {RE}^{n}),$ $\overline{TE} = f_{3} ({TE}^{1}, \dots, {TE}^{n}),$ $\overline{AE} = f_{4} ({AE}^{1}, \dots, {AE}^{n}),$ $\overline{RSE} = f_{5} ({RSE}^{1}, \dots, {RSE}^{n}),$ $\overline{TSE} = f_{6} ({TSE}^{1}, \dots, {TSE}^{n}),$ $\overline{ASE} = f_{7} ({ASE}^{1})$

Aggregate technology and aggregate efficiency

The goal of this sub-section is to define and outline characterization of aggregate technology. This aggregate technology will then be used to derive an aggregation scheme for aggregating scale efficiency such that it is coherent with aggregation of other related efficiency measures. To achieve this goal, let us denote the input and output allocations among DMUs within a group of interest by $X = (x^{1}, \dots, x^{n})$ , which is an $N \times n$ matrix, and by $Y = (y^{1}, \dots, y^{n})$ , which is an $M \times n$ matrix.

A critical step here is

Some special cases

It is now worth considering some interesting special cases for our aggregation problem. First of all, note that if all DMUs exhibit (or are to be measured with respect to) the same technology then the formulas derived above can still be applied without any changes.

Secondly, note that for all the types of efficiency measures considered here, except the allocative scale efficiency component, the aggregate efficiency is equal to one (i.e., 100% efficiency of the certain type) if and only if each

Concluding remarks

In this paper we developed a theory for aggregation of scale efficiency measures across DMUs (firms, industries, countries, group of countries, etc.). The derivation is based on assuming optimization behavior, additive aggregation structure on the output sets and the same (e.g., equilibrium) output prices across all DMUs. An advantage of the resulting aggregation scheme (and aggregation weights, in particular) is that it is not ad hoc but derived from certain assumptions coherent with economic

Acknowledgements

The author thanks the editor and the anonymous referees for the very fruitful comments and suggestions—they helped improving the paper substantially

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Interfaces with Other DisciplinesAggregation of scale efficiency

Highlights

Abstract

Introduction

Section snippets

Characterization of individual efficiencies

Key decompositions of individual efficiency

Efficiency aggregation: mathematical approach

Aggregate technology and aggregate efficiency

Some special cases

Concluding remarks

Acknowledgements

Journal of Mathematical Analysis and Applications

European Journal of Operational Research

European Journal of Operational Research

European Journal of Operational Research

European Journal of Operational Research

European Journal of Operational Research

European Journal of Operational Research

European Journal of Operational Research

European Journal of Operational Research

European Journal of Operational Research

European Journal of Operational Research

European Journal of Operational Research

European Journal of Operational Research

European Journal of Operational Research

European Journal of Operational Research

Some models for estimating technical and scale inefficiencies in data envelopment analysis

Management Science

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Journal of Productivity Analysis

Bootstrap methods: Another look at the jackknife

Annals of Statistics

Functional equations in economics

A nonparametric cost approach to scale efficiency

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Notes on some inequalities in economics

Economic Theory

Interfaces with Other Disciplines
Aggregation of scale efficiency