Decomposing profit inefficiency in DEA through the weighted additive model

doi:10.1016/j.ejor.2011.01.054

European Journal of Operational Research

Volume 212, Issue 2, 16 July 2011, Pages 411-416

https://doi.org/10.1016/j.ejor.2011.01.054 Get rights and content

Abstract

An issue that has received little attention in the Data Envelopment Analysis literature is the decomposition of profit inefficiency by means of measures that account all sources of technical inefficiency. In this paper we introduce a new way to measure and decompose profit inefficiency through weighted additive models. All our results are derived from a new Fenchel–Mahler inequality using duality theory.

Introduction

This paper introduces a new way to measure the amount by which the observed profit of a firm deviates from the maximum profit, and decomposes this gap (if it exists) into two components. We are particularly interested in ensuring that the technical inefficiency component we propose accounts for all sources of inefficiency. In particular, the approach is operationalized in a DEA (Data Envelopment Analysis) context through the weighted additive model (Lovell and Pastor, 1995). In some sense, this paper takes up where Cooper et al. (1999, p. 26–35) left off in the decomposition of profit inefficiency by means of additive models.

In for-profit organizations the measurement of profit inefficiency is particularly important and, in fact, is generally the most important objective. A firm is usually interested in changing input and output quantities if this leads to real economic gains. In this sense, profit inefficiency measures how close the firm is to the optimal profit. Obviously, the measurement of this type of inefficiency requires not only quantity but also market price information.

In the Data Envelopment Analysis literature, the profit inefficiency of a firm has been usually decomposed into technical and allocative inefficiency. Technical inefficiency measures how close the firm is to the frontier of the technology. Whereas allocative inefficiency for technically inefficient units measures the loss due to being sub-optimal under given market prices.

An issue that has received little attention in the Data Envelopment Analysis (DEA) literature is the decomposition of the gap between optimal and actual profit through Generalized Efficiency Measures (GEMs). Unlike other efficiency measures, a GEM is non-oriented and non-radial in character and it incorporates all inefficiencies that a DEA model is capable of identifying (see Cooper et al., 1999). Indeed, a GEM corresponds to the Pareto–Koopmans definition of technical efficiency, in contrast to radial measures and directional distance functions (see Ray, 2004, p.95), which ignore the possible existence of slacks associated with the projected points on the production frontier. In fact, this drawback of the usual approaches has previously been criticized by Portela and Thanassoulis (2007, p. 484): “The calculation of technical (in)efficiency through the hyperbolic or directional models … do not account … for all the sources of inefficiency, namely those associated with slacks. This is an important problem in a context where overall efficiency is being measured”. All these features have yielded an increasing interest of researchers in GEMs (see, for example, Silva Portela et al., 2003). However, there exist only a few attempts to develop a decomposition of profit inefficiency by means of GEMs in DEA. This limits in practice the use of this type of technical efficiency measure when information about market prices is available.

In this paper, we show that the usual measure of profit inefficiency can be decomposed into its technical component, measured through a weighted additive model, and its allocative component, derived as a residual. Our findings are all based on a new Fenchel–Mahler inequality using duality theory (see Färe and Grosskopf, 2000).

The remainder of the paper is organized as follows: In Section 2, existing approaches for calculating and decomposing profit inefficiency by means of GEMs are outlined. In Section 3, we briefly detail the definition and properties of the weighted additive models. In Section 4, we present a new measure of profit inefficiency based on the weighted additive models and illustrate its calculation and decomposition through a simple numerical example. In Section 5, we present the conclusions.

Section snippets

Brief review of existing approaches

In standard microeconomic theory, the economic behaviour of the firm is usually characterized by cost minimization, revenue maximization or profit maximization. Obviously, the choice of an economic approach of the firm depends, in part, on what assumptions one is willing to make. Throughout this paper, we assume that firms face exogenously determined output and input prices. We also assume that the objective of each firm is to choose the inputs and outputs combinations that result in the

Weighted additive models

Working in the usual DEA framework, let us consider n decision making units (DMUs) to be evaluated. DMU_j consumes $X_{j} = (x_{1 j}, \dots, x_{mj}) \in R_{+}^{m}$ amounts of input for the production of $Y_{j} = (y_{1 j}, \dots, y_{sj}) \in R_{+}^{s}$ amounts of output. The relative efficiency of each DMU in the sample is assessed with reference to the so-called production possibility set, which can be empirically constructed in DEA from the observations by assuming several postulates (see Banker et al., 1984). Also, to operationalize the approach, we

A new way for measuring and decomposing profit inefficiency

In this section we introduce a new way for measuring and decomposing profit inefficiency of a price taker firm. It is based on the classical difference-form measure, optimal profit minus observed profit. Both the profit inefficiency measure and the decomposition we propose come from duality theory. In particular, they are derived from a new additive Fenchel–Mahler inequality, as we show next.

In order to reach our goals we need first of all to show the dual problem of (4), which is $\begin{matrix} WA (X_{0}, Y_{0}, W^{-}, W^{+}) \end{matrix}$

Conclusions

In a parametric setting, the approach set up by Chambers et al. (1998) based on the directional distance function is a nice tool for measuring and decomposing overall inefficiency. Unfortunately, this approach may be inappropriate in a DEA framework, particularly if non-directional inefficiencies – non-zero slacks – are detected. Therefore, we have shown in this paper how to use the weighted additive model, which is associated with the Pareto–Koopmans definition of technical efficiency, to

Acknowledgements

We thank three anonymous referees for providing constructive comments and help in improving the contents and the presentation of this paper. We are grateful to the Ministerio de Ciencia e Innovacion, Spain, for supporting this research with grant MTM2009–10479.

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Interfaces with Other DisciplinesDecomposing profit inefficiency in DEA through the weighted additive model

Abstract

Introduction

Section snippets

Brief review of existing approaches

Weighted additive models

A new way for measuring and decomposing profit inefficiency

Conclusions

Acknowledgements

Journal of Econometrics

European Journal of Operational Research

European Journal of Operational Research

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

Management Science

Profit, directional distance functions, and Nerlovian efficiency

Journal of Optimization Theory and Applications

RAM: a range adjusted measure of inefficiency for use with additive models, and relations to others models and measures in DEA

Journal of Productivity Analysis

Multi-Output Production and Duality: Theory and Applications

Notes on Some Inequalities in Economics

Economic Theory

Interfaces with Other Disciplines
Decomposing profit inefficiency in DEA through the weighted additive model