شاخص بهره وری مالم کوییست هزینه با در نظر گرفتن عملکرد گروه A cost Malmquist productivity index capturing group performance

abstract This paper develops an index for comparing the productivity of groups of operating units in cost terms when input prices are available. In that sense it represents an extension of a similar index available in the literature for comparing groups of units in terms of technical productivity in the absence of input prices. The index is decomposed to reveal the origins of differences in performance of the groups of units both in terms of technical and cost productivity. The index and its decomposition are of value in contexts where the need arises to compare units which perform the same function but they can be grouped by virtue of the fact that they operate in different contexts as might for example arise in comparisons of water or gas transmission companies operating in different countries. © ۲۰۱۴ Elsevier B.V. All rights reserved. 1. Introduction Efficiency and productivity are major sources of economic development and a thorough understanding of the factors affecting productivity is important for managers, economists and policy makers, especially in difficult times of economic crisis where better performance is paramount for sustainability and progress. It is not surprising, therefore, that in recent decades the measurement and analysis of performance has enjoyed a great deal of interest and has seen major developments from a theoretical, methodological and empirical point of view. The measurement and analysis of ef- ficiency and productivity evolved for a long time as independent scientific fields but in recent years the two have merged in a common framework and in this context often efficiency is incorporated in productivity analysis, which is deemed a better approach by many. We address here the case where operating units are using multiple inputs to secure multiple outputs and input prices are exogenous and available. Further, we address the case where the units in question perform the same function, using the same types of inputs to secure the same kinds of outputs, but are operating in different contexts. One case in point is the increasing need to conduct comparisons across countries. For example see Haney and Pollitt (2012) on the international comparison of electricity transmission companies. Clearly companies performing the same function but in different countries can be grouped as operating in different contexts (e.g. on ∗ Corresponding author. Tel.: +447753620465. E-mail addresses: e.thanassoulis@aston.ac.uk, dea.et@btinternet.com (E. Thanassoulis), khanjani@tabrizu.ac.ir (R.K. Shiraz), nmaniadakis@esdy.edu.gr (N. Maniadakis). prices and regulatory regimes). Even within a given country, however, often operating units performing the same function can differ by context. For example the branches of a bank may differ in terms of scope of activities and types of clientele depending on whether they operate in a rural or urban environment. In such cases input costs, e.g. for labour and capital assets, may differ between groups of units as well as within units of a given group. Comparisons therefore of units need to isolate and measure the impact of group membership on productivity. This issue has already been addressed by a number of authors. The concept of ‘metafrontiers’ has been developed to isolate group membership from ‘managerial’ effects on efficiency and productivity (e.g. see Battese, Rao, & O’Donnell, 2004; O’Donnell, Rao, & Battese, 2008). These approaches assume each group has its own best practice frontier but that there is a metafrontier which envelops all individual group frontiers. This allows one to decompose the evaluated unit’s attainment into a part attributable to the unit itself (i.e. its own management) and a part attributable to group membership. More recently Brennan, Carla, and John (2014) have considered groupings of operating units by environmental context and have developed models for estimating an index to capture the impact on productivity change attributable to the context of each grouping. They have prior notions as to more and less favourable operating contexts. An alternative to the metafrontier approach for comparing groups of units on performance is put forth by Camanho and Dyson (2006). This approach relies on assessing units within existing groupings without recourse to a metafrontier. Recourse to a metafrontier implies that there is an expansion of technology by convexification of existing group technologies. The Camanho and Dyson (2006) approach does not make this assumption. Further, it does not distinguish between http://dx.doi.org/10.1016/j.ejor.2014.09.002 0377-2217/© ۲۰۱۴ Elsevier B.V. All rights reserved. E. Thanassoulis et al. / European Journal of Operational Research 241 (2015) 796–۸۰۵ ۷۹۷ more or less favourable operating contexts. Thus the Camanho and Dyson (2006), and therefore the approach in this paper too, is less demanding of prior assumptions. The Camanho and Dyson (2006) approach compares groups of operating units where the focus is on technical efficiency and prices of inputs or outputs either do not exist or are ignored. Our paper builds on the Camanho and Dyson (2006) approach to address the case where input prices are available and they may differ for units both within and across groups. As in their case we make no recourse to the notion of a metafrontier and make no prior assumptions as to whether operating in one group as opposed to another is necessarily advantageous and whether that holds for all input-output mixes and or scale sizes. The index developed here, as we will see later, offers a number of advantages over more traditional metafrontier based approaches for comparing groups of DMUs on performance. The advantages stem from the fact that the index developed here takes into account both technical and cost efficiency. Further, more than the metafrontier approach, it is decomposable both in cost and technical terms at several levels, enriching the insights that can be gained into group performance. We return to the advantages and drawbacks of the index at the concluding section. Turning to measures of productivity, there are alternative approaches to quantifying productivity and a very popular one is the Malmquist productivity index. Malmquist’s (1953) seminal work stayed unnoticed and without any applications for some time. Caves, Cristensen, and Diewert (1982) reintroduced it to productivity measurement and subsequently, Färe and Grosskopf (1992, 1996), Grosskopf (1993), Färe, Grosskopf, Lindgren, and Roos (1989), Färe, Grosskopf, and Lovell (1994), Färe and Grosskopf (1996), Färe, Grifell-Tatje, Grosskopf, and Lovell (1997), Färe, Grosskopf, and Russell (1998), Portela and Thanassoulis (2006, 2010) further elaborated the approach. A major extension of the index was its decomposition into a measure capturing efficiency change and one capturing technical change over time by Färe, Grosskopf, Norris, and Zhang (1994). We refer to this here as the ‘classical’ Malmquist index. The part measuring efficiency change measures the shift of the individual unit relative to its frontier over time while technical change captures the shift of the production boundary itself over time. The index can be computed in the empirical context using DEA models. Under certain conditions the Malmquist index approximates other popular indices such as the Törnqvist (1936) and the Fisher (1922) index. These two indices are easy to compute and they have been shown to be exact for general forms of technology, but in the presence of inefficiency they may provide biased (see Coelli, Prasada Rao, & Battese, 1998) estimates of productivity and thus the Malmquist index is preferable. The classical Malmquist index Färe et al. (1994) was generally developed for cases where technical efficiency in terms of input–output levels was the focus and input prices either did not exist or were ignored. Later a parallel strand of the literature evolved which takes input prices into account where they are available. In this case an important form of efficiency, namely allocative, is contributory to productivity change in cost terms. Allocative efficiency captures the degree to which an already technically efficient production unit can further reduce its aggregate cost of securing its outputs by selecting an optimal mix of inputs given the exogenously fixed prices at which it can secure its inputs. Allocative efficiency and its change may affect performance significantly and this is important in light of empirical studies which have identified frequent instances of allocative inefficiency at production units. In such cases production units may improve over time their performance by changing the input mix they employ to produce their output. Hence the impact of allocative efficiency change on productivity change should be accounted for Coelli et al. (1998) when input prices are available. In this context, Bauer (1990) and Balk (1998) decomposed, in the econometric and index number framework respectively, productivity change so that allocative efficiency change is captured. Maniadakis and Thanassoulis (2000, 2004) developed a cost Malmquist productivity index, computed through DEA models, which is decomposed into technical change and overall efficiency change which captures costs. The index is defined in terms of cost rather than input distance functions and is applicable when producers can be assumed to be cost minimisers and input–output quantity and input price data are available. This index has seen many applications in various settings including health care, banks, electricity units, real estate, forest product industries, and educational programmes and it has also seen further extensions (Hosseinzadeh, Jahanshahloo, & Akbarian, 2007). Camanho and Dyson (2006) address the case where units can be grouped by operating context. They developed measures, based on the Malmquist index, that enable the decision making unit’s internal inefficiencies to be distinguished from those associated with the group (or program) to which the unit belongs. The present paper extends this idea to show how the cost Malmquist index of Maniadakis and Thanassoulis (2004) can be used to build on the Camanho and Dyson (2006) ideas so as to compare groups of operating units in cost terms. The paper develops an overall index that captures the relative productivity in terms of cost between units belonging to different groups. The index is then decomposed to reveal the impact of technical and allocative efficiency at group level. Information of this type would be useful for managing the performance of groups of units. It would enable managers to identify best practice across groups and this would be both in terms of technical and cost efficiency. The remainder of this paper is organized as follows. Section 2 provides a review of literature on the classical Malmquist index; the cost Malmquist index and group (technical) Malmquist index. Section 3 develops the cost Malmquist index for comparing groups of units on productivity. Section 4 develops the decomposition of the index de- fined in Section 3. Section 5 illustrates the index developed by means of a numerical example. Section 6 concludes