Hicks-Moorsteen versus Malmquist: a connection by means of a radial productivity index

A primal index of productivity change is introduced which decomposes exactly into three components: technical change, technical efficiency change and average scale economies (radial scale change). The productivity index is defined using variations of the distance function along pre-assigned input–output rays and, for this reason, it is deemed a radial productivity index (RPI). It is proven that: first, the RPI index collapses to the Malmquist productivity index when the technology is constant returns to scale (CRS); second the RPI index equals the Hicks-Moorsteen productivity index under homotheticity of technology (and non-CRS). The key to these results is a new definition and measure of the contribution of scale economies to productivity change.     

An aggregate quantity framework for measuring and decomposing productivity change

Total factor productivity (TFP) can be defined as the ratio of an aggregate output to an aggregate input. This definition naturally leads to TFP indexes that can be expressed as the ratio of an output quantity index to an input quantity index. If the aggregator functions satisfy certain regularity properties then these TFP indexes are said to be multiplicatively complete. This paper formally defines what is meant by completeness and reveals that (1) the class of multiplicatively complete TFP indexes includes Laspeyres, Paasche, Fisher, T?rnqvist and Hicks-Moorsteen indexes, (2) the popular Malmquist TFP index of Caves et al. (Econometrica 50(6):1393?C1414, 1982a) is incomplete, implying it cannot always be interpreted as a measure of productivity change, (3) all multiplicatively complete TFP indexes can be exhaustively decomposed into measures of technical change and efficiency change, and (4) the efficiency change component can be further decomposed into measures of technical, mix and scale efficiency change. Artificial data are used to illustrate the decomposition of Hicks-Moorsteen and Fisher TFP indexes.     

Indirect productivity measurement

Caves, Christensen and Diewert [1982a] showed that the Törnqvist productivity index is superlative in a considerably more general sense than had been previously believed. We examine the allocative and technical efficiency hypotheses on which their finding rests. We show that the allocative efficiency hypothesis can be modified, which makes the Törnqvist index superlative in a wider sense than even Caves, Christensen and Diewert showed, since it is consistent with a type of allocative efficiency other than the standard cost minimization and revenue maximization hypotheses considered by Caves, Christensen and Diewert. We also show that if the technical efficiency hypothesis is relaxed, the CCD result may no longer hold, and the distance functions that form the basis of the Malmquist productivity indexes, and hence of the Törnqvist productivity index, must be calculated. We then show how to calculate the underlying distance functions, and we argue that there are real advantages to doing so.The refereeing process of this paper was handled through N.R. Adam.     

The biennial Malmquist productivity change index

In this paper we introduce a new Malmquist productivity index that has three attractive features: it avoids linear programming infeasibilities under variable returns to scale, it allows for technical regress, and it does not need to be recomputed when a new time period is added to the data set. The proposed index is compared to both the adjacent Malmquist index and the global Malmquist index in an empirical example, which highlights the drawbacks of the existing indexes compared to the proposed biennial Malmquist index.Our results show that 13% of the observations in the data set may have to be ignored due to infeasibilities when decomposing the adjacent Malmquist index. Using only this reduced data set does at times lead to quite different results than those generated by applying the proposed biennial Malmquist index to the entire data set. The empirical example also shows that productivity change estimated between two time periods using the global Malmquist index change substantially when a third time period is added to the data set, whereas the proposed biennial Malmquist index is immune to this problem.     

A new decomposition and interpretation of Hicks-Moorsteen productivity index for analysis of Stock Exchange companies: Case study on pharmaceutical industry

This paper deals with the productivity index based on Hicks-Moorsteen (HM) productivity index for analysis of Stock Exchange firms. In this paper, a decomposition of Hicks-Moorsteen productivity index is provided that uses data envelopment analysis (DEA) models. On the other hand, it is offered a new interpretation and decomposition for components of HM productivity index that in previous studies were not performed any interpretation of it. These components have named the output production and the input consumption in terms of changing the efficiency frontier. In addition, by comparing the components of HM productivity index and productivity value, the optimal values of output production and input consumption are determined. Also, the analysis is determined how to identify the optimal value of a firm over a period of time. The proposed approach is applied to 26 pharmaceutical manufacturers that are from Tehran Stock Exchange. Data have been collected for six consecutive years (2009–2014) and have been considered financial ratios for evaluating companies. The results show that the Hicks-Moorsteen productivity index is more able to provide managerial aspects than other productivity indexes such as the Malmquist productivity index.     

Productivity indexes under Hicks neutral technical change

The Malmquist and Hicks–Moorsteen productivity indexes are the two most widely used theoretical indexes for measuring productivity growth of firms, industries and countries. We indicate that these two indexes coincide under constant returns to scale technology and Hicks neutral technical change. While the conditions on production technology under which two indexes coincide have been examined before, this is the first study on the types of technical change under which they coincide. We shed new insight on the relationship between these popular indexes.     

Malmquist Indexes Using a Geometric Distance Function (GDF). Application to a Sample of Portuguese Bank Branches

Traditional approaches to calculate total factor productivity (TFP) change through Malmquist indexes rely on distance functions. In this paper we show that the use of distance functions as a means to calculate TFP change may introduce some bias in the analysis, and therefore we propose a procedure that calculates TFP change through observed values only. Our total TFP change is then decomposed into efficiency change, technological change, and a residual effect. This decomposition makes use of a non-oriented measure in order to avoid problems associated with the traditional use of radial oriented measures, especially when variable returns to scale technologies are to be compared. The proposed approach is applied in this paper to a sample of Portuguese bank branches.     

Parametric Decomposition of a Generalized Malmquist Productivity Index

This paper provides a parametric decomposition of a generalized Malmquist productivity index which takes into account scale economies. Unlike Balk (2001), the contribution of scale economies to productivity change is evaluated without recourse to scale efficiency measures, which are neither bounded for globally increasing, decreasing, or constant returns to scale technologies nor for ray-homogeneous technologies. An empirical application using panel data from Spanish savings banks is included. This application shows the advantages of the suggested method compared to Balk's approach. The results show an increase of total factor productivity which can be mainly attributed to technical progress and the positive effect of returns to scale.     

A global Malmquist-Luenberger productivity index

This paper introduces an alternative environmentally sensitive productivity growth index, which is circular and free from the infeasibility problem. In doing so, we integrated the concept of the global production possibility set and the directional distance function. Like the conventional Malmquist-Luenberger productivity index, it can also be decomposed into sources of productivity growth. The suggested index is employed in analyzing 26 OECD countries for the period 1990–2003. We also employed the conventional Malmquist-Luenberger productivity index, the global Malmquist productivity index and the conventional Malmquist productivity index for comparative purposes in this empirical investigation.     

Efficiency, productivity, and scale economies in the U.S. property-liability insurance industry

The paper examines efficiency, productivity and scale economies in the U.S. property-liability insurance industry. Productivity change is analyzed using Malmquist indices, and efficiency is estimated using data envelopment analysis. The results indicate that the majority of firms below median size in the industry are operating with increasing returns to scale, and the majority of firms above median size are operating with decreasing returns to scale. However, a significant number of firms in each size decile have achieved constant returns to scale. Over the sample period, the industry experienced significant gains in total factor productivity, and there is an upward trend in scale and allocative efficiency. More diversified firms and insurance groups were more likely to achieve efficiency and productivity gains. Higher technology investment is positively related to efficiency and productivity improvements.     

A primal Divisia technical change index based on the output distance function

We derive a primal Divisia technical change index based on the output distance function and further show the validity of this index from both economic and axiomatic points of view. In particular, we derive the primal Divisia technical change index by total differentiation of the output distance function with respect to a time trend. We then show that this index is dual to the Jorgenson and Griliches (1967) dual Divisia total factor productivity growth (TFPG) index when both the output and input markets are competitive; dual to the Diewert and Fox (2008) markup-adjusted revenue-share-based dual Divisia technical change index when market power is limited to output markets; dual to the Denny et al. (1981) and Fuss (1994) cost-elasticity-share-based dual Divisia TFPG index when market power is limited to output markets and constant returns to scale is present; and also dual to a markup-and-markdown-adjusted Divisia technical change index when market power is present in both output and input markets. Finally, we show that the primal Divisia technical change index satisfies the properties of identity, commensurability, monotonicity, and time reversal. It also satisfies the property of proportionality in the presence of path independence, which in turn requires separability between inputs and outputs and homogeneity of subaggregator functions.     

A Quasi-Malmquist Productivity Index

The Malmquist productivity index is based on distance functions, which are reciprocals of radial Debreu-Farrell efficiency measures, and which have a number of desirable properties. Linear programming techniques are frequently employed to calculate the efficiency measures. However these techniques can leave slacks, which constitute a non-radial form of inefficiency which is not incorporated into the analysis. Thus a radial efficiency measure overstates true efficiency, the reciprocal distance function understates the distance to the relevant efficient subset, and the Malmquist productivity index is adversely affected, although in an analytically indeterminate direction. This has led us to consider a new definition of one-sided efficiency, and to develop a new nonradial efficiency measure which incorporates all slacks on the selected side. Replacing conventional radial efficiency measures with our new non-radial efficiency measures generates what we call a quasi-Malmquist productivity index. We illustrate our quasi-Malmquist productivity index with an application to productivity change in Spanish banking.     

Parametric decomposition of the input-oriented Malmquist productivity index: with an application to Greek aquaculture

Using a stochastic frontier approach and a translog input distance function, this paper implements the input-oriented Malmquist productivity index to a sample of Greek aquaculture farms. It is decomposed into the effects of technical efficiency change, scale efficiency change, input-mix and, technical change, which is further attributed to neutral, output- and input-induced shifts of the frontier. Implementable expressions for the aforementioned components are obtained using a discrete changes-approach that is consistent with the usual discrete-form data. Empirical findings indicate that the productivity of the farms in the sample increased during the period 1995–1999 and it was shaped up primarily by the input mix-effect and technical change.     

Cost Malmquist productivity index: an output-specific approach for group comparison

The cost Malmquist productivity index (CMPI) has been proposed to capture the performance change of cost minimizing Decision Making Units (DMUs). Recently, two alternative uses of the CMPI have been suggested: (1) using the CMPI to compare groups of DMUs, and (2) using the CMPI to compare DMUs for each output separately. In this paper, we propose a new CMPI that combines both procedures. The resulting methodology provides group-specific indexes for each output separately, and therefore offers the option to identify the sources of cost performance change. We also define our index when input prices are not observed and establish, in that case, a duality with a new technical productivity index, which takes the form of a Malmquist productivity index. We illustrate our new methodology with a numerical example and an application to the US electricity plant districts.     

A cross-country comparison of productivity growth using the generalised metafrontier Malmquist productivity index: with application to banking industries in Taiwan and China

This paper is an extension of the metafrontier Malmquist productivity index, which takes into account the effect of scale efficiency change in its decomposition for both the non-parametric and parametric frameworks. Meanwhile, the ‘catch-up’ in the index is also disintegrated as two components: pure technological catch-up and frontier catch-up. An empirical application that uses unbalanced panel data of the Taiwanese and Chinese commercial banking industry is also conducted under a parametric framework. The results reveal that the adverse scale efficiency change is the key factor to inducing the inferior productivity growth seen in Chinese banks compared with Taiwanese banks, which spotlights the importance of the scale efficiency change term on productivity measures. It also provides one possible explanation for the recent hot issue about the motives for the two shores of the Taiwan Straits advancing financial openness to each other and mutually signing a banking Memorandum of Understanding.     

Index Number Concepts,Measures and Decompositions of Productivity Growth

This paper explores the definitions and properties of total factor productivity growth (TFPG) indexes, focusing especially on the Paasche, Laspeyres, Fisher, Törnqvist, and implicit Törnqvist ones. These indexes can be evaluated from observable price and quantity data, and certain of these are shown to be measures of TFPG concepts and theoretical indexes that have been proposed in the literature. The mathematical relationships between these and quantity aggregates, financial measures, and price and quantity indexes are explored. Decompositions of the productivity growth indexes are also given. The paper concludes with a brief overview of some limitations on our analysis.     

An Implicit Törnqvist Index of Real GDP

Recently much attention has been devoted to superlative indexes in the context of the national accounts. In this paper we advocate the use of the implicit Törnqvist quantity index to measure real GDP. This index, which has been proposed by Diewert and Morrison (1986), has never received serious consideration in the literature. Yet, compared to the better-known Fisher index, the implicit Törnqvist index of real GDP has a number of advantages. Thus, it can be shown to be exact for the Translog GDP function, it allows for a complete multiplicative decomposition of nominal and real GDP, and it is consistent with state-of-the-art measures of total factor productivity that typically rely on the Törnqvist aggregation. Estimates for a sample of 26 countries are reported. We find that the Laspeyres quantity index still used by the statistical agencies of most countries tends to underestimate real growth. Over the 1960–1996 period, the cumulated shortfall was as much as 13.4% of GDP in the case of Japan.     

The Decomposition of Malmquist Productivity Indexes

Two different Malmquist productivity indexes have been proposed. One I call partially oriented because it is either output- or input-oriented, and the other is simultaneously output- and input-oriented. The partially oriented Malmquist index owes some of its popularity to the fact that it has been decomposed to isolate various sources of productivity change. Conversely, the simultaneously oriented Malmquist index has not achieved popularity in part because it has not been decomposed. In this paper I evaluate alternative decompositions of the partially oriented Malmquist index, and I obtain a new decomposition of the simultaneously oriented Malmquist index. This new decomposition leads me to conclude that the latter index is deserving of greater attention than it has received to date.     

Decomposing hospital productivity changes, 1985–1988: A nonparametric Malmquist approach

Annual data on U.S. hospitals from 1985–1988 are evaluated by ownership type—profit, nonprofit, state and local government, and U.S. Department of Veterans Affairs (VA)—for changes in hospital productivity over time. Distance functions are used to measure Malmquist indices of productivity change, which are then decomposed into indices of efficiency change and technology change. In contrast to previous studies using this approach, we allow for variable returns to scale and use both input and output orientations. We find that changes in technology dominate changes in inefficiency in determining changes in productivity.     

Measurement of Productivity Changes: Two Malmquist Index Approaches

The adjacent Malmquist productivity index is compared to the more recently suggested base period Malmquist productivity index. The two index approaches are evaluated with respect to base period dependency, the circular test, and with respect to a set of additional classical index tests. In addition it is shown that the base period index is independent of base period if and only if the marginal rate of substitution of inputs is independent of time. Finally, the adjacent and the base period indexes are put through a Monte Carlo (bootstrap) test to see if they yield similar results when applied to a panel of Swedish pharmacy data.