Revisiting the decomposition of cost efficiency for non-homothetic technologies: a directional distance function approach

In the early 1980’s Kopp and Diewert proposed a popular method to decompose cost efficiency into allocative and technical efficiency for parametric functional forms based on the radial approach initiated by Farrell. We show that, relying on recently proposed homogeneity and duality results, their approach is unnecessary for self-dual homothetic production functions, while it is inconsistent in the non-homothetic case. By stressing that for homothetic technologies the radial distance function can be correctly interpreted as a technical efficiency measure, since allocative efficiency is independent of the output level and radial input reductions leave it unchanged, we contend that for non-homothetic technologies this is not the case because optimal input demands depend on the output targeted by the firm, as does the inequality between marginal rates of substitution and market prices—allocative inefficiency. We demonstrate that a correct definition of technical efficiency corresponds to the directional distance function because its flexibility ensures that allocative efficiency is kept unchanged through movements in the input production possibility set when solving technical inefficiency, and therefore the associated cost reductions can be solely—and rightly—ascribed to technical-engineering-improvements. The new methodology allowing for a consistent decomposition of cost inefficiency is illustrated resorting to simple examples of non-homothetic production functions.     

Decomposing profit change: Konüs,Bennet and Luenberger indicators

We introduce complementary decompositions of profit change that, relying on the duality between the profit function and the directional distance function, shed light on the different sources of profit growth including measures of technical efficiency, allocative efficiency and technological change. Our decompositions extend the literature on Konüs and Bennet quantity and price indicators to profit change. The first decomposition is ‘exact’ in the sense of Diewert, by completely exhausting the sources of profit change into profit inefficiency change (including technical and allocative inefficiency change), technological change, and output and input price change. The second decomposition equates the Bennet quantity indicator to a productivity measure represented by the Luenberger indicator plus allocative inefficiency change. We deem it ‘complete’ because in contrast to the existing literature, it retains the information on allocative inefficiency change while preventing the existence of residual terms capturing price variations, whose meaningful interpretation has not been addressed until now. Our proposed solution takes advantage of the flexibility of the directional distance function when choosing a suitable directional vector. All decompositions have the same structural form and therefore their components can be compared to each other vis-à-vis, providing alternative measures of equivalent sources of profit growth.     

Determination of the efficiency of the world railway companies by method of DEA and comparison of their efficiency by Tobit analysis

This paper attempts to measure the performance of railway companies that produce passenger and freight services around the world. The data covering 10 years from 2000 to 2009 is analyzed first via the data envelopment analysis method in order to obtain technical efficiency and allocative efficiency scores of 31 railway companies for the purpose of the study. In the analysis conducted by use of the CCR model, while total 17 firms were efficient in the first year, this figure reaches to 18 companies for the last year with one more addition. While only two companies seem efficient in the first year, this figure goes down to one for the last year. With input oriented and variable return analysis conducted by use of the BCC model, the firms having technical efficiency at the beginning of the period were 20 in number. At the end of the period, the figure reaches to 24. Next, the outputs of DEA are correlated by Tobit regression and tried to determine decisiveness of the outputs on the efficiency. It has been seen that the same output composition used with Tobit analysis gives more compliant results with the allocative efficiency scores rather than with the technical efficiency scores.     

Theory and Application of Directional Distance Functions

In 1957 Farrell demonstrated how cost inefficiency could be decomposed into two mutually exclusive and exhaustive components: technical and allocative inefficiency. This result is consequence of the fact that—as shown by Shephard—the cost function and the input distance function (the reciprocal of Farrell's technical efficiency measure) are dual to each other. Similarly, the revenue function and the output distance function are dual providing the basis for the decomposition of revenue inefficiency into technical and allocative components (see for example, Färe, Grosskopf and Lovell (1994)). Here we extend those results to include the directional distance function and its dual, the profit function. This provides the basis for defining and decomposing profit efficiency. As we show, the output and input distance functions (reciprocals of Farrell efficiency measures) are special cases of the directional distance function. We also show how to use the directional distance function as a tool for measuring capacity utilization using DEA type techniques.     

Comparisons of Economic Inefficiency Between Output and Input Measures of Technical Inefficiency Using the Fourier Flexible Cost Function

The current paper constructs a Fourier flexible cost function, which is commonly known to be a more general function form than the typical translog form, and can globally approximate a true (but unknown) cost function. Both allocative and technical inefficiencies are considered using the Fourier function in the context of the parametric approach. The former is modeled using shadow input prices and the latter is formulated either by adding an extra term of scale parameter (when the Farrell's (1957) input technical inefficiency is assumed), or by correcting all the terms involving output quantities by a scale parameter (when the Farrell's output technical inefficiency is assumed). It is found that sample banks could save up to 23% of total costs, within the range of 3 and 69% uncovered by the previous works, in which allocative inefficiency plays a more important role than technical inefficiency. Furthermore, the cost of misallocated labor input alone constitutes more than 80% of total allocative inefficiency. Financial deregulation starting from 1991 in Taiwan appears to have improved economic efficiency of the banking industry.     

Creating an efficient market for nursing home care.

This paper reviews the theoretical foundations of the common Medicaid nursing home reimbursement systems: Reasonable cost related, fixed rate and negotiated rate reimbursement. Each reimbursement system is examined in terms of the four reimbursement system design goals: allocative efficiency, appropriateness of care, quality of care and equity of economic rewards. None of the reimbursement approaches are found to be deficient on the theoretical level, but practical problems of implementation are shown to be very difficult. As an alternative, a competitive binding system is proposed which would bring competitive market efficiency to the allocation of Medicaid funds for nursing home care. A mathematical programming model is developed to process the bidding information and to allocate Medicaid funds to nursing homes.     

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.     

Estimation and decomposition of inefficiency when producers maximize return to the outlay: an application to Norwegian fishing trawlers

This paper deals with estimation of a production technology where endogeneous choice of input and output variables is explicitly recognized. In particular, we assume that producers maximize return to the outlay (RO). For simplicity and tractability we start with a Cobb–Douglas transformation function with multiple inputs and outputs and show how the first-order conditions of RO maximization can be used to derive an estimating equation which is nothing but a partial input productivity equation. This equation does not suffer from the econometric endogeneity problem although the output and input variables are endogenous. First, we consider the case where producers are fully efficient allocatively but technically inefficient. The model is estimated using a single equation stochastic frontier approach. The model is then extended to allow allocative inefficiency and it is estimated as a system using generalized method of moment. Algebraic expressions are derived to decompose the effect of technical and allocative inefficiencies on RO. We also consider translog specifications that are estimated as (1) a single equation frontier model as well as (2) a system. We use a panel of Norwegian fishing trawlers data to estimate the model. Outputs are different species caught while inputs are labor and vessel size. We also control for number of days of operation, age of the vessel and year effects. Empirical results show that the average rate of RO is reduced by about 20 to 30 % due to technical inefficiency. On the other hand, average allocative efficiency is found to be about 78 %. The average overall efficiency is found to be around 60 %.     

Profit efficiency in the European Union banking industry: a directional technology distance function approach

We employ the directional technology distance function approach and present estimates of profit efficiency in the 25 European Union (EU) member states over the period 1998–2008. This method decomposes profit efficiency into its technical and allocative components. We investigate potential efficiency differences across the old EU region and the new EU member states, across countries and across banks of different size. Our results indicate a significant level of profit inefficiency for the EU region, which is predominantly attributed to allocative inefficiency. Our findings also suggest that banks operating in the old EU region are, on average, more profit efficient than credit institutions in the new EU member states. Overall, we observe considerable variation of efficiency scores across countries and different patterns in efficiency change over time, as well as a negative relationship between bank size and efficiency.     

The examination of pseudo-allocative and pseudo-overall efficiencies in DEA using shadow prices

The original Data Envelopment Analysis (DEA) models developed by Charnes et al. (Eur J Oper Res 2:429–444, 1978), Banker et al. (Manag Sci 30:1078–1092, 1984) were both radial models. These models and their varied extensions have remained the most popular DEA models in terms of utilization. The benchmark targets they determined for inefficient units are primarily based on the notion of maintaining the same input and output mixes originally employed by the evaluated unit (i.e. disregarding allocative considerations). This paper presents a methodology to investigate allocative and overall efficiency in the absence of defined input and output prices. The benchmarks determined from models based on this methodology will consider all possible input and/or output mixes. Application of this methodology is illustrated on a model of the financial intermediary function of a bank branch network.     

Relative economic efficiency in Texas nursing facilities: A profit function analysis

The purpose of this analysis is to examine the relative economic efficiency of profit-seeking versus nonprofit nursing facilities. A Cobb-Douglas profit function is used on cross-sectional data to determine the technical efficiency, price efficiency, and overall economic efficiency of Texas nursing facilities. Ownership form influences the profit level of the firm. Profit-seeking firms are more economically efficient than their nonprofit counterparts. Inclusion of price efficiency analyses reinforces profit-seeking firms’ superior technical efficiency. Additionally, nursing facilities are price takers in the output market as well as in resource markets, indicating effective industry regulation.     

The directional profit efficiency measure: on why profit inefficiency is either technical or allocative

The directional distance function encompasses Shephard’s input and output distance functions and also allows nonradial projections of the assessed firm onto the frontier of the technology in a preassigned direction. However, the criteria underlying the choice of its associated directional vector are numerous. When market prices are observed and firms have a profit maximizing behavior, it seems natural to choose as the directional vector that projecting inefficient firms towards profit maximizing benchmarks. Based on that choice of directional vector, we introduce the directional profit efficiency measure and show that, in this general setting, profit inefficiency can be categorized as either technical, for firms situated within the interior of the technology, or allocative, for firms lying on the frontier. We implement and illustrate the analytical model by way of Data Envelopment Analysis techniques, and introduce the necessary optimization programs for profit inefficiency measurement.     

On the Determinants of School District Efficiency: Competition and Monitoring

A number of researchers have asserted that inefficiency in the U.S. school system arises from a lack of incentives for public schools to behave efficiently. This paper uses a Shephard input distance function to model educational production, and a switching-regressions estimation to explore the relationship between school district efficiency and two existing incentive mechanisms—competition and voter monitoring. We find evidence that ease of monitoring enhances both technical and allocative efficiency of urban school districts, and that increased competition reduces allocative inefficiency in communities above a competitive threshold. We find no evidence that competition is related to technical inefficiency.     

Impact of conversion to Critical Access Hospital status on hospital efficiency

This study examines the impact of Critical Access Hospital (CAH) Program on hospital efficiency using a two-stage approach, where data envelopment analysis is used in the first stage to estimate cost, technical, and allocative efficiency scores of a sample of rural hospitals. Densities of efficiency scores of CAHs and prospectively paid rural hospitals are estimated and compared using a nonparametric kernel density estimator and a bootstrap-based test. In the second stage, efficiency scores are regressed on environmental variables using bootstrapped truncated regressions. Density analysis and results from bootstrapped truncated regressions show that CAHs are less cost and allocatively efficient compared to prospectively paid rural hospitals, without being less technically efficient. Relative to their pre-conversion selves, CAHs appear to be slightly less allocatively efficient, while they are slightly more technically efficient and no less cost efficient. Overall, our results suggest that the CAH Program may have decreased the allocative and cost efficiencies of those rural hospitals that converted to CAH status relative to prospectively paid rural hospitals, without significantly increasing their technical efficiency.     

A Note on the Technical and Allocative Efficiency of Local Exchange Carriers in the United States

The issue explored is whether incentive regulation of local exchange carriers in the United States has resulted in an increase in efficiency. After providing an overview of the nature of incentive regulation, the methodology for measuring the effects of incentive regulation on efficiency is reviewed. This methodology is data envelopment analysis and allows for the measurement of both technical efficiency and allocative efficiency of individual local exchange carriers. The results of empirically implementing the DEA approach indicate that there is little change in technical efficiency. In fact average technical efficiency in 1988 was the same as in 2001. Next, while outputs continued to grow at about their historical rate across LECs, the sizeable increase in the two types of capital increased inputs well above their historical average rates for some LECs leading to short run allocative inefficiency. On average, however, allocative efficiency shows no identifiable trend between 1988 and 2001. Finally, in the aggregate, total economic efficiency does not demonstrate any trend between 1988 and 2001.     

Multiple Comparisons with the Best: Bayesian Precision Measures of Efficiency Rankings

A large literature measures the allocative and technical efficiency of a set of firms using econometric techniques to estimate stochastic production frontiers or distance functions. Typically, researchers compute only the precision of individual efficiency rankings. Recently, Horrace and Schmidt (Journal of Applied Economics 15, 1–26, 2000) have applied sampling theoretic statistical techniques known as multiple comparisons with a control (MCC) and multiple comparisons with the best (MCB) to make statistical comparisons of efficiency rankings. As an alternative, this paper offers a Bayesian multiple comparison procedure that we argue is simpler to implement, gives the researcher increased flexibility over the type of comparison, and provides greater, and more intuitive, information content. For these methods and a parametric bootstrap technique, we carry out multiple comparisons of technical efficiency rankings for a set of U.S. electric generating firms, estimated using a distance function framework. We find that the Bayesian method provides substantially more precise inferences than obtained using the MCB and MCC methods.Jel Classification: C11, C32, D24.     

How to optimize the allocation of research resources? An empirical study based on output and substitution elasticities of universities in Chinese provincial level

Although significant progress has been made in China's basic research in recent years, there remains a wide gap between research in China and that from developed countries. How to optimize the allocative efficiency of research resources is of great importance for increasing research output. In this paper, using the fixed effect stochastic frontier model based on the translog production function, we estimate output and substitution elasticities of research and development (R&D) inputs at universities in China's provincial level during 2009–2016. We find that the R&D technical efficiency of China's universities, after a rapid growth, has tended to become relatively stable. Improvements of internationalization degree and exogenous R&D capabilities are conducive to promoting R&D technical efficiency, whereas expenditures from government grants inhibit the promotion of R&D technical efficiency; the effects of R&D capital deepening and internet penetration are not evident. The output elasticity of R&D capital is much higher than that of R&D personnel, suggesting that R&D capital is the main driving force of research output. The substitution elasticity between R&D capital and personnel has experienced a change from substitution to complementary since 2014. To realize sustained growth of research output, we should increase R&D input with positive output elasticity or reduce R&D input with negative output elasticity, making the necessary trade-offs according to the substitution relationship between the two R&D inputs.     

Efficiency measurement with multiple outputs and multiple inputs

This paper discusses modeling technical and allocative inefficiencies in both cost minimizing and profit maximizing frameworks with special emphasis on multiple inputs and multiple outputs. Both primal and dual models are considered for this purpose. In the primal approach we use a separable output and input function (the constant elasticity of transformation output function and Cobb-Douglas input function). The dual models assume translog cost or profit functions. Technical inefficiency is assumed to be random in the cross-sectional models, and fixed firm-specific parameter in the panel data models. Allocative inefficiencies are always treated as input-specific parameters. We derive exact relations linking technical inefficiency and allocative inefficiencies to cost and profit when the underlying technology is represented by a flexible functional form such as the translog. It is shown that appending a one-sided homoscedastic error term to model technical inefficiency, or neglecting technical inefficiency altogether in a translog profit tunciton results in model misspecification and inconsistent parameter estimates.     

Economic Efficiency and Productivity Growth in the Post-Privatization Chilean Hydroelectric Industry

Chiles hydroelectric industry was privatized in 1985, but required to operate within a regulatory framework designed to achieve a competitive outcome. A centralized dispatch center was established to ensure production at minimum cost, subject to constraints on minimum release and minimum reservoir stock. A reluctance to rapidly reduce the industry work force may also have existed. We develop a constrained cost-minimization model for thermal and hydro generation to obtain the shadow price of water and to determine the qualitative effect of these constraints on allocative efficiency. Using panel data from 1986–1997, we assess the economic efficiency of the hydro industry by estimating a stochastic distance frontier and price equations from the dual cost-minimization problem. We find dramatic increases in technical change and productivity change, with positive efficiency change for all years but the last. We also observe a dramatic decline in allocative inefficiencies over our sample period. The share of hydro generation from run-of-river and thermal plants relative to reservoir plants has increased, presumably in reaction to the water release and reservoir stock constraints, reducing the relative over-utilization of capital to water from the pre-1985 regime. Further, the over-utilization of labor to capital and water has fallen over time. However, considerable allocative inefficiencies remain, consistent with our finding of industry-wide scale economies. Substantial cost savings would result if technical and allocative efficiency were eliminated.JEL Classification: L94, D24     

A cross-national comparison and taxonomy of DEA-based hospital efficiency studies

This paper provides the first taxonomy of hospital efficiency studies that uses data envelopment analysis (DEA) and related techniques. We provide a systematic review of 79 such studies published from 1984–2004 that represent 12 countries. Only studies written in English are considered. A cross-national comparison reveals significant differences with respect to important study characteristics such as type of DEA model selected and choice of input and output categories. Compared with US studies, European efforts are more likely to measure allocative rather than technical efficiency, use longitudinal data, and use fewer observations. We take a longitudinal perspective that illustrates the life cycle of this research, as well as its diffusion across disciplines. Our taxonomy can be used by policy makers and researchers to review past, and assemble new, DEA models.