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.     

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.     

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.     

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.     

A Consistent Approach to Cost Efficiency Measurement*

Consistent specifications of the allocative inefficiency function in ‘cost plus input share equations’ systems may be difficult, if not impossible, to find because most plausible ones violate certain reasonable a priori conditions. Moreover, the models to which they lead give rise to highly non‐linear likelihood functions that are very hard to estimate. In an effort to confront these difficulties, this paper adapts an idea first suggested by Greene (1993) that allocative inefficiency ought to be related to input prices and allocative distortions in the input share equations. The system of ‘cost plus input demand equations’ that emerges is estimated by standard seemingly unrelated regression (SUR) techniques using data from private and state firms that operated in Greek manufacturing during the 1979–88 period. Among other findings, the estimates show that overall inefficiency for private and state firms was 63.5% and 102.2%, respectively in comparison with the least inefficient firms in their class. In relative terms these figures imply that state firms were almost 61% less efficient than private firms were. Technical and allocative reasons amounting to 64% and 36%, respectively, accounted for this excess inefficiency of state firms, in addition to differences in the utilization of labour, capital and debt. Lastly, it is found that the magnitudes of technical and allocative inefficiencies depend critically upon a self‐consistent specification of the allocative inefficiency function.     

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.     

Pitfalls in the estimation of a cost function that ignores allocative inefficiency: A Monte Carlo analysis

In the stochastic frontier literature, it is a widely held view that allocative inefficiency can be lumped together with technical inefficiency in the estimation of cost frontiers. Therefore, a one-sided error term in the cost function is believed to capture the cost of overall (technical plus allocative) inefficiency. In this paper we challenge that view through a detailed Monte Carlo investigation. The results show that failure to include the cost of allocative inefficiency explicitly in the cost function biases the estimates of: (i) the cost function parameters, (ii) returns to scale, (iii) input price elasticities, and (iv) cost-inefficiency.     

Estimating DEA technical and allocative inefficiency using aggregate cost or revenue data

In this paper, we address the question of Data Envelopment Analysis (DEA) evaluation of efficiency when aggregate cost or revenue data must be used. We show that the DEA technical inefficiency measure using total revenues as the single output variable or total costs as the single input variable equals the aggregate technical and allocative inefficiency. We employ this result to estimate allocative inefficiency and construct statistical tests of the null hypothesis of no allocative inefficiency analogous to those of the null hypothesis of no scale inefficiency. We illustrate our method using revenue and personnel data for the top U.S. public accounting firms over 1995–1998. Our empirical results indicate the existence of statistically significant allocative inefficiency in the public accounting industry.

Estimating technical and allocative inefficiency relative to stochastic production and cost frontiers

Earlier papers by Aiger, Lovell and Schmidt and by Meeusen and van den Broeck have considered stochastic frontier production functions. this paper extends that work by considering the duality between stochastic frontier production and cost funstions, under the assumtions of exact cost minimization (tecchnical inefficiency only) and of inexact cost minimization (technical and allocative inefficiency). We show how to measure both types of inefficiency, and the associated cost of inefficiency. The techniques are illustrated using data on steam-electric generating plants.     

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 %.     

A test for the existence of allocative inefficiency in firms

This paper is concerned with the problem of testing for the presence of allocative inefficiency in firms for models in which the cost function is stochastic, the possibility of technical inefficiency is allowed, and the efficient input cost share equations are stochastic. The test exploits the relationships among the disturbances in the cost and share equations to derive implications of the existence of allocative inefficiency. This leads to the use of a Wald-type statistic to take account of the one-sided nature of the alternative. Some empirical experience is reported.     

Measuring technical and allocative inefficiency in the translog cost system: a Bayesian approach

In this paper, we propose simulation-based Bayesian inference procedures in a cost system that includes the cost function and the cost share equations augmented to accommodate technical and allocative inefficiency. Markov chain Monte Carlo techniques are proposed and implemented for Bayesian inferences on costs of technical and allocative inefficiency, input price distortions and over- (under-) use of inputs. We show how to estimate a well-specified translog system (in which the error terms in the cost and cost share equations are internally consistent) in a random effects framework. The new methods are illustrated using panel data on U.S. commercial banks.     

Feasible estimation of firm‐specific allocative inefficiency through Bayesian numerical methods

Both the theoretical and empirical literature on the estimation of allocative and technical inefficiency has grown enormously. To minimize aggregation bias, ideally one should estimate firm and input‐specific parameters describing allocative inefficiency. However, identifying these parameters has often proven difficult. For a panel of Chilean hydroelectric power plants, we obtain a full set of such parameters using Gibbs sampling, which draws sequentially from conditional generalized method of moments (GMM) estimates obtained via instrumental variables estimation. We find an economically significant range of firm‐specific efficiency estimates with differing degrees of precision. The standard GMM approach estimates virtually no allocative inefficiency for industry‐wide parameters. Copyright © 2009 John Wiley & Sons, Ltd.     

A Bayesian approach to imposing curvature on distance functions

The estimated parameters of output distance functions frequently violate the monotonicity, quasi-convexity and convexity constraints implied by economic theory, leading to estimated elasticities and shadow prices that are incorrectly signed, and ultimately to perverse conclusions concerning the effects of input and output changes on productivity growth and relative efficiency levels. We show how a Bayesian approach can be used to impose these constraints on the parameters of a translog output distance function. Implementing the approach involves the use of a Gibbs sampler with data augmentation. A Metropolis–Hastings algorithm is also used within the Gibbs to simulate observations from truncated pdfs. Our methods are developed for the case where panel data is available and technical inefficiency effects are assumed to be time-invariant. Two models—a fixed effects model and a random effects model—are developed and applied to panel data on 17 European railways. We observe significant changes in estimated elasticities and shadow price ratios when regularity restrictions are imposed.     

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.     

Homotheticity and Non-Radial Changes

Färe and Mitchell(1992) have shown that cost functions for a multi-output firmobey a particular output-scaling law if and only if the underlyingproduction technology is ray-homothetic. Multi-output firms,however, frequently change their output mix in addition to theirscale. Therefore, it is important to identify technologies thatpossess relatively tractable analytical characteristics whensubjected to non-radial changes in the output vector. This paperconsiders additive and other, more general, changes in the outputvector. One result shows that the cost function obeys an ``output-translationlaw' if and only if the input correspondence is input homothetic,thus suggesting that ``input homotheticity' is more than justan ``input-scaling' property.     

Duality and continuity

This paper fills the one remaining lacuna in (multiple-output) duality theory by provingjoint continuity (in input and output vectors) of cost, benefit, and (input and output) distance functions. Continuity is an important property where measurement error exists, for it provides assurance that small errors of measurement (of quantities or prices) result only in small errors in concepts like minimal cost. We consider continuity not only in prices and quantities, but also in technologies. Continuity in technologies might be more important than continjity in prices or quantities, because production technologies are almost certainly measured (or estimated) with error.     

Measuring technical and allocative efficiencies for banks in the transition countries using the Fourier flexible cost function

The transition economies are known to have quite different market structures from the market economies. State-owned banks accounts for a major part of the financial sector in East European countries before the transition period. Since the input prices of the sector are frequently under the control of those governments, the misallocated resources may incur the loss of economic efficiency. This paper attempts to gauge the technical and allocative efficiency using unbalanced panel data of 340 banks from 14 transition countries under the framework of the Fourier flexible shadow cost function. Accommodating technical and allocative efficiencies simultaneously, as suggested by Atkinson and Cornwell (Int Econ Rev 35:231–243, 1994a) and Kumbhakar and Wang (J Econom 134:317–340, 2006a), avoids potential specification errors and leads to consistent parameter estimates. The average total cost savings resulting from greater technical and allocative efficiency are around 28.31 and 7.13%, respectively. Foreign-owned banks are found to be the most economically efficient. The enforcement of financial liberalization does gradually improve upon the sample banks’ technical efficiency. The allocative inefficiency arises from over capitalization and excess funds. Scale diseconomies appear to prevail in the sample states with a few exceptions.     

Econometric estimation of distance functions and associated measures of productivity and efficiency change

Multi-input multi-output production technologies can be represented using distance functions. Econometric estimation of these functions typically involves factoring out one of the outputs or inputs and estimating the resulting equation using maximum likelihood methods. A problem with this approach is that the outputs or inputs that are not factored out may be correlated with the composite error term. Fernandez et al. (J Econ 98:47–79, 2000) show how to solve this so-called ‘endogeneity problem’ using Bayesian methods. In this paper I use the approach to estimate an output distance function and an associated index of total factor productivity (TFP) change. The TFP index is a new index that satisfies most, if not all, economically-relevant axioms from index number theory. It can also be exhaustively decomposed into a measure of technical change and various measures of efficiency change. I illustrate the methodology using state-level data on U.S. agricultural input and output quantities (no prices are needed). Results are summarized in terms of the characteristics (e.g., means) of estimated probability density functions for measures of TFP change, technical change and efficiency change.