Estimating demand with distance functions: Parameterization in the primal and dual

Our purpose is to investigate the ability of different parametric forms to ‘correctly’ estimate consumer demands based on distance functions using Monte Carlo methods. Our approach combines economic theory, econometrics and quadratic approximation. We begin by deriving parameterizations for transformed quadratic functions which are linear in parameters and characterized by either homogeneity or which satisfy the translation property. Homogeneity is typical of Shephard distance functions and expenditure functions, whereas translation is characteristic of benefit/shortage or directional distance functions. The functional forms which satisfy these conditions and include both first- and second-order terms are the translog and quadratic forms, respectively. We then derive a primal characterization which is homogeneous and parameterized as translog and a dual model which satisfies the translation property and is specified as quadratic. We assess functional form performance by focusing on empirical violations of the regularity conditions. Our analysis corroborates results from earlier Monte Carlo studies on the production side suggesting that the quadratic form more closely approximates the ‘true’ technology or in our context consumer preferences than the translog.     

Hyperbolic Efficiency and Parametric Distance Functions: With Application to Spanish Savings Banks

Distance functions are gaining relevance as alternative representations of production technologies, with growing numbers of empirical applications being made in the productivity and efficiency field. Distance functions were initially defined on the input or output production possibility sets by Shephard (1953, 1970) and extended to a graph representation of the technology by Färe, Grosskopf and Lovell (1985) through their graph hyperbolic distance function. Since then, different techniques such as non parametric-DEA and parametric-SFA have been used to calculate these distance functions. However, in the latter case we know of no study in which the restriction to input or output orientation has been relaxed. What we propose is to overcome such restrictiveness on dimensionality by defining and estimating a parametric hyperbolic distance function which simultaneously allows for the maximum equiproportionate expansion of outputs and reduction of inputs. In particular, we introduce a translog hyperbolic specification that complies with the conventional properties that the hyperbolic distance function satisfies. Finally, to illustrate its applicability in efficiency analysis we implement it using a data set of Spanish savings banks.     

On functional form representation of multi-output production technologies

The introduction of directional distance functions has given researchers an alternative to Shephard distance functions. In this paper we conduct a Monte Carlo study to investigate the performance of distance functions as an approximation for models of technology. Our results indicate that quadratic representations of technology have better approximation properties than translog parameterizations.     

Productivity trends in U.S. manufacturing: Evidence from the NQ and AIM cost functions

In this paper, we take the econometric approach to productivity measurement in United States manufacturing, using KLEM data over the period from 1953 to 2001. We are also interested in technical change bias, price elasticities, and elasticities of substitution in the U.S. manufacturing industry. We present an empirical comparison and evaluation of the effectiveness of four well-known flexible cost functions—the locally flexible generalized Leontief (see Diewert [1971. An application of the Shephard duality theorem: a generalized Leontief production function. Journal of Political Economy 79, 481–507]), translog (see Christensen et al. [1975. Transendendal logarithmic utility functions. American Economic Review 65, 367–364]), and normalized quadratic (see Diewert and Wales [1987. Flexible functional forms and global curvature conditions. Econometrica 55, 43–68])—and the globally flexible asymptotically ideal model (see Barnett et al. [1991. Semi-nonparametric Bayesian estimation of the asymptotically ideal production model. Journal of Econometrics 49, 5–50]), the latter modified to introduce technical change by means of Thomsen's [2000. Short cuts to dynamic factor demand modelling. Journal of Econometrics 97, 1–23] factor-augmenting efficiency index approach.     

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.     

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.     

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.     

Credit risk pricing and the rationality of lending decision-making within dual banking systems: A parametric approach

The reduction of non-performing loans, and making correct provisions for them, plays a primary role in the management and minimization of banking credit risk. However, these actions depend primarily upon the cost at which banks may dispose of these bad loans. Hence, this study aims to perceive the price of banks’ credit risk via estimating the shadow price of non-performing loans. We assess and compare the perceived price of the credit risk of Islamic and conventional banks operating in 9 countries from the Middle East and Asia, using a quadratic directional distance function. Following this, we evaluate the impact of different settings of directional vectors on shadow prices by conducting a risk-sensitivity analysis. Applying bootstrap regression, the factors affecting NPLs’ prices are further investigated. The paper concludes that the estimation of the shadow prices of bad loans can provide important elements in favor of credit risk management and, therefore, credit risk mitigation.     

Undesirable outputs and a primal Divisia productivity index based on the directional output distance function

Despite their great popularity, all the conventional Divisia productivity indexes ignore undesirable outputs. The purpose of this study is to fill in this gap by proposing a primal Divisia-type productivity index that is valid in the presence of undesirable outputs. The new productivity index is derived by total differentiation of the directional output distance function with respect to a time trend and referred to as the Divisia–Luenberger productivity index. We also empirically compare the Divisia–Luenberger productivity index and a representative of the conventional Divisia productivity indexes–the radial-output-distance-function-based Feng and Serletis (2010) productivity index–using aggregate data on 15 OECD countries over the period 1981–2000. Our empirical results show that failure to take into account undesirable outputs not only leads to misleading rankings of countries both in terms of productivity growth and in terms of technological change, but also results in wrong conclusions concerning efficiency change.     

Two Perspectives on DEA: Unveiling the Link between CCR and Shephard

In this paper we contrast the DEA and activity analysis approaches by Charnes, Cooper and Rhodes (1978) and Shephard (1970), respectively. We show that by appropriately normalizing Shephard's output price model the two approaches coincide.     

Estimating farmers’ productive and marketing inefficiency: an application to vegetable producers in Benin

This paper estimates the technical and marketing inefficiency of a sample of urban vegetable producers in Benin. Marketing inefficiency is defined as the failure of farmers to achieve better marketing output and is reflected in lower output price indices. The study proposes a Russell-type measure of inefficiency using a directional distance function that accounts simultaneously for the expansion of outputs and price indices and the contraction of variable inputs. A truncated bootstrap regression is used in the second stage to consistently analyze factors that underlie differences in inefficiency. The first-stage results suggest that vegetable producers are more inefficient with respect to marketing than production. The second-stage results indicate that technical inefficiency is affected by the production environment and private extension services. Marketing inefficiency is affected by the type of marketing arrangements.     

An iterative approach to the estimation of the abatement costs of harmful emissions

The ability to quantify tradeoffs involved in the process of reducing harmful emissions is essential to successful policy-making in the environmental planning area. The approach by Färe et al. (J Econom 126: 469–492, 2005) to computing point estimates of the marginal abatement costs (MACs) of reducing pollution by estimating the directional output distance function has been gaining popularity in recent years. The contribution of this study is to compute MACs as slopes of the iterated parametric production possibilities frontier (PPF) estimated on the basis of the set of efficient projections of observable output combinations obtained from the parameters of directional output distance function. Policy makers are thus provided with the general shape of the production possibilities set for a polluting technology rather than with a set of point estimates of the MACs. We apply our methodology to a balanced panel of seven Korean manufacturing sectors spanning the period between 1999 and 2009, obtaining theoretically consistent concave PPFs based on a large set of directional output distance vectors. Finally, we estimate the parameters of a directional output distance function corresponding to the iterated PPF.     

An interregional policy model for energy-economic-environmental interactions

This article deals with conflicts between energy, pollution and economic issues in an interregional framework. On the basis of an interregional input output model extended with pollution and energy sectors a policy analysis aiming at finding compromise solutions forconflicting objectives is carried out. The rigid input output structure is adjusted by incorporating price effects of inputs via demand relationships based on a translog price possibility frontier. The analysis is illustrated by means of some empirical applications to the Dutch economy.     

Estimation of technical and allocative inefficiency: A primal system approach

The estimation of technical and allocative inefficiencies using a flexible (translog) cost system is found to be quite difficult, especially when both the inefficiencies are random. In this paper we use the alternative primal system consisting of the production function (translog) and the first-order conditions of cost minimization. The estimation of the primal system is more straightforward and it enables us to estimate observation-specific technical and allocative inefficiencies, and their impact on input demand and cost. We use data on steam-electric generating plants from the U.S. to estimate the model using both Cobb–Douglas and translog production functions.     

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.     

Probabilistic characterization of directional distances and their robust versions

In this work, we analyze the performance of production units using the directional distance function which allows to measure the distance to the frontier of the production set along any direction in the inputs/outputs space. We show that this distance can be expressed as a simple transformation of radial or hyperbolic distance. This formulation allows to define robust directional distances in the lines of α-quantile or order-m partial frontiers and also conditional directional distance functions, conditional to environmental factors. We propose simple methods of estimation and derive the asymptotic properties of our estimators.     

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.     

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 directional slacks-based measure of technical inefficiency

Radial measures of efficiency estimated using linear programming (LP) methods can be biased since slack in the constraints defining the technology suggests that at least one input can be reduced, or one output can be expanded, even though a firm is deemed to be “technically efficient.” In this paper, we propose a directional slacks-based measure of technical inefficiency to account for the potential of slack in technological constraints. When no such slacks exist, directional slacks-based inefficiency collapses to the directional technology distance function. Our proposed measure helps to generalize some of the existing slacks-based measures of inefficiency. We examine the financial services provided by Japanese cooperative Shinkin banks, and estimate their inefficiency during the period 2002–2005. This inefficiency declined slightly during the period. We thus propose that slack is an important source of inefficiency which is often not captured by the directional technology distance function.