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Article Excerpt
1. INTRODUCTION
Mainstream neoclassical economic theory assumes that all firms/producers in an economy always operate efficiently. This is rarely the case in practice, however. A firm, using a vector of inputs, is technically inefficient if it fails to produce the maximum possible output, given the technology. This is called output-oriented (OO) technical inefficiency. Alternatively, a technically inefficient firm uses more inputs than the minimum amounts required to produce a given level of output. This is known as input-oriented (IO) technical inefficiency. Inefficiency may also arise from a firm's inability to choose the least-cost input mix to produce a given level of output (allocative/price inefficiency). Farrell (1957), in a seminal article, first introduced these two concepts of inefficiency (IO technical inefficiency and allocative/price inefficiency). If a firm is inefficient technically and/or allocatively, then its cost will be greater than that of an efficient firm producing the same level of output and purchasing the inputs at the same prices. Thus one can define overall cost inefficiency as the percentage increase in cost due to both technical and allocative inefficiencies.
Although several methods are available to measure inefficiency, in this article we concentrate on the stochastic frontier (SF) methodology introduced by Aigner, Lovell, and Schmidt (1977) and Meeusen and van den Broeck (1977). The SF models involve estimation of a parametric production/cost function with a composed error term consisting of a two-sided disturbance term that reflects the usual noise component (exogenous shocks) and a one-sided term that captures technical inefficiency (Bauer 1990; Greene 1993, 2001; Kumbhakar and Lovell 2000; Koop and Steel 2001). Distributional assumptions are made on these error components, and the maximum likelihood (ML) method is usually used to estimate the parameters of the production/cost function. Technical inefficiency is then estimated from the conditional distribution of the one-sided inefficiency term given the estimates of the composed error term. Usually the mean (or mode) of the conditional distribution is used to obtain an observation-specific measure of technical inefficiency (Jondrow, Lovell, Materov, and Schmidt 1982).
But estimating the production function alone, one can estimate only OO technical inefficiency, which measures the potential output loss for the inefficient producers. Because a firm can be inefficient in allocating inputs as well, it is natural to estimate both technical and allocative inefficiency components, as well as increases in cost due to these inefficiencies. Extraction of this information from the data requires a system approach that takes into account input allocation decisions explicitly into the model. The system approach is not widely used in the empirical efficiency literature despite the fact that the theory behind estimating a system of equations is well developed (see, e.g., Schmidt and Lovell 1979; Kumbhakar 1987; Kumbhakar and Lovell 2000, chap. 4), at least for a Cobb-Douglas (CD) production function with no errors in the input allocation decision equations [i.e., first-order conditions of cost minimization (profit maximization)] other than allocative inefficiency. If such errors are introduced, then using the standard sampling theory approach (i.e., the ML method) is almost impossible even for the CD system. The reason is that the error structure comprising noises and technical and allocative inefficiencies becomes so complicated that the likelihood function cannot be derived in closed form.
There are two issues to be addressed when one goes beyond the CD production function and wishes to use a flexible functional form. The first issue is modeling, that is, how to specify, for example, a translog cost function with both technical and allocative inefficiencies. The specification proposed by Kumbhakar (1997) is theoretically consistent and solves the modeling problem. The second issue, estimation, is what we focus on in this article. Joint estimation of technical and allocative inefficiencies in a translog cost system poses difficulties (Greene 1980; Bauer 1990), especially when both technical and allocative inefficiencies are viewed as random variables and the data are cross-sectional. If panel data are available and one is willing to assume that both technical and allocative inefficiencies are time-invariant parameters, then the sampling theory approach can be easily used to estimate these inefficiencies (Kumbhakar and Lovell 2000, chap. 4; Maietta 2002). Because panel data are not always available, we need a technique to estimate both technical and allocative inefficiencies from a system in which these inefficiencies are viewed as random/latent variables.
The translog cost model proposed by Kumbhakar (1997) leads to a nonlinear random-effects system model when cross-sectional data are used. We call it a nonlinear random-effects system because of the following features of the model, especially when it is estimated with cross-sectional data. First, firm-specific technical and both firm- and input-specific allocative inefficiencies must be random (hence the term random effects). Second, they must be separated from the random noise terms appearing in each equation of the system. Finally, the inefficiency terms appear in a highly nonlinear fashion, which helps in separating them from random errors. Estimation of such a model using the sampling theory approach has the following two main problems. First, the likelihood function cannot be expressed analytically, making it difficult to obtain ML estimates of the parameters of the cost function. Second, we need to separate both technical and allocative inefficiencies from the random noises affecting the cost function and the cost share equations, but this seems an impossible task in a non-Bayesian framework. In contrast, the Bayesian approach using Markov chain Monte Carlo (MCMC) procedures comes in handy. That is, although the likelihood function does not have an analytical expression, we can draw from the posterior using the MCMC method. Thus one advantage of the Bayesian approach is the variety of functions of interest about which we can perform inferences. These functions include technical and allocative inefficiencies, as well as the increase in cost resulting from these inefficiencies, among others. Although the model can be estimated using a classical approach organized around simulated maximum likelihood, the computationally efficient schemes to implement the approach are not entirely clear. Moreover, the estimation of moments for the functions of interest would necessitate integrations over a high-dimensional parameter space. Thus for the classical approach, these estimations are difficult, if not impossible, to perform.
Consequently, our objective in this article is to show how Bayesian techniques can be used to draw inferences in a nonlinear random-effects system model with cross-sectional data where firm-specific technical inefficiency and both firm- and input-specific allocative inefficiencies are separated from random noise terms (which were not considered by either Kumbhakar 1997 or Schmidt and Lovell 1979). We also obtain observation-specific estimates of the costs for the technical and allocative inefficiency components, as well as the impact of allocative inefficiencies on input overuse (underuse). Finally, we derive estimate of scale economies for each firm in our sample.
Bayesian analysis of the SF function was first proposed by van den Broeck, Koop, Osiewalski, and Steel (1994) for the single-equation frontier models. Koop, Steel, and Osiewalski (1995) proposed the Gibbs sampler as an effective numerical technique. Systems of equations involving a multiple-output production function and output shares were proposed and estimated by Fernandez, Koop, and Steel (2000). Here we assume cost-minimizing behavior on behalf of the producers, which leads to a nonlinear random-effects system due to the presence of technical and allocative inefficiencies. The translog cost system with both technical and allocative inefficiencies is a nontrivial extension of the system that involves either technical (allocative) inefficiency alone or no inefficiency. Specialized numerical methods are used to provide parameter inferences and measures of technical and allocative inefficiencies. These techniques, which involve the use of MCMC procedures, are applied to a sample of U.S. commercial banks. Given that the banking system plays a unique role in an economy, analyzing the efficiency of the banking system is a worthwhile exercise. Efficiency of U.S. commercial banks has been extensively investigated (Berger and Humphrey 1997), but none of the previous studies has examined both technical and allocative inefficiencies in a theoretically consistent fashion using a system approach.
The rest of the article is organized as follows. Section 2 develops the theoretical and econometric model. Section 3 presents data and discusses results. Section 4 concludes.
2. THE ECONOMETRIC MODEL
2.1 Economic Foundations
To formalize the concepts introduced in the preceding section, let the production technology be specified as [q.sub.i] = f([x.sub.i][e.sup.-[u.sub.i]]), where [q.sub.i] is output and [x.sub.i] is a vector of m inputs for firm i (i = 1,..., n), f(*) is the production function, and [u.sub.i] [greater than or equal to] measures IO technical inefficiency (Farrell 1957). This formulation implies that a technically inefficient producer overuses all of the inputs by u * 100% compared with an efficient producer producing the same output. Consequently, the IO measure of technical inefficiency is useful when addressing the input allocation problem. Note that this formulation is different from the OO technical inefficiency [unless f(*) is linear homogeneous, which is not assumed here] introduced by Aigner et al. (1977) and widely used in the efficiency literature. To address the input allocation problem, we assume that the objective of the producers is to minimize cost for an exogenously given level of output (q). In pursuing this objective, producers may face external constraints that result in misallocation of inputs. Such misallocations are often labeled as allocative distortions that can arise from optimization errors, noncompetitive input markets, government regulations, and so on. The presence of such distortions means that the usual first-order conditions of cost minimization subject to [q.sub.i] = f([x.sub.i][e.sup.-[u.sub.i]]) will not be exactly satisfied. Here we follow Schmidt and Lovell (1979) and Kumbhakar (1997) in modeling allocative distortions by specifying the first-order conditions of cost minimization as [f.sub.j]([x.sub.i][e.sup.-[u.sub.i]])/[f.sub.1]([x.sub.i][e.sup.-[u.sub.i]]) = [w.sub.j,i][e.sup.[[xi].sub.j,i]]/[w.sub.1,i], j = 2,..., m. Two things about these first-order conditions are worth noting. First, unless the production function is homogeneous, the IO technical inefficiency term (u) will not drop out from the foregoing first-order conditions. Second, to allow input misallocations, an extra term (exp([[xi].sub.j,i])) is (multiplicatively) appended to each of the first-order conditions. Thus in this formulation a...
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