Estimating variable returns to scale production frontiers with alternative stochastic assumptions

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Abstract

Two stochastic production frontier models are formulated within the generalized production function framework popularized by Zellner and Revankar (Rev. Econ. Stud. 36 (1969) 241) and Zellner and Ryu (J. Appl. Econometrics 13 (1998) 101). This framework is convenient for parsimonious modeling of a production function with returns to scale specified as a function of output. Two alternatives for introducing the stochastic inefficiency term and the stochastic error are considered. In the first the errors are added to an equation of the form h(logy,θ)=logf(x,β) where y denotes output, x is a vector of inputs and (θ,β) are parameters. In the second the equation h(logy,θ)=logf(x,β) is solved for logy to yield a solution of the form logy=g[θ,logf(x,β)] and the errors are added to this equation. The latter alternative is novel, but it is needed to preserve the usual definition of firm efficiency. The two alternative stochastic assumptions are considered in conjunction with two returns to scale functions, making a total of four models that are considered. A Bayesian framework for estimating all four models is described. The techniques are applied to USDA state-level data on agricultural output and four inputs. Posterior distributions for all parameters, for firm efficiencies and for the efficiency rankings of firms are obtained. The sensitivity of the results to the returns to scale specification and to the stochastic specification is examined.

Introduction

The estimation of stochastic production frontier models is a common procedure for assessing the efficiency of firms within an industry. Several versions of stochastic frontier models have been suggested in the literature, designed to accommodate the varying nature of data and specific characteristics of empirical applications. A typical model relevant for panel data involving observations on a number of firms over time can be written aslogyit=logq(xit,β)-zi+uit.In this equation yit denotes output for the ith firm in the tth time period, xit is a corresponding vector of inputs, β represents a vector of unknown parameters, and q(xit,β) is the deterministic part of the production frontier. It is assumed that the random errors uit capture measurement and/or specification error and that they are independent normal random variables with mean zero and variance ω-1. Each zi is assumed to be a nonnegative random variable that describes the inefficiency of the ith firm in terms of the distance of logyit from the stochastic frontier logq(xit,β)+uit. Alternative distributions that have been suggested in the literature for the zi include the exponential, gamma, truncated normal and half-normal distributions. The inefficiency term is assumed to be constant over time, although this assumption can be relaxed. To measure efficiency (rather than inefficiency), and to make measurement of efficiency across firms comparable, it is conventional to use τi=exp(-zi) to denote the efficiency of the ith firm. Since 0τi1, this measure allows us to say that the ith firm is 100τi% efficient. Also, it has a natural interpretation as the ratio of mean output conditional on the inefficiency of the ith firm (zi) to mean output on the frontier (conditional on zi=0). This interpretation is expressed algebraically asE(yit|zi)E(yit|zi=0)=q(xit,β)exp(-zi)E[exp(uit)]q(xit,β)E[exp(uit)]=exp(-zi)=τi.Reviews by Greene (1997) and Koop and Steel (2001) provide a convenient access to the extensive literature on stochastic production frontiers and its historical development. Greene emphasizes the sampling theory approach while Koop and Steel focus on Bayesian inference.

In this paper we are concerned with Bayesian estimation of stochastic frontier models that exhibit variable returns to scale. One way to specify a model with variable returns to scale is to choose an appropriate form for the function logq(xit,β). For example, a Cobb–Douglas specification is not satisfactory because it exhibits a constant degree of returns to scale. On the other hand, a translog specification for logq(xit,β) yields a returns to scale function that is a linear function of the logs of the inputs; it can exhibit regions of increasing, constant and decreasing returns to scale. The translog model has some disadvantages, however—it does not automatically satisfy the regularity conditions of concavity and monotonicity; functions with several inputs require estimation of a large number of parameters; and the relationship between the substitutability of the inputs and the returns to scale may be unnecessarily complicated. An alternative approach without these problems, and the approach adopted in this paper, is the generalized production function specification pursued by Zellner (1971), Revankar (1971), Zellner (1971, p.176), and Zellner and Revankar, 1969, Zellner and Ryu, 1998. In this model the production function is assumed to be homothetic, implying it can be written asy=g*[f(x,β)]where g* is a monotonic transformation and f(x,β) is a homogeneous function of degree μ. For the moment, we omit the i and t subscripts and the stochastic inefficiency and error terms. Also, we have switched notation from q(x,β) to f(x,β) to avoid potential ambiguities that could arise when the error terms are reintroduced.

In the context of Eq. (3), Zellner (1971) show that the returns to scale (RTS), defined as the proportional change in output resulting from a 1% change in all inputs, can be written as the functionRTS(y)=μdydffy.A ‘generalized production function’ is obtained by specifying functions for RTS(y) and f(x,β), and solving the differential equation in (4) to yield an explicit representation of the production relationship in Eq. (3). Conditions that the production function must satisfy to obtain such a representation are given in Zellner and Revankar (1969). One example, suggested by Zellner (1971), is the RTS functionRTS(y)=μ1+θy.that leads to the production relationshiplogy+θy=logf(x,β).If y is scaled such that y1, and the inequality restrictions θ>0 and μ>1+θ hold, the RTS is greater than one for low outputs and decreases monotonically as output increases, leading to a U-shaped average cost curve that has a minimum when RTS(y)=1. The RTS function in (5) achieves this desirable property with the introduction of only one additional parameter, θ. Also, the production relationship satisfies concavity and monotonicity regularity conditions as long as f(x,β) is chosen to have these properties. Substitutability of the inputs is governed by the function f(x,β).

If functions like (6) are considered desirable for modeling stochastic production frontiers with variable returns to scale, the next question that must be addressed is the way in which stochastic inefficiency and error terms are introduced. One possibility is that adopted by Kumbhakar (1988) and Kumbhakar et al. (1991)—they add the stochastic terms to Eq. (6). This strategy is a natural extension of the assumption employed in Zellner (1971), Van den Broeck et al. (1994) and Zellner and Revankar (1969) where a single normally distributed error is added to Eq. (6). In the panel data context described earlier, Kumbhakar's strategy leads to the modellogyit+θyit=logf(xit,β)-zi+uit.We will refer to models like this one, where the left side of the equation is a parametric function of yit and the error terms are added to logf(xit,β), as ‘f-additive’. In such models the efficiency measure introduced in Eq. (2) can no longer be written as a simple function of zi, but will depend on the input levels xit. Specifically, if uitN(0,ω-1), then the density function for output conditional on the inefficiency term is given byp(yit|zi)=ω2π1/21+θyityitexp-ω2(logyit+θyit-logf(xit,β)+zi-uit)2and, using the definition of efficiency introduced in Eq. (2), the efficiency of the ith firm can be written asE(yit|zi)E(yit|zi=0)=(1+θyit)exp-ω2(logyit+θyit-logf(xit,β)+zi-uit)2dyit(1+θyit)exp-ω2(logyit+θyit-logf(xit,β)-uit)2dyit.Thus, the stochastic assumptions in (7) lead to a complex expression for efficiency involving integrals for which no closed-form solution is apparent. However, despite the complexity of (9), it can be evaluated numerically. Moreover, it is possible to use the cost function to find an alternative definition of inefficiency that is relatively simple. This interpretation was noted by Kumbhakar (1988); we consider it explicitly later in the paper.

A second way to introduce the stochastic inefficiency and error terms is to view them as being added to the solution for logy from Eq. (6). If the solution for logy is given by logy=g[θ,logf(x,β)], then this model can be written aslogyit=g[θ,logf(xit,β)]-zi+uit.We will refer to models like this one, where the error terms are added to g[θ,logf(xit,β)], as ‘g-additive’. Including the stochastic terms in this way preserves τi=exp(-zi) as a measure of the efficiency of the ith firm. In line with Eq. (2) we can writeE(yit|zi)E(yit|zi=0)=g[θ,logf(xit,β)]exp(-zi)E[exp(uit)]g[θ,logf(xit,β)]E[exp(uit)]=exp(-zi)=τi.Specifications like (10) have been overlooked in the literature, probably because of the need to obtain the solution logy=g[θ,logf(x,β)]. However, as we will see, estimation is still possible within the framework of Bayesian inference.

The objectives of this paper are to describe and illustrate Bayesian methodology for estimating stochastic frontiers like those described in Eqs. (7) and (10). Two RTS functions are considered, that in Eq. (5) and one other. These two are a subset of five functions considered by Zellner and Revankar (1969) in their application to the transport equipment industry. We compare estimates of the parameters and the firm efficiencies under the two different RTS specifications and the g- and f-additive stochastic specifications. As far as we are aware, Bayesian estimation of functions like (7) has not appeared in earlier literature, and estimation of functions like (10) has not been attempted using sampling theory or Bayesian inference. Our strategy for estimation, and the prior assumptions that we adopt, are modifications of those described in Koop and Steel (2001), adapted to accommodate the introduction of the RTS function and the different stochastic assumptions. One particular novelty in our work is the ability to estimate the parameters β and θ in Eq. (10), even although no analytical expression for g is available. This problem is overcome by solving for g numerically for every candidate value of the parameters generated within a Markov chain Monte Carlo algorithm. To ensure similar prior information is used for both functions we place a prior density function on the RTS when output is unity, and on the level of output for which average cost is a minimum (RTS=1).

One of the advantages of the Bayesian approach (also noted by Koop and Steel) is the ease with which useful inferences can be made about quantities that are intractable functions from a sampling theory point of view. For example, we can provide posterior density functions for making finite-sample inferences about firm efficiencies, the efficiency ranking of each firm, the output at which average cost is a minimum and the probability that one firm is more efficient than another. We provide examples such as these from our application, thus illustrating the flexibility of the Bayesian approach.

The plan of the paper is as follows. The model and stochastic assumptions are presented in Section 2. Section 3 contains descriptions of the prior density functions. The conditional posterior density functions and the Markov chain Monte Carlo (MCMC) algorithms are described in Section 4. The empirical application and results are presented in Section 5, with concluding remarks being made in Section 6.

Section snippets

Models and assumptions

The two models that we consider can be written asZR:logyit+θyit=logf(xit,β)NR:logyit+γ(logyit)2=logf(xit,β).The function ZR is attributable to Zellner (1971); it was discussed in the Introduction. The function NR is attributable to Nerlove (1963) and Ringstad (1967). It is derived from the RTS functionRTS(y)=μ1+2γlogy.For this function to decrease monotonically from a point above one to a point below one, producing a “U-shaped” average cost curve with a minimum at RTS=1, we require the units of

Prior distributions

The parameters requiring prior distributions are β0,β1,β2,β3,μ,β5,λ and ω for all functions, θ for the ZR models, and γ for the NR models. Caution needs to be exercised if improper priors are used for some of these parameters. For the case where E[logyit] is a linear function of β, Fernández et al. (1997) show that, in the absence of panel data, proper priors on both λ and ω are required to obtain proper posterior densities. The improper prior p(ω)ω-1 can be used when panel data are available,

Estimation

The joint posterior density for the parameters in the stochastic frontier models considered here (and in other stochastic frontier models considered elsewhere in the literature) is intractable in the sense that marginal posterior densities for individual parameters and efficiencies cannot be derived from it analytically. As an alternative, MCMC techniques can be used to draw observations from the joint posterior density and these observations can be used to estimate marginal posterior densities

The application

The Economic Research Service (ERS) of the USDA compiles annual indexes of output, input use, and total factor productivity for the aggregate farm sector and for the individual states. A discussion of the methods and data used to construct the indexes, and some insights into farm production, can be found in Ball and Nehring (1998) and Ball et al. (1999). For our application we use state-level indexes for total output (y) and for the inputs materials (x1), capital (x2), land (x3) and labor (x4)

Concluding remarks

We have demonstrated how Bayesian estimation can be used to make finite-sample inferences about parameters and firm efficiencies in a stochastic production frontier with a returns-to-scale function that depends on output. The stochastic errors in earlier studies of this kind were introduced in a way that facilitates estimation, but no longer retains the same inefficiency interpretation of the one-sided error. We show how the traditional inefficiency interpretation can be retained by adding

Acknowledgements

This paper has benefited from comments from participants at the 2002 Conference on Current Developments in Productivity and Efficiency Measurement held at the University of Georgia and from comments from three anonymous referees. Griffiths acknowledges support from a University of Melbourne, Faculty of Economics and Commerce Research Grant.

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