Stochastic estimation of firm inefficiency using distance functions.

• Author: Atkinson, Scott E.

1. Introduction

   Both the cost function and the distance function are valid representations of multioutput technologies. The arguments of the cost function are output quantities and input prices, while the arguments of the distance function are input and output quantities. The input distance function measures the extent to which the firm is input efficient in producing a given set of outputs, while the cost function assumes cost minimization by a firm which is input efficient. The shadow cost function is a generalization of the cost function and depends on output quantities and shadow (internal to the firm) input prices rather than actual (market) input prices. Thus, the shadow cost function assumes shadow cost minimization but not actual cost minimization. This function has frequently been employed in fixed-effects estimation of allocative efficiency (AE) and technical efficiency (TE). An observed input vector is technically efficient if it is on the isoquant of the observed output vector. An input vector is allocatively eff icient if the radial contraction of the input vector to the isoquant is cost minimizing.

   Identification of parameters that measure AE can be achieved by the estimation of a shadow cost system comprising the actual cost equation and the share or input demand equations, which are expressed in terms of shadow prices and output quantities. Shadow cost systems have been estimated by Atkinson and Halvorsen (1984), Sickles, Good, and Johnson (1986), Eakin and Kniesner (1988), Kumbhakar (1992), and Atkinson and Cornwell (1994), among others. Given panel data, the deviation of shadow from actual input prices can be measured by price-specific inefficiency parameters that may vary across firms and over time. However, the estimation of the effect of allocative inefficiency on input usage is indirect. One must compare the fitted input demands obtained from the estimated share or input demand equations with and without AE imposed.

   The distance function has served mainly as a theoretical device. An important exception has occurred with efficiency measurement, for which the reciprocal of the input distance function, termed the Farrell input measure of TE, has been widely computed nonstochastically using the linear programming techniques discussed in Diewert and Parkan (1983) and Fare, Grosskopf, and Lovell (1985). Stochastic estimation of distance functions has been carried out by Grosskopf, Hayes, and Hirschberg (1995), Coelli and Perelman (1996), Grosskopf et al. (1997), and Atkinson, Cornwell, and Honerkamp (2002). However, these studies estimate a distance equation without the price equations necessary to identify AE, thereby failing to provide a parametric method for the direct estimation of AE.

   As an alternative to both the standard distance function approach and the shadow cost system approach, in this paper we derive and estimate a shadow input distance system using the generalized method of moments (GMM) procedure. This system is a shadow input distance function, which is a function of input shadow quantities and output quantities, and the first-order conditions from the dual shadow cost minimization problem. Since we estimate these equations jointly, we impose the assumption of shadow cost minimization on our estimation problem. Unlike the shadow cost system, however, the shadow input distance system allows one to directly estimate the effects of allocative inefficiency, since the shadow input distance function is defined in terms of output quantities and shadow input quantities. The latter quantities indicate the cost-minimizing ratios of inputs the firm wishes to utilize but cannot because of some constraint. After estimating the shadow distance function, for which shadow input quantities are specified as parametric functions of actual input quantities, we decompose TE from noise. Finally, we compute returns to scale and the cost savings obtained by attaining AE and TE. We illustrate these procedures using panel data on U.S. railroads, an approach that allows us to identify input-specific, firm-specific, and time-varying AE parameters. (1)

   The remainder of this paper is organized as follows. In section 2, we develop the cost-distance function duality assuming AE. In section 3, we add the parametric specification of AE to this dual relationship. In section 4, we discuss issues regarding stochastic estimation of the input distance function, AE, and TE. An application to U.S. railroads is presented in section 5, and conclusions follow in section 6.

2. The Cost-Distance Duality: Assuming Allocative Efficiency

   In this section we assume AE, which implies that shadow prices equal actual prices and that shadow quantities equal actual quantities. Let x = ([x.sub.1], ..., [x.sub.N])' [member of] [R.sup.N.sub.+] denote a vector of N nonnegative inputs and let y = ([y.sub.1], ..., [y.sub.M])' [member of] [R.sup.M.sub.+] denote a vector of M nonnegative outputs. We define the technology set, [L.sup.t], as the set of all feasible input-output vectors in period t, i.e.,

   [L.sup.t] = {(x, y) [member of] [R.sup.N.sub.+] X [R.sup.M.sub.+] : x can produce y in period t} (2.1)

   where t = 1, ..., T. Following Shephard (1970), we denote input requirements sets by

   L[(y).sup.t] = {x [member of] [R.sup.N.sub.+] : (x, y) [member of] [L.sup.t]}, y [member of] [R.sup.M.sub.+]. (2.2)

   We assume that inputs are freely disposable, i.e., if x [member of] L[(y).sup.t]. We also assume that the input requirement set, L[(y).sup.t], is a convex set for all y [member of] [R.sup.M.sub.+].

   The input distance function is defined as

   [D.sub.i](y,x,t) = [[lambda].sup.*] = max{[lambda]: (x/[lambda]) [member of] L[(y).sup.t]} (2.3)

   where [lambda] is a scalar such that 1 [less than or equal to] [lambda]. The input distance function indicates the radial contraction of inputs required to move the input vector x to the isoquant for y. See Figure 1.

   Free disposability of inputs guarantees that x [member of] L[(y).sup.t] if and only if [D.sub.i](y,x) [greater than or equal to] 1. Also, given our assumption of free disposability and convexity, the input distance function is homogeneous of degree one, nondecreasing, and concave in x (see Shephard 1970). The cost function corresponding to the input distance function is defined by

   C(y,p,t) = [min.sub.x] {px : x [member of] L[(y).sup.t]}, (2.4)

   where p ([p.sub.1],..., [p.sub.N]) [member of] [R.sup.N.sub.+] denotes the vector of N nonnegative input prices. An equivalent definition of the cost function is

   C(y,p,t) = [min.sub.x] {px : [D.sub.i](y,x,t,) [greater than or equal to]1}, (2.5)

   since x [member of] L[(y).sup.t] if and only if [D.sub.i](y,x,t) [greater than or equal to] 1. Thus, the input distance function indicates the radial reduction in inputs required to move the firm to the frontier but not necessarily to the cost-minimizing position.

   Assuming that observed cost equals minimum cost and letting [[nabla].sub.y][D.sub.i](y,x) = [[partial][D.sub.i](y,x)/[partial][y.sub.1],...,[partial][D.sub.i](y, x)/[partial][y.sub.M] and [[nabla].sub.y]C(y,p) = [[partial]C(y,p)/[partial][y.sub.1],...,[partial]C(y,p)/[partial][y.s ub.M], Fare and Primont (1995) show that [[eta].sup.C][y,p) = [[eta].sup.[D.sub.i](y,x) wherer

   [[eta].sup.C](y,p,) = C(y,p,t)/[[nabla].sub.y]C(y,p,t)y

   and, in terms of the distance function,

   [[eta].sup.[D.sub.i]](y, x) = -[D.sub.i](y,x)/[[nabla].sub.y][D.sub.i](y,x)y (2.6)

   are measures of returns to scale using the cost and distance functions, respectively. We report scale elasticities for our data set in section 5.

3. The Cost-Distance Duality: Allowing Allocative Inefficiency

   We now wish to introduce parameters to explicitly measure allocative inefficiency. Assuming shadow cost minimization, Atkinson and Cornwell (1994) employ a shadow cost function corresponding to the actual cost function in Equation 2.3,

   C(y,[p.sup.*],t) = [min.sub.x]{[p.sup.*]x : x [member of] [L(y).sup.t]}, (3.1)

   where [p.sup.*] =[[p.sup.*.sub.1],...,[p.sup.*.sub.N]] =[[k.sub.1][p.sub.1],...,[k.sub.N][p.sub.N]] is a (1 X N) vector of shadow prices. That is, [p.sup.*] is the price that makes the optimal input vector, h(y, [p.sup.*], t), equal to the actual input vector, x. The [k.sub.n] parameters, n = 1,..., N, measure the divergence of actual prices from shadow prices for the firm. Atkinson and Comwell (1994) employ panel data to measure both AE and TE assuming fixed effects. Using Equation 3.1, these authors derive and estimate a shadow cost system, consisting of actual costs and N - 1 actual shares, expressed in terms of shadow input prices and output quantities. With panel data, the [k.sub.n] parameters can theoretically be made time- and firm-specific. The extent of parameterization depends on the independent variation in the data.

   Estimation of the shadow cost system yields direct measurement of the divergence between shadow prices and actual prices. However, often the researcher is more concerned with the effect of allocative inefficiency on input utilization. That is, one typically wishes to know the degree to which one input is over- or underutilized relative to another input rather than to know the ratio of their relative shadow prices. Using the shadow cost system, this calculation is indirect. One must first compute the fitted input quantities from the input share or demand equations assuming AE. These estimates are then compared with the fitted input quantities obtained after allowing for allocative inefficiency by estimating the AE parameters. However, by computing the shadow distance system, one directly obtains estimates of the effect of failing to attain AE on input utilization from the estimated AE parameters.

   To derive the shadow input distance function in terms of shadow input quantities, we reverse the roles of shadow input prices and input quantities in the shadow cost minimization problem. Now one assumes shadow cost minimization in terms of shadow input quantities and actual...