Dynamic efficiency measurement: theory and application.

Output is measured by total pounds of milk sold. No information is available on the quantities used for the variable inputs other than hired labor. Miscellaneous variable expenses is available for each farm in each year and incorporates several components such as feed purchased, custom work hired, utilities, gas and oil, fertilizer and lime, veterinary and medicine, machinery repair, and crop and seed supplies. The miscellaneous expenses category is taken as a measure of the farm's variable costs other than hired labor. The implicit farm-specific hourly wage rate is determined as the ratio of annual hired labor expenses to hired labor hours per year.

The total farm assets item involves land, buildings, machinery and equipment, livestock, and cash. Land, buildings, machinery and equipment and livestock are reported as stock variables and the reported values are book values. No market value of the farm's assets is available. These assets can be categorized according to their average useful life. Machinery and equipment can be classified as an intermediate-run asset with an average life ranging between 1 and 10 years, where land and buildings can be considered a long-run asset with an average useful life of more than 10 years.

Several quasi-fixed factors are present in the dairy production. The quasi-fixed factors are land, buildings, machinery and equipment, livestock and family labor. The depreciation rates used for buildings, machinery and equipment and livestock are 3%, 10%, and 20%, respectively.

Total debt consists of debt for farm operation. No information is provided on the allocation of the farm debts among different uses as well as on the possible different rates of interest associated with specific debts. The implicit farm-specific interest rate is determined as the ratio of total interest over total farm debts. The implicit farm-specific rate of interest is used as the rental cost price of capital and assumed to be the same for all quasi-fixed factors except family labor. The farm-specific wage rate is used as the rental cost price of family labor.

Estimation Procedures

A number of implementation issues must be addressed. The data and methodological demands to generate efficiency measures for variable inputs are different from those to generate efficiency measures for all factors of production.

The variable overcost and undercost at each time t, respectively in equations (10) and (13), and the lower and upper bounds on the efficiency measures for variable inputs depend on observed variables ([w.sup.i.sub.t],[y.sup.i.sub.t] [x.sup.i.sub.t],[I.sup.i.sub.t][ks.up.i.sub.t]). Generating lower and upper bounds on short-run efficiency levels involves solving a set of classical linear programming problems.

In contrast, the upper and lower bounds on the actual dynamic cost in equations (5) and (7), respectively depend additionally on the underlying shadow value of capital, which is an endogenously determined variable. Furthermore, the dynamic undercost [equation (7)] and the lower bound on the dynamic technical efficiency measure [equation (18)] depend also on the behavioral shadow value of capital. The estimation procedures involved in the computation of the upper and lower bounds on the dynamic efficiency measures are described separately.

The upper bound on the dynamic technical efficiency measure is generated by solving problem (16) and depends only on observed variables ([y.sup.i.sub.t], [x.sup.i.sub.t], [I.sup.i.sub.t], [k.sup.i.sub.t]). However, the upper bound on the dynamic cost efficiency measure [equation (23)] requires the value of the dynamic overcost function rW(.,[V.sub.I]) at each data point. Consider the dynamic overcost in equation (5). If vector [w.sup.i.sub.kt] were observed, equation (5) is a classic linear programming problem. Given that the actual shadow value of capital measures the impact on the value function due to a small change in the initial capital stock, one must allow for the linkage between the optimal value function, W(*), and the vector [W.sup.i.sub.kt]. In fact, problem in equation (5) indicates that both primal variables ([x.sub.t], [I.sub.t]) and dual variables ([w.sup.i.sub.kt]) are present in the primal form of the minimization problem. If a dual variable is not considered as such, it is difficult (or even impossible) to interpret it in a way consistent with the structure of the primal constraints (Paris 1979a).

The Linear Complementarity Problem (LCP) allows for the presence of both primal and dual variables in the specification of the optimization problem in such a way to guarantee the dual variables are not treated as primal variables in optimization. (11) Consider the Kuhn-Tucker conditions in equation (6) associated with the minimization problem in equation (5) and, in particular, the second condition in equation (6). The variable [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is simply the symmetric of the dual variable [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and the Kuhn-Tucker conditions in equation (6) become

(39) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

l = 1, ..., m;h = 1, ..., o, and j = 1, ..., n.

The Kuhn-Tucker conditions in (39) can be stated as a LCP in the following way:

(40) s = q + Md

d [greater than or equal to] 0

s [greater than or equal to] 0

d's = 0

where M is a square matrix of order (n + 2m + o+2) and q is a (n+2m+o+2)vector. The vectors s and d are the vector of slack variables and the vector of primal and dual variables, respectively. Conditions (40) are the Kuhn-Tucker necessary optimality conditions associated with the dynamic cost minimization problem in equation (5). The LCP consists of finding vectors d and s satisfying equation (40). While there is no objective function to be optimized in an LCE the LCP is equivalent to the problem of finding a stationary point of the following quadratic program

(41) min { Q(d) = q'd + 1/2 d' (M + M')d :

q + M'd [greater than or equal to] O, d [greater than or equal to] 0}.

The function Q(d) is bounded from below on the feasible set F = {d : q + M'd [greater than or equal to]0, d [greater than or equal to] 0}. If F = [empty set], the LCP is not feasible. If F [not equal to] [empty set], two possible cases exist. Either Q([d.sup.*]) = minQ(d) = 0, d* [member of] F, implying d* is the solution of the LCP or minQ(d) > 0 implying the LCP is feasible but has no solution (A1-Khayyal 1987, 1989 Cheng 1984).

The quadratic minimization problem in equation (41) is a convex quadratic program since Q(d) is a convex function suggesting M is a positive semi-definite matrix. A feasible solution d for equation (41) is a Kuhn-Tucker point if d satisfies equation (40). Thus, the LCP in equation (40) is the problem of finding a Kuhn-Tucker point for equation (41). By definition, if d is an optimal solution for equation (41), d must be a Kuhn-Tucker point. Consequently, equation (40) provides the necessary optimality conditions for a feasible solution d of equation (41) to be optimal. Furthermore, since equation (41) is a convex quadratic program, equation (40) provides necessary and sufficient optimality conditions (Al-Khayyal 1987, 1989; Cheng 1984). The solution obtained by solving equation (41) provides the optimal variable input and gross investment vectors solving the minimization problem in equation (5), the value of the dynamic overcost function and the underlying shadow values of the quasi-fixed factors.

Consider now the lower bound on the dynamic technical and cost efficiency measures. The lower bound on the dynamic technical efficiency measure in equation (18) requires information on the behavioral shadow value of the quasi-fixed factors. The behavioral shadow value of the quasi-fixed factors, [W.sup.bi.sub.kt], is estimated using the kernel estimation method and the symmetric of the marginal cost of adjustment evaluated at the observed gross investment vector. Optimal and observed gross investments in quasi-fixed factors differ necessarily for a dynamic cost inefficient firm. The kernel estimation method allows estimation of the marginal cost of adjustment evaluated at the observed levels of gross investment without a parametric specification of the restricted (or variable) cost function.

Let z = ([w.sup.o], I, y, k) and [w.sup.o] = ([w.sub.2]/[w.sub.1], ..., [w.sub.m]/[w.sub.1]). The marginal cost of adjustment of the quasi-fixed factor h, h = 1, ..., o, is estimated as follows: (12)

(42) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [C.sup.i] is the observed variable cost of observation i,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The parameter [d.sub.j] is the estimator of the window-width (or smoothing parameter) for each independent variable in z that minimizes the integrated mean square error (Ullah 1988a, 1988b; Bierens 1987; Pagan and Ullah 1999). A point estimate of the behavioral shadow value of capital for each observation i can be obtained since the kernel estimator of the marginal cost of adjustment for each quasi-fixed factor is a function of z. The marginal cost of adjustment is evaluated at the observed gross investment vector and the optimal and observed gross investment levels differ necessarily for a dynamic cost inefficient firm. However, the kernel estimator of the marginal cost of adjustment in equation (42) assumes a variable cost efficient firm.