Dynamic efficiency measurement: theory and application.

The foundation of the revealed preference approach to production analysis (Farrell 1957; Afriat 1972; Hanoch and Rothschild 1972) and the refinement of this approach to production modelling in the context of static decision making (Diewert and Parkan 1983; Varian 1984; Banker and Maindiratta 1988) provide a consistent static theoretical framework to measure and evaluate production efficiency in a nonparametric fashion. Nonparametric programming methods have been used increasingly in the evaluation of production efficiency (e.g., Fare and Lovell 1978; Banker, Charnes, and Cooper 1984; Fare, Grosskopf, and Lovell 1983; Chavas and Aliber 1993; Caputo and Lynch 1993; Fare, Grosskopf, and Lovell 1983) as well as productivity growth (e.g., Chavas and Cox 1990; Fare et al. 1994; Tauer 1998; Ray 2004).

Theoretical and empirical studies focusing on production efficiency have typically ignored the time interdependence of production decisions and the firm adjustment paths over time. A few studies modeling some dynamic aspects of production in a nonparametric fashion are Sengupta (1995) and Nemoto and Goto (1999, 2003). Sengupta (1995) introduces the first-order conditions of dynamic optimization into data envelopment analysis (DEA) models, while Nemoto and Goto (1999, 2003) treat the stock of capital at the end of the period as an output and incorporate it within the conventional DEA framework.

Silva and Stefanou (2003) develop a nonparametric revealed preference approach to the dynamic theory of production in the context of an adjustment-cost technology and intertemporal cost minimization. Capital as a quasi-fixed factor is managed as an asset where rapid expansion or contraction of the stock of capital is accompanied by adjustment costs. The dynamics are explicitly addressed in the production technology specification as an adjustment cost in the form of the properties of the family of input requirement sets (or the production possibilities set) with respect to the change in quasi-fixed factors (or dynamic factors).

This article develops nonparametric dynamic measures of technical, allocative and economic efficiency in the short- and long-run using the theoretical framework developed by Silva and Stefanou (2003). Lower and upper bounds on each efficiency measure are proposed for each production unit at each point of time. The efficiency measures proposed in this article are temporal in nature by describing the degree of efficiency of the firm at a particular point along its adjustment path. The empirical implementation of these measures is illustrated for a balanced panel data set of Pennsylvania dairy operators during the time period 1986-92.

Technological Information and Dynamic Cost Minimization

The adjustment-cost model of the firm is a dynamic approach to the theory of the firm in which the source of the intertemporal link of production decisions is the adjustment costs associated with changes in the level of quasi-fixed factors. Adjustment costs are usually characterized as either internal or external. Internal adjustment costs may be conceived as output-reducing costs the firm bears by diverting resources from production to investment support activities (e.g., installing the new capital goods, training personnel), implying a trade-off between current production and current growth and future production (e.g., Lucas 1967a; Treadway 1969, 1970). External adjustment costs arise from market forces, such as monopsony in the market for investment goods (e.g., Eisner and Strotz 1963; Lucas 1967b; Gould 1968). External adjustment costs are typically added to the other costs of the firm while internal costs are incorporated in the production technology specification. In this article, capital quasi-fixity is treated as arising from internal adjustment costs.

Let [y.sub.t] denote the maximum output level a firm can produce at time t, given the m-dimensional vector of variable inputs [x.sub.t], the o-vector of gross investment [I.sub.t], and the o-vector of initial capital stocks [k.sub.t] at time t. Let V([y.sub.t] : [k.sub.t]) represent the input requirement set for [y.sub.t] given the initial capital stock vector [k.sub.t]. Internal adjustment costs are incorporated in V(y.sub.t] : [k.sub.t]) and are discussed below.

At any point of time t, the firm is presumed to minimize the discounted flow of costs over time as follows:

(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where w is the vector of current variable input prices, c is the current rental price vector of quasi-fixed factors, y is the current production target, r is the constant discount rate, (1) [??] = dK/dt is the vector of net investment and [delta] is a diagonal (oxo) matrix of the depreciation rates [[delta].sub.h], h = 1, ..., o. The value function W(*) represents the long-run cost function starting at time t.

The optimal current value function W(w, c, y, [k.sub.t]) associated with problem in equation (1) obeys the dynamic programming equation or Hamilton-Jacobi-Bellman equation

(2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where rW(..) is a flow version of the intertemporal cost and [W.sub.k] = [W.sub.k](w, c, y, [k.sub.t]) is the vector of the shadow value of capital. By definition, the shadow value of the quasi-fixed factor h, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], measures the impact on the value function due to a small change in the initial capital stock [k.sub.h]. Consequently, the shadow value of capital is an endogenous price and influenced by input prices (w, c), the production target and initial capital stocks.

Equation (2) states the value function is defined as the discounted present value of the current total cost plus the marginal value of the optimal change in net investment. A solution to the minimization problem in equation (1) at each time period can be obtained by solving equation (2). The optimal policy functions [x.sup.*.sub.t] = [x.sup.*] (w, c, y, [k.sub.t]) and [I.sup.*.sub.t] = [I.sup.*] (w, c, y, [k.sub.t]) provide the levels of the variable inputs and gross investment in quasi-fixed factors at each time t for a dynamic cost efficient firm. The value function resulting from the minimization problem in equation (2) is denoted hereafter as the actual dynamic cost function. Actual dynamic cost functions refer to the perfect minimization of cost (i.e., exactly meeting the optimization conditions). Also, [W.sub.k] = [W.sub.k] (w, c, y, [k.sub.t]) is denoted as the actual shadow value of capital.

Silva and Stefanou (2003) show that a well-behaved technology can be represented by a family of input requirement sets satisfying some regularity conditions where the dynamics are explicitly addressed in the production specification as an adjustment cost. Consider a data series [S.sup.c] = {([y.sup.i.sub.t], [x.sup.i.sub.t], [I.sup.i.sub.t], [k.sup.i.sub.t], [w.sup.i.sub.t], [c.sup.i.sub.t]); i = 1, ..., n; t = 1, ..., T} representing the observed behavior of each production unit i at each time t and including information on w and c for each observation i at each time t. (2) Theorems 2 and 3 in Silva and Stefanou (2003) establish the existence of two families of input requirements sets. By theorem 2, {[V.sub.I]([y.sub.t] : [k.sub.t])} is the tightest inner bound on {V([y.sub.t] : [k.sub.t])} and theorem 3 establishes {[V.sub.o]([y.sub.t] : [k.sub.t])} as the outer bound.

The tightest inner bound on V([y.sub.t] : [k.sub.t]) is the convex monotonic hull of ([x.sup.i.sub.t], [I.sup.i.sub.t]) and can be constructed as (3)

(3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [[lambda].sub.t] is the intensity vector at time t. The inclusion of the gross investment constraint in the nonparametric construction of [V.sub.I]([y.sub.t] : [k.sub.t]) means the maximum output level depends not only on variable and quasi-fixed factors but also on the magnitude of the dynamic factors. This notion of quasi-fixity of capital follows from the assumption that the firm incurs internal adjustment costs whenever there is a change in the quasi-fixed factors. The input requirement set in equation (3) is negative monotonic in [I.sub.t] and reverse nested in [k.sub.t], implying current additions to the capital stock are output decreasing in the current period but increase output in the future by increasing the future stock of capital. Negative monotonicity of [V.sub.I]([y.sub.t] : [k.sub.t]) in [I.sub.t] reflects the adjustment costs associated with gross investment. The set of constraints defining [V.sub.I]([y.sub.t] : [k.sub.t]) assures the underlying production function is concave in ([x.sub.t], [I.sub.t]) given [k.sub.t]. (4) In this case, the more rapidly the quasi-fixed factors are adjusted the greater the cost, leading to sluggish adjustment in the quasi-fixed factors.

The tightest outer bound is constructed using the intertemporal dual relation between the actual dynamic cost function and the underlying adjustment-cost technology. The outer bound is defined as

(4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]