Measuring excess capital capacity in agricultural production.

Stochastic production frontier models have been used extensively to analyze technical efficiency of firms operating in agriculture and other industries (Kumbhakar and Lovell 2000). It is generally found that many firms fail to operate with full technical efficiency. One measure of the extent of an individual firm's technical inefficiency is "excess capacity," defined as the difference between the frontier output for the firm's input vector and actual output of the firm (Fare, Grosskopf, and Kokkelenberg 1989; Morrison Paul 1999). Excess capacity is an output-oriented measure that measures technical inefficiency in terms of all inputs jointly. However, in most production studies "capacity" refers to capital goods (e.g., plant and equipment), not to variable inputs (e.g., materials) that are consumed in one production cycle.

In this article, we focus on capital directly and measure the extent to which firms have "excess capital capacity." In particular, we examine the extent to which technically inefficient firms can produce the same level of output using less capital while holding other inputs constant. This "excess capital capacity" measure of technical inefficiency is clearly input oriented, but we single out capital specifically and measure its overuse.

Measurement of excess capital capacity is useful because it provides different information about the nature of technical inefficiency than other measures. Capital is a major factor of production, and overinvestment in capital may increase costs and decrease competitiveness. Investigating excess capital capacity is particularly relevant for agricultural policy in developed countries where production is often heavily subsidized, and farms have been found to be overinvesting in capital (see, e.g., Guan et al. 2005; Guan and Oude Lansink 2006). Furthermore, better insights into the factors that explain excess capital capacity may suggest ways to improve capital use in agriculture. The importance of capital in agriculture is also highlighted by the vast literature on agricultural investment decisions (e.g., Taylor and Monson 1985; Vasavada and Chambers 1986; Oude Lansink and Stefanou 1997; Pietola and Myers 2000). Our approach is complementary to this literature because we analyze the extent to which capital allocation can be improved. In our empirical application to Dutch cash crop farms, we find evidence of excess capital capacity and analyze the individual farm characteristics contributing to it.

We base our analysis of excess capital capacity on the concept of a stochastic input requirement frontier that gives the minimum amount of an input required to produce a specified level of output, given a fixed level of other inputs used in the production process (Diewert 1974). Clearly, an input requirement frontier is just an alternative representation of the production frontier but is useful for directly measuring the extent to which a particular input of interest can be reduced in a technically inefficient firm, without changing other input levels, to move the firm to its production frontier. To date, stochastic input requirement frontiers have been used primarily to investigate excess labor capacity (Kumbhakar and Hjalmarsson 1995, 1998; Battese, Heshmati, and Hjalmarsson 2000; Heshmati 2001; Kumbhakar, Heshmati, and Hjalmarsson 2002; El-Gamal and Inanoglu 2005). Here, however, we focus on an input requirement frontier for capital.

In specifying the input requirement function, the input of interest is expressed as a function of all other inputs and outputs. This might create an endogeneity problem, especially when the input of interest is jointly determined with output. The presence of endogenous regressors causes a parameter inconsistency problem not only for estimation of the input requirement frontier, but also for maximum likelihood (ML) estimation of stochastic frontier (SF) functions in general. The usual instrumental variable (IV) method used in least distance estimators is not directly applicable in the ML framework which is essential for efficiency estimation in SF models.

In this study, we solve the problem using a two-step method. First, we solve the parameter inconsistency problem caused by the presence of endogenous variables (viz., output in the capital requirement function) by using the generalized method of moments (GMM) approach. Second, we use the residuals from the GMM regression and utilize the ML method to estimate excess capital capacity. The two-step method constitutes a methodological contribution. Although we apply it to a stochastic capital requirement function, the method is general and can be used in other applications where ML estimation is necessary while endogeneity problems may exist. These applications might include a stochastic production function (in which some inputs might be endogenous), a cost function (in which output[s] might be endogenous), or an input (output) distance function (in which input [output] ratios appearing as regressors might be endogenous).

In the next section of this article, we outline the stochastic input requirement frontier model and explain how it can be applied to estimate excess capital capacity. We also discuss how the definition of excess capital capacity relates to earlier literature on excess capacity, asset fixity, and overinvestment in agriculture (Hsu and Chang 1990; Oude Lansink and Stefanou 1997; Pietola and Myers 2000). Following this, we detail our two-step estimation approach and explain how GMM overcomes the endogenous regressor problem. This is followed by an application to Dutch cash crop farms. The final section makes some concluding remarks.

The Stochastic Input Requirement Frontier Model

An input requirement frontier is derived by solving (implicitly) the frontier production function for the input of interest to get:

(1) K = F[(Y, X)e.sup.v]

where K is the input of interest (in our case the capital stock) and F(*) is the minimum amount of capital required to produce the output vector Y given other inputs X. Here, F(*) can also be viewed as the deterministic capital requirement frontier. The error term v makes the frontier stochastic and represents random factors that are not in the control of any firm but influence capital requirements, even for those operating on the frontier. In applying this model to individual firm data, it is possible that some firms use more capital than the minimum required to produce a given level of output, given other input quantities. This is allowed by adding an additional one-sided error u [greater than or equal to] 0. The full model can then be represented as

(2) K = F[(Y, X)e.sup.v+u].

If u = 0, the firm is operating on the frontier, while if u > 0, the firm is inefficient in capital use because it is using more than the minimum amount of capital to produce the same level of output using the same level of other inputs. The model is a purely technical relationship and does not impose any behavioral assumptions on firm managers.

An empirical measure of the extent of inefficiency in capital use in firm i can be derived by noting that the minimum capital capacity required to produce output level [Y.sub.i] given inputs [X.sub.i] and random error [v.sub.i] is

(3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Thus, a measure of "excess capital capacity" for firm i is the difference between the actual capital stock and the minimum capital capacity required to produce the given output level:

(4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

It is also possible to specify an auxiliary model that explains the degree of excess capital capacity in terms of firm and environmental characteristics, [u.sub.i] = g([Z.sub.i]), where [Z.sub.i] is a vector of firm and environmental characteristics that explain excess capital capacity. Estimating the auxiliary model along with the frontier can help identify the causes of excess capital capacity.

Interpreting Excess Capital Capacity

The definition of "excess capital capacity" derived from a capital requirement frontier is closely related to the notion of "excess capacity" in frontier production models (Klein 1960; Fare, Grosskopf, and Kokkelenberg 1989; Morrison Paul 1999; Dupont et al. 2002; Kirkley, Morrison Paul, and Squires 2002, 2004; Felthoven and Morrison Paul 2004). In frontier production models, excess capacity of firm i is defined as the difference between a firm's frontier output (the output they could produce if they kept the same level of inputs but eliminated inefficiency and moved to the frontier) and their actual output (Fare, Grosskopf, and Kokkelenberg 1989; Morrison Paul 1999; Dupont et al. 2002; Kirkley, Morrison Paul, and Squires 2002, 2004; Felthoven and Morrison Paul 2004). "Excess capacity" is therefore an output-based measure of the extent to which firms are operating inside their production frontier. In a capital requirement frontier model "excess capital capacity" [K.sup.e.sub.i] is also a measure of the extent to which a firm is operating inside its production frontier, but the measure is in capital space rather than output space. That is, the distance to the production frontier is measured by the amount by which capital can be reduced without changing output (and other inputs). Hence, although closely related, excess capacity is not the same as excess capital.

The notion of excess capital capacity is also related to another common input-oriented measure of technical inefficiency in terms of all inputs, which can be estimated using either a stochastic frontier model (Kumbhakar and Tsionas 2006) or stochastic input distance function (Coelli et al. 2005). In both of these approaches, however, technical inefficiency is a radial measure in that all inputs (rather than the single capital input) are reduced by the same proportion. This does not allow inference on excess capital capacity because it cannot be separated from overuse of other inputs.

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