Dual Approach

The dual approach is based on dynamic, intertemporal duality theory, established by McLaren and Cooper (1980), formalized by Epstein (1981), and further explored by Epstein and Denny (1983). The method has been extensively applied to the dynamic investment and capital accumulation problems in the late 1980s and early 1990s. In a typical dual approach, a risk neutral entrepreneur is assumed to minimize an expected value of a discounted sequence of production costs conditional on exogenous input prices and output levels. A flexible functional form is then chosen for the optimal value function, and the optimal decision mies are derived by applying the envelope theorem directly to the value function.1

The dual approach is the most flexible in tenns of the generality of the underlying production technology. In particular, a major advantage of the dual approach is that the adjustment matrix and the parameters in the optimal value function are related to each other in a simple fashion and, therefore, it is straightforward to solve the model for closed form decision mies in terms of the underlying structural form parameters (Epstein and Denny 1983). Restrictions on the production technology, such as independent adjustment rates, are not required.

Independence of adjustment rates can be tested rather than assumed. The model can be easily extended to cover an arbitrary number of variable inputs, capital goods, and outputs.

Although the dual approach is flexible with respect to the underlying production technology, it has limitations in identifying price and output expectations separately from the technology parameters (Taylor 1984). With endogenous output and an estimated supply equation, the model has enough overidentification restrictions for identifying and testing simple expectations structures that follow a first order differential equation system (Epstein and Denny 1983). But if an exogenous output is assumed and no supply equation is included in the model, there are no overidentification restrictions for testing how expectations are formed. Therefore, it has been a standard procedure that the technology parameters have been pinned down by imposing a specific expectations structure on the model.

These kinds of applications are, for example, Taylor and Monson (1985), Shapiro (1986), Vasavada and Chambers (1986), Stefanou (1987, 1989), Howard and Shumway (1988, 1989), Weersink and Tauer 1989, Weersink 1990, Fernandez-Comejo et al.

(1992).

Typically, studies have imposed static price and output expectations such that the current prices and outputs have been expected to prevail forever (e.g. Fernandez-Cornejo et al. 1992, Vasavada and Chambers 1986). Some authors, e.g. Vasavada and Chambers (1986), claim that static expectations are realistic in relatively small agricultural firms where goods can be stored easily and frequent acquisition of market information is costly. In European agriculture, static expectations may be realistic because the marketing institutions and agricultural legislation have stabilized and protected farm outputs and farm gate prices (e.g. Thjissen 1994). Thjissen (1996) even concluded that the model with static expectations fits the Dutch dairy farm data well, but the rational expectations model is inconsistent with the theory.

In many duality applications, however, the seemingly restrictive assumptions on the expectations structure can be made without much loss of generality. The expectations assumption does not necessarily alter the most important behavioral economic results. The statistical inference concerning adjustment rates, as well as price and output elasticities, may be independent of the expectation structure as long as they meet the Markov property, in the sense that the probability distribution and expected value of the next period price (or output) is a function of current prices and output. The reason is that, under Markovian expectations, the behavioral equations are functions of current prices and outputs, which include ali relevant information about their future values. Therefore, correctly specified behavioral equations will depict the aggregate effects of expectations and technology even though their separate effects cannot be identified and distinguished.2

Most dynamic duality models of investment and capital accumulation use aggregate data maintaining the assumption of interior solutions with positive gross investment. The assumption can be made without loss of generality only in aggregate data. Studies using firm level data, on the other hand, have typically used the primal approach based on the argument that the dual approach is not applicable, because firm level investment can be zero, or even negative (e.g. Thjissen 1994). But there are no theoretical reasons that prevent relaxing the assumption of positive investments, and applying the dual approach to firm level data. Epstein (1981) concludes that the regularity conditions of the optimal value function with positive gross investment are readily extended to account for negative gross investment too. Nevertheless, it is difficult to construct an optimal value function such that the regularity conditions hold for ali individual firms, because some of the regularity conditions differ qualitatively between the investing and disinvesting firms. For example, consistent shadow prices for installed capital, which are partial derivatives of the optimal value function, must alternate their signs depending whether the firm is investing or disinvesting (Epstein and Denny 1980).

Another alternative is to endogenously stratify the data into positive, zero, and negative investment regimes. One can then specify a consistent value function for each of these regimes, and the regularity conditions for a consistent value function can be tested within each regime. With this approach, the firm level data provide a good opportunity to account for the asymmetries and discrete characteristics in the optimal

2 The claim holds in our model, and this will become clear in the next chapter, where the economic model is derived.

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investment rules. The decision mies can be estimated, for example, by the maximum likelihood technique. Chang and Stefanou (1988) and Oude Lansink (1996) have exploited this idea by modeling positive and negative net investments using an endogenous dummy model, and the two-stage self-selection model of Hecicman (1976). However, the dimensionality problem becomes severe when modeling both discrete and continuous characteristics of the optimal decision mies with a large number of capital goods.

3.3 Preferred Approach

Among the reviewed approaches, there are trade-offs in the flexibility of the underlying technology, identification of expectations, statistical efficiency, computational intensity, and the form of the results. The traditional primal methods with closed form decision mies would produce consistent results in a preferred form (e.g. elasticities), but these methods have problems. The derivation of the optimal decision mies is complex unless the production technology is restricted. If the number of capital goods exceeds two, the technology has to be restricted so that the adjustment matrix is diagonal. In addition, sufficient overidentification restrictions for testing how expectations are formed requires profit maximization conditions so that ali the input demands, the output supplies, and equations on price processes can be used for identification (Epstein and Yatchew 1985). With exogenously determined output, the structure of expectations has to be imposed on the model without testing its validity.

If the Euler equations, on the other hand, are used for estimating the parameters we can allow for more flexible technology, but the closed form decision mies and elasticities can no longer be computed. Further, expectations could not be tested with this approach either, because the profit maximization conditions are not met in our application. The reason is the lack of overidentification restrictions, as explained above. Therefore, by using this method nothing would be gained compared to the dual approach, which allows fiexible technology but has no overidentification restrictions for testing the expectations.

In this study, we use the dual approach because we wish to model more than just one quasi-fixed input and to test for independent adjustment rather than assume it. We are also interested in price, output, and scale elasticities, which are straightforward to estimate when the dual approach is used.

Our dual model is constructed so that the instantaneous production technology is augmented by internal adjustment costs, which has been the standard in the adjustment cost literature (e.g. Fernandez-Cornejo et al). But, in addition, we generalize the model to allow for uncertainty, irreversibility, and more general adjustment costs, like fixed adjustment costs. Therefore, the model allows not only for

investment smoothing but also for an optimal choice of inaction, i.e. an optimal choice of zero investment3.

The adjustment cost literature has focused only on the regime of positive gross investment, with few exceptions. Chang and Stefanou (1988) and Oude Lansink (1996), for example, stratified data into two regimes, contracting and expanding firms, while modeling adjustment costs asymmetric in net investment. But we follow Abel and Eberly (1994), stressing irreversibility of agricultural investments, and incorporate the mixture of discrete and continuous characteristics of investments by identifying and partitioning the following two investment regimes:

a regime of zero gross investment and a regime of positive gross investment.

The regime of negative gross investment was excluded in the analysis because the number of negative gross investments in the data was too small for a complete analYsis. Changes in the labor services, on the other hand, did not include zeros and only the regimes of negative and positive changes in labor services were modeled. We now move on to a detailed derivation of the economic model.

3 An adjustment cost rationalization ofJohnson's (1956) fixed asset theory, as in Hsu and Chang (1990), results in a similar identification of frictions and classification of observed investment demands.

Chapter 4