ESTIMATION RESULTS
7.5 Shadow Prices for Installed Capital and Labor
Recall the consistency conditions (B.ii) in Chapter 4, which conjectured that the shadow price for installed capital is negative if the firm is investing and positive if the firm is disinvesting. In other words, the optimal value function (the discounted present value of the cost stream) is expected to be decreasing in the capital stock when the firm is investing and increasing in the capital stock when the firm is disinvesting. With zero investment, the shadow price can be either negative or positive, but between the shadow prices of the investing and disinvesting firms. Because the approximation to the optimal value function holds only around the sample averages, we correspondingly test the shadow prices around the sample means only. From the conditions (B.ii) we construct the following null hypotheses:
The shadow prices of real estate and machinery, evaluated around the sample means of the state variables, are negative, because the average observed investment is positive.
The shadow price of labor, evaluated at the sample means of the state variables, is positive, because the average change in the labor services is negative.
The estimated shadow prices for both capital goods differ significantly from zero, but they are positive, having opposite signs to what we expected (Table 7.14). The null hypotheses in (1) are therefore rejected and we conclude that the data does not support the view that the optimal value function is decreasing in real estate and machinery
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capital. This result suggests that the discounted present value of the firm's cost stream is expected to increase, if its capital stock is increased ceteris paribus.
Because real estate and machinery are both measured in equal units, their shadow prices should be equal at the margin, provided farmers have had equal access to both capital goods. But given the small standard errors of the shadow price estimates, it is obvious that the difference between the prices is statistically significant. Indeed, the shadow price of machinery is about one third larger than the shadow price for real estate.
If there were internal adjustment costs only, firms could have reduced their costs by equating shadow prices across the capital goods. In other words, they could have decreased costs at the margin by delaying the machinery investments and by investing only in real estate until the marginal benefit from a unit of capital became equal across the goods. The large difference between the shadow prices cannot be explained by intemal adjustment costs. Therefore, the result suggests that farmers have not had equal access to both capital goods or, more likely, they have had external incentives for investing in machinery. Farmers have been more eager to invest in machinery than in real estate either by tastes, or because of higher incentives through tax shields.
Using the equality -afiaki for j = 1,2, the estimated shadow prices imply that adjustment costs exist, but the instantaneous cost function is decreasing with investments, i.e. 0 for 0, j = 1,2. This result supports the view that there are scale effects in investment, such that the larger the investment the smaller the adjustment costs that are realized. This result is plausible, but it contradicts the traditional postulate that adjustment costs are increasing and convex in investment. Our result is also supported by Rothschild (1971), who claims that the arguments for cost functions being increasing in investment are weak. It has to be noted, however, that the result obtained does not meet the sufficient conditions for an extremum in the original maximization problem. We rely, therefore, on the necessary conditions.
Even though the shadow price of installed real estate capital is positive, the instantaneous cost function is decreasing in real estate (Table 7.15). This result suggests that the cost of investing, including adjustment costs, has outweighed the cost reductions resulting from the increased capital stock. Investments in machinery, on the other hand, have not even reduced the instantaneous costs.
Table 7.14. Shadow Prices for Installed Capital and Labor. at sample means. Standard errors are in parentheses.
b The sample average of gross investment is positive.
The sample average of change in the labor services is negative.
n.a. refers to "not available".
Table 7.15. Derivatives of the Instantaneous Cost Function with respect to Capital and Labor. a
The shadow price of labor is positive and significant, as expected, and the null hypothesis (2) is supported by the data. The optimal value function is increasing in labor even though the instantaneous cost function is decreasing in labor (see Tables 7.14 and 7.15). In other words, the sum of expenses, including adjustment costs, to purchase more labor services has exceeded the sum of instantaneous cost reductions provided by increased labor. Net returns to incremental labor services have been negative, and firms have been gradually substituting capital for labor to decrease costs. Further, had the firms been able to reduce their labor input even more, they would have been able to reduce their production costs, ceteris paribus.
Again, using the conditions in (B.ii) and, in particular, the equality
aciaL=
we conclude that the instantaneous cost function is decreasing in changes of labor, i.e.8ciai<
0 forL<
0, as expected. Note that, as the average labor88
change is negative, this result implies that the cost function is increasing in the absolute value of the labor change.
In summary we conclude that, even though instantaneous costs have been decreasing with one out of two capital goods, and with labor services, the sample farms have had excess capital and labor relative to their output levels. At a given output level they could have reduced the discounted present value of their production costs by reducing the capital stock and labor.
7.6 Short-Run Response Probabilities and Elasticities