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124 q{t) + Bit) = 1 + ), (8.25) where e{t) is the credit at time f. The equation for q, (8.24), is unchanged. Equation (8.25) implies that the capital stock is constant when q + 6 = 1. An investment tax credit of therefore shifts the = 0 locus down by ; this is shown in Figure 8.8. If the credit is permanent, q jumps down to the new saddle path at the time it is announced. Intuitively, because the credit mcreases investment, it means that the industrys profits (neglecting the credit) will be lower, and thus that existing capital is less valuable. and q then move along the saddle path to the new longrun equilibrium, which involves higher and lower q. Now consider a temporary credit. From our earlier analysis of a temporary change in output, we know that the announcement of the credit causes q to fall to a point where the dynamics of and q, given the credit, bring FIGURE 8.8 The effects of a permanent investment tax credit invest while the credit is in effect. Our model can be used to investigate this argument. For simplicity, assume that the investment tax credit takes the form of a direct rebate to the hrm of fraction of the price of capital, and assume that the rebate applies to the purchase price but not to the adjustment costs. When there is a credit of this form, the firm invests as long as the value of the capital plus the rebate exceeds the capitals cost. Thus the flrstorder condition for current investment, (8.18), becomes
q = 0 FIGURE 8.9 The effects of a temporary investment tax credit them to the old saddle path just as the credit expires. They then move up that saddle path back to the initial longrun equilibrium. This is shown in Figure 8.9. As the figure shows, q does not fall all the way to its value on the new saddle path; thus the temporary credit reduces q by less than a comparable permanent credit does. The reason is that, because the temporary credit does not lead to a permanent increase in the capital stock, it causes a smaller reduction in the value of existing capital. Now recall that the change in the capital stock, K, depends on q + 6 (see [8.25]). q is higher under the temporary credit than under the permanent one; thus, just as the informal argument suggests, the temporary credit has a larger effect on investment than the permanent credit does. Finally, note that the figure shows that under the temporary credit, q is rising in the later part of the period that the credit is in effect. Thus, after a point, the temporary credit leads to a growing investment boom as firms try to invest just before the credit goes out of effect. Under the permanent credit, in contrast, the rate of change of the capital stock declines steadily as the industry moves towards its new longrun equilibrium. 8.6 The Effects of Uncertainty: An Introduction Our analysis so far assumes that firms are certain about future profitability, interest rates, and tax policies. In practice, they face uncertainty about all
q(t) = e~Et[7r{K{r))]dr (8.26) (see [8.22]). This expression can be used to find how q is expected to evolve over time. Since (8.26) holds at aU times, it imphes that the expectation as of time f of Q at some later time, t + At, is given by Edq(t + At)\ = £, T=f+At J (8.27) T=t+M where the second line uses the fact that the law of iterated projections implies that [ +\ ( is just ( { ))\. Differentiating (8.27) with respect to and evaluating the resulting expression at = 0 gives us Et[m\ = rq(t)7r(K{t)). (8.28) Except for the presence of the expectations term, this expression is identical to the equation for q in the model with certainty (see [8.24]). As before, each firm invests to the point where the cost of acquiring new capital equals the market value of capital. Thus equation (8.23), (f) = f(q{t)), continues to hold. Our analysis so far appears to imply that uncertainty has no direct effect on investment: firms invest as long as the value of new capital exceeds the cost of acquiring it, and the value of that capital depends only on its expected payoffs. But this analysis neglects the fact that it is not quite correct to assume that there is exogenous uncertainty about the future values of 7 { ). Since the path of is determined within the model, what can be taken as exogenous is uncertainty about the position of the () function; the combination of that uncertainty and firms behavior then determines uncertainty about the values of { ). In one natural baseline case, this subtlety proves to be unimportant: if 77(») is linear and C(«) is quadratic and if the uncertainty concerns the of tfiese. This section therefore introduces some of the issues raised by uncertainty. Uncertainty about Future Profitability We begin with the case where there is no uncertainty about the path of the interest rate; for simplicity it is assumed to be constant. Thus the uncertainty concerns only future profitability. In the case, the value of one unit of capital is given by
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