An Example

To see the effects of uncertainty about profitabihty, consider the following example. Suppose that the assumptions of our baseline case are satisfied, and that initially the {) function is constant and the industry is in long-run equilibrium. It then becomes known that the government is considering a change in the tax code that would raise the intercept of the 77() function. The proposal will be voted on after time T, and it has a 50 percent chance of passing. There is no other source of uncertainty.

The effects of this development are shown in Figure 8.10. The figure shows the ir = 0 locus and the = 0 loci and the saddle paths with the initial 77(«) function and the potential new, higher function. Given our assumptions, aU of these loci are straight lines (see Problem 8.9). Initially, and q are at Point E. After the proposal is voted on, they wiU move along the appropriate saddle path to the relevant long-run equilibrium (Point E if the proposal is passed, E if it is defeated). There cannot be an expected capital gain or loss at the time the proposal is voted on. Thus, since the proposal has a 50 percent chance of passing, q must be midway between the points on the two saddle paths at the time of the vote; that is, it must be on the dotted line in the figure. Finally, before the vote the dynamics of and q are given by (8.28) and (8.23) with the initial 7 («) function and no uncertainty about q.

Thus at the time it becomes known that the government is considering the proposal, q jumps up to the point such that the dynamics of and q carry them to the dashed line after time T. q then jumps up or down depending on the outcome of the vote, and and q then converge to the relevant long-run equilibrium.

Irreversible Investment

If 77(«) is not linear or C(«) is not quadratic, uncertainty about the 77(*) function can affect expectations of future values of iriK), and thus can affect current investment. Suppose, for example, that it is more costly for firms to reduce their capital stocks than to increase them. Then if iri*) shifts up, the industry-wide capital stock will rise rapidly, and so the increase in TriK) will be brief; but if {) shifts down, will fall only slowly, and so the decrease in TTiK) will be long-lasting. Thus with asymmetry in adjustment costs, uncertainty about the position of the profit function reduces expectations of future profitability, and thus reduces investment.

intercept of the ttW function, then the uncertainty does not affect investment. That is, one can show that in this case, investment at any time is the same as it is if the future values of the intercept of the 7 {) function are certain to equal their expected values (see Problems 8.9 and 8.10).

FIGURE 8.10 The effects of uncertainty about future tax policy when adjustment costs are symmetric

This type of asymmetry in adjustment costs means that investment is somewhat irreversible: it is easier to increase the capital stock than to reverse the increase. In the phase diagram, irreversibility causes the saddle path to be curved. If exceeds its long-run equilibrium value, it falls only slowly thus profits are depressed for an extended period, and so q is much less than 1. If if is less than its long-run equilibrium value, on the other hand, it rises rapidly, and so q is only slightly more than 1.

To see the effects of irreversibility, consider our previous example, but now with the assumption that the costs of adjusting the capital stock are asymmetric. This situation is analyzed inFigure 8.11. As before, at the time the proposal is voted on, q must be midway between the two saddle paths, and again the dynamics of and q before the vote are given by (8.28) and (8.23) with the initial 77(*) function and no uncertainty about q.

Thus, as before, when it becomes known that the government is considering the proposal, q jumps up to the point such that the dynamics of and q carry them to the dashed line after time T. As the figure shows, however, the asymmetry of the adjustment costs causes this jump to be smaller than it is under symmetric costs. Specifically, the fact that it is costly to reduce capital holdings means that if firms build up large capital stocks before the vote and the proposal is then defeated, the fact that it is hard to reverse the increase causes q to be quite low. This acts to reduce the value of capital before the vote, and thus reduces investment. Intuitively, when investment

FIGURE 8.11 The effects of uncertainty about future tax policy when adjustment costs are asymmetric

is irreversible, there is an option value to waiting rather than Investing. If a Arm does not invest, it retains the possibihty of keeping its capital stock low; if it invests, on the other hand, it commits itself to a high capital stock.

A large recent literature investigates the effects of irreversibility in more detail. Realistically, adjustment costs are likely to be more comphcated than just taking some asymmetric form around = 0. For example, the marginal cost of both the first unit of investment and the first unit of disinvestment maybe strictly positive (so that C(k) is not differentiable at = 0). In this case, there is a range of values of q around 1 for which the firm leaves its capital stock unchanged. The firm increases its capital only if q exceeds some threshold that is strictly greater than 1, and decreases it only if q is below some threshold that is strictly less than 1 (Abel and Eberly, 1994).

In addition, there may be a fixed cost to undertaking any nonzero amount of investment (so that ( ) is discontinuous at = 0). Such a fixed cost increases the range of values of q for which the firm leaves its capital stock unchanged (again, see Abel and Eberly). Investment in the presence of fixed costs and uncertainty is most usefully analyzed using the tools of option-pricing theory from finance; Dixit and Pindyck (1994) develop this approach in detail.

Bernanke (1983a) is an important early paper on irreversibility.