FIGURE 8.3 The phase diagram

then continue to fall indefinitely. Similarly, if q starts too high, the industry eventually moves into the region where both and q are rising and remains there. One can show that the transversality condition fails for these paths, and thus that they can be ruled out.

This discussion suggests why a firms optimal policy must satisfy the transversahty condition. Along the path starting at A, for example, the representative firm is continually building up capital because the value it attaches to the capital is always high. This high value is justified not by large marginal revenue products, but by further increases in the value the firm attaches to the capital (that is, equation [8.19], ( ) = rq - q, holds with a high value of q not because { ) is high, but because q is high). But attaching this high and rising value to capital makes sense only if at some point the capital actually makes large contributions to the firms profits. On the path starting at A, this time never comes. As a result, one can show that the firm can raise the present value of its lifetime profits by lowering the path of its capital holdings. An analogous argument applies to paths where and q are continually falhng.

Thus the unique equilibrium, given the initial value of A, is for q to equal the value that puts the industry on the saddle path, and for and q to then move along this saddle path to E. This saddle path is shown in Figure 8.4.

"For formal demonstrations of tliis, see Abel (1982) and Hayaslii (1982).

FIGURE 8.4 The saddle path

The long-run equihbrium, Point E, is characterized by = 1 (which im plies = 0) and q = 0. The fact that q equals 1 means that the market anc replacement values of capita] are equal; thus firms have no incentive to increase or decrease their capital stocks. And from (8.19), for q to equal zero when q is 1, the marginal revenue product of capital must equal r. Tht means that the profits from holding a unit of capital just offset the for gone interest, and thus that investors are content to hold capital without the prospect of either capital gains or losses.

8.5 Implications

The model developed in the previous section can be used to address manv issues. This section examines its implications for the effects of changes in output, interest rates, and tax policies.

"It is straightforward to modify the model to be one of external rather than interna! adjustment costs. The key change is to replace the adjustment-cost function with a supply curve for new capital goods, = ( ), where g(-) > 0 and where is the relative price of capital. With this change, the market value of firms always equals the replacemen" cost of their capital stocks; the role played by q in the model with internal adjustment costs is played instead by the relative price of capital. See Foley and Sidrauski (1970) anc Problem 8.7.

The Effects of Output Movements

An increase in aggregate output raises the demand for the industrys product, and thus raises proftts for a given capital stock. Thus the natural way to model an increase in aggregate output is as an upward shift of the vi) function.

For concreteness, assume that the industry is initially in long-run equilibrium, and that there is an unanticipated, permanent upward shift of the 77(«) function. The effects of this change are shown in Figure 8.5. The upward shift of the 77(«) function shifts the q = 0 locus up: since profits are higher for a given capital stock, smaller capital gains are needed for investors to be willing to hold shares in firms (see [8.24]). From our analysis of phase diagrams in Chapter 2, we know what the effects of this change are. q jumps immediately to the point on the new saddle path for the given capital stock; and q then move down that path to the new long-run equilibrium at Point E. Since the rate of change of the capital stock is an increasing function of q, this implies that jumps at the time of the change and then gradually returns to zero. Thus a permanent increase in output leads to a temporary increase in investment.

The intuition behind these responses is straightforward. The increase in output raises the demand for the industrys product. Since the capital stock cannot adjust instantly, existing capital in the industry earns rents, and so

FIGURE 8.5 The effects of a permanent increase in output