Constant returns can be introduced by assuming that the adjustment costs take tb; form C(k/k)k, with C(») having the same properties as before. With this assumption, dou biing both and doubles the adjustment costs. Changing our model in this way imphc-that affects profits not only directly, but also through its impact on adjustment costs fee a given level of investment. As a result, it complicates the analysis. The basic messages the same, however. See Problem 8.8 and Sala-i-Martm (1991a).

to denote the value of capital in the previous section. Our analysis implies that what is relevant to investment is margmal -the ratio of the market value of a marginal unit of capital to its replacement cost. Marginal q is hkely to be harder to measure than average -the ratio of the total value of the ftrm to the replacement cost of its total capital stock. Thus it is important to know how marginal q and average q are related.

One can show that in our model, marginal q is less than average q. The reason is that when we assumed that adjustment costs depend only on k, we implicitly assumed diminishing returns to scale in adjustment costs. Our assumptions imply, for example, that it is more than twice as costly for a firm with 20 units of capital to add 2 more than it is for a firm with 10 units to add 1 more. Because of this assumption of diminishing returns, firms hfetime profits, , rise less than proportionally with their capital stocks, and so marginal q is less than average q.

One can also show that if we modify the model to have constant returns in the adjustment costs, average and marginal q are equal (Hayashi, 1982).-The source of this result is that the constant returns in the costs of adjustment imply that q determines the growth rate of a firms capital stock. As a result, all firms choose the same growth rate of their capital stocks. Thus if, for example, one firm initially has twice as much capital as another and if both firms optimize, the larger firm will have twice as much capital as the other at every future date. In addition, profits are linear in a firms capital stock. This implies that the present value of a firms profits-the value of when it chooses the path of its capital stock optimally-is proportional to its initial capital stock. Thus average q and marginal q are equal.

In other models, there are potentially more significant reasons than the degree of returns to scale in adjustment costs that average q may differ from marginal . If a firm faces a downward-sloping demand curve for its product, for example, doubling its capital stock is likely to less than double the present value of its profits; thus marginal q is less than average q. If the firm owns a large amount of outmoded capital, on the other hand, its marginal q may exceed its average q.

8.4 Analyzing the Model

We wiU analyze the model using a phase diagram similar to the one we used in Chapter 2 to analyze the Ramsey model. The two variables we wih focu-on are the aggregate quantity of capital, K, and its value q. As with and . in the Ramsey model, the initial value of one of these variables is given, bu"

K(t) = f{q{t)), 1) = 0, (-)>0,

(8.23)

where f(q) = NC~(q -1). Equation (8.23) implies that is increasing when > 1, decreasing when q < 1, and constant when q = 1. This information is summarized in Figure 8.1.

Equation (8.19) states that the marginal revenue product of capital equals its user cost, rq - q. Rewriting this as an equation for q yields

m = rqit) - 77(1(0).

(8.24)

This expression implies that q is constant when rq = { ), or q = 7 { )/ . Since T7(K) is decreasing in K, the set of points satisfying this condition

(K>0)

{K<0)

FIGURE 8.1 The dynamics of the capital stock

the other must be determmed: the quantity of capital is something that the industry inherits from the past, but its price adjusts freely in the market.

Equation (8.18) states that each firm invests to the point where the purchase price of capital plus the marginal adjustment cost equals the value of capital: 1 + Cil) = q. Since C(I) is increasing in /, this condition implies that / is increasing in q. And since C(0) is zero, it also imphes that / is zero when is 1. Finally, since q is the same for all hrms, all hrms choose the same value of /. Thus the rate of change of the aggregate capital stock, K, is given by the number of hrms, N, times the value of / that satisfies (8.18).

Putting this information together, we can write

The Phase Diagram

Figure 8.3 combines the information in Figures 8.1 and 8.2. The diagram shows how and q must behave to satisfy (8.23) and (8.24) at every point in time given their initial values. Suppose, for example, that and q begin at Point A. Then, since q is more than 1, firms increase their capital stocks: thus is positive. And since is high and profits are therefore low, q can only be high if it is expected to rise; thus q is also positive. Thus and q move up and to the right in the diagram.

As in the Ramsey model, the initial level of the capital stock is given. But the level of the other variable-consumption in the Ramsey model, the market value of capital in this model-is free to adjust. Thus its initial level must be determined. As in the Ramsey model, for a given level of there is a unique level of q that produces a stable path. Specifically, there is a unique level of q such that and q converge to the point where they are stable (Point E in the diagram). If q starts below this level, the industn eventually crosses into the region where both and q are falling, and they

f (>0)

(q<0)

FIGURE 8.2 The dynamics of q

is downward-sloping in (K, q) space. In addition, (8.24) implies that q is increasing in a:; thus q is positive to the right of the q = 0 locus, and negative to the left. This information is summarized in Figure 8.2.