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119 See n. 11 and Problem 8.7. The model presented here is developed by Abel (1982): Hayashi (1982); and Summers (1981b). in the capital stock requires an infinite rate of investment. For the economy as a whole, hovever, investment is limited by the economys output; thus aggregate investment cannot be infinite. The second problem with the model is that it does not identify any mechanism through which expectations affect investment demand. The model implies that firms equate the current marginal revenue product of capital with hs current user cost, without regard to what they expect future marginal revenue products or user costs to be. Yet it is clear that in practice, expectations about demand and costs are central to investment decisions: firms expand their capital stocks when they expect their sales to be growing and the cost of capital to be low, and they contract them when they expect their sales to be falling and the cost of capital to be high. ,. Thus we need to modify the model if we are to obtain even a remotely reasonable picture of actual investment decisions. The standard theory that does this emphasizes the presence of costs to changing the capital stock. Those adjustment costs come in two forms, internal and external (Mussa, 1977). Internal adjustment costs arise when firms face direct costs of changing their capital stocks (Eisner and Strotz, 1963; Lucas, 1967; Gould, 1968). Examples of such costs are the costs of installing the new capital and training workers to operate the new machines. Consider again a discrete cut in interest rates. If the adjustment costs approach infinity as the rate of change of the capital stock approaches infinity, the fall in interest rates causes investment to increase but not to become infinite. As a result, the capital stock moves gradually toward the new desired level. External adjustment costs arise when each firm, as in our baseline model, faces a perfectly elastic supply of capital, but where the price of capital goods relative to other goods adjusts so that firms do not wish to invest or disinvest at infinite rates (Foley and Sidrauski, 1970). When the supph of capital is not perfectly elastic, a discrete change that increases firms desired capital stocks bids up the price of capital goods. Under plausible assumptions, the result is that the rental price of capital does not change discontinuously but merely begins to adjust, and that again investment increases but does not become infinite. 8.2 A Model of Investment with Adjustment Costs We now turn to a model of investment with adjustment costs. For concreteness, the adjustment costs are assumed to be internal; it is straightforward, however, to reinterpret the model as one of extemal adjustment costs. The model is known as the q theory model of investment.
 [TTiKWMt)  )  cum dt. (8.6) where we assume for simplicity that the real interest rate is constant. Each firm takes the path of the industrywide capital stock, K, as given, and chooses its investment over time to maximize given this path. To solve the firms maximization problem, we need to employ the calculus of variations. To understand this method, it is helpful to first consider a discretetime version of the firms ." The evolution of the firms capital stock is now given by Kt+i = , + I,, and the adjustment costs are Note tlial tliese assumptions imply that in the model of Section 8.1, !t(K,Xi,...,X„) takes the form Tr{Xi,...,Xn)K, and so the assumption that < 0 fails. Thus in this case, m the absence of adjustment costs, the firms demand for capital is not well defined: it is infinite if tt{Xi,...,X„) > 0, zero if Tr(Xi,...,X„) < 0, and not defined if ir{Xi,...,X„) = 0. •For more thorough and formal introductions to the calculus of variations, see Kamien and Schwartz (1991) and Dixit (1990, Chapter 10). Assumptions Consider an industry with N firms. A representative firms real profits at time f, neglecting any costs of acquiring and installing capital, are proportional to its capital stock, ( ), and decreasing in the industrywide capital stock, K(ty, thus they take the form ( {1)) ( ), where nW < 0. The assumption that the firms profits are proportional to its capital is appropriate if the production function has constant returns to scale, output markets are competitive, and the supply of all factors other than capital is perfectly elastic. Under these assumptions, if one firm has, for example, twice as much capital as another, it employs twice as much of all inputs; as a result both its revenues and its costs are twice as high as the others. And the assumption that profits are decreasing in the industrys capital stock is appropriate if the demand curve for the industrys product is downwardsloping. The key assumption of the model is that firms face costs of adjusting their capital stocks. The adjustment costs are a convex function of the rate of change of the firms capital stock, k. Specifically, the adjustment costs, C(k), satisfy C(0) = 0, C(0) = 0, and C"C) > 0. These assumptions imply that it is costly for a firm to increase or decrease its capital stock, and that the marginal adjustment cost is increasing in the size of the adjustment. The purchase price of capital goods is constant and equal to 1; thus there are only internal adjustment costs. Finally, for simphcity, the depreciation rate is assumed to be zero; thus k(t) = I{t), where I is the firms investment. A DiscreteTime Version of the Firms Problem These assumptions imply that the firms profits at a point in time are { )  I  (I). The firm maximizes the present value of these profits,
The awkward fact that and q in penod r concern the value of capital in period t will disappear when we consider the continuoustime case. given by CUt). The hrms objective function is therefore = S 77 MKt)Kt It cm. (8.7) r=o We can think of the firm as choosing its investment and capital stock each period subject to the constraint that they are related by Kf+i = Kf + It for each t. Since there are inhnitely many periods, there are infinitely constraints. The Lagrangian for the firms maximization problem is = i 3 Wf)<t   C(/t)] (8.8) + X Af(Kr + It  Kt+i). Af is the Lagrange multiplier associated with the constraint relating Kti and Kt. It therefore gives the marginal value of relaxing the constraint; that is, it gives the marginal impact of an exogenous increase in Kt+i on the lifetime value of the firms profits discoimted to time zero. If we define = (1 + ry\t, Qt therefore shows the value to the firm of an additional unit of capital at time f +1 in timef dollars. With this definition, we can rewrite the Lagrangian as C = i 7717 MKt)Kt It C(It) + QtiKt + It Kt+i)]. (8.9) The firstorder condition for the firms investment in period f is therefore J [ l C(/t) + %] = 0. (8.10 Multiplying both sides by (1 + rY, we obtain l + C{It) = qt. (8.11 To interpret this condition, observe that the cost of acquiring a unit of capital equals the purchase price (which is fixed at 1) plus the marginal adjustment cost. Thus (8.11) states that the firm invests to the point where the cost of acquiring capital equals the value of the capital. Now consider the firstorder condition for capital in period f. The term for period f in the Lagrangian, (8.9), involves both Kt and Kt+i. Thus the
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