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120 (8.12) Multiplying this expression by (1 i rY and rearranging yields TTiKt) = (1 + r)ti  Qt. (8.13) If we define Aq, = qt  qti, we can rewrite the righthand side of (8.13) as (1 + r)(q,  Aq,)  qt, or rq,  Aqt  rAq,. Thus we have iriKt) = rqt  Aqt  rAq,. (8.14) The lefthand side of (8.14) is the marginal revenue product of capital. And the righthand side is the opportunity cost of a unit of capital. Intuitively, owning a unit of capital for a period requires forgoing rqt of real interest and involves offsetting capital gains of , (see [8.4] with the depreciation rate assumed to be zero; in addition, there is an interaction term involving r and Aq that will disappear in the continuoustime case.) For the firm to be optimizing, the returns to capital must equal this opportunity cost. This is what is stated by (8.14). This condition is thus analogous to the condition in the model without adjustment costs that the firm rents capital to the point where its marginal revenue product equals its rental price. The final condition characterizing the firms behavior concerns what happens as t approaches infinity. If the firm has a finite horizon, T, and if it cannot have a negative capital stock, optimality requires that the value of its terminal capital stock is zero. If not, the firm would be better off reducing its terminal capital holdings. The condition that the value of the firms terminal capital holdings is zero is qTKT = 0. (8.15) (1 + r)r The infinitehorizon analogue of this condition is Equation (8.16) is known as the transversality condition. It states that the value of the capital stock must approach zero. If this condition fails, then, loosely speaking, the firm is holding valuable capital forever, and so it can increase the present value of its profits by reducing its capital stock. See Section 8.4 for more on the interpretation of this condition. capital stock in period t, ,, appears in both the term for period f and the term for period t  1. The firstorder condition for Kf is therefore
An alternative approach is to formulate the presentvalue Hamiltonian, HMt),I(t)) = e "lTr{K{t))K{t)  Ht) C{I{t))] + A(t)[/(f)  k(t)]. This IS analogous to using the Lagrangiar (8.8) rather than (8.9). With this formulation, (8.19) is replaced by e "7r{K(t)) = A(t). It istraightforward to check that, since q(t) = A(r)e", these two conditions are equivalent. The ContinuousTime Case We can now consider the case when time is continuous. The firms profitmaximizing behavior m this case is characterized by three conditions that are analogous to the three conditions that characterize its behavior in discrete time, (8.11), (8.14), and (8.16). Indeed, the optimality conditions for continuous time can be derived by considering the discretetime problem where the time periods are separated by intervals of length and then taking the limit as approaches zero. We wiU not use this method, however; instead we will simply describe how to find the optimality conditions, and justify them as necessary by way of analogy to the discretetime case. The firms problem is now to maximize the continuoustime objective function, (8.6), rather than the discretetime objective function, (8.7). The first step in analyzing this problem is to set up the currentvalue Hamiltonian: HMtint)) = 7rmt))K{t)  7(f)  ant)) + qmW  Kit)]. This expression is analogous to the period term m the Lagrangian for the discretetime case (see [8.9]). There is some standard terminology associated with this type of problem. The variable that can be controlled freely (I) is the control variable; the variable whose value at any time is determined by past decisions ( ) is the state variable; and the shadow value of the state variable (q) is the costate variable. The first condition characterizing the optimum is that the derivative of the Hamiltonian with respect to the control variable at each point in time is zero; this is analogous to the condition in the discretetime problem that the derivative of the Lagrangian with respect to I for each t is zero. This condition is 1 + cm)) = q(t). (8.181 This condition is analogous to (8.11) in the discretetime case. The second condition is that the derivative of the Hamiltonian with respect to the state variable equals the discount rate times the costate variable minus the derivative of the costate variable with respect to time. In our case, this condition is MKit)) = rq(t)  qit). (8.19 This condition is analogous to (8.14) in the discretetime problem.
q(t) = emrMr + eqm (8.21) for any T > t? One can show that the transversality condition implies that the second term approaches zero as approaches infinity. Thus we have /•oo q(t) e~77iK{T))dr. (8.22) Expression (8.22) states that the value of a unit of capital at a given time equals the discounted value of its future marginal revenue products. 8.3 Tobins q Our analysis implies that q summarizes all information about the future that is relevant to a firms investment decision, q shows how an additional dollar of capital affects the present value of profits. Thus the firm wants to increase its capital stock if q is high and reduce it if q is low; it does not need to know anything about the future other than the information that is summarized in q in order to make this decision (see [8.18]). q has a natural economic interpretation. A oneunit increase in the firms capital stock increases the present value of the firms profits by q, and thus raises the value of the firm by q. Thus q is the market value of a unit of capital. If there is a market for shares in firms, for example, the total value of a firm with one more unit of capital than another firm exceeds the value of the other by q. And since we have assumed that the purchase price of capital is fixed at 1, is also the ratio of the market value of a unit of capital to its replacement cost. Thus equation (8.18) states that a firm increases its capital stock if the market value of capital exceeds what it costs to acquire it, and that it decreases its capital stock if the market value of the capital is less than what it costs to acquire it. The ratio of the market value to the replacement cost of capital is known as Tobins q (Tobin, 1969); it is because of this terminology that we used q 4o verify that (8.21) follows from (8.19), differenhate (8.21) with respect to f, and then rearrange the resulting expression to obtain (8.19). The final condition is the continuoustime version of the transversality condition. This condition is lime qitMt) = 0. (8.20) foo Equations (8.18), (8.19), and (8.20) characterize the firms behavior. Finally, it is useful to note that we can express q, the value of capital, in terms of capitals future marginal revenue products. Equation (8.19) implies
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