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131

Problems

8.1. Consider a firm that produces output using a Cobb-Douglas combination of capital and labor: = JfLi ",0 < a < 1. Suppose that the firms price is fixed in the short run; thus il takes both the price of its product, P, and the quantity, y, as given. Finally, input markets are competitive; thus the firm takes the wage, W, and the rental price of capital, , as given.

(a) What is the firms choice of L given P, Y, W, and K?

(b) Given this choice of L, what are profits as a function of P, Y, W, and K?

ic) Find the first-order condition for the profit-maximizing choice of If. Is the second-order condition satisfied?

(d) Solve the first-order condition in part (c) for If as a function of P, Y, W, and . How, if at all, do changes in each of these variables affect K?

8.2. Corporations in the United States are allowed to subtract depreciation allowances from their taxable income. The depreciation allowances are based on the purchase price of the capital; a corporation that buys a new capital good at time t can deduct fraction D(s) of the purchase price from its taxable income at time t + s. Depreciation allowances often take the form of straight-line depreciation: D{s) equals 1/ for selO, T], and equals zero for s > T, where T is the tax life of the capital good.

(a) Assume straight-line depreciation. If the marginal corporate income tax rate is constant at and the interest rate is constant at i, by how much does purchasing a unit of capital at a price of reduce the present value of the firms corporate tax Uabilities as a funcfion of T, , i, and ? Thus, what is the after-tax price of the capital good to the firm?

ib) Suppose that i = r - , and that increases with no change in r. How does this affect the after-tax price of the capital good to the firm?

8.3. The major feature of the tax code that affects the user cost of capital in the case of owner-occupied housing in the tJnited States is that nominal interest payments are tax deductible. Thus the after-tax real interest rate relevant to home ownership is r - tz , where r is the pretax real interest rate, z is the nominal interest rate, and is the marginal tax rate. In this case, how does an increase in inflation for a given r affect the user cost of capital and the desired capital stock?

8.4. Using the calculus of variations to solve the social planners problem in the Ramsey model. Consider the social planners problem that we analyzed in Section 2.4: the planner wants to maximize J~o e *4c(f) */(l - )] dt subject to kt) = f{k(t)) - cit) ~{n+ g)k{t).

These papers findings are representative of the findings in tfiis hterature: the evidence consistently suggests that financial-market imperfections are important to investment. Precisely what form those imperfections take, and how important they are quantitatively, remain open questions.



( ) What IS the current value HamiltonianWhat variables are the control van able, the state variable, and the costate variable?

(b) Find the three condUions that characterize optimal behavior analogous to equations (8 18), (8 19), and (8 20), in Section 8 2

(c) Show that the hrst two conditions m part (b), together with the fact that f (kit)) = r(t), imply the Euler equation (equaUon [2 19])

id) let p denote the costate variable Show that [ ( )/ ( )] ~ p = (n + g)~ r(f), and thus that e lnit) is propornonal to e «<« ( +0>< Show that this im plies that the transversahty condition in part (b) holds if and only if the budget constraint, equation (2 12), holds with equality

8.5. Consider the model of investment m Sections 8 2-8 5 Describe the effects of each of the following changes on the = 0 and q =0 loci, on and q at the time of the change, and on their behavior over time In each case, assume that and q are initially at their long run equilibrium values

( ) A war destroys half of the capital stock

(b) The government taxes returns from owning hrms at rate

(c) fhe government taxes investment Specifically, firms pay the govemment for each unit of capital they acquire, and receive a subsidy of for each unit of disinvestment

8.6. Consider the model of investment in Sections 8 2-8 5 Suppose it becomes known at some date that there will be a one time capital levy, specifically, capital holders will be taxed an amount equal to fracUon f of the value of their capital holdings after time Assume the industry is initially m long run equilibrium What happens at the time of this news? How do and q behave between the time of the news and the time the 1 \ is imposed What happens to and q at the time of the levy How do they behave thereafter (Hint is q anticipated to change discontinuously at the time of the levy)

8.7. A model of the housing market. (This follows Poterba, 1984 ) Let H denote the stock of housmg, I the rate of investment, the real price of housing, and R the rent Assume that I is increasmg in , so that I = I(pn), with I () > 0, and that H = I ~ SH Assume also that the rent is a decreasing function of H R = R(H),R() < 0 tinally, assume that rental mcome plus capital gams must equal the exogenous required rate of return, r (R + )/ = r

(a) Sketch the set of pomts in ( , ) space such that H = 0 Sketch the set of pomts such that = 0

(b) What are the dynamics of H and in each region of the resulting diagram Sketch the saddle path

(c) Suppose the market is initially in long rim equilibrium, and that there is an unexpected permanent increase in r What happens to H and at the time of the change How do H, , I, and R behave over time following the change

(d) Suppose the market is initially m long run equilibrium, and that it becomes known that there will be a permanent increase in r time T in the



future. What happens to H and at the time of the news? How do H, pu, I, and R behave between the time of the news and the time of the increase? What happens to them when the increase occurs? How do they behave after the increase?

(e) Are adjustment costs internal or external in this model? Explain. ( Why is the = 0 locus not horizontal in this model?

8.8. Suppose that the costs of adjustment exhibit constant returns in and . Specifically, suppose they are given by { / ) , where C(0) = 0, C(0) = 0, C"() > 0. In addition, suppose capital depreciates at rate S; thus k{t) = I{t) -SkU). Consider the representative firms maximization problem.

(a) What is the current-value Hamiltonian?

ib) Find the three conditions that characterize optimal behavior analogous to equations (8.18), (8.19), and (8.20), in Section 8.2.

(c) Show that the condition analogous to (8.18) implies that the growth rate of each firms capital stock, and thus the growth rate of the aggregate capital stock, is determined by q. In {K,q) space, what is the = 0 locus?

(d) Substitute your result in part (c) into the condition analogous to (8.19) to express q in terms of and q.

(e) In {K,q) space, what is the slope of the = 0 locus at the point where q = l7

8.9. Suppose that ( ) = a - bK and (I) = al42.

(a) What is the q = 0 locus? What is the long-run equilibrium value of K?

ib) What is the slope of the saddle path? (Hint: use the approach in Secfion 2.6.)

8.10. Consider the model of investment under uncertainty with a constant interest rate in Section 8.6. Suppose that, as in Problem 8.9, MK) = a - bK and that {I) = aI/2; in addition, suppose that what is uncertain is future values of a. This problem asks you to show that it is an equilibrium for qit) and Kit> to have the values at each point in time that they would if there were no uncertainty about the path of a. Specifically, let qit + , t) and Kit + r, t) be the paths q and would take after time f if a(r - ) were certain to equal £f[a(t + t)] for all > 0.

ia) Showthatif£,[t?(r-i-t)] = (fH-t,f)forallT > 0,XhexiEt\Kit+r)\ = Kit+T,t for all t > 0.

ib) Use equation(8.26) to show that this implies thatif£,[q(t-t-t)] = (fH-t,t». then Qit) = qit.t), and thus that Kit) = N[qit,t) - l]/a, where N is the number of firms.

8.11. (This follows Bernanke, 1983a, and Dixit and Pindyck, 1994.) Consider a firm that is contemplating undertaking an investment with a cost of I. There are two periods. The investment will pay off 771 in period 1 and 772 in period 2. ;



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