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Problem Set 6 {due May 7} Problem 1 The economy is populated by a continuum of measure one of agents, indexed by 1' and uniformly distributed over the [If]1 l] interval. Agents are risk neutral with utility 12 \"iAki_ki where k.- is the individual investment of agent i, A is the return to investment, and 52: is the cost of investment. Let 1 K = f Jodi n denote the aggegate level of investment. The return to investment is given by A = (l of"? + oK where o E [l], g. The random variable 9 parametrizes the fundamentals of the economy. If o: > I} there is a complementarity in that the return to individual investment is increasing in the aggregate level of investment, and the parameter :1 captures the degree of comple- mentarity. The fundamentals 6 are not known at the time investment decisions are made. Further- more, agents have heterogeneous beliefs about 6'. The common prior is uniform over R. Agent 1' has private information xi = 3 + dx'fi and there is public information 3; = K + eye. The random variables Eh i E [If]1 l] and a are standard normal and independent as well as independent of 6'. The precisions of the two sources ofinformation are denoted as arr = of, and it], = 0172, respectively. Let social welfare be given by a utilitarian awegator 1 w =f midi. c 1. Check that it is an equilibrium for investment to takes the form ki=ms+{1.5)y. Determine the coecient ,5 and describe how it varies with the degree of complemen- tarity o and the two precisions 1r: and a}. Provide an intuitive explanation of your ndings. 2. How do heterogeneity Varfh|,y} and volatility VaerIS) vary with the three param- eters o, in: and Try. Provide intuition. 3. Show that social welfare conditional on fundamentals E[w|9] is a linear function of heterogeneity and volatility. Use your previous results to discuss how social welfare varies with the parameters and provide intuition for your results. 4. Now suppose there is a second source of public information Problem 2 (A. simple Model of Savings} Consider the problem of a consumer who wants to maximize the following program: co c:Hy max E 1 {coon g'llv U 51-. {i} w; = 31+]?th Iii-rI = c+hu1 {5}) (11] 1nd. 1:: [I The endowment shock e; and the interest rate R. are i.i.d. Don't worry about the non- negativity constraint on consumption. 1. Rewrite the problem in recursive form. 2. Without solving for the value function, derive the rst order condition {FUD} and the envelope condition {EC}. Combine the two to obtain the Euler Equation {EE}. 3. Assume in this section that the endowment shock is [l in all periods. Make a guess for a value function. Using this guess], derive the consumption function. Using the EE, solve for the constant. Then replace hack into the Bellman equation, and verify that you indeed found the value function. 2 4. Assume now that the endowment shock is stochastic, and that the interest rate is deterministic and R. = R. Use the EE to analyse consumption growth. What happens if R}? = 1? What is the R that makes expected consumption growth zero? Discuss the implications of uninsurahle risk. 5. Assume that R; is stochastic, and that the consumer is the representative agent of the economy. Assume that endowment is stochastic, and that the asset is in zero net supply. Use the EE to price the asset when there is only aggregate risk. Discuss (but do not solve] the case with only idiosyncratic risk. 5. Now assume that there is no uncertainty, that the interest rate is constant, and the endowment shock I}. Solve for the value mction, and the optimal consumption and wealth path as a function of initial wealth. Using the optimal consumption path that you derived from the recursive approach, derive the value function by replacing consumption in expected utility. What condition in 1- do you need to make sure that the solution is indeed optimal? Discus]. Problems 1. Consider a firm with capital as the only factor of production. Its revenues at time t are R(K(t)) if installed capital is K(t). The accumulation constraint has the usual form, K(t) = I(t) - 6K (t), and the cost of investing I(t) is a function G(I(t)) that does not depend on installed capital (for simplicity, PK = 1). . (a) Suppose the firm aims at maximizing the present discounted value at rate r of its cash flows, F(t). Express cash flows in terms of the func- tions R(.) and G(.), derive the relevant first-order conditions, and char- acterize the solution graphically making specific assumptions as to the derivatives of R(.) and G(.). (b) Characterize the solution under more specific assumptions: suppose revenues are a linear function of installed capital, R(K) = ok, and let the investment cost function be quadratic, G(1) = 1 +bl. Derive and interpret an expression for the steady-state capital stock: what happens if 6 = 0? 2. A firm's production function is Y (t) = avK(t) + BVL(t), and its product is sold at a given price, normalized to unity. Factor _ is not subject to adjustment costs, and is paid w per unit time. Factor K obeys the accumulation constraint K(t) = I(t) - 5K(t) and the cost of investing I is G(1) = 1+212 per unit time (we let pp = 1). The firm maximizes the present discounted value at rate r of its cash flows. (a) Write the Hamiltonian for this problem, derive and discuss briefly the first-order conditions, and draw a diagram to illustrate the solution. 13 (b) Analyze graphically the effects of an increase in 6 (faster depreciation of installed capital) and give an economic interpretation of the adjustment trajectory.3. As a function of installed capital K, a firm's revenues are given by R(K) = K -=K2 The usual accumulation constraint has o = 0.25, so K = 1-0.25k. Investing I costs PG(I) = px (I + $12) . The firm maximizes the present discounted value at rate r = 0.25 of its cash flows. (a) Write the first-order conditions of the dynamic optimization problem, and characterize the solution graphically supposing that PR = 1 (con- stant). (b) Starting from the steady state of the pe = 1 case, show the effects of a 50% subsidy of investment (so that px is halved). (c) Discuss the dynamics of optimal investment if at time t = 0, when pr is halved, it is also announced that at some future time T' > 0 the interest rate will be tripled, so that r(t) = 0.75 for t 2 T. 4. The revenue flow of a firm is given by R(K, N) = 2R1/2 N1/2 where N is a freely adjustable factor, paid a wage w(t) at time t; K is accumulated according to K = 1-6K and an investment flow I costs G(D) = (1+ 212) (note that px = 1, hence q = 1). (a) Write the first-order conditions for maximization of present discounted (at rate r) value of cash flows over an infinite planning horizon. 14 (b) Given r e o constant, write an expression for A(0) in terms of w(t), the function describing the time path of wages. (c) Evaluate that expression in the case where w(t) = w is constant, and characterize the solution graphically. (d) How could the problem be modified so that investment is a function of the average value of capital (that is, of Tobin's average q)
