14.06 Macroeconomics Spring 2003 Problem Set 7 (due on the day of Lecture # 11) Problem 1 Romer Problem 4.3 (First-order autoregressive shocks) Let ln A0 denote the values of A in period 0, and let the beha vior of ln A be given by ¯ ˜ln At = A + gt + At, ˜ ˜At = ρAAt−1 + At, s.t. − 1 <ρA < 1, ˜where A is an input in the production function, kno wn as tec hnology, and A reflects the fact that technology evolves according to a random process, specifically, a first-order au-toregressive process. ¯1. Express ln A1, ln A2,and ln A3 in terms of ln A0,A1,A2,A3, A,g,and ρA. 2. Given that expectations of the A ’s are zero, what are the ex pectations of ln A1, ln A2,and ¯ln A3 given ln A0, A, ρAand g? Problem 2 Augmen ted Rom er Prob lems 4.4 and 4.5 Suppose the period-t utilit y function, ut,is ut =ln ct + b(1 − lt)1−γ /(1 − γ),b> 0,γ > 0. (Note that with γ =1, utility reduces to the form seen in class: ut =ln ct + b ln(1 − lt).) 1. Consider the househ old’s problem of maximizing utility subject to a budget constraint c = wl where is c consumption, l is hours work ed, and w is the wage. Find the first order conditions and solve for the labor supply. How, if at all, does labor supply depend on the wage? 2. Consider an extension to the previous problem. Instead of a static problem, the 1consumer/worker lives two periods (and discoun ts second period utility by 1+ρ .)There is no uncertain ty. (a) Write the lifetime budget constra int. (b) Write down the first order conditions, and solv e for the relative dem an d for leisure in the two periods. (c) How does the relative demand for leisure depend on the relative wage? Sho w that an increase in both w1 and w2 that leaves w1/w2 unchanged does not affect l1 or l2.(d) Suppose output is given by Yt = Ktα(AtLt)1−α , 0 <α< 1. Solve for wt,the wage rate (assume labor is paid its marginal product). Now suppose there is a positive technology shock at tim e 1, A1 = A, A > 1 (assume that at time 2, tec hnology returns to A2 =1). What is the effect, if an y, on the relative wage and on the relative demand for leisure? Does it make sense? (e) How does the relative demand for leisure depend on the interest rate?, on the time preference rate? (f) Explain intuitively why γ affects the responsiveness of labor supply to wages and the in terest rate. (g) Solve for the Euler equation, that is expres s the relationship b etween c1 and c2. What if ρ = r? (h) Now assume that the household has initial wealth of amount Z> 0. Does the Euler equation derived in part g continue to hold? Problem 3 Romer Problem 4.8 (A simplified RBC model with additive technology shocks). Consider an economy consisting of a constant population of infinitely-lived ind ividuals. The representative individual maximizes the expected value of ∞X 1 u(Ct),ρ> 0 (1 + ρ)t t=0 where u(Ct)= Ct − θCt 2,θ> 0 Assume that C is always in the range where u0(C) is positiv e. Output is linear in capital, plus an additive disturbance: Yt = AKt + et. There is no depreciation; thus Kt+1 = Kt + Yt − Ct, and the interest rate is A. Assume A ≡ r = ρ. Finally, the disturbance follows a first-o rder autor egressive process: et = φet−1 + εt,where −1 <φ< 1 and where the ε’s are mean zero, i.i.d shocks. 1. Find the first-order condition (Euler equation) relating Ct and expectations of Ct+1.(Hint: set up the Bellman equation and max im iz e w.r.t Kt+1 after substituting for Ct as functions of Kt,Kt+1 etc.) 2. Guess that consumption takes the form Ct = α + βKt + γet. Given this guess, what is Kt+1 as a function of Kt and et? 3. What values must the parameters α, β, and γ have for the first-order condition in part 1 to be satisifed for all value s of Kt and et? 4. What are the effects of a one-time shock to ε (suppose ∆εt =1) on the paths of Y, K, and C