As shown in the pictures:
w; L, and capital income rik, as follows. max In C1 + #InC2 + 82 InCs (3) subject to Ctl w.4,+ nike, Kit1 = It, Lp 0 and {wen)in are given. Denote the optimal solution to this problem as a function of {wy,}, by Li({wen,}) = 1 (labor supply function), Ki({w.,n,}) (capital supply function), and YP({wyn}) := CP({wen]) + IP({won}) (good demand function). . Representative Firm Maximizes Its Profit: in each period , a representative firm max- imizes its profit given wages and interest rates {wertz- max Y, - w,L - r,K, subject to Y, = KFL)-#, (4) for * = 1,2,3. Denote the optimal solution to this problem as a function of {wnt- by LP ( {were}) (labor demand function), KP({w.,;}) (capital demand function), and Y({went}) (good supply function). . Markets Clear: The equilibrium wages and interest rates {wr, );_ clear labor mar- kets, capital markets, and good markets: LP ((won)) = Li(won)), KP((won:}) = Ki({war:}), and YP({won}) = Yf({won}) fort =1,2,3. (5) We will denote k, = K:/L, and c, = C/L. Note that, in equilibrium, L = 1 so that K, = K, and c = Cholds. Questions: (a) Characterize the utility maximization problem of (3) by first simplifying the maxi- mization problem to an unconstrained maximization problem by substituting L; = 1, It = Ki+1, and Ce = we + niki - Ki+1 and express as a maximization problem of choos- ing only {Ki+1), [Please refer to question (2) above]. Derive the first order condi- tions for this unconstrained maximization problem. (b) Characterize the solution to the firm's profit maximization problem (4) by the first order conditions. Can you explicitly derive LP({wer;}) (labor demand function), Ke ({were}) (capital demand function), and Y? ({w,,r}) (good supply function). (c) Using the first order conditions in (a) and (b) and using the market clearing conditions (5), show that the path of capital is given by ki+1 = ab 1 - (aB)4-(1+1) which is the same as the optimal path of capital (2) in planner's problem. [Hint: firm's first-order conditions implies that w, + nik, = ;. Given this, the first order conditions for (3) will imply the first order conditions for planner's problem (1).]Problem 3 {[Stokey, Lucas Jr, and Prescott(1989}] Based on Exercise 22: Optimal Growth). Corr sider the planner's problem: max tnq+slncz+s21nc3 (1) htkattlttztfa subjectto cr+k+1 g kf for t = 1,2,3 k; =0, k1 )0 ngiVEn, where and E [0,1]. ii is the discount rate. k1 is the initial capital given to the planner. It? is the output produced using capital kt. The output is divided into consumption c, and next period's capital km, where 100 percent depreciation of the current capital is assumed. (1} Argue that the optimal consumption level satises that c, = lq'.' k,+1. (2) Substitute c, = inf k,+1 into the above maximization problem and characterize the rst necessary order conditions with respect to kg and kg. (3) Write these rst order conditions using the change of variables 2; = kt/k','_1 to convert the result into a rstorder difference equation in 2,. Plot 21H against zr and plot the 45 degree line on the same diagram. (4) Noting that Z4 = 0, show that the unique solution is 1 _ 4] em = 2,3. (5) Check that the path for capital _ 4(t+'l) kt+1 = % (2) for t = 1,2 satises the rst order conditions you have derived in (2). (6) Consider the following economy: a Representative Consumer Maximizes Utility: Given the initial capital stock K0 and a sequence of wages and interest rates {w,,r,}=1, a representative consumer supplies one unit of labor L; = 1 and maximizes the discounted sum of her utility by choosing consumption (Cr) and investment Ir+1 given the income that consists of wage income
