Final Exam ARE 261 (2002)Answer all parts of all three questions. The value of the question isgiv e n in p arentheses next to th e q uestion n u mber. Don’t spend more than 3hours o n the exam. If you feel pressed for time, outline your answer withoutproviding details.Questio n 1) (40 points) S is an index of environ m ental qualit y and x isthe flow of pollution. The ev olution of S is giv en by˙S = x − g(S; α),wheregisincreasingandconcaveinS,andα is a parameter of the function g. Theinstantaneous (flow) payoff is x −x22-S22and the instantaneous discount rateis r. You want to maximize the presen t discoun ted integ ral of the flow ofwelfare from time 0 to infinity. The initial cond ition S0is give n.(i) Write the current value Ham iltonian and necessary conditions.(ii) Ske tch the phase portrait in (S, x), space. The portrait should includethe0-isoclines,thedirectionalarrows,andthestablesaddlepath(theoptimalcon trol rule).(iii) Write the equations that determine the optimal steady state.(iv) Explain ho w to find the comparative statics of the steady state withrespect to α. You should set up the comparativ e statics expression and ex-plain how you use the stabilit y of the steady state in findin g the compara tivestatics. (It is not necessary to carry out all of the calculations — just explainclearly how you w ould proceed.)(v) Find the differen tial equation that the optimal control rule, x∗(S),must satisfy. (Write the expression fordx∗dS) Explain one method of nume ri-cally solving this ODE.Questio n 2) (40 points) Let Ktbe the stock of capital in a period, and letπ(Kt) be the restricted profit function. The cost of investmen t in period t isan increasing convex function of in vestment, c(It).Thefirm has a discountfactor of β and it wants to max imize∞Xt=0βt(π(Kt) − c(It))subject to Kt+1= It− δKt, K1giv e n.(i) Write the Dynamic Programming equation for this problem .1(ii) Describe an algorithm for solving the DPE.(iii) Derive the Euler Equatio n associated with this problem(iv) Interp ret the Euler Equation.(v) Find an expression that the steady state stoc k of capital m ust satisfy.(vi) Interpret the condition that determines the steady state stock ofcapital.(vii) Ho w does a decrease in the depreciation rate (an increase in δ)affectthe steady state stock of capital?Pro blem 3 (20 poin ts) Co nsid er the following utilit y maximization prob-lem:MaxcTXt=0βtln (Ct)subject toKt+1=(Kt− Ct)α,α>1.K0given; Kt≥ 0, Ct≥ 0, for all t. Here Ctis the consumption at time t,and Ktis the stock of money at time t. Derive the optim a l consum p tionrule and show ho w it c hanges as the “time to go”, T − t (where t is calenda rtime)
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