Final exam, ARE P 263Fall 2006You ha ve three hours to work on this exam — closed book, closed notes.Please retur n the exam to me or to m y mailbox in 207 Giannini by 5 PM onFriday.Answer all questions. You canno t avo id using m ath em atics to answerthese questions. How ever, DO NO T GET BO G G E D DOWN IN ALGE B RA.It is sufficien t to provid e a clear stateme nt of the algebraic man ipulations thaty ou w ould perform, and the nature of the result that y ou w ould obtain.I hope that there are no typos or ambiguities in the follo w in g questions. Ifyou find a problem of this sort, just be sure to let me kno w what assumption syou have made.Questio n 1. Con sider the following two-sector model, with total stoc k oflabor = 1. T he amount of labor in the manufacturing sector at time t isequal to L(t). Th e va lue of output in the manufacturing sector is Lf(L)with f0(L) > 0 The increasing returns to scale in the sector are external tothe firm, so each worker receiv es the w age w = f(L). The v alue of outputin the agricultura l sector is (1 − L)a; the agricultural wage is the constanta.Assumethatf(0) <a<f(1).Theflow of labor into the manufacturingsector is m. The social cost of migration is γm2/2,andeachworkerwhoc h anges sectors pays the migration cost γ |m|. The constant discount rateis r.a) (Economic interpretation) (i) How would the model c hange if theincreasing returns to scale were in ternal to the firm ? (ii) W ha t is the impli-cation of the assumption that f(0) <a<f(1)?b) Discuss the qualitativ e solution to the optimization problem of thesocial planner who is able to directly choose migration in order to maximizethe present discounted value of output min us adjustment costs.c) Discuss the qualitative solution to the equilibrium prob lem of (non-strategic) wo rkers with rational poin t expectations, each of whom decides(at each poin t in time) whether to migrate. A w orker w ants to maximize1the presen t discoun ted value of wages min us adjustmen t costs. (Your answershould explain what you mean b y “rational point expectations”.)d) Can you design a tax policy that causes the competitive equilibriumto produce the socially optimal program?e) Describe an algorithm (or algorithms) for solving the optimizationproblem and the equilibrium problem . What (if an y ) differences betweenthese tw o problems do y ou need to tak e into accoun t?Sketch of answ ers:a) i) A "stand ar d" com petitiv e equilibriu m does not exist, because if firmshave in ternal IRTS and take the wage as given they want to increase theirsize without bound. ii) The assum ption means that a corner solution is astable steady state.b) I w anted y ou to set up the problem and recognize that this is a "Skibaproblem", and discuss how to solv e it.c) This is just Krugman’s model, with a more general wage function (finstead of a linear function).d) Might be possible, but not ob vio us in the case where there are m ultip lerationa l expectations equilibria.e) For the optimization problem (when there are two candidate trajecto-ries) you could proceed in a couple of w ays. Either approximate the valuefunctions under the t wo candidate decision rules and compare them, or ap-pro ximate the costate v ariable under the t wo candidate decision rules anduse the Skiba technique. For the equilibrium problem y ou need to solv e therational expectations equilibrium that tak es the econo my to either steadystate, and find the domains of these equilibria. Rember that the conditionunder which there are t wo can didate s in the optim iz ation problem, or tworationa l expectations equilibria, are different.Questio n 2 Consider the follo w in g t wo problems:Problem A: You ha v e a three-stage (two decision periods) cake eatingmodel. In each period the utilit y fro m eating c units is U(c). The total sizeof the cak e is 1, so c1+ c2+ c3≤ 1. The one-step ahead discoun t rate is r1and the two-step ahead discount rate is r2. (These are known numbers.) T hecorresponding one-step and t wo-step discount factors are βi=11+ri.Thatis,the present value at time 1 of a unit of cake at time 3 is β1β2; the present2valueattime1ofaunitofcakeattime2isβ1; the present value at time 2of a unit of cak e at time 3 is β1(NO T β2). Th e decision-maker at time 1wan ts to maximize the presen t discounted valu e of the stream of utility.Problem B: You have a three-stage (two decision periods) cake eatingmodel. In each period the utilit y fro m eating c units is U(c). The total sizeofthecakeis1,soc1+ c2+ c3≤ 1. The discount rate in period i isri= ri−1(1 + εi)where εiare independently and identica lly distributed random variables withmean 0 and support (−1, 1).Inperiodi,whenchoosinghowmuchcaketoeat, the decisionmaker know s r1−1. Thepresentvalueattime1ofaunitofcake at time 2 is β1;thepresentvalueattime1ofaunitofcakeattime3is β1β2; the presen t v alue at time 2 of a unit of cak e at time 3 is β2(NOTβ1). The decision-maker at time 1 wants to maxim ize the expected presen tdiscounted value of the stream of utility.a) Discuss the differen ces in assum p tions between these t wo models, andexplain the economic significance of these differen ces. (I see t wo importan tdifferen ces in assump tions. I w ant you to tell me ho w these differences reflectdifferen ces in the underlying story that the models are intended to describe.)b) Explainhowyouwouldfind the decision rules, and the first perioddecision, in these tw o models. (Tell me what problems y ou need to solv e;I’m not asking for an algorithm that produces a numerical solution.)Sketch of answ ersa)InproblemAthediscountrateforaperioddependsontheamountof time un til the period arrives. This model can be used to describe thetempta tion to procrastinate. In this case, the trajectory that is optimal forthe decisionmak er in the first period is tim e inconsistent. The decisionmakerdoes not learn an ything as time goes on; but the amount of cak e that theagen t in period 1 w an ts to ha ve eaten is not the same as the amoun t of cakethat the agen t in period 2 w ants to eat. In problem B the discount rate isstoc ha stic, but the discount rate for a period is not a function of the amou ntof time until that period. In this case, there is no time inconsistency, butthe decision maker should recognize that she will get more informa tion inthe second period.b) For both problems you need to wo rk backwards. In problem A, the3second period problem giv esc∗2(c1)=argmaxU (c2)+β1U