Christiano411, Winter 2005FINAL EXAMAnswer three of the following equally-weighted four questions. If a ques-tion seem s ambiguous, state why, sharpen it up and answer the revised ques-tion. You have 1 hour and 55 minutes. Good luck!1. Consider the Krusell-Rios Rull model, in whic h there are 4 possiblestrategies available to each agent at the beginning of her 3-period life(young, middle age, old): (1) be unskilled in each period, (2) be un-skilled in the same vin tage for the first 2 periods and then be skilled inthat technology in the 3rdperiod, (3) be unskilled in the first period,a fast learner in some vintage in middle age and skilled in that vintagewhen old, (4) be an innovator in the first 2 periods and then skilled inthe new technology in the last period. The availability of strategy (4)depends on whether innovation is, or is not, allowed. Whether innova-tion is allowed is determined by majority rule. There is a mass of 1 foreach generation, so that the total mass of agents at a given date t is 3.Agents discount the future at the rate, 0 <β<1, and maximize theirdiscounted income. The aggregate production function for technologyτ is Cobb-Douglas:fτ(ϕs,τ,ϕu,τ)=Aτϕαs,τϕ1−αu,τ,where ϕs,τand ϕu,τare the number of skilled and unskilled workers,respectiv ely, in vintage τ. P roductivity improves with vintage:AτAτ+1= γ.Consider the follo wing modification to the structure of the economyseen in class. Assume that innovators receiv e a transfer, µt, from thegovernment in each of the 2 periods of their life. The transfer is fi-nanced through labor taxes. If ωt(ϕs,τ,ϕu,τ) is the gross wage paidto the unskilled worker at time t in vintage τ, then the net wage is(1 − δ) ωt(ϕs,τ,ϕu,τ)whereδ is an exogenous constant and δ ∈ [0, 1].1(a) For given ϕs,τ,ϕu,τ, derive the gross wage paid to the unskilledworkers working in the best technology in a given period of time,t. Derive the net profits of a skilled worker in vintage τ at time t.(b) Derivethediscountedlifetimeincomeforthe4possiblestrategies.(c) Conjecture a stationary equilibrium where in any period t acon-stant fraction, ¯ϕ, of the young choose to innovate and develop newtec hnologies and the remainder, 1 − ¯ϕ, choose to remain unskilled.Derive the ratio of skilled to unskilled workers at each period oftime t as a function of ¯ϕ. Derive the wage of unskilled w orkers andthe profits of skilled workers as a function of this ratio. Derive thebudget constraint the government must satisfy in eac h period t.(d) Deriv e the equilibrium value of ¯ϕ as a function (α, γ, β, δ). Discussho w ¯ϕ changes as we increase δ. Provide economic intuition.(e) Derive conditions under which the conjectured equilibrium is ac-tually an equilibrium, i.e., make sure that no one would deviateand choose strategy (2) or (3).(f) Suppose the economy is in a no-innovation equilibrium at time t,and a vote is held on whether to allow innovation. (50% of votesin favor of innovation are needed to make innovation possible.) Ifinnovation is allowed, innovators get a subsidy as abo ve. Discussthe incentives each different agent in the economy has to vote infavor or against inno va tion as a function of her current and futureincome. In particular, how is the subsidy to innovation going toaffect this decision? Explain carefully.2. Suppose there are 2 types of agen ts. In period t, type 1 agent receivesendowment et, and type 2 agent receives endowment 1 − et. Suppose etcantakeononlytwovalues: 3/4and1/4. Let stdenote the exogenousuncertainty in period t, with st= et. Let stdenote the history ofexogenous uncertaint y from the first period, date 0, until period t :st=(s0,s1, ..., st).The probability of a particular history, st, is πt(st) .2A planner chooses a sequence of consumption allocations across the 2types by maximizing the following objective function:∞Xt=0Xstβtπt³st´hθ1u1³c1³st´´+ θ2u2³c2³st´´i,subject to the resource constraint:c1³st´+ c2³st´≤ 1, for all st,and the participation constraints:∞Xj=0Xst+j|stβtπt+j³st+j|st´u1³c1³st+j´´≥∞Xt=0Xstβtπt+j³st+j|st´u1³e³st+j´´≡ V1³st´∞Xt=0Xstβtπt+j³st+j|st´u2³c2³st+j´´≥∞Xt=0Xstβtπt+j³st+j|st´u2³1 − e³st+j´´≡ V2³st´,for each st. The utility functions, u1and u2, are strictly concave andincreasing.(a) Let λi(st) ≥ 0 be the Lagrange multiplier on the ithagent’s par-ticipation constraint and µ (st) be the Lagrange multiplier on theresource constraint in history st. Write down the Lagrangian ver-sion of the planner problem.DefineMi³st´= θi+ λi³s0´+ λi³s1´+ ... + λi³st´,i=1, 2.Rewrite the planner’s lagrangian problem in terms of Mi(st). Usethis formulation to argue that Mi(st) can be interpreted as theweight the planner assigns to agent i.(b) Derive the first order conditions to the planner problem.Definez³st´=M2(st)M1(st),νi³st´=λi(st)Mi(st),i=1, 2.3Derive a law of motion for z (st)asafunctionofz (st−1) ,ν1(st)and ν2(st) . In terpret these results and in particular discuss howthe optimal allocations, c1(st)andc2(st) , change as z (st) changes.What does z (st) measure? Provide economic intuition on the re-lationship between λi(st)andz (st) .(c) Write down the recursive formulation of the problem and definecarefully the state variables. Carefully analyze the role of each ofthem.From now on, consider a particular (low probability!) history withthe property:et=3/4, for all t =0, 1, 2, ..., +∞.(d) Suppose that, given the above realization of etfrom t =0tot = t∗,the social planner has chosen allocations up to time t∗, such thatc1³st∗´< 3/4andc2³st∗´> 1/4. Prov e that the participationconstraint of one of the two agents will not be binding at t∗givenst∗. What about the participation constraint of the other agent?Will it bind? Provide economic intuition.(e) Given your results in (b)-(d), and given the realization of etandc1³st∗´, c2³st∗´specified above, would c1³st∗+1´and c2³st∗+1´begreater or smaller than, respectively, c1³st∗´and c2³st∗´?Providea proof and economic intuition for your answ er. (Hint: study howz (st) changes.)3. Consider the following 2 period economy. The representative householdmaximizesu(c1+ c2,l)subject to the following two budget constraints:c1+ k ≤ ωc2≤ (1 − δ) Rk +(1− τ ) ωl.Here, ω is the w age rate (which corresponds to the marginal productof labor), R denotes the rental rate of capital (its