Christiano411, Fall, 2008FINAL EXAMEach of the four questions is w o rth 25 points. Allocate your time accord-ingly. If a question s eems ambiguous, state why, sharpen it up and answerthe revised question. Good luc k!1. Consider the sequence represen tation of the follo wing two-sector plan-ning problem:max{ct,it,k1,t+1,k2,t+1,l1,t,l2,t}∞t=0∞Xt=0βtu(ct)subject toct≤ ztF (k1,t,l1,t)it≤ qtztF (k2t,l2t).Here, kitand litare capital and labor allocated to sector i, i =1, 2.Assume that factors can be freely moved betw een sectors, subject to:kt≡ k1,t+ k2,t,l1,t+ l2,t= l,where ktis the aggregate stock of capital given at the beginning of timet and l is the (fixed) amount of labor effort supplied by households.Finally, we also requirekt+1,ct≥ 0,k0> 0and the identitykt+1≤ (1 − δ)kt+ it.The sequences {zt,qt}∞t=0are exogenously given. It is assumed that uis continuously differentiable, strictly increasing, and strictly concave,that F is continuously differentiable, strictly increasing in both argu-ments, homogeneous of degree one, and strictly quasiconcav e, and thatδ, β ∈ (0, 1).1(a) Show that a necessary condition for optimization isk1tl1t=k2tl2t, andthat this implies the constraint set above can be replaced b yct+itqt≤ ztF (kt,l)kt+1=(1− δ)kt+ itand k0> 0ct,kt+1> 0.(b) Assume that the two productivity ch ange series follow:zt= γtzand qt= γtq,where γz6= γqare each greater than one. Suppose F has a Cobb-Douglas form, and u(c)=c1−σ/(1 − σ),σ<1. Let a steady-stategrowth path be a situation in which ct,kt, and itare gro wing ata constant rate. Let γkand γcdenote the steady-state gro wthrates of the capital stock and consumption, respectively. Developequations that determine γkand γcas a function of the parametersof the problem.(c) Scale the variables and show that the sequence represen tation ofthe planning problem can be expressed as a problem involving nogrowing variables.(d) Express the non-growing, scaled, representation of the planningproblem in functional equation form.(e) Write the competitiv e equilibrium that corresponds to the scaledplanning problem as a sequence of markets competitive equilib-rium.(f) Write the competitive equilibrium in recursive competitive equi-librium form.22. Consider an econom y in which there are two types of households, cap-italists and workers. Capitalists accumulate capital, but have no laborpower. Workers supply labor, but have no access to capital markets.There are two periods. In the first period, capitalists have an endow-ment, ω, which they can accumulate in the form of capital, k, or theycan consume, c1.Theirfirstperiodbudgetconstraintis:c1+ k ≤ ω.Theirsecondperiodbudgetconstraintis:ck2≤ R(1 − δ)k,where R is the rental rate of capital (an exogenous parameter of themodel) and δ is the capital income tax rate. The lifetime utility ofcapitalists is:uk(c1,ck2)=c1+ ck2We assume that if the capitalist is indifferent between consuming inperiods 1 or 2, then they choose to do all their consumption in period2.Considertheworkers. Inperiod1,theyhavenoutility. Inperiod2,theirbudgetconstraintis:cw2≤ (1 − τ)l,where τ isthelabortaxrateandthewagerateissettounity. Workers’utility function isuw(cw2,l)=cw2−12l2.The social welfare function in this society is:u(uw,uk)=uw+ uk.Suppose the government faces an exogenously determined required levelof spending, g, where(R − 1)ω<g<(R − 1)ω +14The government’s budget constraint is:g ≤ τl+ δRk.3(a) Consider the best equilibrium, relative to the giv en social welfarefunction. Explain carefully why the labor tax rate in the best equi-librium must be positive. Call the policies in the best equilibrium,the Ramsey policies.(b) Suppose the task of administering government policy is given to anadministrator who is benevolent in the sense that he is in terestedin maximizing the social welfare function. Suppose this personmust impose the policy during an ‘administration period’, whichoccurs at the beginning of period 2, before the labor decision hasbeen taken. Prove that this administrator will deviate from theRamsey policies. Provide intuition.(c) Suppose everyone understands that policy will be implemented b ythe benevolent administrator in (b). Define a sustainable equilib-rium for this economy, and explain the outcomes that occur insustainable equilibrium.(d) Suppose the task of administering governm ent policy in the admin-istration period is given to an administrator who is not benevolent.The administrator has preferences:u(uw,uk; λ)=uw+ λuk.An administrator with preferences λ>1 is partial towards cap-italists. Explain why it is that there is a λ>1, such that anadministrator with this value of λ would choose, in the adminis-tration period, not to deviate from the Ramsey policies.(e) Given a choice bet ween the sustainable equilibrium outcomes de-scribed in (c) and the outcomes in (d), is it possible that workersmight prefer an administrator that is partial to capitalists, if theywere aske d at the beginning of period 1? Explain.43. Consider an economy in which the representative household has thefollo wing preferences:∞Xt=0βtu(ct,nt), 0 <β<1,where u satisfies the usual restrictions. The resource constraint is:ct+ kt+1− (1 − δ)kt≤ yt,and 0 <δ<1. Final goods, yt, are produced using the linear homoge-neous technology:yt=∙Z10xt(i)λdi¸1λ,λ>1,where xt(i) is the quantity of the ithintermediate good used. Thetechnology for producing interm ediate goods isxt(i)=kt(i)μnt(i)γ,μ,γ>0, 1 <μ+ γ ≤ ψ for all i ∈ (0, ∞).Here, ψ is a parameter to be discussed below.(a) Decentralize this economy and define a symmetric, sequence ofmarkets equilibrium. In the equilibrium, give the final good tech-nology to a representative competitive firm; give the technology forproducing the ithintermediate good to a monopolist; and supposethe representative household ren ts capital and labor in homoge-neous, competitive markets. Only consider symmetric equilibria,in which all intermediate good firms behave identically. Let pt(i)denote the price of the ithintermediate good and let wtand rtde-note the wage rate and capital rental rate, respectively. All datet prices are denoted in units of the date t consumption good.(b) Derive an expression for the demand curve faced by the monopo-list. What happens to the slope of the demand curve (with pt(i)on the vertical axis and xt(i)onthehorizontal)asλ → 1? Provideintuition.(c) What restriction on ψ is