Eco 504.2 Spring 2003 Chris SimsFINAL EXAM ANSWERSPart II(3) FTPL model: 45 points in total. Consider a model with representative agents who facethe problemmaxC,BE"∞Xt=0βtC1−γt1 − γ#subject to (3.1)Ct+BtQtPt= Yt+Bt−1Pt+Bt−1QtPt− τt(3.2)Bt≥ 0 . (3.3)Here C is consumption, B is the number of consols (perpetual nominal government debt)held, Q is the current nominal yield on consols. One consol pays one dollar per time periodforever. So Q is one divided by the price of a consol. τ is the level of per capita lum sumtaxes and P is the price level. Assume that Ytis i.i.d. and with probability one exceedssome¯Y > 0.The government faces the constraintBtQtPt=Bt−1Pt+Bt−1QtPt− τt. (3.4)Assume the government’s policy is to keep Qt≡¯Q and τt≡ ¯τ.(a) (7 points) Find the Euler equation first order conditions for the agents’ problem.After substituting out the Lagrange multiplier, we arrive at1CγtQtPt= βEt·Qt+ 1Cγt+1Qt+1Pt+1¸.(b) (7 points) Find the transversality condition(s) for the agents’ problem.The objective function is concave (so long as γ > 0, as is conventionally assumed)and the constraint is linear in the choice variables. the Lagrange multiplier is λt= C−γt,which is always positive. And Bt= 0 is always feasible, so long as ¯τ <¯Y . Under theseconditions the TVC is just the conventionalβtE·BtCγtQtPt¸→ 0 .The ¯τ <¯Y condition is quite restrictive. If it does not hold, this TVC is too strong, becauseit assumes a zero-debt path is feasible for the individual, but the conventional condition isstill a sufficient condition for optimality, even though less likely to be necessary.(c) (7 points) Determine conditions (if any) on the parameters of the problem under whichthere is a deterministic steady state solution, with real variables constant and nominalvariables growing at a common fixed exponential rate. The FOC and constraints implythat in such an equilibrium Ct≡¯Y , Bt/(¯QPt) ≡ ¯τ/(β−1− 1), and Pt+1/Pt≡ β(¯Q + 1).c°2003 by Christopher A. Sims. This document may be reproduced for educational and research purposes, so longas the copies contain this notice and are retained for personal use or distributed free.FINAL EXAM ANSWERS 2These equations always have a feasible solution, so long as B > 0. If B = 0, thennecessarily ¯τ = 0.(d) (12 points) Determine conditions (if any) on the parameters of the model under whichthere is a unique solution to the stochastic model in which the price level is determinateand real variables do not explode.Multiplying the budget constraint by 1/Cγtand applying the Et−1operator, we getEt−1Xt= Et−1·(¯Q + 1)Pt−1Cγt−1¯QPtCγt¸Xt−1− Et−1¯τCγt, whereXt=Bt¯QPtCγt.Using the FOC and the social resource constraint Yt= Ct, (obtained as usual by substi-tuting the government budget constraint into the private constraint), this becomesEt−1Xt= β−1Xt−1− E·¯τYγt¸. (∗)This is an unstable difference equation in EtXt+s, whose only stable solution isXt≡ E·¯τ(β−1− 1)Yγt¸=¯X .This stable solution will always exist, since¯Y > 0 implies Y−γt< 1/¯Yγ< ∞. This willdetermine Bt/Ptas a function of Yt, so long as Bt> 0. Then the government budgetconstraint will determine Ptuniquely from Ytand quantities already fixed at t, via¯X =(¯Q + 1)Pt−1Yγt−1PtYγtXt−1+¯τYγt.This equation always has a positive solution for Pt, so long as¯X > ¯τY−γt. Using thedefinition of¯X and some algebra, we can see that this will be true iff1 − βYγt< βE·1Yγt¸.For this to be always true, it must hold with¯Y substituted for Yton the left. This conditionis likely to be met with reasonable distributions for Y and β near 1, but it is not automatic.For any given distribution for Y , we can be sure it will not hold for β’s close enough tozero.(e) (12 points) Can explosive solutions to the Euler equations be ruled out as rationalexpectations competitive equilibria? Explain your answer.The explosive solutions to the difference equation (∗) we derived above make Xtgrowexponentially as β−t, which violates the TVC. This is only suggestive, not decisive, though,since the TVC is sufficient but not in general necessary for a solution. If γ < 1 and ¯τ <¯Y ,a complete argument is easy, since in that case, if B/QP gets large enough, the utilitygain from eating it all up now grows without bound, while the loss from forever thereafterbeing forced to set Ct= Yt− ¯τ has a finite discounted present value that does not growover time. If γ > 1 or ¯τ >¯Y , the situation is quite a bit more complicated. The utilitygain from eating a lot of debt now is bounded, while the utility loss from driving C close tozero at some point in the future is unbounded. No one gave a complete analysis of theseFINAL EXAM ANSWERS 3cases on the exam, and this was not counted against you. Recognizing that these casesexisted was “A” work.(4) Intergenerational tax burden shifting: 45 points in total. In class we discussed a modelwhere agents had a linear investment technology available and debt-financed expenditurecould shift the burden of taxation onto future generations. Here we consider a version ofthis model in which the timing of the taxes that pay off the debt is uncertain.Agents solvemaxC1(t),C2(t+1)Et£log¡C1(t)C2(t + 1)¢¤subject to (4.1)C1(t) + St+ Bt= Yt(4.2)C2(t + 1) = θSt+ RtBt− τt+1(4.3)Bt≥ 0 . (4.4)C1(t) is consumption of generation t while young. C2(t + 1) is consumption of generationt while old. Stis physical savings by generation t. θ > 1 is the rate of return to physicalaccumulation. Btis government debt purchased by generation t. τtis lump-sum taxes paidby generation t − 1 during the second period of life. Population is constant.The government budget constraint isBt+ τt= Rt−1Bt−1+ gt. (4.5)The time path of expenditures gtis given: g0= ¯g, gt= 0 for t > 0. The financingscheme is that τt= 0 at all dates except a single randomly chosen date t∗between 1and T . At dates t < t∗, the probability of a tax next period, given information at t, isP [τt+1> 0 | It] = 1/(T − t). When the tax is non-zero, it is set at τt∗= Rt∗−1Bt∗−1. Thatis, at some random date between 1 and T − 1, there will be a tax imposed on the old thatwipes out the government’s debt obligation.(a) (20 points) Find the equilibrium time paths of C1, C2, and B. Determine what boundson ¯g and T are necessary for existence of equilibrium.The agent’s