Econ 510a (second half)Yale UniversityFall 2006Prof. Tony SmithHOMEWORK #2This homework assignment is due at 5PM on Friday, November 10 in Marnix Amand’s mailbox.1. Consider a consumer with the following sequence of budget constraints:ct+ at+1= Rtat+ wt, t = 0, 1, 2, . . . ,where wt≥ 0 is the consumer’s labor income in period t. The consumer’s consumptioncannot be negative. The period-t gross interest rate Rtis greater than 1 for all t.(a) Find the no-Ponzi-game restriction and use it to derive a consolidated, or in-tertemportal, budget constraint for the consumer. Interpret your answer.(b) Suppose that wt= w > 0 and Rt= R > 1 for all t. Show that the nPg restrictionis equivalent to imposing a constraint that the consumer’s asset holdings never fallbelow a fixed amount B, where B is allowed to be negative. In other words, showthat there is a borrowing limit B such that the s et of f easible consumption levelsdefined by the sequential budget constraints and the nPg restriction is identical tothe set of feasible consumption levels defined by the sequential budget constraintsand the borrowing constraint at+1≥ B for all t. Express B in terms of R and w.(Hint: Imagine a consumer who has zero consumption and whose asset holdingsdo not change ove r time.)(c) Suppose again that wtand Rtare time-varying. What would the borrowingconstraint have to look like in order to obtain a result like the one in part (b)?2. Consider a consumer with the following optimization problem:max{ct, at+1}∞t=0∞Xt=0βtu(ct), given a0> 0,subject to the sequence of budget constraints in the first problem and a no-Ponzi-game restriction. The felicity function u is strictly increasing, strictly concave, twicecontinuously differentiable, and satisfies the Inada condition limc→0u0(c) = ∞.(a) Find the transversality c ondition for this problem. Show that the nPg restrictionis met if the transversality condition and the Euler equation are both satisfied.(b) Modify the proof of Proposition 3.4 on p. 23 of the lecture notes by Per Krusellto prove that a s equence {a∗t}∞t=0that satisfies the transversality condition andthe Euler equation maximizes the consumer’s objective, subject to the sequenceof budget constraints and the nPg restriction. (Note that the proposition inthe lecture notes imposes the requirement that the consumer’s asset holdings benonnegative in each period; in this problem, we are imposing instead the nPgrestriction.) Before considering the general case in which labor income and theinterest vary over time, you might want to study the special case in which theyare constant.3. Consider an exchange economy with two consumers named A and B. The two con-sumers have identical preferences: they each value consumption streams according toP∞t=0βtlog(ct). Consumer i’s endowment of consumption goods is {ωit}∞t=0, i = A, B.Consumption goods are perishable (i.e., they cannot be stored and used for consump-tion in future periods).(a) Carefully define a competitive equilibrium with date-0 trading for this economy.(b) Suppose that ωAt= 4 for all t and ωBt= 1 for all t. Find the competitiveequilibrium allocations and prices.(c) Suppose now that the endowments fluctuate deterministically: consumer A’s en-dowment stream is {4, 1, 4, 1, 4, 1, . . .} and consumer B’s endowment stream is{1, 4, 1, 4, 1, 4, . . .}. Find the competitive equilibrium allocations and prices.(d) In parts (b) and (c) there is no variation in the aggregate endowment acrosstime. Suppose that, as in part (b), consumer A’s endowment is 4 in everyperiod but that consumer B’s endowment fluctuates: his endowment streamis {1/2, 2, 1/2, 2, 1/2, 2, . . .}. Find the competitive equilibrium allocations andprices.(e) The social planning problem for this economy is:max{cAt}∞t=0, {cBt}∞t=0(αA∞Xt=0βtlog(cAt) + αB∞Xt=0βtlog(cBt)),subject to the resource constraint cAt+ cBt= ωAt+ ωBtfor all t. The numbers αAand αBare called Negishi weights. For each of the pairs of endowment streams inparts (b), (c), and (d), show that the consumption allocation chosen by the plan-ner coincides with the allocation that arises in competitive equilibrium, providedthat the weight αiis set equal to the inverse of consumer i’s marginal utility ofincome in competitive equilibrium.(f) Carefully define a competitive equilibrium with sequential trading for this econ-omy. Use your results from parts (b), (c), and (d) to determine the equilibriuminterest rates for each pair of endowment streams. In addition, for each casedetermine how each consumer’s asset holdings vary over time (assume that eachconsumer starts with zero assets in perio d