Econ 510a (second half)Yale UniversityFall 2005Prof. Tony SmithHOMEWORK #3This homework assignment is due at the beginning of class on Wednesday, November 9.1. Consider a consumer with the following optimization problem:max{ct, at+1}∞t=0∞Xt=0βtu(ct), given a0> 0,subject to:ct+ at+1= Rat+ w, t = 0, 1, 2, . . . ,and the no-Ponzi-game (nPg) restriction thatlimt→∞at+1Rt≥ 0.Assume that the gross interest rate R is greater than one, that the wage w is positive,and that the discount factor β ∈ (0, 1). The felicity function u is strictly increasing,strictly concave, twice continuously differentiable, and satisfies an Inada condition:limc→0u0(c) = ∞.(a) Find the transversality condition for this problem. Show that the nPg restrictionis met if the transversality condition and the Euler equation are both satisfied.(b) Modify the proof that we discussed in lecture (see also pp. 23–24 in the lecturenotes) to prove that a sequence {a∗t}∞t=0that satisfies the transversality condi-tion and the Euler equation maximizes the consumer’s objective, subject to thesequence of budget constraints and the nPg restriction.2. This problem considers a two-period economy similar to the one that you studied inthe third problem on Homework #1, but under the assumption that firms, ratherthan consumers, control the capital stock. Firms, in turn, are owned by consumers.In particular, consumers own shares in the firm; each share entitles its owner to afraction of the firm’s profits. Normalize the total number of shares to one, so that eachof the identical consumers (whose total measure is also normalized to one) owns oneshare. In addition, in period 0 consumers can trade riskfree bonds (i.e., claims to theconsumption good in the next period) in a competitive market.The (representative) consumer takes prices as given and seeks to maximize his lifetimeutility of consumption:maxc0, c1, b1u(c0) + βu(c1)subject to:c0+ q0b1= w0+ s0π0+ b0andc1= w1+ b1+ s1π1where q0is the price of a bond in period 0, πtis profits per share in period t, btis theamount of bonds owned by the consumer in period t, and stis the amount of sharesowned by the consumer in period t. Assume that, initially, each consumer holds nobonds: b0= 0. As discussed above, each consumer owns one share in the firm in eachtime period: s0= s1= 1. Consumers are also endowed with unit of time in each timeperiod, which they supply inelastically in a competitive labor market.The (representative) firm takes prices as given and seeks to maximize the net presentvalue of profits:maxn0, n1, k1y0− w0n0− k1+ q0(y1− w1n1),where yt= f(kt, nt) is the firm’s output in period t, ktis the amount of capital ownedby the firm at the beginning of period t, ntis labor demand in period t, and wtis thewage in period t. The firm discounts profits in period 1 using the price q0of period-1consumption goods (relative to the price of period-0 consumption goods). Assumethat each firm is endowed with the same amount of capital in pe riod 0 and that theproduction function f exhibits constant returns to scale.In equilibrium, the markets for labor, bonds, and goods must clear: in particular,nt= 1, b1= 0, and total output in period t is either consumed or invested in period t(note that investment in period 1 is zero because the economy ends).(a) Show that the competitive equilibrium allocation for this economy is identical tothe one that obtains when consumers control the capital stock directly (as in thethird problem on Homework #1). As part of your answer, determine the price q0in terms of the primitives u, f, and k0. (You do not have to explicitly solve forq0, but you have to explain how to compute it.)(b) Suppose now that a market for shares in the firm is introduced into the economy.In particular, a typical consumer’s budget constraints now read:c0+ q0b1+ p0s1= w0+ b0+ (p0+ π0)s0andc1= w1+ b1+ s1π1,where p0is the price of a share in period 0. Each consumer is endowed withone share in period 0 (i.e., s0= 1), but he is free to make any choice for s1. Inequilibrium, the market for shares must clear: s1= 1. Show that the introductionof this additional market does not change the allocation from the one in part (a).As part of your answer, determine the price p0in terms of the primitives u, f ,and k0. Interpret.(c) Suppose now that the government taxes income from profits at a proportionalrate τ and returns the proceeds to consumers in a lump-sum fashion (so that itsbudget balances in each period). Is the competitive equilibrium allocation in thiseconomy Pareto optimal? Explain why or why not.3. Consider an exchange economy with two (types of) consumers. Each type of consumerhas measure one, so that the total population of consumers equals two. The two(types of) consumers have identical preference s: they each value consumption streamsaccording toP∞t=0βtu(ct), where u has constant elasticity of substitution σ−1, i.e.,u(c) = (1 − σ)−1(c1−σ− 1). A consumer of type-i, where i = 1, 2, is endowed with asequence of (perishable) consumption goods denoted by {ωit}∞t=0.(a) Carefully define a competitive equilibrium with date-0 trading for this economy.(b) Suppose that ω1t= 2 for all t and ω2t= 1 for all t. Find the competitive equilibriumallocations and prices. (Hint: Your answers to all of the questions in this problemshould depend only on β and σ. Before you consider the general case, it mightbe helpful to consider the special case σ = 1, i.e., logarithmic utility.)(c) Suppose now that the consumers’ endowments fluctuate deterministically: theendowment stream of a type-1 consumer is {2, 1, 2, 1, 2, 1, . . .} and the endow-ment stream of a type-2 consumer is {1, 2, 1, 2, 1, 2, . . .}. Find the competitiveequilibrium allocations and prices.(d) In parts (b) and (c) there is no variation in the aggregate endowment across time.Suppose that, as in part (b), the endowment of a type-1 consumer is 2 in everyperiod but that the endowment stream of a type-2 consumer fluctuates deter-ministically: it is given by {0, 1, 0, 1, 0, 1, . . .}. Find the competitive equilibriumallocations and prices.(e) State the s ocial planning problem for this economy. For each of the pairs of endow-ment streams in parts (b), (c), and (d), find the Pareto weights (i.e., the weightson each of the two types of consumers) that deliver the competitive equilibriumallocation as the solution to the planning problem. Interpret the (relative) sizesof the weights.(f) Carefully