University of California, BerkeleyEconomics 201AFall 2000 Final ExaminationInstructions: You have three hours to do this examination. The exam isout of a total of 300 points; allocate your time accordingly. Please write yoursolution to each question in a separate bluebook.1. (75 points) Define or state and briefly discuss the importance of eachof the following within or for economic theory:(a) Brouwer’s Fixed Point Theorem(b) a theorem on the existence of approximate Walrasian equilibrium,when preferences are nonconvex(c) Index Theorem(d) a theorem concerning the cores of exchange economies with manyconsumers(e) incomplete markets2. (75 points) Tom Hanks is the sole owner of a firm with access to theproduction technologyY = {(y1,y2):y1≤ 0,y2≤√−y1}Hanks’ endowment is (1, 0). Hanks has the Cobb-Douglas utility func-tion u(x1,x2)=√x1x2. There are no other firms or consumers in theeconomy. Find a Walrasian equilibrium (p∗,x∗,y∗).13. (150 points) Consider a pure exchange economy with aggregate endow-mentω 0 in which every consumer has a “rational,” continuous,strictl y convex and strongly montone preference. A social planner im-poses a price-dependent income transfer scheme of the following form:Ti(p)=αp · ωI− p ·ωi(1)where α ∈ [0, 1] is a constant.1The budget set for agent i is Bi(p)={x ∈ RL+: p · x ≤ p · ωi+ Ti(p)}; the demand is Di(p)={x ∈ Bi(p):x∈ Bi(p) ⇒ x ix}.Letz(p)= Ii=1Di(p) −ω.(a) Show that for every α ∈ [0, 1], there exists p∗such that z(p∗)=0.(b) Show that if z(p∗)=0andx∗is the allocation given by x∗i=Di(p∗), then x∗is Pareto optimal. You m ay do this in either oftwo ways:i. Assume the First Welfare Theorem as stated in class. Youwill need to explain why it applies to this situation, with aprice-dependent income transfer.ii. Adapt the proof of the First Welfare Theorem to this situa-tion. If you choose this approach, you will get full credit forproving that x∗is Pareto optimal in the weak sense, that thereis no exact allocation xsuch that xiix∗ifor all i =1,...,I.(c) Suppose the preferences are such that demand is a C1functionof price and α. Using the transversality theorem, show that foralmost all (α, ω), the re sulting economy is regular.(d) Suppose there are exactly two consumers in the economy. Whatdoes the Second Welfare Theorem tell you about the ability toachieve a given Pareto optimum using a transfer of the type de-scribedinEquation(1)? Hint: you will need to consider α ∈ R,not just α ∈ [0, 1].1This is a bit like an income tax, in the sense that the tax charged is proportional tothe value of each person’s endowment; the reve nue generated by the tax is then rebatedin equal amounts to each individual.2(e) Suppose the social planner imposes the following modified transferschemeTi(p, x)=αp · (ωi− xi)+− Ij=1p · (ωj− xj)+Iwhere x is an allocation and (y+)=max{y, 0}.2Let Bi(p, x),Di(p, x)andz(p, x) be defined as above, substituting Tifor Ti.Ifz(p∗,x∗)=0andx∗i= Di(p∗,x∗), does it follow that x∗is Paretooptimal?2This is closer to the income tax in practice in the United States, in that the tax appliesto the income generated by the sale of your endowment, but the consumption of your ownendowment (for example, living in a house you own that could be rented out, but isn’t; orthe devotion of time to leisure activities) is not taxed. As before, the revenue generatedby the tax is rebated in equal amounts to each individual.3***Note to me: the last question (except for the part about the modifiedtransfer Ti) is just like a situation in which each person is given endowmentαωI+(1−α)ωi. To see this, note that this endowment is always exactly on thebudget frontier for every price. Hence, existence and first welfare theoremfollow immediately from the same results in an exchange economy. Genericregularity follows from the fact that the measure induced on endowments byαωI+(1− α)ωiis mutually absolutely continuous with respect to