University of California, BerkeleyEconomics 201AFall 2001 Second Midterm Test–Dece mber 11, 2001Instructions: You have three hours to do this test. The test is out of a totalof 300 points; allocate your time accordingly. Please write your solution toeach question in a separate bluebook.1. (100 points) Define or state and briefly discuss the importance of eachof the following within or for economic theory:(a) Kakutani’s Fixed Point Theorem(b) Lebesgue measure zero(c) Core of an exchange economy(d) First Welfare Theorem in an Arrow-Debreu economy(e) Index Theorem2. (80 points) Consider an Edgeworth Box economy, whereω1=(2, 1) ω2=(1, 2)u1(x11,x21)=√x11x21u2(x12,x22)=√x12x22(a) Find a Walrasian equilibrium.(b) Show that the allocation x1=(1, 1), x2=(2, 2) is Pareto opti-mal. Without using the Second Welfare Theorem, show that thisallocation is a Walrasian equilibrium with transfers.13. (120 points) Consider the function z :∆0× R → R2defined byz(p, α)=1p1+ α cos(2πp1), −1+αp1cos(2πp1)p2Note that cos(0) = 1 andddxcos x = −sin x.(a) For what values of α does the function zα(p)=z(p, α)satisfytheconditions of the Debreu-Gale-Kuhn-Nikaido Lemma?(b) For what values of α does there exist p ∈ ∆ such that z(p, α)=0?(c) Show that for every α ∈ R and every ε>0, there is an exchangeeconomy with two agents whose excess demand function agreeswith zαon {p ∈ ∆:p1∈ [ε, 1 −ε]}.(d) Show that there is a set A ⊂ R of Lebesgue measure zero such thatfor every α ∈ A, the economy with excess demand zαis
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