201Fall01Exam

201Fall01Exam - 1 3(120 points Consider the function z ∆...

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University of California, Berkeley Economics 201A Fall 2001 Second Midterm Test–December 11, 2001 Instructions: You have three hours to do this test. The test is out of a total of 300 points; allocate your time accordingly. Please write your solution to each question in a separate bluebook. 1. (100 points) DeFne or state and briefy discuss the importance of each of the following within or for economic theory: (a) Kakutani’s ±ixed Point Theorem (b) Lebesgue measure zero (c) Core of an exchange economy (d) ±irst Welfare Theorem in an Arrow-Debreu economy (e) Index Theorem 2. (80 points) Consider an Edgeworth Box economy, where ω 1 =(2 , 1) ω 2 =(1 , 2) u 1 ( x 11 ,x 21 )= x 11 x 21 u 2 ( x 12 ,x 22 )= x 12 x 22 (a) ±ind a Walrasian equilibrium. (b) Show that the allocation x 1 =(1 , 1), x 2 =(2 , 2) is Pareto opti- mal. Without using the Second Welfare Theorem, show that this allocation is a Walrasian equilibrium with transfers.
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Unformatted text preview: 1 3. (120 points) Consider the function z : ∆ × R → R 2 deFned by z ( p, α ) = 1 p 1 + α cos(2 πp 1 ) , − 1 + αp 1 cos(2 πp 1 ) p 2 ! Note that cos(0) = 1 and d dx cos x = − sin x . (a) ±or what values of α does the function z α ( p ) = z ( p, α ) satisfy the conditions of the Debreu-Gale-Kuhn-Nikaido Lemma? (b) ±or 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 exchange economy with two agents whose excess demand function agrees with z α on { p ∈ ∆ : p 1 ∈ [ ε, 1 − ε ] } . (d) Show that there is a set A ⊂ R of Lebesgue measure zero such that for every α 6∈ A , the economy with excess demand z α is regular. 2...
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This note was uploaded on 08/01/2008 for the course ECON 201B taught by Professor Anderson during the Spring '06 term at Berkeley.

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201Fall01Exam - 1 3(120 points Consider the function z ∆...

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