ProblemSet9_2009

# ProblemSet9_2009 - 1> 0 then x 1 2 = 6 and x 2 2 = 9(b...

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Econ 6202, Fall 2009 Dmitry Shapiro Problem Set 9 Due Tuesday December 1 1. (JR 5.11) Consider a two-consumer, two-good exchange economy. Utility functions and endowments are: u 1 ( x 1 ,x 2 ) = ( x 1 x 2 ) 2 and e 1 = (18 , 4) , u 2 ( x 1 ,x 2 ) = ln( x 1 ) + 2ln( x 2 ) and e 2 = (3 , 6) . (a) Characterize the set of Pareto-eﬃcient allocations as completely as possible. (b) Find the Walrasian equilibrium prices and allocations. 2. (JR 5.13) An exchange economy has two consumers with expenditure func- tions: e 1 ( p ,u ) = £ 3(1 . 5) 2 p 2 1 p 2 exp( u ) / 1 / 3 , e 2 ( p ,u ) = £ 3(1 . 5) 2 p 1 p 2 2 exp( u ) / 1 / 3 . Suppose that the initial endowments are e 1 = (10 , 0) and e 2 = (0 , 10). Find the Walrasian equilibrium. 3. Consider a two-consumer two-good exchange economy with externalities. Utility functions and endowments are: u 1 ( x 1 1 ,x 1 2 ,x 2 2 ) = x 1 1 + 2ln( x 1 2 ) + ln( x 2 2 ) and e 1 = (12 , 10) , u 2 ( x 2 1 ,x 2 2 ) = x 2 1 + 2ln( x 2 2 ) and e 1 = (8 , 5) . (a) Prove that if x = (( x 1 1 ,x 1 2 ) , ( x 2 1 ,x 2 2 )) is a Pareto eﬃcient allocation with x 1 1 > 0 and x 2
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Unformatted text preview: 1 > 0 then x 1 2 = 6 and x 2 2 = 9. (b) Find the Walrasian equilibrium prices and allocation. Is the equilib-rium allocation Pareto eﬃcient? Explain. (c) Consider the eﬃcient allocation x * = ((13 , 6) , (7 , 9)). Suppose that the government can impose a lump-sum tax T i to consumer i = 1 , 2. Furthermore, the government can subsidize consumer 2’s consump-tion of commodity 2 with a (Pigouvian) subsidy. Les s denote the per-unit subsidy (i.e. consumer 2 receives s for each unit of commod-ity 2 that he buys). Find T * 1 ,T * 2 and s * such that the Walrasian equi-librium allocation is x * . Specify the Walrasian equilibrium prices. Is the government’s budget constraint satisﬁed with equality? Explain. 1...
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