example PO Walras sol - Class Advanced Microeconomics II...

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Unformatted text preview: Class Advanced Microeconomics II Solution Problem Set 12, Exercise 2 Spring 2008 1 Solution Problem Set 12, Exercise 2 2. Consider an Exchange Economy with three consumers whose consumptions are all ∈ R 2 + . The consumers have the following utility functions: U 1 ( x 1 ) = ln x 1 1 + ln x 1 2 , U 2 ( x 2 ) = q x 2 1 · q x 2 2 , U 3 ( x 3 ) = 7 · x 3 1 · x 3 2 . The consumers endowments are: e 1 = (1 , 6) , e 2 = (3 , 2) , e 3 = (5 , 1). Focus on the following allocations: x 1 = ((2 , 3) , (2 , 4) , (5 , 2)) , x 2 = 31 10 , 31 10 , (3 , 3) , 29 10 , 29 10 , x 3 = 7 2 , 7 2 , 5 2 , 5 2 , (3 , 3) . (a) Check, whether these allocations are Pareto-efficient or not. (b) Compute the WEA. Solution (a) We know, that any (interior) Pareto-efficient allocation/Pareto-optimum (PO) requires: MRS 2 / 1 1 ! = MRS 2 / 1 2 ! = MRS 2 / 1 3 . We compute the three MRS 2 / 1 : 1 MRS 2 / 1 1 = 1 x 1 2 1 x 1 1 = x 1 1 x 1 2 , MRS 2 / 1 2 = (1 / 2)( x 2 1 ) 1 / 2 ( x 2 2 )- 1 / 2 (1 / 2)( x 2 1 )- 1 / 2...
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example PO Walras sol - Class Advanced Microeconomics II...

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