2011HW4

# 2011HW4 - n X i =1 ¯ x i + c (¯ y ) = n X i =1 ω ix . 3....

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Economics 210C Problem Set #4 (Due June 3) Spring 2011 1. Find the Pareto optimal and CE allocations for an economy with one consumer, one ﬁrm, and two goods, in which u ( x ) = ( x 1 + 3) x 2 , ω = ( a, 0) , Y = ± y : y 1 0 , 0 y 2 ( - y 1 ) b b ² , a > 0 , 0 < b < 1 . 2. Suppose there are n individuals. There is a private good which is taken to be the numeraire and a public good. Providing an amount y of the public good costs c ( y ) units of the private good, where c is an increasing and convex cost function satisfying lim y -→ 0 c 0 ( y ) = 0 and lim y -→∞ c 0 ( y ) = . Individual i has a utility function u i ( x i ,y ) = x i + v i ( y ). For each individual i , v i is a strictly increasing and strictly concave function with lim y -→ 0 v 0 ( y ) = and lim y -→∞ v 0 ( y ) = 0. Show that an allocation ((¯ x i ) n i =1 , ¯ y ) is Pareto optimal if and only if ¯ y solves max y : y 0 ,c ( y ) n i =1 ω ix n X i =1 v i ( y ) - c ( y ) and ((¯ x i ) n i =1 , ¯ y ) satisﬁes

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Unformatted text preview: n X i =1 ¯ x i + c (¯ y ) = n X i =1 ω ix . 3. Consider a simple economy with 2 consumers, one private good x , and one public good y . Consumer 1 has endowment ω 1 = ( ω 1 x ,ω 1 y ) = (100 , 0) and utility function u 1 ( x,y ) = 2 √ y + x. Consumer 2 has endowment ω 2 = ( ω 2 x ,ω 2 y ) = (200 , 0) and utility function u 2 ( x,y ) = 2 √ y + x. Each unit of the public good requires one unit of the private good as input. a. Why are there many Pareto optimal allocations even though there is only one Pareto optimal level of the public good? 1 b. Find a Lindal equilibrium and verify that it is Pareto optimal. 4. Exercise 11.B.2. 5. Exercise 11.D.2. 2...
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## This note was uploaded on 12/26/2011 for the course ECON 210C taught by Professor Qin during the Fall '09 term at UCSB.

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2011HW4 - n X i =1 ¯ x i + c (¯ y ) = n X i =1 ω ix . 3....

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