Econ808midans.02

Econ808midans.02 - Department of Economics The Ohio State...

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Department of Economics The Ohio State University Midterm Questions and Answers–Econ 808 Profs. Levin and Peck March 7, 2002 1. (30 points) In the village of Debreuvia, there are 150 consumers and one physical com- modity per state of nature. For i = 1, . .. , 150, consumer i is a von Neumann- Morgenstern expected utility maximizer with (Bernoulli) utility of certain con- sumption given by u i ( x i ) = log( x i ) . For i = 1, . .. , 100, consumer i lives in the highlands, and has an endowment of 1 is all states of nature. For i = 101, . .. , 150, consumer i lives in the ‡ood plain. When a ‡ood occurs, all consumers living in the ‡ood plain receive an endowment of 0 .T h e probability of a ‡ood is 1 10 . When a ‡ood does not occur (probability 9 10 ), all consumers living in the ‡ood plain receive an endowment of 1 . (a) (10 points) If the entire village of Debreuvia trades state-contingent commodities, de…ne a competitive equilibrium for this economy. (b) (20 points) Calculate the competitive equilibrium price vector and allo- cation. Answer: (a) There are two states of nature, corresponding to a ‡ood (state 1) and no ‡ood (state 2). A competitive equilibrium is a price vector, ( p 1 ;p 2 ) ,andan allocation, ( x 1 i ;x 2 i ) 150 i =1 , such that (i) for i = 1, . .. , 100, ( x 1 i 2 i ) solves max[ 1 10 log( x 1 i )+ 9 10 log( x 2 i )] subject to p 1 x 1 i + p 2 x 2 i · p 1 + p 2 ; (ii) for i = 101, . .. , 150, ( x 1 i 2 i ) solves max[ 1 10 log( x 1 i 9 10 log( x 2 i )] subject to p 1 x 1 i + p 2 x 2 i · p 2 ; (iii) markets clear: 150 X i =1 x 1 i · 100 150 X i =1 x 2 i · 150 : 1

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(b) Normalizethepriceofconsumptioninstate2(no‡ood)tobe1, p 2 =1 . First, we calculate the demand function for those living in the highlands (i=1, ... , 100). The …rst order conditions are x 2 i 9 x 1 i = p 1 p 1 x 1 i + x 2 i = p 1 +1 ; which we solve for the demand functions x 1 i = p 1 10 p 1 x 2 i = 9( p 1 +1) 10 : Next, we calculate the demand function for those living in the ‡ood plain (i=101, . .. , 150). The …rst order conditions are x 2 i 9 x 1 i = p 1 p 1 x 1 i + x 2 i ; which we solve for the demand functions x 1 i = 1 10 p 1 x 2 i = 9 10 : Market clearing for consumption in state 2 (no ‡ood) implies 100( 9( p 1 10 )+50( 9 10 )=150 : Solving for p 1 ,wehave p 1 = 1 6 . The C.E. allocation is given by ( x 1 i ;x 2 i )=( 7 10 ; 21 20 ) for i = 1, . .. , 100, ( x 1 i 2 i 3 5 ; 9 10 ) fori=101 ,.
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This note was uploaded on 07/17/2008 for the course ECON 808 taught by Professor Peck during the Spring '02 term at Ohio State.

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Econ808midans.02 - Department of Economics The Ohio State...

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