Solution_Winter2009

# Solution_Winter2009 - Econ 51 Final Exam 1 Uncertainty and...

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Econ 51: Final Exam March 18, 2009 1. Uncertainty and General Equilibrium (30 points) Consider three securities. One unit of security c pays a dollar in June if the Celtics win the NBA Championship, and zero otherwise. One unit of security l pays a dollar in June if the Lakers win the NBA Championship, and zero otherwise. One unit of security f pays a dollar in June if neither the Celtics nor the Lakers win the NBA Championship, and zero otherwise. Consider a market with two agents, Ray and Kobe, each endowed with 10 units of security c , 10 units of security l , and one unit of security f . Security f is not tradable, so agents have to hold their endowment of it. Ray believes that the Celtics will win it all with probability 0.3 and that the Lakers will win it all with probability 0.3 (thus, he believes that a third team will win it with probability 0.4). Kobe is more optimistic about the Lakers’ chances, and thinks that the Celtics have only a 0.2 probability to win, and that the Lakers have a 0.4 probability to win. Both Ray and Kobe have a vNM utility of u ( x ) = log( x ) and they have no other income (except for the proceeds they get from holding on to the securities). Throughout this question, note that all agents have a dollar if neither the Lakers nor the Celtics win, and since log(1) = 0 we can simply ignore this third component of the utility. allocations of securities c and l ; security f is not tradable). Ray’s (expected) utility is u R ( c; l ) = 0 : 3 log( c )+0 : 3 log( l ) and Kobe’s utility is u K ( c; l ) = 0 : 2 log( c ) + 0 : 4 log( l ). Aggregate endowment is (20,20). This is now a standard MRS R = MRS K , that is l R c R = l K 2 c K = (20 l R ) 2(20 c R ) or l R = 20 c R 40 c R for any c R between 0 and 20. b. (5 points) Find Walrasian Equilibrium (prices and allocations of securities c and l ). Normalize p c = 1, and let p denote the price for l . Ray’s and Kobe’s demands for c are given by 1 2 (10 + 10 p ) and 1 3 (10 + 10 p ). To clear the market we must have 1 2 (10 + 10 p )+ 1 3 (10 + 10 p ) = 20, which implies that p = 1 : 4. Allocations are (12 ; 8 : 57) for Ray and (8 ; 11 : 43) for Kobe. c. A third agent, Lebron, shows up to the market. He has the same utility and the same endowment as the other two, but he believes that the Celtics probability of winning it is 0.15 and that the Lakers’ is 0.3 (he simply expects the Cavs to have a good shot at the title). (i) (7 points) Repeat part (b). 1

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{ It’s easy to see that Lebron’s utility is the same as that of Kobe (to see this, all we need is to apply a linear transformation), so it’s really almost the same thing. Market clearing now implies 1 2 (10 + 10 p )+ 1 3 (10 + 10 p )+ 1 3 (10 + 10 p ) = 30, or p = 1 : 57. Allocations are (12 : 86 ; 8 : 18) for Ray, (8 : 57 ; 10 : 91) for Kobe, and (8 : 57 ; 10
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Solution_Winter2009 - Econ 51 Final Exam 1 Uncertainty and...

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