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midtermsolutions_Winter2008

midtermsolutions_Winter2008 - Econ 51 Midterm Solutions 1...

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Econ 51: Midterm Solutions 1. Short questions (32 points) (a) I. Yes, these are vNM preferences with a vNM utility function v ( x ) = log( x ), which represents a risk averse individual. The individual will not take the lottery because (using a calculator) u (0 : 2 ; ° 0 : 8) < u (0 ; 0). II. Yes, these are vNM preferences with a vNM utility function v ( x ) = exp( x ), which represents a risk loving individual. The individual will take the lottery because (using a calculator) u (0 : 2 ; ° 0 : 8) > u (0 ; 0). III. This case represents the same utility function as case II (one is simply a monotone transformation of the other), so the answers are the same. IV. No, these are not vNM preferences. The individual will not take the lottery because (using a calculator) u (0 : 2 ; ° 0 : 8) < u (0 ; 0) (note that although these are not vNM preferences, these are still preferences, so we can compare utilities). (b) (i) The only Pareto e°cient allocation is for each of them to consume his ideal allocation, (5 ; 5). (ii) In any Pareto e°cient allocation Rob and Tom will consume their ideal number of bananas (5), and Friday will consume the rest (Rob and Tom don’t like additional bananas, while Friday doesn’t mind having them). As for apples, any allocation among the three is e°cient, as long as neither Rob nor Tom consume more than 5 apples (if they do, they could give it to Friday and improve both utilities).
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