solutions for ps4

Solutions for ps4 - Solutions for Problem Set 4 Question 1(a By definition the utilitarian solution is the list of h 1 h 2 h n that maximizes n n

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Solutions for Problem Set 4 Question 1 (a) By definition, the utilitarian solution is the list of h 1 , h 2 ,..., h n that maximizes the sum of all farmers’ valuations, i 1 n qh i v i , where q 10 j 1 n h j .I fwede f ine H j 1 n h j , which is the total number of cow-hours of grazing, then q 10 H . Then the sum of all farmers’ valuations becomes 10 H i 1 n h i v i . For a given H , in order to maximize i 1 n h i v i , all the hours must be assigned to the farmer with the highest value. That is, for every i , h i Hi f v i is the highest value ; 0 otherwise Hence the sum of all farmers’ valuations is 10 H Hv m , where v m is the highest value of v 1 , v 2 ,..., v n . The last step is to choose H to maximize H 10 H v m . It is easy to get that H 5. Therefore, the utilitarian solution of h 1 , h 2 ,..., h n is that, for each i , h i 5 if v i is the highest value ; 0 otherwise , which means only the farmer with the highest value will be able to use the land for 5 hours. (b) (i) Suppose the announcement is v ̂  v ̂ 1 , v ̂ 2 ,..., v ̂ n with v ̂ 1 6, v ̂ 2 4 and v ̂ i 4 for i 3,4,. .., n , then according to part (a), the utilitarian alternative x v ̂ is that only farmer 1 is allowed to use the land for 5 hours. First consider farmer 1. The utilitarian alternative for the society consisting of everyone except farmer 1, x v ̂ 1 , is that only farmer 2 is allowed to use the land for 5 hours. (This is because without farmer 1, farmer 2 is the one with the higest value.) Hence, the VCG payment for farmer 1 is, t 1 VCG v ̂ j 1 v ̂ j x v ̂  j 1 v ̂ j x v ̂ 1  0 10 5 5 v ̂ 2 25 4 100. Now consider farmer i 1. The utilitarian alternative for the society without i , x v ̂ i , is that only farmer 1 is allowed to use the land for 5 hours. Hence, x v ̂ i x v ̂ . Then the VCG payment for farmer i is, t i VCG v ̂ j i v ̂ j x v ̂  j i v ̂ j x v ̂ i  0 . (because x v ̂ i x v ̂ ) Hence, in this case, farmer 1 pays 100, and other farmers pay nothing. (ii) Suppose farmer 1’s value is v
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If farmer 1 announce his value truthfully, then from the above part we know that farmer 1 pays 100, and since he is allowed to use the land for 5 hours, his valuation is 10 5 5 6 150 , which means farmer 1’s payoff is 150 100 50. If farmer 1 misreports and announces a value
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This note was uploaded on 01/14/2012 for the course ECON 201 taught by Professor Witte during the Spring '08 term at Northwestern.

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Solutions for ps4 - Solutions for Problem Set 4 Question 1(a By definition the utilitarian solution is the list of h 1 h 2 h n that maximizes n n

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