PS5_sol - Question 1 Part(a To maximize social welfare the...

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Question 1 Part (a) To maximize social welfare, the public good should be produced if and only if the sum of individuals°values for the good is higher than its production cost, i.e. v 1 + v 2 ° c Let°s denote by x ° = 1 the decision that the public good is produced; and x ° = 0 the decision that the public good is not produced The utilitarian decision rule is given by x ° ( v 1 ; v 2 ; c ) = ° 1 if v 1 + v 2 ° c 0 if v 1 + v 2 < c Part (b) No. There is no mechanism that satis±es the following at the same time: 1. implements the utilitartian decision rule x ° 2. provides dominant strategy to tell the truth 3. balances the budget. We have seen in class that even when the production cost c is commonly known, a mechanism that satis±es all of (1) , (2) , and (3) simultaneously does not exist. Adding private information on part of production cost certainly won°t help. To be more precise, we know from class that the only mechanism that sat- is±es (1) and (2) above is a VCG mechanism. The decision rule is given by x ° above and the VCG transfer rule is given by t V CG 1 ( v 1 ; v 2 ; c ) = ° min f v 2 ± c; 0 g if v 1 + v 2 ° c 0 otherwise for individual 1, and t V CG 2 ( v 1 ; v 2 ; c ) = ° min f v 1 ± c; 0 g if v 1 + v 2 ° c 0 otherwise for individual 2,and t V CG s ( v 1 ; v 2 ; c ) = ° v 1 + v 2 if v 1 + v 2 ° c 0 otherwise If the good is produced, i.e. v 1 + v 2 ° c , the total transfer is given by t V CG 1 + t V CG 2 + t V CG s = min f v 2 ± c; 0 g + min f v 1 ± c; 0 g + v 1 + v 2 ° 0 1
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which is strictly positive if v 1 + v 2 > c . Hence a VCG mechanism runs a budget de±cit. Consequently, there is no mechanism that satis±es all of (1) ± (3) above. Part (c) A mechanism that combines the idea of both ±xed-price mechanism and ±xed cost-share mechanism has the following feature: 1. The producer is paid a ±xed price p s whenever the good is produced 2. Individual
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