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Substituting into the budget constraint yields the

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Substituting into the budget constraint yields the following modified reaction function C F = 55 - C M 2 Similarly, we have for Mozart: C M = 55 - C F 2 Solving for an interior solution, the system of two equations in two unknowns, yields: C F = C M = 55 / 3 = 18 . 334 , The total number of concerts that each would enjoy would be 46.667 ( C F + C M + 10), and they would enjoy X = 46 . 667. So, the government solution does not improve on the competitive equilibrium (same private savings, same number of concerts). (c) To find the social optimum is to use the Samuelson condition together with the aggregate resource (budget) constraint. The marginal cost of providing one more concert is equal to 1 (this is stated in the problem). Using our utility function,
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the Samuelson condition is: ∂U F /∂C ∂U F /∂X F + ∂U M /∂C ∂U M /∂X M = MC X M C + X F C = 1 X F + X M = C The aggregate resource constraint is: X F + X M + C = 140 Solving the system of equations for C yields: C = 70 which implies that X M = 35 = X F . We conclude that the competitive equilib- rium is inefficient since it does not provide enough public goods. (d) If a benefactor pays for 10 concerts, Falco chooses C F and X F by solving the following problem: max X F ,C F U = log X F + log( C M + C F + 10) subject to 70 = X F + C F solving this optimization problem yields the following reaction function: C F = 60 - C M 2 Since Falco’s and Mozart’s utility functions are the same, we have: C M = 60 - C F 2 which implies the following equilibrium: C M = C F = 20 Thus, the total number of concerts enjoyed is 40, with Falco and Mozart each retaining 50 units of income for private consumption. The provision is not socially optimal because this provision provides less than the socially optimal number of concerts (70). However individuals are better of in (c) than in (a) or (b).
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4. Efficiency of Public Good Provision: (a) For each i = 1 , 2 , 3, we can compute the marginal rate of substitution as follows: ∂U i ∂G / ∂U i ∂X i = X i G .
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