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B now we know that the marginal cost of the public

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(b) Now, we know that the marginal cost of the public good is 10. Thus, the Samuel- son condition will be 3 X i =1 X i G = 10 , or X 1 + X 2 + X 3 = 10 G . (c) The aggregate resource constraint is X 1 + X 2 + X 3 +10 G = w 1 + w 2 + w 3 (= 100). Thus, substituting the Samuelson condition into the resource constraint, we have 10 G + 10 G = 100 , which implies G * = 5.
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5. Majority Rule: The resource constraint is given by: G = N X i =1 T i = t N X i =1 Y i = tN Y Therefore, we can rewrite the proportional income tax in terms of G T i = G N Y Y i The preferred level of public good provision for a voter with income level Y i can be calculated by solving the following individual optimization problem: max G U i = Y i - G N Y Y i 1 - α G α = Y 1 - α i 1 - G N Y 1 - α G α The first order condition of this problem is: ∂U i ∂G = Y 1 - α i 1 - G N Y 1 - α G α ( (1 - α ) 1 - G N Y - 1 - 1 N Y + αG - 1 ) = Y 1 - α i 1 - G N Y 1 - α G α (1 - α ) - 1 N Y - G + α 1 G = 0 You can find the solution by making the terms inside the bracket to be zero: (1 - α ) 1 N Y - G = α 1 G The individual preferred level is given by: G = αN Y which is the same for all the members of society, and hence it is the outcome of majority rule voting.
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6. Congestion Costs: Figure 1: Commuting Time (a) Proportion of Car Users Commuting Time 0 1 80 70 Bus Car (b) A bus trip and a car trip take the same amount of time if 70 = 20 + 60 x o , or, x o = 5 6 . Suppose x e is the true equilibrium. If x e < x o , then a car trip would be shorter than a bus trip; if this is the case, some bus riders will stop riding the bus and start driving instead, so x e would not really ”stable” (and hence, not an equilibrium). Similarly, if x e > x o , then a bus trip would be shorter than a car trip; if this is the case, some car drivers will stop driving and start taking the bus instead, so x e would not really be ”stable” (and hence, not an equilibrium). (c) To minimize total travel time, we want to find a proportion of drivers, x * , that minimizes the total, or, average, traveling time: min x * x * (20 + 60 x * ) + (1 - x * )70 THE FOC is given by: 20 + 120 x * - 70 = 0 which implies that: x * = 5 12
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(d) x * < x o because nobody in the competitive outcome ( x o ) wants to waste his
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