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Unformatted text preview: (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. 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. 6. Congestion Costs: Figure 1: Commuting Time (a) Proportion of Car Users Commuting Time 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 (d) x * < x o because nobody in the competitive outcome ( x o ) wants to waste his time taking the bus so that others can enjoy a speedy car ride; overall, however, if some citizens volunteer to take the bus, people’s average commute time would be shorter. The competitive equilibrium is inefficient because individuals do not take into account the congestion externality that is associated with driving....
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 Fall '12
 Sieg
 Fiscal Policy, Public Good, Mozart, CF, Falco

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