Muskrat, Ontario, has 1,000 people. Citizens of Muskrat consume only one private good, Labatt’s ale. There is one public good, the town skating rink. Although they may differ in other respects, inhabitants have the same utility function. This function is U(Xi,G) = Xi − 100/G, where Xi is the number of bottles of Labatt’s consumed by citizen i and G is the size of the town skating rink, measured in square meters. The price of Labatt’s ale is $1 per bottle and the price of the skating rink is $10 per square meter. Everyone who lives in Muskrat has an income of $1,000 per year.
(a) Write down an expression for the absolute value of the marginal rate of substitution between skating rink and Labatt’s ale for a typical citizen. ___________ What is the marginal cost of an extra square meter of skating rink (measured in terms of Labatt’s ale)? __________
(b) Since there are 1,000 people in town, all with the same marginal rate of substitution, you should now be able to write an equation that states the condition that the sum of absolute values of marginal rates of substitution equals marginal cost. Write this equation and solve it for the Pareto efficient amount of G.
(c) Suppose that everyone in town pays an equal share of the cost of the skating rink. Total expenditure by the town on its skating rink will be $10G. Then the tax bill paid by an individual citizen to pay for the skating rink is $10G/1, 000 = $G/100. Every year the citizens of Muskrat vote on how big the skating rink should be. Citizens realize that they will have to pay their share of the cost of the skating rink. Knowing this, a citizen realizes that if the size of the skating rink is G, then the amount of Labatt’s ale that he will be able to afford is
(d) Therefore we can write a voter’s budget constraint as Xi + G/100 = 1, 000. In order to decide how big a skating rink to vote for, a voter simply solves for the combination of Xi and G that maximizes his utility subject to his budget constraint and votes for that amount of G. How much G is that in our example?
(e) If the town supplies a skating rink that is the size demanded by the voters will it be larger than, smaller than, or the same size as the Pareto optimal rink?
(f) Suppose that the Ontario cultural commission decides to promote Canadian culture by subsidizing local skating rinks. The provincial government will pay 50% of the cost of skating rinks in all towns. The costs of this subsidy will be shared by all citizens of the province of Ontario. There are hundreds of towns like Muskrat in Ontario. It is true that to pay for this subsidy, taxes paid to the provincial government will have to be increased. But there are hundreds of towns from which this tax is collected, so that the effect of an increase in expenditures in Muskrat on the taxes its citizens have to pay to the state can be safely neglected. Now, approximately how large a skating rink would citizens of Muskrat vote for?
(g) Does this subsidy promote economic efficiency?