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Microeconomics Exercises: Fishing Industry, Public Goods, and VCG Mechanism

1) Consider the fishing industry. Access to the ocean is rival and non-exclusive: Every fisher can go fishing, but once fish is caught, it is the fisher’s private propertySuppose the total value of fish is , where is b is the total number of fishing boats. The cost of sending a boat is .

(a) Suppose each potential fisher decides whether to send a boat independently. What is the equilibrium number of boats?

(b) What is the socially optimal number of boats?

(c) Now suppose the ocean is privatized. Its owner can charge each fishing boat an entry fee of . What is the profit-maximizing level of entry fee? Under this fee, is the equilibrium number of boats socially optimal?

2) 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, , where is the number of bottles of Labatt’s consumed by citizen and 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, . (

b) What is the marginal cost of an extra square meter of skating rink (measured in terms of Labatt’s ale)?

(c) 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 socially efficient amount of .

(d) Suppose that everyone in town pays an equal share of the cost of the skating rink in taxes. Write out each citizen’s (expected) budget constraint for a given size of skating rink

. (e) Given the situation in part d),every year the citizens of Muskrat vote on how big the skating rink should be. What size of the skating rink would each citizen most prefer? (Hint: solve for the optimal given the above budget constraint.) How does this compare to the socially optimal rink size? (f) Given the situation in part

d), 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. Now, approximately how large a skating rink would citizens of Muskrat vote for? Does this subsidy promote efficiency? (Hint: Rewrite the budget constraint for individuals observing that local taxes will be only half as large as before.)

3) Homeowners 1, 2, and 3 live at the end of a badly deteriorated road. Fixing the road would cost $C. The value to Homeowner 1 of fixing the road is $3,000, the value to Homeowner 2 is $5,000, and the value to Homeowner 3 is $8,000. Each homeowner claims that fixing the road is not worth much to him, because each wants the others to pay the cost. The local government suspects that the total value to these homeowners of fixing the road is greater than $C and has decided to require the three homeowners to use the VCG mechanism to determine whether to fix the road: Costs are allocated equally among the three homeowners if the road is provided.. Each homeowner is asked to report his value for fixing the road. If the sum of the reported values is greater than C, the road will be fixed and each homeowner will have to pay $C/3. Furthermore, each homeowner also has to pay an additional tax as calculated by the VCG mechanism (as specified in your lecture notes). Suppose that C = $13, 500, so that each homeowner has to pay $4,500 as his share of the cost.

(a) Will the road be fixed?

(b) What will be the additional tax paid by each homeowner?

(c) How does each homeowner’s welfare change as a result of the VCG mechanism? How does total welfare change?