Intermediate Microeconomics PS6
Fall 2011
1、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 i(Xi, Y) = Xi − 100/Y , where Xi is the number of bottles of Labatt’s consumed by citizen i and Y 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 out the equation implied by the Samuelson conditions.
b) Show that this equation uniquely determines the efficient rink size for Muskrat. What is that size?
2、Consider a consumer with preferences for consumption of a composite good (Y) and leisure (N) given by the following utility function:
Denote the consumer's wage rate by w and total time available for labor and leisure is 24 hours. The price of consumption is one. Denote the amount of labor supplied as N, so that N + H = 24. The consumer also receives an unearned income of I.
a) Write out the budget constraint determining feasible allocations of leisure and consumption.
b) Using substitution method to derive the optimal bundle of labor supply H(w, I) and optimal bundle of consumption Y (w, I) (Hint, only need to solve the interior solution where labor supply and consumption are positive).
c) Derive the consumer's leisure consumption function: N(w, I).
d) Determine the effect of increasing unearned income I on the supply of labor
(that is, compute the relevant partial derivative ).
e) How does unearned income I affect the consumption of the composite good, Y
(that is, compute the relevant partial derivative)?
f) Compute the effects of an increase in wage on labor supply (that is, compute the relevant partial derivative).
3、A good, q, can be produced at MC(q) = q + 4 . The marginal utility of consuming the good is MU(q) = 20- q . Consumption of the good has a marginal external cost of MEC(q) = q / 2 .
a) Calculate the market equilibrium quantity and price.
b) What is the marginal social cost?
c) What is the socially optimal quantity?
d) Calculate the optimal quantity regulation and Pigouvian tax.
4、Langrun Company is the only hirer of workers, and thus it acts as a monopolist. The supply curve for coal workers is given by L=20w, where L is the number of workers hired and w is their hourly wage. Langrun Company’s labor demand
(marginal revenue product) curve is given by .
a) How many workers will Langrun Company hire to maximize its profits and what wage will it pay?
b) Assume now that the government implements a minimum wage law covering all workers. If the minimum wage is set at 50 per hour, derive the marginal expense curve faced by Langrun Company. How many workers will Langrun Company hire and how much unemployment will there be?
c) Graph your results.
d) How does a minimum wage imposed under monopsony differ in results as compared with a minimum wage imposed under perfect competition (assuming the minimum wage is above the market-determined wage?
Good luck for final exams
