Advanced Microeconomics 6QQMN970 Exercise Sheet 1
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Advanced Microeconomics 6QQMN970
Exercise Sheet 1
1 Exercise 1
Two individuals live next door to each other. Individual 1 enjoys the consumption of a private good Y, and listening to music. Individual 1’s joy from the music is represented by the variable h, which measures the volume of the music. Listening to music costs nothing to
individual 1.
Individual 2, however, does not like the music that individual 1 plays, which in fact makes him rather miserable. Individual preferences can be represented by the following utility functions:
U1 (Y1 , h) = Y1 + v1 (h)
U2 (Y2 , h) = Y2 + v2 (h)
Where the first derivatives of the v functions satisfyv
(h) > 0 and v
(h) < 0.
The price of a unit of Y is p_ = 1 and individual incomes are I1 and I2 , respectively. Since the individuals do not spend any resources on h and since p_ = 1, it follows that Y1 = I1 and Y2 = I2 .
1. Find the condition that determines individual 1’s choice of h. What is the condition that a Pareto optimal allocation satisfies?
2. Find the privately-chosen and the Pareto efficient levels of h when:
v1 (h) = 60 + 40h − 4h2
v2 (h) = 5 − h2
3. Consider a Pigouvian Tax on h. What should be the the tax per unit of h, such that the quantity chosen by individual 1 is Pareto efficient?
2 Exercise 2
The fare on the train from Sheffield to Manchester is one pound, and the duration of the train ride is one hour. Assume that there is no limit on the number of passengers on the train, and that the marginal costs per passenger are zero. Each passenger can also take a
private car using the road connecting Sheffield and Manchester. The duration of the car ride t depends on the number of cars on the road, n. Specifically, t = 0.2 + n/20.
The price of time for each passenger is 2 pounds per hour (it is the same for all passengers). Every day, N passengers have to get from Sheffield to Manchester. (For the purposes of this question, please ignore the environmental impacts of transportation).
1. What is the number of passengers (cars) on the road in a socially efficient solution?
2. Absent government intervention, how many people (cars) will use the road and what will be the duration of the trip?
3. Suggest a way for government intervention that might achieve social efficiency.
2022-08-04