ECON GU 4251: Problem set 1
Spring 2024
Problem sets are graded (check/check minus) based on effort, this includes the formatting of your work (typesetting is strongly encouraged). You can work in groups, but each student must upload an individual solution. Submissions need to be in pdf format and submitted via Canvas. We will not accept other formats.
Demand and welfare analysis
1. Consider a market with two goods j = 1, 2. Good j = 1 consists of “green” but slow vehicles, while j = 2 corresponds to polluting yet fast vehicles. There are four types of consumers, differing in terms of their derived utilities from owning a vehicle. A consumer of type t = 1, 2, 3, 4 has preferences described
by the pair Vt = (V1(t), V2(t)). Consumers of type t choose j = 1 if
V1(t) - p1 ≥ V2(t) - p2 , and V1(t) - p1 ≥ 0;
they choose j = 2 if
V2(t) - p2 ≥ V1(t) - p1 , and V2(t) - p2 ≥ 0; and they choose to not purchase either good otherwise.
The following table provides values of Vt for t = 1, 2, 3, 4 and a brief description of the consumers.
|
Type (t)
|
Description
|
V1(t)
|
V2(t)
|
|
1
|
environmental, walks to work
|
10
|
2
|
|
2
|
neutral, walks to work
|
5
|
4
|
|
3
|
environmental, drives to work
|
15
|
9
|
|
4
|
likes speed, drives to work
|
11
|
20
|
a. Briefly (max four sentences) discuss whether the interpretation/label assigned to each type is consistent with the values of Vt.
b. Out of a population of 170 million, there are 30 million type 1 individuals, and 100 million type
3 individuals. You learn that, when the prices of the two vehicles are P = (p1 , p2 ) = (7, 15), q1 (p1 , p2 ) = 130 million, and q2 (p1 , p2 ) = 25 million.
How many individuals are type 4, and how many individuals are type 2?
c. Using your answer in point b. above, and holding fixedp2 = 15, draw q1 (p1, 15) as a function of p1.
d. What would be the effecton quantities and consumer surplus of an increase in the price (e.g. due to a tax) of polluting fast vehicles from P = (p1 , p2 ) = (7, 15) to (7, 25)?
e. What would be the effecton quantities and consumer surplus of a decrease in the price (e.g. due to a subsidy) of green slow vehicles from P = (p1 , p2 ) = (7, 15) to (4, 15)?
f. In less than 50 words, can you comment on how would you choose between a policy (tax) inducing the price change in d. and a policy inducing the price change in e.?
Multiproduct monopoly
2. Suppose a monopolist Fp in the burgers market produces patties (p) at zero cost and the demand for patties is given by:
Qp = 1 - Pp - Pb
where Qp is the quantity of patties demanded, Pp is the price of patties, and Pb is the price of buns, which are available to consumers through a different firm, Fb , at a price of Pb = . The demand for buns if given by:
Qp = 1 - 3/1Pp - Pb
Now suppose Fp acquires Fb and produces both patties and buns at zero cost. Will this acquisition lead to an increase or a decrease in the price of patties? Find the old and new equilibrium quantities and prices.
Price Discrimination
3. Suppose there are two groups of consumers potentially served by a monopolist, one with high income and one with low income, denoted by A and B respectively. Let inverse demand function for group A be PA = 20 - QA , and for group B be PB = 10 - 2QB .
(a) Find the demand function for each group.
(b) Consider the case in which the monopolist is not allowed to price discriminate and it charges a uniform. price. Let its marginal cost be equal to zero. Compute optimal price, quantity, consumer surplus, profits and social welfare.
(c) Suppose now the monopolist can do third-degree price discrimination and charge different prices to each group. What price would the monopolist charge to each group? Compute optimal price, quantity, consumer surplus, profits and social welfare.
(d) Consider again the case in which the monopolist is not allowed to price discriminate and it charges a uniform. price but the marginal cost is c = 8. Would the monopolist serve both groups? Compute optimal price, quantity, consumer surplus, profits and social welfare.