EC332 Problem Set 3
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EC332 Problem Set 3
November 9, 2022
This problem set is due before the lecture on Nov 16. The submission is not compulsory, and will not be graded. However, you are encouraged to finish this problem set to check your progress.
1 (easy)
Suppose consumers receive utility ui = θi s − p from purchasing a product from a monopoly. θi draws uniformly from [0, 1].
a
Write down the demand function P(q,s). The equation should not contain θ .
b
Graphically illustrate the impact of an increase in s in a p-q diagram.
c
Graphically illustrate why the monopolist underprovides quality.
d
Now assume the utility function is ui = θi + (1 − θi )s − p. Repeat part b and c.
2 (medium)
In a linear city, a monopolist can set up as many stores at any location as he likes by incurring a fixed cost f for each store. A consumer purchases one unit of good from only one store and receives utility u = s − pi − tdi where pi is the price of the store he purchases from and di is the distance between the consumer’s location and the store’s location. Assume 
< f < 
.
a
How many stores will the monopolist set up?
b
How many stores will a social planner set up?
3 (medium)
In the quality moral hazard problem with informed consumers, we know high-quality pure strategy equilibrium exists if and only if c1 − c0 < α(p − c0 ). But we also showed that there cannot be an equilibrium if uninformed consumers never purchase. Show that if c1 − c0 > α(p − c0 ), there is a mixed-strategy equilibrium where a proportion γ of uninformed consumers purchase and the monopoly provides high quality with a probability β . You should express γ and β as functions of exogenous parameters and p.
4(easy)
Suppose there are three types of cars: high quality with s = 10000, middle quality with s = 2000, low quality with s = 0. Buyer receives θs − p if he purchases and he thinks a car is equally likely to be one of the three types. Seller receives p if sells, s otherwise.
a
Under what conditions, there is always a sale?
b
Under what conditions, do low and middle car sell but high car withdraw?
c
Under what conditions, do middle and high car sell but low car withdraws?
2022-12-17