Collusion in Prices under a stochastic Demand system

Suppose there are two firms competing through prices in a market. Here the inverse demand function is given
by
P(Q) = 20ψ − 4Q
where ψ is a random variable that may take two values. With probability 0.4, we have ψ = ψh = 10 and with
probability 0.6 we have ψ = ψl = 5. Thus, market demand itself is a random variable that may take two values.
On the other hand, firms are assumed to be symmetric, having the same cost function given by C(q) = 4q.
Both firms discount future expected profits with the same discount factor δ ∈ (0, 1).
a) (3 points) What is the monopolistic collusion price when demand is high (h)?
b) (3 points) What is the monopolistic collusion price when demand is low (l)?
c) (3 points) What is the monopolistic collusion profit when demand is high (h)?
d) (3 points) What is the monopolistic collusion profit when demand is low (l)?
e) (12 points) For which values of δ can we ensure that firms will be able to sustain monopolistic collusion
(πh = π
M
h
and πl = π
M
l
) as a SPE?
Exercise 2: Horizontal Differentiation
Part 1
Consider a two firm differentiated product environment where firms choose their price but not their product
type. The product is differentiated in one dimension: in particular, the good can take on types ranging from zero
1
Econ 4631-Industrial Organization HW 3
to one. As we saw in class, consumers derive an inherent utility from consuming one unit of the good (v¯), pay
a price pi
for it (where i can be either 1 or 2, depending which firm they are buying from) and incur in a linear
adjustment cost (dis-utility) of t = 1 when deviating from their favorite type. Consumer types are identified
with the parameter θ which is distributed uniformly in the [0,1] interval. Suppose firms’ cost functions are as
follows: C1(q1) = αq1 + F and C2(q2) = 2
3
αq2 + F. Suppose firm 1 is forced to locate at point 0 and firm 2 is
forced to locate at point 1.
a) (4 points) What is the optimal price that each firm will set?
b) (4 points) Assume now that α = 1 and F = 0.2. What are the profits for each firm?
Part 2
Suppose now a differentiated product environment where there are N firms, located equidistantly one from the
other and sharing the same cost function: C(qi) = 1
2
qi + 1
10 . Consumers in this market are assumed to behave in
the exact same way as before.
a) (3 points) What is the demand function that a firm i is facing in this market? Provide a clear expression
b) (3 points) Set up firm i’s maximization problem and find the optimal price. Use this to obtain firm’s profits
c) (4 points) Find the equilibrium number of firms in the market (if you obtain a non-integer number, then
consider only the integer part of your number to answer this question). Suppose fixed cost drops to 1
81 .
What is the new number of firms? Provide an explanation to your results.