1. Group Pricing
There is a unit mass of consumers, each demanding one unit of the goods. Consumers' willingness to pay v is uniformly distributed on interval [0, 1]. A monopolist has zero production cost.
a. What is the demand curve (or inverse demand) in this market? (hint: consumers only buy if v ≥ p and v is uniformly distributed.)
b. If the monopolist can only do uniform pricing, what is the optimal uniform price?
c. Given your answer in [b.], calculate the corresponding monopolist's profit, consumer surplus and deadweight loss.
d. Now suppose the consumers can be partitioned into two equal-size groups according their valuations: group 1 in which all consumer have valuations in [0, 1/2], and group 2 in which all consumers have valuationsin [1/2,1]. What are the demand functions for the two groups respectively? (hint: for each group, the demand cannot be bigger than 1/2)
e. The monopolist can distinguish the two groups and price to them separately. What are the optimal group prices?
f. Given your answer in [e.], calculate the corresponding monopolist's profit, consumer surplus and deadweight loss.
g. Compare [c.] and [f.] and comment on the impact of group pricing on monopolist's profit, consumer surplus, and deadweight loss.
2. Cournot Model with Free Entry
A market is characterized by linear inverse demand p = 1 - Q. Suppose N firms compete in quantity. Assume constant marginal cost c < 1.
a. Write down firm i's profit maximization problem when all other firms choose symmetric quantity q.
b. Write down firm i's best response when other firms choose symmetric quantity q
c. Apply full symmetry q∗ = BRi(q∗) and solve for the equilibrium quantity q∗.
d. Denote K > 0 the entry cost. Use the free-entry condition to endoge- nously determine the equilibrium number of firms N∗.
e. What is the expression of total welfare for given N? (hint: write down firms' total profits and consumer surplus for any N, then add them up.)
f. Following [e.], maximize this total welfare by choosing N and get the efficient number of firms Ne. Is Ne higher or lower than N∗.
g. Explain in words the result you found in [f.].
3. Collusion with Market Growth
Two firms, 1 and 2, compete in price. Market demand in period t is given by D(t) = AtD(p) with A > 0. The common discount factor is δ ∈ (0, 1). Suppose the firms use trigger strategies to collude at the monopoly price pm = arg max(p - c)(A)tD(p) ≡ (A)tπm (note that pm does not depend on A and t due to the function form). Suppose the punishment after deviation is returning to marginal cost pricing forever. If the firms collude, they set the same prices and evenly split the profits.
- What are firms' collusive profits in period t?
- If a firm undercuts below pm in period t, what are the (optimal)
- deviating price and deviating profit.
- Write down the no-deviating condition in period t?
- Simplify the no-deviating condition and derive the critical discount factor δ.
- Compared to when the market is shrinking (A < 1), does a expanding market (A > 1) make collusion easier?
- Explain in words your finding in [e.]