What are firms collusive profits in period t if a firm

Question

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.

1. What are firms' collusive profits in period t?
2. If a firm undercuts below pm in period t, what are the (optimal)
3. deviating price and deviating profit.
4. Write down the no-deviating condition in period t?
5. Simplify the no-deviating condition and derive the critical discount factor δ.
6. Compared to when the market is shrinking (A < 1), does a expanding market (A > 1) make collusion easier?
7. Explain in words your finding in [e.].