Question

? = 10 − 1/4Q

4. (Collusion) Now suppose firms face the same market demand as in Problem 3. But now there are three firms (firm 1, firm 2, and firm 3) where Q = q1 + q2 + q3. All of them bear the same production marginal cost of c1 = c2 = c3 = 4 per one gallon of water. Lastly, the game among these firms is repeated indefinitely in each period t = 1, 2, 3, ... . Let δ ∈ (0, 1) denote the firms’ common time discount factor. (a) Find the static (joint) monopolist’s quantity, price and profit when they collude. (b) Solve the static Cournot game. That is, find q1 * , q2 * , q3 * , p * , π1 * , π2 * , π3 * . (c) Now find the static profit of firm 1 when he solely deviates from collusion.(d) Calculate the lifetime value of firm 1 when all of the firms maintain collusion forever (Vcol). Also find the lifetime value of firm 1 when he solely betrays at t = 1 and is detected by others at t = 2 (Vdev). (e) Compute the minimum threshold value of δ that would make collusion sustainable.

Answer 1