Economic Question (collusion)

4. (Collusion) Now suppose firms face the same market demand as in Problem 1. But now there are three firms (firm 1, firm 2, and firm 3) where Q = q; + q; + (313. All of them bear the same production marginal cost of c; = 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 6 E (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 q;"', q2*, q3*, p', 111', 112*, 113'. (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 (Veal). 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 0er that would make collusion sustainable