Question

1. (Static Cournot Model) In Long Island there are two suppliers of distilled water, labeled firm 1 and firm 2. Distilled water is considered to be a homogenous good. Let p denote the price per gallon, q1 quantity sold by firm 1, and q2 the quantity sold by firm 2. Firm 1 bears the production cost of c1 = 4, and firm 2 bears c2 = 2 per one gallon of water. Long Island’s inverse demand function for distilled water is given by P = 10 − 1/4Q where Q = q1 + q2 denotes the aggregate industry supply of distilled water in Long Island. Solve the following problems: (a) Suppose the firms compete in quantities (production levels). Compute each firm’s quantity best response function and conclude how much each firm produces in a Cournot-Nash equilibrium. (b) Compute the price and the profit of each firm in a Cournot-Nash equilibrium. 3 2. (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 = 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. 4 (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