Suppose that a market is composed of two firms that simultaneously choose quantities. Firm 1 has a cost function C1(q1) = q1^2. Firm 2 has a cost function C2(q2) = 2q2^2. The inverse demand equation

is p(Q) = 100 Q

Suppose for simplicity that each firm can choose among only two levels of output: the Cournot level found in part (c), or the collusion level found in part (e). The objective is to write down a payoff matrix that represents the single period game in which each firm chooses the Cournot or the collusion quantity. To do that, you first need to:

(f) Find each firm's profits when firm 1 chooses the Cournot quantity while firm 2 chooses the collusion quantity. (7 points) (Hint: you need to first find aggregate quantity and price in order to derive profits).

(g) Find each firm's profits when firm 2 chooses the Cournot quantity while firm 1 chooses the collusion quantity. (7 points)

(h) Using the profits found in parts (c), (e), (f), and (g) as payoffs, write down a payoff matrix. Round profits to the closest integer. There should be two players, firm 1 and firm 2, and two possible actions for each player, namely producing the Cournot quantity and producing the collusion quantity. Solve for the Nash equilibrium in pure strategies. (10 points)

(i) Now assume that the firms play the previous game for infinitely many periods. In each period, they simultaneously choose quantities. They both discount future profits with a discount factor where0<1sothattotalprofitsare1+2+23 =.... Suppose that each firm follows the "grim trigger" strategy, that is:

(i) Each firm produces the collusion quantity as long as the other firm does the same. (ii) Each firm produces the Cournot quantity in period t + 1 and thereafter if the other firm produces the Cournot output at time t. What is the value of such that collusion forever is an outcome of the repeated game? (12 points)