2. The following is a version of the Cournot competition game with four firms, in which the firms simultaneously choose nonnegative quantities of a good to produce. When the firms’ choices are the profile (q1, q2, q3, q4), the price at which the firms sell their output is α − q1 − q2 − q3 − q4 if q1 +q2 +q3 +q4 < α; otherwise, the price of the good they sell is zero. Each firm i has a cost of producing qi units of output of ci ×qi, where ci ≥ 0; different firms can have different cost functions. A firm’s payoff is its profit, the revenue it gets from selling its output qi minus its cost.
1. (a) Write down the expression for the payoff firm 1, as function of all of the firms’ choices of quantities, as well as of α and any of the marginal costs ci, for the case in which q1+q2+q3+q4 < α. [Note: other firms’ payoffs will look similar.] (4 points)
2. (b) Find firm 1’s best response function for this game, for values of (q2, q3, q4) such that q2 + q3 + q4 + c1 < α. [Note: other firms’ best-response functions will look similar.] (6 points)
3. (c) Suppose that firm 1 has a cost function such that c1 = 0; and that firms 2, 3 and 4 have the same cost function: c2 = c3 = c4 = c. Find a Nash equilibrium of this game in which firms 2, 3, and 4 choose the same quantity as each other (but not, in general, the same quantity as firm 1), with all quantities expressed as functions of α and c. You may assume that each firm chooses a positive quantity in the equilibrium (and, as such, determines its best response as in part (b)), which will be true as long as α is sufficiently greater than c. Remember that since firms 2, 3, and 4 are assumed to choose the same output as each other, you are really only solving for two outputs. (10 points)
4. (d) Suppose that firms 2, 3 and 4’s cost per unit c increases, while firm 1’s remains at 0. Say whether this makes firm 1’s equilibrium quantity greater or smaller; whether it makes firm 2, 3 and 4’s equilibrium quantities greater or smaller; whether it makes the total quantity produced by all firms greater or smaller; and whether it increases or decreases the price of the good that consumers pay in the equilibrium. (7 points)