2.- Each of two firms, firms 1 and 2, has a cost function C(q) = 1/2 q; the demand function for the firms' output is Q = 3/2 p, where Q is the total output. Firms compete in prices. That is, firms choose simultaneously what price they charge. Consumers will buy from the firm offering the lowest price. In case of tying, firms split equally the demand at the (common) price. The firm that charges the higher price sells nothing. (Bertrand model.)

(a) Formally argue that there could be no equilibrium in prices other than p1 = p2 = 1/2 .

That is, consider all cases: [p1> p2>0.5, p1 >0.5 > p2] , [.5> p1 > p2, p1 = p2 >.5] , and [p1= p2 <0.5].

You do not need to consider the cases with 1 and 2 swapped, do you?

Now, suppose that firm 1 has a capacity constraint of 1/3. That is, no matter what demand it gets, it can serve at most 1/3 units. (Suppose that these units are served to the consumers who are willing to pay the most.) Thus, even if firm 2 sets a price above that of firm 1, firm 2 may be able to sell some output.

(b) Obtain the (residual) demand of firm 2 (as a function of its own price, p2) assuming that the other firm sets price equal to 5/6.

(c) Find the optimal price for firm 2, and so its output