Consider a duopoly where each of the two (identical) firms have the cost function

C(y_i) =y_i^2/2

fori= 1,2.

The market demand is

p(y)=ay, a >0

whereyis the total quantity sold in the market i.e.y=y₁+y₂.

(a)[5 points] Write the best response functions for each firm.

(b)[5 points] What is the Nash equilibrium outcome?

(c)[5 points] What is the optimal output level if only one of the two firms operated in this market (and was, therefore, a monopolist in this market)? [Hint: If you like, you can just use the best response function from part (a) to find this.]

(d)[5 points] Find firms' profits in duopoly and the profit of the monopolist.

(e)[5 points] Suppose the industry starts off as a monopoly, with firm 1 as the only operational firm. Firm 2 is deciding whether it should enter the market. If it enters, both firms will simultaneously choose the level of output to produce. Firm 1 can impose and entry cost ofFon firm 2, i.e. firm 2 will have to payFif it enters the market. What is the minimum value ofFthat firm 1 must set so as to remain a monopoly?

(f)[5 points] Once again, suppose the industry starts off as a monopoly, with firm 1 as the only operational firm. Firm 2 is deciding whether it should enter the market. However, now the firms will compete in prices ala Bertrand. Suppose amounts can only be in whole number dollars (no cents allowed). What is the minimum fixed cost the incumbent monopolist has to impose so as to deter entry of the new firm?