Question

Suppose two firms compete by selecting quantities q₁ and q₂, respectively, with the market price given by p = 1000-3q₁ -3q₂. Firm 1 (the incumbent) is already in the market. Firm 2 (the potential entrant) must decide whether or not to enter and, if she enters, how much to produce. First the incumbent commits to its production level q₂. The potential entrant, having seen q₁, decides whether to enter the industry. If firm 2 chooses to enter, then it selects its production level q₂. Both firms have the cost function c(qi) = 100q_i+F, where F is a constant fixed cost. If firm 2 decides not to enter, then it obtains a payoff of 0. Otherwise, it pays the cost of production, including the fixed cost. Note that firm i in the market earns a payoff of pq_i-c(q_i).

(a) What is firm 2’s optimal quantity as a function of q₁, conditional on entry?
(b) Suppose F = 0. Compute the subgame perfect Nash equilibrium of this game. Report equilibrium strategies as well as the outputs, profits, and price realized in equilibrium. This is the Stackelberg or entry-accommodating outcome.
(c) Now suppose F > 0. Compute, as a function of F, the level of q₁ that would make entry unprofitable for firm 2. This is called the limit quantity.
(d) Find the incumbent’s optimal choice of output and the outcome of the game in the following cases: (i) F = 18; 723, (ii) F = 8; 112, (iii) F = 1; 728, and (iv) F = 108. It will be easiest to use your answers from parts (b) and (c) here; in each case. compare firm 1’s profit from limiting entry with its profit from accommodating entry.

I'm especially stuck on parts c and d. Thanks for helping!

Answer 1