Given the exercise extends the analysis of the Stackelberg duopoly game to include fixed costs of production. The analysis produces a theory of limit quantity, which is a quantity the incumbent firm can produce that will induce the potential entrant to stay out of the market.
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, q1.
Then 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) = 100qi + 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 pqi - c(qi).
(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 entryaccommodating 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.