In the text, we demonstrated the equilibrium that emerges when two oligopolists compete on price when there are no fixed costs and marginal costs are constant. In this exercise, continue to assume that firms compete solely on price and can produce whatever quantity they want.
A: We now explore what happens as we change some of these assumptions. Maintain the assumptions we made in the text and change only those referred to in each part of the exercise. Assume throughout that costs are never so high that no production will take place in equilibrium, and suppose throughout that price is the strategic variable.
(a) First, suppose both firms paid a fixed cost to get into the market. Does this change the prediction that firms will set p =MC?
(b) Suppose instead that there is a recurring fixed cost FC for each firm. Consider first the sequential case where firm 1 sets its price first and then firm 2 follows (assuming that one of the options for both firms is to not produce and not pay the recurring fixed cost). What is the sub game perfect equilibirum? (If you get stuck, there is a hint in part (f).)
(c) Consider the same costs as in (b). Can both firms produce in equilibrium when they move simultaneously?
(d)What is the simultaneous move Nash Equilibrium? (There are actually 2.)
(e) True or False: The introduction of a recurring fixed cost into the Bertrand model results in p = AC instead of p =MC.
(f) You should have concluded above that the recurring fixed cost version of the Bertrand model leads to a single firm in the oligopoly producing. Given how this firm prices the output, is this outcome efficient—or would it be more efficient for both firms to produce?
(g) Suppose next that, in addition to a recurring fixed cost, the marginal cost curve for each firm is upward sloping. Assume that the recurring fixed cost is sufficiently high to cause AC to cross MC to the right of the demand curve. Using logic similar to what you have used thus far in this exercise, can you again identify the sub game perfect equilibrium of the sequential Bertrand game as well as the simultaneous move (pure strategy) Nash equilibria?
B: Suppose that demand is given by x (p) = 100−0.1p and firm costs are given by c(x) = FC +5x2.
(a) Assume that FC = 11,985. Derive the equilibrium output xB and price pB in this industry under Bertrand competition.
(b) What is the highest recurring fixed cost FC that would sustain at least one firm producing in this industry?