In exercise 25.1, we checked how the Bertrand conclusions (that flow from viewing price as the strategic variable) hold up when we change some of our assumptions about fixed and marginal costs. We now do the same for the case where we view quantity as the strategic variable in the simultaneous move Cournotmodel.
A: Again, maintain all the assumptions in the text unless you are asked to specifically change some of them.
(a) First, suppose both firms paid a fixed cost to get into the market. Does this change the predictions of the Cournotmodel?
(b) Let xC denote the Cournot equilibrium quantities produced by each of two firms in the oligopoly as derived under the assumptions in the text. Then suppose that there is a recurring fixed cost FC for each firm (and FC does not have to be paid if the firm does not produce). Assuming that both firms would still make non-negative profit by each producing xC , will the presence of FC make this no longer a Nash equilibrium?
(c) Can you illustrate your conclusion from (b) in a graph with best response functions that give rise to a single pure strategy Nash equilibrium with both firms producing xC?
(d) Can you illustrate a case where FC is such that both firms producing xC is one of 3 different pure strategy Nash equilibria?
(e) Can you illustrate a case where FC is sufficiently high such that both firms producing xC is no longer a Nash equilibrium? What are the two Nash equilibria in this case?
(f) True or False: With sufficiently high recurring fixed costs, the Cournot model suggests that only a single firm will produce and act as a monopoly.
(g) Suppose that, instead of a recurring fixed cost, the marginal cost for each firm was linear and upward sloping—with the marginal cost of the first unit the same as the constant marginal cost assumed in the text. Without working this out in detail, what do you think happens to the best response functions — and how will this affect the output quantities in the Cournot equilibrium?
B: Suppose that the cost function for both firms in the oligopoly have the cost function c(x) = FC + (cx2/2), with demand given by x (p) = A−α p (as in the text).
(a) Derive the best response function x1(x2) (of firm 1’s output given firm 2’s output) as well as x2(x1).
(b) Assuming that both firms producing is a pure strategy Nash equilibrium, derive the Cournot equilibrium output levels.
(c) What is the equilibrium price?
(d) Suppose that A = 100, c = 10 and α = 0.1. What is the equilibrium output and price in this industry assuming FC = 0?
(e) How high can FC go with this remaining as the unique equilibrium?
(f) How high can FC go without altering the fact that this is at least one of the Nash equilibria?
(g) For what range of FC is there no pure strategy equilibrium in which both firms produce but two equilibria in which only one firm produces?
(h)What happens if FC lies above the range you calculated in (g)?