(Bertrand’s duopoly game with different unit costs) Consider Bertrand’s duopoly game under a…

(Bertrand’s duopoly game with different unit costs) Consider Bertrand’s duopoly game under a variant of the assumptions of Section 3.2.2 in which the firms’ unit costs are different, equal to c1 and c2, where c1 2. Denote by pm 1 the price that maximizes (p − c1)(α − p), and assume that c2 1)(α − p) is increasing in p up to pm 1 .

a. Suppose that the rule for splitting up consumers when the prices are equal assigns all consumers to firm 1 when both firms charge the price c2. Show that (p1, p2) = (c2, c2) is a Nash equilibrium and that no other pair of prices is a Nash equilibrium.

b. Show that no Nash equilibrium exists if the rule for splitting up consumers when the prices are equal assigns some consumers to firm 2 when both firms charge c2.