(Nash equilibrium of Cournot’s game with small firms) Suppose that there are infinitely many…

(Nash equilibrium of Cournot’s game with small firms) Suppose that there are infinitely many firms, all of which have the same cost function C. Assume that C(0) = 0, and for q > 0 the function C(q)/q has a unique minimizer q; denote the minimum of C(q)/q by p. Assume that the inverse demand function P is decreasing. Show that in any Nash equilibrium the firms’ total output Q∗ satisfies

(That is, the price is at least the minimal value p of the average cost, but is close enough to this minimum that increasing the total output of the firms by q would reduce the price to at most p.) To establish these inequalities, show that if P(Q∗) ∗ + q) > p then Q∗ is not the total output of the firms in a Nash equilibrium, because in each case at least one firm can deviate and increase its profit.