(Cournot’s game with many firms)

Consider Cournot’s game in the case of an arbitrary number n of firms; retain the assumptions that the inverse demand function takes the form (56.2) and the cost function of each firm i is c(q) = (c/2) *q^2, with c < a. Find the best response function of each firm and set up the conditions for (q ? 1 , . . . , q? n ) to be a Nash equilibrium, assuming that there is a Nash equilibrium in which all firms outputs are positive. Solve these equations to find the Nash equilibrium. (First show that in an equilibrium all firms produce the same output, then solve for that output. If you cannot show that all firms produce the same output, simply assume that they do.) Find the price at which output is sold in a Nash equilibrium and show that this price decreases as n increases as the number of firms increases without bound.

P(Q) = a - Q if Q <= a

P(Q) = 0 if Q >a (56.2)