The case in which n = 2 is considered in another example. We use the same procedure to find a Nash equilibrium as we did in that case.
y1(120 y1 y2 ... yn) 30y1 = y1(120 y1 Y1) 30y1,where Y1 = y2 + ... + yn, the total output of all the firms except firm 1. Taking the derivative of this profit with respect to y1 (holding all the other outputs constant) and setting the derivative equal to zero we obtain
120 2y1 Y1 30 = 0,or
y1 = (90 Y1)/2.Thus the best response function of firm 1 is given by
b1(y2,...,yn) = (90 Y1)/2.The other firms' best response functions are the same, since all the firms' cost functions are the same.
y* = b1(y*,...,y*), y* = b2(y*,...y*), ..., y* = bn(y*,...,y*).Since all the best response functions are the same, all these equations are the same; we need to solve only one of them. Each one is
y* = (90 (n1)y*)/2,or
y* = 90/(n + 1).
In this equilibrium, the total output of the firms is
n(90/(n+1)) = 90n/(n+1)and the price is
120 90n/(n+1).As n increases, the total output thus approaches 90 and the price approaches 30, the total output and price in the long run competitive equilibrium. That is, if there is a large number of firms then the outcome in a Nash equilibrium of Cournot's model is close to the long run competitive equilibrium.