## Examples and exercises on Cournot's oligopoly model and the competitive model

### Example

Each of n firms has the cost function TC(y) = 30y; the inverse demand function for the firms' output is p = 120  Q, where Q is the total output. What are the firms' outputs in a Nash equilibrium of Cournot's model?

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.

• First find the firms' best response functions. Firm 1's profit is
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.
• We now need to find a list (y1, ..., yn) of outputs with the property that every firm's output is a best response to all the other firms' outputs. Since all the firms are identical, it is reasonable to suppose that there is a Nash equilibrium in which all their outputs are the same: y1 = y2 = ...  = yn. (Note, however, that it does not follow that every Nash equilibrium must take this form.) Denote the common output in this case y*. Then for a Nash equilibrium we need
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).
The conclusion is that a Nash equilibrium when there are n firms has each firm producing 90/(n+1) units of output.

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.