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

### Example

Each of *n* firms has the cost function TC(*y*) = 30*y*; 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
*y*_{1}(120 *y*_{1} *y*_{2} ... *y*_{n})
30*y*_{1} = *y*_{1}(120 *y*_{1} *Y*^{1}) 30*y*_{1},

where *Y*^{1} = *y*_{2} + ... + *y*_{n}, the total output of all the firms *except* firm 1. Taking the derivative of this profit with respect to *y*_{1} (holding all the other outputs constant) and setting the derivative equal to
zero we obtain
120 2*y*_{1} *Y*^{1} 30 = 0,

or
*y*_{1} = (90 *Y*^{1})/2.

Thus the best response function of firm 1 is given by
*b*_{1}(*y*_{2},...,*y*_{n}) = (90 *Y*^{1})/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 (
*y*_{1}, ..., *y*_{n}) 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:
*y*_{1} = *y*_{2} = ... = *y*_{n}. (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** = *b*_{1}(*y**,...,*y**), *y** = *b*_{2}(*y**,...*y**), ..., *y** = *b*_{n}(*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 (*n*1)*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)) = 90*n*/(*n*+1)

and the price is
120 90*n*/(*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.

Copyright © 1997 by Martin J. Osborne