## Examples and exercises on comparisons of the Nash equilibrium of Cournot's model, the competitive output, and the monopoly output

Each of two 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. Compare the Nash equilibrium of Cournot's model with the competitive outcome and the monopoly outcome.
We know that in the unique Nash equilibrium each firm's output is 30.

In a long run competitive equilibrium the price is equal to the minimum average cost, which is 30. Thus the total output of the firms is *Q* such that 30 = 120 *Q*, or *Q* = 90.

A monopolist chooses the output for which MC is equal to MR. We have TR(*y*) = *y*(120 *y*), so MR(*y*) = 120 2*y*. Thus the monopolist's output *y* satisfies MC(*y*) = 30 = 120 2*y*, or
*y* = 45.

In summary, the three outcomes are given in the following table.

| Total output | Price |

Competition | 90 | 30 |

Duopoly | 60 | 60 |

Monopoly | 45 | 75 |

Copyright © 1997 by Martin J. Osborne