The Classical Cournot Model - Nonlinear Oligopolies: Stability and Bifurcations - page 17

where z k

is the solution of the equation

A 2B z k BQ k M k .Q k / D 0

inside the interval .0;L k /.Thatis,

A BQ k M k .Q k /
2B

z k D :

We mention here that for an arbitrary value of Q k , the profit of each firm k is zero
with x k D 0, so the payoff at the best response also must be non-negative. Hence at
any equilibrium the firms have non-negative profit values.
If M k .Q k / is a linear function, then z k is also linear in Q k ,soR k .Q k / is a
piece-wise linear function similar to Example 1.1. If we assume that M k .Q k / is a
quadratic function, then z k is also quadratic in Q k . Thus if we write

M k .Q k / D ˛ k C ˇ k Q k C k Q k ;

then

.A ˛ k / C . B ˇ k /Q k k Q k
2B

z k

D :

Let k >1be a given constant and select

˛ k D A; ˇ k D B.1 C 2 k / and k D 2B k ;

then we have the relatively simple form

z k

D k Q k .1 Q k /:

Example 1.7. Consider again the oligopoly of the previous example with the only
difference being that the marginal cost of each firm k is a hyperbola of the form

c k
1 C k Q k :

M k .Q k / D

In this case R k .Q k / has the same structure as in the previous example with

A BQ k :

A BQ k M k .Q k /
2B

1
2B

c k
1 C k Q k

z k D D

In Chap. 3 we will give a detailed analysis of this example.

In our last example we show an oligopoly for which no equilibrium exists.

Example 1.8. Consider the case of two firms, N D 2, with capacity limits L 1 D L 2 D

0:5, linear price function f.Q/ D 1 Q with Q D P kD1 x k , and discontinuous cost