The Classical Cournot Model - Nonlinear Oligopolies: Stability and Bifurcations - page 16

and by substituting it into the above expression for x k , the equilibrium output of
firm k is given by

A.N 1/ 2 c k
P lD1 c l 2 ;

.N 1/A
P lD1 c l

x k D

and the equilibrium profit of firm k is given by

P l D 1 c l
N 1 c k

!

D A 1

! 2

Ax k
Q c k x k d k

.N 1/c k
P lD1 c l

' k D D x k d k d k :

In order to guarantee that all equilibrium outputs of the firms are positive, we have
to assume that

c k < P l ¤ k c l

N 2 ;
that is, the marginal costs cannot be too high.

The examples above considered the case in which the cost function of a firm
depends only on its own output. We will next present two particular examples
including cost externalities, with linear price and cost functions, where the fixed
costs are equal to zero and the marginal cost of each firm depends on the output of
the rest of the industry.

Example 1.6. In the case ofN firms assume a linear price function f.Q/ D A BQ,
and furthermore assume that the marginal cost of each firm is a function of the
output of the rest of the industry, M k .Q k /: If zero fixed cost is assumed, then the
cost function of firm k is given as (see Howroyd and Russell (1984), Russell et al.
(1986) and Furth (2009))

C k .x k ;Q k / D x k M k .Q k /;

so the profit of firm k is

x k .A Bx k BQ k / x k M k .Q k /;

by assuming that x k C Q k A=B. Notice that this function is strictly concave in
x k , so in the case of sufficiently small capacity limits there is a unique best response
function given by

8
<

0 if A BQ k M k .Q k / 0;

R k .Q k / D

L k if A 2BL k BQ k M k .Q k / 0;

:

z k

otherwise;