The Classical Cournot Model - Nonlinear Oligopolies: Stability and Bifurcations - page 11

x 2

L 2

A c 1 .B C e 1 /L 1

B

x 1

L 1

A c 2 .B C e 2 /L 2
B

Fig. 1.7 Example 1.2; the Cournot model with linear price function and quadratic cost function in
the case of duopoly .N D 2/. The figure shows case (iv) when e k < B and the existence of two

equilibria with convex profit functions

In the duopoly case .N D 2/ the Nash equilibrium is at the intersection of
the two best response functions. The number of equilibria can be 1, 2 or 3
depending on the relative order of magnitude of the values (A c k .B C
e k /L k )/B and L l .l ¤ k/. In Fig. 1.7 we show the case of two equilibria .L 1 ;0/
and .0;L 2 /.
Notice that in all cases at x k D 0 the profit of firm k is zero, therefore at the best
response it has to be non-negative. Hence, at any equilibrium the profit of each firm
is also non-negative.

Example 1.3. Consider again the duopoly in which N D 2, furthermore take L 1 D
L 2 D 1:5, C 1 .x 1 / D 0:5x 1 , C 2 .x 2 / D 0:5x 2 and assume that the price function is
given by

8
<

1:75 0:5Q if 0 Q 1:5;
2:5 Q

f.Q/ D (1.13)

if 1:5 Q 2:5;

:

0 if Q 2:5:

Notice that the cost functions are linear but that the price function is piece-wise
linear. Because of the kink in the price function the profit functions are not differen-
tiable at Q D 1:5. By calculating and comparing the left and right hand derivatives
of the profit function, it is easy to show that there are infinitely many equilibria and
they form the set

X Df .x 1 ; x 2 / j 0:5 x 1 1; 0:5 x 2

1; x 1 C x 2 D 1:5 g :

Notice that the total output of the two firms is unique, satisfying x 1 C x 2 D 1:5,
but this total output can be divided between the two firms in infinitely many