The Classical Cournot Model - Nonlinear Oligopolies: Stability and Bifurcations - page 18

functions

( 10 if x k D 0;
10x k C 5 if 0<x k

C k .x k / D (1.15)

1
2 :

The higher costs at zero reflect exit barriers, which do not occur when the firms start
producing. We will show that this oligopoly has no equilibrium. On the contrary,
assume that .x 1 ; x 2 / is an equilibrium. Assume first that x 1 >0;then

' 1 .x 1 ; x 2 / D x 1 .1 x 1 x 2 / .10x 1 C 5/ D x 1

.9x 1 C x 1 x 2 C 5/

with derivative

@' 1
@x 1 .x 1 ; x 2 / D 2x 1 9 x 2 <0:
Therefore' 1 is strictly decreasing in x 1 . Assume next that x 1 D 0.Then' 1 .0; x 2 / D
10 with lim x 1 !0 C ' 1 .x 1 ; x 2 / D 0 f.Q/ 5 D 5>' 1 .0; x 2 / showing that at
x 2 ,firm1 has no best response. Hence no equilibrium exists.

1.2 Dynamic Adjustment Processes

In this section dynamic adjustment processes in the Cournot model will be intro-
duced. If all firms simultaneously select the corresponding output levels of an
equilibrium, then none of the firms can change unilaterally its output level and
increase profit. So without coordination and cooperation between the firms, the out-
put level of all firms will remain steady at the equilibrium levels. If the selected
output levels do not form an equilibrium, then at least one firm is able to increase
its profit by changing its output level unilaterally. Since the firms are rational, all
firms will do the same. Since the firms change their output levels simultaneously,
they cannot reach their best response levels, because the competitors simultaneously
move away from their previously assumed output levels at the same time. In this way
the firms usually would not reach an equilibrium, so output changes are again under-
taken, and a dynamic process develops. The model of the resulting process depends
on the assumed nature of the time scales and on the way the firms adjust output
levels, which in turn depends on their expectation formation.
In the discrete time case let t D 0;1;2 denote the time periods, then here
we shall assume that in each time period each firm changes its output level to the
best response based on its latest belief of the total production level of the rest of the
industry. This process can be written as
x k .t C 1/ D R k Q k .t C 1/ ; (1.16)

where Q k .t C 1/ is the total output of the rest of the industry expected by firm
k for the next time period t C 1. We emphasize here the fact that expectation is
not meant in its probabilistic sense, rather it is a deterministic predicted value. The