The Classical Cournot Model - Nonlinear Oligopolies: Stability and Bifurcations - page 36

where (see the reaction function in case (i) of Example 1.2)

8
<

z <0; 0 if

R..N 1/x/ D

z >L; L if

:

z

otherwise;

with z D A c B.N 1/x = 2.B C e/ :
Observe that the number of firms N enters as a parameter, so we can study the
stability conditions as N is increased. The positive equilibrium is given by

A c
B.N C 1/ C 2e

x D

and the map T is a contraction provided that j T 0 .x/ j <1,thatis

0<a BN C B C 2e
2.B C e/

<2.

This implies that the positive equilibrium is always asymptotically stable for suffi-
ciently small values of the adjustment speed a. Moreover, given 0<a 1,asymp-
totic stability is obtained for

N< .4 a/B C 2.2 a/e
aB

:

In the case of best reply dynamics, a D 1, the stability condition reads N<.3B C
2e/=B. In the case of linear costs, e D 0, we obtain the result by Theocharis stating
that asymptotic stability is obtained for N<3.
In the semi-symmetric case .N 1/ firms are assumed to be identical, whereas
one firm differs with regard to its production costs and/or initial production quantity.
Let firms 2;:::;N be identical, then their production choices will coincide in each
period, that is x k

D x 2 for all k 2. Let us denote the production quantity of firm

1byx 1 ,then

Q 1 D .N 1/x 2 and Q 2 D x 1 C .N 2/x 2 : (1.44)

By using the reaction functions R 1 and R 2 D D R N , we obtain a two-
dimensional system with state variables x 1 and x 2 . In (1.23) we set c 2 D D c N ,
e 2 D D e N ;a 2 D D a N ,andL 2 D D L N : Then the 2-dimensional
model that governs the behavior of firm 1 and the common behavior of the identical
firms 2;:::;N becomes

x 1 .t C 1/ D .1 a 1 /x 1 .t/ C a 1 R 1 ..N 1/x 2 .t//;
x 2 .t C 1/ D .1 a 2 /x 2 .t/ C a 2 R 2 .x 1 .t/ C .N 2/x 2 .t//;

T N W

where (again refer to the reaction function in case (i) of Example 1.2)