General Oligopolies - Nonlinear Oligopolies: Stability and Bifurcations - page 109

where denotes the cost ratio between the firms. In addition,

.N 1/ C .3 2N/
2.N 1/

1 .N 1/
2.N 1/ :

r 1 D and r 2 D

The dynamic process can be written as

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//;

so the Jacobian has the special form
1 a 1 a 1 r 1 .N 1/
a 2 r 2 1 a 2 C a 2 r 2 .N 2/

D R 0 1 D R 0 2

where r 1 and r 2 at the equilibrium. The characteristic equation of this

matrix can be written as

.1 a 1 /.1 a 2 C a 2 r 2 .N 2/ / a 1 a 2 r 1 r 2 .N 1/ D 0;

which can be simplified to

2

C . 2 C a 1 C a 2 C .2 N/a 2 r 2 / C .1 a 1 a 2 C .N 2/a 2 r 2
C a 1 a 2 .1 C .2 N/r 2 C .1 N/r 1 r 2 // D 0:

Using results from Appendix F we know that the roots are inside the unit circle if
and only if

a 1 C a 2 ..N 2/r 2 1/ C a 1 a 2 .1 C .2 N/r 2 C .1 N/r 1 r 2 /<0; (3.6)

1 C .2 N/r 2 C .1 N/r 1 r 2 >0; (3.7)

4 2a 1 C a 2 . 2 C .2N 4/r 2 / C a 1 a 2 .1 C .2 N/r 2 C .1 N/r 1 r 2 />0: (3.8)
The form of the stability region for .a 1 ;a 2 / depends on the number of firms and the
actual values of the derivatives r 1 and r 2 . Inserting the expressions for the derivatives
r 1 and r 2 given above, the stability conditions can be written in terms of the cost
ratio D c 2 =c 1 , the number of firms N, and the adjustment coefficients a 1 and a 2
as

4a 1 .N 1/ C a 1 a 2 .1 C .N 1// 2

C 2a 2 . 2 C N.1 C N// < 0; (3.9)

.1 C .N 1// 2 >0; (3.10)

8. 2 C a 1 /.N 1/ C a 1 a 2 .1 C .N 1// 2

C 4a 2 . 2 C N.1 C N// > 0:
(3.11)