General Oligopolies - Nonlinear Oligopolies: Stability and Bifurcations - page 123

Condition (3.5) for k D 1 and k D 2 is

c 1 C .N 1/c 2
N 1

c 1 C .N 1/c 2
N 1

; 2 c 1 :

The second inequality always holds, the first can be rewritten as

c 2
c 1

N 2
N 1 :

(3.17)

By introducing again the notation D c 2 =c 1 we have

N 2
N 1 ;

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

1 .N 1/
2.N 1/ :

r 1 D and r 2 D

The two-dimensional system for the adjustment of firms' outputs has the form

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

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

with Jacobian matrix

a 1 a 1 r 1 .N 1/
a 2 r 2 a 2 .r 2 .N 2/ 1/

:

The characteristic equation can be written as

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

or

2 C Œa 1 C a 2 .1 C r 2 .2 N// C a 1 a 2 Œ1 C .2 N/r 2 .N 1/r 1 r 2 D 0: (3.18)

Clearly,

1 .N 1/ N 2
N 1
2.N 1/

3 N
2.N 1/ :
Notice first that the linear coefficient of (3.18) is always positive since r 2 0. With
the new variable

r 2 D

K D .N 1/, the multiplier of a 1 a 2 in the constant term of (3.18)

has the form

1 C .2 N/ .1 K

/

.N 1/ .

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

1 K
2 2 K K

K C 1/ 2
4

1
4

.

D Œ4 K C .4 2N/.1 K / .1 K /. K C 3 2N/ D >0:

K K