Concave Oligopolies - Nonlinear Oligopolies: Stability and Bifurcations - page 72

Therefore

@R 1
@x 1 D 0 and

@R 1
@x 2

N 1
2

is either 0 or :

Similarly,

@R 2
@x 1

1
2 ;

is either 0 or

and

@R 2
@x 2

N 2
2

is either 0 or :

Therefore the Jacobians in the different regions have the forms
1 a 1

! 1 a 1 ! ;

a 1 .N 1/
2

a 1 .N 1/

D

2

a 2 a 2 .N 2/ a 2 a 2 N

2

1 a 2 1

2

1 a 1 !

a 1 .N 1/
2

a 2

1 a 2

and a form in which one or both of the off-diagonal elements are equal to zero.

x0
01

we see that this

Using the row norm generated by the diagonal matrix P D

norm of all possible Jacobians is below one if

a 1 .N 1/x
2

1 a 1 C <1;

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ

a 2
2x C

a 2 N
2

1 <1;

and a 2
2x C 1 a 2 <1:
The first inequality can be rewritten as

2
N 1 ;

x<

and the third condition can be simplified to

x> 1

2 :

The second inequality is equivalent to

x> 1
N

and a 2 <x.4 Na 2 /: