General Oligopolies - Nonlinear Oligopolies: Stability and Bifurcations - page 116

.k/ where the map T has different expressions.
As observed in Chap. 2, the curves that divide these regions are curves of non-
differentiability, and these curves may play the role of folding curves (or critical
curves, following the terminology used in Mira et al. (1996)). In order to write the
expression of the map T in the different regions

D D Œ0;L 1 Œ0;L 2 into regions D

.k/ , notice that for the set of
parameters considered, in the expression of the reaction curve (3.12) of firm 1 we
have

D

z 1 <0 for x 2 > 16

5.N 1/ ;
whereas the constraint z 1 >L 1 is ineffective since L 1 >A=.4c 1 /. Likewise, for the
reaction function R 2 of firms 2;:::;N,wehave

z 2 <0 for x 1 C .N 2/x 2 > 8=3

and the constraint z 2 >L 2 is ineffective since L 2 >A=.4c 2 /. The lines x 2 D

16
5.N

83x 1
3.N

D D D

.1/ ,

and x 2 divide the strategy space into 4 regions. In region

1/ 2/

16
5.N

and x 2 < 8 3x 1
3.N

where x 2 <

2/ ,wehave 1/

x 1 .t C 1/ D .1 a 1 /x 1 .t/ C a 1 q 16.N 1/x 2 .t /

5 .N 1/x 2 .t/ ;
x 2 .t C 1/ D .1 a 2 /x 2 .t/ C a 2 h q 16.x 1 .t / C .N 2/x 2 .t //
6

8
<

T j D .1/ W

x 1 .t/

:

.N 2/x 2 .t/ i :

and x 2 > 8 3x 1
3.N

.2/ ,wherex 2 < 16

5.N

In region D

2/ ,themapis 1/

8
<

x 1 .t C 1/ D .1 a 1 /x 1 .t/ C a 1 q 16.N 1/x 2 .t /
5

.N 1/x 2 .t/ ;

T j D .2/ W

:

x 2 .t C 1/ D .1 a 2 /x 2 .t/:

16
5.N

and x 2 > 8 3x 1
3.N

D

.3/ ,wherex 2 >

In region

2/ ,themapis 1/

x 1 .t C 1/ D .1 a 1 /x 1 .t/;
x 2 .t C 1/ D .1 a 2 /x 2 .t/:

T j D .3/ W

.4/ ,wherex 2 > 16

5.N

and x 2 < 8 3x 1
3.N

In region D

2/ ,wehave 1/

8
< x 1 .t C 1/ D .1 a 1 /x 1 .t/;

x 2 .t C 1/ D .1 a 2 /x 2 .t/ C a 2 h q 16.x 1 .t / C .N 2/x 2 .t //
6

x 1 .t/

T j D .4/ W

.N 2/x 2 .t/ i :

: