Concave Oligopolies - Nonlinear Oligopolies: Stability and Bifurcations - page 82

x 2

2

L 2 + A - c 2 -2 L 2

N −2

L 2

D (3) D (4) D (5)

A − c 1
N −1

D (2) D (1) D (6)

2

L 1 + A - c 1 -2 L 1

N −1

D (9) D (8) D (7)

x 1

L 1

2
L 2 + A - c 2 -2 L 2

A − c 2

Fig. 2.16 Example 2.5; quadratic price and linear cost functions. The regions of the piece-wise
map in the semi-symmetric case

Also in this case the strategy space D D Œ0;L 1 Œ0;L 2 can be subdivided into

.i / , as shown in Fig. 2.16. In each of these regions the
map T is then differentiable. For example, in regions

up to nine different regions D

.1/ and .2/

D D we have

8
<

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

C a 1 3 q .N 1/ 2 x 2 C 3.A c 1 / 2.N 1/x 2 ;

T j D .1/ W x 2 .t C 1/ D .1 a 2 /x 2 .t/

C a 2 3 h q .x 1 C .N 2/x 2 / 2

:

C 3.A c 2 /

2.x 1 C .N 2/x 2 / i ;

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

C a 1 3 q .N 1/ 2 x 2 C 3.A c 1 / 2.N 1/x 2 ;
x 2 .t C 1/ D .1 a 2 /x 2 .t/ C a 2 L 2 ;
and the corresponding Jacobian matrices are given by

T j D .2/ W

:

0
@

1/

1
A

1/x 2
3 p .N 1/ 2 x 2 C 3.A c 1 /

.N

2
3

1 a 1 a 1 .N

J .1/

a 2

D ;

j .1/
22

x 1 C .N 2/x 2
3 p .x 1 C .N 2/x 2 / 2

2
3

c 2 /

C 3.A

D 1 a 2 C a 2 .N 2/

:

where j .1/
22

x 1 C .N 2/x 2

3 p .x 1 C.N 2/x 2 / 2 C3.Ac 2 /

2
3