Modified and Extended Oligopolies - Nonlinear Oligopolies: Stability and Bifurcations - page 171

A
S

β

D (2) : z * < 0

S

b 0

D (1)

z 2

b

β

S

z 1
S

b 1

β

D (3) : z *> L

D (1)

x

Fig. 4.12 Example 4.11; the oligopoly model with intertemporal demand interaction and adaptive
adjustment in the discrete time case. The phase space for the N-firm symmetric model with iso-
elastic price function and linear cost functions.

.3/ is empty if L>A=.4c/. In this case the capacity
constraint is ineffective since it is larger than the maximum value attained by z
(see also Fig. 1.9). In Fig. 4.12 these regions are shown for the case L<A=.4c/.
This kind of subdivision is important for the computation of the equilibrium
points. In fact, the map in the different regions is given by the expressions

Notice that the region D

x 0 D .1 a/x C a q c ..N 1/x C ˇ S S/ ..N 1/x C ˇ S S/ ;
S 0 D ˇ S S C NŒ.1 a/x
C a q c ..N 1/x C ˇ S S/ ..N 1/x C ˇ S S/ ;

8
<

T j D .1/ W

:

( x 0 D .1 a/x;
S 0 D ˇ S S C N.1 a/x;

T j D .2/ W

( x 0 D .1 a/x C aL;
S 0 D ˇ S S C NŒ.1 a/x C aL:

T j D .3/ W