Modified and Extended Oligopolies - Nonlinear Oligopolies: Stability and Bifurcations - page 172

It is straightforward to calculate that T j D .3/ has the unique fixed point

L; ;

NL
1 ˇ S

E 3 D

.3/ ,

which is an equilibrium for the dynamical system (4.47) provided that E 3 2 D

that is if z .E 3 />L. This condition is equivalent to

L< A.1 ˇ S /.N 1 C ˇ S /
cN 2

:

.2/ , so it is not

T j D .2/ has the unique fixed point O D .0;0/ which is not inside D

an equilibrium of the dynamic process.
The fixed points of T j D .1/ are the solutions of the algebraic system

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

x D

From the second equation we get x D .1 ˇ S /S=N and after substituting this
expression into the first equation we get

A
Nc ..N 1/.1 ˇ S /S C Nˇ S S/ D S 2 :

Using these relations it is easy to see that T j D .1/ has the unique fixed point

E 1 D x 1 ; S 1 with

A.1 ˇ S /.N 1 C ˇ S /
N 2 c

A.N 1 C ˇ S /
Nc

S 1 D

x 1 D

, .

.1/ ,
in other words if 0< z .E 1 /<L. Using the fact that z .E 1 / D x 1 , this condition
becomes

2 D

This fixed point is also an equilibrium of the map T provided that E 1

L> A.1 ˇ S /.N 1 C ˇ S /
cN 2

:

Since

A.1 ˇ S /.N 1 C ˇ S /

cN 2 < A

4c
always holds, being equivalent to .N 2.1 ˇ S // 2 >0, we can summarize the
existence results for an equilibrium as follows:

If L> A

.3/ is empty and the unique steady state is E 1 .

4c , then the region

D