Modified and Extended Oligopolies - Nonlinear Oligopolies: Stability and Bifurcations - page 181

Theorem 4.2. If 1<r k 0 and a k >0 for all k , and furthermore ˇ S ; S >0 ,
then all roots of the eigenvalue equation (4.58) have negative real parts implying
the local asymptotic stability of the equilibrium of the system (4.31)-(4.32).

Proof. Let D A C iB denote a root and suppose that A 0.Ifg./ and h./
denote the left and right hand sides of (4.58), respectively, then

N N

X X

a k r k
A C iB C a k .1 C r k / D

a k r k .A C a k .1 C r k / iB/
.A C a k .1 C r k // 2

g.A C iB/ D

C B 2 ;

k D 1 k D 1

which has a non-positive real part under the stated assumptions. Similarly,

A C iB C S
A C iB C S C ˇ S D

.A C S C iB/.A C S C ˇ S iB/

h.A C iB/ D ;

.A C S C ˇ S / 2 C B 2

the real part of which is given by

.A C S /.A C S C ˇ S / C B 2
.A C S

>0:

C ˇ S / 2 C B 2

The contradiction implies that A<0must hold.

In the isoelastic case there is no guarantee that the r k values are non-positive. In
this case the analysis can be performed in a similar fashion to the case shown earlier
in Sect. 3.1.3. The details are not presented here.
In introducing time lags into the model (4.31)-(4.32) we assume that the firms
react to delayed information about the value of S, and that the firms have identical
delays. The more general non-symmetric case, when all information on the firms
own outputs as well as on the outputs of the competitors are also delayed, can be
discussed similarly, but the analysis becomes much more complicated.
In the simple case of delayed information about S the system (4.31)-(4.32)
becomes

0 0

@ X

1 1

x l .t/ C ˇ S Z t
0

@ R k A x k .t/ A ; (4.59)

x k .t/ D ˛ k w .t s;T;m/S.s/ds

l ¤ k

X

N

S.t/ D

x l .t/ S S.t/;

l D 1

where the weighting function is selected in the same way as in Sect. 2.6 for concave
oligopolies. Linearizing the system we obtain