Concave Oligopolies - Nonlinear Oligopolies: Stability and Bifurcations - page 98

2˛ 4 T 2 .2aT C 2/ C 4˛ 2 T.˛ 2 T a/.aT C 2/
. 2˛ 2 T 2 / 2

Re P

D :

C 4˛ 2 .aT 2 C 2T/ 2

The numerator can be simplified to 4˛ 2 T.˛ 2 T Ta 2

2a/. Here the first factor is

positive and the second factor can be rewritten as

1 T 2 a 2
T

.1 C 2aT/T
T 2

Ta 2

2a D ¤ 0

since it is easy to see that

.aT/ 1 <1<.aT/ 2 :
Hence the conditions for a Hopf bifurcation are satisfied, giving the possibility of
the birth of limit cycles around the equilibrium.

If we consider larger values of m then (2.58) leads to higher order equations and
therefore the stability analysis becomes far more complicated, and would usually
require the use of computational methods.
However we can show that if r> 1
N

1 , then all roots of (2.58) have negative
real parts, so the equilibrium is asymptotically stable. On the contrary assume that
Re 0.Then

j C a j a and ˇ ˇ 1 C

p ˇ ˇ 1;

T

so that

ˇ ˇ . C a/ T

p C 1 mC1

ˇ ˇ a> ar.N 1/ Dj ar.N 1/ j ;

hence cannot be root of (2.58).
So far we have made the simplifying assumption that S D 0 in equation (2.57).
In order to illustrate a case when there are lags in the information on both the rivals'
and own outputs consider (2.57) with positive S and T and with m D l D 0.The
cubic equation

ST 3 C 2 .S C T/ C .1 C aT .N 1/arS/ C .a .N 1/ar/ D 0

is then obtained. All coefficients are positive, and the Routh-Hurwitz criterion
implies that the roots have negative real parts if and only if

.S C T/.1 C aT .N 1/arS/ > ST .a .N 1/ar/;

which can be rewritten as

S C T C aT 2 .N 1/arS 2 >0:

This inequality always holds, since r 0. Hence the equilibrium is always locally
asymptotically stable.