Oligopolies with Misspecified and Uncertain Price Functions, and Learning - Nonlinear Oligopolies: Stability and Bifurcations - page 260

for all k. By applying relation (E.3) we see that the determinantal equation simplifies
to

( N C 1
a k

m k C1 )

C 1 1 C

Y

N

T k
p k

kD1

2 3

X

N

1

4 5 D 0:

1 C

(5.128)

N C 1
a k

C 1 1 C

m k C1

T k
p k

k D 1

The roots of the first part of the left hand side are

a k
N C 1

p k
T k ;

D and D

for k D 1;2;:::;N, all are negative, therefore we should examine only the roots of
the second part. So we turn our attention to the equation

X

N

1

1 C

D 0:

(5.129)

N C 1
a k

C 1 1 C

m k C1

T k
p k

kD1

This is clearly equivalent to a polynomial equation, the roots of which can be
determined by using computational methods in the general case. In order to obtain
analytical results we will consider the case of symmetric firms, when the initial
states are identical, a k

a;T k T;m k m and so p k p. In this case (5.129)

is reduced to

N C 1
a

C 1 1 C

mC1

T
p

C N D 0: (5.130)

Assume first that T D 0, that is, there is no time delay. Then (5.130) assumes the
linear form

N C 1
a

C 1 C N D 0

with the only root D a, so the steady state is globally asymptotically stable.
Note that this result is a special case of Theorem 5.7. Assume next that T>0.
Forlargervaluesofm computational methods are needed to locate the eigenvalues
and check stability. For m D 0 and m D 1; analytical methods are available, and
the following theorem can be proved.

Theorem 5.8. If m D 0 , then the steady state is always globally asymptotically
stable. In the case of m D 1 we have the following possibilities:

(i) If N<8; then the steady state is always globally asymptotically stable,