Oligopolies with Misspecified and Uncertain Price Functions, and Learning - Nonlinear Oligopolies: Stability and Bifurcations - page 258

and

T

b D .1;:::;1/:

By using again the results of Appendix E we can see that the eigenvalue equation of
the coefficient matrix is

N C 1 " 1 C

#

1

N N a k

N

Y X

a k

C 1

D 0: (5.121)

a k
N

1

C 1

k D 1 k D 1

By repeating the proof of Theorem 2.1 we can easily show the following result.

Theorem 5.6. The steady state is globally asymptotically stable if and only if for
all k ,

a k <2.N C 1/

(5.122)

and

X

N

a k
2.N C 1/ a k

<1: (5.123)

k D 1

In the symmetric case a k a, relations (5.122) and (5.123) reduce to

a<2.N C 1/

and

a<2; (5.124)

where the second inequality is the stronger of the two. Therefore the steady state is
globally asymptotically stable if and only if (5.124) holds.
Consider next the continuous time model (5.92). Its coefficient matrix is J with
eigenvalue equation

N C 1 " 1 C

#

N N a k

N

Y X

a k

C 1

D 0: (5.125)

a k
N

C

C 1

k D 1 k D 1

By repeating the proof of Theorem 2.2 the following stability result is obtained.

Theorem 5.7. The steady state is always globally asymptotically stable.

Global asymptotic stability means that regardless of how inaccurate the initial
estimations of parameter A are, as t !1 ; the estimates always converge to the true
value of A.
We will next show that this nice stability property may be lost, if the firms obtain
delayed price information. Assuming continuously distributed time lags and using
the same weighting functions as in Sect. 2.6, the dynamical system (5.92) becomes