Oligopolies with Misspecified and Uncertain Price Functions, and Learning - Nonlinear Oligopolies: Stability and Bifurcations - page 221

5.1.3 Continuous Time Models

Consider the continuous model (5.7) and assume that conditions (A)-(C) hold,
furthermore h k >0. We will first prove the following result.

Theorem 5.2. Under assumption (A)-(C) and by assuming that h k >0 and a k >0
for all k , the equilibrium is always locally asymptotically stable under the continu-
ous adjustment process (5.7).

Proof. The Jacobian of system (5.7) can be written as

0 1

a 1 Œr 1 .h 1 1/ 1

a 1 r 1 h 1 a 1 r 1 h 1

@ A

a 2 Œr 2 .h 2 1/ 1

a 2 r 2 h 2 a 2 r 2 h 2

(5.34)

: : : : : : : : :

: : :

a N r N h N a N r N h N a N Œr N .h N 1/ 1

which is a straightforward extension of the Jacobian (2.46) of the full information
case, since if f k D H k f for all k,thenH k is the identity map with h k D 1.

The eigenvalue equation has now the form

" 1 C #

N N

Y X

a k r k h k
a k .1 C r k /

Πa k .1 C r k / D 0: (5.35)

k D 1 k D 1

This equation is equivalent to (2.48) with the only difference that the j values are
now the sums of numerators a k r k h k with identical denominators.
Under conditions (A)-(C) of Sect. 5.1.1, Theorem 2.2 remains true, that is, the
equilibrium is locally asymptotically stable.

The case of isoelastic price function also can be examined in the same way as
was demonstrated in Chap. 3 for the full information case, and the conclusions are
also identical.
The global asymptotic stability of the equilibrium based on Lyapunov functions
can be similarly discussed to the full information case. The details are omitted.
We will only examine the effect of delayed information on the stability of the
equilibrium.
Assume next that there is a time delay in obtaining and implementing information
on the market price that is used in (5.2) by the firms to form their expectations on the
output of the rest of the industry. By assuming the same type of weighting function
as in Sect. 2.6 for the full information case, the dynamic model (5.7) becomes

Z t

w .t s;T k ;m k /f N x l .s/ ! ds ! x k .t/ ! x k .t/ ! ;

X

f 1
k

R k

x k .t/ D ˛ k

0

l D 1

(5.36)