Concave Oligopolies - Nonlinear Oligopolies: Stability and Bifurcations - page 90

whereweassumethat j ¤ 0 for all j. This is equivalent to a polynomial equation
of degree s,sothereares real or complex roots. Let g./ denote the left hand side,
then clearly

g./ D 1; g./ D˙1 ;

lim lim

!˙1 ! a j .1 C r j / ˙ 0

and

X

s

j
.a j .1 C r j / C / 2 <0:

g 0 ./ D

j D1

The graph of g./ is the same as the one shown earlier in Fig. 2.1 with the only
difference being that the poles are now the values a j .1 C r j /.j D 1;2;:::;s/.
Since all poles are negative, all roots have to be real and negative. This observation
implies the assertion.

The result of this theorem can also be obtained directly from the proof of Theo-
rem 2.1. Notice that the Jacobian (2.46) can be written as H I ,where H is given
in (2.20) and I is the identity matrix. Therefore the eigenvalues of the Jacobian of
the continuous time case can be obtained by subtracting one from the eigenvalues of
the discrete time case. Since the eigenvalues in the discrete time case are less than
unity, all eigenvalues of the continuous time case have to be negative.
So far conditions (A)-(C) (or (A),(B),(C') and (D)) have been (see Sect. 2.1)
assumed to hold in the entire feasible set of x k and Q k , and they have implied the
existence of a Nash equilibrium. If the oligopoly has an interior equilibrium, then
these conditions need to be satisfied only in a neighborhood of this equilibrium in
order to guarantee its local asymptotic stability. In comparing Theorems 2.1 and 2.2
we notice that in the discrete time case asymptotic stability can be lost if one or
more firms change their adjustment schemes so that the conditions of Theorem 2.1
no longer hold. In the continuous case the equilibrium is always locally asymptot-
ically stable, thus we see that the asymptotic behavior of the equilibrium is much
richer in the discrete case. In the continuous case asymptotic stability cannot be
lost by changing adjustment schemes, however - as we will demonstrate in the next
session - it can be lost if the firms have only delayed information to which to react,
or they respond to certain averaged past information.
Finally we mention that several linear extensions and modifications of the main
result of this section can be found in Okuguchi and Szidarovszky (1999) especially
for multiproduct oligopolies. Al-Nowaihi and Levine (1985), Dixit (1986) and Furth
(1986) introduced adjustment processes based on the marginal profits of the firms
by requiring that x k for all times must have the same sign as the marginal profit.
These gradient adjustment processes have been briefly discussed at the end of the
previous chapter. Bellman (1969) offers a comprehensive background in the stability
theory of ordinary differential equations. Global analysis of the asymptotic stability
of continuous time systems is usually based on Lyapunov theory (see Appendix A)