Modified and Extended Oligopolies - Nonlinear Oligopolies: Stability and Bifurcations - page 160

Notice that the condition g.0/>0 can be rewritten as

X

N

r k
1 C r k

<1;

kD1

which does not depend on the adjustment scheme, it depends only on the derivatives
of the best response functions.

Example 4.7. In the special case introduced earlier in Example 4.1 we have seen
that

1

2 <R 0 k <0,so j <0for all j. Therefore the equilibrium is always locally
asymptotically stable.

Example 4.8. Consider next the case of Example 4.3, where we have infinitely many
equilibria. By using the special form of r k given in (4.19) it can easily be proved that
zero is always an eigenvalue, so we cannot establish local asymptotic stability of the
equilibrium. Clearly this should be the case, since if the initial state is selected close
to any given equilibrium on the ray (4.18), then the state will remain there for all
future times and will not converge back to the original equilibrium. However Li et al.
(2003) have proved that the ray (4.18) is a strongly attracting set, meaning that any
point near the ray is attracted (that is, the trajectory starting at this point converges)
to some particular point on the ray. The basin of attraction contains a cone which is
centered at the ray. In order to prove this result the theory of differentiable manifolds
was used (see for example, Hirsch et al. (1977)), a topic the discussion of which
would take us beyond the scope of this topic.

Models with continuously distributed time lags can be discussed in the same way
as was done in Chap. 2. The only difference being that there is no sign restriction on
the r k values.

Example 4.9. Consider again the symmetric case described by characteristic equa-
tion (2.58). The case 1<r<0has been examined in Sect. 2.6. In the case of r D 0,
the eigenvalues are a and .p=T/, both of which are negative implying the local
asymptotic stability of the equilibrium. That leaves us to consider the case r>0.
If T D 0 then (2.58) becomes

C a .N 1/ar D 0

with solution

D ..N 1/r 1/a;
which is negative if r <1=.N 1/ implying the local asymptotic stability of the
equilibrium. If r >1=.N 1/ then the equilibrium is unstable.
If T>0and m D 0, then (2.58) becomes the quadratic equation

2 T C .1 C aT/ C a.1 .N 1/r/ D 0: