General Oligopolies - Nonlinear Oligopolies: Stability and Bifurcations - page 110

It is clear that the second inequality is always fulfilled. The properties of the stability
region for .a 1 ;a 2 / depend on the number of firms and the ratio of the firms' unit
costs.
Instead of giving a complete analysis in general we reconsider the duopoly case
of Example 3.2, where a 1

D a 2 D a. In this special case the conditions (3.6)-(3.8)

further simplify to

2a C a 2 .1 r 1 r 2 /<0;
1 r 1 r 2 >0;

and

4 4a C a 2 .1 r 1 r 2 />0:
The second and third inequalities are always satisfied, since in Example 3.2 we have
shown that r 1 r 2 <0:The first relation holds if and only if

a.1 r 1 r 2 /<2:

This condition is the same as the one that was obtained earlier in Example 3.2.

The case of linear cost functions is examined in detail in the topic of Okuguchi
and Szidarovszky (1999) and Puu (2003).

3.1.2 Global Dynamics of Discrete Time Models

As we have seen in the discussion in Chap. 2 on concave oligopolies, the conditions
for global asymptotic stability are very restrictive. In most cases of isoelastic price
functions this is true as well.
Under condition (D) of Sect. 3.1, for at most one firm r k >0, and for all other
firms, 1<r k

0.Ifallr k values are non-positive, then the global stability condi-
tions are still given by (2.31). However if one r k is positive, this condition can no
longer be used, it has to be modified accordingly.
We also notice that the global stability condition given in Theorem B.3 cannot
be applied either. At Q k D 0,firmk has no best response, which is clear from its
definition given in Example 1.5 and in the first part of Sect. 3.1. Therefore the set
where the dynamical system

x k .t C 1/ D x k .t/ C ˛ k .R k .Q k .t// x k .t//; .k D 1;2;:::;N/;

is defined is not closed, so the contraction mapping theorem (upon which the proof
of Theorem B.3 relies) cannot be used. If we consider the continuous extension
by defining R k .0/ D 0, then in addition to the Nash equilibrium the zero output
vector also becomes a steady state of the above dynamical system, so the presence
of multiple steady states excludes the possibility of global asymptotic stability.