Concave Oligopolies - Nonlinear Oligopolies: Stability and Bifurcations - page 62

Therefore the equilibrium is globally asymptotically stable if for all k and all
feasible output levels,

1 a k .1 C r k .N 1// q<1

(2.31)

for some positive q. All subregions of the feasible output set are compact and
on each of them the functions R k and ˛ k are continuously differentiable, so the
function 1 a k .1 C r k .N 1// attains its maximum value in each subregion.
Since there are only finitely many subregions, condition (2.31) can be weakened
by assuming that for all k, and all feasible output levels,

1 a k .1 C r k .N 1// < 1;

that is,

1
N 1 :

r k > (2.32)

Notice that this is a sufficient condition for global asymptotic stability.
By assuming no cost externalities and using (2.5), this sufficient condition holds
for all feasible output levels if and only if

.N 2/.f 0 C R k f 00 / C .C 0 k f 0 />0: (2.33)

Conditions (B) and (C) imply that the first term is always non-positive and the
second term always positive. In the case of duopoly N D 2, then (2.33) holds, so
the equilibrium is globally asymptotically stable. If we have a triopoly N D 3,then
(2.33) can be written as

R k f 00 C C 0 k >0;

which is not guaranteed to be satisfied. If N becomes larger, then the first term
on the left hand side of (2.33) converges to negative infinity if f 0 C R k f 00 is not

identically zero, so with a larger number of firms condition (2.33) is violated.
In such cases we might try to apply different matrix norms, for example row or
column norms generated by special diagonal matrices, similar to the examples dis-
cussed in the previous chapter. The choice of an appropriate norm depends on the
problem and its existence is not guaranteed. This fact raises the need to develop and
apply more sophisticated methods for the global analysis of the nonlinear oligopoly
models that we will encounter in this topic. The need for more advanced meth-
ods combining numerical, analytical, and geometrical arguments is also underlined
by the fact that neither local analysis, nor the above described sufficient global
asymptotic stability condition can be used in the case of non-differentiable best
responses.