General Oligopolies - Nonlinear Oligopolies: Stability and Bifurcations - page 124

Then Lemma F.2 implies that the equilibrium is always locally asymptotically
stable.

Example 3.6. Assume in the previous example that N D 2, a 1 D a 2 D a.Then

(3.18) simplifies to

C 2a C a 2 .1 r 1 r 2 / D 0:
From Example 3.2 we know that r 1 r 2 <0if c 1 ¤ c 2 . In this case both the linear and
constant coefficients are positive (as in the general case of the previous example),
and the discriminant is

2

4a 2 4a 2 .1 r 1 r 2 / D 4a 2 r 1 r 2 <0:

So both roots are complex, showing that there is no guarantee that the eigenvalues
are real, contrary to the case of concave oligopolies discussed in Sect. 2.5.

The topic by Okuguchi and Szidarovszky (1999) contains some stability results
in the case of linear cost functions. A detailed stability analysis is presented by
Chiarella and Szidarovszky (2002) for the general nonlinear case. Models with con-
tinuously distributed time lags are identical to the concave case, so the derivations
and the similar results are not duplicated here.

3.2 Cost Externalities and Multiple Interior Nash Equilibria

In Chap. 2 we demonstrated that under some standard assumptions on the demand
function and on the cost functions of the oligopolists, the reaction functions of
the firms are decreasing. However, there are several situations where the microe-
conomic fundamentals of an oligopoly model lead to reaction functions which
are non-monotonic. For example, in the previous subsection we have shown that
with isoelastic price functions the reaction functions are increasing over the range
where the expected aggregate quantity of the other players is small, otherwise it is
decreasing (see also Example 1.5 and Bulow et al. (1985 b )). Using non-monotonic
reaction functions, several authors have considered the best response dynamics and
the partial adjustment towards the best response and have demonstrated that such
adjustment processes may lead to non-convergence with complicated, but bounded
fluctuations of the production sequences (for example, Rand (1978), Dana and
Montrucchio (1986), Witteloostuijn and Lier (1990) and Puu (1991)). The focus of
these contributions has been mainly towards questions of local stability of the Nash
equilibria and the creation of complex attractors if convergence to an equilibrium
fails. The emphasis of the analysis is, in this case, on the delineation of a trapping
region in the space of production quantities, where the asymptotic dynamics of the
oligopoly game are ultimately bounded.
In the present subsection we will turn our attention to externalities in the cost
functions, which might also give rise to non-monotonic reaction functions (see
Example 1.6, Kopel (1996), Puhakka and Wissink (1995), Bischi and Lamantia