General Oligopolies - Nonlinear Oligopolies: Stability and Bifurcations - page 111

In this subsection we start to investigate the kinds of dynamic behavior that
we can observe when the restrictive conditions for global stability are not satis-
fied. A characterization of the global dynamics is not trivial, since we are dealing
with an N-dimensional piecewise differentiable dynamical system. Therefore, our
study is based on a combination of analytical, geometrical and numerical arguments.
As has been demonstrated in previous chapters, qualitative changes of the dynam-
ics are often caused by contacts between singularities known as critical sets (see
Appendix C), lines of non-differentiability, and basin boundaries. In general such
contacts can only be revealed numerically, since the equations of the curves which
are involved in such contacts cannot be analytically expressed in terms of elemen-
tary functions. Hence, an analysis of global bifurcations is, in general, carried out by
using both theoretical and numerical methods. The occurrence of such bifurcations
is shown by computer-assisted proofs, and is based on the knowledge of the prop-
erties of the singularities involved and their graphical representation (see Mira et al.
(1996) for many examples and see also Brock and Hommes (1997)). This “modus
operandi” is quite common in the study of the global properties of nonlinear two-
dimensional discrete dynamical systems. However an extension of such methods
to higher-dimensional dynamical systems is obviously limited. A practical problem
which arises is that the visualization of objects in a phase space of dimension greater
than two and the detection of contacts between surfaces may become very difficult.
Consequently, in the examples that follow we will (again) restrict ourselves to the
case of duopoly or the semi-symmetric case of an oligopoly. It should be mentioned
that in the case of isoelastic demand, the non-negativity of prices is always guar-
anteed. So, in contrast to the oligopolies with for example linear or quadratic price
functions as considered before, we do not need to ensure this property by selecting
the values of the model parameters carefully. On the other hand, we still need to
look at the profits along the sequence of quantity decisions in order to see if the
long-run dynamics are viable from an economic point of view. Although the prob-
lem of negative profits is regularly neglected in the literature on complex dynamics
in oligopolies, it is a crucial element of the analysis of an adjustment type model.
The dynamical system just represents the firms' individual production decisions,
but does not directly tell us if the firms are profitable as a result of the collective
outcome.

Example 3.4. We consider again the reaction functions in the model with isoelastic
demand and linear cost functions derived at the beginning of this chapter, which in
the current example becomes

8
<

A
c k ;

z k 0, i.e., Q k

0 if

2L k

c k Q k C

z k L k , i.e., Q k C

A

L k 0;

R k .Q k / D

L k if

q AQ k
c k

:

z k D Q k otherwise;

(3.12)