Overview and Directions for Future Research - Nonlinear Oligopolies: Stability and Bifurcations - page 268

Chapter 6
Overview and Directions for Future Research

In Chap. 1 we introduced the classical Cournot model and after setting up the general
framework we focused on a number of specific examples involving combinations of
linear and hyperbolic price functions and linear and quadratic cost functions, also
taking careful account of capacity constraints. These examples illustrated the vari-
ety of reaction functions that can occur and the various types of equilibria (possibly
multiple) both in the interior of the domain of interest and on its boundaries. We
then went on to introduce the various types of adjustment processes that under-
pin the dynamic processes, the study of the local and global dynamics of which
has occupied much of the space in this topic. In particular we considered discrete
time and continuous time versions of partial adjustment towards the best response
with naive expectations and adaptive expectations as well as the gradient adjust-
ment process. We then introduced some of the basic tools for the analysis of global
dynamics via some examples involving duopoly or symmetric and semi-symmetric
oligopolies. We introduced the important concept of basins of attraction of different
equilibria and the important tool of the critical curve and the concept of border colli-
sion bifurcations. Already with the simple examples considered we see the types of
complexity that can arise in oligopoly models under the type of dynamic adjustment
processes we consider here.
In the second chapter we considered the widely studied class of concave
oligopolies. We first obtained the properties of the reaction function both with and
without cost externalities, and then used these to study the local and global dynam-
ics of discrete time and continuous time concave oligopolies under the various
best response processes of Chap. 1. We made use of the determinantal relation in
Appendix E to obtain results on local stability. The full array of the tools for the
analysis of the global dynamics were brought to bear to obtain interesting results in
a number of special cases of price and cost functions, as well as on the nature of the
oligopoly, such as whether it is a duopoly or semi-symmetric. We saw in particular
the important role of border collision bifurcations in determining the global dynam-
ics and how the number of firms in the oligopoly, the capacity constraints of firms
and their speeds of adjustment are all important bifurcation parameters. The chapter
concluded with a study of the local dynamics of continuous time oligopolies with
continuously distributed informational time lags, and we saw how such time lags
can have a strong influence of local bifurcation behavior.