Overview and Directions for Future Research - Nonlinear Oligopolies: Stability and Bifurcations - page 269

Chapter 3 considered general oligopolies and started with an analysis of the case
of isoelastic price functions under both continuous time and discrete time adjust-
ment processes. Again the results of Appendix E were invoked to analyze the local
stability and we saw that for semi-symmetric oligopolies in both the discrete time
and continuous time cases this is determined qualitatively by the same set of graphs,
though of course quantitatively the two cases differ. With regard to the global anal-
ysis of the discrete time model we saw the richness of the local bifurcations with
respect to the number of firms, the cost ratio and speeds of adjustment in the semi-
symmetric case. We found also that the speeds of adjustment played a role in the
generation of border collisions and hence global bifurcations. The remainder of the
chapter considered the role of cost externalities that are captured by the assump-
tion of a certain type of non-monotonic reaction function. Here we focused on the
duopoly case and saw that such models can generate situations of several coexist-
ing equilibria that are locally stable, each having its own basin of attraction. In the
case of identical speeds of adjustment we were able to analyze and understand in
some detail the way in which the different equilibria can be born and the way in
which the structure of their basins of attraction change with key parameters, due
mainly to the occurrence of contact bifurcations. Some numerical examples of the
non-identical speed of adjustment situation illustrate how the basins can become
even more complex in this case. The disconnected nature of the basins of attraction
means that the outcome (in the sense of to which equilibrium the game converges) of
oligopolies with cost externalities is highly path dependent. These examples convey
in a very clear way the important distinction between local bifurcations and global
bifurcations.
In Chap. 4 we apply the analysis of the first three chapters to a number of models
that are an extension of the basic oligopoly set-up or are dynamic economic games
that essentially reduce to classical oligopolies. These are market share attraction
games, labor-managed oligopolies, oligopolies with intertemporal demand inter-
action, oligopolies with production adjustment costs and oligopolies with partial
cooperation amongst the firms. Such extensions of the basic oligopoly model and
dynamic economic games exhibit the range of behaviors observed in the basic
oligopolies of the previous chapters.
Finally in Chap. 5 we considered learning behavior under incomplete knowledge
of the demand relationship. We started by considering oligopolies in which firms
have misspecified price functions but otherwise we still use the adjustment pro-
cesses of Chap. 1. Now the possibility of subjective equilibria arises, the local and
global dynamics of which are studied through some examples that illustrate how
such subjective equilibria are born, their local stability properties and the (some-
times complicated) nature of their basins of attraction. Next we assume that firms
use some kind of approximate learning procedure to resolve their incomplete knowl-
edge of the price function. Using a number of specific examples we study the local
stability of the equilibria and how its loss can give rise to fluctuating attractors as
various parameters change. Global analysis indicates how the learning scheme can
affect the basin of attraction of a stable equilibrium. Next we study other types
of learning schemes by firms as they try to determine the true shape of the price