Overview and Directions for Future Research - Nonlinear Oligopolies: Stability and Bifurcations - page 271

effect at higher levels of output. Uniqueness of the equilibrium could be lost in such
situations or there may be no equilibrium at all. Also many of the local and global
stability results we have derived rely heavily on the special analytical properties of
the cost functions, so we might expect to see a much richer set of dynamic outcomes.
With regard to the modified and extended oligopolies of Chap. 4 a number of
extensions can be envisaged. The market share attraction games are equivalent
to oligopolies with isoelastic price functions, so all the remarks of the previous
paragraph apply to this class of model as well. In our analysis of labor-managed
oligopolies we assumed very special forms for the labor demand functions, but both
equilibrium results as well as the dynamic analysis will change if we consider more
general forms for these demand functions. The models with intertemporal demand
interaction were analyzed under the assumptions of the concave oligopolies so again
the relaxation of the assumptions on the price and cost functions will lead to a richer
set of outcomes for the equilibria and the dynamics. In the models with production
adjustment costs we have assumed that this additional cost component depends on
the output change from the previous period. A more realistic assumption might be
to make this cost depend on a state variable related to the capacity limit that adjusts
dynamically in such a way that the firm increases it if output needs to go beyond it.
The analysis of oligopolies under partial cooperation also relies very much on the
concavity assumptions being satisfied by the profit functions. Here also it would be
of interest to study the situations in which these concavity conditions are relaxed. It
would also be interesting to include partial cooperation into some of the extensions
described earlier in this chapter.
In the learning schemes in the models with misspecified and uncertain cost func-
tions we have adopted various assumptions on the learning behavior of the firms,
from remaining statically with the same misspecification over every time period,
to updating their estimate of it based on the most recently observed price. There
is now a vast literature on learning in dynamic economic models, see for example
Fudenberg and Levine (1998), and many of these ideas could be brought into the
problems considered in Chap. 5. Many of these schemes are probabilistic in nature
so this strand of research will involve the analysis of economic models evolving
dynamically under random influences, this is an area into which research has barely
begun as it involves bringing together the theory of dynamical systems and the
theory of stochastic processes.