Overview and Directions for Future Research - Nonlinear Oligopolies: Stability and Bifurcations - page 270

function. We focus on the case of linear price and cost functions and consider three
scenarios in which firms have different types of partial information about some
parameters of the price function and seek to learn about the remaining parameters
by some adjustment process. Again via specific examples we see that these learning
schemes can generate the type of local and global bifurcations seen in the previ-
ous chapters. Finally we conclude this chapter with a brief discussion of uncertain
price functions which brings us to the edge of the field of statistical learning, which
presents a whole different field of research.
From the point of view of nonlinear dynamical systems the topic has introduced
the still relatively new (at least for economists) concept of border collisions and
illustrated its use in a number of examples. The examples have emphasized how
there are in fact two types of complexity of importance in dynamic economic mod-
els. The first is the familiar one arising as a result of local bifurcations, which
frequently occur when equilibria lose local stability via Hopf or flip bifurcations
and local stability of an equilibrium gives way to some sort of fluctuation around
it. The other, less familiar one, arises when a border collision occurs, and basins of
attraction of different equilibria undergo a change in their structure. Also there may
be co-existing attractors within the same basin of attraction. A typical result of such
bifurcations is that the outcome of the economic adjustment process under consid-
eration may be highly sensitive to initial conditions. Future research in economic
applications in this area will probably focus on the systematic description of the dif-
ferent sources of such bifurcations, the elaboration of the types of examples where
such border collision bifurcations can occur and the typical sorts of behavior that
can emerge from them. It would also be useful to try to understand the economic
origins of the different types of such bifurcations.
With regard to the specific models we have studied in this topic, a number of
issues are likely to occupy the attention of researchers in the years ahead.
Considering first the case of concave oligopolies, we have derived most of our
results under the assumptions (A)-(C) in Sect. 2.1 (or their modifications in different
model types) which we recall placed restraints on the inverse demand function and
cost functions so as to guarantee the concavity of the profit function and in the
concave case the monotonicity of the best response functions. An important task
for future research will be to study the implications of relaxing any one of these
assumptions. We have seen in Example 1.2 that just by relaxing the condition (C)
how more complicated equilibrium situations can arise.
A number of our examples involved the isoelastic price function, which is widely
used in the literature on oligopoly because it affords a lot of analytical tractability.
However this price function has the disadvantage that it has no reservation price (or
rather the reservation price is infinite) and this is rather unrealistic. Future research
should try to introduce a reservation price into this price function, either by using a
translated hyperbola, or by truncating the function at some (presumably high) price.
The isoelastic price function has also been used frequently in conjunction with a
convex cost function, but what would happen if we were to allow a certain amount
of concavity into the cost function? This could arise for instance if there were an
increasing return to scales effect at low outputs and a decreasing returns to scale