The Classical Cournot Model - Nonlinear Oligopolies: Stability and Bifurcations - page 25

x k .t/ D a k B X

l

x l .t/ 2Bx k .t/ C A c k

¤ k

D 2Ba k

;

2 X
l¤k

A c k
2B

1

x l .t/ x k .t/ C

which is the same as system (1.26) with a k replaced by 2Ba k .

The dynamical behavior of these adjustment process systems largely depends on
the type and the parameters of the adjustment schemes as well as on the analytical
properties of the best response functions, which in turn depend on the shapes of the
price and cost functions.
There has been some criticism of the modeling of boundedly rational firms in
dynamic oligopoly models using the previously discussed adjustment processes (see
for example, Friedman (1977, 1982)). The essence of the criticism is that the firms
ignore the fact that their current actions will have an impact on the future actions
of the competitors (that is the limit of the adjustment process itself may not be an
equilibrium of the repeated game). Therefore, it has been suggested that it would be
more reasonable to assume that firms operating in markets over many time periods
would seek to maximize a discounted stream of profits over a finite or infinite time
horizon taking the strategic behavior of their competitors into account. Beside the
fact that such an approach necessarily assumes a high degree of information and
rationality on the part of the firms, one justification for the interest in models of
the type studied in this topic is given by more recent results demonstrating that
myopic play is (approximately) optimal if the discount factor is very small (see Dana
and Montrucchio (1986, 1987)). Moreover, non-equilibrium adjustment processes
like the adjustment processes presented above can be shown to implicitly rely on a
combination of “lock-in” and impatience, and this may serve as a further explanation
for the players' myopia (see Fudenberg and Levine (1998), and Tirole (1988)). In
any case, in this topic we follow the argument that the kind of adjustment processes
introduced above can “... be interpreted as a crude way of expressing the bounded
rationality of agents” (Vives (1999), p. 49). Readers interested in dynamic games
where players are more rational and forward-looking might want to consult the topic
by Dockner et al. (2000) who present a variety of models and summarize many
interesting results. In this topic we will mainly concentrate on best response based
dynamic processes.

1.3 An Introduction to the Analysis of Global Dynamics

The purpose of this section is to introduce the main concepts and tools for the analy-
sis of the global properties of a discrete time dynamical system. In order to do so we
will use the example of a simple Cournot oligopoly with linear inverse demand and
quadratic costs. This example has already been introduced in Sect. 1.1 (see Exam-
ple 1.2), where we denoted the linear price function as p D f.Q/ D A BQ and