The Classical Cournot Model - Nonlinear Oligopolies: Stability and Bifurcations - page 35

the following issues: the quasi-competitiveness of the economy, that is the question
as to whether output increases and the market price decreases with an increasing
number of firms in the industry; the asymptotic stability of the equilibrium if entry
occurs; the question as to whether perfect competition is obtained in the limit as the
number of competitors is increased. The interested reader should consult for exam-
ple Frank (1965), Ruffin (1971), Howrey and Quandt (1968), Okuguchi (1976), or
more recently, Seade (1980) and Amir and Lambson (2000) to get an impression
of the variety of interesting results obtained concerning this issue. In this section
we focus on asymptotic stability issues and we try to answer the question: is local
asymptotic stability obtained when the number of firms increases? Furthermore, we
also address the topic of global dynamics, that is we look at the changes in the basins
of attraction of the stable equilibria. Clearly, a discussion of these issues becomes
more complicated when the model is nonlinear, since increasing the number of play-
ers means increasing the dimension of the dynamical system. This is so since such
increases lead to greater complexity in the dynamics of nonlinear systems, whereas
in the case of linear systems no new dynamic phenomena arise.
In order to keep the mathematical analysis tractable, but at the same time to
also shed some light on the relation between asymptotic stability and the number
of firms, in what follows we will consider both the symmetric and semi-symmetric
models. Recall that in the symmetric case it is assumed that all firms are identi-
cal, so that they have identical cost functions and all firms start from the same
initial production quantities. Since the cost and demand parameters are identical
for all firms, the reaction functions R k will be identical, say R k D R for each k.
Consequently, the quantities will be identical for all periods, and the dynamics are
governed by a 1-dimensional system. If we let x.t/ denote the common output of
the representative firm, then the one-dimensional model in the symmetric case is
obtained by setting Q k D .N 1/x for each k. It is worth noting that the symmetric
case may be structurally unstable, that is the outcome obtained for the representa-
tive firm in the symmetric case may be completely different from the outcome of the
model with almost identical, but nevertheless heterogeneous firms (the firms might
differ in their production costs or might select slightly different initial quantities).
Therefore, the insights obtained from the symmetric model need to be accepted with
some caution. In order to derive some results which can be compared with the exist-
ing literature, we reconsider the partial adjustment towards the best response process
given by (1.23).
The symmetric case is obtained if we assume N players with identical quadratic
cost functions (as in Example 1.2), that is c 1 D c 2 D D c N D c and e 1 D
e 2 D D e N D e, identical adjustment speeds, that is a 1 D a 2 D :::;a N D a,
and identical capacity limits L 1 D L 2 D D L N D L. It is also assumed that
B C e>0, so the payoff functions of the firms are strictly concave in their strategies.
Then from (1.23) the 1-dimensional model which summarizes the common behavior
of all identical firms starting from identical initial condition x 1 .0/ D x 2 .0/ D D
x N .0/ D x.0/ is
x.t C 1/ D T.x.t// .1 a/x.t/ C aR..N 1/x.t//;