The Classical Cournot Model - Nonlinear Oligopolies: Stability and Bifurcations - page 34

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' 1 x 1 ;x 2 D .A c 1 /Œe 2 .A c 1 / C B.c 2 c 1 /=.4.B C e 1 /.B C e 2 //,and

' 2 x 1 ;x 2 D .A c 2 /Œe 1 .A c 2 / C B.c 1 c 2 /= .4.B C e 1 /.B C e 2 //.
This shows that for at least one of the firms, profits are negative along the cycle.
Moreover, if B<e k <0and c 1 D c 2 , then we have negative profits for both firms,
a situation which is not sustainable for any firm. As a consequence of these consid-
erations, what our analysis of the global dynamics reveals is that for some initial
production choices an economically infeasible situation will emerge for the firms.
Notice that this important result can only be obtained through a global study of the
structure of the basins of attraction.
We also would like to draw the reader's attention to a global bifurcation which is
responsible for the drastic change in the dynamics obtained in this simple duopoly
model. In a situation where marginal costs are decreasing strongly and x k <x k ,
we obtain three coexisting attractors: two boundary equilibria and a 2-cycle. Notice
that the limiting quantities x k are located on a line where the map is not differen-
tiable. Consider now what happens if marginal costs increase. At a certain point, a
boundary equilibrium x k will collide with x k , and if marginal costs are increased
even further, then the interior equilibrium becomes globally stable. This is actu-
ally a first example of a border collision bifurcation, a global bifurcation occurring
whenever a qualitative change in the phase diagram (that is, creation/destruction of
invariant sets and/or stability change of existing ones) is due to a contact (and cross-
ing) of an invariant set with a border where the map is not differentiable separating
regions where it is differentiable. In this case the boundary that separates regions
D

.1/ is the one involved in the contact, and such a border is due to the
presence of non-negativity constraint. This kind of global (or contact) bifurcations,
specific to piece-wise differentiable dynamical systems, will be examined in more
detail in Chap. 2, in particular in Examples 2.3 and 2.4.

.5/

and D

1.3.2 A Cournot Oligopoly Game

In his seminal paper, Theocharis (1960) studied the asymptotic stability of the
Cournot-Nash equilibrium under discrete-time best reply dynamics with naive
expectations. For this quantity-setting model with linear demand and linear costs,
he found that the (unique) equilibrium is asymptotically stable only in the case of
two competitors. It is marginally stable (see definition (A.1) in Appendix A) for
three firms and unstable for more than three firms. Among others, McManus and
Quandt (1961) and Fisher (1961) demonstrated that this result depends on the type
of adjustment process the firms use to determine their production quantities. They
showed that for certain adjustment processes in continuous-time the equilibrium is
stable no matter what the number of firms is. These facts will be later discussed
in Chap. 2. Despite this result Fisher (1961, p.125) notes that “... the tendency to
instability does rise with the number of sellers for most of the processes consid-
ered”. These early papers gave rise to a lively discussion that has endured until the
present day. One of the main topics in this body of literature is the relation between