The Classical Cournot Model - Nonlinear Oligopolies: Stability and Bifurcations - page 31

which is below one if

a 2 B
2.B C e 2 /x <1;

1 a 2 C

since 0<a k 1 for k D 1;2. This relation can be rewritten as

B
2.B C e 2 / ;

x>

so a feasible positive x exists. From the local stability result of Appendix B we
conclude that the monopoly equilibrium .0;x 2 / is locally asymptotically stable.
The stability of the other monopoly equilibrium .x 1 ;0/can be proved similarly.
This provides a first conclusion with regard to the equilibrium selection problem,
because even if we obtain three Nash equilibria, from an evolutionary perspective
a stability argument suggests that the interior equilibrium will not be selected. It
remains an open question, however, as to which one of the two monopoly equi-
libria is more likely to be observed in the long run. The situation is even more
intricate, since in addition to the two asymptotically stable boundary equilibria, in
the strategy space another attracting set might coexist. This can be demonstrated by
considering the best reply dynamics obtained for a k D 1, k D 1;2. In the case when
x 2 >.A c 1 /=B and x 1 >.A c 2 /=B we have .R 1 .0/;R 2 .0// D .x 1 ;x 2 /

and R 1 .x 2 /;R 2 .x 1 / D .0;0/. Therefore, under best reply dynamics the peri-

odic cycle C 2 D ˚ .0;0/ I x 1 ;x 2 coexists with the two stable monopoly equi-
libria. It is also easy to see that C 2 is stable, so it may even occur that an adjustment
process fails to converge towards any Nash equilibrium in the long run. In such a
situation, where several attractors coexist, the question of which attractor will be
reached in the long run crucially depends on the initial conditions and the observed
outcome becomes path dependent. Each of these long run outcomes has its own
basin of attraction (see Appendix C for definitions of these concepts from the qual-
itative theory of dynamical systems) and any external random factor (a so-called
“historical accident”) that causes a displacement of some of the initial outputs may
cause the trajectory to move across a basin boundary and, consequently, it will
converge to a different attractor.
We can shed some light on this issue by using a mixture of analytical, geometrical
and numerical methods, an approach which is typically used in the study of the
global dynamical properties of nonlinear systems of dimension greater than one
(see for example Mira et al. (1996), Brock and Hommes (1997) and Puu (2003)).
To get a better feeling for the global dynamics of our duopoly game where firms
use partial adjustment towards the best response, we numerically compute the basins
of attraction for the coexisting attractors. Let the reservation price be A D 450 and
the slope of the linear inverse demand function be B D 30.Forthesakeofsim-
plicity, we consider identical firms with cost parameters c 1 D c 2 D c D 275 and
e 1 D e 2 D e D 17, so that production costs are increasing, but marginal costs are
decreasing. (Similar values were chosen by Cox and Walker (1998) in an experimen-
tal setup). In order to guarantee non-negative prices, we select L 1 D L 2 D 7:5,which
ensures that L 1 C L 2

A=B. For these parameter values condition (1.38) is fulfilled