The Classical Cournot Model - Nonlinear Oligopolies: Stability and Bifurcations - page 32

and the interior equilibrium is unstable. In addition, A c k 2.B k C e k /L k <0
implying that the output space Œ0;L 1 Œ0;L 2 is divided into only four regions
rather than the nine shown in Fig. 1.10).
In Fig. 1.11a the basins of attraction of E 1 ;E 2 ; and the coexisting 2-cycle C 2
are shown for the best reply dynamics, namely for a 1 D a 2 D 1. The basin of
attraction of E 1 is represented by the light-grey region, the basin of E 2 by the
dark-grey region, and the basin of the cycle C 2 by the white region. The peculiar
rectangular-shaped structure of the basins is related to the particular structure of
the best reply process, x 1 .t C 1/ D R 1 .x 2 .t//, x 2 .t C 1/ D R 2 .x 1 .t//, where next
period's output of firm i only depends on the current output of the other firm. This
implies that the eigenvectors associated with the unstable equilibrium E (that
belongs to the basin boundaries) are parallel to the coordinate axes. Moreover,
the map which generates the dynamics transforms vertical lines into horizontal
lines and vice versa. Hence, the invariant sets associated with the unstable node
E, that form the boundaries of the basins, are formed by vertical and horizontal
lines (on this point see also Bischi et al. (2000b)).
If the speeds of adjustment are smaller than 1, important differences can be
observed in the global dynamics. For example, Fig. 1.11b has been obtained with
a 1 D 0:97, a 2 D 0:98, leaving all the other parameters unchanged. Now the stable
2-cycle has both periodic points characterized by positive coordinates, namely
C 2 D f .0:19;0:13/ I .6:39;6:38/ g , and the structure of the basins is different, in
particular the basin of the cycle C 2 is smaller. The rectangular shape of the basins
is lost since in the case of partial adjustment the eigenvectors associated with E
are no longer parallel to the coordinate axes.
If the speeds of adjustment are even further decreased, the basin of the cycle C 2
shrinks; see Fig. 1.11c obtained with a 1 D 0:93;a 2 D 0:95. The periodic points of
C 2 approach the boundary of its basin and after a contact with such a boundary,
the cycle C 2 becomes unstable. As a consequence, the whole strategy space is
shared by the basins of the two asymptotically stable boundary Nash equilibria
E 1 and E 2 , as depicted in Fig. 1.11d obtained with a 1 D 0:9, a 2 D 0:92.
Our analysis suggests the following insights. First, the basins of the Nash equi-
libria E 1 and E 2 are always simply connected. We emphasize this fact since later on
we will encounter examples where the basins will not have such a simple structure.
Second, whereas the local asymptotic stability of the boundary Nash equilibria does
not depend on the adjustment speeds, the shape of the basins changes significantly
when adjustment speeds become smaller. If the players' speeds of adjustment are
lower, then the size of the basins of the equilibria is larger. As far as local asymptotic
stability is concerned, it is well-known in the literature that decreasing the speeds of
adjustments usually stabilizes the system (see for instance Fisher (1961), McManus
and Quandt (1961) and some results to be presented in Chap. 2). Here, however,
we emphasize that (in the present example) this also holds for the global dynam-
ics. Finally, since the firm with the smaller adjustment speed has the larger basin,
this firm is more likely to achieve the role of the monopolist, if initial production
quantities are selected randomly from a close to uniform distribution.