Concave Oligopolies - Nonlinear Oligopolies: Stability and Bifurcations - page 85

for N>9the boundary equilibrium point 0; p 8=.N 2 1/ is the unique (global)

attractor. Then, as N is increased, a 2-cycle appears with periodic points located in
the regions

.4/ respectively, coexisting with the stable boundary equilib-
rium with the stable boundary equilibrium point. Each attractor has its own basin
of attraction, and then the basin of the boundary equilibrium point shrinks, until the
equilibrium becomes unstable. For N D 10 the stable 2-cycle remains the unique
(global) attractor. However, this cycle is not meaningful as a description of a long
run solution of the game that we are considering here. The reason is that a careful

.2/

D and D

analysis should check if profits, given by ' k D x 1 A c 1 .x 1 C .N 1/x 2 / 2 ,
are positive. The two periodic points of the stable cycle for N>10have coordi-
nates c 2

' .0:0889;0:3058/ and c 2

' .0;0445;0:2429/. Hence, the profits of firm

1 along this cycle are ' 1 c 2 D 0:18 and ' 1 c 2 D 0:046, and the correspond-

ing profits of firm 2 are ' 1 c 2 D 0:006 and ' 1 c 2 D 0:13. Consequently, as
expected firm 1 has no incentive to resume production again since the average profit
along the stable 2-cycle is negative. With a higher number of firms more compli-
cated non-equilibrium dynamics can be observed. For example, let us consider the
bifurcation diagram obtained with parameters N D 12, A D 144, c 1 D 10, c 2 D 8,
L 1 D L 2 D 1, a 2 D 0:7 and increasing values of the speed of adjustment a 1 2 .0:1
(Fig. 2.18). In this case chaotic oscillations of large amplitude dominate, a situa-
tion that may imply that firms have great difficulty in forecasting, so that naive
expectations may in fact represent a reasonable assumption.
However, the shape of the chaotic attractor in the strategy space (see Fig. 2.19a,
obtained with the same parameters as those used in Fig. 2.18 and a 1 D 0:8) reveals
a certain degree of correlation among the production quantities of firm 1 and the

1

x 1

0.5

1

x 2

0.5

0 1

a 1

Fig. 2.18 Example 2.5; quadratic price and linear cost function. The semi-symmetric case. Bifur-
cation diagrams of x 1 , x 2 with respect to a 1 , the speed of adjustment of firm 1. The number of
firms is held equal to N D 12