Modified and Extended Oligopolies - Nonlinear Oligopolies: Stability and Bifurcations - page 175

1.6

x

0

20

S

0

0.5 0.75 1 1.25 1.5 1.75 L 2

Fig. 4.13 The oligopoly model with intertemporal demand interaction and adaptive adjustment
in the discrete time case. The N-firm symmetric model with iso-elastic price function and linear
cost functions. Bifurcation diagrams with respect to L when the number of firms N 4.Other

parameter values are A D 2;c D 0:15;ˇ S D 0:6;a D 0:5. The bifurcation value is L D 1:2

D

value a simple change from the stable equilibrium E 3 to the stable equilibrium E 1
(that is independent of L) is observed.
In contrast to this, in the bifurcation diagram of Fig. 4.14, obtained with one
more firm (N D 5)anda D 1 (the case of best reply dynamics) the bifurcation,
now occurring at L D 3:68=3:75 ' 0:98, leads to the creation of a stable cycle
of period 2. Indeed, by slight changes of the parameters, the creation of stable
cycles of several different periods can be observed, as well as the sudden 3 cre-
ation of a chaotic attractor. This is shown in the bifurcation diagram of Fig. 4.15,
obtained with parameters N D 6, A D 2, c D 0:1, ˇ S D 0:6,anda D 0:7 with bifur-
cation value L D 4:48=3:6 ' 1:24. However, we are mainly interested in the effect
of the inertia parameter ˇ S on the dynamic behavior of the model. The bifurcation
diagram of Fig. 4.16 shows the role of increasing values of ˇ S , varying in the range
Œ0;1, with the other parameters fixed at the values N D 3, A D 1, c D 0:15, L D 2
and a D 1. The equilibrium E 1 is stable for low values of ˇ S , then it loses stability
and a stable cycle of period 2 appears. The amplitude of the oscillations increases
for increasing values of ˇ S , until the lower periodic point reaches the constraint at
x D 0. It is also interesting to study the impact of the number of firms in the
market on the bifurcation with respect to ˇ S . This can be seen from the bifurcation
diagram of Fig. 4.17 obtained with the same parameter values as in Fig. 4.16, but

3 By “sudden” here we mean without the usual sequence of period doubling bifurcations.