Concave Oligopolies - Nonlinear Oligopolies: Stability and Bifurcations - page 77

1.5

x 1

0.6

1

x 2

0.3

6 7 8 9 10 11 12 N

Fig. 2.13 Example 2.4; linear inverse demand and cost functions, the case of semi-symmetric
capacity constrained firms. Bifurcations of x 1 (output of firm 1)andx 2 (output of all other firms)
with respect to N (the number of firms)

the equilibrium point is unstable. In fact, as already noted in the study of the sym-
metric case, the attracting sets created after the bifurcation of a piecewise linear
map depend on the global properties of the map and are influenced by the borders
between the different regions

.i / , where the map is not differentiable. In order to
illustrate this point, we consider in Fig. 2.13 the numerically computed bifurcation
diagram of outputs obtained with parameters A D 14, B D 1, a 1

D

D 0:5, a 2 D 0:4,

c 1 D 6, c 2 D 6, L 1 D 2, L 2 D 1 and values of the bifurcation parameter N in the

range Œ6;13.
Analogously to the symmetric one-dimensional model in Example 2.3 we give
a mathematical description of the bifurcations involved in the present situation in
order to understand the peculiar properties of piecewise linear dynamical systems.
However, also here we should point out that asymptotic behavior characterized by
a low-periodic stable cycle is not realistic from an economic point of view, because
presumably firms would detect such a simple periodicity and change their naive
expectations. Moreover, also in this case the profits are negative in the upper peri-
odic point. For N<7the point

.1/ , with
x 2 >L 2 , hence it is not an equilibrium of the dynamical system (2.34). For
N D 7,

x given in (2.35) is outside the region D

.1/ as N is
further increased (Fig. 2.14b, obtained with N D 8), and it is locally asymptotically
stable for N<N b .0:5;0:4/ D 8:714. At this bifurcation value the equilibrium x
becomes unstable and a stable cycle of period 2 appears, clearly visible in the bifur-
cation diagram of Fig. 2.13, with periodic points located in different regions (see
Fig. 2.14c, obtained with N D 10, where the two stable periodic points are labelled
by c .i 2 ).

x D .1;1/ (see Fig. 2.14a) and it then enters the region D