Concave Oligopolies - Nonlinear Oligopolies: Stability and Bifurcations - page 74

border where the map is not differentiable, the eigenvalues of the equilibrium (or of
the periodic cycle) may suddenly change. Such transitions are again accompanied
by border collision bifurcations, which we shall demonstrate below. The equilib-
rium x D .x 1 ; x 2 / in the semi-symmetric case can be obtained from the result of
Sect. 1.3.2 with e 1 D e 2 D 0, and has the form

A Nc 1 C .N 1/c 2
B.N C 1/

A 2c 2 C c 1
B.N C 1/ :

x 1 D ; x 2 D (2.35)

If this equilibrium is interior, then its local asymptotic stability is determined by the
eigenvalues of the Jacobian matrix

1 a 1 ! :

a 1 .N 1/

J .1/

2

D

a 2 Na 2

2

1

The characteristic polynomial of this matrix is the quadratic equation

2

C p C q D 0

with

Na 2
2 ;

p D 2 C a 1 C

and

q D .1 a 1 / 1

Na 2
2

a 1 a 2 .N 1/
4

:

Simple calculation shows that the stability conditions q<1, p C q C 1>0and
p C q C 1>0(see Appendix F) are satisfied if and only if

16 a 1 .8 a 2 /
a 2 .4 a 1 /

N<N b .a 1 ;a 2 / D :

The right hand side is decreasing in a 1 , so the global asymptotic stability condition
is obtained by selecting the smallest right hand side value for a 1 D 1. As expected,
also in this case, an increasing number of firms in the oligopoly leads to instabil-
ity, and the bifurcation value N b depends on the speeds of adjustment. As for the
one-dimensional symmetric case in Example 2.3, it is not easy to predict what kind
of asymptotic dynamics are obtained when the equilibrium point is unstable. 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 dif-
ferent regions

.i / , where the map is not differentiable. In order to illustrate this
point, we consider a case of border collision bifurcation where the border crossing
has a remarkable qualitative effect. The bifurcation diagram of outputs in Fig. 2.11 is
obtained for N D 21, A D 16, B D 1, a 1 D 0:2, a 2 D 0:3, c 1 D 6, c 2 D 6, L 1 D 2
and the capacity limit L 2 is the bifurcation parameter in the range Œ0:4;0:6.Ata
capacity level of L 2

D

' 0:45 the equilibrium x crosses the boundary from region