Concave Oligopolies - Nonlinear Oligopolies: Stability and Bifurcations - page 76

0.5

x

Negavtive profits

2

T ( F )

Positive profits

x

TF

3

()

F

2

TF ()

L = 0.58

0.35

2

0.3 0.8

x

Fig. 2.12 Example 2.4; linear inverse demand and cost functions, the case of semi-symmetric
capacity constrained firms. Computing the bounds on the chaotic attractor in the space of x 1 (out-
put of firm 1)andx 2 (output of all other firms). Also shown is the line above which profits are
negative

obtained in the symmetric case we noted that the kinks (the local maxima and min-
ima, where the map is non-differentiable) can be used to determine bounds for
the asymptotic dynamics of the dynamical system. In the two-dimensional model
describing the semi-symmetric case the lines of non-differentiability which separate
the different regions may play the role of folding curves. That is, they may act as
critical curves of noninvertible maps. As explained in Appendix C, the images of the
curves where the Jacobian determinant changes sign can be used to bound trapping
regions, within which the asymptotic dynamics are confined. Indeed, if we repre-
sent the chaotic attractor obtained for a capacity level of L 2 D 0:58 (see Fig. 2.12),
we notice that it is crossed by the line F of non-differentiability, the equation of
which is x 1

A c 2
B

.1/ .
This line acts as a folding line and its images of increasing rank, say F 1 D T.F/,
F 2 D T.F 1 / D T 2 .F/, give the upper and lower boundaries of the output sequences
along the chaotic attractor. In Fig. 2.12 the line x 1 C 20x 2

.7/

C .N 2/x 2 D 2L 2 , separating the regions D and D

D 10 is also displayed

(thin line). Above this line profits of all firms are negative.
Obviously, as the trajectory of production quantities evolves along the chaotic
attractor, some time periods exist in which the profits are negative. For the set of
parameters used to obtain the bifurcation diagram of Fig. 2.11, this problem only
occurs for L 2 >0:5, so that when L 2 <0:5the chaotic attractor is entirely
included in the region of the strategy space where profits are positive. Also in this
case, it is not easy to prove what kind of asymptotic dynamics are obtained when