Concave Oligopolies - Nonlinear Oligopolies: Stability and Bifurcations - page 69

bifurcation, sometimes called degenerate or critical flip bifurcation , occurs in our
piecewise linear map. Although the cycle emerging after the equilibrium has lost
its stability is probably not a plausible description of agents' behavior from an
economic point of view (see below), we present this case mainly to discuss some
mathematical properties of dynamical systems involving piecewise differentiable
maps. Since piecewise linear functions are often used in models of economic sys-
tems, a study of their peculiar dynamic features may be useful as a reference in other
circumstances.
Let us consider the set of parameters A D 16, B D 1, a D 0:4, c D 8 and L D 1,
and take N as a bifurcation parameter. If N<7then A c>.N C 1/BL,sothe
equilibrium is x D L, and it is stable as was shown earlier. For N>7the equilibrium

Ac
B.N

x D

1/ is stable for N<4=a 1 D 9,andforN>9it is unstable. For a linear
map instability of the equilibrium means divergence of all the trajectories starting
arbitrarily close to it, however this is not the case for our piecewise linear model, as
its trajectories are bounded. Indeed, as shown in the numerically computed bifurca-
tion diagram of Fig. 2.8, after the bifurcation occurring at N D 9, a stable cycle C 2
of period 2 suddenly appears. We note that the amplitude of the oscillations along
the newly born stable cycle is of finite amplitude from the moment of its creation .
This suggests that such a “hard” bifurcation is different from what is usually
called a flip (or period doubling) bifurcation. The appearance of the stable cycle is
not due to local properties around the equilibrium but is related to the global shape
of the map generating the dynamical system. It should be mentioned, however, that
the non-equilibrium dynamics emerging after the equilibrium has lost its stability do

C

1

x = 8 / N

x

Negative profits

Positive profits

0.3

7 8 9 10 11 12 13 14 15 N

Fig. 2.8 Example 2.3; linear inverse demand and cost functions and identical capacity constrained
firms. Bifurcation of output with respect to number of firms N when stability of equilibrium is lost
because T 0 ./ becomes less than 1, and a 2-cycle emerges. Note that profits become negative

above the dotted line