Concave Oligopolies - Nonlinear Oligopolies: Stability and Bifurcations - page 70

not seem to be a plausible description of real-world economic behavior for at least
two reasons. First, it seems reasonable to assume that such low-periodic dynamics
would be detected by the firms. As a result, they most probably would change their
expectations. Second, for profits to be positive, A c BNx>0 has to hold. How-
ever, for the parameter values considered here, profits are negative for x>8=N,that
is above the dotted line depicted in Fig. 2.8. Therefore, for N>10the production at
the upper periodic points involve negative profits, that is all firms would cyclically
make a loss every other time period. Again firms would most likely change their
expectations.
In what follows we give a brief mathematical description of the mechanism
which leads to such a bifurcation. Let us consider Figs. 2.9a, b, where the graph
of the map T.x/is shown for N D 8 and N D 10 respectively (the other parameters
have the same values as in Fig. 2.5). In Fig. 2.9a the equilibrium is stable (a typ-
ical trajectory is shown), whereas in Fig. 2.9b it is unstable, with a stable 2-cycle
around it and having periodic points located on different branches of the piecewise
linear map. The multiplier of this 2-cycle is given by the product of the derivatives

1

1

x

x 0

0 x 1 0 x 1

(a) (b)

1

1

0 x 1 0 x 1

(c) (d)

Fig. 2.9 Example 2.3; linear inverse demand and cost functions and identical capacity constrained
firms. Examining the bifurcation of Fig. 2.5 in more detail. ( a )ForN D 8 the equilibrium is stable.

( b )ForN D 10 atwocycleemerges.( c )ForN D 13 the two cycle becomes unstable and a stable

four cycle is born. ( d )ForN D 15 a stable two cycle reappears