Concave Oligopolies - Nonlinear Oligopolies: Stability and Bifurcations - page 84

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

' 2 D

As a numerical example, consider A D 16;c 1 D 10 > 8 D c 2 D ::: D c N .Then
using the expressions given in (2.37) we obtain the unique equilibrium

7 N
p 8N 2

9

x 1 D

C 14N 4 ;

x 2 D ::: D x N D p 8N 2

C 14N 4 ;

which shows that for N>7 firm 1 stops producing and we have a boundary
equilibrium

r 8
N 2

1 :
For N D 7, total equilibrium industry output becomes Q D p 6 and the correspond-
ing equilibrium price is f. Q/ D 10; which obviously equals the marginal cost of
firm 1. However this is not the end of the story. Figure 2.17 shows a bifurcation dia-
gram of outputs obtained for the model (2.36) with speeds of adjustment a 1 D 0:5,
a 2 D 0:4, capacity limits L 1 D L 2 D 0:4 and bifurcation parameter N in the range
Œ3;10 (notice that L 1 C .N 1/L 2

x 1 D 0 and x 2 D ::: D x N D

A in the whole range, so that non-negativity of
prices is ensured). The bifurcation diagram of Fig. 2.17 confirms that x 1 goes to zero
for N>7.However,forN D 10 a positive stable cycle of period 2 characterizes the
long-run dynamics and it appears that firm 1 resumes production. Mathematically,
this stable cycle is created through a border collision bifurcation between N>9
and N>10, and at its creation it coexists with the stable boundary equilibrium. So,

0.4

x 1

0

0.4

x 2

0

3 4 5 6 7 8 9 10

N

Fig. 2.17 Example 2.5; quadratic price and linear cost function. The semi-symmetric case. Bifur-
cation diagrams of x 1 , x 2 with respect to the number of firms N. Illustrating how x 1 can go to zero
for some values of N