General Oligopolies - Nonlinear Oligopolies: Stability and Bifurcations - page 128

3.2.1 Identical Speeds of Adjustment

We first assume that the speeds of adjustment are identical for the two firms, that is

a 1 D a 2

D a:

Under this assumption, in contrast to the previous examples, the singularities that
are involved in global bifurcations can be given in closed form. Moreover, the
exact values for the parameters at which global bifurcations occur can be explicitly
determined (see Bischi and Kopel (2001) for further details).
In this case, it is obvious that the steady states of this system correspond to the
Nash equilibria of the game and are independent of the adjustment speed a. A proper

study of the two-dimensional map T W .x 1 ;x 2 / ! x 0 1 ;x 0 2 defined by

x 0 1 D .1 a/x 1 C ax 2 .1 x 2 /;
x 0 2 D .1 a/x 2 C ax 1 .1 x 1 /;

T W (3.23)

should provide some answers to the questions stated above. Since we restrict our-
selves to 2 .1;4, the strategy space S D f Œ0;1 Œ0;1 g is trapping for each

value of a 2 .0;1 and for each initial value of production quantities in . 3 S In other

words, any sequence of production quantities which starts inside S remains feasible

for all t 0.
We first turn to the question of local stability of the interior Nash equilibria and
provide a characterization of the corresponding stability regions (see also Fig. 3.11).
Proposition 3.1. Let D ˚ .;a/ 2 R

j 1< 4;0< a 1 denote the appro-
priate region in the parameter space. Then the following holds.

2

(i) The symmetric Nash equilibrium E S D f 1 1=;1 1= g exists for all .;a/
2 . It is locally asymptotically stable for .;a/ 2 ,if 1<<3 .

(ii) The Nash equilibria E i , i D 1;2 , given in (3.21) exist for >3 .Theyare
locally asymptotically stable for .;a/ 2 ; if a<a h ./ D 2=. 2 2 3/ .

(iii) In the set

( .;a/ 2 j >3;a h ./ > a>a p ./ D ) ;

6 p 12. 2/
3 C 2 2

s .E i ;C 2 / D

(3.24)
the two stable Nash equilibria E i , i D 1;2 , given in (3.21) coexist with a stable
cycle of period two

3 This is so since the maxima of the reaction functions R k occur at k =4, and here we have 1 D
2 D with 0< 4.