General Oligopolies - Nonlinear Oligopolies: Stability and Bifurcations - page 129

Fig. 3.11 Oligopolies with linear inverse demand function and cost externalities. The case of
duopoly with identical speeds of adjustment. Multiple Nash equilibria in the .;a/ plane. Note
that E S is unique and stable for <3.For>3E S becomes unstable and two stable equilibria
E 1 ;E 2 occur

C 2 D f .p 1 ;p 1 /;.p 2 ;p 2 / g 2 ;

(3.25)

with coordinates

q a 2 . 1/ 2

a. 1/ C 2 4

p 1 D ;

2a

and

q a 2 . 1/ 2

a. 1/ C 2 C 4

p 2 D :

2a

For the interested reader it should be mentioned that for .;a/ 2 with
a>a h ./, more complicated dynamics might be observed. The proof of this propo-
sition is based on a standard analysis of the eigenvalues of the Jacobian matrix and
is given in detail in Bischi and Kopel (2001).