Oligopolies with Misspecified and Uncertain Price Functions, and Learning - Nonlinear Oligopolies: Stability and Bifurcations - page 240

1
2 <1

r kQ 0 and 0<r kx r kQ

by assuming that C 0 k 0 for all k. Therefore all results derived previously for
the concave case remain valid. If there is a dominant firm, then the method shown
above for the concave case cannot be applied, since the monotonicity of the function
in (5.79) cannot be guaranteed. However the sufficient stability conditions (5.72)-
(5.74) remain applicable without any changes.

Example 5.11. Consider finally the case of a general duopoly, when N D 2.From
the special form of the Jacobian we see that the characteristic polynomial is

.1 C a 1 .r 1x 1/ /.1 C a 2 .r 2x 1/ / a 1 a 2 r 1Q r 2Q

which becomes the quadratic equation

2

C . 2 C a 1 .1 r 1x / C a 2 .1 r 2x //
C .1 C a 1 .r 1x 1//.1 C a 2 .r 2x 1// a 1 a 2 r 1Q r 2Q D 0:

By using the stability condition introduced in Appendix F we see that the roots are
inside the unit circle if and only if

.1 C a 1 .r 1x 1//.1 C a 2 .r 2x 1// a 1 a 2 r 1Q r 2Q <1;

2 C a 1 .1 r 1x / C a 2 .1 r 2x /
C .1 C a 1 .r 1x 1//.1 C a 2 .r 2x 1// a 1 a 2 r 1Q r 2Q C 1>0;

2 a 1 .1 r 1x / a 2 .1 r 2x /
C .1 C a 1 .r 1x 1//.1 C a 2 .r 2x 1// a 1 a 2 r 1Q r 2Q C 1>0:

Instead of examining this in the general case, assume that a 1 D a 2 D 1.Thenwe

have the conditions

r 1x r 2x r 1Q r 2Q <1;

1 r 1x r 2x C r 1x r 2x r 1Q r 2Q >0;
1 C r 1x C r 2x C r 1x r 2x r 1Q r 2Q >0;

showing that

r 1x r 2x 1<r 1Q r 2Q < min f 1 r 1x r 2x C r 1x r 2x ;1 C r 1x C r 2x C r 1x r 2x g :