Modified and Extended Oligopolies - Nonlinear Oligopolies: Stability and Bifurcations - page 198

stability conditions. Before turning to special cases, three sufficient conditions for
stability will be presented for the general case.

Theorem 4.6. Assume that for all k , a k >0 and

0
@ X
l

1

A R 0 kS > 1;

.N 1/R 0 kQ C

kl (4.102)

¤ k

then the equilibrium is locally asymptotically stable.

Proof. Notice that condition (4.102) implies that the Jacobian is strictly diagonally
dominant in every row with negative diagonal elements. Then the Gerschgorin-
cycle theorem (see for example, Szidarovszky and Yakowitz (1978)) implies that
all eigenvalues have negative real parts.

Applying this result to the transpose of the Jacobian one easily obtains the
following theorem.

Theorem 4.7. Assume that for all k , a k >0 and
a k C X
l

a l .R lQ C lk R lS />0;

(4.103)

¤ k

then the equilibrium is locally asymptotically stable.

The application of Theorem B.7 given in Appendix B also offers a sufficient
stability condition. Notice that the Jacobian matrix (4.101) can be factored as

A . R Q C R S G / (4.104)

D diag.R 0 1S ;R 0 2S ;:::;R 0 NS /,

where A D diag.a 1 ;a 2 ;:::;a N /, R S

0 1

0 1

1R 0 1Q ::: R 0 1Q
R 0 2Q 1 ::: R 0 2Q
: : : : : : : : : : : :
R 0 NQ R 0 NQ ::: 1

0 12 ::: 1N
21 0 ::: 2N
: : : : : : : : :
N1 N2 ::: 0

@ A

@ A

R Q D and G D ;

and observe that A is positive definite, if a k >0for all k. The above considerations
make it possible to assert the following theorem:

Theorem 4.8. Assume that a k >0 for all k , . R Q C R S G / C . R Q C R S G / T

is

negative definite. Then the equilibrium is locally asymptotically stable.

T
Q is negative definite and

The condition of this theorem is satisfied, if R Q C R

T

R S G C G R S is negative semi-definite.

Consider now the special case when kl k , that is, the cooperation levels of
each firm are identical toward its competitors. In this case let