Oligopolies with Misspecified and Uncertain Price Functions, and Learning - Nonlinear Oligopolies: Stability and Bifurcations - page 261

(ii) If N D 8; then it is globally asymptotically stable if and only if LT ¤ 1;
where L D a=.N C 1/;
(iii) If N 9; then the steady state is globally asymptotically stable if and only
if either LT < .LT/ 1 or LT > .LT/ 2 ; where .LT/ 1 and .LT/ 2 are given
by (5.134). At these critical values Hopf bifurcations occur giving rise to the
possibility of the birth of limit cycles.

Proof.
Let T>0and m D 0. Then (5.130) becomes the quadratic equation

C T C 2 T.N C 1/

a

N C 1
a

C .N C 1/ D 0:

Since all coefficients are positive, all roots have negative real parts (see Lemma F.2
in Appendix F) implying global asymptotic stability.
Consider next the case of T>0and m D 1. Then (5.130) becomes a cubic
equation, and with the notation L D a=.N C 1/ it has the form

. C L/.1 C 2T C 2 T 2 / C NL D 0;

that is,

T 2 3

C 2 .2T C LT 2 / C .1 C 2LT/ C L.N C 1/ D 0:

(5.131)

Since all coefficients are positive, the Routh-Hurwitz stability condition implies that
all eigenvalues have negative real parts if and only if

.2T C LT 2 /.1 C 2LT/ > T 2 L.N C 1/;

(5.132)

which can be rewritten as a quadratic inequality in LT , namely

2.LT/ 2

C .LT/.4 N/ C 2>0: (5.133)

The discriminant of the left hand side is
.4 N/ 2

16 D N.N 8/:

Depending on the number of roots of the left hand side of (5.133) we have to
consider the following cases.

Case 1. If N<8, then the discriminant is negative, so (5.133) always holds and
the steady state is always globally asymptotically stable.

Case 2. If N D 8, then there is a unique real root

N 4
4 D 1;

LT D