Concave Oligopolies - Nonlinear Oligopolies: Stability and Bifurcations - page 94

or

C a S

q C 1 .lC1/

.N 1/ar T

p C 1 .mC1/

D 0;

which can be rewritten as the polynomial equation

S

q C 1 lC1 T p C 1 mC1

C a T

p C 1 mC1

.N 1/ar S

q C 1 lC1

D 0:
(2.57)
Assume first that the firm's own information lag S is much smaller than T ,the
information lag about rival firms. Making the simplest assumption that S D 0,
(2.57) becomes

. C a/ T

p C 1 mC1

.N 1/ar D 0: (2.58)

Consider first the special case of T D 0, when there is no information lag about
rival firms. Then (2.58) reduces to the linear equation

. C a/ .N 1/ar D 0;

with solution

D .N 1/ar a<0;

so the equilibrium is locally asymptotically stable. This case was discussed under
much more general conditions in Theorem 2.2 where the same conclusion was
reached.
Consider next the case when T>0and m D 0. Then (2.58) becomes the
quadratic,

2 T C .1 C aT/ C a.1 .N 1/r/ D 0:

Since all coefficients are positive, both roots are negative or have negative real
parts (see Appendix F), so again the equilibrium is locally asymptotically stable.
In the case of m D 1, (2.58) reduces to the cubic equation

3 T 2

C 2 .aT 2

C 2T/ C .1 C 2aT/ C a.1 .N 1/r/ D 0:

(2.59)

All coefficients are positive and the Routh-Hurwitz criterion (see Szidarovszky and
Bahill (1998)) implies that all roots have negative real parts if and only if

.aT 2 C 2T/.1 C 2aT/ >T 2 a.1 .N 1/r/:

(2.60)

This inequality can be rewritten as a quadratic inequality in the variable aT in the
form

2.aT/ 2 C aT.4 C r.N 1// C 2>0: (2.61)