Oligopolies with Misspecified and Uncertain Price Functions, and Learning - Nonlinear Oligopolies: Stability and Bifurcations - page 224

Since all coefficients are positive by assuming again that Nh 1, the equilibrium is
locally asymptotically stable.
Consider next the case of T>0and m D 1. Then (5.39) becomes the cubic
equation

3 T 2 C 2 .a.r C 1/T 2 C 2T/ C .1 C 2a.r C 1/T/ C a.1 C r.1 Nh// D 0: (5.40)

All coefficients are positive if Nh 1, and the Routh-Hurwitz stability criterion
shows that the roots have negative real parts if and only if

.a.r C 1/T 2 C 2T/.1 C 2aT.r C 1// > T 2 a.1 C r.1 Nh//; (5.41)

which is equivalent to the quadratic inequality

2.r C 1/ 2 .aT/ 2 C .aT/.4.r C 1/ C Nhr/ C 2>0: (5.42)

The discriminant of the left hand side of (5.42) is

.4.r C 1/ C Nhr/ 2 16.r C 1/ 2 D Nhr.Nhr C 8.r C 1//:

The first factor is negative, so we have the following cases:

Case 1. If Nhr C 8.r C 1/ > 0, then the discriminant is negative, so (5.42) always
holds and the equilibrium is locally asymptotically stable.
Case 2. If Nhr C 8.r C 1/ D 0, then (5.42) holds for all values of aT except the sin-
gle root of the quadratic polynomial. So the equilibrium is locally asymptotically
stable unless

4.r C 1/ Nhr
4.r C 1/ 2

4.r C 1/
4.r C 1/ 2

1
r C 1 D 1 C

8
Nh :

aT D D D

Case 3. If Nhr C 8.r C 1/ < 0, then the quadratic polynomial (5.42) has two real

roots,

4.r C 1/ Nhr ˙ p Nhr.Nhr C 8.r C 1//
4.r C 1/ 2

.aT/ 1;2 D : (5.43)

Since

4.r C 1/ Nhr D .Nhr C 8.r C 1// C 4.r C 1/ > 0;

both roots are positive. Hence the equilibrium is locally asymptotically stable if

aT < .aT/ 1 or aT > .aT/ 2