Concave Oligopolies - Nonlinear Oligopolies: Stability and Bifurcations - page 95

The discriminant of the left hand side is

.4 C r.N 1// 2 16 D r.N 1/Œr.N 1/ C 8:

The first factor, r.N 1/, is negative, so we have the following cases.
Case 1. If r.N 1/ C 8>0, then the discriminant is negative, so (2.61) always
holds.
Case 2. If r.N 1/ C 8 D 0, then (2.61) holds for all values of aT except the single
root of the quadratic polynomial. So the equilibrium is locally asymptotically
stable unless

4 C r.N 1/
4

8 C r.N 1/
4

aT D

D C 1 D 1:

Case 3. If r.N 1/ C 8<0, then the quadratic polynomial (2.61) has two real roots,

4 r.N 1/ ˙ p r.N 1/Œr.N 1/ C 8
4

.aT/ 1;2 D : (2.62)

Since 4 r.N 1/ D .8 C r.N 1// C 4>0, both roots are positive. Hence
the equilibrium is locally asymptotically stable if

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

where .aT/ 1 <.aT/ 2 . The equilibrium is unstable if

.aT/ 1 <aT<.aT/ 2 :

Summarizing these results the stability region is shown as the shaded area in
Fig. 2.20.
From the above analysis we can draw the following interesting conclusions. If
N 9,thenr.N 1/ C 8>0, so Case 1 always occurs and the equilibrium is always
locally asymptotically stable. Assume next that N>9. Then Case 1 occurs if r>

8
N 1

resulting in the local asymptotic stability of the equilibrium. Case 2 occurs if

8

r D

1 , so the equilibrium is locally asymptotically stable unless aT D 1.Case
3 is obtained when r<

N

8

1 , in which case local asymptotic stability occurs if
aT is either sufficiently small (less than .aT/ 1 / or sufficiently large (greater than
.aT/ 2 /. The asymptotic stability does not depend on the individual values of a and
T , it depends on only the product of aT . This property shows a certain kind of
compensation between the speed of adjustment and the average information delay.
If the average delay T is given, then in Case 2 the firms must not select a D

N

1

T ,

and in Case 3 they should select either a small a< .aT / 1
T

or a large a> .aT / 2
T

value of a in order to stabilize the equilibrium.
Assume next that Case 3 occurs, that is,

8

1 .IfaT <.aT/ 1

1<r< or

N

aT >.aT/ 2 then the equilibrium is locally asymptotically stable, and if aT is