Oligopolies with Misspecified and Uncertain Price Functions, and Learning - Nonlinear Oligopolies: Stability and Bifurcations - page 225

aT

( aT ) *

2

−8
1+ Nh

( aT ) *

1

r

−1

8
8+ Nh

−

Fig. 5.3 The continuous time model with symmetric firms and misspecified demand. The stability
region ( shaded )inthe.r;aT/ plane. The parameter r is the slope of the reaction function at the
steady state, a D ˛ 0 .0/ and T is the time lag in the weighting function. Notice how stability is lost
then regained as aT increases along the dashed vertical line. Hopf bifurcations may occur at the
points .aT/ 1 and .aT/ 2

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

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

8

Nh ,
then the equilibrium is asymptotically stable with small and large values of aT .
With fixed such values of r,ifaT is gradually increasing from a very small value
and crosses .aT/ 1 , then stability is lost. The instability holds until the value of
aT reaches .aT/ 2 , and after crossing this value stability is regained. We will next
show, that at these critical points Hopf bifurcations occur. Select T as the bifurcation
parameter similarly to the full information case. Then the eigenvalues are functions
of T ,so D .T/. At the critical values of T , the inequality (5.41) becomes an
inequality, so the cubic equation (5.40) can be rewritten as

The stability region is shown in Figure 5.3. Assume now that 1<r<

8 C