If mfund=mma=0, then because Cfund≥ Cma≥ 0, we have that qfund ≤ qma. Thus, if no lower bound is imposed on the discrete choice probabilities, then at the steady state the fundamental belief is chosen with smaller probability than the moving average belief if the costs of the fundamental belief are higher than the costs of the moving average belief. Because qfund+qma=1, the steady state probabilities are equal to
|
|
| qfund= |
|
⎛
⎜
⎜
⎝ |
1+ (mfund-mma) + (1-mfund-mma) tanh |
⎛
⎜
⎜
⎝ |
|
(Cfund-Cma) |
⎞
⎟
⎟
⎠ |
⎞
⎟
⎟
⎠ |
, |
|
| qma =1-qfund. |
|
6.5.2 Local stability of the steady state
The local behavior of the dynamical system
zt=Φ(
zt-1) around the steady state
z is equivalent to the behavior of the linearized system
| (zt-z)= |
|
⏐
⏐
⏐
⏐ |
|
(zt-1-z) = J (zt-1-z), |
if none of the eigenvalues of the Jacobian matrix
J lies on the unit circle. Hence we can study the dynamical behavior of the system for different parameter values by calculating the eigenvalues of
J. The steady state
z is locally stable if all eigenvalues lie within the unit circle and becomes unstable if one of the eigenvalues crosses the unit circle. At this point a bifurcation, a qualitative change in dynamical behavior, occurs.
A straightforward computation shows that the Jacobian matrix of Φ at the steady state z is equal to
|
|
| J= |
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣ |
|
0 |
0 |
|
0 |
0 |
0 |
0 |
| 1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
| 0 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
| µ |
0 |
0 |
1-µ |
0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
1 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
0 |
0 |
η |
0 |
| 0 |
0 |
0 |
0 |
0 |
0 |
0 |
η |
|
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦ |
|
|
.
|
259