The characteristic polynomial of J evaluated at the steady state is equal to
|
|
| p(ξ)= |J-ξ I|= |
| ξ4 (η-ξ)2
|
⎛
⎜
⎜
⎝ |
ξ2-( 1-µ + |
|
+ |
|
|
)ξ +
(1-µ) |
|
+ |
|
|
|
⎞
⎟
⎟
⎠ |
. |
|
|
(42) |
Thus the eigenvalues of
J are 0 (with algebraic multiplicity 4), η (with algebraic multiplicity 2) and the roots ξ
1, ξ
2 of the quadratic polynomial in the last bracket. Note that these roots satisfy the relations
|
ξ1+ξ2=1-µ + |
|
+ |
|
|
and
ξ1 ξ2=(1-µ) |
|
+ |
|
|
,
(43) |
because
(ξ-ξ1)(ξ-ξ2)=ξ2-(ξ1+ξ2)ξ + ξ1 ξ2.
(44)
Also note that because memory cannot be infinite in the steady state (i.e. 0 ≤ η <1), the stability of the steady state is entirely determined by the absolute values of ξ
1 and ξ
2. Furthermore, if there is no difference in costs between implementing the fundamental or moving-average strategy, that is
Cma-
Cfund=0, then
qma and
qfund are independent of the intensity of choice parameter β and hence the local stability of the steady state of the heterogeneous agents model is independent of β.
We have seen in (6.39) that if the risk aversion, a, or if the expected dispersion of the return of the fundamental belief, σ2, goes to zero, then the equilibrium price is entirely determined by the fundamental belief. Now, using (6.42), we find
|
|
|
|
p(ξ) = |
|
p(ξ)=ξ4 (η-ξ)2 (ξ- |
|
) (ξ-(1-µ)),
(45) |
so that the eigenvalues are equal to 0, η,
v/
R and (1-µ). Because
R>
v all eigenvalues lie within the unit circle. Hence, in this limiting case, the fundamental steady state is locally stable, because near the fundamental steady state fundamental traders exploit all profit opportunities, driving the price back to the fundamental value.
6.5.3 Bifurcations
A bifurcation is a qualitative change in the dynamical behavior of the system when varying the value of one of the parameters. Bifurcations occur for example, if one of the eigenvalues of the linearized system in the steady state crosses the unit circle. We are now going to investigate local bifurcations of the steady state.
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