|
EtbtW+1 |
= BW btW , |
btW |
WW = mmct χt |
^w ^w^∣ | ||
|
(1 — α) — λi (μ-1)[а+л2(1- |
-α)] |
—α(1 — α) |
(μ- |
-i)[α+Λ2(i-α)] |
0 | |
|
βΛ2 | ||||||
|
_1 _ (μ-i)Λι(1-Λ2) |
α |
(μ-i)(i-Λ2) |
0 | |||
|
BW ≡ |
1 βΛ2 |
βΛ2 | ||||
|
Λι |
0 |
i |
0 | |||
|
- — ( K )σ |
(1 |
-α) K + σα K |
0 |
1 + K. | ||
One eigenvalue of the system is given by 1 + C > 1, while another eigenvalue is zero.
Consequently, equilibrium determinacy requires that the two remaining eigenvalues of BW
are outside the unit circle. Then by Proposition C.1 of Woodford (2003) the following
result is obtained:
Proposition 4 Suppose that monetary policy is characterized by a forward-looking inter-
est rate rule. Then a necessary and sufficient condition for determinacy of the aggregate
system is
1 <μ< 1 + min∣ΓA ≡ ⅛ ≡ ' ' . ) (39)
1 αΛ1 2 Λ1 [α(2 - Λ2) + Λ2]
where Λ1 = (i-ψ)ψi-βψ) and Λ2 = 1 — β(1 — δ).
As discussed by Carlstrom and Fuerst (2005) the regions of determinacy for a closed-
economy are remarkably narrow under a forward-looking rule with capital.18 Again sup-
pose that α = 0.36, β = 0.99 and δ = 0.025. Then the upper bound for determinacy is
1.01124 = Γ1A < Γ2A . Propositions 5 and 6 below show that in an open-economy the range
of determinacy is even smaller.
4.1.2 Difference System
The set of linearized conditions for cross-country differences yields a system of the form:
18 Carlstrom and Fuerst (2005) present a necessary condition for determinacy, whereas Proposition 4 provides
a necessary and sufficient condition for determinacy.
21
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