Investment and Interest Rate Policy in the Open Economy

		Etbt^W+1	= B^W b_t^W ,	bt^W	WW = mmc_t ^χ_t	^w ^w^∣ ^πt Kt J
	(1 _— α) _— ^λi (^μ^-^1)[^а+^л2⁽¹^-	-α)]	—α(1 — α)	(μ-	-i)[α+Λ2(i-α)]	0
	(1 _— α) _— ^λi (^μ^-^1)[^а+^л2⁽¹^-		—α(1 — α)		βΛ2	0
	_1 _ (μ-i)Λι(1-Λ₂)		α		(μ-i)(i-Λ2)	0
_BW _≡	¹ βΛ2		α		βΛ2	0
	Λι β		0		i β	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 B^Ware 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 = Γ₁^A < Γ₂^A . 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.

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Investment and Interest Rate Policy in the Open Economy

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