Investment and Interest Rate Policy in the Open Economy

four eigenvalues are determined by the upper left 4 × 4 submatrix of A_C^R_{P I} , denoted by

rR , , . . . . rR .

ACPI. Then the characteristic equation of ACPI is

r r³ + a3r² + a2r + a1 = 0

where

a3 = -1

1 Λ1

ββ

1 α(1 - Λ2)(2a - 1)∙

. Λ2 .

— 2(1 — a)μ

¹^λ-1Λ ₁ ^α(¹^-^Л2⁾⁽2^а^-¹)

a² = β ⁺ — ^{1 +}-----Λ2-----,

_ι αΛ₁(2a — 1) _ι 2(1 — a)μ(1 + β)

⁺ Λ2β ⁺ β

μΓ Λ'^α(²a ^-¹)

^a1 = ^- β ²⁽¹^-^a^{) +}----Λ----

Hence one eigenvalue is zero and the three remaining eigenvalues are the solutions to the
cubic equation r³ + a3r² + a2r + a1 = 0. Determinacy requires two eigenvalues to lie inside
the unit circle and the other three eigenvalues to lie outside the unit circle. First suppose
the eigenvalue eK is outside the unit circle |eK | > 1, which requires either a > 0.5 or
0.5 > a > 2-1δ [1 — δ — 1 =]. Then using Proposition C.2 of Woodford (2003), two of the
remaining three eigenvalues are outside the unit circle and one eigenvalue is inside the
unit circle if and only if: (Case I)

1 + a3 + a2 + a1 > 0 ⇔

(^μ ɪ^¹ [1 — (2a — 1)α] < 0,
β

(B1)

— 1+a3—a2+a1 > 0 -⇔ —2(1+β)—4μ(1—a)(1+β) — (μ+1)Λι

₁ I ⁽²^a^—¹)^α(²^— ^) 1 ₀

¹⁺ Λ2 J >⁰^,

(B2)

or (Case II)

1 + a3 + a2 + ɑɪ > 0 ⇔ ———¹ [1 — (2a — 1)α] > 0, (B3)

— 1+a3—a2+a1 > 0 ⇔ —2(1+β)—4μ(1—a)(1+β) — (μ+1)Λι

and either

1 (2a — 1)α(2 — Λ2)'
. ⁺ Λ2 .

< 0,

(B4)

a₁ — a1a3 + a2 — 1 > 0, (B5a)

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