Investment and Interest Rate Policy in the Open Economy

|a3| < -3.

Assume μ > 1, or otherwise the aggregate system would be indeterminate, then Case

I is not relevant since condition (B1) is violated by assumption. For Case II, condition
(B3) is satisfied ∀μ > 1. If a > 0.5 then by inspection condition (B4) is automatically
satisfied and either (B5a) or (B5b) is required for determinacy. If a < 0.5 condition (B4)
is automatically satisfied provided (^1-2a)Λα₂(2^-ʌ²) < 1. Otherwise the following condition
is required: Λ₁(1 + μ) [α(1 — 2a)(2 — Λ₂) — Λ₂] < 2Λ2(1 + β)[1 + 2μ(1 — a)]. In addition
either (B5a) or (B5b) is required for determinacy.

Now suppose that the eigenvalue eK is inside the unit circle |eK | < 1, which re-
quires a < 1—δ [1 - δ — 11 C] < 0.5. Determinacy then requires that the remaining three
eigenvalues be outside the unit circle. From the characteristic equation of ARPI this

implies that r(1) = (^μ 1^)A1 [1 + α(1 — 2a)] > 0 given the assumption that μ > 1 and
β

r(0) = —^μ [2(1 — a) — ʌɪɑʌ^-2^]. This has to be positive r(0) > 0 otherwise there would

be (at least) one stable root, which requires Λ1α(1 — 2a) > 2(1 — a)Λ2 . Therefore if
r(—1) > 0 then the three roots, either real or complex, are outside the unit circle. Since
r(—1) = — 2(1+β)4μ(1~а)(1+в)— Λ₁(μ+1) [1 — ^α(¹~ 2^(^2-ʌ² )], then r(—1) > 0 provided
α(1 — 2a)(2 — Λ₂) > Λ₂ andΛ₁(1 + μ) [α(1 — 2a)(2 — Λ₂) — Λ₂] > 2Λ2(1+ β)[1 + 2μ(1 — a)].

This completes the proof. ■

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