Investment and Interest Rate Policy in the Open Economy

a₀ — a0a2 + a1 — 1 > 0,                           (A5a)

|a2| < —3.

Assume μ > 1, since otherwise the aggregate system would be indeterminate. Then Case
I is not relevant since condition (A1) is violated by assumption. For Case II, condition
(A3) is satisfied ∀ μ > 1 since 1 — α(2a — 1) > 0. If a > 0.5 then by inspection condition
(A4) is automatically satisfied and either (A5a) or (A5b) is required for determinacy. If
a < 0.5 then condition (A5a) can be derived as:

^λi^^{1 —} 2a)^{μ λ}√^{1 — 2}a)(^{μ —} 1)         .      ,      . .

-----+—+---- --------+--+ ^λ1 + ^a2^{e + (1 — 2a})^αΛ 1

Λ2β             Λ2

which is always satisfied by inspection and thus condition (A5b) does not apply. Fi-
nally condition (A4) is automatically satisfied provided (^1-2a)α(²-^2) < 1. Otherwise the
following upper bound on μ is required: μ < л^^-'а'+в—Лг'-Лг] — 1.

Now suppose that the eigenvalue e_κ is stable, |e_K | < 1 which requires a < 2-g [1 — δ — 2 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(0) = ^Л1^Л2/з^-2а) > 0.

If μ > 1 then r(1) = (^μ-1λ- [1 + α(1 — 2a)] > 0. Therefore if r( — 1) > 0 then

β

the three roots, either real or complex, are outside the unit circle. Since r(—1) =
—2(1+ β) — Λ₁(μ +1) [1 — ^а(1-2Л)₂^(2-Л2)], then r( — 1) > 0 provided α(1 — 2a)(2 — Λ₂) > Λ₂
and 1 + μ > _{λ l α}(₁ ''(¹,,)., 2''⁴.2₂, _Л2 . This completes the proof. ■

A.2 Proof of Proposition 3

If monetary policy targets current-looking consumer price inflation then one eigenvalue of

the coefficient matrix ARPI is given by e_κ ≡ 1 + (_2a¹_-1)

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