Investment and Interest Rate Policy in the Open Economy

A Appendix

A.1 Proof of Proposition 2

As shown in the main text, if monetary policy targets current-looking domestic price
inflation this results in a four dimensional system [mιcR xR πR(^{h f} ) kR , where KR
is the only predetermined endogenous variable. Determinacy of the difference system
thus requires one eigenvalue to lie inside the unit circle and the other three eigenvalues
to lie outside the unit circle. One eigenvalue of the coefficient matrix A_P^R_{P I} is given by
e_κ ≡ 1 + (2_a— 1) [KC + 2δ(1 — °)], which modulus can be greater or less than one. The
remaining three eigenvalues are determined by the upper left 3 × 3 submatrix of A_P^R_PI,
denoted by ARPI. Then the characteristic equation of ARPI is

r³ + a2r² + a1r + a0 = 0

where

1 Λ1 α(1 - Λ2)(2a - 1)
a² = ^-1 — β — J [1⁺       Λ2      _

1 _ι Λιμ    _ι α(1 — Λ2)(2a — 1) _ι αΛι(2a — 1)

β ⁺ ~ [^{1 +}     Λ2     J ⁺   Λ2β

Λιμα(2a — 1)

^a0 =      Λ2β

First suppose the eigenvalue eK is unstable, |eK | > 1, which requires either a > 0.5 or
0.5 > a > 2—δ [1 — δ — 1 =J. 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 + a₂ + aι + ao > 0 ⇔ ———-—¹ [1 — (2a — 1)α] < 0,

or (Case II)

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