5.5. Solution for (φ,a,μ,n) in the Small Open Economy
The general stable solution to the differential equation system (3.9a) is represented by the
following expressions:
φ = φ + M1eθrt + M2eθ2t (A13a)
a = a ■ A1eθ1t + A2eθ2t (A13b)
μ = μ + P1eθ1t + P2eθ2t (A13c)
n = n + Q1eθ1t + Q2eθ2t
(A13d)
where Mj, Ni, Pi and Qi, i = 1, 2 are constants (eigenvectors) corresponding to the stable
eigenvalues θι and θ2 of the small open economy and where (φ, a, μ, n) are the steady-
state solutions derived from the system (3.8a)-(3.8d). Since only two of these constants
are independent, the first step in obtaining the complete solution is to solve Mi, Ni, Pi in
terms of Qi, i = 1, 2. These relationships, using (3.9a) and (A13a)-(A13d), are calculated
from the homogeneous system:
(14a)
(Jsoe - θl)x =
( (1 + β + δ) - θi 0 -1 0 |
M Mi |
0 "ʌ | ||
-cμ - [(1 + δ) + θi] cμ 0 |
Ni |
0 | ||
0 0 -θi -μα |
Pi |
0 | ||
-cμ -1 - ( ʌ''lμ - cμ} (в + α'n} - θi J |
Qi J |
10 J |
where x = (Mi, Ni, Pi, Qi),and where the constants in the small open economy (soe) for
i = 1, 2 are written as:
Mi =
Pi
(1 + β + δ) - θi
Pi
Φsoe(θi)
-μ<QQi.
θiΦ≡oe(θi),
33
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