homogeneous system:
(J — ωl)x =
where
|
/ |
(1 + β + S) — ωi |
0 |
—1 |
0 |
A Ai |
∖ | |
|
-cμ |
— [(1 + δ) + ωi∖ |
cμ |
0 |
Bi | |||
|
0 |
0 |
d33 — ωi |
⅛4 |
Ci | |||
|
∖ |
cμ |
1 |
d43 |
d44 — ωi / |
Di |
|
0 | |
|
0 |
(A3a) |
|
∖0 J |
⅛3 = -lμμFkl > 0, ⅛4 = — μ(Fkk + Fkllk) > 0, ⅛ = Fllμ-cμ > 0, d44 = β+Fllk > 0.
From (A3a), the constants Ai, Bi, Ci and Di, i = 1, 2 are written as:
= Ci = Ci = —⅜4Di
(1 + β + δ) — ωi Φ(ωi) (θ33 — ωi)Φ(ωi) ’
B cμd34
(θ33 — ω⅛) [(1 + δ) + ωi]
(⅛Φγ^) Di = Ω. (⅛≡ Di,
V Φ(ωi) J V Φ(ωi) J
(A3b)
where
''' ω = (1 + β + δ) - ωi, ⅛ ω = (θ33 — ωi)i[(13+ ^)+ ^i] ’
To solve for the constants Di and complete the solution of (A3a), we use the fact that
the stock of durable goods and physical capital evolve continuously from their initial
conditions, α(0) = ao and к (0) = ko∙ From (A2b, d) this gives us the two equation system
a + Ω1 (ω1) f1 — .φ^ D1 + Ω2 (ω2) P — D2 = ao
V Φ(ωι) J 4 Φ(ωi) J
к + Di + D2 = ко
(A4a)
from which we solve for D1, D2 :
φ(wJφ(^ [—(à — ao) + ω2 (ω2) ^1 φφ2^(к — М]
Ωι (u1) Φ(ω2) [1 — Φ(ω1)] — Ω2 (ω2) Φ(ω1) [1 — Φ(ω2)],
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