determinacy alter substantially with the inclusion of capital.
3.1 Aggregate System
The set of linearized equations for the world aggregates, given in Table 1, can be reduced
to the following four-dimensional system:
AW ≡
|
EtbtW+1 = AWbtW, |
W WW bt = mntt xt |
^w ^w^l | |||
|
(1 — a) + λi [∣ + |
α(i-Λ2) ] Λ2 J |
—a(1 — a) |
μμ |
— β ) [1 + αi1ΛΛ)- ] |
0 |
|
Λ1(1-Λ2) |
1 |
a |
(*—β) [ 22] |
0 | |
|
Λ2β | |||||
|
Λι |
0 |
1 |
0 | ||
|
— ( KK )σ |
(1-α)Z . C K + σa K |
0 |
1 + K . | ||
Since the dynamics of mc, x, and π are independent of the capital stock dynamics, one
eigenvalue of the system is 1 + KK > 1. Consequently, given that capital is the only
predetermined variable in the column vector bW , equilibrium determinacy requires that
two of the remaining eigenvalues of AW are outside the unit circle and one eigenvalue is
inside the unit circle. Then by Proposition C.2 of Woodford (2003) the following result
is obtained:
Proposition 1 Suppose that monetary policy is characterized by a current-looking inter-
est rate rule. Then a necessary and sufficient condition for determinacy of the aggregate
system is μ > 1 and either
(2β - 1)Λ2 <Λ1[1-β(1-δ)(1-α)]
(27)
or
μΛια Λια(μ — 1)
^!2β [ Λ2
— 1 — Λ1 (1 — α)
+ 1 — β + Λιμ(1 — α) I—-— > 0,
Λ2
(28)
where Λι = (1-ψ)ψ1-βψ) and Λ2 = 1 — β(1 — δ).
The determinacy conditions summarized in Proposition 1 are isomorphic to the conditions
obtained by Carlstrom and Fuerst (2005) for the closed-economy. Suppose α = 0.36,
β = 0.99 and δ = 0.025. For these parameter values (27) is violated only if ψ ≥ 0.75.
Thus if prices are sufficiently sticky, indeterminacy (of order two) can arise for some values
of the difference system.
13
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