1 Γ C 1
(Case II) 0.5 > a > 2—δ 1 - δ — к 2
and
2(1+ β)Λ2
Λι [(1 — 2a)α(2 — Λ2) — Λ2]
if
2α(1— 2a)
1 + α(1 — 2a)
> Λ2;
(Case III)
a<
1 — δ — Cɪ] < 0.5
K 2
(i)
2(1+ β)Λ2
Λι [(1 — 2a)α(2 — Λ2) — Λ2]
and
and
2α(1 — 2a)
(ii) ---ʒʒ > Λ2.
1 + α(1 — 2a)
(31)
(32)
where Λι = (1 ψ)ψ1 βψ), Λ2 = 1 — β(1 — δ) and Λ4 = Λια(2a — 1).
Proof. See Appendix A.1. □
Cases I and II of Proposition 2 show the regions of determinacy when the root associ-
ated with the capital stock dynamics is unstable, whereas Case III shows the regions of
determinacy when this root is stable. We illustrate these determinacy conditions using
the following parameter values. Suppose α = 0.36, β = 0.99, δ = 0.025, λ = 7.66 and
ψ = 0.75. Figure 1 depicts the regions in the parameter space (a, μ) that are associated
with determinacy (D), first-order indeterminacy (I1) and second-order indeterminacy (I2)
around the neighborhood of the steady state.14 If the degree of trade openness is suf-
ficiently low (a > 0.5) then second-order indeterminacy can arise in the open-economy.
This follows from the violation of conditions (29) and (30) of Case I in Proposition 2.
Indeed comparing (29) with condition (27) of Proposition 1 yields
(2β — 1)Λ2 < Λι [1 — β(1 — δ)(1 — (2a — 1)α)] < Λχ [1 — β(1 — δ)(1 — α)],
which by inspection implies that a higher degree of price stickiness is required in the
open economy to prevent the emergence of second-order indeterminacy. Furthermore, as
depicted in Figure 1, if the degree of trade openness is sufficiently high then first-order
indeterminacy can also exist. This arises from the violation of condition (32) of Case III
in Proposition 2. First consider condition (31). Under the assigned parameter values,
14 Recall that with these parameter values indeterminacy is not possible in the closed-economy (aggregate
system) ∀μ > 1.
15