α ξ(1 —α)
= ξ-a /adʌ ξ-α a⅛Eπ-δ⅛⅛L' P (1 - δ> (1 - . ʌ • (21)
ξ ∖ξ J ∖ (1 + β - 2βδ)2 J
α(ξ + θ) —ξ αη α(ξ+η + θ) —ξ
• (α (ξ + θ) - ξ) α-ξ (an)α-ξ (α (ξ + П + θ) - ξ) ξ-α
β ξζ — a τ 1 ʌ aad∖ ξ-α λ-L- ,. ξ(1-α) τ- α(1-ξ) ββ (1 — δ) (1 — βδ)δ ξ-“
= L--L - 1 + all A ξ-α E (1—δ)(ξ-α) L ξ-α (22- (22)
1 — β V ξ /U/ к (1 + β — 2βδ)2 )
ξ-α(ξ + θ) αη α(ξ+η + θ) —ξ
• (a (ξ + θ) - ξ) ξα (an)a—ξ (a (ξ + n + θ) - ξ) ξα
The following result can then be stated:
Proposition 2 Provided the following restrictions on the parameters hold:
1 1 + β .
δ < — and
2β
ξ
<—< < a < ξ if n > 0
ξ + θ
ξ
a < ξ+θ < ξ if n < 0
there exists a unique steady state of the model with 0 <T< 1 and 0 <ρ< 1 where
the values of the different variables are given by the expressions (15)-(22).
Proof. The steady-state values of the variables are obtained in Appendix 7.2.
Concerning the restrictions on the parameters, given the expression obtained for T :
T 1 - βδ
1 + β - 2βδ
the fact that 0 < T < 1 implies 0 < 1+β-2βδ < 1, and since the numerator is positive
(because 0 <β< 1 and 0 <δ < 1) we must have (for the first inequality to hold):
1+β
1 + β - 2βδ> 0 ⇒ δ< +~p
2β
while the second inequality is always verified (being δ<1). Given the expression
obtained for ρ:
an
P a (ξ + n + θ) - ξ
then, the fact that 0 < ρ < 1 implies 0 < α(ξ+α+θ)-ξ < 1. At this point it is necessary
to distinguish the case n> 0 and the case n< 0.Ifn> 0 the restriction 0 <ρ< 1
requires:
ξ
a (ξ + n + θ) - ξ> 0 ⇒ a> -------
ξ+ n + θ
and also:
ξ
an<a (ξ + n + θ) - ξ ⇒ a> ξ+θ
17