using ( 55) we have:
Le = (1 - а)в N
(63)
0 (1 - α)β +(1 - β) N0.
Substituting ( 62) in ( 63) and solving the equation for w* we obtain:
w* = (1 - α)[(1
- α)β +(1 - β) AO- Ko
(1 - α)β No
(64)
that is:
* ∩ u(1 - α)β + (1 - β) KO iα {(ir∖
ωo = <1 - a)[—.— A, ∙ (65)
From ( 53) we obtain : st* = β(1 - α)wt* and defining the equilibrium unem-
ployment benefit per unit of efficient labor, σ*, as σ* = At we also have
σto = β(1 - α)ωto∙ (66)
Substituting ( 58) and ( 61) in ( 66) we obtain:
σ* = β(1-α)(1 - α)(2-α)[⅛-]α;
(1 + n)
(67)
and
σ*-R = β(1 - α)2-α (—C—)1-α . (68)
- (1 + n)
Finally, substituting ( 61) in ( 23) we obtain that yLo -R = (1 + g)(1 + n) - 1
and the rate of growth of employment is always equal to the rate of population
growth because the rate of unemployment is constant with time.
Proof of Proposition 6. 1 We define equation 31 as F (ωt, ωt-1 , ut) = 0 and
equation 32 as G(ωt, ωt-1, ut-1, ut) = 0 which implicitly define the system:
ωt = f1 (ωt-1, ut-1), ut = f2 (ωt-1 , ut-1). Linearizing this system around the
steady state and simplifying we obtain the fol lowing jacobian matrix:
J=
∂fl ∂f2
∂ωt-ι ∂ut-ι
∂f2 ∂fι
∂ωt-ι ∂ut-ι
Y+[(1-a)(1+g)-Y] U
y+ α [(1-αX1+5)-γ] U
1 + α (1-u)
ɪ γ
γ+ α [(1-α)(1+g)-γ] U
-[(1-α)(1+S)-Y] u(ι-u) ^
γ+α [(1-α)(1+g)-γ] U
.
_________γ_________
γ+α [(1-a)(1+g)-Y] и? -
(69)
The eigen values, λ, are obtained computing the equation det(λI - J) = 0. If
γ = (1 - α)(1 + g) then u = 1 and J =
, which means a unique eigen
value λ = 1 and, then, the system does not converge to the steady state. If
γ < (1 - α)(1 + g) then 0 < u < 1. In this case one can compute that the eigen
values will be real numbers if and only if
α1
(70)
γ< 4(Γ+Σ)[(1 - a)(1 + g) - γ]U∙
23