GROWTH, UNEMPLOYMENT AND THE WAGE SETTING PROCESS.



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)[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 t-1, ut-1), ut = f2 t-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 [(1X1+5)-γ] U
1 + α (1-u)
ɪ γ

γ+ α [(1-α)(1+g)-γ] U


-[(1-α)(1+S)-Y] u-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



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