proposition 3.1.1 and theorem 4.1 we have:
* - st
-t (1 - α)β ;
substituting st* using proposition 5.1 we obtain:
-* = 1 (1 - β)-*Le ;
t (1 - α)β Nt - Le
and simplifying wt* equation 54 becomes:
Lte
(1 - α)β N
(1 - α)β +(1 - β) t;
that is,
ut =
1 - β
1 - αβ
From ( 22) we have:
Lte
1-α
cAt α wt-1 Le-1
(53)
(54)
(55)
(56)
(57)
1-α
Substituting ( 55) in ( 57), if there is unemployment in period t - 1 solving for
wt* we obtain:
* 4l-α∕1 vcw*-1Nt-1 1a 4l-α∕1 v cw*-1 lα ∕κcλ
wt = cAt (1 - α)[----N----] = At (1 - α)[(1 + n) ] . (58)
Diving ( 58) by At we obtain:
wt--1
w* c ~ï---
— = (1 - α)[-----A—----]a. (59)
At ( )[(1 + n)(1+ g)] ( )
Ifwe call at the equilibrium wage per unit of efficient labor ω*, that is, ω* = A-
equation ( 59) becomes:
ω* = (1 - α)[ ---——----]a. (60)
t ( )[(1 + n)(1+ g)] ()
Solving ( 60) for ωL* -R when ωt* = ωt*-l = ωL* -R we obtain :
ωL в = (1 - α)1-α (-------—:-------)1-α . (61)
l-r ( ) ((1 + n)(1 + g)) ()
It is easy to check, drawing the phase diagram that the wage per unit of efficient
labour converges to ωL* -R. From ( 22) we have in period zero
e
L0
1-α
A0 α Ko ,
[ twl ] α ;
l-α
(62)
22