A Appendix.
Proof of Proposition 2.3. 1 We know that if 1 + Rt = R(wt) then Yts
F(Kd,AtLd) and AKt d = k(wt). The competitive wage, wtc, implies:
At Lt
Ktd = Kts and Ltd = Nt, (33)
and, then, wtc satisfies:
Ks
■ = ANY 3,.
Substituting ( 2) and ( 6) in ( 34), and solving the equation for wtc we obtain:
Ss
Wtc = (1 - α)(At)1-α['.. ]α.
If wt > wtc we obtain:
7/ ʌ
k (wt) =
Ktd
AtLd
Kts
> . t,r
AtNt
St-1
AtNt ’
(35)
because k(wt) is increasing in wt. If Kd = St-ι, then, from ( 35), we obtain:
1-α
d d κd At~α^ -t_ 1 d
Ld = . ■-1—- = -------t— < Nt. If, on the contrary, Ld = Nt, then, from
t Atk(wt) [ wt ]— t , ,, t t, , J
[ 1 — α ]
( 35), we have: Ktd = kk(wt)At Nt > Kts .
If wt < wtc we have:
k (wt)
Kd K = St-A
AtLd AtNt AtNt ■
(36)
If Ld = Nt, then, from ( 36), we have: Kd = k(wt)AtNt = [ɪwt-] — —N—— < Kt.
1 α At —
d t d Kd
If, on the. contrary, Kt = St-ι, then, from ( 36), we obtain: Ld = - t^ > > Nt.
Proof of Proposition 3.1. 1 First note that the wage set by the union will
never be lower than the competitive wage. From proposition 2.3.1, if wt < wtc
then Lte = Nt . Thus setting the wage equal to the competitive wage we still have
Lte = Nt and a greater wage. Now we solve the program of the union assuming
A ɪ -s
that Le = -------t-1. It is easy to show that the wt that maximizes the program
t [ιwtα]α
of the union is the wage that maximizes the function:
_1
(37)
St = ((1 - τt)wt - ht)wt α ;
where ht = γt(1-τt-1)wt-1+(1-γt)st. The first order condition for a maximum
of St is:
1 (1 - τt )wt - ht
(1 - τt) - “ ʌ-----W ----t = 0. (38)
Solving ( 38) for wt we obtain:
ht = Yt(1 - τt-1)wt-1 + (1 - Yt)st (39)
(1 - α)(1 - τt) (1 - α)(1 - τt)
18
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