GROWTH, UNEMPLOYMENT AND THE WAGE SETTING PROCESS.



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 w
t < wtc
then Lte = Nt . Thus setting the wage equal to the competitive wage we still have
L
te = 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 S
t 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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