GROWTH, UNEMPLOYMENT AND THE WAGE SETTING PROCESS.

A Appendix.

Proof of Proposition 2.3. 1 We know that if 1 + R_t = R(w_t) then Y_t^s

F(Kd,A_tLd) and AKt _d = k(w_t). The competitive wage, wtc, implies:
A_t L_t

K_t^d = K_t^s and L_t^d = Nt,                      (33)

and, then, w_t^c satisfies:

K^s

■    = ANY                      3,.

Substituting ( 2) and ( 6) in ( 34), and solving the equation for w_t^c we obtain:
S^s

Wt^c = (1 - α)(At)^1-α['.. ]^α.

If w_t > w_t^c we obtain:

because k(w_t) is increasing in w_t. If Kd = St_-ι, then, from ( 35), we obtain:
1-α

_{d d}       κ^d       A_t~α^ -t_ _{1                                    d}

Ld =  . ■-1—- = -------t— < N_t. If, on the contrary, Ld = N_t, then, from

^t      A_tk(w_t)         [ wt ]—            ^{t ,                        ,,    t         t,        ,} J

^[ 1 — α ^]

( 35), we have: K_t^d = k^k(w_t)A_t N_t > K_t^s .

If w_t < w_t^c we have:

If Ld = N_t, then, from ( 36), we have: Kd = k(w_t)A_tN_t = [ɪwt-] — —N—— < Kt.

^{1 α}  At —

d t                                        d       ^Kd

If, on the. contrary, Kt = St_-ι, then, from ( 36), we obtain: Ld =   - ^t^ > > N_t.

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 < w_t^c
then L_t^e = N_t . Thus setting the wage equal to the competitive wage we still have
L_t^e = 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 w_t that maximizes the program

t        [ι^wtα]^α

of the union is the wage that maximizes the function:

_1

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 - τ_t )w_t - h_t

(1 - τ_t) - “ ʌ-----W ----t = 0.                   (38)

Solving ( 38) for wt we obtain:

^ht        ₌ Yt(^{1 - τ}t-1)wt-1 + ^{(1 -} Yt)st          ₍₃₉₎

^{(1 - α})(^{1 - τ}t)            ^{(1 - α})(^{1 - τ}t)

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