GROWTH, UNEMPLOYMENT AND THE WAGE SETTING PROCESS.

∂w^t
∂Y^t
∂s_t
∂γt

∂w_t

∂β
∂st

∂βt

1

∂Wt ∂st

∂βt ∂wt

∂Hit

∂γt

∂st ∂w>t

∂wt ∂γt

∂wt I ∂wt ∂st
∂βt ⁺ ∂st ∂βt
∂s_t ∂W_t I ∂s_t
∂wt ∂βt ^^l^ ∂βt

(44)

It is easy to prove that if (1 — τ_t)w_t-ι > (<)s_t then ^wt > (<)0. We know that
ywt < 0 and ^wt > 0. We also know that N- < 0 and yst < 0. Substituting the
∂βt             ∂st .                          ∂wt            ∂βt .               g

sign of this partial derivatives in ( 44) in we obtain, when (1 — τ_t)w_t-1 > (<)s_t

the following matrix:

∂wt
∂^γt
∂st
∂γt

∂wt
∂βt
∂st
∂βt

+(—) —

This means that, by the

sign of the partial derivatives, we do not know the sign of ∂β^t. In order to have
these partial derivative positive, by ( 44), we need:

∂st ∂wt ∂st

∂wt ∂βt > ∂βt.

(45)

Computing the partial derivatives of ( 45) and substituting the labor demand
elasticity by — -1 we obtain:

(1 — βt)Ld^α-(Nt — Ld) — (1 — βt)α(Ld)² _wt      wtLd

(Nt — Ld)^{2                  (} βt )    (Nt — Ld).

(46)

Simplifing ( 46) we obtain:

—— — wt^Ld^(Nt ^{— L}d^{) +} 1^-β^βt wt^(Ld⁾²     wtLd

(Nt — Ld)²                > (Nt — Ld).

(47)

Inequality ( 47) is true if and only if (1 — β_t)L_t^d > (α + β_t — 1)(N_t — L_t^d), that
is, if and only if: u_t < ^1-J^βt.

Proof of Proposition 5. 1 If βt = β for all t from the proof of theorem 4.1 we
w *

have wt_t = w” _t = (1--1). Then theorem 4.1 becomes: If wt: < (γ-1 there exists
w^*

a unique equilibrium with unemployment. If wtC ≥ (1-^-1) there exists a unique
w^*

equilibrium with full employment. Finally, it is easy to check that wtc < (ι-α)

holds if and only if Nt

1 — α        2

cA. ^α (1-α) α Le ₁

t                    t—1

we have w∩ = wξ.

K*

1-α

1) ^α

= Nt . Note also that in period zero

20