figure 1.c there is an equilibrium with full employment. Looking at figure 2.a it
is obvious that there is a unique equlibrium with unemployment. Looking at fig-
ures 2.b and 2.c there is an equilibrium with unemployment and an equilibrium
with full employment.
Proof of Theorem 4. 2 If there is unemployment, by 20, we have τt(wt,βt) =
1 — βt and then ∂∂βt < 0 and dʒt = 0. We can then write the best reply function
of the union as: wt(wt-ι,τt-ι, βt, st,γt). By the implicit function theorem we
have:
∂w^t
∂Yt
∂st
∂γt
∂wt
∂β
∂st
∂βt
1
∂Wt ∂st
∂βt ∂wt
∂Hit
∂γt
∂st ∂w>t
∂wt ∂γt
∂wt I ∂wt ∂st
∂βt + ∂st ∂βt
∂st ∂Wt I ∂st
∂wt ∂βt ^l^ ∂βt
(44)
It is easy to prove that if (1 — τt)wt-ι > (<)st 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 — τt)wt-1 > (<)st
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)Ldα-(Nt — Ld) — (1 — βt)α(Ld)2 _wt wtLd
(Nt — Ld)2 ( βt ) (Nt — Ld).
(46)
Simplifing ( 46) we obtain:
—— — wtLd(Nt — Ld) + 1-ββt wt(Ld)2 wtLd
(Nt — Ld)2 > (Nt — Ld).
(47)
Inequality ( 47) is true if and only if (1 — βt)Ltd > (α + βt — 1)(Nt — Ltd), that
is, if and only if: ut < 1-Jβt.
Proof of Proposition 5. 1 If βt = β for all t from the proof of theorem 4.1 we
w *
have wtt = 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