GROWTH, UNEMPLOYMENT AND THE WAGE SETTING PROCESS.

Now, w_t > w_t^c implies, using ( 39):

^γt(^{1 - τ}t-i)wt-i + (1 ^{- γ}t)^st    ∕₁      1- α ∖1-α rSt-1 iα

------->----VTi-----ʌ------- > (^{1 - α})(At)    ʒ-

(1 - α)(1 - τt )                           Nt

If γt = 1 inequality 40 becomes:

If 0 ≤ γt < 1 solving 40 for st we obtain:

At                                '..

(^{1 -} γt)

^γthu-  ^γf ^{γ γ}t ^γ W( ^γ∏V^-' Wt+ — ^Yt(ɪ  ʃt ¹)w^{t  1 +} 0^  ^γt)^st    _?..C   Tf Q, <5 Q, j^T)P^f∩ Q∩P

u ∣uSθj jj st > st we avec∙ `wt — (i α)(i τ )        > wt .  ∙' st ≤ ðt enw we

have Le — Nt in which case wt — wtc. Finally we compute when st < 0. From
( 42) this inequality holds ijand only ij:

_ιc < Yt(1 - τt-i)wt-i
^t (^{1 - α})(^{1 - τ}t)

Note from (41) and ( 43) that w_t ≥ 0.

Proof of Theorem 4. 1 First we compute the best reply junction oj the union

when τt — 1 - βt. Using the proof of proposition 3.1.1 we obtain: If w_t^c <
w”  _— ^Yt(i-Tt-1)w^t-1 then _w. _— ^Yt(i-Tt-1)w^t-1+(i-Yt)st > wc IJ wc ≥ w”

^w t ^— (i_-α)β_t then ^wt ^—          (i_-α)β_t          > ^wt . I^{j w}t ≥ ^w t

and γ_t — 1 then w_t — w_t^c. If w_t^c ≥ w”_t and 0 ≤ γ_t < 1 then there ex-
ists s”_t — (^{i α})^βtw^t (_i-Yi) ^τt-1)w^t-1 ≥ 0 such that if s_t > s”_t then w_t —
^{Yt(i τ}t~(i-α^-)β+(^{i Yt})s^t > wc and if s_t ≤ s” _t then w_t — wtc. We denote this
best reply function of the union as w_t(w_t-i, τ_t-i, 1 — βt, s_t, γ_t).

Now we compute the best reply function of the union when τt — 0. Using the
proof of proposition 3.1.1 we obtain: If wC < wt — ^Yt(i (i' /.)w' ¹ then w_t —
^γt(ɪ__^τt^-1)^wt^{-1 + (}ɪ__^γt)^st /∕^,c Jf /∕^,c '> 7∕/ nr∣d ^,y, — 1 then cιn — cn^c Jf cn^{c γ} tιd

(i  α)              > ɛwt.  Jj ^wt ≥ 'w^!t ann∣  tt — ɪ t∣ven `wt — 'w^!t.  Jj 'w^!t ≥ 'wⁱt

and 0 ≤ γ_t < 1 then there exists st — ^{(i α})w^{t γ}t^{t(i τt-1})w^t-1 ≥ 0 such that if
γt                                 t                 (i-γt)

s_t > st then w_t — ^{Yt(i τt-1})W-^-,¹)⁺(^{i Yt})s^t > wt and if s_t ≤ s'_t then w_t — wf. We

denote this best reply function of the union as w_t(w_t-i,τ_t-i, 0, s_t,γ_t).

Functions wt(wt-i,τt-i, 1 — βt, st,γt) and wt(wt-i,τt-i, 0, st, γt) when γt —

1 and wc < wt, wt ≤ wc < w” _t and w” _t ≤ wc are represented in figures 1.a, 1.b
and 1.c respectively. Functions w_t(w_t-i, τ_t-i, 1—β_t, s_t, γ_t) and w_t(w_t-i, τ_t-i, 0, s_t, γ_t)
when 0 ≤ γ_t < 1 and wc < wt, wt ≤ wc < w” t and w” _t ≤ wc are represented
in figures 2.a, 2.b and 2.c respectively. On the other hand, we can also repre-
sent function s_t in all figures. Looking at figure 1.a it is obvious that there is a
unique equlibrium with unemployment. Looking at figure 1.b there is an equilib-
rium with unemployment and an equilibrium with full employment. Looking at

19

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