either the expected income of the worker or the sum of all workers income,
which are the most common utility functions used for unions. The solution to
Program Ut is given by the following proposition:
Proposition 3.1.1 There exists wt ≥ 0 such that if wtc < wt then
wt =
γt(1 - τt-1)wt-1 + (1 - γt)st
(1 - α)(1 - τt)
> wtc .
(18)
If wtc ≥ wt and γt = 1 then wt = wtc. If wtc ≥ wt and 0 ≤ γt < 1 then there
exists st ≥ 0 such that if st > st then
wt =
γt(1 - τt-1)wt-1 + (1 - γt)st
(1 - α)(1 - τt)
> wtc ;
(19)
and if st ≤ st then wt = wtc.
Proposition 3.1.1 gives the best reply function of the union which is denoted
by: wt(wt-1 ,τt-ι,τt, st ,γt). Proposition 3.1.1 says that if the competitive wage
is low enough then the union always sets a wage that implies unemployment. If
the competitive wage is high and γt = 1 the union always sets the competitive
wage. If the competitive wage is high and 0 ≤ γt < 1, then there are low values
of the unemployment benefit where the union sets the competitive wage. It is
easy to check from equations ( 18) and ( 19) that, if there is unemployment, wt
is increasing in τt and when 0 ≤ γt < 1 it is increasing in st . Note also that
if γt = 1 and τt-1 = τt then wt = (wt-1 that is, the growth rate of the real
wage is given by: (1-1 α) — 1. If γt = 0 then wt = (ɪ-Oχ1-T). In this last case
if st = (1 - α)(1 - τt)wt-1 then we have wt = wt-1, that is, real wage rigidity
along time. We have also real wage rigidity in this case if st and τt are constant
over time.
3.2 The Government
We assume that if there is unemployment in period t the government sets the
tax rate and the unemployment benefit in order to maximize a utility function
that depends on the income of employed and unemployed workers. We assume
also that the government balances its budget at period t.
The formal program of the government is the following:
Program Gt : Given wt > wtC and Lte, choose τt and st in order to maximize:
Gt = ((1 — τt)wtLe)βt (st(Nt — Le))1-βt,
subject to:
st (Nt — Lte) = τtwtLte .
We assume, then, that the government maximizes a social welfare function
that depends on the income of people affected by its actions. This utility func-
tion is similar to the used in the standard social planner program where the