5 Special Cases
In general it is not possible to compute the equilibrium w*u, τ*u and s*u except
in the special cases: γt = 1 and βt = β for all t, and γt = 0, βt = β for all t.
These two extreme cases have an easy interpretation. In the first case, γt = 1
and βt = β for all t, the union only takes into account the wage after taxes of
the previous period, when setting the wage. In the second case the union only
takes into account the unemployment benefit of the period. In both cases the
government gives always the same weight to the welfare of employed workers.
When γt = 1 and βt = β for all t we can prove the following proposition:
Proposition 5.1 There exists Nt such that if Nt > (≤)Nt then there exists
a unique equilibrium with unemployment (full employment) in period t and
I*
1-αα. If there is unemployment in each period t (Nt > INt for
all t) and g < y-α^, the rate of growth of output, yt, and the rate of growth
of employment, lt, decrease with time and there is some period t' such that for
all t ≥ t' the rate of unemployment, ut, increases. If g = y-α^, yt and lt are
constant. If lt < n then ut increases with time. If lt = n then ut is constant. If
lt > n then ut decreases. If g > y-α^, yt and lt increase with time and there is
some period t” such that for all t ≥ t' the rate of unemployment, ut, decreases.
Proposition 5.1 says that if the rate of growth of wages is less than the rate
of productivity growth then there is an increase in the rates of growth of output
and employment and the rate of unemployment decreases after some time. This
condition implies that the rate of growth of the wage per unit of effective labour
is negative, that is, the wage per unit of effective labor decreases with time. The
bad situation is when the rate of growth of wages is greater than the rate of
productivity growth, i. e. when the wage per unit of effective labour increases,
in this case growth decreases and unemployment increases. From the proof of
proposition 5.1 it is easy to see that we can have situations with a positive rate
of growth of output and a negative rate of growth of employment.
When γt = 0 and βt = β for all t, we can prove the following proposition:
Proposition 5.2 If γt = 0 and βt = β for all t there always exists an equi-
librium with unemployment and an equilibrium with full employment. The un-
employment rate at the equilibrium with unemployment is the same for all t
and is given by ut = ι^--ββ. Moreover if there is always unemployment, the
wage per unit of effective labour, ω* = AWt, and the unemployment benefit per
unit of effective labour, σ* = -AA-, converge to a long run wage and unemploy-
1 ι 1 α
ment benefit per unit of effective labour, ω',L R = (1 — α)1—α ( (i+n)(i+g) )1—α and
σA-R = β(1 — α)1—α ( (i+nc(i+g) )1—a , the long run rate of growth of output is
equal to (1 + n)(1 + g) — 1 and the long run rate of growth of employment is
equal to n.
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