social planner maximizes also the utility of a representative agent. Because in
this model we have heterogeneity of workers (employed and unemployed) the
income of both app ears in the utility function. We use as argument income
instead of utility (log of income) because it simplies the solution. The param-
eter βt weights the welfare of employed and unemployed workers in the utility
function of the government. If βt increases we say that the government becomes
more proemployed workers. We assume that 0 ≤ βt ≤ 1. Note that the income
of the old people in period t, (1 + Rt)cwt-1Lte-1, do not appear in the utility
function of the government. This is because we assume that the government
only cares for the welfare of the people affected by its actions when there is
unemployment. However, one can check, that the solution of the government’s
program will not change if we introduce this term as a factor.
The approach of the social welfare function for the government with different
weights has been used in other models with heterogeinity of agents (Przeworski
and Wallerstein [9]). The other alternative standard approach, not followed in
this paper, is to assume that the government represents the median voter.
It is easy to check that the solution to Program Gt is given by:
τt = 1 - βt ,
(20)
and
st =
(1 - βt)wtLe
Nt - Le
(21)
If there is full employment the government does nothing, which means τt = 0
and st = 0.
Note that the government decides both the tax rate and the unemployment
benefit and its optimal choice implies a constant tax rate and the amount of
the unemployment benefit changes depending on the amount of employment.
How the unemployment benefit is set is really important and it will affect the
dynamics of the model. Other assumptions used in the literature is to set
the unemployment benefit proportional to wages (z = λw, Pissarides [8]) or
proportional to aggregate output ( s = σy, Daveri and Tabellini [3]).
The behaviour of the government is expressed by the best reply functions of
the government, τt(wt,βt) and st(wt,βt). Note that, if there is unemployment,
the taxe rate does not depend on the wage and that it is decreasing in β . The
variable decided by the government that depend on the wage is the unemploy-
ment benefit. It is easy to check that st is decreasing in wt and βt. Note also,
from ( 21), that when there is few unemployment, the unemployment benefit is
really big, thus, it can be the case, that the real wage after taxes is less than
10