2
Problem 1:
Elasticity of Labor Supply and Fiscal Policy. (40 points).
Consider the static
(i.e., one-period) consumption-leisure framework.
In quantitative and policy applications
that use this framework, a commonly-used utility function is
!"
1 1/
168
1 1/
( , )
ln
ul
c l
c
#
$
%
&
%
’&
,
in which
c
denotes consumption,
l
denotes the number of hours (in a week) spent in leisure,
and
and
(the Greek letters “psi” and “theta,” respectively) are
constants
(even
though we will not assign any numerical value to them) in the utility function.
The
representative individual has no control over either
or
, and both
0
(
and
0
(
.
Labor
is measured as
n
= 168 –
l
, the
real wage
is denoted by
w
, and the labor-income tax
rate is denoted by
t.
The
take-home rate
(the fraction of labor income that an individual
keeps) can thus be defined as
S
= (1-
t
).
Finally, expressed in real terms, the representative
individual’s budget constraint is
(1
)
c
t wn
’
&
.
Thus, this is
exactly
the consumption-leisure framework studied in Chapter 2, with now a
particular functional form for
u
(
c,l
).
a.
(5 points)
For the given utility function, set up the consumption-leisure optimality
condition, and then solve it for
n
,
expressing the solution in terms of (among other
things)
S
,
rather than
t
.
(that is, your final expression here should be of the form
n
= …,
and on the right-hand side the take-home rate
S
should appear, but the tax rate
t
should
not appear.).
(
Note:
You do
not
need to set up a Lagrangian in order to analyze this
problem; you may go straight to the consumption-leisure optimality condition.
Be very
careful in deriving the result, because the rest of the analysis is based on this.)