2
Problem 1:
Intertemporal Consumption Leisure Framework – A Numerical
Analysis (20 points).
Consider the two-period intertemporal consumption-leisure
economy.
Suppose the representative consumer’s lifetime utility function is given by
1
1
2
2
1
1
2
2
( , ,
, )
ln( )
ln( )
v c l c l
c
l
c
l
!
"
"
"
, where, as usual,
c
denotes consumption and
l
denotes hours of leisure (hence 168 –
l
is an individual’s hours of labor during a week).
Assume that the representative consumer begins period 1 with zero assets.
The period-1,
period-2, and lifetime budget constraints in this model, expressed in real terms, are thus
given, respectively, by
1
1
1
1
1
2
2
1
2
2
2
(1
)
(168
)
)
)
(168
)
c
a
t
w
l
c
a
r
a
t
w
l
"
!
#
$
$
#
"
!
"
$
"
#
$
$
#
2
2
2
2
1
1
1
1
)
(168
)
)
(168
)
11
c
t w
l
c
t w
l
rr
#
#
"
!
#
#
"
""
The tax rates in the two periods are
12
0.5
tt
!
!
, and the real wages in the two periods are
1
20
w
!
and
2
22
w
!
.
Note that you are NOT given a numerical value for the real
interest rate (you will solve for this in part b below).
a.
(10 points)
Is it possible to solve numerically for the representative consumer’s
optimal choices of consumption in each of the two periods?
If so, do so, showing any
important steps in your logic/computation.
If not, briefly describe the economic
and/or mathematical issue(s) that prevents doing so.
(
Note
:
If you can solve without
setting up and solving a Lagrangian, you may do so.)
Solution:
Construct the three optimality conditions in the model.
To do this, you needed
to compute the four marginal utility functions, which are:
1
1
1
c
v
c
!
,
2
2
1
c
v
c
!
,
1
1
l
v
!
, and
2
1
l
v
!
.
With these, the three optimality conditions are
1
1
2
2
1
2
1
22
2
1
2
1
)
1/
1
)
1
l
c
l
c
c
c
v
tw
vc
v
v
c
r
!
!
#
!
!
#
!
! "
The first two of these expressions are the optimal consumption-leisure margins within
period
1
and
period
2,
respectively,
while
the
third
expression
is
the
optimal
consumption-savings margin linking period 1 and period 2.
Note that leisure
1
l
and
2
l
do
not
appear in any of these optimality conditions (this observation is critical for part c
below).