1
Problem 1:
Consumption and Savings in the Two-Period Economy (25 points).
Consider a
two-period economy (with no government), in which the representative consumer has no control
over
his
income.
The
lifetime
utility
function
of
the
representative
consumer
is
!"
1
2
1
2
,
ln
ln
u c c
c
c
#$
, where
ln stands for the natural logarithm.
We will work here in purely
real terms:
suppose the consumer’s
present discounted value of ALL lifetime REAL income
is 26.
Suppose that the real interest rate between period 1 and period 2 is zero (i.e.,
r
= 0), and
also suppose the consumer begins period 1 with zero net assets.
a.
(17 points)
Set up the lifetime Lagrangian formulation of the consumer’s problem, in order
to answer the following:
i)
is it possible to numerically compute the consumer’s optimal
choice of consumption in period 1?
If so, compute it; if not, explain why not.
ii) is it
possible to numerically compute the consumer’s optimal choice of consumption in period 2?
If so, compute it; if not, explain why not.
iii) is it possible to numerically compute the
consumer’s real asset position at the end of period 1?
If so, compute it; if not, explain why
not.
Solution:
We know that with zero initial assets, the LBC of the consumer is
22
11
,
cy
rr
$
#
$
$
$
where the notation is standard from class.
The Lagrangian is thus
1
2
1
1
( ,
)
yc
u c c
y
c
%
&
’
$
$
(
(
)
*
$$
+
,
,
where
of course is the Lagrange multiplier (note there’s only one multiplier since this is the
lifetime formulation of the problem not the sequential formulation of the problem).
The first-
order conditions with respect to
1
c
and
2
c
(which are the objects of choice) are, as usual:
1
1
2
2
1
2
1
( ,
)
0
( ,
)
0
1
u c c
u c c
r
(
#
(
#
$
(And of course the FOC with respect to the multiplier just gives back the LBC.)
Also as usual,
these
FOCs
can
be
combined
to
give
the
consumption-savings
optimality
condition,
1
1
2
1
2
1
2
( ,
)
1
( ,
)
u c c
r
u c c
.
With the given utility function, the marginal utility functions are
1/
uc
#
and
#
, so the consumption-savings optimality condition in this case becomes
2
1
1
/1
c
c
r
#
$
.