1
Problem 1:
Consumption and Savings in the TwoPeriod Economy (20 points).
Consider a
twoperiod economy (with no government and hence no taxes), 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 nominal terms:
suppose the consumer’s
present discounted value of ALL lifetime
NOMINAL income is 52.
For part a of this problem, suppose also the following:
1.
The nominal interest rate between period 1 and period 2 is zero (i.e.,
i
= 0, which is
roughly what the nominal Federal Funds interest rate is currently).
2.
The consumer begins period 1 with zero net assets.
3.
Nominal prices of consumption in the two periods are
1
2
P
#
and
2
2
P
#
.
a.
(14 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
nominal
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 nominal LBC of the consumer is
2 2
2
2
1 1
1 1
,
11
P c
P y
Pc
P y
ii
$
#
$
$
$
where the notation is standard from class.
The Lagrangian is thus
2
2
2 2
1
2
1 1
1 1
( ,
)
P y
P c
u c c
P y
ri
i
%
&
’
$
$
(
(
)
*
$
$
+
,
,
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
1
2
2
1
2
( ,
)
0
( ,
)
0
1
u c c
P
P
u c c
i
(
#
(
#
$
(And of course the FOC with respect to the multiplier just gives back the LBC.)
Proceeding
exactly
as we did in class, these FOCs can be combined into
1
1
2
2
1
2
2
( ,
)
1
1
( ,
)
1
u c c
i
r
u c c

$
#
$
, where the
last equality follows by the Fisher Equation.
Note that because we know both
i
and inflation