1
Problem 1:
Two-Period Economy (25 points).
Consider a two-period 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
±²
12
1
2
,l
n
ucc
c c
³
,
where ln stands for the natural logarithm (that is not a typo – it is only
1
c
that is inside a ln(.)
function,
c
2
is
not
inside a ln(.) function).
Suppose the following numerical values: the
nominal
interest rate is
0.02
i
, the nominal price
of period-1 consumption is
1
100
P
, the nominal price of period-2 consumption is
2
102
P
, and
the consumer begins period 1 with zero net assets.
a.
(3 points)
Is it possible to numerically compute the
real
interest rate (
r
) between period one
and period two?
If so, compute it; if not, explain why not.
Solution:
The inflation rate is easily computed as
2
2
1
101
11
0
.
0
1
100
P
P
S
´
´
.
Then, using the
exact Fisher equation,
2
.
0
1
.
0
1
i
r
³
³
³
, so that
0
r
.
b.
(14 points)
Set up a
sequential
Lagrangian formulation of the consumer’s problem, and
compute first-order conditions 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.
Solution:
The sequential Lagrangian for this problem (here cast in real terms, but you could
have case it in nominal terms as well) is
11 1
1
2 2
1 2
(, )
[
]
[
(
1 )
]
uc c
y c a
y
ra c
O
³´
´
³
³
³
´
,
where
1
and
2
are the multipliers on the period-1 and period-2 budget constraints.
The first-
order condition with respect to
1
c
is
112
1
0
´
, with respect to
2
c
212
2
0
´
,
and with respect to
1
a
is
(1
)
0
r
´³
.
The third FOC allows us to conclude
)
r
³
.
Substituting this into the FOC on
1
c
gives
2
(
r
³
.
Next, the FOC on
2
c
allows us
to
obtain
22
1
2
.
Substituting
this
into
the
previous
expression
gives
us
(
r
³
, or
1
r
³
, which of course is the usual consumption-
savings optimality condition.
Using the given functional form, the consumption-savings
optimality condition for this problem can be expressed as
1
1/
1
1
c
r
³
, which immediately
allows us to conclude
1
1
c
r
³
, which completes part i.
However,
2
c
cannot
be