1
Problem 1:
Consumption and Savings in the TwoPeriod Economy (25 points).
Consider a
twoperiod
economy
in
which
the
government
co
l
l
e
c
t
s
on
ly
lump

sum
t
ax
e
s
f
rom
the
representative consumer, and in which the representative consumer has no control over his pre
tax
income.
The
lifetime
utility
function
of
the
representative
consumer
is
±²
12
1
2
,l
n
l
n
ucc
c
c
³
, where, as usual,
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
pretax income is 26, and the present discounted value of ALL lifetime tax payments is 6.
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
2
1
22
11
,
1
cy
t
r
t
rr
³
´
³´
³
³³
in which the right hand side is the present value of lifetime
disposable
(i.e., aftertax) income
(the notation is standard from Chapter 7).
The Lagrangian is thus
1
1
2
1
(, )
1
yc
uc c
y
c
t
t
r
O
ªº
´
´
´
´
«»
¬¼
³
,
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:
112
212
0
0
1
r
´
´
³
(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
consumptionsavings
optimality
condition,
1
r
³
.
With the given utility function, the marginal utility functions are
1/
uc
and