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Department of Applied Economics
Johns Hopkins University
Economics 602
Macroeconomic Theory and Policy
Midterm Exam Suggested Solutions
Professor Sanjay Chugh
Summer 2010
NAME:
Each problem’s total number of points is shown below.
Your solutions should consist of some
appropriate combination of mathematical analysis, graphical analysis, logical analysis, and
economic intuition, but in no case do solutions need to be exceptionally long.
Your solutions
should get straight to the point –
solutions with irrelevant discussions and derivations will be
penalized.
You are to answer all questions in the spaces provided.
You may use one page (doublesided) of notes.
You may
not
use a calculator.
Problem 1
/ 26
Problem 2
/ 20
Problem 3
/ 32
Problem 4
/ 22
TOTAL
/ 100
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Problem 1:
TwoPeriod Economy (26 points).
Consider a twoperiod economy (with no
government), in which the representative consumer has no control over his income.
The lifetime
utility function of the representative consumer is
( )
12
1
2
,l
n
ucc
c c
=
+
, where ln stands for the
natural logarithm (that is not a typo – it is only
1
c
that is inside the ln operator).
Suppose the following numerical values: the
nominal
interest rate is
0.05
i
=
, the nominal price
of period1 consumption is
1
100
P
=
, the nominal price of period2 consumption is
2
105
P
=
, and
the consumer begins period 1 with zero net assets.
a.
(4 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
105
1
1
0.05
100
P
P
π
=−
=
−
=
.
Them using the
exact Fisher equation,
2
11
.
0
5
.
0
5
i
r
+
+=
=
=
+
, so that
0
r
=
.
b.
(14 points)
Set up a
sequential
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.
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
λ
+−
−
+
+
+
−
,
where
1
and
2
are the multipliers on the period1 and period2 budget constraints.
The first
order condition with respect to
1
c
is
112
1
0
−
=
, with respect to
2
c
is
212
2
0
−
=
,
and with respect to
1
a
is
(1
)
0
r
−+
+ =
.
The third FOC allows us to conclude
)
r
=+
.
Substituting this into the FOC on
1
c
gives
2
(
r
=
+
.
Next, the FOC on
2
c
allows us
to obtain
22
1
2
=
.
Substituting this into the
previous expression gives us
(
r
, or
1
r
=
+
, which of course is the usual consumption
savings optimality condition.
Using the given functional form, the consumptionsavings
optimality condition for this problem can be expressed as
1
1/
1
1
c
r
=
+
, which immediately
allows us to conclude
1
1
c
r
==
=
+
, which completes part i.
However,
2
c
cannot
be
computed here because you are given no information regarding income, either in presentvalue
or periodbyperiod form.
2
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This note was uploaded on 03/08/2011 for the course ECON 602 taught by Professor Chugh during the Spring '11 term at Johns Hopkins.
 Spring '11
 chugh
 Economics

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