Department of Economics
University of Maryland
Economics 325
Intermediate Macroeconomic Analysis
Problem Set 1 Suggested Solutions
Professor Sanjay Chugh
Spring 2009
Instructions
:
Written (typed is strongly preferred, but not required) solutions must be
submitted no later than 11:00am on the date listed above (either in class or in the
Economics Department Main Office, Tydings Hall 3105).
Your solutions, which likely
require some combination of mathematical derivations, economic reasoning, graphical
analysis, and pure logic, should be thoroughly presented and not leave the reader (i.e.,
your TAs and I) guessing about what you actually meant.
You must submit your own independentlywritten solutions.
You are permitted (in
fact, encouraged) to work in groups to think through issues and ideas, but your “writing
up” of solutions should be done independently of anyone else.
Under no circumstances
will multiple verbatim identical solutions be considered acceptable.
There are three problems in total, each with multiple subparts.
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2
Problem 1:
Optimal Choice in the ConsumptionSavings Model During a Credit
Crunch:
A Numerical Analysis.
Consider a twoperiod economy (with no government and
hence neither government spending nor taxation), 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 real income in period 1 is
y
1
= 10 and the
consumer’s real income in period 2 is
y
2
= 22.
Suppose that the real interest rate between
period 1 and period 2 is ten percent (i.e.,
r
= 0.10), and also suppose the consumer begins
period 1 with
real
net wealth (inclusive of interest) of (1+
r
)
a
0
= 2.
Set up the lifetime Lagrangian formulation of the consumer’s problem, and use it to answer
part a, b, and c.
Show all steps in your logic/arguments.
Solution:
The Lagrangian in real terms, using the given functional form for utility, is
2
2
1
2
1
0
1
ln
ln
(1
)
1
1
y
c
c
c
y
r a
c
r
r
%
&
’
$
$
$
$
$
(
(
)
*
$
$
+
,
,
where the term in square brackets (when set equal to zero) is simply the LBC in real terms.
The firstorder conditions of this problem are (recognizing that the FOC with respect to
!
is
simply the LBC in real terms):
1
2
1
0
1
0
1
c
c
r
%
%
(
#
(
#
$
Combining these two equations to eliminate the multiplier as usual gives the
consumptionsavings optimality condition for this particular utility function:
2
1
1
c
r
c
#
$
.
In what follows, you must use the consumptionsavings optimality condition along with
the LBC (which together constitute two equations in the two unknowns
c
1
and
c
2
) to
proceed.
a.
I
s
it possible to numerically compute the consumer’s optimal choice of consumption in
period 1?
If so, compute it; if not, explain why not.
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 Spring '08
 chugh
 Economics, Public Finance, Period, FISCAL STIMULUS, representative, credit crunch

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