Department of Economics
University of Maryland
Economics 325
Intermediate Macroeconomic Analysis
Practice Problem Set 4 Suggested Solutions
Professor Sanjay Chugh
Spring 2011
1.
Optimal Choice in the ConsumptionSavings Model with Credit Constraints:
A
Numerical Analysis.
Consider our usual twoperiod consumptionsavings model.
Let preferences of the representative consumer be described by the utility function
12
1
2
(, )
,
uc c
c
c
E
±
where
1
c
denotes consumption in period one and
2
c
denotes consumption in period
two.
The parameter
is known as the subjective discount factor and measures the
consumer's degree of impatience in the sense that the smaller is
, the higher the
weight the consumer assigns to present consumption relative to future consumption.
Assume that
1/1.1.
For this particular utility specification, the marginal utility
functions are given by
112
1
1
2
ucc
c
and
212
2
2
c
.
The representative household has initial
real
financial wealth (including interest) of
0
(1
)
1
ra
±
The household earns
1
5
y
units of goods in period one and
2
10
y
units in period two.
The real interest rate paid on assets held from period one to
period two equals 10% (i.e.,
0.1
r
).
a.
Calculate the equilibrium levels of consumption in periods one and two. (
Hint:
Set up the Lagrangian and solve.)
b.
Suppose now that lenders to this consumer impose
credit constraints
on the
consumer.
Specifically, they impose the tightest possible credit constraint – the
consumer is not allowed to be in debt at the end of period one, which implies that
the consumer’s real wealth at the end of period one must be nonnegative (
1
0
a
t
).
What is the consumer’s choice of periodone and periodtwo consumption under
this credit constraint?
Briefly explain, either logically or graphically or both.
c.
Does the credit constraint described in part b enhance or diminish welfare (i.e.,
does it increase or decrease lifetime utility)?
Specifically, find the level of utility
under the credit constraint and compare it to the level of utility obtained under no
credit constraint.
Suppose now that the consumer experiences a temporary increase in real income in
period one to
1
9
y
, with real income in period two unchanged.
d.
Calculate the effect of this positive surprise in income on
1
c
and
2
c
, supposing
that there is no credit constraint on the consumer.
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e.
Finally, suppose that the credit constraint described in part b is back in place.
Will it be binding?
That is, will it affect the consumer’s choices?
Solution:
a.
The consumer’s problem is to maximize lifetime utility (given by
12
(, )
uc c
subject
to the LBC.
The Lagrangian for this problem is thus
22
0
1
1
(, ,) (, )
(
1 )
11
y
c
Lc c
ra
y
c
rr
OO
§·
±
±
±
±
²
²
¨¸
±
±
©¹
,
where we must include the nonzero initial real wealth (inclusive of interest)
0
(1
)
±
.
The firstorder conditions with respect to
1
c
and
2
c
are
112
212
0
0
1
ucc
r
O
²
²
±
C
o
m
b
i
n
i
n
g
t
h
e
s
e
,
w
e
g
e
t
t
h
e
u
s
u
a
l
c
o
n
s
u
m
p
t
i
o
n

s
a
v
i
n
g
s
o
p
t
i
m
a
l
i
t
y
c
o
n
d
i
t
i
o
n
,
(, ) (
1 ) (, )
rucc
±
(ie, the MRS equals the slope of the LBC).
Using the
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 Spring '08
 chugh
 Economics, Utility, credit constraint

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