Homework 3, Spring 10
(Total 20 points)
Due on Feb. 16, Tue.
Q1
: (3
×
2): (a) When the interest rate is 4%, what is the present value of an asset
that gives you $300 this period and $400 next period?
(b) When the interest rate is 6%, what is the present value of an asset that perpetually
gives you $300 every period?
Answer
: (a) 300 +
1
1
.
04
·
400
≈
684
.
6
(b)
300
1

1
1
.
06
=
1
.
06
0
.
06
·
300 = 5300
Q2
:
(4):
Consider the twoperiod consumption/saving problem.
The consumer’s
preference over consumption streams is represented by
u
(
x
1
, x
2
) = ln
x
1
+ 0
.
95 ln
x
2
.
When her earning stream is (40
,
30) and the interest rate is 4%, how much is her
saving in the first period?
Answer
This is a special case of the standard consumption choice problem where
p
1
= 1,
p
2
=
1
1+
r
,
m
=
ω
1
+
1
1+
r
ω
2
.
Since
MU
1
=
1
x
1
and
MU
2
=
0
.
95
x
2
, we have
MRS
=
x
2
0
.
95
x
1
.
Since
MRS
=
p
1
p
2
= 1 +
r
holds at optimality, we have
x
2
0
.
9
x
1
= 1 +
r
= 1
.
04, which
results in
x
2
= 1
.
04
·
0
.
95
x
1
.
Plug this into the budget constraint
x
1
+
1
1
.
04
x
2
= 40 +
1
1
.
04
·
30 = 68
.
85, the we have
x
1
+
1
1
.
04
·
1
.
04
·
0
.
95
x
1
= 1
.
95
x
1
= 68
.
85. Hence
x
1
= 30
.
31,
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 Spring '10
 d
 Certainty Equivalent, Oskar, gross return rate

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