Economics 11: Microeconomic Theory 1
Professor Christian Hellwig
Exercises Week 6 ANSWERS
1)
In a two period world, Mark
has an income of $50 today and $20 tomorrow. He has
the following production technology: Q
2
=20*I
1
1/2
. Where I
1
is the investment
today and Q is the gross return of the investment.
a)
Suppose that Mark wants to consume $60 tomorrow. How much is his
consumption and investment today?.
Answer
If
40
60
2
=
⇒
=
Q
C
F
,
I need to invest today:
I
Q
⋅
=
=
20
40
2
then
4
2
20
40
=
⇒
=
⇒
=
I
I
I
b)
Suppose that the interest rate is 100% and his utility function is
U(C
1
,
C
2
)=C
1
1/2
C
2
1/2
.
Find the optimal investment and consumption.
Answer
Initial situation (A)
Wealth without investment
60
60
10
50
1
1
20
50
1
min
2
1
min
=
⇒
=
+
=
+
+
=
+
+
=
W
r
C
C
W
A
A
first stage: Optimal Investment
r
C
C
W
Max
+
+
=
1
2
1
I
I
C
C
I
I
f
C
C
a
s
A
A
−
=
−
=
⋅
+
=
+
=
50
20
20
)
(
:
.
.
1
1
2
2
r
I
I
r
I
f
C
I
C
MaxW
A
A
+
⋅
+
+
−
=
+
+
+
−
=
1
20
20
50
1
)
(
2
1
Solving for the optimal I
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/
1
2
/
1
2
/
1
'
10
1
1
10
1
0
1
20
2
1
1
1
)
(
1
)
1
(
−
−
−
⋅
=
+
⇒
+
⋅
=
⇒
=
+
⋅
⋅
+
−
=
+
+
−
=
∂
∂
I
r
r
I
r
I
r
I
f
I
W
()
1
10
1
()
/
10
1
10
1
2
2
/
1
2
2
2
/
1
2
/
1
=
⇒
=
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 Spring '08
 cunningham
 Economics, Wmax

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