Econ 101A — Problem Set 3
Due in class on Th 12 March. No late Problem Sets accepted, sorry!
This Problem set tests the knowledge that you accumulated mainly in lectures 10 to 13, but it builds on
the work of the previous weeks. It is focused on choice under uncertainy and timeinconsistency. General
rules for problem sets: show your work, write down the steps that you use to get a solution (no credit for
right solutions without explanation), write legibly. If you cannot solve a problem fully, write down a partial
solution. We give partial credit for partial solutions that are correct. Do not forget to write your name on
the problem set!
Problem 1. Relative and Absolute Risk aversion
(6 points) In class we introduced the concepts of
relative and absolute risk aversion, but we have not used them. This exercise introduces you to two useful
classes of utility functions.
1. Consider the exponential utiliy function
−
exp (
−
ρc
)
.
Show that it is increasing
(
u
0
>
0)
and concave
(
u
00
<
0)
for all
c
as long as
ρ >
0
, that is, as long as the agent is riskaverse. Show that this function
has constant absolute risk aversion coe
ﬃ
cient
r
A
given by
ρ.
(2 points)
2. Consider the power utiliy function
c
1
−
ρ
1
−
ρ
for
ρ
6
= 1
.
Show that it is increasing
(
u
0
>
0)
and concave
(
u
00
<
0)
for all
c >
0
as long as
ρ >
0
.
Show that this function has constant relative risk aversion
coe
ﬃ
cient
r
R
given by
ρ.
(2 points)
3. Consider the log utility function
ln (
c
)
.
Show that it is increasing
(
u
0
>
0)
and concave
(
u
00
<
0)
for all
c >
0
.
Show that this function has constant relative risk aversion coe
ﬃ
cient
r
R
equal to
1
.
(in fact, it
is possibile to show
lim
ρ
→
1
c
1
−
ρ
−
1
1
−
ρ
= ln (
c
)
— you are not required to prove this) (2 points).
Problem 2.
Investment in Risky Asset
(26 points) We consider here a standard problem of in
vestment in risky assets, similar to the one that we covered in class.
The agent can invest in bonds or
stocks.
Bonds have a return
r >
0
.
(in class we asumed
r
= 0
) Stocks have a stochastic return,
r
+
> r
with probability
p,
and
r
−
< r
with probability
1
−
p.
In expectations, the stocks outperform bonds, that is,
pr
+
+ (1
−
p
)
r
−
> r.
The agent has income
w
and utility function
u,
with
u
0
(
x
)
>
0
and
u
00
(
x
)
<
0
for all
x
. The agents wants to decide the optimal share
α
of his wealth to invest in stocks. The agent maximizes
max
α
(1
−
p
)
u
(
w
[(1
−
α
) (1 +
r
) +
α
(1 +
r
−
)]) +
+
pu
(
w
[(1
−
α
) (1 +
r
) +
α
(1 +
r
+
)])
s.t.
0
≤
α
≤
1
or, after some sempli
fi
cation,
max
α
(1
−
p
)
u
(
w
[1 +
r
+
α
(
r
−
−
r
)]) +
pu
(
w
[1 +
r
+
α
(
r
+
−
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 Spring '08
 Staff
 Utility, ice cream, Tim, ﬁrst order conditions

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