ECN 712
Fall 2008
Professor Schlee
Final Examination
Instructions
: Attempt all questions and write your answers in the blue books provided. There
are 110 points on the exam, one for each minute of the exam period.
Allocate your time carefully.
1. (25 points) Consider the twoasset portfolio problem,
max
α
∈
R
+
Z
u
(
w
+
αx
)
dF
t
(
x
)
,
where
w >
0 is the initial wealth,
α
is wealth invested in a risky asset with realized rate
of return,
x
, and, for
t
= 0
,
1,
F
t
is a cumulative distribution function (cdf) with bounded
support.
The safe asset has a rate of return of 0.
(Note that investment in the safe asset
can be negative–the asset can be sold short.) The vNM utility
u
on
R
is thrice continuously
differentiable with
u
0
>
0 globally.
For
t
= 0
,
1, let
α
t
solve this problem and assume throughout that
Z
xdF
t
(
x
)
>
0
.
(a) (10) Show that
α
t
>
0 (for either
t
= 0 or
t
= 1).
(b) (15) Suppose that
F
0
is
strictly riskier than
F
1
.
1
i. Identify a condition on
u
which ensures that
α
1
≥
α
0
(a decrease in risk raises
investment). For a vNM utility satisfying your condition, prove that indeed
α
1
≥
α
0
ii. Consider a vNM utility with
u
00
(
z
)
<
0 and
u
000
(
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 Spring '10
 schlee
 Microeconomics, Game Theory, subgame perfect Nash, perfect Nash equilibrium, vNM utility

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