Economics 206
Spring 2007 (Prof. G. Loury)
Assignment 2: Risk and Information
1. Concerning Measures of Risk Aversion:
Let
u
i
:
< → <
, i
= 1
,
2
,
be two strictly increasing, strictly concave, twice differentiable
Bernoulli utility functions; and, let
F
:
< →
[0
,
1] be a cumulative distribution function
for some realvalued random variable, ˜
y
, which is the (positive or negative) reward from
accruing some risky situation. So, agent
i
’s preferences over gambles can be represented as
follows:
U
i
(
F
) =
R
u
i
(
y
)
dF
(
y
)
, i
= 1
,
2. Define
c
(
F
;
u
i
) to be the certainty equivalent of
the gamble represented by
F
for agent
i
:
c
(
F
;
u
i
)
≡
u

1
i
[
R
u
i
(
y
)
dF
(
y
)]
.
We say that agent
1 is “more risk averse than agent 2” if and only if:
∀
F
:
c
(
F
;
u
1
)
5
c
(
F
;
u
2
). Prove that
the following three claims are equivalent:
(a) Agent 1 is more risk averse than agent 2.
(b) There exists an increasing concave function
v
:
< → <
such that,
∀
y
∈ <
:
u
1
(
y
) =
v
(
u
2
(
y
))
.
(c)
∀
y
∈ <
:

u
00
1
(
y
)
u
0
1
(
y
)
=

u
00
2
(
y
)
u
0
2
(
y
)
2. Concerning Blackwell’s Theorem:
Let
C
=
{
c
1
, ...c
N
}
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 Spring '07
 G.LOURY
 Microeconomics, Utility, Convex function

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