1
Lecture XIX
Increasing Risk and Ranking Functions
I.
Implications of Definitions
A.
Theorem 1.
Consider cumulative distributions
( )
Fx
and
()
Gx
, then there
is an increasing continuous function
( )
rx
such that:
[]
Fx d
y
y
() ()
,
[,]
,
−≥
∀
∈
∫
0
00
1
if and only if
is at least as risky as
( )
by Definition 1.
1. Proof:
Assume there exists an increasing twice differentiable
( )
such that
y
y
,
.
∀
∈
∫
0
1
Let
zrx
=
and define functions
( )
*
.
G
and
*
.
F
by
() ( )
*1
Gz G
r z
−
=
and
( ) ( )
( )
Fz F
−
=
.
Then,
Grx Frx d
y
Gz Fzd
z
y
y
r
ry
**
(()
)
) ()
,
,
∀
∈
∀
∈
∫
∫
0
0
1
1
or
We know by previous proof (here attributed to Hadar and Russell) that
Gz Fzuzd
z
u
z
r
r
() '
⇒−
≥
∫
∫
0
0
0
0
A couple of notes on this point:
a.
If you made the assumption that
( )
is a onetoone mapping, we
have simply changed the definition of the cumulative distribution,
defining it on the variable
z
which is related to the original
variable
x
by
( )
=
.
This assumption would be implied by the
imposition
( )
1
x
rz
−
=
.
b.
Along the same lines, we really haven’t changed the bounds of
integration.
We have only mapped them into the variable
z
.
2.
Mapping the transformation back to
x
by
(
)(
)
( )
vx urx
=
, the integral
becomes
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View Full DocumentAEB 6182–Agricultural Risk Analysis and Decision Making
Fall 2004
Professor Charles Moss
Lecture 19
2
[]
Gx
Fx d
vx
()
() ()
−≥
∫
0
1
0
Integrating this relationship by parts yields
vxdFx
vxdGx
0
1
0
1
∫∫
≥
for all such
rx
.
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 Fall '08
 Weldon
 Utility, Cumulative distribution function, risk averse, Charles Moss II

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