This preview shows page 1. Sign up to view the full content.
Pareto Optimality
Let
A
be a set (of alternatives) and
{±
i
}
I
i
=1
a family of preference relations on
A
. ˆ
a
∈
A
is
Pareto
Optimal
if there is no
a
0
∈
A
with
a
0
±
i
ˆ
a
for
i
= 1
,...,I
and
a
0
²
j
ˆ
a
for some
j
∈ {
1
,...,I
}
.
Suppose that, for each
i
= 1
,...,I
,
±
i
is represented by a utility function
u
i
on
A
.
Lemma 1
Any solution to
max
a
∈
A
I
X
i
=1
u
i
(
a
)
(1)
is Pareto Optimal.
Proof
: Suppose that ˆ
a
is not Pareto Optimal. Then there is an
a
0
∈
A
with
u
i
(
a
0
)
≥
u
i
(ˆ
a
)
(2)
for
i
= 1
,...,I
and
u
j
(
a
0
)
> u
j
(ˆ
a
)
(3)
for some
j
∈ {
This is the end of the preview. Sign up
to
access the rest of the document.
Unformatted text preview: 1 ,...,I } . Adding (2) over all i , and using (3), we get I X i =1 u i ( a ) > I X i =1 u i (ˆ a ) so that ˆ a does not solve (1). ± More generally, let W : R I → R be a strictly increasing function. Any maximizer of W ( u 1 ( a ) ,...,u I ( a )) on A is Pareto optimal . The lemma is the special case in which W = ∑ u i . 1...
View
Full
Document
This note was uploaded on 11/19/2010 for the course ECON 202 taught by Professor Schlee during the Spring '10 term at ASU.
 Spring '10
 schlee
 Microeconomics

Click to edit the document details