University of California, Davis
ARE/ECN 200A Fall 2008
PROBLEM SET 1 ANSWER KEY
1. From Textbook
1.B.3
Let
x
2
X
and
y
2
X:
Since
u
(
:
)
represents
; x
y
if and only if
u
(
x
)
±
u
(
y
)
:
Since
f
(
:
)
is strictly increasing,
u
(
x
)
±
u
(
y
)
if and only if
v
(
x
)
±
v
(
y
)
:
Hence,
x
y
if and only if
v
(
x
)
±
v
(
y
)
:
Therefore,
v
(
:
)
represents
:
1.B.4
Suppose ±rst
x
y:
If, furthermore,
y
x;
then
x
²
y
and hence
u
(
x
) =
u
(
y
)
:
If, on
the contrary, we do not have
y
x;
then
x
³
y:
Hence,
u
(
x
)
> u
(
y
)
:
Thus, if
x
y;
then
u
(
x
)
±
u
(
y
)
:
Suppose conversely that
u
(
x
)
±
u
(
y
)
:
If, furthermore,
u
(
x
) =
u
(
y
)
;
then
x
²
y
and hence
x
y:
If, on the contrary,
u
(
x
)
> u
(
y
)
;
then
x
³
y;
and hence
x
y:
Thus, if
u
(
x
)
±
u
(
y
)
;
then
x
y:
So
u
(
:
)
represents
:
1.B.5
Lemma
. Let
be a transitive preference relation, and let
N
be a positive integer
±
3
:
Then
x
1
x
2
x
3
:::
x
N
1
x
N
)
x
1
x
N
:
Proof: Because
x
1
x
2
and
x
2
x
3
;
transitivity implies that
x
1
x
3
;
which together with
x
3
x
4
;
implies
x
1
x
4
:
Iterating the argument, we obtain
x
1
x
N
:
Now, turn to the question:
(1) We consider ±rst the case where there is no indi²erence, i.e.,
x
i
x
j
,
x
i
³
x
j
:
(1)
Thus, by the completeness of the preference relation, given
x
i
;x
j
in
X;
either x
i
³
x
j
or x
j
³
x
i
:
(2)
Let
Y
be a nonempty subset of
X;
and let
N
be the number of elements of
Y:
We want to
show that there is an element of
Y
that is preferred to all other elements of
Y:
The proof is by
1