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Answers to sample questions 1
1. MWG, 1.B.3
Proof: We say that
u
(
·
)
represents
±
on choice set
X
if
for all
x, y
∈
X
,
x
±
y
if and only if
u
(
x
)
≥
u
(
y
)
(1)
To complete the proof note that if
f
(
·
)
is a strictly increasing function de
f
ned on
the real numbers then
u
(
x
)
≥
u
(
y
)
if and only if
f
(
u
(
x
))
≥
f
(
u
(
y
))
.
MWG, 1.B.4
Proof: Denote
for all
x, y
∈
X
,
x
∼
y
if and only if
u
(
x
)=
u
(
y
)
(2)
for all
x, y
∈
X
,
x
"
y
if and only if
u
(
x
)
>u
(
y
)
(3)
Given that (2) and (3) are true we need to prove that (1) is true. Note that
x
±
y
implies either
x
"
y
in which case
u
(
x
)
(
y
)
or
x
∼
y
in which case
u
(
x
u
(
y
)
;
combining the two options then
x
±
y
implies
u
(
x
)
≥
u
(
y
)
. Conversely,
u
(
x
)
≥
u
(
y
)
implies either
u
(
x
)
(
y
)
in which case
x
"
y
or
u
(
x
u
(
y
)
in which case
x
∼
y
.
So combining the two options then
u
(
x
)
≥
u
(
y
)
implies
x
±
y
.
2. (a) WARP holds in some choice structure
(
B
,C
(
·
))
if:
B,B
±
∈
B
,
{
x, y
}
⊂
B,
{
x, y
}
⊂
B
±
:
x
∈
C
(
B
)
,y
∈
C
(
B
±
)
⇒
x
∈
C
(
B
±
)
.
The only way to contradict WARP here is for some problem to arise with budget
sets that don’t contain
d.
But there is none of these with two or more elements
in common, so the supposition in the de
f
nition of WARP is not satis
f
ed and thus
WARP must be true. Remember, anything is true of the empty set.
(b) No, because
a
"
b
"
c
but
c
"
a,
so transitivity doesn’t hold.
(c) Try
C
(
{
a, b, c
}
{
a
}
;
but
C
(
{
c, a
}
{
c
}
so WARP is violated.
Try
C
(
{
a, b, c
}
{
b
}
;
but
C
(
{
a, b
}
{
a
}
so WARP is violated.
C
(
{
a, b, c
}
{
c
}
;
but
C
(
{
b, c
}
{
b
}
so WARP is violated.