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1
Sample test 1 questions
1. Let
X
=
{
a, b, c, d
}
,
B
=
±
{
a, b
}
,
{
a, c
}
,
{
a, d
}
,
{
b, c
}
,
{
b, d
}
,
{
c, d
}
,
{
d, b, c
}
,
{
a, b, d
}
,
{
a, c, d
}
²
and
(
B
,C
(
·
))
be a choice structure. Suppose
C
(
·
)
is such that
d
is the best choice whenever
d
is
available and
C
(
{
a, b
}
)=
{
a
}
(
{
b, c
}
{
b
}
,
and
C
(
{
c, a
}
{
c
}
.
(a) Does
C
(
·
)
satisfy WARP?
WARP holds in some choice structure
(
B
(
·
))
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) Is there a rational preference relation which rationalizes
C
(
·
)
on
B
?
No, because
a
"
b
"
c
but
c
"
a,
so transitivity doesn’t hold.
(c) If
{
a, b, c
}
were another budget set could
C
(
·
)
satisfy WARP? Defend your
answers to (a), (b) and (c) carefully.
Try
C
(
{
a, b, c
}
{
a
}
;
but
C
(
{
c, a
}
{
c
}
so WARP is violated.
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.
So the only way to satisfy WARP is to set
C
(
{
a, b, c
}
the empty set, which is
not allowed.
2. You are given the following information about a consumer’s purchases. Goods
1and2aretheon
lygoodsconsumed
.
Year 1
Year 2
Quantity Price Quantity Price
Good 1
100
100
120
80
Good 2
100
100
?
100