1
21st April 2007
ANSWERS
Instructions
: Answer 7 of the following 9 questions. All questions are of equal
weight. Indicate clearly on the
f
rst page which questions you want marked.
1. Let
X
be a choice set and
(
B
,C
(
·
))
be a choice structure on
X
.
(a) Using mathematics, de
f
ne the weak axiom of revealed preference (WARP).
*****
Let
B
1
,B
2
∈
X
,and
{
x, y
}
⊂
B
1
∩
B
2
.
Then WARP says
x
∈
C
(
B
1
)
and
y
∈
C
(
B
2
)
⇒
x
∈
C
(
B
2
)
.
(b) Suppose
X
=
{
a, b, c
}
,
B
=
{{
a, b
}
,
{
b, c
}
,
{
c, a
}
,
{
a, b, c
}}
,
C
(
{
a, b
}
)=
{
a
}
,
C
(
{
b, c
}
)=
{
b
}
,
and
C
(
{
c, a
}
)=
{
c
}
. Assuming the choice rule must pick at least
one element from each budget set so that
C
(
{
a, b, c
}
)
cannot be the empty set, can
this choice structure satisfy WARP? Justify your answer.
*****
Suppose
a
were amongst the best elements of
{
a, b, c
}
,
that is,
a
∈
C
(
{
a, b, c
}
)
.
Then, according to WARP,
a
would have to be amongst the best elements of
{
c, a
}
but this is contradicted by our assumption above that
C
(
{
c, a
}
)=
{
c
}
,
so
a/
∈
C
(
{
a, b, c
}
)
. Given the perfect symmetry in this question between
a
and
b
and
c
,it
is clear
b/
∈
C
(
{
a, b, c
}
)
and
c/
∈
C
(
{
a, b, c
}
)
.S
ince
C
(
{
a, b, c
}
)
cannot be the empty
set, there is no way to make this choice structure consistent with WARP.
2. Consider a risk-averse individual with initial wealth
w
0
and VNM utility func-
tion
u
(
·
)
who must decide whether and for how much to insure his car. Assume the
probability that he will not have an accident is
π
. In the event of an accident, he
incurs a loss of
$
L
in damages. Suppose that insurance is available at an actuarially
fair price, that is, one that yields insurance companies zero expected pro
f
ts; denote
the price of
$1
worth of insurance coverage by
p.