1
8th April 2011
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. Answer both parts.
(a) What does it mean to say that a utility function,
u
(
·
)
, represents a preference
relation on some choice set
X
?P
r
o
v
et
h
a
ti
f
u
(
·
)
represents preference relation
±
,
this preference relation must be complete and transitive.
(b) Now suppose
X
=
R
2
+
and
(
x
1
1
,x
1
2
)
"
(
x
2
1
,x
2
2
)
when
x
1
1
>x
2
1
,
or
x
1
1
=
x
2
1
and
x
1
2
>x
2
2
.
Is this preference relation complete, transitive and continuous? Defend your
answers.
ANSWER
(a) A utility function
u
:
X
→
R
represents a preference relation
±
on the choice
set
X
if
∀
x, y
∈
X, x
±
y
⇔
u
(
x
)
≥
u
(
y
)
.
Completeness: We need to show that
∀
x, y
∈
X
,e
ither
x
±
y
or
y
±
x
or both.
Now note that
u
(
x
)
and
u
(
y
)
are real numbers so either
u
(
x
)
≥
u
(
y
)
in which case
x
±
y
or
u
(
y
)
≥
u
(
x
)
in which case
y
±
x
;i
f
u
(
x
)=
u
(
y
)
we know both are true.
Transitivity: Suppose
x,y,z
∈
X
,and
x
±
y
and
y
±
z.
We need to show
x
±
z.
Since
u
represents
±
,u
(
x
)
≥
u
(
y
)
and
u
(
y
)
≥
u
(
z
)
. Since these are real numbers
u
(
x
)
≥
u
(
z
)
⇒
x
±
z.
(b) This is an example of lexicographic preferences where good 1 is the dominant
good. Lexicographic preferences are complete and transitive but not continuous.
Completeness: Consider any two distinct points in
R
2
+
–
(
x
1
1
,x
1
2
)
(point
a
)and
(
x
2
1
,x
2
2
)
(point
b
). If
x
1
1
>x
2
1
,a
"
b.
If
x
2
1
>x
1
1
,b
"
a.
If
x
1
1
=
x
2
1
,
since
a
and
b
are
distinct, it must be that either
x
1
2
>x
2
2
in which case
a
"
b
or
x
2
2
>x
1
2
in which case
b
"
a.
Transitivity: Let
a,b,c
∈
R
2
+
where
a
=(
x
1
1
,x
1
2
)
,b
=(
x
2
1
,x
2
2
)
and
c
=(
x
3
1
,x
3
2
)
.
Suppose
a
"
b
and
b
"
c.
We want to prove
a
"
c.
Now notice that
a
"
b
and
b
"
c
⇒
x
1
1
≥
x
3
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x
1
1
=
x
2
1
=
x
3
1
in which case it must be that
x
1
2
>x
2
2
>x
3
2
, and thus
x
1
2
>x
3
2
which
means
a
"
c.
In case (ii)
x
1
1
>x
3
1
andthenimmed
iate
lyweknow
a
"
c.
Continuity: Suppose
{
x
n
}
⊂
R
2
+
and
lim
n
→∞
x
n
=
x
and
{
y
n
}
⊂
R
2
+
and
lim
n
→∞
y
n
=
y
and
x
n
"
y
n
,
∀
n.
Then if
"
were continuous we would be able to deduce that
x
"
y.
Here is one counterexample. Let
x
n
=(1
/n,
0)
and
y
n
=(0
,
1)
.
Then
x
=(0
,
0)
,
y
=(0
,
1)
,x
n
"
y
n
∀
n
but
y
"
x.
So lexicographic preferences are complete and
transitive but they violate continuity.
2. This question applies what we have done in class this term to think about a car
bon tax. Suppose a typical Canadian household currently buys 200 litres of gasoline
per month at $1 per litre and spends $1800 per month buying goods (and services)
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 Fall '08
 Burbidge,John
 Utility, Transitivity, p1

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