ECON 2100: Advanced microeconomic theory I
Problem Set 3  suggested solutions
Prepared by David Klinowski
September 19, 2011
Problem 1
Construct a preference relation on
R
that is not continuous, but admits a utility
representation.
Solution
Let
u
(
x
)=
1
x<
0
−
2
x
=0
−
1
x>
0
represent
x
°
y
⇐⇒
x<
0
or
x>
0
and
y
≥
0
or
x
=
y
=0
°
is not continuous because the upper contour set is not closed. To see this,
take
x
=
−
1
. Its upper contour set is
°
(
−
1) = (
−∞
,
0)
, which is not a closed
set.
Problem 2
Prove that if
u
represents
°
,then
:
(1)
°
is weakly monotone if and only if
u
is nondecreasing;
°
is strictly monotone if and only if
u
is strictly increasing.
(2)
°
is convex if and only if
u
is quasiconcave;
°
is strictly convex if and only if
u
is strictly quasiconcave.
(3)
°
is quasilinear if and only if it admits a quasilinear representation.
(4)
a continuous
°
is homothetic if and only if it is represented by a
utility function that is homogeneous of degree one.
Solution
(1)
<if>
Suppose
u
is nondecreasing and
x
≥
y
.
u
(
x
)
≥
u
(
y
)
since
u
is nondecreasing
x
°
y
by
u
representation
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Suppose
°
is weakly monotone and
x
≥
y
.
x
°
y
by weak monotonicity
u
(
x
)
≥
u
(
y
)
by
u
representation
<if>
Suppose
u
is strictly increasing and
x
≥
y
,
x
°
=
y
.
u
(
x
)
>u
(
y
)
since
u
is strictly increasing
x
±
y
by
u
representation
<only if>
Suppose
°
is strictly monotone and
x
≥
y
,
x
°
=
y
.
x
±
y
by strict monotonicity
u
(
x
)
>u
(
y
)
by
u
representation
(2)
<if>
Suppose
u
is quasiconcave and
u
(
x
)
≥
u
(
y
)
.
∀
t
∈
[0
,
1]
u
(
tx
+(1
−
t
)
y
)
≥
u
(
y
)
by quasiconcavity of
u
∀
t
∈
[0
,
1]
tx
+(1
−
t
)
y
°
y
by
u
representation
<only if>
Suppose
°
is convex and
x
°
y
.
∀
t
∈
[0
,
1]
tx
+(1
−
t
)
y
°
y
by convexity of
°
∀
t
∈
[0
,
1]
u
(
tx
+(1
−
t
)
y
)
≥
u
(
y
)
by
u
representation
<if>
Suppose
u
is strictly quasiconcave and
u
(
x
)
≥
u
(
y
)
,
x
°
=
y
.
∀
t
∈
[0
,
1]
u
(
tx
+(1
−
t
)
y
)
>u
(
y
)
by strict quasiconcavity of
u
∀
t
∈
[0
,
1]
tx
+(1
−
t
)
y
±
y
by
u
representation
<only if>
Suppose
°
is convex and
x
°
y
,
x
°
=
y
.
∀
t
∈
[0
,
1]
tx
+(1
−
t
)
y
±
y
by strict convexity of
°
∀
t
∈
[0
,
1]
u
(
tx
+(1
−
t
)
y
)
>u
(
y
)
by
u
representation
(3)
<if>
Suppose
°
admits a quasilinear representation of the form
u
(
x, m
)=
v
(
x
)+
m
.
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 çAKMAK
 Microeconomics, Utility, Convex function, P1 P2, P1 P2 P1, p1 p1 p1, P2 P2 P2, p2 p1 p2

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