Answers to some old prelim questions:
1. Ms Blue and Mr. Green
A Ms Blue has same preferences as Mr. Green, since her utility function is
a monotone increasing function of his
U
B
=
e
U
G
B Preferences are homothetic if (
x,y
)
I
(
x
0
,y
0
) implies (
tx,ty
)
I
(
tx
0
,ty
0
) for
all
t >
0. His preferences are not homothetic. To see this, we note for
example that
U
(0
,
1) = 3
/
2 =
U
(3
/
2
,
0). But
U
(0
,
2) = 2
6
=
U
(3
,
0) = 3.
C
y
(
p,w
) = 2

p
if
p
≤
2:
y
(
p,w
) = 0 if
p >
2.
x
(
p,w
) =
w

py
(
p,w
).
D If
p
≤
2,
v
(
p,w
) = ln (
w

p
(2

p
) + 2(2

p
)

(2

p
)
2
/
2) = ln (
w
+ (2

p
)
2
/
2)
.
If
p >
2,
v
(
p,w
) =
w
.
E Roy’s law requires that
y
(
p,w
) =

∂v
(
p,w
)
∂p
∂v
(
p,w
)
∂w
.
In this instance if
p <
2,

∂v
(
p,w
)
∂p
∂v
(
p,w
)
∂w
=

(2

p
)(
w
+ (2

p
)
2
/
2)
w
+ (2

p
)
2
/
2
=

y
(
p,w
)
.
If
p >
0

∂v
(
p,w
)
∂p
∂v
(
p,w
)
∂w
= 0 =

y
(
p,w
)
.
F Mr. Green’s expenditure function satisﬁes the equation
u
=
v
(
p,e
(
p,u
)) = ln (
e
(
p,u
) +
(2

p
)
2
2
)
and hence
e
(
p,u
) =
e
u

(2

p
)
2
2
.
G If
p <
2,
h
(
p,u
) = 2

p
. If
p >
2,
h
(
p,u
) = 0.
1