Now to prove lower hemicontinuity. Consider an open set
V
such that
V
∩
B
p,w
6
=
∅
. If
w
= 0,
then
0
n
∈
V
because
V
has nonempty intersection with
B
p,w
=
{
0
n
}
.
But then
V
∩
B
p
0
,w
0
is
nonempty for all
p
0
, w
0
because
0
n
∈
B
p
0
,w
0
.
So, assume without loss of generality that
w >
0.
Since
V
is an open set and
B
p,w
has an interior (as
w >
0), we can also find a point
y
∈
V
such
that
y
is in the interior of
B
p,w
. But this means
p
·
y < w
. But this relationship is maintained in
a neighborhood of (
p, w
). So
p
0
·
y < w
0
for sufficiently close (
p
0
, w
0
). But then
y
∈
B
p
0
,w
0
. In other
words,
y
∈
V
and
y
∈
B
p
0
,w
0
, so
V
∩
B
p
0
,w
0
is nonempty.
Problem 5.19.
Recall the nonlinear pricing structure of Exercise 5.3. Now, suppose the price of
food is
p
1
, the price of the first
M
units of electricity is
p
2
, and the price of each unit of electricity
over
M
is
q
2
.
M
is fixed and does not change. Then the budget set is parameterized as
B
p
1
,p
2
,q
2
,w
,
which can also be expressed as a correspondence Γ :
R
3
++
×
R
+
⇒
R
2
+
. Prove that the Walrasian
demand is upper hemicontinuous in (
p
1
, p
2
, q
2
, w
).
Problem 5.20.
Suppose the consumer can choose to move to either Philadelphia or Queens. The
price of
n
goods in Philadelphia is
p
and the price in Queens is
q
.
She may consume at either
the Philadelphia price or the Queens price, whichever makes her better off (so she gets to decide
where to move). All of her shopping must take place in one city; she cannot purchase some goods
in Philadelphia and other goods in Queens. Suppose the consumer’s utility function
u
:
R
n
+
→
R
is continuous. Prove the following: The Walrasian demand correspondence, which is now defined
on
two
prices and wage,
x
*
:
R
n
++
×
R
n
++
×
R
+
⇒
R
n
+
, defined by
x
*
(
p, q, w
) = arg
max
x
∈
B
p,q,w
u
(
x
)
28
is upper hemicontinuous; and the indirect utility function,
v
:
R
n
++
×
R
n
++
×
R
+
→
R
, defined by
v
(
p, q, w
) =
max
x
∈
B
p,q,w
u
(
x
)
is continuous.
Theorem 5.21
(Implicit Function Theorem)
.
Suppose
A
is an open set in
R
n
+
m
and
f
:
A
→
R
n
is continuously differentiable. Let
D
x
f
refer to the
n
×
n
derivative matrix of
f
with respect to its
first
n
arguments, i.e.
(
D
x
f
)
ij
=
∂f
i
∂x
j
.
If
f
(¯
x,
¯
q
) =
0
n
and
D
x
f
(¯
x,
¯
q
)
is nonsingular, then there
exists a neighborhood
B
of
¯
q
in
R
m
and a unique continuously differentiable
g
:
B
→
R
n
such that
g
(¯
q
) = ¯
x
and
f
(
g
(
q
)
, q
) =
0
n
for all
q
∈
B
.
Moreover,
D
q
g
(¯
q
) =

[
D
x
f
(¯
x,
¯
q
)]

1
D
q
f
(¯
x,
¯
q
)
.
Proposition 5.22.
Suppose
u
is twice continuously differentiable, locally nonsatiated, strictly qua
siconcave and that there exists
ε >
0
such that
x
*
i
(¯
p,
¯
w
)
>
0
if and only if
x
*
i
(
p, w
)
>
0
, for all
(
p, w
)
such that
k
(¯
p,
¯
w
)

(
p, w
)
k
< ε
.
13
Let
m
=
{
i
:
x
*
i
(¯
p,
¯
w
)
>
0
}
and reindex
R
n
so the
m
dimensions
come first. If the bordered Hessian
H
of
u
with respect to its first
m
dimensions, i.e.
H
=
"
0
(
D
x
u
)
>
D
x
u
D
2
xx
u
#
=
0
u
1
u
2
· · ·
u
m
u
1
u
11
u
21
· · ·
u
m
1
u
2
u
12
u
22
· · ·
u
m
2
.