S
: F
E
500, 8:00
December 18, 2008
Question 1
Utility is quasilinear, i.e.,
u
(
x
)
=
v
(
x
1
)
+
x
2
. Thus,
v
′
(
x
1
)
=
p
1
. Thus,
20
−
4
p
1
=
x
1
implies
p
1
=
5
−
(1
/
4)
x
1
. Hence,
v
′
(
x
1
)
=
5
−
(1
/
4)
x
1
, and therefore
v
(
x
1
)
=
5
x
1
−
(1
/
8)
x
2
1
. Thus,
u
(
x
1
,
x
2
)
=
5
x
1
−
(1
/
8)
x
2
1
+
x
2
Question 2
To produce
q
units of output, we must have
z
1
+
2
z
2
=
q
2
. Thus, costs are
c
(
w,
q
)
=
min
{
w
1
,
0
.
5
w
2
}
q
2
. The first order conditions for profit maximization is
p
=
min
{
w
1
,
0
.
5
w
2
}
2
q
. Thus,
q
=
p
min
{
2
w
1
, w
2
}
.
Hence profit is
p
2
min
{
2
w
1
, w
2
}
−
min
{
w
1
,
0
.
5
w
2
}
p
2
4 min
{
w
1
,
0
.
5
w
2
}
2
=
p
2
min
{
2
w
1
, w
2
}
−
p
2
2 min
{
2
w
1
, w
2
}
.
Then the firm’s profit function is
π
(
w
1
, w
2
,
p
)
=
p
2
2 min
{
2
w
1
,w
2
}
Question 3
Suppose that a firm can operate at two locations, using one input at each
location (the input also has the same price at the two locations). The production