S
: F
E
500, 8:00
December 12, 2007
Question 1
The production function is
f
(
z
)
=
2 min
{
z
1
,
z
2
}
+
z
3
.
Clearly
z
1
=
z
2
. In order to produce
q
units of output, one can use
z
1
=
z
2
=
q
/
2 or
z
3
=
q
, whichever has lower costs. Thus,
z
1
=
z
2
=
q
/
2 will be used if
w
1
+
w
2
< w
3
,
resulting in costs (
w
1
+
w
2
)
q
and
z
3
=
q
will be used if
w
1
+
w
2
> w
3
, resulting in
costs
w
3
q
. Thus,
c
(
w,
q
)
=
q
min
b
w
1
+
w
2
2
, w
3
B
.
Question 2
Suppose a production function
f
(
z
1
,
z
2
) has the following properties.
1. The marginal product of input
i
only depends on the level of input
i
, not on
that of the other input.
2. Changing both inputs by a factor of
α
changes output by
α
2
.
Let
∂
f
(
z
1
,
z
2
)
∂
z
i
=
g
i
(
z
i
). Then it follows that
f
(
z
1
,
z
2
)
=
G
1
(
z
1
)
+
G
2
(
z
2
), where
G
′
i
=
g
i
(the integration constant
k
can be included into
G
i
). We can also define
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 Spring '08
 Staff
 Konrad Zuse, Z1, 2 min

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