4]) are all closed sets (as the inverse image
of closed intervals). The production possibility set is closed because it is the
intersection of 3 closed sets.
The production set is bounded because
y
2
≤
4 and
y
≥
0 imply 0
≤
y
≤
2 and
0
≤
x
≤
4.
b
) Since the utility is continuous (we know all polynomials are continuous) and the
production set is compact (closed and bounded), the Weierstrass Theorem applies
to yield a maximum.
4. Suppose a ﬁrm’s production function is
Q
=
K
1
/
3
L
2
/
3
and that
K
= 1000 and
L
= 125.
a
) How much can the ﬁrm produce?
b
) What are the marginal products of capital (
K
) and labor (
L
)?
c
) Suppose that the available capital falls by 2 units, while labor increases by 5 units.
Without plugging the new numbers for
K
and
L
into the production function,
compute approximately how much the ﬁrm can now produce.
Answer:
a
) Maximum production is
Q
= (1000)
1
/
3
(125)
2
/
3
= 250.
b
) Now
MP
K
=
∂Q/∂K
=
1
3
K

2
/
3
L
2
/
3
and
MP
L
=
∂Q/∂L
=
2
3
K
1
/
3
L

1
/
3
. Using
K
= 1000 and
L
= 125 yields
MP
K
=
1
12
and
MP
L
=
4
3
.
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 Spring '08
 STAFF
 Economics, Topology, Eigenvalue, eigenvector and eigenspace, Compact space, limit point

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