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Final Exam Answer Key
University of California, Los Angeles
September 10, 2008
Write your name and ID number on your exam booklet.
Exams without
identi±cation will not be graded. Handheld calculators are permitted.
You have two hours to complete the exam. Each question is worth 20 points,
for a total of 100. Good luck!
1. Let
q
=
k
1
3
l
1
3
.
Assume that both factors are variable, and the price of
capital is
v
, while the price of labor is
w
.
(a) Derive the cost function. Find the marginal and average cost func
tions.
ANSWER:
MRTS
kl
=
l
k
=
v
w
q
=
k
1
3
v
w
k
±
1
3
=
v
w
±
1
3
k
2
3
k
(
q
)
=
w
v
±
1
2
q
3
2
l
(
q
)
=
v
w
±
1
2
q
3
2
The cost function is
C
(
q
)
=
vk
(
q
) +
wl
(
q
)
=
v
w
v
±
1
2
q
3
2
+
w
v
w
±
1
2
q
3
2
=
2 (
vw
)
1
2
q
3
2
so the marginal and average cost functions are
MC
(
q
)
=
3 (
vw
)
1
2
q
1
2
AC
(
q
)
=
2 (
vw
)
1
2
q
1
2
(b) Show that the marginal cost function intersects the minimum of the
average cost function.
ANSWER: The minimum of the average cost curve occurs where
dAC
(
q
)
dq
=
(
vw
)
1
2
q
±
1
2
= 0
q
=
0
1
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View Full DocumentMarginal cost is equal to average cost where
MC
(
q
)
=
AC
(
q
)
3 (
vw
)
1
2
q
1
2
=
2 (
vw
)
1
2
q
1
2
9
q
=
4
q
which only occurs where
q
= 0
.
(c) Find the amount of
k
and
l
necessary to produce
q
= 400
when
w
= 4
and
v
= 1
.
ANSWER: We use our demand functions from part
(
a
)
k
(400)
=
(4)
1
2
(400)
3
2
= 16
;
000
l
(400)
=
1
4
±
1
2
(400)
3
2
= 4
;
000
(d) If the output price
P
= 3
,
w
= 4
and
v
= 1
with this production technology choose to produce?
P
=
MC
(
q
)
, which
is
P
=
3 (
vw
)
1
2
q
1
2
3
=
3 (4)
1
2
q
1
2
q
=
1
2
±
2
=
1
4
2. The tools you have learned in this class may help you plan your life. Let
us split your life into two periods: period one, where you work, and period
two, where you are retired. Suppose that in period one, you are able to
earn an income from your labor of
y
1
= $10400
. In period two, you retire
(
y
0
= 0
) and consume any savings or investments you made in period one.
Your utility function is
U
(
c
1
;c
2
) =
c
1
2
1
c
1
2
2
, where
c
1
is your consumption
in period one and
c
2
is your consumption in period two.
(a) Suppose that in period one, you must pay
$400
in Social Security tax.
of
$400
.
Assuming you make no additional savings
of any kind
,
calculate your consumption in each period and your utility level.
ANSWER: Since there is no possibility of savings,
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 Spring '08
 Brown

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