ECO 301: Intermediate Microeconomics
Marco A Castaneda
Homework 5: Answers
1.
ANSWER
For this production function, we have
2
/
1
L
L
1
MP
=
and
2
/
1
K
K
1
MP
=
a)
The problem can be stated as
max
50q – (10K + 5L)
K,L
q = 2(K)
1/2
+ 2(L)
1/2
Therefore, the necessary conditions for the optimal quantities (K*, L*) of the
inputs are
L
L
w
pMP
=
⇒
5
L
1
50
2
/
1
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
(1)
K
K
w
pMP
=
⇒
10
K
1
50
2
/
1
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
(2)
q
=
F
(
K
,
L
)
⇒
q = 2(K)
1/2
+ 2(L)
1/2
(3)
Equations (1) and (2) imply
L* = 100
and
K* = 25
Therefore, the optimal quantity of output is
q* = 2(K*)
1/2
+ 2(L*)
1/2
= 2(25)
1/2
+ 2(100)
1/2
= 30
Finally, we have
C = w
K
K* + w
L
L*
= 750
R = pq*
=1500
Π
* = pq*  [w
K
K* + w
L
L*]
= 100(15) – [10(25) + 5(100)]
=
7
5
0
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View Full Documentb)
The problem can be stated as
min
10K + 5L
K,L
30 = 2(K)
1/2
+ 2(L)
1/2
Therefore, the optimal quantities (K*, L*) of the inputs are determined by the
equations
)
L
,
K
(
F
q
w
w
MP
MP
K
L
K
L
=
=
⇒
2
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 Spring '08
 Castaneda
 Microeconomics, Substitution Method, Expression, Necessary and sufficient condition, L*

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