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Economics 3203 – Fall 2010
Problem Set # 3: Labor Demand (2)
ANSWER KEY
1.
a.
We can set MC = MR or MRP
L
= ME
L
in order to maximize profits.
Both will give the same result.
The second one requires more math but gets to the employment level directly.
First method: MC = MR:
Find MR:
Total Revenue = P * Q = 60Q – 2Q
2
MR = dTR/dQ = 60 – 4Q
Find MC:
TC = W * L = 2*L
We need to take the derivative of TC with respect to Q in order to find MC.
This means we need TC in
terms of Q instead of L.
We have the production function given to us that can be solved for L and plugged
into TC:
Q = [2L200]
1/2
>
Q
2
= 2L – 200
>
Q
2
+ 200 = 2L
>
L = (Q
2
/2) + 100
So TC = 2*[(Q
2
/2) + 100] = Q
2
+ 200
Then MC = dTC/dQ = 2Q
MR = MC yields:
60 – 4Q = 2Q
>
Q=10
and
P=$40.
This level of Q can be plugged into the production function
to find L = 150.
Second Method:
MRP = ME
L
(it’s longer, but it works)
Note that the wage is given, so ME
L
= W = $2.
MRP = ME
L
MP
L
* MR = 2
Find MR:
Total Revenue = P * Q = 60Q – 2Q
2
MR = dTR/dQ = 60 – 4Q
Find MP
L
:
MP
L
= dQ/dL = (1/2)*(2L200)
(1/2)
* 2
(note that this used the chain rule – see your calc review sheet if this is unfamiliar)
MP
L
=(2L200)
(1/2)
Now plug these back in to the profitmaximizing equation (MRP=ME
L
)
(2L200)
(1/2)
* (60 – 4Q) = 2
We still have an L and a Q here, so we need to substitute in for Q using the production function.
(2L200)
(1/2)
* (60 – 4(2L200)
(1/2)
) = 2
1
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View Full DocumentMultiply this out:
60*(2L200)
(1/2)
– 4(2L200)
(1/2)
* (2L200)
(1/2)
= 2
The two ugly pieces of the second term cancel out.
Then we just have:
60*(2L200)
(1/2)
– 4 = 2
60*(2L200)
(1/2)
= 6
Get L alone on one side, with the other numbers on the other side:
(2L200)
(1/2)
= 6/60
>
(2L200)
(1/2)
= .1
Square both sides:
(2L200)
1
= .01
Multiply both sides by (2L200) to get rid of negative exponent:
(2L200)
1
*(2L200) = .01*(2L200)
1 = .02L – 2
Æ
3 = .02L
Æ
3/.02 = L
Æ
L = 150
To find price, use the demand curve – but first we need Q to use that.
Q = (2L200)
1/2)
Q = (300200)
(1/2)
Æ
Q = 100
(1/2)
Æ
Q = 10
Plug into demand curve:
P = 60 – 2Q
Æ
P = 60 – 20
Æ
P = $40
b.
This price ceiling is below the equilibrium price, so it is binding (in other words, it affects the decisions of
the firm because it restricts them from choosing their profitmaximizing price and quantity).
The firm will
not be allowed to sell its output for $40; the price will have to be less than or equal to $36.
First consider P = $36, and see what Q maximizes profit if the firm assumes it can sell as much as it wants at
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 Fall '10
 HAMERSMA

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