ISL233-MİKROİKTİSAT-UYGULAMA DERSİ-7
29.11.2011
38. Acme Container Corporation produces egg cartons that are sold to egg distributors. Acme has estimated this
production function for its egg carton division:
Q = 25L0.6K0.4,
where Q = output measured in one thousand carton lots,
L = labor measured in person hours, and K = capital measured in machine hours. Acme currently pays a
wage of $10 per hour and considers the relevant rental price for capital to be $25 per hour. Determine the
optimal capital-labor ratio that Acme should use in the egg carton division.
Solution:
K 0.4
L0.4
L0.6
= 10 0.6
K
MPL = .6(25)L−0.4 K 0.4 = 15
MPK = .4(25)L0.6 K -0.6
MRTS =
MPL
MPK
K 0.4
0.4
0.6
L0.4 = 1.5 K • K
MRTS =
L0.6
L0.4 L0.6
10 0.6
K
K
MRTS = 1.5
L
15
Equate MRTS to
w
.
r
K 10
=
L 25
K
1.5 = 0.4
L
1.5
1.5K = 0.4L; K=0.266L
39. Davy Metal Company produces brass fittings. Davy's engineers estimate the production function represented
below as relevant for their long-run capital-labor decisions.
Q = 500L0.6K0.8,
where Q = annual output measured in pounds,
L = labor measured in person hours,
K = capital measured in machine hours.
The marginal products of labor and capital are:
MPL = 300L-0.4K0.8
MPK = 400L0.6K-0.2
Davy's employees are relatively highly skilled and earn $15 per hour. The firm estimates a rental charge of
$50 per hour on capital. Davy forecasts annual costs of $500,000 per year, measured in real dollars.
a.
b.
c.
Determine the firm's optimal capital-labor ratio, given the information above.
How much capital and labor should the firm employ, given the $500,000 budget? Calculate
the firm's output.
Davy is currently negotiating with a newly organized union. The firm's personnel manager
indicates that the wage may rise to $22.50 under the proposed union contract. Analyze the
effect of the higher union wage on the optimal capital-labor ratio and the firm's employment of
capital and labor. What will happen to the firm's output?
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Solution:
a.
K 0.8
L0.4
L0.6
= 400 0.2
K
MPL = 300L−0.4 K 0.8 = 300
MPK = 400L0.6 K −0.2
K 0.8
0.8
0.2
L0.4 = 0.75 K K
MRTS =
L0.6
L0.4 L0.6
400 0.2
K
K
MRTS = 0.75
L
w 15
Equate to = .
r 50
K 15
0.75 =
L 50
K
0.75 = 0.3
L
K
= 0.4; K = 0.4L
L
300
b.
C = 500,000
C = wL + rK
500,000 = 15L + 50K
K = 0.4L from optimal ratio
500,000 = 15L + 50(0.4L)
500,000 = 15L + 20L
500,000 = 35L
L = 14,285.71 or 14,286 hours
Substitute to solve for K.
500,000 = 15(14286) + 50K
500,000 = 214,290 + 50K
285,710 = 50K
K = 5714.20
or K = 5714
Q = 500(14,286)0.6(5,714)0.8
Q = 157,568,191
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c.
K
L
w 22.5
New =
= 0.45
r
50
w 22.5
Equating MRTS to =
.
r
50
K
0.75 = 0.45
L
K
= 0.6
L
K = 0.6L
MRTS = 0.75
Substitute into C:
500,000 = 22.50L + 50K
K = 0.60L
500,000 = 22.50L + 50(0.6L)
500,000 = 22.50L + 30L
500,000 = 52.50L
L = 9,523.8 or 9,524
L fell from 14,286 to 9,524. Substitute to solve for K.
500,000 = 22.50(9,524) + 50K
285,710 = 50K
K = 5,714.20 or 5,714
K remains constant.
Q = 500(9524)0.6(5714)0.8
Q = 123,541,771.8
Output fell from 157,568,202.5 to 123,541,771.8.
40.
a. Homer's boat manufacturing plant leases 50 hydraulic lifts and produces 25 boats per period. Homer's shortq5
run cost function is: C ( q, K ) = 15 5 + 200 K , where q is the number of boats produced and K is the
K 2
10
number of hydraulic lifts. Homer's long-run cost function is: CLR ( q ) = 173.5578q
7
. At Homer's current
short-run plant size, calculate Homer's short-run average total cost of production. If Homer would lease 11
more hydraulic lifts in the short run, will his short-run average total cost of producing 25 boats increase or
decrease? Does Homer's long-run cost function exhibit increasing, constant, or decreasing returns to scale?
Solution: At Homer's current short-run plant size, Homer's short-run average total cost of production is:
 15 ( 25 )5


+ 200 ( 50 ) 
5
 ( 50 ) 2

 = 731.46. If Homer leases an additional 11 hydraulic
ATC (25,50) = 
25
 15 ( 25 )5


+ 200 ( 61) 
5
 ( 61) 2

 = 689.62. We
lifts, short-run average total costs become: ATC (25, 61) = 
25
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see that Homer's short-run average total costs decrease if he uses 11 additional hydraulic lifts.
CLR ( q )
10
3
173.5578q 7
= 173.5578q 7 . Since
q
q
long-run average costs increase as output increases, Homer's production process has
decreasing returns to scale.
Homer's long-run average costs are: ACLR ( q ) =
=
b. Marge's Hair Salon uses 15 hair dryers to produce 10 units of output per period. Marge's short-run cost
function is: C ( q, K ) =
15q 2
+ 12 K , where q is the number of units produced and K is the number of hair
K
dryers Marge leases. Marge's long-run cost function is: CLR ( q ) = 26.8q. If Marge used 4 fewer hair dryers
in the short-run, would short-run average total costs increase or decrease? Does Marge's long-run cost curve
exhibit increasing, constant, or decreasing returns to scale?
Solution:
 15 (10 ) 2


+ 12 (15 ) 
 15

 = 28.00.
Currently, Marge's short-run average costs are: SRATC (10,15 ) = 
10
If Marge uses 4 fewer hair dryers in the short run, her short-run average total costs become:
 15 (10 ) 2


+ 12 (11) 
 11

 = 26.84. If Marge uses 4 fewer dryers and produces 10
SRATC (10,11) = 
10
units, here short-run average total costs decrease. Marge's long-run average costs are:
C ( q ) 26.8q
LRAC = LR
=
= 26.8. We see that Marge's long-run average costs are constant.
q
q
This implies that Marge's cost curve exhibit constant returns to scale.
c. Apu leases 2 squishy machines to produce 40 squishies in the short run. Apu's short-run cost function is:
q2
C ( q, K ) = 0.85 2 + 0.5K , where q is the number of squishies produced and K is the number of squishy
K
2
machines used. Apu's long-run cost function is: CLR ( q ) = 1.13q 3 .
If Apu decides to lease 7 squishy
machines, what happens to Apu's short-run average total cost of producing 40 squishies? Does Apu's longrun cost function exhibit increasing, constant, or decreasing returns to scale?
Solution:
With 2 squishy machines, Apu's short-run average total costs are:
2

( 40 ) + 0.5 2 
 0.85
( )
2

( 2)

 = 8.525. If Apu leases 7 squishy machines, his shortSRATC ( 40, 2 ) =
40
2

( 40 ) + 0.5 7 
 0.85
( )
2

(7)

 = 0.78. Leasing 5
run average total costs become: SRATC ( 40, 7 ) =
40
additional squishy machines lowers Apu's short-run average total cost by 91%. Apu's long-run
1.13q
average cost curve is: LRAC ( q ) =
q
2
3
1.13
. Since Apu's long-run average costs
1
q 3
decrease as output increases, Apu's cost curve exhibit increasing returns to scale.
=
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41. The following table contains information for a price taking competitive firm. Complete the table and
determine the profit maximizing level of output (round your answer to the nearest whole number).
0
1
2
3
Total
Cost
5
7
11
17
4
5
6
27
41
61
Output
Marginal
Cost
Fixed
Cost
Average
Cost
Total
Revenue
0
10
20
30
Average
Revenue
Marginal
Revenue
40
50
60
Solution:
Output
0
1
2
3
4
5
6
Total
Cost
5
7
11
17
27
41
61
Marginal
Cost
–
2
4
6
10
14
20
Fixed
Cost
5
5
5
5
5
5
5
Average
Cost
–
7
5.5
6
7
8
10
Total
Revenue
0
10
20
30
40
50
60
Average
Revenue
–
10
10
10
10
10
10
Marginal
Revenue
–
10
10
10
10
10
10
The profit maximizing level of output is either 3 or 4. Note that at Q=4 the profit-maximizing
condition MR=MC is satisfied. Since this problem is discrete, the profit at Q=3 happens to be the
same as the profit at Q=4, so either of these answers is correct.
42. Homer's Boat Manufacturing cost function is: C ( q ) =
MC ( q ) =
75 4
q + 10, 240 . The marginal cost function is:
128
75 3
q . If Homer can sell all the boats he produces for $1,200, what is his optimal output?
32
Calculate Homer's profit or loss.
Solution: The profit maximizing output level is where the market price equals marginal cost (providing
the price exceeds the average variable cost). To determine the optimal output level, we need to
75 3
first equate marginal cost to the market price. That is, MC ( q ) =
q = P = 1, 200 ⇔ q = 8.
32
75 ( 512 )
75
3
The average variable cost at this output level is: AVC ( 8 ) =
= 300. Since
(8) =
128
128
P > AVC ( 8 ) , Homer will maximize profits at 8 units.
Homer's profits are:
 75 ( 8) 4

+ 10, 240  = −3, 040.
 128

π = Pq − C ( q ) = 1, 200 ( 8 ) − 
Homer will produce and make a
loss as losing $3,040 is better than not producing and losing $10,240.
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43. Sarah's Pretzel plant has the following short-run cost function: C ( q, K ) =
wq 3
+ 50 K , where q is
3
1000 K 2
Sarah's output level, w is the cost of a labor hour, and K is the number of pretzel machines Sarah leases. Sarah's
3wq 2
short-run marginal cost curve is MC ( q, K ) =
. At the moment, Sarah leases 10 pretzel machines, the
3
100 K 2
cost of a labor hour is $6.85, and she can sell all the output she produces at $35 per unit. If the cost per labor
hour rises to $7.50, what happens to Sarah's optimal level of output and profits?
Solution: First, we need to determine Sarah's optimal output and profits before the increase in the wage
rate. The profit maximizing output level is where the market price equals marginal cost
(providing the price exceeds the average variable cost). To determine the optimal output level,
we need to first equate marginal cost to the market price.
That is,
2
3wq
MC ( q, K ) =
= P = 35 ⇔ q = 232.07. The average variable cost at this output level
3
1000 ( k ) 2
is:
wq 2
AVC ( 232.07,10 ) =
1000 K
Sarah
will
3
=
2
6.85 ( 232.07 )
1000 (10 )
3
2
= 11.67.
Since
P > AVC ( 232.07,10 ) ,
2
maximize
profits
at
232.07
units.
Sarah's
3
 6.85 ( 232.07 )



π = Pq − C ( q,10 ) = 35 ( 232.07 ) − 
+ 50 (10 )  = 4,915.08.
3
2
 1000 (10 )

profits
are:
To determine the optimal output level at the higher wage rate, we need to first equate marginal
3 ( 7.50 ) q 2
= P = 35 ⇔ q = 221.79. The
cost to the market price. That is, MC ( q, K ) =
3
1000 (10 ) 2
average
variable
AVC ( 221.79,10 ) =
will
wq
cost
2
3
1000 K 2
maximize
profits
=
at
7.50 ( 221.79 )
1000 (10 )
3
this
output
level
is:
2
= 11.66.
Since P > AVC ( 221.79,10 ) , Sarah
2
at
221.79
units.
Sarah's
profits
are:
 7.50 ( 221.79 )3



π = Pq − C ( q,10 ) = 35 ( 221.79 ) − 
+ 50 (10 )  = 4, 675.11. The higher wage
3
2
 1000 (10 )

rate causes Sarah to reduce output and her profits also fall. In this case, profits fall by 4.9%
when the wage rate rises by 9.5%.
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