A firm has the following short-run production function Q-50L 61-0.5 where Q= Qua
ID: 367864 • Letter: A
Question
A firm has the following short-run production function Q-50L 61-0.5 where Q= Quantity of output per week L = Labor (number of workers) a. When does the law of diminishing returns take effect? b. Calculate the range of values for labor over which Stages I, II, and III occur. c. Assume each worker is paid $10 per hour and works a 40-hour week. How many work- ers should the firm hire if the price of the output is $10? Suppose the price of the output falls to $7.50. What do you think would be the short-run impact on the firm's production? The long-run impact?Explanation / Answer
a. At what value of L will Diminishing Returns take effect?
For L=4, Diminishing Returns take effect as Diminishing Returns starts after maximum level of Marginal production
Labor (number of workers), L
Quantity of output per week, Q (Q= 50L + 6L^2 – 0.5L^3)
Marginal Production MP (Q/L)
1
55.50
55.50
2
120.00
64.50
3
190.50
70.50
4
264.00
73.50
5
337.50
73.50
6
408.00
70.50
7
472.50
64.50
8
528.00
55.50
9
571.50
43.50
10
600.00
28.50
11
610.50
10.50
12
600.00
-10.50
13
565.50
-34.50
14
504.00
-61.50
15
412.50
-91.50
b. Calculate the range of values for labor over which Stages I, II, and III occur.
Stages I occur from starting to Maximum MP Level
Stage II starts at maximum AP (AP=68),
And Stage III starts when negative MP occurs to infinity
Labor (number of workers), L
Quantity of output per week, Q (Q= 50L + 6L^2 – 0.5L^3)
Marginal Production MP (Q/L)
Average Production AP (Q/L)
1
55.50
55.50
55.50
Stage I
2
120.00
64.50
60.00
3
190.50
70.50
63.50
4
264.00
73.50
66.00
5
337.50
73.50
67.50
6
408.00
70.50
68.00
Stage II
7
472.50
64.50
67.50
8
528.00
55.50
66.00
9
571.50
43.50
63.50
10
600.00
28.50
60.00
11
610.50
10.50
55.50
12
600.00
-10.50
50.00
Stage III
13
565.50
-34.50
43.50
14
504.00
-61.50
36.00
15
412.50
-91.50
27.50
c. Assume each worker is paid $10 per hour and works a 40-hour week. How many workers should the firm hire if the price of the output is $10?
The profit is maximum at L=9 therefore firm should hire 9 labors if the price of the output is $10
Labor (number of workers), L
Quantity of output per week, Q
Marginal Production MP (Q/L)
Average Production AP (Q/L)
Labor cost per week @ $10/hour for 4o hours
Product's Revenue per week at $10
Profit/Loss (revenue -cost)
1
55.50
55.50
55.50
400
555
155
2
120.00
64.50
60.00
800
1200
400
3
190.50
70.50
63.50
1200
1905
705
4
264.00
73.50
66.00
1600
2640
1040
5
337.50
73.50
67.50
2000
3375
1375
6
408.00
70.50
68.00
2400
4080
1680
7
472.50
64.50
67.50
2800
4725
1925
8
528.00
55.50
66.00
3200
5280
2080
9
571.50
43.50
63.50
3600
5715
2115
10
600.00
28.50
60.00
4000
6000
2000
11
610.50
10.50
55.50
4400
6105
1705
12
600.00
-10.50
50.00
4800
6000
1200
13
565.50
-34.50
43.50
5200
5655
455
14
504.00
-61.50
36.00
5600
5040
-560
15
412.50
-91.50
27.50
6000
4125
-1875
Suppose the price of the output falls to $7.50. What do you think would be the short-run impact on the firm’s production? The long-run impact
The short-run impact on the firm’s production is that it should hire only 8 labors at this price as the profit is maximum at L=8
Labor (number of workers), L
Quantity of output per week, Q
Marginal Production MP (Q/L)
Average Production AP (Q/L)
Labor cost per week @ $10/hour for 4o hours
Product's Revenue per week at $7.5
Profit/Loss (revenue -cost)
1
55.50
55.50
55.50
400
416.25
16.25
2
120.00
64.50
60.00
800
900.00
100.00
3
190.50
70.50
63.50
1200
1428.75
228.75
4
264.00
73.50
66.00
1600
1980.00
380.00
5
337.50
73.50
67.50
2000
2531.25
531.25
6
408.00
70.50
68.00
2400
3060.00
660.00
7
472.50
64.50
67.50
2800
3543.75
743.75
8
528.00
55.50
66.00
3200
3960.00
760.00
9
571.50
43.50
63.50
3600
4286.25
686.25
10
600.00
28.50
60.00
4000
4500.00
500.00
11
610.50
10.50
55.50
4400
4578.75
178.75
12
600.00
-10.50
50.00
4800
4500.00
-300.00
13
565.50
-34.50
43.50
5200
4241.25
-958.75
14
504.00
-61.50
36.00
5600
3780.00
-1820.00
15
412.50
-91.50
27.50
6000
3093.75
-2906.25
The variables can change in long-run.
Labor (number of workers), L
Quantity of output per week, Q (Q= 50L + 6L^2 – 0.5L^3)
Marginal Production MP (Q/L)
1
55.50
55.50
2
120.00
64.50
3
190.50
70.50
4
264.00
73.50
5
337.50
73.50
6
408.00
70.50
7
472.50
64.50
8
528.00
55.50
9
571.50
43.50
10
600.00
28.50
11
610.50
10.50
12
600.00
-10.50
13
565.50
-34.50
14
504.00
-61.50
15
412.50
-91.50
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