A firm has the following short-run production function (where L = Labor and Q =
ID: 1168982 • Letter: A
Question
A firm has the following short-run production function (where L = Labor and Q = Output):
Q = 10L – 0.5L2
Suppose that the output can be sold for $10 per unit. Further assume that the firm can obtain as much of the variable input (L) as it needs at $20 per unit.
a) Determine the marginal revenue function (5 points)
b) Determine the value of L that maximizes profits (5 points)
A company uses two variable inputs, labor (L) and materials (M), to produce its output. At the company’s current level of output:
CL = $10 / unit MPL = 25
CM = $2 / unit MPM = 4
a) Determine whether the firm is operating efficiently, given that its objective is to minimize the cost of producing the given the level of output (5 points)
b) Determine what changes (if any) in the relative proportions of labor and materials are needed to operate efficiently (5 points)
A company has a cost structure described in the table below. Using your knowledge of the types of costs involved in a production process, fill in the blanks in the table.
Quantity produced
Total Cost
Total Fixed Cost
Total Variable Cost
Average Total Cost
Average Fixed Cost
Average Variable Cost
Marginal Cost
0
100
0
-
-
-
-
1
160
2
80
3
65
4
10
5
120
6
40
7
180
8
30
9
421
10
100
Quantity produced
Total Cost
Total Fixed Cost
Total Variable Cost
Average Total Cost
Average Fixed Cost
Average Variable Cost
Marginal Cost
0
100
0
-
-
-
-
1
160
2
80
3
65
4
10
5
120
6
40
7
180
8
30
9
421
10
100
Explanation / Answer
1.
a.
Q = 10L – 0.5L2
Revenue= PQ, Price (P)=10
Marginal revenue is change in the quantity of revenue when quantity of product is increased by 1 unit.
MR=dR/dQ=P=10
b.
Cost to firm = WL
Revenue= PQ= 100L-5L2
Profit = revenue-cost= 100l-5L2 – 20L
Profit maximisation where dP/dL=0 = 100-10L-20=0
L=8
2.
a.
CL = $10 / unit MPL = 25 , Cl/MPL = 10/25=.4
CM = $2 / unit MPM = 4, Cm/MPm = .5
Cost of input to marginal productivity of input gives the average cost of employing a additional unit of input. Since labour is more cheaper compared to material, so firm must employ more labour to minimize cost.
b.
In efficiency cost to productivity ratio will be equal. By increasing the number of labour, its cost will be same but MP will start decreasing. So after some point it will equalize to the cost to MP ratio of other input.
3.
TC=TFC+TVC
ATC=TC/Q
AVC=TVC/Q
MC=TVC2-TVC1
Quantity produced
Total Cost
Total Fixed Cost
Total Variable Cost
Average Total Cost
Average Fixed Cost
Average Variable Cost
Marginal Cost
0.00
100.00
100.00
0.00
-
-
-
-
1.00
160.00
100.00
60.00
160.00
100.00
60.00
60.00
2.00
180.00
100.00
80.00
90.00
50.00
40.00
20.00
3.00
195.00
100.00
95.00
90.00
33.33
260.00
60.00
4.00
205.00
100.00
105.00
51.25
25.00
26.25
60.00
5.00
220.00
100.00
120.00
44.00
20.00
24.00
15.00
6.00
240.00
100.00
140.00
40.00
16.67
23.33
20.00
7.00
280.00
100.00
180.00
40.00
14.29
25.71
40.00
8.00
340.00
100.00
240.00
42.50
12.50
30.00
60.00
9.00
421.00
100.00
321.00
46.78
11.11
35.67
81.00
10.00
521.00
100.00
421.00
52.10
10.00
42.10
100.00
Quantity produced
Total Cost
Total Fixed Cost
Total Variable Cost
Average Total Cost
Average Fixed Cost
Average Variable Cost
Marginal Cost
0.00
100.00
100.00
0.00
-
-
-
-
1.00
160.00
100.00
60.00
160.00
100.00
60.00
60.00
2.00
180.00
100.00
80.00
90.00
50.00
40.00
20.00
3.00
195.00
100.00
95.00
90.00
33.33
260.00
60.00
4.00
205.00
100.00
105.00
51.25
25.00
26.25
60.00
5.00
220.00
100.00
120.00
44.00
20.00
24.00
15.00
6.00
240.00
100.00
140.00
40.00
16.67
23.33
20.00
7.00
280.00
100.00
180.00
40.00
14.29
25.71
40.00
8.00
340.00
100.00
240.00
42.50
12.50
30.00
60.00
9.00
421.00
100.00
321.00
46.78
11.11
35.67
81.00
10.00
521.00
100.00
421.00
52.10
10.00
42.10
100.00
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