Managerial Economics
Assignment No. 5
Dr. Fakhry Eldin Elfiky
Name:
Mahmoud Mohamed Raafat
Group:
C
( MBA )
Question (1)
A firm has the following short-run production function:
Q = 50 L +6 L² – 0.5L³
a- when does the law of diminishing returns take effect?
The law of diminishing returns takes effect when the marginal production
reaches its maximum,
That means the slope of MPL curve = 0
So we can get this point by the following :
MPL = 50 + 12L - 1.5L²
Max MPL is when d MPL = zero
dL
d MPL = 12- 3L = 0
dL
12 = 3L
L =4
b- Calculate the values for labor over which stages I , II , III
occur.
Stage 1
-
Starts from the origin that means L = 0
-
End when APL = MPL and that happens when APL is at its maximum,
That means when the slope of APL curve = 0
APL = (50 L +6 L² – 0.5L³ )
L
= 50 + 6 L – 0,5 L²
Page 1
dAPL = 6 - L = 0
dL
So
L=6
Stage 2
-
Begins from the point of end of stage 1 when L = 6
-
Ends when MPL = 0
MPL = 50 + 12L - 1.5L² = 0
L= b±√ b2 - 4ac
2a
= 12±√144 - (4×-1.5×50)
3
L = 11
Stage 3
-
Begins from the point of end of stage 2 when L = 11
-
Ends when L = ∞
To summarize :
Stage 1 : from
L=0
to
L=6
Stage 2 : from L = 6
to
L = 11
Page 2
Stage 3 : from L = 11
to
L = infinity
C- Assume each worker is paid LE 10 per hour and works a 40 hour week,
How many workers should the firm hire if the price of the output is
LE10 per unit ? Suppose the price of the output falls to LE 7.50. What
do you think would be the short run impact on the firm’s production ?
The long run impact?
The optimal number of worker is the number who achieves the maximum profits
and that will happened when MRP = MLC
When the price = 10
MRP = MPL × P = 10 (50 + 12L - 1.5L² ) = 500+ 120L -15 L²
MLC = 10 × 40 = 400
So
500+ 120L -15 L² = 400
100 + 120L - 15 L² = 0
L= b±√ b2 - 4ac
2a
= 120±√14400 - (4×-15×100)
30
L = 8.76 = 9
Page 3
When the price = 7.5
MRP = MPL × P = 7.5 (50 + 12L - 1.5L² ) = 375 + 90 L -11.25 L²
MLC = 10 × 40 = 400
So
375 + 90 L -11.25 L² = 400
-25 + 90 L -11.25 L² = 0
L= b±√ b2 - 4ac
2a
= 90 ±√8100 - (4×-11.25×-25)
22.5
L = 7.71 = 8
The short run impact on the firm’s production
Q = 50 L +6 L² – 0.5L³
Q = (50 × 9) + ( 6 × 81 ) – (0.5 × 729 ) = 571.5 units
Q = (50 × 8) + ( 6 × 64 ) – (0.5 × 512 ) = 528 units
The production decreased by 43.5 units as because the number of worker
changed from 9 workers to 8 workers as a result of changing in price from LE
10 to LE 7.5
Page 4
The long run impact on the firm’s production
There is no impact because we are still in stage 2 also we have only one
variable until now .
Question (2)
2- Following are different algebraic expression of the production
function. Decide whether each has constant, increasing, or decreasing
return to scale .
0.25 0.75
1- Q = 75 L K
0.25 + 0.75 = 1
(CRTS ) constant return to scale
0.15 0.40 0.45
2- Q = 75 L K
C
0.15 + 0.40 + 0.45 = 1
(CRTS ) constant return to scale
0.70 0.60
3- Q = 75 L K
0.70 + 0.60 = 1.30
(IRTS ) increasing return to scale
4- Q = 100 + 50 L + 50 K
If we use 10 units of each input,
Q = 100 +( 50×10)+( 50×10) =1100 unit
Page 5
And if we assume that each input will be doubled,
Q = 100 +( 50×20)+( 50×20) =2100 unit
So ∆Q < ∆ L,K
So EQ < 1
(DRTS ) Decreasing return to scale
5- Q = 50 L² + 50 K²
If we use 2 units of each input,
Q = (50 ×4)+( 50×4) =400 unit
And if we assume that each input will be doubled,
Q = ( 50×16)+( 50×16) =1600 unit
So ∆Q > ∆ L,K
So EQ > 1
(IRTS ) increasing return to scale
6- Q = 50 L + 50 K + 50 LK
If we use 10 units of each input,
Q = ( 50×10)+( 50×10)+ ( 50×10 ×10) =6000 unit
And if we assume that each input will be doubled,
Q = ( 50×20)+( 50×20) + ( 50×20 ×20) =22000 unit
So ∆Q > ∆ L,K
So EQ > 1
(IRTS ) increasing return to scale
Page 6
Question (3)
3- The owner of a car wash is trying to decide on the number of people
to employ based on the following short production function:
Q = 6 L – 0.5 L²
Where Q = No. of car washes per hour
L = No. of workers
L
TPL
MPL
APL
0
1
2
3
4
5
6
7
8
9
10
11
0
5.5
10
13.5
16
17.5
18
17.5
16
13.5
10
5.5
0
5.5
4.5
3.5
2.5
1.5
0.5
-0.5
-1.5
-2.5
-3.5
-4.5
5.5
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
20.00
8.0018.00
6.0016.00
14.00
4.0012.00
2.0010.00
0.00 8.00
6.00
-2.00 4.001
-4.00 2.00
-6.00 0.00
TP
MP,AP
a- Generate a schedule showing TP,AP,MP., then graph it.
-8.00
2
1
3
2
4
3
5
4
5
6
7
6
7
8
8
L
L
Page 7
MP
AP
9
9
10
10
11
11
12
12
b- Suppose the price for a basic car wash (no under coating ,no wax treatments
etc.) in his area of business is LE 5. How many people should he hire if he
pays each worker LE 6 / he.?
L
TP
0
1
2
3
4
5
6
7
8
9
10
11
12
0
5.5
10
13.5
16
17.5
18
17.5
16
13.5
10
5.5
0
MP
5.5
4.5
3.5
2.5
1.5
0.5
-0.5
-1.5
-2.5
-3.5
-4.5
-5.5
AP
TRP
5.5
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
0
27.5
50
67.5
80
87.5
90
87.5
80
67.5
50
27.5
0
MRP
TLC
27.5
22.5
17.5
12.5
7.5
2.5
-2.5
-7.5
-12.5
-17.5
-22.5
-27.5
0
6
12
18
24
30
36
42
48
54
60
66
72
MLC
TRP-TLC
MRP-MLC
6
6
6
6
6
6
6
6
6
6
6
6
0
21.5
38
49.5
56
57.5
54
45.5
32
13.5
-10
-38.5
-72
0
21.5
16.5
11.5
6.5
1.5
-3.5
-8.5
-13.5
-18.5
-23.5
-28.5
-33.5
If the price of labor equals 6LE / h , the firm should hire5 workers to reach its
maximum profitability of 57,5LE .
 Beyond that point the firm would pay more in additional labor cost than it would
receive in additional revenue.

C- Suppose he consider hiring students on a part time basis for LE 4 \ hr. . Do
you think he should hire more students at this lower rate ? Explain.
L
0
1
2
3
4
5
6
7
8
9
10
11
12
TP
0
5.5
10
13.5
16
17.5
18
17.5
16
13.5
10
5.5
0
MP
AP
5.5
4.5
3.5
2.5
1.5
0.5
-0.5
-1.5
-2.5
-3.5
-4.5
-5.5
5.5
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
TRP
0
27.5
50
67.5
80
87.5
90
87.5
80
67.5
50
27.5
0
MRP
27.5
22.5
17.5
12.5
7.5
2.5
-2.5
-7.5
-12.5
-17.5
-22.5
-27.5
Page 8
TLC
0
4
8
12
16
20
24
28
32
36
40
44
48
MLC
4
4
4
4
4
4
4
4
4
4
4
4
TRP-TLC
0
23.5
42
55.5
64
67.5
66
59.5
48
31.5
10
-16.5
-48
MRP-MLC
0
23.5
18.5
13.5
8.5
3.5
-1.5
-6.5
-11.5
-16.5
-21.5
-26.5
-31.5
If the price of labor equals 4 LE / h , the firm should also hire 5 workers to
reach its maximum profitability.
 Beyond that point the firm would pay more in additional labor cost than it would
receive in additional revenue.
 But if the pay per worker is 4 LE rather than 6LE the firm maximum profit will
be 67.5LE inside of 57.5LE and this is due to the saving in labor cost.

Page 9