Managerial Economics Numerical's and Solutions
CHAPTER
2
OPTIMIZATION
TECHNIQUES AND NEW MANAGEMENT TOOLS
1. A firm's demand function is defined
as Q
= 14 − 2P. Use this function to calculate total revenue
when price is equal to 3 and when price is equal to 4. What is marginal revenue
equal to between P = 3
and P
= 4?
Solution:
Q = 14
− (2)(4) = 6 so total revenue is (6)(4) = $24
Q = 14
− (2)(3) = 8 so total revenue is (8)(3) = $24
Marginal revenue is 0.
2. A firm's demand function is defined
as Q
= 30 − P. Use this function to calculate total revenue
when price is equal to 5 and when price is equal to 6. What is marginal revenue
equal to between P = 5
and P
= 6?
Solution:
Q = 30
− 5 = 25 so total revenue is (25)(5) = $125
Q = 30
− 6 = 24 so total revenue is (24)(6) = $144
Marginal revenue is $19.
3. A firm's demand function is defined
as Q
= 30 − 2P. Use this function to calculate total revenue
when price is equal to 10 and when price is equal to 11. What is marginal revenue
equal to between P = 10
and P
= 11?
Solution:
Q = 30
− (2)(10) = 10 so total revenue is (10)(10) = $100
Q = 30
− (2)(11) = 8 so total revenue is (8)(11) = $88
Marginal revenue is −$6.
4. Use the production relationship
between total product (Q) and units of labor (L) employed that is presented in the
table below to calculate the average and marginal product of labor when L = 5.
L 1 2 3 4 5
6 7 8 9
Q 3 7 13 17 20 22
23 23 22
Solution:
Average product = 20/5 = 4
Marginal product = 20 − 17 = 3
5. Use the total cost (TC) schedule that is presented in the
table below to calculate average total cost, average variable cost, average
fixed cost, and marginal cost when output (Q) is equal to 5.
Q 0 1 2 3 4 5
6 7 8
9
TC 5 7 8 10 14 20
28 38 50 72
Solution:
Average total cost = 20/5 Average
variable cost = (20 − 5)/5 = 3
Average fixed cost = 5/5 = 1 Marginal
cost = (20 − 14)/(5 − 4) = 6
6. Use the total cost (TC) schedule that is presented in the
table below to determine the optimal rate of production when the firm can sell
all of the output it produces at a price of $10 per unit. Also determine the
level of profit (or loss) that the firm will experience at this level of
output.
Q 0 1 2 3 4 5 6
7 8 9
TC 5 7 8 10 14 20 28 38 50 72
Solution:
Q 1 2 3 4 5 6 7 8 9
MC 2 1 2 4 6 8 10 12 22
The firm should produce Q = 7. Its profit will be (7)(10) − 38 =
$32.
7. Use the total cost (TC) schedule that is presented in the
table below to determine the optimal rate of production when the firm can sell
all of the output it produces at a price of $6 per unit. Also determine the
level of profit (or loss) that the firm will experience at this level of
output.
Q 0 1 2
3 4 5
6 7 8
9
TC 15 17 18 20 24 30 38 48 60 82
Solution:
Q 1
2 3 4 5 6 7 8 9
MC 2 1 2 4 6 8 10 12 22
The firm should produce Q = 5. Its profit will be (5)(6) − 30 =
0.
8. Use the demand schedule that is
presented in the table below to determine the optimal rate of production and
price when the firm has a constant marginal cost of $10 per unit.
Quantity 1 2
3 4 5 6 7 8 9 10
Price 80 60 48 40 34 29 25 20 15 10
Solution:
Quantity 1 2 3 4 5 6 7 8 9 10
TR 80 120 144 160
170 174 175
160 135 100
MR 80 40 24 16 10
4 1 -15 -25
-35
The firm should produce Q = 5.
9. Use the demand schedule that is
presented in the table below to determine the optimal rate of production and
price when the firm has the following marginal cost function:
MC = 1
+ Q/2.
Quantity 1 2
3 4 5 6 7 8 9 10
Price 80 60 48 40 34 29 25 20 15 10
Solution:
Quantity 1 2 3 4 5 6 7 8 9 10
TR 80 120 144 160 170 174 175 160 135 100
MR 80 40 24
16 10 4 1 -15
-25
-35
MC 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
The firm should produce Q = 6.
10. A firm's demand function is defined
as Q
= 20 − 2P. Use this function to calculate total revenue
when price is equal to 3 and when price is equal to 4. What is marginal revenue
equal to between P = 3
and P
= 4?
Solution:
Q = 20
− (2)(4) = 12 so total revenue is (12)(4) = $48
Q = 20
− (2)(3) = 14 so total revenue is (14)(3) = $42
Marginal revenue is (48 − 42)/(12 − 14)
= −3.
11. A firm's demand function is defined
as Q
= 40 − P. Use this function to calculate total revenue
when price is equal to 5 and when price is equal to 6. What is marginal revenue
equal to between P = 25
and P
= 26?
Solution:
Q = 40
− 25 = 15 so total revenue is (15)(25) = $375
Q = 40
− 26 = 14 so total revenue is (14)(26) = $364
Marginal revenue is (375 − 364)/(15 −
14) = 11.
12. A firm's demand function is defined
as Q
= 100 − 5P. Use this function to calculate total revenue
when price is equal to 10 and when price is equal to 12. What is marginal revenue
equal to between P = 10
and P
= 12?
Solution:
Q =
100 − (5)(10) = 50 so total revenue is (10)(50) = $500.
Q =
100 − (5)(12) = 40 so total revenue is (12)(40) = $480.
Marginal revenue is (500 − 480)/(50 −
40) = 2.
13. Use the production relationship
between total product (Q) and units of labor (L) employed that is presented in the
table below to calculate the average and marginal product of labor when Q = 4.
L 1
2 3 4 5 6
7 8 9
Q 2 5 11 16 20 22 23 23 22
Solution:
Average product = 16/4 = 4
Marginal product = 16 − 11 = 5
14. Use the total cost (TC) schedule that is presented in the
table below to calculate average total cost, average variable cost, average
fixed cost, and marginal cost when output (Q) is equal to 4.
Q 0 1 2 3 4 5 6
7 8 9
TC 3 6 8 11 15 20
26 34 55 70
Solution:
Average total cost = 15/4 = 4.25
Average variable cost = (14 − 3)/4 = 2.75
Average fixed cost = 3/4 = 0.75
Marginal cost = (15 − 11)/(4 − 3) = 4
15. Use the total cost (TC) schedule that is presented in the
table below to determine the optimal rate of production when the firm can sell
all of the output it produces at a price of $6 per unit. Also determine the
level of profit (or loss) that the firm will experience at this level of
output.
Q 0 1 2 3 4
5 6 7
8 9
TC 3 6 8 11 15 20 26 34 55 70
Solution:
Q 1 2 3 4 5 6 7 8 9
MC 3 2 3 4 5 6 8 11 15
The firm should produce Q = 6. Its profit will be (6)(6) − 26 =
$10.
16. Use the total cost (TC) schedule that is presented in the
table below to determine the optimal rate of production when the firm can sell
all of the output it produces at a price of $8 per unit. Also determine the
level of profit (or loss) that the firm will experience at this level of
output.
Q 0 1 2 3
4 5 6
7 8 9
TC 15 17 18 20 24 30 38 48 60 82
Solution:
Q 1 2 3 4 5 6 7 8 9
MC 2 1 2 4 6 8 10 12 22
The firm should produce Q = 6. Its profit will be (6)(8) − 38 =
10.
17. Use the demand schedule that is
presented in the table below to determine the optimal rate of production and
price when the firm has a constant marginal cost of $16 per unit.
Quantity 1 2
3 4 5
6 7 8
9 10
Price 80 60 48 40 34 29 25 20 15 10
Solution:
Quantity 1 2 3 4 5 6 7 8 9 10
TR 80 120 144 160
170
174 175 160
135 100
MR 80 40 24 16 10 4 1 −15
−25 −35
The firm should produce Q = 4.
18. Use the demand schedule that is
presented in the table below to determine the optimal rate of production and
price when the firm has the following marginal cost function:
MC = 1
+ Q.
Quantity 1
2 3 4
5 6 7
8 9 10
Price 80 60 48 40 34 29 25 20 15 10
Solution:
Quantity 1 2 3 4 5 6 7 8 9 10
TR 80 120 144 160
170 174 175
160 135 100
MR 80 40 24 16 10 4 1 −15 −25 −35
MC 2 3 4 5 6 7 8 9 10 11
The firm should produce Q = 5.
19. A firm's demand function is Q = 16 − P and its total cost function is defined
as TC
= 3 + Q + 0.25Q2. Use these two functions to form the firm's profit function
and then determine the level of output that yields the profit maximum. What is
the level of profit at the optimum?
Solution:
TR = 16Q − Q2 so profit = (16Q − Q2) −
(3 + Q
+ 0.25Q2) = −3 + 15Q − 1.25Q2
Using calculus: dProfit/dQ = 15 − 2.5Q = 0 implies Q = 6 and the second derivative is −2.5,
which implies that Q = 6 is a maximum.
Profit = −3 + (15)(6) − (1.25)(36) = 42
An alternative method of solution can
be applied by noting that MC = 1 + Q/2 and MR = 16 − 2Q and then setting the two equal to each other.
20. A firm's demand function is Q = 40 − 2P and its total cost function is defined
as TC
= 100 + 2Q + 0.25Q2. Use these two functions to form the firm's profit function
and then determine the level of output that yields the profit maximum. What is
the level of profit at the optimum level of output?
Solution:
TR = 20Q − 0.5Q2 so
Profit = (20Q − 0.5Q2) − (100 + 2Q + 0.25Q2) =
−100 + 18Q
− 0.75Q2
Using calculus: dProfit/dQ = 18 − 1.5Q = 0 implies Q = 12 and the second derivative is −1.5,
which implies that Q = 12 is a maximum.
Profit = −100 + (18)(12) − (0.75)(144)
= 8
An alternative method of solution can
be applied by noting that MC = 2 + Q/2 and MR = 20 − Q and then setting the two equal to each other.
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