39. Profit Maximization: Equations. Woodland Instruments, Inc. operates in the highly competitive
electronics industry. Prices for its R2-D2 control switches are stable at $100 each. This means that P = MR =
$100 in this market. Engineering estimates indicate that relevant total and marginal cost relations for the R2-D2
model are:
TC
= $500,000 + $25Q + $0.0025Q2
MC
= TC/ Q = $25 + $0.005Q
A.
Calculate the output level that will maximize R2-D2 profit.
B.
Calculate this maximum profit.
40. Profit Maximization: Equations. Austin Heating & Air Conditioning, Inc., offers heating and air
conditioning system inspections in the Austin, Texas, market. Prices are stable at $50 per unit. This means that
P = MR = $50 in this market. Total cost (TC) and marginal cost (MC) relations are:
TC
= $1,000,000 + $10Q + $0.00025Q2
MC
= TC/ Q = $10 + $0.0005Q
A.
To find the profit-maximizing level of output, set MR = MC and solve for Q:
= MC
$100
= $25 + $0.005Q
0.005Q
= 75
Q
= 15,000
B.
The total revenue function for Woodland is:
TR
= P ´ Q = $100Q
Then, total profit is:
p
= TR – TC
= $62,500
A.
Calculate the output level that will maximize profit.
B.
Calculate this maximum profit.
41. Profit Maximization: Equations. Jewelry.com is a small but rapidly growing Internet retailer. A popular
product is its standard 14k white gold diamond anniversary rings (1/4 ct. tw.) that retail for $250. Prices are
stable, so P = MR = $250 in this market. Total and marginal cost relations for this product are:
TC
= $3,250,000 + $70Q + $0.002Q2
MC
= TC/ Q = $70 + $0.004Q
A.
Calculate the output level that will maximize profit.
B.
Calculate this maximum profit.
A.
To find the profit-maximizing level of output, set MR = MC and solve for Q:
A.
To find the profit-maximizing level of output, set MR = MC and solve for Q:
= MC
= $10 + $0.0005Q
0.0005Q
= 40
Q
= 80,000
B.
The total revenue function is:
TR
= PQ = $50Q
Total profit is:
p
= TR – TC
= $50Q – $1,000,000 – $10Q – $0.00025Q2
= $600,000
42. Profit Maximization: Equations. Virus Soft, Inc., operates in the highly competitive virus detection and
protection software industry. Prices for its basic software are stable at $30 each. This means that P = MR = $30
in this market. Engineering estimates indicate that relevant total and marginal cost relations for this product are:
TC
= $750,000 + $20Q + $0.00002Q2
MC
= TC/ Q = $20 + $0.00004Q
A.
Calculate the output level that will maximize profit.
B.
Calculate this maximum profit.
A.
To find the profit-maximizing level of output we set MR = MC and solve for Q:
= MC
= $20 + $0.00004Q
0.00004Q
= 10
Q
= 250,000
= MC
$250
= $70 + $0.004Q
0.004Q
= 180
Q
= 45,000
B.
The total revenue function is:
= PQ = $250Q
Total profit is:
p
= TR – TC
= $800,000
43. Profit Maximization: Equations. Lone Star Insurance offers mail-order automobile insurance to
preferred-risk drivers in the state of Texas. The company is the low-cost provider of insurance in this market
with fixed costs of $18 million per year, plus variable costs of $750 for each driver insured on an annual basis.
Annual demand and marginal revenue relations for the company are:
P
= $1,500 – $0.005Q
MR
= TR/ Q = $1,500 – $0.01Q
A.
Calculate the profit-maximizing activity level.
B.
Calculate the company’s optimal profit and return-on-sales levels.
A.
Set MR = MC and solve for Q to find the profit-maximizing activity level:
= MC
$1,500 – $0.01Q
= $750
0.01Q
= 750
Q
= 75,000
= PQ = $30Q
Then, total profit is:
p
= TR – TC
= $500,000
44. Profit Maximization: Equations. Dot.com Products, Inc., offers storage containers for fine china on the
Internet. The company is the low-cost retailer of these quilted boxes with fixed costs of $480,000 per year, plus
variable costs of $30 for each box. Annual demand and marginal revenue relations for the company are:
P
= $70 – $0.0005Q
MR
= TR/ Q = $70 – $0.001Q
A.
Calculate the profit-maximizing activity level.
B.
Calculate the company’s optimal profit and return-on-sales levels.
B.
p
= PQ – TC
-$750(75,000)
= PQ
= $84,375,000
= $10,125,000/$84,375,000
45. Profit Maximization: Equations. Steam Cleanin, Inc., offers professional carpet cleaning to home owners
in Huntsville, Alabama. The company is the low-cost provider in this market with fixed costs of $168,750 per
year, plus variable costs of $10 per room of carpet cleaning. Annual demand and marginal revenue relations for
the company are:
P
= $40 – $0.001Q
MR
= TR/ Q = $40 – $0.002Q
A.
Calculate the profit-maximizing activity level.
B.
Calculate the company’s optimal profit and return-on-sales levels.
= MC
$70 – $0.001Q
= $30
0.001Q
= 40
Q
= 40,000
B.
p
= PQ – TC
– $30(40,000)
= PQ
= $2,000,000
= $320,000/$2,000,000
46. Optimal Profit. Hardwood Cutters offers seasoned, split fireplace logs to consumers in Toledo, Ohio. The
company is the low-cost provider of firewood in this market with fixed costs of $10,000 per year, plus variable
costs of $25 for each cord of firewood. Annual demand and marginal revenue relations for the company are:
P
= $225 – $0.125Q
MR
= TR/ Q = $225 – $0.25Q
A.
Calculate the profit-maximizing activity level.
B.
Calculate the company’s optimal profit and return-on-sales levels.
= MC
$40 – $0.002Q
= $10
0.002Q
= 30
Q
= 15,000
B.
p
= PQ – TC
– $10(15,000)
= PQ
= $375,000
= $56,250/$375,000
47. Not-for-Profit Analysis. The Indigent Care Center, Inc., is a private, not-for-profit, medical treatment
center located in Denver, Colorado. An important issue facing Dr. Kerry Weaver, ICC’s administrative director,
is the determination of an appropriate patient load (level of output). To efficiently employ scarce ICC resources,
the board of directors has instructed Weaver to maximize ICC operating surplus, defined as revenues minus
operating costs. They have also asked Weaver to determine the effects of two proposals for meeting new state
health care regulations. Plan A involves an increase in costs of $100 per patient, whereas plan B involves a
$20,000 increase in fixed expenses. In her calculations, Weaver has been asked to assume that a $3,000 fee will
be received from the state for each patient treated, irrespective of whether plan A or plan B is adopted.
In the calculations for determining an optimal patient level, Weaver regards price as fixed; therefore, P = MR =
$3,000. Prior to considering the effects of the new regulations, Weaver projects total and marginal cost relations
of:
TC
= $75,000 + $2,000Q + $2.5Q2
MC
= TC/ Q = $2,000 + $5Q
where Q is the number of ICC patients.
A.
Before considering the effects of the proposed regulations, calculate ICC’s optimal patient and operating surplus levels.
B.
Calculate these levels under plan A.
C.
Calculate these levels under plan B.
= MC
$225 – $0.25Q
= $25
0.25Q
= 200
Q
= 800
B.
p
= PQ – TC
= $225(800) – $0.125(8002) – $10,000 – $25(800)
= $70,000
TR
= PQ
= $225(800) – $0.125(8002)
= $100,000
Return on Sales
= p/TR
= $70,000/$100,000
= 70%
48. Average Cost Minimization. Commercial Recording, Inc., is a manufacturer and distributor of reel-to-reel
recording decks for commercial recording studios. Revenue and cost relations are:
TR
= $3,000Q – $0.5Q2
MR
= TR/ Q = $3,000 – $1Q
TC
= $100,000 + $1,500Q + $0.1Q2
MC
= TC/ Q = $1,500 + $0.2Q
A.
Calculate output, marginal cost, average cost, price, and profit at the average cost-minimizing activity level.
B.
Calculate these values at the profit-maximizing activity level.
C.
Compare and discuss your answers to parts A and B.
And,
= $1,500 + $0.2(1,000)
= $1,700
= $1,700
To find the average cost-minimizing level of output, set MC = AC and solve for Q:
49. Average Cost Minimization. Better Buys, Inc., is a leading discount retailer of wide-screen digital and
cable-ready plasma HDTVs. Revenue and cost relations for a popular 55-inch model are:
TR
= $4,500Q – $0.1Q2
MR
= TR/ Q = $4,500 – $0.2Q
TC
= $2,000,000 + $1,500Q + $0.5Q2
MC
= TC/ Q = $1,500 + $1Q
A.
Calculate output, marginal cost, average cost, price, and profit at the average cost-minimizing activity level.
B.
Calculate these values at the profit-maximizing activity level.
C.
Compare and discuss your answers to parts A and B.
To find the average cost-minimizing level of output, set MC = AC and solve for Q:
And,
= $1,500 + $1(2,000)
= $3,500
= $3,500
50. Revenue Maximization. Restaurant Marketing Services, Inc., offers affinity card marketing and monitoring
systems to fine dining establishments nationwide. Fixed costs are $600,000 per year. Sponsoring restaurants are
paid $60 for each card sold, and card printing and distribution costs are $3 per card. This means that RMS’s
marginal costs are $63 per card. Based on recent sales experience, the estimated demand curve and marginal
revenue relations for are:
P
= $130 – $0.000125Q
MR
= TR/ Q = $130 – $0.00025Q
A.
Calculate output, price, total revenue, and total profit at the revenue-maximizing activity level.
B.
Calculate output, price, total revenue, and total profit at the profit-maximizing activity level.
C.
Compare and discuss your answers to parts A and B.
A.
To find the revenue-maximizing level of output, set MR = 0 and solve for Q:
= 0
$130 – $0.00025Q
= 0
0.00025Q
= 130
Q
= 520,000
P
= $130 – $0.000125Q
= $130 – $0.000125(520,000)
= $65
= PQ
= $65(520,000)
= $33,800,000
p
= TR – TC
= $33,800,000 – $600,000 – $63(520,000)
= $440,000
B.
To find the profit-maximizing level of output, set MR = MC and solve for Q:
