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11
Pricing Practices
OVERVIEW
Earlier chapters discussed output and pricing decisions under some very
simplistic assumptions. It was assumed, for example, that profit-maximizing
firms produce a single good or service, that production takes place in a
single location, that these firms sell their products in a well-defined market,
that the managements of these firms have perfect information about its production, revenue, and cost functions, and that these firms sell their output
at a uniform price to all customers. In reality, these assumptions are rarely
satisfied. This chapter considers alternative pricing practices, which in some
cases are derivatives of the more general cases already encountered, was
required.
Price discrimination is the situation where a firm sells identical products
in two or more markets at different prices. Economists have identified three
degrees of price discrimination. First-degree price discrimination occurs
when a firm charges each buyer in the market a different price that is based
on what he or she is willing to pay. In practice, first-degree price discrimination is virtually impossible.
Second-degree price discrimination, often referred to as volume discounting, involves charging different prices for different blocks of units, or
bundling different products and sold at a package price. An example of
second-degree price discrimination is block pricing, which involves charging different prices for different blocks of goods and services. Seconddegree price discrimination requires that a firm be able to closely monitor
the level of services consumed by individual buyers.
Managerial Economics: Theory and Practice
167
Copyright © 2003 by Academic Press.
All rights of reproduction in any form reserved.
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Third-degree price discrimination, which is by far the most frequently
practiced type of price discrimination, occurs when firms segment the
market for a particular good or service into easily identifiable groups, and
then charging each group a different price. Such market segregation may
be based on such factors as geography, age, product use, income, etc. For
third-degree price discrimination to be successful, firms must be able to
prevent resale of the good or service across segregated markets.
Cost-plus pricing, also known as mark-up or full-cost pricing, is an
example of non-marginal pricing. Firms that engage in non-marginal pricing
are unable or unwilling to devote the resources required to accurately estimate the total revenue and total cost equations, or does not have sufficient
knowledge about demand and cost conditions to determine the profitmaximizing price and output levels. Cost-plus pricing occurs when a firm
sets the selling price of its product as a markup above it’s fully-allocated
per unit cost of production. One criticism of cost-plus pricing is that it is
insensitive to demand conditions. In practice, however, the size of a firm’s
mark up tends to be inversely related to the price elasticity of demand for
a good or service.
Multi-product pricing involves optimal pricing strategies of firms producing and selling more than one good or service. Firms that independently
produce two products with interrelated demands will maximize its profits
by producing where marginal cost is equal to the change in total revenue
derived from the sale of the product itself, and the change in total revenue
derived from the sale of the related product. A profit-maximizing firm
jointly producing two goods with independent demands that are jointly produced in variable proportions, will equate the marginal revenue generated
from the sale of each good to the marginal cost of producing each product.
Finally, a profit-maximizing firm that jointly produces two goods in fixed
proportions with independent demands will equate the sum of the marginal
revenues of both products expressed in terms of one of the products with
the marginal cost of jointly producing both products expressed in terms of
the same product.
Peak-load pricing occurs when a profit-maximizing firm charges a higher
price for a service when capacity is fully utilized and a lower price when
capacity is under utilized. Off-peak prices are determined by equating marginal revenue to marginal operating costs. Peak prices, on the other hand,
are determined by equating marginal revenue to the marginal cost of
increasing capacity.
Price leadership occurs when an oligopolist establishes a price that is
quickly followed by other firms in the industry. There are two types of price
leadership: Barometric price leadership and dominant price leadership.
Barometric price leadership occurs when a price change by one firm in an
oligopolistic industry, usually in response to perceived changes in macroeconomic or market conditions, is quickly followed by price changes by
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169
other firms in the industry. Dominant price leadership occurs when the
largest firm in the industry establishes the industry price as a result of its
profit-maximizing behavior. Once the industry price is established, the
remaining firms become price takers in the sense that they face a perfectly
elastic demand curve for their output.
Other important pricing practices include transfer pricing, price
skimming, penetration pricing, prestige pricing, and psychological pricing.
Transfer pricing is a method of correctly pricing a product as it is
transferred from one stage of production to another.
Price skimming is the practice of charging a higher price for a new
product than is justified by economic analysis because competition is weak.
During the time when competitors are trying to catch up, the firm may have
monopoly pricing power. Penetration pricing occurs when a firm entering
a new market charges a lower price than its competitors to gain a foothold
in the industry. Prestige pricing occurs when a firm charges a higher price
for its product in the belief that demand will be higher because of the prestige that ownership bestows on the buyer. Finally, psychological pricing is
a marketing ploy designed to create the illusion in the mind of the consumer that a product is being sold at a significantly lower price when, in
fact, the price differential is inconsequential.
MULTIPLE CHOICE QUESTIONS
11.1
The practice of charging different prices to different individuals or
groups for the same good or service by profit-maximizing firms
with market power is called:
A. Differential pricing.
B. Block pricing.
C. Price discrimination.
D. Two-part pricing.
11.2
First-degree price discrimination occurs when a firm:
A. Charges different prices for different quantities sold.
B. Charges each buyer a different price for each individual unit
purchased.
C. Charges each buyer a different price based on an ability to pay.
D. Charges each buyer a different prices based on a willingness to
pay.
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11.3
Consumer surplus may be defined as:
A. The difference between benefits received from the purchase of
a good or service and the benefits actually paid for.
B. The benefits received from purchasing bulk quantities of a good
or service.
C. The difference between the quantities of goods or service
purchased and the quantities of goods and services consumed.
D. The difference between consumer expenditures calculated using
wholesale versus retail prices for final goods and services.
11.4
Suppose that an individual’s demand for an good is Q = 15 - 0.2P.
If the product price is $50, then the dollar value of total benefits
received by the individual is:
A. $62.50.
B. $250.00.
C. $312.50.
D. $375.00.
11.5
Suppose that an individual’s demand for an good is Q = 15 - 0.2P.
If the product price is $50, then the dollar value of total benefits
paid for by the individual is:
A. $62.50.
B. $250.00.
C. $312.50.
D. $375.00.
11.6
Suppose that an individual’s demand for an good is Q = 15 - 0.2P.
If the product price is $50, then the dollar value of total benefits
received by not paid for is:
A. $62.50.
B. $250.00.
C. $312.50.
D. $375.00.
11.7
Firms are able to capture the entire amount of consumer surplus
by practicing:
A. First-degree price discrimination.
B. Second-degree price discrimination.
C. Third-degree price discrimination.
D. Differential pricing.
E. Two-part pricing.
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11.8
The practice of charging the highest price that each buyer is willing
to pay for each unit purchased is called:
A. Differential pricing.
B. First-degree price discrimination.
C. Second-degree price discrimination.
D. Third-degree price discrimination.
E. Market power.
11.9
Second-degree price discrimination occurs when:
A. Transfers some producer surplus to the consumer in the form of
increased utility.
B. Transfers all producer surplus to the consumer in the form of
increased utility.
C. Transfers all consumer surplus to the producer in the form of
higher profits.
D. Transfers no consumer surplus to the producer in the form of
higher profits.
E. Transfers some consumer surplus to the producer in the form of
higher profits.
11.10 Second-degree price discrimination occurs when a firm:
A. Charges different prices for different quantities sold.
B. Charges each buyer a different price for each individual unit
purchased.
C. Charges each buyer a different price based on an ability to pay.
D. Sell its product in “blocks” or “bundles” rather than one unit at
a time.
E. Charges each buyer a different prices based on a willingness to
pay.
11.11 Firms that charge different prices for the same product in different
markets practice:
A. First-degree price discrimination.
B. Second-degree price discrimination.
C. Third-degree price discrimination.
D. Differential pricing.
E. Spatial pricing.
11.12 Second-degree price discrimination includes:
A. Block pricing.
B. Two-part pricing.
C. Commodity pricing.
D. All of the above.
E. None of the above.
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11.13 Block pricing involves:
A. Charging a fixed fee for the right to use or purchase a good or
service, plus a per-unit charge.
B. Selling a product in fixed quantities.
C. Combining of two or more different products into a single
package, which is sold at a single price.
D. Charging a higher price for a service when demand is high and
capacity is fully utilized and a lower price for the service when
demand is low and capacity is under utilized.
E. Adding a predetermined “mark-up” to a firm’s estimated per
unit cost of production when setting the selling price of its
product.
11.14 Two-part pricing involves:
A. Charging a fixed fee for the right to use or purchase a good or
service, plus a per-unit charge.
B. Selling a product in fixed quantities.
C. Combining of two or more different products into a single
package, which is sold at a single price.
D. Charging a higher price for a service when demand is high and
capacity is fully utilized and a lower price for the service when
demand is low and capacity is under utilized.
E. Adding a predetermined “mark-up” to a firm’s estimated per
unit cost of production.
11.15 Commodity bundling involves:
A. Charging a fixed fee for the right to use or purchase a good or
service, plus a per-unit charge.
B. Selling a product in fixed quantities.
C. Combining of two or more different products into a single
package, which is sold at a single price.
D. Charging a higher price for a service when demand is high and
capacity is fully utilized and a lower price for the service when
demand is low and capacity is under utilized.
E. Adding a predetermined “mark-up” to a firm’s estimated per
unit cost of production when setting the selling price of its
product.
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11.16 Third-degree price discrimination occurs when:
A. Firms charge the maximum price that each consumer is willing
to pay.
B. Firms charge different prices for different “bundles” of goods
and services.
C. Firms charge different prices for the same product based on
geography, income, sex, age, etc.
D. All of the above.
E. None of the above.
11.17 Third-degree price discrimination is only possible when:
I. The price elasticity of demand in each market is the same.
II. Markets may be segmented on the basis of group
characteristics.
III. Member of one group purchasing a product at a lower price
are not able to resell that product to a member of another
group paying the higher price.
Which of the following is correct?
A. I only.
B. II only.
C. III only.
D. I and II only.
E. II and III only.
11.18 Movie theaters that offer a 20 percent discount to senior citizens
are practicing:
A. First-degree price discrimination.
B. Second-degree price discrimination.
C. Third-degree price discrimination.
D. Differential pricing.
11.19 A firm that practices third-degree price discrimination will charge a
higher price in the market:
I.
II.
III.
IV.
With the highest price elasticity of demand.
Where product demand is greatest.
With the lowest price elasticity of demand.
Where product demand is lowest.
Which of the following is correct?
A. I only.
B. III only.
C. I and II only.
D. III and IV only.
E. I and IV only.
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11.20 Cost-plus pricing is a form of:
A. First-degree price discrimination.
B. Second-degree price discrimination.
C. Third-degree price discrimination.
D. Differential pricing.
E. Non-marginal pricing.
11.21 Cost-plus pricing involves:
A. Charging a fixed fee for the right to use or purchase a good or
service, plus a per-unit charge.
B. Selling a product in fixed quantities.
C. Combining of two or more different products into a single
package, which is sold at a single price.
D. Charging a higher price for a service when demand is high and
capacity is fully utilized and a lower price for the service when
demand is low and capacity is under utilized.
E. Adding a predetermined “mark-up” to a firm’s estimated per
unit cost of production.
11.22 A profit-maximizing firm sells its product for $10 and the
price elasticity of demand is -2.5. The firm’s marginal cost of
production is:
A. $6.
B. $10.
C. $12.
D. $15.
11.23 Suppose that a profit-maximizing firm charges $32 for its product.
If the firm’s marginal cost of production is $16, the price elasticity
of demand for this product is:
A. -2.5.
B. -2.
C. -1.
D. -0.5.
11.24 Suppose that a firm practices cost-plus pricing. If the average total
cost of producing the product is $50 and the markup is 15 percent,
then the selling price of the product will be:
A. $7.50.
B. $42.50.
C. $65.50.
D. None of the above.
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11.25 Suppose that a firm practices cost-plus pricing. If the firm prices its
product at $45.50 and the average total cost of producing the
product is $35, then the markup is:
A. 15 percent.
B. 20 percent.
C. 25 percent.
D. 30 percent
E. None of the above.
11.26 Suppose that a firm practices cost-plus pricing. If the price elasticity
of demand for a firm’s product is -3.5, then the percentage markup over the fully-allocated, per-unit cost of production is:
A. 10 percent.
B. 25 percent.
C. 40 percent.
D. 60 percent.
11.27 Multi-product pricing involves:
I. Optimal pricing of two or more products with interdependent
demands and independent production.
II. Optimal pricing of two or more products with independent
demands jointly produced in variable proportions.
III. Optimal pricing of two or more products with independent
demands jointly produced in fixed proportions.
Which of the following is correct?
A. I only.
B. II only.
C. III only.
D. I, II and III are correct.
E. I, II and III are incorrect.
11.28 Suppose that a firm produces two products with interdependent
demands, but which are independently produced. To maximize
profits, the firm should:
A. Produce where the marginal cost of producing each product is
equal to marginal revenue from the sale of the product plus the
change in total revenue from the sale of the related product.
B. Produce where marginal revenue from the sale of each product
equals the marginal cost of producing each product.
C. Produce where the sum of the marginal revenues of both
products expressed in terms of one of the two products is equal
to the marginal cost of jointly producing the two expressed in
terms of the same product.
D. None of the above are correct.
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11.29 Suppose that a firm produces two products with independent
demands, but which are jointly produced in variable proportions.
To maximize profits, the firm should:
A. Produce where the marginal cost of producing each product is
equal to marginal revenue from the sale of the product plus the
change in total revenue from the sale of the related product.
B. Produce where marginal revenue from the sale of each product
equals the marginal cost of producing each product.
C. Produce where the sum of the marginal revenues of both
products expressed in terms of one of the two products is equal
to the marginal cost of jointly producing the two expressed in
terms of the same product.
D. None of the above are correct.
11.30 Suppose that a firm produces two products with independent
demands, but which are jointly produced in fixed proportions. To
maximize profits, the firm should:
A. Produce where the marginal cost of producing each product is
equal to marginal revenue from the sale of the product plus the
change in total revenue from the sale of the related product.
B. Produce where marginal revenue from the sale of each product
equals the marginal cost of producing each product.
C. Produce where the sum of the marginal revenues of both
products expressed in terms of one of the two products is equal
to the marginal cost of jointly producing the two expressed in
terms of the same product.
D. None of the above are correct.
11.31 Peak-load pricing involves:
A. Charging higher prices in markets with the highest price
elasticity of demand.
B. Charging higher prices in markets where product demand is
greatest.
C. Charging a higher price for a service when capacity is fully
utilized.
D. Adding a “mark-up” to the product price when product
demand is high.
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11.32 Peak-load pricing:
A. Suggests that commuter railroad riders be charged a higher fare
during peak hours than during non-peak hours.
B. Is practiced when the marginal cost of providing a service is
constant until the firm’s capacity is fully utilized.
C. Is the practice of charging a higher price when the price
elasticity of demand for a service is higher during capacity
utilization.
D. A, B, and C are correct.
E. None of the above.
11.33 Transfer pricing:
A. Is only possible when an external market exists for the output
of one division of a firm that is also used to product the output
of another division of the same firm.
B. Is only possible when an external market does not exists for the
output of one division of a firm that is also used to product the
output of another division of the same firm.
C. Is possible regardless of whether an external market exists for
the output of one division of a firm that is also used to product
the output of another division of the same firm.
D. Is only possible for firms that practice non-marginal pricing of
the output of two divisions of the same firm.
11.34 The pricing strategy in which one firm in the industry establishes
the market price for all firms in the industry is called:
A. Collusion.
B. Price leadership.
C. Predatory pricing.
D. Price fixing.
11.35 Price leadership occurs:
A. When a price change initiated by one firm in an industry is
followed by other firms in the same industry.
B. When a price increase initiated by one firm in an industry is
followed by other firms in the industry, but a price reduction is
not followed.
C. When a price decrease initiated by one firm in an industry is
followed by other firms in the industry, but a price increase is
not followed.
D. When an oligopolistic firm charges a price that attracts new
firms into the industry.
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11.36 Price leadership:
A. Is similar to prestige pricing.
B. May be dominant or barometric.
C. Is similar to psychological pricing.
D. Is a strategy that initially results in a high product price, but
which eventually declines as market demand is satisfied.
E. Can only practice by monopolies.
11.37 Barometric price leadership occurs when:
A. A price change by one firm is quickly followed by price
changes by other firms in the industry.
B. The dominant firm in the industry establishes the industry
price as a result of its profit-maximizing behavior.
C. A firm introduces a new product to extract consumer surplus
through differential pricing before other firms in the industry
develop close substitutes.
D. A firm changes the price of its product the public is
psychologically prepared for the change.
11.38 The practice of initially charging a high price for a new product,
and then lowering the price incrementally to attract additional
customers to extract consumer surplus is called:
A. Predatory pricing.
B. Price leadership.
C. Price skimming.
D. Penetration pricing.
E. Psychological pricing.
11.39 The practice of charging a price that is lower than the prevailing
market price to gain a foothold in the industry is called:
A. Predatory pricing.
B. Price leadership.
C. Price skimming.
D. Penetration pricing.
E. Psychological pricing.
11.40 The practice of charging a higher price for a product to exploit the
belief by some consumers that a higher price means better quality
is called:
A. Predatory pricing.
B. Prestige pricing.
C. Price skimming.
D. Psychological pricing.
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Shorter Problems
11.41 Suppose that a firm charges a price of $39.99 for its product instead
of $40.00. This firm is practicing:
A. Predatory pricing.
B. Prestige pricing.
C. Price skimming.
D. Psychological pricing
E. Price discounting.
SHORTER PROBLEMS
11.1
Suppose that an individual’s demand equation is:
Pi = 100 - 2Qi
Where Pi is the price of the product and Qi the quantity
demanded. Suppose that the market price of the product is $90.
A. Approximate the value of this individual’s consumer surplus
when DQ = 1.
B. What is value of consumer surplus as DQ Æ 0?
11.2
The market demand for a product is given by the equation:
Q = 16,000 - 40P
Suppose that the price of the product is $25. Calculate the value of
total consumer surplus in this market.
11.3
The Paradise Hotel and Casino in Las Vegas, Nevada is considering
offering a special three-day vacation package that includes hotel
accommodations and membership in the hotel’s health spa.
Paradise has identified three groups of vacationers that would be
interested in the package. Although Paradise is not able to identify
individual members of each group, the hotel estimates that there
are approximately 30, 40, and 45 hotel guests in Groups 1, 2, and 3,
respectively. The following table summarizes how the members of
each group value each item in the package.
Group
Hotel Accommodations
Health Spa
1
2
3
$300
$250
$220
$50
$80
$90
Suppose that the per unit cost to Paradise of providing hotel
accommodations and a health spa accommodations are $150 and
$40, respectively.
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A. If each component of the package separately, then how much
profit would Paradise earn per group?
B. How much should Paradise sell a hotel accommodations/health
spa package? At this package price, how much profit will
Paradise earn?
11.3
What is the estimated percentage mark-up over the fully-allocated
per-unit cost of production for the following price elasticities of
demand?
A. ep = -10
A. ep = -5
B. ep = -3.5
C. ep = -1
LONGER PROBLEMS
11.1
Cry Havoc Adventures estimated the following demand equation
for the average guest to its theme park.
Q = 24 - 3P
where Q represents the number or rides per guest, and P the price
per ride. The total cost of providing a ride to each is guest is
TC = 2 + Q
A. How much should Cry Havoc charge each guest per ride to
maximize profits? At this price, what is the Cry Havoc’s total
profit per customer?
B. Suppose that Cry Havoc decides to adopt a Pay-One-Price
policy. What admission fee will maximize the park’s profit?
What is the estimated average profit per guest?
11.2
Bijou Cineplex is considering a policy of offering matinee
discounts. A matinee is defined as all daily movie showings before
6:00 p.m. The total weekly cost equation per customer is:
TC = 2 + 0.5Q
where Q represents the average weekly total number of visits per
customer per week. The average weekly demand equation for
evening movies is:
QE = 2 - 0.1PE
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The average weekly demand equation for matinee movies is:
QM = 3 - 0.25PM
A. If Bijou Cineplex is a profit-maximizer, then how much should
the theater charge for matinees? How much should the Bijou
charge?
B. What is the price-elasticity of demand for matinee movies?
What is the price-elasticity of demand for evening movies?
C. What is Bijou Cineplex’s average weekly profit per customer?
11.3
The Solipsis Company produces two styles of mood rings: The
Know Thyself and the Touchy Feely. Solipsis’ market researchers
have estimated the followng demand equations for its mood rings:
QK = 1,500 - 2PK - 3QT
QT = 2,000 - 4PT - 4QK
where the subscripts K and T represent Know Thyself and Touchy
Feely, respectively. The total cost equations for the two styles of
mood rings are:
TCK = 50 + 5QK2
TCT = 80 + 4QT2
A. Suppose that Solipsis is a profit maximizer. What are the profitmaximizing prices and output levels?
B. How much profit is Solipsis earning?
11.4
The Electraline Company produces electric golf carts (QG) and
motorized wheel chairs (QW) on the same assembly line. The
demand equations for these products are:
QG = 150 - 0.01PG
QW = 250 - 0.005PW
where PG and PW are the prices of electric golf carts and motorized
wheel chairs, respectively. Electraline’s total cost equation is
TC = 5,000 + 2QG2 + QW2 + QGQW
A. Determine the profit maximizing prices and output levels for
electric golf carts and motorized wheel chairs.
B. Calculate Electraline’s total profit at the profit-maximizing
output levels.
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11.5
Pricing Practices
Digicomp Inc. produces state-of-the-art personal computers (C)
and high-definition, flat-screen monitors (M) in fixed proportions,
i.e., QC = QM. The demand equations for Digicomp computers and
monitors are:
QC = 2,498 - 0.5PC
QM = 1,600 - 0.4PM
Digicomp’s total cost equation is:
TC = 1,500 - 0.5Q2
where Q = QC + QM.
A. What are the profit-maximizing prices and output levels for
Digicomp computers and monitors.
B. At the profit-maximizing output levels, calculate Digicomp’s
profit.
11.6
Suppose that in previous problem that Digicomp Inc. decided to
produce 2 high-definition, flat-screen monitors for each state-ofthe-art personal computers (C) produced., i.e., QM = 2QC.
where Q = QC + QM.
A. What are the profit-maximizing prices and output levels for
Digicomp computers and monitors.
B. At the profit-maximizing output levels, calculate Digicomp’s
profit.
11.7
The Rock Island Commuter Express Railway (RICER) has
estimated the following demand equations for tickets:
Peak:
RP = 2,000 - 100FP
Off-Peak:
RO = 450 - 125FO
where RP and RO represent the number of peak and off-peak rides,
respectively, and FP and FO represent peak and off-peak fares,
respectively. RICER’s maximum train capacity is 1,000 riders. The
total cost equations for peak and off-peak service are:
Peak:
TCP = 2,000 + RP
Off-Peak:
TCO = 100 + QO
A. Suppose that RICER is a profit-maximizer. What fare should
RICER charge peak and off-peak riders?
B. Suppose that RICER’s capacity is 800 riders. What fare should
RICER charge peak and off peak riders?
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Solutions to Shorter Problems
183
ANSWERS TO CHAPTER 11 MULTIPLE
CHOICE QUESTIONS
11.1
11.2
11.3
11.4
11.5
11.6
11.7
11.8
11.9
11.10
11.11
11.12
11.13
11.14
11.15
11.16
11.17
11.18
11.19
11.20
11.21
C.
B.
A.
C.
B.
A.
A.
B.
E.
D.
D.
D.
B.
A.
C.
C.
E.
C.
C.
E.
E.
11.22
11.23
11.24
11.25
11.26
11.27
11.28
11.29
11.30
11.31
11.32
11.33
11.34
11.35
11.36
11.37
11.38
11.39
11.40
11.41
A.
B.
D.
D.
C.
D.
A.
B.
C.
C.
D.
C.
B.
A.
B.
A.
C.
D.
B.
D.
SOLUTIONS TO SHORTER PROBLEMS
11.1
A. CS = Si=1Æn(b0 + b1Qi)DQ - PnQn
Qn = 50 - 0.5(90) = 5
For Pn = $90 and DQ = 1 this equation becomes
CS = Si=1Æn(100 - 2Qi) - 90(5)
For values of Qi from 0 to 5 this becomes
CS = (100 - 2) + (100 - 4) + (100 - 6) + (100 - 8) + (100 - 10)
- 450 = $20
The value of consuming 5 units of this good is $470 dollars. If
the consumer pays $5 ¥ 90 = $450 for 5 units of the good. The
approximate dollar value of total benefits received is $450 + $20
= $470. The approximate value of consumer surplus is $20.
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B. CS = 0.5(b0 - Pn)Qn = 0.5(100 - 90)5 = $25
Alternatively, the value of consumer surplus can be determined
by first calculating the value of total benefits received as the
area under the demand equation between output levels 0 and 5,
and then subtracting the total value of total benefits purchased.
The total value of total benefits received may be obtained by
integrating the demand equation.
Ú50Pi(Qi)dQi = Ú50(100 - 2Qi)dQi
5
=|0[100Qi - Qi2 + c]
= [100(5) - (5)2 + c] - [100(0) - (0)2 + c]
= 500 - 25 = $475
The total value of benefits paid for is $5 ¥ 90 = $450 for 5 units
of the good. Thus, the value of consumer surplus is $475 - $450
= $25.
11.2
P = 400 - 0.025Q
where b0 = $400. For a linear demand equation, consumer surplus
may be calculated as:
CS = 0.5(b0 - P)Q
= 0.5[400 - 25][16,000 - 1,000]
= 0.5(375)(15,000) = $2,812,500
Alternatively, the value of consumer surplus can be determined by
first calculating the value of total benefits received as the area
under the demand equation between output levels 0 and 15,000,
and then subtracting the total value of total benefits purchased.
The total value of total benefits received may be obtained by
integrating the demand equation.
P(Q)dQ = Ú15,000
(400 - 0.025Q)dQ
Ú15,000
0
0
15,000
= |0 [400Q - 0.0125Q2 + c]
= [400(15,000) - 0.0125(15,000)2 + c] - [400(0)
- 0.0125(0)2 + c]
= $6,000,000 - 2,812,500 = $3,187,500
The total value of benefits paid for is $25 ¥ 15,000 = $375,000. Thus,
the value of consumer surplus is $3,187,500 - $375,000 =
$2,812,500.
11.3
A. Paradise will sell hotel three-day hotel accommodations for
$220. The reason for this is that while per unit profit is only
$220 - 150 = $70, Paradise will sell to members of each group
for a total profit of $70(115) = $8,050. By contrast, if it sells
hotel accommodations for $250, it will only sell to members of
Group 1 and 2 for a total profit of ($250 - $150)70 = $7,000.
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Solutions to Longer Problems
185
Finally, if Paradise sells hotel accommodations for $300, then its
profit will be ($300 - $150)30 = $4,500. By analogous reasoning,
if Paradise sells health spa memberships for $50, $80 or $90,
then its total profits will be ($50 - $40)(115) = $1,150, ($80 $40)(75) = $3,000, and ($90 - 40)45 = $2,250, respectively. By
pricing health spa memberships at $80, the hotel will sell spa
memberships to Groups 2 and 3, but not to Group 1.
B. If Paradise sell the three-day vacation package for $300, as
suggested in part A, then Paradise’s total profit will be ($300 $190)115 = $12,650. The reason for this is that the price of this
package is lower than the price of any other package and will
purchased by members of every group. On the other hand, a
package price of $220 + $90 = $310 will still be attractive to
members of every group since it is the least expensive of any
other package. In this case, Paradise’s total profit will be ($310
- $190)115 = $13,800. At this package price, Paradise will be
able to extract total consumer surplus from Group 3, and at
least some consumer surplus from the remaining two groups.
11.4
A. m = -1/(ep + 1) = -1/(-10 + 1) = 0.1111 or a 11.11
percent mark-up.
B. m = -1/(ep + 1) = -1/(-5 + 1) = 0.25 or a 25 percent mark-up.
C. m = -1/(ep + 1) = -1/(-2.5 + 1) = 0.6666 or a 66.67 percent
mark-up.
D. m = -1/(ep + 1) = -1/(-1 + 1) = 0.00 or a 0 percent mark-up.
SOLUTIONS TO LONGER PROBLEMS
11.1
A. P = 8 - (1/3)Q
TR = PQ = 8Q - (1/3)Q2
p = TR - TC = [8Q - (1/3)Q2] - [2 + Q] = -2 + 7Q - (1/3)Q2
dp/dQ = 7 - (2/3)Q = 0, i.e., the first-order condition for p
maximization.
d2p2/dQ22 = -2/3 < 0, i.e., the second-order condition for p
maximization is satisfied.
Solving the first-order condition for Q we obtain
Q* = 10.5 rides
P* = 8 - (1/3)(10.5) = $4.50
p* = -2 + 7(10.5) - (1/3)(10.5)2 = $34.76
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B. CS = 0.5(b0 - P)Q = 0.5[8 - (8 - (1/3)Q]Q
= 0.5(1/3)Q2
p = TR - TC = PQ + CS - TC
= [8 - (1/3)Q]Q + 0.5(1/3)Q2 - (2 + Q) = -2 + 7Q - (1/6)Q2
dp/dQ = 7 - (1/3)Q = 0, i.e., the first-order condition for p
maximization.
d2p2/dQ22 = -1/3 < 0, i.e., the second-order condition for p
maximization is satisfied.
Solving the first-order condition for Q we obtain
Q* = 21 rides
P* = 8 - (1/3)(21) = $1.00 = MC
p* = -2 + 7(21) - (1/6)(21)2 = $71.50
Admission fee = MC ¥ Q + CS
= MC ¥ Q + 0.5(1/3)Q2
= 1(21) + 0.5(1/3)(21)2 = $94.50
11.2
A. PM = 12 - 4QM
PE = 20 - 10QE
TRM = 12QM - 4QM2
TRE = 20QE - 10QE2
p = TRM + TRE - TC
= 12QM - 4QM2 + 20QE - 10QE2 - 2 - 0.5(QM + QE)
= -2 + 11.5QM - 4QM2 + 19.5QE - 10QE2
∂p/∂QM = 11.5 - 8QM = 0
QM* = 1.4375
∂p/∂QE = 19.5 - 20QE = 0
QE* = 0.975
PM* = 12 - 4(1.4375) = $6.25 for a matinee movie
PE* = 20 - 10(0.975) = $10.25 for an evening movie
B. MRM = PM(1 + 1/eM)
MRE = PE(1 + 1/eE)
eM = (∂QM/∂PM)(PM/QM) = -0.25(6.25/1.4375) = -1.09
eE = (∂QE/∂PE)(PE/QE) = -0.1(10.25/0.975) = -1.05
C. pM* = TRM - TCM = PMQM - (2 + 0.5QM)
= 6.25(1.4375) - 2 - 0.5(1.4375)
= $6.27 average profit per matinee goer
pE* = TRE - TCE = PEQE - (2 + 0.5QE)
= $10.25(0.975) - 2 - 0.5(0.975)
= $7.51 average profit per evening movie goer
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Solutions to Longer Problems
187
11.3
A. PK = 750 - 0.5QK - 1.5QT
PT = 500 - 0.25QT - QK
TRK = PKQK = 750QK - 0.5QK2 - 1.5QTQK
TRT = PTQT = 500QT - 0.25QT2 - QKQT
p = TRK(QK, QT) + TRT(QT, QK) - TCK(QK) - TCT(QT)
p = (750QK - 0.5QK2 - 1.5QTQK) + (500QT - 0.25QT2
- QKQT) - (50 + 5QK2) - (80 + 4QT2)
= -130 + 750QK + 500QT - 5.5QK2 - 4.25QT2 - 2.5QTQK
The first-order conditions for p maximization are:
∂p/∂QK = 750 - 11QK - 2.5QT = 0
∂p/∂QT = 500 - 8.5QT - 2.5QK = 0
The second-order conditions for p maximization are satisfied,
i.e.,
∂2p/∂QK2 = -11 < 0
∂2p/∂QT2 = -8.5 < 0
(∂2p/∂QK2)(∂2p/∂QT2) - (∂2p/∂QK∂QT) = (-11)(-8.5) - (2.5)2
= 82.75 > 0
Solving the first-order conditions simultaneously we obtain the
profit-maximizing output levels.
QK* = 58.74
QT* = 41.55
The profit-maximizing prices are:
PK* = 750 - 0.5(58.74) - 1.5(41.55) = $658.31
PT* = 500 - 0.25(41.55) - (58.74) = $430.87
B. p* = -130 + 750(58.74) + 500(41.55) - 5.5(58.74)2 - 4.25(41.55)2
- 2.5(41.55)(59.74)
= $32,180.17
11.4
A. PG = 15,000 - 100QG
PW = 50,000 - 200QW
TRG = 15,000QG - 100QG2
TRW = 50,000QW - 200QW2
p = TRC(QC) + TRW(QW) - TC(QC, QW)
= (15,000QG - 100QG2) + (50,000QW - 200QW2) - (5,000
+ 2QG2 + QW2 + QGQW)
= -5,000 + 15,000QG - 102QC2 + 50,000QW - 201QW2 - QCQW
The first-order conditions for p maximization are:
∂p/∂QC = 15,000 - 204QC - QW = 0
∂p/∂QW = 50,000 - 402QW - QC = 0
The second-order conditions for p maximization are satisfied,
i.e.,
∂2p/∂QC2 = -204 < 0
∂2p/∂QW2 = -402 < 0
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(∂2p/∂QC2)(∂2p/∂QW2) - (∂2p/∂QC∂QW) = (-204)(-402) - (1)2
= 82,005 > 0
Solving the first-order conditions simultaneously we obtain the
profit-maximizing output levels.
QC* = 72.92
QW* = 124.20
The profit-maximizing prices are:
PG* = 15,000 - 100(72.92) = $7,708
PW* = 50,000 - 200(124.20) = $25,160
B. p* = -5,000 + 15,000(72.92) - 102(72.92)2 + 50,000(124.2)
- 201(124.2)2 - (72.92)(124.2) = $3,656,822.40
11.5
A. PC = 4,996 - 2QC
PM = 4,000 - 2.5QM
TRC = PCQC = 4,996QC - 2QC2
TRM = PMQM = 4,000QM - 2.5QM2
TC = 1,500 - 0.5(QC + QM)2
p = TRC + TRM - TC
= (4,996QC - 2QC2) + (4,000QM - 2.5QM2)
- [1,500 - 0.5(QC + QM)2]
Since QC = QM, the total profit equation becomes
p = (4,996QC - 2QC2) + (4,000QC - 2.5QC2)
-[1,500 - 0.5(QC + QC)2]
= -1,500 + 8,996QC - 6.5QC2
dp/dQC = 8,996 - 13QC = 0, i.e., the first-order condition for p
maximization.
d2p/dQC2 = -13 < 0, i.e., the second-order condition for p
maximization is satisfied.
Solving the first-order condition for QC we obtain:
QC* = 692
Since QC and QM are produced in fixed proportions, then
QM* = 692
B. p* = -1,500 + 8,996(692) - 6.5(692)2 = $3,111,116
11.6
A. PC = 4,996 - 2QC
PM = 4,000 - 2.5QM
TRC = 4,996QC - 2QC2
TRM = 4,000QM - 2.5QM2
TC = 1,500 - 0.5(QC + QM)2
p = TRC + TRM - TC
= (4,996QC - 2QC2) + (4,000QM - 2.5QM2)
- [1,500 - 0.5(QC + QM)2]
Since QM = 2QC, the total profit equation becomes
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Solutions to Longer Problems
189
p = (4,996QC - 2QC2) + [4,000(2QC) - 2.5(2QC2)]
- [1,500 - 0.5(QC + 2QC)2]
= -1,500 + 12,996QC - 16.5QC2
dp/dQC = 12,996 - 33QC = 0, i.e., the first-order condition for p
maximization.
d2p/dQC2 = -33 < 0, i.e., the second-order condition for p
maximization is satisfied.
Solving the first-order condition for QC we obtain:
QC* = 393.82
QM* = 2QC* = 787.64
B. p* = -1,500 + 12,996(393.82) - 6.5(393.82)2 = $4,108,472.47
11.7
A. To determine the profit-maximizing number of peak rides and
peak fare, we begin by solving the demand equation for FP.
FP = 20 - 0.01RP
TRP = FPRP = 20RP - 0.01RP2
pP = TRP - TCP = (20RP - 0.01RP2) - (2,000 + RP)
= -2,000 + 19RP - 0.01RP2
dp/dRP = 19 - 0.02RP = 0, i.e., the first-order condition for p
maximization.
d2p/dRP2 = -0.02 < 0, i.e., the second-order condition for p
maximization is satisfied.
Solving the first-order condition for RP we obtain
RP* = 950
Since rider demand is less than capacity, then RICER should
charge a peak fare of
FP* = 20 - 0.01(950) = $10.50 per ride
To determine the profit-maximizing number of off-peak rides
and off-peak fare, we begin by solving the demand equation for
FO.
FO = 3.6 - 0.008RO
TRO = FORO = 3.6RO - 0.008RO2
pO = TRO - TCO = (3.6RO - 0.008RO2) - (1,000 + RO)
= -100 + 2.6RO - 0.008RO2
dp/dRO = 2.6 - 0.016RO = 0, i.e., the first-order condition for p
maximization.
d2p/dRO2 = -0.016 < 0, i.e., the second-order condition for p
maximization is satisfied.
Solving the first-order condition for RO we obtain
RO* = 162.5
Since rider demand is less than capacity, then RICER should
charge an off-peak fare of
FO* = 3.6 - 0.008(162.5) = $2.30 per ride
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B. If capacity is 800 rides, then RICER should charge a peak fare
of
FP = 20 - 0.01(800) = $12 per ride
RICER should continue to charge an off-peak fare of $1.80.