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Economics of Strategy (ECON 4550) – Maymester 2015 “Advanced Topics in Pricing: Multi-Product Pricing, Price Discrimination, Versioning, and Bundling” Reading: “Versioning: The Smart Way to Sell Information” (ECON 4550 Coursepak, Page 81) Definitions and Concepts: Price Discrimination – the practice of charging a different price for different units sold of an
essentially identical good (with the difference in price not being a result of differences in cost of production)
“First Degree Price Discrimination” or “Perfect Price Discrimination” – a practice in
which a firm charges every consumer an amount exactly equal to buyer’s reservation price for every unit sold
“Second Degree Price Discrimination” or “Menu Pricing” – a practice in which a firm
presents all consumers with different “pricing menus” and allows each consumer to choose the menu which they prefer
“Third Degree Price Discrimination” or “Segmented Pricing” – a practice under which a
firm separates consumers into different market segments, and then charges each different segment of consumers a different constant per unit price for each unit purchased
Versioning – the practice of selling different variations of a product
Examples of versioning: air travel – “first class” vs. “coach”; computer software/apps – “full feature version” vs. “bare bones version”; computer printers – IBM LaserPrinter vs. LaserPrinter E
Product Differentiation – differences in the characteristics of two similar products that result in a consumer having different values of reservation price for the two products
Vertical Product Differentiation – product differentiation for which all consumers agree on
which characteristics are “more desirable” but differ with respect to how much they value the more desirable characteristic e.g., “Kia Forte” versus “BMW 328i”
Horizontal Product Differentiation – product differentiation for which all consumers do
not agree on which characteristics are “more desirable” (due to fundamental differences in tastes/preferences across consumers) e.g., “Cherry Coca-Cola,” versus “Vanilla Coca-Cola,” versus “Coca-Cola with Lime”
Issues to keep in mind when “versioning”… 1. The “best prices” are not always obvious
as in the example above, you often can’t simply charge each “intended segment of consumers” a price equal to their “reservation price for the intended version”
2. The firm often has a choice over “how many different versions to offer” and a choice over the “specific characteristics/features of each different version” => the best choices in these matters… …depend critically upon “how costly it is to create an additional version” …may at first seem “counterintuitive” (see “free version” and “value subtracted
version” below) 3. It may make sense to offer a “free version” => particularly if…
having a “network” of users matters (e.g., Adobe gives away a free version of Acrobat Reader so that “more people use PDF files”)
benefits of your product are difficult for consumers to know before consuming/using the product (e.g., offer a free version of a video game with only “initial levels” unlocked, and then charge for “all levels”)
there is the potential for “follow-on sales” (offer a “low frills version” for free, with the intention of selling extensions, upgrades, subscription fees, and/or support services down the road => e.g., AOL used to give away the “software” to run their web services for free on a CD)
4. It may make sense to create a “value subtracted versions,” even if positive costs must be incurred to do so IBM LaserPrinter (10 pages per minute) vs. LaserPrinter E (5 pages per minute) LaserPrinter E essentially manufactured by taking a LaserPrinter and inserting
an extra chip to slow down the printing speed => it actually cost MORE to produce the LOWER QUALITY version!
if we think about “what a firm needs to do” in order to have different segments of consumers self-select different versions, these examples make sense. Intuitive example: an airline that offers “first class service” and “coach service”… fixing the quality difference between “first class” and “coach,” each consumer
could determine a “premium” he would be willing to pay for “first class” What could we do to make more people fly first class?
o lower the price of the first class ticket or raise the price of the coach ticket o increase quality of the “first class experience” (e.g., offer a wider selection
of premium snacks, serve drinks in glassware) or decrease quality of the “coach experience” (e.g., seats closer together, no choice of snack)
To get people to fly “first class,” on some level the airline has to make “coach” sufficiently unbearable for a segment of potential consumer
Bundling – the practice of selling multiple distinct products together as a package Pure Bundling – the practice of bundling, while not allowing consumers the option of
purchasing the bundled items separately (i.e., outside of the bundle) Mixed Bundling – the practice of bundling, while also allowing consumers the option of
purchasing the bundled items separately (i.e., outside of the bundle)
Multi-Product Pricing: Consider a firm producing two different products. If we assume constant marginal costs for
each product, then: bbaabbaa qcqcqpqp
If we conceptually think about maximizing profit by choosing prices, then there are now two “first order conditions for profit maximization”
1. 0
a
bb
a
aa
a
bb
a
aaa
a p
qc
p
qc
p
qp
p
qpq
p
2. 0
b
bb
b
aa
b
bbb
b
aa
b p
qc
p
qc
p
qpq
p
qp
p
Focus on “Condition 2” (by the symmetric nature of the problem, similar insights would follow from an examination of “Condition 1”)…
0
b
bb
b
aa
b
bbb
b
aa p
qc
p
qc
p
qpq
p
qp
0
bb
bbb
b
aaa q
p
qcp
p
qcp
Three distinct cases:
I. 0
b
a
p
q => “independent demands”
to maximize profit the firm must operate where:
0
bb
bbb q
p
qcp 0
bb
b
b
bbb p
q
p
p
qcp
bpbb
bb
p
cp
,
1
that is, when pricing products with “independent demands” (along with our assumption of “independent costs”) the firm simply applies the previously derived IEPR [intuitive result]
II. 0
b
a
p
q => “substitutes”
to maximize profit the firm must operate where:
0
bb
bbb q
p
qcp 0
bb
b
b
bbb p
q
p
p
qcp
bpbb
bb
p
cp
,
1
when pricing “substitutes” (again, with “independent costs”) the firm must operate at a point where…
…“price/cost markup as a percentage of price” is greater than the “absolute value of the inverse of own price elasticity of demand”
Intuitive result: you should be “more willing” to charge a higher than otherwise price on ‘Product B,’ since some of the customers you lose from doing so are ‘lost to yourself’ (since they purchase ‘Product A’)
(“Multi-Product Pricing” continued)
III. 0
b
a
p
q => “complements”
to maximize profit the firm must operate where:
0
bb
bbb q
p
qcp 0
bb
b
b
bbb p
q
p
p
qcp
bpbb
bb
p
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,
1
when pricing “complements” (again, with “independent costs”) the firm must operate where…
…“price/cost markup as a percentage of price” is less than the “absolute value of the inverse of own price elasticity of demand”
Intuitive result: you should be “less willing” to charge a higher than otherwise price on ‘Product B,’ since when you do so you are not only losing sales of ‘Product B’ but also losing sales of ‘Product A’
First Degree or Perfect Price Discrimination: Again, the “knowledge required by the seller” to actually engage in such behavior can almost
never be known => no good “real world examples” of 1st Degree P.D. However, it is still insightful to figure out “what a firm would do” if they were able to engage
in such pricing Start by recalling what a “traditional monopolist” (i.e., one who sets a common price for all
customers) would do:
For the “traditional monopolist,” in order to “sell the next unit” he must “drop price on all
units sold” => qqPqR D )()( and therefore
)()()()()( qPqPqqPqRqMR DDD (i.e., “marginal revenue is LESS THAN price”)
(Consumers’ Surplus)=(purple) (Producer’s Surplus)=(orange) (Deadweight-Loss)=(yellow)
$
0
0
quantity
MC(q)
Demand
$
0
0
quantity
MC(q)
Demand
MR(q)Q*
P*
(“First Degree Price Discrimination” continued) If the monopolist can engage in “First Degree” (or “Perfect”) Price Discrimination, then they
will sell to each consumer at a price exactly equal to his reservation price => to “sell to the next unit” they do NOT need to “drop price on all units sold”
That is, the firm charges each consumer an amount exactly equal to reservation price =>
“Marginal Revenue” is simply equal to “height of demand”
Firm would sell (QFDPD) units => extract the “entire area below the demand curve, up to the
quantity sold” as Revenue => (Revenue)=(“blue” + “yellow”) (“yellow”) = (Variable Costs) => (PS) = (“blue”) Note that (CS) = (0) => Consumers are “worse off” (compared to “traditional monopoly”) (DWL) = (0) => Social Surplus is larger (compared to “traditional monopoly”) Price Discrimination can actually increase both Producers’ Surplus and Social Welfare!
$
0
0
quantity
MC(q)
Demand = MR
QFDPD
Second Degree Price Discrimination or Menu Pricing: Firm offers “multiple pricing menus” => all menus are available to all consumers => each
consumer is free to “self select” the menu which they most prefer At least some of the “pricing menus” must be “more complicated than simply a constant per
unit price,” or else all consumers would simply choose the option with the lowest price Very often the “pricing menus” consist of “different marginal prices at different levels of
consumption Example: consider a cell phone company that offers two plans as follows…
o Plan A: $49.99 per month; 1,800 minutes for “free”; 10¢ per minute for each “additional minute”
o Plan B: $19.99 per month; 60 minutes for “free”; 20¢ per minute for each “additional minute”
Firm simply needs to know that there are “fundamental differences across consumers,” but need not be able to identify the “particular type of any specific consumer”
“Plan A” tailored toward “cell phone junkies” and “Plan B” tailored toward “emergency users” => each consumer “self-selects” the plan that “best fits their own needs”
Offering the two plans (instead of only “Plan A”) allows the seller to: Attract “emergency users” as customers Without having to sacrifice the “large revenue” which is being collected from each of the
“cell phone junkies” Other examples (which are closely related to “versioning,” which we will discuss later in this
topic): (i) first class airfare versus coach, (ii) premium cable package versus basic cable Third Degree P.D. or Segmented Pricing: Firm separates “all potential consumers” into different “segments,” and then sets a unique
constant per unit price for each segment Must be difficult (or “relatively costly”) for resale to take place across segments “Segments” are commonly “geographic markets” (in which case the consumers are
physically separated from one another) or “based upon observable (or easily verifiable) characteristics of consumers”
Segmented Pricing is probably the most common form of price discrimination Two important insights which can be gained from examining the behavior of a firm engaging
in “Segmented Pricing”... 1. Price Discrimination may allow a firm to earn a positive profit when traditional
pricing does not 2. Consumer’s Surplus can possibly be higher when firms are able to engage in price
discrimination
(“Third Degree Price Discrimination” continued) Example 1:
“Segment A” consists of 20 “wealthy consumers,” each with a reservation price of 50wbr
for a unit of the good; “Segment B” consists of 50 “poor consumers,” each with a reservation
price of 10pbr for a unit of the good
Suppose that the producer must incur “Fixed Costs” of $1,200 in order to develop the product These costs can be avoided if product is not developed These are the only production costs => both Marginal Costs and Variable Costs are zero
Suppose the firm was legally required to charge a “common price across both segments” (Price)=(50) => sell only to “Segment A” consumers => (Revenue) = (50)(20) = 1,000
=> would lead to a “profit” of (1,000)–(1,200) = –(200) => best to NOT develop product (Price)=(10) => sell to both segments => (Revenue) = (10)(70) = 700 => would lead to a
“profit” of (700)–(1,200) = –(500) => best to NOT develop product Thus, PS = CS = (Social Surplus) = (0)
If allowed to engage in “Segmented Pricing” (Price)=(50) in “Segment A” and (Price)=(10) in “Segment B” => sell to both segments
=> (Revenue) = (50)(20)+(10)(50) = 1,500 => would lead to a profit of (1,500)–(1,200) = (300) => best to develop product
Thus, PS = (Social Surplus) = (300) Price Discrimination allowed the firm to earn a positive profit when traditional pricing did
not!
(“Third Degree Price Discrimination” continued) Example 2:
“Segment A” has demand given by the linear inverse function qqP AD 100)(
“Segment B” has demand given by the linear inverse function qqPBD 20)(
Firm has constant marginal costs of 4MC If these two segments were treated as “one single market,” then
Profit maximized by setting price of $52, selling 48 units in “Segment A” and completely
ignoring “Segment B” “Segment A”: (CS) and (PS) both positive “Segment B”: (CS) and (PS) both zero
If the firm is allowed to engage in “Segmented Pricing”: set price of $52 in “Segment A” => sell 48 units in “Segment A” set price of $12 in “Segment B” => sell 8 units in “Segment B”
Outcome in “Segment A” is identical to what is was before But now, in “Segment B”: (CS) and (PS) are both positive instead of zero Thus, in this example, when the firm is able to practice “Segment Pricing” (as opposed to
restricting the firm to charging a common price across both segments): Producer’s Surplus is larger Total Consumers’ Surplus is larger Therefore, Total Social Surplus is larger (DWL is smaller) Nobody is “worse off” and some people are “strictly better off”
Price Discrimination led to an increase in Total Consumers’ Surplus!
$
0
0
quantity
MC(q)
MR(q)
Demand
‐20
‐60
20
52
Example to illustrate how “versioning” increases profit… consider a simple scenario in which a software company is potentially serving two
different types of consumers: students vs. businesses => developed a “high end version” and a “low end version” of their product
suppose consumer reservation prices are Students Business Users
low end version $60 $100 high end version $75 $190
note: vertical differentiation (both place a higher value on the “high end version” than on the “low end version”)
suppose there are an equal number of “students” and “business users” (1 of each) $10 marginal cost for producing any unit of either version
Options available to firm: offer only “low end,” offer only “high end version,” or “offer both versions”…
I. If the firm offers only the “low end version,” they can either: Sell to both types, by charging a price of $60 => profit of: (60-10)(2) = $100 Sell to only business users, by charging a price of $100 => Profit of: (100-10)(1) = $90 Of these two, the better choice is clearly to charge a price of $60 and sell to both types
II. If the firm offers only the “high end version,” they can either: Sell to both types, by charging a price of $75 => profit of: (75-10)(2) = $130 Sell to only business users, by charging a price of $190 => Profit of: (190-10)(1) = $180 Of these two, the better choice is clearly to charge a price of $190 and sell to only
“business users” Further, this (“sell only the ‘high end version’ at a price of $190”) is the best of the four
options considered thus far III. What if the firm sells both versions?
Would want the “students” to buy the “low end version” and the “business users” to buy the “high end version”…
Charge a price of $60 for the “low end version” => selling this version to the students at this price gives a profit of: (60-10)(1) = $50
What price to charge for the “high end version”? What if we charged $190 for the “high end version”?
Charge the highest price you can for the “high end version” for which the intended segment (i.e., “business users”) prefers to buy that version o “business users” can get a surplus of (100-60) = $40 from buying the “low end
version” for $60 o The most you could charge for the “high end version” and still sell to the “business
users” is $150 (so that their surplus from this purchase is (190-150) = $40)
Thus, set prices of $60 for the “low end version” and $150 for the “high end version” o “students” will buy the “low end version” o “business users” will buy the “high end version”
Profit is: (60-10)+(150-10) = 50+140 = $190 Offering two versions allowed the firm to realize greater profit
Determining consumer purchasing decisions under “Simple Monopoly Pricing” (i.e., no bundling), “Pure Bundling,” and “Mixed Bundling”…
Consider a firm selling two different goods (“good 1” and “good 2”)
Denote Consumer i ’s reservation price for the two goods be denoted by iR1 and iR2 (assume “unit demand” for each)
Each consumer is at a specific point in ),( 21 RR -space
I. Simple Monopoly Pricing – firm offers good 1 for sale at a price of 1p and offers good
2 for sale at a price of 2p
Each consumer assesses the purchase of each good separately => buy good 1if and
only if 11 pRi ; buy good 2 if and only if 22 pRi
Thus, the behavior of all consumers can be illustrated as
0
0
iR2
iR1
2R
1R
0
0
2p
1p
2R
1R
=(buy both goods)
=(buy good 2 but not good 1)
=(buy neither good)
=(buy good 1 but not good 2)
(“Bundling” continued) II. Pure Bundling – firm offers the bundle for sale at a price of bp (and consumers do not
have the option of buying the goods separately)
Consumer i ’s surplus from purchasing the bundle is bii pRR 21 => purchase
bundle if and only if this is positive (i.e., if and only if bii pRR 21 )
Thus, the behavior of all consumers can be illustrated as
III. Mixed Bundling – firm offers the bundle for sale at a price of bp , but consumers also
have the option of separately buying good 1 at a price of 1p and good 2 at a price of 2p
For anyone to ever buy “the bundle,” the firm must set 21 pppb
Consumer i ’s surplus from purchasing o the bundle is b
ii pRR 21
o only good 1 is 11 pRi
o only good 2 is 22 pRi o neither good is 0
0
0
bp
bp
2R
1R
=(buy bundle)
=(do not buy bundle)
=(buy bundle)
=(buy neither good)
0
0
2p
1p
2R
1R
bp
bp
=(buy only good 2)
=(buy only good 1)
1ppb
2ppb
(“Bundling” continued) Potential advantage of “mixed bundling” => allows the seller to: extract a very high price of 1p from those consumers with “relatively high iR1 but
relatively low iR2 ” extract a very high price of 2p from those consumers with “relatively high iR2 but
relatively low iR1 ” while still continuing to serve those consumers with “intermediate valuations for both
goods” (i.e., those people who don’t have a high value of iR1 or iR2 , but still have a high
value of ii RR 21 ) by selling them the bundle at a lower combined price of 21 pppb
Some intuition on profitability of the different options… => Recognize, “mixed bundling”
can cover both of the options of “simple monopoly” and “pure bundling”… That is: Mixed Bundling with 21 pppb would give the same outcome as Simple
Monopoly pricing with 1p and 2p (since any customer that wanted both goods would still never buy “the bundle”)
Mixed Bundling with },min{ 21 pppb would give the same outcome as Pure
Bundling with a price of bp (since buying “the bundle” is less expensive than buying
either good individually) Thus… The maximum profit from Mixed Bundling has to be at least as large as the maximum
profit from Simple Monopoly => **SMMB
The maximum profit from Mixed Bundling has to be at least as large as the maximum
profit from Pure Bundling => **PBMB
But… these inequalities need not be “strict” (i.e., the best Mixed Bundle might be the one
that replicates Simple Monopoly pricing, in which case **SMMB )
we could have either **PBSM or **
PBSM
(“Bundling” continued) Example 1: (simple example to illustrate that Pure Bundling can yield greater profit than Simple Monopoly)
firm that sells movies on DVD: (good 1) = “Movie F” (good 2) = “Movie W”
two types of consumers, with reservation prices of: Movie F Movie W sum
Type A 32 8 40 Type B 12 23 35
Bundling is most advantageous when the different types of consumers have reservation prices that are negatively correlated with each other…
suppose marginal costs of $5
straightforward to verify that the best Simple Monopoly prices are 32fp and
23wp => sell only “Movie F” to “Type A” and only “Movie W” to “Type B” =>
45* SM
what if we offer a bundle that includes a copy of each? Charge $35 and sell bundle to
each consumer => 50)2)(1035(* PB Example 2:
monopolist selling two goods suppose a constant marginal cost of $10 for each good three different consumers with reservation prices of
Good 1 Good 2 Sum Consumer A 75 15 90 Consumer B 50 60 110 Consumer C 20 80 100
Graphically illustrate reservation prices as:
2R
602 BR
751 AR
201 CR
152 AR
1R
0
501 BR
0
802 CR
(“Bundling” continued)
Optimal “Simple Monopoly” prices…? o For good 1:
201 p => 30)3)(1020(1
501 p => 80)2)(1050(1
751 p => 65)1)(1075(1 o For good 2:
152 p => 15)3)(1015(2
602 p => 100)2)(1060(2
802 p => 70)1)(1080(2
o Thus, the best Simple Monopolist choice is 501 p and 602 p =>
18010080* SM
Optimal “Pure Bundling” price…? o Price of bundle:
90bp => 210)3)(2090(
100bp => 160)2)(20100(1
110bp => 90)1)(20110(1
o Thus, the best Pure Bundle choice is 90bp => 210* PB o In this example, Pure Bundling results in greater profit than Simple Monopoly pricing
Optimal “Mixed Bundling” prices…? Recall:
best choice
best choice
best choice
1R
2R
802 CR
0
0
501 BR 201 CR
751 AR
602 BR
152 AR
(“Bundling” continued)
The optimal Mixed Bundle prices are such that:
o Thus, the best Mixed Bundle prices are 110bp for the bundle, 751 p for
good 1, and 802 p for good 2… “B” buys the bundle, “A” buys only good 1, and “C” buys only good 2
o This choice yields profit of:
225706590)1080()1075()20110(* MB
*11 75 pR A
201 CR
1R
2R
0
0
=(“B” buys the bundle)
=(“A” buys only good 1)
=(“C” buys only good 2)802*2 CRp
602 BR
152 AR
501 BR
Multiple Choice Questions: 1. ________________ refers to the general practice of charging a different price for
different units sold of an essentially identical good. A. Versioning B. Price Discrimination C. Bundling D. Product Differentiation 2. Alpha Industries sells two goods: “good x” and “good y.” The cross price elasticity of
demand for “good x” with respect to the price of “good y” and the cross price elasticity of demand for “good y” with respect to the price of “good x” are both positive. It follows that when maximizing profit, this firm should be operating where
A. apaa
aa
p
cp
,
1
.
B. apaa
aa
p
cp
,
1
.
C. apaa
aa
p
cp
,
1
.
D. apaaa cp ,
3. “Vanilla Coca-Cola,” “Cherry Coca-Cola,” and “Coca-Cola with Lime” are examples of
goods that are A. vertically differentiated. B. horizontally differentiated. C. sold by a firm with no market power. D. None of the above answers are correct. 4. Consider a firm that sells two products (“product a” and “product b”) with interdependent
demands. The profit of the firm is generally given by the function Fqcqcpqpq bbaabbaa . Inverse demand for “product a” and “product b”
are respectively given by aba qqp 43
4150 and bab qqp 2
130 . Further,
11ac and 4bc . As a result, when the firm maximizes profit by choosing quantities
of output, they obtain the two first order conditions: 01150 23
43 ab qq and
04230 43 ba qq . Which of the following levels of output maximize profit?
A. 0aq and 13bq .
B. 26aq and 0bq .
C. 26aq and 13bq .
D. 40aq and 28bq .
5. John has reservation prices for “good 1” and “good 2” of 461 R and 182 R . He has
the option to buy “good 1” at a price of 401 p , buy “good 2” at a price of 202 p , or
a bundle consisting of both “good 1” an “good 2” at a price of 55bp . Given this
options, John should choose to A. purchase only “good 1.” B. purchase only “good 2.” C. purchase the bundle consisting of both “good 1” and “good 2.”
D. None of the above answers are necessarily correct (since additional information is needed in order to determine his optimal purchasing decision).
6. Edna sells sunglasses. She offers customers a 10% discount if they show her a valid
student I.D. at the time of purchase. This behavior by Edna is an example of A. First Degree Price Discrimination (or Perfect Price Discrimination).
B. Second Degree Price Discrimination (or Menu Pricing) C. Third Degree Price Discrimination (or Segmented Pricing). D. Fifth Degree Price Discrimination (or Walter Murphy Pricing).
7. Consider a firm that sells two different products. If this firm engages in “Pure Bundling,”
then each consumer A. only has the option of purchasing the two goods separately.
B. has the option of either purchasing the two goods separately or purchasing together (as a package).
C. only has the option of purchasing the two goods together (as a package). D. must purchase a positive quantity of both goods (even if doing so gives him a
negative Consumer’s Surplus). 8. Which of the following statements regarding the profitability of different types of
bundling is correct? A. “The maximum profit from Mixed Bundling must always be at least as large as
the maximum profit from Simple Monopoly pricing.” B. “The maximum profit from Pure Bundling must always be at least as large as the
maximum profit from Simple Monopoly Pricing.” C. “The maximum profit from Pure Bundling must always be exactly equal to the
maximum profit from Mixed Bundling.” D. More than one (perhaps all) of the above answers is correct. 9. Consider a firm producing two goods that are complements to each other. When
maximizing profit, they should price in a way so that A. for each good “price/cost markup as a percentage of price” is less than “the
absolute value of the inverse of own price elasticity of demand.” B. for each good “price/cost markup as a percentage of price” is greater than “the
absolute value of the inverse of own price elasticity of demand.” C. for each good “price/cost markup as a percentage of price” is equal to “the
absolute value of the inverse of own price elasticity of demand.” D. “price/cost markup as a percentage of price” is equal to one for each good.
10. Consider a monopolist selling a good for which inverse demand is given by qqPD 20
140)( and cost of production are 750,98)( qqC . If this seller is able
to engage in “Perfect Price Discrimination,” then she would A. sell 640 units of output. B. charge a per unit price of $24 for every unit sold. C. choose to produce less than the efficient quantity of output. D. More than one (perhaps all) of the above answers is correct. Problem Solving or Short Answer Questions: 1. Justin owns a company called “Old Skool Web,” which offers dial-up Internet service in
the Topeka, KS metro area. “Old Skool Web” offers three different monthly service plans as follows:
Fixed Fee “Free” Minutes Price for Additional Minutes Plan A $10 90 20¢ Plan B $20 500 10¢ Plan C $40 unlimited (not applicable)
1A. If Kendra expects to use the Internet for a total of exactly 120 minutes per month, which of these three plans would be best for her?
1B. If Amanda expects to use the Internet for a total of exactly 600 minutes per month, which of these three plans would be best for her?
1C. If David expects to use the Internet for a total of exactly 800 minutes per month, which of these three plans would be best for him?
1D. Determine an exact range of monthly usage for which consumer expenditures are lowest under “Plan B.”
2. Consider a firm facing inverse demand of qqPD 500120)( . Suppose that the
production costs of this firm are 000,352)( 2000,11 qqqC .
2A. Suppose that this firm must charge all consumers a common price. Determine the resulting: profit maximizing quantity of output, profit maximizing price, value of Deadweight-Loss, value of Total Consumers’ Surplus, value of Producer’s Surplus, and value of Profit.
2B. Suppose that this firm can engage in Perfect Price Discrimination. Determine the resulting: profit maximizing quantity of output, value of Deadweight-Loss, value of Total Consumers’ Surplus, value of Producer’s Surplus, and value of Profit.
2C. How do the values of Deadweight-Loss, Total Consumers’ Surplus, Producer’s Surplus, and Profit compare to each other across the two scenarios of “Simple Monopoly Pricing” and “First Degree Price Discrimination”?
3. Consider a firm selling two goods in a market in which there are three different types of consumers. The table below provides a summary of the reservation price of each type of consumer for each good (as well as notation denoting the number of each type of consumer in the market).
Consumer Type
Reservation Price for Good X
Reservation Pricefor Good Y
Number of Consumers
Type A 165 45 A Type B 150 50 B Type C 135 75 C
The marginal cost of producing each unit of “Good X” and each unit of “Good Y” is a constant $10, while Fixed Costs are equal to zero. Consider three different scenarios (in regards to the size of each market segment) as follows:
Scenario 1 Scenario 2 Scenario 3 A 25 425 25 B 50 25 425 C 425 50 50
3A. Suppose that the seller is restricted to “Simple Monopoly Pricing” (i.e., choosing
xp and yp , in order to sell each good separately). Determine the profit
maximizing price for each good and the resulting profit of the seller in each of the three different scenarios.
3B. Suppose that the seller is restricted to “Pure Bundling” (i.e., selling the two goods
together as a package, for a price of bp ). Determine a general condition in terms
of A , B , and C which specifies when the seller would want to sell the bundle to all three consumer types (versus selling the bundle to only types A and C ). In
which of the three scenarios will the seller set a price of bp for which all three
consumer types purchase the bundle? 3C. Does “Pure Bundling” lead to greater profit than “Simple Monopoly Pricing”?
Explain. 4. “Sterling Microsystems” has developed a new piece of software that is of value to both
Professional Financial Analysts and Finance Students. The potential market for this software consists of N Professionals and N10 Students. Each Professional user has a value of $5,675 for the software, while each Student user has a value of only $275. The marginal cost of producing each unit of the product is $5 (all other production costs are sunk costs). 4A. What price should “Sterling Microsystems” charge for this program? When
setting this price, which segment of consumers purchase the product and how much profit is “Sterling Microsystems” able to earn?
4B. “Sterling Microsystems” recently hired two exceedingly intelligent MBAs named Heidi and Spencer. Spencer is in charge of new product development, and Heidi is in charge of market research and product pricing. Heidi thinks that “Sterling Microsystems” could increase its profit by offering two versions of their software
– a “Deluxe Edition” (with all of the features available in the original version, targeted at Professional users) and a “Basic Edition” (with fewer features, targeted at Student users). Spencer has proposed a subset of features that could be included in such a “value subtracted version” (which would also have a marginal cost of $5), and Heidi has estimated the following reservation prices for each version for each type of user.
Reservation Price for Deluxe Edition
Reservation Price for Basic Edition
Number of each type of customer
Student Users $275 $215 10N Professional Users $5,675 $2,270 N
In order to sell the “Deluxe Edition” to Professionals and the “Basic Edition” to Students, what price should Heidi set for each version?
4C. When setting the prices determined in part (4B), how much profit is “Sterling Microsystems” able to earn? Is this profit greater than what they earned before introducing the second version?
4D. Now suppose that in order to introduce this second version, “Sterling Microsystems” would have to incur product development costs of $630,000 (these costs must be incurred if the new version is developed, but can be completely avoided if the new version is not developed). Should “Sterling Microsystems” develop the second version? Explain.
5. Consider a firm selling a product in two different markets, “Market A” and “Market B.”
The firm has production costs of 000,8CQ (where Q denotes the total quantity of
output produced, and C is a positive constant). Demand in “Market A’ is
ppDA 20000,2)( , while demand in “Market B’ is ppDB 80800,4)( . 5A. Derive and graphically illustrate a function which specifies Total Demand across
the two market segments. 5B. Derive the “Inverse Demand Function” which corresponds to the Demand
Function which you derived in part (5A). 5C. Supposing that the firm must charge the same constant per unit price across the
two markets, determine an expression for Marginal Revenue of the firm. 5D. Suppose that 48C . If the firm must charge the same constant per unit price
across the two markets, what price will the firm set, how much output will they sell, and how much profit will they earn?
5E. Continuing to suppose 48C , if the firm can engage in Third Degree Price Discrimination, what price will they set and how much output will they sell in each market? How much profit is the firm able to earn? How does this outcome compare to that from part (5D) in terms of realized Consumer Surplus in each market? Explain.
5F. Suppose that 28C . If the firm must charge the same constant per unit price across the two markets, what price will the firm set, how much output will they sell, and how much profit will they earn?
5G. Continuing to suppose 28C , if the firm can engage in Third Degree Price Discrimination, what price will they set and how much output will they sell in
each market? How much profit is the firm able to earn? How does this outcome compare to that from part (5F) in terms of realized Consumer Surplus in each market? Explain.
6. Consider a firm selling two goods in a market in which there are three different types of
consumers. The table below provides a summary of the reservation price of each type of consumer for each good, along with a specification of the number of each type of consumer in the market.
Consumer Type
Reservation Price for Good X
Reservation Pricefor Good Y
Number of Consumers
Type A 30 90 100 Type B 50 60 700 Type C 80 20 200
The marginal cost of producing each unit of “Good X” and each unit of “Good Y” is a constant $5, while Fixed Costs are equal to $50,000. 6A. Suppose that the seller is restricted to “Simple Monopoly Pricing” (i.e., choosing
xp and yp , in order to sell each good separately). Determine the profit
maximizing price for each good and the resulting profit of the seller. 6B. Suppose that the seller is restricted to “Pure Bundling” (i.e., selling the two goods
together as a package, for a price of bp ). Determine the optimal value of bp and
the resulting profit of the seller. 6C. Suppose that the seller is able to engage in “Mixed Bundling” (i.e., set a price of
bp for purchasing the goods as a package, in addition to prices of xp and yp
for purchasing the goods separately). If the seller wishes to have type A consumers purchase only “Good Y,” type B consumers purchase the bundle, and type C consumers purchase only “Good X,” what prices should he set? At these prices, how much profit does the seller earn?
6D. How do the maximum profits from “Simple Monopoly Pricing,” “Pure Bundling,” and “Mixed Bundling” compare to each other? Explain.
Answers to Multiple Choice Questions:
1. B 2. A 3. B 4. D 5. C 6. C 7. C 8. A 9. A 10. A
Answers to Problem Solving or Short Answer Questions: 1A. Let m denote minutes used per month. In general, consumer expenditure under “Plan
A” is
90)90)(2(.10
9010
mifm
mifEA . Similarly, expenditure under “Plan B”
is
500)500)(1(.20
50020
mifm
mifEB . Finally, expenditure under “Plan C” is
simply 40CE (for all 0m ). For 120m : 16)90120)(2(.10 AE ,
20BE , and 40CE . Thus, “Plan A” is best for this consumer.
1B. For 600m : 112)90600)(2(.10 AE , 30)500600)(1(.20 BE ,
and 40CE . Thus, “Plan B” is best for this consumer.
1C. For 800m : 152)90800)(2(.10 AE , 50)500800)(1(.20 BE ,
and 40CE . Thus, “Plan C” is best for this consumer.
1D. For “very low values of m ,” consumer expenditures are lowest under “Plan A.” For very “very high values of m ,” consumer expenditures are lowest under “Plan C.” For “intermediate values of m ,” consumer expenditures are lowest under “Plan B.” The specific value of m above which expenditures are lower under “Plan B” than under “Plan A” is the value for which:
BA EE
20)90)(2(.10 m
2.10)90( m
140m The specific value of m below which expenditures are lower under “Plan B” than under “Plan C” is the value for which:
CB EE
40)500)(1(.20 m
1.20)500( m
700m Thus, consumer expenditures are lowest under “Plan B” so long as 700140 m . 2A. If this firm must charge all consumers the same price, then the revenue of the firm is
2500120)( qqqR . It follows that marginal revenue is qqMR 250
120)( .
Recognize that marginal costs are 2)( 5001 qqMC . Thus, the profit maximizing
quantity must satisfy: 220 500
12501 qq
q500318
000,3* q
Further, the firm will charge a price of 1420*)(* 500000,3 qPp D to each customer.
Graphically,
Note that the efficient quantity of trade is the level at which 2)( 5001 qqMC are
equal to qqPD 500120)( . That is, 500,4Eq . From here it follows that:
Deadweight-Loss is equal to 500,4)814)(000,3500,4(21 DWL
Total Consumers’ Surplus is equal to 000,9)1420)(000,3(21 CS
Producer’s Surplus is equal to 000,27)814)(000,3()28)(000,3(21 PS
and Profit is equal to 000,8000,35000,27 FPS 2B. If instead the firm can engage in Perfect Price Discrimination, then they are able to sell
each unit for an amount exactly equal to buyer’s reservation price. Profit is maximized by selling up to the level at which buyer’s reservation price is equal to marginal costs of production. For the given functions, this is 4,500 units of output. The seller is able to extract the entire area below the demand curve as revenue. Thus,
Deadweight-Loss is equal to 0DWL Total Consumers’ Surplus is equal to 0CS
$
0
0
quantity
MC(q)
Demand
MR(q)q*=3,000
p*=14
MC(3,000)=8
=PS
=CS
=DWL
Producer’s Surplus is equal to 500,40)220)(500,4(21 PS
and Profit is equal to 500,5000,35500,40 FPS 2C. Compared to Simple Monopoly Pricing, under Perfect Price Discrimination: Deadweight-
Loss is smaller; Total Consumers’ Surplus is smaller; Producer’s Surplus is larger; and Profit is larger. Further note that Profit is not positive under Simple Monopoly Pricing but is positive under Perfect Price Discrimination.
3A. For Simple Monopoly Pricing, the firm essentially has three different prices to consider
for each good. This is because once the firm decides which types of consumers to sell to, they will want to charge the highest price possible in order to sell to these chosen consumers. For example, for “Good X,” the firm will want to charge a price of either $165 (and sell only to Type A), $150 (and sell to both Type A and Type B, but not Type C), or $135 (and sell to all three types). Focusing on Scenario 1, these three different prices would yield profits from “Good X” of
135xp 500,62)500(125)4255025)(10135( x
150xp 500,10)75(140)5025)(10150( x
165xp 875,3)25(155)25)(10165( x
Clearly the best choice is 135xp , which leads to a profit of 500,62x . By
similar logic, it can be shown that in each of the two markets, across all three scenarios, the best choice of prices and resulting profits are:
xp yp x y yx
Scenario 1 135 75 62,500 27,625 90,125 Scenario 2 165 45 65,875 17,500 83,375 Scenario 3 150 50 63,000 19,000 82,000
3B. If restricted to “Pure Bundling,” the maximum price that the seller could set and still sell
to all three types of consumers is 200bp . If he instead wanted to sell to only types
A and C , he could charge a price up to 210bp . The former price leads to profit of
)(180))(20200( CBACBAabcPB , while the latter price leads to profit
of )(190))(20210( CACAacPB . Thus, selling to all three consumer types
is the better choice if and only if acPB
abcPB
)(190)(180 CACBA
)(10180 CAB
BCA 18)( Applying this condition to each of the three scenarios, if restricted to Pure Bundling, the
seller would sell to all consumer types in Scenarios 1 and 3 (but not in Scenario 2).
3C. Pure Bundling results in a profit of 000,90)500(180)(180 CBAabcPB in
Scenario 1, which is not larger than the profit of $90,125 from Simple Monopoly Pricing.
Pure Bundling results in a profit of 250,90)475(190)(190 CAacPB in
Scenario 2, which is larger than the profit of $83,375 from Simple Monopoly Pricing.
Pure Bundling results in a profit of 000,90)500(180)(180 CBAabcPB in
Scenario 1, which is larger than the profit of $82,000 from Simple Monopoly Pricing. 4A. If they serve both Professional users and Student users, then they would want to charge a
price of $275. If instead they choose to only serve Professional users, then they would want to charge a price of $5,675. Setting a price of $275 gives a profit of
NNNN 970,2)11)(270()10)(5275( While setting a price of $5,675 gives a profit of
NN 670,5))(5675,5( .
Since NN 970,2670,5 for all 0N , profit is maximized by selling to only the Professional users.
4B. When selling the “Basic Edition” to Student users, Heidi would want to charge a price of $215, in order to extract all surplus from these consumers. But note, at this price, a Professional user could realize a surplus of 055,2215270,2 by purchasing the Basic Edition. Therefore, in order to sell the Deluxe Edition to professional users, Heidi must set a price for the Deluxe Edition for which a Professional user would realize a surplus of at least 2,055 from purchasing this version. That is,
055,2675,5 Dp
620,3Dp
Thus, the profit maximizing prices are 215Bp for the Basic Edition and 620,3Dp for the Deluxe Edition.
4C. When charging 215Bp and 620,3Dp , Professional users choose to purchase the Deluxe Edition and Student users choose to purchase the Basic edition. As a result, the firm earns profit of
NNNNN 715,5615,3100,2))(5620,3()10)(5215(
Since NN 670,5715,5 for all 0N , profit is greater from offering the two different versions.
4D. The difference in profit from offering the two versions versus offering only one version (and optimally selling only to Professional users) is NNN 45670,5715,5 . If they need to incur product development costs of $630,000 to create this second version, then they would only want to do so if 000,63045 N . This condition can be simplified to
000,14N . Thus, when facing these up front product development costs, they should
create and sell the second version if and only if 000,14N . 5A. Total Demand (across the two market segments) can be obtained by summing the
quantities demanded in each market at every possible price. In terms of the given
demand functions, this amount to )()()( pDpDpD BA . The given demand
functions are ppDA 20000,2)( and ppDB 80800,4)( . However, since quantity demanded is never negative, what we technically have in “Market A” is
1000
100020000,2)(
pfor
pforppDA
and what we technically have in “Market B” is
600
60080800,4)(
pfor
pforppDB
Thus, Total Demand is
1000
1006020000,2
600100800,6
)()()(
pfor
pforp
pforp
pDpDpD BA
Graphically: 5B. The corresponding Inverse Demand Function is
800,60
800,6800)01(.68
8000)05(.100
)(
qfor
qforq
qforq
qPD
$
0
0
quantity
Demand
100
60
800 6,800
5C. If the firm must charge the same price for all units sold across both market segments, then Marginal Revenue is
800,60
800,6800)02(.68
8000)1(.100
)(
qfor
qforq
qforq
qMR
Graphically 5D. First note that for 000,8CQ , marginal costs are equal to C . As can be seen from the
graph above, for any value of 5820 C , there will be two intersection of marginal revenue and marginal cost (and further, at each intersection marginal revenue intersects marginal cost “from above,” so that each intersection is identifying a “local maximum”
for profit). With 48C , these two intersections occur at 520aq and 000,1abq .
The corresponding prices are 74ap and 58abp , and the corresponding levels of
profit are 520,5000,8520,13 a and 000,2000,8000,10 ab . Thus,
the best choice for the firm is to charge a price of 74* p , sell 520* q units of
output (all in “Market A”), and realize a profit of 520,5* . 5E. If instead the firm can set different prices in each market, they would choose to charge
74Ap in and sell 520
Aq units (in order to earn a Producer’s Surplus of
520,13APS ) in “Market A” and would choose to charge 54
Bp in and sell
480Bq units (in order to earn a Producer’s Surplus of 880,2
BPS ) in “Market B.”
This results in profit of 400,8000,8880,2520,13* . Note that the firm is not altering its behavior in “Market A,” so that the realized Consumers’ Surplus in this market is exactly what it was in part (5D). However, the firm now chooses to sell some output in “Market B” (whereas previously “Market B” was completely ignored). Thus,
$
0
0
quantity
Demand
100
60
800 6,800
58
20
6,800
Marginal Revenue
Consumers’ Surplus in “Market B” is now positive (more precisely, it takes on an exact
value of 440,1)5460)(480(21* BCS ) instead of being equal to zero.
5F. Recall, for any value of 5820 C , there will be two intersection of marginal revenue and marginal cost (and further, at each intersection marginal revenue intersects marginal cost “from above,” so that each intersection is identifying a “local maximum” for profit).
With 28C , these two intersections occur at 720aq and 000,2abq . The
corresponding prices are 64ap and 48abp , and the corresponding levels of profit
are 920,17000,8920,25 a and 000,32000,8000,40 ab . Thus, the
best choice for the firm is to charge a price of 48* p , sell 000,2* q units of output
( 040,1* Aq in “Market A” and 960* Bq in “Market B”), and realize a profit of
000,32* . 5G. If instead the firm can set different prices in each market, they would choose to charge
64Ap in and sell 720
Aq units (in order to earn a Producer’s Surplus of
920,25APS ) in “Market A” and would choose to charge 44
Bp in and sell
280,1Bq units (in order to earn a Producer’s Surplus of 480,20
BPS ) in “Market
B.” This results in profit of 400,38000,8480,20920,25* . Note that when engaging in Third Degree Price Discrimination (in comparison to Simple Monopoly Pricing), the firm now chooses to charge a higher price and seller a lower quantity in “Market A” and chooses to charge a lower price and sell a higher quantity in “Market B.” Thus, realized Consumers’ Surplus will be lower in “Market A” and higher in “Market B” than under Simple Monopoly Pricing.
6A. When restricted to Simple Monopoly Pricing, for “Good X” the firm will want to charge
either: 30xp 000,25)000,1)(530( xPS
50xp 500,40)900)(550( xPS
80xp 000,15)200)(580( xPS
The best choice is 50xp (at which the firm sells to types B and C, but not type A),
which results in a Producer’s Surplus of 500,40xPS from “Good X.” Similarly, for
“Good Y” the firm will want to charge either: 20yp 000,15)000,1)(520( yPS
60yp 000,44)800)(560( yPS
90yp 500,8)100)(590( yPS
The best choice is 60yp (at which the firm sells to types A and B, but not type C),
which results in a Producer’s Surplus of 000,44yPS from “Good Y.” Together,
these prices yield profit of 500,34000,50000,44500,40 M .
6B. If the seller is only able to sell by way of Pure Bundling, then he would want to set either: 100bp 000,90)000,1)(10100( PS
110bp 000,80)800)(10110( PS
120bp 000,11)100)(10120( PS
The best of these options is to charge 100bp and sell the bundle to all three of the
different customer types. This leads to a profit of 000,40000,50000,90 PB .
6C. With Mixed Bundling, if the seller wants to set prices for which type A consumers purchase only “Good Y,” type B consumers purchase the bundle, and type C
consumers purchase only “Good X,” the best prices to set are 80xp , 90yp , and
110bp . These prices result in profit of
000,50)200)(580()100)(590()700)(10110( MB
000,50)200)(75()100)(85()700)(100(
000,50000,15500,8000,70
500,43 . 6D. Comparing the profits from the answers to parts (6A), (6B), and (6C), we see that in this
case, MBPBM . That is, Mixed Bundling results in a strictly greater profit than Pure Bundling, which in turn results in a strictly greater profit than Simple Monopoly Pricing.
