Applied Microeconomics

1
Applied Microeconomics

• Price Discrimination II

2
Outline

• First-degree price discrimination Capturing all
  of consumer surplus
• Third-degree price discrimination
  Discrimination by group membership
• Second-degree price discrimination
  Discrimination by self-selection
• Bundling

3
Readings

• Kreps Chapter 7
• Perloff Chapter 12
• Zandt Chapter 7
• Tirole Chapter 3 (in course package)
• Indiscriminate pricing (1998)
• A Bundle of Trouble (2001)

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Screening

• If the firm knows that buyers have different
  valuations, but cannot tell who has which
  preferences, it can still price discriminate by
  offering bundles that will appeal to different
  customers
• This is referred to as second-degree price
  discrimination, or screening
• The firm must avoid personal arbitrage that a
  consumer to whom a given bundle is directed
  chooses a different bundle

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Example

• Suppose a firm is producing a good at zero cost
  and that it has two potential clients, but cannot
  tell one from the other
• Al has reservation prices 2.5E for the first unit
  of the good, and 1E for the second unit
• Beth has reservation prices 2E for the first unit
  of the good and 0E for the second
• The profit maximizing uniform price is __, giving
  a profit of __

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Example Non-Linear Pricing

• However, the firm can do better by offering two
  different packages
• 2 units of the good for __
• 1 unit of the good for __
• With these prices Al will weakly prefer the first
  package (to one or two of the second) and Beth
  the second
• Profit in this case is __

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Example Two-Part Tariff

• Alternatively, the firm could offer a two-part
  tariff
• A fixed fee of __ for buying from the firm
• A per unit price of __
• With these price, Al will weakly prefer to
  consume two units and Beth will weakly prefer to
  consume one unit
• Profit in this case is __

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Screening

• Different types of screening
• Block pricing
• Quantity discounts
• Two-part pricing
• Tie-in sales
• Nonlinear pricing
• Quality discrimination

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Screening

• Examples
• Telephone, gas, electricity quantity discounts
• Cell phone contracts
• Amusement parks
• Taxi fares
• Computer chips

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Basic Model

• Suppose a firm has constant unit cost c
• Consumers are of two types, a and b (where 0ltaltb)
• Consumers of type a have quasi-linear demand
  function Ua(x,T)av(x)-T, where x is quantity
  consumed of the good, and T is the cost of
  consumption
• Consumers of type b have quasi-linear demand
  function Ub(x,T)bv(x)-T
• We assume that v(0)0, v(x)gt0, v(x)lt0

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Indifference Curves
Tariff
bs indifference curves
as indifference curves
0
x
0
12
Basic Model

• The demand functions of the two types are given
  by Da(p) and Db(p) where Da(p)gtDb(p)
• The consumer surpluses at price p are Sa(p)p?8
  Da(s)ds and Sb(p)p?8 Db(s)ds respectively
• This means that Sa(p)-Da(p)
• The share of a in the population is q and the
  share of b is 1-q
• Aggregate demand is D(p)qDa(p)(1-q)Db(p)

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Perfect Price Discrimination

• If the firm can practice perfect price
  discrimination, it charges each consumer j the
  price pc and a fixed fee ASj(c)
• This makes each consumer indifferent between
  consuming or not and implies a profit of
  p1qSa(c)(1-q)Sb(c)

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Uniform Pricing

• Suppose there is full arbitrage between consumers
  so the firm is forced to charge a uniform price
• The firm solves maxp (p-c)D(p) with first-order
  condition (pu-c)D(pu)D(pu)0
• This gives a profit of pu

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Second Degree P.D. Two-Part Tariff

• Suppose the firm cannot separate the two types,
  but uses a two-part tariff Apx to screen them
• Suppose also that it is optimal to serve both
  types of consumers (this may not be true!)
• The highest fee that both types will pay at price
  p is ASa(p) (since altb)
• This means that the firm solves maxp Sa(p)
  (p-c)D(p) with first-order condition
  -Da(ps)(ps-c)D(ps)D(ps)0

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Illustration of Two-Part Tariff
0
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Example Two-Part Tariff

• Suppose there are equal shares of two types of
  consumers a has demand function Da(p)1-2p and b
  has demand function Db(p)2-4p
• As consumer surplus at price p is given by
  (1-2p)(0.5-p)/2(0.5-p)2
• The firm has unit cost c1/3 and solves maxp
  (0.5-p)2(3-6p)(p-1/3)/2
• First-order condition2pS-13/2-3pS-3pS1-4pS3/2
  0, pS3/8
• A(4/8-3/8)21/64, profit 1/32 (profit serving
  only market b is 1/36)

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Prices and Profits

• The price under perfect price discrimination is
  equal to marginal cost, which is lower than the
  price under screening, which in turn is lower
  than under uniform pricing cltpsltpm
• Since p1 gives maximal profit and any uniform
  price can be accomplished with a two-part tariff,
  it also holds that p1pS pm

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Tie-In Sales

• Customers who buy one product from a firm are
  required to make all their purchases of another
  product from the firm
• Examples
• Xerox copiers and paper or service
• Printers and toners
• The previous model extends to this setting

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Tie-In Sales

• Suppose two types of consumers demands one unit
  of the basic good (copier) and x units of the
  complementary good (paper)
• Suppose also that the market for the basic good
  is a monopoly, but the market for the
  complementary good is competitive
• The firm produces the basic good at cost C and
  the complementary good at unit cost c

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Tie-In Sales

• Given that both types are served, the magnitude
  of C is irrelevant and the two-part tariff
  solution is achieved
• The price of the complementary good is pSgtc
• The price of the basic good is lower than if sold
  separately with perfect price discrimination
  ASa(pS)ltSa(c)

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Two-Part Tariffs with More than Two Types

• A higher fixed fee implies
• Higher revenue per customer
• Less customers willing to pay
• Setting an optimal two-part tariff means trading
  off the two forces
• With knowledge about demand functions, this can
  easily be done using numerical methods

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Non-Linear Pricing

• However, we can achieve an even better outcome
  than the two-part tariff if we use non-linear
  pricing
• This means offering two different price-quantity
  packages (Ta,xa) and (Tb,xb) directed at the two
  consumers

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Non-Linear Pricing
Tariff
G
A
pq
bs
indifference
curves
F
Isoprofit
lines
E
as
indifference
curve
A
x
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Non-Linear Pricing Problem

• This implies solving the following problem
  maxTa,Tb,xa,xb0 qTa(1-q)Tb-c(qxa(1-q)xb)
  subject to the following four constraints
• av(xa)-Ta av(xb)-Tb (Incentive Constraint a)
• bv(xb)-Tb bv(xa)-Ta (Incentive Constraint b)
• av(xa)-Ta 0 (Participation Constraint a)
• bv(xb)-Tb 0 (Participation Constraint b)
• Note that PCb is implied by ICb and Pca

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Non-Linear Pricing Solution

• Low-demand consumer derive no net surplus (their
  p.p. holds with equality), while high demand
  consumers derive a positive surplus
• The second binding constraint is to prevent
  high-demand consumers from buying the low demand
  consumers bundle
• High demand consumer gets the socially optimal
  optimal quantity, xbDb(c), but low demand
  consumers less, xaltDa(c)
• If q small and b large compared to a could be
  more profitable serving only type B

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Quality Discrimination

• Screening can also be done among consumers with
  different tastes for quality by offering product
  with different price/quality characteristics
• Can be modeled using the same technique as for
  non-linear pricing, interpreting x as a measure
  of quality and c as the cost of quality
• Examples
• Train and airline classes
• Insurance companies
• Computer chips

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Bundling

• Selling different products together in packages
• Makes sense when
• Heterogeneous demands some consumers have
  relatively high reservation price for good 1,
  others for good 2
• Negative correlation between demands if a
  consumer has a high reservation price for good 1,
  he is likely to have a low reservation price for
  good 2
• The firm cannot price discriminate perfectly

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Bundling

• Examples
• Automobile packages
• Vacation travel
• Cable TV
• Restaurants
• Financial bundling capital good financing
• Legal impediments
• Microsoft Internet Explorer and Windows
• Honeywell/GE (aircraft engines/financing)

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Setting

• Suppose a firm is selling two goods, 1 and 2
• A consumers reservation prices for the two goods
  are given by R1 and R2
• Without bundling, the firm sets the prices of the
  goods P1 and P2
• With bundling, the firm sets a price PB per
  bundle containing one unit of each good

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Consumption Without Bundling
For any prices P1 and P2, consumers can be
categorized according to whether they have higher
reservation prices R1 and R2 for one of the
goods, none of the goods, or both of the goods
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Pure Bundling

• With pure bundling only bundles, and not separate
  goods, are offered
• Consumers will buy the good if and only if R1
  R2 PB

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Mixed Bundling

• With mixed bundling both bundles and separate
  goods are offered
• In this way, some consumers who would not have
  bought anything under pure bundling will buy
• The drawback is that some consumers who would
  have bought the bundle now only buys one of the
  goods
• Profit is at least as high as under pure bundling

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Example

• Suppose the firm produces two goods, 1, and 2, at
  unit cost c0.5
• There are two consumers
• James has reservation prices R1J1 and R2J2
• Karen has reservation prices R1K2 and R2K1

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Example No Bundling or Pure Bundling

• Without bundling the firm should set P12 and
  P22 and sell one unit of each good
• This would give a profit of 22-20.53
• With pure bundling, the firm could charge PB3
  for each bundle
• This would give profit 23-40.54

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Example Mixed Bundling

• Suppose there are two more consumers
• Al has reservation prices R1A2 and R2A0.5 and
  Beth has reservation prices R1B0.5 and R2B2
• Without bundling the optimal prices would be
  P12, P22 giving a profit of 42-40.56
• With pure bundling neither Al nor Beth would buy
  the bundle at price PB3

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Example Mixed Bundling

• However, if the goods were also sold separately
  at prices P12 and P22, then Al would buy good
  1, Beth would buy good 2, and James and Karen
  would buy the bundle
• This would give a profit of 2322-60.57

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Conclusion

• Second-degree price discrimination or screening
  is a way of price discrimination when the firm
  cannot distinguish between different types of
  consumers
• Two-part tariffs is a simple example that
  generally gives a per unit price higher than
  marginal cost but lower than under uniform
  pricing
• Screening can also be done using quality
  discrimination and tie-in sales
• Bundling is useful when consumers have
  heterogeneous and negatively correlated
  preferences
• Under pure bundling only bundles are available,
  under mixed bundling also separate goods