Price discrimination and monopoly:

1

• Price discrimination and monopoly
• Nonlinear pricing

2
Introduction

• Annual subscriptions generally cost less in total
  than one-off purchases
• Buying in bulk usually offers a price discount
• these are price discrimination reflecting
  quantity discounts
• prices are nonlinear, with the unit price
  dependent upon the quantity bought
• allows pricing nearer to willingness to pay
• so should be more profitable than third-degree
  price discrimination
• How to design such pricing schemes?
• depends upon the information available to the
  seller about buyers
• distinguish first-degree (personalized) and
  second-degree (menu) pricing

3
First-degree price discrimination 1

• Monopolist can charge maximum price that each
  consumer is willing to pay
• Extracts all consumer surplus
• Since profit is now total surplus, find that
  first-degree price discrimination is efficient

4
First-Degree Price Discrimination

• First-degree price discrimination occurs when the
  seller is able to extract the entire consumer
  surplus
• suppose that you own five antique cars and you
  meet two collectors
• each is willing to pay 10,000 for one car,
  8,000 for a second car, 6,000 for a third car,
  4,000 for a fourth and 2,000 for a fifth
• sell the first two cars at 10,000, one to each
  buyer
• sell the second two cars at 8,000, one to each
  buyer
• sell the fifth car to one of the buyers at 6,000
• total revenue 42,000
• Highly profitable but requires
• detailed information
• ability to avoid arbitrage
• Leads to the efficient choice of output since
  price equals marginal revenue and MR MC

5
First-degree price discrimination (cont.)

• The information requirements appear to be
  insurmountable
• No arbitrage is less restrictive but potentially
  a problem
• But there are pricing schemes that will achieve
  the same output
• non-linear prices
• two-part pricing as a particular example of
  non-linear prices

6
Two-Part Pricing

Take an example
V
Jazz club
n identical consumers
Demand is P V - Q
Cost is C(Q) F cQ
c
Marginal Revenue is
MC
MR V - 2Q
MR
Marginal Cost is
V
Quantity
MC c
7
Two-Part Pricing
Charging an entry fee increases profit by (V -
c)2/8 per consumer
What if the seller can charge an entry fee?

With a uniform price profit is maximized by
setting marginal revenue equal to marginal cost
V
The maximum entry fee that each consumer will be
willing to pay is consumer surplus
(Vc)/2
V - 2Q c
c
MC
So Q (V - c)/2
MR
P V - Q
V
So P (V c)/2
(V-c)/2
Quantity
Profit to the monopolist is n(V - c)2/4 - F
Consumer surplus for each consumer is (V - c)2/8
8
Two-Part Pricing
Is this the best the seller can do?

V
This whole area is now profit from each consumer
(Vc)/2
Lower the unit price
c
MC
This increases consumer surplus and so
increases the entry charge
MR
V
(V-c)/2
Quantity
9
Two-Part Pricing
What is the best the seller can do?

V
The entry charge converts consumer surplus into
profit
Using two-part pricing
increases the monopolists profit
Set the unit price equal to marginal cost
c
MC
MR
This gives consumer surplus of (V - c)2/2
V
V - c
Quantity
Set the entry charge to (V - c)2/2
10
Two-part pricing (cont.)

• First-degree price discrimination through
  two-part pricing
• increases profit by extracting all consumer
  surplus
• leads to unit price equal to marginal cost
• causes the monopolist to produce the efficient
  level of output
• What happens if consumers are not identical?
• Assume that consumers differ in types and that
  the monopolist can identify the types
• age
• location
• some other distinguishing and observable
  characteristic
• We can extend our example

11
Two-part pricing with different consumers

• There is an alternative approach block pricing

So the seller can charge an entry fee of 72 t o
each older customer and 32 to each younger one
Younger Consumers
Older Consumers

• Offer older customers entry plus 12 units for
  120

Demand P 16 - Q
Demand P 12 - Q
This converts all consumer surplus into profit

• and younger customers entry plus 8 units for 64

And for the younger customers consumer surplus is
32
If unit price is set at 4 older customers each
buy 12 units
16
Assume that marginal cost is constant at 4 per
unit
Consumer surplus for the older customers is 72
And younger customers each buy 8 units
12
72
72
32
32
48
32
16
12
12
8
Quantity
Quantity
12
Second-Degree Price Discrimination (menu pricing)

• What if the seller cannot distinguish between
  buyers?
• perhaps they differ in income (unobservable)
• Then the type of price discrimination just
  discussed is impossible
• High-income buyer will pretend to be a low-income
  buyer
• to avoid the high entry price
• to pay the smaller total charge
• Confirm from the diagram

13
The example again
High-Demand Consumers
Low-Demand Consumers
Could the seller prevent this by limiting the
number of units that can be bought?
Demand P 16 - Q
Demand P 12 - Q
NO! If a high-demand consumer pays the lower fee
and gets the lower quantity he gets 32 of
consumer surplus
If a high-demand consumer pays the lower fee and
buys 12 units he gets 40 of consumer surplus


16
12
32
8
32
32
8
16
32
32
8
16
12
12
8
Quantity
Quantity
14
Second-Degree Price Discrimination

• The seller has to compromise
• A pricing scheme must be designed that makes
  buyers
• reveal their true types
• self-select the quantity/price package designed
  for them
• This is the essence of second-degree price
  discrimination
• It is like first-degree price discrimination
• The seller knows that there are buyers of
  different types
• But
• the seller is not able to identify the different
  types
• A two-part tariff is ineffective
• allows deception by buyers
• Use quantity discounting
• Examples subscription (theaters, football
  matches, newspapers), ski resorts, supermarkets

15
The example again
High-Demand
Low-Demand
So any other package offered to
high-demand consumers must offer at least 32
consumer surplus
The low-demand consumers will be willing to buy
this (64, 8) package
So will the high- demand consumers because the
(64, 8) package gives them 32 consumer surplus
This is the incentive compatibility constraint
Low demand consumers will not buy the (88,
12) package since they are willing to pay only
72 for 12 drinks
These packages exhibit quantity discounting
high- demand pay 7.33 per unit and low-demand
pay 8
So they can be offered a package of (88, 12)
(since 120 - 32 88) and they will buy this
High demand consumers are willing to pay up to
120 for entry plus 12 drinks if no other package
is available


Offer the low-demand consumers a package of entry
plus 8 drinks for 64
Profit from each high- demand consumer is 40
(88 - 12 x 4)
And profit from each low-demand consumer is 32
(64 - 8x4)
16
12
32
8
32
32
32
40
64
8
24
16
32
32
8
8
16
12
12
8
Quantity
Quantity
16
The example again
The monopolist does better by reducing the number
of units offered to low-demand consumers since
this allows him to increase the charge to
high-demand consumers
Can the club- owner do even better than this?
A high-demand consumer will pay up to 87.50 for
entry and 7 drinks
High-Demand
Low-Demand
So buying the (59.50, 7) package gives him 28
consumer surplus
Suppose each low-demand consumer is offered 7
drinks
So entry plus 12 drinks can be sold for 92 (120
- 28 92)
Each consumer will pay up to 59.50 for entry and
7 drinks


Profit from each (92, 12) package is 44 an
increase of 4 per consumer
16
Yes! Reduce the number of units offered to
each low-demand consumer
Profit from each (59.50, 7) package is 31.50 a
reduction of 0.50 per consumer
12
28
87.50
31.50
44
59.50
92
28
28
48
16
12
12
8
7
7
Quantity
Quantity
17
Second-degree price discrimination (cont.)

• Will the monopolist always want to supply both
  types of consumer?
• There are cases where it is better to supply only
  high-demand
• high-class restaurants
• golf and country clubs
• Take our example again
• suppose that there are Nl low-income consumers
• and Nh high-income consumers

18
Second-degree price discrimination (cont.)

• Suppose both types of consumer are served
• two packages are offered (57.50, 7) aimed at
  low-demand and (92, 12) aimed at high-demand
• profit is 31.50xNl 44xNh
• Now suppose only high-demand consumers are served
• then a (120, 12) package can be offered
• profit is 72xNh
• Is it profitable to serve both types?
• Only if 31.50xNl 44xNh gt 72xNh ? 31.50Nl gt
  28Nh

Nh
31.50
This requires that
lt
1.125
Nl
28
There should not be too high a proportion of
high-demand consumers
19
Second-degree price discrimination (summary)

• Characteristics of second-degree price
  discrimination
• extract all consumer surplus from the
  lowest-demand group
• leave some consumer surplus for other groups
• the incentive compatibility constraint
• offer less than the socially efficient quantity
  to all groups other than the highest-demand group
• offer quantity-discounting
• Second-degree price discrimination converts
  consumer surplus into profit less effectively
  than first-degree
• Some consumer surplus is left on the table in
  order to induce high-demand groups to buy large
  quantities

20
The incentive compatibility constraint

• Any offer made to high demand consumers must
  offer them as much consumer surplus as they would
  get from an offer designed for low-demand
  consumers.
• This is a common phenomenon
• performance bonuses must encourage effort
• insurance policies need large deductibles to
  deter cheating
• piece rates in factories have to be accompanied
  by strict quality inspection
• encouragement to buy in bulk must offer a price
  discount