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Chapter 5: Price Discrimination: Linear Pricing

Price Discrimination and Monopoly: Linear Pricing

Chapter 5: Price Discrimination: Linear Pricing

Introduction

Prescription drugs are cheaper in Canada than the United States
Textbooks are generally cheaper in Britain than the United States
Examples of price discrimination
presumably profitable
should affect market efficiency: not necessarily adversely
is price discrimination necessarily bad – even if not seen as “fair”?

Chapter 5: Price Discrimination: Linear Pricing

Feasibility of price discrimination

Two problems confront a firm wishing to price discriminate
identification: the firm is able to identify demands of different types of consumer or in separate markets
easier in some markets than others: e.g tax consultants, doctors
arbitrage: prevent consumers who are charged a low price from reselling to consumers who are charged a high price
prevent re-importation of prescription drugs to the United States
The firm then must choose the type of price discrimination
first-degree or personalized pricing
second-degree or menu pricing
third-degree or group pricing

Chapter 5: Price Discrimination: Linear Pricing

Third-degree price discrimination

Consumers differ by some observable characteristic(s)
A uniform price is charged to all consumers in a particular group – linear price
Different uniform prices are charged to different groups
“kids are free”
subscriptions to professional journals e.g. American Economic Review
airlines
the number of different economy fares charged can be very large indeed!
early-bird specials; first-runs of movies

Chapter 5: Price Discrimination: Linear Pricing

Third-degree price discrimination 2

The pricing rule is very simple:
consumers with low elasticity of demand should be charged a high price
consumers with high elasticity of demand should be charged a low price

Chapter 5: Price Discrimination: Linear Pricing

Third degree price discrimination: example

Harry Potter volume sold in the United States and Europe
Demand:
United States: PU = 36 – 4QU
Europe: PE = 24 – 4QE
Marginal cost constant in each market
MC = $4

Chapter 5: Price Discrimination: Linear Pricing

The example: no price discrimination

Suppose that the same price is charged in both markets
Use the following procedure:
calculate aggregate demand in the two markets
identify marginal revenue for that aggregate demand
equate marginal revenue with marginal cost to identify the profit maximizing quantity
identify the market clearing price from the aggregate demand
calculate demands in the individual markets from the individual market demand curves and the equilibrium price

Chapter 5: Price Discrimination: Linear Pricing

The example (npd cont.)

United States: PU = 36 – 4QU

Invert this:

QU = 9 – P/4 for P < $36

Europe: PU = 24 – 4QE

Invert

QE = 6 – P/4 for P < $24

Aggregate these demands

Q = QU + QE = 9 – P/4 for $36 < P < $24

At these prices only the US market is active

Q = QU + QE = 15 – P/2 for P < $24

Now both markets are active

Chapter 5: Price Discrimination: Linear Pricing

The example (npd cont.)

Invert the direct demands

P = 36 – 4Q for Q < 3

P = 30 – 2Q for Q > 3

$/unit

Quantity

Marginal revenue is

MR = 36 – 8Q for Q < 3

MR = 30 – 4Q for Q < 3

Demand

Set MR = MC

Q = 6.5

P = $17

6.5

Price from the demand curve

Chapter 5: Price Discrimination: Linear Pricing

The example (npd cont.)

Substitute price into the individual market demand curves:

QU = 9 – P/4 = 9 – 17/4 = 4.75 million

QE = 6 – P/4 = 6 – 17/4 = 1.75 million

Aggregate profit = (17 – 4)x6.5 = $84.5 million

Chapter 5: Price Discrimination: Linear Pricing

The example: price discrimination

The firm can improve on this outcome
Check that MR is not equal to MC in both markets
MR > MC in Europe
MR < MC in the US
the firms should transfer some books from the US to Europe
This requires that different prices be charged in the two markets
Procedure:
take each market separately
identify equilibrium quantity in each market by equating MR and MC
identify the price in each market from market demand

Chapter 5: Price Discrimination: Linear Pricing

The example: price discrimination 2

Demand in the US:

PU = 36 – 4QU

$/unit

Quantity

Demand

Marginal revenue:

MR = 36 – 8QU

MC = 4

Equate MR and MC

QU = 4

Price from the demand curve

PU = $20

Chapter 5: Price Discrimination: Linear Pricing

The example: price discrimination 3

Demand in the Europe:

PE = 24 – 4QU

$/unit

Quantity

Demand

Marginal revenue:

MR = 24 – 8QU

MC = 4

Equate MR and MC

QE = 2.5

Price from the demand curve

PE = $14

2.5

Chapter 5: Price Discrimination: Linear Pricing

The example: price discrimination 4

Aggregate sales are 6.5 million books
the same as without price discrimination
Aggregate profit is (20 – 4)x4 + (14 – 4)x2.5 = $89 million
$4.5 million greater than without price discrimination

Chapter 5: Price Discrimination: Linear Pricing

No price discrimination: non-constant cost

The example assumes constant marginal cost
How is this affected if MC is non-constant?
Suppose MC is increasing
No price discrimination procedure
Calculate aggregate demand
Calculate the associated MR
Equate MR with MC to give aggregate output
Identify price from aggregate demand
Identify market demands from individual demand curves

Chapter 5: Price Discrimination: Linear Pricing

The example again

Applying this procedure assuming that MC = 0.75 + Q/2 gives:

4.75

Price

(a) United States

Quantity

1.75

Price

(b) Europe

Quantity

6.5

Price

Quantity

Chapter 5: Price Discrimination: Linear Pricing

Price discrimination: non-constant cost

With price discrimination the procedure is
Identify marginal revenue in each market
Aggregate these marginal revenues to give aggregate marginal revenue
Equate this MR with MC to give aggregate output
Identify equilibrium MR from the aggregate MR curve
Equate this MR with MC in each market to give individual market quantities
Identify equilibrium prices from individual market demands

Chapter 5: Price Discrimination: Linear Pricing

The example again

Applying this procedure assuming that MC = 0.75 + Q/2 gives:

Price

(a) United States

Quantity

Price

(b) Europe

Quantity

1.75

Price

Quantity

6.5

Chapter 5: Price Discrimination: Linear Pricing

Some additional comments

Suppose that demands are linear
price discrimination results in the same aggregate output as no price discrimination
price discrimination increases profit
For any demand specifications two rules apply
marginal revenue must be equalized in each market
marginal revenue must equal aggregate marginal cost

Chapter 5: Price Discrimination: Linear Pricing

Price discrimination and elasticity

Suppose that there are two markets with the same MC
MR in market i is given by MRi = Pi(1 – 1/hi)
where hi is (absolute value of) elasticity of demand
From rule 1 (above)
MR1 = MR2
so P1(1 – 1/h1) = P2(1 – 1/h2) which gives

Price is lower in the market with the higher demand elasticity

(1 – 1/2)

(1 – 1/1)

12 – 1

12 – 2

Chapter 5: Price Discrimination: Linear Pricing

Third-degree price discrimination 2

Often arises when firms sell differentiated products
hard-back versus paper back books
first-class versus economy airfare
Price discrimination exists in these cases when:
“two varieties of a commodity are sold by the same seller to two buyers at different net prices, the net price being the price paid by the buyer corrected for the cost associated with the product differentiation.” (Phlips)
The seller needs an easily observable characteristic that signals willingness to pay
The seller must be able to prevent arbitrage
e.g. require a Saturday night stay for a cheap flight

Chapter 5: Price Discrimination: Linear Pricing

Product differentiation and price discrimination

Suppose that demand in each submarket is Pi = Ai – BiQi
Assume that marginal cost in each submarket is MCi = ci
Finally, suppose that consumers in submarket i do not purchase from submarket j
“I wouldn’t be seen dead in Coach!”
“I never buy paperbacks.”
Equate marginal revenue with marginal cost in each submarket

Ai – 2BiQi = ci 

Qi = (Ai – ci)/2Bi 

Pi = (Ai + ci)/2

 Pi – Pj = (Ai – Aj)/2 + (ci – cj)/2

It is highly unlikely that the difference in prices will equal the difference in marginal costs

Chapter 5: Price Discrimination: Linear Pricing

Other mechanisms for price discrimination

Impose restrictions on use to control arbitrage
Saturday night stay
no changes/alterations
personal use only (academic journals)
time of purchase (movies, restaurants)
“Crimp” the product to make lower quality products
Mathematica®
Discrimination by location

Chapter 5: Price Discrimination: Linear Pricing

Discrimination by location

Suppose demand in two distinct markets is identical
Pi = A = BQi
But suppose that there are different marginal costs in supplying the two markets
cj = ci + t
Profit maximizing rule:
equate MR with MC in each market as before
 Pi = (A + ci)/2; Pj = (A + ci + t)/2
 Pj – Pi = t/2  cj – ci
difference in prices is not the same as the difference in prices

Chapter 5: Price Discrimination: Linear Pricing

Third-degree rice discrimination and welfare

Does third-degree price discrimination reduce welfare?
not the same as being “fair”
relates solely to efficiency
so consider impact on total surplus

Chapter 5: Price Discrimination: Linear Pricing

Price discrimination and welfare

Suppose that there are two markets: “weak” and “strong”

MR1

MR2

ΔQ1

ΔQ2

Price

Quantity

The discriminatory

price in the weak

market is P1

The discriminatory

price in the strong

market is P2

The uniform

price in both

market is PU

The maximum gain in surplus in the weak market is G

The minimum loss of surplus in the strong market is L

Chapter 5: Price Discrimination: Linear Pricing

Price discrimination and welfare

It follows that ΔW < G – L

= (PU – MC)ΔQ1 + (PU – MC)ΔQ2

= (PU – MC)(ΔQ1 + ΔQ2)

Price discrimination

cannot increase

surplus unless it

increases aggregate

output

MR1

MR2

ΔQ1

ΔQ2

Price

Quantity

Chapter 5: Price Discrimination: Linear Pricing

Price discrimination and welfare 2

Previous analysis assumes that the same markets are served with and without price discrimination
This may not be true
uniform price is affected by demand in “weak” markets
firm may then prefer not to serve such markets without price discrimination
price discrimination may open up weak markets
The result can be an increase in aggregate output and an increase in welfare

Chapter 5: Price Discrimination: Linear Pricing

New markets: an example

Demand in “North” is PN = 100 – QN ; in “South” is PS = 100 - QS

$/unit

North

South

Aggregate

Quantity

100

100

Marginal cost to supply either market is $20

Demand

Chapter 5: Price Discrimination: Linear Pricing

New Markets: the example 2

Aggregate demand is P = (1 + )50 – Q/2 provided that both markets are served

Equate MR and MC to get equilibrium output QA = (1 + )50 - 20

Get equilibrium price from aggregate demand P = 35 + 25

$/unit

Aggregate

Quantity

Demand

Chapter 5: Price Discrimination: Linear Pricing

New Markets: the example 3

Now consider the impact of a reduction in 

Aggregate demand changes

Marginal revenue changes

MR'

It is no longer the case that both markets are served

The South market is dropped

Price in North is the monopoly price for that market

$/unit

Aggregate

Quantity

Demand

Chapter 5: Price Discrimination: Linear Pricing

The example again

Previous illustration is too extreme

So there are potentially two equilibria with uniform pricing

MC cuts MR at two points

At Q1 only North is served at the monopoly price in North

At Q2 both markets are served at the uniform price PU

Switch from Q1 to Q2:

decreases profit by the red area

increases profit by the blue area

If South demand is “low enough” or MC “high enough” serve only North

$/unit

Aggregate

Demand

Quantity

Chapter 5: Price Discrimination: Linear Pricing

Price discrimination and welfare Again

In this case only North is served with uniform pricing

But MC is less than the reservation price PR in South

So price discrimination will lead to South being supplied

Price discrimination leaves surplus unchanged in North

But price discrimination generates profit and consumer surplus in South

So price discrimination increases welfare

Quantity

$/unit

Aggregate

Demand

Chapter 5: Price Discrimination: Linear Pricing

Price discrimination and welfare One more time

Suppose only North is served with a uniform price
Also assume that South will be served with price discrimination
Welfare in North is unaffected
Consumer surplus is created in South: opening of a new market
Profit is generated in South: otherwise the market is not opened
As a result price discrimination increases welfare.

Chapter 5: Price Discrimination: Linear Pricing

Directed reading in Industrial Organization

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