John Riley 27 May 2015

Price Discrimination

1. Direct price discrimination 1

2. Direct Price Discrimination using two part pricing 2

3. Indirect Price Discrimination with two part pricing 5 4. Optimal indirect price discrimination 20 5. Key Insights 32

32 pages

1

1. Direct Price Discrimination Consider a firm selling a product for which demand is not perfectly elastic. If the product can

be easily resold, it is difficult for the firm to charge different people different prices. The firm

thus exploits its monopoly power by setting a single price p. This is depicted below. The profit

of the firm is maximized at the point where marginal revenue and marginal cost are equal.

Figure 1: Demand price and marginal revenue

The price elasticity of demand is the percentage change in quantity when the price rises by one

percent. More precisely, it is the proportional rate of change in quantity with respect to a

proportional increase in price.

( , ) ( ) / ( )p dq dq dp

q pq dp q p

How much a monopoly charges above marginal cost depends on the demand elasticity.

1

( ) ( ) (1 ) (1 )d dp q dp

MR qp q p q q p pdq dq p dq

.

A monopoly maximizes profit by equating MR and MC,

MC

2

1

(1 ) , hence1

1

MCp MC p

.

Differences in price elasticity lead to different prices. The bigger the price elasticity of demand

function the lower is the price.

2. Direct Price Discrimination with two part pricing

For a wide range of services, resale is prohibitively expensive. A monopoly can further

exploit its monopoly power by adopting a two part pricing scheme. The phone company, for

example, charges a monthly access fee and a charge per minute on many plans.

Let the fixed access fee (per month, for example) be K . In addition to the access fee

there is a use fee per unit p . Then if a customer purchases q units her total payment is

R K pq .

A type customer has consumer surplus ( , )CS p . We have already seen that the demand

function of such a consumer is

( , ) ( , )d

q p CS pdp

We will focus on the quadratic case

21( ( ) ) , ( )

2 ( )( , )

0, ( )

a p p abCS p

p a

.

Then

1

( , ) ( , ) ( ( ) )( )

q p CS p a pp b

, for all ( )p a

Inverting yields the demand price function

( , ) ( ) ( )p q a b q .

3

Consider the following figure. The area under the demand price function is the total

benefit ( , )B q . If the price is Ap , then the consumer chooses ( , )Aq p and so pays

( , )A Ap q p . The net gain to the consumer after paying the price per unit Ap (the “consumer

surplus” ) is the area of the shaded region.

Suppose that the price rises from Ap to Bp . The reduction in consumer surplus is the heavily

shaded region. From the figure, this lies between ( , )( )A B Aq p p p and ( , )( )B B Bq p p p .

That is

( , )( ) ( , )( )A B A B B Aq p p p CS q p p p

Then

( , ) ( , )A B

CSq p q p

p

Taking the limit as B Ap p ,

( , ) ( , )A A

dCSq p q p

dp .

4

With two part pricing the monopoly can extract all the consumer surplus by charging an access

fee equal to this surplus.

( ) CS( , )K p .

Then the total payment to the monopoly (the sum of the access and use fees is equal to the

total benefit. Subtracting the total cost yields the profit ( ) ( , ( , )) ( , )MU p B q p cq p . This

is the shaded region below.

Note that this is maximized by setting the price equal to marginal cost. The consumer

surplus is then ( ,c)CS The monopoly covers its costs with the use fee, p c and charges an

access fee K equal to the consumer surplus.

5

3. Indirect price discrimination with two part pricing

The results of the previous section hinge on three assumptions (i) resale is prohibitively

costly (ii) the monopoly can identify the different types of buyer and (iii) there are no legal

constraints to excluding different classes of customer from offers. Here we suppose that either

assumption (ii) or assumption (iii) is not satisfied. Suppose there are T different types of

customer. We label them so that higher types have higher demands. Formally,

Assumption 1: Higher types have higher demand functions

For all s and t s if ( , ) 0sp q then ( , ) ( , )t sp q p q

For simplicity we will focus on linear demand price functions.

1 2( ( ) ) / ( ), { , ,...}q a p b

Then Assumption 1 holds, for example, if there are two types and the demand functions are

11 2

( , ) 10q p q and 12 3

( , ) 15q p p .

Suppose that the monopoly offers a set of alternative plans {( , ),..., ( , )}a a n np K p K . Each

consumer picks one of these alternatives or purchases nothing. Note that the alternative

0 0( , ) ( ,0)Tp K a is equivalent to purchasing nothing. For there is no access fee and, at the

price Ta , no consumer wishes to make a purchase. It is helpful to add this to the set of plans so

that the augmented set is

0 0{( , ),( , ),..., ( , )}a a n np K p K p K

Next define ( ( ), ( ))s sp K to be the choice of type s from this set of plans. We will write the

set of choices as

1 1{( ( ), ( ),..., ( ( ), ( ))}T Tp K p K .

Since ( ( ), ( ))s sp K is the choice of a type s s buyer,

( , ( ), ( )) ( , ( ), ( )), for alls s s s t t tu p K u p K in the set of choices.

6

Key Insight: Higher types are willing to accept a higher access fee in return for a lower price per

unit.

Consider two plans 1 1( , )p K and 2 2( , )p K where 2 1p p .

The increase in consumer surplus if switching to the second plan is

2 1( , ) ( , )CS p CS p

This is depicted in the Figure. Higher types have higher demand price functions. Thus the

increase in consumer surplus is greater for higher types.

We now argue that higher types will never choose a plan with a higher use fee.

Principle 1: Monotone choices

Let ( ( ), ( )s sp K and ( ( ), ( )t tp K be the choices of type s and type t s .

Then ( ) ( )t sp p and ( ) ( )t sK K

Consider any plan ( , )A Ap K where ( )A sp p . Since the latter is the choice of type s this

type has revealed that he was better off paying the higher access fee and getting the lower unit

price. That is, his increase in consumer surplus outweighed the difference in access fees. All

higher types have a greater increase in consumer surplus when switching to a lower prices plan

so they must strictly prefer ( ( ), ( ))s sp K over ( , )A Ap K .

7

We have therefore shown that no higher type will purchase a plan with a higher unit price that

( )sp

We can strengthen this result to obtain the second key principle.

Principle 2:

If type is indifferent between 2 plans ( , )A Ap K and ( , )B Bp K and A Bp p then (i) all higher

types strictly prefer ( , )B Bp K and (ii) all lower types strictly prefer ( , )A Ap K .

The argument is almost the same.

If type switches from plan A to plan B his change in payoff is

ˆ ˆ ˆ ˆ( , , ) ( , , ) [ ( , ) ] [ ( , ) ]B B A A B B A Au p K u p K CS p K CS p K

2 1

ˆ ˆ( , ) ( , )B ACS p CS p K K

By hypotheses this type is indifferent so the increase in consumer surplus exactly offsets the

increase in the access fee.

Higher types have higher increases in consumer surplus when switching to a plan with a lower

unit price so

( , , ) ( , , ) ( , ) ( , ) 0B B A A B A B Au p K u p K CS p CS p K K for all ˆ

Lower types have smaller increases in consumer surplus when switching to a plan with a lower

unit price so

( , , ) ( , , ) ( , ) ( , ) 0B B A A B A B Au p K u p K CS p CS p K K for all ˆ .

Profit maximizing plans

Participation Constraints

The type 1 buyer cannot be forced to participate so we require that

8

1 1 1 1( ) CS( , ( )) ( ) 0U p K (3.1)

Since a higher type has a higher consumer surplus it follows that

2 1 1 1 1 1CS( , ( )) ( ) CS( , ( )) ( ) 0p K p K

So if (3.1) holds, then the higher type is not being forced to participate either.

Revealed preference constraints (incentive constraints)

1. No gain to switching to the plan with the closest higher use fee

1 1 1 1 1( , ( ), ( )) CS( , ( )) ( )tu p K p K

1 1 1 1CS( , ( )) ( ) ( , ( ), ( ))t t t t t tp K u p K . (3.2)

2. No gain to switching to any other plan with a higher use fee.

1 1 1 1 1( , ( ), ( )) CS( , ( )) ( )tu p K p K

CS( , ( )) ( ) ( , ( ), ( ))t s s t s sp K u p K for all s t

3. No gain to switching to a plan with a lower use fee

( , ( ), ( )) CS( , ( )) ( )t t t t t tu p K p K

CS( , ( )) ( ) ( , ( ), ( ))t s s t s sp K u p K for all s t .

Approach: Solve a “relaxed” problem

1. Fix the prices for the plans to satisfy monotonicity:

1( ) ... ( )Tp p

2. Ignore all but the local revealed preference constraint and choose the access fees to

maximize revenue.

3. Then show that the last constraint is satisfied.

Consider the two type case.

The constraints for the relaxed problem ((3.1) and (3.2)) can be rewritten as follows:

9

1 1 1CS( , ( )) ( )p K

2 2 2 1 2 1CS( , ( )) CS( , ( )) ( ) ( )p p K K

If the first constraint is binding we can increase 1( )K and 2( )K by the same amount (so the

second constraint still holds) until the first constraint is binding. Then increase 2( )K until the

second constraint is binding.

Appealing to Principle 2, since type 2 is indifferent between the two plans, the lower type

strictly prefers the plan with the higher price and lower access fee. Thus the other revealed

reference constraint is satisfied.

Principle 3: The profit maximizing monopolist chooses 1 1 1( ) ( , )K CS p so that the payoff of

a type 1 customer is zero.

Principle 4: With two types the monopolist chooses 2( )K so that a type 2 customer is

indifferent between plan 2 and plan 1, that is 2 2 2 2 1 1( , ( )) ( ) ( , ( )) ( )CS p K CS p K

We have therefore shown that the revenue maximizing access fees are

1 1 1( ) ( , ( ))K CS p

2 1 2 1 1 1( ) ( ) ( , ( )) ( , ( ))K K CS p CS p

Given these principles we can use Solver to solve for the profit maximizing plans.

Total revenue from plan t is

1( ) ( ) ( ) ( , ( ))t t t tr K p q p

Total cost is ( ) ( , ( ))t t tC cq p

Let ( )tf be the number of customers of type t . Then the monopoly chooses the prices to

maximizes total profit

1 1 1 2 2 2( )[ ( ) ( )] ( )[ ( ) ( )]MU f r C f r C

10

You should download the spread-sheet TwoPartPricing.xlsm and carefully review all the

formulas. Note the formula for demand. If the price is so high that ( ) / 0q q p b the

demand is actually zero. Thus MAX(0,(q-p)/b) selects th correct quantity, Similarly wity

consumer surplus, the quadrtic formula is only correct for p < a sothe correct formula is

=0.5*(MAX(0,C3-B8))^2/D3

With two types there is a single constraint. The price for plan 2 must be less than or equal to

the price for plans 1.

The spread-sheet above depicts the solution. Note that a type 2 customer has a use fee equal

to marginal cost and therefore consumes exactly the same as with direct price discrimination.

However a type 1 customer has a use fee so high that he makes no purchase. All the sonsumer

surplus can then be squeezed for high types.

In the next figure the intercept parameter for type 1 is higher. Now the profit maximizing

strategy is to have two plans.

11

To understand these results consider the figure below. The slopes of the two demand curves

are equal.

12

In the left diagram the dark shaded region is revenue from the use fee less the cost of

production. The dotted triangle is the consumer surplus of a type 1 customer. Since he is

charged an access fee equal to his consumer surplus the profit of the firm on the type 1

customer is the sum of the shaded and dotted regions.

If a type 2 customer chooses plan 1 his consumer surplus is the area under his

demand price function and above the line 1p p . Since the area of the dotted triangle is the

access fee, the residual striped area is 2 1 1( , )u p K , the payoff to a type 2 customer if he chooses

plan 1. If a type 2 customer chooses plan 2 the shaded area in the right hand figure is the

revenue from plan 2 less the cost of production. His consumer surplus is the sum of the dotted

and striped areas. But the striped area is the minimum payoff that he must get or he will switch

to plan 1. Thus the access fee is the dotted area.

Consider the right hand diagram. As long as 2p c the monopoly can lower 2p and so

increase the sum of dotted and shaded areas, that his, the total profit on a type 2 customer.

Then it is optimal for the monopoly to set 2p c . The revised figure is shown below.

13

The final step is to ask what happens as 1p declines to 1p . This is depicted below.

The increase in profit, 1S , on each type 1 customer rises by the area in the left

diagram bounded by the heavy lines. This is the increase in social surplus for a type 1 buyer.

The monopoly appropriates this by raising the access fee. The reduction in profit, 2K , on

each type 2 customer falls by the area in the right diagram bounded by the heavy lines. The net

increase in profit is then

1 1 2 2n S n K (3.3)

Note that as 1p gets close to c the increase in profit approaches zero while the loss is

bounded away from zero. Thus to maximize profit 1 2p c p . Note also that any change in

parameter that increases the first term or decreases the second term increases the payoff to

lowering the price. Thus, for example, if 1n rises or 2n falls, the profit maximizing use fee , 1p ,

must rise.

14

Solving analytically

Since 1 1 1p a q and 1 1 1ˆ ˆp a q it follows that 1 1 1 1

ˆ ˆp p q q . Then if the change in price is

small,

1 1q p .

To a first approximation,

1 1 1( ) ( c)p c q p p .

and

2 2 1 1 1( ( ) ( ))q p q p p .

Substituting these expression into (3.3)

1 1 2 2 1 1 1[ ( ) ( ( ) ( ))]n p c n q p q p p

Dividing by 1p and taking the limit,

1 1 2 2 1 1 1 1 1 2 2 1

1

( ) ( ( ) ( )) ( ) ( )n p c n q p q p n p c n a ap

.

This is negative if 1p c is sufficiently small. The monopoly raises 1p until either the marginal

profit from increasing 1p is zero or type 1 buyers are excluded from the market.

Exercise: For what parameter values is the profit-maximizing plan for the low types below 1a ?

15

Three or more types of customer

The analysis of the two type case can be extended directly if there are three or more

types of buyer. Again we consider the relaxed problem and only consider the following three

“local downward” constraints:

1 1 1 1( ) CS( , ( )) ( ) 0U p K (3.4)

No gain to switching to a plan with a higher use fee

1 1 1 1 1 1( , ( ), ( )) CS( , ( )) ( )t t t t t tu p K p K

1 1CS( , ( )) ( ) ( , ( ), ( ))t t t t t tp K u p K . (3.5)

We can rewrite these three constraints as follows:

1 1 1CS( , ( )) ( )p K (1)

2 2 2 1 2 1CS( , ( )) CS( , ( )) ( ) ( )p p K K (2)

3 3 3 2 3 2CS( , ( )) CS( , ( )) ( ) ( )p p K K (3)

Arguing as before, first increase all access fees by the same amount until the first constraint is

binding. The other two constraints are unaffected. Next increase 2( )K and 3( )K by the

same amounts until constraint 2 is binding. Finally increase 3( )K so that the third is binding.

Solving with all the constraints

The payoff to buying nothing is zero. This is equivalent to accepting a plan in which the use for

0p is higher than any of the buyer’s reservation prices and the access fee is zero. Given the very

high price per unit, the customer consumes nothing and pays nothing.

Then we can think of the relaxed problem as one with three local downward constraints.

1 1 1 1 1 0 0( ) : ( , ( ), ( )) ( , , ) 0LCD u p K u p K

2 2 2 2 2 1 1( ) : ( , ( ), ( )) ( , ( ), ( ))LCD u p K u p K

2 3 3 3 3 2 2( ) : ( , ( ), ( )) ( , ( ), ( ))LCD u p K u p K

16

As we have just argued, for the relaxed problem revenue is maximized by making all three

constriants bindinf. That is, a type t customer is indifferent between the plan designed for

him ( ( ), ( ))p K and the plan designed for tteh immediatrly lower type. Economists often

express indifference a follows:

1 1( ( ), ( )) ( ( ), ( ))t

t t t tp K p K

If type t has a higher utility under ( ( ), ( ))t tp K then under some other plan ( , )p K we say

that type t prefers ( ( ), ( ))t tp K and write

( ( ), ( )) ( , )t

t tp K p K

If type t has a strictly higher utility under ( ( ), ( ))t tp K then under some other plan ( , )p K

we say that type t strictly prefers ( ( ), ( ))t tp K and write

( ( ), ( )) ( , )t

t tp K p K

The four plans are shown below. We now argue that we can replace the six question

marks by appealing to Principle 2. Note that, by Principle 1,

0 1 2 3( ) ( ) ( )p p p p

plan 0 plan 1 plan 2 plan 3

1 0 0( , )p K 1

1 1( ( ), ( ))p K ? 2 2( ( ), ( ))p K ? 3 3( ( ), ( ))p K

2 0 0( , )p K ? 1 1( ( ), ( ))p K 2

2 2( ( ), ( ))p K ? 3 3( ( ), ( ))p K

3 0 0( , )p K ? 1 1( ( ), ( ))p K ? 2 2( ( ), ( ))p K 3

3 3( ( ), ( ))p K

Consider the two shaded question marks. The price on plan 1 is lower than on plan 0.

For revenue maximization under the relaxed constraints, the lowest type is indifferent between

the two plans. Then by Principle 2, both of the higher types prefer the higher plan. Thus we can

revise the table as shown below.

17

plan 0 plan 1 plan 2 plan 3

1 0 0( , )p K

1

1 1( ( ), ( ))p K ? 2 2( ( ), ( ))p K ? 3 3( ( ), ( ))p K

2 0 0( , )p K

2

1 1( ( ), ( ))p K

2

2 2( ( ), ( ))p K ? 3 3( ( ), ( ))p K

3 0 0( , )p K

3

1 1( ( ), ( ))p K ? 2 2( ( ), ( ))p K

3

3 3( ( ), ( ))p K

Next consider the two question marks in the comparison of plan 2 and plan 3. Since the highest

type is indifferent, both of the lower types at least weakly prefer the lower of the two plans.

Thus we can further revise the table as shown below.

plan 0 plan 1 plan 2 plan 3

1 0 0( , )p K 1

1 1( ( ), ( ))p K ? 2 2( ( ), ( ))p K

1

3 3( ( ), ( ))p K

2 0 0( , )p K 2

1 1( ( ), ( ))p K

2

2 2( ( ), ( ))p K 2

3 3( ( ), ( ))p K

3 0 0( , )p K 3

1 1( ( ), ( ))p K ? 2 2( ( ), ( ))p K

3

3 3( ( ), ( ))p K

It is left to the reader to consider the final two question marks. Using Principle 2 yields the

following table.

plan 0 plan 1 plan 2 plan 3

1 0 0( , )p K 1

1 1( ( ), ( ))p K

1

2 2( ( ), ( ))p K

1

3 3( ( ), ( ))p K

2 0 0( , )p K 2

1 1( ( ), ( ))p K

2

2 2( ( ), ( ))p K 2

3 3( ( ), ( ))p K

3 0 0( , )p K 3

1 1( ( ), ( ))p K

3

2 2( ( ), ( ))p K

3

3 3( ( ), ( ))p K

18

It follows that no customer type is better off switching to another plan. Thus all of the

incentive constraints are satisfied.

While we will not consider more than three types, it is intuitively clear that the

argument made above is general. Hence we have the following further principle.

Principle 5:

If the local downward constraints are all binding, that is 1 1( , ) ( , )t t t t t tU p K U p K , then no type

can gain by switching to another plan.

How many plans will be offered?

With three types it may be optimal to sell to all three types but only have two plans.

Using a spread sheet you should consider a case in which the difference in demand price

functions for type 1 and 2 is small and the proportion of type 2 is small relative to type 1 .

Start with three plans. Then reduce 2( )q . The benefit to type 2 is lower so the cost of plan 2

must be lowered. But with a lower 2( )q plan 2 becomes less attractive to the high type. Thus

the payment by the high type can be realized. The smaller the proportion of type 2 the smaller

is the loss relative to the gain. As you will find, eventually profit is maximized by setting

2 1( ) ( )q q .

Exercise 3.1: Parametric changes

Consider the two type case with linear demand price functions.

(a) What is the effect on the profit maximizing use fee if (i) 1a rises (ii) 1n rises (iii) 1b falls.

(b) In which case dies it follow that 2F rises so type 2 customers are worse off.

19

Technical Note (limitations of Solver)

Solver stops searching for a better solution when there are no neighboring solutions that raise

profit. As you will find, if you start with 1 1( ) ( )p a , Solver may find a maximum by changing

the other prices only. This may or not be the solution. To check, you should then start with

1 1( ) ( )p a and compare the two solutions. One may be what is called a local maximum but

not the global maximum.

20

4. Optimal indirect price discrimination

We now argue that the monopoly can do better by offering “cell phone” plans rather

than 2 part pricing plans. A cell phone plan offers a fixed number of minutes (or data) q for a

total fee r. For a type t customer the benefit from the q units is the area under her demand

curve ( )tB q . Thus her payoff (or utility) is

( , , ) ( , )u q r B q r .

The benefit is the sum of the dotted and shaded areas below, that is,

( , ) ( , ( , )B q p q CS q . In the linear case the consumer surplus is easily calculated since it is

half the area of the rectangle.

( , ) ( ( ) ( , ))CS q a p q q

More generally, the extra benefit from an increase in output from q to q q is bounded

from above by ˆ( , )p q q and bounded from below by ˆ( , )p q q q . That is

ˆ ˆ( , ) ( , )p q q B p q q q .

Then

Figure 4.1: Demand price and total benefit

21

ˆ ˆ( , ) ( , )B

p q p q qq

Taking the limit,

ˆ ˆ( , ) ( , )B

p q p qq

Thus the demand price function is the marginal benefit:

( , ) ( , )B

MB q p qq

Re-integrating, the total, benefit is the area under the consumer’s demand price function.

ˆ

ˆ( , ) ( , )

q

o

B q p q dq

Preferences over outcomes

Regardless of the method that a monopoly uses to extract revenue from a customer, the

two things that th customer ultimately cares about are the quantity consumed and how much

ˆ( , )p q q

ˆ( , )p q

q

p

( , )p q

q

q q q

22

this costs. We call this the “outcome” for the consumer ( , )q r . Given such an outcome the

consumer’s utility is

( , , ) ( , )u q r B q r

It is helpful to think about the challenge for the monopoly be depicting the consumer’s

indifference curves over the space of outcomes, that is, all ( , )q r combinations.

Consider an indifference curve for a type customer through the outcome ˆ ˆ( , )q r

ˆ ˆ ˆ( , , ) ( , ) ( , , )u q r B q r u u q r

Rearranging yields the following equation for the total payment

ˆ( , )r B q u (4.1)

Note that ( ,0) 0B therefore the intersection of the indifference curve with the horizontal

axis is the utility of the consumer u .

To find the slope we differentiate the equation for the indifference curve.

( , ) ( , ) ( , )dr B

q MB q p qdq q

Since the benefit is the integral of the demand price function, the marginal benefit is the

demand price function. Note that increasing the utility holding q constant does not chnge the

slope. (Of course r has to decline.) Thus the gap between two indifference curves is constant.

Example: Quadratic preferences

2( , ) 2 2B q q q ( ) 2 4MB q q

( , , ) ( , )u q r B q r

The slope of an indifference curve is ( , ) 2p q q .

The indifference curves ( , , ) 0u q r and ( , , ) 10u q r are depicted below.

23

If a monopoly simply sets a price p per unit, then the possible outcomes are ˆ( , ) ( , )q r q pq

These are depicted below. The consumer then chooses her consumption ˆ( , )q p so that

marginal benefit (her demand price) is equal to the market price. This is depicted below.

With two part pricing the total payment is ˆ ˆr K pq so the revenue line shifts upwards and

extracts greater revenue.

24

The next figure depicts the indifference curve of a high type 2 with a higher demand price

2( , )p q and that of a low demander with lower demand price 1( , )p q . Since the slope of each

indifference curve is the consumer’s demand price function. The indifference curve of the high

demander is everywhere steeper.

In the figure, if the seller offers the two depicted outcomes or “plans” the two types of

customer would choose different plans.

Exercise 4.1: Efficient price discrimination

The allocation to a type buyer is efficient if the buyer’s marginal benefit is equal to marginal

cost. That is *( , ( ))p q c . If the monopoly simply sets a price, it must therefore set a price

p c in order to achieve efficiency. This depicted below.

(a) Suppose that 1( , ) 20p q q , 2( , ) 30p q q and marginal cost 10c . Solve for the

efficient allocations.

(b) If the monopoly uses two-part pricing, ˆ( , ) ( , )p K c K , what is the maximum access fee

that achieves an efficient outcome?

(c) Explain why the monopoly can extract and still achieve efficiency by offering two plans

* *

1 1 2 2{( ( ), ( )), ( ( ), ( ))}q r q r .

25

(d) What are those plans and what is the monopoly profit.

Comparison of two part pricing and optimal selling schemes

To see why using “plans” rather than two-part pricing is better for the monopolist

consider the low access fee high unit price plan 1 with two-part pricing. This is depicted below.

The monopoly extracts all the consumer surplus from type 1 by charging an access fee 1( )K

equal to 1 1( , ( ))CS p .

Now consider a type 2 customer who switches and purchases plan 1. Her consumer

surplus is the sum of the dotted and striped areas. She also has to pay the access fee 1( )K .

However, as we have just argued, the dotted area is equal to 1( )K . Then the payoff to a type

2 buyer if she switches is the striped area.

26

Note that the total payment by type 1 is the sum of the shaded and dotted areas.

1 1 1 1 1( ) ( )* ( , ( )) ( )r p q p K .

Suppose that the monopoly replaces plan 1 with a “cell phone” plan that offers

1 1 1( ) ( , ( ))q q p units for a monthly payment of 1( )r . This is exactly the same outcome for a

type 1 buyer and the monopoly. But with the new cell phone plan a type 2 buyer only gets

1( )q units so his consumer surplus is the sum of the green striped and dotted areas. Since he

must pay the sum of the dotted and shaded areas his payoff is the green striped area. Note that

this is smaller than the striped area under two-part pricing. So switching is less attractive and it

is possible to “squeeze” a type 2 buyer by charging her more.

Fig. 4.2: Payoff to a type 2 buyer with 2 part pricing

27

Rather than offering a plan in which the consumer chooses the number of units, suppose that

the firm offers a fixed number of units and a total payment per month. This is the way cell

phones are sold. So think of the plans as cell phone plans. The firm offers a set of alternative

plans

{( , ),..., ( , )}a a n nq r q r

A customer has the option of not participating. In this case the outcome is 0 0( , ) (0,0)q r . We

will call this “plan 0”. Then the set of alternatives available to a customer is

0 0{( , ),( , ),..., ( , )}a a n nq r q r q r

Let ( ( ), ( ))s sq r be the choice of a type 2 customer and define

1 1{( , ),..., ( , )}T Tq r q r

to be the set of choices of all the different types.

Figure 4.3: Payoff to switching declines

28

As in section 3 we have two simple but important results.

Key Insight: Higher types are willing to accept a higher payment (per month) in return for a

plan with a higher quantity.

Consider two plans 1 1( ( ), ( ))q r and 2 2( ( ), ( ))q r where 2 1( ) ( )q q .

The increase in benefit if switching to the second plan is

2 1( , ( )) ( , ( ))B q B q

This is depicted in the Figure. Higher types have higher demand price functions. Thus the

increase in benefit is greater for higher types.

We now argue that higher types will never choose a plan with a higher use fee.

Principle 1 : Monotone choices

Let ( ( ), ( )s sq r and ( ( ), ( )t tq r be the choices of type s and type t s .

Then ( ) ( )t sq q and ( ) ( )t sr r

29

Consider any plan ( , )A Aq r where ( )A sq q . Since the latter is the choice of type s this type

has revealed that he was better off paying the higher monthly payment and having th ehigher

quantity. That is, his increase in benefit outweighed the difference in monthly payment. All

higher types have a greater increase in benefit when switching to a plan with a bigger quantity

so they must strictly prefer ( ( ), ( ))s sq r over ( , )A Aq r .

We have therefore shown that no higher type will purchase a plan with a smaller quantity than

( )sq .

Proposition 2 :

If type s is indifferent between 2 plans ( , )a aq r and ( , )b bq r and a bq q then (i) all higher types

strictly prefer ( , )b bq r and all lower types strictly prefer ( , )a aq r

We have already established (i) in the discussion of Principle . To see that (ii) is also true we

proceed in essentially the same fashion.

Profit maximizing plans

Participation Constraints

The type 1 buyer cannot be forced to participate so we require that

1 1 1 1( ) ( , ( )) ( ) 0U B q r (4.2)

Since a higher type has a higher consumer surplus it follows that

1 1 1 1 1( , ( )) ( ) ( , ( )) ( ) 0tB q r B q r

So if (4.2) holds, then the higher type is not being forced to participate either.

30

Revealed preference constraints (incentive constraints)

1. No gain to switching to the plan with the closest lower quantity.

( , ( ), ( )) ( , ( )) ( )t t t t t tu q r B q r

1 1 1 1( , ( )) ( ) ( , ( ), ( ))t t t t t tB q r u q r . (4.3)

2. No gain to switching to any other plan with a lower quantity.

( , ( ), ( )) ( , ( )) ( )t t t t t tu q r B q r

( , ( )) ( ) ( , ( ), ( ))t s s t s sB q r u q r for all s t

3. No gain to switching to a plan with a lower use fee

( , ( ), ( )) ( , ( )) ( )t t t t t tu q r B q r

( , ( )) ( ) ( , ( ), ( ))t s s t s sB q r u q r for all s t .

Approach: Solve a “relaxed” problem

(a) Fix the prices for the plans to satisfy monotonicity:

1( ) ... ( )Tq q

(b) Ignore all but the local revealed preference constraint and choose the access fees to

maximize revenue.

(c ) Then show that the last constraint is satisfied.

Consider the two type case.

The constraints for the relaxed problem ((3.1) and (3.2)) can be rewritten as follows:

1 1 1( , ( )) ( )B q r

2 2 2 1 2 1( , ( )) ( , ( )) ( ) ( )B q B q r r

If the first constraint is binding we can increase 1( )r and 2( )r by the same amount (so the

second constraint still holds) until the first constraint is binding. Then increase 2( )r until the

second constraint is binding.

31

Appealing to Principle 2, since type 2 is indifferent between the two plans, the lower type

strictly prefers the plan with the smaller quantity. Thus the other revealed reference constraint

is satisfied.

Principle 3: The profit maximizing monopolist chooses 1 1 1( ) ( , ( ))r B q so that the payoff of

a type 1 customer is zero.

Principle 4: With two types the monopolist chooses 2( )r so that a type 2 customer is

indifferent between plan 2 and plan 1, that is 2 2 2 2 1 1( , ( )) ( ) ( , ( )) ( )B q r B q r

We have therefore shown that the revenue maximizing access fees are

1 1 1( ) ( , ( ))r B q

2 1 2 1 1 1( ) ( ) ( , ( )) ( , ( ))r r B q B q

Given these principles we can use Solver to solve for the profit maximizing plans.

The total cost of plan t is ( ) ( , ( ))t t tC cq p

Let ( )tf be the number of customers of type t . Then the monopoly chooses the prices to

maximizes total profit

1 1 1 2 2 2( )[ ( ) ( )] ( )[ ( ) ( )]MU f r C f r C

The structure of the spread sheet is almost identical to that for two part pricing. This is

illustrated on the next page.

Note that the prices for the two-part pricing approach are chosen so that the quantities

purchased are the same as in the profit-maximizing plans. Note also that the marginal

information rent (MIR) is higher with two part pricing (as discussed above). Thus the

monopolist is able to capture more of the surplus using plans where the quantity sold is fixed.

32

Exercise: Satisfying all of the incentive constraints

In the discussion above we considered only the “relaxed problem.”

Consider the case of three types and three plans. Use an argument very similar to that for two-

part pricing to show that if the “local downward constraints” are all binding, then all the

incentive constraints must hold.

33

Solving analytically for a simple example

Suppose that marginal cost is 10. The demand price function is ( , )p q q so that total

benefit to a customer purchasing q units is 212

( , )B q q q . The types are 1 20 and

2 25 . Then if the two consumers purchase the same plan ( , )q r , the difference in benefit is

2 1 2 1[ ( , ) ] [ ( , ) ] ( , ) ( , ) 5B q r B q r B q B q q

This difference is called the marginal information rent of a type 2 buyer.

Now consider any pair of plans. The plan chosen by type 1 , 1( )q must be smaller then 2( )q

We consider only the local downward constraints. Squeeze the low type down to zero.

1 1 1 1 1 1( , ( ), ( )) ( , ( )) ( ) 0u q r B q r .

Therefore

211 1 12

( ) 20 ( ) ( )r q q .

As argued above, with the low type having a payoff of zero, the high type has a payoff of 15 ( )q

if he chooses plan 1. Thus the monopoly can squeeze the high type till his payoff is 15 ( )q .

2 2 2 2 2 2 1( , ( ), ( )) ( , ( )) ( ) 5 ( )u q r B q r q .

Rearranging,

212 2 2 12

( ) 25 ( ) ( ) 5 ( )r q q q

The profit on the two plans is therefore as follows:

211 1 1 12

( ) ( ) 10 ( ) ( )r cq q q

212 2 2 2 12

( ) ( ) 15 ( ) ( ) 5 ( )r cq q q q

The total profit of the monopoly is

0 1 2 1 1 1 2 2 2( , ) ( )[ ( ) ( )] ( )[ ( ) ( )]u q q n r cq n r cq

It is then a simple matter to differentiate and so solve for the profit-maximizing plans.

Note that 2( )q only appears in the profit on plan 2. The solution is 2( ) 15q , that is, the

quantity where the demand price is equal to marginal cost. As you can easily confirm, the size

of plan1 one falls as the ratio 2 1( ) / ( )n n rises and eventually drops to zero.

34

5. Key Insights

1. Of all the participation and incentive constraints, the only ones that we need to focus on are

the local downward constraints. This is because higher types are willing to pay more for a higher

quantity.

2. The monopoly sets the total payments so that the local downward constraints are binding. For

type 1 this is the constraint that he must be willing to participate.

3. By increasing the quantity sold to type 1 (and so increasing the benefit to type 1, the

monopolist can increase profit until the local downward constraint is again binding. The higher

type 1 quantity raises the incentive for type 2 to switch. Thus 2r must be reduced until type 2 is

again indifferent. The net profit is therefore 1 1 1 2 2( )n B c q n r . The monopoly sets the

quantity where marginal profit is zero.

4. With more than two types the monopoly must lower the total payment on all higher quantity

plans to maintain the local downward constraint. Thus marginal profit is

1 1 1 2 2( ... )Tn S n n r

5. If the ratio 2 1:n n is sufficiently large, type 1 customers will be squeezed out of the market.

6. For the highest type, changing the quantity does not affect incentive constraints. Therefore if

it is optimal to maximize social surplus and thus set ( )T Tp q c