Pricing Examples. Bundling In marketing, product bundling offers several products for sale as one combined product. This is common in the software business.

Pricing Examples

Bundling

• In marketing, product bundling offers several products for sale as one combined product. This is common in the software business (for example: bundle a word processor, a spreadsheet, and a database into a single office suite), in the cable television industry (for example, basic cable in the United States generally offers many channels at one price), and in the fast food industry in which multiple items are combined into a complete meal.

Office suite example 1Willingness to pay

User Type No. of users Word processor spreadsheet

A 40 $50 $30

B 40 $30 $50

Questions based on example 1

• Assume that the production costs are zero• What is the revenue maximizing price if you

can only sell word processor and spreadsheet separately? What is the corresponding revenue?

• What is the revenue maximizing price if you can sell a bundle of word processor and spreadsheet? What is the corresponding revenue?



A 40 $50 $50

B 40 $30 $30


• What is the revenue maximizing price if you can only sell word processor and spreadsheet separately? What is the corresponding revenue?

• What is the revenue maximizing price if you can sell a bundle of word processor and spreadsheet? What is the corresponding revenue?

• Under what circumstances, would bundling work?



A 40 $50 $30

B 40 $30 $50

C 40 $60 $13


• What is the revenue maximizing price if you can only sell word processor and spreadsheet separately?

• What is the revenue maximizing price if you can sell a bundle of word processor and spreadsheet?

• What is the revenue maximizing price if mixed-bundling is available?

Bundling

• The key advantage of bundling is to remove the downward sloping feature of the demand curve, so that you don’t have to lower the price to sell more.

• But bundling works for this purpose only if different types of consumers’ willingness-to-pay are negatively correlated across products (e.g., type A people value word processor much higher than spreadsheet, but type B people value spreadsheet much higher than word processor.)

Types of Price Discrimination

• First Degree Price Discrimination– Charge every customer a different price based on

their value or willingness-to-pay.

• Second Degree Price Discrimination– Different prices for different product characteristics:

versioning, volume discounts, etc.

• Third Degree Price Discrimination– Different prices based on observable characteristics

(age, etc.)

Two-part tariffs (1)

• Consider a situation where there is only one firm (monopoly). Marginal cost is a constant. Everyone has the same willingness-to-pay (WTP), i.e., everyone has the same demand curve.

• Now let’s figure out what this monopoly should charge for a fixed part(f) and a variable part(p).

• Example: theme park, cell phone plan, etc.


• Consider a linear demand curve, and constant marginal cost (mc).

• Forget about the fixed part for now. The optimal price, p*, for a monopoly to charge is to set MR=MC.

Two-part tariffs figure 1P

Q

MC

Q*

P*

A

B

MR


• Given this monopoly price, the profit is the area B (the rectangle above mc), the consumer surplus (CS) is the area A (the triangle above B).

• So if the monopoly is allowed to charge a fixed fee in addition to per unit price, at most they can charge f=A. So the total profits = A+B

Two-part tariff (4)

• Now let’s consider charging a lower price, p’<p*. Before imposing the fixed part of the fee, the new profit is area B’, and the consumer surplus is A’. Therefore, the new fixed fee that a monopoly can set is f’=A’. Now the total new profits under two-part tariffs = A’+B’. Note that A’+B’ > A+B.

Two-part tariffs figure 2

P

Q

MC

Q'

P* A'

B'

P'

Two-part tariff (5)

• It should be clear that the lower the price, the higher the total profits. Let CS(p) be the consumer surplus associated with p (prior to paying the fixed fee). So the optimal price should be p**=mc, the firm should set f**=CS(mc). In this case, the monopoly can extract the maximum consumer surplus, and achieve the highest possible profits. The outcome is the same as the first degree price discrimination.

3rd degree price discrimination under two-part tariff

• The previous section assumes all consumers are the same. Let’s consider an extension where there are two types of consumers: “light” users and “heavy” users.

• If we observe consumer type, then the monopoly can practice 3rd degree price discrimination. Suppose that the demand curves for “light” users and “heavy” users look like the following.

Two-part tariff (6)P

Q

MCA

B

D1 (Light) D2 (Heavy)

3rd d degree price discrimination under two-part tariff

• Under this situation, it is clear that the monopoly should offer two pricing plans.

• Plan 1: p1 = mc, f1 = area A

• Plan 2: p2 = mc, f2 = area A+B• Assume that there is one consumer for each

type. The total profits from these two consumers will be (2*A + B).

2nd degree price discrimination under two-part tariff

• Under 2nd degree price discrimination, the monopoly cannot discriminate consumers based on observable characteristics. It can only offer a menu of pricing plans for the consumers. The consumers will select the pricing plan that suits them best.


• Consider the same plans that a monopoly offers under the 3rd degree price discrimination. It is clear that no one will buy plan 2. When both consumers choose plan 1, the profits that a monopoly makes is 2*A. This serves as the benchmark profits that we want to beat.


• What can we change the parameters of the pricing plans so that “light” and “heavy” users will choose the one that is designed for them? More importantly, we want to make more profits than 2*A.


• First, we want to make sure that the “heavy” users will not pretend to be the “light” users and take plan 1. So, we need to somehow make plan 1 less attractive to them. In order to do so, we increase the per unit price for plan 1 to p1’> mc.

• We recreate figure 3 to the following figure. Note that A = A1+A2+A3, B=B1+B2.


P

Q

MCA2

B1

D1 (Light) D2 (Heavy)

A1

B2A3

p1'

Q1'


• When we set p1 = p1’>mc, the fixed fee for plan 1 should be adjusted to A1. That is,

• Plan 1’: p1 = p1’, f1 = A1• Now, suppose that the heavy user chooses

plan 1’. Then his net consumer surplus becomes B1. So this is still more attractive than plan 2 (which gives him zero net consumer surplus).


• How can we modify plan 2 so that the heavy user will not choose plan 1’?

• We can reduce the fixed fee for plan 2 such that his net consumer surplus will also be B1.

• If we keep p2’ = mc, we should then set f2’=A+B2. That will make the heavy user indifferent between plan 1’ and plan 2’ (to make sure the heavy user chooses plan 2’, we can make f2’ one cent lower).


• Now, let’s figure out the profits.• From plan 1’, profits = A1 + (p1’- mc)*Q1’ = A1

+ A2 = A – A3. (Recall that A = A1+A2+A3)• From plan 2’, profits = A + B2.• Total profits = (A – A3) + (A+B2) = 2*A + (B2 –

A3).• As long as B2 > A3, we beat the benchmark

profits of just selling plan 1, which is 2*A.


• Note that the profit under the 2nd degree price discrimination is lower than that under the 3rd degree price discrimination. The lower profit represents the price that the firm needs to pay in order to get consumers to self-select the right plan when their type is not observable.

Determinants for using yield management

• Capacity is limited and perishable.• Commitments need to be made when future demand is

uncertain.• Customers will reserve ahead of time.• The firm knows that there are different segments of

consumers. Each segment has a different demand curve. – How can we tell the differences of these consumers? (say for

airline)• The firm can sell the units at different prices (e.g., fare

classes), and the same unit of capacity can be used to deliver many different products or services.

An example of yield management (Dynamic Pricing)

• A hotel has established two fare classes: full price and discount price.

• It has 210 rooms available on Apr 4.• To simplify the problem, let’s assume that leisure

demand occurs first, and then business demand occurs.

• Need to determine how many rooms we need to protect (i.e., reserve) for the full price payers.

• If too many rooms are protected, then there may be empty rooms when Apr 4 arrives.

• If too few rooms are protected, then the hotel forgoes the extra revenue it may received from business customers.

Booking limits and Protection levels

• Booking limit: the max. number of rooms that may be sold at the discount price.

• Protection level: the number of rooms that we will not sell to leisure customers.

• Booking limit = Capacity – Protection level.• In the above example,

– Booking limit = 210 – Protection level.

How to determine the optimal protection level?

• Suppose that the current protection level is Q+1.

• 210 – (Q+1) rooms have already been sold. That is, the booking limit has been used up.

• A customer calls and wants to reserve a room at the discount price.

• Should the hotel lower the protection level from Q+1 to Q to accommodate him?

Example (cont’d)

• It depends on:– The magnitude of full and discount prices.– The anticipated demand for full price rooms.

• Discount price = $105, full price = $159.• Let D be a random variable that represents the

anticipated demand for rooms at the full price.• Let’s assume that the demand is derived directly from

123 days of historical demand (see Table 1 of Netessine and Shumsky http://archive.ite.journal.informs.org/Vol3No1/NetessineShumsky/NetessineShumsky.pdf )

http://archive.ite.journal.informs.org/Vol3No1/NetessineShumsky/NetessineShumsky.pdf



Booking Limit Decision with Data

• F(Q) = Prob(D≤Q), is the cumulative probability. • If we decide to keep the protection level =Q+1,

then we may or may not be able to sell the (Q+1)th room.

• If we keep the protection level unchanged:– What is the probability that we cannot sell the

(Q+1)th room? Why?– What is the probability that we can sell the (Q+1)th

room?

• Protecting Q+1 rooms has an expected value: (1-F(Q))*159 + F(Q)*0 = (1-F(Q))*159.

• Lowering the protection level from Q+1 to Q has the expected value = 105.

• So, we should lower the protection level to Q as long as: (1-F(Q))*159 ≤ 105.

• Or, F(Q) ≥ (159-105)/159 = 0.339.• Suppose Q+1 = 85. Should we lower the protection

level?• What is the optimal protection level? A: 79

A general formula

• The analysis above has been generalized by Belobaba (1989). The technique is called “expected marginal seat revenue”.

• Two fare classes: Rl and Rh, where Rh > Rl • The optimal protection level is determined by:

F(Q*) ≥ (Rh - Rl)/ Rh and F(Q*-1) < (Rh - Rl)/ Rh

• Why does an airline overbook a flight?

Pricing Examples. Bundling In marketing, product bundling offers several products for sale as one combined product. This is common in the software business.

Documents

optimal price

corresponding revenue

bundle of word processor

types of price discrimination

people value word processor

office suite example

product bundling

single office suite