Dynamic Pricing &

Revenue Management
PGDM 2, Term IV

Operations & Analytics Elective

 3. Differential Pricing
The Basic Price Optimization Problem Tactics for Price Differentiation
Optimality Conditions Group Pricing
Channel Pricing
Applying Basic Price Optimization
Regional Pricing
Marginality Test
Couponing and Self-selection
Maximizing Revenue
Product versioning
Weighted Combinations of Revenue &
Time-based differentiation
Contribution
Product versioning or Group pricing
Price Differentiation
Volume discounts
The Economics of Price Differentiation
Calculating differentiated prices
Limits to Price Differentiation Optimal Pricing with Arbitrage
Optimal Pricing with Cannibalization
Finding the best segmentation
 The basic price optimization problem

𝑈𝑛𝑖𝑡 𝑀𝑎𝑟𝑔𝑖𝑛𝑖 = 𝑈𝑛𝑖𝑡 𝑃𝑟𝑖𝑐𝑒𝑖 - 𝐼𝑛𝑐𝑟𝑒𝑚𝑒𝑛𝑡𝑎𝑙 𝐶𝑜𝑠𝑡𝑖

Total contribution= σ𝑛𝑖=0 𝑈𝑛𝑖𝑡 𝑚𝑎𝑟𝑔𝑖𝑛

For a supplier selling a single product at a single price, her total

contribution will be m(p) = (p - c)d(p). Hence, the basic price
optimization problem is 𝒎𝒂𝒙
𝒑 𝒑 − 𝒄 𝒅(𝒑)

Often m(p) is uni-modal or even concave. If so, solve the

optimization problem by setting m’(p) = 0, or 𝑑 ′ 𝑝 𝑝 − 𝑐 +
𝑑 𝑝 =0
Optimality Conditions

▪Total contribution is maximized in the basic price optimization

problem at the price at which marginal revenue(MR) equals
marginal cost(MC).

▪At the optimal price, the contribution margin ratio is equal to the
reciprocal of elasticity
Applying Basic Price Optimization

A widget maker is looking to set the price of

widgets for the current month. If the widget
maker’s unit production cost c is a constant
$5.00 per widget and that his demand for
the current month is governed by the linear
price-response function
 Margin
Price Demand Unit margin
Marginality Test
contribution
$8.74 3008 $3.74 $11,249.92
$8.75 3000 $3.75 $11,250.00
$8.76 2992 $3.76 $11,249.92
30800 31200 31200 30800
30000 30000
28800
27200
25200
22800

20000

16800

13200
10800 11200 11200 10800
10000 10000
8800 8800 9200
7200 7200
6000 5600 5200 4800 5200 4800
4400 4000 3600 3200
2800 2800 2400 2800
2000 1600 1200 800
0 400 0
5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5
Demand Revenue Profit
Maximizing Revenue
The incremental cost is almost zero for some services like movie theatres, sporting events, .
Maximizing total contribution is almost like maximizing total revenue:

When R(p) is uni-modal or even concave, we resort to optimality condition

Revenue-maximizing price occurs when elasticity is equal to 1.

Example
The CEO of the widget-making company decides that the firm’s goal for the next month will
be to maximize revenue from widget sales as part of the long-term strategy to increase
market share.
The revenue-maximizing price can be found

▪The resulting revenue-maximizing price equals $6.25, with corresponding sales of 5,000 units.
The corresponding per-unit margin is $1.25;

▪The total contribution margin is $5,000* $1.25 = $6,250.

▪We can compare this to the maximum margin contribution of $11,250 and conclude that

▪ To maximize total revenue, the company needs to give up $11,250 -$6,250 =$5,000 of
contribution margin
Weighted Combinations of Revenue & Contribution
Sometimes, a company might wish to maximize a weighted combination of total
contribution and revenue. The most common approach to combining ,revenue contribution
is to use a weighting parameter 𝛼 with 0 ≤ 𝛼 ≤ 1 resulting in a weighted objective function
𝑍 𝑝 = 𝛼 𝑝 − 𝑐 𝑑 𝑝 + 1 − 𝛼 𝑝 𝑑(𝑝)
= 𝑝 − 𝛼𝑐 𝑑 𝑝

The price that maximizes a weighted combination of revenue and contribution is greater
than or equal to the revenue-maximizing price and less than or equal to the contribution-
maximizing price.
Price Differentiation

First degree
❑Practice of charging different prices to different
customers, either for exactly the same good or for slight
different versions of the same good.
Second degree
❑Usage of nomenclature as price differentiation in lieu
of price discrimination
Third degree
Types of price differentiation & their implementation forms
Price Differentiation
Market

High WTP Low WTP

segments segments

Higher prices Lower prices

There is a variety of ways, depending on
❑the characteristics of the market

❑the competitive environment

❑the character of the goods/ services being sold

Necessary conditions for successful discrimination

ABILITY TO SEGMENT THE MARKET

• The firm must be able to identify different market segments

(domestic users and industrial users)

SEGMENTS SHOULD HAVE DIFFERENT PRICE ELASTICITIES

• E.g. refer to the figure, Household appliance brand in Ten European

regions

SEPERATION BY TIME, PHYSICAL DISTANCE OR NATURE

OF USE
• E.g. Microsoft Office ‘Schools’ edition which is only available to
educational institutions, at a lower price
Necessary conditions for successful discrimination

➢There must be no seepage between the two markets, which means that a consumer cannot

purchase at the low price in the elastic sub-market, and then re-sell to other consumers in the

inelastic sub-market, at a higher price

➢The firm must have some degree of monopoly power

The Economics of Price Differentiation

One way to look at the price-response function

10000
d(p) = (10000-800p) is that,

Demand
6000 d(p) = 10000 – 800 p
• 10,000 customers are willing to pay 0 or more, C
3000
• 5,000 customers are willing to pay $6.25 or more, A B
• 3,000 customers are willing to pay $8.75 or more, and p = $5.00 p = $8.75 p = $12.50
Price

• 1,200 customers are willing to pay $11 or more

The marketing department might see an opportunity to improve profitability.

What if the seller could determine the maximum amount that each customer would be willing
to pay and could charge that amount to everyone willing to pay> $5?
 More Speculation
At the current optimal single price $8.75, total profit is
(8.75- 5)* 3,000 = $11,250

If charging everyone the price he is wtp (first-degree price discrimination),

Total profit is (12.5-5) * 6,000/2 = $22,500

Suppose the market can be segmented into those

wtp > $7 & those wtp =< $7

We have
d1(p) = min {4,400; (10,000 - 800p) }
d2(p) = (5,600 - 800p)
 Two-market HIGH Willingness to Pay

Segmentation d1(p1) = min[4400,(10000 – 800 p1)]

Demand
4400

Price $12.50

Suppose the seller can perfectly identify customer as belonging to one group or the
other and then offer each customer the appropriate price, without any opportunity
for resale or arbitrage between the two groups.
For the above-7 segment, total contribution
• m1(p) = 4,400(p -5) when 5 <= p <= 7 and
• m1(p) = (10,000 - 800p)(p - 5) when 7 <= p <= 12.5.
• When maximizing m1(p), we find (p1)* = $8.75 and m1(p1)* = $11,250.
 Two-market LOW Willingness to Pay

Segmentation

Demand
5600
d2(p2) = (5600 – 800 p2)

$7.00
Price

For the below-7 segment, the total contribution

• m2 (p) = (5,.600-800p )(p - 5) when 5 <= p <= 7
• When maximizing m2(p), we find (p2)* = $6 and m2(p2)* = $800
This $800 is extra from price differentiation; also, 800 more customers get to
purchase.
It is a win-win situation
 Limits to Price Differentiation Arbitrage
Price differentials create a strong
incentive for third-party
arbitrageurs to buy the product at
the low price and resell to high wtp
customers below the market price,
keeping the difference to
themselves
.

Cannibalization
Under differential pricing, customers
in high-price segments are highly
motivated to find a way to pay the
lower price.
Imperfect Segmentation When 10% of the above-seven
The brain-scan technology customers can pay the low price $6,
required to determine the the benefit of price differentiation
precise willingness to pay of would be eliminated
each customer has not yet .
been developed
.
Tactics for Price Differentiation

group pricing ❑Allows sellers to increase profitability by

charging different prices to customers with
channel pricing different wtp

❑Imperfect segmentation, cannibalization,

regional pricing
or arbitrage can destroy or even reverse
couponing and self-selection the benefits of price differentiation

❑Though with strong incentive to position

product versioning
and price their products differently, sellers
need to plan and manage their pricing
time-based differentiation
carefully
volume discounts
Group Pricing

Examples
Group pricing is the tactic of • Favourable terms offered to large
offering prices to different customers like Wal-Mart,
groups of customers for exactly • Lower prices offered to
the same product. government, educational, and
non-profit organizations.

For group pricing to succeed, four criteria must hold:

• An unambiguous indicator of group membership(student ID)
• A strong correlation between group membership and price sensitivity
• Product or service not easily traded among purchasers
• Culturally and legally acceptable segmentation.
 Group Pricing in
Reality
The criteria for group pricing are so stringent that its pure form rarely appears
in direct consumer sales. It is more common in services

▪Services are often sold directly by the supplier so that Disneyworld, for
example, can check a customer's age before selling a child's ticket

▪Many service, such as health care and haircuts, are intrinsically non-
transferable, so arbitrage is not an issue

Pure group pricing is also common in B2B sales

Channel Pricing Online
vs
Outlets

Here are some examples:

• Special “Web-only" fares for airline tickets are
Channel pricing is the practice of available through the Internet but not travel agencies
• Many fashion and home furnishing merchants offer
selling the same product for lower prices through mail-order catalogues than retail
outlets
different prices through different • Causes for channel pricing are differences in costs
(selling via travel agencies is more costly than through
distribution channels. a web site) and differences in price sensitivities (for
personal loans, customers inquiring online are more
price-sensitive than those contacting a call center)
Regional Pricing

In certain places, McDonalds sells hamburgers for higher prices in wealthy neighbourhoods
than in poorer ones

A roundtrip New York-Tokyo ticket purchased in Japan will usually cost more than the
same ticker purchased in the United States

A glass of beer costs more at an airport bar than at the corner bar

Price differentiation here is based on the supplier's desire to exploit differences in price
sensitivity between locations.
 Couponing and Self-selection

Group pricing is both difficult because it requires customer categorization based on price sensitivity
and unpopular because it often seems “unfair" to consumers.
It is more convenient to differentiate prices in ways that allow customers to self-select. Both the list
price and a discounted price are available to all customers, but it takes additional time, effort, of
flexibility to obtain the discounted price.
Examples

Retailers offer discount coupons Movies theatres charge the lower

through newspapers, direct mail, prices for a weekday matinee than for
and magazines a Saturday night show

Brand-name retailers such as Ralph

Manufacturers offer mail-in
Lauren and Gap operate outlet stores
rebates for purchases of a good in somewhat out-of-the-way locations
•Developing an “inferior" variant and/or “superior" variant of an existing
product.
For inferior goods, consider the following examples:
• Exxon/Mobil and Shell sell excess gasoline in bulk at low prices to so-called “off -brand"
independent dealers who resell it under their own brands.
• A well-known premium wine producer sells some of its production under a different label at
about half the price.
• Apple has its iPad Mini following the success of iPad.
Productapplication
Complex Versioning software packages such as SCM or ERP software are often
sold at different prices, depending on the number of features involved.
Superior Goods and Product Lines

Spendrup, Sweden's largest brewery,

marketed its Old Gold as a premium brand A product line is a series of similar products
and maintained a price 25% to 50% higher serving the same general market but sold at
different prices.
than its other brands, even the former did
Dell PCs form a vertical product line, where
not stand out in comparative taste tests. each product has higher performance than the
product just below it in the line.
Proctor-Silex priced its top model much
differently than the next brand, exploiting The QuickBook financial software come in
different versions.
the segment of the market that wants to
buy the best, despite the cost.
Product-line Pricing

A hotel that charges more for an ocean-view room than a parking-lot-view room.

Hertz one-day rental offers six different products at Seattle airport, ranging from economy
rental to one involving a luxury car.

For service company like Hertz, creation of a product line creates opportunities for
upgrading. A rental car company or a cruise line has the ability to oversell lower-quality car
types and upgrade customers into higher-quality types.

Not only does this provide the company with greater flexibility to manage its inventory, but
being upgraded is usually viewed favourably by customers.

Product line has the advantage of being perceived to be fair by customers.

Time-based Differentiation

Time-based differentiation is a common form a product versioning

• Amazon offers 5- to 9-day “Super-saver" shipping free while charging $3.97 for “standard shipping"

• Passenger airlines offer discount rates to customers who book a week or more prior to departure

• Software- and hardware-support contracts charge more for “two-hour response" than for “two-day
response"

• Fashion goods cost more during the beginning of the season and are marked down toward the end
of the season

Time-based differentiation plays a very important role at passenger airlines, hotels,

and rental car companies, in which time of booking is used as an indicator of whether
or not a potential customer is traveling for leisure or business.
Product Versioning or Group Pricing?

Airline pricing like many successful examples of price differentiation includes

elements of both group pricing and product versioning

The airlines consciously created restricted discounted fares as an inferior

product. They did so, however, as a way to enable them to offer different fares to
different customer groups: lower fares to leisure travelers and higher fares to
business travellers

Pure group pricing is very difficult to pull off in consumer markets

Volume Discounts

➢Volume discount is a time-honoured tactic

Reasons for volume discounts
➢Buy more to save more

➢A six-pack beer costs less than 6 times the • Transaction or order costs
cost of a single bottle • Decreasing marginal utility
➢Larger boxes of laundry detergent cost less
• Increasing price sensitivity
per kg than smaller boxes
• Bargaining powers
➢Volume discount is prevalent in wholesale and
B2B selling too • Seller's risk aversion
Arbitrage
Optimal Pricing with Arbitrage
Regional pricing is subject to arbitrage whenever a product can be purchased in a
low-price region and transported cheaply to be resold at a higher price elsewhere.

For this reason, global companies often set price bands for various markets to
avoid resales from low-price countries that would cannibalize sales in higher-price
countries.

A computer chip manufacturer finds that the contribution-maximizing prices for

his chips are $2.54 in the United States and $2.43 in Brazil.

However, if it costs $0.08 per unit to ship chips from Brazil to the United States, he
will not be able to charge those prices due to the potential for arbitrage between
the two countries.
Mathematical Formulation

A manufacturer sells a common product in n different locations. The

delivered cost in country i is ci and the price-response curve faced by the
supplier in the country is di(.). The cost for an arbitrageur to transport the
product from country i to country j is aij
To avoid arbitrage, the key is to ensure pj <= pi + aij for all (i; j)-pairs.
To find optimal prices pi in regions i, we solve the following:
 12000 16000

Exercise
14000
10000
12000
8000 10000

6000 8000
6000
4000
4000
2000 2000

0 0
5 6 7 8 8.75 9 10 11 12 12.5 5 6 7 8 9 10 10.5 11 12 13 14 15 16

A supplier is selling hammers in two cities, Pleasantville and Happy Valley. It costs him $5.00 per
hammer delivered in each city. Let p1 be the price of hammers in Pleasantville and p2 be the price of
hammers in Happy Valley. The price-response curves in each city are:
Pleasantville: d1(p1) = 10,000 - 800p1
Happy Valley: d2(p2) = 8,000 - 500p2
a. Assuming the supplier can charge any prices he likes, what prices should he charge for hammers in
Pleasantville and Happy Valley to maximize total contribution? What are the corresponding demands
and total contributions?

Pleasentville Happy valley Supplier

Price $ 8.75 $ 10.50
Demand 3000 2750 5750
Total contribution $ 11,250 $ 15,125 $ 26,375
b. An enterprising arbitrageur discovers a way to transport hammers from Pleasantville to Happy Valley
for $0.50 each. He begins buying hammers in Pleasantville and shipping them to Happy Valley to sell.
Assuming the supplier does not change his prices from those given in part a, what will be the optimal
price for the arbitrageur to sell hammers in Happy Valley? How many will he sell? What will his total
contribution be? (Assume that Happy Valley customers will buy hammers from the cheapest vendor.)
What will happen to the total sales and contribution for the supplier? (Remember that he is now selling
to the arbitrageur too.)
Arbitrageor
$ 0.50 $ 9.25
Pleasentville Transport cost Price
Price Demand Revenue Cost Contribution Pleasentville Happy valley Supplier
$ 8.75 3000 $ 26,250.00 $ 15,000.00 $ 11,250.00 Price $ 8.75 $ 10.50
Demand 3000 2750 5750
Happy valley Total contribution $ 11,250.00 $ 15,125.00 $ 26,375.00
Price Demand Revenue Cost Contribution
$ 10.50 2750 $ 28,875.00 $ 13,750.00 $ 15,125.00 with arbitrage Pleasentville Happy valley Supplier
Supplier $ 10.49 2755 $ 24,106.25 $ 13,775.00 $ 10,331.25 Price $ 8.75 $ 10.50
Arbitrageor $ 10.49 2755 $ 28,899.95 $ 25,483.75 $ 3,416.20 Demand 5755 0 5755
Total contribution $ 21,581.25 $ - $ 21,581.25
c. The supplier decides to eliminate the arbitrage opportunity by ensuring that his selling price in
Happy Valley is no more than $0.50 more than the selling price in Pleasantville (and vice versa). What
is his new selling price in each city? What are his corresponding sales and total contribution?

Price Demand Revenue Cost Contribution

Pleasentville 9.23 2,615 $ 24,142.01 $ 13,076.92 $ 11,065.09 i
Happy valley 9.73 3,135 $ 30,502.22 $ 15,673.08 $ 14,829.14 j
Total 5750 Total $ 25,894.23

Arbitrageor
$ 0.50
Transport cost
d. From among the Pleasantville buyers, the Happy Valley buyers, and the seller, who wins and who
loses from the threat of arbitrage?

without arbitrage Pleasantville Happy valley Supplier

Price $ 8.75 $ 10.50
Demand 3000 2750 5750
Total contribution $ 11,250.00 $ 15,125.00 $ 26,375.00
with arbitrage Pleasantville Happy valley Supplier
Price $ 8.75 $ 10.50
Demand 5755 0 5755
Total contribution $ 21,581.25 $ - $ 21,581.25
smart supplier Pleasantville Happy valley Supplier
Price $ 9.23 $ 9.73
Demand 2615 3135 5750
Total contribution $ 11,065.09 $ 14,829.14 $ 25,894.23
Optimal Pricing with Cannibalization
Morals from Cannibalization
Total contribution as a function of the cannibalization fraction
α
Finding the Best Segmentation
Consumer Surplus
Consumer Surplus

Price differentiation can be good for sellers. But is it good for buyers? This practice can
be viewed as unfair and hence unpopular among certain population.

A differential pricing scheme that improves total consumer welfare is more

defensible to both the public and potential regulators.

The key concept is consumer surplus. The surplus of an individual consumer is the
difference between his WTP and the price at which he purchases, if he purchases, and is
zero if he does not purchase.

The total consumer surplus is the sum of individual surpluses

Price
Differentiation
& Consumer Under perfect differentiation, there will be
zero consumer surplus
Welfare

Price differentiation can be good for both

seller and consumers

Most schemes result in both winners and

losers among consumers
Profit and consumer surplus from the two-price case
Review
The Basic Price Optimization Problem
Optimality Conditions, Applying Basic Price
Optimization, Marginality Test
Maximizing Revenue, Weighted Combinations of
Revenue & Contribution

Price Differentiation
The Economics of Price Differentiation, Limits to Price
Differentiation

Tactics for Price Differentiation

Group Pricing, Channel Pricing, Regional Pricing, Couponing
and Self-selection, Product versioning, Time-based
differentiation, Product versioning or Group pricing

Volume discounts

Calculating differentiated prices

Optimal Pricing with Arbitrage
Optimal Pricing with Cannibalization
Finding the best segmentation