Basic Price Optimization

Basic Price Optimization
Lecture 5
The Price Response Curve, d(p)
• How demand for a product varies as a function of
• A function for each element in the PRO cube –
each combination of product, market-segment
and channel
• Similar to market demand function in economics
• E.g. different book sellers – effectiveness of
marketing campaigns, perceived customer
differences in quality, product differences,
location etc.
• E.g. wheat – a commodity, indifference among
offerings, perfect knowledge about prices and
will buy only from the lowest-price seller
• Each seller is relatively small
• Must only worry about how much to produce
• No need for Pricing and Revenue Optimization
Properties of price-response curve
• PRO decisions have time associated with them
– minutes/hours in ecommerce, days/weeks in
retail, longer as in long-term contract pricing
• Nonnegative
• Continuous – no gaps or jumps, thus invertible
• Differentiable – can take a derivative
• Downward sloping
For the rest we will ignore:
• Giffen goods – a student with an $8 budget per
week for dinner; burger $1 and steak $2, the
latter being eaten on Sundays; the price of burger
goes up to $1.1 – and its demand goes up to 7;
VERY rare in reality
• Price as an indicator of quality – wine: a market
with many alternatives and some „lazy“ buyers,
who use price as a proxy
• Conspicuous consumption – a rock star drinking
$300 bottles of Cristal champagne and drive
Price sensitivity: slope, a local estimator
• Semiconductors, current price $0.13 per chip,
slope 1000 chips/week per cent; 2 cent
increase -= 2000 chips per week; 3 cent
decrease += 3000 chips per week; depends on
units – pounds and cents vs. tons and dollars
Price sensitivity: elasticity, a local
estimator, arc and point elasticity
• Percentage change in demand to the
percentage change in price; does not depend
on units
• Semiconductors, current price $0.13 per chip,
at 10000 chips per month; thinks his price
elasticity is 1.5
• Thus, a 15% increase in price, from $0.13 to
$0.15, would lead to a decrease in demand of
about 1.5*15%=22.5%: from 10000 to 7750
Short and long run elasticity
• Gasoline – short run 0.2, long run 0.7; similar
milk – 20c price rise and long run response
• Durables (such as automobiles and washing
machines) – long run elasticity is lower than
• Level of analysis – market or company
Short and long run elasticity
Willingness to pay
• Maximum for one customer– the reservation
• A uniform
corresponds to a
linear priceresponse function
• Partitions the price response function into a totaldemand component and a willingness-to-pay
• E.g. total demand varies seasonally, while the wtp
distribution remains constant over the period of time
• We decompose our goal into estimating total demand
and estimating the price response
• An advertising campaign will not increase the total
population, but will shift wtp distribution
• When we open a new store, the total demand will be
determined by the population served, but we can use
wtp from similar demographics populations
• Wtp can change – e.g. temperature sensitive Coca-Cola
vending machines
• A sudden windfall or a big raise may increase an
individual’s maximum wtp – if these are uncorralated
population will be unchanged as a whole, but
systematic changes will have the price-response
function shifting
• Does not incorporate additional induced demand –
price reduction may have one buying two pairs of socks
• Wtp framework is better for „big ticket“ consumer
items and industrial goods
Linear wtp
• Is not a realistic global model
• If a competitor offers a close substitute, it will
only work close to the market price
C is a parameter: d(1) = C
These two – 3.10 from above, second from wtp slide nr 13
Constant elasticity price-response
• Not a realistic global model – in reality we expect
elisticity to change as price changes
• If e<1 (inelastic demand) R’(p)>0, seller can
increase revenue by increasing price
• If e>1, seller can increase revenue by decreasing
• If e=1 then price change does not change revenue
• Wtp function is highly concentrated near zero
• Assumes, that the distribution of wtp drops
steadily as price increases, but only approaches 0
What is a realistic price-response
Car with market price of $13000
At $20k a few loyal customer will buy
… and there is no big change from $20k->$21k
If we are selling at $9k, almost everyone, who
wants to buy a compact car, will buy from us
(except the few very loyal to a competitor)
• … and there is no big change from $9k->$8k
• But at about the market price, elasticity is high
– At market price, many more will buy from us, when
we are selling at $250 below, and even more will shift,
if we ask $500 below the market price
The logit price response functions
b =0.0005; 0.001; 0.01; p=$13000;
Price sensitivity is the highest at p^=-(a/b)
We will fix p^=$13000, thus a=-$13000*b
C is the market size=20000
As b grows, the market is more price sensitive,
approaching perfect competition
• Empirical research has shown the logit
function to be present on a wide range of
Price response with competition
1. Incorporating Competiton in the PriceResponse Function
• Often the prices are not available at the time
• In business-to-business markets they are never
made visible to the competition
• Retailers do not have the time or resources to do
exhaustive research on a daily basis
• BUT price-response function will be based on
history and thus it already includes „typical“
competitive pricing
2) Consumer-Choice Modeling
• E.g. Online Petroleum Information System
– Different grades of petroleum offered by different
sellers in all major markets in USA and Canada
• Different wtp’s represent the values that are
placed on features and „brand value“
associated with each product
• Surplus is the difference between the price
and the wtp
Let p(p1,p2,..,pn) – vector of prices
Let μi be the market share of product i
μi=fi(p) for all i
Each alternative has a share between 0 and 1
Each buyer chooses one product
Increase in price decreases the market share
Increasing the price of a product increases the
market shares of all competitors (substitutes)
The multinominal logit, b being price
• Note that Koshiba is highly sensitive and gets a
low market share when compared to Cacophonia,
which is twice as expensive
The multinominal logit and the logit
price-response function
• When other prices are constant the MNL
reduces to LPRF
• Assume that other prices are constant, we set:
• Where a=ln(k) and we know that: 1/K=e^(-lnK)
To state the important
• If competitive prices are stable MNL provides
little predictive value over LPRF
• There is a vast literature on consumer-choice
modeling; Statistical packages such as SAS
include procedures for estimating the
parameters of the logit and probit marketshare functions
• But there are weaknesses, namely: ….
• Assumption: everyone purchases one
alternative; what if some don’t purchase at all
– since their wtp is below all prices? Thus we
cannot say that market size D is independent
of the prices offered: an aggressive discount
not only siphons customers away from
competitors, but will also make someone pay,
who might not have purchased at all
• We theoretically need information on all
alternatives, whereas usually only some major
competitors are considered
• A solution here is to derive a competitive
index price by weighting the prices of major
competitors and using this as a single
competitive price in multinominal logit
Price response with competition
3)Anticipating competitive response
• More strategically: if we drop a price, will the
competitors match; if we raise a price – once
again, what will the competitors do?
• The approaches here fall under decision
analysis and game theory – and there is a vast
literature on the use of game theory in
strategic pricing, but we are concidering
tactical decisions of PRO
• For example if customers in a market choose a
supplier based on MNL price response model,
then our action is to maximize our expected
contribution given the competitors prices by
setting our price
• The philosophy of pricing and revenue
optimization is to make money by many small
adjustments, searching for and vacuuming up
small and transient puddles of profit as they
appear in the marketplace
• Many of the price adjustments will fall below the
radar screen of the competition, and will not
trigger any explicit response
• But potential retaliations by competitors should
still be considered: e.g. Hertz in the early 1990s
by communicating its upcoming sophisticated
PRO system in the price war eager car rental
industry – it was able to generate additional
revenue through thousands of small adjustments
to prices, that the competitors were unable to
Incremental costs
• The incremental cost of a customer
commitment is the difference between the
total costs a company would experience if it
makes the commitment and the total cost it
would experience if it doesn’t.
• An airline: additional meal and fuel cost +
commissions or fees paid for booking
• A retailer – buying a stock of fashion goods: zero
• A drugstore – ordering a number of bottles of
shampoo weekly: wholesale unit cost
• A distributor – bidding for a yearly contract with a
hospital: expected cost of purchases, but also
cost of customer service, operating costs and
holding costs due to variance in orders and the
The incremental costs are (closely
related to ABC):
• Forward looking: some costs are sunk, others have still
to be made
• Marginal: made for this customer, may not be the same
as the average cost of similar past commitment
• Not fully allocated: not the overhead or fixed cost of
staying in business
• Can depend on the type, size and duration of the
commitment: for a multiunit order setup costs are
allocated across all the units
• May be uncertain: Roadway Express, a trucking
company – covering all the freight tendered by the
customer at an agreed-on tariff
The basic price optimization problem,
to maximize the margin, m(p) total
contribution; c is incremental cost
• Marginal revenue (derivative of total revenue
with respect to price) should equal marginal
cost (always negative) – total contribution is
Price-response function d(p)=10000-800p
Incremental cost $5
Marginal revenue R’(p)=10000-1600p
Marginal cost = -$4000
• If marginal revenue is greater than marginal
cost, contribution can be increased by
increasing price
• If marginal revenue is lower than marginal
costs, price should be decreased – in order to
increase contribution
Optimal contribution margin and
• We can rewrite 3.17 as 3.20, where e(p) is
point elasticity; since the second term in 3.20
is always positive:
• If elasticity at our price is <1, we can increase
total contribution by increasing price
• If in 3.20 m’(p) = 0 then we get 3.21
• (p-c)/p is the margin per unit expressed as a
fraction of price – ‘the contribution margin
ratio’ – if you purchase at $150 and sell at
$200, it is ($200-$150)/$150=0.33
• Thus, we derive a relationship between price
elasticity, contribution margin and the optimal
price as on the next page
At the optimal price, the contribution
margin ratio is equal to the reciprocal
of elasticity
• Let us derive p* from 3.21
• At the current price, an electronics goods
retailer seems to face a constant-elasticity
price-response function, e=2.5, for a TV set
• Thus by 3.21 1/2.5=40% for a contribution
margin; and for p*=(2.5/1.5)*180=$300 in
order to maximize total contribution
And… we have imputed elasticity
• A seller thinks, he is pricing optimally, and his
contribution margin ratio is 20%
• This can only be true, if the price elasticity is 5
• Imputed price elasticity is a good ‘reality
check’ on the credibility of a company’s
current pricing
An example – a widget maker
• Unit production cost is constant at $5
• Demand is given by a linear price-response fn:
– d(p)=10000-800p
• Thus, demand will be 0 for prices above $12.5
An example – a widget maker
• Demand is given by a linear price-response fn:
– d(p)=10000-800p
• Thus, d’(p)=800
• Using 3.18, we see that the optimal price must
solve: 10000-800p*=800(p*-$5)
• Which is 1600p*=14000; or p*=8.75
An example – a widget maker
Total sales:10000-800*$8.75=3000 units
Total revenue: 3000*$8.75=$26250
Total contribution: 3000*($8.75-$5)=$11250
TC is 0 at p=c=$5, max at p*=$8.75 and 0 at
An example – a widget maker
• Marginality check
Maximizing revenue
• Sometimes (e.g. in order to ‘buy’ more market
share), we want to maximize total revenue
(not total contribution) (or when c = 0)
• Revenue maximizing price occurs where the
elasticity of the price-response function is 1
• Maximum revenue – marginal revenue =0
• Maximum contribution – marginal revenue =
marginal cost
• As in the figure, marginal revenue is a decreasing
function of price, at least in the region of optimal
• As we see, revenue maximizing price is lower
than the contribution maximizing price
Widget company ‘buys’ more market share
From 3.23 we see that:
-800*p*+10000-800p*=0; thus p*=$6.25
and from d($6.25) = 10000-800*p->d=5000
Per unit margin is $1.25 and total contribution
margin is 5000*$1.25=$6250
• Maximum was $11250 – thus we gave up $5000
• And ‘bought’ an additional 2000 units of
• We should at least check for the constraint that
states price > incremental costs, to be out of red
Weighted combinations of revenue
and contribution
• A company might want to maximize a
weighted combination of total contribution
and revenue, using 0<= α <=1 for that
• α=1 means total contribution maximization;
α=0 means revenue maximization
• 3.26 means maximization of contribution with
a discounted cost
• The price that maximizes a weighted
combination of revenue and contribution is
greater than or equal to the revenuemaximizing price and less than or equal to the
contribution-maximizing price.
• The core problem of PRO is a constrained
optimization problem, where the objective
function is to maximize total contribution
• The constraints can be business rules (e.g.
desire to maintain a minimum market share)
or constraints on capacity or inventory
• The constraints may overrule the optimal
price found as a result of calculations
• A key input is the price-response function
• There are three approaches to incorporating
competitive pricing into price optimization;
the first assumes competitive pricing is
already present in the price-response function
– it is correctly used in a stable environment,
or when prices are not available; in a more
volatile environment competitive prices are
included in a broader price-response model;
the final approach tries to anticipate
competitive response to one’s actions – this is
usually done only when a strategic pricing
change is contemplated

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