Basic Price Optimization

Lecture 5

The Price Response Curve, d(p)

• How demand for a product varies as a function of price

• 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 Bentleys

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 short-run

• Level of analysis – market or company

Short and long run elasticity

Willingness to pay

• Maximum for one customer– the reservation price

• A uniform willingness-to-pay distribution corresponds to a linear price-response function

• Partitions the price response function into a total-demand component and a willingness-to-pay component

• 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 functions

• 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 price• 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 function?

• 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 markets

Price response with competition

1. Incorporating Competiton in the Price-Response 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 sensitivity

• 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 market-share functions

• But there are weaknesses, namely: ….

Weaknesses

• 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 competition3)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 match

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.

Examples

• 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 returns

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 maximized

• Price-response function d(p)=10000-800p• Incremental cost $5• Marginal revenue R’(p)=10000-1600p• Marginal cost = -$4000• P*=$8.75

• 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 elasticity

• 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• p*=8.75• 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 p=$12.5

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 price• 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

demand• 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 revenue-maximizing price and less than or equal to the contribution-maximizing price.

…moreover

• 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