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Dynamic pricingDynamic pricing

By: Dr. A. H. Kashan [email protected] Department of IE, Tarbiat Modares UniversityBy: Dr. A. H. Kashan [email protected] Department of IE, Tarbiat Modares University

By: Dr. Ali Hosseinzadeh Kashan

IntroductionIntroductionIntroductionIntroductionHere, we look at settings in which prices rather than quantity controls are the primary variables used to manage demand.Th di ti ti b t tit d i The distinction between quantity and price controls is not always sharpClosing the availability of a discount class can be Closing the availability of a discount class can be considered equivalent to raising the product’s price to that of the next highest classp gHere the techniques are distinguished by their explicit use of price as the control variable

By: Dr. A. H. Kashan [email protected] Department of IE, Tarbiat Modares University

IntroductionIntroductionIntroductionIntroductionVarying prices is often the most natural mechanism for revenue management.To respond to market fluctuations and uncertainty in demand firms To respond to market fluctuations and uncertainty in demand firms use various forms of dynamic pricing:

personalized pricing, markdowns markdowns, promotionscoupons,didiscounts,clearance sales, auctions and price negotiations: B2B sales are often conducted th h RFP hi h ll fi t d t i i through a RFP process, which allows firms to determine prices on a transaction basis.

How to make such price adjustments in a way that maximizes


How to make such price adjustments in a way that maximizes revenues?

Price or quantity based RM Price or quantity based RM Price or quantity based RM Price or quantity based RM Some firms have more price flexibility than quantity flexibilityIn apparel retailing, firms commit to order quantities well in advance of a sales season and may even commit to certain stocking levels in each store. stocking levels in each store. Often, it is impossible (or very costly) to reorder stock or reallocate inventory from one store to another. It is easier (though not costless) for most retailers to change prices, This may require only changing signage and making data This may require only changing signage and making data entries into a point-of-sale system. Online retailers in particular enjoy tremendous price


p j y pflexibility because changing prices is almost costless.

PricePrice--based RM is the preferred option!based RM is the preferred option!PricePrice based RM is the preferred option!based RM is the preferred option!Quantity-based RM operates by rationing the quantity sold to different products or to the quantity sold to different products or to different segments of customers.But it involves reducing sales by limiting But it involves reducing sales by limiting supply.If one has price flexibility, we can reduce If one has price flexibility, we can reduce sales by increasing price.The same quantity-reducing function is The same quantity reducing function is achieved; but with more profitably By increasing price, we both reduce sales


y g p ,and increase revenue at the same time.

Price or quantity based RM Price or quantity based RM Price or quantity based RM Price or quantity based RM

Practical business constraints dictate which of price/quantity based RM or a mixture of both is most appropriate.pp p


ExamplesExamplesppStyle-Goods Markdown Pricing

Retailers f st le and seas nal ds se Retailers of style and seasonal goods use markdown pricing to clear excess inventory before the end of the seasonbefore the end of the season.E.g., apparel, sporting goods, high-tech, and

i h bl f dperishable foodsFirms have an incentive to sell inventory while they can, even at a low price, rather than salvage it.


Explanations for markdown pricingExplanations for markdown pricingExplanations for markdown pricingExplanations for markdown pricing

Firms set high prices for all items initially because they d hi h d ill b ldo not now which products will become popularPopular products are the ones for which customers have high reservation prices, have high reservation prices, So, these sell out at the high initial price. The remaining items are identified as low-reservation-gprice products and marked down.

Wiki di I Wikipedia: In micro economics reservation price is the highest price a buyer is willing to pay or the smallest price at which a seller is willing to sell. Reservation


p gprices are commonly used in auctions


Customers who purchase early have higher willingness to pay, Because they can use the product for a full

season or Because there is some cache to being the first to own itown itMarkdown pricing then serves as a segmentation mechanism to separate price-insensitive mechanism to separate price insensitive customers from those price-sensitive customers willing to defer consumption to get a lower


price.


On holidays and during peak-shopping y g p pp gperiods customers can search for the lowest prices more efficiently p yThey are actively engaged in search, making many shopping trips over a concentrated y pp g pperiod of time.Demand during peak periods is more price-Demand during peak periods is more pricesensitive and retailers respond by running “sales” during these periods


sales during these periods.

ExamplesExamplesExamplesExamplesDiscount Airline Pricing

N ll d i i i i l i d iNot all dynamic pricing involves price reductionsDiscount airlines use primarily price-based RM, but with prices often going up over time.p g g pThese airlines typically offer only one type of ticket on each flight, a non-refundable, one-way fare without advance-purchase restrictions.advance purchase restrictions.They offer these tickets at different prices for different flights, D i h b ki i d f h fli h i During the booking period for each flight, vary prices dynamically based on capacity and demand for that specific departure.


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ExamplesExamplesExamplesExamplesConsumer-Packaged Goods Promotions

P i h i Promotions are short-run, temporary price reductions (soap, diapers, coffee, yogurt, and so on).Customers are aware of past prices and past p p ppromotions, So running promotions too frequently may condition customers to view the brand as a condition customers to view the brand as a frequently discounted product, cutting into brand equity in the long run. Because customers are aware of past prices Because customers are aware of past prices, promotions impact their subjective “reference price”—or sense of the “fair” price—for products.


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Modeling Dynamic PriceModeling Dynamic Price--Sensitive DemandSensitive DemandAny dynamic-pricing model requires a model of how d d i h i di id l d demand, either individual or aggregate, responds to changes in price.In dynamic-pricing problems some additional factors In dynamic pricing problems some additional factors must be considered.Customers behave over time

what factors influence their purchase decisions and how sophisticated their decision-making process is

Th f k dThe state of market conditionsthe level of competition andthe size of the customer population


the size of the customer population.

MyopicMyopic--Versus StrategicVersus Strategic--Customer ModelsCustomer ModelsCustomer ModelsCustomer Models

Most of the models we assume myopic customers

Myopic customers : those who buy as soon as the offered price is less than their willingness to pay

Myopic customers do not refuse to buy in the hope of lower prices in the future.

Strategic customers will optimize their own purchase behavior in response to the pricing strategies of the firmsbehavior in response to the pricing strategies of the firms.

The pricing problem becomes modeling a game between the


firm and its customers, in which we must analyze the equilibrium using game-theoretic tools.

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MyopicMyopic--Versus StrategicVersus Strategic--Customer ModelsCustomer Models

Such a demand model makes the pricing problem a i d li d b h strategic and complicated game between the customers

and the firm.In many situations, customers are sufficiently In many situations, customers are sufficiently spontaneous in making decisions that one can ignore their strategic behavior.Customers often do not have the sufficient time or information to behave very strategically. However the more expensive and durable the However, the more expensive and durable the purchase, the more important it becomes to model strategic customer behavior (for example, automobile b i i h h d f d l )


buyers waiting to purchase at the end of a model year).

One common defense of the myopic One common defense of the myopic iiassumptionassumption

Assuem the most price-sensitive customers tend to adopt a strategy of postponing their purchases until end-of-season clearance sales.The estimated price sensitivity in these later periods will tend to appear much higher than in earlier periods. We do not model the strategic behavior directly, however our forecasting models indirectly capture the correct price response.


InfiniteInfinite--Versus FiniteVersus Finite--Population ModelsPopulation ModelsAn important assumption in demand modeling: the population of potential customers is finite or infinite?

Infinite-population model: we are sampling with replacement when observing customers.

We can view this as a case where customers immediately consume their purchase and then reenter the population (nondurable-goods assumption in economics)

Result: 1. The distribution of the number of customers and the distribution

f h i illi i ff d b h hi f of their willingness to pay is not affected by the past history of observed demand.

2. One does not need to retain the history of demand (or a suitable sufficient statistic) as a state variable in a pricing-optimization


sufficient statistic) as a state variable in a pricing-optimization problem.

InfiniteInfinite--Versus FiniteVersus Finite--Population ModelsPopulation ModelsThe finite-population model assumes a random process without replacement.

There are a finite (possibly random) number of customers with heterogeneous willingness to pay

l values.

If one of the customers in the population purchases If one of the customers in the population purchases, the customer is removed from the population of potential customers.

We can consider it as a case where the good being purchased is consumed over a long period of time (d bl d )


(durable-goods assumption).

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InfiniteInfinite--Versus FiniteVersus Finite--Population ModelsPopulation ModelsInfiniteInfinite Versus FiniteVersus Finite Population ModelsPopulation ModelsSuppose we assume a price p(t) is offered in period tand all customers who value the item at more than p(t) p( )purchase in period t (myopic behavior).

Under a finite population model we know that after Under a finite-population model, we know that after period t the remaining customers all have valuations less than p(t).

The future distribution of willingness to pay is conditioned on the values being less than p(t). g p( )

In formulating a dynamic-pricing problem, we have to keep track of past pricing decisions and their effect on


keep track of past pricing decisions and their effect on the residual population of customers.

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Infinite Infinite vsvs FiniteFiniteInfinite Infinite vsvs FiniteFiniteThe infinite-population model is reasonable when:1.There is a large population of potential

customers 2.The firm’s demand represents a relatively

ll f f h l small fraction of this population 3. The impact of the firm’s past sales on the

b f d h di ib i number of customers and the distribution of their valuations is negligible.

It i l bl f bl dBy: Dr. A. H. Kashan [email protected] Department of IE, Tarbiat Modares University

It is also reasonable for consumable goods.

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Infinite Infinite vsvs FiniteFiniteInfinite Infinite vsvs FiniteFiniteThe finite-population model is reasonable p pwhen:1. The firm’s demand represents a large . e s e a ep ese ts a a ge

fraction of the potential pool of customers 2 The product is a durable good 2. The product is a durable good, 3. Past sales will have a more significant

impact on the statistics of future demandimpact on the statistics of future demand,


Infinite Infinite vsvs FiniteFiniteInfinite Infinite vsvs FiniteFiniteThe two models lead to quite different pricing policies.Finite-population models typically lead to price ki i ti l t t skimming as an optimal strategy, Price skimming: prices are lowered over time in such a way that high valuation customers pay such a way that high-valuation customers pay higher prices earlier while low valuation customers pay lower prices in later periods.p y p pThis creates, segmenting customers with different values for the good and charging


differential prices over time

Infinite Infinite vsvs FiniteFiniteInfinite Infinite vsvs FiniteFiniteIn infinite-population models, there is no p pprice-skimming incentive. The distribution of customer valuations e st but o o custo e va uat o s does not shift over time, The same price that yields a high revenue in The same price that yields a high revenue in one period will yield a high revenue in later periods periods, A firm has no incentive to deviate from this revenue maximizing price


revenue-maximizing price.

Monopoly, Oligopoly, and PerfectMonopoly, Oligopoly, and Perfect--Competition ModelsCompetition Models

Another key assumption in dynamic-pricing y p y p gmodels concerns the level of competition the firm faces.Monopoly models: the demand a firm faces is assumed to depend only on its own price and p y pnot on the price of its competitors.The model does not explicitly consider the The model does not explicitly consider the competitive reaction to a price change.


Monopoly, Oligopoly, and PerfectMonopoly, Oligopoly, and Perfect--Competition ModelsCompetition Models

Oligopoly models: the equilibrium price response of competitors is explicitly modeled and computed.

Competition ModelsCompetition Models

p p y p

The assumption that firms behave rationally may result in a poor predictor of their actual price result in a poor predictor of their actual price response.

The difficulty in collecting competitor data to estimate the models accurately has made the models less popular in practice.p p p

The assumption of no competitive response is sometimes better than the approximating


sometimes better than the approximating assumption of preexisting optimal behavior

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Monopoly, Oligopoly, and PerfectMonopoly, Oligopoly, and Perfect--C titi M d lC titi M d l

Perfectly competitive models: M f l d l

Competition ModelsCompetition Models

1.Many competing firms supply an identical commodity.

2 The output of each firm is assumed to be 2.The output of each firm is assumed to be small relative the market size,

3.Each firm is offering identical commodities4.A firm cannot influence market prices5.Each firm is essentially a price taker—able to

ll h it t t th ili sell as much as it wants at the prevailing market price but unable to sell anything at higher prices.


g p

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Monopoly, Oligopoly, and PerfectMonopoly, Oligopoly, and Perfect--

The assumption that firms have no pricing Competition ModelsCompetition Models

power means that the results are not that useful for price-based RM.N th l th d l l i titNevertheless, they do play a role in quantity-based RM.One can interpret the capacity control models One can interpret the capacity-control models of Chapters 2 and 3.Firms take the price for their various products Firms take the price for their various products as given (set by competitive market forces), and control only the quantity they supply at these


competitive prices.

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SingleSingle--Product Dynamic Pricing Product Dynamic Pricing Wi h R l i hWi h R l i hWithout ReplenishmentWithout Replenishment

There is a single product over a finite sales There is a single product, over a finite sales horizon given a fixed inventory at the start of the sales horizon of the sales horizon. The firm is a monopolist, customers are myopic and there is no replenishment of myopic, and there is no replenishment of inventory.Th d l i f h The models are representative of the type used in style and seasonal goods retail RM.


SingleSingle--Product Dynamic Pricing Product Dynamic Pricing Wi h R l i hWi h R l i hFor such retailers, production and ordering Without ReplenishmentWithout Replenishment

p gcycles are typically much larger than the sales season, The main challenge is to determine the price path of a particular style at a particular store p p y plocation, given a fixed set of inventory at the beginning of the season.g gDemand is a function solely of time and the current price


current price

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SingleSingle--Product Dynamic Pricing Product Dynamic Pricing Wi h R l i hWi h R l i hp(t):The price decision at each time t.Without ReplenishmentWithout Replenishment

d(t,p): demand rate Ωp:The set of allowable pricesp pΩd:The set of achievable demand rates.The demand functions are continuously ydifferentiable and strictly decreasing,The demand functions are bounded above and below and tend to zero for sufficiently high prices.


SingleSingle--Product Dynamic Pricing Without Product Dynamic Pricing Without R l i hR l i hReplenishmentReplenishment

The revenue functions r(t,p)=pd(t,p) or ( p) p ( p)equivalently r(t,d)=dp(t,d) are finite for all p∈Ωpp p

The marginal revenue as a function of demand d,d,

is strictly decreasing.


SingleSingle--Product Dynamic Pricing Without Product Dynamic Pricing Without R l i hR l i h

The demand function can also be expressed

ReplenishmentReplenishment

pas d(t,p)=Nt(1-F(t,p)) Nt is the market-size parameter t s t e a et s e pa a ete F(t,p) is the fraction of the market with willingness to pay less than pwillingness to pay less than p.x(t) denote the inventory at time t=1,…,T


Deterministic ModelsDeterministic ModelsDeterministic ModelsDeterministic ModelsGiven an initial inventory x(0)=C select a y ( )sequence p(t) of prices (inducing demand rates of d(t,p(t)) that maximize total revenues.( p( ))


Deterministic ModelsDeterministic ModelsDeterministic ModelsDeterministic ModelsLet π* be the Lagrange multiplier on the inventory constraintconstraint,The first-order necessary conditions for the optimal rates d*(t) and multiplier π* are

Since r(t,d) is concave, these conditions are also sufficient.The Lagrange multiplier has the interpretation as the


The Lagrange multiplier has the interpretation as the marginal opportunity cost of capacity.

Deterministic ModelsDeterministic ModelsDeterministic ModelsDeterministic ModelsThe first condition says that the marginal y grevenue should be equal the marginal opportunity cost of capacity in each period.pp y p y pThe complementary slackness condition says that the opportunity cost cannot be positive pp y pif there is an excess of stock.


ExampleExampleExampleExampleConsider a two-period selling horizon,The first period demand is d1=-p1+100The second period demand is d2=-2p2+120Customers in the second period are more price-sensitivePurchase behavior is assumed to be myopic.Considered separately, the revenue-maximizing price for the first and second periods are


ExampleExampleExampleExampleSuppose the firm’s capacity is 40. pp p yHow should it divide the sale between the two periods?two pe o s?r(1,d1)=p1d1=(100-d1)d1 J(1,d1)=-2d1+100r(2 d )=p d =(60 d /2)d J(2 d )= d +60r(2,d2)=p2d2=(60-d2/2)d2 J(2,d2)=-d2+60


ExampleExampleExampleExample


The greedyThe greedy allocationallocation algorithmalgorithmThe greedyThe greedy allocationallocation algorithmalgorithm

The total revenue is maximized at the point where the marginal values for the two periods are approximately the same If they were not equal, the firm would reallocate capacity to the higher marginal-value period.The greedy allocation algorithm, works based on the observation that the marginal revenues in all periods are equal at optimalityDiscretize the capacity C into M units of size δ


each, C=Mδ

The greedyThe greedy allocationallocation algorithmalgorithmThe greedyThe greedy allocationallocation algorithmalgorithmThe greedy algorithm proceeds by allocating demand in d l h l discrete amounts so as to equalize the marginal revenue




The greedyThe greedy allocationallocation algorithmalgorithmThe greedyThe greedy allocationallocation algorithmalgorithm

Assign units to the period with highest marginal revenue until all six units are used up. The algorithm terminates with and all six units are allocated and the marginal revenues

li dare equalized

P id d h i l i d i Provided the marginal revenue is decreasing in each period, the greedy procedure produces an optimal (discretized) solution


produces an optimal (discretized) solution.

the Timethe Time--Homogenous CaseHomogenous Casethe Timethe Time Homogenous CaseHomogenous CaseWhen demand is time-homogenous, i.e., gd(t,p)=d(p) for all t, the optimal price J(p*)=π*

is the same in each period. p

This shows that prices fluctuate from period This shows that prices fluctuate from period to period in the deterministic model, only as a result of changes in the demand function over result of changes in the demand function over time.

See the details in the book


See the details in the book

Discrete PricesDiscrete PricesWe often would like to choose prices from a discrete set.For example, prices such as $24.99 or $149.99 or fixed percentage markdowns (such as 25% ff 50% ff)off or 50% off)

denotes the sales rate at time t when using the price p


when using the price pi

Discrete PricesDiscrete PricesDiscrete PricesDiscrete PricesThe discreteness of the prices imposes p ptechnical complications Because the problem is no longer continuous ecause t e p ob e s o o ge co t uous or convex. One can overcome this difficulty by relaxing One can overcome this difficulty by relaxing the problem to allow the use of convex combinations of the discrete prices (or combinations of the discrete prices (or demand rates).


Discrete PricesDiscrete PricesDiscrete PricesDiscrete PricesIn most periods, the optimal solution will be p pto use only one of the discrete prices; In the remaining periods, the solution has t e e a g pe o s, t e so ut o as the interpretation of allocating a fraction of time to each of several prices.p


Discrete PricesDiscrete PricesDiscrete PricesDiscrete Prices

This is a linear program in the variables α(t) so it is easy to solve numerically.t s easy to so ve u e ca y.To relate the solution to the unconstrained price case, introduce a dual variable on the


pcapacity constraint

Discrete PricesDiscrete PricesDiscrete PricesDiscrete PricesThe optimal solution α*(t) in each period is th h t i d b l ithen characterized by solving

Where π*>=0 and α(t*) are convex weights ( ) gsatisfying the complementary slackness condition

The objective function is linear


The objective function is linear

Discrete PricesDiscrete PricesIf there is a unique index i* for which is greatest then the optimal solution is greatest, then the optimal solution is Which corresponds to using the discrete price pi*.

If there is more than one such value i* then there If there is more than one such value i*, then there will be multiple solutions.Determining which is optimal can be resolved by Determining which is optimal can be resolved by appealing to the complementary slackness condition Such a choice could result in a fractional solution in Such a choice could result in a fractional solution in which αi(t)>0 for two or more valuesUse the price i for a fraction α (t) of period t


Use the price i for a fraction αi(t) of period t

ExampleExampleExampleExampleConsider a two week selling seasong

The firm is constrained to offer prices in the set 40 50 70set 40, 50, 70.




Maximum Concave EnvelopeMaximum Concave EnvelopeMaximum Concave EnvelopeMaximum Concave EnvelopeCertain discrete prices may never be optimal to use and can in fact be eliminated from the use and can in fact be eliminated from the problem.If for a given price pj there exist convex weights If for a given price pj there exist convex weights ai(t) such that

a convex combination of other prices a convex combination of other prices produces strictly higher revenue

bi i f h i a convex combination of other prices consumes no more capacity than using pj

Th th i p i ti l t ti t


Then the price pj is never optimal at time t

Maximum Concave EnvelopeMaximum Concave EnvelopeMaximum Concave EnvelopeMaximum Concave Envelope

di(t)


InventoryInventory--Depletion EffectDepletion EffectInventoryInventory Depletion EffectDepletion EffectA practical factor affecting dynamic pricing is the adverse effects of low inventory levels.This is called a broken-assortment effect.Consider an item contains several stock keeping units (SKU)-such as color-size combinations ( )When inventories run low, certain SKUs may be out of stock even though there is a positive g pinventory for the item as a whole.The resulting reduction in alternatives naturally


g yreduces the sales rate at any given price.

InventoryInventory--Depletion EffectDepletion EffectInventoryInventory Depletion EffectDepletion EffectThe inventory-depletion effects can be y pmodeled by making the demand rate a function of inventory as well as of price and y ptime,We can use a variety of functional formsy


InventoryInventory--Depletion EffectDepletion EffectInventoryInventory Depletion EffectDepletion EffectOne must keep track of the inventory at p yeach time in the optimization problem.


See stochastic models in th t t b kthe text book


SingleSingle--Product Dynamic Product Dynamic Pricing with ReplenishmentPricing with ReplenishmentInventory can be replenished at a cost in each period,Both pricing and inventory decisions need to p g ybe made; Pricing decisions are used to control demand,gReplenishment decisions are used to control supply.pp yThe central problem is to optimally coordinate these demand and supply decisions.


pp y

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Deterministic ModelsDeterministic ModelsDeterministic ModelsDeterministic ModelsAssume a single good with an end-of-period g g pinventory, denoted x(t) that can be replenished over time.ht : per-unit holding cost for inventory in period t pct: unit cost for replenishmenty(t): the amount ordered in period ty(t): the amount ordered in period t


Unconstrained CapacityUnconstrained CapacityUnconstrained CapacityUnconstrained CapacityThere is no capacity constraint on the p yamount ordered in each period.


Solution methodSolution methodFor s<=t define: γ is the cost of satisfying demand in period t with γst is the cost of satisfying demand in period t with supply from period s.The lowest cost for supplying period t is:The lowest cost for supplying period t is:

s*(t): an index that achieves the minimum on the s (t): an index that achieves the minimum on the right-hand side above.The optimal sales rate in any period t, d(t*) is The optimal sales rate in any period t, d(t ) is determined by equating the marginal revenue to the lowest marginal cost,


Solution methodSolution methodSolution methodSolution methodThe optimal quantity to order in period s is simply determined by adding up the sales rates from later periods t whose lowest-cost supply is from period s.


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Capacity Constraints on Capacity Constraints on OrderingOrderingThere are capacity constraints on the order p yquantities: y(t)<=bt

While the unconstrained capacity model can W e t e u co st a e capac ty o e ca be solved as a nonlinear program with these added capacity constraints, there is a simpler p y , papproach.


A greedy algorithmA greedy algorithmA greedy algorithmA greedy algorithmFor discretizes the sales quantities, we can qsolve the problem under the assumption that the marginal revenue in each period is g pdecreasing.For a fixed vector of rates

Where g(d) is the minimum cost for meeting th l t d


the sales rates d,

A greedy algorithmA greedy algorithmA greedy algorithmA greedy algorithm

Th f(d) i th ti l fit i th Thus, f(d) is the optimal profit given the demand rates d.Computing f(d) is efficient because the Computing f(d) is efficient because the minimization problem to determine g(d), is a minimum-cost network-flow problem


minimum cost network flow problem.

A greedy algorithmA greedy algorithmA greedy algorithmA greedy algorithm

Th t ith 1 i th t d i ll th


et: The vector with a 1 in the component and a zero in all other components.

A greedy algorithmA greedy algorithmA greedy algorithmA greedy algorithmAt each stage the algorithm simply adds an g g p yincrement δ of demand to the period that yields the largest net gain and stops when y g g pno period produces a positive net gain.


See stochastic models in th t t b kthe text book


Multiproduct, Multiproduct, MultiresourceMultiresourcePricingPricing

Two fundamental factors typically link the yp ypricing decisions for multiple products.First, demand for products may be correlated. st, e a o p o ucts ay be co e ate . When products are substitutes or complements the price charged for one complements, the price charged for one product effects the demand for other related productsproducts.Second, products may be linked by joint capacity constraints


capacity constraints.

Multiproduct, Multiproduct, MultiresourceMultiresourcePricingPricingAn increase in the price of a packet of p ppotato chips will not just cause a drop in demand for potato chips but will likely also p p yincrease the demand for corn chips. At the same time, these products may , p yoccupy the same limited shelf space, Stocking more of one product may require Stocking more of one product may require stocking less (or none) of other products.


Multiproduct, Multiproduct, MultiresourceMultiresourcePricingPricingDue to the complexity issues, many commercial

li i f d i i i d l applications of dynamic-pricing models assume unrelated products and independent demands. They solve a collection of single-product models as They solve a collection of single product models as an approximation.When cross-elasticity or resource-constraint effects are strongProducts are only slightly differentiated, C st mers are er rice sensiti e Customers are very price-sensitive, Joint capacity constraints are tight

Then ignoring multiproduct effects can be severely


Then ignoring multiproduct effects can be severely suboptimal.

Deterministic Models WithoutDeterministic Models WithoutReplenishmentReplenishmentThere are n products indexed by jp y jThere are m resources, indexed by iThere is a horizon of T periods with each There is a horizon of T periods, with each period indexed by t.

The revenue-rate function satisfies the regularity conditions (See slides 30 and 31).


Deterministic Models WithoutDeterministic Models WithoutReplenishmentReplenishmentProduct j uses a quantity aij of resource i.j q y ij

The matrix A=[aij] describes the bill of materials for all n products.ate a s o a p o ucts.

: Capacities for m resources


Deterministic Models WithoutDeterministic Models WithoutR l i h tR l i h tReplenishmentReplenishment

r(t,d) is concave in d, and the following Kuhn-Tucker conditions are necessary and sufficient

π* is the optimal dual price on the joint- capacity constraints, having the interpretation as the


, g pvector of marginal opportunity costs

Deterministic Models WithoutDeterministic Models WithoutReplenishmentReplenishmentReplenishmentReplenishmentThe marginal revenue for each product should

l th i l t it t f th equal the marginal opportunity cost of the resources used by product jExample: For a given path j on the network, the Example: For a given path j on the network, the revenue function is time homogeneous and log-linear

Where p- is interpreted as a reference price for Where p j is interpreted as a reference price for itinerary j, aj is the demand rate at the reference price, and єj is the magnitude of the elasticity of d d h f i


demand at the reference price.





Deterministic Models with Deterministic Models with ReplenishmentReplenishment

where x(t) is an m-vector of inventory levels at the end of period ty(t) is an order of quantities in period t


ActionAction--Space ReductionsSpace ReductionsActionAction Space ReductionsSpace ReductionsOne simplification for multiproduct dynamic pricing problems: Express the problem in terms of resource-

i h h h d d dconsumption rates rather than the demand rates dThis yields an equivalent formulationWi h f l d d di i li With often a greatly reduced dimensionality that can be much easier to solve.C id th f th d t i i ti d l Consider the case of the deterministic model Where there is only one resource but nr d cts


products.

ActionAction--Space ReductionsSpace ReductionsActionAction Space ReductionsSpace ReductionsThis could be a situation similar to the single gresource problem of Chapter 2 But we control the demand for each product j, ut we co t o t e e a o eac p o uct j, dj by adjusting its price pj.The deterministic problem:The deterministic problem:


ActionAction--Space ReductionsSpace ReductionsActionAction Space ReductionsSpace ReductionsExpress the problem in terms of the aggregate p p gg gdemand rate rather than the individual demand rates

For a given define:


ActionAction--Space ReductionsSpace ReductionsActionAction Space ReductionsSpace ReductionsIf r(t,d) is jointly concave in d, then will be concave in

The solution proceeds in two steps: First solve the above model to determine the optimal aggregate sales rate, Then solve the previous model at each time to disaggregate


p gg gthis optimal aggregate rate into a optimal vector of sales rates (equivalently prices) for each product.

ActionAction--space reduction for general space reduction for general l i d l il i d l i bl blmultiproduct multimultiproduct multi--resource problemresource problem

The problem can be reduced to one with only p ym demand rates (one for each resource) rather than the original n rates (one for each product).a


ActionAction--space reduction for general space reduction for general l i d l il i d l i bl blmultiproduct multimultiproduct multi--resource problemresource problem

Reductions show that the complexity of the p ymultiproduct, multi-resource dynamic pricing problem is caused not by the number of p yproducts but by the number of resources


FiniteFinite--Population Models and Population Models and Price SkimmingPrice SkimmingA finite-population model assumes that we p psample customers without replacement from a finite number of potential customers. pThe history of demand affects the distribution of both the number and valuations of the remaining customers


Myopic CustomersMyopic CustomersMyopic CustomersMyopic CustomersCombined with the finite-population assumption, thi b h i b l it d b th fi t this behavior can be exploited by the firm to achieve price skimmingThe firm seeks to price discriminate among the The firm seeks to price discriminate among the finite population of customers.Assume there is a finite population size N Customers in this population have valuations v that are uniformly distributed on the intervalAs an approximation assume that sales can As an approximation, assume that sales can occur in fractions, so the population can be regarded as continuous.


Myopic CustomersMyopic CustomersMyopic CustomersMyopic CustomersConsider a firm that sells a fixed capacity C of a product to the population over T time-periods. The firm is free to set different prices in each period. Wh h l ?What is the optimal pricing strategy?The optimal prices are decreasing over time, The only customers left at time t are those with values less than the minimum price ff d i i d 1 t 1


offered in periods 1,…,t-1

Myopic CustomersMyopic CustomersMyopic CustomersMyopic CustomersThe firm will sell nothing if it posts a price in g p pperiod t that is higher than the minimum price offered in the past.p p

If p(t)<=p(t-1) for all t, the revenue generated

in t is:


Myopic CustomersMyopic CustomersMyopic CustomersMyopic CustomersAssume for simplicity that C > N,

The first constraints are redundant.

Th fi t d diti i l th ti l The first-order conditions imply the optimal unconstrained solution must satisfy ( )


Myopic CustomersMyopic CustomersMyopic CustomersMyopic CustomersThe optimal pricing strategy effectively segments the customers into T+1 groups based on their valuations.W it th ti l i i i d t We can write the optimal price in period t as

The first term on the right, is simply the single-period revenue maximizing priceperiod revenue-maximizing price.Because the remaining customers in period t have values uniformly distributed on [0,p*(t-1)]


have values uniformly distributed on [0,p (t 1)]

Myopic CustomersMyopic CustomersMyopic CustomersMyopic CustomersTherefore, the optimal price in period t is p p phigher than the single-period revenue-maximizing price for period tg p pIf the number of periods T increases, the firm’s revenues increaseThe optimal total revenue for T periods is


Myopic CustomersMyopic CustomersMyopic CustomersMyopic CustomersAs T tends to infinity, the firm achieves yperfect price discrimination because the firm can continuously lower prices from ū down y pto zero over the interval [0, T].Each customer ends up paying a price p p y g parbitrarily close to his valuation.


Myopic CustomersMyopic CustomersMyopic CustomersMyopic CustomersThe solution will not be feasible if C<N.T/(T +1).The optimal price is modified so that only those customers above a lower limit are segmentedg


PromotionsPromotionsPromotionsPromotionsPromotions are short-run, temporary price changes that are frequently applied to replenishable and consumable goodsP ti ith b th f t Promotions are run either by the manufacturer (trade promotions) or by retailers (retail promotions or consumer promotions).promotions or consumer promotions).Manufacturers are interested in increasing sales or profits for their brand, pRetailers are interested in overall sales or profits for an entire product category.


PromotionsPromotionsPromotionsPromotionsA promotion generally increases sales to both the retailer and manufacturer,,Promotions cause an increase in demand because:Higher household inventories lead to fewer stockouts and therefore an increase in consumption;therefore an increase in consumption;

With higher inventories customers don’t have to worry about replacing the inventory at higher prices.L h h ld i t f t t if Larger household inventory causes faster usage rates if product usage occasions are flexible (snack foods), products need refrigeration, or products occupy a prominent place in the pantryprominent place in the pantry.

Customers switching from nondiscounted brands to the discounted brands C ( il f d d l ) k ili k


Customers (or retailers for trade deals) stockpiling to take advantage of the low price (forward buying).

PromotionsPromotionsPromotionsPromotionsOne dominating factor behind the demand gincrease is the type of product. Products such as yogurt and potato chips tend o ucts suc as yogu t a potato c ps te to see an increase because of increased consumption.pFor products such as tomato ketchup, diapers, and toilet paper the sales Increase is primarily and toilet paper, the sales Increase is primarily because of brand switching or stockpiling.


PromotionsPromotionsPromotionsPromotionsPromotions, in the framework of RM, can be thought of as either:

(1) a manufacturer using price to dispose ( ) a a u actu e us g p ce to spose excess inventory(2) a manufacturer trying to gain market (2) a manufacturer trying to gain market share to induce customers to try out its productsproducts(3) retailers experimenting with price to find optimal price points


find optimal price points

PromotionsPromotionsPromotionsPromotions(4) Separating price-sensitive customers, ( ) p g pwho are willing to use coupons or who wait for deals, (5) Retailers trying to increase store traffic, as customers once inside the store are likely yto purchase other, nonpromotional items, (6) A tactic for store brands or small firms (6) A tactic for store brands or small firms to compete against the large advertising budgets of the established brands


budgets of the established brands.

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