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Outline

Introduction

The Model

Results

Multi-ChannelPricing

Computational

Analysis

Conclusionsand OngoingResearch

Optimal Pricing and Production Planning forSubscription-Based Products

W. T. Huh S. Kachani A. Sadighian

Department of Industrial Engineering and Operations Research,Columbia University, New York

MSOM Annual Meeting - Beijing, ChinaJune 18-19, 2007

Huh, Kachani, Sadighian Optimal Pricing and Production Planning

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Outline

Introduction

The Model

Results


Computational

Analysis


Outline

1 Introduction

MotivationLiterature Review

2 The ModelDynamics of the model

DP Representation3 Results

One Stage RepresentationMonopolyDuopoly

4 Multi-Channel Pricing5 Computational Analysis

Symmetric FirmsAsymmetric Firms

6

Conclusions and Ongoing ResearchHuh, Kachani, Sadighian Optimal Pricing and Production Planning

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Outline

IntroductionMotivation

LiteratureReview

The Model

Results

Multi-Channel

Pricing

ComputationalAnalysis


MotivationSubscription-Based Products (Magazines)

Specific business model:

Subscriber BaseAdvertisement Sales vs. Product Sales

Example: Magazine Industry - Columbia Journalism

Review (CJR) one of the leading Journalism Magazines inUS

Sources of Revenue:

Newsstand single copy SalesSubscription Revenue

Advertisement Sales

Circulation revenue for 2006 > $10 billion

Advertisement revenue: 54% of total salesMagazine sales through subscription: 32% of total sales


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Outline


LiteratureReview

The Model

Results

Multi-Channel

Pricing



Literature Review - Quantity and Pricing Models

Environment: Both price and quantity decisions

Inventory Replenishment Problem

Main theme: Choice of correct quantity to

maximize(minimize) revenue(cost) in multiple periodsOne special case: Newsvendor

Models for join quantity-price decision making

Petruzzi and Dada (1999) Price-setting newsvendorFedergruen and Heching (1999) single retailer withstochastic demand, comprehensive model on monopolist(refer for further literature review)


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Outline


LiteratureReview

The Model

Results

Multi-Channel

Pricing



Literature Review - Competition Models

Competitive Newsvendor

Parlar(1988), later Lippman and McCardle (1997), Firmscompeting over the overflow of the unmet demand of theother firms

General Existence results

Kirman and Sobel (1974): No spill-over, backlog or lost,demand as a function of priceBernstein and Federgruen (2004): Demand as a functionof price and a second measure service level

Ahn and Olsen (2005): Magazine problem, built onsupermodular games, prove existence of a Markov PerfectEquilibrium


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Outline


LiteratureReview

The Model

Results

Multi-Channel

Pricing



Goals

Develop a model for joint pricing and production planningfor subscription-based products

Model should capture both stand-price andsubscription-price decisions

Model should yield either closed-form or computationallytractable solutions

Extend our model to the duopoly setting

Extend our model to address multi-channel pricing of

subscription-based products with consumers choiceTest our results numerically and apply our work to themagazine industry in New York area


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Outline

Introduction

The Model

Dynamics of themodel

DPRepresentation

Results




1 Introduction


2 The ModelDynamics of the modelDP Representation

3 ResultsOne Stage RepresentationMonopolyDuopoly



6 Conclusions and Ongoing Research


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Outline

Introduction

The Model


DPRepresentation

Results




Dynamics of the Model

Firm selling its product in multiple periods

Independent demand in each period

Decisions in each period

Newsstand Quantity (qt)

Subscription Price (st)Newsstand Price (pt)

Subscription Mechanism

Conversion of newsstand sales to subscribers t()new subscribers = tE[min(qt, Dt)]

Attrition rate t()leaving subscribers = (1 t)nt

Subscription dynamicsnt(qt, nt1) = (1 t)nt1 + tE[min(qt, Dt)]


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Outline

Introduction

The Model


DPRepresentation

Results




DP Representation

Revenue and Cost Structure

Variable Cost: c(nt + qt)Advertisement and Subscription Revenue: (st + m)nt (madvertisement revenue)

Newsstand Revenue: ptE[min(qt, Dt)]

Profit in each period

E[ptmin(qt, Dt) c(qt + nt)] +t

k=1(sk + m)ntk

Analysis done for finite and infinite horizon cases

Finite horizon case has a linear salvage value v + nvChallenge: The profit generated in period t depends onprevious decisions


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Outline

Introduction

The Model

Results

One StageRepresentation

Monopoly

Duopoly




1 Introduction








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Outline

Introduction

The Model

Results


Monopoly

Duopoly




One Stage Representation

Key Observation: Revenue in each time period is aseparable function of previous decision

Proposition: We can transform the problem to a series ofseparable independent problems


M l

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Outline

Introduction

The Model

Results


Monopoly

Duopoly




MonopolyQuantity Decision

Decision variables:1 Production Quantity qt

The problem can be reduced to a newsvendor problem in

each period with no carry over of inventoryRevenue: pt +

t(st+mc)1(1t)

Cost: c

Structure in the optimal quantity values:

Decreasing subscription prices leads to decreasing optimal

production quantityIncreasing planning horizon increases the optimalproduction quantity


M l

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Outline

Introduction

The Model

Results


Monopoly

Duopoly




MonopolyQuantity and Subscription Price Decision (1)


2 Subscription Price st

Attrition rate t() and conversion rate t() are notconstant anymoret = t(st) and t = t(st)

Two interpretations for t

Macro: Percentage of people subscribingMicro: Probability of one person joining


M l

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Outline

Introduction

The Model

Results


Monopoly

Duopoly




MonopolyQuantity and Subscription Price Decision (2)

Under certain regularity condition for t and t, theoptimal subscription price can be determinedindependently of optimal production quantity

If advertisement revenue is strictly positive, it is possiblethat the optimal subscription price is 0 but in absence ofadvertisement revenue optimal subscription price ispositive

After determining optimal st

finding optimal productionquantity reduces to the previous case


M l Q tit

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Outline

Introduction

The Model

Results


Monopoly

Duopoly




Monopoly - Quantity,Subscription Price and Stand Price Decision (1)


2 Subscription Price st

3 Stand Price pt

In this new setting conversion rate t() is assumed to beboth a function of stand price pt and subscription price st

The same one-stage representation can be achieved

Optimal Subscription Price:Assumption: Dependency of subscription price on stand

price and subscription price is of the multiplicative formt(st, pt) = t1 (st)t2 (pt)

Proposition: the choice of optimal subscription price isthe same as monopoly case without stand price decision


M l Q tit

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Outline

Introduction

The Model

Results


Monopoly

Duopoly




Monopoly - Quantity,Subscription Price and Stand Price Decision (2)

Transformation for one stage representation

Concept of Virtual Price pt :

pt pt + t(st,pt)(st+mc)1(1t(st))

Type of random demand on newsstand:Additive: t(pt) = a bpt +

Multiplicative: (p)t = aptb

Assuming certain functional forms for (), demand as a

function of Virtual price can be written in additive ormultiplicative form too. Thus, the problem is reduced to aprice setting newsvendor problem.


Duopoly

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Outline

Introduction

The Model

Results


Monopoly

Duopoly




Duopoly -General Setting

Two firms competing over unsatisfied demand

New demand function:

Dti = ti + i(

tj q

tj )

+

Each firm has a loyal customer base tiSecond part of demand function: the portion of unsatisfieddemand that switch to a new product

The same micro and macro level interpretation exists for i

In this setting the demand for firm i is a function of(pti , qt

j) and indirectly pt

j


Duopoly

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Outline

Introduction

The Model

Results


Monopoly

Duopoly




Duopoly -Quantity Decision (1)

Proposition: The one stage representation of the problemstill holds with modification of demand function

Expected value of sales will be adjusted to

Emin(qt1,t

1+ 1(

t

2 qt

2)+)

We will assume uniform demand to derive closed formsolution

Using the one stage representation the problem ofcompeting firms can be reduced to the problem of two

competing newsvendorsWe use results from Lippman and McCardle (97) tocharacterize the equilibrium


Duopoly

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Outline

Introduction

The Model

Results


Monopoly

Duopoly




Duopoly -Quantity Decision (2)

Notation i = ci/pi

The equilibrium point for the two competing newsvendoris the pair (q1 , q

2) iff (q

1, q

2) satisfy

P(Di > qi) = i

Notice that i is the ratio of cost to the virtual price

Using the above result we derive the equilibrium point


Duopoly Quantity Decision

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Outline

Introduction

The Model

Results


Monopoly

Duopoly




Duopoly - Quantity DecisionTypes of the Quantity Game Equilibria (1)

Type of resulting equilibrium depends on and parameters

If the cost structure of the two firms is not very different

the emerging equilibrium is symmetric (They use thesame type of best response functions)

Types of equilibria (type of best response functions):

Type I (symmetric): The competing firms produce withinthe support of their loyal customers demand (i)Type II (asymmetric): One of the competing firm producesmore than the upper bound of its loyal customers demand


Duopoly -

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Outline

Introduction

The Model

Results


Monopoly

Duopoly




Duopoly -Types of the Quantity Game Equilibria (2)

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 11

2 Type I(Symmetric)

Type II

Type II

(i constant)


Duopoly -

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Outline

Introduction

The Model

Results


Monopoly

Duopoly




Duopoly Types of the Quantity Game Equilibria (3)

Interpretation of value as degree of separation ofmarkets:

Recall Dti = ti + i(

tj q

tj )

+

Proposition: For any combination of (c1, p1) and (c2, p2)there exists a = 1 = 2 such that resulting equilibria issymmetric for all

If firms operate separate markets the symmetric type of

equilibrium will emerge, meaning they will use the sametype of best response


Duopoly - Convergence of

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Outline

Introduction

The Model

Results


Monopoly

Duopoly




Duopoly Convergence ofBest Response Functions in Quantity Game

Proposition: If the firms start from a non-equilibriumpoint a tatonnement scheme using their best responsefunctions will eventually converge to the equilibrium point

Moreover the convergence rate when the best responsefunctions are symmetric is linear

Implication of the above result:

If the firms start with a limited view of what they shouldproduce (one might dismiss the possibility of producing

above its loyal customer base demand) the best responsefunctions will eventually lead them to the equilibriumpoint!


Duopoly -

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Outline

Introduction

The Model

Results


Monopoly

Duopoly




Duopoly Quantity and Subscription Price Quantity Game (1)

As in monopolist case we consider the case werecompetitors decide on their subscription price too

The equation governing optimal choice of subscription

price is:1(s1)

s1+m1c1

1 (s1)1(s1)1(11(s1))

+ 1(s1)

= 0

The optimal subscription price can be determined beforeentering the competition

Optimal subscription price independent of quantity decisionOptimal subscription price independent of other firmsquantity and subscription price decision


Duopoly -

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Outline

Introduction

The Model

Results


Monopoly

Duopoly




Duopoly Quantity and Subscription Price Quantity Game (2)

After setting the subscription rates the problem reduces tothe previous case

Proposition: Two firms competing with quantity andsubscription price first set their subscription prices like amonopolist and then choose their production level as theprevious case. This decision will determine the uniqueequilibrium point.


Duopoly - Effect of

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Outline

Introduction

The Model

Results


Monopoly

Duopoly




uopo y ect oSubscription Price on the Equilibrium Point

Observation: Recall

Type of emerging equilibria depends on = c/p

pp+ (s,p)(s+mc)1(1(s))

Although choice of s can be done without entering thecompetition, however it will have an effect on the type ofemerging equilibrium.


Multi-Channel Pricing

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Outline

Introduction

The Model

Results


Computational

Analysis


gGeneral Setting (1)

One publishing firm offering the magazine throughdifferent channels (traditional paper-based, electronic,combination)

Consumers enter the system and based on a consumer

choice model (for example multinomial logit choose theirpreferred product)

In each period the consumers re-evaluate their subscriptiontype

There are different types of customers (i.e. different utilityfunction co-efficient with regard to different subscriptiontypes)



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Outline

Introduction

The Model

Results


Computational

Analysis


gGeneral Setting (2)

Decision

Print

Online

Print

+

Online

ot

ct

srt

(Pst

)

ort(Po

t)

crt(Pc

t)

Pst

Pot

Pct

st

Demand

Exit

Nnt

Nrt

Nt

e

t



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Outline

Introduction

The Model

Results


Computational

Analysis


gChallenges

Number of subscribers for each product depends on theprice decisions in all previous periods.

The dynamic programming formulation is highly non-lineareven when simple linear functions are chosen for utilityfunctions.

Because of limited pool of customers not only the pricebut production quantity decision will also impact the

future states directly.


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Outline

IntroductionThe Model

Results


Computational

AnalysisSymmetricFirms

AsymmetricFirms


1 Introduction








Effect of cost on value of considering the

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Outline

IntroductionThe Model

Results


Computational


AsymmetricFirms


competition - Case of similar firms

0%

10%

20%

30%

40%

0 1 2 3 4 5Cost

%o

fProfitImprovementby

ConsideringtheCompetition i= 1 i= 0.8

i = 0.6

Increase in cost of production increases the value ofconsidering the competition

m as bs ap bp a b p

13 0.1 0.005 0.3 0.001 0.05 0.001 0.9 6


Effect of stand price on value of considering the

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Outline

Introduction

The Model

Results


Computational


AsymmetricFirms


competition - Case of similar firms

Increase in stand price decreases the value of consideringthe competition

m as bs ap bp a b c

13 0.1 0.005 0.3 0.001 0.05 0.001 0.9 3


Who benefits more? Asymmetric firms

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Outline

Introduction

The Model

Results


Computational


AsymmetricFirms


Asymmetry in cost

Firms differ in their costs

Considering thecompetition has a highervalue for the firms whichare operating with highercost

m as bs ap bp13 0.1 0.005 0.3 0.001a b c1 p

0.05 0.001 0.9 0.8 3 6


Who benefits more? Asymmetric firms

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Outline

Introduction

The Model

Results


Computational


AsymmetricFirms


Asymmetry in customer attraction power

Firms differ in their values, interpret ascustomer attraction power

Considering thecompetition has a highervalue for the firms otherfirms customers are lessloyal

m as bs ap bp13 0.1 0.005 0.3 0.001a b 1 c p

0.05 0.001 0.7 0.8 3 6


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Outline

Introduction

The Model

Results




1 Introduction








Conclusions and Ongoing Research

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Outline

Introduction

The Model

Results




Conclusions and Ongoing Research

Presented a model for joint pricing and productionplanning for subscription-based products

Model captured both stand-price and subscription-pricedecisionsModel yielded closed-form and computationally tractable

solutions that brought insight into the problemExtended our model to the duopoly setting

Presented encouraging numerical results

Currently implementing our work at Columbia Journalism

Review (CJR)Will enhance the model to incorporate differentdistribution channels (e.g. online) and effect ofpromotional activities on subscription base


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Outline

Introduction

The Model

Results




Optimal Pricing and Production Planning forSubscription-Based Products

W. T. Huh S. Kachani A. Sadighian

Department of Industrial Engineering and Operations Research,Columbia University, New York

MSOM Annual Meeting - Beijing, ChinaJune 18-19, 2007


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