Supply chain management in the presence of strategic consumers and consumption externalities

[1991] find that when the vaccine supply is unlimited, thesupply chain inefficiency is due to the positive externality, i.e., each consumer considers only direct ben-efits from vaccinati

SUPPLY CHAIN MANAGEMENT IN THE PRESENCE OF STRATEGIC

CONSUMERS AND CONSUMPTION EXTERNALITIES

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SUPPLY CHAIN MANAGEMENT IN THE PRESENCE OF STRATEGIC CONSUMERS AND

CONSUMPTION EXTERNALITIES

KENAN AR˙IFO ˘GLU

In this dissertation, I study the management of supply chains in which consumers’ utilities from theproduct depend on the total consumption (consumption externalities) and all parties involved, consumers

as well, maximize their own well-being (strategic consumers) I consider two practical motivations:vaccine for seasonal influenza (flu) epidemic and high-fashion luxury goods

In Chapter 2, I study the impact of yield uncertainty (supply side) and self-interested consumers(demand side) on the inefficiency in the influenza vaccine supply chain Previous economic studies,focusing on demand side, find that the equilibrium demand is always less than the socially optimaldemand since self-interested individuals do not internalize the social benefit of protecting others viareduced infectiousness (positive externality) In contrast, I show that the equilibrium demand can begreater than the socially optimal demand after accounting for the limited supply due to yield uncertaintyand manufacturer’s incentives The main driver for this result is a second (negative) externality: self-interested individuals ignore that vaccinating people with high infection costs is more beneficial forthe society when supply is limited I show that the extent of the negative externality can be reducedthrough more efficient and less uncertain allocation mechanisms In order to investigate the relativeeffectiveness of government interventions on supply and demand sides under various demand and supply

characteristics, I construct two partially centralized scenarios, where the social planner (government)intervenes either on the demand side or the supply side but not both, and conduct an extensive numericalanalysis.

In Chapter 3, I develop a mechanism which coordinates a supply chain with consumption nalities and a profit-maximizing manufacturer having uncertain production process in the presence ofrational consumers This mechanism includes tax/subsidy payments on the demand side (individual con-sumers) and a transfer payment on the supply side (manufacturer) I show that, under the mechanism,the manufacturer is ex ante better off; moreover, expected total utility of all individuals are higher I alsoshow that the mechanism allows arbitrary division of ex ante total social welfare between individuals andthe manufacturer

exter-Chapter 4 develops an analytical model to study the impact of snobbish (exclusivity-seeking) sumer behavior on a firm’s price and quantity decisions I consider a profit-maximizing monopoly firmselling a product over two periods to two segments of consumers (with high and low valuation of theproduct), who are forward-looking and snobbish, i.e., their valuation of the product decreases as morepeople in the population consume it This modeling framework enables me to explain the heterogeneity

con-in priccon-ing of snob appeal products observed con-in practice Specifically, I fcon-ind that markdown priccon-ing is timal when the fraction of high-value consumers is small whereas uniform pricing is optimal when thisfraction is large Thus, snobbish consumer behavior provides another motivation for markdown pricing.When selling to snobbish consumers, inter-temporal price discrimination increases product exclusivityand hence consumers’ willingness to pay, in addition to the usual effect of attracting consumers withdifferent valuations Similar to normal products, I find that advance purchase discounts are optimal onlywhen consumers do not know their true valuations in the first period However, I show that snobbishconsumer behavior coupled with uncertain valuations may lead to discounts in advance I use this mod-eling framework to formally distinguish between scarcity and exclusivity, where the former is definedwith respect to the demand whereas the latter is defined with respect to the entire population I find that

op-snobbish consumer behavior leads to scarcity, but not necessarily to exclusivity, i.e., the product may beexclusive even when consumers are not snobbish Finally, contrary to intuition, I find that the productmay become more exclusive when the price is marked down and hence price markdowns need not always

be associated with excess inventory when selling to snobbish consumers

I am very grateful to Prof Izak Duenyas for his guidance and support during my research antship at the University of Michigan He was a great resource for me I enjoyed working with himand engaging in academic discussions I thank my committee member Prof Diego Klabjan for histhought-provoking questions and suggestions.

assist-I thank Prof Martin Larivieri for always being there to discuss research problems His suggestionsgreatly improved this dissertation I thank Prof Barıs¸ Ata for his mentorship

I have been very fortunate to know many people during my doctoral study at Northwestern sity I am thankful to all of my professors, classmates, and officemates My special thanks go to Linlin,Luis, Mustafa, Neda, and Yan I have benefited a lot from my conversations with them I also would like

Univer-to thank Ahmet, Betul, Can, G¨okhan, G¨urkan, Kezban, Koray, Muzaffer, and Yasin They were a constantsource of encouragement and support; it would have been very boring without their companionship

Last, but definitely not the least, I am immensely grateful to my family, especially to my parents Itwas their unconditional love, care, and encouragement which made this dissertation possible Withouttheir support, I do not think that I could overcome the difficulties during these years.

To my parents,

Ali and Radet Arifo˘glu,

who made all of this possible,

for their love and continuous support

Chapter 2 Consumption Externality and Yield Uncertainty in the Influenza Vaccine Supply Chain:

Chapter 3 Coordinating Supply Chains with Uncertain Yield and Consumption Externalities 79

Chapter 4 Pricing and Strategic Rationing When Selling to Snobbish Consumers 106

4.5 Optimal Selling Strategy with Deterministic Consumer Valuations 122

List of Tables

4.1 Equilibrium capacity with potentially-optimal APD when the population is

4.2 Equilibrium price, capacity and demand with potentially-optimal UP when the

population is homogenous and consumer valuations are uncertain 1194.3 Equilibrium capacity with potentially-optimal MP when the population is homogeneous

4.4 Equilibrium price, capacity and demand with potentially-optimal UP when the

population is heterogeneous and consumer valuations are deterministic 1324.5 Equilibrium price, capacity and demand with potentially-optimal MP when the

population is heterogeneous and consumer valuations are deterministic 1334.6 Equilibrium price, capacity and demand with potentially-optimal UP when the

population is homogenous and consumer valuations are deterministic 1504.7 Equilibrium capacity with potentially-optimal MP when the population is homogenous

List of Figures

2.2 Marginal social benefit when social planner decides the demand, where B (δ) is given

2.3 Value of centralized solution, and supply-side and demand-side interventions for

2.4 Sensitivity analysis of Availability and Negative Externality Effects when R0= 3 The

socially optimal β in each case is pointed by an arrow 492.5 (a) Ex post demand in the decentralized equilibrium and social optimum; (b) The

probability density function (PDF) and cumulative distribution function (CDF) of yield 53

2.6 (a) Ex post demand in the decentralized equilibrium and social optimum; (b) The

probability density function (PDF) and cumulative distribution function (CDF) of yield 54

3.1 The additional gain/loss from vaccination under the mechanism (tnv(Qr) − tv(Qr)) 94

3.2 Transfer payment from the government to the manufacturer (T (Qr)) 97

4.1 Firm’s selling strategy when consumer valuations are deterministic, i.e., α = 1

and β = 0 For a fixed vH, the plane (λ, θ) is divided into four (I-IV) regions

The ˜θ functions defining the boundary of regions are given by (4.30)-(4.32)

123

4.2 Firm’s capacity and the discount rate as a function of sensitivity to consumption (λ)

when consumer valuations are deterministic, and vH = 15, vL= 5, c = 1, θ∗ ' 0.286

4.3 Firm’s optimal selling strategy when N = 1000, vH = 15, vL= 5 and c = 1 136

CHAPTER 1

Introduction

According to Global Supply Chain Forum,

Supply chain management (SCM) is the integration of key business processes across the

supply chain for the purpose of creating value for customers and stakeholders [Cooper

in operations management [Dana and Petruzzi, 2001; Su and Zhang, 2008; Liu and van Ryzin, 2008;Cachon and Swinney, 2009] study forward-looking (strategic) consumer behavior These papers assumethat consumers are no longer myopic and choose the time of their purchase so as to maximize their utility.They show that strategic consumer behavior affects the pricing policy and capacity decision of a firm.For example, a firm selling to strategic consumers should set a lower capacity when the price is markeddown over time since this discourages strategic consumers from waiting for lower prices [Su and Zhang,2008; Liu and van Ryzin, 2008]

A consumer’s purchasing behavior is not independent from that of other consumers For example,

if a consumer decide to buy the product, she decreases the product availability for all other consumers,

as a result, it imposes negative consumption externality among consumers who are willing to purchasethe product This, in turn, discourages the consumers with low valuations from buying which results

in lower aggregate demand Operations management papers modeling strategic consumer behavior alsoconsider the dependence between product availability and consumers’ purchasing decisions (see Shenand Su [2007] and Aviv et al [2009] for an overview of these papers)

However, consumption externalities affect the purchasing behavior of consumers not only throughproduct availability In fact, consumers value certain products based on the total consumption For ex-ample, if too many people are vaccinated against an infectious disease, probability of infection is verylow for each unvaccinated individual, so is the value of the vaccine Similarly, a BMW on every drive-way may dilute the value of the car to consumers [Amaldoss and Jain, 2005a] Except two recent studies[Agrawal et al., 2011; Tereya˘go˘glu and Veeraraghavan, 2012], the dependence of consumer valuations onthe total consumption is not studied in operations literature Both Agrawal et al [2011] and Tereya˘go˘gluand Veeraraghavan [2012] assume that consumers are exclusivity-seeking (snobbish), i.e., the product isless valuable if too many people purchase it Agrawal et al [2011] analyze product design and introduc-tion decisions of a firm selling a durable good to snobbish consumers They show that firms selling tosnobbish consumers should design products that undergo slow value erosion, and should introduce them

at high-price and low-volume Tereya˘go˘glu and Veeraraghavan [2012] analyze the pricing and tion decisions of a firm selling to a heterogeneous market which also includes snobbish consumers Theyshow that, depending on the market composition and consumers’ sensitivity to consumption, the firmmay charge a higher price and set a lower capacity when selling to snobbish consumers

produc-Certainly, a model ignoring strategic consumer behavior and consumption externalities cannot curately represent the supply chain of certain products The solution of such a model may exacerbate

ac-the total supply chain inefficiency Thus, in this research, we study ac-the supply chains with strategic sumer behavior and consumption externalities Our research is particularly motivated by the vaccine forseasonal influenza (flu) epidemic and luxury goods.

con-In Chapter 2, we study the influenza (flu) vaccine supply chain in the USA The influenza vaccine

in the USA is produced by private companies through a complex and unreliable production process andconsumers make their vaccination decisions in a rational and self-interested way [Pauly, 2005; Vietri

et al., 2008] The supply chain of vaccines, especially that of influenza vaccine, has long been studied

by researchers in health economics (i.e., Brito et al [1991]), epidemiology (i.e., Bauch and Earn [2004];Reluga et al [2006]; Galvani et al [2007]) and operations management (i.e., Chick et al [2008]; Deo andCorbett [2009]) Papers in health economics and epidemiology literatures focus only on the rational de-cision making by consumers and assume that the vaccine supply is unlimited whereas those in operationsmanagement literature study only the supply side and ignore the rational decision making by consumers.However, we consider both supply and demand sides and show that the interaction between them leads

to novel insights For example, Brito et al [1991] find that when the vaccine supply is unlimited, thesupply chain inefficiency is due to the positive externality, i.e., each consumer considers only direct ben-efits from vaccination but ignore the social benefits due to protecting others via reduced infectiousness,and hence the demand is lower in equilibrium than in the socially-optimal solution In contrast, we showthat when supply is endogenized, the supply chain inefficiency may be due to the negative externality,i.e., when the vaccine supply is limited, each consumer searching for the vaccine reduces the vaccineavailability for other consumers, which results in equilibrium demand being higher than socially-optimaldemand Problems related to flu vaccine supply chain in the recent past have generated a lot of public

as well as scholarly reactions advocating a need for a more active role by the government [Kilbourne,1991; Harris, 2006; Hinman, 2005a] However, there is considerable debate about the exact nature of thisintervention Opinions range from interventions focusing solely on the demand side (to induce demand)such as increased insurance coverage, improving accessibility and public awareness campaigns [Institute

of Medicine, 2004; Hinman, 2005b], to those focusing solely on the supply-side (to ensure more ply) such as subsidizing the manufacturers and even owning production facilities [Grady, 2004; Hinman

sup-et al., 2005] To address this policy debate, in Chapter 2, we also analyze the relative effectiveness ofgovernment interventions in only one (demand or supply) side of the supply chain

In Chapter 3, motivated by flu vaccine supply chain, we design a mechanism which eliminates theentire inefficiency in a supply chain with consumption externalities and a profit-maximizing manufac-turer having uncertain production yield This mechanism aligns consumers’ incentives with social op-timum through tax and subsidies on the demand side To eliminate the inefficiency due to negativeexternality, the mechanism curbs the demand by taxing/subsidizing vaccinated/unvaccinated consumersfor low realizations of production quantity On the other hand, for high realization of production quan-tity, it taxes/subsidizes unvaccinated/vaccinated consumers to induce the demand and to decrease theinefficiency due to positive externality On the supply side, our mechanism requires a transfer paymentbetween the manufacturer and social planner Through this transfer payment, the social planner sharesthe manufacturer’s risk due to uncetain yield

In Chapter 4, motivated by high-fashion luxury products, we develop a model to study the pricingpolicy and capacity decision of a monopolist firm selling a product over two periods to strategic andsnobbish consumers who value a product less as more people consume it The literature has identi-fied demand uncertainty [Rodriguez and Locay, 2002; Nocke and Peitz, 2007], risk-averse attitude ofconsumers [Liu and van Ryzin, 2008] and uncertain consumer valuations [M¨oller and Watanabe, 2010]

as some reasons why firms use inter-temporal pricing (e.g., price markdowns or advance purchase counts) and why they create scarcity intentionally We complement the literature by studying the implica-tions of snobbish consumer behavior on firm’s pricing policy and capacity decision We show that firmsselling snob-appeal products intentionally create scarcity due to exclusivity-seeking consumer behavior[Rosenbloom, November 19, 2009] We also find that the snobbish consumer behavior itself leads to theoptimality of inter-temporal pricing when the number of consumers valuing the product high is low Due

dis-exclusivity-seeking consumer behavior, the firm creates scarcity to charge a higher price which createsrationing risk Consumers with high valuations can pay more not to be rationed Therefore, the firmsells the product at higher price and only consumers with high valuations purchase Later, she reducesthe price and satisfy demand coming from consumers with lower valuations When the number of con-sumers with high valuations is very large (i.e., the firm has high brand image), the firm does not mark theprice down and always charges a very high uniform price Consistent with this result, we observe thatbrands with high image, e.g., Louis Vuitton and Hermes, never discount [Sherman, July 10, 2009].Lastly, we summarize our main findings and discuss about the ideas for future research in Chapter 5.

CHAPTER 2

Consumption Externality and Yield Uncertainty in the Influenza Vaccine

Supply Chain: Interventions in Demand and Supply Sides

2.1 IntroductionUnlike pediatric vaccines, supply chain for the influenza (flu) vaccine in the U.S is highly decen-tralized Profit maximizing firms decide the production quantity and bring the produced vaccine to themarket while individuals decide whether or not to get vaccinated in a rational and self-interested mannerbased on the availability of the vaccine, severity of the infection, and vaccination costs [Vietri et al.,2008] The role of the government and its agencies such as the Center for Diseases Control and Pre-vention (CDC) is limited to deciding the vaccine composition and recommending priority groups forvaccination [Institute of Medicine, 2004; Government Accountability Office, 2004] On the supply side,potentially insufficient incentives of manufacturers together with a long production process (six to eightmonths) and the significant uncertainty in the production yield have contributed to shortages in the re-cent past [Government Accountability Office, 2004] On the demand side, positive externality effectassociated with vaccination—each vaccinated individual decreases the infection risk of her close con-tacts but fails to internalize this value while making her own vaccination decision [Brito et al., 1991]—has arguably contributed to vaccination coverage rates that are lower than the socially desired target rates[Harper et al., 2005]

In this chapter, we develop an integrated model of production (yield) uncertainty and rational1sumer behavior, and investigate the interaction between these two potential sources of inefficiency Mo-tivated by the current practice, we study a decentralized system where the manufacturer determines theproduction quantity and individuals make their vaccination decisions We consider a single flu seasonand a population comprising two priority groups based on the infection disutility of its individuals First,given the design of priority groups, the manufacturer decides the production quantity and incurs the re-lated production cost A random fraction of this production quantity is realized and brought to the market.The supply received at the beginning of the flu season is first allocated to the group with higher infectiondisutility, and the remaining supply, if any, is then allocated to the group with lower infection disutil-ity Then, knowing the design of priority groups and observing the realized production, self-interestedindividuals decide whether to spend time and money to search (e.g., physical travel or information col-lection) for the vaccine Due to limited supply, individuals that decide to search are not guaranteed to getthe vaccine Unvaccinated individuals, irrespective of whether they search for the vaccine or not, may beinfected with some probability that depends on the vaccinated fraction of the population Every infectedindividual incurs some infection disutility (e.g., payment for drugs, health care costs, lost wages, death).Vaccinated individuals become immune against influenza but may incur some vaccination disutility (e.g.,side effects, vaccine price, administration costs).

con-We also analyze a benchmark centralized system where a social planner decides the demand andthe production quantity to maximize the total social welfare We find that the production quantity inthe centralized setting is always greater than that in the decentralized setting in accordance with previoussupply-chain models of yield uncertainty However, contrary to existing economic models of vaccination,

we find that the expected demand in the decentralized setting can be higher than that in the centralizedsetting

1 We use the term “rational” or “self-interested” interchangeably to refer to an individual that maximizes her own expected net utility.

This result is driven by the negative externality effect—when the vaccine supply is limited, eachindividual searching for the vaccine reduces the vaccine availability for some individuals with higherinfection disutility but does not internalize this cost The negative externality effect causes inefficiency(i.e., lower social welfare) on the demand side under limited supply for two main reasons: (i) not allindividuals searching for the vaccine can get it and hence wasteful search costs are incurred, and (ii)those who need vaccine the most (individuals with high infection disutility) may not get it The firstcomponent is due to the uncertainty/randomness in the vaccine allocation, i.e., if individuals who failed

to get the vaccine knew this a priori, they would not seek vaccination so that the equilibrium demandwill be equal to the available supply The second component is due to the inefficiency in the allocation ofvaccines, i.e., if the vaccine was allocated to individuals in decreasing order of their infection disutility,individuals with higher infection disutility would be vaccinated first

In order to understand the impact of prioritization, we analyze a model with a completely randomallocation mechanism and a single group of individuals We show that prioritization makes the vaccineallocation more efficient and less uncertain by increasing the chance of obtaining the vaccine for individ-uals with high infection disutility As a result, we find that the negative externality effect is smaller underthe priority scheme with two groups than under a complete random allocation with no priority groups.Increasing the number of priority groups reduces the negative externality effect further by decreasing theinefficiency and uncertainty in the allocation mechanism

In addition, we study how individuals accounting for the limited supply affect the total social welfare.For this purpose, we define availability effect as the difference in the equilibrium outcomes between thedecentralized system and a reference case, where the fill rate is 1 and where the infection probability isbased on unlimited supply We find that this availability effect is beneficial to the society when searchcosts are high and infection disutilities among individuals are low In such cases, accounting for limitedavailability of vaccines decreases the incentive for individuals with low infection disutility to search for

vaccines compared to the reference case, which results in lower wasteful search costs and better vaccineallocation for individuals with high infection disutility.

We also construct partially centralized models to inform the public policy debate regarding the tive effectiveness of government intervention on the demand side [Institute of Medicine, 2004; Hinman,2005b] vs supply side [Grady, 2004; Hinman et al., 2005] Under the demand-side intervention, themanufacturer chooses profit maximizing production quantity but the social planner chooses the demand

rela-to maximize the rela-total social welfare Under the supply-side intervention, the social planner decides thewelfare maximizing production quantity but the demand is determined by self-interested individuals

We find that the demand-side intervention in some cases can provide “too much” incentive (expectedmarginal benefit) to the manufacturer and results in production quantity that is greater than that in thecentralized solution Similarly, the production quantity under supply-side intervention can be higher thanthat in the social optimum, especially at lower yield realizations, to compensate for the self-interestedbehavior of individuals When a large fraction of the population has high infection disutility, the demand-side inefficiency due to negative externality effect is high and the manufacturer underproduces due to highsupply risk and lower marginal benefits In such cases, we find that the social planner should intervene

in the supply side to increase the production levels and consequently decrease the negative externalityeffect On the contrary, when a large fraction of population has low infection disutility and the epidemic

is less infectious, individuals have less willingness to search for the vaccine In such cases, the socialplanner can improve total social welfare more by intervening on the demand side

2.2 Literature Review

In this section, we outline our contribution to three distinct streams of literature

2.2.1 Operations management literature

Our consumer model contributes to the growing literature on supply-chain management in the presence

of strategic consumer behavior [Dana and Petruzzi, 2001; Su and Zhang, 2008; Cachon and Swinney,2009] Most papers in this stream assume that the valuation of the product for rational consumers isexogenously specified To our knowledge, ours is among the first papers to consider a supply chainwith consumption externality, i.e., consumers’ valuation for the product is determined as a function

of the fraction of market consuming the product in equilibrium In a recent paper, Tereya˘go˘glu andVeeraraghavan [2012] analyze a model with consumption externality similar to ours However, oursdiffers significantly in various modeling elements such as yield uncertainty, heterogeneity of consumervaluations, allocation mechanism, and partially centralized scenarios

Flu vaccine supply chain has drawn considerable attention from OM researchers recently [Wu et al.,2007; Kornish and Keeney, 2008; Chick et al., 2008; Deo and Corbett, 2009; Cho, 2010] Deo andCorbett [2009], using a model of Cournot competition with endogenous entry, show that yield uncer-tainty can explain the high concentration of the U.S flu vaccine market Chick et al [2008] propose

a cost-sharing contract to eliminate the supply-side inefficiency, i.e., a profit-maximizing manufacturerproduces less vaccine compared to the socially-optimal solution We complement them by incorporatingrational consumer behavior, examining the interaction between the demand side and supply side, andderiving relevant policy implications

2.2.2 Health economics literature

Several papers in health economics analyze the externality arising from vaccination and find that thedemand for vaccines is lower in equilibrium than in the socially-optimal solution [Brito et al., 1991;Geoffard and Philipson, 1996; Philipson, 2000] In contrast, we show that when supply is endogenized,

it can result in equilibrium demand that is higher than the socially-optimal demand due to the negativeexternality arising from limited and stochastic supply This effect acts counter to the positive externality

effect from herd immunity, which results in lower than socially-optimal demand For some instances ofyield uncertainty, the latter effect dominates the former effect, thereby resulting in higher demand.

2.2.3 Epidemiology literature

Empirical evidence suggests that the decision to vaccinate is also strongly correlated with the vaccineeffectiveness, flu severity, and vaccine side effects [Chapman and Coups, 1999] Several studies [Bauchand Earn, 2004; Reluga et al., 2006; Galvani et al., 2007] combine epidemiologic models and deductivegame-theoretic models to analyze rational vaccination decisions by individuals in the presence of thesefactors These models are also consistent with the health benefit model [Janz and Becker, 1984], whichhas been shown to explain vaccination decisions of individuals [Larson et al., 1979] Given the strongempirical support, we also adopt a deductive2game approach but also incorporate the impact of supply-side factors such as yield uncertainty, which is missing from the earlier models

2.3 Model and Assumptions

In this section, we describe various components of our model

2.3.1 Supply model

Injectable vaccines produced via embryonated chicken eggs still constitute majority (over 97%) of fluvaccine produced and administered [Danzon et al., 2005; Palese, 2006; Government Accountability Of-fice, 2008] An important characteristic of this process is yield uncertainty due to uncertain growthcharacteristics of virus strains inside the chicken eggs and the possibility of bacterial contamination Inaccordance with the models proposed in the literature [Palese, 2006; Chick et al., 2008; Deo and Corbett,2009], we assume that the obtained number of vaccine doses Qris a stochastic proportion of the plannedegg production Q (see Yano and Lee [1995] for models with proportional yield) That is, Qr = U Q,

2 For an alternative inductive game formulation, where individuals base their decisions on past experience only, see Breban et al [2007].

where yield U ∈ [0, ∞) is a random variable with mean µ, standard deviation σ, and coefficient of ation CV = σ/µ We assume that the yield U has a continuous, differentiable and strictly increasingprobability distribution M (·) In addition, we let w denote the manufacturer’s unit price for a dose ofvaccine and c denote the cost of the manufacturer per unit of planned egg We assume that the salvagevalue of any vaccine remaining at the end of the season is zero, since it cannot be used in subsequentseasons Motivated by the FDA’s regulatory requirements for all biological products [Food and DrugAdministration, 2008], we assume that accurate information regarding the quantity of vaccine produced

vari-is available before vaccination

2.3.2 Demand model

We consider a population of N individuals,3where each healthy individual enjoys utility V The healthoutcome of an infected individual may fall in one of the following categories : (i) no medical care sought,(ii) outpatient visit, (iii) hospitalization, and (iv) death, each with some probability We use infectiondisutility δ to represent the expectation of all direct and indirect costs that an infected individual incurs inthese four scenarios [Meltzer et al., 1999; Galvani et al., 2007] It has probability density function g (·)and cumulative probability distribution G (·) , which is strictly increasing over the finite interval0, ¯δand is common knowledge Each individual searching for the vaccine also incurs a disutility θ, whichcaptures the time and money spent by the individual while searching.4Individuals receiving the vaccineincur some additional disutility, r ≥ w, which includes vaccine price, administration costs, morbidityrisk and side effects from the vaccine [Meltzer et al., 1999; Galvani et al., 2007] We assume that θ and

r are constant across the population for analytical tractability and because they are small compared to δ

3 We assume that each individual has negligible impact on the aggregate decision of population, which is referred to as the

“fluid” or “continuous” approximation whose solution converges to that of the real system as the population size (N ) becomes larger [Aumann, 1964; Brito et al., 1991; Gallego and S¸ahin, 2010, and references therein].

4 Earlier work [Meltzer et al., 1999; Galvani et al., 2007] does not consider stochastic availability and hence implicitly assumes that all costs incurred by an individual are due to receiving the vaccine However, a large fraction (over 35%) of these costs are incurred before an individual knows if the vaccine is available [Meltzer et al., 1999] We label these costs as search costs here.

low-at the beginning of the season and is sequentially alloclow-ated, first to individuals with δ ≥ β and then tothose with δ < β Hence, an individual’s ex-post probability of obtaining the vaccine (φ) depends notonly on the obtained number of doses but also on his/her infection disutility and β We assume that β isexogenously given, common knowledge and does not depend on the number of vaccine doses produced

Qr This corresponds to the ACIP practice of announcing the priority groups before the realization ofsupply [Institute of Medicine, 2004]

For analytical tractability, we assume that all individuals in the same priority group have an equalchance of getting the vaccine, i.e., complete random allocation within each priority group It is con-ceivable that allocation within a priority group may be more efficient, for instance, those with higherdisutility might be more perseverant in their search However, CDC data show that this is not the casedue to the problems in vaccine production and distributions [Government Accountability Office, 2008].For example, only about 33%-65% of high-risk individuals received vaccination in 2008-09 flu seasonwhile about 11%-35% of low risk individuals were vaccinated in the same year [Fiore et al., 2010]

5 The mortality costs dominate total costs in all other scenarios in which an infected individual may end up [Meltzer et al., 1999; Galvani et al., 2007] Thus, people having high mortality risk also have high infection disutility As a result, our model also prioritizes the risk groups being prioritized in practice For example, our model prioritizes elderly individuals over young individuals since they have higher infection disutility ($4163 vs $567) The practice also prioritizes elderly individuals since they have higher mortality and morbidity risks [Galvani et al., 2007].

2.3.4 Epidemiology model

We consider an SIR (susceptible-infected-recovered) epidemic model in a closed homogenous tion with fixed size (N ) in which vaccination is followed by the onset of (instantaneous) infections fromexogenous sources [Chick et al., 2008] We assume that the vaccine is perfectly effective, i.e., all vac-cinated individuals are immunized against the infection [Brito et al., 1991] Unvaccinated individualsmay be infected with probability p (h) , that is continuous and nonincreasing in h ∈ [0, 1], the vaccinatedfraction of population Similar to Brito et al [1991], we assume that p(·) is common knowledge Inaddition, we make following technical assumptions for our subsequent analysis:

popula-Assumption 2.1 (i) The individual with largest infection disutility searches for the vaccine when

nobody is vaccinated, i.e., ¯δp (0) > r + θ

(ii) The infection probability is continuously differentiable and strictly decreasing for allh < hzr,i.e.,p0(h) is continuous and negative for all h < hzr, which we term as the zero-risk vaccina-tion fraction and is given by:

(iii) The expected number of infections (N (1 − h) p (h)) is convex in h for all h ∈ [0, 1]

Assumption 2.1 (i) is sufficient to guarantee an interior solution in the decentralized equilibrium.Assumption 2.1 (ii) ensures the uniqueness of the solution By Assumptions 2.1 (ii) and (iii), the number

of infections is strictly convex for hzr = 1 and piecewise-linear convex for hzr ∈ [0, 1] , and p (h) isconstant for h < hzr These two special cases are considered in Chick et al [2008]

2.4 Equilibrium in the Decentralized System

We model a decentralized supply chain consisting of a profit-maximizing manufacturer that producesthe vaccine and sells directly to rational utility-maximizing individuals who make their own vaccinationdecisions First, the manufacturer decides the production quantity Second, each individual observesthe obtained number of doses and decides whether to search for the vaccine or not We use backwardinduction to characterize the subgame perfect equilibrium of this two-stage game

2.4.1 Individual’s problem

In the second stage of the game, given the priority groups and obtained number of doses, each individualdetermines whether to search for the vaccine or not An individual with infection disutility δ who decidesnot to search for the vaccine will remain healthy with probability 1 − p (h) and will enjoy utility V She will be infected with probability p (h) and will enjoy utility V − δ, yielding an expected utility of

vns(h) = V − δp (h) On the other hand, if she decides to search for the vaccine, she will incur asearching disutility θ Moreover, she will be vaccinated with probability φ and will obtain a utility equal

to V − r − θ; she will not be vaccinated with probability 1 − φ and will obtain a net utility equal to

V − δp (h) − θ, yielding an expected net utility of vs(h, φ) = V − δp (h) + φ [δp (h) − r] − θ Thus,

an individual with infection disutility δ searches for the vaccine if

(2.2) vs> vns⇔ θ < φ [δp (h) − r]

The consumer model in Dana and Petruzzi [2001] is a special case of our model since individuals’valuation of the vaccine, δp (h) , is dependent on the consumption level and decreasing in it Whileour immediate motivation is flu vaccines, this model is applicable to a wider class of luxury productscalled “conspicuous goods” or “snob goods”, where consumers value the good not only for its intrinsic

functionality but also for its prestige value that depends on the level of consumption [Amaldoss and Jain,2005a].

Lemma 2.1 In equilibrium, for a given Qr, if an individual with infection disutility ˆδ does not searchfor the vaccine, then none of the individuals withδ < ˆδ search for the vaccine

All proofs are presented in Appendix 2.A.3 Lemma 2.1 implies the existence of a marginal vidual who is indifferent between searching and not searching such that all individuals with infectiondisutility higher than that of marginal individual search for the vaccine We define δE(β, Qr) as theinfection disutility of the marginal individual in equilibrium In what follows, we suppress the arguments

indi-of δE(β, Qr) to improve readability The vaccine demand is given by NG δ¯ E
 and the vaccinatedfraction is given by:

where ¯G (·) = 1 − G (·) In addition, the probability of being vaccinated for an individual with infectiondisutility δ ≥ δE is given by:

θ = 0, we assume that δE = r/p (0) to ensure the continuity of δE in Qr Proposition 2.1 characterizesthe equilibrium fraction of population searching for the vaccine for Qr> 0.

Proposition 2.1 Define δzras follows:

wherehzr∈ [0, 1] is the zero-risk vaccination fraction given by (2.1)

(i) For allQr > 0, there exists a unique threshold value δE ∈0, ¯δ and the equilibrium fraction

of population searching for the vaccine is given by:

2 (β, Qr) satisfy:

δ1Ep ¯G δ1E

− r = θ(2.7)

Figure 2.1 illustrates the equilibrium demand in the decentralized setting for β ∈0, ¯δ and a β ∈

0, ¯δ
as a function of the obtained number of doses Qr For Qr ≥ N [ ¯G(δE

1)], the demand is less than

or equal to the available supply, and hence everyone searching for a vaccine gets it Thus, individuals’decision is independent of the available supply in this region For Qr < N [ ¯G(δE

1)], vaccination ismore valuable for each individual than that for Qr ≥ N [ ¯G(δE

1)] since the infection risk is higher due

to the limited supply Hence, the equilibrium demand is more than the available supply as shown byProposition 2.1 (ii) Depending on the relative impact of the increase in infection risk and decrease invaccine availability, the demand for high obtained number of vaccine doses may be more or less thanthat for low obtained number of vaccine doses

Figure 2.1 Demand in the decentralized and centralized systems

Proposition 2.1 (i) implies that the ex-post vaccinated fraction in decentralized equilibrium is alwaysless than the zero-risk vaccination fraction (hzr) since δ1E ∈ δzr, ¯δ
For some functional forms ofinfection probability used in the literature [Bauch and Earn, 2004; Breban et al., 2007; Chick et al., 2008],the zero-risk vaccination fraction in (2.1) is the same as the critical vaccination fraction in epidemiologythat reduces the reproductive number, i.e., average number of individuals infected by an infected person,below 1 [Hill and Longini, 2003] This implies that the critical level of vaccination is not attained withself-interested individuals [Breban et al., 2007]

Proposition 2.1 shows that the equilibrium demand is independent of the choice of β for Qr >

N ¯G δE

1
Thus, prioritizing individuals with high infection disutility (δ) cannot affect individual sion making when all individuals searching for the vaccine can get it However, when supply is limited,prioritization increases the availability of vaccine for individuals in high priority group This in turn maydiscourage some individuals in low priority group from searching, thereby decreasing the equilibriumdemand as shown by Figure 2.1 Proposition 2.2 formalizes this intuition

deci-Proposition 2.2 For all β ∈ 0, ¯δ and Qr < NG δ¯ E

1
 , the ex-post equilibrium demand withtwo patient groups is less than or equal to that in equilibrium with one group However, the ex-postvaccinated fraction of population is the same in both equilibria

Proposition 2.2 implies that, when the supply is limited, prioritization does not affect manufacturer’stotal sale (profit) but improves the social welfare by reducing wasteful search costs and increasing theprobability that individuals with high infection disutility get the vaccine Although some individuals

in the low priority group may be worse off, total expected utility loss due to infection decreases underprioritization We analyze the magnitude of this benefit as a function of various problem parameters inSection 2.8.2

Next, we consider an efficient allocation mechanism, where individuals are ordered in descendingorder of their infection disutility and an individual is vaccinated if and only if demand coming fromindividuals with greater infection disutility is satisfied

Proposition 2.3 Under efficient rationing, the ex-post equilibrium demand is equal to the availablesupply forQr≤ NG δ¯ E

1
 and is equal to N  ¯G δE1
 for Qr> NG δ¯ 1E


Proposition 2.3 implies that the efficient allocation mechanism completely eliminates wasteful searchesand ensures that the right people get the vaccine when the supply is limited Unfortunately, this allocationmechanism is not practical as it comprises large number of priority groups, each with one individual

Proposition 2.4 Recall the definition of δ1E from(2.7) The unique optimal production quantity QE isgiven by:

manu-2.5 Optimal Solution of the Centralized System

In the centralized system, a social planner such as the government or one of its agencies coordinatesthe production and vaccination to maximize the total expected social welfare This centralized solutionwill serve as a benchmark to understand the sources and extent of inefficiencies in the decentralizedsetting First, given the priority groups β, the planner determines the optimal egg production Q tomaximize the total expected social welfare Second, observing the obtained number of doses Qr, theplanner determines the demand to maximize the ex-post total social welfare Here, the ex-post utility per

individual is given by:

(2.11) W (δ, β, Qr) = V − θG (δ) − rh (δ, Q¯ r) − p (h (δ, Qr))

Z δ 0

zdG (z) − p (h (δ, Qr)) ×

"

(1 − φ (max {δ, β} , δ))

Z ¯ max{δ,β}

zdG (z) + (1 − φ (min {δ, β}) , δ)

Z β min{δ,β}

zdG (z) + δp ¯G (δ)
− (r + θ − w) ∀ δ ∈0, ¯δ

(i) There exists a uniqueδC ∈δzr, ¯δ
for all Qr ≥ 0 The fraction of population searching forthe vaccine in the centralized system is:

in the utility loss of all unvaccinated individuals while the rest of the expression is the direct benefits

to the society when the individual with infection disutility δ is vaccinated Observe that the net gain ofthe society from vaccination of an individual with searching disutility δ > δC1 is positive and increasing

by Assumption 2.1 (iii) and (2.16) Hence, in the case of unlimited supply, it is optimal for the socialplanner to vaccinate individuals with infection disutility greater than δC1 However, if the supply islimited, i.e., Qr < NG δ¯ C1
, it is optimal for the society that all individuals with infection disutility

δ ≥ ¯G−1(Qr/N ) search for the vaccine so that all the vaccine is utilized and everyone who searches forthe vaccine gets it

Proposition 2.5 implies that the socially-optimal vaccination fraction may be less than the zero-riskvaccination fraction, which coincides with the critical vaccination for some basic epidemiological mod-els when supply is unlimited and when limδ→δ+

zrB (δ) < 0 [Bauch and Earn, 2004; Breban et al., 2007;Chick et al., 2008] This result is different from those previously available in the literature because of

important differences in the underlying modeling assumptions For instance, Hill and Longini [2003]show that critical vaccination fraction is socially-optimal when the objective is minimization of number

of infections instead of welfare maximization or cost minimization Chick et al [2008], using an jective similar to ours (minimization of total social costs), show that the critical vaccination fraction issocially-optimal However, they assume that all individuals have identical infection disutility and obtainpositive expected net benefit from vaccination when coverage is less than the critical fraction Also,since demand is determined before yield realization in their model, it may be higher than the criticalvaccination fraction to compensate for low yield realizations

ob-Remark 2.1 (Positive Externality Effect) Observe from Figure 2.1 that the demand as well as number

of vaccinated individuals in the centralized system is higher than these in the decentralized system for

Qr ≥ NG δ¯ C

1
 For such obtained number of vaccine doses, the demand-side inefficiency is due

to the positive externality of vaccination, i.e., the government considers the impact of an individual’svaccination on the likelihood of infection for others but the individuals do not ForNG δ¯ E1
 < Qr<

NG δ¯ C1
 , the demand and number of vaccinated individuals in centralized system are still higherthan those in equilibrium but the gap between the two is lower and dependent onQr

Remark 2.2 (Negative Externality Effect) Figure 2.1 shows that for sufficiently low obtained number

of vaccine doses, i.e.,Qr< NG δ¯ 1E
, the equilibrium demand is actually higher than the demand inthe centralized system, whereas number of vaccinated individuals is equal in both systems The demand-side inefficiency in this region is due to the negative externality effect, i.e., when the vaccine supply islimited, each individual searching for the vaccine reduces the vaccine availability for individuals withhigher infection disutility and hence their expected net utility but does not internalize it The negativeexternality effect reduces the social welfare due to two characteristics of the allocation mechanism: (i)uncertainty/randomness, i.e., not everyone searching can get the vaccine and hence wasteful searchcosts are incurred, and (ii) inefficiency, i.e., those who need vaccine the most (consumers with highδ)

may not get it Note that (i) is due to the uncertainty/randomness in the vaccine allocation (i.e, onlyindividuals who will get the vaccine would search if they knew that they will get it) while (ii) is due to theinefficiency in the allocation of vaccines (i.e., individuals with higher infection disutility are vaccinatedfirst if the vaccine is allocated to individuals in decreasing order of their infection disutility) The secondobservation indicates that the demand-side inefficiency forQr < NG δ¯ E

1
 would still exist if searchcosts were zero Also, Propositions 2.2 and 2.3 show that the demand-side inefficiency due to negativeexternality effect can be substantially reduced by either appropriate choice ofβ for two priority groups

or by increasing the number of priority groups We explore this issue further in Section 2.8.2

Remark 2.3 (Availability Effect) If the consumers ignore the fact that supply is limited, similar to Brito

et al [1991], the equilibrium demand is equal toNG δ¯ E1
 for all obtained number of vaccine doses.However, when individuals consider limited supply, the equilibrium demand is not necessarily equal to

NG δ¯ E1
 and it depends on the available supply We label the gap between the two demands asthe availability effect Figure 2.1 shows that the equilibrium demand forQr < NG δ¯ 1E
, which isalways more than the available supply, is less thanNG δ¯ 1E
 for some values of Qr but higher forothers Thus, the availability effect is ex post beneficial for the society for some cases and harmful forother cases As a result, ex ante, the availability effect could be beneficial or harmful depending on theyield distribution Proposition 2.3 implies that under efficient rationing the availability effect is alwaysbeneficial for society since it completely eliminates wasteful searches and does not change number ofvaccinations We further explore the impact of allocation mechanism (through the choice ofβ) on themagnitude of availability effect in Section 2.8.2

Now, returning to the first stage, the social planner chooses the egg production Q to maximize thetotal expected social welfare Hence, social planner’s first stage problem is given by:

Q≥0WC(Q) := N EUW δC(U Q) , β, U Q
 + EUR δC(U Q) , U Q
 − cQ

Proposition 2.6 characterizes the optimal production quantity in the centralized system.

Proposition 2.6 (i) The optimal production quantity for the centralized system (QC) is unique

whereδ2C(Qr) = ¯G−1(min {1, Qr/N }) and B (δ) is given by (2.14)

(ii) The optimal production quantity in the centralized system (QC) is greater than that in thedecentralized system (QE)

To get an intuitive understanding of result (ii), note from Figure 2.2 that the marginal benefit to the ciety is always greater than w, which is the marginal benefit to the manufacturer, for Qr≤ NG δ¯ E

so-1
.This together with δ1E > δ1C imply that at any Qr ≤ QE, the expected marginal benefit to the soci-ety (LHS of (2.19)) is greater than the expected marginal benefit to the manufacturer (LHS of (2.10)).However, marginal costs to the society and to the manufacturer are the same Hence, the government

in the centralized system targets a higher production quantity than the manufacturer in the decentralizedsystem

Proposition 2.5 shows that ex-post demand can be higher in equilibrium for some values of therealized quantity Hence, it is conceivable that for some values of the parameters, QE and QC are suchthat the expected demand is greater in the decentralized setting Due to analytical intractability, weverified this numerically For an illustration, see Example 2.1 in the Appendix 2.A.2 This observation iscontrary to the existing results on products with positive externality with unlimited supply, which statethat the consumers demand less than the socially-optimal level [Brito et al., 1991]

Figure 2.2 Marginal social benefit when social planner decides the demand, where

B (δ) is given by (2.14)

2.6 Government InterventionsDespite general agreement on the need for a more active government role in coordinating the U.S.flu vaccine supply chain [Kilbourne, 1991; Harris, 2006], experts disagree on whether the governmentshould modify consumer behavior on the demand side [Institute of Medicine, 2004; Sloan et al., 2004;Hinman, 2005a] or provide adequate incentives for the manufacturers on the supply side [Pauly, 2005;Hinman, 2005a] We model these two broad policy suggestions as alternative intermediate scenarios ofpartial centralization

In the demand-side intervention, the government decides the socially-optimal demand, given that theproduction quantity is determined by a profit maximizing manufacturer In the supply-side intervention,the government decides the welfare maximizing production quantity while individuals decide whether tovaccinate themselves or not Our objective is to characterize relative effectiveness of these interventions

as a function of different parameters and to isolate the inefficiency on each side We do not aim toformulate concrete mechanisms for implementing them in the decentralized supply chain Table 2.1categorizes these scenarios depending on who makes supply and demand decisions