Operational Issues and Network Effects in Vaccine Markets

Transcription

1 Operational Issues and Network Eects in Vaccine Markets Elodie Adida Debabrata Dey April 8, 212 Hamed Mamani Abstract One o the most important concerns or managing public health is the prevention o inectious diseases Although vaccines provide the most eective means or preventing inectious diseases, there are two main reasons why it is oten diicult to reach a socially optimal level o vaccine coverage: (i) the emergence o operational issues (such as yield uncertainty) on the supply side, and (ii) the existence o negative network eects on the consumption side In particular, uncertainties about production yield and vaccine imperections oten make manuacturing some vaccines a risky process and may lead the manuacturer to produce below the socially optimal level At the same time, negative network eects provide incentives to potential consumers to ree ride o the immunity o the vaccinated population In this research, we consider how a central policy-maker can induce a socially optimal vaccine coverage through the use o incentives to both consumers and the vaccine manuacturer We consider a monopoly market or an imperect vaccine; we show that a ixed two-part subsidy is unable to coordinate the market, but derive a two-part menu o subsidies that leads to a socially eicient level o coverage Keywords: Vaccine coverage, negative network eect, random yield, vaccine pricing, vaccine subsidy 1 Introduction Prevention, cure, and control o inectious diseases are important concerns or modern health care systems The World Health Organization (1999) reports that inectious diseases are the world s biggest killer o young adults and children, and describes them as a crisis o global proportions threatening hard-won gains in health and lie expectancy Every one in two deaths in developing countries is caused by an inectious disease; globally, inectious diseases account or over 13 million deaths every year In 2, HIV/AIDS, TB, and malaria, together, killed 57 million people and caused debilitating illnesses in millions more (World Health Organization, 22) Inluenza and pneumonia claim about 25, to 5, lives every year globally, and about 36, deaths in the US alone (World Health Organization, 25) In addition to the emergence o new inectious diseases, such as Severe Acute Respiratory Syndrome or SARS (World Health Organization, 23), some o the diseases that were once eliminated resurace in populations earlier considered ree o the disease Old inections, such as tuberculosis and diphtheria, have occurred in large epidemics School o Business Administration, University o Caliornia, Riverside, CA USA (corresponding author) Foster School o Business, University o Washington, Seattle, WA USA

2 2 in Europe and other industrialized countries The 1996 outbreak o polio in Albania, Greece, and the then Federal Republic o Yugoslavia indicates that an inectious disease can easily resurace, i immunization coverage is allowed to drop (World Health Organization, 1999) Facing these grim realities, variety o epidemic control models have been proposed and used in practice including vaccination programs, prevention programs (such as changing risky behavior), and treatment programs (such as uarantine and use o antivirals); see (Brandeau et al 24) or more details Among a variety o intervention strategies, we ocus on vaccination programs in this research, as they are well-known or both their eiciency and cost-eectiveness In act, the World Health Organization (2) considers vaccination to be the ultimate weapon against inection and drug resistance For inluenza (or lu, more commonly), Germann et al (26) argue that vaccination, among other control programs such as social distancing, school closure, and use o antivirals, remains the most eective tool to eliminate the epidemic They show that even i a lu vaccine is poorly matched to the circulating strains, it can still drastically slow down the spread o the disease The global vaccination market is a large one, with annual sales o US$217 and US$253 billion in 29 and 21, respectively, and is projected to grow at a compound annual rate o 93%, reaching an extraordinary igure o US$395 billion in 215 (Carlson 211, 212) Despite this seemingly large size o the global market and the clear beneits o vaccination programs, vaccine uptakes in populations have typically been low (Blue 28) Such undesirably low vaccine coverage can be attributed to two main actors: Supply-Side Issues: Operational issues on the supply side, such as yield uncertainty, oten lead proit-maximizing manuacturers to under-produce, resulting in supply shortages, shortage o inluenza vaccines being a prime example In act, several papers in the operations management literature argue that the operational risk borne by a manuacturer is one o the main reasons or the shortage o inluenza vaccine in the market (Chick et al 28, Deo and Corbett 29) Negative Network Eect: A relatively low vaccine coverage can also arise rom the negative network eect aced by consumers As the raction o vaccinated individuals grows, the chance o contacting an inection diminishes Thereore, the willingness to pay or the vaccine reduces, an eect contrary to the positive network eect, in which the willingness to pay or a product increases with the size o the network (Katz and Shapiro 1985) In the end, a traditional ree market or vaccines may not be socially eicient On one hand, let to its proit-maximizing ways, a supplier may under-produce, resulting in shortages On the other, an individual may choose to ree ride o the herd immunity, and, conseuently, voluntary vaccination may not reach a socially optimal level in the population We are, thereore, interested in studying the ollowing research uestions that arise in this context: In the ace o yield uncertainty and network eects, what is the socially optimal level o vaccine coverage?

3 3 Is it possible, rom a contract theory perspective, to induce the manuacturer and the consumers to achieve that level? How does the eectiveness o a vaccine inluence the above decisions? What roles can governments or global non-proit organizations play here in coordinating the market? Each o the above issues has been examined in an isolated manner in prior research Some research in the operations management area has been devoted to the role o supply uncertainty and its eect on the supply chain outcome For example, Chick et al (28) argue that the production risk due to unknown vaccine yields, which is assumed by vaccine manuacturers, is the primary reason or an insuicient supply o inluenza vaccine in the market They design a variant o a cost sharing contract which provides incentives to manuacturers, as well as to governments that purchase vaccines, so that a supply chain achieves an optimal balance between or-proit manuacturer incentives and public health incentives Deo and Corbett (29) examine the role o production yield in explaining the limited number o players in the inluenza vaccine market However, these works do not consider the negative network eect and the consumers willingness to pay or the vaccine On the other hand, the demand-side issue has started receiving some attention in more recent literature in operations management For example, Cho (21) studies consumers willingness to pay, and Arioglu et al (212) address network externality eects in a similar context However, the objective o these articles are uite dierent rom the current one, as we are concerned with designing contracts to coordinate the vaccine market Epidemic modeling and health economics literature has primarily ocused on the negative network externality acing the consumers as a key driver or vaccination levels that are lower than what is socially desirable For example, Bauch and Earn (24) provide a game-theoretic analysis o vaccine coverage that considers the negative network eect as well as the necessary epidemiological details They show that a voluntary vaccination program without any government intervention ails to achieve the vaccination level necessary to eliminate the epidemic Brito et al (1991) provide a more comprehensive model in their economic analysis o the situation, but lack the necessary epidemiological details Geoard and Philipson (1997) also consider vaccine subsidies with the goal o complete eradication o the disease, but do not consider the cost o production and do not discuss an eicient coordination o the vaccine market They ind that a subsidy program would not be eective at achieving eradication because o network externalities In a more recent article, Althouse et al (21) study how public subsidies can help bring the vaccine uptake to a more eicient level in the case o a disease that is subject to network externalities, 1 but they consider only the consumers their subsidy is designed to maximize the aggregate consumer utility Cook et al (29) analyze epidemiological and economic ield data rom two sites in Kolkata, India, 1 The role o subsidies in market coordination has also been studied in other contexts that exhibit network eects For example, within the context o security, Zhuang (21) and Zhuang et al (27) show that providing ully subsidized security to targeted agents (with heterogeneous time preerences) can reduce the total social cost o security and improve the perormance o the system

4 4 to estimate the socially optimal subsidy or a cholera vaccine (disease subject to similar network externalities as inluenza) In most o the prior work, however, vaccine production is either neglected (eg, Bauch and Earn 24, Bauch et al 23, Geoard and Philipson 1997) or considered as deterministic and exogenous (eg, Brito et al 1991) Recently, Mamani et al (212) consider the role o governments in market coordination through subsidies in the presence o multiple vaccine manuacturers, but they do not take into account the yield uncertainty inherent in the production process, which, as shown here, has major implications in the subsidy program In particular, they ind that a constant one-part subsidy coordinates the market in a deterministic setting In extending their results, one would naturally expect that a constant two-part subsidy should align the incentives o dierent parties under the presence o production uncertainty Our results, however, indicate that, when the yield is stochastic, a constant two-part subsidy is just not suicient to align both consumer demand and production uantity with the irst-best outcome; rather, a menu o subsidies is necessary The purpose o this research is to bridge this conspicuous gap in the literature by considering yield uncertainty and network externalities together in order to develop a more comprehensive analysis and to provide a more practical coordination scheme The remainder o the paper is organized as ollows We discuss related prior work in Section 2 Section 3 develops the modeling ramework and examines the market euilibrium In Section 4, we employ a total social welare unction to identiy the socially optimal outcome Then, in Section 5, we derive a two-part subsidy scheme that induces this outcome Section 6 concludes the paper and oers directions or uture research 2 Literature Review The topic o network externalities and consumers willingness to pay in the context o public goods in general, and vaccines in particular, is gaining traction in the operations management community Cho (21) considers the issue o consumers willingness to pay and price elasticity to determine the socially optimal policy or selecting strains that are to be included in an inluenza vaccine This threshold policy is obtained based on a trade-o: On one hand, retaining the current composition o inluenza strains could lead to an ineective vaccine i new strains were to emerge On the other, including new strains in the vaccine composition could increase the production yield uncertainty Arioglu et al (212) combine consumption-side externalities with the supply uncertainty, and show that the limited availability o vaccines lead to an inlated demand They uantiy the level o ineectiveness o the decentralized model with partially centralized scenarios In this paper, we, too, combine negative network eects with yield uncertainty to identiy the level o ineiciency in a market-based system However, unlike Arioglu et al (212), we investigate this ineiciency with the lens o contract theory to design incentive mechanisms that can induce consumers and suppliers to make decisions that are aligned with the social optimum Our work is closely related to the recent work by Mamani et al (212); similar to theirs, our model also includes consumers ree-riding behavior, the production cost borne by the vaccine

5 5 manuacturer, and a central planner s desire to achieve the social optimum using subsidy In addition, however, we also consider the production yield uncertainty aced by the manuacturer In our model, the yield is stochastic, and hence the vaccine coverage is uncertain Conseuently, the uantity decision is made by the manuacturer under yield uncertainty, whereas the demand decisions are made by consumers acing both the supply uncertainty and the negative network eect These two decisions are not necessarily aligned with the social objective, and a central planner has a role in aligning them towards the social optimum This modeling dierence has critical conseuences on the euilibrium solution, social optimum, and coordination mechanisms In particular, in (Mamani et al 212), where the yield is deterministic, a one-part subsidy is shown to be suicient in coordinating the market In our paper, unsurprisingly, the stochastic yield makes a one-part subsidy ineectual or market coordination Perhaps more unexpectedly, even a two-part ixed subsidy scheme cannot coordinate the market The coordinating subsidy is a two-part menu o subsidies that depends on the coverage This undamental dierence in the euilibrium outcome is essentially a result o the signiicant dierences in the modeling contexts, reuiring dierent analyses Recent literature studies contract design issues when considering customers willingness to pay or public interest goods For example, Taylor and Yadav (211) investigate two orms o contracts or public products (eg, essential medicines): a purchase subsidy (to the retailers) and a sales subsidy (to the consumers) They show that when consumers are homogeneous in their valuation o the product, the subsidy provider (donor) would preer a purchase subsidy, whereas a sales subsidy is more preerable or heterogeneous consumers A key assumption in (Taylor and Yadav 211) is that the retailer has reedom to set prices ater market uncertainties are resolved and the product is acuired This is a valid assumption or their model as they consider populations in developing countries where the retailer can change prices based on market conditions In our model, however, the uncertainty exists with respect to the yield (not the demand), and pricing and production decisions are made prior to the realization o the yield The manuacturer, once committed to a ixed price, cannot change it easily Ovchinnikov and Raz (212) study a number o mechanisms to coordinate price and uantity decisions in a market or public goods where these decisions are made simultaneously They show the eectiveness o subsidies compared to rebates in coordinating either price or uantity decisions They also show that a joint rebate-subsidy mechanism can coordinate both decisions However, both the aorementioned papers assume a deterministic supply but a random demand We, on the other hand, consider a scenario where supply is uncertain and demand is deterministic (eg, the case o pre-orders or inluenza vaccine) Finally, we ocus on a speciic class o products vaccines rather than a generic product This allows us to derive a demand unction that is based not only on economic parameters such as the vaccine price, but also on other epidemiological actors such as the population s perception o the vaccine eectiveness and the dynamics o the inection In summary, the major contributions o our paper are two old: First, we build on a stream o work that combines concepts rom three distinct bodies o literature operations management, economics, and epidemiology to generate insights about public policy issues related to vaccine

6 6 pricing, coverage, and subsidies in the context o potentially imperect vaccines Second, our model contributes to the operations management literature by considering contracting issues within a production process with random supply and price-dependent demand While others have examined single-period models with random and price-dependent demand (Petruzzi and Dada 1999), as well as supply contracts in that setting (Cachon 23), we show that eects o random supply are dierent than those o random demand when considering price-sensitive consumers, and thus the resulting coordinating mechanism must dier accordingly More speciically, we show that a menu o two-level incentive mechanism is needed to coordinate the market, and a simple two-part ixed subsidy is not suicient to align both the uantity and pricing decisions simultaneously Our results have important implications in terms o the role o governments and global health organizations in managing vaccine programs and provide directions or appropriate incentive schemes that can work towards a greater good 3 The Model In this section, we develop a model o the vaccine market with a single manuacturer supplying a speciic vaccine or a ixed population We consider the ollowing seuence o events in this seuential game At the beginning o the production season, the manuacturer decides the vaccine price and the production volume, and incurs the corresponding production cost Next, consumers ollow with their choice o whether (or not) to get immunized, which determines the demand or the vaccine This decision is based on the consumers expected utility that takes into account their probability o inection, disutility rom the inection, and the price they pay or the vaccine Finally, the total vaccine coverage is determined based on the realized production yield and the demand Any letover vaccine is discarded and any unmet demand is lost We note that Arioglu et al (212) consider a dierent model o consumer behavior in which individuals make the decision to search or vaccines based not only on the vaccine price but also the availability o vaccines There, the demand or vaccines is determined based on the vaccine price and the realized production yield in order to incorporate the availability eect in their model In this paper, we assume that vaccine production uantity or yield does not signiicantly aect the demand or vaccines This assumption is justiied in the ollowing contexts: There are situations where the realized (or targeted) production uantity is not public inormation, or simply when its eect on consumer behavior is not as signiicant as the vaccine price In some cases, a signiicant portion o the demand or vaccines is based on pre-orders received rom healthcare providers that are placed well in advance o the production yield realization; or example, see (Fine 24, CDC 211a) or the case o inluenza vaccines 2 In this case while consumers do not directly purchase the vaccine rom the manuacturer, the total demand rom 2 In case o inluenza vaccines, such pre-orders happen between January and March (CDC 211a) While lu vaccines can be purchased at a later time closer to the lu season, Fine (24) reports that suppliers emphasis and preerence is on pre-booking

7 7 a healthcare provider or a certain vaccine price should eectively mirror the actual demand rom consumers based on demand and price elasticity inormation rom prior periods 31 Characterization o Consumer Behavior We now provide a description o the consumer model, also borrowed rom (Mamani et al 212) We consider consumers to be heterogeneous, in the sense that direct and indirect costs o inection varies rom one person to another This is captured by indexing consumers by a parameter u that is assumed to be uniormly distributed over the interval (, 1); u can be viewed as the relative loss suered by an individual rom an inection This assumption allows us to model individuals who perceive the vaccine as dangerous and are thus more willing to risk inection rather than be vaccinated (low value o u), as well as individuals like children and the elderly who, or example, are more prone to complications when contracting inluenza and would thus experience more disutility rom an inection (high value o u) The absolute cost to consumer u rom an inection can then be expressed as Lu, where L is a constant Because o dierent inection dynamics or the vaccinated and unvaccinated ractions o a population, the inection probabilities are also dierent or each group Following Mamani et al (212), by p() and P(), we denote the inection probabilities or the vaccinated and unvaccinated ractions, respectively, or an overall vaccine coverage o ; P() must be strictly greater than p() or the vaccine to have any value The inection probability or the entire population, r(), can also be viewed as the overall raction o inected individuals in the population: r() = p() + (1 )P() Let the price o the vaccine be W Since vaccines are not ully eective, an immunized person incurs a vaccine cost o W and may still get inected Thereore, W + Lup() represents the total expected cost o getting vaccinated On the other hand, a consumer who has not been immunized does not incur the cost o W but has a higher chance, P(), o becoming inected The expected cost incurred by a unimmunized individual is thus LuP() Thereore, or the indierent consumer, the average cost o getting vaccinated and the cost o an inection are the same Let ū denote such a marginal consumer, then we have: LūP() = W +Lūp(), or W = Lū(P() p()) Any consumer with u ū gets immunized, whereas anyone with u < ū does not Thereore, ū = 1, as u is uniormly distributed over (, 1) Substituting this and using the act that P() p() = r() p() 1, we get: w = W L = r() p() T() (1) Euation (1) tells us that consumers make their vaccination decisions in such a manner that the normalized price, w, o a vaccine represents the marginal reduction it induces in the probability o inection As discussed earlier, we do not consider the eect o production uncertainty and possible production shortage on consumers demand decision O course, this modeling ramework can be easily adapted to the case where an aggregated pre-order is placed by healthcare providers prior to the

8 8 production season, when there is simply no inormation about the vaccine yield or production shortage This is because, as illustrated in the next section, the consumer choice model in (1) is euivalent to the one where the total orders placed by consumers (or healthcare provider) ollows a piece-wise linear orm, which is a standard assumption in many price-demand models 32 Inection Probability In order to complete the characterization o consumer behavior, we need to ind the unctional orms o r(), p(), and T() In a large part o the literature on vaccine programs, it is assumed that an individual receiving a vaccine becomes completely immune rom the underlying inection, that is, the vaccine has been assumed to be perect This assumption, however, may not be valid or a number o inectious diseases o interest, such as seasonal inluenza and HIV 3 In this regard, we ollow Mamani et al (212) and assume that vaccines may indeed be imperect a vaccinated consumer, albeit less likely, can still get and transmit the inection There are two ways in which an imperect vaccine aects the transmission o the disease: First, imperect vaccines can reduce the probability o becoming inected i exposed to an inectious contact, which is known as the vaccine eect on susceptibility Second, imperect vaccines can also aect the probability o disease transmission when a vaccinated individual gets inected; this is known as the vaccine eect on inectiousness (Datta et al 1998, Hill and Longini 23) To combine these two eects, we take an approach that is similar to the one adopted by Longini et al (1978) Let < φ 1 be the vaccine eicacy parameter that relects the combined eect o vaccine on transmission, which includes both susceptibility and inectiousness eects Note that φ = 1 corresponds to the perect vaccine case Following Mamani et al (212), we express r() as (see Appendix A in the Electronic Companion to this paper or more details): r() = {, i > F = R 1 φr 1 φ 1 R, otherwise, where R, the basic reproduction number, is deined as the total number o secondary inections caused by an inectious individual in an otherwise susceptible population Thereore, R is a measure that indicates whether the introduction o a single initial case o inection into a susceptible population will result in a nontrivial outbreak It must be noted that, i R 1, the inection probability is zero, and the disease transmission dies out Hence, or the remainder o the paper, we only consider the interesting case o R > 1 We note that F = R 1 φr is the minimum vaccination raction that drops the overall inection probability, r(), to zero In epidemiology, this threshold is called the critical vaccination raction; it represents the minimum level o vaccine coverage necessary or providing the so-called herd immunity eect a situation that arises when immunization level in the population is suiciently 3 Seasonal inluenza strains constantly mutate over time making a vaccine an imperect match to the circulating strains in a community (Gross et al 1995, Hill and Longini 23, Sullivan 1996) Within the context o potential HIV vaccines, many experts believe that, even i a vaccine were to become available in the uture, it would likely provide only a partial protection rom the disease (Datta et al 1998, Hillier et al 25, Longini et al 1996)

9 9 high so that the disease is eliminated entirely rom the population It should be noted that F could be greater than one, in which case the disease cannot be completely eradicated by a vaccination program alone Thereore, we introduce F min {F, 1} Beore we discuss the unctional orm o p(), and hence that o T(), it is necessary to see how the negative network eect maniests itsel in this context The willingness to pay or consumer u is given by WTP u = Lu(P() p()) The act that WTP u depends on, the market coverage, is an evidence o the presence o a network eect Now, note that, when φ = 1, p() becomes zero as the vaccine is perect; i, in addition R, then r() (1 ) and P() 1; this is because, when the reproduction number becomes large, an unvaccinated individual is almost certain to contract the inection In this case, P() p() = 1, and WTP u becomes independent o the coverage level Thereore, in this limiting case, the actions o other individuals become irrelevant As a result, the externality eect completely disappears the only way to enjoy the beneits o the vaccine is to acuire it In contrast, i R < or φ < 1, the term P() p() is lower than 1 and is no longer independent o This implies that, or the same consumer, WTP u is lower than in the limiting case considered above, and its speciic value depends on the overall coverage level Thereore, the desire to purchase depends not only on a consumer s own preerence, but also on decisions made by other consumers This happens because an unvaccinated individual could beneit rom the (partial) immunity o the vaccinated population Put dierently, unlike typical products or services, in the vaccine market, a consumer does gain some utility rom the vaccine, without having to actually purchase it Hence, the coverage level has an impact on the purchase decision o an individual Finally, in order to ind an expression or p(), we note that it must abide by two properties First, the probability o inection or vaccinated individuals must be less than that or the general population, ie, p() < r(), or all Second, when the vaccine is ully eective, a vaccinated individual will not contract the disease, implying p() = i φ = 1 The ollowing unctional orm abides by these two conditions and provides a good approximation o the exact inection probability unction (see Appendix B in the Electronic Companion to this paper or details): p() = η(1 φ)r(), where η 1 1 φ is a constant; in this paper, we use this unctional orm or p() Using this approximation, we ind that WTP u = Luθr() 1, where θ = 1 η(1 φ) Thereore, the externality eect, which can be viewed as the strength o the dependence o WTP u on, can be written as: EF u = WTP u =, i > F = R 1 φr, Luθ (1 ) 2 1 φ 1 R, otherwise It is now easy to see that, when < F, EF u depends on both R and φ; it approaches zero when R and φ 1, but is non zero otherwise Finally, T() can be written as: T() = θr() = (1 η(1 φ))r() (2)

10 1 33 Production Decision As mentioned earlier, we consider a production process with uncertain yield This is particularly relevant in the context o inluenza vaccines, where uncertainties in the manuacturing process have been shown to play a key role in decisions across the entire vaccine supply chain (Chick et al 28); our model reduces nicely or vaccines with deterministic yield (Mamani et al 212) We consider a stochastically proportional yield or the inal production output (eg, Yano and Lee 1995) Speciically, let be the target production level normalized with respect to the size o the population; then, we assume that the inal production output is U where < U < Ū is a random variable with an average E[U] = µ, a probability distribution unction J( ), and a probability density unction j( ), where j(x) is positive only i x (, Ū) and zero otherwise; the possibility that U > 1 is not excluded in our model and represents a situation in which the yield is higher than the targeted production uantity We analyze a market where the vaccine price is determined beore the production campaign and the realization o the random yield, such as the inluenza vaccine market; CDC (211b), or example, lists vaccine prices as early as January An underlying assumption o our model is that, once committed to a price, the manuacturer cannot simply change it based on the production yield or other market conditions This is a plausible scenario or markets such as the US (CDC 211b) Taylor and Yadav (211) study an alternative model where prices are set ater the market conditions are realized This is the case when the supplier has the authority to change prices based on the realized demand and available uantity and is perhaps a better representation o situations in developing countries In a market where pricing decisions are made prior to random yield realization, the vaccine price does not necessarily clear the market and the manuacturer can have vaccine shortage or excess We assume that the manuacturer is subject to a constant (normalized with respect to L) marginal production cost o c per unit For a production yield U, a normalized vaccine price w, and a production uantity, the manuacturer s proit is: Π(, w, U) = w min{u, } c 34 Market Euilibrium In a monopoly market, the manuacturer chooses his vaccine price and production uantity in order to maximize his expected proit, anticipating the eect o his decision on the inal production output and consumers reaction: π(, w) = E U [Π(, w, U)] = E U [w min{u, } c] = w xdj(x) + w dj(x) c,

11 11 where is given by Euation (1) and can be substituted to rewrite the manuacturer s expected proit, π(, w), as a unction o and : π(, ) = T() xdj(x) + T() dj(x) c (3) To solve the manuacturer s problem, we irst ind the optimal production uantity or a ixed demand, denoted by m (), and then we solve or the euilibrium demand, m Let be ixed such that < F Taking the derivative with respect to, using Leibnitz s rule, and rearranging terms, we get the ollowing result Lemma 1 Let be the uniue solution o: xdj(x) c T() = Then, m () is given by: m () = {, i T() c µ,, otherwise (4) Note that when the production cost exceeds the expected revenue rom the vaccine (µt() < c), the manuacturer would simply set m () = { } x Example 1 I U ollows a uniorm distribution on [a, 2µ a], then J(x) = min 1, 2(µ a) In that case: m () = 2 T() a 2 T()+4c(µ a), i T() µ c,, otherwise Figure 1 illustrates the behavior o m () or two dierent values o φ, when U ollows a uniorm distribution on [,2] and R = 2, c = 1, and η = 33 This igure shows that m () euals zero when T() > c µ This implies that, given a φ, there exists a threshold or a point o discontinuity always below the critical vaccination raction, F beyond which the production uantity is zero Indeed, beyond this point, the vaccine price becomes so low that production costs outweigh expected revenues and the manuacturer chooses not to enter the market This threshold occurs at a lower value o when the vaccine is less perect because the euilibrium price o a vaccine with a lower φ is also lower We also notice that or a ixed coverage level, the lower the value o φ, the lower is the uantity produced by the manuacturer, since the vaccine price decreases as φ decreases Finally, we note that m () is not necessarily monotonic in, even to the let o the threshold; this can be clearly seen in the plot corresponding to φ = 1 There, the uantity irst increases in as a larger demand provides incentives to supply higher uantities However, beyond a certain value, the price o the vaccine, w = T(), becomes too low and the supplier, in turn, chooses to reduce the

12 12 m () 25 R = 2 φ 2 = 1 c = 1 η = φ = Figure 1: Optimal Monopoly Production Quantity with U Uniorm[, 2] uantity produced This observation is consistent with observations in traditional markets where a monopolist is known to under-produce Using Lemma 1, we can reduce the objective unction to a unction o a single variable,, by substituting or m () rom Euation (4) into Euation (3) For the Newsvendor problem with price-dependent and random demand, this method was introduced by Whitin (1955) We ollow a similar process when the supply is random It turns out that the proit unction π( m (), ) is not concave; however, as indicated by the next theorem, we can still prove the existence and uniueness o an optimal solution or a large class o random variables, including the increasing ailure rate (IFR) distributions or supply uncertainty: Theorem 1 Suppose, or all z, the distribution unction J(z) satisies the ollowing condition: 2ρ(z)(1 J(z)) + z xdj(x) + z z dj(x) dρ(z), dz where ρ(z) = j(z) 1 J(z) is the hazard rate Then, the euilibrium monopoly price is w m = θr( m ), and the monopoly production uantity is m ( m ) units, where m () is obtained rom Euation (4), and m is the uniue in the region m = otherwise (, F c µφθ ) that satisies dπ(m(),) d = i F c µφθ, and It is interesting to observe that, despite the similarities between our model and the Newsvendor model with price-dependent demand, the condition o the hazard rate unction in Theorem 1 is dierent than the similar condition or the latter model (eg, Petruzzi and Dada 1999) In classical Newsvendor models, the price-dependent demand is either (linear) additive or (exponential) multiplicative In our model, however, the supply uncertainty is multiplicative in nature, and, at the same time, the demand unction is nonlinear Thus, the objective unction in our model is aected by the supply uncertainty in a dierent manner than in those models where demand is uncertain

13 13 Example { 2 Once } again, let us assume that U is uniormly distributed over [a, 2µ a], ie, J(x) = x min 1, 2(µ a) Then, rom Euation (3) and Example 1, we get: π() = ( (2µ a)t() ) (a 2 T() + 4c(µ a))t() 2(µ a) which can be maximized using Theorem 1 to obtain m Figure 2 represents the euilibrium in the case o a perect vaccine (φ = 1) or a range o values o c when U ollows a uniorm distribution on [,2] and η = 33 In order to see the impact o yield uncertainty, we also plot the value o m or the case o deterministic yield (ie, U = 1), m or m m (U = 1) m 1 m c (a) R = 2 m or m m (U = 1) 2 m m c (b) R = 4 φ = 1; η = 33 Figure 2: Euilibrium Production and Coverage or a Perect Vaccine (U Uniorm[, 2]) Comparing the two cases in the igure, it is clear that the presence o yield uncertainty implies a lower vaccine coverage at euilibrium This inding is in conormity with the results rom previous studies (on operational risks born by inluenza vaccine suppliers), which conclude that manuacturing uncertainties help explain low vaccine supplies and sometimes even shortages (Fine 24) Moreover, high externality eects (low R ) lead to lower coverage as a single vaccination helps protect more individuals against the disease We also observe that when the per unit cost is low, the manuacturer has an incentive to produce high uantities o the vaccine (compared to the euilibrium coverage); as the cost increases, however, the yield uncertainty makes it too risky to produce extra vaccines, and the uantity drops below the euilibrium demand In order to see the impacts o vaccine eectiveness φ and the distribution o U, we also plot the euilibrium outcome or φ = 8 in Figure 3, or two values o R and two dierent distributions o U The comparison o Figure 3 with Figure 2 makes the role o vaccine eectiveness clear First, or low values o the production cost, the vaccine coverage at euilibrium decreases with vaccine eectiveness, but this trend reverses or high production cost This illustrates that a more

14 14 ineective vaccine reuires more coverage in the population to achieve the same level o protection, which is possible only when the vaccine is cheap enough to produce When it is more expensive, a lack o eectiveness makes the vaccine less worthwhile to purchase, and coverage drops Moreover, the euilibrium uantity is lower than when the vaccine is perect The impact o R in Figure 3 is similar to the one see in Figure 2 m or m m 1 m c (a) R = 2 m or m m m c (b) R = 4 φ = 8; η = 33 Figure 3: Euilibrium Production and Coverage or an Imperect Vaccine (U Uniorm[, 2]) 4 First Best Solution In order to get a more complete picture o whether or not the existence o negative network eects and yield uncertainties in the production process leads to market ineiciencies, we need to solve the irst best problem, whereby the social planner maximizes the total social welare, which is the sum o all the surpluses or the manuacturer and the consumers (vaccinated and unvaccinated) In addition, we also account or the societal surplus that may accrue rom a lower inection rate Manuacturer surplus: The manuacturer surplus is the net proit earned by the manuacturer when the target production uantity is, and a raction g = min{u, } o the population is vaccinated Manuacturer surplus = σ M = wg c Vaccinated consumers surplus: When individuals are vaccinated, their probability o inection reduces and the expected cost rom inection is lower The aggregate normalized cost or purchasing the vaccine is wg I these individuals were not vaccinated, the total vaccinated raction would be zero and their probability o getting inected would be r(); i they do get inected, they incur a normalized inection cost u [1 g, 1) The surplus gained rom

15 15 vaccination or this raction o the population is then the dierence between the costs with and without vaccination Vaccinated surplus = σ V = 1 1 g x(r() p(g))dx wg = [ ] 1 (1 g)2 (r() p(g)) wg 2 2 Unvaccinated consumers surplus: One o the key dierences between vaccines and most other products is that an individual can enjoy some surplus even i (s)he does not purchase or acuire the product, because o the immunity generated by other vaccinated individuals that leads to the negative network externality eect The unvaccinated consumers surplus is determined as the dierence in utility between when the remaining consumers get vaccinated and when they do not Each individual in the unvaccinated raction, with a normalized inection cost u (, 1 g), has a probability P(g) o becoming inected when the other raction is vaccinated, and a probability r() o becoming inected when the other raction is not vaccinated Calculating the dierence in total cost between these two situations yields the total surplus gained or this raction o the population rom the presence o the vaccine Thereore, the unvaccinated raction (1 g) gets a surplus o: Unvaccinated surplus = σ N = 1 g x (r() P(g))dx = (r() P(g)) (1 g)2 2 Societal surplus: To obtain the total social welare, we also consider the indirect cost to the society we assume that every individual who gets inected poses a normalized loss o λ on the society; this cost could include a societal loss o work time and healthcare costs Ovchinnikov and Raz (212) consider a similar term in their social welare unction called the externality eect that represents the public interest component o the product Since, a raction r(g) is likely to get inected in an expected sense, we write the societal surplus as: Societal surplus = σ S = λr(g) Combining all these together, we get the normalized total social welare: Total Social Welare = σ M + σ V + σ N + σ S = r() 2 (1 g)r(g) + gp(g) 2 λr(g) c, where, as beore, g = min{u, } is the actual raction o the population that get vaccinated Maximizing this total social welare then is euivalent to minimizing the ollowing social cost: SC(,, U) = = (2λ + 1)r(g) gt(g) + c 2 (2λ + 1)r(min{U, }) min{u, }T(min{U, }) + c 2

16 16 Thus, the central planner s objective would be to minimize this expected social cost unction: Γ(, ) = E U [SC(,, U)] = E U [ 1 2 ((2λ + 1)r(min{U, }) min{u, }T(min{U, })) + c ] = 1 Ū 1 ((2λ + 1)r(x) xt(x))dj(x) + ((2λ + 1)r() T())dJ(x) + c 2 2 Taking the partial derivative o Γ with respect to and leads to: Γ Γ = = ( φ(2λ + 1) + T(x) + xt (x) ) xdj(x) + c, (5) 2 ( φ(2λ + 1) + T() + T () ) dj(x) (6) The next result characterizes the irst best solution: Theorem 2 Let and be the socially optimal vaccine coverage and production uantity Furthermore, let be the solution o: F 1 ( φ(2λ + 1) + T(x) + xt (x) ) xdj(x) c = (7) 2 Then, = {, i c µ (, otherwise, φλ + φ+t() 2 ), and = min{ F, Ū } Theorem 2 states that, when the production cost (c) is low, or when the inection cost or the society (λ) is high, it is socially optimal to set the demand at its maximum (the critical vaccination raction or the maximum possible production) However, this does not imply that it is socially optimal to always reach this high vaccine uptake Having the demand at its maximum ensures that the demand will not be the bottleneck in determining the coverage The actual coverage is the minimum o the demand and the realized production, and will thus be determined by the yield realization and uantity Example 3 Suppose U is uniormly distributed over [a, 2µ a] Let be the uniue positive solution o: ( ( ) ) 2 φ F (1 + 2λ + Fθ) a 2 Fθφ ( ( ) ) 2 F a 3 F = 2c(µ a) 4 3

17 17 Then, rom Theorem 2, we get: = (, i c < φµ 3(φµ(1+2λ+ Fθ) 2c) φµ, i 2φθ((2µ a) 2 +2aµ) 2, otherwise, = min{ F, (2µ a)} λ Fθ 3 ) a2 Fθφ 3(2µ a), ( ) 1 + 2λ Fθ 3 a2 F θφ 3(2µ a) c φµ 2 ( 1 + 2λ + Fθ ), and Furthermore, i a = and µ = 1, that is, i U ollows a uniorm distribution on [,2], then = F φ(3+6λ Fθ) 2 6c For that case, the irst two panels in Figure 4 depicts the socially optimal vaccine coverage and production uantity or two dierent values o each φ and R, when λ = 1 and η = 33 Similar to the market euilibrium case, or low values o c, the targeted production or c (a) U Uniorm[,2]; φ = 8 or c (c) U Truncated-Normal[, 4]; φ=8 λ = 1 η = 33 or c (b) U Uniorm[,2]; φ = 1 or c (d) U Truncated-Normal[, 4]; φ=1, R = 2, R = 4 Figure 4: Socially Optimal Production and Coverage

18 18 uantity exceeds the demand so as not to risk inventory shortages However, as the cost parameter, c, increases, the targeted production alls below the demand to avoid over-producing expensive vaccines that are likely to go unsold The impact o vaccine eectiveness is also clear rom Figure 4 For low values o c, the optimal production and coverage levels both increase as the eectiveness, φ, decreases This is to be expected as the vaccine eectiveness goes down, it becomes necessary to immunize a larger raction o the population to obtain the same level o immunity and to produce more to meet that higher coverage level However, at high values o c, maintaining a high level o immunity is no longer desirable There, the trend reverses Both optimal production and coverage level decrease uickly and reduce to zero as the vaccine eectiveness decreases It is not optimal to keep producing an ineective vaccine at a high production cost It is also clear rom these plots that the socially optimal production and coverage levels increase with R This is expected as R increases, the higher level o inectiousness makes it necessary to produce more vaccines in order to immunize a larger raction o the population In order to veriy the robustness o our results with respect to the distribution o U, we have numerically tested them with other distributions as well In particular, in the last two panels in Figure 4, we report the results when U ollows a truncated normal distribution on [, 4]; or the sake o easy comparison, the mean and the standard deviation o the truncated normal distribution, in this as well as other illustrative plots in this paper, are kept the same as those or the uniorm distribution used in the irst two panels in Figure 4 4 The similarity o the last two panels with the irst two clearly illustrates the robustness o our results with respect to the distributional assumptions about U 5 Incentive Mechanism to Coordinate the Market This section studies the eects o a subsidy program as a coordinating mechanism to align the incentives o dierent entities in a vaccine market We introduce and motivate the subsidy scheme used to coordinate the market and show that a menu o two-part subsidies can indeed achieve this goal We also examine the eiciency o partial subsidy programs, where a one-part mechanism is implemented by only subsidizing vaccines to the consumers 51 Subsidy scheme Comparing the results rom Sections 3 and 4 or more speciically comparing Figures 2 and 3 with Figure 4 we can easily see that negative network eect and yield uncertainty can lead to large market ineiciencies An important thrust o this paper is to understand whether, and how, a social planner could intervene in the vaccine market to align the euilibrium outcome with the social optimum O course, as amply demonstrated in Figure 4, achieving the herd immunity may not always be optimal or the society as a whole (i, or example, the vaccine is very expensive to 4 We set the mean and standard deviation o the underlying normal distribution at and , respectively; this way the mean and the standard deviation o the truncated distribution matches those o Uniorm[,2], which are 1 and 1 3, respectively

19 19 produce and an inection causes little harm or disutility to consumers) Government intervention should thus aim at not necessarily achieving a vaccine coverage eual to the critical vaccination raction, but rather to some raction that optimizes a measure o the societal beneit In this section, we propose an incentive mechanism that can eliminate the market ineiciencies and achieve the social optimum As mentioned earlier, we consider a situation where the vaccine prices are set well beore the production yield is realized, sometimes as early as a year beore production starts We ocus on subsidies to simultaneously coordinate price and uantity decisions Furthermore, since both price and uantity decisions are made beore resolving the yield uncertainty, a simple one-parameter subsidy is not suicient to coordinate both decisions at the same time This is due to the act that a subsidy that can provide incentives to the individuals to ensure a suiciently high level o vaccine coverage (coordinate price) does not necessarily align the manuacturer s production decision, under yield uncertainty, with the central planner s objective (coordinate uantity) A similar behavior can be observed in the classical Newsvendor model with price-dependent demand (eg, Cachon 23) Thereore, we consider a two-parameter incentive mechanism (s, e), in which a subsidy s is provided to the consumers (to coordinate the vaccine price or euivalently vaccine coverage, ), and, under a cost sharing proposal, the central planner agrees to pay e to the manuacturer or every production unit (to coordinate production uantity, ) 5 The cost sharing component in this contract is analogous to the purchase subsidy, and the subsidy component o our contract is similar to the sales subsidy in (Taylor and Yadav 211) One way to implement the contract, among several other possible alternatives, would be to oer both incentives as a package The manuacturer could purchase the raw materials or the production directly rom the government at a per unit cost o c e In return, the manuacturer would get a payment o s or each vaccine sold to the consumers, whereas the customers would only pay w s to the manuacturer at the time o purchase Thus, the eective production cost to the manuacturer would become c e, and the demand-price relationship can be written as: w = s + T() The manuacturer s proit can be written as ollows π(, ) = (T() + s) xdj(x) + (T() + s) dj(x) (c e) In what ollows, we irst obtain the optimal production uantity or a ixed demand, denoted by m (), and then ind the euilibrium demand, m Let be ixed such that F Taking the derivative with respect to, we can determine the optimal production uantity m () 5 This is in direct contrast with the case where the vaccine price can be set ater the production yield is realized I that were indeed the case, then one can modiy (Taylor and Yadav 211) using the approach presented in this paper to account or the production uncertainty and show that this market can be coordinated with just a single-parameter subsidy

20 2 Lemma 2 Let be the uniue solution o: xdj(x) c e T() + s = Then, m () is given by: m () = {, i T() c e µ s,, otherwise (8) Example 4 I U ollows a uniorm distribution on [a, 2µ a], then: 2 (T()+s) c e, i T() m () = a 2 (T()+s)+4(c e)(µ a) µ s,, otherwise Using the irst order optimality conditions, we can ind the euilibrium or the decentralized vaccine market More speciically, or a pair (s, e), Euation (8) allows us to reduce the objective to a unction o only by substituting m () into π(, ) Beore we derive the subsidy pair (s, e), it is important to note that it is not possible to coordinate the market with ixed values o s: Proposition 1 Let s > be a ixed subsidy provided to each vaccinated consumer and e the unit production payment provided to the manuacturer Then, there is no contract with ixed payments (s, e) that can coordinate the market Proposition 1 tells us that the consumer subsidy must be given as a menu o subsidies, that is, s should be a unction o the coverage level This is in direct contrast with the results in (Mamani et al 212), where not only is the subsidy one-part because yield is deterministic, but it is also ixed and does not change with the coverage level The ollowing result characterizes a subsidy/cost-sharing scheme (s, e) that can coordinate the vaccine market Theorem 3 Let and be the socially optimal demand and production uantity Let e and s() be given by: e = s() = c (T( ) + s( )), =, θ, <, θ, < F,, F xdj(x), >, and (9) Then, the contract (s(), e), in which a subsidy o s() is provided to each vaccinated consumer and a unit production payment o e is provided to the manuacturer, coordinates the market (1)