Optimal Vaccine Subsidies for Epidemic Diseases

TECHNOLOGIES such as vaccines, condoms, and mosquito nets that protect individuals against infectious diseases can generate positive externalities by reducing transmission to others. While textbook economic models provide a justification for public subsidies of such preventive technologies, the appropriate magnitude of these subsidies remains understudied. Without a deeper understanding, it is difficult for economists to provide guidance—even at a conceptual level—on the optimal level of government subsidies for infectious-disease control and how such subsidies should vary across diseases. Furthermore, much existing work by economists has been oriented toward policies combating endemic diseases playing out over generations, such as vaccination campaigns to eradicate polio or circumcision campaigns to reduce the spread of HIV. The COVID-19 pandemic, however, has underscored the urgency of quelling global outbreaks of novel diseases.

To address these questions, we construct a tractable model that integrates epidemiological and economic considerations. For concreteness, we focus on the market for a vaccine, but the analysis applies to other preventive technologies. Consumers and producers base their economic decisions on rational expectations of disease dynamics based on a susceptible-infected-recovered (SIR) model, standard in the epidemiology literature. We analyze a vaccine campaign introduced at a single point in time into a population without turnover, modeling choices intended to capture relevant features of epidemics like COVID-19 that may rise and fall within a generation, calling for a concentrated policy response.

A key finding is that the positive externality that the marginal consumer's vaccination exerts on others reaches its peak for intermediate rather than the highest values of $R0$ (the disease's basic reproductive number, a widely used measure of infectiousness). This finding holds robustly across market structures ranging from perfect competition to Cournot competition to monopoly. For low values of $R0$⁠, the marginal externality is low because there is little disease transmission among people. For high values of $R0$⁠, vaccinating a given consumer does not provide much protection to others since they are almost certain to contract the disease from someone else anyway. To be sure, a consumer’s vaccination provides a substantial social benefit when $R0$ is extremely high, but most of that benefit is internalized by the consumer. The optimal subsidy, which corrects for the marginal externality, likewise peaks for intermediate values of $R0$⁠.

At certain intermediate values of $R0$⁠, free riding can be so extensive that an increase in $R0$⁠, by reducing some of this free riding, can perversely reduce equilibrium infections. This epidemiological version of the Peltzman (1975) effect arises when consumers compensate for increased risk by increasing vaccination to such an extent that it more than offsets the direct increase in infectiousness.

Our results go beyond nonmonotonicities. Whether universal vaccination can be a viable business strategy is explored in section V. Previous game-theoretic analyses have suggested that a perfectly effective vaccine would never be universally purchased at a positive price because with all other consumers protected, the marginal consumer obtains no private benefit (Geoffard & Phillipson, 1997; May, 2000; Bauch & Earn, 2004). In our model, however, universal vaccination of susceptibles with a perfectly effective vaccine can be profitable. The risk of contracting the disease from those infected before the arrival of the vaccine but not yet recovered preserves a positive willingness to pay for the marginal consumer even if all other susceptibles are protected. Universal vaccination with a perfectly effective vaccine is not just possible but guaranteed in equilibrium for sufficiently low cost and sufficiently high infectiousness.

In typical economic settings, benefit functions are concave, leading to decreasing returns, but epidemiological effects may lead to disproportionate benefits of vaccination, as folk wisdom suggests may happen around the threshold for herd immunity. Section VI explores this issue formally by providing conditions under which vaccination exhibits increasing social returns. The key indicator is $R0S^0$⁠, the product of the basic reproductive number $R0$ (technically, the number of secondary cases an infectious individual would hypothetically transmit in a fully susceptible population) and the proportion of susceptibles in the relevant population $S^0$ (here, at date 0, when the vaccine is introduced). The product $R0S^0$⁠, known as the effective reproductive number, tells us the number of secondary cases an infectious individual transmits in the relevant population. If $R0S^0<1$⁠, primary infections lead to fewer secondary infections from the start, implying the infection rate falls throughout the epidemic even absent a vaccine. For infections to rise to a peak before falling in the classic epidemic pattern requires $R0S^0>1$⁠. If the effective reproductive number is yet higher—we show $R0S^0>2$ is sufficient—then social returns to vaccination are initially increasing. With such a high effective reproductive number, the epidemic is so explosive that a small amount of vaccine does little to slow it; to make a measurable dent in the epidemic requires concentrating supplies in one region. If $R0S^0$ exceeds a yet higher threshold, vaccination exhibits increasing social returns for all capacity levels, meaning that it is efficient to fully vaccinate one region before moving to the next.

Section VII compares the results to a market for a drug that is similar in all ways to the vaccine except that it treats symptoms but does nothing to reduce disease transmission from treated individuals. We show that a monopoly always prefers to develop the drug, but parameters exist for which social welfare is higher with the vaccine. Consistent with nonmonotonicities found elsewhere, the monopoly’s bias toward a drug peaks for intermediate values of $R0$⁠.

Section VIII calibrates the model to the COVID-19 pandemic. While too stylized for quantitative policy guidance, the calibrations provide qualitative insights into the magnitude of distortions caused by externalities and market power and allow us to assess whether the theoretical conditions behind results such as the Peltzman effect, increasing social returns, and a bias toward drugs versus vaccines are practically relevant.

Our analysis is intentionally built on a basic epidemiological model, sacrificing realism to obtain rigorous propositions involving interpretable economic conditions rather than results based on isolated simulations or structural estimates tied to current circumstances. The SIR model omits features required for quantitative forecasting in a real-world epidemic such as heterogeneous agents (Ellison, 2020; Acemoglu et al., 2021), transmission along networks (Newman, 2002; Fajgelbaum et al., 2021), and macroeconomic dynamics (Eichenbaum, Rebelo, & Trabandt, 2020). Perhaps the key omission is endogenous social distancing, which would flatten the epidemic’s path relative to SIR predictions. A growing literature has advanced increasingly sophisticated models to approximate the dynamic behavior of rational, forward-looking agents, seeking to reduce risk by curtailing activity.¹ We do not address all of these omissions but address some in several pieces of additional work. Online appendix A.4 allows consumers to be heterogeneous in harm. Online appendix A.5 extends the model to allow consumers to purchase a second preventive in addition to, or instead of, the vaccine. While not capturing a continuously updating distancing decision, the extension could capture fixed investments in masks or lifestyle changes. Our companion paper (Goodkin-Gold et al., 2022) maintains the SIR framework but adopts alternative modeling assumptions suited to an endemic disease, incorporating population turnover and continuous vaccination of arriving newborn cohorts. The analysis focuses on the steady-state equilibrium in which the effective reproductive number is always 1, which Gans (2020) suggests is a reasonable shortcut for modeling endogenous social distancing. All of the additional work in the online appendixes and companion paper finds that marginal externalities and optimal subsidies are highest for intermediate values of $R0$⁠, increasing confidence in the robustness of the results to modeling assumptions.

Our paper contributes to the theoretical literature analyzing vaccine externalities.² Economists have long observed that vaccines may provide positive externalities that could affect consumers’ and firms’ decisions.³ Boulier, Datta, and Goldfarb (2007) use a standard epidemiological model alone (i.e., neither interacted with consumer decisions nor a supply-side model of firm behavior) to examine properties of vaccination externalities that arise solely due to epidemiological concerns. Geoffard and Philipson (1997) use an epidemiological model similar to ours to show that a vaccine producer with market power will not choose to eradicate the disease in the steady state. Galeotti and Rogers (2013) model vaccination choices in a heterogeneous population and consider the effect of network structures in determining optimal vaccine allocation. Manski (2021), building on a series of the author’s earlier papers, provides guidance on optimal vaccine policies (including mandates) when the extent of externality is unknown. Avery (2021) is a recent ambitious attempt to integrate social distancing and vaccinations in a tractable model. We contribute a precise characterization of the nonmonotonicity of externalities and optimal subsidies as a function of disease infectiousness. We also contribute by formally modeling the supply side of the vaccine market, allowing firms to have market power.

Our paper is perhaps closest to two companion papers in the operations research literature: Mamani, Adida, and Dey (2012) and Adida, Dey, and Mamani (2013). They also analyze optimal subsidies for various degrees of supplier market power. Their focus is on consumers with uniformly distributed harm. While we also examine consumer heterogeneity (see online appendix A.4), our analysis focuses on homogeneous consumers, allowing us to derive more definitive expressions for equilibrium variables, which in turn afford additional insights, allowing us to analyze the comparative-static effect of increases in $R0$⁠. Our central result on the nonomontonicity of optimal subsidies in $R0$ and other comparative statics are novel in our paper. We also provide calibrations, results on increasing social returns, and a comparison between drugs and vaccines not found in their papers.⁴

The epidemiology literature previously recognized the possibility that the nonlinear nature of epidemics may dictate optimal policy concentrating a scarce stockpile in one population rather spreading across them (Keeling & Shattock, 2012; Keeling & Ross, 2015; Nguyen & Carlson, 2016; and Enayati & Özaltin, 2020).⁵ This literature has the appeal of studying increasingly rich epidemiological models but with the drawback of having to simulate results in numerical examples. We contribute a formal conceptualization of initial and eventual increasing social returns and aid understanding by providing a necessary and sufficient condition for these outcomes in analytical form.

Our paper also contributes to the economic literature responding to the COVID-19 pandemic. Scholars sought to apply detailed models to forecast the course of the pandemic (Atkeson, Kopecky, & Zha, 2020), to recommend lockdown and testing protocols (Alvarez, Argente, & Lippi, 2021), and to recommend policies for prioritizing scarce vaccine supplies among heterogeneous consumers (Buckner, Chowell, & Springborn, 2021). The focus of our COVID-19 calibration is different: on optimal subsidies in a decentralized market rather than optimal strategies for a central planner. Our overall goal is also different: obtaining general principles in a stylized model rather than quantitative results in a more complex model.

We model the vaccine market by first specifying the disease epidemiology, which determines the probability of infection for an unvaccinated consumer. We then bring rational economic actors into the vaccine market. On the demand side, the probability of infection factors into consumers’ willingness to pay for the vaccine. Profit-maximizing firms operate on the supply side under either perfect competition or monopoly.

The epidemiological model is designed to suit the current COVID-19 pandemic. We assume the epidemic is of relatively short duration, rising and falling within a human generation, against which an intensive vaccine campaign is waged to quickly mitigate the harm experienced by the current population, with future generations spared much damage by the epidemic’s natural decline. For pedagogical purposes, we adopt the extreme assumption that all doses of vaccine that will ever be administered are administered in a single instant. The model contrasts the alternative analyzed in our companion paper (Goodkin-Gold et al., 2022) in which generations of newborns are continuously vaccinated against an endemic disease with a constant steady-state infection rate. The two analyses together help provide a more complete understanding of vaccine markets for different diseases.

The foundation of our analysis is the standard susceptible-infected-removed (SIR) epidemiological model due to Kermack and McKendrick (1927). Continuous time indexed by $t$ begins with the arrival of the vaccine at $t=0$⁠. Assume for simplicity that the disease is nonfatal and that there are no births or deaths within the short time frame considered, leaving the population size constant over time. Anticipating their role in the vaccine market, we call individuals in this population consumers. Assume consumers are homogeneous in harm, disease spread, and all other dimensions (relaxed in online appendix A.4).

A susceptible consumer is assumed to contract the disease from an infected consumer at rate $β>0$⁠, embodying the rate of contact between people and the rate at which a contact leads to infection. Assuming the infection rate is linear in the number of infected consumers, a single susceptible consumer is infected with probability $βIt$⁠. The mass of susceptibles $St$ generates $βItSt$ new infections. Equation (2) indicates that the susceptible population falls by the number of newly infected and newly vaccinated. Equation (3) indicates that the infected population is increased by the number of newly infected and reduced by the mass $αIt$ of previously infected consumers who recover, where $α∈(0,1)$ denotes the recovery rate. This $αIt$ mass flows into $Rt$⁠, as indicated by equation (4). Under the assumption that recovered individuals cannot be reinfected, this is the only change to $Rt$⁠. Equation (5) reflects the instantaneous nature of the vaccination campaign, with no further vaccine administered after the initial tranche at date 0.⁶ We assume that if the initial vaccine course is not effective for a person, a further course will not be effective for them either. Under that assumption, administering all vaccine in the first instant is both the profit-maximizing and welfare-maximizing strategy.⁷

In subsequent notation, $Q$ is appended as an argument to equilibrium variables to emphasize their dependence on that key variable to be endogenized later. Limiting compartment values at the end of the epidemic are denoted by $S∞(Q)$⁠, $I∞(Q)$⁠, and $R∞(Q)$⁠. For example, $S∞(Q)=limt↑∞St(Q)$⁠.

The following series of lemmas, which characterize $St(Q)$ and $It(Q)$ for finite and limiting values of $t$⁠, helps streamline the subsequent analysis. Many of the proofs are sketched in Martcheva (2015); online appendix A.1 provides full details.

Intuitively, the infection rate is always positive in finite time because, if not increasing, infections are at worst declining at a proportional rate less than 100% each instant, which can never force the infection rate to 0. The infection rate does asymptote to 0 as the stock of susceptibles is depleted and recovery takes over as the dominating force, reducing the stock of infecteds. Turning to results for the population of susceptibles, with an imperfectly effective vaccine (⁠$θ<1$⁠), even a universal vaccination campaign cannot eliminate the stock of susceptibles at date 0. The stock of susceptibles is never forced to 0 after because the proportional decline is less than 100% each instant. The stock of susceptibles strictly decreases over time since it is subject to outflows but not inflows.

According to lemma 3, the path of infections over the epidemic has two possible shapes: monotonically decreasing or hump shaped, expanding up to a peak and declining thereafter. The shape of the path hinges on the product $R0S0(Q)$⁠. Multiplying $R0$⁠, the expected number of secondary cases an infectious individual transmits in a fully susceptible population, by $S0(Q)$⁠, the proportion of susceptibles in the relevant population, yields the expected number of secondary cases an infectious individual transmits in the relevant population, called the effective reproductive number. If $R0S0(Q)≤1$⁠, there are fewer secondary infections than primary infections in the initial population, leading the infection rate to fall initially. Otherwise, the infection rate rises initially.

Lemma 5 says that the effective reproductive number cannot exceed 1 at the end of the epidemic. If $R0S∞(Q)>1$⁠, infections would be increasing, implying a growing, not waning, epidemic.

The term $S∞(Q)$—the proportion of people who remain healthy throughout the epidemic despite not being successfully immunized—plays a key role in the subsequent analysis, factoring into private and social benefits, thus determining equilibrium outcomes and the efficiency of these outcomes. While no closed-form solution exists for $S∞(Q)$⁠, the next lemma expresses it as an implicit function of other model parameters. The lemma also provides an expression for $S∞(Q)$ in terms of the principal branch of the Lambert W function, here denoted $L¯$⁠.¹⁰

Consumers are homogeneous and risk neutral. Consistent with the short-run perspective adopted in this paper, assume that agents do not discount the future. Let $H$ denote the total expected harm suffered by a consumer who contracts the disease over the spell before recovery.

The $S^0$ individuals in the susceptible compartment when the vaccine is introduced are potential consumers. They make their demand decisions by comparing the vaccine’s price $P$ to their marginal private benefit, which can be written $MPB(Q)=θHΦ(Q)$⁠, where $Φ(Q)$ denotes the probability a susceptible contracts the disease during the epidemic.

We analyze two different market structures in the text: perfect competition and monopoly. Online appendix A.3 provides results from a more general model of Cournot competition among $n$ firms that nests these extremes.

Assume firms produce at constant marginal and average cost $c>0$ per vaccine course (where a course involves multiple doses when needed to provide immunity). Under perfect competition, vaccine supply is perfectly elastic at price $c$⁠. Under monopoly, the firm sets a price-maximizing industry profit $Π$ from date-0 sales.

Let $Q**$ denote the first-best quantity, maximizing $W(Q)$⁠. If $Q**$ is not a corner solution, involving either no vaccination or universal vaccination, it is an interior solution solving the social planner’s first-order condition $MSB(Q**)=c$⁠.

Equilibrium values of variables are distinguished with stars and a subscript indicating the relevant market structure. Under perfect competition, the equilibrium price is $Pc*=c$ and profit is $Πc*=0$⁠. The remaining equilibrium variables can be computed using straightforward algebra applied to the supplied equations. Table 1 reports the equilibrium values of selected variables as a function of $R0$⁠.

To visualize how the variables in table 1 vary with $R0$⁠, figure 1 graphs a selection of them as functions of $R0$⁠. Focus for now on the dotted curves representing equilibrium under perfect competition. Vaccine quantity $Qc*$⁠, graphed in the first panel, rises throughout case ii from its value of 0 in case i to the first-best value $Q**$ in cases iii and iv. It is unsurprising that the equilibrium quantity is weakly increasing in the infectiousness of the disease measured by $R0$⁠. Other equilibrium variables also display expected comparative statics in $R0$⁠. $MPBc*$ is weakly increasing and $Wc*$ is weakly decreasing in $R0$⁠. It is noteworthy that $MPBc*$ levels off at $c$ in case ii. Given that some consumers purchase in this case, consumers must be indifferent between purchasing and not, implying that the equilibrium price $Pc*=c$ must extract the entire marginal private benefit, implying $MPBc*=c$ over the whole interval.

Other variables display interesting nonmonotonicities. Cumulated infections over the epidemic $R∞(Qc*)$ initially increase in case i due to the epidemiological effects of the higher $R0$⁠. In case ii, when consumers begin purchasing vaccine, $R∞(Qc*)$ reverses course, sloping downward in $R0$⁠. The counterintuitive downward slope can be explained by the risk-compensation effect à la Peltzman (1975): the direct effect of an increase in infectiousness is more than offset by consumers’ behavioral response in the form of increased vaccine purchases.¹¹ In cases iii and iv, $R∞(Qc*)$ again rises with $R0$ because the direct effect of an increase in infectiousness cannot be offset by an increase in vaccine purchases given that all susceptibles are vaccinated. The marginal externality $MEXc*$ exhibits an even more complex nonmonotonic pattern. The interplay between increasing infectiousness and increasing vaccine quantity generates two local maxima in the figure, with the global maximum occurring at the boundary between cases i and ii.

Proposition 1 summarizes the comparative-static effects of an increase in $R0$ on the steady-state equilibrium under perfect competition, showing that the observations from figure 1 are quite general. Online appendix A.1 provides proofs for results not obvious from table 1.

Note that the Peltzman effect identified for cumulated infections, whereby an increase in $R0$ leads to an equilibrium reduction in $R∞(Qc*)$⁠, does not extend over the whole range of $R0$ but just over case ii. Note further that the reductions in infections in case ii do not translate into an increase in welfare, which is weakly decreasing for all $R0$⁠. Increased consumer spending on vaccines more than offsets the reduction in cumulated infections.

Since price equals cost under perfect competition, but a monopoly charges a markup above cost, price is weakly higher under monopoly and quantity weakly lower. It follows that in case i, in which $Qc*=0$⁠, we have $Qm*=0$⁠. Case i is thus trivially identical across perfect competition and monopoly. In the remaining cases, perfectly competitive firms are able to make positive sales at price $c$⁠. By continuity, the monopoly can make positive sales at some small markup above $c$⁠, implying $Qm*>0$ for $R0$ in cases ii and above.

The proof in online appendix A.1 fills in details omitted from the sketch preceding the proposition, including verifying that the condition $MR(S^0)≥c$ is satisfied for sufficiently high $R0$⁠.

The values of $Qm*$ and other equilibrium variables are cataloged in table 2. The entry for $Qm*$ (and by extension, the other equilibrium values expressed in terms of $Qm*$⁠) are not provided in analytic form, let alone in closed form, in cases ii and iii. This need not preclude definitive comparative-statics results; one could apply the implicit function theorem to the condition $MR(Qm*)=c$ to determine how $Qm*$ changes with $R0$ in those cases. However, this approach still does not deliver a definitive sign. We can be sure that $Qm*$ increases in $R0$ for some $R0$ in cases ii and iii—since $Qm*$ must rise from 0 to the first-best quantity $S^0$ somewhere in that set by continuity—but we have not been able to rule out the possibility that the monopoly responds to an increase in $R0$ in some subintervals by reducing output in order to extract an even larger price increase than otherwise. Despite these challenges, we are able to derive definitive comparative-statics results for some equilibrium variables, reported in the next proposition, proved in online appendix A.1.

Equilibrium values of selected variables under monopoly are graphed as functions of $R0$ as the solid curves in figure 1. The two market structures overlap in case i, neither generating any vaccine output. The two market structures overlap again in case iv, both generating the first-best quantity $Q**=S^0$⁠. In between—in cases ii and iii—the two market structures diverge, with monopoly generating strictly lower output, entailing more total infections over the epidemic, higher marginal private benefit, and lower welfare. The large gap between the dotted and solid curves for intermediate values of $R0$ suggests that the distortion arising from the monopoly’s exercise of its market power is worst for moderate levels of infectiousness. The marginal externality can be considerably higher under monopoly for some $R0$ but can be slightly lower for some $R0$ as the lower monopoly output generates higher marginal private benefit, leaving less residual externality.

The graph of $W*$ under monopoly illustrates the remarkable possibility that increasing $R0$ can increase welfare, impossible under perfect competition according to proposition 1. Under monopoly, not only do consumers fail to consider the external benefit their vaccination provides other consumers, but the monopoly compounds this by placing negative value on consumption to the extent it reduces others’ willingness to pay for a vaccine. An increase in $R0$ can mitigate this compound underconsumption problem, providing such a large indirect benefit that it swamps the direct harm from increased infectiousness, leading to an increase in social welfare.

We have seen that free riding can lead to inefficiently low vaccination under both perfect competition and monopoly. This naturally raises the question of whether the government can intervene to correct the market failure. In this section, we characterize the optimal government subsidy and determine its comparative-static properties.

Assume a benevolent government with the objective of maximizing social welfare commits to a per course subsidy $G≥0$ at the outset of the game. According to standard public finance results, the economic incidence is the same whether consumers or firms are the statutory target of a tax or subsidy (Fullerton & Metcalf, 2002). We adopt the accounting convention that $G$ is paid to firms, in which case the subsidy is equivalent to a reduction in firms’ marginal cost from $c$ to $c-G$⁠. Since social welfare is maximized by the first-best quantity $Q**$⁠, the first-best subsidy $G**$ is that implementing $Q**$⁠. To accommodate cases in which the government is indifferent among a possibly open set of subsidies maximizing social welfare, we take $G**$ to be the infimum of the set (effectively assuming that the government has lexicographic preferences over welfare and expenditure savings).

It is straightforward to establish a set of broad results for any market structure. Since the marginal vaccine externality is nonnegative by equation (23), equilibrium output $Q*$ never exceeds the first best $Q**$⁠. If $Q**=0$⁠, then $Q*=Q**=0$ as well, implying $G**=0$ since the first best can be achieved without a subsidy. The proof of the next proposition shows that $Q**=0$ for all $R0$ in a neighborhood above 0, implying that $G**=0$ in this neighborhood for any market structure.

We can also draw broad conclusions about the optimal subsidy for high values of $R0$⁠. By proposition 2, $Qm*=S^0$ for sufficiently high values of $R0$⁠. For such $R0$⁠, $S^0=Qm*≤Q**≤S^0$⁠, implying $Qm*=Q**$⁠, in turn implying $Gm**=0$ since the first best can be achieved without a subsidy under monopoly. The result that $G**=0$ for sufficiently high $R0$ immediately extends to perfect competition or any market structure involving weakly higher output than monopoly.

Having established that $G**=0$ for intervals of low and high values of $R0$ for general market structures, if it can be shown that $G**>0$ for some intermediate value of $R0$⁠, it is immediate that $G**$ is nonmonotonic, attaining a global maximum for some interior $R0∈(0,∞)$ as the next proposition states. The proof provided in online appendix A.1 fills in this and other omitted details.

As proposition 4 indicates, the optimal subsidy is not monotonically increasing in $R0$ as might be inferred based solely on epidemiological considerations but is maximized for an interior value of $R0$⁠. The difficulty in addressing a disease depends not only on its infectiousness but also on consumers’ response to this infectiousness. Free riding on the vaccination of others disappears with extremely infectious diseases; moderately infectious diseases provide consumers more leeway to free-ride, requiring a higher optimal subsidy to address.

We conclude the section with a more precise precise characterization of the optimal subsidy under perfect competition and monopoly provided by the next proposition, proved in online appendix A.1.

Under both market structures, equilibrium attains universal vaccination of susceptibles for a nonempty set of parameters. Two reservoirs of infection motivate the marginal consumer to purchase at a positive price even when all other consumers also purchase. With an imperfectly effective vaccine (⁠$θ<1$⁠), some vaccinated consumers remain susceptible and able to transmit the disease to others. Even in the limit of perfect efficacy $θ↑1$⁠, however, the $I^0$ initially infected individuals remain a reservoir. Previous game-theoretic analyses finding that a perfectly effective vaccine would never be universally purchased at a positive price (Geoffard & Phillipson, 1997; May, 2000; Bauch & Earn, 2004) omitted this feature of the SIR model.

It is obvious that equilibrium under perfect competition must attain universal vaccination if the disease is infectious enough. Even a small reservior of infecteds $I^0$⁠, when combined with a sufficiently high $R0$⁠, generates high enough infection risk to motivate the marginal consumer to purchase at any fixed $c<H$⁠. It is less obvious that universal vaccination is attained in a monopoly equilibrium. Given that its monopoly price is endogenous, not fixed, the monopoly might respond to an increase in $R0$ by raising price, keeping output short of universal vaccination. For sufficiently high $R0$⁠, the marginal consumer is almost certain to contract the disease from the reservior of $I^0$ infecteds irrespective of how many susceptibles are vaccinated. The monopoly serves all consumers at a price approaching the marginal private benefit of being protected against certain infection, leaving little room for any further price increase without losing most customers.

The next proposition, proved in online appendix A.1, provides a simple necessary and sufficient condition for universal vaccination with a perfectly effective vaccine to obtain in equilibrium under each market structure.

It can be shown that both conditions hold for sufficiently high $R0$⁠: the factors on the left-hand side of both conditions equal 1 in the limit $R0↑∞$ and $1>c˜$ by assumption (19). It is also obvious that neither condition holds for any finite $R0$ when $I^0=0$⁠, reflecting the fact that an initial stock of infecteds is required to generate demand for a perfectly effective vaccine under a universal vaccination program.