Monopoly Pricing in the Market for Vaccines∗ Sebastian Kessing
Robert Nuscheler
Freie Universit¨ at Berlin and EUI
WZB and Freie Universit¨ at Berlin
2002-02-22
Abstract We study the market for vaccinations considering income heterogeneity on the demand side and monopoly power on the supply side. A monopolist has an incentive to exploit the external effect of vaccinations and leave the poor susceptible in order to increase the willingness to pay of the rich. If the income differential is large, then the problem of vaccination discrimination is large. Discrimination disappears when income is distributed more equally. Pigouvian subsidies may even make things worse. If the poor are covered in a mandatory vaccination program, the disease can be eradicated. Keywords: Monopoly Pricing, Vaccines, Externality, Vaccination Discrimination. JEL classification numbers: D42, D62, H23, I11, I18.
∗
Correspondence: Kessing: Freie Universit¨at Berlin, Boltzmannstrasse 20, 14195 Berlin, Ger-
many, phone: +49 30 838-55229, e-mail:
[email protected]; Nuscheler: Wissenschaftszentrum Berlin f¨ ur Sozialforschung (WZB), unit: Market Processes and Governance, Reichpietschufer 50, 10785 Berlin, Germany, phone: +49 30 25491-430, e-mail:
[email protected]. We thank Kai A. Konrad and the participants of the Microeconomic Colloquium at the Freie Universit¨ at Berlin for helpful comments. The usual caveat applies.
1
1
Introduction
Traditionally, vaccinations were regarded as one of the prime examples of positive externalities. Consequently, government intervention in the form of mandatory vaccinations and Pigouvian subsidies were considered to be appropriate policy responses to the distortions caused by the externality. More recently, this traditional view has been challenged by various contributions that have produced a number of somewhat conflicting results about the form and optimality of government intervention in the market for vaccines (see e.g. Brito et al. (1991), Francis (1997), and Geoffard and Philipson (1997)). These results typically depend on the specific assumptions made in the respective models about agent heterogeneity, market structure and dynamics. This paper contributes to this literature by analyzing optimal government policy in the context of two hitherto neglected dimensions. First, individuals are assumed to differ with respect to income. Second, we consider monopoly power on the supply side. In the existing theoretical literature agent heterogeneity is usually introduced, if at all, through the assumption of varying disutility of vaccinations. Empirically, such disutility is difficult to observe. Empirical studies of individual vaccination decisions usually find a clear positive relationship between income and the probability of being vaccinated. Philipson (1996, table 2, p. 624) reports a positive income effect on the probability of measles vaccination for children in the U.S., for example. England et al. (2001, p.19) report that, if there is a fee, as with hepatitis B in China, “poorer people are more likely to go without essential immunization”. Since the evidence indicates the importance of income differences, it is natural to introduce agent heterogeneity through income into the theoretical analysis. Moreover, since government action usually affects individuals’ incomes, such an analysis promises to approximate the consequences of different policy measures better. We derive the effects of income heterogeneity by modelling the individual vaccination decision in an insurance-theoretic framework. The individual either decides
2
to get full coverage by vaccination or no coverage by doing without it. Since this decision is between a certain outcome and a risky one, the individual willingness to pay for vaccination amounts to the individual risk premium. We subsequently derive a sufficient condition for individual utility and loss functions to conform with the empirical evidence. In the main part of the paper we analyze the consequences of income-dependent demand with monopoly power on the supply side. Biotechnological progress, together with the current patent policy where living organisms can be protected, monopoly power will be a major issue in the near future. We concentrate on the case where the monopolist does not price discriminate. All qualitative results are robust when perfect price discrimination is considered. Vaccination discrimination is in fact reduced, but may be still present. On the demand side, we consider the case where the population has to pay for the vaccinations, i.e. the costs are not covered by health insurance companies or the state. Consequently, there is no bargaining between either insurance companies or state agencies.1 We summarize both the income dependence of the individual willingness pay and the external effect of a reduced infection probability due to a higher number of vaccinated individuals with a simple linear aggregate demand schedule faced by the monopolist. The decisive element in this setting is the monopolist’s strategic incentive for a greater share of the population to remain susceptible in order to increase the infection probability and to increase the willingness to pay for vaccinations. This strategic incentive is most easily analyzed in a static environment. But the results will also apply in a dynamic framework, since the importance of the external effect increases.2 In the monopoly solution the share of vaccinated individuals increases with the 1
For an empirical analysis testing whether price discrimination or bargaining is present in the
U.S. vaccine market see Kauf (1999). 2 Francis (1997) showed that the externality disappears in his dynamic setting. The allocation is efficient. But this result does not hold in general with heterogenous individuals.
3
parameter expressing income dependence and decreases with the size of the external effect. Both these results follow directly from the reduced prevalence elasticity of demand. It follows that an increase in average income also increases protection in the population. More interesting is the effect of income inequality: if income inequality increases, the share of vaccinated individuals falls. The higher incomes of the rich increase their willingness to pay. Therefore, the strategic incentive for the monopolist grows, leading him to set a higher price and thus reducing demand. Accordingly, income redistribution will help mitigate the discrimination problem. In the theory of public goods, the problem of under-provision can be eliminated by Pigouvian subsidies. Although vaccinations are an example of privately provided public goods, subsidies do not work very well. At first demand increases, since the individual price is reduced. However, this increase lowers the infection probability and thus reduces the willingness to pay. This counteracting effect limits the effect of these subsidies (see Geoffard and Philipson (1997, p. 225)). We show that subsidies may make things even worse. We assume that the price subsidy is financed by lump-sum taxation. The tax reduces the willingness to pay for vaccinations. If the income effect is sufficiently large, the positive price effect is overcompensated and lower share of vaccinated individuals results. This contrasts with the classical regulation arguments for Pigouvian subsidies and strengthens Philipson (2000) who states that “Pigouvian subsidies traditionally seen as resolving the under-provision problem of vaccines can be short-run, or out of steady state, arguments” (p. 1777), since they may even fail in static settings. Another public policy usually suggested is mandatory vaccination. If mandatory vaccination programs do not cover the whole population, the vaccinated individuals lower the infection probability of the susceptible. The willingness to pay is reduced, i.e., mandatory demand crowds out voluntary demand. This is a standard argument for why it is difficult to eradicate a disease by mandatory vaccination if not the entire population is included in the program (see Philipson (2000, p. 1781)). In our model such a program is much more effective: since the individuals differ in income, the 4
public program can cover the poor and the monopolist the rich. Of course the willingness to pay of the rich is reduced, but it remains relatively high due to the income effect. Full vaccination can be achieved with a mandatory participation rate that is strictly smaller than one. The approach presented here is related to Brito et al. (1991). They consider a static model with a continuum of individuals that differ in the amount of disutility they incur by vaccination. Since vaccines are provided at no charge, price discrimination cannot be studied in their setting. The first-best outcome can be implemented by subsidizing those who decide to vaccinate, or by taxing those without immunization. But when the subsidy has to be financed by taxation, the first-best is attained only under the strong assumption of identical marginal utility of income across individuals. In their dynamic model Geoffard and Philipson (1997) address the question of disease eradication. Both price subsidies and mandatory vaccination programs are limited in their impact, since the positive effects of the respective policies are partly offset by the negative effect of the externality. The current paper is also related to the literature on network externalities, e.g. Bensaid and Lesne (1996), Cabral et al. (1999), and Mason (2000). The main difference is the sign of the network effect which is positive in those models while negative in ours, leading to completely different results. With a positive network effect, introductory pricing may occur to built up a certain critical network size. With vaccinations it is the other way round: a critical mass will never be exceeded in order to prevent the market shrinking or disappearing. The paper is organized as follows: in section 2 we present a microeconomic foundation of the reduced form approach applied in the rest of the paper. The monopolist’s price setting problem and the comparative static properties of this solution are studied in section 3. In section 4 we discuss public policy issues that may be applied to reduce discrimination and thus increase social welfare. Section 5 concludes.
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2
The vaccination decision
Consider an individual with an original income of a, which reflects individual productivity or wage-earning abilities, and preferences which obey to the von NeumannMorgenstern axioms. The individual is exposed to the threat of infection with a transmittable disease. The probability of infection is given by π ∈ (0, 1], which is taken to be exogenous to the individual. Of course, in equilibrium, this probability will depend on the number of susceptible individuals. The monetary loss from infection depends on income, β = β(a). It is sensible to assume that β ∈ [0, 1], since illness will lead to absence from work for a certain time. Hence, a high income individual will lose more than a low income individual. A vaccine is available that yields perfect protection against the disease and the price for being vaccinated is denoted p. Then, the utility for a vaccinated individual with income a is given by u = u(a − p).
(1)
The individual decides to vaccinate, if and only if the utility exceeds the expected utility of remaining without protection which is given by Eu = πu(a − β(a)) + (1 − π)u(a).
(2)
Empirical evidence shows that the willingness to pay for vaccinations is likely to be increasing in income (see Philipson (1996, p. 624) and England et al. (2001, p. 19)). We derive a sufficient condition on the utility function and the loss function for the individual willingness to pay to be globally increasing in income. The decision to vaccinate amounts to the choice between the certain outcome and the original risky outcome. Thus, the willingness to pay for vaccinations p(a) coincides with the risk premium. Applying the approximation formula for the risk premium derived by
(EX) V ar(X) Arrow and Pratt (see Pratt (1964)), we have p(a) ≈ − uu (EX) , where EX = 2
a − πβ (a) and V ar(X) = π (1 − π) [β(a)]2 . Then, for a given infection probability, the willingness to pay globally increases in income if for all a > 0 the following 6
condition holds:
2β u u ≥ − . (1 − πβ ) β u u
(3)
In the case of constant absolute risk aversion, the right hand side is zero, implying a strictly increasing willingness to pay for vaccinations if β > 0. For an illustration see figure 1. Consider an individual with low income facing a lottery described by a point such as P1 . If the individual decides to vaccinate, he is immune and therefore will have a certain income. Depending on the price, the individual ends up on the certainty line somewhere in between the points Q1 and R1 . If the vaccine is provided free of charge, the individual finds himself at Q1 , and if the price equals the risk premium, he is located at R1 . The individual willingness to pay pL is given by the horizontal difference between Q1 and R1 . Now, consider an individual with a higher income and constant absolute risk aversion. Obviously, if the loss is constant in income, as at point P2 , the willingness to pay is unchanged. But, if the loss increases in income such that the lottery is given by a point such as P3 , the willingness to pay increases, i.e. pH > pL . If the utility function exhibits constant relative risk aversion, β must be sufficiently large for the willingness to pay be non-decreasing in income. If condition (3) holds, this has a clear implication for aggregate demand. If the income distribution is continuous on an interval [aL , aH ], then, for a given price p, there will be a critical income acrit ∈ [aL , aH ], such that all individuals with an income higher than acrit will decide to vaccinate. The poor will stay without vaccination. Hence, the aggregate demand for vaccinations falls in price and those who vaccinate are the rich.3 To complete the analysis we have to show that acrit is the only solution to the problem u(acrit − p) = Eu(acrit ; π). The critical income is linked to the share of vaccinated individuals, θcrit = θ(acrit ). The externality of vaccinations is modelled 3
It is also possible to construct the theoretical case that the willingness to pay decreases with
income, e.g. if β = 0 and constant relative risk aversion is assumed. But this is contradicted by empirical evidence.
7
In c o m e if in fe c te d c e rta in ty lin e
Q
R
1
1
P P P
p L
2
3
1
p H
In c o m e if n o t in fe c te d
Figure 1: Constant absolute risk aversion and a proportional loss imply that the willingness to pay increases in income. as π = π(θcrit ) with π < 0. The probability of falling ill is lower when more people are immune, since they can no longer communicate the disease. This is a common way of modelling this interaction between the prevalence of a disease and the risk of infection (see e.g. Francis (1997)). Now suppose that acrit ∈ (aL , aH ) is an equilibrium. Then the expected infection probability must be π e = π(θ(acrit )) and u(acrit − p) = Eu(acrit ; π e ). Note that expectations on θ directly translate into changes in π and vice versa. Suppose that all individuals expect that nobody will vaccinate, i.e. θe = 0. Then u(acrit − p) > Eu(acrit ; π(0)), and the individual with income acrit will prefer to vaccinate. This also applies to all individuals with income a > acrit contradicting θe = 0. Now assume that θe = 1. Then u(acrit − p) < Eu(acrit ; π(1)), and the individual with income acrit does better without vaccination. 8
Again, this leads to a contradiction since this applies to all individuals with income lower than acrit . A similar reasoning applies to all expected infection probabilities different from π(θ(acrit )) and establishes acrit as the unique solution to the above problem. Thus, although the current model has similarities with models studying network effects, the opposite sign of the network externality eliminates the problem of multiple equilibria. Consider a positive network effect, then, for a given price, everybody may expect that only the rich buy. This expectation will be confirmed in equilibrium. But if everybody expects that almost all individuals will demand the network good, an equilibrium with a larger network size will be established. The individuals who decided not to buy in the first equilibrium now have a higher willingness to pay for the network good, since the network size increased. This cannot happen in the model presented here: the higher the network size, the lower the willingness to pay of those who remain unprotected.
3
A reduced form approach
In principle, assuming particular forms for the utility functions, the loss functions and the distribution of income, by solving u(a) = Eu(a) for given p and inverting (if possible), explicit aggregate demand schedules can be derived. However, these are highly specific. Since none of the many potential combinations complying with condition (3) lends itself as a natural candidate, we approximate the individual willingness to pay by a general linear scheme. This may also be interpreted as the linear approximation to a particular non-linear willingness to pay scheme resulting from a specific combination. It exhibits the essential features of the situation and keeps the analysis tractable, p(θ) = zθ (1 − θ) + za a.
(4)
θ ∈ [0, 1] denotes the share of individuals who vaccinate. Hence, 1 − θ is the share of individuals that are susceptible. The external effect of these susceptible individuals 9
on the willingness to pay is assumed to be linear in 1 − θ with parameter zθ > 0. As argued in the preceding section, the infection probability is lower, the higher the rate of immunization, i.e.
∂p ∂θ
< 0. Furthermore, in line with our reasoning above,
the willingness to pay increases in income. The effect is assumed to be linear in income with rate za ∈ (0, 1). The lower bound of za follows directly from condition (3). The upper bound is justified by normality. Observe that an exogenous risk of infection is assumed since p(1) = za a > 0. This is justified by exogenous sources of infection either from outside the market under study4 or by accidental laboratory outbreaks or terrorist attacks. The population is a continuum of individuals with mass 1. For simplicity, we now assume that income is uniformly distributed on [aL , aH ], where 0 < aL ≤ aH . Since the willingness to pay increases in income we know that θcrit =
aH −acrit . aH −aL
Solving for
acrit yields acrit = aH − θcrit [aH − aL ]. Substituting this into (4) and solving for θ gives the aggregate demand θ = θ(p).
3.1
Monopoly pricing
The monopolist’s objective function may now be written as Π(θ) = p(θ)θ = (zθ (1 − θ) + za (aH − θ[aH − aL ]))θ.
(5)
We suppose constant marginal costs of zero. The socially optimal policy is thus always given by θ = 1. The first order condition is derived by differentiation:5 θ∗ =
zθ + za aH . 2(zθ + za [aH − aL ])
(6)
Without the externality the monopolist would face the demand schedule p = zθ + za (aH + θ[aH − aL ]) yielding an optimal supply of
zθ +za aH 2za [aH −aL ]
< θ∗ . Facing the exter-
nality, the monopolist has an incentive to reduce supply to increase the willingness 4
Actually, smallpox is the only disease ever completely eradicated through vaccinations world-
wide. 5 To avoid θ∗ exceeding one it is assumed that zθ + za (aH − 2aL ) ≥ 0.
10
to pay and thus profit. The price corresponding to θ∗ is 1 p∗ = zθ + za aH . 2
(7)
Except aL , the price increases in all exogenous parameters.
3.2
Comparative static results
Now we derive the comparative static effects of the model: first, note that
dθ∗ dza
> 0.
With an increasing income effect, the relative importance of the external effect of vaccinations is reduced and with it the incentive to cut the supply. More interesting is the effect of a change in the external effect parameter zθ which is clearly negative, i.e.
dθ∗ dzθ
< 0. The higher the external effect of susceptible individuals on the
willingness to pay, the higher the monopolists’ incentive to exploit this effect, i.e. to reduce the amount of vaccines sold. To study the effect of income inequality on equilibrium let aH = a + ∆ and aL = a − ∆, where ∆ ≥ 0. Then θ∗ =
zθ +za (a+∆) . 2(zθ +2za ∆)
The income inequality effect is
observed by differentiation with respect to ∆ yielding
dθ∗ d∆
< 0. The more unequally
the income is distributed among the population, the more severe the problem of vaccination discrimination. Note that the average income a is not affected by changes in ∆. Now consider that the population becomes richer as a whole, but (absolute) inequality remains unchanged:
dθ∗ da
> 0. The income effect becomes more important
relative to the discrimination effect. Consequently, a higher share of the population decides to vaccinate. To summarize, vaccination discrimination is more likely to occur in societies that are poor or are facing substantial income inequality.
4
Public policy
Obviously, within our setting of zero marginal cost, the socially optimal policy would be to have the monopolist cover the whole market. We now discuss the consequences
11
of various public policies. To evaluate their respective benefits, we analyze their potential to increase the degree of immunization in society.
4.1
Price subsidies
Consider a policy of paying the monopolist a per unit subsidy of size s. This is usually a standard tool for alleviating the inefficiency caused by monopoly and a Pigouvian cure for the vaccination externality. If the subsidy is financed by a head tax of size T , the income distribution shifts to [aL − T, aH − T ]. The government budget constraint is given by T = sθ. Thus the monopolist now faces a willingness to pay of p(θ; s) = zθ (1 − θ) + za (aH − sθ − θ [aH − aL ]) + s.
(8)
If there is a positive subsidy, the monopolist chooses demand such that θs =
zθ + za aH + s . 2(zθ + za s + za [aH − aL ])
(9)
To decide on the effectiveness of the subsidy we have to compare θs with the laissezfaire share θ∗ of equation (6): θs − θ∗ =
s zθ (1 − za ) + za [aH − aL ] − za2 aH . 2 (zθ + za s + za [aH − aL ]) (zθ + za [aH − aL ])
(10)
Since the denominator of the right hand side of equation (10) is always positive, the sign of θs −θ∗ is determined by the numerator: if za = 0 the numerator turns out to be zθ > 0. The problem of discrimination is reduced by Pigouvian subsidies, since there is no income effect to offset the positive effect of the subsidy. This coincides with the result when a public budget constraint is not considered. More interesting, if za = 1, then θs < θ∗ . Discrimination is further increased by subsidizing vaccines. This is due to the income effect caused by financing the subsidy. By continuity, there exists some value zacrit ∈ (0, 1) such that the subsidies have no effect. In this case, the positive price effect of the subsidy on demand is exactly offset by the negative financing effect. If the income effect is sufficiently large, i.e. za > zacrit , a subsidy makes things even 12
worse. This is in contrast to the classical regulatory arguments, where Pigouvian subsidies are applied to correct for the inefficiencies due to the externality. This extreme effect was not previously known in the theory of vaccinations. Philipson (2000, p. 1777), for example, states that subsidies are limited in their impact in dynamic settings, but that they may have an effect in the short-run or out of steady state. We show, strengthening this result, that subsidies may have no effect or that they may even have a negative effect in a static environment.
4.2
Income redistribution
As mentioned before, income heterogeneity is a new aspect in studying the market for vaccines. It allows the effects of public redistribution politics on the vaccination outcome to be addressed. Hence, we assume that the social planner can observe income, while the monopolist cannot, or does not, use this information. Although formally not necessary, this assumption eases the presentation of the main ideas. As mentioned above, perfect price discrimination yields the same results qualitatively. Since vaccination discrimination is lower with perfect price discrimination, the optimal public health intervention will be lower in size, but will still be necessary. It follows directly from the comparative static effect
dθ∗ d∆
< 0 that a redistribution
of income increases vaccination. This does not necessarily imply that income must be equally distributed for complete vaccination. The income span corresponding to full market coverage is given by aH − aL =
2(za a−zθ ) . 3za
The effectiveness of income
redistribution hinges on the importance of the external effect relative to the income effect. If the external effect is large, zθ > za a, even an egalitarian income distribution will not result in full immunization.
4.3
Mandatory vaccination
Another public policy usually suggested is mandatory vaccination. If mandatory vaccination programs do not cover the whole population, the vaccinated individuals 13
reduce the infection probability of the susceptible. The willingness to pay is reduced, i.e. mandatory demand crowds out voluntary demand. This is a standard argument for why it is difficult to eradicate a disease by mandatory vaccination, if not the entire population is included in the program (see Philipson (2000, p. 1781)). This argument also applies to the model presented here if the social planner has no information about individual income levels. But consider that income is observable, then a program is much more effective than usual: since the individuals differ in income the public program may only cover the poor and the monopolist the rich. Let m ∈ [0, 1] be the share of mandatory vaccinated individuals. Consider that these are the 100 times m percent poorest in the society. The willingness to pay is now given by p(θ; m) = zθ (1 − θ − m) + za (aH − θ [aH − aL ]) .
(11)
The optimum is obtained by differentiation with respect to θ and is attained if: θ (m) =
(1 − m) zθ + za aH . 2(zθ + za [aH − aL ])
(12)
The overall share of vaccinated individuals is given by min{1, m + θ (m)}. If m + θ(m) < 1, then the effect of extending the mandatory vaccination program on the share of vaccinated individuals is clearly positive: eradicated if m ≥
zθ +2za (aH /2−aL ) . zθ +2za (aH −aL )
d(m+θ(m)) dm
> 0. The disease is
In line with our comparative static results on
income inequality, the share of the population to be included in the program is higher, the higher income inequality. If a society faces a high amount of inequality, this is accompanied by a serious amount of vaccination discrimination. Thus, the mandatory vaccination program must be relatively large for full immunization. Note that the entire market is covered at values of m that are strictly smaller than one. Mandatory vaccination is more effective than in other models, e.g. Geoffard and Philipson (1997), since the negative effect of the externality is reduced by the still high income effect. Of course, if income inequality is relatively low, there is no need for a public vaccination program, since the monopolist serves the entire market as 14
we learned from the preceding subsection. Again, with perfect price discrimination the size of the public intervention is smaller than without price discrimination, i.e., to cover the entire market fewer people have to be included in the mandatory vaccination program. If the social planner is not informed about the individual income levels, he is left with two policy alternatives: Pigouvian subsidies and mandatory vaccination. Unfortunately, subsidies are of limited use since the positive price effect is opposed by two negative effects, the prevalence effect and the income effect. In some cases subsidies make things even worse. As usual, mandatory vaccination programs fail to eradicate the disease if not the entire population is included in the program. Things change dramatically when the social planner is informed about individual income. The public health interventions may then be conditional on income. A mandatory program covering the poor only yields full vaccination at participation rates that are strictly smaller than one. Income redistribution also reduces vaccination discrimination. But with a substantial externality, this policy fails to eradicate the disease.
5
Conclusion
We presented a simple static model to study the effects of monopoly power on the supply side in the market for vaccines. We highlighted the importance of income inequality when analyzing the monopolist’s incentive to exploit the external effect of vaccinations to maximize its profits. Since empirical evidence shows that the poor are more likely to remain without essential immunization, we derived, using an insurance theoretic framework, a simple condition on the utility function and the loss function for the willingness to pay to be increasing in income. In the monopoly solution the poor are discriminated, i.e., remain susceptible, to increase the willingness to pay of the rich. Discrimination was found to be more severe if the prevalence elasticity of demand is high, i.e., when the income effect is 15
low or the impact of the external effect is high. Societies with low average wealth or high income inequality are left with a high share of susceptible individuals. We studied several public policies to counter the strategic behavior of the monopolist. A standard tool for curing the distortions caused by the externality is a subsidy as in the theory of privately provided public goods. But in the market for vaccines, the effect of such a subsidy is limited, since the reduced price increases demand and thus reduces the willingness to pay. The positive effect is opposed by the negative prevalence effect. In our model the negative effect is even larger: if the subsidy is to be financed by taxation, then the income effect further reduces the willingness to pay and therefore demand. We showed that a subsidy may make things even worse, raising doubts about whether Pigouvian subsidies are appropriate at all. Since the monopolist’s incentive to discriminate against the poor is reduced when income is distributed more equally, the social planner may redistribute income to improve welfare. When the external effect is not too strong, discrimination disappears if income inequality is sufficiently reduced. Another policy, that may be conditional on income, is a mandatory vaccination program. Usually these programs fail to eradicate the disease, since the infection probability, and thereby the willingness to pay, is reduced by a mandatory program. Using the information on income levels, the social planner can include only the poor in the program. The monopolist serves the rich. Although the willingness to pay of the rich is reduced, it will remain relatively high due to the still high income effect. Thus, if the social planner is informed about individual income levels, there is a tool for success in eradicating the disease via mandatory programs at participation shares that are well below one. As our public policy analysis showed, one cannot be sure whether Pigouvian subsidies are appropriate for increasing the immunization rate. Thus, in the presence of monopoly power on the supply side, the social planner should resort to public health interventions that make use of individual income levels. Although both policies are successful in improving welfare, mandatory vaccination dominates income 16
redistribution, since the disease can always be eradicated by including the poor in the program.
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