Optimal tax/subsidy combinations for the flu season

https://doi.org/10.1016/j.jedc.2003.08.001Get rights and content

Abstract

This paper uses a dynamic susceptible-infected-recovered (SIR) epidemic model to identify optimal vaccination policy mixes for the flu season. It begins with the solution to the relevant optimal control problem and then gives the derivation of a perfect-foresight equilibrium outcome for vaccination behavior in an unregulated market. Both the optimal and market outcomes are characterized by switching curves in SI state space; exact expressions for these are derived. The final section discusses taxes and subsidies that can be used to attain the optimal outcome.

Introduction

This paper is concerned with determining taxation and subsidy policies to attain the optimal time path of a one-time disease outbreak. The analysis is based on a dynamic contagion model of the sort that is used to characterize influenza epidemics: the disease is assumed to be vaccine-preventable and to spread and decline sufficiently rapidly that population turnover through births and deaths can be ignored.

This work draws on several different strains of the literature. In its use of dynamic epidemic models populated by rational agents it is in the spirit of the economic epidemiology literature summarized in Philipson (2000). However, this paper is intended to take a more overtly normative approach than Philipson takes. In its overall direction, if not its analytical technique, this paper has something in common with static analyses such as Brito et al. (1991) and various discussions of a more qualitative nature going farther back in time. The optimal control problem solution is drawn from an area of the applied mathematics literature that was very active in the 1970s but less so since then. Relative to the economics literature, this analysis is a little unusual in that it addresses and solves an explicitly off-steady-state problem. Although there are some papers that have considered such situations, the epidemic dynamics involved in those analyses are generally more simplistic than those considered here, and they usually stop short of an explicit characterization of the disease trajectory.1

The structure of the paper is as follows. The next section is devoted to the epidemic model and its mathematical characteristics. Section 3 gives the solution to the optimal vaccination policy problem, which is due to Morton and Wickwire (1974). Section 4 is devoted to the behavior of an unregulated market for vaccinations and the epidemic time path that results. The solution to this problem, as with the optimal vaccination policy in the previous section, is described by a partitioning of the state space into vaccinate and no-vaccinate regions. Not surprisingly, the market partitioning is not in general the same as the optimal one, although there are boundary states where they coincide. One result that comes out of the analysis is that a population of homogeneous rational individuals with perfect foresight may initially buy vaccinations but then cease purchasing them before an epidemic reaches its peak. (For such a population, that outcome is always sub-optimal.)

Section 5 presents taxation and subsidy policies that work through the market behavior of section four to achieve the outcomes of section three. There are several combinations of tax and subsidy that could achieve the optimal outcome while maintaining revenue neutrality, if that is desired. Previous writers have suggested taxing unvaccinated individuals as well as subsidizing vaccinations (see Brito et al., 1991); here I discuss some other possibilities, including the alternative of taxing infected individuals.

Section snippets

Epidemic model

The standard model for a one-shot epidemic is given by the following equations:2Ṡ(t)=−βS(t)I(t)N−r(t),İ(t)=βS(t)I(t)N−γI(t),Ṙ(t)=r(t)+γI(t).

The variables and parameters are defined as follows:

S(t)number of susceptible (non-infected and non-immune) individuals at time t
I(t)number of infected individuals at time t
R(t)number of immune individuals at time t
r(t)

Optimal vaccination policies

We turn first to optimal vaccination policies for the epidemic environment described above. In order to define the problem, we must introduce some economic parameters into the model. We will assume that all individuals are identical, although I will offer one comment about potential implications of heterogeneity later. Individuals are assumed to have infinite lifetimes, or, perhaps more sensibly, we assume that all relevant effects play themselves out rapidly enough that mortality is not an

Unregulated market outcome

We next turn to the question of what the outcome would be for this model when no policy is imposed and individuals are free to purchase immunizations at the stated cost θ. The other parameters from above remain, and we assume perfect foresight of the overall course of the epidemic. The method is to determine the locus of points in the state space for which an individual would be indifferent between vaccinating and not doing so if he thought that no one else would. Those points define a market

Optimal taxes and subsidies

We are working with a model that involves three states and three individual economic parameters. (The parameter w is a function of those individual economic parameters—ū, and θ—as well as the recovery rate γ.) On the assumption that the economic parameters are things that can be affected by government, we now examine policies to attain the optimal outcome.8

Concluding remarks

This paper presents a simple worked example of the welfare economics associated with a dynamic, continuous-time, off-steady-state epidemic environment in which individual vaccination choice is explicitly accommodated. The vaccination externality has been characterized in terms of individual responses to the state variables of the system, those being the number of infecteds (i.e., prevalence) and the number of susceptibles. As such, these results can be viewed as an expansion of the notion of

Acknowledgements

This paper represents the views of the author only, and does not necessarily represent the views of the CNA Corporation. I am indebted to Don Birchler, Bryan Boulier, an anonymous referee, and seminar participants at the Eastern Economic Association 2001 meetings and the Western Economic Association 2001 meetings for comments on earlier drafts of this paper. All remaining errors are of course my responsibility.

References (16)

There are more references available in the full text version of this article.

Cited by (70)

  • Effects of official information and rumor on resource-epidemic coevolution dynamics

    2022, Journal of King Saud University - Computer and Information Sciences
  • On the management of population immunity

    2022, Journal of Economic Theory
  • Optimal prevention and elimination of infectious diseases

    2021, Journal of Mathematical Economics
    Citation Excerpt :

    Most articles in the literature adopt a positive approach focusing on private behaviors, such as individual choices on self-exposure to the risk (as in Geoffard and Philipson, 1996; Kremer, 1996), health expenditure (Momota et al., 2005), human capital accumulation (Bell and Gersbach, 2013; Corrigan et al., 2005; Boucekkine et al., 2016), fertility (Young, 2005), income distribution (Boucekkine and Laffargue, 2010) or biodiversity (Bosi and Desmarchelier, 2020). Some papers analyze the effect of public policies on private behavior (e.g. Geoffard and Philipson, 1997 ), while others consider optimal policy correcting for the obvious externalities produced by infectious diseases (e.g. Gersovitz and Hammer, 2005; Francis, 2004; Gersovitz, 2013; Bethune and Korinek, 2020 or Eichenbaum et al., 2020). While few papers in economics adopt a normative perspective, one observes there exists an abundant mathematical epidemiology literature, which dates back from Bernoulli (1760)!

View all citing articles on Scopus
View full text