The Mechanics of Individually- and Socially-Optimal Decisions during an Epidemic

Guillaume Vandenbroucke

doi:10.1515/bejm-2021-0072

Published by De Gruyter September 14, 2021

The Mechanics of Individually- and Socially-Optimal Decisions during an Epidemic

Guillaume Vandenbroucke

From the journal The B.E. Journal of Macroeconomics

https://doi.org/10.1515/bejm-2021-0072

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Abstract

I present a model where work implies social interactions and the spread of a disease is described by an SIR-type framework. Upon the outbreak of a disease reduced social contacts are decided at the cost of lower consumption. Private individuals do not internalize the effects of their decisions on the evolution of the epidemic while the planner does. Specifically, the planner internalizes that an early reduction in contacts implies fewer infectious in the future and, therefore, a lower risk of infection. This additional (relative to private individuals) benefit of reduced contacts implies that the planner’s solution feature more social distancing early in the epidemics. The planner also internalizes that some infectious eventually recover and contribute further to a lower risk of infection. These mechanisms imply that the planner obtains a flatter infection curve than that generated by private individuals’ responses.

Keywords: SIR model; epidemic; social optimum; social distance; contact rate

JEL Classification: E1; H1; I1

Corresponding author: Guillaume Vandenbroucke, Research Division, Federal Reserve Bank of St. Louis, P.O. Box 442, St. Louis, MO, 63166, USA, E-mail: guillaumevdb@gmail.com

Acknowledgments

The views expressed in this article are mine and do not necessarily reflect the views of the Federal Reserve Bank of St. Louis or the Federal Reserve System. I thank the editor and an anonymous referee for comments that improved the paper. I have also benefited from discussions with B. Ravikumar and Chris Waller, as well as comments from the Brown-Bag Seminar participants at the St. Louis Fed.

Appendix A: The Flow of New Infections

Suppose contact rates, that is the number of people met by a given person in a period, are group specific: Λ_I, Λ_S and Λ_R. Let p _ir be the fraction of contacts made by a member of group P _I that is with a member of group P _R. Define p _sr, p _rr etc… similarly. We have

∑ y ∈ { i , s , r } p x y = 1 for x ∈ { i , s , r } .

The number of contacts by members of P _I with members of P _S is then Λ_I P _I p _is, and it must equal the number of contacts by members of P _S with members of P _I which is Λ_S P _S p _si. Thus, we have the following consistency conditions

P I ↔ P S : Λ I P I p i s = Λ S P S p s i P I ↔ P R : Λ I P I p i r = Λ R P R p r i P S ↔ P R : Λ S P S p s r = Λ R P R p r s

Define

p x y = Λ y P y Λ I P I + Λ S P S + Λ R P R .

It is then immediate that the consistency conditions above are satisfied:

Λ I P I p i s = Λ I P I Λ S P S Λ I P I + Λ S P S + Λ R P R = Λ S P S p s i , Λ I P I p i r = Λ I P I Λ R P R Λ I P I + Λ S P S + Λ R P R = Λ R P R p r i , Λ S P S p s r = Λ S P S Λ R P R Λ I P I + Λ S P S + Λ R P R = Λ R P R p r s .

New infections result from a susceptible meeting with an infectious. The number of meetings made by a susceptible is Λ_S P _S and a fraction p _si are with other infectious. Let ϕ denote the probability that a meeting results in an infection. The number of new infections in a period is then

new infections = Λ S ϕ P S p s i = Λ S ϕ P S Λ I P I Λ I P I + Λ S P S + Λ R P R

If meeting rates were identical, i.e., Λ_S = Λ_I = Λ_R = Λ, the number of new infections in a period would be

Λ ϕ P S P I P I + P S + P R .

If susceptible and infectious people have the same meeting rate, Λ_I = Λ_S = Λ, and if the meeting rate of recovered is different, Λ_R ≠ Λ, then the number of new infections in a period would be

Λ ϕ P S P I P I + P S + P R Λ R / Λ .