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

Guillaume Vandenbroucke

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.

JEL Codes: 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:

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: ΛI, ΛS and ΛR. Let pir be the fraction of contacts made by a member of group PI that is with a member of group PR. Define psr, prr etc… similarly. We have

y{i,s,r}pxy=1forx{i,s,r}.

The number of contacts by members of PI with members of PS is then ΛIPIpis, and it must equal the number of contacts by members of PS with members of PI which is ΛSPSpsi. Thus, we have the following consistency conditions

PIPS:ΛIPIpis=ΛSPSpsiPIPR:ΛIPIpir=ΛRPRpriPSPR:ΛSPSpsr=ΛRPRprs

Define

pxy=ΛyPyΛIPI+ΛSPS+ΛRPR.

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

ΛIPIpis=ΛIPIΛSPSΛIPI+ΛSPS+ΛRPR=ΛSPSpsi,ΛIPIpir=ΛIPIΛRPRΛIPI+ΛSPS+ΛRPR=ΛRPRpri,ΛSPSpsr=ΛSPSΛRPRΛIPI+ΛSPS+ΛRPR=ΛRPRprs.

New infections result from a susceptible meeting with an infectious. The number of meetings made by a susceptible is ΛSPS and a fraction psi 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=ΛSϕPSpsi=ΛSϕPSΛIPIΛIPI+ΛSPS+ΛRPR

If meeting rates were identical, i.e., ΛS = ΛI = ΛR = Λ, the number of new infections in a period would be

ΛϕPSPIPI+PS+PR.

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

ΛϕPSPIPI+PS+PRΛR/Λ.

Appendix B: Robustness

Figure 3: Trajectory of the epidemic – private response, various σ.Source: Author’s calculations.

Figure 3:

Trajectory of the epidemic – private response, various σ.

Source: Author’s calculations.

Figure 4: Trajectory of the epidemic – planner’s response, various σ.Source: Author’s calculations.

Figure 4:

Trajectory of the epidemic – planner’s response, various σ.

Source: Author’s calculations.

Figure 5: Trajectory of the epidemic – private response, various ρ.Source: Author’s calculations.

Figure 5:

Trajectory of the epidemic – private response, various ρ.

Source: Author’s calculations.

Figure 6: Trajectory of the epidemic – planner’s response, various ρ.Source: Author’s calculations.

Figure 6:

Trajectory of the epidemic – planner’s response, various ρ.

Source: Author’s calculations.

Figure 7: Trajectory of the epidemic – private response, various γ.Source: Author’s calculations.

Figure 7:

Trajectory of the epidemic – private response, various γ.

Source: Author’s calculations.

Figure 8: Trajectory of the epidemic – planner’s response, various γ.Source: Author’s calculations.

Figure 8:

Trajectory of the epidemic – planner’s response, various γ.

Source: Author’s calculations.

Figure 9: Trajectory of the epidemic – private response, various R0.Source: Author’s calculations.

Figure 9:

Trajectory of the epidemic – private response, various R0.

Source: Author’s calculations.

Figure 10: Trajectory of the epidemic – planner’s response, various R0.Source: Author’s calculations.

Figure 10:

Trajectory of the epidemic – planner’s response, various R0.

Source: Author’s calculations.

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Received: 2021-03-29
Revised: 2021-09-01
Accepted: 2021-09-03
Published Online: 2021-09-14

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