Abstract
We propose a model, which nests a susceptible-infected-recovered-deceased (SIRD) epidemic model into a dynamic macroeconomic equilibrium framework with agents’ mobility. The latter affect both their income and their probability of infecting and being infected. Strategic complementarities among individual mobility choices drive the evolution of aggregate economic activity, while infection externalities caused by individual mobility affect disease diffusion. The continuum of rational forward-looking agents coordinates on the Nash equilibrium of a discrete time, finite-state, infinite-horizon Mean Field Game. We prove the existence of an equilibrium and provide a recursive construction method for the search of an equilibrium(a), which also guides our numerical investigations. We calibrate the model by using Italian experience on COVID-19 epidemic and we discuss policy implications.
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References
Acemoglu, D., Chernozhukov, V., Werning, I., Whinston, M. D.: A multi-risk SIR model with optimally targeted lockdown. National Bureau of Economic Research (2020)
Alvarez, F., Argente, D., Lippi, F.: A simple planning problem for COVID-19 lock-down, testing, and tracing. Am. Econ. Rev. Insights 3(3), 367–82 (2021)
Atkeson, A.: What will be the economic impact of COVID-19 in the US? Rough estimates of disease scenarios, National Bureau of Economic Research (2020)
Avery, C., Bossert, W., Clark, A., Ellison, G., Ellison, S.F.: An economist’s guide to epidemiology models of infectious disease. J. Econ. Perspect. 34(4), 79–104 (2020)
Bandyopadhyay, S., Chatterjee, K., Das, K., Roy, J.: Learning versus habit formation: optimal timing of lockdown for disease containment. J. Math. Econ. 93, 102452 (2021)
Bethune, Z. A., Korinek, A.: Covid-19 infection externalities: trading off lives vs. livelihoods. National Bureau of Economic Research (2020)
Bhattacharya, J., Chakraborty, S., Yu, X.: A rational-choice model of covid-19 transmission with endogenous quarantining and two-sided prevention. J. Math. Econ. 93, 102492 (2021)
Bisin, A., Moro, A.: Spatial-SIR with Network Structure and Behavior: lockdown Rules and the Lucas Critique. National Bureau of Economic Research (2021)
Bonnans, F., Lavigne, P., Pfeiffer, L.: Discrete-time mean field games with risk-averse agents. ESAIM: Control Optim. Calculus Var. 27:44 (2021)
Boppart, T., Harmenberg, K., Hassler, J., Krusell, P., Olsson, J.: Integrated epi-econ assessment. National Bureau of Economic Research (2020)
Brotherhood, L., Kircher, P., Santos, C., Tertilt, M.: An economic model of the Covid-19 epidemic: the importance of testing and age-specific policies. CESifo Working Paper (2020)
Bulow, J.I., Geanakoplos, J.D., Klemperer, P.D.: Multimarket oligopoly: strategic substitutes and complements. J. Polit. Econ. 93(3), 488–511 (1985)
Cardaliaguet, P., Porretta, A.: An introduction to mean field game theory. In: Cardaliaguet, P., Porretta, A., Santmbrogio, F. (eds.) Mean Field Games: Cetraro, Italy 2019, pp. 1–158. Springer, Berlin (2020)
Carmona, R., Delarue, F.: Probabilistic Theory of Mean Field Games with Applications, vol. 84. Springer, Berlin (2018)
Chowell, G., Sattenspiel, L., Bansal, S., Viboud, C.: Mathematical models to characterize early epidemic growth: a review. Phys. Life Rev. 18, 66–97 (2016)
Cochrane, J.: An SIR model with behavior. The “Grumpy Economist” Blog, Monday May, 4:2020 (2020)
Cooper, R., John, A.: Coordinating coordination failures in Keynesian models. Q. J. Econ. 103(3), 441–463 (1988)
Day, M.: Covid-19: identifying and isolating asymptomatic people helped eliminate virus in Italian village. BMJ Br. Med. J. (Online), 368 (2020)
Doncel, J., Gast, N., Gaujal, B.: Discrete mean field games: existence of equilibria and convergence. J. Dyn. Games 6(3), 221–239 (2019)
Eichenbaum, M.S., Rebelo, S., Trabandt, M.: The macroeconomics of epidemics. Rev. Financial Stud. 34(11), 5149–5187 (2021)
Eichenbaum, M.S., Rebelo, S., Trabandt, M.: The macroeconomics of testing and quarantining. J. Econ. Dyn. Control, 104337 (2022)
Engle, S., Keppo, J., Quercioli, E. K.M., Smith, L., Wilson, A.: The Behavioral SIR Model, with Applications to the Swine Flu and COVID-19 Pandemics. Working Paper, University of Wisconsin (2020)
Farboodi, M., Jarosch, G., Shimer, R.: Internal and external effects of social distancing in a pandemic. J. Econ. Theory 196, 105293 (2021)
Fenichel, E.P.: Economic considerations for social distancing and behavioral based policies during an epidemic. J. Health Econ. 32(2), 440–451 (2013)
Ferguson, N., Laydon, D., Nedjati Gilani, G., Imai, N., Ainslie, K., Baguelin, M., Bhatia, S., Boonyasiri, A., Cucunuba Perez, Z., Cuomo-Dannenburg, G., et al.: Report 9: Impact of non-pharmaceutical interventions (NPIs) to reduce COVID19 mortality and healthcare demand. Technical report, MRC Centre for Global Infectious Disease Analysis (2020)
Flaxman, S., Mishra, S., Gandy, A., Unwin, H.J.T., Mellan, T.A., Coupland, H., Whittaker, C., Zhu, H., Berah, T., Eaton, J.W., et al.: Estimating the effects of non-pharmaceutical interventions on COVID-19 in Europe. Nature 584(7820), 257–261 (2020)
Gollier, C.: Cost-benefit analysis of age-specific deconfinement strategies. J. Public Econ. Theory 22(6), 1746–1771 (2020)
Gomes, D.A., Mohr, J., Souza, R.R.: Discrete time, finite state space mean field games. Journal de mathématiques pures et appliquées 93(3), 308–328 (2010)
Goolsbee, A., Syverson, C.: Fear, lockdown, and diversion: comparing drivers of pandemic economic decline 2020. J. Public Econ. 193, 104311 (2021)
Granas, A., Dugundji, J.: Fixed Point Theory. Springer, Berlin (2003)
Hadikhanloo, S., Silva, F.J.: Finite mean field games: fictitious play and convergence to a first order continuous mean field game. Journal de Mathématiques Pures et Appliquées 132, 369–397 (2019)
Huang, J., Wang, H., Xiong, H., Fan, M., Zhuo, A., Li, Y., Dou, D.: Quantifying the economic impact of COVID-19 in mainland China using human mobility data. (2020) arXiv preprint arXiv:2005.03010
Huang, M., Malhamé, R.P., Caines, P.E.: Large population stochastic dynamic games: closed-loop McKean–Vlasov systems and the Nash certainty equivalence principle. Commun. Inf. Syst. 6(3), 221–251 (2006)
Hung, I.F., Poland, G.A.: Single-dose Oxford-AstraZeneca COVID-19 vaccine followed by a 12-week booster. The Lancet 397(10277), 854–855 (2021)
Jones, C., Philippon, T., Venkateswaran, V.: Optimal mitigation policies in a pandemic: social distancing and working from home. Rev. Financial Stud. 34(11), 5188–5223 (2021)
Jovanovic, B., Rosenthal, R.W.: Anonymous sequential games. J. Math. Econ. 17(1), 77–87 (1988)
Kaplan, G., Moll, B., Violante, G.L.: The great lockdown and the big stimulus: tracing the pandemic possibility frontier for the US. National Bureau of Economic Research (2020)
Laibson, D., Lee, S.C.L., Maxted, P., Repetto, A., Tobacman, J.: Estimating discount functions with consumption choices over the lifecycle. National Bureau of Economic Research (2007)
Lasry, J.-M., Lions, P.-L.: Jeux à champ moyen. I – Le cas stationnaire. Comptes Rendus Mathématique, 343(9):619–625 (2006a)
Lasry, J.-M., Lions, P.-L.: Jeux à champ moyen. II – Horizon fini et contrôle optimal. Comptes Rendus Mathématique, 343(10):679–684 (2006b)
Lasry, J.-M., Lions, P.-L.: Mean field games. Jpn. J. Math. 2(1), 229–260 (2007)
Manski, C.F.: Economic analysis of social interactions. J. Econ. Perspect. 14(3), 115–136 (2000)
Meloni, S., Perra, N., Arenas, A., Gómez, S., Moreno, Y., Vespignani, A.: Modeling human mobility responses to the large-scale spreading of infectious diseases. Sci. Rep. 1(1), 1–7 (2011)
Moser, C., Yared, P.: Pandemic lockdown: The role of government commitment. Review of Economic Dynamics (2022)
Nouvellet, P., Bhatia, S., Cori, A., Ainslie, K.E., Baguelin, M., Bhatt, S., Boonyasiri, A., Brazeau, N.F., Cattarino, L., Cooper, L.V., et al.: Reduction in mobility and COVID-19 transmission. Nat. Commun. 12(1), 1–9 (2021)
OCDE.: evaluating the initial impact of Covid-19 containment measures on economic activity. Technical report, OCDE Publishing Paris, France (2020)
Piguillem, F., Shi, L., et al.: The optimal Covid-19 quarantine and testing policies. Einaudi Institute for Economics and Finance (EIEF) (2020)
Rosen, S.: The value of changes in life expectancy. J. Risk Uncertain. 1(3), 285–304 (1988)
Salje, H., Kiem, C.T., Lefrancq, N., Courtejoie, N., Bosetti, P., Paireau, J., Andronico, A., Hozé, N., Richet, J., Dubost, C.-L., et al.: Estimating the burden of SARS-CoV-2 in France. Science 369(6500), 208–211 (2020)
Steedman, I.: Consumption Takes Time: Implications for Economic Theory. Routledge, Abingdon (2001)
Toxvaerd, F.: Equilibrium social distancing. Faculty of Economics, University of Cambridge, Preprint (2020)
Verity, R., Okell, L.C., Dorigatti, I., Winskill, P., Whittaker, C., Imai, N., Cuomo-Dannenburg, G., Thompson, H., Walker, P.G., Fu, H., et al.: Estimates of the severity of coronavirus disease 2019: a model-based analysis. Lancet. Infect. Dis 20(6), 669–677 (2020)
Voinsky, I., Baristaite, G., Gurwitz, D.: Effects of age and sex on recovery from COVID-19: analysis of 5769 Israeli patients. J. Infect. 81(2), e102–e103 (2020)
Vollmer, M.A., Mishra, S., Unwin, H.J.T., Gandy, A., Mellan, T.A., Bradley, V., Zhu, H., Coupland, H., Hawryluk, I., Hutchinson, M., et al.: Report 20: Using mobility to estimate the transmission intensity of COVID-19 in Italy: a subnational analysis with future scenarios. Technical report, Cold Spring Harbor Laboratory (2020)
Wiecek, P.: Discrete-time ergodic mean-field games with average reward on compact spaces. Dyn. Games Appl. 10(1), 222–256 (2020)
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The work of Giorgio Fabbri is partially supported by the French National Research Agency in the framework of the “Investissements d’Avenir” program (ANR-15-IDEX-02) and of the center of excellence LABEX MME-DII (ANR-11-LABX-0023-01). Salvatore Federico, Davide Fiaschi, and Fausto Gozzi were supported by the Italian Ministry of Education, University and Research (MIUR), in the framework of PRIN projects 2015233N54 006 Deterministic and Stochastic Evolution Equations and 2017FKHBA8 001 “The Time-Space Evolution of Economic Activities: Mathematical Models and Empirical Applications”.
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Fabbri, G., Federico, S., Fiaschi, D. et al. Mobility decisions, economic dynamics and epidemic. Econ Theory 77, 495–531 (2024). https://doi.org/10.1007/s00199-023-01485-1
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DOI: https://doi.org/10.1007/s00199-023-01485-1