The social cost of contacts: Theory and evidence for the first wave of the COVID-19 pandemic in Germany

Building on the epidemiological SIR model, we present an economic model with heterogeneous individuals deriving utility from social contacts creating infection risks. Focusing on social distancing of individuals susceptible to an infection we theoretically characterize the gap between private and social cost of contacts. Our main contribution is to quantify this gap by calibrating the model with unique survey data from Germany on social distancing and impure altruism from the beginning of the COVID-19 pandemic. The optimal policy is to drastically reduce contacts at the beginning to almost eradicate the epidemic and keep them at levels that contain the pandemic at a low prevalence level. We find that also in laissez faire, private protection efforts by forward-looking, risk averse individuals would have stabilized the epidemic, but at a much higher prevalence of infection than optimal. Altruistic motives increase individual protection efforts, but a substantial gap to the social optimum remains.

S1 File S1 Appendix Derivations of theoretical results. The Lagrangian for the problem to optimize (9), subject to the epidemiological dynamics given by (1), reads 838 L = j T t=0 δ t j (S jt + I jt ) u s j (c jt ) + Q jt u q j + R jt u r j + D jt u d j + δ j λ s jt −S j,t+1 + S jt − β c jt S jt l I lt + δ j λ i jt −I j,t+1 + β c jt S jt l I lt + δ j λ q jt −Q j,t+1 + (1 − α q j − γ q j ) Q jt + I jt + δ j λ r jt −R j,t+1 + R jt + γ q j Q jt + δ j λ d jt −D j,t+1 + α q j Q jt . (20) The first-order necessary conditions (10a) to (10f) for optimal social distancing are S2 Appendix Calibration of epidemiological parameters.
We estimate COVID-19 mortality rate α q j for each population group j as follows: We use the daily number of new known infections and new deaths in Germany reported by the Robert Koch Institute [1]. Here, we consider the time period from January 6 to April 26, 2020. and aggregate them to the weekly level. We use (1) and replace θ j I jt and α q j Q jt with the time series of reported new known infections ∆Q j,t+1 and new deaths ∆D j,t+1 respectively. We suppose Q j,1 = 0, since the estimation relies on data ∆Q j,t , making Q j,1 unimportant and because reported cases in January and early February were probably still affected by imported cases [2,3]. We then estimate α q j by means of ordinary least squares subject to Q j,t+1 = (1 − γ) Q jt + ∆Q j,t+1 − ∆D j,t+1 . This gives α q j as shown in 844 S1 Table. 845 Our estimate for θ j is obtained from the time period until an infection is detected, 846 on average, which we assume to be 5 days, which corresponds to an exponential detection rate is θ j = 1 − e −7/5 = 0.75 per week. The initial basic reproduction number R 0 was estimated to be between 2 and 3 [4][5][6][7], although higher values with a range of 3 to 851 852 12 [8] or up to 14.8 [9] have been suggested. We calibrate our model to the more conservative and widely accepted value of R 0 = 3 for the German population during the initial phase of the pandemic. From this we obtain β = 2.25. 853 We calibrate γ q j such that 90% of COVID-19 patients recover within two weeks after 854 the infection is detected. In our discrete time model the recovery rate is γ q j = 1 − e −7/6 855 per week. The assumption of an exponential distribution captures the fact that some 856 individuals need much longer to recover.

857
This implies the infection fatality ratios IFR j = α q j α q j +γ q j as reported in S1 Table. The 858 estimated mean IFR of 3.86 percent for the German population is close to the fatality 859 rates reported by the WHO [10]. S1  Here we present some extra information about the data we used in our empirical model. S2 Table presents

S4 Appendix Numerical solution method and AMPL programming code
Numerically, the problem to solve the set of dynamic equations that describe 899 epidemiological dynamics, and the optimality conditions, (1), (10a), (10b), and (11), 900 along with initial conditions for the number of susceptibles, infected, and recovered from 901 all groups of individuals, and transversality conditions, is usually a more difficult task 902 than solving an optimization problem. In our numerical approach to compute the 903 utilitarian optimum, we thus use the following lemma: 904 Lemma 1. The social optimization problem (9) is equivalent tô wherec jt is the individual choice of physical social contacts in Nash equilibrium and V q jt 905 is the individual present value of becoming a COVID-19 patient, V q jt , as determined 906 by (5).

911
In the online Supporting Information we provide the AMPL code for computing the 912 utilitarian optimal epidemiological dynamics, as shown in Figure 1.

913
To compute the Nash equilibrium with selfish individuals, we solve the system of 914 equations (7), (8) and epidemiological dynamics (1) for t = 0, . . . , T with T = 20 weeks. 915 This is done by means of a code similar to the one below, except that the objective 916 function (9) is dropped (i.e. there is no opimization) and that equations (7) and (8)