## wpiea2020277-print-pdf ## Source details **Canonical URL:** [wpiea2020277-print-pdf](https://www.imf.org/-/media/files/publications/wp/2020/english/wpiea2020277-print-pdf.pdf) ## Other formats - [Markdown version](/-/media/files/publications/wp/2020/english/wpiea2020277-print-pdf.pdf.md) - [Structured JSON version](/-/media/files/publications/wp/2020/english/wpiea2020277-print-pdf.pdf.json) --- ### 2.1 Epidemiological dynamics - Model structure - Two agent types: low-risk (y) and high-risk (o), health status X ∈ {S, I, R, D}. - P^J_t = S^J_t + I^J_t + R^J_t + D^J_t for J = {y, o}. - Agents may be contained (proportion m^j_t) or working (proportion 1 − m^j_t). Retirees (fraction or) do not work, receive fixed pension income, and are not subject to containment. - Infection channels: consumption, work, and non-activity (home) contacts. - Formal infection flows (compact) - Newly infected working individuals T^j_w_t given by three additive terms with sensitivities π^S_1 (consumption), π^S_2 (work), π^S_3 (home), and involving contacts across I^o_w_t, I^o_k_t, I^y_w_t, I^y_k_t, I^o_r_t. - Newly infected contained individuals T^j_k_t analogous with N^Sjk_t ≡ 0 for the work channel. - Newly infected retirees T^o_r_t analogous with retiree-specific Ψ_o and χ_o in the home channel. - Aggregation: T^y_t = T^y_w_t + T^y_k_t; T^o_t = T^o_w_t + T^o_k_t + T^o_r_t. - Transmission sensitivity and contact structure - π^S_1, π^S_2, π^S_3: sensitivities for consuming, working, and being at home. - Assortative mixing: χ_y is share of low-risk contacts with low-risk; for high-risk the within-group share is 1 − χ_o. - Ψ_j captures contact heterogeneity (Ψ_y > Ψ_o). - Population transitions - S^J_{t+1} = S^J_t − T^J_t - I^J_{t+1} = I^J_t − (π^R_J + π^D_J_t) I^J_t + T^J_t - R^J_{t+1} = R^J_t + π^R_J I^J_t - D^J_{t+1} = D^J_t + π^D_J_t I^J_t - P^J_{t+1} = P^J_t − π^D_J_t I^J_t - Mortality and healthcare overload - π^D_J_t = π^D_J + κ (I_t)^2 (mortality rises convexly with aggregate infected I_t). - Initial conditions - Initial infected fraction I_0 = ε. - Initial susceptible S_0 = 1 − ε (population normalized to one). ### 2.2 Agents’ optimization problems - Preferences - Lifetime utility: U_{Xj t} = E_t ∑_{τ=t}^{∞} β_{j,τ−t} u_t(C_{Xj τ},N_{Xj τ}) - Period utility: u_t(C_{Xj t},N_{Xj t}) = ln(C_{Xj t}) − (θ/2) (N_{Xj t})^2 - β_j varies with risk category j (accounts for non-COVID-19 mortality). - Non-optimizing groups - Contained individuals: C_{X j k t} = Γ_{k t} (consumption financed by government lump-sum transfer Γ_{k t}). - Retired agents: C_{X o r} = Γ_r (fixed pension income). - Non-contained workers (optimize) - Budget: C_{X j w t} = W_t N_{X j w t} − Γ_{w t} - Government containment policy m_{j t} and lump-sum tax Γ_{w t} taken as given. - Susceptible agents' problem - Maximize U_{S j w t} with infection probability τ_{j w t} ≡ T_{j w t} / S_{j w t} composed of π_S1 (consumption-related term), π_S2 (work-related term), and π_S3 Ψ_J [ χ_j I_{y t} P_{y t} + (1−χ_j) I_{o t} P_{o t} ]. - Budget: C_{S j w t} = W_t N_{S j w t} − Γ_{w t} - First-order condition (w.r.t. N_{S j w t}) equates marginal utility of consumption and disutility of labor adjusted by infection externality term (full expression provided in source). - Infected agents - Productivity: φ_I < 1 = φ_S = φ_R - Maximize U_{I j w t} with budget W_t φ_I N_{I j w t} = C_{I j w t} + Γ_{w t} - First-order condition: φ_I W_t / (φ_I W_t N_{I j w t} − Γ_{w t}) − θ N_{I j w t} = 0 - Recovered agents - Budget: W_t N_{R j w t} = C_{R j w t} + Γ_{w t} - First-order condition analogous to susceptible/recovered specification (full expression in source). ### 2.3 Government’s optimization problem - Externalities making decentralized equilibrium inefficient - Infected individuals’ consumption and work raise susceptible infection risk. - Low-risk susceptibles (lower mortality) have stronger incentives to work/consume, affecting high-risk individuals. - Large infections congest healthcare and increase mortality. - Government objective and instruments - Utilitarian social welfare with equal weights across agents. - Initial-time welfare expression: W1 = (Sy1 U_Sy1 + Iy1 U_Iy1) + (Sot1 U_Sot1 + Iot1 U_Iot1) at t = 1 with D1 = R1 = 0. - Policy instruments: - Containment path {mjt}∞t=1 ∈ [0,1], differentiated by risk profile j. - Lump-sum taxes on workers {Γwt}∞t=1 to finance transfers {Γkt}∞t=1 to contained agents. - Government budget constraint: - Γkt (Sok t + Syk t + Iok t + Iyk t + Rok t + Ryk t) + Γwt (Sow t + Syw t + Iow t + Iyw t + Row t + Ryw t) = 0 - Parametric approximation of policy paths - Containment mjt approximated by generalized logistic: - mjt = γj0 / [1 + exp(−γj1 (t − γj2))] - γj0 controls level of containment at t = 0 - γj1 governs when containment is reduced - γj2 commands swiftness of reduction - As t → ∞, mjt → 0 - Transfer Γkt approximated by scaled generalized logistic: - Γkt = φ0 · λ / [1 + exp(−φ1 (t − φ2))] - Γwt determined by government budget constraint. - Optimization chooses {γy0, γy1, γy2, γo0, γo1, γo2, φ0, φ1, φ2}; optimal paths evaluated for t = 1..250. - Homogeneous policy: γyi = γoi for i ∈ {0,1,2}. - Numerical approach - Parametric approach reduces optimization arguments from 750 to 9; non-parametric approach would require computing 250 containment and transfer values and is numerically unstable. - Calibration (selected parameter values) - Population shares and mortality - Low-risk (age 0–54): y = 0.71 of population; mortality pDy = 0.002 (0.2 percent) - High-risk (age 55+): ot = 0.29 of population; mortality pDot = 0.047 (4.7 percent) - Share of retirees within high-risk: or = 0.48 (47.9 percent of 55+) - Epidemiological timing - After 18 days infected person either recovers or dies → πDJ + πRJ = 7/18 - Recovery rates: Ry = 7/18 − πDy ; Rot = 7/18 − πDot - Infection sensitivity parameters - πS1 = 8.4792×10^−8 (consumption) - πS2 = 1.6376×10^−4 (work) - πS3 = 0.1293 (other) - Contact rates (Prem et al. (2017)) - Ψy = χyy + χyo = 4.3 - Ψo = χoo + χoy = 1.1 - χy = 3.3/(3.3 + 1.0) - χo = 0.2/(0.9 + 0.2) - Economic parameters - Discount factor low-risk βy = 0.96^(1/52) (reported as 0.9992) - Non-COVID-19 mortality for high-risk ι = 0.05 → βot = βy · (1 − ι)^(1/52) (reported as 0.9982) - Disutility of labor θ = 0.001275 - Productivity parameters: A = 39.835; φI = 0.8; φS, φR = 1 - Hours per week N = 28 - Salary per week W = 58,000/52 - Retirement per week Γr = 0.744W - Targets for πS1, πS2, πS3 - (i) 16 percent of infections from consumption activities - (ii) 17 percent from work activities - (iii) 60 percent of population either recovers or dies by end of pandemic - Policy scenarios - Targeted policy: containment heterogeneous across low- and high-risk individuals - Blanket policy: identical containment across low- and high-risk individuals - No-policy: no containment - Optimal containment computed over 250 weeks for low- and high-risk individuals - Key quantitative epidemiological results (250-week horizon unless noted) - Optimal containment bans from working - Blanket policy: 33.9 percent of people (percent excludes retirees) - Targeted policy: 38.9 percent of people; restrictions held about a month and a half longer than blanket policy - Under targeted policy containment composition: almost all high-risk individuals contained; 27.1 percent of low-risk individuals contained - Infection peaks (percent of initial population; week noted where applicable) - No-policy: infections peak during week 27 at 5.7 percent - Blanket policy: infections peak at 3.8 percent - Targeted policy: infections peak at 3.9 percent (a month and a half after no-policy peak) - Deaths (percent of initial population) - No-policy: deaths account for 0.55 percent - Blanket policy: deaths equal 0.40 percent - Targeted policy: deaths equal 0.34 percent - Risk-group-specific epidemiology - High-risk infected peak: - Blanket: 0.34 percent (of initial population) - Targeted: 0.15 percent - High-risk deaths (share of initial population): - Blanket: 0.25 percent - Targeted: 0.11 percent - Aggregate lives saved (250 weeks, U.S. 2019 population basis, Table 2) - Low-risk deaths - Blanket: 475,687 - Targeted: 478,110 - Difference: −2,423 (targeted loses 2,423 low-risk lives relative to blanket) - High-risk deaths - Blanket: 813,056 - Targeted: 643,044 - Difference: 170,012 (targeted saves 170,012 high-risk lives) - Total deaths - Blanket: 1,288,743 - Targeted: 1,121,154 - Difference: 167,589 (targeted saves 167,589 total lives relative to blanket) - Interpretation - Targeted policy substantially reduces infections and deaths among high-risk individuals while producing slightly worse outcomes for low-risk individuals. - Targeted policy saves about 167,589 lives relative to a blanket policy in the calibrated U.S. population example (250 weeks horizon). - Macroeconomic impacts - Consumption (percent deviation from steady state) - No-policy: recession driven by voluntary self-distancing; consumption trough coincides with infection peak. - Blanket policy: immediate abrupt plunge; consumption trough is a 33 percent collapse. - Targeted policy: immediate abrupt plunge; consumption trough is a 31 percent collapse; consumption remains subdued longer. - Reasons targeted policy generates larger recession: larger share contained for longer (optimal containment ~60 weeks vs 45 weeks under blanket); herd immunity achieved earlier under blanket, especially among high-risk. - Hours worked: patterns resemble consumption due to linear production and transfers; deviations smaller because retirees continue consuming from pension income. - Behavior of non-contained individuals - Under blanket: low-risk individuals work less when infections rise; consumption falls then recovers as infections decline. - Under targeted: low-risk non-contained individuals initially work more (high-risk are banned) to offset taxes; aggregate consumption and hours of non-contained individuals collapse because almost all contained individuals are banned from working. - Welfare: present value of social welfare highest under targeted policy despite a more prolonged recession; government prefers targeted containment because reduction in deaths (and increase in leisure) outweighs larger consumption losses. - Extensions: household structure and increased interactions can reduce benefits of targeted containment; results revisited by raising contact rates and increasing cross-risk contact shares. ### 5.1 Higher contact rates - Setup - Each group increases average contact rate by 10 percent. - New total contact rates: Ψy = 4.73 and Ψo = 1.21. - Assortativity maintained; χj calibrated as in Table 1. - Alternative exercise (section 5.2) sets χj = 0.71 to raise cross-risk contact shares. - Optimal containment outcomes (central findings) - Blanket policy: 45.1 percent of the population banned from working. - A 10 percent increase in contact rates prompts government to ban an additional 11.2 percent of people relative to the lower-contact setting. - Targeted policy: 51.7 percent of the population banned from working (about 12.8 percentage points higher than in lower-contact setting). - Containment applies to all high-risk individuals and 42.3 percent of low-risk individuals (15.2 percentage points more than with lower contact rates). - Targeted containment becomes less targeted with increased contact rates, reducing benefits of separating low- and high-risk individuals. - Epidemiological outcomes (appendix quantitative results) - Infection peaks: 4.4 percent (blanket) and 4.5 percent (targeted) of initial population. - Death rates: 0.44 percent (blanket) and 0.38 percent (targeted) of initial population. - Blanket policy produces more recovered individuals; targeted policy leaves more susceptible individuals. - Targeted containment saves about 167,000 more lives than a blanket one (aggregate result reported in Conclusions). - Macroeconomic outcomes - Aggregate dynamics under blanket and targeted policies become extremely similar with higher contact rates. - Distributional differences by risk-type remain large: - Targeted: initial drop in consumption and hours reflects containment of all high-risk and 42.3 percent of low-risk. - Blanket: containment applied to low-risk higher than under targeted; containment applied to high-risk lower. - Recovery in hours and consumption reflects infection evolution and lifting of containment; targeted recession lasts longer because containment is held longer and herd immunity achieved later. - Effect of household structure and cross-risk contacts - Increasing share of contacts between low- and high-risk (χj = 0.71) while holding total contacts constant reduces within-group contacts. - A sufficiently high share of cross-risk contacts makes optimal targeted approach equivalent to optimal blanket approach. - Intuition: work bans reduce workplace infections but cannot prevent cross-risk contamination within households; benefits of targeting vanish when cross-risk household contacts are high. - Policy implications (summary) - Targeted containment can substantially reduce deaths among high-risk individuals and overall mortality (saves about 167,000 more lives versus blanket) but typically requires containing a larger share of the population and sustaining containment longer, deepening and prolonging the recession. - Effectiveness and advantage of targeted containment diminish as: - Aggregate contact rates rise (targeted containment becomes less targeted). - Household interactions increase or share of cross-risk contacts rises (cross-risk household contamination undermines targeting and can make targeted and blanket policies equivalent). *Source: wpiea2020277-print-pdf — 2.1–5.1 (selected sections)* ### 2.1 Epidemiological dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ### 2.1 Epidemiological dynamics ### Overview - The model extends the SIR framework to two agent types: low-risk (y) and high-risk (o), each categorized by health status X ∈ {S, I, R, D} with - P^J_t = S^J_t + I^J_t + R^J_t + D^J_t for J = {y, o}. - Agents may be contained (proportion m^j_t) or working (proportion 1 − m^j_t). Retirees (fraction or) do not work, receive fixed pension income, and are not subject to containment. - Infection dynamics include three distinct channels: consumption, work, and non-activity contacts (e.g., at home). ### Infection channels (formal representation) - Newly infected working individuals T^j_w_t: - T^j_w_t = π^S_1 (S^j_w_t C / S^j_w_t) (I^o_w_t C / I^o_w_t + I^o_k_t C / I^o_k_t + I^y_w_t C / I^y_w_t + I^y_k_t C / I^y_k_t + I^o_r_t C / I^o_r_t) + π^S_2 (S^j_w_t N / S^j_w_t) (I^o_w_t N / I^o_w_t + I^y_w_t N / I^y_w_t) + π^S_3 S^j_w_t Ψ_j [ χ_j I^y_t / P^y_t + (1 − χ_j) I^o_t / P^o_t ] - Newly infected contained individuals T^j_k_t: - T^j_k_t = π^S_1 (S^j_k_t C / S^j_k_t) (I^o_w_t C / I^o_w_t + I^o_k_t C / I^o_k_t + I^y_w_t C / I^y_w_t + I^y_k_t C / I^y_k_t + I^o_r_t C / I^o_r_t) + π^S_2 (S^j_k_t · [N^Sjk_t ≡ 0]) (I^o_w_t N / I^o_w_t + I^y_w_t N / I^y_w_t) + π^S_3 S^j_k_t Ψ_j [ χ_j I^y_t / P^y_t + (1 − χ_j) I^o_t / P^o_t ] - Newly infected retirees T^o_r_t: - T^o_r_t = π^S_1 (S^o_r_t C / S^o_r_t) (I^o_w_t C / I^o_w_t + I^o_k_t C / I^o_k_t + I^y_w_t C / I^y_w_t + I^y_k_t C / I^y_k_t + I^o_r_t C / I^o_r_t) + π^S_2 (S^o_r_t · [N^Sor_t ≡ 0]) (I^o_w_t N / I^o_w_t + I^y_w_t N / I^y_w_t) + π^S_3 S^o_r_t Ψ_o [ χ_o I^y_t / P^y_t + (1 − χ_o) I^o_t / P^o_t ] - Aggregation: - T^y_t = T^y_w_t + T^y_k_t - T^o_t = T^o_w_t + T^o_k_t + T^o_r_t ### Transmission sensitivity parameters and contact structure - π^S_1, π^S_2, π^S_3: sensitivities of new infections due to interactions while consuming, working, and being at home, respectively. - Assortative mixing and contact heterogeneity: - χ_y: share of contacts of a low-risk individual with low-risk individuals. - For high-risk individuals the within-group share is 1 − χ_o. - Ψ_j captures heterogeneity in the number of contacts (Ψ_y > Ψ_o), acknowledging young people tend to have more contacts than older people. - The third channel (home/non-activity) infection flow depends on π^S_3, S^j_t, Ψ_j, χ_j, and group infection prevalence I^J_t / P^J_t. ### Population transition equations - Susceptibles: - S^J_{t+1} = S^J_t − T^J_t - Infected: - I^J_{t+1} = I^J_t − (π^R_J + π^D_J_t) I^J_t + T^J_t - π^R_J: probability of recovery (time-invariant here). - π^D_J_t: time-varying probability of death (see mortality function below). - Recovered: - R^J_{t+1} = R^J_t + π^R_J I^J_t - Deaths: - D^J_{t+1} = D^J_t + π^D_J_t I^J_t - Population: - P^J_{t+1} = P^J_t − π^D_J_t I^J_t ### Mortality as a function of healthcare overload - Mortality rate specification: - π^D_J_t = π^D_J + κ (I_t)^2 - Mortality rises convexly with the aggregate number of infected I_t to capture health system overload. ### Initial conditions - Initial infected fraction I_0 = ε. - Population normalized to one implies initial susceptible S_0 = 1 − ε. *Source: wpiea2020277-print-pdf — 2.1 Epidemiological dynamics* ### 2.2 Agents’ optimization problems ### 2.2 Agents’ optimization problems ### Preferences and lifetime utility - Discounted lifetime utility: - U_{Xj t} = E_t \sum_{\tau=t}^{\infty} β_{j,τ−t} u_t(C_{Xj τ},N_{Xj τ}) - Period utility: - u_t(C_{Xj t},N_{Xj t}) = ln(C_{Xj t}) − (θ/2) (N_{Xj t})^2 - β_j is the discount factor that varies with risk category j (accounts for probability of dying from factors other than COVID-19). ### Contained and retired agents (non-optimizing by construction) - Contained individuals: - Consumption financed by government lump-sum transfer Γ_{k t}: - C_{X j k t} = Γ_{k t} - Retired agents: - Consume fixed pension income Γ_r: - C_{X o r} = Γ_r ### Non-contained individuals (optimize consumption and labor) - Budget constraint for non-contained workers: - C_{X j w t} = W_t N_{X j w t} − Γ_{w t} - Take as given: government containment policy m_{j t} and lump-sum tax Γ_{w t}. ### Susceptible individuals - Susceptibles maximize: - Max U_{S j w t} = u(C_{S j w t},N_{S j w t}) + β_j { (1−τ_{j w t}) [ (1−m_{j t+1}) U_{S j w t+1} + m_{j t+1} U_{S j k t+1} ] + τ_{j w t} [ (1−m_{j t+1}) U_{I j w t+1} + m_{j t+1} U_{I j k t+1} ] } - Probability of getting infected: - τ_{j w t} ≡ T_{j w t} / S_{j w t} = - π_S1 (C_{S j w t}) ( I_{o w t} C_{I o w t} + I_{o k t} C_{I o k t} + I_{y w t} C_{I y w t} + I_{y k t} C_{I y k t} + I_{o r t} C_{I o r t} ) - + π_S2 (N_{S j w t}) ( I_{o w t} N_{I o w t} + I_{y w t} N_{I y w t} ) - + π_S3 Ψ_J [ χ_j I_{y t} P_{y t} + (1−χ_j) I_{o t} P_{o t} ] - Budget constraint: - C_{S j w t} = W_t N_{S j w t} − Γ_{w t} - First-order condition with respect to N_{S j w t} (after substituting (6) and (7)): - W_t / (W_t N_{S j w t} − Γ_{w t}) − θ N_{S j w t} + β_j { [ (1−m_{j t+1}) U_{I j w t+1} + m_{j t+1} U_{I j k t+1} ] − [ (1−m_{j t+1}) U_{S j w t+1} + m_{j t+1} U_{S j k t+1} ] } × { π_S1 W_t [ I_{o w t} C_{I o w t} + I_{y w t} C_{y o w t} + I_{o k t} C_{I o k t} + I_{y k t} C_{I y k t} ] + π_S2 [ I_{y w t} N_{I y w t} + I_{o w t} N_{I o w t} ] } = 0 ### Infected individuals - Productivity: - φ_I < 1 = φ_S = φ_R - Infected maximize: - Max U_{I j w t} = u(C_{I j w t},N_{I j w t}) + β_j { (1−m_{j t+1}) [ (1−π_{R j} − π_{D j t}) U_{I j w t+1} + π_{R j} U_{R j w t+1} ] + m_{j t+1} [ (1−π_{R j} − π_{D j t}) U_{I j k t+1} + π_{R j} U_{R j k t+1} ] } - Budget constraint: - W_t φ_I N_{I j w t} = C_{I j w t} + Γ_{w t} - First-order condition with respect to N_{I j w t} (after substituting (9) into (8)): - φ_I W_t / (φ_I W_t N_{I j w t} − Γ_{w t}) − θ N_{I j w t} = 0 ### Recovered individuals - Recovered maximize: - Max U_{R j w t} = u(C_{R j w t},N_{R j w t}) + β_j [ (1−m_{j t+1}) U_{R j w t+1} + m_{j t+1} U_{R j k t+1} ] - Budget constraint: - W_t N_{R j w t} = C_{R j w t} + Γ_{w t} - First-order condition with respect to N_{R j w t} (after substituting (11) into (10)): - W_t / (W_t N_{I j w t} − Γ_{w t}) − θ N_{I j w t} = 0 *Source: wpiea2020277-print-pdf - 2.2 Agents’ optimization problems* ### 2.3 Government’s optimization problem ### 2.3 Government’s optimization problem ### Externalities and objective - Three sources of externalities make the decentralized equilibrium inefficient: - Infected individuals’ consumption and working decisions increase the probability of infection of susceptible agents. - Low-risk susceptible individuals (lower mortality rate) have stronger incentives to work and consume than high-risk susceptible individuals (higher mortality rate); the former’s decisions affect the latter. - Large numbers of infections congest the healthcare system, increasing the mortality rate. - Government objective: - Utilitarian social welfare function with equal weights across all agents. - At time t = 1, with D1 = R1 = 0, welfare: - W1 = (Sy1 U_Sy1 + Iy1 U_Iy1) + (Sot1 U_Sot1 + Iot1 U_Iot1) - Policy instruments: - Containment policy {mjt}∞t=1 ∈ [0,1], differentiated by risk profile j. - Lump-sum taxes on workers {Γwt}∞t=1 to finance lump-sum transfers to contained agents {Γkt}∞t=1. - Government takes individual optimal responses and epidemiological dynamics into account. - Government budget constraint: - Γkt (Sok t + Syk t + Iok t + Iyk t + Rok t + Ryk t) + Γwt (Sow t + Syw t + Iow t + Iyw t + Row t + Ryw t) = 0 ### Parametric approximation of policy paths - Containment path approximated by generalized logistic functions (following Glover et al. (2020)): - mjt = γj0 / [1 + exp(−γj1 (t − γj2))] - γj0 controls level of containment at t = 0 - γj1 governs when containment is reduced - γj2 commands swiftness of reduction - As t → ∞, mjt → 0 - Transfer path Γkt approximated by scaled generalized logistic: - Γkt = φ0 · λ / [1 + exp(−φ1 (t − φ2))] - Γwt determined by the government budget constraint and equation (15) - Optimization chooses parameters {γy0, γy1, γy2, γo0, γo1, γo2, φ0, φ1, φ2} - Optimal policy paths {Γkt, myt, mot}250t=1 obtained by evaluating parametric functions at optimal parameters - Homogeneous policy: set γyi = γoi for i ∈ {0,1,2} ### Numerical approach - Parametric approach reduces number of optimization arguments from 750 to 9 - Non-parametric approach would require computing 250 containment policies for each type and 250 transfers and is numerically unstable - Glover et al. (2020) report similar results for parametric and non-parametric approaches --- ### Model calibration and alternative scenarios - Calibration to the US; parameters reported in Table 1 (selected values reproduced below): - Shares and mortality: - Low-risk individuals (age 0–54): y = 0.71 of population; mortality pDy = 0.002 (0.2 percent) - High-risk individuals (age 55+): ot = 0.29 of population; mortality pDot = 0.047 (4.7 percent) - Share of retirees within high-risk: or = 0.48 (47.9 percent of 55+) - Epidemiological timing: - After 18 days infected person either recovers or dies → πDJ + πRJ = 7/18 - Recovery rates: Ry = 7/18 − πDy ; Rot = 7/18 − πDot - Infection sensitivity parameters (authors’ calculations): - πS1 = 8.4792×10^−8 (consumption) - πS2 = 1.6376×10^−4 (work) - πS3 = 0.1293 (other) - Contact rates (Prem et al. (2017)): - Ψy = χyy + χyo = 4.3 - Ψo = χoo + χoy = 1.1 - χy = 3.3/(3.3 + 1.0) - χo = 0.2/(0.9 + 0.2) - Economic parameters: - Discount factor low-risk βy = 0.96^(1/52) (reported as 0.9992) - Non-COVID-19 mortality for high-risk ι = 0.05 → βot = βy · (1 − ι)^(1/52) (reported as 0.9982) - Disutility of labor θ = 0.001275 - Productivity parameters: A = 39.835; φI = 0.8; φS, φR = 1 - Hours per week N = 28 - Salary per week W = 58,000/52 - Retirement per week Γr = 0.744W - Targets for πS1, πS2, πS3: - (i) 16 percent of infections from consumption activities - (ii) 17 percent from work activities - (iii) 60 percent of population either recovers or dies by end of pandemic - Scenarios analyzed: - Targeted policy: containment heterogeneous across low- and high-risk individuals - Blanket policy: identical containment across low- and high-risk individuals - No-policy: no containment - Optimal containment computed over 250 weeks for low- and high-risk individuals --- ### Key quantitative results — containment and epidemiology - Optimal containment bans from working: - Blanket policy: 33.9 percent of people (percent excludes retirees) - Targeted policy: 38.9 percent of people; restrictions held about a month and a half longer than blanket policy - Under targeted policy containment composition: - Almost all high-risk individuals contained - 27.1 percent of low-risk individuals contained - Infection peaks (percent of initial population, week noted where applicable): - No-policy: infections peak during week 27 at 5.7 percent - Blanket policy: infections peak at 3.8 percent - Targeted policy: infections peak at 3.9 percent (a month and a half after no-policy peak) - Deaths (percent of initial population): - No-policy: deaths account for 0.55 percent - Blanket policy: deaths equal 0.40 percent - Targeted policy: deaths equal 0.34 percent - Risk-group-specific epidemiology: - High-risk infected peak: - Blanket: 0.34 percent (of initial population) - Targeted: 0.15 percent - High-risk deaths (share of initial population): - Blanket: 0.25 percent - Targeted: 0.11 percent - Aggregate lives saved (250 weeks, US 2019 population basis): - Table 2 death counts after 250 weeks: - Low-risk: - Blanket: 475,687 - Targeted: 478,110 - Difference: −2,423 (targeted loses 2,423 low-risk lives relative to blanket) - High-risk: - Blanket: 813,056 - Targeted: 643,044 - Difference: 170,012 (targeted saves 170,012 high-risk lives) - Total: - Blanket: 1,288,743 - Targeted: 1,121,154 - Difference: 167,589 (targeted saves 167,589 total lives relative to blanket) - Interpretation: - Targeted policy substantially reduces infections and deaths among high-risk individuals while producing slightly worse outcomes for low-risk individuals. - Targeted policy saves about 167,589 lives relative to a blanket policy in the calibrated U.S. population example (250 weeks horizon). --- ### Macroeconomic impacts - Consumption dynamics (percent deviation from steady state): - No-policy: recession driven by voluntary self-distancing; consumption trough coincides with infection peak. - Blanket policy: immediate abrupt plunge; consumption trough is a 33 percent collapse. - Targeted policy: immediate abrupt plunge; consumption trough is a 31 percent collapse; consumption remains subdued longer. - Reasons targeted policy generates larger recession: - Larger share of people contained for longer (optimal containment ~60 weeks vs 45 weeks under blanket). - Herd immunity achieved earlier under blanket policy, especially among high-risk individuals. - Hours worked: - Patterns resemble consumption due to linear production function and transfers to contained individuals. - Deviations smaller for hours worked than for consumption because retirees continue consuming out of pension income. - Behavior of non-contained individuals: - Under blanket policy: low-risk individuals work less when infections rise; consumption falls and then recovers as infections decline. - Under targeted policy: low-risk non-contained individuals initially work more (high-risk are banned), and try to offset taxes to sustain consumption. - Aggregate consumption and hours of non-contained individuals collapse in targeted scenario because almost all contained individuals are banned from working. - Welfare: - Present value of social welfare highest under targeted policy despite a more prolonged recession. - Government prefers targeted containment because reduction in deaths (and increase in leisure) outweighs larger consumption losses. --- ### Extensions — increased interactions and household structure - Concern: government-imposed restrictions (e.g., banning high-risk from working) could increase low–high-risk contacts at home, raising exposure for high-risk individuals. - Household composition heterogeneity matters: higher share of low–high-risk cohabitation could reduce benefits of targeted containment. - The paper reassesses targeted benefits by (i) raising contact rates and (ii) increasing the share of contacts between low- and high-risk individuals (results in next section). *Source: wpiea2020277-print-pdf - 2.3 Government’s optimization problem* ### 5.1 Higher contact rates ### 5.1 Higher contact rates ### Setup - Each group of agents increases its average contact rate by 10 percent relative to the initial settings. - New total contact rates: Ψy = 4.73 and Ψo = 1.21. - Assortativity of social interactions is maintained; parameters χj calibrated as in Table 1. - Alternative exercise (section 5.2) sets χj = 0.71 so that the share of contacts of high-risk individuals with low-risk individuals increases from 0.24 to 0.71 and the share of contacts of low-risk individuals with high-risk individuals increases from 0.18 to 0.71. ### Optimal containment outcomes (central findings) - Blanket policy: - 45.1 percent of the population gets banned from working. - A 10 percent increase in contact rates prompts the government to ban an additional 11.2 percent of people from working relative to the lower-contact setting. - Targeted policy: - Optimal containment bans 51.7 percent of the population from working, about 12.8 percentage points higher than in the lower-contact setting. - Containment applies to all high-risk individuals and extends to 42.3 percent of low-risk individuals, which is 15.2 percentage points more than with lower contact rates. - The targeted containment becomes less targeted with increased contact rates, reducing the benefits of separating low-risk from high-risk individuals. ### Epidemiological outcomes - Qualitative dynamics follow previous figures (Figure 3 and 4); targeted containment continues to save more high-risk lives than blanket containment. - Appendix A quantitative results (higher contact rates): - Infection peaks at 4.4 and 4.5 percent of initial population under a blanket policy and a targeted one, respectively. - Death rates: 0.44 percent of initial population under a blanket policy and 0.38 percent of initial population under a targeted policy. - More recovered individuals under the blanket policy; more susceptible individuals under the targeted policy. - Targeted containment saves about 167,000 more lives than a blanket one (aggregate result reported in Conclusions). ### Macroeconomic outcomes - Aggregate dynamics under blanket and targeted policies become extremely similar with higher contact rates. - Distributional differences by risk-type are large: - Under targeted policy, initial drop in consumption and hours worked reflects containment of all high-risk individuals and 42.3 percent of low-risk individuals. - Under blanket policy, containment applied to low-risk individuals is higher than under targeted policy, while containment applied to high-risk individuals is lower. - Non-contained individuals under the targeted policy are mainly low-risk and have less incentive to voluntarily reduce labor supply, but their reduction in labor supply is larger under the blanket policy. - Recovery in hours worked and consumption reflects infection evolution and lifting of containment; under targeted policy the recession lasts longer because containment is held for longer and herd immunity is achieved later. ### Effect of household structure and contact shares (section 5.2) - Increasing the share of contacts between low- and high-risk individuals (χj = 0.71) while holding total contacts constant reduces within-group contacts. - A sufficiently high share of cross-risk contacts makes the optimal targeted approach equivalent to the optimal blanket approach. - Intuition: work bans contain infections of targeted individuals but cannot prevent cross-risk contamination within households; hence benefits of targeting vanish when cross-risk household contacts are high. ### Policy implications (summary) - Targeted containment can substantially reduce deaths among high-risk individuals and overall mortality (saves about 167,000 more lives versus blanket), but it typically requires containing a larger share of the population and sustaining containment longer, which deepens and prolongs the recession. - The effectiveness and advantage of targeted containment diminish as: - Aggregate contact rates rise (targeted containment becomes less targeted). - Household interactions increase or the share of contacts between low- and high-risk individuals rises (cross-risk household contamination undermines targeting and can make targeted and blanket policies equivalent). *Source: wpiea2020277-print-pdf - 5.1 Higher contact rates* --- _Source: https://www.imf.org/-/media/files/publications/wp/2020/english/wpiea2020277-print-pdf.pdf_