## 1. Known recovered unconstrained agent’s problem ## Source details **Canonical URL:** [1. Known recovered unconstrained agent’s problem](https://www.imf.org/-/media/files/publications/wp/2020/english/wpiea2020188-print-pdf.pdf) ## Other formats - [Markdown version](/-/media/files/publications/wp/2020/english/wpiea2020188-print-pdf.pdf.md) - [Structured JSON version](/-/media/files/publications/wp/2020/english/wpiea2020188-print-pdf.pdf.json) --- ### Stationarity and information - Recovered agents know they have developed immunity through: - recovery from sickness; - testing positive in a viral test while asymptomatic; - testing positive in an antibody test. - Once knowledge of immunity is acquired, agents are free to optimally adjust time allocation without fear of infection. - Because the only time-varying features of the model relate to the pandemic’s progress, a known recovered agent’s value function is stationary. ### Flow utility and technology - Flow utility: - U(c, Ll, Lh) = u(c) + h(Ll) + uh × Lh - c is consumption of the numeraire good. - Ll is time spent in leisure activities. - Lh is time spent at home. - u(·) and h(·) are continuous, twice differentiable, strictly increasing concave functions. - uh > 0 is a positive parameter indicating the flow benefit of staying home (can be interpreted as home production). - Agents: - have no savings and no wealth endowments. - face perfectly competitive labor markets. - Production and time endowment: - Firms produce consumption goods with a linear production function using hours worked as the sole input. - c = w Lw, where w is the equilibrium wage rate and Lw is time allocated to work. - Total time endowment per period is L, so Lw + Ll + Lh = L and Li ≥ 0 for all i in {l, w, h}. ### Known recovered agent’s value function - Value function (stationary): - R_K = [ u(w×Lw^K) + h(Ll^K) + uh × ( L − Lw^K − Ll^K ) ] / (1−β) (2) 11 ### First-order conditions and interior solution - First-order conditions from optimal time allocation: - (Lw^K): w u′(w × Lw^K) − uh = 0 - (Ll^K): h′(Ll^K) − uh = 0 - Assuming u(·) and h(·) are invertible and parameters satisfy an interior solution, optimal allocations are: - Lw^K = 1/w u′−1 ( uh / w ) - Ll^K = h′−1 ( uh ) - Lh^K = L − 1/w u′−1 ( uh / w ) − h′−1 ( uh ) (3) *Source: wpiea2020188-print-pdf - 1. Known recovered unconstrained agent’s problem* ### 1. Known recovered unconstrained agent’s problem. . . . . . . . . . . . . . 10 ### 1. Known recovered unconstrained agent’s problem. ### Major sections (by heading and page) - 1. Known recovered unconstrained agent’s problem. . . . . . . . . . . . . . . . . 10 - 2. Potentially susceptible unconstrained agent’s problem. . . . . . . . . . . . . 11 - C. Time-constrained agents. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 - D. Equilibrium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 - IV. Quantitative Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 - A. Taking the model to the data. . . . . . . . . . . . . . . . . . . . . . . . . . 19 - B. Parameter values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 - V. Quantitative Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 - A. Testing and its effects on the epidemic and the economy. . . . . . . . . 21 - B. What if the proportion of asymptomatic cases was different?. . . . . . 23 - C. Effects of the epidemic for different agents. . . . . . . . . . . . . . . . . 24 - D. An economy with a larger fraction of time-constrained agents. . . . . . 25 - E. Testing when the fraction of time-constrained agents is larger. . . . . 26 - F. What if mortality is higher?. . . . . . . . . . . . . . . . . . . . . . . . . 27 - G. Government-enforced lockdown. . . . . . . . . . . . . . . . . . . . . . . . 28 - H. Effectiveness of lockdowns when health systems may collapse. . . . . . 30 - VI. Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ### Tables (listed) - 1. COVID-19 Cases in NYC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 - 2. Calibrated Parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ### Figures (listed) - 1. Change in the Number of Dine-in Customers: Open Table.com. . . . . . . . . 5 - 2. Viral Testing: Rich vs. low-income economies. . . . . . . . . . . . . . . . . . 5 - 3. Movements across state spaces – Calibrated Model. . . . . . . . . . . . . . . 33 - 4. Pandemic’s pattern: With and without testing. . . . . . . . . . . . . . . . . . . 35 - 5. Pandemic’s pattern: Testing only sick agents. . . . . . . . . . . . . . . . . . 36 - 6. Pandemic’s pattern: Different levels of asymptomatic. . . . . . . . . . . . . 37 - 7. Impact of testing given levels of asymptomatic. . . . . . . . . . . . . . . . . . 37 - 8. Pandemic’s pattern: Unconstrained vs. time-constrained agents - Benchmark. . . . 38 - 9. Pandemic’s pattern: Different levels of time-constrained agents. . . . . . . . . 38 - 10. Pandemic’s pattern: Unconstrained vs. time-constrained agents - 50 percent con- strained. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 - 11. Pandemic’s pattern: With and without testing - 50 percent time-constrained. . . 39 - 12. Pandemic’s pattern: High vs. low mortality rates. . . . . . . . . . . . . . . . . 40 - 13. Hours worked by susceptible unconstrained workers. . . . . . . . . . . . . . . . 40 - 14. Pandemic’s pattern: Effect of lockdowns. . . . . . . . . . . . . . . . . . . . . 41 - 15. Hours worked by susceptible unconstrained workers. . . . . . . . . . . . . . . . 41 - 16. Pandemic’s pattern: Effect of lockdowns - 50 percent time-constrained. . . . . 42 - 17. Hours worked by susceptible unconstrained workers. . . . . . . . . . . . . . . . 42 - 18. Pandemic’s pattern: Effect of lockdowns – exhaustible ICU beds. . . . . . . . 43 ### Document emphasis (from section and figure titles) - The document contains model-based formulation and analysis of agents categorized as: known recovered unconstrained, potentially susceptible unconstrained, and time-constrained. - It develops equilibrium conditions and takes the model to the data (Quantitative Analysis), including specified parameter values. - Quantitative Results focus on: - Testing and its effects on both the epidemic and the economy. - Sensitivity to the proportion of asymptomatic cases. - Heterogeneous effects across agent types (unconstrained vs. time-constrained). - Scenarios varying the fraction of time-constrained agents, mortality rates, and government-enforced lockdowns. - The role of health system capacity (exhaustible ICU beds) in lockdown effectiveness. - Empirical elements include COVID-19 case data for NYC and calibrated parameters; figures track pandemic patterns, testing impacts, hours worked, and mobility across state spaces. *Source: wpiea2020188-print-pdf (IMF working paper PDF), table of contents and lists of tables and figures as provided.* ### 19. Hours worked by susceptible unconstrained workers. . . . . . . . . . . . . . . . . 43 ### 19. Hours worked by susceptible unconstrained workers. . . . . . . . . . . . . . . . . . . . . 43 ### Abstract and high-level findings - The paper introduces two types of agent heterogeneity in a calibrated epidemiological search model: (i) agents who cannot afford staying home (time-constrained agents) and (ii) heterogeneity in developing symptoms (asymptomatic vs symptomatic). - Poor (time-constrained) agents bear most of the epidemic’s health costs; having more agents who do not change behavior during the pandemic can lead to a deeper recession. - Diseases with higher share of asymptomatic cases, even if less lethal, lead to worse health and economic outcomes because asymptomatic and pre-symptomatic agents spend more time outside infecting others. - Viral and antibody testing are particularly beneficial in economies with larger shares of poor (time-constrained) agents; lockdowns lose effectiveness when part of the population takes precautions independent of government actions. ### Quantitative results and key statistics - If 80 percent of the infections are asymptomatic, increasing testing from 0 to 10 percent represents a cumulative gain of around 6.3 percent of GDP after two years. - Calibration (emerging market economy, absent health system collapse): among time-constrained agents - 85 percent would be infected at some point. - 0.18 percent would die. - In contrast, among unconstrained agents: 21 percent infected; less than 0.05 percent dead. - Economic sensitivity to time-constraint share: - An increase of 20 percentage points in the share of time-constrained agents increases GDP losses by 3.7 percentage points. - Testing impacts in an economy with 50 percent time-constrained agents: - Moving from 0 percent to 10 percent incidence of tests implies a reduction of 90,000 in the number of dead from the disease. - Compared to an economy with no testing, a 10 percent testing rate increases GDP by 2.8 percentage points. - Lockdowns (benchmark calibration): - Introduced lockdowns reduce the number of deaths due to COVID-19 by 11,500 in the first year and fewer than 5,000 in the second year. - GDP growth falls by 5 percentage points in the first year compared to the no-lockdown benchmark, with no difference in the second year. - Cost per life saved (benchmark): $26 million for a comprehensive short lockdown; $54 million for a mild long lockdown. - Extreme cases and developing-economy comparisons: - If all agents are time-constrained, a mild and long lockdown would save 325,000 lives at a cost of $2.5 million per life. - Calibration closer to an average emerging economy (around 50 percent time-constrained agents) yields a cost per life of around $14 million (this figure does not include the costs of preventing time-constrained agents from going to work nor required transfers). ### Mechanisms and model structure (selected elements preserved exactly) - Two agent types: unconstrained agents can optimally allocate weekly time endowment L across work (Lw), leisure (Ll), and time at home (Lh); time-constrained agents have very limited control and either allocate time as if no pandemic or spend all time at home if sick or quarantined. Fraction of time-constrained agents is θ. - Agent health states: healthy but susceptible (h), infected (i), sick (s), recovered (r), or dead (d). Transitions across states are not necessarily observed by agents. - Infection arrival rate (as written in the source): - λt(Lwt, Llt) = mw(THwt, TIwt) THwt Lwt + ml(THwt, TIw) THlt Llt. - THwt and THlt are the aggregate times in period t where agents healthy but susceptible are at work and in leisure activities respectively. TIwt and TIlt are the aggregated times where infected agents are at work and in leisure activities in period t. mw(THwt, TIwt) and ml(THlt, TIlt) are the infection matching functions for labor markets and leisure activities, respectively. λt(Lwt, Llt) is strictly increasing in both arguments. - Infected agents may be asymptomatic or symptomatic. Arrival rate of the asymptomatic shock is γ. Infected agents who are not asymptomatic become sick with probability μ. Sick agents face a flow disutility of c and die at rate δt, with δt = δ(St ICUt) and δ′(St ICUt) ≥ 0 (Poisson arrival rate weakly increasing in ratio of sick agents to ICU units). - Asymptomatic infected agents recover at rate ˜θA. Infected agents are tested with probability τIt; antibody tests arrive with probability τRt. Positive viral tests force infected agents to stay at home (reducing infection); positive antibody tests reveal acquired immunity, allowing unconstrained agents to optimally adjust time allocation and boost labor supply. ### Policy analysis and recommendations - Testing (viral and antibody) is necessary to reduce both the spread of the disease and the economic cost of the epidemic; its importance rises with the share of asymptomatic infections and the share of time-constrained agents. - Viral testing reduces infection by quarantining asymptomatic and pre-symptomatic agents and lowers casualties while allowing unconstrained agents to increase labor supply. - Antibody testing does not materially affect virus dynamics but boosts economic activity by informing recovered-asymptomatic agents of immunity, enabling more labor supply. - Lockdowns alone (without further measures) delay progression of the virus but are not an effective mechanism to reduce human and economic costs in many contexts, particularly where unconstrained agents already optimally reduce exposure. Lockdowns can be more useful in developing economies with high shares of time-constrained agents, but their cost-effectiveness depends strongly on the share of time-constrained agents and on transfers/support needed to make lockdowns viable. *Source: "Information and Inequality in the Time of a Pandemic", June 2020 (excerpt).* ### 1. Known recovered unconstrained agent’s problem ### 1. Known recovered unconstrained agent’s problem ### Stationarity and information - Recovered agents know they have developed immunity through: - recovery from sickness; - testing positive in a viral test while asymptomatic; - testing positive in an antibody test. - Once knowledge of immunity is acquired, agents are free to optimally adjust time allocation without fear of infection. - Because the only time-varying features of the model relate to the pandemic’s progress, a known recovered agent’s value function is stationary. ### Flow utility and technology - Flow utility: - U(c, Ll, Lh) = u(c) + h(Ll) + uh × Lh - c is consumption of the numeraire good. - Ll is time spent in leisure activities. - Lh is time spent at home. - u(·) and h(·) are continuous, twice differentiable, strictly increasing concave functions. - uh > 0 is a positive parameter indicating the flow benefit of staying home (can be interpreted as home production). - Agents: - have no savings and no wealth endowments. - face perfectly competitive labor markets. - Production and time endowment: - Firms produce consumption goods with a linear production function using hours worked as the sole input. - c = w Lw, where w is the equilibrium wage rate and Lw is time allocated to work. - Total time endowment per period is L, so Lw + Ll + Lh = L and Li ≥ 0 for all i in {l, w, h}. ### Known recovered agent’s value function - Value function (stationary): - R_K = [ u(w×Lw^K) + h(Ll^K) + uh × ( L − Lw^K − Ll^K ) ] / (1−β) (2) 11 ### First-order conditions and interior solution - First-order conditions from optimal time allocation: - (Lw^K): w u′(w × Lw^K) − uh = 0 - (Ll^K): h′(Ll^K) − uh = 0 - Assuming u(·) and h(·) are invertible and parameters satisfy an interior solution, optimal allocations are: - Lw^K = 1/w u′−1 ( uh / w ) - Ll^K = h′−1 ( uh ) - Lh^K = L − 1/w u′−1 ( uh / w ) − h′−1 ( uh ) (3) *Source: wpiea2020188-print-pdf - 1. Known recovered unconstrained agent’s problem* ### 2. Potentially susceptible unconstrained agent’s problem ### 2. Potentially susceptible unconstrained agent’s problem ### Bellman equations and state definitions - Healthy but susceptible agent Bellman equation at time t (equation (4)): - H_t(L_wt,L_lt,L_ht) = u(wL_wt) + h(L_lt) + u_h × L_ht + β{ λ_t(L_st,L_lt)[ γ I_AU_{t+1}(...) + (1−γ) I_SU_{t+1}(...) ] + [1−λ_t(L_st,L_lt)] H_{t+1}(...) } - Key definitions: - β is the discount rate. - λ_t(L_st,L_lt) is the arrival rate of infection. - γ is the probability of an asymptomatic infection. - I_AU_{t+1} and I_SU_{t+1} are Bellman equations for infected asymptomatic and eventually symptomatic agents, with subscript U indicating infection not yet known. - Assumption: susceptible agents are not being tested for viral genetic material or antibodies (arrival rate of test results = zero for healthy but susceptible agents). - Infected asymptomatic without viral test (equation (5)): - I_AU_t(L_wt,L_lt,L_ht) = u(wL_wt) + h(L_lt) + u_h × L_ht + β{ τ_I_t I_AK_{t+1} + ˜θ_AU R_U_{t+1}(...) + [1−τ_I_t − ˜θ_AU] I_AU_{t+1}(...) } - τ_I_t is the arrival rate of viral tests. - I_AK_{t+1} is Bellman for asymptomatic with positive viral test. - ˜θ_AU is arrival rate of recovery. - R_U_{t+1} is Bellman for recovered with unknown immunity. - Note: I_AK_{t+1} is not a function of time allocation because agents with a positive viral test are required to stay home. - Asymptomatic with positive viral test (equation (6)): - I_AK_t = u_h × L + β{ ˜θ_AK R_K(L_wK,L_lK,L_hK) + (1−˜θ_AK) I_AK_{t+1} } - ˜θ_AK is the arrival rate of recovery. - R_K is Bellman for recovered with known immunity (equation (2)). - This Bellman equation is stationary ⇒ I_AK_t = I_AK for every t. - Eventually symptomatic without viral test (equation (7)): - I_SU_t(L_wt,L_lt,L_ht) = u(wL_wt) + h(L_lt) + u_h × L_ht + β{ τ_I_t I_SK_{t+1} + μ_U S_{t+1} + [1−τ_I_t − μ] I_SU_{t+1}(...) } - μ_U is the arrival rate of symptoms. - I_SK_{t+1} and S_{t+1} are not functions of time allocation (quarantine required if test positive or symptomatic). - Eventually symptomatic with positive viral test (equation (8)): - I_SK_t = u_h × L + β{ μ_K S_{t+1} + (1−μ_K) I_SK_{t+1} } - μ_K is the arrival rate of symptoms. - I_SK_t is non-stationary because S_{t+1} depends on time. - Sick agent Bellman equation (equation (9)): - S_t = u_h × L − c + β{ δ_t (−c_D) + θ_S_t R_K(L_wK,L_lK,L_hK) + [1−δ_t − θ_S_t] S_{t+1} } - c is the flow cost of sickness. - δ_t is the death arrival rate. - c_D is the one-time cost of death before exiting the environment. - θ_S_t is the arrival rate of recovery. - S_t is non-stationary because δ_t and θ_S_t vary over time due to over-utilization of health services. - Recovered with unknown immunity (equation (10)): - R_U_t(L_wt,L_lt,L_ht) = u(wL_wt) + h(L_lt) + u_h × L_ht + β{ τ_R_t R_K(L_wK,L_lK,L_hK) + [1−τ_R_t] R_U_{t+1}(...) } - τ_R_t is the arrival rate of antibody tests. ### Optimal time allocation for potentially susceptible unconstrained agents - Agents in states H_t, I_AU_t, I_SU_t, and R_U_t face the same time allocation problem because infection is unobservable. - Time allocation problem (equations (11) and (12)): - max_{L_wt,L_lt,L_ht} π_Ht H_t + π_IAt I_AU_t + π_ISt I_SU_t + π_Rt R_U_t - subject to L_wt + L_lt + L_ht = L - π_Ht, π_IAt, π_ISt, π_Rt are beliefs about being in each state (formed on aggregate measures). - Define λ_t = λ_t(L_w*_t, L_l*_t) where L_w*_t, L_l*_t, L_h*_t are optimal allocations for potentially susceptible agents. ### Laws of motion for unconstrained agents (LM1–LM9) - Denote h_t as measure of unconstrained healthy but susceptible agents. - h_{t+1} − h_t = −λ_t h_t (LM1) - Since recovered agents obtain immunity, h_t declines only at infection rate λ_t. - Asymptomatic infected without positive test (i_AU_t): - i_AU_{t+1} − i_AU_t = λ_t γ h_t − (˜θ_AU + τ_I_t) i_AU_t (LM2) - γ is share of infections that stay asymptomatic. - ˜θ_AU is recovery rate for untested asymptomatics. - τ_I_t is viral testing rate in period t. - Asymptomatic infected with positive viral test (i_AK_t): - i_AK_{t+1} − i_AK_t = τ_I_t i_AU_t − ˜θ_AU i_AU_t (LM3) - τ_I_t is viral testing rate; ˜θ_AK is rate at which asymptomatic tested agents recover (notation in text). - Eventually symptomatic untested (i_SU_t): - i_SU_{t+1} − i_SU_t = λ_t (1−γ) h_t − (μ_U + τ_I_t) i_SU_t (LM4) - μ_U is rate untested eventually symptomatic agents become sick. - Eventually symptomatic tested pre-symptomatic (i_SK_t): - i_SK_{t+1} − i_SK_t = τ_I_t i_SU_t − μ_K i_SK_t (LM5) - μ_K is rate tested eventually symptomatic agents become sick. - Sick unconstrained agents (s_t): - s_{t+1} − s_t = μ_U i_SU_t + μ_K i_SK_t − (δ_t + θ_S_t) s_t (LM6) - δ_t death rate; θ_S_t recovery rate. - Recovered unconstrained with unknown immunity (r_U_t): - r_U_{t+1} − r_U_t = ˜θ_AU i_AU_t − τ_R_t r_U_t (LM7) - τ_R_t is antibody testing rate. - Dead unconstrained agents (d_t): - d_{t+1} − d_t = δ_t s_t (LM8) - Recovered with known immunity (r_K_t): - r_K_{t+1} − r_K_t = θ_S_t s_t + τ_R_t r_U_t (LM9) ### Time-constrained agents: behavior, Bellman equations, and laws of motion - Time-constrained agents (subscript c) cannot optimally adjust time allocation; they are assumed to follow choices of knowingly recovered agents (Section III.B.1). - L_wC,t = 1/w u'^{-1}(u_h / w) - L_lC,t = h'^{-1}(u_h) - L_hC,t = L − 1/w u'^{-1}(u_h / w) − h'^{-1}(u_h) (equation (13)) - These allocations remain time dependent because agents can be forced to quarantine if sick or tested positive (in which case L_hC,t = L). - Infection rate for time-constrained agents λ_Ct (equation (14)): - λ_Ct(TH_wt, TI_wt) = m_w(TH_wt, TI_wt) TH_wt L_wC,t + m_l(TH_wt, TI_w) TH_lt L_lC,t - TH_wt, TI_wt, TH_lt, TI_lt are aggregated total hours by healthy susceptible and infected agents in work and leisure. - Note: λ_Ct ≥ λ_t since time-constrained agents do not account for infection risk. - Bellman equations for time-constrained agents mirror unconstrained counterparts with fixed time allocation: - Healthy susceptible time-constrained (equation (15)): - H_C,t = u(w×L_wC,t) + h(L_lC,t) + u_h × L_hC,t + β{ λ_Ct[ γ I_AU_{C,t+1} + (1−γ) I_SU_{C,t+1} ] + (1−λ_Ct) H_C,t+1 } - Infected asymptomatic untested time-constrained (equation (16)): - I_AU_{C,t} = u(w×L_wC,t) + h(L_lC,t) + u_h × L_hC,t + β{ τ_I_t I_AK_{C,t+1} + ˜θ_AU R_U_{C,t+1} + [1−τ_I_t − ˜θ_AU] I_AU_{C,t} } - Asymptomatic with positive viral test time-constrained (equation (17)): - I_AK_{C} = u(b) + u_h × L + β{ ˜θ_AK R_K + (1−˜θ_AK) I_AK_{C} } (stationary) - Eventually symptomatic untested time-constrained (equation (18)): - I_SU_{C,t} = u(w×L_wC,t) + h(L_lC,t) + u_h × L_hC,t + β{ τ_I_t I_SK_{C,t+1} + μ_U S_{C,t+1} + [1−τ_I_t − μ_U] I_SU_{C,t+1} } - Eventually symptomatic tested time-constrained (equation (19)): - I_SK_{C,t} = u(b) + u_h × L + β{ μ_K S_{C,t+1} + (1−μ_K) I_SK_{C,t+1} } (non-stationary via S_{C,t+1}) - Sick time-constrained (equation (20)): - S_{C,t} = u(b) + u_h × L − c_S + β{ δ_t (−c_D) + θ_S_t R_{K,C,t} + [1−δ_t − θ_S_t] S_{C,t+1} } - For time-constrained agents: R_U_{C,t} = R_K_{C,t} = R_K_{C} = R_K = [ u(w×L_wC,t) + h(L_lC,t) + u_h × L_hC,t ] / (1−β) (equation (21)). - Laws of motion for time-constrained agents (LM_C1 – LM_C9) mirror unconstrained ones with λ_t → λ_Ct: - h_{C,t+1} − h_{C,t} = −λ_Ct h_{C,t} (LM_C1) - i_AU_{C,t+1} − i_AU_{C,t} = λ_Ct γ h_{C,t} − (˜θ_AU + τ_I_t) i_AU_{C,t} (LM_C2) - i_AK_{C,t+1} − i_AK_{C,t} = τ_I_t i_AU_{C,t} − ˜θ_AU i_AU_{C,t} (LM_C3) - i_SU_{C,t+1} − i_SU_{C,t} = λ_Ct (1−γ) h_{C,t} − (μ_U + τ_I_t) i_SU_{C,t} (LM_C4) - i_SK_{C,t+1} − i_SK_{C,t} = τ_I_t i_SU_{C,t} − μ_K i_SK_{C,t} (LM_C5) - s_{C,t+1} − s_{C,t} = μ_U i_SU_{C,t} + μ_K i_SK_{C,t} − (δ_t + θ_S_t) s_{C,t} (LM_C6) - r_{U,C,t+1} − r_{U,C,t} = ˜θ_AU i_AU_{C,t} − τ_R_t r_{U,C,t} (LM_C7) - d_{C,t+1} − d_{C,t} = δ_t s_{C,t} (LM_C8) - r_{K,C,t+1} − r_{K,C,t} = θ_S_t s_{C,t} + τ_R_t r_{U,C,t} (LM_C9) ### Equilibrium definition and beliefs - A rational-expectations equilibrium with initial measure of agents n consists of a sequence {λ_t}_{t≥0} such that at every period: 1. Potentially susceptible unconstrained agents optimally solve time allocation (equations (11) and (12)). 2. Known recovered unconstrained agents optimally solve time allocation (equation (2)). 3. Laws of motion (LM1–LM9) and (LM_C1–LM_C9) generate the sequence {λ_t}_{t≥0}. - Rational expectations imply beliefs coincide with objective probabilities: - h_t / (h_t + i_AU_t + i_SU_t + r_U_t) = π_Ht (22) - i_AU_t / (h_t + i_AU_t + i_SU_t + r_U_t) = π_IA_t (23) - i_SU_t / (h_t + i_AU_t + i_SU_t + r_U_t) = π_IS_t (24) - r_U_t / (h_t + i_AU_t + i_SU_t + r_U_t) = π_Rt (25) - Solution method: time-stacking algorithm using sparsity of Jacobian blocks in a gradient-based method to solve equilibrium paths for t = 0,...,500. ### Calibration: functional forms and matching functions - Utility functional form imposed (equation (26)): - U(c,L_l,L_h) = α ln(1 + c) + (1−α) ln(1 + L_l) + u_h × L_h - Time allocation restrictions as in Section III (c = w L_w, L_w + L_l + L_h = L, L_i ≥ 0). - Matching functions (Cobb-Douglas): - Workplace infections (equation (27)): - m_w(TH_wt, TI_U_wt) = A_w (TH_wt)^ζ (TI_U_wt)^{1−ζ} - TH_wt = h_C,t L_wC,t + h_t L_wt (total hours at work by healthy susceptible agents). - TI_U_wt = (i_AU_t + i_SU_t) L_wt + (i_AU_{C,t} + i_SU_{C,t}) L_wC,t (total hours at work by infected untested agents). - A_w and ζ are parameters to calibrate. - Leisure infections (equation (28)): - m_l(TH_lt, TI_U_lt) = A_l (TH_lt)^χ (TI_U_lt)^{1−χ} - TH_lt = h_C,t L_lC,t + h_t L_lt. - TI_U_lt = (i_AU_t + i_SU_t) L_lt + (i_AU_{C,t} + i_SU_{C,t}) L_lC,t. - A_l and χ are parameters to calibrate. - Given constant returns to scale in matching functions, infection rate per unit time exposed: - m_s(TH_U_s_t, TI_U_s_t) / TH_s_t = A_s (TI_U_s_t / TH_s_t)^{1−ζ} = A_s (i_U_t / h_t)^{1−ζ} - m_l(TH_U_l_t, TI_U_l_t) / TH_l_t = A_l (TI_U_l_t / TH_l_t)^{1−χ} = A_l (i_U_t / h_t)^{1−χ} ### Parameter choices and calibration targets - Time period: one week. - Recovery or death average: 14 days ⇒ θ_S + δ = 7/14. - Incubation period parameter μ calibrated so incubation ≈ 10 days. - Death rate based on Fernández-Villaverde and Jones (2020), matching early serology tests in Heinsberg, Germany and Iceland; robustness reported using death rates of 1 percent. - Fraction asymptomatic = 50 percent (from Iceland voluntary screening). - Initial number infected e = 0.001. - Initial population and wage normalized to one. - Fraction time-constrained agents = 30 percent (initial calibration from Laxton and others (2010)); alternative values used for developing economies. - u_h and α set so steady-state time spent at home, work, and outside leisure matches American Time Use Survey. - Matching function parameters set so number of infected agents approaches zero after two years. ### Quantitative implication highlighted - Model predicts large health and economic benefits from increasing testing (transition to quantitative results in Section V). *Source: wpiea2020188-print-pdf - 2. Potentially susceptible unconstrained agent’s problem* ### 4. The solid line is for the economy without testing, the dashed line represents the economy ### 4. The solid line is for the economy without testing, the dashed line represents the economy ### Testing: overall effects and quantitative impacts - Benchmark testing: 10 percent of infected persons and the same percentage of persons who recovered without ever knowing they were infected are tested per week (adjust both τI and τR). - With this level of testing, by the end of the second year: - Deaths reduced by more than 60,000 compared to no-testing scenario (Figure4d). - Cumulative gain around 3.4 percent of GDP (Figure4a). - Partial antibody-testing policy (increase τR from 0 percent to about 5 percent): - Minimal effects on virus dynamics (Figures5b,5c,5d). - Cumulative output gain after two years: 0.1 percent of GDP. ### Two channels through which testing affects the economy - Stop spreading channel (affects epidemic dynamics): - Early testing removes infected agents from contact, reducing infection (see Figure4b), sick (Figure4c), and dead (Figure4d). - Mechanisms on economy: - Quarantining infected people reduces labor supply (negative). - Unconstrained agents increase labor supply due to reduced infection risk (positive). - Net effect: second mechanism larger, so testing is net positive to GDP. - Information channel (purely informational): - Testing reveals immune status of agents recovered without knowing they were infected, changing their labor supply choices without altering epidemic dynamics. - Antibody testing of unknowingly recovered agents improves GDP by raising labor supply of those agents. ### Sensitivity: proportion of asymptomatic cases (γ) - Benchmark γ calibrated from Iceland; robustness explored between 80 percent and 20 percent asymptomatic cases while holding expected death constant (adjust δ). - Key mechanisms when γ changes: - More asymptomatic → fewer visibly sick (raises utility of infected), but asymptomatic remain in contact longer (increases spread). - More unknowingly recovered agents behave as susceptible (prolongs economic costs). - Quantitative outcomes: - Higher fraction of asymptomatic cases → peak infections and total infected higher (Figure6b); overall deaths higher because more people infected (Figure6d). - Economic path: - After 10 weeks, recession is substantially deeper when more asymptomatic cases (agents restrict labor supply more). - Recovery: economy rebounds and forms more of a V-shaped recovery when asymptomatic fraction is higher; with fewer asymptomatic cases, recession milder but more prolonged. - Testing value by asymptomatic calibration (Figure7): - After two years, cumulative GDP gain from testing 10 percent of infected: - 1.8 percentage points when few asymptomatic cases. - around 6.3 percentage points when more asymptomatic individuals. ### Heterogeneity: unconstrained vs. time-constrained agents - Incidence and mortality by group (benchmark): - Less than 13 percent of unconstrained agents ever infected over two years. - Almost 60 percent of time-constrained agents ever infected over two years (Figure8a). - Deaths: 0.03 percent of unconstrained agents die vs. 0.12 percent of time-constrained agents (Figure8b). - Interpretation: time-constrained agents bear greatest health burden; unconstrained agents reduce outside exposure more, contributing to economic slowdown. ### Economy with larger fraction of time-constrained agents (50 percent vs. benchmark 30 percent) - Epidemic timing and scale: - Peak reached at 34 weeks (vs. 42 weeks benchmark). - Almost all infections occur in the first year. - Total infected peak at 2.9 percent of total population (vs. less than 1 percent benchmark) (Figure9b). - Around 85 percent of time-constrained agents get infected; 0.18 percent die (Figures10a and10b). - Just over 21 percent of unconstrained agents ever infected; less than 0.05 percent die. - Economic effects: - GDP bottoms at about -8.7 percent in benchmark vs. over -19 percent in alternative calibration. - Present value of cumulative losses about 3.7 percent of GDP for economy with more time-constrained agents. - Recovery is faster and more V-shaped in the economy with more time-constrained agents; after one year GDP per capita higher than benchmark despite larger initial fall. ### Testing in economies with larger share of time-constrained agents - Testing effectiveness increases with more time-constrained agents: - Testing reduces peak infections by almost 1.5 percentage points (vs. less than 0.5 percentage points in benchmark) (Figure11b). - Lives saved in the first year: around 30,000 in the higher time-constrained calibration; by end of epidemic, testing saves around 14,000 more lives in the benchmark calibration (Figure11d). - GDP troughs: without testing, about -19 percentage points below steady state; with testing, falls by 13 percentage points (Figure11a). - By end of first year, testing improves GDP by: - 2.8 percentage points in higher time-constrained calibration. - 1.9 percentage points in benchmark calibration. ### Sensitivity: higher mortality scenarios (case mortality rate) - Benchmarks and comparisons: - Initial case mortality thought as high as 3.4 percent; serology tests suggested as low as 0.3 percent; Fernández-Villaverde and Jones(2020) suggest between 0.8 percent and 1.2 percent. - Figure12 compares case mortality rate at 1 percent with benchmark δ calibration. - Epidemiologic effects: - Peak of infections lower when mortality is higher (Figure12a); epidemic paths diverge around week 37; peak at week 42 in both cases. - About 300,000 more people would die in the US after two years when mortality is higher (Figure12d). - Economic effects: - At trough, labor supply drops by more than 35 percent when mortality higher (vs. less than 15 percent in benchmark). - First-year GDP per capita falls by 12.5 percentage points when mortality is higher vs. 4.8 percentage points in benchmark. - Testing more effective in smoothing GDP contraction when mortality is higher. ### Government-enforced lockdowns: strategies, effects, and costs - Lockdown scenarios: - Lockdown A: short and comprehensive — all agents forced to shelter at home for no less than an extra 25 percent of their time for one month. - Lockdown B: mild and long — all agents mandated to shelter at home for no less than an extra 10 percent of their time for 26 weeks. - Lockdown start assumption: starts in the 11th week of the epidemic. - Epidemiologic impacts: - Lockdowns slow and delay epidemic in first year but have almost no effect by end of second year (Figure14b). - Lives saved in first year: 11,400 (Lockdown A) and 9,500 (Lockdown B). - By end of second year, difference less than 5,000 lives saved (Figure14d). - Economic impacts and cost-per-life (counterfactual = benchmark model where agents independently reduce outside time): - GDP growth in first year falls by 5.5 percentage points (Lockdown A) and 6.6 percentage points (Lockdown B) vs. contraction of around 4 percentage points in benchmark. - Cost per life: - Lockdown A: around $26 million. - Lockdown B: around $54 million. - Alternate counterfactual (agents do not change behavior at all): Lockdown B would save 325,000 lives at a cost of $2.5 million per life. - Lockdowns in poorer economies (more time-constrained agents): - Lockdowns more cost-effective: lockdown B saves almost 17,000 lives after first year. - Average GDP falls by around 10 percentage points in first year (vs. 8.6 percentage points if no lockdown). - Cost per life in lockdown B: around $14 million (less than half the cost per life when more unconstrained agents present). - Long lockdown B becomes ineffective for unconstrained agents after week 24 when unconstrained agents voluntarily stay home longer than mandated (Figure17). ### Effectiveness of lockdowns when health systems may collapse (exhaustible ICU beds) - Model extension: time-varying mortality δt = δ + ((Hot / ICUt)^2) / K, where: - δ is benchmark mortality. - Hot is number of hospitalizations. - ICUt is number of ICU beds. - K calibrated so probability of dying is four times larger than benchmark at epidemic peak. - Assumptions for hospitalizations and ICU: - 20 percent of sick people need to be hospitalized. - 5 percent of hospitalized require ICU treatment. - ICU beds: 34.7 beds per 100,000 inhabitants (source McCarthy(2020) / National Center of Biology Information). - Effects: - Lockdowns A and B reduce first-year deaths by 15,000 and 16,000, respectively (Figure18d). - Endogenous behavior: awareness of ICU constraints makes unconstrained agents more careful even without lockdowns, reducing marginal impact of lockdowns on lives saved. - Economic contractions with strained health systems: - Without lockdown: economy contracts by 5.6 percentage points. - Lockdown A: contracts by 6.5 percentage points. - Lockdown B: contracts by 7.7 percentage points (Figure18a). - Cost per life with strained systems: - Lockdown A: $13.4 million. - Lockdown B: $27.7 million. ### Conclusion: key qualitative findings and policy implications - Information is highly valuable during epidemics, especially when many infections are asymptomatic. - Heterogeneity matters: not everyone can forego income to avoid infection; time-constrained agents face higher health costs and poorer economies face larger health and economic impacts. - Testing has two distinct benefits: - Changes epidemic dynamics by removing infectious agents (stop spreading channel). - Improves economic outcomes by revealing immune status (information channel); this channel is particularly important with many asymptomatic cases. - Assessment of containment measures must use a realistic counterfactual that accounts for voluntary behavioral responses; lockdowns save fewer lives relative to a benchmark where agents already reduce outside activity. - Lockdowns and testing are more cost-effective in economies with larger shares of time-constrained agents; testing pays off more at epidemic onset in poorer economies. - Overall distributional insight: the poor (time-constrained) bear most health costs; the rich (unconstrained) drive much of the economic slowdown by voluntarily reducing outside activity. *Source: IMF working paper chapter text (figures and tables referenced within the unit).* ### REFERENCES ### REFERENCES ### Epidemiological models and dynamics - Acemoglu, Daron, Victor Chernozhukov, Iván Werning, and Michael D Whinston, 2020, “A multi-risk SIR model with optimally targeted lockdown,” Techn. rep., National Bureau of Economic Research. - Bai, Yan, Lingsheng Yao, Tao Wei, Fei Tian, Dong-Yan Jin, Lijuan Chen, and Meiyun Wang, 2020, “Presumed asymptomatic carrier transmission of COVID-19,”Jama, Vol. 323, No. 14, pp. 1406–1407. - Fernández-Villaverde, Jesús, and Charles I Jones, 2020, “Estimating and Simulating a SIRD Model of COVID-19 for Many Countries, States, and Cities,” Techn. rep., National Bureau of Economic Research. - 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REFERENCES* --- _Source: https://www.imf.org/-/media/files/publications/wp/2020/english/wpiea2020188-print-pdf.pdf_