## 3.1 Epidemic Model: A-SIR

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---

### Overview and key modeling choices
- Classic SIR modified:
  - Probabilities of being infected depend on economic activity (following Eichenbaum et al. (2020b)).
  - Infected split into symptomatic and asymptomatic; asymptomatic less infectious (following Prem et al. (2020)).
- Model integrates A-SIR epidemic block with a New Keynesian business-cycle framework with staggered pricing.
- Model does not take into account the possibility of a vaccine becoming available; a robustness check indicates main results remain broadly unchanged if mortality declines gradually after a year.

### Population groups
- Five types:
  - Susceptible: S_t
  - Infected asymptomatic: A_t
  - Infected symptomatic: I_t
  - Formerly asymptomatic recovered: V_t
  - Formerly symptomatic recovered: R_t
- Supposedly susceptible (behave identically): ˜S_t = S_t + A_t + V_t

### Infection transmission channels and probabilities
- Three channels:
  1. Consumption channel: depends on individual consumption c^S_t(i), aggregate consumption of symptomatic I_t c^I_t, and aggregate consumption of asymptomatic A_t c^A_t.
  2. Labor channel: depends on individual hours n^S_t(i) and aggregate hours of asymptomatic A_t n^A_t. Symptomatic either do not work or work remotely and do not transmit via labor.
  3. Other channels (schools, family meetings, etc.): depends on total infected (I_t + κ A_t) and lockdown variable τ_t.
- Key parameters:
  - κ: asymptomatic infectiousness relative to symptomatic, 0≤κ<1.
  - ζ: scaling of symptomatic infectiousness to allow for isolation, 0≤ζ≤1.
  - $c, $n > 0: constants controlling relative importance of consumption and labor channels.
- Infection probability for susceptible i:
  - $I,t(i) = $c c^S_t(i)(ζ I_t c^I_t + κ A_t c^A_t) + $n n^S_t(i) κ A_t n^A_t + $t (I_t + κ A_t)
- Probability that a supposedly susceptible becomes symptomatic infected:
  - ˜$I,t(i) = ρ $I,t(i) S_t / ˜S_t

### Exact discrete-time transition equations
- Susceptible:
  - S_{t+1} = (1 − $I,t) S_t
- Symptomatic infected:
  - I_{t+1} = (1 − $R) I_t + ρ $I,t S_t
- Asymptomatic infected:
  - A_{t+1} = (1 − $R) A_t + (1 − ρ) $I,t S_t
- Formerly asymptomatic recovered:
  - V_{t+1} = V_t + $R A_t
- Formerly symptomatic recovered:
  - R_{t+1} = R_t + ($R − $D,t) I_t
- Deceased:
  - D_{t+1} = D_t + $D,t I_t

### Behavioral implications and first-order conditions for supposedly susceptible
- Asymptomatic behave identically to susceptible and formerly asymptomatic recovered.
- Budget constraint:
  - (1 + τ_{c,t}) P_t ̃c_t(i) + ̃B_{t+1}(i) = (1 − τ_{n,t}) W_t ̃n_t(i) + I_{t−1} ̃B_t(i) + P_t Γ_t
- Recursive expected-utility problem:
  - ̃U_t(̃b_t(i)) = max_{̃c_t(i),̃n_t(i),̃B_{t+1}(i),̃$I,t(i)} [ log ̃c_t(i) + θ log(1 − ̃n_t(i)) + β(1 − ̃$I,t(i)) ̃U_{t+1}(̃b_{t+1}(i)) + β ̃$I,t(i) U_{I,t+1}(̃b_{t+1}(i)) ]
- First-order conditions (omitting index i):
  - 1 / ̃c_t = ̃λ_{S,t} (1 + τ_{c,t}) − ̃λ_{$,t} ρ S_t / ̃S_t $c (ζ I_t c^I_t + κ A_t ̃c_t)
  - θ / (1 − ̃n_t) = ̃λ_{S,t} (1 − τ_{n,t}) w_t + ̃λ_{$,t} ρ S_t / ̃S_t $n κ A_t ̃n_t
  - ̃λ_{$,t} = β [ U_{I,t+1}(̃b_{t+1}) − ̃U_{t+1}(̃b_{t+1}) ]
  - ̃λ_{S,t} = β [ (1 − ̃$I,t) ̃λ_{S,t+1} + ̃$I,t λ_{I,t+1} ] I_t / π_{t+1}
- Interpretation:
  - Last terms in consumption and labor FOCs internalize infection risk (discounted utility loss from infection times infection risk).
  - ̃λ_{$,t} equals discounted utility loss due to infection.
  - ̃λ_{S,t} follows an Euler equation incorporating probability of remaining supposedly susceptible.

### Symptomatic infected: constraints and FOCs
- Remote work productivity parameter: 0≤ξ≤1.
- Bond return adjusted to I_t / (1 − $D,t) to account for deaths.
- Budget constraint:
  - (1 + τ_{c,t}) P_t c^I_t(i) + B^I_{t+1}(i) = W_t ξ n^I_t(i) + I_{t−1} B^I_t(i) / (1 − $D,t−1) + P_t Γ_t
- Recursive utility:
  - U_{I,t}(b^I_t) = max_{c^I_t(i), n^I_t(i), B^I_{t+1}(i)} [ log c^I_t(i) + θ log(1 − n^I_t(i)) + β(1 − $R) U_{I,t+1}(b^I_{t+1}(i)) + β($R − $D,t) U_{R,t+1}(b^I_{t+1}(i)) + β $D,t U_D ]
- FOCs (omitting index i):
  - 1 / c^I_t = λ_{I,t} (1 + τ_{c,t})
  - θ / (1 − n^I_t) = ξ λ_{I,t} w_t
  - λ_{I,t} = β [ (1 − $R − $D,t) λ_{I,t+1} + $R λ_{R,t+1} ] I_t / (π_{t+1} (1 − $D,t))
- Interpretation:
  - Choices reflect reduced productivity ξ and adjusted intertemporal valuation due to recovery and death probabilities ($R and $D,t).

---

### 3.4 Symptomatic recovered individuals

### Preferences and budget
- Recovered not at risk of reinfection; choose consumption cR_t(i), labor nR_t(i), and bond holdings BR_{t+1}(i).
- Budget constraint:
  - (1 + τ_{c,t}) P_t cR_t(i) + BR_{t+1}(i) = (1 − τ_{n,t}) W_t nR_t(i) + I_{t−1} BR_t(i) + P_t Γ_t
- Recursive problem:
  - UR_t(bR_t(i)) = max_{cR_t(i), nR_t(i), BR_{t+1}(i)} [ log cR_t(i) + θ log(1 − nR_t(i)) + β UR_{t+1}(bR_{t+1}(i)) ]
- "Standard first-order conditions follow."

### Firms: structure and pricing
- Retail firms in perfect competition aggregate intermediate goods y_t(ι) into final good y_t at price P_t.
- Aggregation and demand:
  - y_t = [ ∫_{ι∈[0,1]} y_t(ι)^{(ε−1)/ε} dι ]^{ε/(ε−1)}
  - y_t(ι) = ( P_t(ι) / P_t )^{−ε} y_t
  - P_t = [ ∫_{ι∈[0,1]} P_t(ι)^{1−ε} dι ]^{1/(1−ε)}
- Intermediate firm technology:
  - y_t(ι) = Z n_t(ι) with Z > 0
  - Marginal cost: mc_t = w_t / Z
- Calvo pricing:
  - Firms adjust prices with probability 1 − δ and choose ̃P_t(ι) to maximize discounted profits:
    - max_{̃P_t(ι), {y_{t+j}(ι)}_{j≥0}} E_t ∑_{j=0}^∞ (β δ)^j Λ_{t,t+j} ( ̃P_t(ι)/P_{t+j} − mc_{t+j} ) y_{t+j}(ι)
  - Absent signal price remains unchanged P_{t+1}(ι) = P_t(ι).
  - Λ_{t,t+j} is a weighted average of marginal utility of consumption across household types.

### Government, central bank, and health-care system
- Government:
  - Uses τ_{c,t} and τ_{n,t} to restrict activity; revenue rebated to households; transfers firm profits to households.
  - Period-by-period budget balance:
    - τ_{c,t} P_t c_t + τ_{n,t} W_t n_t + P_t y_t − W_t n_t = ( ˜S_t + I_t + R_t ) Γ_t
- Lockdown operational rule:
  - τ_{c,t} = Φ_c I_t
  - τ_{n,t} = Φ_n I_t
  - Φ_c, Φ_n > 0
  - Lockdown reduces transmission via other channel:
    - ω_t = ω (1 − τ_{c,t})^{Φ_ω} with ω, Φ_ω > 0
- Central bank (Taylor-type) rule possibly responding to infections:
  - I_t / ̄I = ( π_t / ̄π )^{Φ_π} ( y_t / y^f_t )^{Φ_y} exp(I_t)^{Φ_I}
  - Φ_π, Φ_y, Φ_I ≥ 0; y^f_t denotes flexible-price output.
- Health-care system mortality rule:
  - $D,t = min[ (1 + I_t / ν_0) $D, ν_1 $D ] with ν_0, ν_1 > 0

### Market clearing and asset evolution
- Population partition: S_t, I_t, R_t.
- Final goods market clearing:
  - ∫_{i∈S_t} ̃c_t(i) di + ∫_{i∈I_t} cI_t(i) di + ∫_{i∈R_t} cR_t(i) di ≡ c_t = y_t
- Labor market clearing:
  - ∫_{i∈S_t} ̃n_t(i) di + ∫_{i∈I_t} nI_t(i) di + ∫_{i∈R_t} nR_t(i) di ≡ n_t = ∫_{ι∈[0,1]} n_t(ι) dι
- Aggregate production with price dispersion:
  - ∆_t y_t = z_t n_t
  - ∆_t = ∫_{ι∈[0,1]} ( P_t(ι) / P_t )^{−ε} dι
- Assets by agent type evolve:
  - ˜S_{t+1} ̃B_{t+1} = (1 − ˜$I,t) ∫_{i∈S_t} ̃B_{t+1}(i) di
  - I_{t+1} B_{I,t+1} = (1 − $R) ∫_{i∈I_t} B_{I,t+1}(i) di + ˜$I,t ∫_{i∈S_t} ̃B_{t+1}(i) di
  - R_{t+1} B_{R,t+1} = ∫_{i∈R_t} B_{R,t+1}(i) di + ($R − $D,t) ∫_{i∈I_t} B_{I,t+1}(i) di
- Bond market clearing:
  - ̃S_t ̃B_t + I_t B_{I,t} + R_t B_{R,t} = 0

---

### Calibration, simulations, and main quantitative findings

### Calibration highlights (pandemic and macro blocks)
- Pandemic block targets:
  - Each of the two economic channels (consumption and labor) accounts for one-sixth of transmission absent containment; about two-thirds of population infected before pandemic dies out.
  - Terminal share recovering or dying consistent with herd immunity levels of 60-70%.
  - Time to recover or die: 18 days (i.e., 7/18 periods in weekly model).
  - Infection fatality rate used: 0.6%.
  - Share of symptomatic agents among infected: 0.6.
  - Relative infectiousness of asymptomatic: 0.5.
  - Relative productivity of infected agents: 0.8.
  - Mortality doubles when infected exceeds 1% of population; maximum mortality set to 3 · $D.
- Macroeconomic block:
  - Discount factor based on standard quarterly value 0.99 (converted to weekly).
  - Weight on leisure targets 40% of time spent at work-related activities.
  - Elasticity of substitution set to obtain product markup of 20%.
  - Degree of price stickiness from quarterly Calvo probability of 0.75 expressed in weekly units.
  - Monetary policy reaction to inflation and output gap set to 1.5 and 0.5 (converted to weekly) respectively.
- Lockdown/disutility calibration:
  - Sweden: Φ_c = Φ_n = 0; disutility of dying set so model-implied recession matches Sweden fallout 3.4%.
  - Euro area: Φ_c, Φ_n, Φ_ω chosen jointly to match lost output 6.5%, change in inflation (inflation declines by 0.78%), and death rate (≈ 0.23% excess deaths).

### Baseline parameter values (selected from Table 1)
- A. Epidemics block:
  - $c = 0.212
  - $n = 1.185
  - $t = 0.570
  - $R = 0.389
  - $D = 7 18 ·0.006
  - ρ = 0.6
  - κ = 0.5
  - ζ = 1
  - ξ = 0.8
  - U_d = -3863
  - ν_0 = 0.01
  - ν_1 = 3
- B. Households:
  - β = 0.99 1/13
  - θ = 1.447
- C. Firms:
  - Z = 2
  - ε = 6
  - δ = 0.75 1/13
- D. Policy:
  - Φ_π = 1.5
  - Φ_y = 0.5/52
  - Φ_I = 0
  - Φ_c = 7.65
  - Φ_n = 3.8
  - Φ_ω = 3.8

### Simulation method
- Deterministic simulations of full nonlinear model.
- Homothetic preferences allow aggregation within groups.
- Value functions off-equilibrium evaluated using linear Taylor expansion.

### Key simulation findings: epidemic and containment measures (first year outcomes)
- Baseline laissez-faire (Φ_c = Φ_n = 0):
  - Output declines by approximately 4.1%
  - Inflation declines by 0.3%
  - Final fatalities ≈ 0.61% of population
- Baseline calibrated lockdown:
  - Output declines on average by 6.7%
  - Fatality rate ends up slightly less than 0.26% of population
- Total isolation of visibly infected (ζ = 0) — pure isolation only:
  - First-year output declines by 1.29%
  - Death toll limited to 0.42% of population
- Isolation coupled with economy-wide lockdown:
  - Fatalities = 0.16%
  - Output decline = 3.39%
- Strict lockdown (baseline Φ_c and Φ_n multiplied by three):
  - Economy nearly frozen for over two years
  - Ultimate death toll slightly below 0.17%
  - Lowest death toll after 6 quarters (attractive in vaccine-development context)
- Trade-offs:
  - Isolation relatively low-cost economically but less effective than lockdown alone.
  - Combination of lockdown and isolation achieves low fatalities with moderate output loss.

### Inflation dynamics across scenarios
- Consumption decline → deflationary pressure; labor supply decline → upward pressure.
- Scenario outcomes:
  - Baseline lockdown: strongest deflationary effect.
  - Laissez-faire: inflation almost flat.
  - Isolation of infected: strongest inflationary effect.

### Welfare costs (percent of lifetime consumption) and policy effectiveness (Table 2 summaries)
- Welfare measure = percent of steady-state consumption a susceptible agent would forgo to avoid the epidemic.
- Selected scenario outcomes:
  - No measures introduced:
    - Baseline monetary policy: Welfare cost = 1.240% ; Deaths = 0.606% ; Φ_I = 0.0625
    - Optimized monetary policy: Welfare cost = 1.230% ; Deaths = 0.594%
    - Standard monetary policy: Welfare cost = 1.247% ; Deaths = 0.610%
  - Isolation of infected:
    - Baseline monetary policy: Welfare cost = 0.845% ; Deaths = 0.418% ; Φ_I = 0.0629
    - Optimized monetary policy: Welfare cost = 0.834% ; Deaths = 0.409%
    - Standard monetary policy: Welfare cost = 0.850% ; Deaths = 0.420%
  - Lockdown (baseline):
    - Baseline monetary policy: Welfare cost = 0.555% ; Deaths = 0.263% ; Φ_I = -0.0096
    - Optimized monetary policy: Welfare cost = 0.554% ; Deaths = 0.264%
    - Standard monetary policy: Welfare cost = 0.564% ; Deaths = 0.268%
  - Strict lockdown:
    - Baseline monetary policy: Welfare cost = 0.383% ; Deaths = 0.169% ; Φ_I = -0.0272
    - Optimized monetary policy: Welfare cost = 0.382% ; Deaths = 0.169%
    - Standard monetary policy: Welfare cost = 0.456% ; Deaths = 0.171%
  - Lockdown & isolation:
    - Baseline monetary policy: Welfare cost = 0.333% ; Deaths = 0.162% ; Φ_I = -0.0093
    - Optimized monetary policy: Welfare cost = 0.333% ; Deaths = 0.162%
    - Standard monetary policy: Welfare cost = 0.339% ; Deaths = 0.163%
- Welfare implications:
  - Laissez-faire: highest welfare cost = 1.24% of lifetime consumption.
  - Baseline lockdown: cuts welfare cost by more than half relative to laissez-faire.
  - Most restrictive policies reduce welfare cost to about one quarter of laissez-faire value.
  - Even with containment, welfare cost remains large relative to standard cyclical fluctuations.

---

### 5.2 Monetary policy

### Role and trade-offs of monetary policy during the pandemic
- Central banks adopted expansionary stances (deep interest rate cuts and quantitative easing) to avoid collapse of economic and financial systems.
- Pandemic recession differs from standard recessions because endogenous behavior and administrative restrictions limit interactions; accommodative monetary policy can accelerate the epidemic and increase fatalities.
- Two externalities:
  - New Keynesian aggregate demand externality (nominal rigidities) → monetary accommodation desirable when activity contracts.
  - Epidemiological externality: agents do not internalize how their actions affect disease spread → monetary accommodation can increase infections and fatalities.
- Policy question: which externality dominates; should monetary policy be contractionary or expansionary?

### Policy rules analyzed
- Standard monetary policy: reacts to deviation of output from trend (commonly used in practice).
- Optimized pandemic-aware monetary policy: reacts to output gap and chooses Φ_I to maximize social welfare (10); rule links interest rates to number of infected.

### Main findings from policy experiments
- Standard monetary reaction is more expansionary than baseline flexible-price output-gap rule because it ignores pandemic's large negative effect on natural output.
- Applying the standard monetary reaction during the pandemic increases fatalities: monetary stimulus reduces the decline in output (output can increase initially) but raises fatalities.
- Standard monetary reaction is detrimental for welfare across baseline and containment scenarios; direction is unequivocal even if magnitudes are modest.
- Optimized monetary policy:
  - When containment absent or weak (isolation), optimized Φ_I > 0: monetary policy should be contractionary (raise real interest rates sharply), deepening the recession to limit disease spread and reduce fatalities.
  - When sufficiently strong containment in place, optimized Φ_I < 0: monetary policy should be more expansionary (lower real interest rates), improving output and inflation.
  - Even when optimized to be expansionary under strong containment, the impact on smoothing cyclical fluctuations is relatively small compared with the recession size.

### Trade-offs and efficiency
- Monetary policy faces a trade-off: stabilizing the economy increases social interactions and fatalities.
- Figure 7 policy frontiers (consumption loss vs life loss) show:
  - Lockdowns are more efficient than monetary policy for saving lives per unit of consumption loss (lockdowns reduce transmission via three channels).
  - Monetary expansion raises fatalities; contraction reduces fatalities. Effect stronger with no containment due to health-care capacity constraints.
  - Monetary policy trade-off is relatively flat: achieving significant fatality reductions via monetary policy requires a very deep recession, so monetary policy is not an efficient epidemic containment tool.
  - Because the health cost of monetary expansion is relatively small when containment is present, central banks have some freedom to support growth at a relatively small health cost — explaining expansionary optimized policy under strong containment.

### Policy implications and recommendations
- Monetary policy should not react to sharp deviations of output from trend as in standard business cycles; doing so reduces welfare irrespective of containment measures.
- Optimal monetary stance depends on containment:
  - With sufficient containment: monetary policy can play usual stabilization role and provide stimulus.
  - Without appropriate containment: monetary policy should be contractionary because the life-saving motive dominates (a third-best outcome given instrument limitations).
- Monetary policy is not an effective primary tool for epidemic containment; lockdowns are more efficient at saving lives per unit of economic cost.
- Given the relatively small health cost of monetary expansion when containment exists, central banks can support fiscal rescue packages; formal coordination modeling is left for further research.

*Source: wpiea2021274-print-pdf — 3.1 Epidemic Model: A-SIR; 3.4 Symptomatic recovered individuals; 5.2 Monetary policy — https://www.imf.org/-/media/files/publications/wp/2021/english/wpiea2021274-print-pdf.pdf*

### 3.1    Epidemic Model:  A-SIR   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .6

### 3.1    Epidemic Model:  A-SIR

### Overview and key modeling choices
- The classic SIR model is modified along two dimensions:
  - Probabilities of being infected depend on economic activity (following Eichenbaum et al. (2020b)).
  - Infected people are split into symptomatic and asymptomatic types; asymptomatic infected are less infectious than symptomatic (following Prem et al. (2020)).
- The model integrates the A-SIR epidemic block with a New Keynesian business cycle framework where monetary policy can affect real allocations via staggered pricing.
- The model does not take into account the possibility of a vaccine becoming available; a noted robustness check indicates main results remain broadly unchanged if mortality declines gradually after a year.

### Population groups (types of individuals)
- Five types of individuals:
  - Susceptible: S_t
  - Infected asymptomatic: A_t
  - Infected symptomatic: I_t
  - Formerly asymptomatic recovered: V_t
  - Formerly symptomatic recovered: R_t
- Supposedly susceptible (behave identically in decisions):  ̃S_t = S_t + A_t + V_t

### Infection transmission channels and probabilities
- Three channels for infection:
  1. Consumption channel: infection probability depends on individual consumption c^S_t(i), aggregate consumption of symptomatic I_t c^I_t, and aggregate consumption of asymptomatic A_t c^A_t.
  2. Labor channel: infection probability depends on individual hours n^S_t(i) and aggregate hours of asymptomatic A_t n^A_t. Symptomatic infected either do not work or work remotely and do not transmit via labor.
  3. Other channels (schools, family meetings, etc.): infection probability depends on total infected (I_t + κ A_t) and on lockdown variable $t.
- Parameters controlling infectiousness and channels:
  - κ: asymptomatic infectiousness relative to symptomatic, 0≤κ<1.
  - ζ: scaling of symptomatic infectiousness to allow for isolation, 0≤ζ≤1.
  - $c, $n > 0: constants controlling relative importance of consumption and labor channels.
- Probability that a susceptible individual i becomes infected (with or without symptoms) is given by:
  - $
I,t
(i) = $
c c^S_t(i)(ζ I_t c^I_t + κ A_t c^A_t) + $n n^S_t(i) κ A_t n^A_t + $t (I_t + κ A_t)
- Supposedly susceptible agents face probability of becoming symptomatic infected:
  - ̃$
I,t
(i) = ρ $
I,t
(i) S_t ̃S_t
  (expression as in text; note ̃S_t = S_t + A_t + V_t)

### Infection and transition dynamics (exact discrete-time equations)
- Susceptible evolution:
  - S_{t+1} = (1 − $
I,t
) S_t
- Symptomatic infected evolution:
  - I_{t+1} = (1 − $
R
) I_t + ρ $
I,t
 S_t
- Asymptomatic infected evolution:
  - A_{t+1} = (1 − $
R
) A_t + (1 − ρ) $
I,t
 S_t
- Formerly asymptomatic recovered evolution:
  - V_{t+1} = V_t + $
R
 A_t
- Formerly symptomatic recovered evolution:
  - R_{t+1} = R_t + ($
R
 − $
D,t
) I_t
- Deceased evolution:
  - D_{t+1} = D_t + $
D,t
 I_t

### Behavioral implications
- Asymptomatic infected behave the same as susceptible and formerly asymptomatic recovered (they do not realize they are infected).
- Agents become heterogeneous once the virus spreads; decisions on consumption and labor are affected by infection risk.

### Supposedly susceptible individuals: budget, problem and first-order conditions
- Supposedly susceptible agents choose ̃c_t(i), ̃n_t(i), ̃B_{t+1}(i) with nominal bond return I_t; real bond holdings defined as ̃b_t = ̃B_t / P_{t−1}.
- Budget constraint:
  - (1 + τ_{c,t}) P_t ̃c_t(i) + ̃B_{t+1}(i) = (1 − τ_{n,t}) W_t ̃n_t(i) + I_{t−1} ̃B_t(i) + P_t Γ_t
  - τ_{c,t}: consumption tax rate; τ_{n,t}: labor income tax rate. Taxes model administrative restrictions (lockdowns); tax revenue is rebated to households as Γ_t.
- Recursive expected-utility problem:
  - ̃U_t(̃b_t(i)) = max_{̃c_t(i),̃n_t(i),̃B_{t+1}(i),̃$
I,t
(i)} [ log ̃c_t(i) + θ log(1 − ̃n_t(i)) + β(1 − ̃$
I,t
(i)) ̃U_{t+1}(̃b_{t+1}(i)) + β ̃$
I,t
(i) U_{I,t+1}(̃b_{t+1}(i)) ]
  - ̃$
I,t
(i) enters via the probability of infection and the associated continuation utilities.
- First-order conditions (omitting individual index i):
  - 1 / ̃c_t = ̃λ_{S,t} (1 + τ_{c,t}) − ̃λ_{$,t} ρ S_t / ̃S_t $c (ζ I_t c^I_t + κ A_t ̃c_t)
  - θ / (1 − ̃n_t) = ̃λ_{S,t} (1 − τ_{n,t}) w_t + ̃λ_{$,t} ρ S_t / ̃S_t $n κ A_t ̃n_t
  - ̃λ_{$,t} = β [ U_{I,t+1}(̃b_{t+1}) − ̃U_{t+1}(̃b_{t+1}) ]
  - ̃λ_{S,t} = β [ (1 − ̃$
I,t
) ̃λ_{S,t+1} + ̃$
I,t
 λ_{I,t+1} ] I_t / π_{t+1}
- Interpretation:
  - Consumption and labor choices of supposedly susceptible agents internalize infection risk from activities: the last terms in the first two FOCs represent the discounted utility loss from infection multiplied by the infection risk during consumption and labor activities.
  - ̃λ_{$,t} equals the discounted utility loss due to infection.
  - ̃λ_{S,t} follows an Euler equation incorporating the probability of remaining supposedly susceptible.

### Symptomatic infected individuals: budget, problem and first-order conditions
- Symptomatic infected can work remotely with reduced productivity factor 0≤ξ≤1.
- Their bond return is adjusted to I_t / (1 − $
D,t
) to account for a fraction $
D,t
 dying each period.
- Budget constraint:
  - (1 + τ_{c,t}) P_t c^I_t(i) + B^I_{t+1}(i) = W_t ξ n^I_t(i) + I_{t−1} B^I_t(i) / (1 − $
D,t−1
) + P_t Γ_t
- Recursive expected-utility problem:
  - U_{I,t}(b^I_t) = max_{c^I_t(i), n^I_t(i), B^I_{t+1}(i)} [ log c^I_t(i) + θ log(1 − n^I_t(i)) + β(1 − $
R
) U_{I,t+1}(b^I_{t+1}(i)) + β($
R
 − $
D,t
) U_{R,t+1}(b^I_{t+1}(i)) + β $
D,t
 U_D ]
  - U_D denotes disutility associated with dying.
- First-order conditions (omitting individual index i):
  - 1 / c^I_t = λ_{I,t} (1 + τ_{c,t})
  - θ / (1 − n^I_t) = ξ λ_{I,t} w_t
  - λ_{I,t} = β [ (1 − $
R
 − $
D,t
) λ_{I,t+1} + $
R
 λ_{R,t+1} ] I_t / (π_{t+1} (1 − $
D,t
))
- Interpretation:
  - Symptomatic infected choices reflect reduced productivity (ξ) and adjusted intertemporal valuation due to nonzero recovery and death probabilities ($R and $D,t).

*Source: wpiea2021274-print-pdf — 3.1    Epidemic Model:  A-SIR — https://www.imf.org/-/media/files/publications/wp/2021/english/wpiea2021274-print-pdf.pdf*

### 3.4    Symptomatic recovered individuals

### 3.4    Symptomatic recovered individuals

### Recovered individuals: preferences and budget
- Recovered individuals are not at risk of getting infected and "choose consumption cR_t(i), labor supply nR_t(i), and bond holdings BR_{t+1}(i)."
- Budget constraint:
  - (1 + τ_{c,t}) P_t cR_t(i) + BR_{t+1}(i) = (1 − τ_{n,t}) W_t nR_t(i) + I_{t−1} BR_t(i) + P_t Γ_t
- Recursive problem:
  - UR_t(bR_t(i)) = max_{cR_t(i), nR_t(i), BR_{t+1}(i)} [ log cR_t(i) + θ log(1 − nR_t(i)) + β UR_{t+1}(bR_{t+1}(i)) ]
  - Subject to the budget constraint above.
- "Standard first-order conditions follow."

### Firms: structure and pricing
- Retail firms in perfect competition buy intermediate goods y_t(ι) at price P_t(ι) and combine them into final goods y_t sold at price P_t.
- Profit maximization:
  - Maximize P_t y_t − ∫_{ι∈[0,1]} P_t(ι) y_t(ι) dι subject to technology.
- Aggregation and demand:
  - y_t = [ ∫_{ι∈[0,1]} y_t(ι)^{(ε−1)/ε} dι ]^{ε/(ε−1)}
  - Demand for intermediate goods: y_t(ι) = ( P_t(ι) / P_t )^{−ε} y_t
  - Aggregate price level: P_t = [ ∫_{ι∈[0,1]} P_t(ι)^{1−ε} dι ]^{1/(1−ε)}
- Intermediate firm technology and marginal cost:
  - y_t(ι) = Z n_t(ι) with Z > 0
  - Marginal cost: mc_t = w_t / Z
- Price setting under Calvo:
  - Each period a firm receives a signal to adjust prices with probability 1 − δ and resets to ̃P_t(ι) to maximize discounted profits:
    - max_{̃P_t(ι), {y_{t+j}(ι)}_{j≥0}} E_t ∑_{j=0}^∞ (β δ)^j Λ_{t,t+j} ( ̃P_t(ι)/P_{t+j} − mc_{t+j} ) y_{t+j}(ι)
  - Absent the signal price remains unchanged P_{t+1}(ι) = P_t(ι).
  - Λ_{t,t+j} is a weighted average of marginal utility of consumption across household types.

### Government, central bank, and health-care system: policies and rules
- Government:
  - Uses consumption and labor tax rates τ_{c,t} and τ_{n,t} to restrict activity and slow virus spread.
  - Collected revenue rebated to households; transfers firms’ profits to households.
  - Assumes implicit redistribution financed without distortions (e.g., lump sum taxes); budget balanced every period:
    - τ_{c,t} P_t c_t + τ_{n,t} W_t n_t + P_t y_t − W_t n_t = ( ̃S_t + I_t + R_t ) Γ_t
- Lockdown policy operational rule:
  - τ_{c,t} = Φ_c I_t
  - τ_{n,t} = Φ_n I_t
  - Φ_c, Φ_n > 0
  - Lockdown reduces transmission via a third channel:
    - ω_t = ω (1 − τ_{c,t})^{Φ_ω} with ω, Φ_ω > 0
- Central bank (Taylor-type) rule possibly responding to infections:
  - I_t ̄I = ( π_t ̄π )^{Φ_π} ( y_t / y^f_t )^{Φ_y} exp(I_t)^{Φ_I}
  - Φ_π, Φ_y, Φ_I ≥ 0; y^f_t denotes flexible-price output.
- Health-care system: mortality depends on strain on system via piecewise linear relationship:
  - $D,t = min[ (1 + I_t / ν_0) $D, ν_1 $D ] with ν_0, ν_1 > 0
  - Mortality increases with infections but levels off beyond a certain point.

### Market clearing and asset evolution
- Population partitioned: susceptible S_t, symptomatically infected I_t, symptomatically recovered R_t.
- Final goods market clearing:
  - ∫_{i∈S_t} ̃c_t(i) di + ∫_{i∈I_t} cI_t(i) di + ∫_{i∈R_t} cR_t(i) di ≡ c_t = y_t
- Labor market clearing:
  - ∫_{i∈S_t} ̃n_t(i) di + ∫_{i∈I_t} nI_t(i) di + ∫_{i∈R_t} nR_t(i) di ≡ n_t = ∫_{ι∈[0,1]} n_t(ι) dι
- Aggregate production with price dispersion:
  - ∆_t y_t = z_t n_t
  - ∆_t = ∫_{ι∈[0,1]} ( P_t(ι) / P_t )^{−ε} dι
- Assets by agent type evolve:
  - ̃S_{t+1} ̃B_{t+1} = (1 − ̃$_{I,t}) ∫_{i∈S_t} ̃B_{t+1}(i) di
  - I_{t+1} B_{I,t+1} = (1 − $R) ∫_{i∈I_t} B_{I,t+1}(i) di + ̃$_{I,t} ∫_{i∈S_t} ̃B_{t+1}(i) di
  - R_{t+1} B_{R,t+1} = ∫_{i∈R_t} B_{R,t+1}(i) di + ($R − $D,t) ∫_{i∈I_t} B_{I,t+1}(i) di
- Bond market clearing:
  - ̃S_t ̃B_t + I_t B_{I,t} + R_t B_{R,t} = 0

### Calibration highlights
- Pandemic block targets:
  - Each of the two economic channels (consumption and labor) accounts for one-sixth of transmission absent containment; about two-thirds of population infected before pandemic dies out.
  - Terminal share recovering or dying consistent with herd immunity levels of 60-70%.
  - Time to recover or die: 18 days (i.e., 7/18 periods in weekly model).
  - Infection fatality rate used: 0.6%.
  - Share of symptomatic agents among infected: 0.6.
  - Relative infectiousness of asymptomatic: 0.5.
  - Relative productivity of infected agents: 0.8.
  - Mortality doubles when infected exceeds 1% of population; maximum mortality set to 3 · $D.
- Macroeconomic block standard calibrations:
  - Discount factor based on standard quarterly value 0.99 (converted to weekly).
  - Weight on leisure targets 40% of time spent at work-related activities.
  - Elasticity of substitution set to obtain product markup of 20%.
  - Degree of price stickiness from quarterly Calvo probability of 0.75 expressed in weekly units.
  - Monetary policy reaction to inflation and output gap set to 1.5 and 0.5 (converted to weekly) respectively.
- Lockdown/disutility calibration procedure:
  - Disutility of dying and Φ_c, Φ_n, Φ_ω set using empirical fallout:
    - Sweden: keeping Φ_c = Φ_n = 0, disutility of dying set so model-implied recession matches data (Sweden fallout 3.4%).
    - Euro area: Φ_c, Φ_n, Φ_ω chosen jointly to match lost output (6.5%), change in inflation (inflation declines by 0.78%), and death rate (≈ 0.23% excess deaths).

### Simulation method
- Deterministic simulations accounting for full nonlinear model.
- Homothetic preferences allow aggregation within groups.
- Value functions off-equilibrium evaluated using linear Taylor expansion.
- Equilibrium conditions in aggregate listed in online Appendix.

### Key simulation findings: epidemic and containment measures
- Baseline laissez-faire scenario (Φ_c = Φ_n = 0):
  - Only administrative restriction: sick-leave for visibly infected; remote work allowed with lower productivity.
  - Over first year:
    - Output declines by approximately 4.1%
    - Inflation declines by 0.3%
    - Final fatalities ≈ 0.61% of population
  - Mechanism: large fractions of population (susceptible, asymptomatic) reduce consumption and work due to infection risk; these groups dominate contraction.
- Baseline calibrated lockdown (red dashed line):
  - Authorities impose taxes on consumption and labor tied to visibly infected.
  - First-year outcomes:
    - Output declines on average by 6.7%
    - Fatality rate ends up slightly less than 0.26% of population
  - Lockdown is more costly economically but reduces fatalities sharply.
- Total isolation of visibly infected (ζ = 0) — two variants:
  - Pure isolation only (yellow dashed line):
    - First-year output declines by 1.29%
    - Death toll limited to 0.42% of population
    - Less successful epidemiologically than lockdown in model.
  - Isolation coupled with economy-wide lockdown (purple dash-dotted line):
    - Mix is highly successful:
      - Fatalities amount to 0.16%
      - Output decline 3.39%
- Strict lockdown (green dotted line): baseline Φ_c and Φ_n multiplied by three
  - Economy nearly frozen for over two years
  - Ultimate death toll slightly below 0.17%
  - Attractive in vaccine-development context (lowest death toll after 6 quarters).
- Trade-offs:
  - Isolation is relatively low-cost economically but less effective than lockdown alone.
  - Combination of lockdown and isolation achieves low fatalities with moderate output loss.

### Inflation dynamics
- Inflation response depends on relative declines in consumption demand vs labor supply:
  - Consumption decline → deflationary pressure
  - Labor supply decline → upward pressure on prices
- Outcomes by scenario:
  - Baseline lockdown: strongest deflationary effect (calibrated to match declining inflation).
  - Laissez-faire: inflation almost flat (consumption and labor reductions similar, balancing demand and supply).
  - Isolation of infected: strongest inflationary effect (susceptible households less afraid to consume, raising inflationary pressure).

### Welfare costs and policy effectiveness
- Welfare measure: difference in aggregate welfare at time 0 between epidemic and non-epidemic world, expressed as percent of steady-state consumption a susceptible agent would forgo to avoid the epidemic.
- Findings (Table 2 referenced):
  - Laissez-faire scenario: highest welfare cost = 1.24% of lifetime consumption.
  - Baseline lockdown: cuts welfare cost by more than half relative to laissez-faire.
  - Most restrictive policies (strict lockdown, mix of lockdown and isolation): welfare cost declines to about one quarter of laissez-faire value.
  - Even with containment, welfare cost remains large relative to standard cyclical fluctuation costs.

*Source: wpiea2021274-print-pdf - 3.4    Symptomatic recovered individuals*

### 5.2    Monetary policy

### 5.2 Monetary policy

### Role of monetary policy during the pandemic
- Central banks worldwide assumed an expansionary policy stance in response to the COVID-19 crisis, manifesting in deep interest rate cuts and subsequent rounds of quantitative easing.
- An important goal of these interventions was to avoid a collapse of the economic and financial system and alleviate pressure on governments implementing large rescue plans.
- The pandemic recession differs from a standard recession: it combines endogenous reactions and administrative policy measures limiting social and economic interactions, so an accommodative monetary policy could be counterproductive by accelerating the epidemic and increasing fatalities.
- Two externalities determine monetary policy trade-offs:
  - A New Keynesian aggregate demand externality associated with nominal rigidities, suggesting monetary accommodation when economic activity contracts.
  - An externality from agents failing to internalize how their actions affect disease spread, implying monetary accommodation can increase infections and fatalities.
- The question is which externality dominates and whether monetary policy should be contractionary or expansionary during the pandemic.

### Policy experiments and main findings
- Two monetary-policy rules are analyzed in the scenarios:
  - Standard monetary policy: authorities react to deviation of output from the steady state (more common in practice because natural (flexible-price) level output is unobservable).
  - Optimized pandemic-aware monetary policy: policy reacts to the output gap (baseline assumption) with the policy parameter Φ_I chosen to maximize the social welfare function (10); the rule relates interest rates to the number of infected agents.
- Findings from Figures 2 to 6 and Table 2:
  - The standard monetary reaction function yields a more expansionary stance than the baseline flexible-price output-gap rule because it does not account for the pandemic’s large negative effect on the natural level of output.
  - The difference between standard and baseline rules is weakest in the isolation variant and strongest (at least in the first year) in the baseline scenario.
  - Applying the standard monetary reaction during the pandemic increases fatalities: monetary stimulus reduces the decline in output (output even increases initially) but raises the number of fatalities.
  - Using the standard monetary reaction is detrimental for welfare across baseline and containment scenarios; effects are not large in relative terms but the direction is unequivocal.
- Optimized monetary policy results:
  - Optimized reaction parameters (first column of optimized policy panel in Table 2) differ from zero, so policy has a role.
  - When containment is absent or weak (isolation), the optimized Φ_I is positive: monetary policy should be contractionary (raise real interest rates sharply), deepening the recession to limit disease spread and reduce fatalities (evident in Figures 2 and 5).
  - When sufficiently strong containment policy is in place, the optimized monetary policy is more expansionary (coefficient on number of infected is negative): optimized policy lowers real interest rates, improving output and inflation (shown in Figures 3, 4 and 6).
  - Even when optimized to be expansionary under strong containment, the impact of monetary policy on smoothing cyclical fluctuations is relatively small compared with the size of the recession.

### Trade-offs between stabilizing the economy and containing the epidemic
- Monetary policy faces a trade-off that differs dramatically from usual objective dilemmas: stabilizing the economy increases social interactions, infections, and fatalities.
- Figure 7 (efficient policy frontiers) summarizes trade-offs:
  - Horizontal axis: cumulative consumption loss during the first 2 years of the epidemic.
  - Vertical axis: percentage of deceased agents.
  - Lockdown frontier (solid blue line): efficient combinations of Φ_c and Φ_n in equations (30) and (31), assuming monetary policy follows baseline Taylor rule (33) with Φ_I = 0.
  - Monetary policy frontiers: yellow dash-dotted (laissez-faire) and red dashed (baseline lockdown) lines for various Φ_I.
- Key trade-off insights:
  - Saving lives occurs at an economic cost of foregone consumption in all cases.
  - Lockdowns have a much steeper profile and are more efficient: they reduce transmission via all three contagion channels, including social contacts, while monetary policy affects transmission only via consumption and work.
  - Monetary expansion raises fatalities; contraction reduces fatalities. This effect is stronger with no containment measures because of higher probability of dying due to limited health care capacity in laissez-faire.
  - The monetary policy trade-off is relatively flat: significant reductions in fatalities via monetary policy would require engineering a very deep recession, so monetary policy is not an efficient tool to contain the epidemic.
  - The relatively flat trade-off implies monetary expansion is not very harmful (especially with other containment policies in place), giving central banks some freedom to support economic growth at a relatively small health cost—explaining expansionary optimized policy under some scenarios.

### Policy implications and recommendations
- Monetary policy should not react to sharp deviations of output from trend as in standard business cycles; doing so reduces welfare irrespective of containment measures.
- The optimal monetary stance depends on containment measures:
  - If sufficient containment measures are in place, monetary policy can act in its usual role of stabilizing the business cycle and provide monetary stimulus to support a deeply recessing economy.
  - If appropriate containment measures are not in place, monetary policy should be contractionary because the life-saving motive dominates; this is a third-best outcome since central bank instruments are better at steering the economy than at reducing fatalities.
- Monetary policy is not an effective primary tool for epidemic containment; lockdowns are more efficient at saving lives for a given economic cost.
- Because the health cost of monetary expansion is relatively small when containment is present, central banks can support fiscal authorities implementing rescue packages for affected businesses and households; formalizing this coordination requires different modeling and is left for further research.

*Source: 5.2 Monetary policy, wpiea2021274-print-pdf*

### References

### wpiea2021274-print-pdf - References

### Key empirical and modeling literature cited
- Papers on targeted lockdowns, optimal mitigation, and macro-epidemiological interactions:
  - Acemoglu, Daron, Victor Chernozhukov, Iván Werning, and Michael D. Whinston (2020) ‘Optimal Targeted Lockdowns in a Multi-Group SIR Model.’ NBER Working Papers 27102, May.
  - Alvarez, Fernando E., David Argente, and Francesco Lippi (2020) ‘A simple planning problem for COVID-19 lockdown.’ NBER Working Papers 26981.
  - Eichenbaum, Martin S., Sergio Rebelo, and Mathias Trabandt (2020a,b,c) NBER Working Papers 27430, 26882, 27104.
  - Jones, Callum J., Thomas Philippon, and Venky Venkateswaran (2020) NBER Working Papers 26984.
  - Kaplan, Greg, Benjamin Moll, and Gianluca L. Violante (2020a,b) mimeo and NBER Working Papers 27794.
  - Krueger, Dirk, Harald Uhlig, and Taojun Xie (2021) ECONtribute Discussion Papers Series 075.
- Epidemiological and clinical studies referenced for infection, asymptomatic rates, household transmission, and mortality:
  - Bi, Qifang et al. (2020) ‘Household Transmission of SARS-COV-2: Insights from a Population-based Serological Survey.’ medRxiv.
  - Byambasuren, Oyungerel et al. (2020) ‘Estimating the extent of asymptomatic COVID-19...’ Official Journal of the Association of Medical Microbiology and Infectious Disease Canada 5(4), 223–234.
  - O’Driscoll, Megan et al. (2020) ‘Age-specific mortality and immunity patterns of SARS-CoV-2.’ Nature pp. 1–6.
  - Zhou, Fei et al. (2020) ‘Clinical course and risk factors for mortality...’ The Lancet 395(10229), 1054–1062.
  - Institute for Health Metrics and Evaluation (2021) ‘Estimation of excess mortality due to COVID-19.’ Accessed: 2021-05-15.
- Monetary policy and macro references relevant to the DSGE-SIR and monetary response:
  - Clarida, Richard, Jordi Gali, and Mark Gertler (1999) Journal of Economic Literature 37(4), 1661–1707.
  - Gertler, Mark, and Peter Karadi (2011) Journal of Monetary Economics 58(1), 17–34.
  - Lepetit, Antoine, and Cristina Fuentes-Albero (2020) SSRN.
  - Levin, Andrew T., and Arunima Sinha (2020) NBER Working Papers 27748.
- Other notable modeling and empirical contributions referenced:
  - Ferguson, Neil M. et al. (2006) Nature 442(7101), 448–452; Imperial College COVID-19 Response Team (2020) working paper on NPIs.
  - Gomes, M Gabriela M et al. (2020) ‘Individual variation ... lowers the herd immunity threshold.’ MedRxiv.
  - Prem, Kiesha et al. (2020) The Lancet Public Health on control strategies reducing social mixing.

### Baseline parameter values (Table 1)
- A. Epidemics block
  - $c$ = 0.212 — Parameter governing infection through consumption activity
  - $n$ = 1.185 — Parameter governing infection through labor activity
  - $ $ = 0.570 — Parameter governing infection through other activity
  - $R$ = 0.389 — Probability of becoming removed (either death or recovery)
  - $D$ = 7 18 ·0.006 — Basic probability of dying
  - ρ = 0.6 — Probability of being symptomatic conditional on infection
  - κ = 0.5 — Infectiousness of asymptomatic relative to symptomatic
  - ζ = 1 — Non-isolation of infected
  - ξ = 0.8 — Relative productivity of infected households
  - U_d = -3863 — Disutility of death
  - ν_0 = 0.01 — Parameter governing capacity constraint on health-care system
  - ν_1 = 3 — Parameter governing capacity constraint on health-care system
- B. Households
  - β = 0.99 1/13 — Discount factor
  - θ = 1.447 — Weight on labor in utility
- C. Firms
  - Z = 2 — Productivity
  - ε = 6 — Elasticity of substitution between product varieties
  - δ = 0.75 1/13 — Calvo probability
- D. Policy
  - Φ_π = 1.5 — Interest rate reaction to inflation
  - Φ_y = 0.5/52 — Interest rate reaction to output gap
  - Φ_I = 0 — Interest rate reaction to infected
  - Φ_c = 7.65 — Consumption channel lockdown
  - Φ_n = 3.8 — Work channel lockdown
  - Φ_ω = 3.8 — Elasticity of other activities channel to lockdown

### Cost of the epidemic (Table 2)
- Note: The welfare cost of the epidemic is expressed in per cent of lifetime consumption. Baseline monetary policy refers to the calibration presented in Table 1. Optimized monetary policy responds additionally to the number of symptomatic infected agents. Standard monetary policy responds to inflation (like baseline case) but to output deviations from trend rather than from its level under flexible prices.
- Scenario outcomes (Welfare cost, Deaths, Φ_I, and comparisons across monetary-policy regimes):
  - No measures introduced
    - Baseline monetary policy: Welfare cost = 1.240% ; Deaths = 0.606% ; Φ_I = 0.0625
    - Optimized monetary policy: Welfare cost = 1.230% ; Deaths = 0.594%
    - Standard monetary policy: Welfare cost = 1.247% ; Deaths = 0.610%
  - Isolation of infected
    - Baseline monetary policy: Welfare cost = 0.845% ; Deaths = 0.418% ; Φ_I = 0.0629
    - Optimized monetary policy: Welfare cost = 0.834% ; Deaths = 0.409%
    - Standard monetary policy: Welfare cost = 0.850% ; Deaths = 0.420%
  - Lockdown (baseline)
    - Baseline monetary policy: Welfare cost = 0.555% ; Deaths = 0.263% ; Φ_I = -0.0096
    - Optimized monetary policy: Welfare cost = 0.554% ; Deaths = 0.264%
    - Standard monetary policy: Welfare cost = 0.564% ; Deaths = 0.268%
  - Strict lockdown
    - Baseline monetary policy: Welfare cost = 0.383% ; Deaths = 0.169% ; Φ_I = -0.0272
    - Optimized monetary policy: Welfare cost = 0.382% ; Deaths = 0.169%
    - Standard monetary policy: Welfare cost = 0.456% ; Deaths = 0.171%
  - Lockdown & isolation
    - Baseline monetary policy: Welfare cost = 0.333% ; Deaths = 0.162% ; Φ_I = -0.0093
    - Optimized monetary policy: Welfare cost = 0.333% ; Deaths = 0.162%
    - Standard monetary policy: Welfare cost = 0.339% ; Deaths = 0.163%

### Figures — reported outcomes and notes
- Figure notes consistently report:
  - Horizontal axis in weeks, vertical in percent.
- Figures illustrate dynamics across scenarios:
  - Epidemic containment policies (Figure 1): time paths for Susceptibles, Infected (with asymptomatic), Recovered (with asymptomatic), Deceased, GDP, Output gap, Inflation (Y-o-Y), Interest rate (1Y), Ex ante real interest rate (1Y) across: no containment measures, baseline lockdown, isolation of infected, isolation & lockdown, strict lockdown.
  - Figures 2–6: Comparisons of baseline, standard and optimized monetary policy under different containment regimes (no measures, baseline lockdown, strict lockdown, isolation, isolation & lockdown) showing impacts on epidemiological states and macro variables (GDP, output gap, inflation, interest rates).
  - Figure 7: Policy frontiers — trade-off between Consumption loss and Life loss:
    - Axes scaled in percent.
    - Consumption loss is calculated as average percent deviation from steady state over the first two years of the epidemic.
    - Life loss refers to the final death rate.
    - Curves shown: lockdowns (with baseline monetary policy), monetary policy (without containment measures), monetary policy (with baseline lockdown).

### Appendix — model structure and key equations
- Epidemic block (discrete-time evolution)
  - Evolution of susceptible individuals: S_{t+1} = (1−$I,t) S_t (A.1)
  - Evolution of symptomatic infected: I_{t+1} = (1−$R,t) I_t + ρ $I,t S_t (A.2)
  - Evolution of asymptomatic infected: A_{t+1} = (1−$R,t) A_t + (1−ρ) $I,t S_t (A.3)
  - Evolution of formerly asymptomatic infected: V_{t+1} = V_t + $R,t A_t (A.4)
  - Evolution of recovered individuals: R_{t+1} = R_t + ($R,t − $D,t) I_t (A.5)
  - Evolution of deceased individuals: D_{t+1} = D_t + $D,t I_t (A.6)
  - Supposedly susceptible individuals: ˜S_t = S_t + A_t + V_t (A.7)
- Infection and death probabilities
  - Probability of susceptible individuals becoming infected (A.8) — full functional form shown in source.
  - Probability of supposedly susceptible agent becoming symptomatic infected: ˜$I,t = ρ $I,t S_t / ˜S_t (A.9)
  - Death probability: $D,t = min[(1 + I_t / ν_0) $D, ν_1 $D] (A.10)
- Household, infected, recovered agent optimization
  - Utility and budget constraints for supposedly susceptible individuals (A.11–A.12), and first-order conditions (A.13–A.16).
  - Utility and budget constraints for symptomatic infected (A.17–A.18), with optimality conditions (A.19–A.21).
  - Symptomatic recovered utility, budget constraint, and optimality conditions (A.22–A.26).
- Firms and price setting
  - Optimal price set by reoptimizing firms: ˜p_t = Ω_t / Υ_t (A.27)
  - Auxiliary functions Ω_t and Υ_t defined (A.28–A.29).
  - Price index and dispersion equations (A.30–A.31).
- Government and central bank policy rules
  - Containment policies linked to symptomatic infected:
    - τ_{c,t} = Φ_c I_t (A.32)
    - τ_{n,t} = Φ_n I_t (A.33)
    - ω_t = ω(1−τ_{c,t})^{Φ_ω} (A.34)
  - Monetary policy rule (A.35): I_t / ̄I = (π_t / ̄π)^{Φ_π} (y_t / y^f_t)^{Φ_y} exp(I_t)^{Φ_I}
- Market clearing conditions
  - Final goods market (A.36), labor market (A.37), aggregate output (A.38), bond market (A.39).
- Flexible price block
  - Flexible price block is the same set of equations with δ = 0; per capita flexible-price output y^f_t enters the monetary policy rule.

*Source: wpiea2021274-print-pdf - References — https://www.imf.org/-/media/files/publications/wp/2021/english/wpiea2021274-print-pdf.pdf*

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_Source: https://www.imf.org/-/media/files/publications/wp/2021/english/wpiea2021274-print-pdf.pdf_