## 2.1 Households (and related model sections in the chapter)

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---

### Background and model setup
- Integrates climate-induced, left-tail “rare disaster” shocks into a non-linear New Keynesian DSGE model with nominal and real rigidities, price and wage stickiness, and wages indexed to past inflation.
- Disasters occur with probability p and cause a significant fall in productivity, affecting capital, labor, and potential output.
- Monetary and fiscal policies are set according to Taylor-style policy rules in which the nominal interest rate reacts to changes in inflation and the growth rate of output (rather than the output gap).
- Higher-order approximations (third-order) are used to capture non-linearities and large deviations from steady state introduced by left-tail shocks.

### Calibration and empirical anchors
- Barro (2006): annual 1.7 percent chance of occurrence with associated output declines between 15 percent and 64 percent, averaging 29 percent.
- Nakamura et al. (2013): rare disasters can reduce consumption by 15 percent.
- Policy experiments compare inflation targets of 2 percent and 4 percent.
- Key model calibration parameters (preserved exactly as in source):
  - Structural parameters
    - β 0.994 Discount factor
    - σ 1 Risk aversion
    - 1/η 0.5 Frisch elasticity
    - ε_p 0.6 Calvo probability (Price)
    - ε_w 0.75 Calvo probability (Wage)
    - θ_p 10 Elasticity of subs. (goods)
    - θ_w 10 Elasticity of subs. (labor)
    - δ 0.0025 Depreciation rate
    - b_y 0.7 the steady-state debt-to-GDP ratio
    - g_y 0.0025 the share of government in the economy
  - Policy parameters
    - φ_π 1.5 Inflation response coefficient
    - φ_y 0.5 Output response coefficient
    - γ_1 0.04 Fiscal rule coefficient on debt
    - γ_2 0.3 Tax smoothing parameter
    - g_y 0.2 Steady-state share of government spending
    - b_y 0.4 Steady-state share of bonds
  - Shock parameters
    - ρ_a 0.9 Persistence of the productivity shock
    - std(ε_{at}) 0.0075 Standard deviation of the shock
    - p 0.0045 the probability of a disaster occurring (quarterly)
    - φ −38.67 Given a disaster, parameterised by φ, productivity drop by 29%
- Additional calibration notes:
  - Capital share α assumed to be 21 percent.
  - Disaster frequency set at p = 0.45 percent for each quarter (i.e., p 0.0045 quarterly).

### Household preferences and constraints (preserved equations and conditions)
- Expected lifetime utility: E_t Σ_{t=1}^∞ β^t [ C_t^{1−σ}/(1−σ) − N_t^{1+η}/(1+η) ] (Equation (3)).
- Intertemporal budget constraint (Equation (4)): P_t C_t + I_t + Σ_{s_{t+1}} Q(s_{t+1}|s_t) B^s_{t+1} ≤ B_t + R_k K_t + W_t N_t + Π_t + T_t.
- Capital accumulation (Equation (5)): K_{t+1} = (1−δ)K_t + [1−S(I_t/I_{t−1})] i_t.
- Investment adjustment cost (Equation (6)): S(I_t/I_{t−1}) = κ/2 (I_t/I_{t−1} − 1)^2.
- First-order conditions include:
  - λ_t = C_t^{−σ} (7)
  - Q(s_{t+1}|s_t) = 1/R_t (8)
  - λ_t = β E_t[λ_{t+1} R_{k,t} / π_{t+1}] (9)
  - Additional conditions for q_t and investment pricing (10)–(11).

### Wage setting and labor supply
- Households supply differentiated labor N^h_t; composite labor:
  - N_t = [∫_0^1 N_{ht}^{θ_w−1/θ_w} dh]^{θ_w/(θ_w−1)} (12).
  - Demand for type-h labor: N_t(h) = (W_{ht}/W_t)^{−θ_w} N_t (13).
  - Aggregate wage index: W_t = [∫_0^1 W_{ht}^{1−θ_w} dh]^{1/(1−θ_w)} (14).
- Calvo wage-setting with hazard 1−γ_w; real reset wage and indexation reflected in equations (15)–(18).

### Firms, prices, and aggregate production
- Firm production: Y_{it} = A_t N_{it}^{1−α} K_{it}^{α} (19).
- Aggregation (Dixit-Stiglitz): Y_t = [∫_0^1 Y_{it}^{(ε−1)/ε} di]^{ε/(ε−1)} (20); price index P_t = [∫_0^1 P_{it}^{1−ε} di]^{1/(1−ε)} (21).
- Demand for firm i: Y_{it} = (P_{it}/P_t)^{−ε} Y_t (22).
- Marginal cost (23): mc_t = [(1/(1−α))^{1−α} (1/α)^{α} w_t^{1−α} (R_{k,t})^{α}] / A_t.
- Calvo price setting with hazard 1−γ_p, reset price equations (24)–(26), price index evolution (27), and price dispersion (28).
- Aggregate output (29): Y_t = A_t N_t^{1−α} K_t^{α} s_t, where s_t denotes price dispersion and reduces aggregate output.

### Government and monetary policy rules
- Government budget evolution (30): b_t = b_{t−1} R_{t−1} / π_{t−1} + g_t − T_t.
- Tax rule (31): T_t / y_t = τ^* + γ_1 (b_{t−1}/y_{t−1} − b^*) + γ_2 (b_{t−1}/y_{t−1} − b_{t−2}/y_{t−2}).
- Central bank Taylor-style interest rate rule in logs (32): r_t − r = φ_π (π_t − π) + φ_y (y_t − y_{t−1}), with x_t = ln(X_t).

---

### 3 Results — moments, policy trade-offs, and welfare

### Summary of moments (standard vs. disaster cases)
- Table 2 reported moments (means, standard deviations, skewness):
  - a. Standard
    - Inflation: Mean 1.00, Std. Dev. 0.01, Skewness -0.08
    - Output: Mean 2.15, Std. Dev. 0.04, Skewness 0.04
    - labor: Mean 0.94, Std. Dev. 0.00, Skewness 0.51
    - Wages: Mean 1.62, Std. Dev. 0.01, Skewness 0.09
    - Marginal cost: Mean 0.90, Std. Dev. 0.01, Skewness 0.41
    - Capital: Mean 47.57, Std. Dev. 0.21, Skewness -0.17
    - Rental Rate: Mean 0.01, Std. Dev. 0.00, Skewness 0.38
    - Productivity: Mean 1.00, Std. Dev. 0.01, Skewness -0.03
    - Welfare: Mean 35.90, Std. Dev. 0.17, Skewness -0.17
  - b. Disaster
    - Inflation: Mean 1.00, Std. Dev. 0.00, Skewness 4.63
    - Output: Mean 2.14, Std. Dev. 0.03, Skewness -4.14
    - labor: Mean 0.94, Std. Dev. 0.00, Skewness 9.22
    - Wages: Mean 1.62, Std. Dev. 0.01, Skewness -3.34
    - Marginal cost: Mean 0.90, Std. Dev. 0.01, Skewness 7.54
    - Capital: Mean 47.47, Std. Dev. 0.23, Skewness -0.90
    - Rental Rate: Mean 0.01, Std. Dev. 0.00, Skewness -0.29
    - Productivity: Mean 1.00, Std. Dev. 0.01, Skewness -5.13
    - Welfare: Mean 35.81, Std. Dev. 0.19, Skewness -1.81
- Key interpretations:
  - Mean values of most variables remain stable despite marked changes in skewness.
  - Inflation skewness increases to 4.63 while output skewness falls to -4.14.
  - Marginal cost skewness rises sharply to 7.54.
  - Productivity skewness declines to -5.13; wages skewness falls to -3.34.
  - Labor supply skewness increases to 9.22, interpreted as households increasing labor to offset welfare losses.
  - Welfare skewness decreases modestly to -1.81, indicating higher labor supply helps mitigate disaster impacts on welfare.

### Persistence and impacts on mean output
- Benchmark persistence parameter: 0.9.
- Annualized output loss relative to no-disaster scenario at persistence 0.9: almost 2 percent.
- If persistence increased to 0.98, the annualized output loss increases to 7 percent.
- Figure 1 persistence-to-loss mapping (reported values preserved):
  - 0.90 → 1.7%
  - 0.92 → 2.1%
  - 0.94 → 2.7%
  - 0.96 → 3.8%
  - 0.98 → 6.9%

### The policy trade-off imposed by rare disasters
- Core result: rare disasters move price stability and output stability in opposite directions, creating a policy trade-off.
- Frequency experiment findings:
  - Increasing disaster frequency (up to benchmark 1.7 percent) for fixed shock size:
    - Makes output skewness more negative.
    - Increases inflation skewness substantially.
  - Central bank actions to stabilize output exacerbate inflation skewness; higher disaster likelihood worsens the trade-off.
- Taylor-inspired skewness curves:
  - Vary φ_y from 0.5 to 1.1 and plot resulting skewness of inflation and output to characterize the trade-off.
  - Mechanism: productivity drops reduce output and raise marginal costs, causing inflation and output to move in opposite directions (mc_t equation preserved).

### Welfare-focused policy experiments
- Welfare definition used (as in source): Welfare_t = E_t Σ_{t=1}^∞ β^t U_t (= U_t + βE_t Welfare_{t+1}), implied by third-order approximation.
- Vary φ_y in Taylor rule r_t − r = φ_π(π_t − π) + φ_y(y_t − y_{t−1}) with φ_π = 1.5.
- Main results:
  - Increasing emphasis on output stabilization reduces welfare skewness up to a threshold at φ_y = 1.1.
  - Beyond φ_y = 1.1, welfare skewness becomes more negative; there is an optimal φ_y from households’ perspective.
  - Strict-inflation-focused policy is not effective for climate-induced shocks because responding to growth rates influenced by lagged output creates periods of above-target inflation while smoothing welfare distributional outcomes.

### Increasing the inflation target: 2 percent → 4 percent
- Experiment: raise inflation target from 2 percent to 4 percent.
- Findings:
  - Higher inflation target shrinks the inflation gap when disasters cause deviations, partially reducing needed output adjustments and interest rate changes.
  - Taylor-inspired skewness curves show the 4 percent target shifts the policy frontier left, reducing the cost of stabilizing output versus the 2 percent target; improvement is limited.
  - Welfare implications: a 4 percent target reduces the cost of output stabilization but produces a more negatively skewed welfare measure.
  - Mechanisms:
    - Firms set higher price mark-ups and larger price adjustments to protect prices over reset intervals.
    - Larger price adjustments increase price dispersion and reduce potential (long-run) output.
  - Consistent finding: the Taylor rule with a 1.2 coefficient on output minimizes welfare skewness across targets in the experiments, but higher inflation target carries a potential lasting welfare cost.

### Long-run determinacy and the Woodford/Taylor principle in presence of rare disasters
- Long-run Taylor rule expression (as in source): r = φ_π π + φ_y y (Equation 49).
- Modified Taylor principle and determinacy condition (preserved expressions):
  - ∂r/∂π = φ_π + φ_y ∂y/∂π > 1 (Equation 50).
  - Equilibrium determinacy condition: φ_π > 1 − φ_y ∂y/∂π (Equation 51).
- Effects of rare disasters:
  - Disasters reduce long-run output; negatively skewed output lowers ∂y/∂π and thus reduces φ_y ∂y/∂π.
  - Maintaining determinacy requires a larger φ_π (stronger inflation response).
- Policy tension:
  - After a disaster central banks may want to increase φ_y to stabilize output immediately.
  - Simultaneously, because long-run output is lower, central banks must ensure φ_π increases sufficiently to satisfy the generalized Taylor/Woodford principle and maintain equilibrium determinacy.

---

### 4 Optimal monetary policy and debt-to-GDP implications

### Debt-to-GDP ratio and Taylor-rule weights
- Baseline: φ_y = 0.5 (standard Taylor rule) exhibits pronounced asymmetry in the debt-to-GDP ratio under rare disasters.
- Under a 2 percent inflation target:
  - Increasing φ_y to 1.2 reduces the ratio’s skewness to 1.42.
  - Minimum skewness for the debt-to-GDP ratio occurs at φ_y = 1.2.
  - Beyond φ_y = 1.2, diminishing returns and possible counterproductive effects.
- Under a 4 percent inflation target:
  - Prioritizing output stabilization beneficial up to φ_y = 1.2.
  - When φ_y > 1.2, the skewness of the debt-to-GDP ratio worsens.
- Interpretation: raising emphasis on output stabilization tends to stabilize the debt-to-GDP ratio, but raising the inflation target can negatively impact skewness and output, increasing debt-to-GDP.

### Robustness to shock persistence (reported results)
- Shock persistence values considered: 0.8, 0.9 (benchmark), 0.95 with φ_π = 1.5.
  - Persistence 0.8: Optimal φ_y = 1.5; Reduction in skewness 16%.
  - Persistence 0.9: Optimal φ_y = 1.2; Reduction in skewness 9%.
  - Persistence 0.95: Optimal φ_y = 1.1; Reduction in skewness 4%.
- Interpretation:
  - Main conclusion unaffected: balanced focus on output and inflation is crucial.
  - Higher persistence corresponds to lower optimal φ_y and smaller reductions in skewness.
  - Less persistent shocks yield larger welfare gains from output stabilization.

### Robustness to milder disasters (reported results)
- Alternative disaster size: 15 percent fall in productivity (versus 29 percent benchmark).
  - Persistence 0.8: Optimal φ_y = 1.4; Reduction in skewness 19%.
  - Persistence 0.9: Optimal φ_y = 1.2; Reduction in skewness 11%.
  - Persistence 0.95: Optimal φ_y = 1.1; Reduction in skewness 5%.
- Interpretation:
  - Milder disasters yield larger gains from optimal policy and larger reductions in skewness than benchmark severity.

### Variations in φ_π (inflation coefficient)
- Baseline φ_π = 1.5 in robustness checks.
- Varying φ_π does not change main qualitative conclusion: welfare skewness is minimized when the Taylor rule balances output and inflation stabilization.
- As φ_π increases, marginal gains from emphasizing output stabilization diminish.

### Policy-relevant findings and recommendations (concise)
- Main findings:
  - Climate-induced rare disasters create asymmetric risks: inflation exhibits upward skewness; output exhibits downward skewness.
  - A Taylor rule that places greater emphasis on output stabilization than the standard rule mitigates welfare losses from rare disasters and stabilizes the debt-to-GDP ratio.
  - Optimal φ_y is empirically around 1.2 in the benchmark; optimal φ_y declines with increased shock persistence.
  - Increasing the inflation target from 2 percent to 4 percent can partly alleviate the policy trade-off but reduces household welfare (via higher price mark-ups and price dispersion) and contributes to a higher debt-to-GDP ratio.
  - Policymakers must account for entire outcome distributions (skewness), not only means and standard deviations.
- Policy recommendations and considerations:
  - Adopt a Taylor rule that places substantial weight on output stabilization (targeting around φ_y ≈ 1.2 in the benchmark) while maintaining a credible inflation response.
  - Recognize diminishing returns to increasing φ_y beyond the optimal threshold; avoid over-tuning the Taylor rule toward output stabilization.
  - Consider higher inflation targets cautiously: they can reduce the trade-off but carry lower household welfare and higher debt-to-GDP.
  - Account for changes in long-run determinants: disasters can reduce long-run output and the slope of the Phillips curve, implying the need for more aggressive inflation responses to ensure equilibrium determinacy.
  - Preserve flexibility for unconventional measures (e.g., QE) at the time of shocks; detailed design of unconventional measures requires further research.
  - Incorporate full distributional risks (including skewness) in monetary policy design in disaster-prone environments.

*Italic: Monetary Policy Design with Recurrent Climate Shocks — Working Paper No. WP/2023/243 (excerpted content unit "wpiea2023243-print-pdf").*

### 2.1    Households    .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .   13

### 2.1    Households

### Background and model setup
- The paper integrates climate-induced, left-tail “rare disaster” shocks into a non-linear New Keynesian DSGE model with nominal and real rigidities, price and wage stickiness, and wages indexed to past inflation.
- Disasters occur with probability p and cause a significant fall in productivity, affecting capital, labor, and potential output.
- Monetary and fiscal policies are set according to Taylor-style policy rules in which the nominal interest rate reacts to changes in inflation and the growth rate of output (rather than the output gap).
- Higher-order approximations are used to capture non-linearities and large deviations from steady state introduced by left-tail shocks.

### Calibration and empirical anchors (as cited)
- Barro (2006): annual 1.7 percent chance of occurrence with associated output declines between 15 percent and 64 percent, averaging 29 percent.
- Nakamura et al. (2013): rare disasters can reduce consumption by 15 percent.
- Policy experiments compare inflation targets of 2 percent and 4 percent.

### Key findings
- Climate-induced shocks generate increased skewness in inflation and more negative skewness in output, fattening the left tail of outcome distributions.
- A policy trade-off emerges for central banks between inflation stabilization and output stabilization; this trade-off is meaningfully characterized using Taylor-inspired skewness curves.
- A policy that stabilizes both inflation and output mitigates adverse impacts on household welfare and helps maintain a stable debt-to-GDP ratio.
- Increasing the inflation target from 2 percent to 4 percent eases the policy trade-off but results in lower household welfare.
  - Higher inflation leads to higher price mark-ups and greater price dispersion (due to sticky prices), both contributing to lower output and a higher debt-to-GDP ratio.
- A Taylor rule that places greater weight on output stabilization helps stabilize the debt-to-GDP ratio in the presence of climate-induced disasters.
- When p = 0 (no disasters), the model simplifies to standard formulations (Smets and Wouters (2007); Schmitt-Grohe and Uribe (2007b)), and strict inflation targeting performs similarly to the proposed policy; the inclusion of climate-induced shocks materially changes policy conclusions.

### Mechanisms and interpretation
- Climate-induced productivity shocks reduce potential output, creating a positive output gap (demand-driven), which raises inflation.
- Monetary policy that reacts to the growth rate of output and uses lagged output introduces inertia; gradual closure of the output gap causes sustained above-target inflation but smooths welfare skewness by avoiding abrupt policy shifts.
- Long-run considerations: repeated rare disasters may lower long-run output and could flatten or make negative the slope of the long-run Phillips curve, implying a need for a stronger nominal interest rate response to inflation to preserve equilibrium determinacy (per Woodford (2003)).

### Policy implications and recommendations
- Central banks should integrate the likelihood of high-impact climate-induced disasters into policy-rule formulation rather than treating such shocks only as reactive supply events.
- Robust monetary policy in the presence of rare disasters requires placing weight on both output and inflation stabilization in the objective function.
- Raising the inflation target to 4 percent is not recommended as a welfare-improving policy despite easing the policy trade-off; it reduces household welfare and can raise the debt-to-GDP ratio through lower output.
- A Taylor-style rule that increases emphasis on output stabilization is effective at reducing welfare skewness and stabilizing debt dynamics amid climate-induced disaster risk.

*Source: IMF Working Paper (excerpt: Introduction and table of contents).*

### 2. The Model

### 2. The Model

### Model overview
- The model is the standard new Keynesian model, as in Christiano et al. (2005) and Smets and Wouters (2007), with a key difference: productivity (A_t) is subjected to rare disaster shocks that can occur with probability p.
- Given the left-tail nature of the shock, the model is solved by taking a third-order approximation using Dynare.
- Four agents: households, firms, the government, and the central bank. Output is divided between consumption, investment and government spending.

### Productivity and rare-disaster process
- Productivity in logs: a_t = ρ_a a_{t−1} + ε_{at} (where a_t = ln A_t).
- Shock innovation ε_{at} is:
  - N(0,1) with probability 1−p
  - φ with probability p
- Negative values of φ indicate left-tail innovations (rare disasters).
- The paper notes ε_{at} was added directly to the capital stock equation but omission does not affect results.

### Households
- Representative households indexed h ∈ [0,1].
- Expected lifetime utility: E_t Σ_{t=1}^∞ β^t [ C_t^{1−σ}/(1−σ) − N_t^{1+η}/(1+η) ] (Equation (3)).
- Intertemporal budget constraint (Equation (4)):
  P_t C_t + I_t + Σ_{s_{t+1}} Q(s_{t+1}|s_t) B^s_{t+1} ≤ B_t + R_k K_t + W_t N_t + Π_t + T_t.
- Capital accumulation (Equation (5)): K_{t+1} = (1−δ)K_t + [1−S(I_t/I_{t−1})] i_t.
- Investment adjustment cost (Equation (6)): S(I_t/I_{t−1}) = κ/2 (I_t/I_{t−1} − 1)^2.
- First-order conditions include:
  - λ_t = C_t^{−σ} (7)
  - Q(s_{t+1}|s_t) = 1/R_t (8)
  - λ_t = β E_t[λ_{t+1} R_{k,t} / π_{t+1}] (9)
  - Additional conditions for q_t and investment pricing (10)–(11).

### Wage setting
- Households supply differentiated labor N^h_t; composite labor:
  N_t = [∫_0^1 N_{ht}^{θ_w−1/θ_w} dh]^{θ_w/(θ_w−1)} (12).
- Demand for type-h labor: N_t(h) = (W_{ht}/W_t)^{−θ_w} N_t (13).
- Aggregate wage index: W_t = [∫_0^1 W_{ht}^{1−θ_w} dh]^{1/(1−θ_w)} (14).
- Calvo wage-setting: fraction 1−γ_w can reset wages each period to a common reset wage W^*_t (real reset wage w^*_t = W^*_t / P_t). Real reset wage equation (15) with auxiliary variables f_{1t} (16) and f_{2t} (17).
- Average real wage evolution with wage indexation (18).

### Firms and price setting
- Firm production: Y_{it} = A_t N_{it}^{1−α} K_{it}^{α} (19).
- Aggregation (Dixit-Stiglitz): Y_t = [∫_0^1 Y_{it}^{(ε−1)/ε} di]^{ε/(ε−1)} (20) and price index P_t = [∫_0^1 P_{it}^{1−ε} di]^{1/(1−ε)} (21).
- Demand for firm i: Y_{it} = (P_{it}/P_t)^{−ε} Y_t (22).
- Marginal cost (23): mc_t = [(1/(1−α))^{1−α} (1/α)^{α} w_t^{1−α} (R_{k,t})^{α}] / A_t.
- Calvo price setting: reset price equations (24)–(26) with hazard rate 1−γ_p, price index evolution (27), and price dispersion (28).
- Aggregate output (29): Y_t = A_t N_t^{1−α} K_t^{α} s_t, where s_t denotes price dispersion and reduces aggregate output.

### Government
- Government budget evolution (30): b_t = b_{t−1} R_{t−1} / π_{t−1} + g_t − T_t, where b_t is real government debt.
- Tax rule (31): T_t / y_t = τ^* + γ_1 (b_{t−1}/y_{t−1} − b^*) + γ_2 (b_{t−1}/y_{t−1} − b_{t−2}/y_{t−2}).
- Fiscal policy stabilizes the debt ratio target b^* via changes in the tax rate τ_t.

### Central bank
- Taylor-style interest rate rule in logs (32): r_t − r = φ_π (π_t − π) + φ_y (y_t − y_{t−1}), with x_t = ln(X_t) for generic X_t.

### Steady-state relationships
- Reset price steady-state (33): p̄^*_ = [1−γ_p π^{θ_p−1}]^{1/(1−θ_p)}.
- Price dispersion steady-state (34): s = (1−γ_p) [1/π^*]^{θ_p}.
- Marginal cost steady-state (35): mc = (θ_p−1)/θ_p p̄^* [ (1−β γ_p π^{θ_p}) / (θ_p (1−β γ_p π^{θ_p−1})) ].
- Real wage steady-state (36): w = (1−α) [ mc^{α} R^{α} ]^{1/(1−α)}.
- Labor and capital steady-state relations (37)–(41), and resource constraint y = c + g + i (42).
- Steady-state tax relation (43): T = G + [R/π − 1] b.

### Choice of parameters (calibration)
- Model calibrated at quarterly frequency using common business cycle values. Values listed below are preserved exactly as in source:
  - Structural parameters
    - β 0.994 Discount factor
    - σ 1 Risk aversion
    - 1/η 0.5 Frisch elasticity
    - ε_p 0.6 Calvo probability (Price)
    - ε_w 0.75 Calvo probability (Wage)
    - θ_p 10 Elasticity of subs. (goods)
    - θ_w 10 Elasticity of subs. (labor)
    - δ 0.0025 Depreciation rate
    - b_y 0.7 the steady-state debt-to-GDP ratio
    - g_y 0.0025 the share of government in the economy
  - Policy parameters
    - φ_π 1.5 Inflation response coefficient
    - φ_y 0.5 Output response coefficient
    - γ_1 0.04 Fiscal rule coefficient on debt
    - γ_2 0.3 Tax smoothing parameter
    - g_y 0.2 Steady-state share of government spending
    - b_y 0.4 Steady-state share of bonds
  - Shock parameters
    - ρ_a 0.9 Persistence of the productivity shock
    - std(ε_{at}) 0.0075 Standard deviation of the shock
    - p 0.0045 the probability of a disaster occurring (quarterly)
    - φ −38.67 Given a disaster, parameterised by φ, productivity drop by 29%

- Additional calibration notes from the source:
  - Discount factor β set to 0.994.
  - Relative risk aversion σ = 1.
  - Both elasticities of substitution ε_p and ε_w set to 10.
  - Frisch elasticity 1/η set to 0.5.
  - Capital share α assumed to be 21 percent.
  - Depreciation rate δ specified at 0.0025.
  - Calvo price stickiness ε_p is 0.6, implying a hazard rate of 0.4.
  - Calvo wage stickiness ε_w is 0.75.
  - Monetary policy rule parameters: φ_π = 1.5, φ_y = 0.5.
  - Fiscal feedback parameters: γ_1 = 0.04 and γ_2 = 0.3.
  - Productivity shock persistence 0.9 and standard deviation 0.0075.
  - Disaster calibration: φ = −38.67 implies disasters induce a 29 percent decline in productivity (as in Barro (2006)); disaster frequency set at p = 0.45 percent for each quarter.

*Source: 2. The Model (from the provided chapter content).*

### 3. Results

### 3. Results

### Summary of moments (standard case vs. disaster)
- Table 2: Summary of Moments (means, standard deviations, skewness)
  - a. Standard
    - Inflation: Mean 1.00, Std. Dev. 0.01, Skewness -0.08
    - Output: Mean 2.15, Std. Dev. 0.04, Skewness 0.04
    - labor: Mean 0.94, Std. Dev. 0.00, Skewness 0.51
    - Wages: Mean 1.62, Std. Dev. 0.01, Skewness 0.09
    - Marginal cost: Mean 0.90, Std. Dev. 0.01, Skewness 0.41
    - Capital: Mean 47.57, Std. Dev. 0.21, Skewness -0.17
    - Rental Rate: Mean 0.01, Std. Dev. 0.00, Skewness 0.38
    - Productivity: Mean 1.00, Std. Dev. 0.01, Skewness -0.03
    - Welfare: Mean 35.90, Std. Dev. 0.17, Skewness -0.17
  - b. Disaster
    - Inflation: Mean 1.00, Std. Dev. 0.00, Skewness 4.63
    - Output: Mean 2.14, Std. Dev. 0.03, Skewness -4.14
    - labor: Mean 0.94, Std. Dev. 0.00, Skewness 9.22
    - Wages: Mean 1.62, Std. Dev. 0.01, Skewness -3.34
    - Marginal cost: Mean 0.90, Std. Dev. 0.01, Skewness 7.54
    - Capital: Mean 47.47, Std. Dev. 0.23, Skewness -0.90
    - Rental Rate: Mean 0.01, Std. Dev. 0.00, Skewness -0.29
    - Productivity: Mean 1.00, Std. Dev. 0.01, Skewness -5.13
    - Welfare: Mean 35.81, Std. Dev. 0.19, Skewness -1.81
- Key findings from the moments
  - Mean values of most variables remain stable despite marked changes in skewness.
  - Inflation skewness increases to 4.63 while output skewness falls to -4.14.
  - Marginal cost skewness rises sharply from approximately zero to 7.54.
  - Productivity skewness declines to -5.13; wages skewness falls to -3.34.
  - Labor supply skewness increases to 9.22, interpreted as households increasing labor to offset welfare losses.
  - Welfare skewness decreases modestly to -1.81, indicating that higher labor supply helps mitigate disaster impacts on welfare.

### Persistence of shocks and impacts on mean output
- Sources of the stability of means:
  - Negative-skew disasters are modeled as left-tail, extreme events with large immediate impact but not necessarily long-lasting output effects in the baseline calibration.
- Role of persistence in translating skewness into lower means:
  - Benchmark persistence parameter in shock process: 0.9.
  - Annualized output loss relative to no-disaster scenario at persistence 0.9: almost 2 percent.
  - If persistence increased to 0.98, the annualized output loss increases to 7 percent.
- Figure 1 (persistence vs. annualized percentage drop in output)
  - Persistence values and associated annualized percentage drops shown: 0.90 → 1.7%; 0.92 → 2.1%; 0.94 → 2.7%; 0.96 → 3.8%; 0.98 → 6.9%.

### The policy trade-off imposed by rare disasters
- Core result:
  - Rare disasters move the central bank’s two objectives—price stability and output stability—in opposite directions, creating a policy trade-off.
- Frequency experiment
  - Disaster frequency raised from zero up to benchmark 1.7 percent (Barro (2006) suggested frequency).
  - For fixed shock size, increases in disaster frequency:
    - Make output skewness more negative.
    - Increase inflation skewness substantially.
  - Implication: central bank actions to stabilize output (via nominal interest rate adjustments) exacerbate inflation skewness; higher disaster likelihood worsens the trade-off.
- Taylor-inspired skewness curves (Figure 3)
  - Procedure: vary the output coefficient in the Taylor rule from 0.5 to 1.1 and plot resulting skewness of inflation and output.
  - Intuition via marginal cost equation (reproduced as in source):
    - mc_t = [ (1/(1−α))^(1−α) (1/α)^α ] w_t^(1−α) (R^k_t)^α A_t   (Equation 45 as presented)
  - Mechanism: productivity drops during disasters reduce output and raise marginal costs, causing inflation and output to move in opposite directions.

### Navigating the trade-off: welfare-focused policy
- Objective: identify Taylor-rule parameterization that minimizes welfare skewness in the presence of rare disasters.
- Welfare definition (as in source)
  - Welfare_t = E_t Σ_{t=1}^∞ β^t U_t  (= U_t + βE_t Welfare_{t+1})
  - Welfare measure is implied by the third-order approximation of the model.
- Experiment: vary the coefficient on output (φ_y) in the Taylor rule; Taylor rule assumed is r_t − r = φ_π(π_t − π) + φ_y(y_t − y_{t−1}), with φ_π = 1.5.
- Main results (Figure 4)
  - Increasing emphasis on output stabilization reduces welfare skewness up to a threshold at φ_y = 1.1.
  - Beyond φ_y = 1.1, welfare skewness becomes more negative, implying an optimal φ_y from households’ perspective.
  - Strict-inflation-focused policy is not effective for climate-induced shocks because it introduces policy inertia: responding to growth rates influenced by lagged output creates periods of above-target inflation but smooths welfare distributional outcomes.

### Increasing the inflation target (2 percent → 4 percent)
- Policy experiment: raise inflation target from 2 percent to 4 percent.
- Rationale: higher target shrinks the inflation gap when rare disasters cause deviations, potentially reducing needed output adjustments and interest rate changes.
- Taylor-inspired skewness curves (Figure 5)
  - Comparison between 2 percent and 4 percent inflation targets:
    - Higher inflation target shifts the policy frontier left, reducing the cost of stabilizing output versus the 2 percent target.
    - Improvement is limited.
- Welfare implications (Figure 6)
  - While a 4 percent target reduces the cost of output stabilization, it produces a more negatively skewed welfare measure.
  - Mechanisms:
    - Firms set higher price mark-ups (larger price adjustments) to protect prices over reset intervals.
    - Larger price adjustments increase price dispersion.
    - Combined effect: fall in potential (long-run) output.
  - Consistent finding: the Taylor rule with a 1.2 coefficient on output minimizes welfare skewness across targets, but higher inflation target carries a potential lasting welfare cost.

### Implementation of monetary policy and the Woodford/Taylor principles in the era of rare disasters
- Long-run Taylor rule expression (as in source):
  - r = φ_π π + φ_y y  (Equation 49 as presented)
- Modified Taylor principle and equilibrium determinacy
  - ∂r/∂π = φ_π + φ_y ∂y/∂π > 1  (Equation 50 as presented)
  - Equilibrium determinacy condition:
    - φ_π > 1 − φ_y ∂y/∂π  (Equation 51 as presented)
  - Interpretation: required inflation response φ_π depends on the slope of the long-run Phillips curve via the term φ_y ∂y/∂π.
- Effects of rare disasters on determinacy
  - Disasters reduce long-run output; negatively skewed output lowers ∂y/∂π and thus reduces φ_y ∂y/∂π.
  - As a result, maintaining determinacy requires a larger φ_π (stronger inflation response).
- Policy tension:
  - After a disaster central banks may want to increase φ_y to stabilize output immediately.
  - Simultaneously, because long-run output is lower, central banks must ensure φ_π increases sufficiently to satisfy the generalized Taylor/Woodford principle and maintain equilibrium determinacy.

*Italic: Source — Content unit "wpiea2023243-print-pdf - 3. Results" from the provided PDF.*

### 4. Optimal Monetary Policy and Debt to GDP Ratio in the Pres-

### 4. Optimal Monetary Policy and Debt to GDP Ratio in the Presence of Climate-Induced Disasters

### Debt-to-GDP implications of alternative Taylor-rule weights
- Baseline: standard Taylor rule with φy = 0.5 exhibits pronounced asymmetry in the debt-to-GDP ratio under rare disasters.
- Under a 2 percent inflation target:
  - Increasing the coefficient on output to φy = 1.2 reduces the ratio’s skewness to 1.42.
  - The minimum skewness for the debt-to-GDP ratio occurs at φy = 1.2.
  - Beyond φy = 1.2, diminishing returns in skewness reduction become evident; further increases can be counterproductive.
- Under a 4 percent inflation target:
  - Prioritizing output stabilization remains beneficial up to φy = 1.2.
  - When the coefficient on output exceeds φy = 1.2, the skewness of the debt-to-GDP ratio worsens.
- Key interpretation:
  - A Taylor rule that places greater emphasis on output stabilization tends to stabilize the debt-to-GDP ratio.
  - Raising the inflation target can negatively impact skewness and output, thereby increasing the debt-to-GDP ratio.

### Robustness — role of shock persistence (Table 3 results)
- Shock persistence values considered: 0.8, 0.9 (benchmark), 0.95.
- Reported outcomes (φπ = 1.5):
  - Persistence of the shock 0.8: Optimal coefficient on output in Taylor rule 1.5; Reduction in skewness 16%.
  - Persistence of the shock 0.9 (benchmark): Optimal coefficient on output in Taylor rule 1.2; Reduction in skewness 9%.
  - Persistence of the shock 0.95: Optimal coefficient on output in Taylor rule 1.1; Reduction in skewness 4%.
- Interpretation:
  - Main conclusion unaffected by persistence variations: balanced focus on output and inflation is crucial.
  - Higher shock persistence corresponds to a lower optimal φy and smaller reductions in skewness.
  - Less persistent shocks yield larger welfare gains from output stabilization.

### Robustness — milder disasters (Table 4 results)
- Alternative assumed disaster shock size: 15 percent fall in productivity (versus 29 percent in benchmark).
- Reported outcomes:
  - Persistence of the shock 0.8: Optimal coefficient on output in Taylor rule 1.4; Reduction in skewness 19%.
  - Persistence of the shock 0.9 (benchmark severity reduced): Optimal coefficient on output in Taylor rule 1.2; Reduction in skewness 11%.
  - Persistence of the shock 0.95: Optimal coefficient on output in Taylor rule 1.1; Reduction in skewness 5%.
- Interpretation:
  - With milder disasters, greater gains from an optimal policy arise and reductions in skewness are larger (e.g., 11% vs. 9% in the benchmark comparison).
  - As with other experiments, lower persistence magnifies the benefits of output stabilization.

### Variations in the inflation coefficient
- Baseline inflation coefficient used in robustness checks: φπ = 1.5.
- Figure 8 outcome summary:
  - Varying the inflation coefficient does not change the main conclusion: the skewness of welfare is minimized when the Taylor rule balances output and inflation stabilization.
  - As the coefficient on inflation increases, the marginal gains from emphasizing output stabilization diminish.
- Conceptual note:
  - Variations in φπ can be interpreted as a central banker’s “trembling hand”; involuntary deviations do not alter the qualitative optimality of balancing output and inflation.

### Summary of policy-relevant findings and recommendations
- Main findings:
  - Climate-induced rare disasters create asymmetric risks: inflation exhibits upward skewness; output exhibits downward skewness.
  - A Taylor rule that places greater emphasis on output stabilization than the standard rule mitigates welfare losses from rare disasters and stabilizes the debt-to-GDP ratio.
  - Optimal φy is empirically around 1.2 in the benchmark; optimal φy declines with increased shock persistence.
  - Increasing the inflation target from 2 percent to 4 percent can partly alleviate the policy trade-off but reduces household welfare (via higher price mark-ups and price dispersion) and contributes to a higher debt-to-GDP ratio.
  - Policymakers must account for entire outcome distributions (skewness), not only means and standard deviations.
- Policy recommendations and considerations:
  - Adopt a Taylor rule that places substantial weight on output stabilization (targeting around φy ≈ 1.2 in the benchmark) while maintaining a credible inflation response.
  - Recognize diminishing returns to increasing φy beyond the optimal threshold; avoid over-tuning the Taylor rule toward output stabilization.
  - Consider higher inflation targets cautiously: they can reduce the trade-off but come with lower household welfare and higher debt-to-GDP.
  - Account for changes in long-run determinants: disasters can reduce long-run output and the slope of the Phillips curve, implying the need for more aggressive inflation responses to ensure equilibrium determinacy.
  - Preserve flexibility for unconventional measures (e.g., QE) at the time of shocks; detailed design of unconventional measures requires further research.
  - Incorporate full distributional risks (including skewness) in monetary policy design in disaster-prone environments.

*Monetary Policy Design with Recurrent Climate Shocks — Working Paper No. WP/2023/243*

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_Source: https://www.imf.org/-/media/files/publications/wp/2023/english/wpiea2023243-print-pdf.pdf_