## wpiea2023174-print-pdf - Section 3 discusses the results.  Section 4 concludes.

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### Model structure and purpose
- Purpose: quantify how carbon pricing policy shocks affect corporate default risks and transition dynamics via a parsimonious continuous-time model that mainly uses corporate balance sheet data and does not rely on market-based data.
- Economy: continuum of firms of mass one and a financial sector represented by a bank.
- Carbon pricing: modeled as a shock to firm cash flows via a carbon price parameter P and captured compactly by δ = 1 − θP.
- Key structural elements and notation:
  - Firms’ idiosyncratic cash-flow shocks: Brownian motion Wf t with volatility σ.
  - Emission intensity θ: emissions proportional to output via E = θμ.
  - Firms’ debt expense parameter γ; firms’ discount rate ρ.
  - Firm objective: V(x) = max{ct, ̄αt} E ∫0∞ e−ρt ct dt with net worth dynamics dXt = (̄αt(μδ − γ) − ct) dt + ̄αt σ dWt.
  - Default time τ = inf{t > 0 : Xt ≤ 0}; default rate φ is the killing rate of the diffusion and equals the average outflow (and birth) rate in steady state.
- Risk management: firms choose ̄αt ∈ [0,1] (exposure kept) trading off mean income ̄αt(μ − γ) against volatility ̄αt σ.
- Cross-sectional dynamics: Kolmogorov Forward equation ∂t gt(x) = L∗ g(x) + φ ̄g(x), with L∗ g(x) = −∂x(η(x) gt(x)) + 1/2 ∂xx(gt(x) ˆσ2) − φ gt(x).
- Financial sector: representative bank with net worth dynamics dXb t = ( ̃μ + R Xb t − cb t) dt + ̃σ dWb t and HJB ρ V(xb) = max cb [ cb + (ψ + ̃μ − cb) V′(xb) + 1/2 ̃σ2 V′′(xb) ].
- Equilibrium closure (partial-equilibrium focus): aggregate corporate debt expenses Tf = −E[̄α γ], bank income Tb, and market clearing Tb + Tf = 0 used to map corporate risk changes into implied cost of capital R for comparative statics.

### Calibration (2012–19 US sector cases)
- Sectors calibrated: Transportation (SIC group 5, codes 4000–4800) and Manufacturing (SIC group 3, codes 2000–4000).
- Data: Datastream; asset-weighted averages for 2012-19. Net worth X normalized to [0,1]. Variables winsorized at 5% and 95%.
- Calibration mappings:
  - μ: average EBIT; γ calibrated from R = Interest Expenses / Total Debt and γ = R × B.
  - m: payout rate m = average dividend payout ratio × μ.
  - ̄x: average shareholder equity of dividend-paying firms.
  - σ: set to match fraction of firms with ICR < 1; sectoral default-rate proxies: 16% (transportation), 36% (manufacturing).
  - δ: captures carbon-price impact via δ = 1 − θP.
- Baseline calibrated parameters (2012-19 sample):
  - μ: Transportation 0.1615; Manufacturing 0.1350
  - γ: Transportation 0.0250; Manufacturing 0.0183
  - σ: Transportation 0.1977; Manufacturing 0.2886
  - m: Transportation 0.0344; Manufacturing 0.0140
  - ̄x: Transportation 0.2738; Manufacturing 0.2578
- Carbon price scenarios considered: $25, $50, $75 per tCO2e.
- Calibrated δ and θP shocks:
  - Transportation:
    - Baseline: δ = 1 (θP = 0%)
    - $25: δ = 0.92 (θP = -8%)
    - $50: δ = 0.84 (θP = -16%)
    - $75: δ = 0.76 (θP = -24%)
  - Manufacturing:
    - Baseline: δ = 1 (θP = 0%)
    - $25: δ = 0.9948 (θP = -0.52%)
    - $50: δ = 0.9896 (θP = -1.04%)
    - $75: δ = 0.9844 (θP = -1.56%)
- Emission intensity ratio: transportation’s emission intensity is about 15 times higher than manufacturing.
- Mitigation experiments: ∆θ = −10% used to assess the effect of emission-intensity reductions.

### Carbon price shocks and default risks — closed-form results and quantitative implications
- Optimal payout policy (Proposition 1):
  - c* = 0 whenever x ≤ ̄x and c* = m whenever x > ̄x.
- Closed-form default probabilities (exact expressions):
  - For x ≤ ̄x:
    - P = m+ (μδ−γ−m)e^{2(μδ−γ) / σ^2 (̄x−x)} / m+ (μδ−γ−m)e^{2(μδ−γ) ̄x}
  - For x > ̄x:
    - P = (μδ−γ)e^{−2(μδ−γ−m)(x−̄x) / σ^2} / m+ (μδ−γ−m)e^{2(μδ−γ) ̄x / σ^2}
- Key analytic implications:
  - Default probability P decreases with higher net worth x.
  - Carbon price shocks enter non-linearly via δ and affect P asymmetrically across the net worth distribution.
  - If m → (μδ−γ), then P → 1.
- Transportation sector quantitative effects (averages and heterogeneity):
  - Baseline average default risk P = 0.16 (16%).
  - Under $25, $50, $75 shocks: average default rates 0.1826, 0.2121, 0.2517.
  - Changes ∆P: 0.0266, 0.0521, 0.0917.
  - Panel B heterogeneity (transportation):
    - Baseline bottom 10%: 0.7098; top 10%: 0.0055.
    - Bottom 10% under $25/$50/$75: 0.7356, 0.7632, 0.7929 (∆P: 0.0258, 0.0534, 0.0831).
    - Top 10% under $25/$50/$75: 0.0102, 0.0191, 0.0359 (∆P: 0.0047, 0.0137, 0.0304).
  - Mitigation ∆θ = −10% (transportation average ∆P): 0.0201, 0.0456, 0.0785 for $25/$50/$75.
    - Bottom 10% with ∆θ = −10%: ∆P = 0.0231, 0.0477, 0.0740.
    - Top 10% with ∆θ = −10%: ∆P = 0.0041, 0.0114, 0.0242.
  - Interpretation: transportation experiences large, nonlinear increases in default risk; mitigation reduces but does not eliminate elevated risks, especially under larger shocks.
- Manufacturing sector quantitative effects:
  - Baseline average default risk P = 0.36 (36.0%).
  - Under $25, $50, $75 shocks: average default rates 0.3620, 0.3639, 0.3659.
  - ∆P: 0.0020, 0.0039, 0.0059.
  - Panel B heterogeneity (manufacturing):
    - Baseline bottom 10%: 0.8687; bottom 10% under shocks: 0.8695, 0.8703, 0.8711 (∆P: 0.0008, 0.0016, 0.0024).
    - Baseline top 10%: 0.0931; top 10% under shocks: 0.0947, 0.0962, 0.0978 (∆P: 0.0015, 0.0031, 0.0046).
  - Mitigation ∆θ = −10% (manufacturing average ∆P): 0.0018, 0.0035, 0.0053.
  - Interpretation: manufacturing shows marginal sensitivity of default risk to carbon prices relative to transportation.

### Optimal downsizing (risk reduction) and emission implications
- Closed-form optimal exposure (Proposition 2):
  - ̄α = (μδ−γ) x / [σ^2 (1−β)] for β ∈ (0,1).
- Implications:
  - ̄α linear in net worth x: higher x → higher ̄α → firms retain more risk; low-net-worth firms downsize more.
  - Critical threshold ˆx = σ^2 (1−β) / (μδ−γ): firms with x ≥ ˆx retain all risks (̄α → 1).
  - Larger carbon shocks (lower δ) → lower ̄α and higher α (more downsizing) → production and emissions decline for affected firms.
  - Policy implication: uniform carbon pricing induces emission reductions concentrated among low-net-worth firms; high-net-worth firms may not reduce production/emissions, limiting aggregate emissions reductions and imposing disproportionate burdens on lower-net-worth firms.

### Implied cost of capital / funding-cost results (implicit calibration)
- Method: calibrate corporate borrowing rate R (with δ = 1) so that changes in R reproduce model-implied average default-rate increases; R interpreted as the funding cost investors would demand.
- Transportation sector (Panel A):
  - Baseline funding cost R = 4.51%.
  - Implied R under $25, $50, $75 shocks: 6.76%, 9.01%, 11.26%.
  - ∆R: 224 bps, 449 bps, 674 bps.
  - ∆R with ∆θ = −10%: 202 bps, 405 bps, 607 bps.
  - Interpretation: substantial increases in funding costs for transportation; mitigation reduces but does not eliminate increases.
- Manufacturing sector (Panel B):
  - Baseline funding cost R = 4.05%.
  - Implied R under $25, $50, $75 shocks: 4.21%, 4.36%, 4.52%.
  - ∆R: 16 bps, 31 bps, 47 bps.
  - ∆R with ∆θ = −10%: 15 bps, 28 bps, 42 bps.
  - Interpretation: small increases in funding costs for manufacturing; mitigation has limited effect.
- Broader implication: sectoral heterogeneity implies substantially different investor risk compensation and corporate spreads across sectors under carbon pricing.

### Transition dynamics — operator-theory approach and speed of adjustment
- Objective: characterize time evolution of cross-sectional net worth distribution g_t(x) to new steady state after carbon price shocks.
- Dynamics used: KF equation ∂_t g_t(x) = L* g(x) + φ ̄g(x) with L* g(x) = −η ∂_x g_t(x) + 1/2 σ^2 ∂_xx g_t(x) − φ g_t(x), η = (μδ − γ − m).
- Speed of convergence (Proposition 3, closed-form):
  - S = φ + (μδ − γ − m)^2 / (2σ^2) + π^2 σ^2 / (2 u^2)
  - Half-life t_1/2 = log(2) / S.
- Theoretical implications:
  - Higher default rate φ → faster transition speed S.
  - Carbon price shocks affect S non-linearly via (μδ − γ − m)^2 and via φ.
  - Tradeoff: larger carbon price shocks increase default risk but accelerate transition to new steady state; mitigation reduces default risk but slows transition.
- Sectoral half-life results (exact reported values):
  - Transportation half-lives (direct shocks):
    - $25 → t_1/2 = 1.4500 years
    - $50 → t_1/2 = 1.4411 years
    - $75 → t_1/2 = 1.3919 years
  - Transportation with ∆θ = −10%:
    - $25 → t_1/2 = 1.4488 years
    - $50 → t_1/2 = 1.4457 years
    - $75 → t_1/2 = 1.4113 years
  - Manufacturing half-lives (direct shocks):
    - $25 → t_1/2 = 0.8295 years
    - $50 → t_1/2 = 0.8284 years
    - $75 → t_1/2 = 0.8273 years
  - Manufacturing with ∆θ = −10%:
    - $25 → t_1/2 = 0.8296 years
    - $50 → t_1/2 = 0.8287 years
    - $75 → t_1/2 = 0.8276 years
- Observations:
  - Transportation has slower baseline transition and shows larger non-linear acceleration of transition speed as shocks increase.
  - Manufacturing settles faster and shows almost linear changes in transition speed across shocks.
- Exit-from-lower-tail dynamics (Proposition 4, closed-form):
  - For interval [x1,x2] and κ = 2(μδ − γ − m)/σ^2,
    - P(τ_{x2}) = (e^{−κ x1} − e^{−κ x}) / (e^{−κ x1} − e^{−κ x2})
  - Larger carbon shocks reduce P(τ_{x2}) — firms in the lower tail face lower probability of escaping to higher net worth; mitigation increases escape probabilities.

### Policy-relevant synthesis and implications
- Key tradeoffs and findings:
  - Carbon pricing raises default risks in carbon-intensive sectors (transportation) substantially, while effects on less carbon-intensive sectors (manufacturing) are marginal for the shock sizes considered.
  - Larger carbon price shocks induce faster structural adjustment (shorter half-lives) but at the cost of higher corporate insolvency rates and elevated implied funding costs.
  - Emission-intensity mitigation (e.g., ∆θ = −10%) lowers default risk and implied funding-cost increases but tends to slow the transition to the new steady state.
  - Uniform carbon pricing can produce asymmetric intra-sectoral burdens: low-net-worth firms downsize more and bear larger failure-risk increases; high-net-worth firms may retain production/emissions, limiting aggregate emissions reductions.
  - Implied increases in cost of capital are large in transportation (∆R up to 674 bps for a $75 shock) versus small in manufacturing (∆R up to 47 bps), indicating sector-specific risk-premium adjustments.
- Use cases:
  - Closed-form solutions and transition characterizations enable analytically tractable climate-risk stress testing and financial-stability analysis that incorporate balance-sheet heterogeneity.
- Limitations noted in analysis:
  - The paper does not make welfare statements; transition analysis pertains to evolution of the net worth distribution under exogenous carbon price shocks and a partial-equilibrium closure rather than a fully green/decarbonized general-equilibrium economy.

*Source: wpiea2023174-print-pdf - Section 3 discusses the results.  Section 4 concludes.*

### Section 3 discusses the results.  Section 4 concludes.

### wpiea2023174-print-pdf - Section 3 discusses the results.  Section 4 concludes.

### Model structure and purpose
- Purpose: quantify how carbon pricing policy shocks affect corporate default risks and transition dynamics via a parsimonious continuous-time model that mainly uses corporate balance sheet data and does not rely on market-based data.
- Economy: continuum of firms of mass one and a financial sector represented by a bank.
- Carbon pricing policy modeled as a shock to firm cash flows via a carbon price parameter P.
- Notation and key definitions:
  - Probability space (Ω,F,F,P) with filtration F = {F}t≥0.
  - Firms indexed by f ∈ [0,1].
  - Emission intensity θ: ratio of total carbon emissions E to output; emissions proportional to output via E = θμ.
  - Define δ = 1 − θP so that μ − θPμ = μδ.
  - Firms’ idiosyncratic cash-flow shocks: Brownian motion Wf t with volatility σ (σ captures productivity dispersion).
  - Firms’ debt expense parameter γ (can be represented as γ = R × B where R is interest rate and B is total debt).
  - Firms’ discount rate ρ.

### Firm cash-flow and net worth dynamics
- Cash-flow diffusion:
  - dZf t = (μ − θPμ − γ)dt + σ dWf t (equation (1))
  - Compact form using δ: dZf t = (μδ − γ)dt + σ dWf t (equation (2))
- Net worth / equity dynamics (with control variables consumption/payout ct and risk exposure ̄αt):
  - Firms maximize V(x) = max{ct, ̄αt} E ∫0∞ e−ρt ct dt (equation (3))
  - Net worth evolution: dXt = (̄αt(μδ − γ) − ct) dt + ̄αt σ dWt (equation (4))
- Dividend/payout constraints and managerial interpretation:
  - Cumulative payout Cf t = ∫0t cf u du; Cf t is non-negative, non-decreasing.
  - Two assumptions: firms pay dividends only when net worth reaches beyond threshold ̄x; payout rate bounded above: cf t ≤ m < ∞.
  - Consumption and payout used interchangeably; managers/insiders control payouts.

### Default/insolvency definition and steady-state flows
- Default time τ defined as first passage time Xt ≤ 0:
  - τ = inf{t > 0 : Xt ≤ 0}
  - Failure probability P(Xt ≤ 0) ≡ P(τ < ∞)
- Practical assumption: μ > γ (otherwise probability of default is one).
- Default/exit mechanics:
  - Defaulted/insolvent firms exit the market; default rate φ is the average outflow (and birth) rate.
  - After insolvency firms are replaced by new firms drawn from ergodic distribution ̄g(x).
  - The default rate φ is the killing rate of the diffusion process; process killed at τ and sent to cemetery state ∂.

### Risk control (proportional risk-reduction) and trade-offs
- Firms can reduce balance-sheet risk by shrinking exposure (shutting down portion of business), captured by control ̄αt ∈ [0,1]:
  - Reducing risk reduces volatility term and proportionally reduces income: firms keep ̄αt Zt of stochastic cash flow.
  - Trade-off: giving up a fraction of net income ̄αt(μ − γ) in exchange for reduced exposure ̄αt σ dWt.
- HJB characterization of firm optimization:
  - ρ V(x) = max{c, ̄α} [ c + η(x) V′(x) + 1/2 ˆσ2 V′′(x) ] (equation (5))
  - Where η(x) = (̄α(μδ − γ) − c) and ˆσ = ̄α σ.

### Cross-sectional distribution and dynamics
- Let gt(x) denote time-t cross-sectional distribution of firms over net worth; ̄g(x) denote ergodic distribution.
- Kolmogorov Forward (KF) equation for evolution of gt(x):
  - ∂t gt(x) = L∗ g(x) + φ ̄g(x) (equation (6))
  - Where L∗ g(x) = −∂x(η(x) gt(x)) + 1/2 ∂xx(gt(x) ˆσ2) − φ gt(x).

### Financial sector and equilibrium closure
- Bank problem (financial sector representative bank):
  - Bank maximizes V(xb) = max cb t E ∫0∞ e−ρb t cb t dt (equation (7)); bank discount rate ρb.
  - Bank net worth dynamics: dXb t = ( ̃μ + R Xb t − cb t) dt + ̃σ dWb t (equation (8))
    - R Xb t is income from lending at interest rate R; ̃μ income from other sources; ̃σ dWb t bank balance-sheet risk (dWb t independent of dWt).
  - Bank HJB:
    - ρ V(xb) = max cb [ cb + (ψ + ̃μ − cb) V′(xb) + 1/2 ̃σ2 V′′(xb) ] (equation (9))
- Equilibrium closure (partial-equilibrium focus):
  - Aggregate corporate debt expenses Tf = −E[̄α γ].
  - Total bank income from corporate lending Tb.
  - Market clearing in simplified closure: Tb + Tf = 0 and optimal decisions by firms and bank.
  - This simplified closure permits comparative statics with the cost of capital R to reflect heightened corporate risks even absent explicit carbon price shocks (forward-looking investor responses).

### Modeling of carbon price shocks and interpretation
- Carbon price P acts as a negative shock to firm cash flows by raising effective costs proportional to emissions θ.
- δ = 1 − θP captures the net proportional effect on mean cash flows; lower δ implies larger carbon price shock for a given θ.
- The formulation allows exploration of mitigation via reductions in θ (e.g., greener technologies, renewable energy), since emissions are proportional to output and θ represents emission intensity.
- The framework enables tracing effects of varying P on default probabilities, ergodic distributions, and implied changes in cost of capital R via comparative statics.

*Source: wpiea2023174-print-pdf - Section 3 discusses the results.  Section 4 concludes.*

### 2.1  Calibration

### 2.1  Calibration

### Calibration scope and rationale
- Model calibrated on two broad US economy sectors:
  - Transportation: SIC industry group 5 (SIC codes 4000–4800).
  - Manufacturing: SIC industry group 3 (SIC codes 2000–4000).
- Sectors chosen because they emit a large share of aggregate GHG emissions and exhibit heterogeneous emission intensities (emissions per-unit output).
- Calibration uses publicly available sector-level GHG emissions and sectoral output data; model can be applied to any sector or firm level given reliable data.

### Corporate data and parameter construction
- Data source: Datastream; focus on asset-weighted averages of relevant variables for the 2012-19 sample period.
- Main corporate variables used: earnings before interest and taxes (EBIT), total shareholder’s equity (proxy for net worth X), interest expenses, dividend payout ratio, and total debt.
- Normalization and preprocessing:
  - Net worth X normalized to lie on [0,1]; other variables scaled accordingly.
  - Variables winsorized at 5% and 95% to remove outliers.
- Parameter definitions and calibration steps:
  - μ: average EBIT for each sector (EBIT as proxy for free cash flow; excludes capital structure effects).
  - γ: calibrated from average/effective interest rate R and total debt B using R = Interest Expenses / Total Debt and γ = R × B. This allows mapping carbon-price-equivalent impacts to implicit increases in cost of debt.
  - m: payout rate calibrated as m = average dividend payout ratio × μ.
  - ̄x: average shareholder equity value of companies that paid positive dividends.
  - σ: volatility calibrated to match the average fraction of firms with interest coverage ratio ICR = EBIT / Interest Expenses below 1 (ICR < 1 implies financial distress). Sectoral default-rate proxies: 16% for transportation, 36% for manufacturing.
  - δ: shock parameter capturing the impact of carbon price on firm earnings (additional cost of carbon emissions).

### Calibrated baseline parameters (2012-19 sample)
- Parameters for Transportation and Manufacturing (Baseline):
  - μ: Transportation 0.1615; Manufacturing 0.1350
  - γ: Transportation 0.0250; Manufacturing 0.0183
  - σ: Transportation 0.1977; Manufacturing 0.2886
  - m: Transportation 0.0344; Manufacturing 0.0140
  - ̄x: Transportation 0.2738; Manufacturing 0.2578
- Note: Calibrated parameter values use US corporate data for the two sectors over the 2012-19 sample period.

### Carbon emissions, price shocks, and emission intensity
- Carbon price shocks considered: $25, $50, and $75 per tCO2e, reflecting a plausible range of future carbon prices for scenario analysis.
- Emissions data source: Ritchie and Roser (2020); emissions reported in tCO2e. Industry-level GDP/output data obtained from Bureau of Economic Analysis (BEA) for 2012-19.
- Emission intensity θ computed as average emissions per-unit of output over the sample period for each sector.
- δ captures the earnings shock from a carbon price via δ = 1 − θP (where θP denotes the proportional earnings reduction from carbon price).

### Calibrated carbon-shock multipliers δ and percent shocks θP
- Transportation:
  - Baseline (no shock): δ = 1 (θP = 0%)
  - $25 shock: δ = 0.92 (θP = -8%)
  - $50 shock: δ = 0.84 (θP = -16%)
  - $75 shock: δ = 0.76 (θP = -24%)
- Manufacturing:
  - Baseline (no shock): δ = 1 (θP = 0%)
  - $25 shock: δ = 0.9948 (θP = -0.52%)
  - $50 shock: δ = 0.9896 (θP = -1.04%)
  - $75 shock: δ = 0.9844 (θP = -1.56%)
- Interpretation: Transportation faces substantially larger earnings shocks for the same carbon prices due to higher sectoral emission intensity.

### Additional calibration notes and implications
- Emission intensity ratio: transportation sector’s emission intensity is about 15 times higher than manufacturing.
- Any mitigation measures that reduce emission intensity θ will lessen the impact of shock δ proportionally.
- σ is chosen so that model-implied default probabilities match sectoral proxies of 16% (transportation) and 36% (manufacturing), consistent with ICR-based distress measures.
- The model can also be calibrated using historical bankruptcy rates where available, but sectoral granularity and data availability limit this approach; Chava and Jarrow (2004) report a historical bankruptcy rate of 37% in manufacturing (for a broader aggregated SIC group), which is close to the 36% proxy used here.
- Once baseline is established, carbon price shocks are applied per the model equations to analyze impacts on default risk and related outcomes.

*Source: wpiea2023174-print-pdf - 2.1  Calibration*

### 3.1  Carbon Price Shocks and Default Risks

### 3.1  Carbon Price Shocks and Default Risks

### Closed-form default probability and implications
- Proposition 1 (Default Risk): For optimal consumption payout rate c*, with cutoff net worth ̄x,
  - c* = 0 whenever x ≤ ̄x and c* = m whenever x > ̄x.
  - Default probability P(X_t ≤ 0) for firms with net worth x ≤ ̄x is given by equation (10):
    - P = m+ (μδ−γ−m)e^{2(μδ−γ) / σ^2 (̄x−x)} / m+ (μδ−γ−m)e^{2(μδ−γ) / σ^2 ̄x}
  - Default probability P(X_t ≤ 0) for firms with net worth x > ̄x is given by equation (11):
    - P = (μδ−γ)e^{−2(μδ−γ−m)(x−̄x) / σ^2} / m+ (μδ−γ−m)e^{2(μδ−γ) ̄x / σ^2}
- Key analytical implications:
  - Firms with higher (lower) net worth x face lower (higher) failure probability P.
  - If m → (μδ−γ), then P → 1, implying maximal consumption rate cannot exceed available funds (μδ−γ) without producing unit failure probability.
  - Carbon price shocks enter via δ and affect P non-linearly and asymmetrically across the net worth distribution.

### Quantitative effects: Transportation sector (figures and Table 3 excerpts)
- Baseline (no shock) average default risk P = 0.16 (16%).
- Under carbon price shocks of $25, $50, and $75:
  - New average default rates: 0.1826, 0.2121, 0.2517.
  - Changes in average default rate ∆P: 0.0266, 0.0521, 0.0917.
- Effects on distribution:
  - Figure 1: larger carbon price shocks increase default risk and show highly non-linear effects across net worth; $75 shock produces non-trivial increases even for high-net-worth firms.
  - Figure 2: distribution of failure probabilities P shifts right with larger shocks, with substantial mass increase in the right tail under $75 shock.

### Heterogeneity within transportation sector and mitigation (∆θ=−10%)
- Panel B, transportation sector:
  - Baseline default rates:
    - Bottom 10% of firms: 0.7098 (70.98%).
    - Top 10% of firms: 0.0055 (0.55%).
  - Under $25, $50, $75 shocks, bottom 10% default rates: 0.7356, 0.7632, 0.7929.
    - ∆P for bottom 10%: 0.0258, 0.0534, 0.0831.
  - Under $25, $50, $75 shocks, top 10% default rates: 0.0102, 0.0191, 0.0359.
    - ∆P for top 10%: 0.0047, 0.0137, 0.0304.
- Mitigation via 10% reduction in emission intensity (∆θ = −10%):
  - Transportation sector average ∆P with ∆θ = −10%: 0.0201, 0.0456, 0.0785 (corresponding to $25, $50, $75 shocks).
  - Bottom 10% with ∆θ = −10%: default rates change by 0.0231, 0.0477, 0.0740.
  - Top 10% with ∆θ = −10%: default rates change by 0.0041, 0.0114, 0.0242.
- Interpretation:
  - Mitigation (10% reduction in θ) lowers default risk but reductions are not always material, especially except under the $75 shock.
  - Carbon shocks produce asymmetric intra-sectoral impacts; large shocks can significantly raise default risk even for top 10% firms.

### Manufacturing sector results (Table 4 excerpts)
- Baseline average default risk P = 0.36 (36.0%).
- Under $25, $50, $75 shocks:
  - New average default rates: 0.3620, 0.3639, 0.3659.
  - ∆P: 0.0020, 0.0039, 0.0059.
- Panel B, manufacturing heterogeneity:
  - Baseline bottom 10%: 0.8687; bottom 10% under shocks: 0.8695, 0.8703, 0.8711.
    - ∆P bottom 10%: 0.0008, 0.0016, 0.0024.
  - Baseline top 10%: 0.0931; top 10% under shocks: 0.0947, 0.0962, 0.0978.
    - ∆P top 10%: 0.0015, 0.0031, 0.0046.
- Mitigation (∆θ = −10%) in manufacturing:
  - ∆Pwith ∆θ=−10% (average): 0.0018, 0.0035, 0.0053.
  - Effects from mitigation are marginal relative to transportation sector; manufacturing shows smaller sensitivity of default risk to carbon price shocks.

### Firm-level optimal downsizing and emission implications (Proposition 2)
- Proposition 2 (Optimal Risk Reduction):
  - Optimal risk reduction/downsizing α_t = 1 − ̄α_t where
    - ̄α = (μδ−γ) x / [σ^2 (1−β)]  (equation (12)), for β ∈ (0,1).
- Implications:
  - ̄α is linear in net worth x: higher x → higher ̄α → firms keep more risk and downsize less.
  - Critical net worth threshold ˆx = σ^2 (1−β) / (μδ−γ): firms with x ≥ ˆx retain all risks (̄α → 1); low-net-worth firms downsize more.
  - Larger carbon shock δ → lower ̄α and higher α (more downsizing) → production and emissions decline for affected firms.
  - Policy implication: direct carbon pricing induces emission reductions primarily through downsizing among low-net-worth firms; high-net-worth firms beyond ˆx may not reduce production/emissions, limiting aggregate emissions reductions.
  - Uniform carbon pricing may thus have limited effectiveness and impose disproportionate burdens on low-net-worth firms.

### Impact on Cost of Capital (implicit calibration, Table 5 excerpts)
- Method: calibrate corporate borrowing rate R (with δ = 1) to match the average default rate increases induced by carbon price shocks; resulting R reflects implicit increase in cost of capital investors would demand.
- Transportation sector (Panel A):
  - Baseline funding cost R = 4.51%.
  - Implied R under $25, $50, $75 shocks: 6.76%, 9.01%, 11.26%.
  - ∆R: 224 bps, 449 bps, 674 bps.
  - ∆Rwith ∆θ=−10%: 202 bps, 405 bps, 607 bps (resulting funding costs corresponding to mitigation case).
  - Interpretation: substantial increases in cost of capital for transportation; mitigation reduces but does not eliminate the elevated funding costs.
- Manufacturing sector (Panel B):
  - Baseline funding cost R = 4.05%.
  - Implied R under $25, $50, $75 shocks: 4.21%, 4.36%, 4.52%.
  - ∆R: 16 bps, 31 bps, 47 bps.
  - ∆Rwith ∆θ=−10%: 15 bps, 28 bps, 42 bps.
  - Interpretation: increases in funding costs are mild for manufacturing; mitigation has limited effect on funding cost reduction.
- Broader implications:
  - Large implicit shocks to cost of capital in transportation relative to manufacturing imply sector-specific increases in corporate spreads/risk premia.
  - Heterogeneous, non-linear impacts across sectors suggest risk compensation needs will differ sharply by sector.
  - Current asset-pricing estimates may underestimate the impact of carbon emissions/disclosures on risk premium and corporate spreads if they do not account for a range of potential carbon shocks.

*Source: wpiea2023174-print-pdf - 3.1  Carbon Price Shocks and Default Risks*

### 3.3  Transition Dynamics

### 3.3  Transition Dynamics

### Objectives and approach
- Goal: understand the transition dynamics (time evolution to a new long-run steady state) after carbon price shocks, complementing comparative-statics default-risk results.
- Key methodological challenge: characterizing time evolution of the entire cross-sectional net worth distribution g_t(x).
- Approach used: operator theory / Sturm–Liouville spectral methods (an operator-theory approach related to Gabaix et al. (2016), with details in the appendix), with simplifying assumptions including net worth normalized to [0,1] (u = 1) and reflecting boundaries on [0,u].
- Evolution of the net worth distribution (KF / Kolmogorov–Forward form, restated):
  - ∂_t g_t(x) = L* g(x) + φ ̄g(x) (equation (13))
  - L* g(x) = −η ∂_x g_t(x) + 1/2 σ^2 ∂_xx g_t(x) − φ g_t(x), with η = (μδ − γ − m)

### Transition speed: Proposition 3 and interpretation
- Definition: Let ||g_t(x) − ̄g(x)|| denote the L1-norm distance between the distribution at time t and the stationary distribution. Let S denote the average speed of convergence.
- Closed-form expression (Proposition 3, equation (14)):
  - S = φ + (μδ − γ − m)^2 / (2σ^2) + π^2 σ^2 / (2 u^2)
- Key theoretical implications:
  - The higher the default rate φ, the faster the transition speed S toward the new steady state.
    - Economic intuition: higher φ acts like a higher discount rate, lowering the value of future payoffs and making value functions and distributions settle faster; higher φ induces more agitation/turnover in the distribution (insolvencies occur faster).
  - Carbon price shocks δ and implicit shocks to the cost of capital enter non-linearly through (μδ − γ − m)^2 term and via their effect on φ; magnitudes of φ relative to other terms determine whether S behaves approximately linearly in φ or shows non-linear patterns.
  - Tradeoff for policymakers: larger carbon price shocks increase corporate default risk but also induce faster transition to the new steady state; smaller carbon shocks generate slower transition and smaller increases in default risk.
  - Speed is characterized as rate of convergence in L1-norm (robust to outliers).

### Half-life representation and sectoral heterogeneity (transportation vs manufacturing)
- Half-life definition: t_1/2 = log(2) / S (time to halve the L1 distance to steady state).
- Transportation sector (Figure 3 / Table 6, Panel A):
  - Direct carbon shocks:
    - $25 shock → Half-life t_1/2 = 1.4500 years
    - $50 shock → Half-life t_1/2 = 1.4411 years
    - $75 shock → Half-life t_1/2 = 1.3919 years
  - With ∆θ = −10% (mitigation, emission intensity reduced by 10%):
    - $25 shock → Half-life t_1/2 = 1.4488 years
    - $50 shock → Half-life t_1/2 = 1.4457 years
    - $75 shock → Half-life t_1/2 = 1.4113 years
  - Observations:
    - Larger carbon price shocks lead to faster transition (smaller half-life).
    - Mitigation (∆θ = −10%) reduces transition speed (increases half-life).
    - Non-linear impact is pronounced between $50 and $75 shocks in transportation.
- Manufacturing sector (Figure 4 / Table 6, Panel B):
  - Direct carbon shocks:
    - $25 shock → Half-life t_1/2 = 0.8295 years
    - $50 shock → Half-life t_1/2 = 0.8284 years
    - $75 shock → Half-life t_1/2 = 0.8273 years
  - With ∆θ = −10%:
    - $25 shock → Half-life t_1/2 = 0.8296 years
    - $50 shock → Half-life t_1/2 = 0.8287 years
    - $75 shock → Half-life t_1/2 = 0.8276 years
  - Observations:
    - Manufacturing displays an almost linear impact of carbon shocks on transition speed; half-lives are substantially smaller than transportation half-lives (manufacturing settles faster).
    - Relative increase in speed (decline in half-life) is less pronounced in manufacturing because non-linear effects from carbon shocks in equation (14) are smaller.

### Exit-from-lower-tail dynamics: Proposition 4
- Setup: Consider interval [x1, x2] near the lower tail of net worth. Let P(τ_{x2}) = P(τ_{x2} < τ_{x1}) denote probability that X_t starting at x ∈ (x1, x2) reaches upper boundary x2 before lower boundary x1. Define κ = 2(μδ − γ − m) / σ^2.
- Closed-form exit probability (Proposition 4, equation (15)):
  - P(τ_{x2}) = (e^{−κ x1} − e^{−κ x}) / (e^{−κ x1} − e^{−κ x2})
- Quantitative illustration (Figure 5, transportation sector; lower tail [x1,x2] = [0.05, 0.15], x = 0.1):
  - Larger carbon price shocks reduce the probability of exiting the low-net-worth states (P(τ_{x2}) falls), consistent with higher default risk and larger mass in the lower tail.
  - Mitigation (reducing emission intensity) increases P(τ_{x2}), allowing firms a higher chance to escape the lower tail and lowering default risk.

### Synthesis of implications
- Larger carbon price shocks:
  - Increase default risks in carbon-intensive sectors.
  - Increase the transition speed to the post-shock steady state, with generally non-linear dependence on emission intensity and shock size.
- Sectoral heterogeneity:
  - Transportation (high emission intensity): slower baseline transition, larger non-linear acceleration of transition speed as carbon shocks increase.
  - Manufacturing: faster baseline transition, approximately linear speed responses to shocks.
- Policy-relevant insights:
  - There is a tradeoff between inducing faster structural adjustment (faster transition) and raising corporate default risk; mitigation measures can reduce default risk but may slow transition.
  - Closed-form solutions for default risk and transition dynamics enable analytically tractable policy analysis (e.g., climate risk stress testing, financial stability analysis).
- Caution: the paper does not make welfare statements; transition statements pertain specifically to evolution of the net worth distribution across firms under exogenous carbon price shocks, not to a fully green/decarbonized economy.

*Source: Section 3.3, "Transition Dynamics", from wpiea2023174-print-pdf*

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*Default Risk and Transition Dynamics with Carbon Shocks — Working Paper No. WP/2023/174*

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_Source: https://www.imf.org/-/media/files/publications/wp/2023/english/wpiea2023174-print-pdf.pdf_