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### Empirical model and estimation
- Baseline STVAR specification:
  - Yt = (1 − F(zt)) AL(l) Yt−1 + F(zt) AH(l) Yt−1 + εt, εt ∼ N(0, Ωt)
  - Ωt = (1 − F(zt)) ΩL + F(zt) ΩH
  - F(zt) = exp[−γzt] / (1 + exp[−γzt]), γ > 0
- Endogenous variables in Yt (order): government spending (Gt); average personal income tax rate (APITRt); average corporate income tax rate (ACITRt); personal income tax base (PBt); corporate income tax base (CBt); output (GDPt); fiscal news; debt-to-GDP ratio.
- Data and sample:
  - Quarterly sample from 1950Q1 to 2006Q4.
  - ACITRt = TCPt / CPt.
  - APITRt = (PTt + SIt) / PTIt.
  - GDP, corporate and personal tax bases in logarithms and real terms.
  - Fiscal news: Ramey’s defense spending news.
- State variable and calibration:
  - zt = detrended debt-to-GDP via third-order polynomial (deviations above/below trend).
  - Transition smoothness calibrated to γ = 2.1.
  - VAR lag length = 3 (Akaike criterion).
- Identification and estimation:
  - Estimation via MCMC algorithm (Chernozhukov and Hong (2003)).
  - Structural Cholesky-type identification.
  - GIRFs following Koop et al. (1996).
  - Shock normalization: 1 percentage point reduction in the corporate income tax rate.
  - Forecast horizon for GIRFs: 20 quarters.
- Key methodological choices:
  - Disaggregation into corporate and personal income taxes (Mertens and Ravn (2013)); focus on corporate tax as proxy for capital income taxes.
  - GIRFs use all histories of a given debt state; 90-percent confidence bands for inference.
- Exact technical settings to preserve:
  - Calibration: γ = 2.1.
  - Sample: 1950Q1–2006Q4.
  - VAR lags: 3.
  - Shock normalization: 1 percentage point corporate tax cut.
  - GIRF horizon: 20 quarters.

### Interpretation of nonlinear setup
- Continuous, debt-dependent regime switching via F(zt) — observations contribute with varying weights rather than a binary dummy.
- State-contingent variance-covariance matrix Ωt with ΩL in low-debt and ΩH in high-debt periods, allowing different shock volatility across debt states.

### Estimation results (STVAR and VAR GIRFs)
- Linear VAR GIRF: persistent output increase that then stabilizes.
- Nonlinear STVAR GIRFs: reversal dynamics and heterogeneous fiscal responses; distinct high-debt (red) and low-debt (blue) state responses.
- State-dependent responses (90‑percent bands):
  - Corporate income tax cut is expansionary in low-debt state; less expansionary and sometimes contractionary in high-debt state.
  - Short-run heterogeneity emerges after two quarters between states.
  - Low-debt state features:
    - Stronger increases in personal and corporate tax bases.
    - More pronounced output increase.
    - Larger increases in government spending and average personal income tax rate relative to high-debt state.
- Multipliers (construction):
  - Tax revenue response: TRgirft = ACITRgirft * ACITR + CBgirft, where ACITR is mean average corporate income tax rate.
  - Eight-quarter cumulative output multipliers:
    - Low-debt state: −0.36 (statistically significant at the 8-quarter horizon).
    - High-debt state: −0.14.
- Literature comparison (verbatim reported estimates):
  - Romer and Romer (2010): long-run output multiplier of −3.
  - Blanchard and Perotti (2002): twenty-quarter multiplier of −0.8 (−1.3) under deterministic (stochastic) trend specification.
  - Mountford and Uhlig (2009): five-year cumulative multiplier of −5.
  - Barro and Redlick (2011): tax multiplier around −1.1.
  - Mertens and Ravn (2013): multiplier at about −2.5 to −3.
- Endogenous transition dynamics:
  - High likelihood of remaining in high-debt state; switching to low-debt may occur after 15 quarters in baseline.

### DSGE simulation summary (mechanism and quantitative examples)
- Two policy regimes: tax cut regime (no fiscal adjustment) and fiscal adjustment regime (government increases primary surplus to stabilize debt).
- Regime switching probability linked to current debt and fiscal limits (Bi (2012)); switching probability = Pr(current debt exceeds a randomly drawn fiscal limit).
- Simulation of persistent capital income tax cut similar to 2017 TaxAct reduction.
- Baseline fiscal adjustments: transfer reduction and policy reversal (increase in capital income tax); alternative: consumption tax increases (Section 5.5).
- Selected quantitative simulation results (expectation channel only; adjustments do not actually occur during horizon):
  - With expected policy reversal and initial U.S. federal net debt = 45 percent of GDP:
    - Impact output multiplier: −0.5
    - Five-year cumulative output multiplier: about −1
  - With initial debt = 80 percent of GDP:
    - Impact output multiplier: −0.4
    - Five-year cumulative output multiplier: −0.7
  - Same tax cut turns contractionary when debt rises to 120 percent of GDP.
- Mechanism: higher probability of policy reversal lowers expected after-tax return to current investment ⇒ reduces investment incentive ⇒ lower capital and marginal product of labor ⇒ smaller absolute output multipliers at higher debt.

### Robustness checks (empirical)
- Use narratively constructed, exogenous corporate income tax series (Mertens and Ravn (2013)) ordered first.
  - Result: debt-dependent effects persist — output response mostly positive in low-debt and mostly negative in high-debt state.
- Additional checks (conducted but not fully presented):
  1) Adding aggregate narratively constructed tax news measures (Mertens and Ravn (2012)).
  2) Adding unanticipated exogenous aggregate tax changes (Mertens and Ravn (2012)).
  3) Different orderings of variables.
- Conclusion: finding that capital income tax cuts are more expansionary in low-debt than high-debt states remains robust.

### Theoretical baseline model — purpose and mechanism preview
- Use neoclassical DSGE growth model with regime-switching fiscal policy to explain debt-dependent capital income tax effects.
- Baseline fiscal adjustments jointly via policy reversal (increase capital income tax) and transfer reduction; Section 5.5 allows consumption tax increase + transfer reduction.
- Key mechanism: expectations of fiscal adjustments (with higher probability in worse fiscal states) reduce expected after-tax return to investment, weakening expansionary effect of tax cuts when debt is high.

### The private sector (model setup)
- Representative agent maximizes:
  - E_t ∑_{t=0}^{∞} d_t ( c_t^{1−σ}/(1−σ) − l_t^{1+φ}/(1+φ) ), with d_t = ∏_{j=1}^{t} β_j, d_0 ≡ 1.
  - β_t follows AR(1): ln(β_t/β) = ρ_β ln(β_{t−1}/β) + ε^β_t, ε^β_t ∼ N(0, σ^2_β).
- Budget constraint:
  - c_t + i_t + q_t b_t = (1−τ^l_t) w_t l_t + (1−τ^k_t) r^k_t k_{t−1} + b_{t−1} + z_t.
- Capital accumulation with adjustment costs:
  - k_t = (1−δ) k_{t−1} + i_t − κ/2 ( i_t/k_{t−1} − δ )^2 k_{t−1}.
- Firms: Cobb-Douglas y_t = a_t k_{t−1}^α l_t^{1−α}; TFP a_t AR(1): ln(a_t/a) = ρ_a ln(a_{t−1}/a) + ε^a_t, ε^a_t ∼ N(0, σ^2_a).
- Shocks: ε^β_t demand shocks; ε^a_t supply shocks — used in fiscal limit simulations.

### Government — regimes and fiscal rule
- Budget constraint:
  - tax_t + q_t b_t = g_t + b_{t−1} + z_t, tax_t ≡ tax^l_t + tax^k_t.
- Assumptions:
  - g_t = g, τ^l_t = τ^l ∀t; initial τ^k_t = τ^k.
- Regimes after tax cut shock ε^τ_k at t=0:
  - R1 (tax cut): R_t = 1, τ^k_t = τ^k − ε^τ_k, z_t = z.
  - R2 (fiscal adjustment): R_t = 2, τ^k_t = τ̄^k + γ_τ (b_{t−1} − b̄), z_t = z̄ − γ_z (b_{t−1} − b̄), γ_τ, γ_z > 0.
- Regime switching:
  - Transition matrix [[p_{1,t}, 1−p_{1,t}],[1−p_2, p_2]].
  - Switching R1→R2 occurs with probability 1−p_{1,t}; 1−p_{1,t+1} = Pr(debt_{t} exceeds drawn fiscal limit).
  - Very high debt ⇒ p_{1,t+1} ≈ 0.

### Calibration and solution method (parameters and computation)
- Quarterly frequency; structural parameters:
  - β = 0.99, δ = 0.025, α = 0.36, κ = 1.7, a = 1.
  - σ = 1.38, φ = 1.83 (Smets and Wouters (2007)).
- Baseline fiscal steady state (U.S. federal data, 1980–2017 averages):
  - τ^k = 0.208, τ^l = 0.204.
  - Debt-to-annual output b_{4y} = 0.45.
  - g/y = 0.08.
  - Transfers-to-output implied = 0.11.
- Fiscal adjustment targets and parameters:
  - Target debt-to-output = 0.7 (alternative 0.9).
  - γ_τ = 0.005 and γ_z = 0.04 (baseline); alternative γ_τ = 0.01.
  - Targeted transfers reduced from 0.11 to 0.09 (consistent with τ^k returning to pre-cut).
- Regime duration:
  - p_2 = 0.9375 ⇒ average fiscal adjustment lasts 4 years.
- Solution:
  - Nonlinear solution using monotone mapping method (Coleman (1990), Davig (2004)).

### Fiscal limits — definition and simulation
- Fiscal limit (Bi (2012)):
  - b_{t−1}^∼ = ∑_{i=0}^{∞} d_{t+i}/d_t E_t [ λ^{max}_{t+i}/λ^{max}_t ( tax^{max}_{t+i} − g − z_{t+i} ) ].
- Baseline simulation inputs:
  - τ^k,max = 0.240, τ^l,max = 0.292.
  - Future transfers per CBO projection (2019–2038): rise from 12.7 percent to 16.5 percent of output at end of 20 years; thereafter AR(1) returning to 11 percent.
  - TFP parameters: ρ_a = 0.9, σ_a = 0.01.
  - Discount factor persistence: ρ_β = 0.8, σ_β = 0.005.
- Simulation procedure:
  - 10,000 simulations × 1,000 periods each; draw ε^a and ε^β sequences; compute discounted maximum fiscal surpluses; collect {b_j}_{j=1}^{10,000} to form fiscal limit distribution.
- Key simulated finding:
  - At net federal debt of 80 percent (baseline discussion), probability that debt exceeds a drawn fiscal limit is virtually zero; risk emerges when net federal debt exceeds 100 percent of GDP (per CBO projections).

### Effects of capital income tax cuts — simulation experiments
- Experiment: persistent capital income tax cut ≈ two percentage points (CBO comparisons).
- Setup:
  - t = 0 unexpected cut; agents form expectations about switching at t = 1 via probability debt exceeds drawn fiscal limit.
  - Regime assumed not to switch for first five years; agents unaware ex ante of switch timing.
  - Initial debt comparisons: 45%, 80%, 120% of output.
  - Responses reported as percent deviations from no-cut path; x-axis in quarters.
- Baseline tax cut magnitude:
  - τ^k cut from 0.208 to 0.188 (two-percentage-point cut).

### Baseline analysis — nonlinear impulse responses and fiscal-state dependence
- Initial debt 45%:
  - Tax cut expansionary: higher after-tax return ⇒ increased saving/investment ⇒ capital accumulation ⇒ higher marginal product of labor ⇒ higher labor ⇒ output rises.
  - Consumption initially falls, then turns positive.
  - Fiscal outcome: tax revenues fall; debt-to-output rises from 45% to 49% five years after cut.
- Initial debt 80%:
  - Tax cut less expansionary.
  - Agents place small switching probability which rises to 1.3% five years after cut as debt rises from 80% to 93%.
  - Higher expected future taxes weaken stimulative effect.
- Initial debt 120%:
  - Tax cut contractionary.
  - Debt rises to 144% five years after cut.
  - Switching probability rises to almost 50%; agents expect large future capital income tax increase to stabilize debt at 70%.
  - Anticipated future tax hikes dominate stimulative effect ⇒ contractionary outcome.
- Interpretation: expansionary only when unlikely to trigger future fiscal adjustment; high initial debt can reverse effect (Sutherland (1997) intuition).

### Linear vs nonlinear comparison
- Linear fiscal rule: τ^k_t = γ^lin_τ (b_{t−1} − b) + ε^k_t with γ^lin_τ = 0.0001.
- Policy function differences:
  - Realized τ^k paths similar, but expected tax rates differ for high debt:
    - Linear: gradual reversal expectation.
    - Nonlinear: expected τ^k constant when debt < 100%; increases rapidly when debt > 120% due to fiscal adjustment probability.
- Impulse responses:
  - At 45% initial debt: linear and nonlinear predict similar positive responses.
  - At 120%: nonlinear predicts contractionary effects; linear implies debt-independent effects.
- Conclusion: nonlinearity in regime switching and expectations crucial; linear approximations can be biased when debt deviates substantially.

### Revenue feedback effects (definition and key quantitative results)
- Revenue feedback effect:
  - x_t ≡ 1 − (tax_dynamic scoring_t − tax_no tax cut_t) / (tax_static scoring_t − tax_no tax cut_t).
- Interpretation: x_t = 0 if no expansionary effect on tax base (tax_dynamic = tax_static).
- Key findings:
  - Government debt rises after capital income tax cut regardless of initial debt.
  - Expanded tax base offsets some revenue loss at initial debt = 45% and 80%.
  - Revenue feedback effects are fiscal state-dependent.
- Baseline quantitative results:
  - Initial debt = 45%:
    - Impact revenue feedback effect = 0.10.
    - Five-year revenue feedback effect = 0.25.
  - Initial debt = 80%:
    - Five-year revenue feedback effect = 0.17.
- Comparison:
  - Much smaller than Mankiw and Weinzierl (2006) finding of 53 percent self-financing.
  - Debt-financed income tax cuts have smaller revenue feedback effects than those financed by lump-sum taxes (Leeper and Yang (2008) vs Mankiw and Weinzierl (2006)).

### Tax multipliers (method and baseline cumulative multipliers)
- Cumulative tax multipliers:
  - ∑_{j=0}^l (∏_{i=0}^j r_{t+i}^{−1}) Δy_{t+j}  /  ∑_{j=0}^l (∏_{i=0}^j r_{t+i}^{−1}) Δtax^k_{t+j}, with r_t = 1/q_t.
- Baseline cumulative multipliers (Table 3):
  - Initial debt = 45%:
    - impact output = −0.53
    - 1 year = −0.61
    - 5 years = −0.95
    - consumption multipliers: impact 0.94; 1 year 0.86; 5 years 0.50
    - investment multipliers: impact −1.48; 1 year −1.47; 5 years −1.44
  - Initial debt = 80%:
    - impact output = −0.45
    - 1 year = −0.51
    - 5 years = −0.71
    - consumption: impact 0.79; 1 year 0.71; 5 years 0.35
    - investment: impact −1.24; 1 year −1.22; 5 years −1.06
  - Initial debt = 120%:
    - impact output = 0.13
    - 1 year = 0.17
    - 5 years = 0.42
    - consumption: impact −0.24; 1 year −0.25; 5 years −0.27
    - investment: impact 0.37; 1 year 0.42; 5 years 0.70
- Interpretation:
  - At 45% initial debt, five-year cumulative output multiplier ≈ −1, consistent with VAR evidence.
  - Output multipliers decline in absolute value as initial debt rises; at 80% five-year = −0.7.
  - At 120% debt, multipliers become positive (impact 0.1; five-year 0.4) — tax cut contractionary.
  - With current net federal debt ≈ 80%, the 2017 Tax Act cut is simulated expansionary but with multiplier < 1 in absolute value; credible reform could prevent reduction in absolute multiplier.

### Fiscal adjustment via consumption tax increases (Section 5.5)
- Modified fiscal adjustment regime:
  - R1: R_t = 1, τ^k_t = τ^k − ε_{τ^k,z_t} = z^R
  - R2: R_t = 2, τ^c_t = τ̄^c + γ_τ (b_{t−1} − b̄), z_t = z̄ − γ_z (b_{t−1} − b̄)
  - τ̄^c = 0.10; b̄ consistent with targeted debt-to-output = 150%.
- Reference: Japan calibration (2000–2017):
  - Tax revenue-to-output = 0.31.
  - Initial steady-state debt-to-output = 1.19.
  - τ^k = 0.32, τ^c = 0.07, τ^l = 0.24.
  - Transfers-to-output = 0.036; government purchases-to-output = 0.23.
  - Consumption tax upper bound τ^c,max = 0.15.
  - Fiscal limit simulation: at net government debt = 153% of GDP probability ≈ 0.1 of switching to fiscal adjustment.
- Impulse responses and multipliers (Table 4) for persistent two-percentage-point capital tax cut:
  - Initial debt = 119%:
    - impact output −0.56; 1 year −0.64; 5 years −1.00
    - consumption impact 0.82; 1 year 0.74; 5 years 0.38
    - investment impact −1.38; 1 year −1.38; 5 years −1.38
  - Initial debt = 150%:
    - impact output −0.40; 1 year −0.45; 5 years −0.61
    - consumption impact 0.59; 1 year 0.52; 5 years 0.21
    - investment impact −0.99; 1 year −0.97; 5 years −0.82
  - Initial debt = 200%:
    - impact output 0.19; 1 year 0.24; 5 years 0.53
    - consumption impact −0.29; 1 year −0.29; 5 years −0.26
    - investment impact 0.48; 1 year 0.53; 5 years 0.79
- Mechanism:
  - Expected future consumption tax increase lowers Tobin’s q ⇒ reduces value of installed capital ⇒ crowds out investment; when debt very high, negative investment response can dominate, making tax cut contractionary.
  - Real interest rate affected by weaker demand and expected higher future consumption tax; with higher initial debt, expected consumption tax increases drive real interest rate up more.

### Sensitivity analysis
- 6.1 Fiscal adjustment speed and debt target (Table 5):
  - Baseline: γ_τ = 0.005, debt target 0.7.
  - Alternatives: γ_τ = 0.01; debt target = 0.9.
  - Results for initial debt = 80%:
    - impact output: baseline −0.45; γ_τ = 0.01 → −0.42; debt target 0.9 → −0.46
    - 1 year: baseline −0.51; γ_τ = 0.01 → −0.47; debt target 0.9 → −0.53
    - 5 years: baseline −0.71; γ_τ = 0.01 → −0.62; debt target 0.9 → −0.75
  - Results for initial debt = 120%:
    - impact output: baseline 0.13; γ_τ = 0.01 → 0.41; debt target 0.9 → 0.04
    - 1 year: baseline 0.17; γ_τ = 0.01 → 0.50; debt target 0.9 → 0.07
    - 5 years: baseline 0.42; γ_τ = 0.01 → 1.00; debt target 0.9 → 0.25
  - Interpretation: higher γ_τ reduces expansionary effect; higher debt target (weaker adjustment) leads to more expansionary tax cut given initial debt.
- 6.2 Average duration of fiscal adjustment (Table 6):
  - Baseline p_2 = 0.9375 (average 4 years); alternative p_2 = 0.975 (average 10 years).
  - Initial debt = 120%:
    - Baseline one-year = 0.17; five-year = 0.42.
    - p_2 = 0.975 → one-year = 0.26; five-year = 0.55.
    - Longer expected duration makes tax cut slightly more contractionary when debt high.
  - Initial debt = 80%:
    - Baseline one-year −0.51; five-year −0.71.
    - p_2 = 0.975 → one-year −0.49; five-year −0.66 (small changes).
- 6.3 Alternative fiscal limit distributions:
  - Alternative: transfers-to-output stabilize at 0.165 (end of 20-year CBO projection), lowering mean fiscal limit from 1.44 to 0.90.
  - With lower fiscal limits:
    - Higher switching probability to fiscal adjustment for given debt.
    - Even at initial debt = 80%, capital income tax cut becomes barely expansionary: one-year cumulative output multiplier = −0.05.
    - At initial debt = 120%, tax cut becomes more contractionary: one-year multiplier = 0.43 (vs. 0.17 baseline).

### Conclusion — principal results and broader implications
- Principal results:
  - Macroeconomic effects of capital income tax cuts are fiscal state-dependent: expansionary in low-debt states; less expansionary or contractionary in high-debt states.
  - U.S. simulation results:
    - Five-year cumulative output multiplier = −1 at initial government debt = 45% of GDP.
    - Five-year cumulative output multiplier = −0.7 at initial government debt = 80% of GDP.
    - Multiplier turns contractionary when debt reaches 120%.
  - Results robust when expected fiscal adjustment instrument is consumption tax increases (Japan simulation).
  - Nonlinearities driven by expectations of future fiscal adjustments dampen expansionary effects as fiscal state deteriorates.
- Broader implications:
  - Tax policy effects are highly nonlinear; fiscal state is important when assessing fiscal policy effects.
  - As federal fiscal position deteriorates, expectations of future fiscal adjustments can reduce growth and investment responses to corporate tax reductions.

### 2.1  Empirical Model and Estimation. . . . . . . . . . . . . . . . . . . . . . . .    7

### 2.1 Empirical Model and Estimation

### Empirical model specification
- Baseline STVAR specification:
  - Yt = (1 − F(zt)) AL(l) Yt−1 + F(zt) AH(l) Yt−1 + εt, εt ∼ N(0, Ωt)
  - Ωt = (1 − F(zt)) ΩL + F(zt) ΩH
  - F(zt) = exp[−γzt] / (1 + exp[−γzt]), γ > 0
- Endogenous variables in Yt, in the given order:
  - government spending (Gt), the average personal income tax rate (APITRt), the average corporate income tax rate (ACITRt), the personal income tax base (PBt), the corporate income tax base (CBt), output (GDPt), fiscal news, and the debt-to-GDP ratio.
- Data and sample:
  - Quarterly sample from 1950Q1 to 2006Q4.
  - Corporate tax rate construction: ACITRt = TCPt / CPt (taxes on corporate profits divided by corporate profits).
  - Personal income tax rate: APITRt = (PTt + SIt) / PTIt (personal current taxes plus contributions for government social insurance divided by personal tax income).
  - GDP, corporate and personal tax bases are in logarithms and real terms.
  - Fiscal news: Ramey’s defense spending news.
- State variable construction and calibration:
  - zt constructed by de-trending the debt-to-GDP series through a third-order polynomial and considering deviations above/below trend.
  - Transition smoothness calibrated to γ = 2.1 to match the frequency and duration in a high-debt state.
  - VAR lag length set to three based on the Akaike criterion.
- Identification and estimation:
  - Estimation via Monte-Carlo Markov-chain (MCMC) algorithm (Chernozhukov and Hong (2003)).
  - Structural Cholesky-type identification.
  - Generalized impulse response functions (GIRFs) derived following Koop et al. (1996).
  - All impulse responses correspond to a reduction in the corporate income tax rate normalized to one percentage point, over a 20-quarter forecast horizon.

### Interpretation of the nonlinear setup
- The model captures continuous, debt-dependent regime switching via F(zt), allowing all observations to contribute with varying weights rather than imposing a binary high-debt dummy.
- The variance-covariance matrix is state-contingent (ΩL in low-debt periods, ΩH in high-debt periods), permitting different shock volatility across debt states.

### Key methodological choices
- Disaggregation of income taxes into corporate and personal components following Mertens and Ravn (2013); focus on corporate income tax as a proxy for capital income taxes.
- GIRFs are constructed using all histories of a given debt state and 90-percent confidence bands are used for inference.

### Exact technical settings and numerical facts to preserve
- Calibration: γ = 2.1.
- Sample: 1950Q1–2006Q4.
- VAR lags: 3.
- Shock normalization: 1 percentage point reduction in the corporate income tax rate.
- Forecast horizon for GIRFs: 20 quarters.

### References used within the specification (preserved terminology)
- Chernozhukov and Hong (2003); Koop et al. (1996); Auerbach and Gorodnichenko (2012); Caggiano et al. (2015); Mertens and Ravn (2013).

---

### 2.2 Estimation Results

- Linear versus nonlinear dynamics:
  - Linear VAR GIRF: persistent output increase that then stabilizes (black line in Figure 2).
  - Nonlinear STVAR GIRFs: reversal dynamics and heterogeneous responses across fiscal variables, with distinct high-debt (red line) and low-debt (blue line) state responses.

- State-dependent responses (STVAR GIRFs, 90-percent bands):
  - A corporate income tax cut is expansionary in a low-debt state but less expansionary, and in some cases contractionary, in a high-debt state.
  - Short-run heterogeneity emerges after two quarters between high- and low-debt states.
  - In the low-debt state:
    - Stronger increases in personal and corporate income tax bases.
    - More pronounced output increase.
    - Larger increases in government spending and the average personal income tax rate compared to the high-debt state.

- Multipliers (construction and key values):
  - Tax revenue response used in multipliers: TRgirft = ACITRgirft * ACITR + CBgirft, where ACITR is the mean average corporate income tax rate.
  - Eight-quarter cumulative output multipliers:
    - Low-debt state: −0.36 (statistically significant at the 8-quarter horizon).
    - High-debt state: −0.14.
  - Comparison with the literature (reported estimates for context, preserved verbatim):
    - Romer and Romer (2010): long-run output multiplier of −3.
    - Blanchard and Perotti (2002): twenty-quarter multiplier of −0.8 (−1.3) under deterministic (stochastic) trend specification.
    - Mountford and Uhlig (2009): five-year cumulative multiplier of −5.
    - Barro and Redlick (2011): tax multiplier around −1.1.
    - Mertens and Ravn (2013): multiplier at about −2.5 to −3.

- Endogenous transition dynamics:
  - The likelihood of remaining in a high-debt state is high and persistent; switching to a low-debt state may occur after 15 quarters in the baseline specification.

- Model-based DSGE simulation summary (mechanism and quantitative examples):
  - Two policy regimes in the DSGE: tax cut regime (no fiscal adjustment) and fiscal adjustment regime (government increases primary surplus to stabilize debt).
  - Regime switching probability linked to current debt level and the government’s debt repayment capacity, captured by fiscal limits (in the sense of Bi (2012)); switching probability equals the probability that current debt exceeds a randomly drawn fiscal limit from a simulated distribution.
  - Simulation of a persistent capital income tax cut similar in magnitude to the reduction in the corporate income tax rate in the 2017 TaxAct.
  - Baseline fiscal adjustments implemented via a transfer reduction and a policy reversal (increase in the capital income tax rate); alternative adjustments via consumption tax increases considered in Section 5.5.
  - Selected quantitative simulation results (expectation channel only; adjustments do not actually occur during analyzed horizon):
    - With expected policy reversal and initial U.S. federal net debt of 45 percent of GDP (the average federal debt held by the public from 1980 to 2018):
      - Impact output multiplier: −0.5
      - Five-year cumulative output multiplier: about −1
    - With initial debt at 80 percent of GDP (roughly the net federal debt level in 2019):
      - Impact output multiplier: −0.4
      - Five-year cumulative output multiplier: −0.7
    - Same tax cut turns contractionary when debt rises to 120 percent of GDP.
  - Mechanism: higher probability of policy reversal lowers the expected after-tax return to current investment, reducing the incentive effect of the capital income tax cut; lower investment reduces capital and the marginal product of labor, producing smaller (in absolute value) output multipliers at higher debt.

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### 2.3 Robustness Checks

- Addressing endogeneity and foresight:
  - Use of narratively constructed, exogenous unanticipated corporate income tax series from Mertens and Ravn (2013) as an alternative identification; the unanticipated series ordered first in the estimation.
  - Result: debt-dependent effects persist — output response mostly positive in low-debt state and mostly negative in high-debt state (Figure 5).

- Additional robustness checks (conducted but not fully presented in text):
  - 1) Adding aggregate narratively constructed tax news measures in Mertens and Ravn (2012) to the baseline specification.
  - 2) Adding the unanticipated, exogenous aggregate tax changes narratively constructed in Mertens and Ravn (2012).
  - 3) Different orderings of the variables in specification (1).
  - Summary conclusion: The empirical finding that capital income tax cuts are more expansionary in a low-debt state than in a high-debt state remains robust across these alternative estimations.

---

### Theoretical baseline model (preview)

- Purpose:
  - Use a neoclassical DSGE growth model with regime-switching fiscal policy to explain theoretical channels behind debt-dependent capital income tax effects observed empirically.

- Baseline modeling choices:
  - Fiscal adjustments in the baseline are jointly implemented through a policy reversal (increase in the capital income tax rate) and a transfer reduction.
  - Section 5.5 extends the model to allow adjustments jointly implemented by a consumption tax increase and a transfer reduction.

- Key conceptual mechanism retained from empirical findings:
  - Expectations about fiscal adjustments (triggered with higher probability in deteriorated fiscal states) reduce the expected after-tax return to investment, thereby weakening the expansionary effect of capital income tax cuts when debt is high.

---

*Source: https://www.imf.org/-/media/files/publications/wp/2020/english/wpiea2020071-print-pdf.pdf*

### 3.1  The Private Sector

### 3.1  The Private Sector

### Model setup — preferences and constraints
- Representative agent chooses private consumption (c_t), labor (l_t), gross investment (i_t), and capital (k_t) to maximize expected utility:
  - E_t ∑_{t=0}^{∞} d_t ( c_t^{1−σ}/(1−σ) − l_t^{1+φ}/(1+φ) )
  - Time-varying discount factor: d_t = ∏_{j=1}^{t} β_j for t>0, d_0 ≡ 1.
  - β_t follows AR(1): ln(β_t/β) = ρ_β ln(β_{t−1}/β) + ε^β_t, ε^β_t ∼ N(0, σ^2_β).
- Budget constraint:
  - c_t + i_t + q_t b_t = (1−τ^l_t) w_t l_t + (1−τ^k_t) r^k_t k_{t−1} + b_{t−1} + z_t,
  - where b_t is a one-period government bond with price q_t and pays one unit at t+1; τ^l_t labor income tax rate; w_t real wage; τ^k_t capital income tax rate; r^k_t rental rate; z_t government transfers.
- Capital accumulation with adjustment costs:
  - k_t = (1−δ) k_{t−1} + i_t − κ/2 ( i_t/k_{t−1} − δ )^2 k_{t−1}.
- Firms: perfectly competitive, Cobb-Douglas production:
  - y_t = a_t k_{t−1}^α l_t^{1−α}.
- Total factor productivity (TFP) a_t follows AR(1):
  - ln(a_t/a) = ρ_a ln(a_{t−1}/a) + ε^a_t, ε^a_t ∼ N(0, σ^2_a).
- Discount factor shocks (ε^β_t) represent demand shocks; TFP shocks (ε^a_t) represent supply shocks. These shocks are used in simulation of fiscal limit distributions.

### Government — regimes and fiscal rule
- Government budget constraint:
  - tax_t + q_t b_t = g_t + b_{t−1} + z_t,
  - tax_t ≡ tax^l_t + tax^k_t, tax^l_t = τ^l_t w_t l_t, tax^k_t = τ^k_t r^k_t k_{t−1}.
- Assumptions:
  - g_t = g and τ^l_t = τ^l ∀t (government purchases and labor income tax rate fixed at steady-state).
  - Initial steady state: τ^k_t = τ^k.
  - After a tax cut shock ε^τ_k at t=0, τ^k_t follows one of two regimes from t=1:
    - R1 (tax cut regime): R_t = 1, τ^k_t = τ^k − ε^τ_k, z_t = z.
    - R2 (fiscal adjustment regime): R_t = 2, τ^k_t = τ̄^k + γ_τ (b_{t−1} − b̄), z_t = z̄ − γ_z (b_{t−1} − b̄),
      - where γ_τ, γ_z > 0; τ̄^k and z̄ are targeted fiscal policy values; agents know targets.
- Interpretation:
  - R1: no fiscal instrument stabilizes debt; debt can grow exponentially.
  - R2: policy reversal implemented to stabilize debt toward target b̄ consistent with τ̄^k and z̄; transfers allowed as a separate instrument.
- Regime switching:
  - R_t evolves with transition matrix [[p_{1,t}, 1−p_{1,t}],[1−p_2, p_2]].
  - Switching from R1 to R2 occurs with probability 1−p_{1,t}.
  - 1−p_{1,t+1} is set equal to the probability that government debt at end of t exceeds a randomly drawn fiscal limit from its distribution.
  - Very high debt ⇒ p_{1,t+1} ≈ 0 (almost certain switch to fiscal adjustment).

### Calibration and solution method — parameter values and computation
- Quarterly frequency; structural parameters:
  - β = 0.99, δ = 0.025, α = 0.36, κ = 1.7, steady-state TFP a = 1.
  - σ = 1.38, φ = 1.83 (Smets and Wouters(2007) U.S. estimates).
- Baseline fiscal steady-state calibration (U.S. federal data, 1980–2017 averages):
  - τ^k = 0.208, τ^l = 0.204 (average federal capital and labor income tax rates).
  - Debt-to-annual output ratio b_{4y} = 0.45 (average federal debt held by public as share of GDP).
  - Government purchases-to-output g/y = 0.08 (historical average discretionary outlays).
  - Transfers-to-output ratio implied = 0.11 (matches average mandatory spending-to-GDP).
- Fiscal adjustment regime targets and parameters:
  - Targeted debt-to-output ratio = 0.7 (post-crisis 2009–2017 average).
  - Sensitivity analysis: alternative targeted debt-to-output ratio = 0.9.
  - γ_τ = 0.005 and γ_z = 0.04 (baseline); alternative γ_τ = 0.01 in sensitivity analysis.
  - Targeted transfers reduced from 0.11 to 0.09 as share of output (consistent with return of τ^k to pre-cut level and τ^l, g constant).
- Regime duration:
  - p_2 = 0.9375 ⇒ on average fiscal adjustment continues for four years (Devries et al. (2011) evidence).
  - p_{1,t} is determined endogenously based on position of current debt in fiscal limit distribution.
- Solution:
  - Solve model nonlinearly using monotone mapping method (Coleman (1990), Davig (2004)). Appendices list equilibrium equations and algorithm.

### Fiscal limits — definition and simulation
- Fiscal limit defined (following Bi (2012)) as the maximum sustainable debt:
  - b_{t−1}^∼ = ∑_{i=0}^{∞} d_{t+i}/d_t E_t [ λ^{max}_{t+i}/λ^{max}_t ( tax^{max}_{t+i} − g − z_{t+i} ) ],
  - where superscript max denotes values under τ^l,max_{t+i} and τ^k,max_{t+i} ∀i; λ^{max}_t is marginal utility of consumption under maximum tax rates.
- Baseline simulation inputs:
  - Maximum implementable average tax rates: τ^k,max = 0.240, τ^l,max = 0.292 (Bi et al. (2019) calibration).
  - Future transfers from Congressional Budget Office extended baseline projection (2019–2038): rising from 12.7 percent to 16.5 percent of output at end of 20 years; thereafter assumed to follow AR(1) returning slowly to steady state 11 percent of output.
  - TFP shock parameters: ρ_a = 0.9, σ_a = 0.01.
  - Discount factor persistence ρ_β = 0.8, σ_β = 0.005 (Fernández-Villaverde et al. (2015) guidance and literature range).
- Simulation procedure:
  - 10,000 simulations with 1,000 periods each.
  - In each simulation draw sequences of ε^a_{t+i} and ε^β_{t+i} for i = {1,2,...,1000} starting from initial steady state under assumption labor and capital tax rates always at their maximum values.
  - Compute discounted sum of maximum fiscal surpluses per (11); collect {b_j}_{j=1}^{10,000} to form fiscal limit distribution.
- Key simulated finding:
  - At current net federal debt of 80 percent (baseline discussion), the fiscal limit implies the probability that this debt level can exceed a randomly drawn fiscal limit is virtually zero — current debt can be well supported by future primary surpluses.
  - Risk to debt sustainability may emerge when net federal debt exceeds 100 percent of GDP (per CBO projections).

### The effects of capital income tax cuts — overview of experiments
- Experiment: persistent capital income tax cut equal to the change in effective marginal tax rate on capital income estimated by Congressional Budget Office comparisons (roughly two percentage points reduction).
- Setup for simulations:
  - At t = 0: unexpected capital income tax cut; economy in tax cut regime.
  - Agents form expectations about switching to fiscal adjustment regime at t = 1 according to probability that debt at t = 0 can exceed a drawn fiscal limit.
  - Regime is assumed not to switch for first five years after the tax cut (analysis horizon). Agents are unaware ex ante when the regime will switch.
  - Comparison across initial debt levels: 45 percent, 80 percent, 120 percent of output.
  - Responses reported as percent deviations from path without tax cut; x-axis in quarters.

### Baseline analysis — nonlinear model impulse responses and fiscal-state dependence
- Tax cut magnitude in simulations:
  - Capital income tax rate cut from 0.208 to 0.188 (two-percentage-point cut).
- Findings by initial debt level:
  - Initial debt 45 percent of output (steady state):
    - Capital income tax cut is expansionary.
    - Mechanism: higher after-tax return to investment ⇒ increased saving and investment ⇒ capital accumulation ⇒ higher marginal product of labor ⇒ increased labor input ⇒ output rises.
    - Consumption initially falls (more investment raises real interest rate and opportunity cost of consumption), then turns positive as income rises.
    - Real wage falls short run due to increased labor supply, then rises with higher labor productivity.
    - Fiscal outcome: tax revenues fall; debt-to-output rises from 45 percent to 49 percent five years after the tax cut.
  - Initial debt 80 percent of output:
    - Same tax cut is less expansionary than 45 percent case.
    - Agents place a small probability on switching to fiscal adjustment regime which increases over time to 1.3 percent five years after the tax cut as debt rises from 80 to 93 percent of output.
    - Higher expected future tax raises expected return discounting, weakening stimulative effect on investment and output.
  - Initial debt 120 percent of output:
    - Capital income tax cut becomes contractionary.
    - Debt rises to 144 percent of output five years after the tax cut.
    - Switching probability rises substantially to almost 50 percent; agents expect a large future capital income tax increase to stabilize debt at 70 percent of output.
    - Anticipated future tax hikes dominate stimulative effect and generate contractionary outcome.
- Interpretation:
  - Tax cut expansionary to the extent it is unlikely to result in future fiscal adjustment; high initial debt can reverse the effect.
  - Mechanism resembles Sutherland (1997) intuition: when debt is sufficiently high, probability of tax hikes within agents' horizon is high, offsetting benefits of current tax cut.

### Comparison with a linear model — importance of nonlinearity
- Linear-model fiscal rule used for comparison:
  - τ^k_t = γ^lin_τ (b_{t−1} − b) + ε^k_t, z_t = z,
  - Calibrated γ^lin_τ = 0.0001 (minimum to maintain debt sustainability).
  - Fiscal reaction is one-period lag on debt deviation; periodic adjustment.
- Policy function comparison:
  - Realized τ^k paths in nonlinear and linear models are similar (nonlinear: τ^k stays at two-percentage-points below steady-state in R1; linear: small systematic response to debt).
  - Expected tax rates E_t τ^k_{t+1} differ substantially for high debt:
    - Linear model: agents expect gradual reversal as debt grows.
    - Nonlinear model: expected tax rate remains constant when debt < 100 percent of output, but increases rapidly when debt > 120 percent due to nontrivial probability of fiscal adjustment.
- Impulse response comparison (two-percentage-point cut, initial debt 45% vs 120%):
  - At 45 percent initial debt: linear and nonlinear models predict similar positive responses in output, investment, and labor.
  - At 120 percent initial debt: nonlinear model predicts contractionary effects driven by higher expected τ^k; linear model implies effects independent of debt levels.
- Conclusion:
  - Nonlinearity in fiscal regime switching and agents’ expectations is crucial; linear approximation can produce biased or qualitatively different results when government debt deviates substantially from steady state.

*Source: wpiea2020071-print-pdf - 3.1  The Private Sector*

### 5.3  Revenue Feedback Effects

### 5.3  Revenue Feedback Effects

### Definition and measurement
- Revenue feedback effect defined as
  x_t ≡ 1 − (tax_dynamic scoring_t − tax_no tax cut_t) / (tax_static scoring_t − tax_no tax cut_t).  
- tax_static scoring_t is tax revenue under static scoring; tax_dynamic scoring_t accounts for macroeconomic effect on the tax base.
- Interpretation: if a capital income tax cut has no expansionary effect on the tax base, tax_dynamic scoring = tax_static scoring, so x_t = 0 (none of the revenue loss is self-financed).

### Key findings
- Government debt rises following a capital income tax cut regardless of initial debt levels.
- Expanded tax base offsets some revenue loss for initial debt of 45 and 80 percent of output.
- Revenue feedback effects are fiscal state-dependent.

### Quantitative results (baseline simulation)
- With initial debt ratio at 45 percent:
  - Impact revenue feedback effect = 0.10 (10 percent of revenue loss is self-financed).
  - Five-year revenue feedback effect rises to 0.25.
- With initial debt ratio at 80 percent:
  - Five-year revenue feedback effect = 0.17.
- Comparison:
  - These effects are much smaller than Mankiw and Weinzierl (2006) finding that 53 percent of revenue loss is self-financing.
  - Debt-financed income tax cuts have smaller revenue feedback effects than those financed by lump-sum taxes (Leeper and Yang (2008) vs. Mankiw and Weinzierl (2006)).

---

### 5.4  Tax Multipliers

### Method
- Cumulative tax multipliers computed as:
  ∑_{j=0}^l (∏_{i=0}^j r_{t+i}^{−1}) Δy_{t+j}  /  ∑_{j=0}^l (∏_{i=0}^j r_{t+i}^{−1}) Δtax^k_{t+j},
  where Δy and Δtax^k are level changes in output and capital tax revenue relative to the path without a capital income tax cut, r_t = 1/q_t, and when j = 0 the product term ≡ 1. For consumption and investment multipliers, replace Δy with Δc and Δi.

### Baseline cumulative multipliers (Table 3)
- Initial debt = 45% of output:
  - impact output multiplier = −0.53
  - 1 year = −0.61
  - 5 years = −0.95
  - consumption multipliers: impact 0.94; 1 year 0.86; 5 years 0.50
  - investment multipliers: impact −1.48; 1 year −1.47; 5 years −1.44
- Initial debt = 80% of output:
  - impact output multiplier = −0.45
  - 1 year = −0.51
  - 5 years = −0.71
  - consumption multipliers: impact 0.79; 1 year 0.71; 5 years 0.35
  - investment multipliers: impact −1.24; 1 year −1.22; 5 years −1.06
- Initial debt = 120% of output:
  - impact output multiplier = 0.13
  - 1 year = 0.17
  - 5 years = 0.42
  - consumption multipliers: impact −0.24; 1 year −0.25; 5 years −0.27
  - investment multipliers: impact 0.37; 1 year 0.42; 5 years 0.70

### Interpretation and implications
- With initial debt at 45 percent, five-year cumulative output multiplier ≈ −1, consistent with VAR evidence.
- Output multipliers fall in absolute value as initial debt increases; at 80 percent five-year multiplier = −0.7.
- At 120 percent debt, impact and five-year output multipliers turn positive (0.1 and 0.4 respectively), indicating a contractionary capital income tax cut.
- Conclusion: as current net federal debt ratio is about 80 percent, the capital income tax cut in the 2017 Tax Act is simulated to be expansionary but with multiplier less than one in absolute value. Credible reform could prevent reduction in absolute multiplier relative to low-debt case.

---

### 5.5  Fiscal Adjustment via Consumption Tax Increases

### Model modification and calibration
- Fiscal adjustment regime modified as:
  R1: R_t = 1, τ^k_t = τ^k − ε_{τ^k,z_t} = z^R
  R2: R_t = 2, τ^c_t = τ̄^c + γ_τ (b_{t−1} − b̄), z_t = z̄ − γ_z (b_{t−1} − b̄),
  with γ_τ, γ_z > 0 and τ̄^c = 0.10; b̄ consistent with targeted debt-to-output ratio of 150 percent.
- Modified budget constraints (equations (16) and (17)) include consumption tax τ^c_t and assume τ^l_t = τ^l and g_t = g.
- Reference country: Japan. Fiscal limit distribution simulated using Japan fiscal data 2000–2017.
  - Tax revenue-to-output ratio set to 0.31.
  - Initial steady state debt-to-output ratio set to 1.19.
  - τ^k = 0.32, τ^c = 0.07 (reflects average 2010–2019), τ^l = 0.24.
  - Transfers-to-output = 0.036; government purchase-to-output ratio implied = 0.23.
- Consumption tax upper bound τ^c,max = 0.15 based on IMF (2018) recommendation.
- Simulated fiscal limit distribution for Japan implies at net government debt of 153 percent of GDP there is probability ≈ 0.1 government will switch to fiscal adjustment regime.

### Impulse responses and multipliers (Figure 11, Table 4)
- Simulated persistent two-percentage-point cut in capital income tax under initial debt levels 119%, 150%, 200% of output.
- Table 4 cumulative multipliers:
  - Initial debt = 119%:
    - impact output −0.56; 1 year −0.64; 5 years −1.00
    - consumption impact 0.82; 1 year 0.74; 5 years 0.38
    - investment impact −1.38; 1 year −1.38; 5 years −1.38
  - Initial debt = 150%:
    - impact output −0.40; 1 year −0.45; 5 years −0.61
    - consumption impact 0.59; 1 year 0.52; 5 years 0.21
    - investment impact −0.99; 1 year −0.97; 5 years −0.82
  - Initial debt = 200%:
    - impact output 0.19; 1 year 0.24; 5 years 0.53
    - consumption impact −0.29; 1 year −0.29; 5 years −0.26
    - investment impact 0.48; 1 year 0.53; 5 years 0.79

### Mechanism
- Higher expected future consumption tax lowers Tobin’s q, reducing the value of installed capital and crowding out investment in favor of consumption.
- When current debt sufficiently high, the negative effect on investment can dominate the investment incentives from a capital income tax cut, making current investment respond negatively.
- Real interest rate responses are affected both by weaker aggregate demand and by expected higher future consumption tax rates; with higher initial debt, expected consumption tax increases drive the real interest rate up more.

---

### 6  Sensitivity Analysis

### 6.1 Fiscal Adjustment Speed and Debt Target (Table 5)
- Baseline: γ_τ = 0.005, debt target 0.7.
- Alternative: γ_τ = 0.01 (faster adjustment) and debt target 0.9 (higher target).
- Results for initial debt = 80%:
  - impact output: baseline −0.45; γ_τ = 0.01 → −0.42; debt target 0.9 → −0.46
  - 1 year: baseline −0.51; γ_τ = 0.01 → −0.47; debt target 0.9 → −0.53
  - 5 years: baseline −0.71; γ_τ = 0.01 → −0.62; debt target 0.9 → −0.75
- Results for initial debt = 120%:
  - impact output: baseline 0.13; γ_τ = 0.01 → 0.41; debt target 0.9 → 0.04
  - 1 year: baseline 0.17; γ_τ = 0.01 → 0.50; debt target 0.9 → 0.07
  - 5 years: baseline 0.42; γ_τ = 0.01 → 1.00; debt target 0.9 → 0.25
- Interpretation: higher γ_τ reduces expansionary effect; higher debt target (weaker adjustment) leads to more expansionary tax cut conditional on initial debt.

### 6.2 Average Duration of Fiscal Adjustment (Table 6)
- Baseline consolidation length 4 years (p_2 = 0.9375); alternative p_2 = 0.975 (average 10 years).
- Initial debt = 120%:
  - Baseline one-year cumulative output multiplier = 0.17; five-year = 0.42.
  - p_2 = 0.975 → one-year = 0.26; five-year = 0.55.
  - Longer expected duration makes tax cut slightly more contractionary when debt high.
- Initial debt = 80%:
  - Baseline one-year −0.51; five-year −0.71.
  - p_2 = 0.975 → one-year −0.49; five-year −0.66 (small changes).

### 6.3 Alternative Fiscal Limit Distributions
- Baseline fiscal limit distribution uses CBO projection with transfers gradually reduced.
- Alternative assumes transfers-to-output stabilize at 0.165 (end of 20-year CBO projection), lowering mean fiscal limit from 1.44 to 0.90.
- With lower fiscal limits:
  - Agents place higher switching probability to fiscal adjustment regime for given debt.
  - Even at initial debt = 80%, capital income tax cut becomes barely expansionary: one-year cumulative output multiplier = −0.05.
  - At initial debt = 120%, tax cut becomes quite contractionary: one-year multiplier = 0.43 (vs. 0.17 baseline).

---

### 7  Conclusion

### Summary of principal results
- Macroeconomic effects of a capital income tax cut are fiscal state-dependent: expansionary in low-debt states, less expansionary or contractionary in high-debt states.
- Simulation results for U.S. capital income tax cut:
  - Five-year cumulative output multiplier = −1 with initial government debt at 45 percent of GDP.
  - Five-year cumulative output multiplier = −0.7 with initial government debt at 80 percent of GDP.
  - Multiplier turns contractionary when debt reaches 120 percent.
- Results are robust when expected fiscal adjustment instrument is consumption tax increases (Japan simulation).
- Nonlinearities driven by expectations of future fiscal adjustments dampen expansionary effects as fiscal state deteriorates.

### Broader implications
- Tax policy effects are highly nonlinear; the fiscal state is important when assessing fiscal policy effects.
- As federal fiscal position deteriorates, expectations of future fiscal adjustments can reduce the growth and investment responses to corporate tax reductions.

*Source: IMF Working Paper — 5.3 Revenue Feedback Effects (wpiea2020071-print-pdf).*

### References

### References

### Bibliography
- The section lists cited works on fiscal policy, fiscal multipliers, taxation, DSGE and STVAR methods, narrative fiscal shock identification, and related empirical and theoretical literature. Authors and publications include (but are not limited to) Ajello (2016); Alesina, Azzalini, Favero, and Giavazzi (2018); Arin, Koray, and Spagnolo (2015); Auerbach and Gorodnichenko (2012, 2013); Barro and Redlick (2011); Bertola and Drazen (1993); Bi (2012, 2017, 2013, 2016, 2019); Blanchard and Perotti (2002); Braun (1994); Bureau of Economic Analysis (2018); Caggiano et al. (2015); Candelon and Liebman (2015); Canzoneri et al. (2016); Chamley (1986); Chernozhukov and Hong (2003); Coleman (1990); Congressional Budget Office (2017, 2018a, 2018b); Corsetti, Meier, and Müller (2012); Davig (2004); Davig and Foerster (forthcoming); Davig and Leeper (2011); De Mooij and Saito (2014); Demirel (2016); Devries et al. (2011); Erceg and Lindé (2014); Eskandari (2019); Favero and Giavazzi (2007, 2012); Fernández‑Villaverde et al. (2015); Forni, Monteforte, and Sessa (2009); Fotiou (2019); Gourio (2012); Gravelle and Marples (2019); Huidrom et al. (2016); Ilzetzki, Mendoza, and Végh (2013); International Monetary Fund (2018, 2019); Ireland (1994); Jones (2002); Judd (1985, 1987); Kirchner, Cimadomo, and Hauptmeier (2010); Koop, Pesaran, and Potter (1996); Kopp et al. (2019); Leeper, Plante, and Traum (2010); Leeper, Walker, and Yang (2013); Leeper and Yang (2008); Mankiw and Weinzierl (2006); McGrattan (1994); Mertens and Ravn (2012, 2013); Mountford and Uhlig (2009); Nickel and Tudyka (2014); Office of Management and Budget (2018); Perotti (2008); Plante, Richter, and Throckmorton (2018); Ramey (2019); Ramey and Zubairy (2018); Romer and Romer (2010); Sims and Wolff (2018); Smets and Wouters (2007); Sutherland (1997); Tersvirta, Tjostheim, and Granger (2010); Traum and Yang (2011); Yang (2009); Zubairy (2014).

### Key bibliographic characteristics
- Literature covers empirical identification methods (narrative tax measures, news series), nonlinear and regime-switching dynamics (STVAR, Markov switching), medium-scale DSGE modeling, and robustness of fiscal multiplier estimates across debt states and macro regimes.

---

### Appendix A — Data Sources
- Output: Real GDP (NIPA Table 1.1.3 line 1).
- Government spending: Real federal government consumption expenditures and gross investment (NIPA Table 1.1.3 line 22).
- Personal income tax base: Personal income (NIPA Table 2.1 line 1) less government transfers (NIPA Table 2.1 line 17) plus contributions for government social insurance (NIPA Table 3.2 line 11), deflated by the GDP deflator (NIPA Table 1.1.9 line 1).
- Corporate income tax base: Corporate profits (NIPA Table 1.12 line 13) less Federal Reserve Bank profits (NIPA Tables 6.16 B-C-D), deflated by the GDP deflator (NIPA Table 1.1.9 line 1).
- Average personal income tax rate: Sum of federal personal current taxes (NIPA Table 3.2 line 3) and contributions for government social insurance (NIPA Table 3.2 line ) divided by the nominal personal income tax base.
- Average corporate income tax rate: Federal taxes on corporate income excluding Federal Reserve banks (NIPA Table 3.2 line 9) divided by the nominal corporate income tax base.
- Debt: Federal debt is total public debt as percent of GDP (FRED, Federal Reserve Bank of Saint Louis).
- Defense spending news: Valerie Ramey’s news series (e.g., Ramey, 2019).
- Narrative corporate income taxes: Mertens and Ravn narrative series (Mertens and Ravn, 2013).

---

### Appendix B — Empirical Results: Robustness Checks
- Purpose: Assess robustness of baseline STVAR specification and whether corporate income tax cuts have more expansionary effects in low-debt periods.
- Framework: STVAR with alternative fiscal news measures, variable ordering changes, and narratively identified corporate tax shock treated as the shock.

Findings from robustness checks:
- a) Using the aggregate narratively identified tax news measure of Mertens and Ravn (2012):
  - No strong evidence of debt-dependent effects overall (see Figure B.1).
  - Output responses are debt-dependent around the 5th quarter, with output more expansionary in low-debt periods than in high-debt periods.
- b) Using the aggregate narratively identified unanticipated tax measure:
  - Results are very close to the baseline specification (see Figure B.2).
- c) Using the corporate narratively identified unanticipated income tax cut series, ordered first in the VAR and treated as the shock:
  - In both debt states the output response is contractionary, but not statistically significantly different in the short run (see Figure 5).
  - After 4 quarters, the effect on output in the high-debt state remains negative, but turns positive for the low-debt state.
- d) Changing the ordering of variables (government spending ordered 5th):
  - The effect on output is contractionary in a high-debt state (see Figure B.4).

Figures and confidence intervals:
- Figures B.1–B.4 present Generalized Impulse Response Functions (GIRFs) for federal corporate income tax cuts.
- Red (blue) lines correspond to GIRFs when federal government net debt is at a high- (low-) debt state.
- Dashed lines are the 90-percent confidence intervals.

---

### Appendix C — Model Solution

#### C.1 Equilibrium Conditions
- Key definitions and equations (equation labels preserved):
  - λ_t = c_t^(−σ) (C.1)
  - l_t^φ = λ_t (1−τ_l) w_t (C.2)
  - Define Tobin’s Q as TQ_t = ξ_t / λ_t.
  - 1 = TQ_t [1 − κ (i_t / k_{t−1} − δ)] . (C.3)
  - TQ_t = β E_t [λ_{t+1} / λ_t { (1−τ_{k,t+1}) r_{k,t+1} + TQ_{t+1} [ (1−δ) + κ (i_{t+1} / k_t − δ) (i_{t+1} / k_t) − κ^2/2 (i_{t+1} / k_t − δ) ] } ] (C.4)
  - k_t = (1−δ) k_{t−1} + i_t − κ/2 (i_t / k_{t−1} − δ)^2 k_{t−1} (C.5)
  - λ_t q_t = β_t+1 E_t λ_{t+1} (C.6)
  - y_t = a_t k_{t−1}^α l_t^{1−α} (C.7)
  - (1−α) y_t = w_t l_t (C.8)
  - α y_t = r_{k,t} k_{t−1} (C.9)
  - y_t = c_t + i_t + g_t (C.10)
  - τ_l w_t l_t + τ_{k,t} r_{k,t} k_{t−1} + q_t b_t = g_t + b_{t−1} + z_t (C.11)
  - τ_{l,t} = τ_l (C.12)
  - g_t = g (C.13)

- Fiscal policy regimes and stabilization:
  - Before a capital income tax cut:
    - τ_{k,t} = τ_k (C.14)
    - z_t = z − γ_z (b_{t−1} − b) (C.15)
  - With a capital income tax cut (Markov switching):
    - τ_{k,t} = τ_k − ε_{τ_k}, z_t = z for R_t = 1;
    - τ_{k,t} = ̄τ_k + γ_τ (b_{t−1} − ̄b), z_t = ̄z − γ_z (b_{t−1} − ̄b) for R_t = 2. (C.16)
  - Regime index R_t evolves with transition matrix:
    - ( p_{1,t}    1−p_{1,t}
        1−p_2        p_2 ) . (C.17)

- State variables:
  - Minimum set S_t = {a_t, b_{t−1}, k_{t−1}, R_t}. (Footnote: β_t is dropped from the state space in the simulation exercise to facilitate convergence.)

#### C.2 The Numerical Solution Method
- Overview:
  - Discretize the state space and find fixed points for decision rules mapping state variables into endogenous choices.
  - Decision rules: l_t = f_l(S_t) and b_t = f_b(S_t).

- Iterative solution steps:
  1. Define grid points by discretizing the state space. Make initial guesses for f_l^0 and f_b^0 over the state space.
  2. At each grid point, solve the nonlinear model and obtain updated rules f_l^i and f_b^i using previous rules f_l^{i−1} and f_b^{i−1}:
     - (a) Derive τ_{k,t} and z_t using (C.14) and (C.15) before the tax cut; use (C.16) with given R_t when the tax cut is present.
     - (b) Derive y_t, w_t, λ_t, and c_t in terms of a_t, k_{t−1}, and l_t using (C.1), (C.2), (C.7), and (C.8).
     - (c) Compute k_t, i_t, TQ_t, and k_t from (C.3), (C.5), (C.9), and (C.10).
     - (d) Derive bond price q_t from (C.11).
     - (e) Use the simulated fiscal limit and current debt level b_t to derive the switching probability to the fiscal adjustment regime in the next period, (1 − p_{1,t+1}).
     - (f) Use linear interpolation to obtain f_l^{i−1}(S_{t+1}) and f_b^{i−1}(S_{t+1}) where S_{t+1} = {a_{t+1}, b_t, k_t, R_{t+1}}. Then solve λ_{t+1}, τ_{k,t+1}, r_{k,t+1}, TQ_{t+1}, and i_{t+1} for R_{t+1} = 1 and R_{t+1} = 2.
     - (g) Update decision rules f_l^i and f_b^i using (C.4) and (C.6), where expectations are evaluated conditional on the current tax regime R_t:
       - E[x_{t+1} | R_t = 1] = p_{1,t+1} E[x_{1,t+1} | R_{t+1} = 1] + (1 − p_{1,t+1}) E[x_{2,t+1} | R_{t+1} = 2]
       - E[x_{t+1} | R_t = 2] = (1 − p_2) E[x_{1,t+1} | R_{t+1} = 1] + p_2 E[x_{2,t+1} | R_{t+1} = 2] (C.18)
  3. Convergence check:
     - If |f_l^i − f_l^{i−1}| or |f_b^i − f_b^{i−1}| is above the desired tolerance (set to 1e−6), return to step 2.
     - Otherwise f_l^i and f_b^i are the decision rules.

- Numerical notes:
  - The method uses linear interpolation for off-grid next-period states.
  - Solutions converge to functions mapping the minimum state set into endogenous variables.

*Source: wpiea2020071-print-pdf — References, Appendices A–C*

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_Source: https://www.imf.org/-/media/files/publications/wp/2020/english/wpiea2020071-print-pdf.pdf_