## wpiea2019090

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---

### Introduction and motivation
- Focus: fiscal implications of interest rate normalization (raising the federal funds rate and reducing the Federal Reserve’s holdings of longer term securities) against a backdrop of elevated government debt.
- Historical observations:
  - From 2008 to 2017, average net interest payments were 1.4 percent of GDP with an average federal debt (held by the public)-to-GDP ratio of 66.2 percent.
  - From 1990 to 2007, interest payments averaged 2.4 percent of GDP with an average federal debt of 44.0 percent of GDP.
  - Congressional Budget Office projection: interest payments in 2038 would be three times the 2017 level (4.2 percent vs. 1.4 percent) under the slowest debt growth projection.
- Key literature points:
  - Bohn(2008): U.S. debt historically satisfies a sufficient sustainability condition.
  - Blanchard and Zettelmeyer(2017): fiscal crises are unlikely even with rising interest rates, absent risky macroeconomic policies.
  - Long-term liabilities (Social Security, Medicare, Medicaid) pose threats to debt sustainability (Hagist and Kotlikoff(2008); Davig et al.(2010); Kotlikoff(2015); Cao et al.(2018)).
  - Elmendorf and Sheiner(2017): federal budget on unsustainable path but low interest rates justify higher federal debt and public investment and delayed fiscal consolidation.
- Analytical approach: New Keynesian model with sovereign default risk and regime-switching transfers, incorporating a fiscal limit distribution following Bi(2012) and a fully non-linear analysis.

### Model overview
- Household and technology:
  - Representative household maximizes ∑_{t=0}^∞ β^t (c_t^{1−σ}/(1−σ) − χ n_t^{1+φ}/(1+φ)).
  - Budget constraint: P_t c_t + P_t i_t + B_t R_t = (1−Δ_t) B_{t−1} + P_t[(1−τ^l_t) w_t n_t + (1−τ^k_t)(r^k_t k_{t−1} + Υ_t)] + P_t z_t.
  - Capital law: k_t = (1−δ) k_{t−1} + ν_t [ i_t − κ/2 (i_t/k_{t−1} − δ)^2 k_{t−1} ].
  - Investment efficiency shock: ln(ν_t/ν) = ρ_ν ln(ν_{t−1}/ν) + ε^ν_t, ε^ν_t ∼ N(0, σ^2_ν).
  - Final goods aggregator, monopolistic competition, Rotemberg price adjustment costs.
- Government and fiscal instruments:
  - Government budget constraint: B_t R_t + P_t[tax_t] = (1−Δ_t) B_{t−1} + P_t g_t + P_t z_t, where tax_t ≡ τ^l_t w_t n_t + τ^k_t (r^k_t k_{t−1} + Υ_t).
  - Fiscal adjustment: τ^l_t = τ^l + γ_l (b_{t−1} − b); τ^k_t = τ^k + γ_k (b_{t−1} − b).
  - Government purchases g_t assumed constant: g_t = g ∀t.
- Default mechanism:
  - Effective fiscal limit b^{max}_t drawn from a fiscal limit distribution each period.
  - Default indicator Δ_t = 0 if b_{t−1} < b^{max}_t; Δ if b_{t−1} ≥ b^{max}_t.
  - Default probabilities rise nonlinearly with government debt burden.
- Transfers:
  - Regime-switching transfers z_t with two regimes:
    - If i^z_t = 1: (1−ρ_z) z + ρ_z z_{t−1} + ε^z_t with ρ_z < 1.
    - If i^z_t = 2: μ z_{t−1} + ε^z_t with μ > 1.
  - Regime transitions: matrix (p^z_1, 1−p^z_1; 1−p^z_2, p^z_2), calibrated highly persistent.
- Monetary policy:
  - Distinguish risky debt return R_t and risk-free policy rate R^f_t.
  - Policy rule: R^f_t = max[ (R^f_{t−1})^{ρ_{Rf}} [ R^f (π_t/π)^{α_π} ]^{1−ρ_{Rf}} e^{ε^{Rf}_t}, 1 ] with π_t = P_t/P_{t−1} and α_π > 1 indicating policy activeness.

### Calibration and solution
- Quarterly calibration; fiscal steady-state values from 1970–2017 Historical Tables (Office of Management and Budget(2018)).
- Key calibrated parameters and steady states (exact values preserved):
  - δ = 0.025; α = 0.36; σ = 1.38; φ = 1.83; κ = 1.7; θ = 7.67 (markup 15 percent); ψ = 78.2; π = 1.
  - Government interest payments-to-output = 0.02.
  - Discretionary outlays-to-output g/y = 0.08; mandatory outlays-to-output z/y = 0.11.
  - Real interest (R^f/π) = 1.008 ⇒ quarterly discount factor β = 0.992.
  - Steady-state debt-to-annual output implied to be 0.605.
  - Steady-state tax rates: τ^l = 0.203; τ^k = 0.214.
  - Fiscal reaction coefficients: γ_l = 0.02; γ_k = 0.005.
  - Transfer process: ρ_z = 0.96; μ = 1.006; p^z_1 = 0.9944; p^z_2 = 0.9875 (p^z_2 gives average unstable regime length of 20 years).
  - Default haircut: annual 0.28 ⇒ quarterly ∆ = 0.07.
  - Monetary policy response baseline α_π = 1.8; alternative active rule α_π = 2.5.
  - Investment efficiency persistence ρ_ν = 0.9.
  - Interest rate persistence for unexpected policy shock simulations ρ_{Rf} = 0.8; when normalization is endogenous, ρ_{Rf} = 0.
- Solution method:
  - Symmetric equilibrium assumed.
  - Fully non-linear solution using monotone mapping method of Coleman(1990) and Davig(2004).

### Fiscal limit distributions: definition and drivers
- Fiscal limit: expected sum of discounted maximum primary fiscal surplus over an infinite horizon.
- Debt valuation (no-default case) forward iteration:
  - b_{t−1}/π_t = ∑_{i=0}^∞ β^i_t E_t[ λ_{t+i}/λ_t (tax_{t+i} − g_{t+i} − z_{t+i}) ].
- Fiscal limits simulated under maximum implementable tax rates τ^l_{max} and τ^k_{max} conditional on state S_t = {ν_t, k_{t−1}, z_t, i^z_t}:
  - b(S_t)/π^{max}_t(S_t) ∼ ∑_{i=0}^∞ β^i_t E_t{ λ^{max}_{t+i}(S_{t+i}) / λ^{max}_t(S_t) [ tax^{max}_{t+i}(S_{t+i}) − g_{t+i}(S_{t+i}) − z_{t+i}(S_{t+i}) ] }.
- Key drivers of maximum sustainable debt at t:
  1. Inflation (devaluing nominal liabilities increases fiscal limits).
  2. Stochastic discount factor (β^i_t E_t λ_{t+i}/λ_t): higher expected discount factors raise fiscal limits.
  3. Expected future primary surplus at maximum tax rates.

### 4.1 The Baseline Distribution — calibration and baseline results
- Assumptions for baseline simulation:
  - Investment efficiency and transfers follow processes in (4), (11), and (12) after period t.
  - Federal discretionary spending constant in the future.
  - Transfers evolve conditional on transfer regime index i_z_{t+i} and shock ε_z_{t+i}.
- Maximum tax-rate calibrations:
  - τ_l,max = 0.378.
  - τ_k,max = 0.256.
- Rationale:
  - τ_l,max from maximum average marginal federal tax rates on individual income plus Social Security payroll tax rates (Barro and Redlick (2011)).
  - τ_k,max from constructed federal average tax rates on capital income; observed range 0.150–0.256 from 1980 to 2018.
- Baseline distribution results:
  - Cumulative density function shows default probabilities are almost zero for debt below 100 percent of steady-state annual output.
  - Default probability does not rise significantly until debt exceeds 200 percent of output.
- Current context:
  - Current federal debt held by the public is around 80 percent of GDP.
  - Current net federal debt level around 77 percent of GDP (end of 2017).

### Alternative fiscal limit distribution: uncertain future fiscal policies
- Alternative assumptions (dashed line in Figure 4):
  - Lower maximum average tax rates and transfer trajectories without reforms for the next 20 years.
  - τ_k,max = 0.240 and τ_l,max = 0.292 (vs. baseline 0.256 and 0.378).
  - Transfers follow CBO extended baseline projection 2018–2038; after 20 years transfers revert to regime-switching process.
- Justification for τ_k,max = 0.240: CBO (2017) finds Japan at 0.28 among G-7 excluding U.S.; subtracting average 0.04 state and local corporate tax implies federal maximum average rate of 0.24 given tax competition concerns.
- Alternative distribution features:
  - Narrower than baseline (driven by extended period of rising transfer-to-output ratios).
  - Much lower fiscal limits (driven by lower maximum income tax rates).
  - Default probability starts rising quickly when federal debt held by the public exceeds 120 percent of output.
- Key comparative insight: assessment of debt sustainability for a given debt level crucially depends on expected future fiscal policies.

### Simulation sample details used in analyses
- Impulse-response simulations use 20,000 simulations (see Section 5).
- From these 20,000 simulations:
  - 2,690 simulations retained that have debt-to-annual steady state output ratio between 70–80 percent at t = 0 (matching current net federal debt ~77 percent of GDP).
  - 3.2 percent or 640 simulations have debt falling into the range of 140–150 percent of annual output (used in high-debt scenario analyses).

### 5.4 An unexpected acceleration in the policy rate increase — setup and key quantitative responses
- Shock: exogenous monetary policy shock of 25 basis points at the quarterly rate.
- Policy rule: baseline rule in equation (14) with innovation εRf_t capturing unexpected acceleration.
- Initial debt states for simulations:
  - 70–80 percent of output against baseline fiscal limit distribution (solid lines).
  - 140–150 percent of output against alternative distribution with lower fiscal limits (dashed lines).
- Simulations: median responses based on 20,000 simulations with initial debt levels in the ranges of interest.
- On impact, following the policy rule, the policy rate increases by only 7 basis points in response to the contractionary monetary policy shock that lowers inflation.
- Macroeconomic responses (directional and relative magnitudes):
  - Output: declines relative to no-shock; declines bigger with high-debt state.
  - Labor: declines.
  - Investment: declines.
  - Real bond prices: decline.
  - Interest rate on government bonds: rises.
  - Inflation: lowered by the exogenous increase in the policy rate.
  - Real value of government debt: increases from the initial period due to lower inflation.
- Fiscal and sovereign risk outcomes:
  - Government debt burden as share of output: increases (higher interest payments and lower real bond prices).
  - Default probabilities: increase more because of higher debt levels and lower fiscal limits.
  - Risk premia: increase more because of higher debt levels and lower fiscal limits.
- Contrast with endogenous policy rate increase:
  - Exogenous increase lacks initial debt devaluation from unexpected inflation; instead lowers inflation immediately, increasing real value of nominal liabilities.
  - Default probabilities and risk premia increase more under exogenous shock than under endogenous rate increase.

### Interpretation, mechanisms, and policy considerations
- Main mechanisms:
  - Debt valuation: lower inflation under exogenous policy rate increase raises the real burden of nominal debt immediately.
  - Output and tax revenue: expectation of less future inflation reduces goods demand today, producing smaller output response and smaller tax revenue increase to a macro shock, reducing debt-devaluation benefits.
  - More active monetary rule yields smaller inflation response to positive macro shocks, generating smaller debt devaluation benefits and relatively larger long-run increases in debt-to-output.
- Quantitative magnitudes and risk:
  - Baseline simulation: sovereign default risk virtually zero if Treasury debt held by the public is below 100 percent of GDP because baseline fiscal limit distribution has a long left tail.
  - Government net debt-to-output rises around 70–80 percent of GDP under interest rate normalization, but sovereign risk premia remain almost zero because debt is distant from simulated fiscal limits.
  - High-debt scenario: if net debt-to-output is around 140–150 percent of GDP against a left-shifted fiscal limit distribution (lower maximum capital tax and transfers rising per CBO for 20 years), default probability increases to around 20 percent and marginal increase in default probability from an endogenous policy rate increase becomes visible but remains small.
- Caveats and sensitivity:
  - Benign conclusions rely on moderate inflation, current debt far from fiscal limits, and expectations of future fiscal adjustment.
  - Baseline assumes constant haircut ∆ = 0.07; if haircut larger, risk premia and default probability impacts could be larger.
  - Sensitivity analyzed with haircut rate 0.15 for endogenous policy rate increase; authors report it “doesn’t change our key messages.”
- Policy regime considerations and future research:
  - Analysis assumes active monetary policy and passive fiscal policy.
  - Alternative regime-switching (monetary passive, fiscal active) could lead to price level rise that devalues nominal liabilities (fiscal theory of price determination) and alter debt sustainability outcomes.
  - Allowing for policy regime switching proposed as direction for future research.

*italic: wpiea2019090 (content unit).*

### 1.  Introduction .......................................................................................................

### 1. Introduction

### Chapter structure and major components
- 1.  Introduction ....................................................................................................................3
- 2.  The Model Setup ............................................................................................................6
- 3.  Calibration and Solution ..............................................................................................11
- 4.  Fiscal Limit Distributions ............................................................................................13
  - 4.1 The Baseline Distribution ................................................................................14
  - 4.2 Alternative Distribution: Uncertain Future Fiscal Policies ..............................16
- 5.  Fiscal Implications of Rising Policy Rates ..................................................................17
  - 5.1 An Endogenous Policy Rate Increase: the Baseline Analysis .........................18
  - 5.2 An Endogenous Policy Rate Increase against a High-Debt Level  ..................20
  - 5.3 Interactions with a More Active Interest Rate Policy ......................................21
  - 5.4 An Unexpected Acceleration in the Policy Rate Increase ...............................22
- 6.   Conclusion ...................................................................................................................23

### Appendices
- Appendix A. The Equilibrium System...............................................................................25
- Appendix B. The Numerical Solution Method ..................................................................26

### Tables and Figures listed in the chapter
- Tables
  - Table 1. Baseline Calibration .................................................................................30
- Figures
  - Figure 1. Federal Government Debt, Interest Payments, and the Federal Funds Rate ........................................................................................................................31
  - Figure 2. Mandatory and Discretionary Spending of the Federal Government .....32
  - Figure 3. Federal Income Tax Rates ......................................................................32
  - Figure 4. Fiscal Limit Distributions for the Federal Government .........................33
  - Figure 5. Responses of an Endogenous Rising Policy Rate to a Positive Investment Efficiency Shock .................................................................................34
  - Figure 6. Federal Funds Rate vs. Real Interest Rates on Treasury Bonds .............35
  - Figure 7. Responses of an Endogenous Rising Policy Rate to a Positive Investment Efficiency Shock .................................................................................36
  - Figure 8. Responses of an Endogenous Rising Policy Rate to a Positive Investment Efficiency Shock: Different Monetary Policy Activeness ..................37
  - Figure 9. Responses to an Exogenous Monetary Policy Shock .............................38

*Source: wpiea2019090 - 1.  Introduction (chapter/section table of contents and list of tables/figures).*

### References .............................................................................................................

### wpiea2019090 - References

### Introduction and motivation
- After three rounds of quantitative easing since the Great Recession in 2007–09, the U.S. economy is setting the stage for its longest expansion in history.
- Focus: fiscal implications of interest rate normalization (raising the federal funds rate and reducing the Federal Reserve’s holdings of longer term securities) against a backdrop of elevated government debt.
- Historical observations:
  - From 2008 to 2017, average net interest payments were 1.4 percent of GDP with an average federal debt (held by the public)-to-GDP ratio of 66.2 percent.
  - From 1990 to 2007, interest payments averaged 2.4 percent of GDP with an average federal debt of 44.0 percent of GDP.
  - Congressional Budget Office projection: interest payments in 2038 would be three times the 2017 level (4.2 percent vs. 1.4 percent) under the slowest debt growth projection.
- Literature perspectives summarized:
  - Bohn(2008): U.S. debt historically satisfies a sufficient sustainability condition.
  - Blanchard and Zettelmeyer(2017): fiscal crises are unlikely even with rising interest rates, absent risky macroeconomic policies.
  - Long-term liabilities (Social Security, Medicare, Medicaid) pose threats to debt sustainability (e.g., Hagist and Kotlikoff(2008); Davig et al.(2010); Kotlikoff(2015); Cao et al.(2018)).
  - Elmendorf and Sheiner(2017): federal budget on unsustainable path but low interest rates justify higher federal debt and public investment and delayed fiscal consolidation.
- Analytical approach: a New Keynesian (NK) model with sovereign default risk and regime-switching transfers, incorporating a fiscal limit distribution following Bi(2012) and conducting a fully non-linear analysis.

### Model overview
- Core features:
  - Representative household chooses consumption c_t, labor n_t, investment i_t, capital k_t, and one-period nominal government bond B_t to maximize lifetime utility:
    - Utility: sum_{t=0}^∞ β^t (c_t^{1−σ}/(1−σ) − χ n_t^{1+φ}/(1+φ)).
  - Budget constraint (exact form preserved in the source): P_t c_t + P_t i_t + B_t R_t = (1−Δ_t) B_{t−1} + P_t[(1−τ^l_t) w_t n_t + (1−τ^k_t)(r^k_t k_{t−1} + Υ_t)] + P_t z_t.
  - Capital law of motion: k_t = (1−δ) k_{t−1} + ν_t [ i_t − κ/2 (i_t/k_{t−1} − δ)^2 k_{t−1} ].
  - Investment efficiency shock ln(ν_t/ν) = ρ_ν ln(ν_{t−1}/ν) + ε^ν_t with ε^ν_t ∼ N(0, σ^2_ν).
  - Final goods aggregator and intermediate goods monopolistic competition with Rotemberg price adjustment costs.
  - Government budget constraint: B_t R_t + P_t[tax_t] = (1−Δ_t) B_{t−1} + P_t g_t + P_t z_t, where tax_t ≡ τ^l_t w_t n_t + τ^k_t (r^k_t k_{t−1} + Υ_t).
- Default mechanism:
  - Effective fiscal limit in real debt value b^{max}_t drawn from a fiscal limit distribution each period.
  - Default indicator Δ_t = 0 if b_{t−1} < b^{max}_t; Δ if b_{t−1} ≥ b^{max}_t.
  - Default probabilities rise nonlinearly with government debt burden.
- Transfer policy: regime-switching process for transfers z_t with two regimes (stable and unstable) per Davig et al.(2010):
  - z_t conditional law: if i^z_t = 1: (1−ρ_z) z + ρ_z z_{t−1} + ε^z_t with ρ_z < 1; if i^z_t = 2: μ z_{t−1} + ε^z_t with μ > 1.
  - Regime transitions governed by matrix (p^z_1, 1−p^z_1; 1−p^z_2, p^z_2), calibrated to be highly persistent.
- Fiscal adjustment instruments:
  - Income tax rules: τ^l_t = τ^l + γ_l (b_{t−1} − b); τ^k_t = τ^k + γ_k (b_{t−1} − b).
  - Government purchases g_t assumed constant g_t = g ∀t for simplicity.
- Monetary policy:
  - Distinguish risky government debt return R_t and risk-free policy rate R^f_t.
  - Policy rule: R^f_t = max[ (R^f_{t−1})^{ρ_{Rf}} [ R^f (π_t/π)^{α_π} ]^{1−ρ_{Rf}} e^{ε^{Rf}_t}, 1 ] with π_t = P_t/P_{t−1} and α_π > 1 indicating policy activeness.

### Calibration and solution
- Quarterly calibration; fiscal steady-state values from 1970–2017 Historical Tables (Office of Management and Budget(2018)).
- Key parameter calibrations and steady states (exact values preserved):
  - δ = 0.025; α = 0.36; σ = 1.38; φ = 1.83; κ = 1.7; θ = 7.67 (markup 15 percent); ψ = 78.2; π = 1.
  - Government interest payments-to-output = 0.02 (matches 1970–2017 average net interest outlays to GDP).
  - Discretionary outlays-to-output g/y = 0.08; mandatory outlays-to-output z/y = 0.11.
  - Real interest (R^f/π) = 1.008 ⇒ quarterly discount factor β = 0.992.
  - Steady-state debt-to-annual output implied to be 0.605.
  - Steady-state tax rates: τ^l = 0.203; τ^k = 0.214 (1970–2017 averages via Jones(2002) method).
  - Fiscal reaction coefficients: γ_l = 0.02; γ_k = 0.005 (baseline).
  - Transfer process parameters: ρ_z = 0.96; μ = 1.006; p^z_1 = 0.9944; p^z_2 = 0.9875 (p^z_2 gives average unstable regime length of 20 years).
  - Default haircut: annual 0.28 ⇒ quarterly ∆ = 0.07.
  - Monetary policy response baseline α_π = 1.8; alternative active rule α_π = 2.5.
  - Investment efficiency persistence ρ_ν = 0.9.
  - Interest rate persistence for unexpected policy shock simulations ρ_{Rf} = 0.8; when normalization is endogenous, ρ_{Rf} = 0.
- Solution method:
  - Symmetric equilibrium assumed (y_t(i) = y_t, etc.).
  - Fully non-linear solution obtained using monotone mapping method of Coleman(1990) and Davig(2004).

### Fiscal limit distributions: definition and simulation
- Fiscal limit definition: expected sum of discounted maximum primary fiscal surplus over an infinite horizon.
- Debt valuation (no-default case) forward iteration yields:
  - b_{t−1}/π_t = ∑_{i=0}^∞ β^i_t E_t[ λ_{t+i}/λ_t (tax_{t+i} − g_{t+i} − z_{t+i}) ].
- Fiscal limits simulated by computing variables under maximum implementable tax rates τ^l_{max} and τ^k_{max}:
  - Fiscal limit distribution conditional on state S_t = {ν_t, k_{t−1}, z_t, i^z_t}:
    - b(S_t)/π^{max}_t(S_t) ∼ ∑_{i=0}^∞ β^i_t E_t{ λ^{max}_{t+i}(S_{t+i}) / λ^{max}_t(S_t) [ tax^{max}_{t+i}(S_{t+i}) − g_{t+i}(S_{t+i}) − z_{t+i}(S_{t+i}) ] }.
- Key drivers of maximum sustainable debt at t:
  1. Inflation (devaluing nominal liabilities increases fiscal limits).
  2. Stochastic discount factor (β^i_t E_t λ_{t+i}/λ_t): higher expected discount factors (lower expected real interest rates) raise fiscal limits.
  3. Expected future primary surplus at maximum tax rates.

### Key simulation findings on interest rate normalization and debt sustainability
- Baseline simulation (U.S.-calibrated):
  - Sovereign default risk virtually zero if Treasury debt held by public is below 100 percent of GDP because baseline fiscal limit distribution has a long left tail.
  - Position and shape of fiscal limit distribution subject to uncertainties (e.g., maximum income tax rates, transfer trajectories).
- Endogenous policy rate increase (response to expansion):
  - Positive macroeconomic shock increases activity and tax base; unexpected inflation can devalue nominal liabilities.
  - Government debt-to-output declines only in the short run; longer run: higher interest payments and lower real bond prices increase financing needs and debt burden.
  - More active monetary rule (higher α_π) yields smaller responses in output and inflation to a positive shock, implying smaller nominal-devaluation benefits and relatively larger long-run increases in debt-to-output compared to a less active rule—though the effect difference is small for sensible response magnitudes.
- Exogenous policy rate shock (unexpected acceleration in raising policy rate):
  - Government debt burden as share of output rises relative to no-shock path due to higher interest payments and lower real bond prices.
  - Unlike endogenous rate rise, exogenous positive policy shock reduces inflation, increasing the real value of existing nominal liabilities, contributing to rising debt burden.
- Quantitative magnitudes and risk:
  - Government net debt-to-output rises around 70–80 percent of GDP under interest rate normalization, but sovereign risk premia remain almost zero because debt is distant from simulated fiscal limits.
  - If current net debt-to-output ratio is around 140–150 percent of GDP against a left-shifted fiscal limit distribution (driven by lower maximum capital tax and transfers rising per CBO for 20 years), default probability increases to around 20 percent and marginal increase in default probability from an endogenous policy rate increase becomes visible but remains small.
- Conditionality and caveats:
  - Conclusion that policy rate increases of the observed magnitude since 2015 pose no risk to debt sustainability is conditional on moderate inflation, current debt far from fiscal limits, and expectations of future fiscal adjustment.

### Relation to other work
- Close relation to Battistini et al.(2019): both study NK models with sovereign default risk, but current work focuses on interest rate normalization in an expansion and includes capital to let interest rates affect saving decisions between investment and government bonds.
- Approach differs from classic strategic default models (Eaton and Gersovitz(1981); Aguiar and Gopinath(2006); Arellano(2008)) and partial equilibrium empirical approaches (Ghosh et al.(2013); Tanner(2013); Collard et al.(2015)).

_italic: wpiea2019090 - References (source content)

### 4.1  The Baseline Distribution

### 4.1  The Baseline Distribution

### Calibration and key modeling assumptions
- Investment efficiency and transfers follow the stochastic processes in (4), (11), and (12) after period t.
- Federal discretionary spending is assumed constant in the future (government purchases assumed constant).
- Transfers evolve conditional on a transfer regime index (i_z_{t+i}) and a transfer shock (ε_z_{t+i}).
- Maximum tax-rate calibrations:
  - τ_l,max = 0.378 (maximum labor income tax rate).
  - τ_k,max = 0.256 (maximum capital income tax rate).
- Rationale for τ_l,max: maximum average marginal federal tax rates on individual income plus Social Security payroll tax rates from 1912 to 2006 constructed by Barro and Redlick (2011).
- Rationale for τ_k,max: constructed series for federal average tax rates on capital income; observed range 0.150–0.256 from 1980 to 2018; recent drop from 0.19 in 2017 to 0.15 in 2018 after the Tax Cuts and Jobs Act.
- Simulation setup:
  - Baseline distribution simulated conditional on a beginning state S_t at the steady state.
  - Appendix B.1 describes simulation procedures (referenced).
  - Impulse-response simulation details (later sections) use 20,000 simulations (see Section 5).

### Baseline distribution results and interpretation
- Cumulative density function (Figure 4, solid line) of the baseline fiscal limit distribution shows:
  - Default probabilities are almost zero for debt below 100 percent of steady-state annual output.
  - Default probability does not rise significantly until debt exceeds 200 percent of output.
- Current context:
  - Current federal debt held by the public is around 80 percent of GDP (postwar-high but almost risk free by the baseline fiscal limit distribution).
  - Current net federal debt level around 77 percent of GDP (end of 2017).

### Alternative distribution: uncertain future fiscal policies (summary)
- Motivation: large uncertainties around future fiscal parameters and unobserved maximum implementable tax rates for the U.S.
- Alternative distribution assumptions (dashed line in Figure 4):
  - Lower maximum average tax rates and transfer trajectories without reforms for the next 20 years.
  - τ_k,max = 0.240 and τ_l,max = 0.292 (compared to baseline 0.256 and 0.378).
  - Justification for τ_k,max = 0.240: CBO (2017) finds Japan at 0.28 among G-7 excluding U.S.; subtracting an average 0.04 state and local corporate tax implies a maximum average federal rate of 0.24 given tax competition concerns.
  - For transfer policy, transfers follow the extended baseline projection from 2018 to 2038 by the Congressional Budget Office (2018a) (no changes in current law). After 20 years, transfers revert to the regime-switching process in (11) and (12).
- Alternative distribution features:
  - Narrower than baseline (driven by extended period of rising transfer-to-output ratios).
  - Much lower fiscal limits (driven by lower maximum income tax rates).
  - Default probability starts rising quickly when federal debt held by the public exceeds 120 percent of output.
- Key comparative insight: For a given debt level, assessment of debt sustainability crucially depends on expected future fiscal policies.

### Simulation sample details relevant to distributional statements
- From 20,000 simulations used in impulse-response analyses:
  - 2,690 simulations retained that have debt-to-annual steady state output ratio between 70–80 percent at t = 0 (matching current net federal debt ~77 percent of GDP).
  - 3.2 percent or 640 simulations have debt falling into the range of 140–150 percent of annual output (used in high-debt scenario analyses).

*Source: wpiea2019090 - 4.1  The Baseline Distribution.*

### 5.4  An Unexpected Acceleration in the Policy Rate Increase

### 5.4  An Unexpected Acceleration in the Policy Rate Increase

### Simulation setup and shock
- Monetary policy shock: exogenous monetary policy shock of 25 basis points at the quarterly rate.
- Policy rule: baseline monetary policy follows the rule in equation (14) with an innovation term εRf_t capturing unexpected acceleration in the policy rate.
- Initial debt states used in simulations:
  - Initial debt state of 70–80 percent of output against the baseline fiscal limit distribution (solid lines).
  - High-debt state of 140–150 percent of output against the alternative distribution with lower fiscal limits (dashed lines).
- Simulations: median responses based on 20,000 simulations with initial debt levels in the debt ranges of interest.
- On impact, following the monetary policy rule of (14), the policy rate increases by only 7 basis points in response to the contractionary monetary policy shock that lowers inflation.

### Key quantitative responses to the exogenous policy rate increase
- Macroeconomic quantities (directional and relative magnitudes):
  - Output: declines relative to the case without the monetary policy shock; decline magnitudes are bigger with a high-debt state.
  - Labor: declines.
  - Investment: declines.
  - Real bond prices: decline (lower real bond prices).
  - Interest rate on government bonds: rises.
  - Inflation: lowered by the exogenous increase in the policy rate.
  - Real value of government debt: increases from the initial period due to lower inflation.
- Fiscal and sovereign risk outcomes:
  - Government debt burden as a share of output: increases (driven by higher interest payments and lower real bond prices).
  - Default probabilities: increase more because of higher debt levels and lower fiscal limits.
  - Risk premia: increase more because of higher debt levels and lower fiscal limits.
- Contrast with endogenous policy rate increase:
  - No initial debt devaluation effect from unexpected inflation (there is no positive macroeconomic shock that raises inflation).
  - Exogenous increase lowers inflation immediately, which increases the real value of government debt from the initial period.
  - Consequently, default probabilities and risk premia increase more under the exogenous policy rate increase than under the endogenous rate increase scenario depicted in Figure 5.

### Interpretation and caveats
- Overall message:
  - Rising policy rates (endogenous or exogenous) increase government debt burden mainly through higher interest payments and lower real bond prices.
  - Sovereign default risk is subdued whether with an endogenous or exogenous policy rate increase at the recent pace of rate increases in the near term.
- Potential concern:
  - The benign conclusion relies on a constant haircut rate assumption (baseline ∆ = 0.07).
  - If the loss to bond holders can be bigger, risk premia could be higher and the impact on default probabilities from interest rate normalization could be larger.
  - A sensitivity analysis with a haircut rate of 0.15 (vs. 0.07 in the baseline calibration) was pursued with the endogenous policy rate increase and “doesn’t change our key messages.” This result is available upon request.

### Mechanisms highlighted
- Debt valuation effects:
  - Under an exogenous policy rate increase, lower inflation raises the real burden of nominal government debt immediately.
  - Under a more active monetary policy rule, the economy produces less inflation for a given positive macroeconomic shock, generating a smaller debt devaluation effect from unexpected inflation.
- Output and tax revenue channel:
  - Expectation of less future inflation reduces goods demand today, leading to a smaller output response and a smaller tax revenue increase to a macroeconomic shock.
  - Both the smaller debt devaluation effect and the smaller tax revenue response contribute to a larger increase in the debt-to-output ratio for a more active monetary policy rule than for a less active one.

### Policy regime considerations and directions for future research
- Analysis conditioned on regime:
  - The paper’s analysis assumes active monetary policy and passive fiscal policy (monetary authorities raise policy rates sufficiently to control inflation; fiscal authorities raise fiscal surplus sufficiently to stabilize debt).
- Alternative regime-switching:
  - If nominal debt burden continues to rise, monetary authorities could switch to a passive regime and fiscal authorities to an active regime.
  - That combination implies a rise in the price level (as implied by the fiscal theory of price determination), which devalues government nominal liabilities and may alter debt sustainability outcomes.
  - Allowing for policy regime switching may provide a different perspective on debt sustainability of highly indebted economies and is proposed as an avenue for future research.

*Source: 5.4  An Unexpected Acceleration in the Policy Rate Increase (wpiea2019090).*

### References

### References

### Sovereign debt, fiscal limits, and fiscal sustainability
- Aguiar, M.andGopinath, G.(2006). Defaultable debt, interest rates and the current account.Journal of International Economics,69, 64–83.
- Arellano, C.(2008). Default risk and income fluctuations in emerging economies.American Economic Review,98(3), 690–712.
- Bi, H.(2012). Sovereign default risk premia, fiscal limits, and fiscal policy.European Economic Review,56(3), 389–410.
- —(2017). Fiscal sustainability: A cross-country analysis.Economic Review, (Fourth Quarter), 5–35, Federal Reserve Bank of Kansas City.
- —,Leeper, E. M.andLeigh, C.(2018). Sovereign default and monetary policy tradeoffs. International Journal of Central Banking,14(3), 289–324.
- Collard,  F.,Habib,  M.andRochet,  J.-C.(2015). Sovereign debt sustainability in advanced economies.Journal of the European Economic Association,13(3), 381–420.
- Corsetti, G.,Kuester, K.,Meier, A.andMuller, G. J.(2013). Sovereign risk, fiscal policy, and macroeconomic stability.The Economic Journal,123(566), F99–F132.
- Eaton, J.andGersovitz, M.(1981). Debt with potential repudiation: Theoretical and empirical analysis.Review of Economic Studies,48(2), 289–309.
- Ghosh, A. R.,Kim, J. I.,Mendoza, E. G.,Ostry, J. D.andQureshi, M. S.(2013). Fiscal fatigue, fiscal space and debt sustainability in advanced economies.The Economic Journal,123(566), F4–F30.
- Kraay, A.andNehru, V.(2006). When is external debt sustainable?World Bank Economic Review,20(3), 341–365.
- Sturzenegger,  F.andZettelmeyer,  J.(2008). Haircuts: Estimating investor losses in sovereign debt restructurings, 1998-2005.Journal of International Money and Finance,27(5), 780–805.
- Tanner,  E.(2013). Fiscal sustainability: A 21st century guide for theperplexed. IMF Working Paper 13/89, International Monetary Fund, Washington, D.C.

### Monetary policy, central banking, and inflation
- Ascari, G.andRossi, L.(2012). Trend inflation, non-linearities and firms price-setting: Rotemberg vs. Calvo.The Economic Journal,122(563), 1115–1141.
- Ball,  L.  M.andSheridan, N.(2004).Does Inflation Targeting Matter?, pp. 249–276. University of Chicago Press, Chicago.
- Beck, G. W.andWieland, V.(2017). How to normalize monetary policy in the euro area. IMFS Working Paper Series, No. 15, Institute of Monetary andFiscal Stability, Goethe University.
- Blanchard, O. J.(2019). Public debt and low interest rates. American Economic Association Presidential Address, Janauary 4th, Atlanta.
- —andZettelmeyer, J.(2017). Will rising interest rates lead to fiscal crisis? Policy Brief 17-27, Peterson Institute of International Economics.
- Del Negro, M.andSims, C. A.(2015). When does a central bank’s balance sheet require fiscal support?Joural of Monetary Economics,73(C), 1–19.
- Hall, R. E.andReis, R.(2015). Maintaining central-bank financial stability under new-style central banking. NBER Working Paper 21173, National Bureau of Economic Research, Cambridge, MA.
- Leeper, E. M.(1991). Equilibria under ‘active’ and ‘passive’ monetary and fiscal policies.Journal of Monetary Economics,27(1), 129–147.
- —andLeith, C.(2016).Understanding Inflation as a Joint Monetary-Fiscal Phenomenon, pp. 2305–2416. Elsevier.
- Mishkin,  F.  S.(2010).Will  Monetary  Policy  Become  More  of  a  Science?, pp. 81–103. Springer, Berlin, Heidelberg.
- —andSchmidt-Hebbel, K.(2007).Does Inflation Targeting Matter?, pp. 291–372. Central Bank of Chile, Santiago.
- Woodford, M.(2003).Interest and Prices: Foundations of a Theory of Monetary Policy. Princeton, N.J.: Princeton University Press.
- Christensen, J. H.,Lopez, J. A.andRudebusch, G. D.(2015). A probabilty-based stress test of Federal Reserve assets.Journal of Monetary Economics,73(C), 26–43.
- Carpenter, S.,Ihrig,  J.,Klee, E.,Quinn, D.andBoote,  A.(2015). The Federal Reserve’s balance sheet and earnings: A primer and projections.International Journal of Central Banking,11(2), 237–283.
- Cavallo, M.,Del Negro, M.,Frame, W. S.,Grasing, J.,Malin, B. A.andRosa, C.(2018). Fiscal implications of the Federal Reserve’s balance sheet normalization. Staff Report No. 833, Federal Reserve Bank of New York.
- Board  of  Governors  of  the  Federal  Reserve  System(2018a). Data:  Selected interest rates—H. 15. Retrieved from FRED, Federal Reserve Bank of St. Louis, https://fred.stlouisfed.org/series/FEDFUNDS, August 25, 2018.
- Board of Governors of the Federal Reserve System(2018b). FOMC communications related to policy normalization. The Federal Open Market Committee, available on https://www.federalreserve.gov/monetarypolicy/policy-normalization.htm.
- Federal    Reserve    Bank    of    Dallas(2018).  Market  value  of  gross  federal  debt.  Retrieved  from   FRED,   Federal  Reserve  Bank   of   St.   Louis, https://fred.stlouisfed.org/series/MVGFD027MNFRBDAL, December 11, 2018.

### Business cycles, investment, and macroeconomic shocks
- Basu,  S.andFernald,  J.  G.(1995). Are apparent productive spillovers a figment of specification error?Journal of Monetary Economics,36(1), 165–188.
- Gourio, F.(2012). Disaster risk and business cycles.American Economic Review,102(6), 2734–2766.
- Greenwood,  J.,Hercowitz,  Z.andHuffman,  G.  W.(1988). Investment, capacity utilization, and the real business cycle.American Economic Review,78(3), 402–417.
- —,—andKrusell, P.(2000). The role of investment-specific technological change in the business cycle.European Economic Review,44(1), 91–115.
- Jordá, O.,Schularick, M.andTaylor, A. M.(2017).Macrofinancial History and the New Business Cycle Facts, Chicago: University of Chicago Press, vol. 31.
- Justiniano, A.,Pirimiceri, G. E.andTambalotti, A.(2010). Investment shocks and business cycles.Journal of Monetary Economics,57(2), 132–145.
- Smets, F.andWouters, R.(2007). Shocks and frictions in US business cycles: A Bayesian DSGE approach.American Economic Review,97(3), 586–606.
- Jones, J. B.(2002). Has fiscal policy helped stabilize the postwar U.S. economy?Journal of Monetary Economics,49(May), 709–746.
- Gourio, F.(2012). Disaster risk and business cycles.American Economic Review,102(6), 2734–2766.

### Fiscal policy, demographics, and long-term outlooks
- Barro, R. J.andRedlick, C. J.(2011). Macroeconomic effects from government purchases and taxes.Quarterly Journal of Economics,123(1), 51–102.
- Cao, Y.,Badia, M. M.andPeralta-Alva, A.(2018). The cost of waiting: U.S. fiscal policy in 2050. Manuscript, International Monetary Fund, Washington D.C.
- Elmendorf, D. W.andSheiner, L. M.(2017). Federal budget policy with an anging population and persistently low interest rates.Journal of Economic Perspective,31(3), 175–194.
- Hagist,  C.andKotlikoff,  L.  J.(2008). Who’s going broke?  Comparing growth in public healthcare expenditure in ten OECD countries.Hacienda Pública Española/Revista de Economia Pública,188(1), 2009.
- Kotlikoff,  L.  J.(2015). America’s fiscal insolvency and its generational conseuqneces. February 25, Testimony to the Senate Budget Committeee, U.S. Congress, Washington, D.C.
- Qian,  R.(2012). Why do some countries default more often then others?  The role of institutions. Policy Research Working Papers, WPS 5993, World Bank, Washington, D.C.
- The Board of Trustees, Federal OASDI Trust Funds(2018). The 2018 annual report of the board of trustees of the federal old-age and survivors insurance and federal disability insurance trust funds. Social Security Administration, the U.S. Federal Government.
- Congressional Budget Office(2017). International comparisons of corporate income tax rates. U.S. Congress, available at https://www.cbo.gov/publication/52419.
- Congressional Budget Office(2018a). The 2018 long-term budget outlook. The U.S. Congress, Washington, D.C.
- Congressional Budget Office(2018b). The long-term budget outlook under alternative scenarios for fiscal policy. The U.S. Congress, Washington, D.C.
- Office   of   Management   and   Budget(2018).  Historical tables. Retrieved from https://www.whitehouse.gov/omb/historical-tables/, August 25, 2018.
- Department of  the  Treasury(2018). Treasury bulletin. Produced and published by Bureau of the Fiscal Service, U.S. Department of the Treasury.

### Methods, models, and technical contributions
- Battistini,  N.,Callegari,  G.andZavalloni,  L.(2019). Dynamic fiscal limits and monetary-fiscal policy interactions. Manuscript, European Central Bank, Frankfurt, Germany.
- Bohn, H.(2008).The Sustainability of Fiscal Policy in the United States, Cambridge, MA: MIT Press.
- Coleman, W. J., II(1990). Solving the stochastic growth model by policy-function iteration.Journal of Business and Economic Statistics,8(1), 27–29.
- Davig, T.(2004). Regime-switching debt and taxation.Journal of Monetary Economics,51(4), 837–859.
- —,Leeper, E. M.andWalker, T. B.(2010). “Unfunded liabilities” and uncertain fiscal financing.Journal of Monetary Economics,57(5), 600–619.
- Del Negro, M.andSims, C. A.(2015). When does a central bank’s balance sheet require fiscal support?Joural of Monetary Economics,73(C), 1–19.
- Fatás,  A.,Ilian,  M.andRose, A. K.(2007). Quantitative goals for monetary policy. Journal of Money, Credit and Banking,39(5), 1163–1176.
- Cavallo, M.,Del Negro, M.,Frame, W. S.,Grasing, J.,Malin, B. A.andRosa, C.(2018). Fiscal implications of the Federal Reserve’s balance sheet normalization. Staff Report No. 833, Federal Reserve Bank of New York.
- Christensen, J. H.,Lopez, J. A.andRudebusch, G. D.(2015). A probabilty-based stress test of Federal Reserve assets.Journal of Monetary Economics,73(C), 26–43.
- Carpenter, S.,Ihrig,  J.,Klee, E.,Quinn, D.andBoote,  A.(2015). The Federal Reserve’s balance sheet and earnings: A primer and projections.International Journal of Central Banking,11(2), 237–283.

*Content compiled from the document "References."*

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_Source: https://www.imf.org/-/media/files/publications/wp/2019/wpiea2019090.pdf_