## _wp0663

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---

### Introduction
- Topic: relationship between government debt and long-term interest rates.
- Approach:
  - Theoretical: specify and solve a dynamic general equilibrium model that incorporates debt nonneutrality; conduct numerical simulations.
  - Empirical: panel data for 19 industrial countries.
- Key findings:
  - Simulated and estimated interest rate effects of government debt tend to be small.
  - If an increase in government debt is combined with an increase in government consumption, effects on interest rates would be considerably larger.
  - Accumulating government debt can have important economic impacts on the long-run consumption level even if pure crowding out interest-rate effects are limited.

### Literature overview
- Theoretical context:
  - Ricardian equivalence: government debt is neutral.
  - Debt nonneutrality arises in overlapping-generations frameworks (Blanchard (1985); Buiter (1988); Weil (1989)).
  - Partial-equilibrium treatments (Laubach (2003); Engen and Hubbard (2004)) assume crowding out on physical capital; this paper embeds interest-rate determination in a full general equilibrium model usable for calibration and numerical simulation.
- Selected empirical evidence (preserved exactly as presented):
  - United States:
    - A one percentage point increase in the deficit-to-GDP ratio would increase the expected real interest rate by 19-45 basis points.
    - A one percentage increase in the debt-to-GDP ratio would increase the expected real interest rate by 3-5 basis points.
    - Laubach (2003) reports that the estimated effects of a one percentage point increase in the expected deficit-to-GDP ratio is about 5-8 times larger than the effects of a one percentage point increase in the expected debt-to-GDP ratio.
  - Euro area (Faini (2004)):
    - A 5-7 basis point increase in the real interest rate for a one percentage point increase in the public debt-to-GDP ratio of the euro area.
  - Multi-country studies:
    - Ford and Laxton (1995): increase in OECD-wide government debt since the early 1970s was a major factor in the rise in real interest rates in the 1980s and early 1990s.
    - Orr, Edey, and Kennedy (1995): significant long-run effect of an increase in the deficit-to-GDP ratio on long-term real rates for 17 OECD countries.
    - Ardagna, Caselli, and Lane (2004): increase in government debt affects long-term interest rates only for countries with above-average debt levels.
- Methodological note: choice of fiscal variable (deficit-to-GDP vs. debt-to-GDP) affects estimated magnitudes.

### Theoretical analysis: model and mechanism
- Model framework and assumptions:
  - Continuous time, overlapping-generations model extending Buiter (1988).
  - Neoclassical production function (constant returns to scale); utility with constant elasticity of marginal utility.
  - New generation born at constant rate, β; constant probability of death, λ.
  - Population growth rate n ≡ β − λ.
  - Individuals supply fixed labor, receive wages, pay lump-sum tax τ; instantaneous utility with constant elasticity of marginal utility and pure rate of time preference, ρ.
  - Physical capital depreciates at rate δ.
  - Savings accumulate as nonhuman wealth a in physical capital k or government bonds b.
  - Physical capital and government bonds are perfect substitutes; yields on government bonds equal marginal product of physical capital.
- Government:
  - Raises revenue from lump-sum taxes τ; purchases public good g (does not affect individual utility); issues government bonds b.
- Debt nonneutrality mechanism:
  - Government postpones taxation by issuing debt today and committing to higher taxes in the future.
  - Current generations do not fully internalize future tax increases; they perceive part of the debt as net wealth, increase consumption, and run down physical capital.
  - Lower physical capital → higher marginal product of capital → higher interest rates.
- Steady-state characterization (as presented):
  - Steady-state solution for consumption c and physical capital k given government debt b and government consumption g solves:
    - (1) []() 111 '( )1()fkcrkbθρβθλρθ −−− ⎡⎤ −= − ++ + ⎣⎦ ,
    - (2) ()fkcg kδ=−− .
  - Key property: solution depends on b when β > 0; when β = 0, steady state is independent of b.
  - Steady-state interest rate determined by government debt and government consumption:
    - (3) (,,,,,;,)rRbgβλθραδ=

### Numerical calibration and baseline parameters
- Parameters used in simulations:
  - Share of capital income in total output: 0.3 α = ;
  - Intertemporal elasticity of substitution: 3 θ = ;
  - Rate of depreciation of physical capital: 0.05 δ = .
  - Birth and death rates initially: 0.03 β = and 0.02 λ = , implying population growth rate 0.01 n ≡ β − λ = .
  - Pure rate of time preference baseline: 0.06 ρ = (alternative 0.02 ρ = used by Barro and Sala-i-Martin (1995) noted).
  - Simulations consider variations in ρ and deviations in birth and death rates; only cases with β − λ ≥ 0 considered.
  - Lump-sum taxes τ adjust to ensure instantaneous balanced budget; b and g allowed to move independently at steady state.

### Simulation results: steady-state effects (summary)
- Table 1 reports:
  - Column (1): steady-state capital-to-output ratio;
  - Column (2): crowding-out effect on the stock of capital with respect to a unit change in government debt;
  - Columns (3) and (4): change in interest rates, in basis points, with respect to a one percentage point increase in the ratio of government debt or consumption to GDP;
  - Column (5): change in the level of consumption, in percent, with respect to a one percentage point increase in the ratio of government debt to GDP.
- Representative entries from Panel (A) for birth rate 0.03, death rate 0.02:
  - 0.06 ρ = : Capital-output ratio 2.57-0.53 ; Government debt effect on interest rates 1.68.3 ; Long-run effect on consumption -0.06
  - 0.05 ρ = : Capital-output ratio 2.80-0.57 ; Government debt effect on interest rates 1.58.1 ; Long-run effect on consumption -0.06
  - 0.04 ρ = : Capital-output ratio 3.06-0.62 ; Government debt effect on interest rates 1.38.0 ; Long-run effect on consumption -0.05
  - 0.03 ρ = : Capital-output ratio 3.38-0.68 ; Government debt effect on interest rates 1.27.8 ; Long-run effect on consumption -0.05
  - 0.02 ρ = : Capital-output ratio 3.76-0.74 ; Government debt effect on interest rates 1.18.3 ; Long-run effect on consumption -0.04
- Representative entries from Panel (B) for different birth/death combinations (time preference 0.04 ρ = unless noted):
  - 0.04 β =, 0.02 λ = : Capital-output ratio 3.06-0.62 ; Government debt effect on interest rates 1.38.0 ; Long-run effect on consumption -0.05
  - 0.04 β =, 0.01 λ = : Capital-output ratio 4.80-0.39 ; Government debt effect on interest rates 0.43.9 ; Long-run effect on consumption -0.01
  - 0.04 β =, 0.00 λ = : Capital-output ratio 4.03-0.52 ; Government debt effect on interest rates 0.75.6 ; Long-run effect on consumption -0.03
  - 0.04 β =, 0.00 λ = (β = 0.00, λ = 0.00 row): Capital-output ratio 7.500.00 ; Government debt effect on interest rates 0.00.0 ; Long-run effect on consumption 0.00
- Interpretation:
  - Higher ρ → lower steady-state capital-output ratio and larger interest rate effect of government debt.
  - Lower β → smaller interest rate effect of government debt.
  - When β = 0, no crowding-out effect and government debt is neutral.
- Magnitudes emphasized:
  - A one percentage point increase in the government debt-to-output ratio would raise the steady-state real interest rate by up to 2 basis points, depending on parameter values.
  - Interest rate effects of an increase in government consumption are several times larger than effects of government debt.
  - Consumption would be permanently lowered by up to 0.08 percent when the government debt-to-GDP ratio increases by one percentage point (simulated welfare loss).

### Model caveats and interpretation
- Simulated interest rate effect likely a lower bound because:
  - Model captures purely long-run physical crowding-out; uncertainty and risk premia are excluded.
  - Lump-sum tax is available in the model; if distortionary income taxes are used in reality, impacts on real interest rates would be larger.
- Conclusion from simulations: even small interest rate effects can coexist with significant crowding out of physical capital and nontrivial welfare losses.

### Empirical design and estimation
- Data:
  - Panel for 19 OECD countries spanning 1971–2004; five-year averaging used to eliminate short-run fluctuations (each country provides up to seven observations).
- Dependent variable: real long-term interest rates on government bonds (nominal yields minus expected inflation).
- Fiscal variables:
  - Government debt: ratio of current financial liabilities of the general government to nominal GDP (both gross and net liabilities considered).
  - Government consumption: ratio of government final consumption expenditure to nominal GDP.
- Estimated specification motivated by steady-state mapping r = R(b,g; β,λ,θ,ρ,α,δ).
- Estimated equation: itiititit RBGαβγε=+++.

### Estimation results (Table 2)
- No country effect (common intercept):
  - Government debt (net): 0.018 (standard error ( 0.007 )**) using trend inflation; 0.018 ( ( 0.006 )**) using actual inflation.
  - Government debt (gross): 0.023 ( ( 0.009 )**) using trend inflation; 0.024 ( ( 0.008 )**) using actual inflation.
  - Government consumption: 0.25 ( ( 0.06 )**) and 0.19 ( ( 0.06 )**) using trend and actual inflation respectively in first pair; 0.23 ( ( 0.05 )**) and 0.18 ( ( 0.05 )**) in second pair.
  - Number of observations: 111; Adjusted R-squared about 0.15–0.17; Standard error of estimated equation 2.45–2.21.
  - Interpretation: a one percentage point increase in government debt-to-GDP raises real long-term interest rate by about 2 basis points.
  - Government consumption effects around 20–25 basis points per one percentage point increase in government consumption-to-GDP.
- Fixed country effects:
  - Government debt (net): 0.053 ( ( 0.013 )**) using trend inflation; 0.052 ( ( 0.012 ) ) using actual inflation.
  - Government debt (gross): 0.040 ( ( 0.012 )**) using trend inflation; 0.042 ( ( 0.011 ) ) using actual inflation.
  - Government consumption: 0.89 ( ( 0.17 )**) and 0.86 ( ( 0.19 )**) using trend inflation; 0.73 ( ( 0.16 ) )** and 0.68 ( ( 0.17 ) )** using actual inflation.
  - Number of observations: 111; Adjusted R-squared about 0.38–0.42; Standard error of estimated equation 2.03–1.87.
  - Interpretation: with fixed effects, a one percentage point increase in government debt-to-GDP raises real long-term interest rates by about 4–5 basis points; government consumption effects estimated around 90 basis points.
- Authors’ interpretations:
  - Estimated coefficients larger under fixed effects than steady-state numerical simulations.
  - Possible reasons: risk premium on government debt (not in theoretical model) and use of distortionary taxes in reality; suggested model extensions: incorporate uncertainty and income taxation or estimate crowding-out effects more directly.

### Appendix I — core model results and comparative statics
- Production: y = f(k); r = f'(k). (Equation A2)
- Consumer problem and aggregation produce:
  - Individual lifetime utility and budget conditions (Equations A3–A11).
  - Aggregate relationships and per capita dynamics (Equations A12a–A19).
  - Differential equation for per capita consumption: (1/θ) \dot{c}_t / c_t = r_t - ρ - β + m_t. (Equation A20)
  - Evolution of marginal propensity to consume m_t involves r_t, θ, ρ, λ, β (Equation A21).
- Government and resource constraints:
  - Government debt per capita evolution: \dot{b}_t = (r_t - n - λ) b_t + g_t - τ_t. (Equation A22)
  - Balanced budget rule assumed: \dot{b}_t = 0 for every t (Equation A23).
  - Physical capital accumulation: \dot{k}_t = f(k_t) − c_t − g_t − δ k_t. (Equation A24)
- Steady state:
  - Resource condition: f(k) − g − k δ = c. (Equation A26)
  - When β > 0, steady state depends on b; when β = 0, f'(k) = ρ and steady state independent of b.
  - Nonlinear condition determining steady-state k: (Equation A27).
- Comparative statics:
  - ∂k/∂b given by Equation (A28).
  - ∂k/∂g given by Equation (A29).
  - Change in interest rate due to changes in b and g: Equation (A30): dr = (dr/dk) [ (∂k/∂b) db + (∂k/∂g) dg ].

### Appendix II — data sources and definitions
- Data source: OECD Analytical Database, supplemented by national sources where noted.
- Key variable definitions:
  - Real long-term interest rates: nominal long-term interest rate minus expected inflation. Inflation expectations determined by year-over-year percentage change in trend and actual CPI. Trend CPI computed with Hodrick-Prescott filter on quarterly CPI with λ value of 1600.
  - Nominal long-term interest rates: yields on 10-year government bonds for listed countries, with country-specific maturity adjustments for Germany, Austria, Belgium, Italy, Norway, Spain, Switzerland as detailed in the source.
  - Government debt: gross and net financial liabilities of the general government.
  - Government consumption: final consumption of the general government.
- Sample availability (exact year ranges preserved):
  - Australia: 1971-2004 / 1988-2003 / 1971-2004
  - Austria: 1971-2004 / 1980-2003 / 1971-2004
  - Belgium: 1971-2004 / 1971-2003 / 1971-2004
  - Canada: 1971-2004 / 1971-2004 / 1971-2004
  - Denmark: 1971-2004 / 1988-2002 / 1988-2004
  - Finland: 1971-2004 / 1975-2002 / 1971-2004
  - France: 1971-2004 / 1977-2003 / 1971-2004
  - Germany: 1971-2004 / 1971-2002 / 1971-2004
  - Ireland: 1985-2004 / 1974-2003 / 1971-2004
  - Italy: 1971-2004 / 1971-2002 / 1971-2004
  - Japan: 1971-2004 / 1971-2002 / 1971-2004
  - Netherlands: 1971-2004 / 1971-2003 / 1971-2004
  - New Zealand: 1971-2004 / 1992-2002 / 1971-2004
  - Norway: 1971-2004 / 1979-2002 / 1971-2004
  - Spain: 1971-2004 / 1990-2002 / 1971-2004
  - Sweden: 1971-2004 / 1971-2002 / 1971-2004
  - Switzerland: 1971-2004 / 1971-2002 / 1971-2004
  - United Kingdom: 1971-2004 / 1971-2003 / 1971-2004
  - United States: 1971-2004 / 1971-2004 / 1971-2004

### Main conclusions and policy-relevant implications
- Theoretical model clarifies channels: interest rate effects of government debt depend on structural parameters, notably birth rate and time preference.
- Numerical simulations indicate interest rate effects of government debt alone tend to be small; combined increases in government consumption and debt produce considerably larger effects.
- Panel estimation using OECD data supports view that an increase in government debt-to-GDP ratio has a small impact on real long-term interest rates; if the debt increase is driven by government consumption, the effect is considerably larger.
- Even small interest rate effects can accompany significant crowding out of productive capital and permanent welfare losses.
- Recommended extensions: incorporate uncertainty and income taxation in theoretical work; estimate crowding-out effects more directly in empirical work.

*Source: _wp0663 - References (source PDF content provided).*

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

### _wp0663 - References

### Introduction
- Topic: relationship between government debt and long-term interest rates.
- Motivation: renewed interest due to recent developments in major industrial countries (United States, Japan, euro area, United Kingdom, Canada).
- Approach:
  - Theoretical: specify and solve a dynamic general equilibrium model that incorporates debt nonneutrality; conduct numerical simulations.
  - Empirical: panel data for 19 industrial countries.
- Key high-level findings:
  - Simulated and estimated interest rate effects of government debt tend to be small.
  - If an increase in government debt is combined with an increase in government consumption, effects on interest rates would be considerably larger.
  - Accumulating government debt can have important economic impacts on the long-run consumption level even if pure crowding out interest-rate effects are limited.

### Literature overview
- Theoretical context:
  - Ricardian equivalence: government debt is neutral — changing the path of government debt does not affect the real equilibrium and thus has no impact on interest rates.
  - Debt nonneutrality arises in overlapping-generations frameworks (Blanchard (1985); Buiter (1988); Weil (1989)).
  - Partial-equilibrium treatments (Laubach (2003); Engen and Hubbard (2004)) assume crowding out on physical capital; present paper embeds interest-rate determination in a full general equilibrium model usable for calibration and numerical simulation.
- Empirical evidence (selected findings preserved exactly as presented):
  - United States (Canzoneri, Cumby, and Diba (2002); Laubach (2003); Engen and Hubbard (2004)):
    - A one percentage point increase in the deficit-to-GDP ratio would increase the expected real interest rate by 19-45 basis points.
    - A one percentage increase in the debt-to-GDP ratio would increase the expected real interest rate by 3-5 basis points.
    - Laubach (2003) reports that the estimated effects of a one percentage point increase in the expected deficit-to-GDP ratio is about 5-8 times larger than the effects of a one percentage point increase in the expected debt-to-GDP ratio.
    - Advantage of using projected government debt and expected future interest rates: isolates long-run fiscal influences from business-cycle factors.
  - Euro area (Faini (2004)):
    - The debt ratio of the euro area is found to affect significantly the real long-term interest rate: a 5-7 basis point increase in the real interest rate for a one percentage point increase in the public debt-to-GDP ratio of the euro area.
    - Deterioration in the cyclically adjusted primary fiscal balance in one country would boost both the euro area real long-term interest rate and the spread between domestic and euro area interest rates, with a much smaller magnitude for the latter.
  - Multi-country studies:
    - Ford and Laxton (1995): data for nine OECD countries; conclude the increase in OECD-wide government debt since the early 1970s was a major factor in the rise in real interest rates in the 1980s and early 1990s.
    - Orr, Edey, and Kennedy (1995): pooled time-series regressions for 17 OECD countries; find a significant long-run effect of an increase in the deficit-to-GDP ratio on long-term real rates.
    - Ardagna, Caselli, and Lane (2004): panel data of 16 OECD countries; find that an increase in government debt affects long-term interest rates only for countries with above-average debt levels.
- Methodological note: choice of fiscal variable (deficit-to-GDP vs. debt-to-GDP) affects estimated magnitudes.

### Theoretical analysis: model and mechanism
- Model framework:
  - Continuous time, overlapping-generations model extending Buiter (1988) with:
    - Neoclassical production function (constant returns to scale).
    - Utility function with constant elasticity of marginal utility.
  - Demographics and individual behavior:
    - New generation born at constant rate, β, of the existing population.
    - Constant probability of death, λ, regardless of age.
    - Population grows at a constant rate, nβλ=−.
    - Individuals supply a fixed amount of labor, receive wages, and pay a lump-sum tax τ; wage and tax rates identical regardless of age.
    - Instantaneous utility characterized by constant elasticity of marginal utility and constant pure rate of time preference, ρ.
    - Individuals choose consumption and saving paths to maximize expected lifetime utility.
  - Production and capital:
    - Physical capital and labor are inputs; physical capital depreciates at constant rate, δ.
    - Savings accumulate as nonhuman wealth, a, in physical capital, k, or government bonds, b (only financial asset considered).
    - Assumption: physical capital and government bonds are perfect substitutes so that yields on government bonds (interest rates) are always equal to the marginal product of physical capital.
- Government:
  - Raises revenue from lump-sum taxes, τ; purchases public good g (does not affect individual utility); issues government bonds b to finance balance.
- Debt nonneutrality mechanism:
  - Government postpones taxation by issuing debt today and committing to higher taxes in the future.
  - Future generations bear part of the future tax increase; current generations do not fully internalize this burden and perceive part of the debt as net wealth.
  - Current generations increase consumption and run down physical capital; capital accumulation is depressed and the interest rate rises.
- Steady-state characterization (as presented):
  - Given government debt (b) and government consumption (g), steady-state solution solves for consumption (c) and physical capital (k) using:
    - (1) []() 111 '( )1()fkcrkbθρβθλρθ −−− ⎡⎤ −= − ++ + ⎣⎦ ,
    - (2) ()fkcg kδ=−− .
  - Key property: the solution depends on b when the birth rate, β, is positive. When the birth rate is zero, equation (1) reduces to '( )rfkρ==, and the steady state is independent of government debt (government debt is neutral).
  - Steady-state interest rate is determined by government debt and government consumption:
    - (3) (,,,,,;,)rRbgβλθραδ=
- Modeling emphasis: the model links the level of government debt and government consumption one-to-one to physical capital and to interest rates through the perfect-substitute assumption.

### Empirical design and scope
- Data: panel for 19 industrial countries (used in estimation reported later in the paper).
- Estimation specification motivated by steady-state mapping r = R(b,g; β,λ,θ,ρ,α,δ) (equation (3)).

### Key implications and conclusions (from the provided content)
- Simulated and estimated interest-rate effects of government debt are generally small.
- Effects increase substantially when government debt increases jointly with government consumption.
- Even if pure crowding out on interest rates is limited, accumulating government debt can significantly affect the long-run level of consumption.

*Source: _wp0663 - References (source PDF content provided).*

### Appendix I, we also derive an equation (A30), relating a change in interest rates to changes

### _wp0663 - Appendix I, we also derive an equation (A30), relating a change in interest rates to changes

### Mechanism: how government debt affects interest rates
- When the birth rate is positive, government debt affects equilibrium interest rates by crowding out physical capital through a wealth effect.
- Simple government budget rule assumed: when debt changes, the government adjusts the lump-sum tax to maintain a balanced budget.
- Generations currently alive do not fully internalize future tax increases associated with higher debt because future generations share part of the tax burden; current generations therefore perceive part of the debt as net wealth and increase consumption by running down physical capital.
- Key condition for debt nonneutrality: positive birth rate (entry of new generations).
- Higher government debt → lower physical capital stock → higher marginal return on capital → higher equilibrium interest rates.

### Numerical calibration and baseline parameters
- Share of capital income in total output: 0.3α=;
- Intertemporal elasticity of substitution: 3θ=;
- Rate of depreciation of physical capital: 0.05δ=.
- Birth and death rates initially set at 0.03β= and 0.02λ=, which imply population growth rate 0.01 nβλ≡−=.
- Pure rate of time preference is a free parameter; baseline chosen 0.06 ρ= (instead of 0.02 ρ= used by Barro and Sala-i-Martin (1995)) to match observed capital-output ratios.
- Simulations consider variations in ρ and deviations in birth and death rates; only cases with βλ≥ (positive or no population growth) are considered.
- Lump-sum taxes, τ, adjust to ensure instantaneous balanced budget; b (government debt) and g (government consumption) allowed to move independently at steady state.

### Simulation results: steady-state effects (summary)
- Steady-state interest rate effects computed by changing steady-state value of government debt, b, and government consumption, g.
- Table 1 (Panel A and B) reports:
  - Column (1): steady-state capital-to-output ratio;
  - Column (2): crowding-out effect on the stock of capital with respect to a unit change in government debt (1/ Change in the stock of capital with respect to a unit increase in government debt.);
  - Columns (3) and (4): change in interest rates, in basis points, with respect to a one percentage point increase in the ratio of government debt or consumption to GDP;
  - Column (5): change in the level of consumption, in percent, with respect to a one percentage point increase in the ratio of government debt to GDP.
- Representative entries from Panel (A) for different ρ (birth rate 0.03, death rate 0.02):
  - 0.06 ρ= : Capital-output ratio 2.57-0.53 ; Government debt effect on interest rates 1.68.3 ; Long-run effect on consumption -0.06
  - 0.05 ρ= : Capital-output ratio 2.80-0.57 ; Government debt effect on interest rates 1.58.1 ; Long-run effect on consumption -0.06
  - 0.04 ρ= : Capital-output ratio 3.06-0.62 ; Government debt effect on interest rates 1.38.0 ; Long-run effect on consumption -0.05
  - 0.03 ρ= : Capital-output ratio 3.38-0.68 ; Government debt effect on interest rates 1.27.8 ; Long-run effect on consumption -0.05
  - 0.02 ρ= : Capital-output ratio 3.76-0.74 ; Government debt effect on interest rates 1.18.3 ; Long-run effect on consumption -0.04
- Representative entries from Panel (B) for different birth/death combinations (time preference 0.04 ρ= unless noted):
  - 0.04 β=, 0.02 λ= : Capital-output ratio 3.06-0.62 ; Government debt effect on interest rates 1.38.0 ; Long-run effect on consumption -0.05
  - 0.04 β=, 0.01 λ= : Capital-output ratio 4.80-0.39 ; Government debt effect on interest rates 0.43.9 ; Long-run effect on consumption -0.01
  - 0.04 β=, 0.00 λ= : Capital-output ratio 4.03-0.52 ; Government debt effect on interest rates 0.75.6 ; Long-run effect on consumption -0.03
  - 0.04 β=, 0.00 λ= (β=0.00, λ=0.00 row): Capital-output ratio 7.500.00 ; Government debt effect on interest rates 0.00.0 ; Long-run effect on consumption 0.00
- Interpretation from simulations:
  - When ρ is higher, steady-state capital-output ratio is lower and the interest rate effect of government debt is larger.
  - When β is lower, the interest rate effect of government debt is smaller (due to higher steady-state capital-output ratio and smaller crowding-out effect).
  - When birth rate β is zero, there is no crowding-out effect and government debt is neutral.
- Magnitudes emphasized:
  - A one percentage point increase in the government debt-to-output ratio would raise the steady-state real interest rate by up to 2 basis points, depending on parameter values.
  - The interest rate effects of an increase in government consumption are several times larger than the effects of government debt in the model.
  - Consumption would be permanently lowered by up to 0.08 percent when the government debt-to-GDP ratio increases by one percentage point (simulated welfare loss).

### Model caveats and lower-bound interpretation
- Simulated interest rate effect of government debt likely a lower bound because:
  - It captures purely the long-run (steady-state) physical crowding-out effect.
  - No uncertainty is modelled, so risk premia are excluded.
  - Lump-sum tax is available in the model; if instead the government raises distortionary income taxes, the impact on real interest rates would be larger.
- Conclusion from simulations: even small interest rate effects can coexist with significant crowding out of physical capital and nontrivial welfare losses.

### Empirical evidence: panel estimation using OECD data (1971–2004)
- Data: panel of 19 OECD countries spanning 1971–2004; five-year averaging used to eliminate short-run fluctuations (each country provides up to seven observations).
- Dependent variable: real long-term interest rates on government bonds (nominal yields minus expected inflation).
- Government debt measure: ratio of current financial liabilities of the general government to nominal GDP (both gross and net liabilities considered).
- Government consumption: ratio of government final consumption expenditure to nominal GDP.
- Estimated equation: itiititit RBGαβγε=+++  where R is real long-term interest rate, B government debt-to-GDP, G government consumption-to-GDP.

### Estimation results (Table 2)
- No country effect (common intercept):
  - Government debt (net): 0.018 (standard error ( 0.007 )**) using trend inflation; 0.018 ( ( 0.006 )**) using actual inflation.
  - Government debt (gross): 0.023 ( ( 0.009 )**) using trend inflation; 0.024 ( ( 0.008 )**) using actual inflation.
  - Government consumption: 0.25 ( ( 0.06 )**) and 0.19 ( ( 0.06 )**) using trend and actual inflation respectively in first pair; 0.23 ( ( 0.05 )**) and 0.18 ( ( 0.05 )**) in second pair.
  - Number of observations: 111; Adjusted R-squared about 0.15–0.17; Standard error of estimated equation 2.45–2.21.
  - Interpretation: a one percentage point increase in government debt-to-GDP raises real long-term interest rate by about 2 basis points (close to upper bound of simulation).
  - Government consumption effects around 20–25 basis points per one percentage point increase in government consumption-to-GDP.
- Fixed country effects:
  - Government debt (net): 0.053 ( ( 0.013 )**) using trend inflation; 0.052 ( ( 0.012 ) ) using actual inflation.
  - Government debt (gross): 0.040 ( ( 0.012 )**) using trend inflation; 0.042 ( ( 0.011 ) ) using actual inflation.
  - Government consumption: 0.89 ( ( 0.17 )**) and 0.86 ( ( 0.19 )**) using trend inflation; 0.73 ( ( 0.16 ) )** and 0.68 ( ( 0.17 ) )** using actual inflation.
  - Number of observations: 111; Adjusted R-squared about 0.38–0.42; Standard error of estimated equation 2.03–1.87.
  - Interpretation: with fixed effects, a one percentage point increase in government debt-to-GDP raises real long-term interest rates by about 4–5 basis points; government consumption effects estimated around 90 basis points.
- Authors’ interpretations:
  - Estimated coefficients larger under fixed effects than steady-state numerical simulations.
  - Possible reasons: risk premium on government debt (not in theoretical model) and use of distortionary taxes (income taxes) in reality instead of lump-sum taxes; these could increase observed interest rate responses.
  - Suggested model extensions: incorporate uncertainty and income taxation, or estimate crowding-out effects more directly.

### Main conclusions (paper’s summary)
- Theoretical model with debt nonneutrality clarifies channels: interest rate effects of government debt depend on structural parameters, notably birth rate and time preference.
- Numerical simulations indicate interest rate effects of government debt alone tend to be small, but combined increases in government consumption and debt produce considerably larger effects.
- Panel estimation using OECD data supports view that an increase in government debt-to-GDP ratio has a small impact on real long-term interest rates; if the debt increase is driven by government consumption, the effect is considerably larger.
- Even small interest rate effects can accompany significant crowding out of productive capital and permanent welfare losses.
- Further theoretical and empirical research recommended.

*Source: _wp0663 - Appendix I, we also derive an equation (A30), relating a change in interest rates to changes*

### References

### _wp0663 - References

### Key references cited
- Empirical and theoretical papers on fiscal policy, interest rates, and debt dynamics, including:
  - Ardagna, Silvia, Francesco Caselli, and Timothy Lane, 2004, “Fiscal Discipline and the Cost of Public Debt Service: Some Estimates for OECD Countries,” NBER Working Paper No. 10788.
  - Barro, Robert J., and Xavier Sala-i-Martin, 1995, Economic Growth.
  - Bernheim, B.D., 1987, “Ricardian Equivalence: An Evaluation of Theory and Evidence,” NBER Macroeconomics Annual, pp. 263-303.
  - Blanchard, Olivier J., 1985, “Debt, Deficits and Finite Horizons,” Journal of Political Economy, Vol. 93, No. 2, pp. 223-93.
  - Buiter, Willem H., 1988, “Death, Birth, Productivity Growth and Debt Neutrality,” Economic Journal, Vol. 98, pp. 279-93.
  - Engen, Eric M., and R. Glenn Hubbard, 2004, “Federal Government Debt and Interest Rates,” NBER Macroeconomics Annual, pp. 83-138.
  - Laubach, Thomas, 2003 and 2004, on effects of budget deficits and debt on interest rates.
  - Pesaran, M.H., and Ron Smith, 1995, “Estimating Long-Run Relationships from Dynamic Heterogeneous Panels,” Journal of Econometrics, Vol. 68, pp. 79-113.
  - Seater, John, 1993, “Ricardian Equivalence,” Journal of Economic Literature, Vol. 31, pp.241-54.
  - Weil, Phillipe, 1989, “Overlapping Families of Infinitely-Lived Agents,” Journal of Public Economics, Vol. 38, pp. 183-98.
- Policy- and country-specific analyses from OECD, European Commission, Bank of Italy, and national sources are cited for data and comparative studies.

### Appendix I — Derivation of dynamic equilibrium conditions: model structure and core results
- Model basis and extensions:
  - Model based on Buiter (1988), extended with a neoclassical production function and a utility function with constant elasticity of marginal utility.
- Production side:
  - Output per unit of labor: y = f(k) where k is physical capital per unit of labor and f satisfies f'(k) > 0 and f''(k) < 0.
  - Rental rate of capital (real interest rate) r equals marginal product: r = f'(k). (Equation A2)
- Consumer problem:
  - Consumers face constant probability of death λ; born at time s ≤ t.
  - Lifetime utility: maximize ∫_{t}^{∞} e^{-(ρ+λ)(z-t)} u(c_{s,z}) dz. (Equation A3)
  - Instantaneous utility: u(c) = [c^{1-θ}-1]/(1-θ) if θ ≠ 1; u(c) = ln(c) if θ = 1. (Equations A4a–A4b)
  - Flow budget identity: ḃa_{s,t} = (r_t - λ)a_{s,t} + w_t - τ_t - c_{s,t}. (Equation A5; variables age/notation as in text)
  - Solvency constraint: lim_{z→∞} ∫_t^z exp(-∫_t^ζ r_d d d + λζ) a_{s,ζ} dζ = 0. (Equation A6)
  - First-order necessary condition for consumption growth: d/dz c_{s,z}^{-θ} = (θ)(ρ + λ - r_z) c_{s,z}^{-θ}. (Equation A8)
  - Individual consumption function: c(s,z) = m_t [a_{s,t} + h_{s,t}], where m_t is marginal propensity to consume from total wealth and h_{s,t} is human wealth. (Equation A9)
  - Marginal propensity to consume m_t depends on entire future path of real interest rates through an integral expression. (Equation A10)
  - Human wealth h_{s,t} defined as present discounted value of future real wages net of lump-sum taxes. (Equation A11)
- Aggregation:
  - Population aggregate X_t from individual x_{s,t} depends on birth rate β and death rate λ; formulas given for β > 0 and β = 0. (Equations A12a–A12b)
  - Aggregate human wealth H_t = h_t N_t with N_t = e^{n t} and n = β − λ. (Equation A13)
  - Aggregate relationships:
    - Aggregate consumption: C_t = m_t [A_t + H_t]. (Equation A14)
    - Motion of nonhuman wealth: \dot{A}_t = A_t r_t + W_t - T_t - C_t. (Equation A15)
    - Motion of human wealth: \dot{H}_t = H_t (r_t + β) + W_t − T_t. (Equation A16)
  - Per capita variables x_t = X_t e^{-n t} yield:
    - c_t = m_t [a_t + h_t]. (Equation A17)
    - \dot{a}_t = (r_t - n) a_t + w_t - τ_t - c_t. (Equation A18)
    - \dot{h}_t = (r_t - λ) h_t + w_t - τ_t. (Equation A19)
  - Differential equation for per capita consumption:
    - (1/θ) \dot{c}_t / c_t = r_t - ρ - β + m_t. (Equation A20; exact form as in text)
  - Evolution of m_t given by an integral/differential relation involving r_t, θ, ρ, λ, and β. (Equation A21)
- Government sector and resource constraint:
  - Government debt per capita evolves as: \dot{b}_t = (r_t - n - λ) b_t + g_t - τ_t. (Equation A22)
  - Balanced budget rule assumed: \dot{b}_t = 0 for every t, implying τ_t = (r_t + β + λ) b_t + g_t − ... (Equation A23 as given)
  - Physical capital accumulation: \dot{k}_t = f(k_t) − c_t − g_t − δ k_t where δ is depreciation. (Equation A24)
  - Equilibrium path determined by differential equations for c_t, m_t, k_t, with balanced budget relation.
- Steady state characterization:
  - Steady state conditions set \dot{c} = \dot{m} = \dot{k} = 0 leading to:
    - Equation linking r, c, k, and parameters: (Equation A25) 1/(1−θ) [ ... exactly as in text ] (preserves notation and dependencies).
    - Resource condition in steady state: f(k) − g − k δ = c. (Equation A26)
    - When birth rate β > 0, steady state depends on government debt b; when β = 0, steady state yields f'(k) = ρ and is independent of b.
    - Steady-state k determined by nonlinear condition (Equation A27) involving f, f', g, k, b, θ, ρ, δ, β, λ.
- Comparative statics:
  - Partial derivative ∂k/∂b given by Equation (A28), expressed using f', f'', g, k, δ, β, θ, ρ, λ and auxiliary functions Ψ and Ω defined in text.
  - Partial derivative ∂k/∂g given by Equation (A29) with analogous structure.
  - Change in interest rate due to debt and spending changes expressed by (A30): dr/dk times (∂k/∂b db + ∂k/∂g dg).

### Appendix II — Data sources and definitions
- Data source:
  - All data are from the OECD Analytical Database, supplemented by national sources where noted.
- Key variable definitions:
  - Real long-term interest rates: nominal long-term interest rate minus expected inflation. Inflation expectations are determined by year-over-year percentage change in trend and actual CPI. Trend CPI computed with Hodrick-Prescott filter on quarterly CPI with λ value of 1600.
  - Nominal long-term interest rates: yields on 10-year government bonds for the United States, Japan, France, the United Kingdom, Canada, Australia, Denmark, Finland, Ireland, Netherlands, New Zealand, Sweden. Country-specific maturities vary as follows:
    - Germany: yields on 9- to 10-year public sector bonds.
    - Austria: yields on public sector bonds with maturity of more than 1 year up to 1989, and 10-year government bonds from 1990 onwards.
    - Belgium: yields on central government bonds with maturity of more than 5 years.
    - Italy: yields on government bonds with maturity of more than 2 years up to 1991, and 9- to 10-year government bonds from 1992 onwards.
    - Norway: yields on 6- to 10-year government bonds up to 1992, and 10-year government bonds from 1993 onwards.
    - Spain: yields on government bonds with maturity more than 2 years.
    - Switzerland: yields on confederation government bonds with maturity more than 5 years.
  - Government debt: gross and net financial liabilities of the general government.
  - Government consumption: final consumption of the general government.
- Sample availability table (exact year ranges preserved)
  - Real interest rates / Government debt / Government consumption:
    - Australia: 1971-2004 / 1988-2003 / 1971-2004
    - Austria: 1971-2004 / 1980-2003 / 1971-2004
    - Belgium: 1971-2004 / 1971-2003 / 1971-2004
    - Canada: 1971-2004 / 1971-2004 / 1971-2004
    - Denmark: 1971-2004 / 1988-2002 / 1988-2004
    - Finland: 1971-2004 / 1975-2002 / 1971-2004
    - France: 1971-2004 / 1977-2003 / 1971-2004
    - Germany: 1971-2004 / 1971-2002 / 1971-2004
    - Ireland: 1985-2004 / 1974-2003 / 1971-2004
    - Italy: 1971-2004 / 1971-2002 / 1971-2004
    - Japan: 1971-2004 / 1971-2002 / 1971-2004
    - Netherlands: 1971-2004 / 1971-2003 / 1971-2004
    - New Zealand: 1971-2004 / 1992-2002 / 1971-2004
    - Norway: 1971-2004 / 1979-2002 / 1971-2004
    - Spain: 1971-2004 / 1990-2002 / 1971-2004
    - Sweden: 1971-2004 / 1971-2002 / 1971-2004
    - Switzerland: 1971-2004 / 1971-2002 / 1971-2004
    - United Kingdom: 1971-2004 / 1971-2003 / 1971-2004
    - United States: 1971-2004 / 1971-2004 / 1971-2004

*Content derived from the document _wp0663 - References (appendices and references sections) as provided.*

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