## Costly Increases in Public Debt when r < g (wpiea2024010-print-pdf)

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### Abstract and key quantitative findings
- The paper quantifies the costs of a permanent increase in the debt-to-GDP ratio using a deterministic overlapping generations (OLG) model with two assets (public debt and private productive capital) and no default.
- Calibration to U.S. data yields:
  - An increase in the debt ratio from 60 to 120 percent of GDP is associated with:
    - a reduction in the capital stock of about 15 percent;
    - a reduction in steady-state GDP of about 8 percent.
- When about 30 percent of debt is held by foreign investors, quantitative impacts are reduced to:
  - a permanent decline of 4 percent in GDP.

### Core mechanism and interpretation
- Higher public debt crowds out private capital because households allocate a larger share of their savings to government bonds rather than private capital.
- As private capital falls:
  - output declines;
  - the return on capital rises, which raises the interest rate on government debt.
- The crowding-out operates irrespective of the sign of r − g; the model assumes the marginal product of capital m is greater than growth g and the spread m − r is exogenous.
- Government debt return specification: 1 + i_b^t = (1 + i_k^t − γ) where γ is an exogenous convenience yield.
- Portfolio rule: households save in a fund that allocates an exogenous fraction 1 − φ_t of savings to government debt and φ_t to firms’ capital; in equilibrium φ_t = K_t/(K_t + B_t).

### Model structure and analytic assumptions
- Discrete time; individuals live N periods; population grows at rate n > 0 with L_t = (1 + n)^t L_0.
- Production: Y_t = F(K_t, L_t) with constant returns to scale; per-capita f(k) = F(k,1); wages w = (1 − α) f(k).
- Preferences: Σ_{j=0}^{N−1} β^j u(c_{t+j}) with 0 < β.
- Fiscal framework: initial debt B_0; predetermined expenditures {G_t} > 0; taxes limited to consumption taxes τ_t in analytic model.
- Asset market clearing (full depreciation): K_{t+1} + B_{t+1} = S_t.
- Two-period steady-state special case (log utility u = log(c), constant consumption tax):
  - steady-state savings per capita s = β w / (1 + β);
  - market clearing: (1 + n) k* + (1 + n) b* = (1 − α) β f(k*) / (1 + β).

### Analytical steady-state results (lemmas)
- Lemma 1 (crowding out): At the stable steady state, government bond always crowds out private capital.
  - Stability condition: αβ(1−α)/(1+β) k^{α−1} ≤ 1 + n.
  - Sign of crowding out: ∂b∗/∂k∗ = −1 + αβ(1−α)/[(1 + n)(1 + β)] k∗^{α−1} = 1/(1 + n) [−1 + αβ(1−α)/((1 + n)(1 + β)) k∗^{α−1}] < 0.
- Lemma 2 (finite support): There is a maximum amount of debt b_max and a minimum amount of capital k_min the economy can support.
  - k_min = [ (1 + n)(1 + β) / (αβ(1−α)) ]^{1/(α−1)}.
  - b_max = −k_min + β(1−α) / [(1 + β)(1 + n)] k_min^α.
- Lemma 3 (hump-shaped primary deficit): When r − g < 0, the sustainable primary deficit pd has a maximum at intermediate debt.
  - pd = −(r − g) b = −(α k^{α−1} − (1 + n) − γ) b.
  - pd as a function of b ∈ (0, b_max) is hump-shaped (inverted U), with maximum determined by first-order condition; intuition: rising debt crowds out capital, raises interest rates, makes r − g less negative, producing the inverted-U relation.

### Two-period illustrative grounding (parameters and illustrative steady states)
- Calibration choices:
  - A period ≈ 40 years.
  - Population growth rate = zero percent.
  - β = 0.987 40 (as stated).
  - Cobb-Douglas capital share α = 0.484.
  - Convenience yield = 4% annually to guarantee r − g < 0.
- Illustrative quantitative figures:
  - Maximum debt-to-GDP the model can support: b_max = 396 (396 percent of GDP).
  - The inverted-U attains a maximum at debt-to-GDP = 7 percent, with pd_max = 7 (primary deficit, percent).
  - Illustrative steady state: 212 percent of GDP public debt can be sustained together with a 7 percent maximum sustainable deficit.
  - Numerical comparative exercises:
    - Increasing debt-to-GDP from 60 percent to 120 percent: economy can finance a 2.3 percent higher deficit and this translates into an 8.0 percent decline in output.
    - Increasing debt-to-GDP from 120 percent to 212 percent: output declines further by 13.3 percent.

### Quantitative model overview and calibration (selected parameter values and aggregates)
- Model basis: McGrattan and Prescott (2017) augmented with convenience yield.
- Households: OLG with age j up to J, survival probabilities σ_j^t, measures n_j^t, labor efficiency heterogeneity.
- Composite financial asset: fraction φ_t of government debt and 1 − φ_t of claims to private capital; φ_t computed in equilibrium.
- Firms: two sectors (Schedule C corporations and non-corporate); final good aggregator Y_t = Y_{1t}^{θ1} Y_{2t}^{θ2}, θ1 + θ2 = 1.
- Government budget law: B_{t+1} = B_t + i_b^t B_t + G_t − Σ_j n_{j,t} T_j^t(w_t ℓ_{j,t} ε_j) − τ_c^t C_t − τ_π1^t Π_{1t} − τ_d1^t D_{1t} − τ_d2^t D_{2t}.
- Calibration highlights:
  - Demographics and preferences:
    - Growth rate of population (η) = 0.
    - Work life = 45 years.
    - Number of workers per retiree = 3.93.
    - Disutility of leisure (α) = 1.185.
    - Discount factor (β) = 0.987.
    - Growth rate of technology (γ) = 2 (percent).
  - Technology and capital shares:
    - θ1 = 0.500; θ1T = 0.182; θ1I = 0.190; θ2T = 0.502; θ2I = 0.095.
  - Depreciation rates:
    - δ1T = 0.050; δ1I = 0.050; δ2T = 0.015; δ2I = 0.050.
  - Initial government shares and interest calibration:
    - Government spending (φ_G) = 0.080.
    - Government debt (φ_B) = 0.600 (ratio in 2004).
    - Convenience yield (CY) = 4.0 percent.
    - Interest rate on government bonds resulting from CY: 0.79 percent.
    - Real interest rate in 2005 (comparison): around 0.93 percent.
  - Tax parameters:
    - τ_π1 = 0.330; τ_d1 = 0.144; τ_d2 = 0.382.
    - Baseline consumption tax (residual to balance budget) = 8.2 percent.
  - Model vs data aggregates (2000–2010 averages, relative to adjusted GNP):
    - Total Adjusted Income: Data 1.000, Model 1.000.
    - Labor Income: Data 0.585, Model 0.584.
    - Capital Income: Data 0.415, Model 0.416.
    - Consumption: Data 0.745, Model 0.714.
    - Tangible investment: Data 0.211, Model 0.208.
    - Government Deficit*: Data 0.019, Model 0.018.
    - Tangible Capital: Data 4.117, Model 4.045.
    - Intangible Capital: Data 1.700, Model 1.700.
    - Note: Government deficit calculated using the average of 2001–2005.

### Quantitative experiment: staged permanent increase in public debt
- Policy experiment:
  - Annual increase in government expenditures by 3.5 percent over 20 years, holding tax rates constant.
  - Government issues debt to finance spending; after 20 years, spending is readjusted to initial steady state and debt-to-GDP stabilized using a consumption tax.
  - Demographics and other factors held constant.
- Quantitative impacts:
  - Capital declines by 9 percent over the initial 20-year period and by 15 percent in the long run.
  - Associated output declines are consistent with analytical scenarios (e.g., 8.0 percent decline when debt-to-GDP increases from 60 to 120).
- Caveat: model keeps tax policy fixed and abstracts from strategies (e.g., subsidizing capital) that could mitigate crowding out.

### Robustness exercises and alternative specifications
- A. Foreign holdings of public debt:
  - If 30 percent of U.S. government debt is held by foreign investors:
    - GDP declines by about 4 percent in the initial 20-year period and by slightly moreover the long run.
    - Mechanism: foreign absorption of newly-issued debt mitigates domestic crowding out.
  - Emphasized role: elasticity of demand for safe assets in quantifying crowding out.
- B. Tax financing vs debt financing:
  - Financing the baseline spending path via consumption taxes while keeping debt-to-GDP stable:
    - Results in small effects on GDP relative to the baseline debt-financing experiment.
    - Rationale: consumption taxes are less distortionary for marginal decisions in steady state than alternative tax instruments.

### Transition dynamics and additional quantitative results
- Return on capital (MPK) rises slightly: +0.2 percentage points in first 20 years and +0.4 percentage points in the long run.
- GDP declines: over 4 percent in the initial 20-year period and over 8 percent in the long run.
- Labor supply remains relatively stable.
- i_b − g stays negative throughout the analyzed period, initially increases over the first 20 years due to high interest rates and negative GDP growth, then gradually declines toward the new steady state.
- Empirical note: Rachel and Summers (2019) estimate that a 1 percentage point increase in debt-GDP raises interest rates by 3–4 basis points, larger than the paper’s simulations; one possible explanation is erosion of the convenience yield.

### Key takeaways and policy-relevant implications
- Structural result: At steady state, higher government debt crowds out private capital; there exists a finite b_max and associated k_min beyond which equilibrium cannot be sustained.
- Non-monotone fiscal space: When r − g < 0, the sustainable primary deficit as a function of debt is hump-shaped—debt initially increases sustainable deficits but beyond a point reduces them (inverted-U / Laffer-like).
- Quantitative magnitudes (illustrative):
  - Two-period illustration: b_max = 396 (396 percent of GDP); pd_max = 7 (percent) at debt-to-GDP = 7 percent.
  - Moving debt-to-GDP from 60 to 120 allows a 2.3 percent higher deficit but produces an 8.0 percent decline in output; moving from 120 to 212 produces an additional 13.3 percent output decline.
- Policy caution: High steady-state debt levels that the model can mathematically sustain are not normative targets; large debts have high costs in foregone investment and economic potential. The model abstracts from default risk and bond-market instability, which would tighten sustainable limits in practice.

*Source: Cao, Yongquan, Vitor Gaspar, and Adrian Peralta-Alva. 2024. Costly Increases in Public Debt when r < g, IMF Working Paper No. 2024/10.*

### Section 1

### Costly Increases in Public Debt when r < g

### Abstract and key quantitative findings
- The paper quantifies the costs of a permanent increase in the debt-to-GDP ratio using a deterministic overlapping generations (OLG) model with two assets (public debt and private productive capital) and no default.
- Calibration to U.S. data (following McGrattan and Prescott (2017) methodology) yields:
  - An increase in the debt ratio from 60 to 120 percent of GDP is associated with:
    - a reduction in the capital stock of about 15 percent;
    - a reduction in steady-state GDP of about 8 percent.
- When about 30 percent of debt is held by foreign investors, qualitative results are unchanged and the quantitative impacts are reduced to:
  - a permanent decline of 4 percent in GDP.

### Main mechanism and interpretation
- Core mechanism: higher public debt crowds out private capital because households allocate a larger share of their savings to government bonds rather than private capital.
  - As private capital falls, output declines and the return on capital rises.
  - The higher return on private capital raises the interest rate on government debt.
- This crowding-out operates irrespective of the sign of r−g (i.e., whether r−g is positive or negative). The model assumes the marginal product of capital m is greater than growth g, and the spread between m and r is exogenous.
- The return on government debt is modeled as the return on private capital minus an exogenously given convenience yield γ: 1 + ibt = (1 + ikt − γ).
- Households save in a fund that allocates an exogenous fraction 1−φt of savings to government debt and φt to firms’ capital. In equilibrium φt = Kt/(Kt + Bt).

### Model structure and assumptions (analytic model)
- Time is discrete; individuals live N periods; population grows at constant rate n > 0 with L_t = (1 + n)^t L_0.
- Production: Y_t = F(K_t, L_t) with constant returns to scale; per-capita production f(k) = F(k,1); wages equal w = (1 − α) f(k).
- Preferences: time-separable utility Σ_{j=0}^{N−1} β^j u(c_{t+j}) with 0 < β.
- Fiscal framework: government has initial debt B_0, predetermined expenditures {G_t} > 0, taxes limited to consumption taxes τ_t in the analytic model.
- Asset markets clearing (with full depreciation assumed): K_{t+1} + B_{t+1} = S_t.
- In the two-period steady-state special case with log utility u = log(c) and only a constant consumption tax, the steady-state savings per capita satisfies s = β w / (1 + β) and market clearing implies:
  - (1 + n) k* + (1 + n) b* = (1 − α) β f(k*) / (1 + β).

### Comparative statics and steady-state implications
- The steady-state analysis (graphically described and analogous to McCandless and Wallace (1991) Figure 9.2) shows:
  - The maximum attainable steady-state capital occurs at zero public debt (k_max).
  - Any positive level of public debt reduces the steady-state capital stock relative to k_max (crowding out).
- Relationship between debt ratio and sustainable primary deficit:
  - Starting from zero debt, increasing the debt ratio initially increases the sustainable primary deficit.
  - As debt ratios rise further, interest rates increase and the maximum sustainable primary deficit is reached and then declines.
  - When r − g = 0 the sustainable primary deficit is zero; for even larger debt ratios r − g becomes positive and constant public debt ratios require primary surpluses.
  - The mapping from debt ratio to sustainable primary deficit has an inverted U-shape (analogous to a Laffer curve, with debt ratio on the horizontal axis).

### Transitional dynamics (quantitative model)
- In the calibrated quantitative model for the United States:
  - Following a permanent increase in debt from 60 to 120 percent of GDP, transition paths show:
    - GDP exhibits a monotonous decline toward the new lower steady state.
    - Interest rates rise during the transition, even while r − g remains negative.
  - The primary driver of the decline in output is the roughly 15 percent fall in private capital stock due to portfolio reallocation toward government bonds.

### Robustness and extensions
- Allowing for foreign holdings of government debt (~30 percent held by foreign investors) preserves the qualitative crowding-out results but halves (approximately) the long-run GDP impact (from ~8 percent to ~4 percent decline).
- The paper abstracts from sovereign default risk and multiple equilibria involving debt attacks; it focuses exclusively on crowding out as the mechanism for costs of higher debt.
- The convenience yield γ is taken as exogenously given in this analysis; related literature explores endogenous convenience yields and reaches qualitatively similar conclusions about crowding out and costliness of higher debt.

### Relation to existing literature
- Complements and extends:
  - Blanchard (2019, 2023): highlights low r − g and debates on fiscal costs of debt; this paper shows higher debt may change future interest rates and be costly.
  - Reis (2021): emphasizes resource scarcity and fiscal constraints when r < g; this paper quantifies long-run GDP effects under U.S. calibrations.
  - Aguiar et al. (2022): incomplete markets models with convenience yields—this paper shares the mechanism whereby increased government debt can raise interest rates and reduce private capital.
  - Acharya and Dogra (2022): overlapping-generations models with endogenous convenience yields and nominal rigidities—crowding out effects in that work mirror the costly investment reductions found here, though that study is more qualitative whereas this paper provides U.S.-calibrated quantification.

*Source: Cao, Yongquan, Vitor Gaspar, and Adrian Peralta-Alva. 2024. Costly Increases in Public Debt when r < g, IMF Working Paper No. 2024/10.*

### Section 2

### wpiea2024010-print-pdf - Section 2

### Analytical results on debt, capital, and primary deficits
- Lemma 1: At the stable steady state, government bond always crowds out private capital.
  - Stability condition: αβ(1−α)/(1+β) kα−1 ≤ 1 + n.
  - Sign of crowding out: ∂b∗/∂k∗ = −1 + αβ(1−α)/[(1 + n)(1 + β)] k∗α−1 = 1/(1 + n) [−1 + αβ(1−α)/((1 + n)(1 + β)) k∗α−1] < 0.
  - Implication: whenever government bond increases, private capital decreases. This occurs even when r−g is negative.

- Lemma 2: At the stable steady state, there is a maximum amount of debt, b_max, and a minimum amount of capital, k_min, the economy can support.
  - k_min = [ (1 + n)(1 + β) / (αβ(1−α)) ]^{1/(α−1)}.
  - b_max = −k_min + β(1−α) / [(1 + β)(1 + n)] k_min^α.
  - Reasoning: db∗/dk∗ = −1 + αβ(1−α)/[(1 + n)(1 + β)] k∗α−1 is decreasing in k∗; set derivative to zero to obtain maximum debt; associated k is k_min.

- Lemma 3: When r − g < 0, there is a maximum of the primary deficit the economy can support at the stable steady state.
  - Primary deficit at steady state: pd = −(r − g) b = −(α kα−1 − (1 + n) − γ) b.
  - First derivative: ∂pd/∂k = −(α kα−1 − (1 + n) − γ) ∂b/∂k − α(α−1) kα−2 b.
  - At b = b_max and k = k_min, ∂b/∂k = 0, hence ∂pd/∂k |_{b=b_max,k=k_min} = −(α(α−1) kα−2_min) b_max > 0.
  - At b = 0 and r−g < 0, ∂pd/∂k |_{b=0} = −(α kα−1 − n − γ) ∂b/∂k < 0.
  - Second derivative ∂^2 pd/∂k^2 < 0, so pd as a function of b ∈ (0, b_max) is hump-shaped with a maximum determined by setting the first-order condition to zero.
  - Intuition: rising debt can crowd out private capital, raise interest rates, make r−g less negative, and produce an inverted-U (Laffer-curve-like) relationship between sustainable primary deficit and government debt.

- Existence of maximum per-capita debt compatible with equilibrium is emphasized: there is a finite b_max and associated k_min beyond which equilibrium cannot be supported.

### Illustrative two-period grounding and illustrative steady states (parameters used in the two-period model)
- Calibration choices for two-period illustrative grounding:
  - A period ≈ 40 years.
  - Zero percent growth rate for population.
  - β = 0.987 40 (as stated in this section).
  - Cobb-Douglas capital share α = 0.484.
  - Convenience yield set to 4% annually to guarantee r − g < 0.
- Quantitative implications reported for the two-period illustrative economy:
  - Maximum debt-to-GDP the model economy can support: b_max = 396 (396 percent of GDP).
  - The inverted-U (pd vs debt-to-GDP) attains a maximum at debt-to-GDP = 7 percent, with pd_max = 7 (primary deficit, percent).
  - Illustrative steady state figures reported: 212 percent of GDP for public debt can be sustained together with a 7 percent maximum sustainable deficit — noted as an order of magnitude higher than reasonable.
  - Numerical comparative exercises:
    - Increasing debt-to-GDP from 60 percent to 120 percent: the economy is still on the increasing part of the hump; the economy can finance a 2.3 percent higher deficit and this translates into an 8.0 percent decline in output.
    - Increasing debt-to-GDP from 120 percent to 212 percent (up to the illustrative maximum sustainable deficit): output declines further by 13.3 percent.
  - The large negative effects on GDP occur independently of the sign of i_b − g.

### Quantitative model structure (overview)
- Model basis: McGrattan and Prescott (2017) framework augmented with a convenience yield for government debt (government bond interest rate = private asset interest rate − convenience yield).
- Households: OLG structure with cohorts arriving as working-age agents; age j up to maximum age J; survival probabilities σ_j^t; measures n_j^t; labor efficiency heterogeneity.
- Composite financial asset: households hold a composite financial asset consisting of a fraction φ_t of government debt and 1 − φ_t of claims to private firms’ capital; φ_t computed in equilibrium for portfolio consistency.
- Prices and policy sequences:
  - Asset returns: {i_b^t, i_k^t}; wage {w_t}.
  - Policy sequences: tax rates τ = {τ_c^t, τ_d1^t, τ_d2^t, τ_π1^t}; net tax schedules {T_w^t(·), T_r^t(·)}; government debt {B_t}; public good consumption {G_t}.
- Firms and production:
  - Two business sectors: Schedule C corporations (sector 1, subject to corporate tax) and non-corporate sector (sector 2, not subject to corporate profit tax).
  - Final good aggregator: Y_t = Y_{1t}^{θ1} Y_{2t}^{θ2}, θ1 + θ2 = 1.
  - Sector production: Y_{it} = K_{iTt}^{θ_{iT}} K_{iIt}^{θ_{iI}} (Ω_t L_{it})^{1−θ_{iT}−θ_{iI}} for i = 1,2.
  - Capital accumulation with constant depreciation rates.
- Government budget law of motion:
  - B_{t+1} = B_t + i_b^t B_t + G_t − Σ_j n_{j,t} T_j^t(w_t ℓ_{j,t} ε_j) − τ_c^t C_t − τ_π1^t Π_{1t} − τ_d1^t D_{1t} − τ_d2^t D_{2t}.

### Calibration (selected parameter values and aggregates)
- Demographics and preferences (Table 2 and text):
  - Growth rate of population (η) = 0.
  - Work life in years = 45.
  - Number of workers per retiree = 3.93.
  - Disutility of leisure (α) = 1.185.
  - Discount factor (β) = 0.987.
  - Growth rate of technology (γ) = 2 (percent).
- Technology and capital shares:
  - Income share, Schedule C corporations (θ1) = 0.500.
  - Tangible capital, Schedule C (θ1T) = 0.182.
  - Intangible capital, Schedule C (θ1I) = 0.190.
  - Tangible capital, other business (θ2T) = 0.502.
  - Intangible capital, other business (θ2I) = 0.095.
- Depreciation rates:
  - δ1T = 0.050; δ1I = 0.050.
  - δ2T = 0.015; δ2I = 0.050.
- Initial government shares and interest calibration:
  - Government spending (φ_G) = 0.080 (initial steady state).
  - Government debt (φ_B) = 0.600 (initial parameter, ratio of U.S. government debt to GDP in 2004).
  - Convenience yield (CY) = 4.0 percent (calibrated).
  - Interest rate on government bonds resulting from CY: 0.79 percent.
  - Real interest rate in 2005 (comparison): around 0.93 percent.
- Tax parameters:
  - Profits, Schedule C corporations (τ_π1) = 0.330.
  - Distributions, Schedule C corporations (τ_d1) = 0.144.
  - Distributions, other business (τ_d2) = 0.382.
  - Consumption tax in baseline (residual to balance budget) = 8.2 percent.
- Model vs data macro aggregates (averages relative to adjusted GNP, 2000–2010; Table 1):
  - Total Adjusted Income: Data 1.000, Model 1.000.
  - Labor Income: Data 0.585, Model 0.584.
  - Capital Income: Data 0.415, Model 0.416.
  - Consumption: Data 0.745, Model 0.714.
  - Tangible investment: Data 0.211, Model 0.208.
  - Government Deficit*: Data 0.019, Model 0.018.
  - Tangible Capital: Data 4.117, Model 4.045.
  - Intangible Capital: Data 1.700, Model 1.700.
  - Note: Government deficit calculated using the average of 2001–2005.

### Quantitative experiment: permanent increase in public debt via staged government spending
- Policy experiment design:
  - Implement an annual increase in government expenditures by 3.5 percent over a span of 20 years, holding tax rates constant.
  - The government issues debt to finance the additional government spending.
  - After 20 years, government spending is readjusted to the initial steady state and the debt-to-GDP ratio is stabilized using a consumption tax.
  - Demographics and other factors held constant to isolate debt effects.
- Path and qualitative resemblance:
  - The resulting debt path closely resembles the long-term trend observed in US data (figure referenced in text).
- Quantitative impacts reported:
  - Crowding out: capital declines by 9 percent over the initial 20-year period and by 15 percent in the long run.
  - These declines are associated with reduced output in the scenarios discussed in the analytical section (e.g., 8.0 percent decline in output when debt-to-GDP increases from 60 to 120).
- Caveats and cross-references:
  - Aguiar et al. (2022) observation: increasing revenue through debt issuance can be used to subsidize capital, mitigate crowding-out effects and support a Pareto improvement when r − g is negative. (Mentioned as context; model here keeps tax policy fixed and does not implement such subsidies.)

### Key takeaways and policy-relevant implications
- Structural result: At steady state, higher government debt crowds out private capital; there exists a finite b_max and associated k_min beyond which equilibrium cannot be sustained.
- Non-monotone fiscal space: When r − g < 0, the sustainable primary deficit as a function of debt is hump-shaped — increasing debt initially allows larger deficits but beyond a point reduces sustainable deficits (inverted-U / Laffer-like relationship).
- Quantitative magnitudes (illustrative):
  - Two-period calibrated illustration yields b_max = 396 (396 percent of GDP) and a pd_max = 7 (percent) at debt-to-GDP = 7 percent.
  - Moving debt-to-GDP from 60 to 120 allows a 2.3 percent higher deficit but produces an 8.0 percent decline in output; moving from 120 to 212 produces an additional 13.3 percent output decline.
- Policy caution: High steady-state debt and deficit levels the model can mathematically sustain are not normative targets; large debts come with high costs in terms of foregone investment and economic potential. The model abstracts from default risk and bond-market instability, which in practice would tighten sustainable limits.

*Italic: Source — wpiea2024010-print-pdf - Section 2*

### Section 3

### Section 3

### Key quantitative results on debt, capital, and output
- The return on capital (MPK) experiences a slight increase over time, rising by 0.2 percentage points in the first 20 years and 0.4 percentage points in the long run.
- GDP experiences a decline of over 4 percent in the initial 20-year period and over 8 percent in the long run.
- Labor supply remains relatively stable throughout this period.
- i_b − g, which represents the return on capital net of GDP growth and convenience yield, initially increases over the first 20 years due to high interest rate and the negative GDP growth, but gradually declines towards the new steady state as GDP growth returns from negative to zero. The i_b − g value remains negative throughout the analyzed period.
- The increase in government debt from 60 to 120 percent of GDP over the last 20 years could lead to a permanent reduction in long-term GDP by about 8 percentage points.
- Note referencing empirical comparison: Rachel and Summers (2019) suggest that a debt-GDP ratio increase of 1 percentage point of GDP raises interest rates by 3-4 basis points, which is larger than the results obtained from our simulations; one possible explanation offered is the erosion of the convenience yield.

### Debt and capital dynamics (model calibration and paths)
- Debt to GDP: model reproduces a path comparable to data for 2004-2022 with initial steady-state debt around 60 percent of GDP and subsequent increases (figures indicate debt up to around 120 percent of GDP).
- Capital stock: initial steady state capital set to 100; capital evolves over time (capital consists of both tangible and intangible capital).
- Output: initial steady-state GDP set to 100; output declines relative to initial steady state along the simulated transition path.
- Interest-growth differential: r − g (i_b − g) plotted over transition shows negative values throughout the horizon simulated.

### VI. Alternative specifications
- Purpose: test robustness and extend applicability of main results via two exercises: foreign ownership of U.S. public debt, and financing mechanism (debt vs tax).

A. Foreigners also hold public debt
- Alternative simulation assumption: 30 percent of U.S. government debt is held by foreign investors, "as has been the case in recent years."
- Under this assumption, consequences for GDP are consistent with initial findings but exhibit a smaller magnitude.
- Quantitative outcome reported: GDP declines by about 4 percent in the initial 20-year period, and "by slightly moreover the long run." (text as in source)
- Mechanism: crowding-out effect is mitigated because a portion of newly-issued government debt is absorbed by foreign investors, reducing the domestic impact.
- The analysis emphasizes the importance of the elasticity of demand for safe assets as a key variable in quantifying the crowding-out effect and notes that the model’s elasticity of interest rates on government debt in response to changes in the debt-to-GDP ratio aligns with empirical estimates, particularly falling within the lower bound region.

B. Tax financing vs Debt Financing
- Question: given the baseline spending path, would debt financing or tax financing be preferred from the perspective of minimizing negative impact on GDP?
- Modeling choice: use consumption taxes to finance the spending path while keeping debt-to-GDP at its original steady state level.
- Rationale: in steady state, consumption taxes would not be distorting directly any marginal decisions; consumption taxes are known to have lower macro impact than others (e.g., increasing labor taxes).
- Result: financing through consumption taxation while keeping public debt to GDP stable would have small effects on GDP, in contrast to the baseline experiment based on debt financing and debt stabilizing only in the distant future.

### VII. Conclusion (section highlights)
- The paper examines macroeconomic costs of increasing public debt in a general equilibrium overlapping generations framework, abstracting from default and uncertainty.
- Core mechanism: crowding-out captured with two assets; public debt pays interest equal to net return on private capital minus an exogenously determined convenience yield.
- Feasibility constraints for all agents, including the government, are an essential aspect of the model.
- Endogeneity of the real interest rate (r) and the economic growth rate (g) is central; these variables are influenced by policy decisions and interact with elevated public debt levels.
- Main quantitative takeaway: increasing public debt can produce highly substantial quantitative impacts—example cited is increase in public debt from 60 to 120 percent of GDP over the last 20 years potentially leading to a permanent reduction in long-term GDP by about 8 percentage points.
- Policy implication: while fiscal costs of public debt may appear manageable in low-interest-rate environments, macroeconomic costs associated with higher public debt can be notably significant, warranting nuanced understanding of public debt dynamics.

*Costly Increases of Public Debt when r < g — Working Paper No. WP/2024/010 (Section 3)*

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_Source: https://www.imf.org/-/media/files/publications/wp/2024/english/wpiea2024010-print-pdf.pdf_