## _wp14198 - introduction of non-defaultable debt is short lived. We also show that allowing governments in default to

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---

### Introduction — motivation and main quantitative findings
- Objective: quantify gains from introducing a limited non-defaultable financing option into a sovereign-default framework à la Eaton and Gersovitz (1981).
- Policy context: informs proposals for common euro-area sovereign bonds (Eurobonds); analysis is partial and does not study costs or political/moral-hazard difficulties of introducing Eurobonds.
- Key modeling assumption: government can issue long-term defaultable debt and one-period non-defaultable debt; non-defaultable issuances are restricted by an exogenous limit (baseline limit: 10 percent of trend aggregate income).
- Short-lived spread reduction:
  - Introducing non-defaultable debt up to 10 percent of trend income reduces the interest rate spread from 2.9 percent to 1.4 percent at introduction.
  - The spread returns to pre-introduction levels in less than five years.
  - Mechanism: initial substitution into non-defaultable debt lowers defaultable debt and spreads; after the new source is exhausted the government increases defaultable debt and default risk reverts.
- Welfare impacts (equivalent permanent consumption changes):
  - Introduction of non-defaultable debt (10 percent limit): welfare gain equivalent to a permanent consumption increase of 0.9 percent.
- Modalities and quantitative outcomes:
  - Buyback (forced use of non-defaultable debt to buy back defaultable debt):
    - Non-defaultable issuances 10 percent of trend income allow buybacks of defaultable debt up to 12 percent of trend income.
    - Spread reduction at introduction: 1.6 percent (versus 1.5 percent when not forced to buy back).
    - Welfare gain: 0.7 percent.
  - Voluntary debt exchange (government captures bondholder gains):
    - Government reduces defaultable debt by 14 percent of trend income.
    - Spread reduction at introduction: 1.7 percent.
    - Welfare gain: 1.1 percent.
  - Non-defaultable issuances plus defaultable-debt limit of 55 percent of trend income:
    - Government reduces defaultable debt by 16 percent of trend income to 50 percent of trend income.
    - Immediate ability to finance a consumption increase of 5 percent of trend income.
    - Welfare gain: 1.4 percent.
- Additional findings:
  - Allowing governments in default to increase non-defaultable debt is damaging at introduction and inconsequential in the medium term.
  - Initial improvements in sovereign spreads and debt positions are transient in the baseline environment with long-term defaultable debt.

### Model — environment and debt instruments
- Economy:
  - Small open economy with stochastic endowment y ∈ Y ⊂ R++ following a Markov process.
  - Government maximizes E_t ∑_{j=t}^∞ β^{j−t} u(c_j).
- Timing: government observes income, decides whether to default on defaultable debt, then chooses borrowing/issuance subject to default decision constraints.
- Defaultable long-term bond:
  - Promises infinite stream of coupons that decrease at constant rate δ.
  - Bond issued in period t pays (1−δ)^{j−1} units in period t+j for j ≥ 1.
  - Debt law: b_{t+1} = (1−δ) b_t + l_t.
- Non-defaultable bond:
  - One-period bond delivering one unit next period.
  - Exogenous limit to number issued; limit may depend on default status and level of non-defaultable debt.
  - Baseline: governments in default can rollover existing non-defaultable bonds but cannot increase non-defaultable debt while in default.
- Default costs and re-access:
  - Default triggers exclusion from defaultable-debt market for a stochastic number of periods and income cost: income is y − φ(y) during exclusion.
  - With probability ψ ∈ [0,1], starting the first period after the default period government may regain access by repaying with defaultable bonds of size b_R.
- Foreign investors: competitive, risk-neutral, discount at rate r; bond prices satisfy zero-expected-profit.

### Recursive formulation — value functions and pricing (notation preserved)
- Non-defaulting government's value:
  - V(b,e,y) = max{ V_R(b,e,y), V_D(e,y) }.
  - Repaying:
    - V_R(b,e,y) = max_{b′≥0,e′≥0,c≥0} { u(c) + β E_{y′|y} V(b′,e′,y′) }.
    - s.t. c = y − b − e + q(b′,e′,y) [ b′ − (1−δ) b ] + e′ / (1 + r); e′ ≤ ē_R; b′ > (1−δ) b only if q(b′,e′,y) ≥ q̄.
  - Defaulting:
    - V_D(e,y) = max_{e′≥0,c≥0} { u(c) + β E_{y′|y} [ (1−ψ) V_D(e′,y′) + ψ V(b_R,e′,y′) ] }.
    - s.t. c = y − φ(y) − e + e′ / (1 + r); e′ ≤ ē_D(e).
- Decision rules: default rule d̂(b,e,y); next-period defaultable debt b̂; non-defaultable bonds ê_R and ê_D; consumption ĉ_R and ĉ_D.
- Bond pricing (risk-neutral investors):
  - q(b′,e′,y) (1 + r) = E_{y′|y} { [1 − d̂(b′,e′,y′)] [ 1 + (1−δ) q(b′′,e′′,y′) ] + d̂(b′,e′,y′) q^D(b′,e′′_D,y′) }.
  - q^D(b,e′,y) (1 + r) = E_{y′|y} { [ (1−ψ) + ψ d̂(b_R,e′,y′) ] q^D(b,e′′_D,y′) + ψ [1 − d̂(b_R,e′,y′)] ( b_R / b ) [ 1 + (1−δ) q(b′′_R,e′′_R,y′) ] }.
- Interpretation: investors equate selling-and-investing risk-free to expected payoffs given default probabilities and pricing in default states.

### Calibration and numerical solution
- Preferences and endowment:
  - u(c) = (c^{1−γ} − 1)/(1−γ), γ̸=1.
  - log(y_t) = (1−ρ)μ + ρ log(y_{t−1}) + ε_t, ε_t ∼ N(0, σ_ε^2).
- Default cost: φ(y) = max{0, d_0 y + d_1 y^2}.
- Calibration (replicates Spain features; period = quarter):
  - r = 1%
  - γ = 2
  - β = 0.97
  - ψ = 0.282
  - ρ = 0.8
  - σ_ε = 1%
  - μ = (-1/2) σ_ε^2
  - δ = 0.0275 (average bond duration 6 years)
  - q_0 = 0.8 (implies sovereign cannot issue when spread > 3.6 percent; binding in 1.4 percent of simulation periods)
  - d_0 = −1.9
  - d_1 = 2.25
  - b_R = 0.034 (average haircut in simulations 66 percent)
- Solution method:
  - Spline interpolation over debt levels; linear interpolation over endowment levels.
  - Solve finite-horizon economy, increase horizon until first- and second-period functions converge; use first-period equilibrium as infinite-horizon equilibrium.

### Results — A. Effects of introducing non-defaultable bonds (baseline experiment)
- Policy experiment:
  - Start with ê_R = ê_D(e) = 0. Then unanticipated announcement: ê_R = 0.4 and ê_D(e) = e (interpreted as non-defaultable bonds up to 10 percent of trend annual income if not in default).
- Immediate spread and welfare impacts (debt level = 66 percent of trend income; three initial risk states):
  - Spread before introduction: Highest risk 2.9%; Medium risk 1.9%; Lowest risk 1.6%.
  - Spread after introduction: Highest risk 1.4%; Medium risk 1.3%; Lowest risk 1.2%.
  - Welfare gain: Highest risk 0.9%; Medium risk 0.9%; Lowest risk 0.8%.
- Dynamics:
  - Lenders' expectations about future declines in defaultable debt drive immediate spread reductions.
  - Defaultable-debt-to-income ratio declines on average in initial periods; non-defaultable debt rises up to imposed limit.
  - Introduction can prevent imminent defaults; it never triggers additional defaults.
- Welfare metric formula preserved from source notation.

### Results — B. Debt buybacks (non-defaultable issuance used to buy back defaultable debt)
- Buyback mechanism:
  - Government issues non-defaultable bonds up to limit and finances buyback; defaultable debt after buyback satisfies:
    - b_B(b, ê_R, y) = b − ê_R/(1 + r) [1 + (1 − δ) q(b_B(b, ê_R, y), ê_R, y)].
- Quantitative outcomes (debt = 66 percent of trend income; three initial risk states):
  - Spread before buyback: Highest risk 2.9%; Medium risk 1.9%; Lowest risk 1.6%.
  - Spread after buyback: Highest risk 1.3%; Medium risk 1.2%; Lowest risk 1.2%.
  - Defaultable debt bought back: Highest risk 11.8%; Medium risk 11.7%; Lowest risk 11.7% (percent of trend income).
  - Welfare gain: Highest risk 0.7%; Medium risk 0.8%; Lowest risk 0.8%.
- Interpretation:
  - Issuing 10 percent of trend income in non-defaultable bonds allows buyback of ~12 percent of trend income in defaultable debt.
  - Post-buyback increases in sovereign bond prices are minimal because lenders anticipate future increases in defaultable debt (Bulow and Rogoff-style result).
  - Welfare gains are lower when buyback is mandated because it restricts government options.

### Results — C. Voluntary debt exchanges (government captures bondholder gains)
- Design:
  - Voluntary exchange is Pareto-improving: reduces government indebtedness with a positive face-value haircut while leaving market value of bondholders' claims unchanged.
  - Extreme case: government captures all bondholders' capital gains from introducing non-defaultable debt.
  - Equilibrium condition for post-exchange defaultable bond quantity bE preserved as in source (equation (6) notation).
- Quantitative effects (Table 4; three initial risk states):
  - Highest risk:
    - Spread before exchange: 2.9%
    - Spread after exchange: 1.2%
    - Reduction in defaultable debt: 13.8% (of trend income)
    - Welfare gain: 1.1%
  - Medium risk:
    - Spread before: 1.9%
    - Spread after: 1.2%
    - Reduction: 13.3%
    - Welfare gain: 1.1%
  - Lowest risk:
    - Spread before: 1.6%
    - Spread after: 1.1%
    - Reduction: 12.8%
    - Welfare gain: 1.1%
- Comparison with buyback:
  - Voluntary exchange achieves an additional reduction in defaultable debt of between one and two percent of trend income versus buyback.
  - Additional gains are larger when pre-exchange default risk is higher.

### Results — D. Long-run effect of non-defaultable bonds on default risk
- Dynamics:
  - Initial large reductions in default risk and spreads, but the effect declines and is not significant after five years.
  - Typical path: first year government refrains from issuing defaultable debt; after ~one year non-defaultable financing is exhausted; within five years defaultable debt returns to pre-introduction levels.
- Cost detail:
  - Interest payment on non-defaultable bonds represents 0.4 percent of trend income (a 4 percent interest rate on debt equal to 10 percent of trend income).
  - This payment is made irrespective of default and therefore does not significantly alter default decisions.
- Constraint timing:
  - Less than two years after introduction the government hits the 10-percent limit on non-defaultable bonds; thereafter borrowing opportunities resemble those without non-defaultable bonds.

### Results — E. Voluntary exchanges with a defaultable-debt limit (fiscal rule)
- Policy studied:
  - Introduce non-defaultable bonds equal to 10 percent of trend income to finance a voluntary debt exchange while immediately imposing a 55 percent of trend income limit on defaultable debt.
- Rationale for 55 percent limit:
  - Non-defaultable bonds suffice to finance the reduction needed to implement the limit immediately.
  - The 55 percent limit is sufficient to eliminate default risk in the model.
- Quantitative outcomes (Table 5; three initial risk states):
  - Highest risk:
    - Spread before exchange and limit: 2.9%
    - Spread after exchange and limit: 0.0%
    - Reduction in defaultable debt: 15.5% (of trend income)
    - Welfare gain: 1.4%
  - Medium risk:
    - Spread before: 1.9%
    - Spread after: 0.0%
    - Reduction: 15.1%
    - Welfare gain: 1.3%
  - Lowest risk:
    - Spread before: 1.6%
    - Spread after: 0.0%
    - Reduction: 14.7%
    - Welfare gain: 1.3%
- Post-implementation:
  - Government finances a consumption expansion by increasing defaultable debt up to the 55 percent limit; in the long run the government remains at the limit and sovereign spread stays at zero.

### Results — F. Issuances of non-defaultable bonds during defaults
- Policy question: allowing governments in default to issue non-defaultable debt (ê_D(e) = e) versus baseline ê_D(e) = 0.4.
- Findings:
  - Different assumptions on ê_D have negligible effects on mean outcomes: mean defaultable-debt-to-income ratio = 66 percent in both cases; mean spread = 2.0 percent in both cases.
  - Reason: after introduction governments rapidly build non-defaultable debt until hitting the constraint (e.g., 0.4), making the ability to issue in default inconsequential soon after introduction.
  - At the moment of introduction, changing ê_D assumption alters spread reduction by less than 0.1 percent.
  - Conclusion: allowing issuance of non-defaultable bonds during defaults has negligible effects in this model.

### Conclusions — policy-relevant takeaways
- Small issuances of non-defaultable bonds can produce:
  - Significant immediate reductions in sovereign risk.
  - Substantial welfare gains (exact values preserved above).
- Modalities (buyback vs voluntary exchange) yield only small additional differences in sovereign risk reduction.
- Non-defaultable bonds alone do not lower sovereign risk in the long run unless combined with fiscal rules that limit issuance of defaultable debt.
- Policy implication: it could be worthwhile to address implementation costs and difficulties of introducing non-defaultable bonds, but this analysis does not model those costs and therefore does not endorse introduction.

*Source: IMF Working Paper content unit _wp14198 - introduction of non-defaultable debt is short lived. We also show that allowing governments in default to (PDF chapter/section).*

### introduction of non-defaultable debt is short lived. We also show that allowing governments in default to

### _wp14198 - introduction of non-defaultable debt is short lived. We also show that allowing governments in default to

### Introduction — motivation and main quantitative findings
- Objective: quantify gains from introducing a limited non-defaultable financing option into a sovereign-default framework à la Eaton and Gersovitz (1981).
- Policy context: informs proposals for common euro-area sovereign bonds (Eurobonds) that could be virtually non-defaultable; analysis is partial and does not study costs or political/moral-hazard difficulties of introducing Eurobonds.
- Key modeling assumption: government can issue both long-term defaultable debt and one-period non-defaultable debt; non-defaultable issuances are restricted by an exogenous limit (baseline limit used in experiments: 10 percent of trend aggregate income).
- Short-lived spread reduction:
  - Introducing non-defaultable debt up to 10 percent of trend income reduces the interest rate spread from 2.9 percent to 1.4 percent.
  - The spread returns to pre-introduction levels in less than five years.
  - Mechanism: initial substitution into non-defaultable debt lowers defaultable debt and spreads, but after the new source is exhausted the government increases defaultable debt and default risk reverts.
- Welfare impacts (expressed as equivalent permanent consumption changes):
  - Introduction of non-defaultable debt (10 percent limit): welfare gain equivalent to a permanent consumption increase of 0.9 percent.
- Modalities of introduction studied:
  - Buyback (government forced to use non-defaultable debt to buy back defaultable debt):
    - Non-defaultable issuances of 10 percent of trend income would allow buybacks of defaultable debt up to 12 percent of trend income.
    - Spread reduction at introduction: 1.6 percent (versus 1.5 percent when not forced to buy back).
    - Welfare gain: 0.7 percent (lower than 0.9 percent because forcing the buyback reduces government options).
  - Voluntary debt exchange (no capital gains/losses for holders, allows larger reduction of defaultable debt):
    - Government reduces defaultable debt by 14 percent of trend income.
    - Spread reduction at introduction: 1.7 percent.
    - Welfare gain: 1.1 percent.
  - Combining non-defaultable issuances with a defaultable-debt limit:
    - Assumed limit: 55 percent of trend income.
    - With non-defaultable bonds and the debt limit, government reduces debt by 16 percent of trend income to 50 percent of trend income.
    - This allows implementation of the debt limit immediately and without a consumption sacrifice; with debt at 50 percent and limit 55 percent, the government can finance a consumption increase of 5 percent of trend income.
    - Welfare gain: 1.4 percent (versus 1.1 percent without the debt limit).
- Additional findings:
  - Allowing governments in default to increase non-defaultable debt is damaging at the time non-defaultable debt is introduced and inconsequential in the medium term.
  - Overall, initial improvements in sovereign spreads and debt positions are transient in the baseline environment with long-term defaultable debt.

### Model — environment and debt instruments
- Economy: small open economy with stochastic endowment y ∈ Y ⊂ R++ following a Markov process.
- Government objective: maximize E_t ∑_{j=t}^∞ β^{j−t} u(c_j), where β is subjective discount factor and u(c) is strictly increasing and concave.
- Timing each period: government learns income → decides whether to default on defaultable debt → chooses borrowing/issuance subject to default decision constraints.
- Defaultable long-term bond structure:
  - A defaultable bond issued in period t promises an infinite stream of coupons that decrease at constant rate δ.
  - A bond issued in period t promises to pay (1−δ)^{j−1} units in period t+j, for all j ≥ 1.
  - Debt law of motion: b_{t+1} = (1−δ) b_t + l_t, where b_t is number of defaultable coupons due at beginning of period t and l_t is number of defaultable long-term bonds issued in period t.
- Non-defaultable bond structure:
  - Each non-defaultable bond promises to deliver one unit of the good in the next period (one-period bonds).
  - There is an exogenous limit to the number of non-defaultable bonds that can be issued; limit may depend on default status and level of non-defaultable debt.
  - Governments in default can rollover existing non-defaultable bonds but cannot increase their level of non-defaultable debt while in default (in baseline experiments).
- Default costs and re-access:
  - Default triggers exclusion from defaultable-debt market for a stochastic number of periods and an income cost: income is y − φ(y) during exclusion.
  - With constant probability ψ ∈ [0,1], starting the first period after the default period the government may have an opportunity to regain access by repaying with defaultable bonds of size b_R (captures debt exchanges after default).
- Foreign investors:
  - Competitive market of risk-neutral foreign investors who discount at rate r; bond prices satisfy zero-expected-profit conditions.
- Constraint to avoid pre-default consumption booms:
  - Government cannot sell defaultable bonds with price lower than q̄; chosen q̄ eliminates consumption booms before defaults and allows replication of debt and spread levels observed in data.

### Recursive formulation — value functions, constraints, and pricing
- Non-defaulting government's value function:
  - V(b,e,y) = max{ V_R(b,e,y), V_D(e,y) }, (equation (1) in source)
  - Value of repaying:
    - V_R(b,e,y) = max_{b′≥0,e′≥0,c≥0} { u(c) + β E_{y′|y} V(b′,e′,y′) }, (equation (2) in source)
    - Subject to:
      - c = y − b − e + q(b′,e′,y) [ b′ − (1−δ) b ] + e′ / (1 + r)
      - e′ ≤ ē_R
      - b′ > (1−δ) b only if q(b′,e′,y) ≥ q̄
- Value of defaulting:
  - V_D(e,y) = max_{e′≥0,c≥0} { u(c) + β E_{y′|y} [ (1−ψ) V_D(e′,y′) + ψ V(b_R,e′,y′) ] }, (equation (3) in source)
  - Subject to:
    - c = y − φ(y) − e + e′ / (1 + r)
    - e′ ≤ ē_D(e)
- Decision rules (solution objects):
  - Default rule d̂(b,e,y) ∈ {0,1}; next-period defaultable debt b̂; next-period non-defaultable bonds when repaying ê_R and when in default ê_D; consumption policies ĉ_R and ĉ_D.
- Bond pricing (risk-neutral investors):
  - q(b′,e′,y) (1 + r) = E_{y′|y} { [1 − d̂(b′,e′,y′)] [ 1 + (1−δ) q(b′′,e′′,y′) ] + d̂(b′,e′,y′) q^D(b′,e′′_D,y′) }, (equation (4) in source)
  - q^D(b,e′,y) (1 + r) = E_{y′|y} { [ (1−ψ) + ψ d̂(b_R,e′,y′) ] q^D(b,e′′_D,y′) + ψ [1 − d̂(b_R,e′,y′)] ( b_R / b ) [ 1 + (1−δ) q(b′′_R,e′′_R,y′) ] }
  - Where b′′ = b̂(b′,e′,y′), e′′ = ê_R(b′,e′,y′), b′′_R = b̂(b_R,e′,y′), e′′_R = ê_R(b_R,e′,y′), e′′_D = ê_D(e′,y′).
- Interpretation: investors equate the value of selling a defaultable bond and investing risk-free (LHS) to the expected payoff of holding the bond given default probabilities and recovery/pricing in default states (RHS).

### Results overview (as reported in the source)
- Short-run vs medium-run:
  - Immediate significant reduction in sovereign spread upon introduction of limited non-defaultable debt; effect dissipates within five years as defaultable debt levels revert.
- Welfare and debt-level magnitudes (exact values preserved from source):
  - Non-defaultable debt allowed up to 10 percent of trend income:
    - Spread reduction: from 2.9 percent to 1.4 percent at introduction.
    - Welfare equivalent consumption gain: 0.9 percent.
  - Non-defaultable issuances used for a buyback:
    - Buyback capacity: 12 percent of trend income.
    - Spread reduction at introduction: 1.6 percent (versus 1.5 percent when not forced).
    - Welfare gain: 0.7 percent.
  - Voluntary debt exchange:
    - Defaultable-debt reduction: 14 percent of trend income.
    - Spread reduction at introduction: 1.7 percent.
    - Welfare gain: 1.1 percent.
  - Non-defaultable issuances plus a defaultable-debt limit of 55 percent of trend income:
    - Defaultable-debt reduction: 16 percent of trend income (resulting debt level: 50 percent of trend income).
    - Immediate ability to finance a consumption increase of 5 percent of trend income without sacrifice.
    - Welfare gain: 1.4 percent.
- Robust characterization: many of the beneficial effects of introducing limited non-defaultable debt arise immediately but are transient in the baseline setup with long-term defaultable debt and inability to expand non-defaultable debt while in default.

*Source: IMF Working Paper content unit _wp14198 - introduction of non-defaultable debt is short lived. We also show that allowing governments in default to (PDF chapter/section).*

### 1. a set of value functionsV,V

### _wp14198 - 1. a set of value functionsV,V

### Model setup and equilibrium conditions
- Model objects:
  - A set of value functions V, V_R, and V_D.
  - Rules for default ^d, next-period defaultable debt ^b, next-period non-defaultable bonds ^e_R and ^e_D, and consumption ^c_R and ^c_D.
  - A bond price function q.
- Equilibrium requirements:
  - i. Given a bond price function q, {V, V_R, V_D, ^d, ^b, ^e_R, ^e_D, ^c_R, ^c_D} solve Bellman equations (1), (2), and (3).
  - ii. Given policy rules {^d, ^b, ^e_R, ^e_D}, the bond price function q satisfies condition (4).

### Calibration
- Preferences and endowment:
  - Utility u(c) = (c^{1−γ} − 1)/(1−γ), with γ̸= 1.
  - Endowment process: log(y_t) = (1−ρ)μ + ρ log(y_{t−1}) + ε_t, with |ρ|<1 and ε_t ∼ N(0, σ_ε^2).
- Default cost specification:
  - Income loss during default φ(y) = max{0, d_0 y + d_1 y^2}, a quadratic loss as in Chatterjee and Eyigungor (2012).
- Calibration choices and interpretation:
  - Model calibrated without non-defaultable bonds (e_R = e_D(e) = 0) to replicate features for Spain.
  - A period equals a quarter.
  - Risk-free interest rate r = 1 percent.
  - Risk aversion γ = 2.
  - Discount factor β = 0.97.
  - Probability of regaining market access ψ = 0.282.
  - Income process estimated from Spain's GDP 1970Q1–2012Q3.
  - Debt duration parameter δ = 2.75% (0.0275), implying average bond duration of 6 years in simulations.
  - Price cap q_0 = 0.8; implies sovereign cannot issue when spread > 3.6 percent; binding in 1.4 percent of simulation periods.
- Income-cost-of-default parameters targeted to match mean debt and sovereign spread:
  - d_0 = −1.9
  - d_1 = 2.25
  - With these, simulations yield mean debt-to-income ratio = 66 percent and mean spread = 2.0 percent.
- Default haircut:
  - b_R = 0.034, implying average haircut in simulations of 66 percent (consistent with average haircut 65 percent reported by Cruces and Trebesch (2013) for reductions in face value).
- Table 1 (parameter summary, as in source):
  - Risk-free rate r: 1%
  - Risk aversion γ: 2
  - Discount factor β: 0.97
  - Probability of reentry after default ψ: 0.282
  - Income autocorrelation coefficient ρ: 0.8
  - Standard deviation of innovations σ_ε: 1%
  - Mean log income μ: (-1/2) σ_ε^2
  - Debt duration δ: 0.0275
  - Price cap q_0: 0.8
  - Income cost of defaulting d_0: -1.9
  - Income cost of defaulting d_1: 2.25
  - Default haircut b_R: 0.034

### Numerical solution method
- Solution approach:
  - Spline interpolation over debt levels and linear interpolation over endowment levels.
  - Solve finite-horizon economy and increase horizon until value and bond-price functions for first and second periods are sufficiently close; use first-period equilibrium functions as infinite-horizon equilibrium.

### Results — A. Effects of introducing non-defaultable bonds
- Policy experiment:
  - Start from e_R = e_D(e) = 0. Then unanticipated announcement: constraints change to e_R = 0.4 and e_D(e) = e.
  - Interpretation: government can issue non-defaultable bonds up to 10 percent of trend annual income if not in default; when in default, can rollover existing non-defaultable bonds but cannot increase them.
  - Rationale for 10 percent limit: lower end of Eurobond proposals and sufficient to reduce sovereign risk markedly and to finance debt reduction needed to implement an immediate defaultable-debt limit that eliminates sovereign risk (assuming full commitment).
- Immediate spread and welfare impacts (Table 2 results for three initial risk states with debt level = 66 percent of trend income):
  - Spread before introduction of non-defaultable bonds: Highest risk 2.9%; Medium risk 1.9%; Lowest risk 1.6%.
  - Spread after introduction of non-defaultable bonds: Highest risk 1.4%; Medium risk 1.3%; Lowest risk 1.2%.
  - Welfare gain from introduction of non-defaultable bonds: Highest risk 0.9%; Medium risk 0.9%; Lowest risk 0.8%.
- Mechanisms and dynamics:
  - Lenders' expectations about future declines in defaultable debt drive immediate spread reductions because long-term defaultable debt prices depend on expected future debt and default decisions.
  - Figure 1 findings (described): defaultable-debt-to-income ratio declines on average in initial periods after introduction of non-defaultable debt; non-defaultable debt rises up to the imposed limit.
  - Introduction of non-defaultable bonds can prevent imminent defaults: there exist (income, defaultable debt) combinations where government would default without non-defaultable bonds but would not default after introduction; introduction never triggers additional defaults.
- Welfare metric:
  - Welfare gains measured as constant proportional change in consumption leaving consumer indifferent between economies, given by ( (V_E(b,0,y) / V_N(b,y))^{1/(1−·)} − 1 ), where V_E and V_N denote value functions with and without non-defaultable bonds (notation preserved from source).
- Additional remarks:
  - Model likely underestimates gains from lowering sovereign spread because it omits positive productivity/allocation effects of lower interest rates found in other studies.

### Results — B. Debt buybacks (non-defaultable issuance used to buy back defaultable debt)
- Buyback setup:
  - Government issues non-defaultable bonds up to limit and uses proceeds to finance buyback of defaultable debt.
  - Defaultable debt level after buyback given by:
    - b_B(b, e_R, y) = b − e_R/(1 + r) [1 + (1 − δ) q(b_B(b, e_R, y), e_R, y)].
  - Interpretation: e_R/(1 + r) denotes proceeds from issuances; 1 + (1 − δ) q(...) denotes lender's willingness-to-sell price for a defaultable bond.
- Quantitative outcomes (Table 3 results for three initial risk states, debt = 66 percent of trend income):
  - Spread before the buyback: Highest risk 2.9%; Medium risk 1.9%; Lowest risk 1.6%.
  - Spread after the buyback: Highest risk 1.3%; Medium risk 1.2%; Lowest risk 1.2%.
  - Defaultable debt bought back (% of trend income): Highest risk 11.8%; Medium risk 11.7%; Lowest risk 11.7%.
  - Welfare gain from the buyback: Highest risk 0.7%; Medium risk 0.8%; Lowest risk 0.8%.
- Mechanisms and interpretation:
  - Issuing 10 percent of trend income in non-defaultable bonds allows buyback of almost 12 percent of trend income in defaultable debt.
  - The model exhibits a Bulow and Rogoff-style buyback boondoggle: the immediate additional increase in sovereign bond prices from buyback is minimal because lenders anticipate government will increase defaultable debt after the buyback.
  - Because defaultable debt is long-term, bond prices hinge on expected future debt paths; post-buyback defaultable-debt dynamics are similar to the case without forced buybacks (Figure 3).
  - Welfare gains are lower when buyback is mandated because the government is constrained (cannot benefit from the option to choose).

*Source: _wp14198 - 1. a set of value functionsV,V (PDF chapter).*

### introduction of non-defaultable bonds through a defaultable debt buyback, for samples without

### introduction of non-defaultable bonds through a defaultable debt buyback, for samples without defaults

### Limitations of debt buybacks
- Debt buybacks are often sensible only if the government obtains some compensation from bondholders; Bulow and Rogo (1988) conclude that "Buybacks can be justied only if the country negotiates substantial concessions or compensation for undertaking the repurchase."
- The paper contrasts buybacks (where bondholders keep capital gains from the decline of defaultable debt) with voluntary exchanges (where the government can capture concessions or compensation).

### C. "Voluntary" debt exchanges — design and mechanics
- A "voluntary" debt exchange here is Pareto-improving: it reduces government indebtedness (positive face-value haircut) without affecting the total market value of bondholders' debt claims.
- The extreme case studied: the government captures all bondholders' capital gains from introducing non-defaultable debt.
- Take-it-or-leave-it offer described: for each defaultable coupon due this period, bondholders receive eR/b non-defaultable bonds to be paid next period plus bE/b defaultable bonds that start paying coupons in the current period. If rejected, non-defaultable bonds are not introduced.
- The government chooses bE to make bondholders indifferent between accepting or rejecting, thereby capturing all gains.
- Key equilibrium condition for post-exchange defaultable bond quantity bE:
  - eR b(1 +r) + bE(b,eR,y)[1 + (1−δ) q(bE(b,eR,y),eR,y)] b = 1 + (1−δ) qN(^bN(b,y),y). (equation (6) as presented)
- Comparison with buyback condition:
  - Buyback condition can be written as eR (1 +r) + bB(b,eR,y)[1+(1−δ) q(bB(b,eR,y),eR,y)] = b[1+(1−δ) q(bB(b,eR,y),eR,y)]. (equation (7) as presented)
- Because the introduction of non-defaultable bonds raises the price of defaultable bonds, the debt reduction achievable with a voluntary exchange is larger than with a buyback: bE(b,eR,y) < bB(b,eR,y).

### C — Quantitative effects (Table 4)
- Table 4: Effects of introducing non-defaultable bonds with a "voluntary" debt exchange
  - Highest risk
    - Spread before the exchange: 2.9%
    - Spread after the exchange: 1.2%
    - Reduction in defaultable debt (% of trend income): 13.8%
    - Welfare gain from the exchange: 1.1%
  - Medium risk
    - Spread before the exchange: 1.9%
    - Spread after the exchange: 1.2%
    - Reduction in defaultable debt (% of trend income): 13.3%
    - Welfare gain from the exchange: 1.1%
  - Lowest risk
    - Spread before the exchange: 1.6%
    - Spread after the exchange: 1.1%
    - Reduction in defaultable debt (% of trend income): 12.8%
    - Welfare gain from the exchange: 1.1%
- Additional defaultable debt reduction from voluntary exchange versus buyback is between one and two percent of trend income.
- The additional gains from voluntary exchanges are larger when pre-exchange default risk is higher, because bondholders' capital gains from introducing non-defaultable bonds are larger.

### D. Long-run effect of non-defaultable bonds on default risk
- Introduction of non-defaultable bonds reduces default risk and spreads initially, but the effect declines over time and is not significant after five years.
- Figure 4 (described): annual spread paths for three risk profiles show convergence such that after about five years spreads return to levels similar to economies without non-defaultable bonds.
- Mechanism:
  - In the first year after introduction, government refrains from issuing defaultable debt, causing a decline in defaultable debt level and spreads.
  - After roughly one year, financing via non-defaultable bonds is exhausted and the government increases defaultable debt; on average, in five years defaultable debt returns to pre-introduction levels.
- Quantitative detail on cost of non-defaultable interest:
  - Interest payment on non-defaultable bonds represents 0.4 percent of trend income (a 4 percent interest rate on debt for 10 percent of trend income).
  - This payment is made irrespective of default choice and therefore does not significantly alter default decisions or borrowing opportunities.
- Constraint dynamics:
  - Less than two years after introduction the government hits the 10-percent limit on non-defaultable bonds; borrowing opportunities then resemble those without non-defaultable bonds.

### E. Voluntary debt exchanges with a limit for defaultable debt (fiscal rule)
- Considered policy: introduce non-defaultable bonds equal to 10 percent of trend income to finance a voluntary debt exchange while immediately imposing a 55 percent of trend income limit for defaultable debt.
- Rationale for 55 percent limit:
  - (i) Non-defaultable bonds suffice to finance the defaultable debt reduction necessary to implement the limit immediately.
  - (ii) The 55 percent limit is sufficient to eliminate default risk.
- Table 5: Effects of introducing non-defaultable bonds with a limit for defaultable debt
  - Highest risk
    - Spread before the exchange and the debt limit: 2.9%
    - Spread after the exchange and the debt limit: 0.0%
    - Reduction in defaultable debt (% of trend income): 15.5%
    - Welfare gain from the exchange and the debt limit: 1.4%
  - Medium risk
    - Spread before the exchange and the debt limit: 1.9%
    - Spread after the exchange and the debt limit: 0.0%
    - Reduction in defaultable debt (% of trend income): 15.1%
    - Welfare gain from the exchange and the debt limit: 1.3%
  - Lowest risk
    - Spread before the exchange and the debt limit: 1.6%
    - Spread after the exchange and the debt limit: 0.0%
    - Reduction in defaultable debt (% of trend income): 14.7%
    - Welfare gain from the exchange and the debt limit: 1.3%
- After implementation, the government finances a consumption expansion by increasing defaultable debt up to the 55 percent limit; in the long run the government remains at the limit and sovereign spread stays at zero.

### F. Issuances of non-defaultable bonds during defaults
- Policy question: allowing governments in default to issue non-defaultable (Eurobond) debt — could this mitigate the cost of default and make default more attractive?
- Model experiment: compare simulations assuming eD(e) = 0.4 versus eD(e) = e (the baseline assumption).
- Findings:
  - These different assumptions on eD have negligible effects on mean outcomes: mean defaultable-debt-to-income ratio is 66 percent in both cases; mean spread is 2 percent in both cases.
  - Reason: after introduction, governments rapidly build non-defaultable debt until hitting the constraint (e.g., 0.4), so whether default alters ability to issue non-defaultable bonds is inconsequential soon after introduction.
  - Even at the moment non-defaultable bonds are introduced, assuming eD(e) = 0.4 instead of eD(e) = e changes the reduction in spreads by less than 0.1 percent.
  - Conclusion: allowing issuance of non-defaultable bonds during defaults has negligible effects in this model.

### V. Conclusions — summary of core findings
- Simulations of an equilibrium sovereign risk model indicate that relatively small issuances of non-defaultable bonds can produce:
  - Significant reductions in sovereign risk.
  - Substantial welfare gains.
- Comparing modalities for introducing non-defaultable bonds:
  - Introducing bonds through a buyback or a voluntary debt exchange yields only small additional reductions in sovereign risk relative to one another.
- Long-run effects:
  - Non-defaultable bonds alone do not lower sovereign risk in the long run unless combined with fiscal rules that limit issuance of defaultable debt.
- Policy implication:
  - Results suggest it could be worthwhile to address the implementation costs and difficulties of introducing non-defaultable bonds, but the paper does not consider those costs and therefore does not endorse their introduction.

*Source: _wp14198 - introduction of non-defaultable bonds through a defaultable debt buyback, for samples without defaults*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2014/_wp14198.pdf_