## _wp1170 - References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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---

### Introduction — scope and motivation
- Study focus: effects of sovereign debt dilution caused by the government’s lack of commitment to avoid decreasing the value of debt issued in the past by issuing new debt.
- Context and related literature includes: Bizer and DeMarzo (1992); Bolton and Jeanne (2009); Bolton and Skeel Jr. (2005); Borensztein et al. (2004); Detragiache (1994); Eaton and Fernandez (1995); Kletzer (1984); Niepelt (2008); Sachs and Cohen (1982); Saravia (forthcoming); Tirole (2002); UN (2004).
- Market-practice observations:
  - Corporate and household loans often show seniority; sovereign bonds typically do not show legal seniority.
  - Sovereign contracts commonly include pari passu clauses and negative pledge clauses; these prohibit future issuances of collateralized debt but do not protect against dilution from future borrowing behavior.
  - Weak enforcement in sovereign debt markets motivates proposals for more orderly sovereign debt restructurings.

### Novel modeling approach and contribution
- Framework: baseline sovereign default model a la Eaton and Gersovitz (1981) with extensions; small open economy with stochastic endowment of a single tradable good.
- Government decisions each period:
  - Decide whether to default on previously issued debt.
  - Decide how much to borrow or save by issuing non-contingent long-duration bonds (as in Hatchondo and Martinez (2009)).
- Default cost: endowment loss in default period (quadratic loss function φ(y) = d0 y + d1 y^2 referenced).
- Key inefficiency sources captured:
  - Lack of commitment to future repayment policy.
  - Bond payments not contingent on income shocks.
  - Government can borrow from multiple lenders and cannot commit not to decrease value of previously issued debt (debt dilution).

### Elimination of dilution — modeling device
- Modification: government must pay existing bondholders compensation equal to the reduction in the market value of their bonds caused by the government’s current debt issuance (no compensation for bond price declines not caused by new issuances).
- Economic consequence: investors buying sovereign debt anticipate that future value of their investment is independent of future issuances, thereby eliminating the dilution problem without increasing state-space dimensionality.
- Implementation alternatives discussed: consent of existing bondholders, representative intermediaries, majority clauses, redemption rights, buybacks at market price if no new issuance; interpreted as equivalent to an exclusivity problem.

### Quantitative calibration targets and discipline
- Baseline calibration targets:
  - Default probability.
  - Level of public external debt.
  - Debt duration.
  - Mean and standard deviation of the interest rate spread.
- Calibration parameters (Table 1 as preserved):
  - Borrower’s risk aversion σ2 = 2
  - Interest rate r1 = 1%
  - Output autocorrelation coefficient ρ0.9
  - Standard deviation of innovations σǫ2.7%
  - Mean log output μ(-1/2)σ2ǫ (as specified in source)
  - Duration δ0.0341
  - Discount factor β0.969
  - Default cost d0 -0.69
  - Default cost d1 1.01
  - Risk premium α4
  - Period = quarter.
  - δ = 3.41% implies average bond duration of 4.19 years in baseline simulations.
  - Calibration targets: mean spread of 7.4, standard deviation of spread 2.5, mean debt level 28% of mean quarterly output in pre-default samples, default frequency of 3 defaults per 100 years.

### Key quantitative findings (comparative: with dilution vs. without dilution)
- Defaults per 100 years:
  - With debt dilution: 3.10
  - Without debt dilution: 0.42
  - Implied contribution of dilution to default risk: 86%
- Mean interest rate spread (E(Rs)):
  - With debt dilution: 7.38%
  - Without debt dilution: 0.57%
  - Dilution accounts for 92% of the sovereign spread.
- Standard deviation of spread (σ(Rs)):
  - With debt dilution: 2.45
  - Without debt dilution: 0.72
  - Dilution accounts for 71% of spread volatility.
- Mean debt levels:
  - Mean debt face value without dilution is 36% lower than with dilution.
  - Mean debt market value without dilution is 11% lower than with dilution.
- Business-cycle statistics (Data vs With dilution vs Without dilution — preserved values):
  - Defaults per 100 years: Data 3.00; With debt dilution 3.10; Without debt dilution 0.42
  - Mean debt market value: Data 0.20; With 0.18; Without 0.18
  - Mean debt face value: Data 0.28; With 0.28; Without 0.18
  - E(Rs): Data 7.44; With 7.38; Without 0.57
  - σ(Rs): Data 2.51; With 2.45; Without 0.72
  - σ(y): Data 3.17; With 3.03; Without 3.36
  - σ(c)/σ(y): Data 0.94; With 1.04; Without 1.21
  - ρ(c, y): Data 0.97; With 1.00; Without 0.99
  - ρ(Rs, y): Data -0.65; With -0.80; Without -0.63

### Mechanism emphasized
- Main channel: dilution shifts the government's set of borrowing opportunities (combinations of borrowing levels and spreads).
- Equilibrium combinations of debt and spread levels available without dilution are not part of the government's choice set with dilution.
- For equilibrium debt levels in the economy without dilution, equilibrium spread levels would be about 400 basis points higher in the economy with dilution because lenders price in the government's incentive to increase default probability via future issuances.

### Model structure — technical summary
- 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).
- Government objective:
  - Maximize E[∑_{t=0}^∞ β^t u(c_t)] with u(c) = c^{1−γ}/(1−γ).
- Bond structure:
  - Long-duration bonds promising infinite stream of coupons that decrease at constant rate δ.
  - Bond issued in period t pays 1 in t+1 and (1−δ)^{s−1} in t+s for s ≥ 2.
  - Macaulay duration: D = 1 + r^* / (δ + r^*), where r^* denotes constant per-period yield delivered by the bond.
- Risk pricing:
  - Stochastic discount factor M(y′, y) = exp(−r − α ε′ − 0.5 α^2 σ_ε^2) (discrete-time Vasicek one-factor special case).
- Equilibrium concept:
  - Markov Perfect Equilibrium; solved by computing finite-horizon equilibria and extending horizon until convergence.

### Recursive formulation (selected equations and notation)
- Government value before default decision:
  - V(b, y) = max_{d ∈ {0,1}} {  Ṽ(1, b, y) + (1−d) Ṽ(0, b, y) }.
- Post-decision value:
  - Ṽ(d, b, y) = max_{b′ ≤ 0} { u(c) + β ∫ V(b′, y′) F(dy′ | y) }.
- Consumption identity:
  - c = y − d φ(y) + (1−d) b − q(b′, y) [ b′ − (1−d)(1−δ) b ].
- Bond price functional equation:
  - q(b′, y) = (1/(1 + r)) ∫ M(y′, y) [1 − h(b′′)] F(dy′ | y)
    + ((1−δ)/(1 + r)) ∫ M(y′, y) [1 − h(b′′)] q( g( h(b′, y′), b′, y′ ), y′ ) F(dy′ | y).
- Notation:
  - b: number of outstanding coupon claims at beginning of current period (b < 0 implies net issuer in past).
  - d ∈ {0,1}: default indicator.
  - h(b, y): default rule lenders expect (1 if default).
  - g(d, b, y): future borrowing rule lenders expect.

### A model without debt dilution — recursive formulation highlights
- If government issues ̃b − b′ > 0 bonds, compensation to prior holders is −̃b[q(̃b, y) − q(b′, y)].
- Equilibrium bond price decomposition (equations preserved):
  - First term: expected value of next-period coupon payment.
  - Second term: expected compensations bondholders receive if government issues new debt (ensures lenders price bonds anticipating no dilution).
  - Third term: expected next-period value of bond after compensation (next-period value can be affected by income shock, debt buyback, default, but not by new issuances).
- Government budget constraint without dilution (equation (9)):
  - c = y − dφ(y) + (1−d)b + q(b′, y)(̃b − b′) + ̃b max{0, q(̃b, y) − q(b′, y)}.
- Alternative expression (buyback interpretation, equation (10)):
  - c = y − dφ(y) + (1−d)b + ̃b q(̃b, y) − b′ q(b′, y).

### Numerical solution and simulation design
- Numerical method: value function iteration and interpolation (linear for endowment levels, spline for asset positions) solving for ̃V(1, b, y) and ̃V(0, b, y); convergence in q(b′, y) ensured.
- Simulation design:
  - Extract 500 samples of 32 consecutive periods before a default (pre-default samples) for comparison with Argentine data from Q4 1993 to Q3 2001.
  - Default frequencies computed using all simulation data.
  - Only sample paths where last default at least two periods before beginning of each sample considered.

### Welfare costs of debt-dilution
- Ex-ante welfare gains from moving from economy with dilution to one without dilution presented as consumption compensation (percentage terms).
  - Solid line: welfare gain when government enters period with zero debt.
  - Dashed line: welfare gain when government enters period with mean debt level observed in simulations with dilution.
- Domestic consumers are better off without dilution.
- Welfare gains tend to be lower when initial debt level is higher due to adjustment cost of transitioning to ergodic distribution with lower debt.
- Caveats that may understate welfare gains:
  - Calibration targets a relatively low debt level (28% of quarterly output) relevant for external debt held by foreigners; higher debt levels would increase gains from reducing risk premium.
  - Eliminating dilution may also reduce risk premia on other government and private debt.
  - Model lacks production; cannot capture productivity gains from lower level and volatility of interest rates or allocation effects.

### One-period bonds vs. long-duration bonds without debt dilution
- Comparison of bond-duration specifications without dilution:
  - Long-duration bonds: δ = 0.0341.
  - One-period bonds: δ = 1.
- Business-cycle statistics without debt dilution (Table 3 preserved values):
  - Defaults per 100 years: δ=0.0341 → 0.42; δ=1 → 0.23
  - Mean debt (market value): δ=0.0341 → 0.18; δ=1 → 0.19
  - Mean debt (face value): δ=0.0341 → 0.18; δ=1 → 0.19
  - E(Rs): δ=0.0341 → 0.57; δ=1 → 0.45
  - σ(Rs): δ=0.0341 → 0.72; δ=1 → 0.45
  - σ(y): δ=0.0341 → 3.36; δ=1 → 3.11
  - σ(c)/σ(y): δ=0.0341 → 1.21; δ=1 → 1.23
  - ρ(c, y): δ=0.0341 → 0.99; δ=1 → 0.97
  - ρ(Rs, y): δ=0.0341 → -0.63; δ=1 → -0.73
- Key point: one-period bonds and long-duration bonds without dilution produce different allocations — they are different assets.
- Mechanics: in long-bond economy without dilution, before issuing debt the government must buy back previous issuances at the market price that would be observed without new issuances, making financial needs lower when income is lower; with one-period bonds, obligations do not depend on income level.

### Broader implications, policy relevance, and extensions
- Debt dilution helps account for high and volatile interest rates and high debt levels in emerging economies.
- Connection to literature linking interest-rate level and volatility to amplification of shocks and crisis vulnerability (Neumeyer and Perri (2005); Uribe and Yue (2006); Mendoza and Yue (2008)).
- Policy implications:
  - Eliminating debt dilution should be a central issue in sovereign debt management and international financial architecture.
  - Implementing fiscal rules that commit to future borrowing behavior could exploit gains from eliminating dilution (references IMF (2009) on fiscal rules).
- Suggested model extensions:
  - Introduce production to capture effects of interest rates on factor allocation and productivity.
  - Allow sovereign to change debt duration.
  - Allow risk premium to be affected by variables other than domestic income.
  - Introduce additional shocks (these extensions require additional state variables and significant computation).
- Ongoing research: developing sovereign default frameworks that accommodate effects of interest rates on factor allocation (examples cited: Mendoza and Yue (2008); Sosa Padilla (2010)).

*Content derived from _wp1170 - References . . . . . . . . . . . . . . . . . . . . . . . . .*

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

### _wp1170 - References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

### Introduction — scope and motivation
- Study focus: effects of sovereign debt dilution caused by the government’s lack of commitment to avoid decreasing the value of debt issued in the past by issuing new debt.
- Context and literature: references and comparisons to Bizer and DeMarzo (1992), Bolton and Jeanne (2009), Bolton and Skeel Jr. (2005), Borensztein et al. (2004), Detragiache (1994), Eaton and Fernandez (1995), Kletzer (1984), Niepelt (2008), Sachs and Cohen (1982), Saravia (forthcoming), Tirole (2002), UN (2004), and others.
- Observations on market practice:
  - Corporate and household loans often show seniority; sovereign bonds typically do not show legal seniority.
  - Sovereign contracts commonly include pari passu clauses and negative pledge clauses; these prohibit future issuances of collateralized debt but do not protect against dilution from future borrowing behavior.
  - Weak enforcement in sovereign debt markets motivates proposals for more orderly sovereign debt restructurings.

### Novel modeling approach and contribution
- Modeling goal: provide a measure of the effects of debt dilution on levels of sovereign debt and default risk.
- Framework: a baseline sovereign default model `a la Eaton and Gersovitz (1981) with extensions used in recent studies; small open economy with stochastic endowment of a single tradable good.
- Government decisions (each period):
  - Decide whether to default on previously issued debt.
  - Decide how much to borrow or save by issuing non-contingent long-duration bonds (as in Hatchondo and Martinez (2009)).
- Default cost representation: an endowment loss incurred in the default period (quadratic loss function φ(y) = d0 y + d1 y^2 referenced later).
- Key features in the framework that generate inefficiencies:
  - Lack of commitment to future repayment policy.
  - Bond payments are not contingent on income shocks.
  - Government can borrow from multiple lenders and cannot commit not to decrease value of previously issued debt (debt dilution).

### Elimination of dilution — modeling device
- Proposed modification: government must pay existing bondholders a compensation equal to the reduction in the market value of their bonds caused by the government’s current debt issuance (no compensation for bond price declines not caused by new issuances).
- Economic consequence: investors buying sovereign debt anticipate that future value of their investment is independent of future issuances, thereby eliminating the dilution problem without increasing state-space dimensionality.

### Quantitative calibration targets and discipline
- Baseline model calibration targets:
  - Default probability.
  - Level of public external debt.
  - Debt duration.
  - Mean and standard deviation of the interest rate spread (sovereign bond yield minus the risk-free interest rate).

### Key quantitative findings (comparative: with dilution vs. without dilution)
- Default frequency (number of defaults per 100 years):
  - With debt dilution: 3.10
  - Without debt dilution: 0.42
  - Implied contribution of dilution to default risk in simulations: 86% (i.e., dilution accounts for 86% of the default risk)
- Mean interest rate spread:
  - With debt dilution: 7.38%
  - Without debt dilution: 0.57%
- Standard deviation of spread:
  - With debt dilution: 2.45
  - Without debt dilution: 0.72
- Mean debt levels:
  - Mean debt face value without dilution is 36% lower than with dilution.
  - Mean debt market value without dilution is 11% lower than with dilution.
- Mechanism emphasized:
  - Most important effect of dilution on default risk arises from a shift in the government’s set of borrowing opportunities (combinations of borrowing levels and spreads).
  - Equilibrium combinations of debt and spread levels without dilution are not part of the government’s choice set with dilution.
  - For equilibrium debt levels in the economy without dilution, equilibrium spread levels would be about 400 basis points higher in the economy with dilution (lenders anticipate future dilution raising default probabilities and thus demand higher spreads even at low debt levels).

### Broader implications and relevance
- Debt dilution helps account for high and volatile interest rates and high debt levels in emerging economies.
- Connection to literature on emerging market interest rates and business cycles: Neumeyer and Perri (2005), Uribe and Yue (2006), Mendoza and Yue (2008), and others who link level and volatility of interest rates to amplification of shocks and vulnerability to crises in emerging economies.
- Policy relevance: results support the view that debt dilution should be a central issue in discussions of sovereign debt management and the international financial architecture.

### Model structure — technical summary
- 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).
- Government objective:
  - Maximize E[∑_{t=0}^∞ β^t u(c_t)] with u(c) = c^{1−γ}/(1−γ).
- Bond structure:
  - Long-duration bonds promising infinite stream of coupons that decrease at constant rate δ.
  - Bond issued in period t pays 1 in t+1 and (1−δ)^{s−1} in t+s for s ≥ 2.
  - δ is a fixed parameter (not chosen by the government).
  - Macaulay duration formula noted: D = 1 + r^* / (δ + r^*), where r^* denotes constant per-period yield delivered by the bond (as stated).
- Risk premium and bond pricing:
  - Stochastic discount factor M(y′, y) = exp(−r − α ε′ − 0.5 α^2 σ_ε^2), with r the risk-free rate.
  - SDF formulation: discrete-time Vasicek one-factor model special case.
- Equilibrium concept:
  - Markov Perfect Equilibrium; solved by computing finite-horizon equilibria and extending horizon until convergence of first- and second-period objects to approximate infinite horizon.

### Recursive formulation (selected equations and definitions as provided)
- Government value before default decision: V(b, y) = max_{d ∈ {0,1}} {  Ṽ(1, b, y) + (1−d) Ṽ(0, b, y) }.
- Post-decision value: Ṽ(d, b, y) = max_{b′ ≤ 0} { u(c) + β ∫ V(b′, y′) F(dy′ | y) }.
- Consumption identity: c = y − d φ(y) + (1−d) b − q(b′, y) [ b′ − (1−d)(1−δ) b ].
- Bond price functional equation (components):
  - q(b′, y) = (1/(1 + r)) ∫ M(y′, y) [1 − h(b′′)] F(dy′ | y)
    + ((1−δ)/(1 + r)) ∫ M(y′, y) [1 − h(b′′)] q( g( h(b′, y′), b′, y′ ), y′ ) F(dy′ | y).
- Notation:
  - b: number of outstanding coupon claims at beginning of current period (b < 0 implies net issuer in past).
  - d ∈ {0,1}: current-period default indicator (1 if defaulted).
  - h(b, y): default rule lenders expect (1 if default).
  - g(d, b, y): future borrowing rule lenders expect (determines number of coupons maturing next period).
  - In equilibrium, the optimal default and borrowing rules that solve the Bellman problems must equal h(b, y) and g(d, b, y) for all state values.

*Italic source attribution: Content derived from _wp1170 - References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .*

### 1. a set of value functions

### 1. a set of value functions

### Value functions
- ̃
- V(d, b, y) andV(b, y),

### Decision rules
- a default ruleh(b, y) and a borrowing ruleg(d, b, y),

*Source: _wp1170 - 1. a set of value functions*

### 3. a bond price functionq(b

### _wp1170 - 3. a bond price functionq(b

### A. A model without debt dilution
- Modify baseline model so that each period the government compensates holders of previously issued debt for the decline in the market value of their holdings implied by new issuances (but does not compensate for declines resulting from income shocks).
- Purpose: eliminate debt dilution so investors anticipate that the future value of their investment is independent of future issuances.
- Implementation alternatives discussed (consent of existing bondholders, representative intermediaries, majority clauses, redemption rights, buybacks at market price if no new issuance).
- Interpretation: eliminating dilution equivalent to an exclusivity problem—if a single exclusive lender holds all debt, transfers between lender and government replicate the compensation mechanism described.

### D. Recursive formulation of the framework without debt dilution
- Definitions and notation preserved:
  - q(b′, y): bond price.
  - ̃b ≡ (1−d)(1−δ)b < 0: interim number of next-period coupon obligations.
  - If government issues ̃b − b′ > 0 bonds, compensation to prior holders is −̃b[q(̃b, y) − q(b′, y)].
  - Government buys back bonds at secondary price q(b′, y).
- Equilibrium bond price (equations preserved verbatim):
  - q(b′, y)  =
    1
    1 +r
    ∫
    M(y′,y)
    [
    1−h
    (
    b′
    , y′
    )
    ]
    F
    (
    dy′
    |y
    )
    (6)
    +
    1−δ
    1 +r
    ∫
    M(y′,
    y)
    [
    1−h
    (
    b′
    , y′
    )
    ]
    max
    {
    0, q(b′(1−δ), y′)−q(g(h(b′, y′), b′, y′), y′)
    }
    F
    (
    dy′
    |y
    )
    (7)
    +
    1−δ
    1 +r
    ∫
    M(y′,
    y)
    [
    1−h
    (
    b′
    , y′
    )
    ]
    q(g(h(b′, y′), b′, y′), y′)F
    (
    dy′
    |y
    )
    .(8)
  - Interpretation of terms:
    - First term (equation (8) first term): expected value of next-period coupon payment.
    - Second term: expected compensations bondholders receive if government issues new debt (ensures lenders price bonds anticipating no dilution).
    - Third term: expected next-period value of bond after compensation (next-period value can be affected by income shock, debt buyback, default, but not by new issuances).
- Government budget constraint (equation (9) preserved):
  - c = y − dφ(y) + (1−d)b + q(b′, y)(̃b − b′) + ̃b max{0, q(̃b, y) − q(b′, y)}.(9)
  - Last term: government’s compensation to existing bondholders for issuance of new debt.
- Replacing equations (3) and (5) by (8) and (9) in dynamic programming yields problem without debt dilution.

### E. Discussion of the framework without debt dilution
- Mechanisms to implement transfers that eliminate dilution:
  - Government obtains consent of existing bondholders in exchange for compensation.
  - Use of bondholder representative (Bolton and Jeanne (2009) or G-10 (2002) style) or majority clauses.
  - Right for bondholders to redeem at face value if government issues without consent.
  - Equivalence to buyback of previous issuances at market price that would prevail absent new issuance—budget constraint (10):
    - c = y − dφ(y) + (1−d)b + ̃b q(̃b, y) − b′ q(b′, y).(10)
- Nonexclusivity interpretation: resources obtained from borrowing while compensating existing bondholders equal amount an exclusive lender would accept to buy newly issued bonds.

### III. Results — Overview
- Purpose: compare predictions of models with and without debt dilution using Argentina as case study.
- Numerical solution method: value function iteration and interpolation (linear interpolation for endowment levels, spline for asset positions). Algorithm finds two value functions, ̃V(1, b, y) and ̃V(0, b, y); convergence in q(b′, y) assured.

### A. Calibration (Table 1 parameters preserved)
- Borrower’s risk aversion σ2 = 2
- Interest rate r1 = 1%
- Output autocorrelation coefficient ρ0.9
- Standard deviation of innovations σǫ2.7%
- Mean log output μ(-1/2)σ2ǫ (as specified in source)
- Duration δ0.0341
- Discount factor β0.969
- Default cost d0 -0.69
- Default cost d1 1.01
- Risk premium α4
- Remarks:
  - Period = quarter.
  - δ = 3.41% implies average bond duration of 4.19 years in baseline simulations.
  - Calibration targets: mean spread of 7.4, standard deviation of spread 2.5, mean debt level 28% of mean quarterly output in pre-default samples, default frequency of 3 defaults per 100 years.
  - Discount factor obtained is relatively low but higher than some prior studies (e.g., Aguiar and Gopinath (2006) β = 0.8).

### B. Simulation results — quantitative effects of debt dilution
- Simulation design:
  - Extract 500 samples of 32 consecutive periods before a default (pre-default samples) for comparison with Argentine data from Q4 1993 to Q3 2001.
  - Default frequencies computed using all simulation data.
  - Only sample paths where last default at least two periods before beginning of each sample considered.
- Table 2 business cycle statistics (data vs With dilution vs Without dilution) — preserved values:
  - Defaults per 100 years: Data 3.00; With debt dilution 3.10; Without debt dilution 0.42
  - Mean debt market value: Data 0.20; With 0.18; Without 0.18
  - Mean debt face value: Data 0.28; With 0.28; Without 0.18
  - E(Rs): Data 7.44; With 7.38; Without 0.57
  - σ(Rs): Data 2.51; With 2.45; Without 0.72
  - σ(y): Data 3.17; With 3.03; Without 3.36
  - σ(c)/σ(y): Data 0.94; With 1.04; Without 1.21
  - ρ(c, y): Data 0.97; With 1.00; Without 0.99
  - ρ(Rs, y): Data -0.65; With -0.80; Without -0.63
- Key quantitative findings:
  - Debt dilution accounts for 86% of default risk in baseline simulations (defaults per 100 years falls from 3.10 to 0.42).
  - Debt dilution accounts for 92% of the sovereign spread (mean spread falls from 7.38% to 0.57%).
  - Standard deviation of spread decreases from 2.45 to 0.72 when eliminating dilution.
  - Mean face value of outstanding bonds decreases by 36% without dilution; mean market value decreases by 11% (most decline driven by lower interest rates without dilution).
- Mechanism illustrated in Figure 1 (menu of spreads vs next-period debt b′/(δ+r)):
  - Eliminating dilution significantly shifts government's borrowing opportunity set toward combinations with much lower spreads.
  - With dilution, government cannot borrow paying spreads close to zero; without dilution, government can borrow near zero spreads.
  - For equilibrium debt levels without dilution, equilibrium spread levels would be about 400 basis points higher in economy with dilution.
  - Explanation: with dilution, lenders price in the government’s incentive to increase default probability via future issuances; without dilution, future issuances do not dilute lender value because compensation is paid.
- Consumption volatility:
  - Consumption volatility higher in economy without dilution (σ(c)/σ(y) increases from 1.04 to 1.21).
  - Reason: for low income realizations, issuance levels tend to be lower in economy without dilution, making consumption more sensitive to income changes; with dilution, government issues more and smooths consumption more.

### C. Welfare costs of debt-dilution
- Figure 2 presents ex-ante welfare gain (consumption compensation in percentage terms) from moving from economy with dilution to one without dilution.
  - Solid line: welfare gain when government enters period with zero debt.
  - Dashed line: welfare gain when government enters period with mean debt level observed in simulations with dilution.
  - Domestic consumers are better off without dilution.
  - Welfare gains tend to be lower when initial debt level is higher due to adjustment cost of transitioning to ergodic distribution with lower debt.
- Caveats and possible understatement of welfare gains:
  - Calibration targets a relatively low debt level (28% of quarterly output) relevant for external debt held by foreigners; higher debt levels would increase gains from reducing risk premium.
  - Eliminating dilution may also reduce risk premia on other government and private debt; sovereign spreads can influence private spreads.
  - Model lacks production; therefore cannot capture productivity gains from lower level and volatility of interest rates, or allocation effects emphasized in other literature.

### D. One-period bonds vs. long-duration bonds without debt dilution
- Comparison of model without dilution under two bond-duration specifications:
  - Long-duration bonds: δ = 0.0341.
  - One-period bonds: δ = 1.
- Table 3 business cycle statistics without debt dilution (preserved values):
  - Defaults per 100 years: δ=0.0341 → 0.42; δ=1 → 0.23
  - Mean debt (market value): δ=0.0341 → 0.18; δ=1 → 0.19
  - Mean debt (face value): δ=0.0341 → 0.18; δ=1 → 0.19
  - E(Rs): δ=0.0341 → 0.57; δ=1 → 0.45
  - σ(Rs): δ=0.0341 → 0.72; δ=1 → 0.45
  - σ(y): δ=0.0341 → 3.36; δ=1 → 3.11
  - σ(c)/σ(y): δ=0.0341 → 1.21; δ=1 → 1.23
  - ρ(c, y): δ=0.0341 → 0.99; δ=1 → 0.97
  - ρ(Rs, y): δ=0.0341 → -0.63; δ=1 → -0.73
- Key point: one-period bonds and long-duration bonds without dilution produce different allocations — they are different assets.
- Figure 3 (no-arbitrage spread curves and government optimal choices) highlights:
  - Response of equilibrium spread to negative income shock differs across bond-duration specifications.
  - For “very low” income (2 standard deviations below mean) equilibrium spread higher with long bonds; for “low” income (1 standard deviation below mean) equilibrium spread higher with one-period bonds.
  - For “low” income, government pays lower spread with long bonds because financial needs are weaker with long bonds.

*Italic: Source: _wp1170 - 3. a bond price functionq(b (excerpt).*

### Section 2.2.2, one may think about the economy with long bonds and without dilution as

### _wp1170 - Section 2.2.2, one may think about the economy with long bonds and without dilution as

### Long bonds versus one-period bonds: mechanics and implications
- In the economy with long bonds and without dilution, before issuing debt the government must buy back all previous issuances at the market price that would be observed without new issuances.
- In that economy, the government’s financial needs are lower when its income is lower because the market price of debt is lower when income is lower.
- In contrast, with one-period bonds, government’s obligations do not depend on the income level.
- In the economy with long bonds, for the “very low” income level, it does not pay off to issue new debt because the revenue raised by new debt issuances would not be enough to compensate existing bondholders.
  - Thus, the high spread observed when income is “very low” corresponds to previous-period issuances and it is not paid by the government for new issuances.
  - Footnote (text retained verbatim): “For sufficiently low income levels, initial debt levels in the simulations tend to be on the decreasing part of the sovereign-debt-market-value function−q(b′, y)b′. Thus, if the government chooses b′ < b(1−δ), its issuance revenue would be−q(b′, y)[b′−b(1−δ)], which is lower than the amount it would have to pay existing bondholders,−[q(b(1−δ), y)−q(b′, y)]b(1−δ).”
- With one-period bonds, when income is “very low” the government still issues debt because it has to pay all the debt it issued in the previous period (with long bonds it only has to pay δ= 3.41% of its debt).
- The government chooses to pay a spread lower than the high spread implied by previous-period issuances in the long-bond economy (the government also chooses not to pay this very high spread in the long-bond economy).

### Quantitative findings from the simulations: role of debt dilution
- Debt dilution accounts for:
  - 86% of the default risk.
  - 92% of the mean spread.
  - 71% of the spread volatility.
- Eliminating debt dilution substantially reduces default risk even without commitment to future repayment policies and without contingency of sovereign debt.
- The reduction in default risk from eliminating dilution reflects:
  - A sizable reduction in debt levels equal to 36% of the mean face value.
  - A reduction equal to 11% of the mean market value.
  - The most important effect is a shift in the government’s set of borrowing opportunities.

### Policy implications and extensions
- Analyzing debt dilution in a production economy would allow better quantification of the welfare gains from eliminating dilution.
  - Several studies find a significant effect of interest rates on productivity (through the allocation of factors of production) and of interest rate fluctuations in the amplification of shocks (examples cited in the source include Mendoza and Yue (2008), Neumeyer and Perri (2005), and Uribe and Yue (2006)).
  - The authors note that the welfare cost of debt dilution may be large and identify ongoing research to develop sovereign default frameworks that accommodate effects of interest rates on factor allocation (examples cited: Mendoza and Yue (2008) and Sosa Padilla (2010)).
- Other possible extensions discussed:
  - Allow the sovereign to change the debt duration.
  - Allow the risk premium to be affected by variables other than the domestic income.
  - Introduce other shocks to the economy.
  - The authors note these extensions would require including additional state variables and entail significant computation costs.
- Exploiting gains from eliminating dilution:
  - One could explore the benefits of committing to a future borrowing behavior through fiscal rules, which several countries are introducing (the source references IMF (2009) on fiscal rules).
  - The results indicate eliminating debt dilution should be an important motivation for implementing fiscal rules to reduce significantly the risk of debt crises and the mean and volatility of interest rates.

*Source: _wp1170 - Section 2.2.2, one may think about the economy with long bonds and without dilution as*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2011/_wp1170.pdf_