## Equilibrium with bilateral loans only; Model with bilateral loans and defaultable market debt; Welfare effects of bilateral loans

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### Model setup and solution method (Bilateral loans only)
- Economy: small open economy borrowing exclusively from a monopolist large lender; endowment stream y(z) with state z following an AR(1) process.
- Loans: short-term and effectively continuously renegotiated each period.
- Bargaining: Nash bargaining with lender bargaining power parameter θ. Default/repayment threat point is payment of full amount m.
- Transfers and implicit interest rate: transfer x and new loan size m′ solve the Nash bargaining problem; implicit interest rate r satisfies x = (1/(1 + r)) m′ − m.
- Value functions:
  - v(m,z) = u(y(z) + x(m,z)) + β E[v(m′(m,z), z′) | z]
  - h(m,z) = a − x(m,z) + β_L E[h(m′(m,z), z′) | z]
- Normalization: a = 0, so h(m,z) equals the expected present discounted value of transfers (lender profits).
- Parametrization used to illustrate forces: θ = 0.5; β = β_L.

### Equilibrium lending terms and pricing dynamics (Bilateral loans only)
- Borrower behavior:
  - Deleverages in high-income states and receives positive transfers in low-income states.
- Interest-rate pattern:
  - When both debt m and income z are low, the lender offers subsidized and even negative interest rates.
  - When loan size is large, the lender charges much higher interest rates.
- Mechanism:
  - Subsidized lending at low m induces higher indebtedness, weakening the borrower’s threat point and increasing surplus to be split; once indebtedness is large, the lender extracts profits via higher rates.
- Endogenous features:
  - Convexity in the lender’s value function h(m,z) as indebtedness increases, implying risk-seeking (gambling-for-rescue) incentives by the risk-neutral lender.
  - Anticipation of future high rates disciplines the initial subsidy through the division of surplus determined by θ.

### Simulated dynamics and comparative bargaining power (Bilateral loans only)
- Simulated path (θ = 0.5):
  - Loans are subsidized while debt accumulates; once debt has accumulated, interest rates rise to extract profits; borrower anticipates and internalizes this dynamic.
- Comparative extreme (θ = 0):
  - When the borrower holds all bargaining power (take-it-or-leave-it), it borrows at rate β_L^{-1} at all times; rates do not increase when loans are large and cannot be negative when small, recovering an income-fluctuations problem at the risk-free rate without default.

### Key qualitative takeaways (Bilateral loans only)
- Bargained interest rates can increase strongly when the borrower’s repayment threat point is weak (low income z and high indebtedness m).
- Anticipated future profits create lender surplus today and induce lower interest rates initially, supporting accumulation of larger loan balances m.
- Similar patterns would arise even if default on the bilateral loan were possible, so long as the borrower’s threat-point value decreases with indebtedness m.

---

### Model setup and timeline (Bilateral + defaultable market debt)
- Government owes m to a large lender, b to the market, and observes exogenous state z at start of period t. Economy can be in default (ζ = D) or in repayment (ζ = R).
- Government’s and large lender’s value functions: v(b, m, z) and h(b, m, z) when in good standing with the market.
- Default decision in the morning incorporates Type 1 Extreme Value shocks:
  - v(b, m, z) = max{ v_R(b, m, z) + ε_R, v_D(m, z) + ε_D }.
  - Closed forms:
    - v(b, m, z) = χ log( exp(v_D(m, z)/χ) + exp(v_R(b, m, z)/χ) )
    - P(b, m, z) = exp(v_D(m, z)/χ) / ( exp(v_D(m, z)/χ) + exp(v_R(b, m, z)/χ) )
- Default on market debt fully destroys all bonds outstanding and triggers exclusion from international capital markets. Reaccess to markets is stochastic with constant probability ψ.
- Bilateral loans m cannot be defaulted. While in default on market debt, the maximum bilateral loan is constrained by m′ ≤ Γ(m, z). Baseline sets Γ(m, z) = +∞; variant Γ(m, z) = 0 (Limited) considered.

### Private market debt issuance and pricing
- If no default is chosen, government issues new market debt b′ anticipating bargaining with the monopolist:
  - v_R(b, m, z) = max_b′ w_R(b′, b, m, z).
- Market debt is a perpetuity of geometrically-decaying coupons; parameters κ and δ govern coupon size and decay.
- Debt price:
  - q(b′, b, m, z) = 1/(1 + r⋆) E[ (1 − 1_D(b′, m′, z′)) ( κ + (1 − δ) q(b′′, b′, m′, z′) ) | z ]
  - r⋆ is the international risk-free rate.
  - 1_D(·) denotes perceived default policy by competitive lenders.
  - m′ = m′_R(b′, b, m, z) is the expected result of negotiations with the large lender.

### Bilateral loan bargaining (afternoon)
- Nash bargaining determines transfer x and new loan size m′ depending on default status:
  - max_{m′, x} ℒ_R(...)^θ ℬ_R(...)^{1−θ} in repayment
  - max_{m′, x} ℒ_D(...)^θ ℬ_D(...)^{1−θ} in default
- Large lender’s surplus:
  - ℒ_R(b′, x, m, m′, z) = −x − m + β_L E[ h(b′, m′, z′) − h(b′, 0, z′) | z ]
  - ℒ_D(x, m, m′, z) = −x − m + β_L E[ ψ( h(0, m′, z′) − h(0, 0, z′) ) + (1 − ψ)( h_D(m′, z′) − h_D(0, z′) ) | z ]
- Borrower’s surplus:
  - ℬ_R(...) = u( y(z) + B(b′, b, m, z) + x ) − u( y(z) + B(b′, b, m, z) − m ) + β E[ v(b′, m′, z′) − v(b′, 0, z′) | z ]
  - ℬ_D(...) = u( y_D(z) + x ) − u( y_D(z) − m ) + β E[ ψ( v(0, m′, z′) − v(0, 0, z′) ) + (1 − ψ)( v_D(m′, z′) − v_D(0, z′) ) | z ]
  - y_D(z) = y(z) − ξ(z)
  - B(b′, b, m, z) ≡ q(b′, b, m, z)( b′ − (1 − δ) b ) − κ b
- Bargaining yields loan/transfer functions:
  - { x_R(b′, b, m, z), m′_R(b′, b, m, z) } in repayment
  - { x_D(m, z), m′_D(m, z) } in default

### Consumption and value updates
- Consumption after negotiation:
  - c_R(b′, b, m, z) = y(z) + B(b′, b, m, z) + x_R(b′, b, m, z)
  - c_D(m, z) = y(z) − ξ(z) + x_D(m, z)
- Values:
  - w_R(b′, b, m, z) = u(c_R(b′, b, m, z)) + β E[ v( b′, m′_R(b′, b, m, z), z′ ) | z ]
  - v_D(m, z) = u(c_D(m, z)) + β E[ ψ v(0, m′_D(m, z), z′) + (1 − ψ) v_D(m′_D(m, z), z′) | z ]
- Large lender value:
  - h(b, m, z) = P(b, m, z) h_D(m, z) + (1 − P(b, m, z)) h_R(b′(b, m, z), b, m, z)
  - h_R(b′, b, m, z) = a − x_R(b′, b, m, z) + β_L E[ h( b′, m′_R(b′, b, m, z), z′ ) | z ]
  - h_D(m, z) = a − x_D(m, z) + β_L E[ ψ h(0, m′_D(m, z), z′) + (1 − ψ) h_D( m′_D(m, z), z′ ) | z ]

### Quantitative setup (parametrization)
- Quarterly frequency; calibration choices described (most parameters taken from calibration to the 2001 Argentina default in Roch and Roldán,2023).
- BASELINE PARAMETER VALUES:
  - Sovereign’s discount factor β 0.9504
  - Sovereign’s risk aversion γ 2
  - Preference shock scale parameter χ 0.025
  - Lender’s bargaining power θ 0.5
  - Risk-free interest rate r∗ 0.01
  - Duration of debt δ 0.05
  - Income autocorrelation coefficient ρ_z 0.9484
  - Standard deviation of y_t σ_z 0.02
  - Reentry probability ψ 0.0385
  - Default cost: linear d_0 -0.24
  - Default cost: quadratic d_1 0.3
- κ is set to r∗ + δ so that price of debt equals 1 when government repays with certainty.

### Main quantitative findings (business-cycle statistics)
- Simulation setup: statistics based on model simulations of 35 quarters before a default on market debt; “loans” refers to bilateral loans m; welfare gains reported as equivalent consumption increases, computed averaging ergodic distribution of equilibrium with only market debt, conditional on repayment.
- Table 2 — BUSINESS CYCLE STATISTICS WITH AND WITHOUT BILATERAL LOANS (Columns: Only market | Unrestricted, θ = 0.25 | Unrestricted, θ = 0.5)
  - Avg spread (bps): 7141,6132,105
  - Std spread (bps): 3999271,331
  - σ(c)/σ(y) (%): 113 109 109
  - Debt to GDP (%): 22.5 21.7 21.2
  - Loan to GDP (%): 0 3.4 3.0
  - Loan spread (bps): – -52.5 -429
  - Corr. loan & spreads (%): – 61.7 67.5
  - Default frequency (%): 5.7 21 113
  - Welfare gains (rep): – -0.15% -0.43%
- Interpretation:
  - Availability of bilateral loans increases default frequency and leads to higher and more volatile spreads despite slightly lower market debt and modest bilateral borrowing.
  - With θ = 0.25, government prefers equilibrium without the large lender.
  - Typical simulation (conditional on no default): bilateral borrowing is 3.3% of annual income with standard deviation 1.6%.
  - Bilateral loans are heavily used around default events; the large lender subsidizes initial accumulation with negative interest rates and later raises them as default approaches.
  - When market access is recovered, government immediately issues market debt to pay off bilateral loan.

### Limited lending variant (Γ(m, z) = 0)
- Bilateral loans only while in good standing.
- Table 3 — BUSINESS CYCLE STATISTICS WITH BILATERAL LOANS (Only market | Unrestricted, θ = 0.5 | Limited, θ = 0.5)
  - Avg spread (bps): 7142,1051,038
  - Std spread (bps): 3991,331612
  - σ(c)/σ(y)(%): 113 109 113
  - Debt to GDP (%): 22.5 21.2 22.5
  - Loan to GDP (%): 0 3.0 2.1
  - Loan spread (bps): – -429 536
  - Corr. loan & spreads (%): – 67.5 71.1
  - Default frequency (%): 5.7 21 37.7
  - Welfare gains (rep): – -0.43% -0.2%
- Interpretation:
  - Limiting bilateral loans during default reduces average usage by more than two-thirds and lowers welfare losses relative to Unrestricted case, but Unrestricted case produces larger welfare losses.

### Default probabilities, prices, and distributions
- Ex-post default regions (when m = 0):
  - Unrestricted bilateral loans increase default as government can sustain lower debt levels.
  - Limited loans yield default policies virtually unchanged from no-lender case.
- Debt prices:
  - Higher default probability under Unrestricted loans translates into lower prices q.
  - Limited variant mutes but does not eliminate price reductions; prices remain lower relative to no-lender model, especially when debt is low.
- Ergodic distribution of debt-to-GDP:
  - With bilateral loans (both variants), economy spends more time in regions with large default risk, conditional on repayment.
- Monopolist’s profit/value h(b, m=0, z at mean):
  - Profits increase in market debt b when debt is low and one-period-ahead default probability remains contained.
  - In Limited variant, profits decrease as default becomes likely (bilateral lending must be repaid at face value upon default).
  - In Unrestricted variant, profits increase sharply with higher default because large payments can be extracted during default spells.

### Dynamics and mechanisms: relational overborrowing and Euler equation
- Relational overborrowing:
  - Government chooses market debt b′ before bargaining. If negotiations fail, fallback consumption is y(z) + q(b′, b, m, z)( b′ − (1 − δ) b ) − κ b − m.
  - When government issues low b′, its threat point is weak; transfers x from large lender are valuable and require high implicit interest rates. Conversely, high b′ yields better bargaining terms.
  - Interest rate on bilateral loan responds aggressively to b′.
- Modified Euler equation for market debt b′ (new terms from bilateral lender):
  - u′(c)( q + ∂q/∂b′ · i + ∂x/∂b′ ) = β E[ u′(c′)(1 − 1_D)( κ + (1 − δ) q′ − i′( ∂q′/∂b − ∂q′/∂m · ∂m′/∂b′ ) − ∂x′/∂b + ∂x′/∂m · ∂m′/∂b′ ) ]
  - where i = b′ − (1 − δ) b, and derivatives ∂q′/∂b, ∂q′/∂m, ∂x′/∂b, ∂x′/∂m capture how current issuance affects future prices and transfers via the large lender.
- Three new effects relative to only-market-debt world:
  1. Issuing debt affects current-period transfers x received from large lender.
  2. Issuing debt affects desired future bilateral borrowing m′, which influences future debt prices q′ and transfers x′ (chain-rule effects).
  3. Issuing debt affects future price q′ and transfers x′ through dependence on initial indebtedness b.

---

### Welfare effects and value function (Section 5.3)
- Government’s value function v(b,m,z) (Figure 12) as a function of debt b when it owes m = 0 to the large lender shows the government prefers bilateral loans to be Unavailable during default, except when a default in the current period is very likely.
- Forces combine to produce welfare effects through a relational overborrowing effect that incentivizes risk-taking in debt issuance.

### Programming the large lender — exogenous rules
- Replace bargaining with fixed rules:
  - r(b′, m′) = max { r∗, α0 + αb b′ + αm m′ }
  - Bilateral loan always costs weakly more than r∗; large lender makes non-negative profits.
- Two classes of rules:
  - Risk-inducing rule: α0 > 0, αb < 0, αm = 0 — replicates equilibrium with bargaining and induces relational overborrowing.
  - Size-dependent rule: α0 > 0, αb = 0, αm > 0 — does not load on indebtedness in markets and can reduce default and spreads.
- Objectives: test whether rules can curb sovereign default risk, create welfare gains for the government, and deliver Pareto improvements.

### Quantitative outcomes with exogenous rules (Table 4)
- Table 4: OUTCOMES WITH EXOGENOUS RULES FOR BILATERAL LOANS (Columns: Only market | Size dependent r | Risk inducing r | Limited, θ = 0.5)
  - Avg spread (bps): 714 | 623 | 921 | 1,038
  - Std spread (bps): 399 | 315 | 552 | 612
  - σ(c)/σ(y)(%): 113 | 115 | 115 | 113
  - Debt to GDP (%): 22.5 | 23.5 | 22.8 | 22.5
  - Loan to GDP (%): 0 | 0.7 | 10.9 | 21.0
  - Loan spread (bps): – | 6821,264536
  - Corr. loan & spreads (%): –62.5 | 48.1 | 71.1
  - Default frequency (%): 5.7 | 25.1 | 36.9 | 27.7
  - Welfare gains (rep): –0.21% | 0.079% | –0.2%
- Notes:
  - Statistics from simulations of 35 quarters before a default on market debt.
  - ‘Limited’ means bilateral loan balances must be 0 while in default.
  - ‘Size dependent r’ increases bilateral interest rate with bilateral loan balance; ‘Risk inducing r’ decreases bilateral interest rate with the amount of market debt.
  - Default frequencies computed on the ergodic distribution.
  - Welfare gains are equivalent consumption increases over the ergodic distribution of the model without bilateral loans, conditional on repayment.

### Key mechanism and interpretation (Exogenous rules)
- Risk-inducing rule generates dynamics similar to bargaining equilibrium, producing welfare losses for the government through relational overborrowing.
- Size-dependent rule provides an example of a pricing schedule that lowers default and spreads, improving government welfare while generating profits for the large lender.
- Relational overborrowing arises from an endogenous cross-elasticity of bilateral borrowing terms to outcomes in debt markets, which erodes market discipline embodied in spreads.
- Three key ingredients producing the cross-elasticity in the bargaining model:
  - Loans from the large lender are more costly (or impossible) to default.
  - Terms result from bargaining.
  - Bilateral loans are of shorter maturity than marketable debts.
- The exogenous-terms model shows the cross-elasticity alone suffices to incentivize overborrowing, higher sovereign risk, and lower welfare.

### Policy-relevant findings and recommendations
- Increasing the number of sources of indebtedness is not necessarily beneficial for the borrowing government; bilateral loans can both help fend off default or make default more likely.
- Limiting the use of bilateral debt during defaults is a clear welfare-enhancing policy in the model.
- Fiscal rules that constrain market borrowing yield larger gains for countries with access to the type of bilateral debts described, because relational overborrowing operates through increased default risk.
- Simple test implied by the model:
  - Bilateral loans whose interest rate is expected to be strongly decreasing in the amount (or spreads) of marketable debt will induce relational overborrowing and are thus likely to hurt welfare.
- The model highlights benefits of transparent rules for terms of bilateral debts and provides examples (e.g., size-dependent rules) that can improve government welfare without losses for the large lender.

### Additional illustrative results
- When the borrower holds all bargaining power, the loan interest rate is constant at β^{-1}_L; in the quantitative version with β < β_L and θ = 0 the borrower prioritizes bilateral loans and quickly reaches upper bounds.
- Loans around defaults: conditioning on an exclusion period of 2 years, the economy issues market debt to pay off the bilateral loan as soon as it recovers market access.

*Source: The Perils of Bilateral Sovereign Debt — Working Paper No. WP/2025/235*

### 3.1   Equilibrium with bilateral loans only. . . . . . . . . . . . . . . . . . . . . . . . .  10

### 3.1   Equilibrium with bilateral loans only

### Model setup and solution method
- Economy: small open economy borrowing exclusively from a monopolist large lender; endowment stream y(z) with state z following an AR(1) process.
- Loans: short-term and effectively continuously renegotiated each period.
- Bargaining: Nash bargaining with lender bargaining power parameter θ. Default/repayment threat point is payment of full amount m.
- Transfers and implicit interest rate: transfer x and new loan size m′ solve the Nash bargaining problem; implicit interest rate r satisfies x = (1/(1 + r)) m′ − m.
- Value functions: borrower value v(m,z) and lender value h(m,z) satisfy
  - v(m,z) = u(y(z) + x(m,z)) + β E[v(m′(m,z), z′) | z]
  - h(m,z) = a − x(m,z) + β_L E[h(m′(m,z), z′) | z]
- Normalization: a = 0, so h(m,z) equals the expected present discounted value of transfers (lender profits).
- Parametrization used to illustrate forces: θ = 0.5; β = β_L (to isolate consumption smoothing and bargaining from front-loading motives).

### Equilibrium lending terms and pricing dynamics
- Borrower behavior: deleverages in high-income states and receives positive transfers in low-income states.
- Interest-rate pattern:
  - When both debt m and income z are low, the lender offers subsidized and even negative interest rates.
  - When loan size is large, the lender charges much higher interest rates.
- Mechanism: subsidized lending at low m induces higher indebtedness, which weakens the borrower’s threat point and increases surplus to be split; once indebtedness is large, the lender extracts profits via higher rates.
- Endogenous features:
  - Convexity in the lender’s value function h(m,z) as indebtedness increases, implying risk-seeking (gambling-for-rescue) incentives by the risk-neutral lender.
  - Anticipation of future high rates disciplines the initial subsidy through the division of surplus determined by θ.

### Simulated dynamics and comparative bargaining power
- Simulated path (θ = 0.5): loans are subsidized while debt accumulates; once debt has accumulated, interest rates rise to extract profits; borrower anticipates and internalizes this dynamic.
- Comparative extreme (θ = 0): when the borrower holds all bargaining power (take-it-or-leave-it), it borrows at rate β_L^{-1} at all times; rates do not increase when loans are large and cannot be negative when small, recovering an income-fluctuations problem at the risk-free rate without default.

### Key qualitative takeaways from the bilateral-only model
- Bargained interest rates can increase strongly when the borrower’s repayment threat point is weak (low income z and high indebtedness m).
- Anticipated future profits create lender surplus today and induce lower interest rates initially, supporting accumulation of larger loan balances m.
- Similar patterns would arise even if default on the bilateral loan were possible, so long as the borrower’s threat-point value decreases with indebtedness m.

*Source: IMF working paper chapter section 3.1, “Equilibrium with bilateral loans only.”*

### 4.  MODEL WITH BILATERAL LOANS AND DEFAULTABLE MARKET DEBT

### 4.  MODEL WITH BILATERAL LOANS AND DEFAULTABLE MARKET DEBT

### Model setup and timeline
- Government owes m to a large lender, b to the market, and observes exogenous state z at start of period t. Economy can be in default (ζ = D) or in repayment (ζ = R).
- Government’s and large lender’s value functions: v(b, m, z) and h(b, m, z) when in good standing with the market.
- Default decision in the morning incorporates Type 1 Extreme Value shocks:
  - v(b, m, z) = max{ v_R(b, m, z) + ε_R, v_D(m, z) + ε_D }.
  - Closed forms:
    - v(b, m, z) = χ log( exp(v_D(m, z)/χ) + exp(v_R(b, m, z)/χ) )
    - P(b, m, z) = exp(v_D(m, z)/χ) / ( exp(v_D(m, z)/χ) + exp(v_R(b, m, z)/χ) )
- Default on market debt fully destroys all bonds outstanding and triggers exclusion from international capital markets. Reaccess to markets is stochastic with constant probability ψ.
- Bilateral loans m cannot be defaulted. While in default on market debt, the maximum bilateral loan is constrained by m′ ≤ Γ(m, z). Baseline analysis sets Γ(m, z) = +∞, with a variant Γ(m, z) = 0 (Limited) considered.

### Private market debt issuance and pricing
- If no default is chosen, government issues new market debt b′ to competitive lenders anticipating the bargaining outcome with the monopolist:
  - v_R(b, m, z) = max_b′ w_R(b′, b, m, z).
- Market debt is a perpetuity of geometrically-decaying coupons; parameters κ and δ govern coupon size and decay.
- Debt price faced by borrower:
  - q(b′, b, m, z) = 1/(1 + r⋆) E[ (1 − 1_D(b′, m′, z′)) ( κ + (1 − δ) q(b′′, b′, m′, z′) ) | z ]
  - r⋆ is the international risk-free rate.
  - 1_D(·) denotes perceived default policy by competitive lenders.
  - m′ = m′_R(b′, b, m, z) is the expected result of negotiations with the large lender.

### Bilateral loan bargaining (afternoon)
- Nash bargaining problems determine transfer x and new loan size m′ depending on default status:
  - max_{m′, x} ℒ_R(...)^θ ℬ_R(...)^{1−θ} in repayment
  - max_{m′, x} ℒ_D(...)^θ ℬ_D(...)^{1−θ} in default
- Large lender’s surplus:
  - ℒ_R(b′, x, m, m′, z) = −x − m + β_L E[ h(b′, m′, z′) − h(b′, 0, z′) | z ]
  - ℒ_D(x, m, m′, z) = −x − m + β_L E[ ψ( h(0, m′, z′) − h(0, 0, z′) ) + (1 − ψ)( h_D(m′, z′) − h_D(0, z′) ) | z ]
- Borrower’s surplus (reflects market outcomes via B(b′, b, m, z)):
  - ℬ_R(...) = u( y(z) + B(b′, b, m, z) + x ) − u( y(z) + B(b′, b, m, z) − m ) + β E[ v(b′, m′, z′) − v(b′, 0, z′) | z ]
  - ℬ_D(...) = u( y_D(z) + x ) − u( y_D(z) − m ) + β E[ ψ( v(0, m′, z′) − v(0, 0, z′) ) + (1 − ψ)( v_D(m′, z′) − v_D(0, z′) ) | z ]
  - y_D(z) = y(z) − ξ(z)
  - B(b′, b, m, z) ≡ q(b′, b, m, z)( b′ − (1 − δ) b ) − κ b
- Bargaining yields loan/transfer functions:
  - { x_R(b′, b, m, z), m′_R(b′, b, m, z) } in repayment
  - { x_D(m, z), m′_D(m, z) } in default

### Consumption and value updates
- Consumption after negotiation:
  - c_R(b′, b, m, z) = y(z) + B(b′, b, m, z) + x_R(b′, b, m, z)
  - c_D(m, z) = y(z) − ξ(z) + x_D(m, z)
- Values:
  - w_R(b′, b, m, z) = u(c_R(b′, b, m, z)) + β E[ v( b′, m′_R(b′, b, m, z), z′ ) | z ]
  - v_D(m, z) = u(c_D(m, z)) + β E[ ψ v(0, m′_D(m, z), z′) + (1 − ψ) v_D(m′_D(m, z), z′) | z ]
- Large lender value:
  - h(b, m, z) = P(b, m, z) h_D(m, z) + (1 − P(b, m, z)) h_R(b′(b, m, z), b, m, z)
  - h_R(b′, b, m, z) = a − x_R(b′, b, m, z) + β_L E[ h( b′, m′_R(b′, b, m, z), z′ ) | z ]
  - h_D(m, z) = a − x_D(m, z) + β_L E[ ψ h(0, m′_D(m, z), z′) + (1 − ψ) h_D( m′_D(m, z), z′ ) | z ]

### Quantitative setup (parametrization)
- Quarterly frequency; calibration choices described (most parameters taken from calibration to the 2001 Argentina default in Roch and Roldán,2023).
- Table 1 — BASELINE PARAMETER VALUES:
  - Sovereign’s discount factor β 0.9504
  - Sovereign’s risk aversion γ 2
  - Preference shock scale parameter χ 0.025
  - Lender’s bargaining power θ 0.5
  - Risk-free interest rate r∗ 0.01
  - Duration of debt δ 0.05
  - Income autocorrelation coefficient ρ_z 0.9484
  - Standard deviation of y_t σ_z 0.02
  - Reentry probability ψ 0.0385
  - Default cost: linear d_0 -0.24
  - Default cost: quadratic d_1 0.3
- κ is set to r∗ + δ so that price of debt equals 1 when government repays with certainty.

### Main quantitative findings (business-cycle statistics)
- Table 2 — BUSINESS CYCLE STATISTICS WITH AND WITHOUT BILATERAL LOANS (statistics based on model simulations of 35 quarters before a default on market debt; “loans” refers to bilateral loans m; welfare gains reported as equivalent consumption increases, computed averaging ergodic distribution of equilibrium with only market debt, conditional on repayment):
  - Columns: Only market | Unrestricted, θ = 0.25 | Unrestricted, θ = 0.5
  - Avg spread (bps) 7141,6132,105
  - Std spread (bps) 3999271,331
  - σ(c)/σ(y) (%) 113 109 109
  - Debt to GDP (%) 22.5 21.7 21.2
  - Loan to GDP (%) 0 3.4 3.0
  - Loan spread (bps) – -52.5 -429
  - Corr. loan & spreads (%) – 61.7 67.5
  - Default frequency (%) 5.7 21 113
  - Welfare gains (rep) – -0.15% -0.43%
- Interpretation from Table 2:
  - Availability of bilateral loans increases default frequency and leads to higher and more volatile spreads despite slightly lower market debt and modest bilateral borrowing.
  - With θ = 0.25, government prefers equilibrium without the large lender.
  - Typical simulation (conditional on no default): bilateral borrowing is 3.3% of annual income with standard deviation 1.6% (textual summary).
  - Bilateral loans are heavily used around default events; the large lender subsidizes initial accumulation with negative interest rates and later raises them as default approaches.
  - When market access is recovered, government immediately issues market debt to pay off bilateral loan.

- Limited lending variant (Γ(m, z) = 0) — bilateral loans only while in good standing:
  - Table 3 — BUSINESS CYCLE STATISTICS WITH BILATERAL LOANS (Only market | Unrestricted, θ = 0.5 | Limited, θ = 0.5):
    - Avg spread (bps) 7142,1051,038
    - Std spread (bps) 3991,331612
    - σ(c)/σ(y)(%) 113 109 113
    - Debt to GDP (%) 22.5 21.2 22.5
    - Loan to GDP (%) 0 3.0 2.1
    - Loan spread (bps) – -429 536
    - Corr. loan & spreads (%) – 67.5 71.1
    - Default frequency (%) 5.7 21 37.7
    - Welfare gains (rep) – -0.43% -0.2%
  - Limiting bilateral loans during default reduces average usage by more than two-thirds and lowers welfare losses relative to Unrestricted case, but Unrestricted case produces larger welfare losses.

### Default probabilities, prices, and distributions
- Ex-post default regions (Figure 7) for private debt when m = 0:
  - Solid: no bilateral loans; dotted/dashed: Unrestricted and Limited bilateral loans.
  - Unrestricted loans increase default as government can sustain lower debt levels.
  - Limited loans yield default policies virtually unchanged from no-lender case.
- Debt prices (Figure 8):
  - Higher default probability under Unrestricted loans translates into lower prices q.
  - Limited variant mutes but does not eliminate price reductions; prices remain lower relative to no-lender model, especially when debt is low.
- Ergodic distribution of debt-to-GDP (Figure 9):
  - With bilateral loans (both variants), economy spends more time in regions with large default risk, conditional on repayment.
- Monopolist’s profit/value h(b, m=0, z at mean) (Figure 10):
  - Profits increase in market debt b when debt is low and one-period-ahead default probability remains contained.
  - In Limited variant, profits decrease as default becomes likely (bilateral lending must be repaid at face value upon default).
  - In Unrestricted variant, profits increase sharply with higher default because large payments can be extracted during default spells.

### Dynamics and mechanisms: relational overborrowing and Euler equation
- Relational overborrowing:
  - Government chooses market debt b′ before bargaining. If negotiations fail, borrower’s fallback consumption is y(z) + q(b′, b, m, z)( b′ − (1 − δ) b ) − κ b − m.
  - When government issues low b′, its threat point is weak; transfers x from large lender are valuable and require high implicit interest rates. Conversely, high b′ yields better bargaining terms.
  - Interest rate on bilateral loan responds aggressively to b′ (Figure 11).
- Modified Euler equation for market debt b′ (new terms from bilateral lender):
  - u′(c)( q + ∂q/∂b′ · i + ∂x/∂b′ ) = β E[ u′(c′)(1 − 1_D)( κ + (1 − δ) q′ − i′( ∂q′/∂b − ∂q′/∂m · ∂m′/∂b′ ) − ∂x′/∂b + ∂x′/∂m · ∂m′/∂b′ ) ]
  - where i = b′ − (1 − δ) b, and derivatives ∂q′/∂b, ∂q′/∂m, ∂x′/∂b, ∂x′/∂m capture how current issuance affects future prices and transfers via the large lender.
- Three new effects relative to only-market-debt world:
  1. Issuing debt affects current-period transfers x received from large lender.
  2. Issuing debt affects desired future bilateral borrowing m′, which influences future debt prices q′ and transfers x′ (chain-rule effects).
  3. Issuing debt affects future price q′ and transfers x′ through dependence on initial indebtedness b.

*Source: wpiea2025235-source-pdf - 4.  MODEL WITH BILATERAL LOANS AND DEFAULTABLE MARKET DEBT.*

### 5.3  Welfare effects of bilateral loans

### 5.3  Welfare effects of bilateral loans

### Welfare effects and value function
- The government’s value function v(b,m,z) (Figure 12) as a function of debt b when it owes m = 0 to the large lender shows the government prefers bilateral loans to be Unavailable during default, except when a default in the current period is very likely.
- The forces analyzed combine to produce welfare effects through a relational overborrowing effect that incentivizes risk-taking in debt issuance.

### Programming the large lender — model of exogenous rules
- Bilateral loan terms are replaced by fixed rules of the form:
  - r(b′, m′) = max { r∗, α0 + αb b′ + αm m′ }
  - With these rules the government faces the exogenous interest rate r(b′, m′) on bilateral debt when choosing market debt b′ and bilateral loan m′.
  - Because the bilateral loan always costs weakly more than the risk-free rate r∗, the large lender makes non-negative profits.
- Two classes of rules considered:
  - Risk-inducing rule: α0 > 0, αb < 0, αm = 0 — replicates main properties of the equilibrium with bargaining and induces relational overborrowing.
  - Size-dependent rule: α0 > 0, αb = 0, αm > 0 — does not load on indebtedness in markets and can reduce default and spreads.
- Objectives: understand relational overborrowing dynamics in a simpler setting; explore design questions—can rules curb sovereign default risk, create welfare gains for the government, and deliver Pareto improvements where both government and large lender benefit?

### Quantitative outcomes with exogenous rules (Table 4)
- Table 4: OUTCOMES WITH EXOGENOUS RULES FOR BILATERAL LOANS
  - Columns: Only market | Size dependent r | Risk inducing r | Limited, θ = 0.5
  - Avg spread (bps): 714 | 623 | 921 | 1,038
  - Std spread (bps): 399 | 315 | 552 | 612
  - σ(c)/σ(y)(%): 113 | 115 | 115 | 113
  - Debt to GDP (%): 22.5 | 23.5 | 22.8 | 22.5
  - Loan to GDP (%): 0 | 0.7 | 10.9 | 21.0
  - Loan spread (bps): – | 6821,264536
  - Corr. loan & spreads (%): –62.5 | 48.1 | 71.1
  - Default frequency (%): 5.7 | 25.1 | 36.9 | 27.7
  - Welfare gains (rep): –0.21% | 0.079% | –0.2%
- Notes on Table 4 (as reported):
  - Statistics are based on model simulations of 35 quarters before a default on market debt.
  - ‘Only market’ refers to a model version without bilateral loans.
  - ‘Limited’ refers to the version in which bilateral loan balances must be 0 while in default.
  - Middle columns denote versions with exogenous terms for the bilateral loan: ‘Size dependent r’ has the bilateral interest rate increase with the balance in the bilateral loan, while ‘Risk inducing r’ has the bilateral interest rate decrease with the amount of market debt.
  - Default frequencies are computed on the ergodic distribution.
  - Welfare gains are given as equivalent increases in consumption calculated over the ergodic distribution of the model without bilateral loans, conditional on repayment.

### Key mechanism and interpretation
- The risk-inducing rule generates dynamics similar to the bargaining equilibrium, producing welfare losses for the government through relational overborrowing.
- The size-dependent rule provides an example of a pricing schedule that lowers default and spreads, improving government welfare while generating profits for the large lender.
- The relational overborrowing effect arises from an endogenous cross-elasticity of bilateral borrowing terms to outcomes in debt markets, which erodes market discipline embodied in spreads.
- Three key ingredients producing the cross-elasticity in the bargaining model:
  - Loans from the large lender are more costly (or impossible) to default.
  - Terms result from bargaining.
  - Bilateral loans are of shorter maturity than marketable debts.
- The simpler exogenous-terms model shows the cross-elasticity by itself suffices to incentivize overborrowing, higher sovereign risk, and lower welfare.

### Policy-relevant findings and recommendations
- Increasing the number of sources of indebtedness is not necessarily beneficial for the borrowing government; bilateral loans can both help fend off default or make default more likely.
- Limiting the use of bilateral debt during defaults is a clear welfare-enhancing policy in the model.
- Fiscal rules that constrain market borrowing yield larger gains for countries with access to the type of bilateral debts described, because relational overborrowing operates through increased default risk.
- Simple test implied by the model for practical assessment:
  - Bilateral loans whose interest rate is expected to be strongly decreasing in the amount (or spreads) of marketable debt will induce relational overborrowing and are thus likely to hurt welfare.
- The model highlights benefits of transparent rules for terms of bilateral debts and provides examples (e.g., size-dependent rules) that can improve government welfare without losses for the large lender.

### Additional illustrative results (Figures 13–14)
- When the borrower holds all bargaining power, the loan interest rate is constant at β−1_L; in the quantitative version with β < β_L and θ = 0 the borrower prioritizes bilateral loans and quickly reaches upper bounds.
- Loans around defaults: conditioning on an exclusion period of 2 years, the economy issues market debt to pay off the bilateral loan as soon as it recovers market access (Figure 14 shows loan size and interest rate dynamics around default events).

*Source: The Perils of Bilateral Sovereign Debt — Working Paper No. WP/2025/235*

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_Source: https://www.imf.org/-/media/files/publications/wp/2025/english/wpiea2025235-source-pdf.pdf_