## wpiea2021230-print-pdf

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---

### I. Introduction and background
- State-contingent debt instruments (SCDIs) provide debt service relief in bad times and align debt service with payment capacity, reducing default probability and facilitating creditor coordination.
- Historical and literature context:
  - Early SCDI proposals arose in the 1980s (examples: Bailey (1983); Krugman (1988); Froot, Scharfstein, and Stein (1989); Besley and Powell (1989); Shiller (1993)).
  - Re-examined after increased global financial integration and sudden stops in the 1990s and the 2007 global financial crisis.
  - GDP-linked warrants introduced in restructurings: Argentina (2005), Greece (2012), Ukraine (2015).
- Prior findings summarized:
  - Borensztein and Mauro (2002), Braun and Tomassi (2003), Kim and Ostry (2018) analyze GDP-linked bonds and SCDIs for reducing explosive debt paths, creating fiscal space, and supporting debt sustainability.
  - Mallucci (2020) calibrated SCDIs for Caribbean economies, showing welfare improvements and higher sustainable debt capacity.
- Gap: prior work is ad-hoc about SCDI form and optimality; this paper derives a specific optimal SCDI form from a sovereign lending contracting problem.

### II. Main model setup and methodological innovation
- Economic environment:
  - Eaton-Gersovitz (1981) style economy with sovereign default and exclusion from financial markets upon default (financial autarchy).
  - Borrower lacks precommitment technology, generating time inconsistency in debt issuance.
  - Two agent types: a “poor” domestic population receiving government transfers (welfare function W0 = E0 sum_{t=0}^{\infty} β^{t} v(g_t) with lim_{g→0} v'(g)=+∞) and risk-neutral international investors with access to risk-free return r.
  - Output is stochastic i.i.d. with E_t y_{t+i} = ȳ, y ∈ Y = [y_l, y_u], y>0.
- Constraints and investor internalization:
  - Government participation/default constraint summarized as V(y_t,b_t) ≥ V_a (equations (7)-(10) in source).
  - Investors internalize the participation constraint and a promised-value constraint: E_0 sum_{t=0}^{\infty} β^{t} v(c_t) ≥ V_g.
- Non-recursivity and solution approach:
  - Participation constraint spans the entire future; standard recursive Bellman formulation fails.
  - Use recursive multipliers approach (Marcet and Marimon (2019)) to recover recursiveness via auxiliary state θ_t with θ_{t+1} = θ_t + γ_t and θ_0 = 0.

### III. Optimal sovereign debt contract: endowment economy (key analytical results)
- Equilibrium condition (from investor first-order / Kuhn-Tucker conditions):
  - 1 − (γ_t + θ_t + φ) v'(g_t) = ρ R E_t [ 1 − (γ_{t+1} + θ_t + γ_t + φ) v'(g_{t+1}) ]  (equation (17) in source).
- Propositions and implications:
  - Proposition 1: There is no default in equilibrium and the sovereign risk premium is equal to zero (R_t = 1 + r = R).
  - Proposition 2: The optimal sovereign debt contract offered by investors is state contingent.
  - Proposition 3: When neither the participation nor the promised-value constraint bind, dynamics equal the precommitment case: maximal expenditure smoothing and debt acts as shock absorber (Euler equation (12.2) holds).
  - Proposition 4: When the participation constraint binds (γ_t>0), investors provide net-positive financing to prevent default—additional debt issuance reduces current marginal utility by financing consumption today at the expense of future servicing.
  - Proposition 5: When the participation constraint binds, the contract prescribes a downward adjustment in primary expenditures (fiscal consolidation) so consumption decline is smaller than output decline—result is “partial insurance.”
  - Proposition 6: The sovereign’s stock of debt is lower than under full precommitment; lenders optimally keep indebtedness low ex ante to preserve capacity to provide net-positive financing when needed.
- Summary mechanics:
  - Optimal contract keeps all parties better off than autarchy, eliminates default in equilibrium, and provides full insurance against small shocks and partial insurance against large shocks.

### IV. Extension: production economy with capital investment (key additions)
- Production and fiscal setup:
  - Output y_t = β_t k_t^{α}, with β_t i.i.d., E_t β_{t+i} = β̄, α ∈ (0,1), β ∈ S = [β_l, β_u], k_t is tradable capital; government collects taxes τ_t announced one period in advance; government cannot invest.
  - Government budget: b_{t+1} = R_t b_t + g_t − τ_t y_t.
- Expanded time inconsistency:
  - Government may deviate from announced tax rates and can impose constraints on divestment/capital mobility in autarchy.
- Optimal contract conditions:
  - Equation (21.1) analogous to endowment Euler condition.
  - Equation (21.2) determines required expected return on capital r to match international risk-free return, internalizing government incentives to default and to move to autarchy.
- Propositions for production economy:
  - Proposition 7: If a defaulting government can appropriate returns on assets above announced tax rate, then capital stock is lower than with commitment.
  - Proposition 8: If default constraint binds at time t, investors optimally increase capital investment to raise expected tax revenue and government expenditure next period, keeping government engaged.
  - Proposition 9: The stock of capital is increasing in expected terms due to investors’ anticipatory increases when default risk exists.
- Net effect:
  - Required concurrent financing to avoid default is lower than in endowment economy because additional future output/tax revenue can partly compensate current welfare losses.

### V. Policy implications, instrument design, and market considerations
- Consolidated characteristics of the optimal SCDI-compatible contract:
  - (i) No sovereign default in equilibrium.
  - (ii) No sovereign risk premia; borrowing interest rates remain at international “risk-free” level.
  - (iii) Full insurance via debt issuance for relatively small shocks when participation constraint does not bind.
  - (iv) Net-positive financing is provided when default constraint binds to keep debtor engaged and avoid creditor asset loss.
  - (v) Issuance of SCDIs is optimal at all times, not only during restructurings.
  - (vi) Sovereign interest rate is not state contingent.
  - (vii) Fiscal consolidation is required when default constraint binds; adjustment is minimal to keep participation.
  - (viii) Sovereign must maintain lower debt levels compared to full commitment case.
- Instrument design guidance:
  - SCDIs could include automatic positive net-financing triggers under large, exogenous, recurrent, verifiable shocks (e.g., natural disasters, pandemics).
  - Contingent clauses could trigger for sufficiently large current account shocks and/or sudden stops that are demonstrably part of a global trend.
  - Triggers could be conditioned on verifiable pre-conditions such as a sustainable fiscal/external position assessed by an independent institution (example in text: International Monetary Fund) or participation in a Fund-supported program committing to a sustainable macro framework.
- Critique of pro-cyclical indexation proposals:
  - Pro-cyclical indexation to output/exports/terms of trade/government revenue is suboptimal because it transfers shock and debt-service uncertainty to creditors, preserves default possibility and sovereign risk premia, and prevents maximal expenditure smoothing in good states.
  - By contrast, the optimal contract eliminates default in equilibrium, keeping spreads at zero and allowing maximum smoothing.
- Market constraints and practical adoption:
  - Shock verifiability and transparency are key for market development; SCDIs are more demanding on these fronts than non-contingent instruments.
  - Investor liquidity risk and a coordination “chicken and egg” problem remain barriers: a sufficiently large mass of these bonds, issued by many sovereign borrowers, is needed for appropriate diversification by international investors.
  - Concerted efforts by financial asset trading authorities could coordinate issuance to achieve critical mass and improve tradeability and liquidity.
  - Participation of institutional investors with lower liquidity risk and longer horizons (pension and insurance funds) could facilitate market emergence.

### VI. Model assumptions, theoretical notes, and appendices (technical)
- Assumption on investors:
  - Investors have “deep pockets” and face no liquidity constraints; they can invest and sell assets at will in international financial markets at the risk free international interest rate. This is a simplifying stylized assumption, not a feature of real life where liquidity constraints (e.g., margin calls) matter (footnote 17).
- Theoretical note on low market acceptability:
  - Roch and Roldan (2021) explain low market acceptability via model uncertainty faced by “robust” international creditors; robust investors attach high probability to relatively more pessimistic outcomes, implying SCDIs can be optimal for issuers but come at a premium due to robust investors’ discounting—reducing SCDI desirability when calibrated to the data. Their explanation omits consideration of preference for liquidity typical of financial institutions.
- Appendices (high-level technical content):
  - Appendix 1: Transformation of the Lagrange function for an endowment economy; infinite summation on participation constraint rewritten and simplified, imposing θ_0 = 0 to obtain transformed function H.
  - Appendix 2: Derivation of optimal sovereign debt contract for an endowment economy; Kuhn-Tucker conditions (A2.1)–(A2.6), state/co-state evolution θ_{t+1} = θ_t + γ_t with θ_0 = 0, and combination yielding the optimal recursive contract (equation (17)).
  - Appendix 3: Optimal tax rate under precommitment yields τ_{t+1} = 1 − α = τ and condition 1 − (γ_t + θ_t + φ) v'(g_t) = ρ R E_t [1 − (γ_{t+1} + θ_{t+1} + φ) v'(g_{t+1})] (A2.7).
  - Appendix 4: Optimal investment rules with production; Lagrange function (A4.1), transformed maximand H, Kuhn-Tucker conditions (A4.2)–(A4.11), core investment and lending rules (A4.6, A4.7), equilibrium conditions (A4.12, A4.13), and expression linking r in (A4.14).
  - Appendix 5: Detailed transformation to obtain H by expanding the participation constraint across t = 0, 1, 2, ... and simplifying under θ_0 = 0.

*Source: wpiea2021230-print-pdf - introduction of SDCIs. A key assumption behind this result in the stylized model in this*

### CONCLUSION __________________________________________________________________ 19

### CONCLUSION __________________________________________________________________ 19

### VI .
- VI .

*https://www.imf.org/-/media/files/publications/wp/2021/english/wpiea2021230-print-pdf.pdf*

### REFERENCES__________________________________________________________________ 29

### wpiea2021230-print-pdf - REFERENCES__________________________________________________________________ 29

### I. Introduction and background
- State-contingent debt instruments (SCDIs) provide debt service relief in bad times and align debt service with payment capacity, reducing default probability and facilitating creditor coordination.
- Historical and literature context:
  - Early SCDI proposals arose in the 1980s (example: Bailey (1983); Krugman (1988); Froot, Scharfstein, and Stein (1989); Besley and Powell (1989); Shiller (1993)).
  - Re-examined after increased global financial integration and sudden stops in the 1990s and the 2007 global financial crisis.
  - GDP-linked warrants introduced in restructurings: Argentina (2005), Greece (2012), Ukraine (2015).
- Prior findings summarized:
  - Borensztein and Mauro (2002), Braun and Tomassi (2003), Kim and Ostry (2018) analyze GDP-linked bonds and SCDIs for reducing explosive debt paths, creating fiscal space, and supporting debt sustainability.
  - SCDIs have been used for natural-disaster relief with parametric triggers; Mallucci (2020) calibrated SCDIs for Caribbean economies, showing welfare improvements and higher sustainable debt capacity.
- Gap in literature: prior work is ad-hoc about SCDI form and optimality; this paper derives a specific optimal SCDI form from a sovereign lending contracting problem.

### II. Main model setup and methodological innovation
- Economic environment:
  - Eaton-Gersovitz (1981) style economy with sovereign default and exclusion from financial markets upon default (financial autarchy).
  - Borrower lacks precommitment technology, generating time inconsistency in debt issuance.
  - Two agent types: a “poor” domestic population receiving government transfers (welfare function W0 = E0 sum_{t=0}^{\infty} β^{t} v(g_t) with lim_{g→0} v'(g)=+∞) and risk-neutral international investors with access to risk-free return r.
  - Output is stochastic i.i.d. with E_t y_{t+i} = ȳ, y ∈ Y = [y_l, y_u], y>0.
- Participation/default constraint for government (at each period t):
  - E_t [ sum_{n=0}^{∞} β^{n} v(g_{t+n}) ] ≥ E_t [ sum_{n=0}^{∞} β^{n} v(y_{t+n}) ]  (equations (7)-(10)), summarized as V(y_t,b_t) ≥ V_a.
- Investors internalize:
  - The participation constraint and a promised-value constraint: E_0 sum_{t=0}^{∞} β^{t} v(c_t) ≥ V_g.
- Non-recursivity and solution approach:
  - Participation constraint spans the entire future; standard recursive Bellman formulation fails.
  - Use recursive multipliers approach (Marcet and Marimon (2019)) to recover recursiveness via auxiliary state θ_t (θ_{t+1} = θ_t + γ_t, θ_0 = 0).

### III. Optimal sovereign debt contract: endowment economy (key analytical results)
- Equilibrium characterization (investor first-order / Kuhn-Tucker conditions lead to equation (17)):
  - 1 − (γ_t + θ_t + φ) v'(g_t) = ρ R E_t [ 1 − (γ_{t+1} + θ_t + γ_t + φ) v'(g_{t+1}) ]
- Propositions and implications:
  - Proposition 1: There is no default in equilibrium and the sovereign risk premium is equal to zero (R_t = 1 + r = R).
  - Proposition 2: The optimal sovereign debt contract offered by investors is state contingent.
  - Proposition 3: When neither the participation nor the promised-value constraint bind, dynamics equal the precommitment case: maximal expenditure smoothing and debt acts as shock absorber (Euler equation (12.2) holds).
  - Proposition 4: When the participation constraint binds (γ_t>0), investors provide net-positive financing to prevent default—additional debt issuance reduces current marginal utility by financing consumption today at the expense of future servicing.
  - Proposition 5: When the participation constraint binds, the contract prescribes a downward adjustment in primary expenditures (fiscal consolidation) so consumption decline is smaller than output decline—result is “partial insurance.”
  - Proposition 6: The sovereign’s stock of debt is lower than under full precommitment; lenders optimally keep indebtedness low ex ante to preserve capacity to provide net-positive financing when needed.
- Summary mechanics:
  - Optimal contract keeps all parties better off than autarchy, eliminates default in equilibrium, and provides full insurance against small shocks and partial insurance against large shocks.

### IV. Extension: production economy with capital investment (key additions)
- Production setup:
  - Output y_t = β_t k_t^{α}, with β_t i.i.d., E_t β_{t+i} = β̄, α ∈ (0,1), β ∈ S = [β_l, β_u], k_t is tradable capital; government collects taxes τ_t announced one period in advance; government cannot invest.
  - Government budget with taxes: b_{t+1} = R_t b_t + g_t − τ_t y_t.
- Time inconsistency expands to taxation:
  - Government may deviate from announced tax rates and can impose constraints on divestment/capital mobility in autarchy.
- Optimal contract conditions (special case with financial and trade autarchy) include:
  - Equation (21.1) analogous to endowment Euler condition.
  - Equation (21.2) determining required expected return on capital r to match international risk-free return, including terms that internalize government incentives to default and to move to autarchy.
- Propositions for production economy:
  - Proposition 7: If a defaulting government can appropriate returns on assets above announced tax rate, then capital stock is lower than with commitment.
  - Proposition 8: If default constraint binds at time t, investors optimally increase capital investment to raise expected tax revenue and government expenditure next period, keeping government engaged.
  - Proposition 9: The stock of capital is increasing in expected terms due to investors’ anticipatory increases when default risk exists.
- Net effect: required concurrent financing to avoid default is lower than in endowment economy because additional future output/tax revenue can partly compensate current welfare losses.

### V. Policy implications, instrument design, and market considerations
- Characteristics of the optimal SCDI-compatible contract (consolidated):
  - (i) No sovereign default in equilibrium.
  - (ii) No sovereign risk premia; borrowing interest rates remain at international “risk-free” level.
  - (iii) Full insurance via debt issuance for relatively small shocks when participation constraint does not bind.
  - (iv) Net-positive financing is provided when default constraint binds to keep debtor engaged and avoid creditor asset loss.
  - (v) Issuance of SCDIs is optimal at all times, not only during restructurings.
  - (vi) Sovereign interest rate is not state contingent.
  - (vii) Fiscal consolidation is required when default constraint binds; adjustment is minimal to keep participation.
  - (viii) Sovereign must maintain lower debt levels compared to full commitment case.
- Instrument design guidance:
  - SCDIs could include automatic positive net-financing triggers under large, exogenous, recurrent, verifiable shocks (e.g., natural disasters, pandemics).
  - Contingent clauses could trigger for sufficiently large current account shocks and/or sudden stops that are demonstrably part of a global trend.
  - Triggers could be conditioned on verifiable pre-conditions such as a sustainable fiscal/external position assessed by an independent institution (example in text: International Monetary Fund) or participation in a Fund-supported program committing to a sustainable macro framework.
- Critique of pro-cyclical indexation proposals:
  - Pro-cyclical indexation of debt service to output/exports/terms of trade/government revenue is suboptimal because:
    - It transfers some shock and debt-service uncertainty to creditors, preserving default possibility and sovereign risk premia.
    - It prevents maximal expenditure smoothing in good states; additional borrowing to stabilize consumption today raises future default risk and increases risk premia, yielding only partial insurance.
  - By contrast, the optimal contract eliminates default in equilibrium, keeping spreads at zero and allowing maximum smoothing.
- Market constraint and practical adoption:
  - To be macroeconomically relevant, SCDIs must be issued at sufficient critical mass; market reluctance exists due to concerns about complexity, transparency, and liquidity.
  - Despite theoretical creditor-side optimality, real-world uptake has been limited.

*Source: wpiea2021230-print-pdf - REFERENCES__________________________________________________________________ 29*

### introduction of SDCIs. A key assumption behind this result in  the  stylized  model  in  this

### introduction of SDCIs. A key assumption behind this result in the stylized model in this

### Assumption: investors with “deep pockets” and no liquidity constraints
- The model assumes investors are endowed with “deep pockets” and face no liquidity constraints: they can invest and sell assets at will in international financial markets at the risk free international interest rate.
- The assumption is acknowledged as a simplifying feature of the stylized model, and not a feature of real life where liquidity constraints matter (for example, by the activation of margin calls). (footnote 17 referenced)

### Implications for the development of a market for SCDIs
- Shock verifiability and transparency are identified as key for market development; SCDIs are by design more demanding on these fronts than non-contingent instruments.
- Market development supports:
  - Design of benchmark instruments.
  - Robust market infrastructure.
- Remaining barrier: investor liquidity risk and a potential investor coordination, or “chiken and egg”, problem:
  - A sufficiently large mass of these bonds, issued by many sovereign borrowers, is needed for appropriate diversification by international investors.
  - No country nor investor can achieve this market breadth in isolation.
  - A concerted effort led by financial asset trading authorities in major international financial markets could coordinate issuance to allow a critical mass of creditor/investors to hold SCDIs, increasing global tradeability and liquidity.
- Participation of institutional investors that face less liquidity risk and are relatively more concerned with long-term capital preservation (such as pension and insurance funds) could facilitate market emergence for SCDIs.

### Theoretical note on low market acceptability
- Footnote reference: Roch and Roldan (2021) offer an explanation for low market acceptability for SCDIs based on model uncertainty faced by “robust” international creditors; robust investors attach high probability to relatively more pessimistic outcomes, implying SCDIs can be optimal for issuers but come at a premium due to robust investors’ discounting—reducing SCDI desirability when calibrated to the data. Their explanation omits consideration of preference for liquidity typical of financial institutions (banks, credit unions, securities dealers).

### Appendix 1 — Transformation of the Lagrange function for an endowment economy (high-level)
- Lagrange function for foreign investors given by expression (A1.1) with:
  - Infinite summation on participation constraint rewritten and simplified (A1.2).
  - Imposing 휃휃0 = 0 yields reformulation of the participation constraint (A1.3) that can replace the corresponding term in L to obtain transformed function H.

### Appendix 2 — Derivation of optimal sovereign debt contract for an endowment economy (high-level)
- Modified Lagrange function H recalled (reference to equation (16) in text).
- Kuhn-Tucker conditions for the investor problem under participation constraints are provided, including:
  - Condition (A2.1): 1 − 휆휆푡 ≤ 0 when c_t > 0.
  - Condition (A2.2): −휆휆푡 + 휌휌(1 + r) E_t 휆휆_{t+1} ≤ 0 when x_{t+1} > 0.
  - Condition (A2.3): −휆휆푡 + (훾훾푡 + 휃휃푡 + 휑휑) v′(g_t) + 휌휌 E_t R[ ... ] ≤ 0 when b_{t+1} > 0.
  - Participation constraint form (A2.4) and normalization (A2.5).
  - Budget constraint equality (A2.6).
- State and co-state evolution: 휃휃_{t+1} = 휃휃_t + 훾훾_t with 휃휃0 = 0.
- Combination of these interior-solution conditions yields the optimal recursive sovereign debt contract (same as equation (17) in text).

### Appendix 3 — Optimal tax rate under precommitment
- Government chooses tax rate to maximize expected tax revenues next period with constraint linking tax, capital, and (1 + r).
- First order condition provided; algebraic manipulation yields pre-announced tax rate:
  - 휏휏_{t+1} = 1 − 훼훼 = 휏휏
- Additional condition linking multipliers and expectation given by (A2.7):
  - 1 − (훾훾_t + 휃휃_t + 휑휑) v′(g_t) = 휌휌 R E_t [1 − (훾훾_{t+1} + 휃휃_{t+1} + 휑휑) v′(g_{t+1})].

### Appendix 4 — Optimal investment rules in an economy with production (high-level)
- Lagrange function for investor problem given by (A4.1) with added capital dynamics and multiplier μ_t.
- Transformed recursive maximand H constructed analogously to Appendix 1 (see appendix 5).
- Government transfers evolve according to:
  - b_{t+1} = R_t b_t + g_t + τ_t β_t k_t^α
- Kuhn-Tucker conditions include:
  - State/co-state evolution: 휃휃_{t+1} = 휃휃_t + 훾훾_t and 휃휃0 = 0 (A4.2–A4.3).
  - Conditions (A4.4)–(A4.11) set interiority and complementary slackness for c_t, x_{t+1}, k_{t+1}, and participation constraints.
  - Core investment and lending rules summarized in (A4.6) and (A4.7).
  - Equilibrium conditions (A4.12) and (A4.13), together with 휃휃_{t+1} = 휃휃_t + 훾훾_t, characterize the optimal joint debt and investment contract.
  - A further expression for r appears in (A4.14) linking expectations of returns, multipliers, and derivatives of v and V.

### Appendix 5 — Transformation details to obtain H
- Repeats transformation of infinite summation on participation constraint, showing expansion across t = 0, 1, 2, ... and simplification under 휃휃0 = 0.
- Resulting rewriting of participation constraint as sums involving 훾훾_t and 휃휃_t terms that feeds into the transformed Lagrange function H.

*Italicized source: wpiea2021230-print-pdf - introduction of SDCIs. A key assumption behind this result in the stylized model in this (canonical source PDF).*

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_Source: https://www.imf.org/-/media/files/publications/wp/2021/english/wpiea2021230-print-pdf.pdf_