## _wp0499

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### I. INTRODUCTION — key messages
- The “solvent but illiquid” debtor framework explains debt crises as loss of access to credit that can be self-fulfilling through creditor panic.
- If inability to roll over debt is the sole reason investors refuse to lend, simple voluntary-participation market mechanisms can in theory eliminate self-fulfilling liquidity runs.
- Coordination mechanisms proposed:
  - Allow new debt to be issued through mechanisms where investors condition participation on a large enough amount being successfully issued (contingent bids / underwriting analog).
  - Issue state-contingent securities that force payoffs (large enough to force default) if targeted debt reduction by a deadline is not achieved, otherwise pay zero.
- Implication: debt crises attributed solely to intra-borrower investor panic are less robust, but economy-wide coordination failures (e.g., “twin crises”, balance sheet mismatches) may remain immune to these mechanisms.
- Policy implication: the maturity of public debt should not have first-order effects on vulnerability to investor-panic runs that are purely intra-borrower coordination failures; short-term debt remains undesirable for reasons unrelated to liquidity runs.

### II. THE BASIC ENVIRONMENT — model structure and critical conditions
- Economy and preferences:
  - Small open economy with a continuum of measure one of identical, infinitely lived residents.
  - Preferences: additively separable, instantaneous utility u(), continuous and strictly concave, discount factor β.
- Production and default cost:
  - Output if never defaulted: Y_t = A_t f(I_{t-1}), with A_t i.i.d. productivity shock, f' > 0, f'' < 0.
  - Output after default: α A_t f(I_{t-1}), with α < 1 (reputational/output cost of default taken as given).
- Debt structure and timing:
  - Foreign debt stock B_t; one-period maturity bonds issued each period.
  - World risk-free rate = 1/β; investors risk neutral, atomistic, wealth potentially ω_{j,t}.
  - Country announces B_{t+1}; investors submit bids; proceeds p_t B_{t+1} used to repay B_t.
- Key value functions and constraints (notation preserved):
  - Default value:
    - V_D(Y_t) = max_{I_t} { u(Y_t − I_t) + β E_t[ V_D(α A_{t+1} f(I_t)) ] }.
  - Repay value:
    - V_R(Y_t,B_t) = max_{B_{t+1}} { u(Y_t + p_t B_{t+1} − B_t − I_t) + β E_t[ max[ V_R(A_{t+1} f(I_t),B_{t+1}), V_D(A_{t+1} f(I_t)) ] ] } subject to (2) incentive compatibility for I_t, (3) participation p_t = β Pr_t( V_R(A_{t+1} f(I_t),B_{t+1}) ≥ V_D(A_{t+1} f(I_t)) ), and (4) lim_{s→∞} β^{s−t} E_t[B_s] = 0.
- Critical region for liquidity crisis:
  - Liquidity-constrained value when unable to borrow at t:
    - V_R^{LC}(Y_t,B_t) = max_{I_t} { u(Y_t − B_t − I_t) + β E_t V_R(A_{t+1} f(I_t),0) }.
  - Region of interest (Equation (5)):
    - V_R(Y_t,B_t) ≥ V_D(Y_t) > V_R^{LC}(Y_t,B_t).
  - Lemma 1 interpretation: B_t is low enough that the country prefers to repay when it can roll over enough of it, yet large enough that it will default if forced to repay entire stock without new financing.

### III. COORDINATION PROBLEMS — results and mechanisms
- Coordination failure results:
  - If every investor expects others not to buy new bonds, expectations can be self-fulfilling and trigger default (Proposition 1).
  - Intertemporal coordination failure: if investors today expect future investors not to lend and that implies default on debt offered today, today’s investors refuse to hold debt; no B_{t+1} can satisfy the necessary conditions and default follows (Proposition 2).
  - Participation/pricing when future access is expected to be lost: constraints (6), (7), (8) link B_{t+1}, I_t, and actuarially fair price p_t = β Pr_t( V_R^{LC}(A_{t+1} f(I_{t+1}),B_{t+1}) ≥ V_D(A_{t+1} f(I_{t+1})) ).
- Intra-period coordination mechanisms:
  - Direct underwriting by an investment bank (guarantee to hold unsold portion) can eliminate run risk, subject to scale considerations for large sovereign issues.
  - Contingent bidding mechanism: investors submit bids p_{j,t}(b_{j,t+1},B_{t+1}) contingent on the total issue size B_{t+1}; country selects bids to satisfy participation constraints.
  - Minimum issue B_{t+1} (bar) solves country’s indifference and incentive conditions so issuing at least that amount prevents default.
  - Equilibrium bidding strategy (as in source):
    - For B_{t+1} ≥ B_{t+1}: p_{j,t}(b_{j,t+1},B_{t+1}) = β Pr_t( V_R(Y_{t+1},B_{t+1}) ≥ V_D(Y_{t+1}) ), b_{j,t+1} = ω_{j,t} / p_{j,t}.
    - For B_{t+1} < B_{t+1}: p_{j,t}(·) = 0; country rejects p_{j,t} = 0 bids.
  - Lemma 2: allowing contingent bids prevents a rational investor from refusing to lend because she expects others not to.
  - Lemma 3: extension to step-function bids around a threshold; country sets threshold at optimal B_{t+1}.
  - Lemma 4: an auction that cancels and fully refunds bids if resulting issue B_{t+1} < B_{t+1} achieves the same anti-run feature.
  - Practical note: similar mechanisms used by start-up firms and underwriting markets.
- Inter-period coordination mechanisms:
  - Commitment to lower debt below the inter-period crisis-prone range by time T > t or default eliminates inter-period coordination failure via backward induction.
  - Lemma 5: under the forced path (debt must exit crisis-prone region by T or default), an investor at t cannot rationally refuse to lend because she expects investors at t+1 not to — coordination risk removed and borrowing pricing reflects fundamentals only.
  - Stronger commitment: full repayment by date T (B_{T+1} = 0) discussed; associated value V_R^{LC,T}(Y_t,B_t,T) defined as solution of maximand (1) subject to (2),(3),(4) and (9) B_{T+1} = 0.
- Residual risk:
  - Even after eliminating coordination risk, crises driven by fundamentals (deterioration so debt unsustainable at actuarially fair rate) remain possible.

### IV. KEY LEMMAS ON COMMITMENT, TRANSVERSALITY, AND STATE-CONTINGENT SECURITIES
- Transversality condition (4) implication:
  - For any ε, there exists S large enough such that β^{s−t} B_s ≤ ε for all s ≥ S.
  - Thus, the discounted value at time t of the time T+1 debt becomes arbitrarily small as T becomes arbitrarily large.
- Lemma 6 (preference to repay when repayment can be spread):
  - If V_R(Y_t, B_t) > V_D(Y_t), then V_R^{LC,T}(Y_t, B_t, T) ≥ V_R(Y_t, B_t + ε) > V_D(Y_t) for large enough T.
  - Therefore, if the country strictly prefers to repay its debt to default when it can spread repayments across time, it strictly prefers a plan lowering time T debt below the threshold for which an inter-period coordination failure can occur than to default, provided T is large enough.
- Lemma 7 (self-imposed constraint B_{T+1} = 0 via state-contingent securities):
  - The country can issue securities that pay at time T: zero if B_{T+1} = 0 and a very large amount if B_{T+1} > 0.
  - If payments in states where B_{T+1} > 0 are large enough to force default in those states, no investor will hold the country’s debt at time T because such claims would be defaulted on with certainty—effectively imposing B_{T+1} = 0.
  - A pari passu clause can be included on the state-contingent security contract to prevent future lenders from being senior to holders of that security.

### V. PROPOSITIONAL RESULT, ROBUSTNESS, AND LIMITS
- Proposition 3:
  - If the country strictly prefers to repay its debt when it can spread repayments across time, self-fulfilling liquidity crises caused by a coordination failure among investors cannot occur in equilibrium.
  - Logical chain summarized:
    - Lemma 2: contingent bidding eliminates intra-period coordination failure.
    - Lemma 5: inter-period coordination failure eliminated if constraint B_{T+1} = 0 is imposed.
    - Lemma 7: B_{T+1} = 0 can be self-imposed by issuing state-contingent securities.
    - Lemma 6: issuing such securities is preferable to default for large enough T.
  - Conclusion: an investor at time t will be willing to lend since the country will take measures to ensure success of a second bond offer at time t and subsequent periods even if other investors do not.
- Robustness to informational asymmetries and higher-order beliefs:
  - Informational asymmetries: a failed bond offer may lead investors to update beliefs about fundamentals, raising required premia; for credit to fully dry up, perceived fundamentals must deteriorate enough that no lending can be sustained at the actuarially fair rate.
  - Investors typically know whether they did not buy bonds due to coordination concerns or fundamentals concerns, reducing scope for deterioration; liquidity crises are ruled out provided fundamentals are initially strong enough.
  - Higher-order beliefs: intra-period contingent bidding removes strategic uncertainty by allowing investors to condition decisions on an outcome mapping into others' strategies, making it robust to higher-order belief issues; inter-period solution can be compromised if there is no common knowledge that all investors are forward-looking.
- Limits and aggregate externalities:
  - Balance sheet and contract-enforcement externalities can create strategic complementarities across creditors to different borrowers; economy-wide externalities may prevent implementation of the mechanisms.
  - Investors must be able to condition lending on aggregate lending to the home economy to eliminate intra-period coordination failures—difficult if investors move simultaneously.
  - Inter-period mechanism may not be implementable if small private investors do not internalize aggregate externalities; free-rider problems can arise.
  - Government role:
    - Government could internalize externalities by consolidating private liabilities and using the Section III mechanisms to coordinate toward the good equilibrium.
    - Government issuance to take over private liabilities could be offset with collections from rescued borrowers in the inter-temporal budget constraint.
    - Government willingness to repay can be affected if rescuing private borrowers increases gross public liabilities beyond thresholds for strategic sovereign default; both gross and net liabilities matter.
    - If government cannot borrow against future claims without crossing the default threshold, intervention cannot remove investor coordination failure in a decentralized economy.

### VI. CONCLUSIONS AND BROADER IMPLICATIONS
- Market mechanisms that alter the game to eliminate creditor coordination failures can prevent self-fulfilling debt runs at the level of an individual (aggregate) borrower.
- Self-fulfilling features can remain important in economy-wide coordination failures, which may be difficult to overcome with the proposed mechanisms.
- Many small illiquid borrowers can make an economy more vulnerable to investor panics than a single aggregate illiquid borrower.
- At the individual borrower level, short-term debt per se may be less risky than often suggested because mechanisms can eliminate roll-over risk; short-term debt remains undesirable for other reasons, including vulnerability to changes in roll-over terms.
- Potential application to financial innovation:
  - New instruments can face market thinness and multiplicity of equilibria in participation.
  - The contingent bidding mechanism could allow investors to price bids according to expected market thickness associated with issue sizes, possibly enabling efficient market thickness and encouraging financial innovation.

*Source: _wp0499 - References (content supplied).*

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

### _wp0499 - References.............................................................................................................

### I. INTRODUCTION — key messages
- The “solvent but illiquid” debtor framework explains debt crises as loss of access to credit that can be self-fulfilling through creditor panic.
- If inability to roll over debt is the sole reason investors refuse to lend, simple voluntary-participation market mechanisms can in theory eliminate self-fulfilling liquidity runs.
- Coordination mechanisms proposed:
  - Allow new debt to be issued through mechanisms where investors condition participation on a large enough amount being successfully issued (contingent bids / underwriting analog).
  - Issue state-contingent securities that force payoffs (large enough to force default) if targeted debt reduction by a deadline is not achieved, otherwise pay zero.
- Implication: debt crises attributed solely to intra-borrower investor panic are less robust, but economy-wide coordination failures (e.g., “twin crises”, balance sheet mismatches) may remain immune to these mechanisms.
- Policy implication highlighted: the maturity of public debt should not have first-order effects on vulnerability to investor-panic runs that are purely intra-borrower coordination failures; short-term debt remains undesirable for reasons unrelated to liquidity runs.

### II. THE BASIC ENVIRONMENT — model structure and critical conditions
- Economy:
  - Small open economy with a continuum of measure one of identical, infinitely lived residents.
  - Preferences: additively separable, instantaneous utility u(), continuous and strictly concave, discount factor β.
  - Output if never defaulted: Y_t = A_t f(I_{t-1}), with A_t i.i.d. productivity shock, f' > 0, f'' < 0.
  - Output after default: α A_t f(I_{t-1}), with α < 1 (reputational/output cost of default taken as given).
- Debt and timing:
  - Foreign debt stock B_t; one-period maturity bonds issued each period.
  - World risk-free rate = 1/β; investors risk neutral, atomistic, wealth potentially ω_{j,t}.
  - Country announces B_{t+1}; investors submit bids; proceeds p_t B_{t+1} used to repay B_t.
- Key value functions and constraints (preserve notation exactly as in source):
  - Default value: V_D(Y_t) = max_{I_t} { u(Y_t − I_t) + β E_t[ V_D(α A_{t+1} f(I_t)) ] }.
  - Repay value: V_R(Y_t,B_t) = max_{B_{t+1}} { u(Y_t + p_t B_{t+1} − B_t − I_t) + β E_t[ max[ V_R(A_{t+1} f(I_t),B_{t+1}), V_D(A_{t+1} f(I_t)) ] ] } subject to (2) incentive compatibility for I_t, (3) participation p_t = β Pr_t( V_R(A_{t+1} f(I_t),B_{t+1}) ≥ V_D(A_{t+1} f(I_t)) ), and (4) lim_{s→∞} β^{s−t} E_t[B_s] = 0.
- Critical region for liquidity crisis:
  - Define V_R^{LC}(Y_t,B_t) = max_{I_t} { u(Y_t − B_t − I_t) + β E_t V_R(A_{t+1} f(I_t),0) } (liquidity-constrained value when unable to borrow at t).
  - Region of interest when V_R(Y_t,B_t) ≥ V_D(Y_t) > V_R^{LC}(Y_t,B_t). (Equation (5))
  - Lemma 1: If (5) holds, B_t is low enough that the country prefers to repay when it can roll over enough of it, yet large enough that it will default if forced to repay entire stock without new financing.
- Coordination failure results:
  - If every investor expects others not to buy new bonds, expectations can be self-fulfilling and trigger default (Proposition 1).
  - Intertemporal coordination failure (Alesina, Prati and Tabellini style): if investors today expect future investors not to lend and that implies default on debt offered today, today’s investors refuse to hold debt; no B_{t+1} can satisfy (6),(7),(8) then default follows (Proposition 2).
  - Participation/pricing when future access is expected to be lost: constraints (6), (7), (8) link borrowing amount B_{t+1}, investment I_t, and actuarially fair price p_t = β Pr_t( V_R^{LC}(A_{t+1} f(I_{t+1}),B_{t+1}) ≥ V_D(A_{t+1} f(I_{t+1})) ).

### III. SOLVING THE COORDINATION PROBLEMS — mechanisms and lemmas
- Intra-period coordination problem:
  - Direct underwriting by an investment bank (guarantee to hold unsold portion) can eliminate the run risk, though scale concerns may arise for large sovereign issues.
  - Contingent bidding mechanism: investors submit bids p_{j,t}(b_{j,t+1},B_{t+1}) contingent on the total issue size B_{t+1}; country selects bids to satisfy participation constraints.
  - Define minimum B_{t+1} (denoted B_{t+1} with bar in source) solving the country’s indifference and incentive conditions so that issuing at least that amount prevents default (system of equations provided).
  - Equilibrium bidding strategy (as given in source):
    - For B_{t+1} ≥ B_{t+1}: p_{j,t}(b_{j,t+1},B_{t+1}) = β Pr_t( V_R(Y_{t+1},B_{t+1}) ≥ V_D(Y_{t+1}) ), b_{j,t+1} = ω_{j,t} / p_{j,t}.
    - For B_{t+1} < B_{t+1}: p_{j,t}(·) = 0; country rejects p_{j,t} = 0 bids.
  - Lemma 2: If the country allows investors at time t to bid contingent on B_{t+1}, then it cannot be rational for an investor not to lend because she expects the other investors at t not to.
  - Lemma 3: Result extends if bids are allowed to be step-functions around a threshold; country can set threshold at optimal B_{t+1}.
  - Lemma 4: An auction that cancels and fully refunds bids if resulting issue B_{t+1} < B_{t+1} achieves the same anti-run feature.
  - Practical note: similar mechanisms used by start-up firms and underwriting markets.
- Inter-period coordination problem:
  - If country commits to either lower debt below the inter-period crisis-prone range by time T > t or default, then backward induction eliminates inter-period coordination failure.
  - Lemma 5: Under that forced path (debt must exit crisis-prone region by T or default), an investor at t cannot rationally refuse to lend because she expects investors at t+1 not to — coordination risk is removed and borrowing pricing reflects only fundamentals risk.
  - Stronger commitment: full repayment by date T (B_{T+1} = 0) discussed as a constraint that can be imposed; associated value V_R^{LC,T}(Y_t,B_t,T) defined as solution of maximand (1) subject to (2),(3),(4) and (9) B_{T+1} = 0.
- Residual risk:
  - Even after eliminating coordination risk, crises driven by fundamentals (deterioration so debt unsustainable at actuarially fair rate) remain possible.

*Source: _wp0499 - References (content supplied).*

### introduction of constraint (9). Since it was assumed that a country that has fully repaid its debt

### _wp0499 - introduction of constraint (9). Since it was assumed that a country that has fully repaid its debt

### Key lemmas, mechanisms, and logical implications
- Transversality condition (4) implication:
  - For any ε, there exists S large enough such that β^{s−t} B_s ≤ ε for all s ≥ S.
  - Thus, the discounted value at time t of the time T+1 debt becomes arbitrarily small as T becomes arbitrarily large.
- Lemma 6 (preference to repay when repayment can be spread):
  - If V_R(Y_t, B_t) > V_D(Y_t), then V_R^{LC,T}(Y_t, B_t, T) ≥ V_R(Y_t, B_t + ε) > V_D(Y_t) for large enough T.
  - Therefore, if the country strictly prefers to repay its debt to default when it can spread repayments across time, it strictly prefers a plan lowering time T debt below the threshold for which an inter-period coordination failure can occur than to default, provided T is large enough.
- Lemma 7 (self-imposed constraint B_{T+1} = 0 via state-contingent securities):
  - The country can issue securities that pay at time T, yielding zero if B_{T+1} = 0 and a very large amount if B_{T+1} > 0.
  - If the payment in states where B_{T+1} > 0 is large enough to force default in those states, no investor will hold the country’s debt at time T because such claims would be defaulted on with certainty—thereby effectively imposing B_{T+1} = 0.
  - A pari passu clause can be included on the state-contingent security contract to prevent future lenders from being senior to holders of that security; courts in creditor countries would not allow repayment to one group of equally prioritized creditors while another equal-priority group is not repaid.

### Proposition and equilibrium implication
- Proposition 3:
  - If the country strictly prefers to repay its debt when it can spread repayments across time, self-fulfilling liquidity crises caused by a coordination failure among investors cannot occur in equilibrium.
  - Logical chain:
    - Lemma 2: investors can be allowed to bid on new bonds contingent on the resulting size of the issue, eliminating intra-period coordination failure.
    - Lemma 5: inter-period coordination failure can be eliminated if constraint B_{T+1} = 0 is imposed.
    - Lemma 7: constraint B_{T+1} = 0 can be self-imposed by issuing state-contingent securities.
    - Lemma 6: issuing such securities is preferable to default for large enough T.
  - Conclusion: an investor at time t will be willing to lend since the country will take measures to ensure success of a second bond offer at time t and subsequent periods even if other investors do not.

### Robustness, informational asymmetries, and higher-order beliefs
- Informational asymmetries extension:
  - A failed bond offer may lead some investors to update beliefs about fundamentals, raising required premia.
  - For credit to completely dry out, perceived fundamentals must deteriorate enough that no lending can be sustained at the actuarially fair rate.
  - Investors typically know whether they did not buy bonds due to coordination concerns or fundamentals concerns, reducing scope for deterioration.
  - Liquidity crises are still ruled out provided the country’s fundamentals are strong enough initially.
- Higher-order beliefs (Morris and Shin tradition):
  - The intra-period coordination mechanism removes strategic uncertainty by allowing investors to condition decisions on an outcome mapping into others' strategies, making it robust to higher-order belief considerations.
  - The inter-period solution can be compromised if there is no common knowledge that all investors are forward-looking.

### Discussion: limits, externalities, and aggregate considerations
- Extension to decentralized private borrowers with strategic complementarities:
  - Balance sheet effects can create strategic complementarities across creditors to different borrowers (references: Krugman (1999); Aghion, Bacchetta and Banerjee (2001a, 2001b); earlier work Aghion, Bacchetta and Banerjee 1999).
  - Contract enforcement externalities: an investor’s prospect of recouping a loan may fall with widespread default, generating “rush to the exits.”
  - In presence of such economy-wide externalities:
    - Investors must be able to condition lending on aggregate lending to the home economy to eliminate intra-period coordination failures—difficult if investors move simultaneously.
    - Inter-period mechanism (commitment to lower aggregate debt over long but finite horizon) may not be implementable because small private investors do not internalize aggregate externalities.
    - Free-rider problems can arise if there are private costs to participating in coordination mechanisms.
- Role of government:
  - Government could internalize externalities by consolidating private liabilities and using the Section III mechanisms to coordinate the economy toward the good equilibrium.
  - Government issuance to take over private liabilities could offset eventually with collections from rescued borrowers in the inter-temporal budget constraint.
  - However, government willingness to repay can be affected if rescuing private borrowers increases gross public liabilities beyond thresholds for strategic sovereign default; what matters is gross liabilities as well as net.
  - If government cannot borrow against future claims without crossing the default threshold, intervention cannot remove investor coordination failure in a decentralized economy.

### Conclusions and broader implications
- Market mechanisms that alter the game to eliminate creditor coordination failures can prevent self-fulfilling debt runs at the level of an individual (aggregate) borrower.
- Self-fulfilling features can remain important in economy-wide coordination failures, which may be difficult to overcome with the proposed mechanisms.
- Many small illiquid borrowers can make an economy more vulnerable to investor panics than a single aggregate illiquid borrower.
- At the individual borrower level, short-term debt per se may be less risky than often suggested because mechanisms can eliminate roll-over risk; short-term debt remains undesirable for other reasons, including vulnerability to changes in roll-over terms.
- Potential application to financial innovation:
  - Introduction of new instruments can face market thinness and multiplicity of equilibria in participation.
  - The contingent bidding mechanism of Section III could allow investors to price bids according to expected market thickness associated with issue sizes, possibly enabling efficient market thickness and encouraging financial innovation.

*Source: _wp0499 - introduction of constraint (9). Since it was assumed that a country that has fully repaid its debt*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2004/_wp0499.pdf_