## _wp15223

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---

### 2.1 Model structure — time, production, financing, and crisis mechanics
- Time structure: Three-period model, t∈{0,1,2}.
- Agents: a country, its private creditors, and the international community.
- World risk-free rate r∗ = 0.
- Investment and technology:
  - Time-0 investment: exogenous capital k.
  - Time-2 production: A k output with A > 1.
  - Time-1 liquidation: χ k output with χ < 1.
- Financing:
  - Country issues one-period bonds at time 0, pledging repayment k(1 + r) at time 1. Define R = 1 + r.
- Fiscal shock at time 1:
  - d = 0 with probability p
  - d = D with probability (1 − p)
  - D ∈ [0, D̄] with CDF Φ(·).
- Crisis-prevention effort:
  - Country chooses observable effort e with ∂p/∂e > 0 and convex utility cost z(e).
- Time-1 outcomes:
  - If financing needs Rk + d can be met → production continues to time 2.
  - If not → creditors trigger full liquidation of capital (partial liquidation not an equilibrium).

### 2.2 Financing needs met — instruments, constraints, and costs
- Financing sources at time 1 must satisfy (equation (2)):
  - b + f + i + o = Rk + d
    - b = new borrowing, constrained by b ≤ Ak(1) (borrowing limit stated as b ≤ Ak(1) and elsewhere b ≤ Ak).
    - f = fiscal consolidation; utility cost h(f) = f + α f^2 / 2 with α > 0.
    - i = bail-in net relief to country; associated with efficiency loss parameter ξ ≥ 0 so that an (1 + ξ) haircut yields net relief i.
    - o = official transfers; a transfer of (1 + ξ) o yields net funds o to the country.
- Bail-ins and transfers:
  - Bail-ins generate international spillovers λ i with λ ≥ 0. Non-systemic countries have λ = 0; systemic countries have λ > 0.
  - Official transfers assumed not to generate negative spillovers but have efficiency loss ξ analogous to bail-ins.
  - Transfers avoid spillovers but raise moral hazard concerns.
- If financing constraint satisfied:
  - Production continues; time-2 consumption c = A k − b.
- Key parameters and constraints reiterated:
  - h(f) = f + α f^2 / 2, α > 0.
  - ξ ≥ 0, λ ≥ 0.
  - Borrowing limit: b ≤ Ak(1) (also referenced as b ≤ Ak).
  - Financing needs: Rk + d.

### 2.3 Financing needs not met — disruptive default and liquidation
- Creditor-enforced liquidation if financing needs unmet:
  - Liquidation reduces time-2 output to χ k; creditors claim χ k through litigation.
  - Default efficiency costs treated as ξ; creditors suffer losses (1 + ξ)(R − χ) k; country receives debt relief i = (R − χ) k.
  - Bail-in spillovers remain λ i.
- Consequences of liquidation:
  - b = 0 (no time-1 borrowing).
  - Fiscal consolidation f = d (primary deficit closed entirely by consolidation).
  - Time-2 consumption c = 0.
- Interpretation: liquidation prevents issuance of bonds at time 1 and contracts output from A k to χ k, and can arise in laissez-faire equilibrium despite being socially inefficient.

### 2.4 Welfare definitions and equilibrium interest rate
- Country welfare:
  - Ex-post: U_C^1 = c − h(f)
  - Ex-ante: U_C^0 = −z(e) + E_0[U_C^1]
- Creditors (risk-neutral, competitive):
  - Ex-post: U_L^1 = R k − (1 + ξ) i
  - Ex-ante: U_L^0 = −k + E_0[U_L^1]
  - Break-even condition (equation (3)): E_0[R k − (1 + ξ) i] = k
- International community:
  - Ex-post: U_I^1 = −(λ i + (1 + ξ) o)
  - Ex-ante: U_I^0 = E_0[U_I^1]
- Social welfare:
  - U_S^t = U_C^t + U_L^t + U_I^t
  - With creditors’ ex-ante welfare zero by (3), ex-ante social welfare reduces to aggregate of country and international community utilities.
- Implication: creditors can offset expected bail-in losses (1 + ξ) i by charging a higher R consistent with (3), making them ex-ante indifferent over financing mixes.

### 3 Laissez-faire equilibrium — frictions and strategic timing
- Principal market frictions:
  - Coordination failures among creditors (collective action problems, free riding, holdouts).
  - Repudiation (time-inconsistency) risk: country cannot commit to ex-post fiscal consolidation after borrowing at time 1.
- Country’s time-1 choices after raising b:
  - Repay: set f = max{d + R k − b, 0}; U_C^1 = −h(max{d + R k − b, 0}) + A k − b (equation (4)).
  - Trigger liquidation: set f = max{d − b, 0} and use b to cover the primary deficit; creditors liquidate; U_C^1 = −h(max{d − b, 0}) (equation (5)).
- Equilibrium implications:
  - Limited state-contingency in R can produce credit rationing and capital liquidation in laissez-faire.
  - Model contrasts first-best (complete markets) with reduced state-contingency outcomes.

### 3.1 Complete financial markets (first-best benchmark)
- Assumptions:
  - R fully contingent on realization of d; creditors observe d and enforcement exists.
  - Financing condition without consolidation: d + R k ≤ A k (equation (6)).
  - Creditors’ break-even implies E_0[R] = 1.
- Country ex-ante utility (equation (7)):
  - U_C^0 = −z(e) + (A − 1) k − (1 − p) E_0[D]
  - Optimal effort satisfies z′(e) = p′(e) E_0[D] (equation (8)).
- Under complete markets:
  - No bail-ins or official transfers needed; creditors’ ex-ante welfare zero → U_S^0 = U_C^0.
  - Country’s effort choice is socially efficient; IFIs have no role.

### 3.2 Incomplete financial markets — limited contingency and equilibrium thresholds
- Interest-rate contingency limited to crisis vs non-crisis states (equation (9)):
  - R = { R + π if d = 0; R if d = D } (9)
- Break-even premium π satisfies (equation (11)):
  - pπk = (1−p)E0[(1 + ξ)i] + (1−R)k (11)
- Non-crisis rollover feasible if (equation (10)):
  - (R + π)k ≤ Ak (10)
- Financing thresholds:
  - Borrowing-only feasible if D ≤ D1 where (equation (12)):
    - D1 = (A − R)k (12)
  - If D > D1, borrowing insufficient; creditors may restrict b below Ak to elicit fiscal consolidation.
  - Creditors face debt-overhang at b = Ak.
  - Equilibrium requires b < Rk (Appendix A), implying larger fiscal adjustment may be needed for positive future consumption.
  - Creditors can avoid liquidation via credit rationing up to D2 (equation (13)):
    - D2 = max{ D1, (A − R)k / αRk − Rk / 2 } (13)
    - Cases:
      - If α and R large relative to A → D2 = D1 → liquidation occurs once D > D1.
      - If A large relative to α and R → region D1 < D ≤ D2 where credit restriction avoids liquidation.
  - If D > D2 → laissez-faire leads to liquidation: b = 0, f = D, creditors lose (1 + ξ)(R − χ)k, country receives (R − χ)k relief.

### Planner (IFI) role under incomplete markets — objectives and trade-offs
- Planner coordinates creditors and can commit the country to consolidation; maximizes social welfare subject to participation constraints.
- Ex-post social welfare (equation (14)):
  - US1 = Ak − ( D + α f^2 / 2 + (ξ + λ) i + ξ o ) (14)
- Ex-ante social welfare (equation (15)):
  - US0 = −z(e) + (A − 1)k − (1 − p)E0[ D + α f^2 / 2 + (ξ + λ)i + ξ o ] (15)
- Socially-efficient effort eS solves (equation (16)):
  - z′(eS) = p′(eS) E0[ D + α f^2 / 2 + (ξ + λ)i + ξ o ] (16)
- Country’s preferred effort eC solves (equation (18)):
  - z′(eC) = p′(eC) E0[ D + α f^2 / 2 + ξ i − o ] (18)
- Moral hazard (with z(e) = β e^2 / 2 and p(e) = ν e) (equation (19)):
  - eS − eC = ν/β · E0[ λ i + (1 + ξ) o ] (19)
  - Interpretation:
    - Bail-ins induce moral hazard iff λ > 0.
    - Official transfers induce moral hazard via omission of international cost o(1 + ξ).
    - If λ = 0, bail-ins do not create moral hazard because expected losses are priced into π.

### Planner without commitment (no transfers) — ex-post optimization and instrument ordering
- Planner’s ex-post problem (equation (20)):
  - max_{b,f,i ≥ 0} Ak − ( D + α f^2 / 2 + (ξ + λ) i )
    subject to b + f + i = R k + D; b ≤ Ak (20)
- Planner lends up to Ak; if D > D1 uses fiscal consolidation and bail-ins comparing marginal costs:
  - MC_f^ex-post = α f (21)
  - MC_i^ex-post = ξ + λ (22)
- Optimal ordering:
  - Fiscal consolidation used first up to f̄ where α f̄ = ξ + λ → f̄ = (ξ + λ) / α
  - Remaining gap covered exclusively by bail-ins.
  - f̄ rises with λ: more systemic countries endure greater consolidation to limit spillovers.

### Planner with commitment (no transfers) — ex-ante marginal costs and higher consolidation
- With commitment planner internalizes moral hazard; ex-ante marginal costs:
  - MC_f^ex-ante = α f (24)
  - MC_i^ex-ante = ξ + λ + λ·( 1 − pC / 1 − pS − 1 ) (25)
    - pS and pC are non-crisis probabilities under social and country effort choices.
- Fiscal consolidation’s marginal cost unchanged ex-ante vs ex-post; bail-ins have higher ex-ante marginal cost because of spillover-induced moral hazard.
- Result: commitment increases f̄ relative to no-commitment and reduces reliance on bail-ins.
- Define D3 = D1 + f̄ as threshold above which planner employs bail-ins; D3 increases with λ and is higher under commitment.

### Participation and comparative outcomes
- Planner preserves market access by lending up to Ak.
- Instrument composition depends on λ:
  - Larger λ → less bail-ins, more fiscal consolidation (and, with transfers, more official financing).
- Planner’s solution prevents liquidation and generally leaves creditors better off than laissez-faire liquidation, provided maximum primary deficit D̄ is not implausibly large.

### 4.2 Planner with official transfers, no commitment — ex-post choices and moral hazard
- Planner’s ex-post problem with transfers (equation (26)):
  - max Ak − (D + α f^2 / 2 + (ξ + λ) i + ξ o)
  - subject to b + f + i + o = R k + D ; b ≤ A k
- Ex-post marginal cost of transfers: MC_o_ex-post = ξ (27)
- Optimal ex-post policy:
  - Borrow up to Ak.
  - If D > D1, use fiscal consolidation up to f̄ = min{ MC_i_ex-post , MC_o_ex-post } / α.
  - If λ = 0 → bail-ins and transfers identical ex-post.
  - If λ > 0 → transfers less costly ex-post; planner sets f̄ = ξ / α and covers remaining gap with transfers.
- Welfare and participation:
  - With transfers and λ > 0, country and creditors ex-post better off relative to planner without transfers.
  - Major drawback: severe moral hazard ex-ante because planner cannot commit and relies on transfers, creating a large wedge between eS and eC and potentially lowering ex-ante social welfare (possibly below laissez-faire except for highly systemic cases).

### 4.4 Welfare implications and policy recommendations
- Ex-ante social welfare:
  - If λ = 0 (non-systemic), planner can achieve maximum welfare without transfers or commitment by minimizing ex-post costs.
  - If no commitment, allowing transfers can be counter-productive: transfers without commitment often reduce ex-ante social welfare because of moral hazard proportional to ν/β (equation (19)).
  - Even with commitment, social welfare declines in λ because larger spillovers force limiting bail-ins and using costlier tools.
  - Social welfare lower bound when λ ≥ λ2 where it is efficient to entirely avoid bail-ins.
  - Policy implication: reduce systemic spillovers (e.g., regulation forcing banks to hold greater capital buffers against systemic-country debt).
- Ex-ante country welfare:
  - Country strictly better off under any planner solution than in laissez-faire since planner provides financing up to borrowing capacity Ak.
  - Under transfers and commitment, country welfare need not be monotonic in λ; trade-off between increased fiscal consolidation and transfers determines outcome. Moral hazard strength ν/β critical.
  - Differentiated policy mixes raise evenhandedness concerns; could be addressed by ex-ante taxes/subsidies across countries (not solved here).
- Planner’s optimal framework — recommendations:
  - For non-systemic countries (λ = 0): resolve crises without official transfers and without commitment by minimizing ex-post costs via fiscal consolidation and bail-ins (no moral hazard since creditors price expected bail-in losses).
  - For systemic countries (λ > 0): provide official transfers to limit bail-ins and spillovers, but only under IFI commitment and a pre-determined crisis-resolution framework; complement transfers with stricter fiscal consolidation to contain moral hazard.
  - Without commitment, IFIs will over-rely on transfers and generate severe moral hazard.
  - Mobilize international community for transfers for systemic crises but reserve transfers for highly systemic cases due to moral hazard concerns.
- Quantitative/mechanism notes:
  - Moral hazard from transfers proportional to ν/β (equation (19)).
  - No-commitment transfers can worsen ex-ante welfare relative to laissez-faire except for highly systemic countries.
  - When interest rates constant across states (Appendix C):
    - Bail-ins become more costly because raising bail-ins increases R in the crisis state and tightens financing constraint.
    - Crisis-prevention effort becomes more valuable; efficient solutions increase effort in proportion to shadow cost of financing constraint.
- Concluding policy-relevant points and future research:
  - Differentiate crisis-resolution by systemicity; use transfers for systemic crises but under commitment and stricter conditionality.
  - Implement regulatory/macroprudential measures to reduce spillover risks.
  - Explore ex-ante compensating mechanisms (taxes/subsidies or reserve fund) to equalize expected welfare across countries and avoid perverse incentives for systemicity.

*Source: IMF working paper — sections 2.1–4.4 of _wp15223._*

### 2.1    Model structure .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .

### 2.1    Model structure

### Time structure and agents
- Three-period model, t∈{0,1,2}.
- Agents: a country, its private creditors, and the international community.
- World risk-free rate r∗ is equal to 0.

### Production, investment, and liquidation
- At time 0 the country invests an exogenous level of capital k.
- Capital produces A k units of output at time 2, with A > 1.
- Capital can be liquidated in period 1 and returns χ k output units where χ < 1.

### Financing and notation
- The country finances investment by issuing one-period government bonds at time 0 and pledges a repayment equal to k(1 + r) in time 1, where r is the interest rate.
- For notational convenience, define R = 1 + r.

### Fiscal shocks and crisis probability
- At time 1 the country confronts a random primary fiscal deficit d distributed as:
  - d = 0 with probability p
  - d = D with probability (1 − p)
- D is a random variable distributed between 0 and D̄ > 0 with CDF denoted Φ(·).
- p is the probability of the “non-crisis state”; (1 − p) is the probability of the “crisis state”.

### Country effort, observability, and cost
- The country can exercise a crisis-prevention effort e that increases p:
  - ∂p/∂e > 0
- The crisis-prevention effort entails a convex utility cost z(e).
- The effort e is treated as observable.

### Period-1 decisions and equilibrium scenarios
- At time 1 the country must rollover the debt R k and finance the primary deficit d.
- Two equilibrium scenarios are considered:
  - Financing needs are met and production continues until time 2.
  - Financing needs are not met and creditors trigger full liquidation of capital.
- Partial capital liquidation is not described because it is not an equilibrium.

*Source: IMF working paper — section 2.1 Model structure.*

### 2.2    Financing needs are met and production continues

### 2.2    Financing needs are met and production continues

### Financing options and constraints
- Financing needs Rk + d at time 1 can be covered by a combination of:
  - New borrowing b, subject to the borrowing limit b ≤ Ak(1).
  - Fiscal consolidation f, with utility cost h(f) = f + α f^2 / 2 where α > 0.
  - Bail-in of private creditors i, providing net debt relief i to the country but associated with an efficiency loss parameter ξ ≥ 0 so that an (1 + ξ) haircut on creditors yields net relief i.
  - Official transfers o from the international community, where (1 + ξ) o resources provide net funds o to the country.
- Financing constraint summarized:
  - b + f + i + o = Rk + d   (equation (2))

### Bail-ins, transfers, and international spillovers
- Bail-ins:
  - Imposed on private creditors; generate net relief i to the country but entail efficiency losses captured by ξ ≥ 0.
  - May create negative spillovers to the international community equal to λ i, where λ ≥ 0 controls spillover intensity.
  - Countries are classified as non-systemic if λ = 0 and systemic if λ > 0.
- Official transfers:
  - Implemented through restructuring of official claims or concessional lending; assumed not to generate negative spillovers (they are financed by public balance sheets).
  - Assumed to have the same efficiency losses as bail-ins: a transfer of (1 + ξ) o yields net funds o to the country.
  - Advantage over bail-ins in avoiding spillovers must be weighed against moral hazard consequences.

### Production continuation if financing constraint is satisfied
- If b + f + i + o = Rk + d is satisfied:
  - Production continues to time 2.
  - Time-2 consumption: c = A k − b

### Key model assumptions and functional forms
- Fiscal consolidation cost: h(f) = f + α f^2 / 2, with α > 0.
- Efficiency loss parameter for bail-ins/transfers/default: ξ ≥ 0.
- Spillover intensity parameter: λ ≥ 0.
- Borrowing limit: b ≤ Ak(1).
- Financing needs: Rk + d.

---

### 2.3    Financing needs are not met and capital is liquidated

### Default with capital liquidation (disruptive default)
- If financing needs are not met (inability to cover Rk + d), creditors demand liquidation of capital.
- Liquidation reduces output to χ k, which creditors can claim through litigation.
- Default entails the same efficiency losses ξ (for simplicity); creditors are forcefully bailed-in and suffer losses equal to (1 + ξ)(R − χ) k, while the country receives debt relief i = (R − χ) k.
- Bail-in-generated international spillovers remain λ i.
- Consequences of liquidation:
  - Production at time 2 prevented: country cannot issue bonds at time 1 so b = 0.
  - Primary deficit must be closed entirely through fiscal consolidation: f = d.
  - Time-2 consumption: c = 0.
- Interpretation: disruptive default prevents international borrowing and contracts output from time-2 production A k to time-1 liquidation value χ k. Despite being socially inefficient, liquidation can arise in laissez-faire equilibrium under common market failures.

---

### 2.4    Welfare definitions and equilibrium interest rate

### Welfare components (time-1 ex-post and time-0 ex-ante)
- Country:
  - Ex-post welfare: U_C^1 = c − h(f)
  - Ex-ante welfare: U_C^0 = −z(e) + E_0[U_C^1]
    - z(e) is crisis-prevention effort cost chosen at time 0.
- Creditors (risk-neutral, competitive):
  - Ex-post welfare: U_L^1 = R k − (1 + ξ) i
  - Ex-ante welfare: U_L^0 = −k + E_0[U_L^1]
  - Break-even (competitive) condition: E_0[R k − (1 + ξ) i] = k   (equation (3))
    - Implies expected net return must equal the zero risk-free rate.
- International community:
  - Ex-post welfare: U_I^1 = −(λ i + (1 + ξ) o)
  - Ex-ante welfare: U_I^0 = E_0[U_I^1]
- Social welfare:
  - U_S^t = U_C^t + U_L^t + U_I^t
  - Given creditors’ ex-ante welfare is zero by (3), ex-ante social welfare simplifies to aggregate utility of the country and the international community.

### Implication for interest rate
- Creditors are ex-ante indifferent about bail-ins vs. other financing mixes because expected losses (1 + ξ) i can be offset by charging a higher R consistent with (3).

---

### 3    Laissez-faire equilibrium — setup and market frictions

### Market frictions considered
- Coordination problems among creditors that prevent orderly bail-ins (collective action problems, free riding, holdouts).
- Repudiation (time-inconsistency) risk: country cannot commit to future fiscal consolidation after borrowing at time 1. Country chooses f after issuing b and before repaying R k.
  - After raising b, country options:
    - Repay creditors by setting f = max{d + R k − b, 0}; production continues; country welfare at time 1:
      - U_C^1 = −h(max{d + R k − b, 0}) + A k − b   (equation (4))
    - Trigger liquidation by using b to cover the primary deficit, setting f = max{d − b, 0}; creditors liquidate; country welfare at time 1:
      - U_C^1 = −h(max{d − b, 0})   (equation (5))

### Equilibrium features analyzed
- With limited state-contingency in R, laissez-faire equilibrium can involve credit rationing and capital liquidation.
- The model contrasts the first-best (complete markets) with outcomes under reduced state contingency and market failures.

---

### 3.1    Complete financial markets (first-best benchmark)

### Assumptions for first-best
- R can be made fully contingent on realization of the primary deficit d (creditors observe d and enforcement is available).
- Financing constraint and borrowing limit can be met by an appropriate contingent schedule for R without fiscal consolidation, provided:
  - d + R k ≤ A k   (equation (6))
- Creditors’ break-even in absence of bail-ins implies E_0[R] = 1.

### Country's ex-ante utility and optimal effort
- Country ex-ante utility under complete markets:
  - U_C^0 = −z(e) + (A − 1) k − (1 − p) E_0[D]   (equation (7))
  - Country chooses effort e to satisfy:
    - z′(e) = p′(e) E_0[D]   (equation (8))
- Under complete markets:
  - No bail-ins or official transfers are needed → international community suffers no losses.
  - Creditors’ ex-ante welfare is zero by break-even condition → U_S^0 = U_C^0.
  - Country’s choice of effort e is socially efficient; first-best allocation achieved.
  - Conclusion: under complete contingent contracting, IFIs have no role to play.

---

*Source: _wp15223 - 2.2    Financing needs are met and production continues*

### 3.2    Incomplete financial markets

### _wp15223 - 3.2    Incomplete financial markets

### Incomplete state-contingency in sovereign debt contracts
- Lenders permit interest-rate contingency only across crisis vs non-crisis states but not on the exact value of the primary deficit D within the crisis state. Interest rate specification (equation (9)):
  - R = { R + π if d = 0; R if d = D } (9)
- Premium π > 0 charged in non-crisis state to break even in expectation; π compensates for expected bail-in losses and any deviation of R from the zero risk-free rate. Break-even condition (equation (11)):
  - pπk = (1−p)E0[(1 + ξ)i] + (1−R)k (11)
- Rolling over debt in non-crisis state assumed feasible if (equation (10)):
  - (R + π)k ≤ Ak (10)
- Partial insurance against the primary deficit remains via lower interest payments in crisis state, but model departs from complete markets.

### Laissez-faire equilibrium characterization and financing thresholds
- If primary deficit D is small, financing needs can be covered entirely with new borrowing. Borrowing-only feasible if (equation (12)):
  - D1 = (A − R)k (12)
- If D > D1, financing cannot be covered by borrowing alone; creditors restrict lending b below Ak to elicit fiscal consolidation.
- Creditors face debt-overhang at b = Ak (country has no incentive to prevent liquidation).
- Due to inability to commit, b must fall below Rk in equilibrium (Appendix A result), implying fiscal adjustment larger than primary deficit may be required for positive future consumption.
- Laissez-faire can prevent liquidation via credit rationing up to threshold D2 (equation (13)):
  - D2 = max{ D1, (A − R)k / αRk − Rk / 2 } (13)
- Cases:
  - If α and R large relative to A, D2 = D1 and creditors cannot elicit consolidation beyond D1 → liquidation and loss of market access occur once D > D1.
  - If A sufficiently large relative to α and R, there exists region D1 < D ≤ D2 where appropriate credit restriction avoids liquidation.
- If D > D2, laissez-faire involves capital liquidation: b = 0, fiscal consolidation f = D, creditors suffer losses equal to (1 + ξ)(R − χ)k and country receives debt relief (R − χ)k.

### Planner (IFI) role: objectives and trade-offs
- Planner coordinates creditors and can commit the country to fiscal consolidation; represents IFIs (e.g., IMF, ESM) that can require restructuring as condition for official lending and monitor conditionality.
- Planner maximizes social welfare while satisfying participation constraints for the country and creditors (plan must yield higher ex-post welfare than laissez-faire).
- Planner avoids liquidation because liquidation reduces output from Ak to χk inefficiently.
- Ex-post social welfare (substituting borrowing constraint) (equation (14)):
  - US1 = Ak − ( D + αf^2/2 + (ξ + λ)i + ξo ) (14)
  - Highlights ex-post social costs: quadratic utility cost of fiscal consolidation αf^2/2, efficiency costs ξ of bail-ins and official transfers, and spillovers λ from bail-ins.
- Ex-ante social welfare (substituting lenders’ break-even) (equation (15)):
  - US0 = −z(e) + (A − 1)k − (1 − p)E0[ D + αf^2/2 + (ξ + λ)i + ξo ] (15)
- Socially-efficient crisis-prevention effort eS solves (equation (16)):
  - z′(eS) = p′(eS) E0[ D + αf^2/2 + (ξ + λ)i + ξo ] (16)
- Country’s ex-ante utility (equation (17)):
  - UC0 = −z(e) + (A − 1)k − (1 − p)E0[ D + αf^2/2 + ξi − o ] (17)
  - Country internalizes ξ (efficiency losses from bail-ins) via higher premium π, but does not internalize spillovers λ or the international cost o of transfers.
- Country-preferred effort eC solves (equation (18)):
  - z′(eC) = p′(eC) E0[ D + αf^2/2 + ξi − o ] (18)

### Moral hazard from bail-ins and official transfers
- With simple functional forms z(e) = βe^2/2 and p(e) = νe, difference between socially-efficient and country-preferred effort (equation (19)):
  - eS − eC = ν/β · E0[ λi + (1 + ξ)o ] (19)
- Interpretation:
  - Bail-ins induce moral hazard only if they generate spillovers (λ > 0).
  - Official transfers generate stronger moral hazard because the country neglects the international cost o(1 + ξ).
  - If no spillovers, bail-ins do not create moral hazard as creditors’ losses are priced into π and internalized by the country.

### Planner commitment vs no commitment: implications for optimal instruments
- Two commitment extremes:
  - No commitment: planner minimizes ex-post crisis costs only; ignores ex-ante moral hazard.
  - Full commitment with credible ex-ante conditionality: planner could condition crisis resolution on crisis-prevention effort to elicit eS, eliminating moral hazard; practically difficult to enforce.
- Analysis proceeds assuming planner cannot condition on effort (no ex-ante conditionality), so planner trades off ex-post costs versus ex-ante moral hazard.

### Planner’s solution without official transfers (no commitment)
- Planner maximizes ex-post social welfare (no transfers) subject to financing constraint and b ≤ Ak (problem in (20)):
  - max_{b,f,i ≥ 0} Ak − ( D + αf^2/2 + (ξ + λ)i )
    subject to b + f + i = Rk + D; b ≤ Ak (20)
- Planner allows bonds up to borrowing capacity Ak (lending up to Ak is costless socially).
- If D > D1, planner covers remaining gap with fiscal consolidation and bail-ins by comparing ex-post marginal costs:
  - MC_f^ex-post = αf (21)
  - MC_i^ex-post = ξ + λ (22)
- Optimal ordering and cap:
  - Fiscal consolidation used first; upper bound f̄ where MC_f^ex-post = MC_i^ex-post:
    - f̄ = (ξ + λ) / α
  - Remaining gap beyond f̄ covered exclusively with bail-ins.
  - f̄ rises with λ: more systemic countries (higher λ) endure greater consolidation to limit international spillovers.

### Planner’s solution with commitment (no transfers): ex-ante marginal costs
- With commitment planner internalizes moral hazard and maximizes ex-ante social welfare subject to country’s effort choice (problem in (23)). Country’s first-order condition constraint as in (18).
- Ex-ante marginal costs:
  - MC_f^ex-ante = αf (24)
  - MC_i^ex-ante = ξ + λ + λ·( 1 − pC / 1 − pS − 1 ) (25)
    - where pS and pC are probabilities of non-crisis under socially-efficient and country-chosen effort respectively.
- Interpretation:
  - Fiscal consolidation has same ex-ante and ex-post marginal cost: it does not generate moral hazard (country internalizes its cost).
  - For systemic countries, bail-ins have higher ex-ante marginal cost due to spillover-induced moral hazard; the extra term depends on the percentage increase in crisis probability under country effort relative to social optimum: (1 − pC)/(1 − pS) − 1.
  - Higher ex-ante MC_i implies higher f̄ under commitment:
    - f̄ = MC_i^ex-ante / α

### Comparative outcomes and participation constraints
- Planner lends up to Ak (via IFIs or private lenders) irrespective of systemic status.
- Composition of instruments depends on spillover strength λ:
  - Stronger λ → less bail-ins, more fiscal consolidation.
  - Commitment raises f̄ (more consolidation tolerated) relative to no commitment.
- Define D3 = D1 + f̄ as the primary deficit threshold above which planner employs bail-ins.
  - D3 increases with spillovers λ and is higher under commitment; with sufficiently large f̄ bail-ins may be avoided entirely up to large D.
- Planner’s solution prevents liquidation and thus keeps market access up to Ak; this makes the country better off than laissez-faire where fiscal consolidation needed at least D once D > D1.
- Creditors are better off under planner than under liquidation in laissez-faire as long as maximum primary deficit D̄ is not implausibly large. Under planner, creditor losses ≤ (1 + ξ)(D̄ − D3), generally smaller than liquidation losses provided (A − χ)k + f̄ > D̄.

*Source: _wp15223 - 3.2    Incomplete financial markets*

### 4.2    Planner’s solution with official transfers, but no commitment

### 4.2    Planner’s solution with official transfers, but no commitment

### Setup and planner problem
- Planner maximizes social welfare ex-post but cannot commit; can use official transfers.
- Planner's optimization (variables b,f,i,o ≥ 0):
  - max Ak − (D + α f^2 /2 + (ξ+λ) i + ξ o)
  - subject to (26): b + f + i + o = R k + D ; b ≤ A k
- Ex-post social marginal cost of transfers:
  - MC_o_ex-post = ξ (27)

### Optimal instrument choice ex-post
- Planner allows borrowing up to b ≤ A k.
- If D > D_1 (borrowing capacity insufficient), planner uses fiscal consolidation up to f̄ where its marginal cost reaches the lower of bail-ins and transfers:
  - f̄ = min{ MC_i_ex-post , MC_o_ex-post } / α
- If country is non-systemic (λ = 0):
  - Bail-ins and transfers have identical marginal losses; planner is ex-post indifferent between them.
- If λ > 0:
  - Official transfers are ex-post less costly than bail-ins.
  - Planner sets upper bound on fiscal consolidation:
    - f̄ = ξ / α
  - Any remaining financing gap is covered through transfers alone.

### Welfare and participation consequences
- Compared with planner solution without transfers (Figure 3), when λ > 0:
  - Both the country and creditors are ex-post better off because transfers prevent increases in fiscal consolidation and creditors’ losses.
  - The solution satisfies the country and creditors’ participation constraints.
- Major drawback: severe moral hazard effects.
  - Without commitment, planner neglects ex-ante implications and relies excessively on official transfers.
  - This creates a considerable wedge between socially-efficient and country-preferred levels of crisis-prevention effort.
  - May lead to lower ex-ante social welfare than if planner is prevented from using transfers (to be shown in section 4.4).

*Source: _wp15223 - 4.2    Planner’s solution with official transfers, but no commitment*

### 4.4    Welfare implications of alternative crisis-resolution frameworks

### 4.4    Welfare implications of alternative crisis-resolution frameworks

### Ex-ante social welfare: key findings
- In the absence of spillovers, λ= 0, the planner’s solution achieves the maximum welfare even without transfers and commitment.  IFIs can address the sovereign debt crisis of a non-systemic country by minimizing ex-post costs without providing transfers.
- If IFIs have no commitment, allowing them to use transfers can be counter-productive.  The welfare under the solution with transfers but no commitment depends on the severity of moral hazard effects, which, as shown in equation (19), are proportional to the ratio ν/β.  With the exception of highly systemic countries, ex-ante social welfare is lower relative to the solution without transfers; transfers without commitment may reduce welfare even below the laissez-faire equilibrium because IFIs would use transfers to avoid fiscal consolidation and bail-ins and thereby generate severe moral hazard costs.
- Even under the optimal solution with transfers and commitment, social welfare declines in the size of spillovers λ because larger spillovers require limiting bail-ins and using more costly policy tools (greater fiscal consolidation and transfers).
- Social welfare reaches a lower bound in the case of highly systemic countries, λ≥λ2, for which it is socially efficient to entirely avoid bail-ins.
- Policy implication: the decline in social welfare associated with systemic crises calls for measures to reduce the risk of systemic spillovers or contain their effects (for example, regulation forcing banks to hold greater capital buffers against the debt holdings of systemic countries).

### Ex-ante country’s welfare: key findings
- The country is strictly better off under any of the planner’s solutions than in the laissez-faire equilibrium.  This holds even under the planner’s solution with commitment but no transfers, because IFIs provide financing up to the country’s borrowing capacity (as illustrated in Figure 3 and Figure 4).
- Under the planner’s solution with transfers and commitment, social welfare is monotonically declining in spillovers λ, but individual countries’ ex-ante welfare need not be monotonic in λ.  The shaded area in Figure 8 reflects this ambiguity.
  - Whether a more systemic country is better or worse off depends on whether the reduction in bail-ins is compensated more through greater fiscal consolidation (which hurts the country) or through transfers (which benefit the country).
  - This trade-off hinges on moral hazard strength associated with transfers, proportional to ν/β (equation (19)).  If moral hazard effects are severe, replacing bail-ins with fiscal consolidation reduces systemic countries’ welfare; if moral hazard effects are limited, replacing bail-ins with transfers can make highly systemic countries better off than non-systemic ones.
- Differentiating the policy mix by spillover size produces differences in expected welfare across countries, raising concerns about evenhandedness.  Expected welfare differences may not be large because transfers to systemic countries are accompanied by somewhat greater fiscal consolidation.
- Policy option to equalize welfare (not solved here): design a compensating system of ex-ante taxes and subsidies across countries; e.g., systemic countries that benefit ex-post from transfers could contribute ex-ante to a pool used to finance transfers.

### Planner’s optimal framework: policy recommendations and mechanics
- For non-systemic countries (bail-ins do not generate spillovers):
  - IFIs can efficiently resolve crises without official transfers and without commitment by focusing on minimizing ex-post costs via the least socially costly combination of fiscal consolidation and bail-ins.
  - If creditors price expected bail-in losses into ex-ante borrowing costs, resolution of non-systemic crises entails no moral hazard costs.
- For systemic countries (bail-ins generate spillovers):
  - IFIs should differentiate their policy approach by the degree of systemicity of the country.
  - To limit bail-ins and associated spillovers, IFIs should provide systemic countries with official transfers; otherwise, reductions in bail-ins may require excessive fiscal consolidation.
  - To contain moral hazard from transfers, IFIs should have commitment and operate under a pre-determined crisis-resolution framework so transfers are used only when systemic spillovers are particularly large and are complemented with stricter fiscal consolidation relative to non-recipient countries.
  - Without commitment, IFIs would over-rely on official transfers and generate severe moral hazard.
- The model supports going beyond conventional lending and mobilizing the international community for official transfers (for example, concessional financing or restructuring of debt held by the official sector) when dealing with systemic crises; but transfers should be reserved for highly systemic countries because of moral hazard concerns.

### Quantitative and mechanism notes from the model
- Moral hazard effects associated with transfers are proportional to ν/β (equation (19)); this ratio governs the planner’s trade-off between transfers and fiscal consolidation.
- Social welfare lower bound occurs when λ≥λ2, at which point it is socially efficient to entirely avoid bail-ins.
- Under no-commitment planner solutions, transfers without commitment can worsen ex-ante welfare relative to laissez-faire except for highly systemic countries.
- Under commitment, the planner sets fiscal consolidation, bail-ins, and transfers to respect upper bounds on fiscal consolidation (e.g., expressions for f̄ in the appendices), and proportions τ for bail-ins versus transfers depend on comparisons of expressions involving ξ, λ, and the terms (1−pC)/(1−pS) as shown in the appendix derivations.
- When interest rates are constant across states (Appendix C), two main implications arise:
  - Bail-ins become more costly because raising bail-ins increases R in the crisis state and tightens the financing constraint; planner limits bail-ins and relies more on fiscal consolidation and transfers.
  - Crisis-prevention effort becomes more valuable because it reduces the likelihood of crisis and relaxes the financing constraint in the crisis state by lowering the interest rate; efficient solutions increase effort in proportion to the shadow cost of the financing constraint.

### Conclusions and avenues for future research
- Summary of policy-relevant conclusions:
  - Differentiate crisis-resolution frameworks by systemicity.
  - Use transfers for systemic crises but only under IFI commitment and accompanied by stricter fiscal consolidation to contain moral hazard.
  - Consider regulatory and macroprudential measures to reduce spillover risks (e.g., higher capital buffers for banks holding systemic-country debt).
- Suggested directions for future work:
  - Design measures to prevent or mitigate spillovers, since social costs of resolving crises increase with spillover size.
  - Design ex-ante taxes/subsidies (a reserve fund or compensating scheme) to equalize expected welfare across countries and avoid incentives for countries to become more systemic to obtain transfers; the design of such schemes is left for future research.

*Italic: Source — 4.4    Welfare implications of alternative crisis-resolution frameworks (PDF chapter content).*

### References

### _wp15223 - References

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*Source: _wp15223 - References*

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