## wp18104

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### 2.1  Optimal default and taxation plans under discretion.

- Government problem and environment
  - Time structure: two-period endowment economy with period 1 financial decisions and period 2 realization of output Y taking values YH and YL with probabilities (1 π) and π respectively.
  - Households: risk neutral; invest W0 in public debt B issued at endogenous market price qb and a safe asset K at exogenous price q.
  - Government: benevolent but discretionary (cannot commit ex-ante to a tax schedule); rolls over exogenous financing need B0 by issuing B = B0/qb in period 1 and in period 2 chooses taxes Ti and haircut ρi ∈ [0,1] after observing output realization.
  - Period-2 government budget constraint:
    - Ti − G = (1 − ρi) B + φ ρi B
    - where φ ∈ (0,1) represents a budgetary cost of default proportional to defaulted payments and G is exogenous public spending invariant across states.
  - Deadweight tax distortion: zi(Ti) is strictly convex, with tax distortions larger and growing faster in the low-output state:
    - zL(:) > zH(:)
    - z0L(:) > z0H(:)
  - Fixed default cost: βi = β if ρi > 0, and 0 otherwise.

- Interior solution: first-order condition and implied haircut
  - Interior solution (0 < ρ^i < 1) pins down tax revenues via the first-order condition:
    - z0i( T^i ) = φ / (1 − φ)   (equation (5))
  - Combining the first-order condition with the budget constraint yields the interior haircut:
    - ρ^i = 1/(1 − φ) [ 1 − ( T^i − G ) / B ]   (equation (6))
  - Implication: in an interior solution, tax revenues (T^i) are not affected by the public debt bill B; additional debt increases the size of the haircut ρ^i. The government uses the haircut to smooth tax distortions across states.

- Default condition and minimum haircut
  - Default is optimal only when it increases household welfare (consumption), leading to the default condition:
    - zi(G + B) > zi(Ti) + β + φ ρi B   (equation (7))
  - For interior solutions, define a minimum haircut ρi implicitly by:
    - zi( T^i ) + β = zi( T^i + ( ρi (1 − φ) / (1 − ρi (1 − φ)) ) ( T^i − G ) ) − ( φ ρi / (1 − ρi (1 − φ)) ) ( T^i − G )   (equation (8))
  - Properties:
    - ρi is increasing in the fixed default cost β and independent of B.
    - The minimum haircut is higher in the high-output state: ρL < ρH.
    - If ρ^i < ρi, the interior solution is dominated by the no-default corner with ρi = 0 and Ti = G + B (complete repayment).

- Corner solutions
  - No-default corner (ρi = 0):
    - Ti = G + B (complete repayment).
  - Complete-default corner (ρi = 1) may be optimal when ρ^i > 1 and the corner default condition holds:
    - zi(G + B) > zi(G + φ B) + β + φ B   (equation (9))
    - Taxes set at Ti = G + φ B.
  - In corner solutions, tax revenues become contingent on B rather than marginal tax distortions; taxes cease to be state-contingent when haircut choice is constrained.

- Compact characterization (Table 1 summarized)
  - If ρ^i > 1 and zi(G + B) > zi(G + φ B) + β + φ B:
    - ρi = 1
    - Ti = G + φ B
  - If ρ^i ∈ [ρi, 1]:
    - ρi = ρ^i
    - Ti = T^i = (z0i)−1 [ φ / (1 − φ) ]  (i.e., T^i solves z0i(T^i) = φ/(1−φ))
  - Otherwise:
    - ρi = 0
    - Ti = G + B

### 3.1  A Modigliani-Miller irrelevance result

- Modigliani-Miller irrelevance under an interior haircut
  - If the optimal haircut lies in the interior region (DE < 1), tranching leaves equilibrium boundaries and the government’s debt bill B unchanged.
  - Intuition: the government raises the haircut on junior bonds to fully repay the senior tranche; investors discount junior bond prices q_b, which under risk neutrality exactly offsets the move to a risk-free price q in the senior tranche.
  - Key condition for irrelevance: constancy of the marginal (budgetary) default cost  and hence of the interior primary surplus ^T - G. If  were to increase (decrease) in , tranching would raise (lower) the primary surplus and thus lower (raise) government borrowing costs and incentives to default.

- Mechanics and Proposition 2 (interior case)
  - Proposition 2 defines a minimum senior tranche size ! that leads to a corner solution in the default equilibrium (DE) and shows:
    - @!/@B0 < 0.
    - For all ! < !, T_DE = ^T / (1 - !) * [B0 q - ^T - G] / [(1 - ) B0 q - ^T - G] < 1 (as expressed in the source), with partial derivatives indicating:
      - @_DE/@! > 0 and @/@! > 0.
      - @B_DE/@! = 0 and related derivatives equal 0 (see source expressions).
  - Interpretation: when the equilibrium haircut is interior, tranching simply reallocates payments across tranches without changing total payments to bondholders; hence average funding costs and default incentives are unchanged (Modigliani and Miller (1958) irrelevance).

- Comparative-static intuition (Figure 2 summary)
  - At senior tranche size ! with multiplicity, there is:
    - No default equilibrium (NDE): no haircut, bonds priced at risk-free q, B_NDE = B0 / q.
    - Default equilibrium (DE): haircut _DE on junior bonds, lower junior prices, higher debt bill B_DE.
  - Increasing ! to !0 > !:
    - Pivots market financing schedule B(; B0; !) leftward: more senior debt at price q reduces total debt bill at a given haircut.
    - Increases the minimum haircut  (default base smaller).
    - Shifts the optimal haircut schedule upward (government raises junior haircut to repay a larger senior share).
  - Result: NDE unchanged; DE moves but the higher junior haircut offsets changes in B_DE, leaving equilibrium boundaries unaffected — tranching cannot eliminate debt crises in this interior case.

- Non-neutrality when haircut is at the corner (complete default) — Proposition 3
  - When the optimal haircut reaches the upper corner (_DE = 1), tranching no longer satisfies the Modigliani-Miller irrelevance result.
  - Proposition 3 (for ! > !): in the corner default equilibrium
    - _DE = 1
    - T_DE = G + (! + (1 - !)) B_DE
    - B_DE = B0 q (1 - (1 - !))
    - With @T_DE/@! > 0 and @B_DE/@! < 0.
  - There exists a minimum senior tranche ! (implicitly defined in equation (23) in the source) such that:
    - ! < ! < 1
    - @!/@B0 > 0.
  - Mechanisms when _DE = 1:
    - Government cannot further raise junior haircut; must instead increase tax revenues to fully repay senior tranche, raising total payments to bondholders.
    - Two channels reduce default temptation:
      1. Without a rise in haircut, junior bond prices q_b remain constant as ! increases; because senior bonds trade at q > q_b, tranching reduces average borrowing cost (!q + (1 - !)q_b)^{-1} and total debt bill B, lowering required taxes T = G + B to avoid default.
      2. Higher tax revenues needed under default to repay senior bonds increase tax distortions z(G + (! + (1 - !))B), reducing the benefit of default.
  - Implication: for sufficiently large ! (above !), tranching can eliminate the default equilibrium.

- Risky senior debt and Proposition 4
  - Relaxing the assumption of non-defaultable senior tranche: senior haircut _s and prices
    - q_s = q(1 - _s)
    - q_b = q(1 - )
  - Government budget and market financing (equation (24)):
    - T - G = [!(1 - (1 - ) _s) + (1 - !) ] B
    - B = B0 q [1 -  + !(1 - _s)]
  - In an interior solution senior haircut:
    - ^_s = 1 / [!(1 - )] * [! + (1 - !) - (^T - G) / B * !] (expression (25))
  - Proposition 4 characterizes a senior default equilibrium (SDE) where government defaults on both tranches:
    - It defines _SDE_s (min{..., 1}), _SDE, B_SDE and T_SDE with cases depending on whether _SDE_s = 1.
    - B_SDE = B_DE and T_SDE = T_DE in the regime where the government defaults on the senior tranche; hence MM irrelevance reinstated once senior tranche is defaulted on.
    - There exists an implicit threshold ! such that for all ! > ! the SDE is self-confirming; @!/@_s > 0 and @!/@B0 < 0.
    - For sufficiently high additional fixed senior default cost _s, the set ! ∈ (!, !) is non-empty — an intermediate region where tranching can eliminate default by inducing tax increases rather than senior default.
    - Comparative statics: ! increases with B0 while ! decreases with B0; larger government financing needs require higher _s for tranching to be effective.

- Numerical illustration and calibration (section 5)
  - Calibration:
    - Tax distortion z_i(T_i) = (i) T_i^2 such that interior tax revenues ^T_i =  [2 (i) (1 - )]^{-1}.
    - Set H small and calibrate L > H to attain ^T_L - G = 0:40 (with G = 0).
    - Expected output E[Y] = (1 - ) Y_H +  Y_L normalized to unity; primary surplus interpreted as 40% of expected GDP.
    - Fixed and budgetary default costs calibrated to  =  = 0:10 (10% of expected GDP each when B0 = 1).
    - q = 0:99 (risk-free interest rate of 1%).
    -  = 0:10.
  - Results (summary from Figures described in the source):
    - Without tranching (! = 0), equilibria evolve with B0: unique NDE → multiplicity (NDE and DE) → unique DE.
    - Haircut in state L jumps discretely to a minimum haircut  = 0:76 when switching from NDE to DE and then rises with B0 while interior.
    - Sovereign spreads under DE are around 10% in this calibration.
    - Tranching thresholds:
      - ! (minimum tranche size to reach corner) decreases with B0.
      - ! (minimum senior tranche size to prevent default) increases with B0.
      - Eliminating the bad equilibrium in multiplicity requires roughly a senior tranche size of approximately 50% in the calibration presented.
    - Increasing _s (additional fixed senior default cost) raises !, creating an intermediate region where tranching can prevent default.

- Policy-relevant conclusions
  - Tranching redistributes government revenues from junior to senior bondholders unless:
    - Default costs are higher for senior than junior bonds (_s sufficiently large), and
    - The senior tranche size ! lies in an intermediate region (! ∈ (!, !) where ! < !).
  - In that intermediate region, tranching reduces vulnerability to debt crises because the government, unable to extract more from junior bondholders, must increase tax revenues to pay senior bondholders; higher total payments can reduce borrowing costs under default expectations and eliminate the default equilibrium.
  - Broader implication: mechanisms apply to any setting with fixed default costs and agents able to take costly but non-contractible actions to increase recovery rates; tranching may have a role in corporate debt in sectors with significant moral hazard.

*Source: IMF Working Paper (sections 2.1 and 3.1 as provided in the supplied PDF).*

### 2.1  Optimal default and taxation plans under discretion. . . . . . . . . . . . . .9

### 2.1  Optimal default and taxation plans under discretion.

### Government problem and environment
- Time structure: two-period endowment economy with period 1 financial decisions and period 2 realization of output Y taking values YH and YL with probabilities (1 π) and π respectively.
- Households: risk neutral; invest W0 in public debt B issued at endogenous market price qb and a safe asset K at exogenous price q.
- Government: benevolent but discretionary (cannot commit ex-ante to a tax schedule); rolls over exogenous financing need B0 by issuing B = B0/qb in period 1 and in period 2 chooses taxes Ti and haircut ρi ∈ [0,1] after observing output realization.
- Period-2 government budget constraint:
  - Ti − G = (1 − ρi) B + φ ρi B
  - where φ ∈ (0,1) represents a budgetary cost of default proportional to defaulted payments and G is exogenous public spending invariant across states.
- Deadweight tax distortion: zi(Ti) is strictly convex, with tax distortions larger and growing faster in the low-output state:
  - zL(:) > zH(:)
  - z0L(:) > z0H(:)
- Fixed default cost: βi = β if ρi > 0, and 0 otherwise.

### Interior solution: first-order condition and implied haircut
- Interior solution (0 < ρ^i < 1) pins down tax revenues via the first-order condition:
  - z0i( T^i ) = φ / (1 − φ)   (equation (5))
- Combining the first-order condition with the budget constraint yields the interior haircut:
  - ρ^i = 1/(1 − φ) [ 1 − ( T^i − G ) / B ]   (equation (6))
- Implication: in an interior solution, tax revenues (T^i) are not affected by the public debt bill B; additional debt increases the size of the haircut ρ^i. The government uses the haircut to smooth tax distortions across states.

### Default condition and minimum haircut
- Default is optimal only when it increases household welfare (consumption), leading to the default condition:
  - zi(G + B) > zi(Ti) + β + φ ρi B   (equation (7))
- For interior solutions, define a minimum haircut ρi implicitly by:
  - zi( T^i ) + β = zi( T^i + ( ρi (1 − φ) / (1 − ρi (1 − φ)) ) ( T^i − G ) ) − ( φ ρi / (1 − ρi (1 − φ)) ) ( T^i − G )   (equation (8), written here preserving structure of the source)
- Properties:
  - ρi is increasing in the fixed default cost β and independent of B.
  - The minimum haircut is higher in the high-output state: ρL < ρH.
  - If ρ^i < ρi, the interior solution is dominated by the no-default corner with ρi = 0 and Ti = G + B (complete repayment).

### Corner solutions
- No-default corner (ρi = 0):
  - Ti = G + B (complete repayment).
- Complete-default corner (ρi = 1) may be optimal when ρ^i > 1 and the corner default condition holds:
  - zi(G + B) > zi(G + φ B) + β + φ B   (equation (9))
  - Taxes set at Ti = G + φ B.
- In corner solutions, tax revenues become contingent on B rather than marginal tax distortions; taxes cease to be state-contingent when haircut choice is constrained.

### Compact characterization (Table 1 summarized)
- If ρ^i > 1 and zi(G + B) > zi(G + φ B) + β + φ B:
  - ρi = 1
  - Ti = G + φ B
- If ρ^i ∈ [ρi, 1]:
  - ρi = ρ^i
  - Ti = T^i = (z0i)−1 [ φ / (1 − φ) ]  (i.e., T^i solves z0i(T^i) = φ/(1−φ))
- Otherwise:
  - ρi = 0
  - Ti = G + B

*Source: IMF Working Paper (section 2.1 as provided in the supplied PDF).*

### 3.1  A Modigliani-Miller irrelevance result

### 3.1  A Modigliani-Miller irrelevance result

### Modigliani-Miller irrelevance under an interior haircut
- If the optimal haircut lies in the interior region (DE < 1), tranching leaves equilibrium boundaries and the government’s debt bill B unchanged.
- Intuition: the government raises the haircut on junior bonds to fully repay the senior tranche; investors discount junior bond prices q_b, which under risk neutrality exactly offsets the move to a risk-free price q in the senior tranche.
- Key condition for irrelevance: constancy of the marginal (budgetary) default cost  and hence of the interior primary surplus ^T - G. If  were to increase (decrease) in , tranching would raise (lower) the primary surplus and thus lower (raise) government borrowing costs and incentives to default.

### Mechanics and Proposition 2 (interior case)
- Proposition 2 defines a minimum senior tranche size ! that leads to a corner solution in the default equilibrium (DE) and shows:
  - @!/@B0 < 0.
  - For all ! < !, T_DE = ^T / (1 - !) * [B0 q - ^T - G] / [(1 - ) B0 q - ^T - G] < 1 (as expressed in the source), with partial derivatives indicating:
    - @_DE/@! > 0 and @/@! > 0.
    - @B_DE/@! = 0 and related derivatives equal 0 (see source expressions).
- Interpretation: when the equilibrium haircut is interior, tranching simply reallocates payments across tranches without changing total payments to bondholders; hence average funding costs and default incentives are unchanged (Modigliani and Miller (1958) irrelevance).

### Comparative-static intuition (Figure 2 summary)
- At senior tranche size ! with multiplicity, there is:
  - No default equilibrium (NDE): no haircut, bonds priced at risk-free q, B_NDE = B0 / q.
  - Default equilibrium (DE): haircut _DE on junior bonds, lower junior prices, higher debt bill B_DE.
- Increasing ! to !0 > !:
  - Pivots market financing schedule B(; B0; !) leftward: more senior debt at price q reduces total debt bill at a given haircut.
  - Increases the minimum haircut  (default base smaller).
  - Shifts the optimal haircut schedule upward (government raises junior haircut to repay a larger senior share).
- Result: NDE unchanged; DE moves but the higher junior haircut offsets changes in B_DE, leaving equilibrium boundaries unaffected — tranching cannot eliminate debt crises in this interior case.

### Non-neutrality when haircut is at the corner (complete default) — Proposition 3
- When the optimal haircut reaches the upper corner (_DE = 1), tranching no longer satisfies the Modigliani-Miller irrelevance result.
- Proposition 3 (for ! > !): in the corner default equilibrium
  - _DE = 1
  - T_DE = G + (! + (1 - !)) B_DE
  - B_DE = B0 q (1 - (1 - !))
  - With @T_DE/@! > 0 and @B_DE/@! < 0.
- There exists a minimum senior tranche ! (implicitly defined in equation (23) in the source) such that:
  - ! < ! < 1
  - @!/@B0 > 0.
- Mechanisms when _DE = 1:
  - Government cannot further raise junior haircut; must instead increase tax revenues to fully repay senior tranche, raising total payments to bondholders.
  - Two channels reduce default temptation:
    1. Without a rise in haircut, junior bond prices q_b remain constant as ! increases; because senior bonds trade at q > q_b, tranching reduces average borrowing cost (!q + (1 - !)q_b)^{-1} and total debt bill B, lowering required taxes T = G + B to avoid default.
    2. Higher tax revenues needed under default to repay senior bonds increase tax distortions z(G + (! + (1 - !))B), reducing the benefit of default.
- Implication: for sufficiently large ! (above !), tranching can eliminate the default equilibrium.

### Risky senior debt and Proposition 4
- Relaxing the assumption of non-defaultable senior tranche: senior haircut _s and prices
  - q_s = q(1 - _s)
  - q_b = q(1 - )
- Government budget and market financing (equation (24)):
  - T - G = [!(1 - (1 - ) _s) + (1 - !) ] B
  - B = B0 q [1 -  + !(1 - _s)]
- In an interior solution senior haircut:
  - ^_s = 1 / [!(1 - )] * [! + (1 - !) - (^T - G) / B * !] (as given in the source expression (25)).
- Proposition 4 characterizes a senior default equilibrium (SDE) where government defaults on both tranches:
  - It defines _SDE_s (min{..., 1}), _SDE, B_SDE and T_SDE with cases depending on whether _SDE_s = 1.
  - B_SDE = B_DE and T_SDE = T_DE in the regime where the government defaults on the senior tranche; hence MM irrelevance reinstated once senior tranche is defaulted on.
  - There exists an implicit threshold ! such that for all ! > ! the SDE is self-confirming; @!/@_s > 0 and @!/@B0 < 0.
  - For sufficiently high additional fixed senior default cost _s, the set ! ∈ (!, !) is non-empty — an intermediate region where tranching can eliminate default by inducing tax increases rather than senior default.
- Comparative statics: ! increases with B0 while ! decreases with B0; larger government financing needs require higher _s for tranching to be effective.

### Numerical illustration and calibration (section 5)
- Calibration:
  - Tax distortion z_i(T_i) = (i) T_i^2 such that interior tax revenues ^T_i =  [2 (i) (1 - )]^{-1}.
  - Set H small and calibrate L > H to attain ^T_L - G = 0:40 (with G = 0).
  - Expected output E[Y] = (1 - ) Y_H +  Y_L normalized to unity; primary surplus interpreted as 40% of expected GDP.
  - Fixed and budgetary default costs calibrated to  =  = 0:10 (10% of expected GDP each when B0 = 1).
  - q = 0:99 (risk-free interest rate of 1%).
  -  = 0:10.
- Results (summary from Figures described in the source):
  - Without tranching (! = 0), equilibria evolve with B0: unique NDE → multiplicity (NDE and DE) → unique DE.
  - Haircut in state L jumps discretely to a minimum haircut  = 0:76 when switching from NDE to DE and then rises with B0 while interior.
  - Sovereign spreads under DE are around 10% in this calibration.
  - Tranching thresholds:
    - ! (minimum tranche size to reach corner) decreases with B0.
    - ! (minimum senior tranche size to prevent default) increases with B0.
    - Eliminating the bad equilibrium in multiplicity requires roughly a senior tranche size of approximately 50% in the calibration presented.
  - Increasing _s (additional fixed senior default cost) raises !, creating an intermediate region where tranching can prevent default.

### Policy-relevant conclusions
- Tranching redistributes government revenues from junior to senior bondholders unless:
  - Default costs are higher for senior than junior bonds (_s sufficiently large), and
  - The senior tranche size ! lies in an intermediate region (! ∈ (!, !) where ! < !).
- In that intermediate region, tranching reduces vulnerability to debt crises because the government, unable to extract more from junior bondholders, must increase tax revenues to pay senior bondholders; higher total payments can reduce borrowing costs under default expectations and eliminate the default equilibrium.
- Broader implication: mechanisms apply to any setting with fixed default costs and agents able to take costly but non-contractible actions to increase recovery rates; tranching may have a role in corporate debt in sectors with significant moral hazard.

*Source: wp18104 - 3.1  A Modigliani-Miller irrelevance result*

### References

### References and Appendix (wp18104 - References)

### References
- Beck, T., Wagner, W., and Uhlig, H. (2011). Insulating the Önancial sector from the european debt crisis: Eurobonds without public guarantees. Short article at Voxeu.org. Attained from: https://voxeu.org/article/eurobonds-without-public-guarantees-insulating-Önancial-sector-european-debt-crisis.
- Brunnermeier, M. K., Garicano, L., Lane, P. R., Pagano, M., Reis, R., Santos, T., Thesmar, D., Van Nieuwerburgh, S., and Vayanos, D. (2016). The sovereign-bank diabolic loop and esbies. American Economic Review, 106(5):508ñ12.
- Brunnermeier, M. K., LangÖeld, S., Pagano, M., Reis, R., Van Nieuwerburgh, S., and Vayanos, D. (2017). ESBies: safety in the tranches. Economic Policy, 32(90):175ñ219.
- Calvo, G. A. (1988). Servicing the public debt: The role of expectations. The American Economic Review, 78(4):647ñ661.
- Cole, H. L. and Kehoe, T. J. (2000). Self-fulÖlling debt crises. The Review of Economic Studies, 67(1):91ñ116.
- Corsetti, G. and Dedola, L. (2016). The mystery of the printing press: Monetary policy and self-fulÖlling debt crises. Journal of the European Economic Association, 14(6):1329ñ1371.
- Delpla, J. and von Weizs‰cker, J. (2010). The Blue Bond Proposal. Policy Briefs 403, Bruegel.
- Eij¢ nger, S. C. (2011). Eurobondsñconcepts and implications. Technical report, BrieÖng Note to the European Parliament.
- European Commission (2011). Green paper on the feasibility of introducing stability bonds. Technical report.
- German Council of Economic Experts (2012). After the euro area summit: Time to implement long-term solutions. Technical report.
- Hatchondo, J. C., Martinez, L., and Onder, Y. K. (2017). Non-defaultable debt and sovereign risk. Journal of International Economics, 105(C):217ñ229.
- Lorenzoni, G. and Werning, I. (2013). Slow moving debt crises. Working Paper 19228, National Bureau of Economic Research.
- Modigliani, F. and Miller, M. H. (1958). The cost of capital, corporation Önance and the theory of investment. The American Economic Review, 48(3):261ñ297.
- Muellbauer, J. (2011). Resolving the eurozone crisis: time for conditional eurobonds. CEPR Policy Insight, 59:12.
- Nicolini, J. P., Teles, P., Ayres, J. L., and Navarro, G. (2015). Sovereign Default: The Role of Expectations. Working Papers 723, Federal Reserve Bank of Minneapolis.
- Philippon, T. and Hellwig, C. (2011). Eurobills, not eurobonds. Short article at Voxeu.org. Attained from: https://voxeu.org/article/eurobills-not-euro-bonds.

### Appendix A — Proofs of Propositions (overview)
- Contains formal derivations and equilibrium characterizations for:
  - No default equilibrium (NDE)
  - Default equilibrium (DE), interior and corner cases
  - Coexistence conditions for NDE and DE
  - Comparative statics with respect to the senior tranche size ! and financing need B0
  - Proofs for Propositions 1–4 with implicit function theorem applications

### A.1 Proof of Proposition 1 — No default equilibrium (NDE)
- NDE defined by NDE = 0.
- Interest parity: qNDEb = qBNDE = B0q.
- Condition for absence of default:
  - ^NDE = 1 1  ^T G 1  qBNDE < 
  - Rearranged in terms of government debt:
    - B NDE < ^T G 1 (1 )
  - In terms of initial financing need:
    - B0 < q  ^T G  1 (1 )  (equation (26))
- Sufficient condition for NDE to be self-confirming:
  - B0 < B0  q  ^T G  1 (1 )

### A.1 Proof of Proposition 1 — Default equilibrium (DE)
- Interior DE ( in [;1), with ^DE = 1 1  ^T G (1 )BDE qDEb = q "1  1  1  ^T G BDE!"
- In terms of initial financing needs:
  - ^DE = B0 ^T G q(1 )B0 ^T G
  - qDEb = (1  )B0 q(1 )B0 ^T G
  - BDE = (1 ) B0 q ^T G 1  
- Three interior equilibrium existence conditions (expressed in the source) include:
  - ^DE >  and BDE > ^T G 1 (1 )
  - ^DE < 1 and BDE < ^T G 
- Corner solution with DE+ = 1:
  - qDE+ b = q(1 )
  - BDE+ b = B0 q(1 )
  - Equilibrium condition: ^DE+ 1 $ BDE+  ^T G 
  - In terms of financing needs:
    - B0  1  q  ^T G 
- Coexistence of NDE and DE when financing needs fall in range:
  - B0 B0 < B0 ) 1  1 (1 )  ^T G   B0 q < ^T G 1 (1 )
- Non-empty coexistence requires:
  - 1  1 (1 )  ^T G  < ^T G 1 (1 )
  - This is satisfied when fixed cost of default >0.

### A.2 Proof of Proposition 2 — Comparative statics with respect to !
- For interior DE solution (Appendix B), (32a) indicates:
  - @BDE/@! = 0 as long as ! does not lead to corner solution DE = 1.
- Effect of ! on ^DE:
  - @DE/@! = 1 (1 ) (1 !)2 1  ^T G B !
  - = DE 1 ! > 0  (equation (27))
- Implicit function setup: define f(!; ) such that equation (20) corresponds to f(!; ) = 0.
- Signs of partial derivatives:
  - @f/@! < 0 under condition involving z0 and other terms (equation (28) and subsequent).
  - @f/@ > 0.
- By implicit function theorem:
  - @^/@! = -(@f/@!)/(@f/@) =  1 ! > 0  (equation (29))
  - Therefore @^/@! > @/@! for ^ > .
- Boundaries B0 and B0 are independent of ! in interior solutions:
  - Define g(!;B) and show @B/@! = 0 on boundary; hence @B0/@! = @B0/@! = 0.
- Conclusion: ! has no effect on equilibrium boundaries when haircut falls short of upper corner DE = 1.

### A.3 Proof of Proposition 3 — Properties of ! (minimum senior tranche eliminating DE)
- BDE given by (33); combining with corner default condition (21) yields implicit expression (23) for !.
- Proposition 1 implies ! > !.
- Show ! < 1 by setting ! = 1 in default condition (21): not defaulting leads to strictly higher welfare for any fixed default cost >0.
- ! increases in B0:
  - Define f(!; B0) such that f(!; B0) = 0.
  - Derivatives:
    - @f/@! < 0 and @f/@B0 > 0 under stated sufficient condition.
  - By implicit function theorem:
    - @!/@B0 = -(@f/@B0)/(@f/@!) > 0.

### A.4 Proof of Proposition 4 — Social planner / SDE comparative statics
- Interior equilibrium expressions:
  - B SDE = (1 ) B0 q ^T G 1   SDE s = 1 ! (!+(1 !)) B0 q (1 (1 !)) ^T G (1 ) B0 q ^T G
  - Taxes ^T pinned down by (5).
- Interior equilibrium valid if:
  - B0 q > 1  ^T G; otherwise corner with SDE s = 1 and
    - B SDE = B0 q(1 )
    - T SDE = G +  B SDE
- Default condition in interior equilibrium:
  - z  ^T+ (1 )x(!; B0)  > z  ^T  +  s +  x(!; B0)  where x(!; B0) defined explicitly.
- Define f(B0;  s; !) such that f(B0;  s; !) = 0 (equation (30)).
- Derivatives and signs:
  - @x/@! > 0 and @x/@B0 > 0 under regularity condition 1  > .
  - Given z0 conditions, @f/@! > 0 and @f/@B0 > 0.
- Implicit function theorem yields:
  - @!/ @ s > 0
  - @!/ @B0 < 0
  - Also ! > ! for any  s > 0.
- Corner solution analysis similarly yields:
  - @!/ @ s > 0 and @!/ @B0 < 0
  - And ! > ! for all  s > 0.

### Appendix B — Equilibria under tranching (overview)
- Tranching does not change NDE when both senior and junior bonds are repaid fully:
  -  = 0
  - qNDEb = qBNDE = B0 q
  - TNDE = G + B0 q
- Senior tranche size thresholds:
  - Senior tranches below size !  !NDE  1 1   ^T G B0 = q  ! leave NDE condition identical to no-tranching case.
  - For !  !NDE any deviation to default is at corner; condition for NDE becomes B0  B0(!) implicitly defined by:
    - z  G+ B0 q  = z  G+ (!+ (1 !)) B0 q  +  + (1 !) B0 q  (and @B0/@! > 0 for 8!2   !NDE ;1)
- Default equilibrium (DE) with tranching — interior case:
  - ^DE = 1 1 ! B0 q ^T G (1 ) B0 q ^T G  (equation (31))
  - qDEb = q 1 ! ((1 ) (1 !) ) B0 q + ! ^T G (1 ) B0 q ^T G
  - BDE = B0 !q+ (1 !) qDEb
  - TDE = ^T
- Key mechanism: government raises haircuts on junior bonds to pay senior tranche; qDEb declines anticipating higher haircut. Under risk neutrality, decline in qDEb offsets higher senior prices so total government debt bill remains unchanged:
  - BDE = (1 ) B0 q ^T G 1   (equation (32a))
- Validity of interior DE when:
  - !  !  (1 ) ^T G   B0 q (1 ) B0 q ^T G
- For ! > !, DE is upper corner with DE+ = 1 and:
  - qDE+ b = q(1 )
  - BDE+ = B0 !q+ (1 !) qDE+
  - TDE+ = G+ !+ (1 !)  1 (1 !) B0 q
  - Debt bill decreases in ! in corner: BDE+ = B0 q [1 (1 !)] 1  (equation (33))
- DE is self-confirming for financing need B0  B0 where B0 implicitly defined by (34):
  - z  G+ 1 1 (1 !) B0 q  = z  G+ !+ (1 !)  1 (1 !) B0 q  +  + (1 !) 1 (1 !) B0 q  (equation (34))
  - @B0/@! > 0 for 8!2 (!;1)
- Interpretation: (34) defines a minimum senior tranche ! that eliminates DE; ! > !, increasing in B0 but strictly below 1 at any given B0.

*Source: wp18104 - References (IMF working paper appendix and references).*

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_Source: https://www.imf.org/-/media/files/publications/wp/2018/wp18104.pdf_