## 2. Equilibrium Payoffs and Welfare Loss

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---

### Introduction and economic mechanism
- Short-term debt reliance by banks (e.g., ABCP, repo) creates rollover risk; spikes in uncertainty can freeze short-term funding and cause systemic distress.
- Model focus: interaction between bank investment (asset side) and short-term wholesale financing (liability side) under idiosyncratic and aggregate uncertainty.
- Economic mechanism:
  - Short-term lending emerges from optimal contracting to limit risk-shifting moral hazard.
  - Uncertainty in asset quality + rollover need can cause inefficient liquidation (debt runs) in some states, reducing investment surplus and causing ex ante credit contraction and underinvestment.
- Trade-offs:
  - Short-term debt disciplines bank risk-taking by enabling runs that force liquidation when signaled revenue is low.
  - Runs produce social deadweight loss and contractionary effects, aggravated by aggregate uncertainty.

### Model setup and assumptions
- Agents and timing:
  - Three parties: firm, bank, investor (uninsured wholesale creditor).
  - Dates t = 0; 1; 2. Two-period opportunity cost of capital is R = 1 + r > 1.
- Projects:
  - Two project types: good (g) and bad (b). Technology: invest A at t = 0 to obtain z Y(A) at t = 2, where z ∈ {z_g, z_b} with distributions F_g and F_b.
  - Bad project: lower expected revenue than good:
    ∫_0^1 z dF_g(z) > ∫_0^1 z dF_b(z)
    and higher upside risk by second order stochastic dominance (SOSD): there exists z̄ such that F_g(z̄) = F_b(z̄) and (F_g(z) - F_b(z))(z - z̄) ≥ 0 for all z.
- Signal and liquidation:
  - At t = 1 a public signal perfectly predicts z realized at t = 2.
  - If liquidated at t = 1, capital sold at discount price of ϕ ∈ [0; R) per unit at t = 2.
- Assumptions:
  - Assumption 1: Y(A) strictly concave, increasing, Inada: lim_{A→0} Y'(A) = 1; lim_{A→∞} Y'(A) = 0; and Y'(A) is convex.
  - Assumption 2: For F ∈ {F_g, F_b}, survival function 1 - F is log-concave; hazard rate h(z) = f(z)/(1 - F(z)) is continuous and increasing.

### First-best investment (social planner)
- Planner liquidates when z < z(A), where z(A) solves:
  - z Y(A) = ϕ A.  (equation (1))
- Planner maximizes:
  - max_A ϕ A F_g(z) + ∫_z^1 z Y(A) dF_g(z) - R A.  (equation (2))
- First-order condition:
  - ϕ F_g(z) + ∫_z^1 z Y'(A) dF_g(z) - R = 0.  (equation (3))
- Proposition 1: There exists a unique first-best investment level A_fb ∈ (0;1).

### Contracting framework — bank–firm contracting
- Bank–firm contract characterized by (A; B): bank chooses project type and invests A at t = 0; firm pays B at t = 2 if project continues; liquidation at t = 1 gives bank ϕ A.
- Firm's liquidation threshold under the contract:
  - z̄ Y(A) = B.  (equation (4))
- With full renegotiation power for the bank, ex post renegotiation can restore the efficient threshold z(A) and the bank can capture surplus by making firm indifferent to continuation vs liquidation.
- Bank chooses (A; B) solving:
  - max_{A,B} ϕ A F_g(z̄) + ∫_{z̄}^1 z Y(A) dF_g(z) + B(1 - F_g(z̄)) - R A  subject to firm's participation (PC_F).  (equations (5) and (6))
- Proposition 2: If the bank has all bargaining power in renegotiation, the bank gives the firm the first-best loan A_fb and sets B_fb to satisfy the firm's reservation payoff.

### Contracting framework — bank–creditor contracting and incentive constraints
- Bank finances via collateralized short-term debt (repo): sells assets A for price D at t = 0 and repurchases at t = 2 for ~D; creditor can decide to roll over at t = 1.
- Put-option representation: price of put option on assets with strike ~D denoted p_g(~D; A) for good assets.
- Creditor's expected gross payoff from good project:
  - U_C^g(A) = ~D - p_g(~D; A).
- Bank payoff from good and bad projects:
  - U_B^g(A) = ϕ A F_g(~z) + ∫_{~z}^1 z Y(A) dF_g(z) - U_C^g(A) - U_F^g(A).
  - U_B^b(A) analogous with F_b.
- Incentive compatibility constraint for bank (IC_B): ensure bank chooses good project:
  - Δ U_B(A) ≥ Δ p(~D; A),  (equation (7))
  where
  - Δ U_B(A) = ∫_{z̄}^{~z} z Y(A) d(F_g - F_b) + ϕ A (F_g(~z) - F_b(~z)) - B(F_g(z̄) - F_b(z̄)),
  - Δ p = p_b(~D; A) - p_g(~D; A).
- Lemma 1: If liquidation value ϕ is sufficiently small, IC_B binds and there exists unique ~D_*(A).
- Lemma 2: If ϕ is sufficiently small, there exists unique ~z ∈ (z; z̄) such that ~D_*(A) = ~z Y(A).
- Creditor participation constraint (PCC) and market value of debt D_*:
  - U_C^g(A) = ~D_*(A) - p_g(~D_*; A) ≥ R D, competitive lending pins down D_* solving:
    - R D_* = ~D_*(A) - p_g(~D_*; A).  (equation (9))
- Using ~D_*(A) = ~z Y(A), PCC becomes:
  - R D_* = ϕ A F_g(~z) + ~z Y(A) (1 - F_g(~z)).  (equation (10))
- Lemma 3: There is a unique bank debt-to-asset ratio d_* = ~D_*/A < 1 for given A.

### Equilibrium financing and investment
- Banking equilibrium is quadruplet (A_*; B_*; D_*; ~D_*) where (A_*; B_*) solve bank–firm contracting and (D_*; ~D_*) solve bank–creditor contracting; equilibrium implies liquidation threshold ~z.
- Bank's problem reduces to:
  - max_A ϕ A F_g(z̄) + ∫_{z̄}^1 z Y(A) dF_g(z) - R A - C - ∫_{~z}^{z̄} (z Y(A) - ϕ A) dF_g(z).  (equation (12))
  - Last term represents surplus loss from creditor-induced liquidation in ( ~z; z̄ ).
- Proposition 3: If ϕ is sufficiently small, there exists unique banking equilibrium (A_*; B_*; D_*; ~D_*). Investment is lower than first-best: A_* < A_fb.

### Extensions — aggregate uncertainty, portfolio correlation, endogenous default
- Aggregate uncertainty (mean-preserving spread):
  - Replace F_g by F_m with same mean: ∫_0^1 z dF_g(z) = ∫_0^1 z dF_m(z)
    and F_g(z̄) = F_m(z̄) with (F_g(z) - F_m(z))(z - z̄) ≥ 0.  (equation (13))
  - Proposition 4: Conditional on existence of banking equilibrium (A_*^m; B_*^m; D_*^m; ~D_*^m), higher uncertainty leads to lower credit supply and lower investment: A_*^m < A_*.
  - Intuition: higher uncertainty forces banks to shrink balance sheets to contain risk; socially costly via investment contraction.
- Multiple assets and correlation:
  - Introducing two types of good assets with same mean and variance; increase in correlation ρ increases portfolio variance.
  - Corollary 1: All else equal, an increase in return correlation of the bank's asset portfolio leads to lower credit supply and lower investment.
- Endogenous default and debt maturity:
  - Allowing endogenous default (equity chooses default when z < ~z) preserves prior results: endogenous default is costly and creditors bear cost.
  - Long-term debt vs short-term:
    - If endogenous default is prohibited, long-term debt can increase debt capacity and restore first-best.
    - If endogenous default is possible, advantage of long-term debt disappears; short-term debt remains an optimal instrument for creditors.

### Policy options — efficiency condition and instruments
- Efficiency condition for no welfare loss in banking equilibrium:
  - Proposition 5: Banking equilibrium (A_*; B_*; D_*; ~D_*) incurs no welfare loss iff bank's liquidation value fully repays creditor:
    - ϕ A ≥ ~D.  (equation (14))
  - Interpretation: debt value cannot be too high relative to liquidation value; trade-off between discipline and social efficiency.
- Ex ante policies
  - Capital requirement:
    - Minimum equity-to-asset ratio to satisfy (14) at equality:
      - E / A = (R - ϕ) / R.  (equation (15))
    - Proposition 6:
      - If no equity premium (R_e = R), imposing debt-to-equity ratio ϕ/(R - ϕ) achieves first-best investment.
      - If equity premium exists (R_e > R), investment under such capital requirement remains lower than first-best.
    - Capital requirement eliminates creditor default risk ex post but raises ex ante marginal cost if equity premium exists.
    - Regulatory implication: limiting short-term debt size or imposing net stable funding ratio (NSFR) so long-term funding + equity ratio equals (R - ϕ)/R can achieve efficient funding composition.
  - Liquidity requirement:
    - Require holding liquid assets L funded by equity so that:
      - L + ϕ A ≥ ~D.  (equation (16))
    - Imposing L such that L + ϕ A = ~D guarantees payment to creditor and can restore first-best absent equity premium.
  - Contingent debt:
    - State-contingent liabilities can remedy inefficiency if conversion triggers are based on aggregate variables (not individual outcomes) to preserve bank incentives.
    - Aggregate-triggered conversion that pays ~D if z ≥ ~z and ϕ A if z < ~z preserves creditor payoff and prevents runs in (z; z̄), eliminating welfare loss.
    - Caveat: when banking sector is concentrated ("too big to fail"), aggregation vs idiosyncratic distinction blurs and systemic triggers may distort incentives.
- Ex post policies
  - Monetary policy (interest rate reduction):
    - Creditor required return satisfies:
      - R D_* = ϕ A F_g(~z) + ~z Y(A) (1 - F_g(~z)).
    - Proposition 7: Given committed asset and debt values at t = 0, reduce effective cost of capital from R to
      - R_m = ϕ A_* / D_*  (equation (17))
      at t = 1 to eliminate welfare loss by lowering creditor's liquidation threshold and preventing inefficient runs.
    - Monetary intervention is ex post efficient but does not affect ex ante investment; capital requirements are preferable to the extent they raise ex ante investment toward A_fb.

### Conclusion
- Short-term debt is an optimal discipline device but creates rollover risk that, under asset-quality uncertainty and limited bank capital, leads to inefficient liquidations and ex ante underinvestment.
- Interaction between bank investment decisions and debt capacity is the key channel of uncertainty transmission to the real economy.
- Policy prescriptions that can improve welfare while preserving micro-level discipline include:
  - Ex ante: capital requirements, liquidity requirements, NSFR-style long-term funding mandates, and carefully designed aggregate-trigger contingent debt.
  - Ex post: temporary interest rate reductions to prevent inefficient runs when no adequate ex ante buffer exists.
- Implementation success depends on regulator ability to preserve market discipline while extracting surplus; concentrated banking increases moral hazard when systemic interventions are used.

*Source — "2. Equilibrium Payoffs and Welfare Loss" (contents extracted from _wp1565 - 2. Equilibrium Payoffs and Welfare Loss).*

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

### References

### Contents
- References ................................................................................................................................28

### Figures
- 1. Timing of Events ....................................................................................................................5

*Source: _wp1565 - References; canonical URL: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2015/_wp1565.pdf*

### 2. Equilibrium Payoffs and Welfare Loss ................................................................................

### 2. Equilibrium Payoffs and Welfare Loss

### Introduction and economic mechanism
- Short-term debt reliance by banks (e.g., ABCP, repo) creates rollover risk; spikes in uncertainty can freeze short-term funding and cause systemic distress.
- Model focus: interaction between bank investment (asset side) and short-term wholesale financing (liability side) under idiosyncratic and aggregate uncertainty.
- Economic mechanism:
  - Short-term lending emerges from optimal contracting to limit risk-shifting moral hazard.
  - Uncertainty in asset quality + rollover need can cause inefficient liquidation (debt runs) in some states, reducing investment surplus and causing ex ante credit contraction and underinvestment.
- Trade-offs:
  - Short-term debt disciplines bank risk-taking by enabling runs that force liquidation when signaled revenue is low.
  - Runs produce social deadweight loss and contractionary effects, aggravated by aggregate uncertainty.

### Model setup and assumptions
- Agents and timing:
  - Three parties: firm, bank, investor (uninsured wholesale creditor).
  - Dates t = 0; 1; 2. Two-period opportunity cost of capital is R = 1 + r > 1.
- Projects:
  - Two project types: good (g) and bad (b). Technology: invest A at t = 0 to obtain z Y(A) at t = 2, where z ∈ {z_g, z_b} with distributions F_g and F_b.
  - Bad project: lower expected revenue than good:
    ∫_0^1 z dF_g(z) > ∫_0^1 z dF_b(z)
    and higher upside risk by second order stochastic dominance (SOSD): there exists z̄ such that F_g(z̄) = F_b(z̄) and (F_g(z) - F_b(z))(z - z̄) ≥ 0 for all z.
- Signal and liquidation:
  - At t = 1 a public signal perfectly predicts z realized at t = 2.
  - If liquidated at t = 1, capital sold at discount price of ϕ ∈ [0; R) per unit at t = 2.
- Assumptions:
  - Assumption 1: Y(A) strictly concave, increasing, Inada: lim_{A→0} Y'(A) = 1; lim_{A→∞} Y'(A) = 0; and Y'(A) is convex.
  - Assumption 2: For F ∈ {F_g, F_b}, survival function 1 - F is log-concave; hazard rate h(z) = f(z)/(1 - F(z)) is continuous and increasing.

### First-best investment (social planner)
- Planner liquidates when z < z(A), where z(A) solves:
  z Y(A) = ϕ A.  (equation (1))
- Planner maximizes:
  max_A ϕ A F_g(z) + ∫_z^1 z Y(A) dF_g(z) - R A.  (equation (2))
- First-order condition:
  ϕ F_g(z) + ∫_z^1 z Y'(A) dF_g(z) - R = 0.  (equation (3))
- Proposition 1: There exists a unique first-best investment level A_fb ∈ (0;1).

### Contracting framework — bank–firm contracting
- Bank–firm contract characterized by (A; B): bank chooses project type and invests A at t = 0; firm pays B at t = 2 if project continues; liquidation at t = 1 gives bank ϕ A.
- Firm's liquidation threshold under the contract:
  z̄ Y(A) = B.  (equation (4))
- With full renegotiation power for the bank, ex post renegotiation can restore the efficient threshold z(A) and the bank can capture surplus by making firm indifferent to continuation vs liquidation.
- Bank chooses (A; B) solving:
  max_{A,B} ϕ A F_g(z̄) + ∫_{z̄}^1 z Y(A) dF_g(z) + B(1 - F_g(z̄)) - R A  subject to firm's participation (PC_F).  (equation (5) and (6))
- Proposition 2: If the bank has all bargaining power in renegotiation, the bank gives the firm the first-best loan A_fb and sets B_fb to satisfy the firm's reservation payoff.

### Contracting framework — bank–creditor contracting and incentive constraints
- Bank finances via collateralized short-term debt (repo): sells assets A for price D at t = 0 and repurchases at t = 2 for ~D; creditor can decide to roll over at t = 1.
- Put-option representation: price of put option on assets with strike ~D denoted p_g(~D; A) for good assets.
- Creditor's expected gross payoff from good project:
  U_C^g(A) = ~D - p_g(~D; A).
- Bank payoff from good and bad projects:
  U_B^g(A) = ϕ A F_g(~z) + ∫_{~z}^1 z Y(A) dF_g(z) - U_C^g(A) - U_F^g(A).
  U_B^b(A) analogous with F_b.
- Incentive compatibility constraint for bank (IC_B): ensure bank chooses good project:
  Δ U_B(A) ≥ Δ p(~D; A),  (equation (7))
  where Δ U_B(A) = ∫_{z̄}^{~z} z Y(A) d(F_g - F_b) + ϕ A (F_g(~z) - F_b(~z)) - B(F_g(z̄) - F_b(z̄)),
  and Δ p = p_b(~D; A) - p_g(~D; A).
- Lemma 1: If liquidation value ϕ is sufficiently small, ICB binds and there exists unique ~D_*(A).
- Lemma 2: If ϕ is sufficiently small, there exists unique ~z ∈ (z; z̄) such that ~D_*(A) = ~z Y(A).
- Creditor participation constraint (PCC) and market value of debt D_*:
  U_C^g(A) = ~D_*(A) - p_g(~D_*; A) ≥ R D, competitive lending pins down D_* solving:
  R D_* = ~D_*(A) - p_g(~D_*; A).  (equation (9))
- Using ~D_*(A) = ~z Y(A), PCC becomes:
  R D_* = ϕ A F_g(~z) + ~z Y(A) (1 - F_g(~z)).  (equation (10))
- Lemma 3: There is a unique bank debt-to-asset ratio d_* = ~D_*/A < 1 for given A.

### Equilibrium financing and investment
- Banking equilibrium is quadruplet (A_*; B_*; D_*; ~D_*) where (A_*; B_*) solve bank–firm contracting and (D_*; ~D_*) solve bank–creditor contracting; equilibrium implies liquidation threshold ~z.
- Bank's problem reduces to:
  max_A ϕ A F_g(z̄) + ∫_{z̄}^1 z Y(A) dF_g(z) - R A - C - ∫_{~z}^{z̄} (z Y(A) - ϕ A) dF_g(z).  (equation (12))
  - Last term represents surplus loss from creditor-induced liquidation in ( ~z; z̄ ).
- Proposition 3: If ϕ is sufficiently small, there exists unique banking equilibrium (A_*; B_*; D_*; ~D_*). Investment is lower than first-best: A_* < A_fb.

### Extensions — aggregate uncertainty, portfolio correlation, endogenous default
- Aggregate uncertainty (mean-preserving spread):
  - Replace F_g by F_m with same mean: ∫_0^1 z dF_g(z) = ∫_0^1 z dF_m(z)
    and F_g(z̄) = F_m(z̄) with (F_g(z) - F_m(z))(z - z̄) ≥ 0.  (equation (13))
  - Proposition 4: Conditional on existence of banking equilibrium (A_*^m; B_*^m; D_*^m; ~D_*^m), higher uncertainty leads to lower credit supply and lower investment: A_*^m < A_*.
  - Intuition: higher uncertainty forces banks to shrink balance sheets to contain risk; socially costly via investment contraction.
- Multiple assets and correlation:
  - Introducing two types of good assets with same mean and variance; increase in correlation ρ increases portfolio variance.
  - Corollary 1: All else equal, an increase in return correlation of the bank's asset portfolio leads to lower credit supply and lower investment.
- Endogenous default and debt maturity:
  - Allowing endogenous default (equity chooses default when z < ~z) preserves prior results: endogenous default is costly and creditors bear cost.
  - Long-term debt vs short-term:
    - If endogenous default is prohibited, long-term debt can increase debt capacity and restore first-best.
    - If endogenous default is possible, advantage of long-term debt disappears; short-term debt remains an optimal instrument for creditors.

### Policy options — efficiency condition and instruments
- Efficiency condition for no welfare loss in banking equilibrium:
  - Proposition 5: Banking equilibrium (A_*; B_*; D_*; ~D_*) incurs no welfare loss iff bank's liquidation value fully repays creditor:
    ϕ A ≥ ~D.  (equation (14))
  - Interpretation: debt value cannot be too high relative to liquidation value; trade-off between discipline and social efficiency.
- Ex ante policies
  - Capital requirement:
    - Minimum equity-to-asset ratio to satisfy (14) at equality:
      E / A = (R - ϕ) / R.  (equation (15))
    - Proposition 6:
      - If no equity premium (R_e = R), imposing debt-to-equity ratio ϕ/(R - ϕ) achieves first-best investment.
      - If equity premium exists (R_e > R), investment under such capital requirement remains lower than first-best.
    - Capital requirement eliminates creditor default risk ex post but raises ex ante marginal cost if equity premium exists.
    - Regulatory implication: limiting short-term debt size or imposing net stable funding ratio (NSFR) so long-term funding + equity ratio equals (R - ϕ)/R can achieve efficient funding composition.
  - Liquidity requirement:
    - Require holding liquid assets L funded by equity so that:
      L + ϕ A ≥ ~D.  (equation (16))
    - Imposing L such that L + ϕ A = ~D guarantees payment to creditor and can restore first-best absent equity premium.
  - Contingent debt:
    - State-contingent liabilities can remedy inefficiency if conversion triggers are based on aggregate variables (not individual outcomes) to preserve bank incentives.
    - Aggregate-triggered conversion that pays ~D if z ≥ ~z and ϕ A if z < ~z preserves creditor payoff and prevents runs in (z; z̄), eliminating welfare loss.
    - Caveat: when banking sector is concentrated ("too big to fail"), aggregation vs idiosyncratic distinction blurs and systemic triggers may distort incentives.
- Ex post policies
  - Monetary policy (interest rate reduction):
    - Creditor required return satisfies:
      R D_* = ϕ A F_g(~z) + ~z Y(A) (1 - F_g(~z)).
    - Proposition 7: Given committed asset and debt values at t = 0, reduce effective cost of capital from R to
      R_m = ϕ A_* / D_*  (equation (17))
      at t = 1 to eliminate welfare loss by lowering creditor's liquidation threshold and preventing inefficient runs.
    - Monetary intervention is ex post efficient but does not affect ex ante investment; capital requirements are preferable to the extent they raise ex ante investment toward A_fb.

### Conclusion
- Short-term debt is an optimal discipline device but creates rollover risk that, under asset-quality uncertainty and limited bank capital, leads to inefficient liquidations and ex ante underinvestment.
- Interaction between bank investment decisions and debt capacity is the key channel of uncertainty transmission to the real economy.
- Policy prescriptions that can improve welfare while preserving micro-level discipline include:
  - Ex ante: capital requirements, liquidity requirements, NSFR-style long-term funding mandates, and carefully designed aggregate-trigger contingent debt.
  - Ex post: temporary interest rate reductions to prevent inefficient runs when no adequate ex ante buffer exists.
- Implementation success depends on regulator ability to preserve market discipline while extracting surplus; concentrated banking increases moral hazard when systemic interventions are used.

*Italic: Source — "2. Equilibrium Payoffs and Welfare Loss" (contents extracted from _wp1565 - 2. Equilibrium Payoffs and Welfare Loss).*

### 1. Following (1),lim

### _wp1565 - 1. Following (1),lim

### Existence and uniqueness of optimal investment A (Proposition 1)
- As A → 0:
  - z(A) → 0 and MC(A) → R1; the left-hand side of (19) → +∞.
- As A → ∞:
  - Y0(A) ∈ (0, R); limA→1 MR(A) < limA→1 [Y0(A) h(z) + R1] = R1 by Assumptions 1 and 2.
  - limA→1 z(A) = 1 implying limA→1 MC(A) = 1; left-hand side of (19) → −∞.
- By continuity and the Intermediate Value Theorem, (19) is satisfied at least once → existence of A_fb such that MR(A_fb) = MC(A_fb).
- Monotonicity:
  - @MC(A)/@A = (R) h(z) / (1F g(z)) @z/@A > 0 ⇒ MC(A) increasing.
  - @MR(A)/@A derived and bounded using Lemma 4 and properties of Y(A); under Assumption 1 (Y0(A)=Y(A) strictly convex and decreasing) one obtains @MR(A)/@A < 0 ⇒ MR(A) decreasing.
- Conclusion: uniqueness of A_fb.

### Binding participation constraint for the bank and first-best (Proposition 2)
- Local variational argument: if PCF slack, bank can raise B slightly to increase payoff without violating PCF ⇒ contradiction.
- Therefore, PCF binds under the optimal contract.
- Substituting binding PCF into (5) yields (20):
  - max_A [A F g(z) + ∫_z^1 z Y(A) dF g(z)  R A  C]
  - This equals the planner's problem (2) except for reservation payoff C.
- Since C does not depend on A, solution to (20) is the planner's first-best A_fb.

### Arrow-Debreu pricing, option price difference, and unique debt face value (Lemma 1)
- Price difference of put options for loan (A;B):
  - 
(s;A) = ∫_s^0 (G_b(x;A)  G_g(x;A)) dx where G_g and G_b are densities of revenues for good/bad portfolios.
- For given A, project liquidation for x ≤ A; for x > A, x = z Y(A) monotone in z.
- Under SOSD, there exists x(A) = z Y(A) where G_g(x;A) = G_b(x;A); (G_g  G_b)(x  x) ≥ 0.
- 
(x;A) increases, maximized at x, then decreases for x > x.
- Define asset notional A. Using integration by parts and binding PCF yields expression (23), showing 
(x;A) approaches 
U_B(A) from above as x → A.
- Existence and uniqueness:
  - There exists unique ~D(A) such that 
(~D;A) = 
U_B(A) with positive bank equity.
- Limits:
  - lim_{→0} 
(A;A) = 0.
  - lim_{→R} 
(A;A) → 
(RA;A) > 0.
  - Continuity in a implies existence of a ∈ [0; R) such that for  ≤ a: 
(A;A) < 
(A;A). (Definition of " sufficiently small".)

### Properties and bounds of ~D(A) (Lemma 2 and Lemma B)
- For sufficiently small  ( ≤ a), ~D(A) ∈ (A / z Y(A) ; z Y(A)) equivalently ~D(A) ∈ ( (A) / Y(A) ; z Y(A) ) under monotonicity of z ↦ z Y(A).
- ~D(A) > A / z Y(A) follows since 
(A;A) < 
(A;A) = 
(~D;A).
- ~D(A) < z Y(A) because otherwise bank payoff insufficient to cover debt obligation ⇒ violates participation.
- Lemma B: ~D < R A, and combining with (9) yields RD = ~D  _g(~D;A) < ~D < R A.

### First-order condition, marginal welfare loss, and existence of constrained optimum A' (Proposition 3)
- First-order condition of bank problem (12) is (25):
  -  F_g(z) + ∫_z^1 z Y0(A) dF_g(z)  R  ∫_{~z}^z (z Y0(A)  ) dF_g(z)  (@~z/@A)(~z Y(A)  ) f_g(~z) = 0.
- Interpretation:
  - First three terms = planner FOC (3); last two terms = marginal welfare loss due to contractual constraints.
- Evaluate at A = A_fb:
  - First three terms = 0; marginal welfare loss terms positive ⇒ left-hand side negative at A_fb.
- Behavior as A → 0:
  - First three terms → 1; z, ~z → 0 when  small ⇒ left-hand side → 1 > 0.
- Continuity implies existence of A' < A_fb where (25) holds ⇒ constrained optimum A' < A_fb.
- Once A' is given, (B', D', ~D') solve PCF, ICB, PCC with unique solution.

### Comparative statics with alternative distribution F_m and Lemma 4
- Bank's FOC under F_m given by (27).
- To show A'_m < A', evaluate FOC (27) at A' and show negative by comparing integrals of differences (29).
- Using binding ICB for F_m and F_g at A' (equation (30)) and strict convexity of Y(A), first two terms of (29) are negative.
- Define l_mg() = ∫_{~z}^z (z Y0(A')  ) d(F_m  F_g); continuity and limit arguments show l_mg() > 0 for  ∈ [0; R), making third term positive and overall (29) negative, implying A'_m < A'.

### Long-term vs short-term contracts (long-term contract proof)
- Notation: D_long and ~D_long market and face values for long-term debt; ~z_long liquidation threshold.
- Case 1 (no endogenous default): renegotiated contract pays z Y(A) for z ∈ (z; ~z_long). PCC (31) holds. Backward induction transforms bank problem to:
  - max_A [ A F_g(z) + ∫_z^1 z Y(A) dF_g(z)  R A  C] ⇒ same as planner (2) except C ⇒ solution = A_fb.
- Case 2 (endogenous default): creditor receives liquidation value a A for z ∈ (z; ~z_long). Creditor required payoff:
  - RD_long =  A F_g(~z_long) + ~z_long Y(A) (1  F_g(~z_long)) — identical to short-term PCC (10).
- Conclusion: equilibrium value of long-term contract equals short-term debt.

### Maximal debt-to-asset ratio (Proposition 5)
- When ~z = z and using (1):
  -  A = ~D and substituting into (10) gives RD =  A F_g(z) + z Y(A) (1  F_g(z)) =  A.
- The maximal debt-to-asset ratio follows directly.

### Capital requirements and investment (Proposition 6)
- With capital requirement ensuring no surplus loss, bank-chosen investment maximizes total surplus:
  - max_A [ A F_g(z) + ∫_z^1 z Y(A) dF_g(z)  R_t A  C] where R_t = R + (R_e  R) (R  ) / R.
- If R_e > R, higher marginal cost R_t implies lower investment due to convexity of marginal revenue.

### Rollover, reduced cost of capital, and elimination of welfare loss (Proposition 7)
- Creditor willing to roll over debt for projects with z ≥ ~z_m under reduced cost of capital R_m^D given by:
  - R_m^D =  A F_g(~z_m) + ~z_m Y(A) (1  F_g(~z_m)).
- Eliminating welfare loss requires ~z_m = z, which then implies:
  - R_m^D =  A F_g(z) + z Y(A) (1  F_g(z)) =  A.

*Source: IMF working paper text unit _wp1565 - 1. Following (1),lim*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2015/_wp1565.pdf_