## _wp11193

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---

### Introduction: purpose and main findings
- Purpose: analyze alternative allocations of regulatory functions (lender of last resort, early intervention, short-term bank-liabilities insurance) when regulators have heterogeneous objective functions and when banks face liquidity and solvency shocks and systemic externalities.
- Main findings:
  - i) Under an expanded mandate to explicitly oversee systemic interconnectedness, regulators would be more forbearing towards systemically important institutions, and stricter with non-systematic institutions.
  - ii) In the presence of efficient resolution mechanisms and high political costs for shutting down a financial institution, a unified regulatory arrangement could reduce systemic risk vis-a-vis a multiple regulatory arrangement because:
    - a) forbearance would be reduced,
    - b) when only a subset of regulators have access to private information regarding an institution’s degree of systemic importance, these regulators may have no incentive to share it with other regulators, and
    - c) a multiregulator setting decreases the amount of information gathered even if information can be collected at no cost.

### Model setup (without private information)
- Two banks indexed i ∈ {A, B}.
- Financing: capital K_i and short-term deposits D_i; long-term projects Y_i pay return ̃R_i later.
- Shocks:
  - Liquidity shocks at t = 1: realized deposits ̃D_i; liquidity gap ω_i ≡ D_i − ̃D_i.
  - Solvency shocks: project success probabilities u_A (revealed at t = 1) and u_B (revealed at t = 2). Bank A is systemic: A's failure lowers B’s success probability by γ.
- Timing: four periods t = 0,1,2,3 with no time discounting; A’s investment return realized at t = 2, B’s at t = 3.
- Liquidation and costs:
  - Early liquidation value L_i Y_i, with L_i ∈ (0,1).
  - Bankruptcy cost c_i > 0.
  - Condition (as of time 0): E(u_i)[R + c_i] > 1 + c_i for i = A,B.
- Assumptions preserved exactly:
  - Assumption 1: ̃D_i independent with distribution G(D), and G′(0) > 0.
  - Assumption 2: u_A with support [u_A, u_A] ⊂ [0,1], distribution F_A(u_A); ̃R_A = R with probability u_A, 0 with probability 1 − u_A.
  - Assumption 3: u_B signal = u_B − γχ, u_B with distribution F_B(u_B) with support [u_B, u_B] ⊂ [0,1], γ ∈ [0, u_B]; ̃R_B = R with probability u_B − γχ, 0 with probability 1 − u_B + γχ.

### Regulatory architectures and objective functions
- Architectures compared:
  - Unified regulator (UR) with LoLR and DI functions combined.
  - Multiple regulators: separate lender of last resort (LoLR) and deposit insurer (DI) with early intervention powers.
- Regulators’ objective: maximize private objective functions balancing:
  - Income I(s_i, ̃R_i),
  - Political cost α c_i of closing a bank,
  - Strategic reputational interaction captured by θ indicator (θ = 0 if regulator does not close while other regulator closes; for UR, θ = 1 whenever the regulator chooses to close a bank).
- Decision variables: d_i ∈ {0,1} for DI (d_i = 1 if DI decides not to close), l_i ∈ {0,1} for LoLR (l_i = 1 if LoLR provides emergency funds). s_i ≡ (d_i, l_i).
- Utility specification (preserved exactly):
  - E[I(s_A, ̃R_A) − α c_A χ(s_A, ̃R_A) θ_A + I(s_B, ̃R_B) − α c_B χ(s_B, ̃R_B) θ_B | χ(s_A, ̃R_A), u_B] | u_A.

### Equilibrium — multiple-regulator architecture (closing rules)
- Timing: backward induction (bank B at t = 2, then bank A at t = 1).
- Proposition 1 (bank B closing rule):
  - Bank B is closed at t = 2 if realized u_B < u^m_B(χ), where
    - u^m_B(χ) ≡ max{ u^DI_B(χ), u^LoLR_B(χ) },
    - u^LoLR_B(χ) = ω_B (P + 1) ω_B + α c_B + γ χ,
    - u^DI_B(χ) = L_B Y_B − max{0, ω_B} D_B − max{0, ω_B} + α c_B + γ χ.
- Intuition and comparative patterns:
  - LoLR cutoff u^LoLR_B increases in ω_B (higher loan-loss risk).
  - DI becomes more forbearing as LoLR supplies more liquidity because unpaid debts to LoLR reduce DI’s potential payouts.
  - For non-systemic bank B: higher probability of closing non-systemic institutions in crisis under multiregulator arrangements; LoLR cutoff upward sloping in ω, DI cutoff downward sloping in ω.
  - Presence of systemic risk (γ > 0) makes regulators more forbearing toward systemic institutions and stricter with non-systemic ones.

- Proposition 2 (bank A closing rule in multi-regulator):
  - Bank A is closed if u_A < u^m_A(γ) ≡ max{ u^DI_A(γ), u^LoLR_A(γ) }, where
    - u^LoLR_A(γ) = ω_A (P + 1) ω_A + α c_A + γ [ (P + 1) max{0, ω_B} + α c_B ] Pr{ u_B ≥ u^m_B(χ = 0) },
    - u^DI_A(γ) = L_A Y_A − max{0, ω_A} D_A − max{0, ω_A} + α c_A + γ [ D_B − max{0, ω_B} + α c_B ] Pr{ u_B ≥ u^m_B(χ = 0) }.
- Intuition:
  - Systemic externality (γ > 0) raises the threshold for closing A (more forbearance) because closing A increases B’s liquidation probability and expected downstream costs.

### Equilibrium — unified regulator architecture (non-systemic bank B)
- Proposition 3 (bank B closing rule under UR):
  - Bank B closed at t = 2 if u_B < u^u_B(χ), where
    - u^u_B(χ) ≡ L_B Y_B P max{0, ω_B} + α c_B + D_B + γ χ.
- Comparative insights UR vs multiregulator for bank B:
  - UR internalizes LoLR and DI payoffs and may be less or more forbearing depending on ω:
    - For moderate ω, UR tends to be less forbearing than multiregulator because UR’s payoff when not closing is more weighted toward the failure case (UR must back all deposits if B fails).
    - For sufficiently high ω, UR may be more forbearing than multiregulator because UR’s payoff when not closing is increasing in ω (UR must ultimately bear all deposits), while LoLR’s payoff decreases in ω.
  - Summary: unified and multiregulator cutoffs cross at ω*_B; for ω_B below (above) ω*_B multiregulator is more (less) forbearing than unified.

### Equilibrium — unified regulator architecture (systemic bank A) — qualitative
- Proposition 4 (UR consolidated DI and LoLR closing rule):
  - Bank A is closed for u_A values below
    - u_A(γ)≡ L_A Y_A Pmax{0,ω_A}+D_A+αc_A+γ(Pmax{0,ω_B}+αc_B+D_B)Pr u_B≥u u_B(χ= 0)
- Key qualitative finding:
  - The regulator will be softer on potentially systemic institutions.
  - Consolidating DI and LoLR without tools to prevent institutions from becoming systemic will not eliminate leniency; incentives could be exacerbated.
- Mechanism:
  - Failure of a systemic institution increases likelihood of failures among non-systemic institutions, increasing expected costs and encouraging leniency toward systemic institutions.
  - Distress in the systemic institution reduces survival chances of non-systemic institution, so regulator is less likely to save the non-systemic institution.

### Comparative multiple-vs-unified regulator — propositions and comparative statics
- Non-systemic bank B — Proposition 5 (preserved statements):
  1. If ω_B = 0, then u^u_B(β) = u^m_B(β).
  2. Suppose ω_B > 0.
     - (i) If u^u_B(β) > u^LoLR_B(β) for all ω_B ∈ (0, D_B], then multiregulator is always more forbearing than unified.
     - (ii) If ∃ ω*_B ∈ (0, D_B] such that u^u_B(β) = u^LoLR_B(β), then multiregulator is more forbearing for ω_B ∈ (0, ω*_B) and less forbearing for ω_B ∈ [ω*_B, D_B]. If α is sufficiently large, ω*_B is increasing in L_B, c_B, P, and decreasing in D_B.
- Systemic bank A — Proposition 6 (preserved statements):
  1. If ω_A = 0, then u^u_A(γ) = u^m_A(γ).
  2. Suppose ω_A > 0.
     - (i) If u^u_A(γ) > u^LoLR_A(γ) for all ω_A ∈ (0, D_A], then multiregulator is more forbearing than unified.
     - (ii) If ∃ ω*_A ∈ (0, D_A] such that u^u_A(γ) = u^LoLR_A(γ), then multiregulator is more forbearing for ω_A ∈ (0, ω*_A) and less forbearing for ω_A ∈ [ω*_A, D_A]. If α is sufficiently large, ω*_A is increasing in L_A, L_B, c_A, c_B, D_B, P, γ, and decreasing in D_A.
- Comparative statics summary (preserved):
  - Region where unified regulator is less forbearing than multiple regulators increases in L_B, c_B, P, γ, and D_B, and decreases in D_A.
  - Higher c_B raises expected closure costs of B (via systemic link), increasing forbearance toward A; unified regulator bears full deposit guarantees so its threshold reduction is lower, expanding region where unified is less forbearing.

### Policy-relevant implications (derived from model results)
- Mandates to explicitly monitor systemic interconnectedness change incentives:
  - Regulators become more forgiving of systemic institutions and stricter toward non-systemic ones.
- Institutional design matters:
  - Unified regulatory architecture can reduce systemic risk relative to multiple regulators when political costs of failure are high and resolution mechanisms work efficiently, because it mitigates strategic withholding of information and forbearance externalities.
  - Multiple-regulator architectures can suffer from information fragmentation and reduced incentives to gather or share private information, potentially raising systemic risk.
- Liquidity-provision design:
  - The size of liquidity assistance (ω) and penalty rate (P) crucially affect forbearance thresholds for LoLR, DI, and UR; policy calibration of P and rules governing liquidity support will influence closure decisions and systemic outcomes.

### The model with private information — overview
- Extension: γ (degree of systemic risk imposed by bank A) is private; γ ∼ Z(γ) with support [δ, δ̄] and mean γ ≡ E[γ].
- DI can costlessly observe realized γ; if DI observes, LoLR automatically observes a coarser signal {{γ}|γ < γ*} ∪ {[γ*, 1]}. Denote γ̄ ≡ E[γ | γ ≥ γ*].
- Existence of γ* such that DI observation yields either identical information to LoLR (if γ < γ*) or coarser partition (if γ ≥ γ*).

### Information sharing — incentives and outcomes (private information)
- Strategic reporting case:
  - Consider γ ≥ γ*, u^DI_A(γ) ≤ u_A, and u^LoLR_A(γ) ≤ u_A. DI may wish to send message m so LoLR forms belief γ(m) = E[γ|m] and chooses l_A accordingly; DI can have incentive to misreport.
- Communication effectiveness and equilibria:
  - If γ ≥ γ* and u^DI_A(γ) ≤ u_A, then babbling equilibria exist (m(γ) = m ∀ γ ≥ γ*). Every equilibrium has same outcome as babbling equilibrium — communication is ineffective (Proposition 7).
  - Corollary 1: If γ ≥ γ*, bank A is not liquidated if ∃ ȳ γ ∈ [0,1] such that E[γ | γ* ≤ γ ≤ ȳ γ] = γ*, and is liquidated if otherwise.
- Implication: fully separating equilibria do not exist in relevant parameter regions; competition among regulators can render communication uninformative.

### Information gathering — DI’s incentive to acquire information
- Even if learning γ costless, DI may choose not to learn because revealed low γ could prompt LoLR liquidation that DI would prefer to avoid.
- Proposition 8 (partial preserved statements):
  - Let γ be such that u^DI_A(γ) = u_A. If
    1. max{ u^LoLR_A(γ), u^DI_A(γ) } ≤ u_A,
    2. γ ≤ δ,
    3. ∄ ȳ γ ∈ [0,1] such that E[γ | γ* ≤ γ ≤ ȳ γ] = γ*,
  - (text in source excerpt ends; main equilibrium implication preserved below).
- Main equilibrium result (preserved): If γ∗ ≤ γ∗, then in equilibrium the DI chooses not to observe γ.
- Interpretation: A multiregulator setting decreases amount of information gathered even if information can be collected at no cost. If revealing information makes a regulator worse off, that regulator has less incentive to gather it. Unified regulator gathers and uses information better than a divided one.

### Concluding remarks — regulators’ incentives and forbearance
- Analyses of systemic-risk oversight proposals often omit regulators’ incentives, including regulatory forbearance.
- Adding a systemic-risk monitoring mandate without associated policy tools does not alter basic regulator incentives driving forbearance.
- Regulators often prefer to keep institutions afloat due to political/reputational costs and possibility of recovery.
- Consequence: regulators may be more forgiving with systemically important institutions because of spillover effects.
- Information problems: competing regulators are less likely to gather information and may withhold information once gathered.
- Value of the model: illustrates importance of strategic considerations in optimal allocation of regulatory functions; serves as counterweight to analyses ignoring regulatory incentives.

### Appendix — selected equilibrium conditions and comparative statics highlights
- Regulator utility (t = 2) preserved:
  - Uj(sj, s−j) ≡ E[Ij(sB, ̃RB) − αcB χ(sB, ̃RB) θj(sB) | χ(sA, ̃RA), uB] = uB [Ij(sB, RB) − αcB χ(sB, RB) θj(sB)] + (1 − uB)[Ij(sB, 0) − αcB θj(sB)] − χ(sA, ̃RA) γ [Ij(sB, RB) − Ij(sB, 0) − αcB χ(sB, RB) θj(sB) + αcB θj(sB)].
- Nash-equilibrium structure for regulators j ∈ {lB, dB} (preserved):
  - (s∗j, s∗−j) = (0,1) for uB < ujB(χ),
  - (s∗j, s∗−j) = (1,0) for uB < u−jB(χ),
  - (s∗j, s∗−j) = (1,1) for uB ≥ umB(χ) ≡ max{ujB(χ), u−jB(χ)}.
- Payoff observations (preserved exactly):
  - ILoLR((0,1), 0) = 0,
  - ILoLR((1,1), 0) = −ωB,
  - ILoLR((1,1), RB) = PωB,
  - IDI((0,1), 0) = LB YB − DB,
  - IDI((1,1), 0) = −(DB − max{0, ωB}),
  - IDI((1,1), RB) = 0.
- Unified regulator non-closure thresholds (Propositions 3 and 4 preserved):
  - Unified regulator does not close bank B if:
    - uB ≥ uuB(χ) ≡ [I(0,0) − I(1,0)] / [I(1, RB) − I(1,0) + αcB] + χ(sA, ̃RA) γ,
    - with I(0,0) = LYB − DB, I(1,0) = −DB, I(1, RB) = P max{0, ωB}.
  - Unified regulator does not close bank A if:
    - uA ≥ uuA(γ) ≡ [I(0,0) − I(1,0)] / [I(1, RA) − I(1,0) + αcA] + γ[I(1, RB) − I(1,0) + αcB] Pr(uB ≥ uuB(χ = 0)),
    - with I(0,0) = LYA − DA, I(1,0) = −DA, I(1, RA) = P max{0, ωA}.
- Comparative-static highlights (preserved conclusions):
  - For bank B:
    - If ωB = 0 then uuB(χ = 1) = LB YB / (αcB + DB + γ) = max{uDIB(χ = 1), uLoLRB(χ = 1)} = umB(χ = 1).
    - If ωB > 0:
      - uuB(χ = 1) = LB YB / (PωB + αcB + DB + γ),
      - uLoLRB(χ = 1) = ωB / ((P+1)ωB + αcB + γ),
      - uDIB(χ = 1) = (LB YB − ωB) / (αcB + DB − ωB + γ).
    - Derivatives preserved:
      - ∂uuB/∂ωB < 0,
      - ∂uLoLRB/∂ωB > 0,
      - ∂uDIB/∂ωB < 0 for α big enough.
    - Existence of ω∗B solving uLoLRB = uuB; ∂ω∗B/∂LB > 0, ∂ω∗B/∂DB < 0 for α sufficiently large.
  - For bank A: analogous preserved results for uuA, uLoLRA, uDIA, monotonicities, and ω∗A comparative statics.

*Source: _wp11193 - 1. Non-systemic bank; 2. Systemic bank; 4. Ifγ (PDF chapter/section).*

### 1. Non-systemic bank .................................. .........................................8

### _wp11193 - 1. Non-systemic bank .................................. .........................................8

### Introduction: purpose and main findings
- Purpose: analyze alternative allocations of regulatory functions (lender of last resort, early intervention, short-term bank-liabilities insurance) when regulators have heterogeneous objective functions and when banks face liquidity and solvency shocks and systemic externalities.
- Main findings:
  - i) Under an expanded mandate to explicitly oversee systemic interconnectedness, regulators would be more forbearing towards systemically important institutions, and stricter with non-systematic institutions.
  - ii) In the presence of efficient resolution mechanisms and high political costs for shutting down a financial institution, a unified regulatory arrangement could reduce systemic risk vis-a-vis a multiple regulatory arrangement because:
    - a) forbearance would be reduced,
    - b) when only a subset of regulators have access to private information regarding an institution’s degree of systemic importance, these regulators may have no incentive to share it with other regulators, and
    - c) a multiregulator setting decreases the amount of information gathered even if information can be collected at no cost.

### Model setup (without private information)
- Two banks indexed i ∈ {A, B}.
- Financing: capital K_i and short-term deposits D_i; long-term projects Y_i pay return ̃R_i later.
- Shocks:
  - Liquidity shocks at t = 1: realized deposits ̃D_i; liquidity gap ω_i ≡ D_i − ̃D_i.
  - Solvency shocks: project success probabilities u_A (revealed at t = 1) and u_B (revealed at t = 2). Bank A is systemic: A's failure lowers B’s success probability by γ.
- Timing: four periods t = 0,1,2,3 with no time discounting; A’s investment return realized at t = 2, B’s at t = 3.
- Liquidation and costs:
  - Early liquidation value L_i Y_i, with L_i ∈ (0,1).
  - Bankruptcy cost c_i > 0.
  - Condition (as of time 0): E(u_i)[R + c_i] > 1 + c_i for i = A,B.
- Assumptions preserved exactly:
  - Assumption 1: ̃D_i independent with distribution G(D), and G′(0) > 0.
  - Assumption 2: u_A with support [u_A, u_A] ⊂ [0,1], distribution F_A(u_A); ̃R_A = R with probability u_A, 0 with probability 1 − u_A.
  - Assumption 3: u_B signal = u_B − γχ, u_B with distribution F_B(u_B) with support [u_B, u_B] ⊂ [0,1], γ ∈ [0, u_B]; ̃R_B = R with probability u_B − γχ, 0 with probability 1 − u_B + γχ.

### Regulatory architectures and objective functions
- Architectures compared:
  - Unified regulator (UR) with LoLR and DI functions combined.
  - Multiple regulators: separate lender of last resort (LoLR) and deposit insurer (DI) with early intervention powers.
- Regulators do not maximize social welfare; they maximize private objective functions balancing:
  - Income I(s_i, ̃R_i),
  - Political cost α c_i of closing a bank,
  - Strategic reputational interaction captured by θ indicator (θ = 0 if regulator does not close while other regulator closes; for UR, θ = 1 whenever the regulator chooses to close a bank).
- Decision variables: d_i ∈ {0,1} for DI (d_i = 1 if DI decides not to close), l_i ∈ {0,1} for LoLR (l_i = 1 if LoLR provides emergency funds). s_i ≡ (d_i, l_i).
- Utility specification preserved exactly:
  - E[I(s_A, ̃R_A) − α c_A χ(s_A, ̃R_A) θ_A + I(s_B, ̃R_B) − α c_B χ(s_B, ̃R_B) θ_B | χ(s_A, ̃R_A), u_B] | u_A.

### Equilibrium: multiple-regulator architecture — closing rules
- Solve by backward induction: first bank B (t = 2), then bank A (t = 1).
- Proposition 1 (closing rule for bank B in multi-regulator):
  - At t = 2, once it is known whether bank A has been closed (χ = 0) or not (χ = 1), bank B is closed if realized u_B < u^m_B(χ), where
    - u^m_B(χ) ≡ max{ u^DI_B(χ), u^LoLR_B(χ) },
    - u^LoLR_B(χ) = ω_B (P + 1) ω_B + α c_B + γ χ,
    - u^DI_B(χ) = L_B Y_B − max{0, ω_B} D_B − max{0, ω_B} + α c_B + γ χ.
- Intuition:
  - LoLR requires higher success probability (higher u^LoLR_B) when liquidity injection ω_B is larger because of the loan loss risk.
  - DI becomes more forbearing as LoLR supplies more liquidity, since unpaid debts to LoLR reduce DI’s potential payouts.
- Figure insights preserved:
  - For non-systemic bank B: higher probability of closing non-systemic institutions in crisis under multiregulator arrangements; LoLR cutoff upward sloping in ω, DI cutoff downward sloping in ω.
  - Presence of systemic risk alters cutoffs: regulators become more forbearing towards systemic institutions and stricter with non-systemic ones.

- Proposition 2 (closing rule for bank A in multi-regulator):
  - Bank A is closed for u_A values below u^m_A(γ) ≡ max{ u^DI_A(γ), u^LoLR_A(γ) }, where
    - u^LoLR_A(γ) = ω_A (P + 1) ω_A + α c_A + γ [ (P + 1) max{0, ω_B} + α c_B ] Pr{ u_B ≥ u^m_B(χ = 0) },
    - u^DI_A(γ) = L_A Y_A − max{0, ω_A} D_A − max{0, ω_A} + α c_A + γ [ D_B − max{0, ω_B} + α c_B ] Pr{ u_B ≥ u^m_B(χ = 0) }.
- Intuition:
  - Systemic externality (γ > 0) raises the regulator’s threshold for closing A (more forbearance) because closing A increases the probability that B will be liquidated, raising expected downstream costs.

### Equilibrium: unified regulator architecture — non-systemic bank (specific results)
- Proposition 3 (closing rule for bank B under unified regulator):
  - At t = 2, once it is known whether bank A remains open (χ = 0) or not (χ = 1), bank B is closed if realized u_B < u^u_B(χ), where
    - u^u_B(χ) ≡ L_B Y_B P max{0, ω_B} + α c_B + D_B + γ χ.
- Comparative insights between UR and multi-regulator for non-systemic bank B:
  - UR internalizes LoLR and DI payoffs and may be less or more forbearing depending on shock size:
    - For moderate liquidity shocks, unified regulator tends to be less forbearing than the multiregulator setting because UR’s payoff when not closing is more weighted toward the failure case (UR must back all deposits if B fails), making UR more conservative.
    - For sufficiently high liquidity shocks, unified regulator may be more forbearing than multiregulator because UR’s payoff when not closing is increasing in the liquidity shock (UR must ultimately bear all deposits), while LoLR’s payoff decreases in the liquidity shock.
  - Figure summary: higher probability of closing non-systemic institutions in crisis under single and multiregulator arrangements; UR and DI are less forbearing than LoLR for moderate ω, but UR can become more forbearing as ω grows.

### Equilibrium: unified regulator architecture — systemic bank (qualitative)
- Comparative insights for systemic bank A (illustrated in Figure 5):
  - Presence of systemic risk (γ > 0) increases forbearance toward the systemic bank under both single and multiregulator arrangements.
  - UR may still reduce overall systemic risk relative to multiple regulators when:
    - Efficient resolution mechanisms exist, and
    - Political costs of shutting down institutions are high, because UR internalizes trade-offs and can reduce forbearance stemming from jurisdictional strategic incentives.

### Policy-relevant implications (derived from model results)
- Mandates to explicitly monitor systemic interconnectedness alter incentives:
  - Regulators become more forgiving of systemic institutions and stricter toward non-systemic ones.
- Institutional design matters:
  - Unified regulatory architecture can reduce systemic risk relative to multiple regulators when political costs of failure are high and resolution mechanisms work efficiently, because it mitigates strategic withholding of information and forbearance externalities across agencies.
  - Multiple-regulator architectures can suffer from information fragmentation and reduced incentives to gather or share private information, leading to higher systemic risk in some environments.
- Liquidity-provision design:
  - The size of liquidity assistance (ω) and penalty rate (P) crucially affect forbearance thresholds for LoLR, DI, and UR; policy calibration of P and rules governing liquidity support will influence closure decisions and systemic outcomes.

*Source: _wp11193 - 1. Non-systemic bank (pdf chapter/section).*

### 2. Systemic bank

### 2. Systemic bank

### Proposition 4 — regulator behavior with DI and LOLR consolidated
- Consider the case where a unified regulator has DI and LOLR functions. Then bank A is closed for u_A values below
  - u_A(γ)≡ L_A Y_A Pmax{0,ω_A}+D_A+αc_A+γ(Pmax{0,ω_B}+αc_B+D_B)Pr u_B≥u u_B(χ= 0)
- Key qualitative finding:
  - The regulator will be softer on potentially systemic institutions.
  - Consolidating standard regulatory functions without additional tools to preclude institutions from becoming systemic in the first place will not eliminate regulators’ incentives to be lenient with systemically important institutions; these incentives could be exacerbated.
- Mechanism explained:
  - Failure of a systemically important institution increases the likelihood of failures among non-systemic institutions, increasing expected costs and making any regulator more lenient with a systemically important institution.
  - This is illustrated by a downward shift in lines in Figure 5 (unified regulator more lenient toward systemic bank).
  - Distress in the systemic institution reduces the survival chances of the non-systemic institution, so the regulator is less likely to save the non-systemic institution (upward shift in lines in Figure 4).
  - Presence of systemic risk does not relax regulator’s standards for rescue of non-systemic institutions.

### D. Comparing Multiple-vs-Unified-Regulator Architectures — overview
- Objective: compare multiregulator (agencies LoLR and DI) vs unified regulator (DI and LOLR functions) arrangements in terms of forbearance.
- Approach: derive comparative statics associating levels of forbearance with model primitives.

### 1. Non-systemic bank (bank B) — Proposition 5
- Setup: Given the liquidity shock ω*_B, the liquidation cutoffs of the unified regulator and the multiregulator cross each other; liquidity shocks below (above) this point imply multiregulator arrangement is more (less) forbearing than unified.
- Consequence: increases in ω*_B reduce the “dominance region” (the set of liquidity shocks for which the unified regulator is less forbearing than the multiregulator).
- Proposition 5 (statements preserved):
  1. If ω_B = 0, then u^u_B(β) = u^m_B(β), i.e., the unified regulator and the multiregulator arrangements are equally forbearing.
  2. Suppose ω_B > 0.
     - (i) If u^u_B(β) > u^LoLR_B(β) for all ω_B ∈ (0, D_B], then the multiregulator arrangement is always more forbearing than the unified regulator arrangement.
     - (ii) If ∃ ω*_B ∈ (0, D_B] such that u^u_B(β) = u^LoLR_B(β), then the multiregulator arrangement is more forbearing than the unified regulator arrangement for ω_B ∈ (0, ω*_B) and less forbearing for ω_B ∈ [ω*_B, D_B]. In addition, if α is sufficiently large, then ω*_B is increasing in L_B, c_B, P, and decreasing in D_B.
- Intuition and comparative statics:
  - Increases in liquidation value L_B make the unified regulator less forbearing (higher payoff when bank B is closed) and do not change LoLR behavior → unified better relative to multiregulator.
  - Higher political cost c_B makes both regulators more forbearing, but unified regulator bears a larger share (it must back all deposits when bank B fails) → impact of c_B on forbearance is lower for the unified regulator, reducing the region where multiregulator is less forbearing.
  - Similar logic applies to changes in P.
  - Increases in D_B make the unified regulator more forbearing (higher cost of closing the bank) and do not change LoLR incentives → larger D_B increases relative performance of multiregulator arrangement.

### 2. Systemic bank (bank A) — Proposition 6
- Setup: Suppose ω_B ≤ 0 and let u^m_A(γ) and u^u_A(γ) be liquidation thresholds for bank A under multiregulator and unified regulator respectively.
- Proposition 6 (statements preserved):
  1. If ω_A = 0, then u^u_A(γ) = u^m_A(γ), i.e., the unified regulator and the multiregulator arrangements are equally forbearing.
  2. Suppose ω_A > 0.
     - (i) If u^u_A(γ) > u^LoLR_A(γ) for all ω_A ∈ (0, D_A], then the multiregulator arrangement is more forbearing than the unified regulator arrangement.
     - (ii) If ∃ ω*_A ∈ (0, D_A] such that u^u_A(γ) = u^LoLR_A(γ), then the multiregulator arrangement is more forbearing than the unified regulator arrangement for ω_A ∈ (0, ω*_A) and less forbearing for ω_A ∈ [ω*_A, D_A]. In addition, if α is sufficiently large, then ω*_A is increasing in L_A, L_B, c_A, c_B, D_B, P, γ, and decreasing in D_A.
- Comparative statics summary:
  - The region where the unified regulator is less forbearing than the multiple regulators is increasing in L_B, c_B, P, γ, and D_B, and decreasing in D_A.
  - Results regarding L_A, c_A, P, and D_A parallel those for non-systemic bank.
  - Higher c_B increases expected costs associated with closure of bank B (due to systemic link), increasing regulators’ forbearance toward systemically important bank A; unified regulator bears full deposit guarantees so its liquidation threshold reduction is lower, expanding the dominance region where unified is less forbearing.

### Table 1 — comparative summary (parameter effects on region where multiregulator is less forbearing than unified)
- Notation: L_i is the liquidation value, c_i is the bankruptcy cost, and D_i is the amount of deposits, where i = A, B. P is the penalty rate and γ is the degree of systemic risk.
- Effect signs as presented:
  - Bank B (non-systemic): ∆L_B −−, ∆c_B −−, ∆D_B +−, ∆P −−, ∆γ −
  - Bank A (systemic): ∆L_A −, ∆c_A −, ∆D_A +

### III. The Model with Private Information — overview
- Extension: γ (degree of systemic risk imposed by bank A) is not publicly known; γ has distribution Z(γ) with support [δ, δ̄] and expected value γ ≡ E[γ].
- The DI can costlessly observe realized γ; if DI observes, the LoLR automatically observes a signal from a coarser information set of form {{γ}|γ < γ*} ∪ {[γ*, 1]}. Denote γ̄ ≡ E[γ | γ ≥ γ*].
- There exists γ* such that if DI receives γ < γ* then LoLR receives identical information; if γ ≥ γ* the LoLR only knows γ ∈ [γ*,1].

### A. Information sharing — incentives and outcomes
- Cases:
  - For γ < γ* or for γ ≥ γ* such that u^DI_A(γ) > u_A, prior results apply and DI reporting has no strategic impact.
  - Strategic case considered: γ ≥ γ*, u^DI_A(γ) ≤ u_A (DI does not want to liquidate A). If u^LoLR_A(γ) ≤ u_A then LoLR would provide liquidity. DI can send a message m so that LoLR forms belief γ(m) = E[γ|m] and chooses l_A ∈ {0,1} based on u^LoLR_A(γ(m)). DI may have incentive to misreport to induce LoLR to act per DI preferences.
- Communication effectiveness:
  - If γ ≥ γ̄ ≥ γ*, then any message m yields γ(m) ≥ γ*, implying u_A = u^LoLR_A(γ*) ≥ u^LoLR_A(γ(m)) and bank A is not liquidated.
  - If γ* ≤ γ < γ̄ and if γ* < γ̄ ≤ γ, the DI has incentives to fully reveal information (lower dashed blue line case in Figure 6).
  - If γ̄ ≤ γ* ≤ γ, DI may not reveal information because revelation could induce LoLR to refuse liquidity (upper dashed blue line case).
  - Conclusion: fully separating equilibria do not exist; communication can be ineffective.
- Proposition 7 (preserved):
  - If γ ≥ γ* and u^DI_A(γ) ≤ u_A, then babbling equilibria exist (m(γ) = m ∀ γ ≥ γ*). In addition, every equilibrium has the same outcome as that of a babbling equilibrium, i.e., communication is ineffective.
- Corollary 1 (preserved):
  - If γ ≥ γ*, then bank A is not liquidated if ∃ ȳ γ ∈ [0,1] such that E[γ | γ* ≤ γ ≤ ȳ γ] = γ*, and is liquidated if otherwise.
  - Interpretation: given that posterior belief regarding systemic risk is higher than initial conjecture, the better the bank’s financial condition (lower ω and higher u), the more likely LoLR revises decision to provide liquidity.

### B. Information gathering — DI’s incentive to acquire information
- Even if learning γ is costless, DI might not want to learn because revealed low γ could trigger LoLR liquidation that DI would prefer to avoid.
- Proposition 8 (partial statement preserved):
  - Let γ be such that u^DI_A(γ) = u_A. If
    1. max{ u^LoLR_A(γ), u^DI_A(γ) } ≤ u_A,
    2. γ ≤ δ, i.e., γ is below the lower bound of the support of Z(γ),
    3. ∄ ȳ γ ∈ [0,1] such that E[γ | γ* ≤ γ ≤ ȳ γ] = γ*,
  - (text ends in source excerpt)

*Source: _wp11193 - 2. Systemic bank (excerpt)*

### 4. Ifγ

### 4. Ifγ

### DI's incentive not to observe γ and information gathering
- Main equilibrium result: If γ∗ ≤ γ∗, then in equilibrium the DI chooses not to observe γ.
- Interpretation: A multiregulator setting decreases the amount of information gathered even if information can be collected at no cost.
- Mechanism: If revealing information makes a regulator worse off, that regulator has less incentive to gather the information. Therefore, a unified regulator gathers and uses information better than a divided one.

### Illustration (Figure 6 description)
- Elements shown: LoLR and DI liquidation cutoffs, realizations of (ωA, ωB, uA), and the value γ∗ such that the LoLR’s cutoff goes through the realization (ωA, ωB, uA).
- Incentive pattern: The DI has incentive to report truthfully for realized values of gamma above γ∗ and no incentive for values below γ∗.

### Concluding remarks — regulators’ incentives and forbearance
- Key point: Analyses of systemic-risk oversight proposals often omit the analysis of regulators’ incentives, including "regulatory forbearance"—the incentive to keep institutions afloat when they should be unwound.
- Adding a systemic-risk monitoring mandate without associated policy tools does not alter basic regulator incentives causing shortcomings observed in the crisis.
- Regulators often prefer to keep institutions afloat, since they dislike closures under their watch and some institutions may recover with time.
- Consequence: In the absence of concrete methods to limit institutions’ ability to become systemically important, regulators may be more forgiving with systemically important institutions than with others because of the spillover effects on institutions under their purview.
- Information problems: Competing regulators are less likely to gather information and may have an incentive not to share information once gathered.
- Value of the model: Illustrates the importance of analyzing strategic considerations in the optimal allocation of regulatory functions, serving as a counterweight to analyses ignoring regulatory incentives.

### Appendix — selected equilibrium conditions and comparative statics highlights
- Regulator utility and best responses (key expressions preserved):
  - Utility of regulator j (starting at t = 2): 
    Uj(sj, s−j) ≡ E[Ij(sB, ̃RB) − αcB χ(sB, ̃RB) θj(sB) | χ(sA, ̃RA), uB] = uB [Ij(sB, RB) − αcB χ(sB, RB) θj(sB)] + (1 − uB)[Ij(sB, 0) − αcB θj(sB)] − χ(sA, ̃RA) γ [Ij(sB, RB) − Ij(sB, 0) − αcB χ(sB, RB) θj(sB) + αcB θj(sB)].
  - Best-response difference Uj(1, s−j) − Uj(0, s−j) shown explicitly (see source for full expression).

- Nash equilibria structure (for regulators j ∈ {lB, dB}):
  - (s∗j, s∗−j) = (0,1) for uB < ujB(χ),
  - (s∗j, s∗−j) = (1,0) for uB < u−jB(χ),
  - (s∗j, s∗−j) = (1,1) for uB ≥ umB(χ) ≡ max{ujB(χ), u−jB(χ)}.
- Specific payoff observations used:
  - ILoLR((0,1), 0) = 0,
  - ILoLR((1,1), 0) = −ωB,
  - ILoLR((1,1), RB) = PωB,
  - IDI((0,1), 0) = LB YB − DB,
  - IDI((1,1), 0) = −(DB − max{0, ωB}),
  - IDI((1,1), RB) = 0.

- Analogous results for regulators j ∈ {lA, dA} (starting at t = 1), with symmetric expressions:
  - Nash equilibria:
    - (s∗j, s∗−j) = (0,1) for uA < ujA(γ),
    - (s∗j, s∗−j) = (1,0) for uA < u−jB(γ),
    - (s∗j, s∗−j) = (1,1) for uA ≥ umA(γ) ≡ max{ujA(γ), u−jA(γ)}.
  - Payoff observations:
    - ILoLR((0,1), 0) = 0,
    - ILoLR((1,1), 0) = −ωA,
    - ILoLR((1,1), RA) = PωA,
    - IDI((0,1), 0) = LA YA − DA,
    - IDI((1,1), 0) = −(DA − max{0, ωA}),
    - IDI((1,1), RA) = 0.

- Unified regulator thresholds (Propositions 3 and 4):
  - Unified regulator does not close bank B if:
    uB ≥ uuB(χ) ≡ [I(0,0) − I(1,0)] / [I(1, RB) − I(1,0) + αcB] + χ(sA, ̃RA) γ.
    - With observations I(0,0) = LYB − DB, I(1,0) = −DB, I(1, RB) = P max{0, ωB}.
  - Unified regulator does not close bank A if:
    uA ≥ uuA(γ) ≡ [I(0,0) − I(1,0)] / [I(1, RA) − I(1,0) + αcA] + γ[I(1, RB) − I(1,0) + αcB] Pr(uB ≥ uuB(χ = 0)).
    - With observations I(0,0) = LYA − DA, I(1,0) = −DA, I(1, RA) = P max{0, ωA}.

- Comparative static and critical-point results (selected highlights; preserved formulas and monotonicity statements):
  - Proposition 5 (bank B):
    - If ωB = 0 then uuB(χ = 1) = LB YB / (αcB + DB + γ) = max{uDIB(χ = 1), uLoLRB(χ = 1)} = umB(χ = 1).
    - If ωB > 0:
      - uuB(χ = 1) = LB YB / (PωB + αcB + DB + γ),
      - uLoLRB(χ = 1) = ωB / ((P+1)ωB + αcB + γ),
      - uDIB(χ = 1) = (LB YB − ωB) / (αcB + DB − ωB + γ).
    - Derivative signs:
      - ∂uuB/∂ωB < 0,
      - ∂uLoLRB/∂ωB > 0,
      - ∂uDIB/∂ωB < 0 for α big enough.
    - Existence of ω∗B solving uLoLRB = uuB; ω∗B independent of γ; ∂ω∗B/∂LB > 0, ∂ω∗B/∂DB < 0 for α sufficiently large.
    - Expressions:
      - ∂ω∗B/∂LB = YB [(P+1)ω∗B + αcB] / [P(ω∗B − LB/YB) + Pω∗B − LB/YB + αcB + DB] (positive for α large).
      - ∂ω∗B/∂DB = −ω∗B P / [P(ω∗B − LB/YB) + Pω∗B − LB/YB + αcB + DB] (negative for α large).
      - ∂ω∗B/∂cB and ∂ω∗B/∂P have signs depending on φ(LB) ≡ LB/Y − ω∗B; if φ(LB) < 0 then they are < 0 for α big enough; if φ(LB) > 0 then > 0.
  - Proposition 6 (bank A): analogous comparative statics and critical-point results for ω∗A with preserved formulas:
    - uuA(γ) and uLoLRA(γ), uDIA(γ) definitions preserved.
    - ∂uuA/∂ωA < 0, ∂uLoLRA/∂ωA > 0, ∂uDIA/∂ωA < 0 for α big enough.
    - Existence of ω∗A solving uLoLRA = uuA; for α big enough ∂ω∗A/∂LA > 0, ∂ω∗A/∂DA < 0.
    - Full derivative expressions preserved in source (see appendix).

### Babbling equilibria and DI learning incentives (Propositions 7 and 8)
- Proposition 7 (babbling equilibria existence and logic):
  - Babbling equilibria arise when the LoLR’s belief treats all messages as uninformative, γ(m) = γ, and messages independent of types m(γ) = m for all γ ≥ γ∗.
  - Two cases considered:
    - If ∃ ̄γ ∈ [0,1] such that E[γ | γ∗ ≤ γ ≤ ̄γ] = γ∗, then γ ≥ γ∗ and uLoLRA(γ) ≤ uDIA(γ∗) = u; outcome: LoLR does not liquidate bank A for all γ ≥ γ∗.
    - If no such ̄γ exists, then γ < γ∗ and uLoLRA(γ) > uDIA(γ∗) = u; outcome: LoLR liquidates bank A for all γ ≥ γ∗.
  - Deviation arguments show that equilibria with mixed messaging facing inconsistent beliefs cannot persist; either all messages lead to non-liquidation or all to liquidation in the babbling equilibrium.
- Proposition 8 (DI learning disincentives under support assumptions):
  - If the DI chooses not to learn about γ, bank A is not liquidated since max{uLoLRA(γ), uDIA(γ)} ≤ uA.
  - If the DI chooses to learn and γ lies below the lower bound of the support of Z(γ), then for any observed γ we have uDIA(γ) ≤ uDIA(γ̄) = uA, so DI prefers non-liquidation.
  - If γ ≥ γ∗ and ∄ ̄γ with E[γ | γ∗ ≤ γ ≤ ̄γ] = γ∗, bank A is liquidated and the DI is worse off.
  - If γ < γ∗ and γ∗ ≤ γ∗, the LoLR always liquidates bank A and the DI is worse off if he learns γ.

*Source: _wp11193 - 4. Ifγ (PDF chapter/section).*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2011/_wp11193.pdf_