## _wp16180

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---

### Introduction and research gap
- Motivation:
  - The 2007 financial crisis revealed global financial interconnectedness and drying up of interbank short-term lending; credit crunch triggered a bank run at Northern Rock—"something not seen in the UK for over 140 years and in Western Europe for the last 15 years."
  - Existing contagion literature uses direct balance-sheet linkages and indirect linkages (fire sales, expectations); prior work largely compares network structures rather than endogenizing banks’ formation of interconnections.
- Contribution:
  - Develops a simple stylized model that endogenizes interbank network formation when banks recognize contagion risk.
  - Endogenises the crisis state (rather than treating crises as zero-probability exogenous shocks).
  - Models banks’ optimizing decision to trade off liquidity-sharing benefits versus contagion costs.
  - Emphasizes regulatory frictions: explicit deposit insurance and implicit SIFI guarantees; notes other channels like underestimation of crisis probability and disaster myopia.

### Model setup (Section II): environment and mechanics
- Timeline and agents:
  - Three dates t = 0, 1, 2; two non-overlapping regions A and B.
  - Each region: continuum of identical depositors (live three periods) and a single representative competitive bank (zero profit in equilibrium).
- Depositor types and preferences:
  - Depositor is impatient with probability ψ (consume at date 1) and patient with probability 1−ψ (consume at date 2); depositors learn type privately at date 1.
  - Utility: u(c1,c2) = { c1 with probability ψ; c2 with probability 1−ψ }.
  - Endowment: 1 unit at date 0 only; can invest in liquid asset or deposit at bank; deposit contract allows withdrawal at t = 1 or t = 2.
- Assets and returns:
  - Liquid asset: 1 at t produces 1 at t+1.
  - Long asset: invest at t = 0; returns R > 1 at t = 2; if liquidated at t = 1 produces 0 < r < 1.
- Liquidity states and crisis:
  - Normal times: ψ ∈ {ω + x, ω − x}, negatively correlated across regions, aggregate liquidity ω constant.
  - Crisis state: ψ = 1 (all consumers impatient); crisis occurs with probability γ in each region i.i.d., with γ ≈ 0.
- Banks’ balance sheet and interbank market:
  - Bank chooses fraction y in liquid asset and may exchange interbank deposits of size x at t = 0.
  - Pecking order at t = 1: liquid asset → interbank market → costly liquidation of long asset.
  - Interbank market benefits: liquidity sharing across regions, economizing on liquid asset holdings, increasing c2 for patient depositors in normal times.
  - Contagion channel: interbank market assumed to shut down in crisis state (interbank deposits disappear); contagion arises when high liquidity demand in one region coincides with crisis in the other region, potentially forcing liquidation and triggering runs.

### Social optimality and core comparative statics (Section III)
- No interbank market (optimal y, payoffs):
  - Proposition 1: optimal y = ω + x.
  - Assumption 1: r < R / (2R − 1).
  - Depositor payoffs by liquidity state (Proposition 2):
    - High Liquidity | Probability 1/2 − γ/2 | c1 = 1 | c2 = R
    - Low Liquidity | Probability 1/2 − γ/2 | c1 = 1 | c2 = (2x + (1−ω−x)R) / (1−ω + x)  (< R)
    - Crisis | Probability γ | c1 = ω + x + (1−ω−x)r < 1 | Bank Run (everyone withdraws at date 1)
  - Expected depositor payoff (Proposition 3):
    - Π_NIB = (1/2 − γ/2)[(ω+x)+(1−ω−x)R] + γ[ω+x+(1−ω−x)r] + (1/2 − γ/2)[(ω−x)+(1−ω+x)·((2x+(1−ω−x)R)/(1−ω+x))]
    - Simplified: Π_NIB = R(1−ω−x) + ω + x − (1−ω−x)(R−r)γ
- With interbank market (optimal y, contagion):
  - Proposition 4: optimal y = ω and interbank deposits of size x.
  - Assumption 2 (contagion materiality): x > (1−ω)(R−1) / (R/r − 1) + ε (text presents this condition in words; Corollary 2 uses x > (1−ω)(R−1) / (R/r − 1) to ensure c2 < 1 and hence a run).
  - Depositor payoffs by state (Proposition 5): outcomes depend on whether the other region is in crisis; contagion can turn high liquidity demand states into runs.
  - Expected depositor payoff (Proposition 6):
    - Π_IB = (1/2 − γ/2)(1−γ)[(ω+x)+(1−ω−x)R] + (1/2 − γ/2)γ[ω+(1−ω)r] + (1/2 − γ/2)(1−γ)[(ω−x)+(1−ω+x)R] + γ[ω+(1−ω)r] + (1/2 − γ/2)γ[(ω−x)+(1−ω+x)·((x+(1−ω)R)/(1−ω+x))]
    - Simplified: Π_IB = ω + (1−ω)R − (3/2)(1−ω)(R−r)γ + 1/2(1−ω)(R−r)γ^2
- Participation trade-off and threshold:
  - Proposition 7: Π_IB − Π_NIB is decreasing in γ.
  - Proposition 8: There exists γ^* such that Π_IB > Π_NIB for γ < γ^* and Π_IB < Π_NIB for γ > γ^*.
  - Social planner rule: participate in interbank market if γ ≤ γ^*; do not participate if γ > γ^*.

### Market frictions and private optimality (Section IV)
- Friction categories producing excessive interconnectedness:
  - (i) Underestimation of γ (behavioral biases, disaster myopia).
  - (ii) Correct measurement of γ but frictions—deposit insurance and implicit SIFI guarantees—induce suboptimal private participation.
- Deposit insurance (explicit guarantee):
  - Modeled as guarantee that depositors get c1 = 1 at t = 1 even if bank insolvent; assumed externally funded (ex-post pay-as-you-go).
  - No interbank market with deposit insurance (Proposition 9):
    - Payoffs by state same as no-interbank case except crisis yields c1 = 1 and deposit insurance pays out.
  - Expected depositor payoff without interbank market but with deposit insurance (Proposition 10):
    - Π_NIB_DI = (1/2 − γ/2)[(ω+x)+(1−ω−x)R] + γ + (1/2 − γ/2)[(ω−x)+(1−ω+x)·((2x+(1−ω−x)R)/(1−ω+x))]
    - Simplified: Π_NIB_DI = R(1−ω−x) + ω + x − (1−ω−x)(R−1)γ
    - Expected cost of deposit insurance: C_NIB_DI = (1−ω−x)(1−r)γ = Π_NIB_DI − Π_NIB
  - Interbank market with deposit insurance (Proposition 11 and 12):
    - Π_IB_DI = (1/2 − γ/2)(1−γ)[(ω+x)+(1−ω−x)R] + (1/2 − γ/2)γ + (1/2 − γ/2)(1−γ)[(ω−x)+(1−ω+x)R] + γ + (1/2 − γ/2)γ[(ω−x)+(1−ω+x)·((x+(1−ω)R)/(1−ω+x))]
    - Simplified: Π_IB_DI = ω + (1−ω)R − (3/2)(1−ω)(R−1)γ + 1/2(1−ω)(R−1)γ^2
    - Expected cost of deposit insurance when participating:
      - C_IB_DI = (1/2 − γ/2)γ(1−ω)(1−r) + γ(1−ω)(1−r) = 3/2(1−ω)(1−r)γ − 1/2(1−ω)(1−r)γ^2 = Π_IB_DI − Π_IB
  - Comparative results:
    - Proposition 13: γ_DI > γ^* (deposit insurance raises the private participation threshold).
    - Explicit solution given for γ_DI (ignoring γ_DI = 1 root): γ_DI = 2x / (1 − ω).
    - Proposition 14: For γ^* > γ > γ_DI, banks’ participation in the interbank market is socially sub-optimal because private incentives (under deposit insurance) ignore fiscal cost.
  - Intuition: Deposit insurance reduces depositor downside in crisis, increasing private willingness to accept interbank risk; depositors do not internalize government cost, producing excessive interconnectedness when γ lies between γ^* and γ_DI.
- Implicit SIFI guarantees (too-big-to-fail):
  - If forming interconnections makes a bank a SIFI, implicit government guarantees (bailouts) effectively remove crisis downside for that bank.
  - Result: private incentives always favor interbank participation irrespective of γ, producing excessive interconnectedness even when γ is correctly measured.
- Underestimation of γ (behavioral channel):
  - Availability heuristic and disaster myopia can bias perceived γ downward, leading to excessive interconnectivity even absent explicit/inimplicit guarantees.
  - Long intervals without bank runs historically can contribute to underestimation.

### Policy insights and welfare implications (Section V conclusions)
- Main welfare message:
  - In absence of frictions (deposit insurance, SIFI guarantees, behavioral biases), private and social optima align: banks participate iff γ ≤ γ^*.
  - Explicit deposit insurance and implicit SIFI guarantees create a wedge between private and social optima and can induce excessive interconnectedness and higher systemic risk.
- Policy-relevant observations:
  - Deposit insurance raises private threshold γ_DI above social γ^*, producing socially sub-optimal participation when γ^* < γ < γ_DI.
  - Implicit SIFI guarantees can lead to always-participate equilibria that are socially excessive.
  - Behavioral underestimation of crisis probability is an additional channel to excessive interconnectedness.
- Limitations noted:
  - Framework is stylized; not by itself sufficient for full policy analysis. The paper points to broader literature for more detailed policy evaluation.

### Appendix A: proofs — selected technical results and formulas
- Key exact expressions and conditions preserved from proofs:
  - Assumption 1: r < R/(2R − 1).
  - Optimality without interbank market: y = ω + x (Proposition 1).
  - c2 in low liquidity, no interbank: c2 = [2x + (1 − ω − x)R] / (1 − ω + x) and 1 < c2 < R (Corollary 1).
  - Optimality with interbank market: y = ω, interbank deposits x (Proposition 4); condition for optimality involves comparison of liquidation loss and opportunity cost; near-universally satisfied except when liquidation cost ≈ 0 or R ≈ 1 and γ high.
  - Contagion condition (Corollary 2): run occurs if x > (1 − ω)(R − 1) / (R/r − 1); Assumption 2 ensures this holds.
  - Difference Π_IB − Π_NIB closed-form (Proposition 7):
    - ΠIB − ΠNIB = x(R − 1) − 1/2 (1 − ω + 2x)(R − r)γ + 1/2 (1 − ω)(R − r)γ^2
    - ∂/∂γ (ΠIB − ΠNIB) = −1/2 (1 − ω + 2x)(R − r) + (1 − ω)(R − r)γ
    - Decreasing in γ for γ < 1/2 and generally for relevant parameter ranges.
  - γ^* solves:
    - 0 = x(R − 1) − 1/2 (1 − ω + 2x)(R − r)γ + 1/2 (1 − ω)(R − r)γ^2
    - γ* = [1/2 (1 − ω + 2x)(R − r) − sqrt{1/4 (1 − ω + 2x)^2 (R − r)^2 − 2(1 − ω)(R − r)x(R − 1)}] / [(1 − ω)(R − r)]
  - Deposit-insurance adjusted difference (Proposition 13):
    - ΠIBDI − ΠNIBDI = x(R − 1) − 1/2 (1 − ω + 2x)(R − 1)γ + 1/2 (1 − ω)(R − 1)γ^2
    - γDI explicit (ignoring γDI = 1 root): γDI = 2x / (1 − ω)
  - Proposition 14 logic: for γ in (γDI, γ*), private participation with deposit insurance is higher than privately optimal without it, but social accounting including fiscal cost shows participation is sub-optimal.

*Source: _wp16180 - References and Sections II–V, Appendix A (Proofs).*

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

### _wp16180 - References .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .

### Introduction: motivation and research gap
- The financial crisis that hit the world economy in 2007 revealed the interconnected nature of the global financial system and led to a drying up of interbank short-term lending; the credit crunch triggered a bank run at Northern Rock—"something not seen in the UK for over 140 years and in Western Europe for the last 15 years."
- Existing literature on contagion uses two main approaches: direct balance-sheet linkages and indirect linkages operating through fire sales and expectations.
- Prior work has focused largely on comparing social costs and benefits of different network structures rather than on banks’ endogenous decision to form interconnections.
- This paper fills that gap by developing a simple stylized model to study banks’ decision to form financial interconnections, explicitly endogenizing network formation when banks recognize the risk of contagion.

### Model approach and objectives
- The model:
  - Endogenises the crisis state of the world rather than treating crises as zero-probability exogenous shocks.
  - Models interbank market formation as an optimizing decision by banks that trade off liquidity-sharing benefits against contagion costs.
  - Models contagion through direct balance-sheet linkages and shows how these can result in contagious bank runs, following Bryant (1980) and Diamond and Dybvig (1983).
- Focus of analysis:
  - Investigates market frictions that generate a wedge between socially optimal and privately optimal network formation.
  - Emphasizes two regulatory interventions as key frictions: explicit deposit insurance and implicit guarantees for systemically important financial institutions (SIFI).
  - Notes but does not primarily focus on other sources of suboptimal formation such as underestimation of crisis probability and disaster myopia.

### Key conceptual findings presented in the Introduction
- Deposit insurance and perceptions of "too big to fail" (SIFI implicit guarantees) can create a wedge between social and private optimality.
  - In the presence of these guarantees, competitive banks may optimally participate in the interbank market even when contagion risk is high and social optimality would advise against such participation—leading to excessive interconnections in equilibrium.
- When no externalities or market failures are present, competitive banks’ optimal participation strategy is socially optimal.
- Introducing market frictions (deposit insurance, SIFI implicit guarantees) can cause banks to form interconnections that are excessive from a social planner’s perspective.

### Relationship to existing literature (selected themes and exact citations as presented)
- Network contagion and liquidity-sharing benefits:
  - Allen and Gale (2000): incomplete networks may be more prone to contagion than complete structures; complete structures can be more resilient.
  - Allen, Babus, and Carletti (2009): survey of literature on financial crises.
- Global games and equilibrium selection:
  - Dasgupta (2004); Morris and Shin (2000, 2003); Brusco and Castiglionesi (2007); Rochet and Vives (2004); Goldstein and Pauzner (2005); Chen, Goldstein, and Jiang (2010).
- Complex networks and fragility:
  - Gai and Kapadia (2010): greater connectivity reduces likelihood of widespread default but can increase impact when shocks occur.
  - Acemoglu, Ozdaglar, and Tahbaz-Salehi (2015a): magnitude of negative shocks affects whether dense connectivity is stabilizing or destabilizing.
- Endogenous network formation and optimality:
  - Leitner (2005); Kahn and Santos (2010); Babus (2009); Castiglionesi and Navarro (2007); Acemoglu, Ozdaglar, and Tahbaz-Salehi (2015b).
  - Babus (2009): predicts a connectivity threshold above which contagion does not occur; implicit link costs prevent over-connection.
  - Kahn and Santos (2010): banks generally fail to find the correct degree of interdependence when exogenous liquidity is scarce.
  - Acemoglu, Ozdaglar, and Tahbaz-Salehi (2015b): even with bilateral internalization of counterparty risk, banks can over-lend and under-diversify.

### Paper organization (as stated)
- Section II: sets up the model and outlines the costs and benefits of participating in the interbank market.
- Section III: solves the model for the banks’ optimal participation strategy; competitive banks’ optimal strategy is socially optimal in absence of externalities and market failures.
- Section IV: introduces market frictions and explores how they create a wedge between socially optimal and privately optimal participation strategies, producing excessive interconnectedness in equilibrium.

*Source: _wp16180 - References .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .*

### Section V concludes. The proofs are presented in the appendix.

### _wp16180 - Section V concludes. The proofs are presented in the appendix.

### II. MODEL SETUP
- Timeline and structure:
  - Economy with three dates, t = 0, 1, 2, and two non-overlapping regions A and B.
  - Each region: continuum of ex-ante identical depositors (live three periods) and a single representative competitive bank (zero profit in equilibrium).
  - Single good used for consumption and investment (numeraire).
- Depositor preferences and uncertainty:
  - Depositor is either impatient (consume at date 1) or patient (consume at date 2).
  - Fraction of impatient depositors denoted by ψ; depositors learn type privately at date 1.
  - Utility: u(c1,c2) = { c1 with probability ψ; c2 with probability 1−ψ }.
  - Endowment: 1 unit at date 0 only; can invest in liquid asset or deposit at bank; deposit contract allows withdrawal at date 1 or date 2.
- Assets:
  - Liquid asset: 1 invested at t produces 1 at t+1.
  - Long asset: invest only at t = 0; returns R > 1 at t = 2; if liquidated at t = 1 produces 0 < r < 1.
  - Asset cash flows summarized (Figure 2): liquid asset flows 1 → 1; long asset 1 → r (liquidated at t = 1) or → R (not liquidated at t = 2).
- Deposit contracts:
  - Demand deposit contracts (c1,c2), with focus on conversion at par at t = 1: c1 = 1 (following Dasgupta (2004)).
  - Bank pools assets to insure depositors against liquidity shocks; multiple equilibria possible but paper focuses on the no-run efficient equilibrium where patient depositors wait till t = 2 (deposit insurance ensures no-run is dominant).
- Banks’ balance sheet choice:
  - Bank holds fraction y in liquid asset and (1−y) in long asset; may decide to participate in interbank market by exchanging interbank deposits b at t = 0 (does not affect budget ex-ante).
  - At t = 1: if y ≥ ψ, bank uses liquid asset to service withdrawals; if y < ψ, bank relies on interbank market or liquidates long asset.
  - Pecking order: liquid asset → interbank market → liquidation (costly).
- Liquidity states and crisis:
  - Normal times: ψ ∈ {ω + x, ω − x}; states negatively correlated across regions; both equally likely so aggregate liquidity ω constant.
  - Crisis state: ψ = 1 (all consumers impatient); crisis occurs with probability γ in each region i.i.d., where γ ≈ 0.
  - Figure 1 summarizes possible states (states include combinations: High/Low/Crisis across regions).

### Interbank market: benefits and contagion channel
- Benefits:
  - Allows banks to share liquidity during normal times because shocks are negatively correlated; banks hold average liquidity ω and interbank deposits x to insure regional shocks (Allen and Gale (2000); Dasgupta (2004)).
  - Participation economizes on liquid asset holdings and increases c2 offered to patient depositors in normal times.
- Contagion and crisis modeling:
  - Interbank market assumed to shut down in crisis state (interbank deposits disappear), consistent with evidence of interbank market shutdowns (e.g., post-Lehman).
  - Contagion arises when high liquidity demand in one region coincides with crisis in the other region: a bank reliant on interbank liquidity (y = ω) can be forced to liquidate long assets and suffer a run despite being solvent ex-ante.
  - Contagion takes place only when high liquidity demand coincides with crisis in the other region (model abstracts from other contagion channels like asset fire sales).

### III. MODEL SOLUTION – SOCIAL OPTIMALITY
- No Interbank Market (banks optimize y ignoring crisis state since crisis makes bank bust):
  - Proposition 1: When not participating in the interbank market, the bank optimally holds y = ω + x of the liquid asset.
  - Assumption 1: r < R / (2R − 1) (liquidation sufficiently costly).
  - Depositor payoffs (Proposition 2) when no interbank market:
    - Liquidity State | Probability | c1 | c2
      - High Liquidity | 1/2 − γ/2 | 1 | R
      - Low Liquidity | 1/2 − γ/2 | 1 | (2x + (1−ω−x)R) / (1−ω+x)  (< R)
      - Crisis | γ | ω + x + (1−ω−x)r < 1 | Bank Run, everyone withdraws at date 1
  - Corollary 1: In the low liquidity demand state with no interbank market, 1 < c2 < R.
  - Expected depositor payoff with no interbank market (Proposition 3):
    - Π_NIB = (1/2 − γ/2)[(ω+x)+(1−ω−x)R] + γ[ω+x+(1−ω−x)r] + (1/2 − γ/2)[(ω−x)+(1−ω+x)·((2x+(1−ω−x)R)/(1−ω+x))]
    - Simplified expression given: Π_NIB = R(1−ω−x) + ω + x − (1−ω−x)(R−r)γ
- With Interbank Market:
  - Proposition 4: When participating in the interbank market, the bank optimally holds y = ω of the short asset and exchanges interbank deposits of size x.
  - Assumption 2: x > (1−ω)(R−1) / (R/(r−1)) + ε where ε → 0 (condition ensuring contagion materiality; text states condition as x > (1−ω)(R−1) / (R / (r − 1)) + ε in words; see original for exact formulaic arrangement).
  - Depositor payoffs (Proposition 5) when participating in interbank market:
    - Liquidity State | Other Region | Probability | c1 | c2
      - High Liquidity, No Crisis | (1/2 − γ/2)(1−γ) | 1 | R
      - High Liquidity, Crisis | (1/2 − γ/2)γ | ω + (1−ω)r < 1 | Contagion: Bank Run, everyone withdraws at date 1
      - Low Liquidity, No Crisis | (1/2 − γ/2)(1−γ) | 1 | R
      - Low Liquidity, Crisis | (1/2 − γ/2)γ | 1 | (x + (1−ω)R) / (1−ω + x)  (< R)
      - Crisis | γ | ω + (1−ω)r < 1 | Bank Run, everyone withdraws at date 1
  - Corollary 2: If Assumption 2 holds, contagion results in a run on the bank during the high liquidity demand state.
  - Expected depositor payoff with interbank market (Proposition 6):
    - Π_IB = (1/2 − γ/2)(1−γ)[(ω+x)+(1−ω−x)R] + (1/2 − γ/2)γ[ω+(1−ω)r] + (1/2 − γ/2)(1−γ)[(ω−x)+(1−ω+x)R] + γ[ω+(1−ω)r] + (1/2 − γ/2)γ[(ω−x)+(1−ω+x)·((x+(1−ω)R)/(1−ω+x))]
    - Simplified expression: Π_IB = ω + (1−ω)R − (3/2)(1−ω)(R−r)γ + 1/2(1−ω)(R−r)γ^2
- Participation trade-off and threshold:
  - Proposition 7: Π_IB − Π_NIB is decreasing in crisis probability γ.
  - Proposition 8: There exists γ^* such that Π_IB > Π_NIB for γ < γ^* and Π_IB < Π_NIB for γ > γ^*.
  - Interpretation: Socially optimal strategy: participate in interbank market if γ ≤ γ^*; stop participating if γ > γ^*. In absence of frictions, private and social optimality align.

### IV. MARKET FRICTIONS – PRIVATE OPTIMALITY
- Overview:
  - Two broad friction categories causing excessive interconnectedness:
    - (i) Underestimation of the probability of crisis (γ).
    - (ii) Correct measurement of γ but frictions (deposit insurance, implicit SIFI guarantees) induce suboptimal participation.
  - Focus: deposit insurance and too-big-to-fail / SIFI implicit guarantees.
- Deposit Insurance:
  - Mechanism in model:
    - Deposit insurance modeled as guarantee that depositors get c1 = 1 at t = 1 even if bank insolvent (protects depositor capital but not interest; alternative stronger assumption guaranteeing R at t = 2 would strengthen results).
    - Deposit insurance assumed externally funded (ex-post pay-as-you-go), consistent with some real-world schemes.
  - Effects on payoffs:
    - Proposition 9 (no interbank market with deposit insurance): depositor payoffs by state:
      - High Liquidity: prob 1/2 − γ/2: c1 = 1; c2 = R
      - Low Liquidity: prob 1/2 − γ/2: c1 = 1; c2 = (2x+(1−ω−x)R)/(1−ω+x) < R
      - Crisis: prob γ: c1 = 1; Bank Run, deposit insurance pays out
    - Proposition 10: Expected depositor payoff without interbank market but with deposit insurance:
      - Π_NIB_DI = (1/2 − γ/2)[(ω+x)+(1−ω−x)R] + γ + (1/2 − γ/2)[(ω−x)+(1−ω+x)·((2x+(1−ω−x)R)/(1−ω+x))]
      - Simplified: Π_NIB_DI = R(1−ω−x) + ω + x − (1−ω−x)(R−1)γ
      - Expected cost of deposit insurance: C_NIB_DI = (1−ω−x)(1−r)γ = Π_NIB_DI − Π_NIB
    - Proposition 11 (interbank market with deposit insurance): depositor payoffs by state are identical to interbank case except crisis outcomes now give c1 = 1 (deposit insurance pays out).
    - Proposition 12: Expected depositor payoff with interbank market and deposit insurance:
      - Π_IB_DI = (1/2 − γ/2)(1−γ)[(ω+x)+(1−ω−x)R] + (1/2 − γ/2)γ + (1/2 − γ/2)(1−γ)[(ω−x)+(1−ω+x)R] + γ + (1/2 − γ/2)γ[(ω−x)+(1−ω+x)·((x+(1−ω)R)/(1−ω+x))]
      - Simplified: Π_IB_DI = ω + (1−ω)R − (3/2)(1−ω)(R−1)γ + 1/2(1−ω)(R−1)γ^2
      - Expected cost of deposit insurance when participating: C_IB_DI = (1/2 − γ/2)γ(1−ω)(1−r) + γ(1−ω)(1−r) = 3/2(1−ω)(1−r)γ − 1/2(1−ω)(1−r)γ^2 = Π_IB_DI − Π_IB
  - Key comparative results:
    - For every γ, Π_IB − Π_NIB is strictly greater with deposit insurance; threshold γ_DI > γ^* (Proposition 13).
    - Proposition 14: For γ^* > γ > γ_DI, banks’ participation in the interbank market is socially sub-optimal.
    - Accounting for government cost: Π_IB_DI − C_IB_DI = Π_IB. Thus deposit insurance creates a wedge between private optimal (higher threshold γ_DI) and socially optimal γ^* because depositors do not internalize government cost.
  - Intuition: Deposit insurance reduces depositor losses in crisis, increasing depositors’ willingness to accept network risk; competitive banks must offer higher returns (via interbank participation), producing excessive interconnectedness when γ is between γ^* and γ_DI.
- Systemically Important Financial Institutions (SIFIs) and implicit guarantees:
  - If a bank becomes a SIFI upon forming interconnections, it benefits from an implicit government guarantee (bailout) that prevents failure.
  - Under SIFI assumption: crisis probability effectively zero for the bank (no insolvency allowed), so private incentives always favor interbank participation irrespective of γ.
  - Result: implicit SIFI guarantees can generate excessive interconnectedness even when γ is correctly measured.
- Underestimation of γ (disaster myopia):
  - Behavioral frictions (availability heuristic, disaster myopia) can lead to downwards bias in perceived γ, producing excessive interconnectivity even absent explicit/inexplicit guarantees.
  - Historical precedent: long intervals without runs (e.g., UK over 140 years) can contribute to underestimation of crisis risk.

### V. CONCLUSION
- Framework and contribution:
  - Developed a simplified model endogenizing network formation with an explicit crisis state; banks trade off liquidity sharing benefits of interbank participation against contagion costs.
  - Identified a socially optimal threshold γ^*: participate if γ ≤ γ^*, otherwise do not.
- Market frictions causing excessive interconnectedness:
  - Deposit insurance (explicit government guarantee) raises private threshold γ_DI > γ^*, inducing banks to participate in interbank networks when socially suboptimal (γ^* < γ < γ_DI).
  - Implicit too-big-to-fail guarantees for SIFIs remove crisis downside and induce always-participate behavior, creating excessive interconnectedness.
  - Behavioral underestimation of γ (disaster myopia) is an additional channel for excessive interconnectedness.
- Policy and welfare insight:
  - In absence of frictions, private and social optima align; frictions (explicit or implicit guarantees, behavioral biases) create wedges leading to suboptimal network formation and higher systemic risk.
  - The framework highlights channels through which deposit insurance and SIFI protections can increase systemic interconnectedness, but the paper notes it is not by itself sufficient for full policy analysis and points to broader literature for more detailed policy evaluation.

*Source: _wp16180 - Section V concludes. The proofs are presented in the appendix.*

### REFERENCES

### _wp16180 - REFERENCES

### Key propositions, corollaries, and analytical results (Appendix A: Proofs)
- Proposition 1
  - When not participating in the interbank market, the bank optimally holds y = ω + x of the liquid asset.
  - Deviation by holding y = ω + x − ε (ε → 0) is not optimal because the loss from early liquidation outweighs the gain from a smaller opportunity cost.
  - Comparison leading to optimality uses:
    - Loss due to liquidation: (1 − ω − x)· εR/(1 − ω − x) (1/r − 1)
    - Gain due to smaller y: (1 − ω + x)· ε(R − 1)/(1 − ω + x)
    - Condition reduces to r < R/(2R − 1), which holds by Assumption 1.

- Corollary 1
  - In the low liquidity demand state with no interbank market, 1 < c2 < R, where
    - c2 = [2x + (1 − ω − x)R] / (1 − ω + x)
  - Proof shows c2 > 1 because (1 − ω − x) > 0 and R > 1; c2 < R because x < R and x < 1 with R > 1.

- Proposition 4
  - When participating in the interbank market, the bank optimally holds y = ω of the liquid asset and exchanges interbank deposits of size x.
  - Two-stage argument:
    - Banks hold just enough liquidity (investment in liquid asset + interbank deposits) to pay c1 = 1 at t = 1 during normal times.
    - Given opportunity cost, holding more liquidity than ω + x is suboptimal; holding less either via smaller interbank deposits or smaller y reduces payoffs.
  - Cases analyzed:
    - Case 1(a): y = ω and interbank deposits x − ε ⇒ c2 decreases in ε; optimal ε = 0.
    - Case 1(b): y = ω − ε and interbank deposits x ⇒ liquidation value falls by ε(1 − r); expected gain from reduced y is dominated by expected loss for γ < 1/2.
    - Case 2(a): y = ω − ε and interbank deposits x + ε ⇒ loss due to liquidation; optimal ε = 0.
    - Case 2(b): y = ω + ε and interbank deposits x − ε ⇒ c2 decreases in ε in normal states; although liquidation value rises in run states via ε(1 − r), expected loss when no run typically exceeds expected gain.
  - Condition for optimality (expected loss > expected gain) given:
    - 1 − r
      R − 1
      <
      γ
      2
      − γ + 2
      γ(1 − γ)
    - Holds for most parameter values; violated only when liquidation cost close to 0 and/or R very close to 1 and probability of crisis very high.

- Corollary 2
  - If Assumption 2 holds, contagion results in a run on the bank during the high liquidity demand state.
  - In crisis in region B, region A bank must liquidate l = x/r of the long asset to meet ψ = ω + x withdrawals; remaining assets produce
    - c2 = (1 − ω − x/r)R / (1 − ω − x)
  - For a run, need c2 < 1 which leads to x > (1 − ω)(R − 1) / (R/r − 1); given Assumption 2, this is always true.

- Proposition 7
  - ΠIB − ΠNIB is decreasing in crisis probability γ (the benefits of participating in the interbank market decline as γ increases).
  - Expression:
    - ΠIB − ΠNIB = x(R − 1) − 1/2 (1 − ω + 2x)(R − r)γ + 1/2 (1 − ω)(R − r)γ^2
  - Partial derivative:
    - ∂/∂γ (ΠIB − ΠNIB) = −1/2 (1 − ω + 2x)(R − r) + (1 − ω)(R − r)γ
  - Sign is negative if γ < (1 − ω + 2x) / [2(1 − ω)], always true for γ < 1/2; thus ΠIB − ΠNIB decreases in γ for γ < 1/2.

- Proposition 8
  - There exists γ* such that ΠIB > ΠNIB for γ < γ* and ΠIB < ΠNIB for γ > γ*.
  - Solve ΠIB − ΠNIB = 0 for γ:
    - 0 = x(R − 1) − 1/2 (1 − ω + 2x)(R − r)γ + 1/2 (1 − ω)(R − r)γ^2
    - γ* = [1/2 (1 − ω + 2x)(R − r) − sqrt{1/4 (1 − ω + 2x)^2 (R − r)^2 − 2(1 − ω)(R − r)x(R − 1)}] / [(1 − ω)(R − r)]

- Proposition 13
  - The critical level of γ is higher with deposit insurance: γDI > γ*.
  - Without deposit insurance:
    - ΠIB − ΠNIB = x(R − 1) − 1/2 (1 − ω + 2x)(R − r)γ + 1/2 (1 − ω)(R − r)γ^2
  - With deposit insurance:
    - ΠIBDI − ΠNIBDI = x(R − 1) − 1/2 (1 − ω + 2x)(R − 1)γ + 1/2 (1 − ω)(R − 1)γ^2
  - For every γ, gains from participating are larger with deposit insurance; hence γDI > γ*.
  - Explicit solution for γDI (ignoring root γDI = 1): γDI = 2x / (1 − ω).

- Proposition 14
  - For γ* > γ > γDI, banks' participation in the interbank market is socially sub-optimal.
  - Logic:
    - For γ in (γDI, γ*), privately optimal behavior differs with/without deposit insurance:
      - No deposit insurance: bank optimally does not participate; expected depositor payoff = ΠNIB.
      - With deposit insurance: privately optimal to participate; expected depositor payoff = ΠIBDI.
    - Social optimality requires accounting for deposit insurance cost CIBDI = ΠIBDI − ΠIB.
    - Participation is socially sub-optimal when ΠNIB > ΠIBDI − CIBDI, which reduces to ΠNIB > ΠIB. This holds for all γ > γ* (Proposition 8). Therefore for γ* > γ > γDI participation is socially sub-optimal.

*Source: Appendix A (Proofs) and related propositions as presented in the content unit.*

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