## _wp10170

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---

### I. Introduction — overview and key mechanisms
- Context and motivation:
  - Banks increasingly borrow short-term wholesale funds to supplement retail deposits (Feldman and Schmidt, 2001).
  - Wholesale instruments include large-denomination certificates of deposits, brokered deposits, repurchase agreements, Fed funds, and commercial paper.
  - Traditional literature emphasizes a "bright side": exploiting investment opportunities beyond local deposit supply, market discipline by wholesale financiers (Calomiris, 1999), and refinancing unexpected retail withdrawals (Goodfriend and King, 1998).
  - Recent mortgage banking crisis examples (Acharya et al., 2008; Huang and Ratnovski, 2009) reveal a "dark side": aggressive lending, compromised credit quality, and abrupt withdrawal by wholesale financiers triggering inefficient liquidations.
- Central insight and modelling approach:
  - Wholesale funding is beneficial when financiers are informed, but may lead to inefficient liquidations when uninformed.
  - Bank finances a risky long-term project with retail deposits (sluggish, partly insured) and wholesale funds (sophisticated, roll-over financing).
  - Benchmark: Calomiris and Kahn (1991, CK) — sophisticated wholesale financiers add value through monitoring; monitoring incentives maximized when they are senior at refinancing stage.
- Novel feature:
  - A costless but noisy public signal on bank project quality (e.g., market prices, credit ratings, sector indicators) that wholesale financiers can costlessly process and that is a noisy proxy for bank-specific fundamentals.
- Main qualitative results:
  - Presence of a costless but noisy signal:
    - Lowers incentives of wholesale financiers to monitor.
    - Gives wholesale financiers excess incentives to liquidate banks based on noisy public information.
    - Distortions strengthen when wholesale financiers are more senior claimants to liquidated assets (contrasting with CK).
  - Mechanism:
    - Without noisy public signal, uninformed wholesale financiers typically roll over funding.
    - With noisy public signal, uninformed financiers may liquidate on a negative but imprecise public signal.
    - Senior financiers obtain larger share of liquidation value, reducing payoffs to long-term passive depositors and discouraging private monitoring.
  - Welfare implication:
    - Socially optimal seniority trades off CK monitoring benefit against the liquidation distortion and has an interior optimum.
    - Monitoring incentives increase in seniority for low values (CK effect) but decrease for higher values when seniority translates into more liquidations.
  - Key parameters making higher seniority more attractive socially (exact wording preserved):
    - "The noisy public signal is more precise, yet not as precise as to make liquidation decisions based on it socially optimal;"
    - "The share of passive deposits in bank liabilities is higher."
    - "Liquidation value of bank assets is higher."
    - "Interest rate offered to wholesale financiers in case of success is lower."
  - Cross-sectional prediction:
    - Senior short-term funds beneficial in "traditional" banks holding opaque and nontradable relationship loans (CK applies).
    - "Dark side" significant for banks with large exposures to standardized and tradable arm’s length assets with readily available public information, particularly when short-term wholesale financiers are senior claimants.
  - Private incentives:
    - Arm’s length banks may choose senior short-term wholesale funds because interest rates on them are lowest when assets are marketable and public signals are informative.
  - Net conclusion:
    - Use and high seniority of wholesale funds is not always socially beneficial; with a costless but noisy signal, higher seniority can reduce monitoring and encourage inefficient liquidations. Social welfare is constrained-maximized for an intermediate level of seniority depending on funding structure, signal precision, liquidation value, and interest rates.

### II. The Bright Side of Wholesale Funding — model setup (CK-type benchmark)
- Environment and timing:
  - Three dates (0;1;2), no discounting, everyone is risk-neutral.
- The project:
  - For each unit invested at date 0, at date 2 the project returns X with probability p or 0 with probability 1 p, with a positive net present value: Xp>1.
  - Project may be liquidated at date 1 returning L<1 per unit initially invested.
  - Maximum investment size is 1.
- Funding structure and financiers (retail depositors):
  - Retail deposits: unsophisticated, passive, scarce; never get advance information on date 2 realization; never withdraw before date 2.
  - Interest rate on retail deposits fixed at R_D:1R_D<pX.
  - Bank endowed with fixed deposit base D<1; cannot expand within model horizon.
- Benchmark case: retail deposits only:
  - Spare capacity is inefficient when D<1.
  - Bank always continues until date2; bad projects continue and yield 0 at date2 to preserve liquidation value L.
  - Monetary value of social welfare: Dep = D(pX − 1).  (equation (2))

### III. Wholesale funds — social-welfare maximization (no noisy signal)
- Environment and financier characteristics:
  - Wholesale financier: sophisticated, unlimited supply at date0 against real expected return R̄; can monitor with intensity m (0≤m<1) at private cost C(m) with C(0)=0, C(1)=1, C′(0)=0, C′(m)>0, C′′(m)>0.
  - Monitoring yields precise information with probability m and fails with probability 1−m.
  - Wholesale funding must be rolled over at date1; refusal leads to liquidation with liquidation value L(D+W) shared: financier receives share s (0≤s≤1).
  - Assumption: pW > L (so financier never liquidates solely to receive L(D+W)).
- Social welfare with wholesale funding:
  - Maximize Π = (D+W) (pX + m(1−p)L − 1) − C(m).  (equation (3))
  - Socially optimal choices:
    - W̄ = 1 − D (full investment).
    - Monitoring intensity m̄ satisfying C′(m̄) = (1−p)L.  (equation (4))
  - Benefits relative to retail-only:
    - Higher investment volume (D+W = 1) instead of D.
    - Preservation of some bad banks’ liquidation value at cost C(m̄).
- Wholesale funds — private equilibrium (no noisy signal):
  - Wholesale financier’s payoff: Π_W = pWR + m(1−p)sL(D+W) − C(m).
  - Private first-order condition: C′(m_W) = (1−p)sL(D+W).  (equation (5))
  - Interest rate demanded (zero-profit): R = (Wζ + C(m_W) − m_W(1−p)sL(D+W)) / (W p).  (equation (7))
  - Bank surplus: Π_B = p[D(X − R_D) + W(X − R)].  (equation (6))
  - Lemma: Π_B increases in s and W; bank maximizes surplus with W = 1 − D and s = 1.
  - Benchmark Proposition 1: In "bright side" case, W = W̄, s = s̄ = 1, and m = m̄ lead to socially optimal outcome; only banks known to be bad are liquidated.

### IV. Introduction of a free but noisy public signal (the "dark side")
- Noisy public signal mechanics:
  - Received by the wholesale financier prior to date1 but after choosing m; not observed by retail depositors.
  - Signal: "positive" or "negative", precision parameter ϕ (0 ≤ ϕ ≤ 1; ϕ = 0 complete noise, ϕ = 1 precise).
  - Unconditional probability of positive signal is p.
  - Conditional probabilities:
    - If signal positive: project yields X at date2 with probability [p + ϕ(1−p)], yields 0 with probability [(1−p) − ϕ(1−p)].
    - If signal negative: project yields X with probability [p − ϕp], yields 0 with probability [(1−p) + ϕp].
- Key qualitative effects:
  - Noisy signal lowers incentives to monitor (private information less valuable).
  - Noisy signal can induce excess liquidation based solely on the noisy signal when precision is sufficiently high.

### V. Welfare maximization with noisy signal
- Date1 refinancing rule under monitoring/noisy signal:
  - If monitoring yields precise info: continue if good and liquidate if bad.
  - If monitoring yields no info: use public signal — refinance on positive signal; negative signal may trigger liquidation depending on precision.
- Threshold precision for socially optimal liquidation on negative signal:
  - ϕ̄ = 1 − (L / (p^2 X)).  (equation (8))
  - If ϕ > ϕ̄ and [p − ϕ p] p X < L, social optimum calls for liquidation on negative signal.
- Social welfare when liquidating on negative signal:
  - Π_Liq = (D+W) ( m[pX + (1−p)L] + (1−m) [ p[p + ϕ(1−p)]X + (1−p)L ] − 1 ) − C(m).  (equation (9))
- Optimal wholesale funding and monitoring under high precision:
  - W̄_Liq = 1 − D (= W̄).
  - Monitoring intensity m̄_Liq satisfies C′(m̄_Liq) = p(1−p) (1−ϕ) X.  (equation (10))
  - Note: m̄_Liq < m̄ (monitoring intensity falls when the noisy signal is used).

### VI. Wholesale financier incentives with noisy signal — inefficient liquidations and monitoring
- Private liquidation decision when uninformed but observing noisy signal:
  - Continuation payoff conditional on negative signal: [p − ϕ p] W R.
  - Liquidation payoff: s L (D+W).
  - Financier privately liquidates on negative signal when:
    - s L (D+W) > [1 − ϕ] p W R.  (equation (11))
- Threshold expressions:
  - Threshold precision for financier to follow noisy signal:
    - ϕ_W = 1 − (s L (D+W)) / (p W R).  (equation (12))
  - Threshold seniority for financier to follow noisy signal:
    - s_W = ( (1 − ϕ) p W R ) / ( L (D+W) ).  (equation (13))
  - Interpretation: higher s raises financier’s incentives to liquidate on noisy negative signals.
- Monitoring intensity with noisy signal:
  - If junior (s ≤ s_W): financier disregards noisy signal and uses C′(m_W) = (1−p) s L (D+W).  (equation (5))
  - If senior (s > s_W): financier uses noisy signal and maximizes payoff (equation (14)), with first-order condition:
    - C′(m_W_Liq) = p(1−p) (1−ϕ) W R_Liq.  (equation (15))
  - Interest rate in liquidation regime:
    - R_Liq = ( Wζ + C(m_W_Liq) − (1−p) s L (D+W) ) / ( m_W_Liq W p + (1−m_W_Liq) [ p + ϕ(1−p) ] W p ).  (equation (16))
  - Comparative statics:
    - As s increases, R_Liq decreases, which reduces m_W_Liq; hence m_W_Liq decreases in s.
- Lemma 2 (threshold s_W properties):
  1. s_W decreases in ϕ and L; s_W decreases in D and increases in W (provided D+W = 1).
  2. For s ≤ s_W, financier never liquidates based on noisy public signal and monitoring intensity increases in seniority: ∂m_W/∂s > 0.
  3. For s > s_W, financier liquidates following a negative noisy signal and monitoring intensity decreases in seniority: ∂m_W_Liq/∂s < 0.

### VII. Socially optimal seniority, bank incentives, and numerical demonstration
- Proposition 2 (socially optimal seniority):
  - The socially optimal creditor seniority of the wholesale financier is s = s_W , s_W < 1.
  - Setting s = s_W aligns continuation decisions with social optimum and eliminates inefficient liquidations.
  - s = s_W maximizes monitoring intensity, albeit below the social optimum: m_W (s_W) < m̄.
  - All else equal, incentives for inefficient liquidations are higher (and socially optimal seniority lower) when:
    - precision ϕ is higher;
    - liquidation value L is higher;
    - there are more deposits D serving as buffer for wholesale funds’ exit.
- Bank incentives and potential deviation:
  - Bank chooses creditor seniority to maximize private surplus and may prefer s > s_W because higher seniority reduces interest rate R on wholesale funding.
  - Bank surplus at s = s_W: Π_B s=s_W = p[D(X R_D ) + W(X R s=s_W )].  (equation (17))
  - For s > s_W (liquidation regime): Π_B Liq =  p (1 m W Liq ) p(1 ϕ)(1 p) [D(X R_D ) + W(X R Liq )].  (equation (18))
  - Immediate discrete drop at s_W when early liquidations trigger: Π_B s=s_W   Π_B Liq ; s=s_W  = (1 m_W_Liq) p(1 ϕ)(1 p) [D(X R_D ) + W(X R)].  (equation (19))
  - Derivative of Π_B Liq w.r.t. s (equation (20)) ambiguous: one negative term from dm_W_Liq/ds < 0 and one positive term from dR_Liq/ds > 0.
  - Simplifying restriction: assume C(m) such that m effectively constant, m = m_C ; under this, Π_B Liq linear and increasing in s and global maximum of Π_B is either at s = s_W or s = 1 depending on ΔΠ_B.
  - ΔΠ_B expression under simplification:
    - ΔΠ_B = p[R s=s_W   R Liq ; s=1 ] W   (1 m_C ) p(1 ϕ)(1 p) [D(X R_D ) + W(X R Liq ; s=1 )]
  - Lemma 3 (cross-sectional properties of ΔΠ_B): ΔΠ_B increases in ϕ and L; it increases in D and decreases in W.
- Numerical demonstration and Proposition 3:
  - Numerical exercise parameter values in text:
    - m = 0:5; ϕ = 1; p = 0:90; X = 1:15; R_D = 1:10.
    - W takes values 0:25, 0:5, and 0:75 in three scenarios.
  - Proposition 3: The "dark side" exists — the set of parameter values for which the bank assigns wholesale financier too high seniority (subjecting itself to inefficient liquidation risk) is non-empty.
  - Bank incentives to assign too high seniority stronger when ϕ and L are higher and when there are more deposits D.

### VIII. Comparative statics, model scope, and policy implications
- Comparative statics summary:
  - Banks more likely to assign too high seniority, and wholesale financiers more likely to undertake inefficient liquidations, when ϕ and L are higher.
  - "Traditional" banks (relationship-based small business loans: low ϕ and L) → bright side likely dominates.
  - "Modern" banks (tradable, arm’s length assets with higher ϕ and L) → more vulnerable to dark side.
- Long-term funds and non-depository institutions:
  - "Retail deposits" can metaphorically represent long-term funds (bonds, customer funds) that lose out to short-term wholesale funds when liquidated assets are scrambled.
  - Model can describe short-term vs long-term financier conflict in non-bank institutions (example numbers cited: high-quality liquid assets fell from $18.1 billion on March 10, 2008 to $2 billion three days later).
- Policy implications:
  - Bank’s suboptimal use of senior wholesale funds arises because banks capture private savings via lower interest expense and do not internalize negative externality on depositors.
  - A Pigouvian tax on senior wholesale funds (taxing short-term wholesale funds such as collateralized repo’s) may help align bank incentives with the social optimum—short maturity and over-collateralization proxy for higher effective seniority.
  - Relation to systemic risk tax (Acharya et al., 2010): both internalize negative externalities; Acharya et al.'s proposal is broader and targets overall systemic risk.

### IX. Conclusion
- The paper analyzes the "dark side" of bank wholesale funding: insufficient monitoring and inefficient liquidations by short-term wholesale financiers.
- Wholesale funds can be beneficial in "traditional" banks with opaque, non-tradable loans but can create significant risks in "modern" banks with arm’s length assets and noisy public signals on values.

### X. Proofs and technical results (high-level highlights)
- Lemma 1: Π_B = p[D(X R_D ) + W(X R)] and R = Wζ + C(m_W )   m_W (1 p) s L(D+W) / (W p). Shown Π_B increasing in s and W and maximized for s = 1 and W = 1 D under stated conditions.
- Lemma 2: s_W = (1 ϕ) p W R / ( L (D+W) ). Shown ds_W / dϕ < 0; ds_W / dL < 0; ds_W / dD < 0; under D+W = 1, ds_W / dW > 0. Continuity of switches between m_W and m_W_Liq and between R and R_Liq at s_W established.
- Lemma 3: Detailed expression for ΔΠ_B under m = m_C and C(m_C ) = 0, with sign results: dΔΠ_B / dL > 0; dΔΠ_B / dϕ > 0; dΔΠ_B / dW < 0 and dΔΠ_B / dD > 0. Full derivatives and algebra provided in proofs.

*Source: IMF Working Paper content (excerpt).*

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

### _wp10170 - References .............................................................................................................

### I. Introduction — overview and key mechanisms
- Context and motivation:
  - Banks increasingly borrow short-term wholesale funds to supplement retail deposits (Feldman and Schmidt, 2001).
  - Wholesale instruments include large-denomination certificates of deposits, brokered deposits, repurchase agreements, Fed funds, and commercial paper.
  - Traditional literature emphasizes the "bright side" of wholesale funding: exploiting investment opportunities beyond local deposit supply, market discipline by wholesale financiers (Calomiris, 1999), and refinancing unexpected retail withdrawals (Goodfriend and King, 1998).
  - Recent mortgage banking crisis examples (Acharya et al., 2008; Huang and Ratnovski, 2009) reveal a "dark side": aggressive lending, compromised credit quality, and abrupt withdrawal by wholesale financiers triggering inefficient liquidations.
- Central insight and modelling approach:
  - Wholesale funding is beneficial when informed, but may lead to inefficient liquidations when uninformed.
  - Setup: a bank finances a risky long-term project with two sources of funds: retail deposits and wholesale funds.
    - Retail deposits are sluggish, insensitive to risks (partly insured), and provide stable long-term funding.
    - Wholesale funds are relatively sophisticated, able to acquire information on project quality, but supplied on a rollover basis and must be refinanced before final returns are realized.
  - Benchmark: Calomiris and Kahn (1991, hereafter CK) — "sophisticated" wholesale financiers add value through monitoring and market discipline; monitoring incentives are maximized when they are senior at refinancing stage.
- Novel feature introduced:
  - A costless but noisy public signal on bank project quality (public information that wholesale financiers can costlessly process and that is a noisy proxy for bank-specific fundamentals).
  - Examples: market prices or credit ratings for traded assets (e.g., mortgage-backed securities), performance of other similar banks, market- or sector-wide indicators (e.g., house or energy prices).
- Main theoretical results (qualitative):
  - Presence of a costless but noisy signal:
    - Lowers incentives of wholesale financiers to monitor.
    - Gives wholesale financiers excess incentives to liquidate banks based on noisy public information.
    - These distortions become stronger when wholesale financiers are more senior claimants to liquidated assets (contrasting with CK).
  - Mechanism:
    - Without a noisy public signal, uninformed wholesale financiers typically roll over funding (no news is good news).
    - With a noisy public signal, uninformed financiers may liquidate based on a negative but imprecise public signal.
    - Senior wholesale financiers obtain a larger share of liquidation value, reducing payoffs to long-term passive depositors and discouraging costly private monitoring.
  - Welfare implication:
    - Socially optimal seniority of short-term wholesale funds trades off the CK monitoring benefit against the novel liquidation distortion and has an interior optimum.
    - Monitoring incentives increase in seniority for low values (CK effect) but decrease for higher values when seniority translates into more liquidations.
    - Deviations from the interior optimum to either side reduce monitoring and can increase inefficient liquidations.
  - Key parameters affecting welfare-maximizing seniority (exact wording preserved):
    - "The noisy public signal is more precise, yet not as precise as to make liquidation decisions based on it socially optimal;"
    - "The share of passive deposits in bank liabilities is higher."
    - "Liquidation value of bank assets is higher."
    - "Interest rate offered to wholesale financiers in case of success is lower."
  - Cross-sectional prediction:
    - Senior short-term funds are beneficial in "traditional" banks holding opaque and nontradable relationship loans (CK applies).
    - "Dark side" effects are significant for banks with large exposures to standardized and tradable arm’s length assets with readily available public information, particularly when short-term wholesale financiers are senior claimants.
  - Private incentives:
    - Private incentives may drive arm’s length banks toward actively using senior short-term wholesale funds because interest rates on them are lowest when assets are marketable and public signals are informative.
  - Net conclusion:
    - Use and high seniority of wholesale funds is not always socially beneficial; with a costless but noisy signal, higher seniority can reduce monitoring and encourage inefficient liquidations. Social welfare is constrained-maximized for an intermediate level of seniority depending on funding structure, signal precision, liquidation value, and interest rates.

### II. The Bright Side of Wholesale Funding — model setup (CK-type benchmark)
- Environment and timing:
  - Three dates (0;1;2), no discounting, everyone is risk-neutral.
- The project:
  - For each unit invested at date 0, at date 2 the project returns X with probability p or 0 with probability 1 p, with a positive net present value: Xp>1.
  - The project may be liquidated at date 1 returning L<1 per unit initially invested.
  - The maximum investment size is 1.
- Funding structure and financiers:
  - The bank has no initial capital and needs to borrow to invest.
  - Two types of financiers:
    1. Retail depositors:
      - Unsophisticated, passive, and scarce.
      - Never get advance information on date 2 project realization, and never withdraw before date 2, providing a stable long-term (yet formally demandable) source of funds.
      - Interest rate payable on retail deposits (date 0 to date 2) is risk-insensitive and fixed at R_D:1R_D<pX.
      - The bank is endowed with a fixed deposit base of D<1 and it is prohibitively costly to expand it within the horizon of the model.

*Source: IMF Working Paper content (excerpt).*

### 2.  The "wholesale financier" is sophisticated, has an unlimited supply of funds, but is

### _wp10170 - 2.  The "wholesale financier" is sophisticated, has an unlimited supply of funds, but is

### Model setup and key assumptions
- Wholesale financier:
  - Sophisticated, unlimited supply of funds at date0 against real expected return R̄ (opportunity cost), short-term (funding must be refinanced at date1).
  - Can monitor the bank between dates0 and1 with intensity m (0≤m<1), incurring cost C(m) with C(0) = 0, C(1) = 1, C′(0) = 0, C′(m) > 0, C′′(m) > 0.
  - Monitoring yields precise information on date2 realization with probability m; monitoring fails with probability 1−m.
  - Single wholesale financier (no competition/coordination).
- Bank and funding:
  - Initial project size up to 1; initial retail funding D with D < 1. Wholesale funds amount W with W ≤ 1−D.
  - Maximum investment size is 1, so W ≤ 1−D.
  - Wholesale funding must be rolled over at date1; refusal leads to liquidation.
  - Liquidation value is L(D+W) with L < 1. Wholesale financier receives share s of liquidation value (0 ≤ s ≤ 1).
  - Assumption for tractability: pW > L, which implies pWR > sL(D+W) so the financier never liquidates solely to receive L(D+W).
- Timing:
  - Date0: funding and monitoring intensity choice.
  - Between date0 and date1: monitoring outcome realized for financier.
  - Date1: financier decides refinance or liquidate.
  - Date2: project realization X (or 0) realized.

### Monitoring, liquidation, and seniority mechanics
- Monitoring cost and informational return:
  - Cost function C(m) as above; private monitoring yields precise information with probability m.
- Liquidation and seniority:
  - If wholesale financier refuses rollover at date1, bank is liquidated; division of L(D+W) governed by s.
  - To ensure continuation absent negative information, impose pW > L (equation (1) in source).
- Indifference tie-breaker:
  - Agents prefer continuation when indifferent; only wholesale financier can trigger liquidation at date1.

### Benchmark case: retail deposits only
- With retail deposits only (no wholesale funds):
  - Initial investment D is less than 1 → spare capacity is inefficient.
  - Bank always continues until date2; bad projects continue and yield 0 at date2 to preserve liquidation value L.
  - Monetary value of social welfare:
    - Dep = D(pX − 1).  (equation (2))

### Wholesale funds — social-welfare maximization (no noisy signal)
- Social welfare with wholesale funding:
  - Maximize Π = (D+W) (pX + m(1−p)L − 1) − C(m).  (equation (3))
- Socially optimal choices:
  - W̄ = 1 − D (full investment: D+W = 1).
  - Monitoring intensity m̄ satisfying C′(m̄) = (1−p)L.  (equation (4))
- Benefits relative to retail-only:
  - Higher investment volume (D+W = 1) instead of D.
  - Preservation of some bad banks’ liquidation value at cost C(m̄).

### Wholesale funds — private equilibrium (no noisy signal)
- Wholesale financier’s private maximization (when choosing m):
  - Π_W = pWR + m(1−p)sL(D+W) − C(m).
  - Private first-order condition: C′(m_W) = (1−p)sL(D+W).  (equation (5))
- Interest rate demanded (zero-profit condition):
  - R = (Wζ + C(m_W) − m_W(1−p)sL(D+W)) / (W p).  (equation (7))  [Wζ denotes W times the real expected return placeholder as in source]
- Bank’s surplus and incentives:
  - Bank surplus: Π_B = p[D(X − R_D) + W(X − R)].  (equation (6))
  - Lemma: Π_B increases in s and W, so bank maximizes surplus with W = 1 − D and s = 1.
- Benchmark Proposition 1:
  - In the "bright side" case, bank’s choices W = W̄, s = s̄ = 1, and wholesale financier’s m = m̄ lead to socially optimal outcome; only banks known to be bad are liquidated.

### Introduction of a free but noisy public signal (the "dark side")
- Noisy public signal:
  - Received by the wholesale financier prior to date1 but after choosing m; not received by retail depositors.
  - Signal takes values "positive" or "negative", precision parameter ϕ (0 ≤ ϕ ≤ 1; ϕ = 0 is complete noise, ϕ = 1 is precise).
  - Unconditional probability of positive signal is p (same as project success probability).
  - Conditional probabilities:
    - If signal positive: project yields X at date2 with probability [p + ϕ(1−p)], yields 0 with probability [(1−p) − ϕ(1−p)].
    - If signal negative: project yields X with probability [p − ϕp], yields 0 with probability [(1−p) + ϕp].
- Key qualitative effects:
  - The noisy signal lowers incentives to monitor (private information less valuable).
  - The noisy signal can induce excess liquidation based solely on noisy signal when precision is high.

### Welfare maximization with noisy signal
- Date1 refinancing rule:
  - If monitoring yields precise info: continue if good (X > L) and liquidate if bad (L > 0).
  - If monitoring yields no info: use public signal — refinance on positive signal; negative signal may trigger liquidation depending on precision.
  - Threshold precision for socially optimal liquidation on negative signal:
    - ϕ̄ = 1 − (L / (p^2 X)).  (equation (8))
    - If ϕ > ϕ̄ and [p − ϕ p] p X < L, social optimum calls for liquidation on negative signal.
- Social welfare when using the noisy signal and liquidating on negative signal:
  - Π_Liq = (D+W) ( m[pX + (1−p)L] + (1−m) [ p[p + ϕ(1−p)]X + (1−p)L ] − 1 ) − C(m).  (equation (9))
- Optimal wholesale funding and monitoring under high precision:
  - W̄_Liq = 1 − D (= W̄).
  - Monitoring intensity m̄_Liq satisfies C′(m̄_Liq) = p(1−p) (1−ϕ) X.  (equation (10))
  - Note: m̄_Liq < m̄ (monitoring intensity falls when the noisy signal is used).

### Wholesale financier incentives with noisy signal — inefficient liquidations and monitoring
- Private liquidation decision when uninformed but observing noisy signal:
  - Refinancing continuation payoff conditional on negative signal: [p − ϕ p] W R.
  - Liquidation payoff: s L (D+W).
  - The financier privately liquidates based on negative signal when:
    - s L (D+W) > [1 − ϕ] p W R.  (equation (11))
- Equivalent threshold expressions:
  - Threshold precision for financier to follow noisy signal:
    - ϕ_W = 1 − (s L (D+W)) / (p W R).  (equation (12))
  - Threshold seniority for financier to follow noisy signal:
    - s_W = ( (1 − ϕ) p W R ) / ( L (D+W) ).  (equation (13))
  - Interpretation: higher s raises financier’s incentives to liquidate on noisy negative signals.
- Monitoring intensity choices with noisy signal:
  - If junior (s ≤ s_W), financier disregards noisy signal and uses monitoring intensity m_W as in benchmark: C′(m_W) = (1−p) s L (D+W).  (equation (5))
  - If senior (s > s_W), financier uses noisy signal and maximizes:
    - Π_W = m[p W R + (1−p) s L (D+W)] + (1−m) [ p[p + ϕ(1−p)] W R + (1−p) s L (D+W) ] − C(m).  (equation (14))
  - First-order condition in liquidation regime:
    - C′(m_W_Liq) = p(1−p) (1−ϕ) W R_Liq.  (equation (15))
  - Interest rate in liquidation regime:
    - R_Liq = ( Wζ + C(m_W_Liq) − (1−p) s L (D+W) ) / ( m_W_Liq W p + (1−m_W_Liq) [ p + ϕ(1−p) ] W p ).  (equation (16))  [Wζ denotes W times the real expected return placeholder as in source]
  - Comparative statics:
    - As s increases, R_Liq decreases (wholesale financier receives more in date1 liquidations), which feeds back to reduce m_W_Liq; hence m_W_Liq decreases in s.
- Lemma 2 (summary of threshold s_W properties):
  1. s_W decreases in ϕ and L; s_W decreases in D and increases in W (provided D+W = 1).
  2. For s ≤ s_W, financier never liquidates based on noisy public signal and monitoring intensity increases in seniority: ∂m_W/∂s > 0.
  3. For s > s_W, financier liquidates following a negative noisy signal and monitoring intensity decreases in seniority: ∂m_W_Liq/∂s < 0.

*Source: _wp10170 - 2.  The "wholesale financier" is sophisticated, has an unlimited supply of funds, but is (PDF).*

### 4. Monitoring and interest rate functions are continuous ats

### 4. Monitoring and interest rate functions are continuous ats

### Socially optimal seniority of wholesale funds
- Proposition 2 (case with possible welfare-reducing liquidations: W s=1 < ): the socially optimal creditor seniority of the wholesale financier is s=s W , s W <1.
- Setting s=s W aligns the continuation decision of the wholesale financier with the social optimum, and there are no inefficient liquidations.
- s=s W maximizes the intensity of monitoring, albeit at a level below the social optimum: m W (s W ) < m  .
- All else equal, incentives of the wholesale financier for inefficient liquidations are higher (and thus the socially optimal seniority of wholesale funding is lower) when:
  - the precision of the public signal  is higher;
  - the bank’s liquidation value L is higher;
  - there are more deposits D serving as buffer for wholesale funds’ exit.
- Points W can be thought of as the highest seniority consistent with the "bright side" of wholesale funding. For s > s W the wholesale financier becomes sufficiently senior to undertake inefficient liquidations based on noisy public information, and higher seniority leads to lower monitoring.

### Incentives of the bank
- The bank chooses creditor seniority to maximize its private surplus and may deviate from the social optimum.
- The bank has no incentive to assign creditor seniority below the socially optimal level because for s < s W its surplus Π B (given in (6)) increases in s.
- Private incentive to assign too high seniority (s > s W):
  - Cost: losses when good projects are abandoned in inefficient liquidations (similar to social cost).
  - Benefit: offering higher seniority reduces the interest rate R on wholesale funding, lowering interest expense (R D is fixed) and increasing bank surplus.
  - If the net effect is positive (lower interest expense compensates higher risk of inefficient liquidations), the bank prefers s > s W .
- Bank surplus expressions:
  - At s = s W : Π B s=s W = p[D(X R D ) + W(X R s=s W )] (equation (17)), with R given by (7).
  - For s > s W (liquidation regime): Π B Liq =  p (1 m W Liq ) p(1 )(1 p) [D(X R D ) + W(X R Liq )] (equation (18)), with R Liq given by (16).
  - The immediate discrete drop when early liquidations trigger at s W :
    - Π B s=s W   Π B Liq ; s=s W  = (1 m W Liq ) p(1 )(1 p) [D(X R D ) + W(X R)]: (19)
- Derivative of Π B Liq w.r.t. s (equation (20)):
  - dΠ B Liq / ds = (dm W Liq / ds) p(1 )(1 p) [D(X R D ) + W(X R Liq )]   (dR Liq / ds)  p (1 m W Liq ) p(1 )(1 p) W
  - First term negative since dm L iq / ds < 0 (higher s reduces monitoring → more inefficient liquidations).
  - Second term positive since  dR Liq / ds > 0 (higher s lowers interest rate on wholesale funding).
  - Overall effect is ambiguous.
- Simplifying restriction: assume C(m) such that m is effectively constant, m = m C , in relevant parameter range (sharp J-shape for C(m)). Then:
  - First term in (20) becomes zero; second term constant → Π B Liq linear and increasing in s.
  - Global maximum of Π B achieved either at s = s W when ΔΠ B = Π B Liq ; s=1   Π B s=s W  is positive, or at s = 1 otherwise.
- Expression for ΔΠ B under the simplifying restriction:
  - ΔΠ B = p[R s=s W   R Liq ; s=1 ] W   (1 m C ) p(1 )(1 p) [D(X R D ) + W(X R Liq ; s=1 )]
  - First term: lower interest expense for more senior wholesale funds.
  - Second term: probability of inefficient liquidations.
- Lemma 3 (cross-sectional properties of ΔΠ B): ΔΠ B increases in  and L; it increases in D and decreases in W.
  - Intuition: higher , L, and D reduce cost of early liquidations for wholesale financier → lower interest rate → higher bank surplus. Higher W has the opposite effect since W = 1 D.

### Numerical demonstration and Proposition 3 (bright and dark sides)
- Numerical exercise shows ΔΠ B can be either positive or negative within plausible parameter ranges, validating existence of both "bright" and "dark" sides of wholesale funding (illustrated in Figure 3).
- Simulation parameter values used in the text:
  - m = 0:5;  = 1; p = 0:90; X = 1:15; R D = 1:10.
  - W takes values 0:25, 0:5, and 0:75 in three scenarios.
- Proposition 3: The "dark side" of wholesale funding exists — the set of parameter values for which the bank assigns wholesale financier too high seniority (subjecting itself to inefficient liquidation risk) is non-empty.
  - All else equal, bank incentives to assign too high seniority are stronger when:
    - precision of the public signal  is higher;
    - bank’s liquidation value L is higher;
    - there are more deposits D serving as buffer for wholesale funds’ exit.

### Comparative statics, model scope, and policy implications
- Comparative statics (from Propositions 2 and 3):
  - Banks more likely to assign too high seniority, and wholesale financiers more likely to undertake inefficient liquidations, when  and L are higher.
  - Distinction suggested:
    - "Traditional" banks: primarily relationship-based small business loans (low  and L) → bright side of wholesale funding likely dominates.
    - "Modern" banks: more tradable and arm’s length assets (mortgage loans, securities) with higher  and L → more vulnerable to dark side.
- Long-term funds and non-depository banks:
  - "Retail deposits" in the model can be a metaphor for all long-term funds (bonds, customer funds) that lose out to short-term wholesale funds when liquidated assets are scrambled.
  - Model can describe conflict between short-term and long-term financiers in non-bank financial institutions (example: Bear Stearns run; short-term financiers withdrawal reduced high-quality liquid assets from $18.1 billion on March 10, 2008 to $2 billion three days later).
- Policy implications:
  - Bank’s suboptimal use of senior wholesale funds stems from private savings via lower interest expense; bank does not internalize negative externality on depositors.
  - A Pigouvian tax on senior wholesale funds (taxing short-term wholesale funds such as collateralized repo’s) may help align bank incentives with social optimum—short maturity and over-collateralization are good proxies for higher effective seniority.
  - Relation to systemic risk tax (Acharya et al., 2010): both aim to internalize negative externalities; Acharya et al.'s proposal is broader and targets overall systemic risk.

### Conclusion
- The paper analyzes the "dark side" of bank wholesale funding: insufficient monitoring and inefficient liquidations by short-term wholesale financiers.
- Wholesale funds can be beneficial in "traditional" banks with opaque, non-tradable loans.
- Wholesale funds can create significant risks in "modern" banks with arm’s length assets and noisy public signals on values.

### Proofs and technical results (high-level highlights)
- Lemma 1: Π B = p[D(X R D ) + W(X R)] and expression for R = W  + C(m W )   m W (1 p) s L(D+W) / (W p). Shown Π B increasing in s and W and maximized for s = 1 and W = 1 D under stated conditions.
- Lemma 2: s W = (1 ) p W R L (D+W). Shown ds W / d < 0; ds W / dL < 0; ds W / dD < 0; under D+W = 1, ds W / dW > 0. Continuity of switches between m W and m W Liq and between R and R Liq at s W established.
- Lemma 3: Detailed expression for ΔΠ B under m = m C and C(m C ) = 0, and sign results: dΔΠ B / dL > 0; dΔΠ B / d > 0; dΔΠ B / dW < 0 and dΔΠ B / dD > 0. Full derivatives and algebra provided in proofs.

*Source: _wp10170 - 4. Monitoring and interest rate functions are continuous ats (IMF working paper PDF content).*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2010/_wp10170.pdf_