## wp17285

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### Introduction and research questions
- Motivation:
  - Rapid growth in financial intermediation by non-bank entities combining highly-leveraged, short-term funding with risky long-term investments (e.g., sub-prime mortgage lending).
- Central questions:
  - What circumstances and mechanisms lead to the emergence and expansion of shadow banking?
  - How does shadow banking affect the portfolio and funding strategies of traditional banks?
- Core proposal:
  - Depositors may withdraw deposits early in reaction to crises; these “early withdrawals” are an optimal depositor response to adverse changes in banks’ solvency prospects and become a source of market discipline.
  - When commitment to a safe strategy is costly (commitment cost τ > 0), shadow banking emerges as an alternative: risky portfolio strategy combined with early withdrawals in times of crisis.
- Novelty:
  - First model where shadow and traditional banks coexist and interact without regulatory arbitrage or direct contractual linkages.

### Model structure and core mechanisms
- Economy and timing:
  - Closed economy with households, banks, entrepreneurs, and outside investors; three periods.
  - Period 1: banks collect deposits and invest in cash M and risky assets I originated by entrepreneurs.
  - Period 2: public news signal; with probability q the signal is “bad news”; households decide on early withdrawal; banks trade in secondary market.
  - Period 3: assets yield payoffs σ_h when fundamentals strong and σ_l when weak; following bad news fundamentals weak with conditional probability p.
- Key friction and heterogeneity:
  - Costly commitment τ > 0: some banks pay τ and commit to safe (traditional banking), others do not and adopt shadow banking.
  - Share of shadow banks γ ∈ [0,1] endogenously determined by free entry (free entry condition E[Π_SB] = E[Π_TB] − τ).
- Secondary market and fire-sales:
  - Shadow banks liquidate assets in secondary market during early withdrawals, generating fire-sales that depress prices.
  - Secondary market pricing: P2 = φ[(1−p)σ_h + pσ_l] with φ ∈ [0,1] the fire-sale discount; φ = f(Ĩ) where Ĩ is excess supply and f(.) is continuous and decreasing.
  - Interaction: larger shadow banking sector → greater excess supply Ĩ → lower φ → deeper fire-sales → reduces shadow banking payoff relative to traditional banking → interior equilibrium with coexistence.
- Distinction from classic run models:
  - Early withdrawals here are optimal and not purely self-fulfilling; liquidity risk extension introduces Diamond-Dybvig style bank-runs as an additional channel.

### Key analytical results (no-liquidity baseline)
- Household deposit rates and withdrawal incentives:
  - RTB = 1 + qp(1−V) / (1−qp(1−V))  (equation (13))
  - RSB = 1 + q/(1−q) (1−θ_SB)  (equation (14))
  - RTB decreases with higher V; RSB decreases with higher θ_SB.
  - RTB = 1 when V = 1; RSB > 1 unless θ_SB = 1.
- No-withdrawal constraint and market discipline:
  - Minimum recovery rate for no early withdrawal: V̄ ≡ 1/p (1/RTB − (1−p))  (equation (15)).
  - In the simple model without liquidity risk, equilibrium yields V̄ = RTB = 1 (equation (16)); market discipline can eliminate solvency risk.
- Shadow banking characterization (Lemma 1):
  - θ_SB = (1 + μ) P2 / ((1−q) σ_h + q(1 + μ) P2) < 1 ∀ φ ∈ [0,1]
  - RSB = σ_h / ((1−q) σ_h + q(1 + μ) P2) > 1 ∀ φ ∈ [0,1]
  - E[Π_SB] = (1−q) μ/(1 + μ) σ_h I1SB
  - I1SB = (A α/α)^{1/(1−α)} ((1−q) σ_h/(1 + μ) + q P2)^{α/(1−α)}
- Traditional banking characterization (Lemma 2, under μ ≤ ̄μ and φ > φ_):
  - I1TB = (A α/α)^{1/(1−α)} P2^{α/(1−α)}  (equation (24))
  - I2TB = 0; M2TB = P2 I1TB + M1TB  (equation (25))
  - E[Π_TB] = (1−q) (σ_h − P2) I1TB
  - ∂E[Π_TB]/∂P2 < 0 ∀ P2 > α σ_h
  - Traditional banks liquidate risky assets after bad news and re-allocate to cash to satisfy no-withdrawal constraint.
- Equilibrium and fire-sale externality:
  - Free entry condition E[Π_SB] = E[Π_TB] − τ determines γ.
  - Entry into shadow banking increases excess supply, deepens fire-sales (φ falls), reduces E[Π_SB] and deters further entry: strategic substitutability yields interior equilibrium under parameter restrictions (Proposition 2).
  - Sufficient conditions for interior equilibrium include μ ≤ μ̄, φ ∈ (φ_, φ̄), α < 0.5, p < 1−α, and τ ∈ (τ_, τ̄) with τ̄ > τ_ > 0.

### Motivating evidence and stylized facts (US, 2002–2015 and 2007–2009 crisis)
- Fact 1. Sectoral size dynamics:
  - Shadow bank assets: expanded from 125% of GDP in 2002 to over 180% at peak in 2007; in 2015 shadow bank assets amounted to 145% of GDP.
  - Traditional banks: assets increased from 88% of GDP in 2002 to 118% in 2015; no contraction in traditional banking during the crisis.
- Fact 2. Funding dry-up and asset liquidation:
  - Asset-backed commercial papers (ABCP) peaked at $1.2 trillion in July 2007; ABCP contracted by $350 billion in H2 2007 and a further $400 billion by end-2009.
  - Shadow banks sold $1.5 trillions of debt securities between 2008Q3 and 2010Q1, approximately half mortgage-backed securities.
- Fact 3. Spreads and fire-sales:
  - Spreads between private debt securities and Treasury bonds peaked in last quarter of 2008; lower-rated securities saw larger increases.
- Fact 4. Traditional bank responses:
  - Traditional banks re-allocated toward cash and government securities; mortgage-backed securities ≈ 12% of total assets and unchanged.
  - Funding costs: TED spread increased to 200 basis points in December 2007 and 335 basis points in October 2008.
- Model mapping:
  - Bad news probability q and conditional weak-fundamentals probability p drive creditor withdrawals; shadow-bank liquidations in secondary markets create fire-sale discounts and spread dynamics consistent with observed patterns.

### Extensions with liquidity risk and bank-runs (Section 4 and 4.1)
- Bank-run mechanism:
  - Introduce sunspot εξ ∼ Uniform[0,1] with threshold ̄ξ ∈ [0,1]; probability of bank run ξ = 0 for θ = 1 and ξ = ̄ξ for θ < 1.
  - Bank-run realization assumed idiosyncratic; ξ interpreted as share of traditional banks liquidated in a bank-run.
- Multi-asset secondary market (safe s and risky r traded in period 2):
  - Excess supplies: ̃I(i) = γ I_SB_1(i) + (1−γ)(I_TB_1(i) − (1−ξ) I_TB_2(i)) ≥ 0 ∀ i ∈ {s, r}
  - Common fire-sale discount φ defined implicitly by:
    - φ = 1 / g′(̃E − φ[(1−p)σ_h + pσ_l] ̃I(r) − φ ̃I(s))  (equation (27))
  - Asset-specific prices:
    - P2(s) = 1 for ̃I(s) = 0; P2(s) = φ otherwise.
    - P2(r) = (1−p)σ_h + pσ_l for ̃I(r) ≤ 0; P2(r) = φ[(1−p)σ_h + pσ_l] otherwise.
- Proposition 3 (commitments, deposit rate, and bank-run interaction):
  - Traditional banks commit to:
    - ̄V = 1 − q/(p ξ) (1−θ_TB) / [1−q(1−ξ(1−θ_TB))]  (equation (28))
  - Deposit rate on traditional bank deposits:
    - R_TB = 1 + q/(1−q) ξ (1−θ_TB)  (equation (29))
  - Implications:
    - If P2(s) = 1 → θ_TB = 1 → ξ = 0 → ̄V = R_TB = 1.
    - If P2(s) < 1 sufficiently low → θ_TB < 1 → ξ > 0 → ̄V < 1 < R_TB.
  - Mechanism: fire-sales on safe assets can create liquidity shortfalls, enabling bank-runs and undermining market discipline.

### Numerical calibration and equilibrium behavior (highlights)
- Calibration targets (US/crisis):
  - σ_l = 0.21
  - q = 0.41
  - p = 0.22
  - α = 0.33
  - A = 1.00
  - μ = 0.18
  - ̃E = 2.80
  - z = 10.0
  - ̄ξ = 0.15
  - τ = 0.11
- Calibration notes:
  - σ_h backed out using normalization (1−qp)σ_h + qpσ_l = 1.
  - (q, p) set to match NBER frequencies; ̃E and z match percentage decline in bond prices from peak spread in November 2008; ̄ξ calibrated to match R_TB_2 to average TED spread in last quarter of 2008 via R_TB_2 formula.
  - τ = 0.11 targets pre-crisis shadow bank assets ≈ 63% of total bank assets; τ = 0.11 ≈ just below 7% of value of a traditional bank’s assets in period 1.
- Numerical equilibrium features:
  - Unique and globally stable interior equilibrium γ∗ where shadow and traditional banks coexist.
  - Higher commitment cost τ increases γ∗ and deepens fire-sales (φ falls).
  - Increase in liquidity available to outside investors (rise in ̃E) raises φ at given γ but leads to larger γ∗; fire-sale discount returns to initial value at new γ∗ — thicker secondary markets increase shadow banking size but do not alleviate the fire-sale at equilibrium.
  - Threshold ̄γ: sector size above which safe assets suffer fire-sale; under calibration, high τ leads to γ∗ > ̄γ so safe-asset fire-sale exists in equilibrium.
- Portfolio responses:
  - As γ rises and fire-sale deepens: shadow banks reduce deposits; traditional banks expand balance sheets.
  - Traditional banks re-allocate away from risky assets up to 18% when shadow banking small; in large-shadow equilibrium re-allocation ≈ 4% away from risky assets.
  - Shadow banks skew toward risky assets and do not change composition in response to fire-sales due to limited liability.

### Policy analysis and recommendations
- Asset purchases in secondary market:
  - Government asset purchases I_AP absorb excess safe assets and can eliminate an existing fire-sale at initial γ∗.
  - Anticipation effect: alleviating fire-sales reduces shadow bank funding costs and raises expected shadow-bank payoff → entry into shadow banking until new equilibrium γ∗_AP > γ∗.
  - Conclusion: asset-purchase interventions are ineffective ex-ante under free entry; only surprise (ex-post) purchases avoid time-inconsistency problem.
- Interventions securing traditional banks:
  - Collateralized liquidity provision:
    - Permit traditional banks to borrow L from central bank at risk-free rate in period 2 with full collateralization.
    - Condition to eliminate solvency/liquidity risk: L ≥ L ≡ D_TB − P2(r) I_TB_1(r) − P2(s) I_TB_1(s) − I_1(λ).
    - Outcomes: bank-runs eliminated (ξ = 0); ̄V = 1 and R_TB = 1; traditional banks insulated; fire-sale on safe assets eliminated via arbitrage; shadow banking sector expands but net effect on financial stability positive (stabilizing effect dominates).
  - Deposit insurance combined with regulation:
    - Deposit insurance guarantees eliminate liquidity risk but require replacing market discipline with bank regulation (impose ̄V = 1).
    - Outcome similar to liquidity provision for stabilizing traditional banks, but different in role of market discipline.
- Tax on shadow bank profits (Section 6.3):
  - Introduce constant tax T so free entry becomes E[Π_SB] − T = E[Π_TB] − τ.
  - For 0 < T < τ, tax reduces equilibrium shadow-banking sector to fire-sale threshold ̄γ and can eliminate fire-sale on safe assets; strengthens market discipline and removes solvency risk.
  - Tax effect equivalent to a decrease in τ; alternative via transfer to traditional banks or taxing shadow-bank liabilities (e.g., ABCP) achieves similar outcome.
  - Quantitative indication: necessary transfer amounts to 9% of commitment costs, ≈ 0.6% of value of a traditional bank’s portfolio in period 1.
- General policy design principle:
  - Must account for endogenous adjustments in shadow banking size; interventions that alleviate fire-sales ex-post (asset purchases, liquidity support) can fuel sector growth ex-ante and may be time-inconsistent. Taxes or transfers that reduce expected shadow-bank payoffs can contain sector size and restore market discipline.

### Appendices and robustness
- Appendices provide:
  - Proofs of propositions and lemmas (withdrawal incentives, limited liability, shadow- and traditional-bank optimization).
  - Conditions and explicit expressions for τ bounds (τ_, τ̄) ensuring interior equilibrium.
  - Alternative bank-run specification ξ = ζ(θ) with ζ′(.) ≤ 0 yields qualitatively same equilibria and sharper declines in traditional-bank recovery rates when liquidity deteriorates.
  - Example fire-sale function: g(̃K) = κ^{−1} ln(̃K) yields f(̃I) = κ ̃E /(1 + κ [(1 − p) σ_h + p σ_l] ̃I).
  - Extension to three-asset economy with liquid (λ), safe (s), and risky (r) assets and corresponding first-order conditions and constraints.
- Robustness:
  - Main qualitative results hold when relaxing assumptions (Appendix D.2 and Appendix J); liquidity-risk extension (Section 4) introduces bank-run equilibria that interact with fire-sales and market discipline.

*Source: IMF Working Paper wp17285 - 2017*

### 2017.  The  views  expressed  herein  are  those  of  the  authors  and  do  not  necessarily  represent  those  of  the

### wp17285 - 2017.  The  views  expressed  herein  are  those  of  the  authors  and  do  not  necessarily  represent  those  of  the

### Introduction: motivation and research questions
- Recent decades saw rapid growth in financial intermediation by non-bank entities combining highly-leveraged, short-term funding with risky long-term investments (e.g., sub-prime mortgage lending).
- Two central questions:
  - What circumstances and mechanisms lead to the emergence and expansion of shadow banking?
  - How does shadow banking affect the portfolio and funding strategies of traditional banks?
- Key proposal: depositors may withdraw deposits early in reaction to crises; these “early withdrawals” are an optimal depositor response to adverse changes in banks’ solvency prospects and become a source of market discipline.
- When commitment to a safe strategy is costly, shadow banking emerges as an alternative: risky portfolio strategy combined with early withdrawals in times of crisis.
- Novelty claim: first model where shadow and traditional banks coexist and interact without regulatory arbitrage or direct contractual linkages.

### Model structure and mechanisms (overview)
- Closed economy with households, entrepreneurs, outside investors, and a banking sector.
- Banks collect deposits and choose portfolios of safe, risky, and liquid assets; households lend to banks on terms dependent on solvency prospects; entrepreneurs originate assets.
- Following news signals that revise expected asset returns, households decide whether to withdraw early; banks trade assets with outside investors in a secondary market with cash-in-the-market pricing (Allen and Gale, 1994, 2005).
- Shadow banks liquidate assets in the secondary market to repay depositors during early withdrawals; such liquidations generate fire-sales that depress prices and create externalities.
- Equilibrium relationship between secondary market prices and bank strategies:
  - As shadow banking sector size increases, its liquidation causes deeper fire-sales, reducing shadow banking payoff relative to traditional banking.
  - This leads to an interior equilibrium where shadow and traditional banks coexist.

### Key insights on market discipline, deposit rates, and financial vulnerability
- Low deposit rates strengthen early-withdrawal incentives because depositors forfeit less interest — therefore market discipline on traditional banks is strong when deposit rates are low.
- In absence of fire-sales:
  - Interest rates on deposits are low.
  - Market discipline drives traditional banks to commit to safe portfolio strategies.
- With prospect of fire-sales:
  - Depositors demand higher rates to compensate for liquidity risk.
  - Higher deposit rates weaken early-withdrawal incentives and allow traditional banks to pursue risky portfolios that may leave them insolvent.
- Equilibrium outcome: shadow banking expands to a size where its liquidation triggers a fire-sale and undermines market discipline.

### Policy implications and comparative effects
- Taxation of shadow bank activities:
  - Conducive to financial stability by deterring banks from pursuing shadow strategies and preventing the sector from reaching fire-sale-triggering size.
- Asset purchases in the secondary market:
  - Effective at alleviating fire-sales for a given shadow bank sector size.
  - However, ex-ante anticipation of asset purchases fuels further growth of the sector and can render the intervention completely ineffective (time inconsistency concern).
- Interventions securing traditional banks:
  - Collateralized liquidity provision secures traditional banks from liquidity risk and restores market discipline.
  - Deposit insurance guarantees eliminate liquidity risk but require replacing market discipline with bank regulation.
  - Both measures (collateralized liquidity provision; deposit insurance with regulatory constraints) stabilize traditional banks but diminish the fire-sale and expand the shadow banking sector; net impact on financial stability is positive as the stabilizing effect dominates.
- Policy design must account for endogenous adjustments in shadow banking size when evaluating interventions.

### Relation to empirical facts and literature
- Model built on observations from the 2008 financial crisis:
  - Shadow banks experienced sudden funding dry-ups and asset liquidations that caused fire-sales.
  - Traditional banks did not suffer deposit withdrawals but faced higher funding costs and shifted toward safe, liquid assets.
- Consistency with empirical evidence:
  - Collapse in asset-backed commercial paper and repo markets accounted for a large share of shadow bank funding (Acharya et al., 2013; Gorton and Metrick, 2012; Krishnamurty et al., 2014).
  - Dry-up in shadow bank funding associated with rises in macro-financial risk (Covitz et al., 2013).
  - Empirical evidence supports fire-sales during the crisis (Krishnamurthy, Merrill et al., Mitchell and Pulvino; Shleifer and Vishny review).
- Distinction from related mechanisms:
  - Unlike classic Diamond-Dybvig self-fulfilling runs, early withdrawals here are optimal responses even absent others withdrawing.
  - Different from risk-shifting results (e.g., Diamond and Rajan 2005): illiquidity in this model generates solvency risk where none previously existed through interactions between deposit rates and market discipline.

### Contribution and methodological notes
- Demonstrates that shadow banking can arise as an equilibrium outcome even without:
  - Advantages in intermediation technology,
  - Regulatory arbitrage opportunities.
- Commitment costs cause ex-ante identical banks to split into traditional (pay lump-sum cost to commit to safe portfolios) and shadow strategies.
- Commitment costs can reflect costly actions to resolve asymmetric information (e.g., balance sheet disclosures, avoiding opaque securitization, issuing costly equity with voting rights).
- Provides a free-market benchmark to compare bank regulation and other policy interventions.

*Source: IMF Working Paper wp17285 - 2017*

### Section 3 describes the core mechanisms of the model in a simple framework without liquidity

### Section 3 describes the core mechanisms of the model in a simple framework without liquidity

### Motivating evidence: four stylized facts (US, 2002–2015 and 2007–2009 crisis)
- Fact 1. The shadow banking sector expanded rapidly until its collapse in 2007; the traditional banking sector grew at a slower rate but did not suffer from a collapse.
  - Shadow bank assets: expanded from 125% of GDP in 2002 to over 180% at its peak in 2007; in 2015 shadow bank assets amounted to 145% of GDP.
  - Traditional banks: assets increased from 88% of GDP in 2002 to 118% in 2015; no contraction in traditional banking during the crisis.
  - Model interpretation: size of shadow banking sector is endogenously determined through free entry and may rise with (i) an increase in liquidity available in secondary markets and (ii) a rise in the cost of commitment τ. Expectations that policymakers will lean against a fire-sale increase the equilibrium size of the shadow banking sector in the same manner as the deepening of secondary markets.
- Fact 2. At the onset of the financial crisis, shadow banks experienced a sudden dry-up of funding and liquidated their assets.
  - Asset-backed commercial papers (ABCP) peaked at $1.2 trillion in July 2007; ABCP contracted by $350 billion in H2 2007 and a further $400 billion by end-2009.
  - Shadow banks sold $1.5 trillions of debt securities between 2008Q3 and 2010Q1, approximately half mortgage-backed securities.
  - Model mechanism: adverse public “bad news” signal (probability q) leads creditors to optimally withdraw from shadow banks; shadow banks liquidate assets in the secondary market at an endogenous fire-sale discount.
- Fact 3. Spreads between private debt securities and Treasury bonds increased sharply during the crisis; higher-risk securities experienced greater increases in spreads.
  - Spreads peaked in the last quarter of 2008, coinciding with liquidation of shadow bank assets; lower-rated corporate bonds saw larger increases.
  - Model mechanism: cash-in-the-market pricing during liquidation causes fire-sales and large discounts on illiquid assets; portfolio re-allocation by traditional banks from risky to safe/liquid assets contributes to the fire-sale.
- Fact 4. Traditional banks re-allocated portfolios toward safe and liquid assets and were able to increase liabilities; funding costs rose.
  - Traditional banks increased liabilities during the crisis (driven by Federal Reserve funding and a rise in insured deposits); no decline in large time deposits.
  - Portfolio change: re-allocation from bank credit (risky, illiquid) to cash assets and government securities (liquid, safe); mortgage-backed securities ≈ 12% of total assets and unchanged.
  - Funding costs: TED spread increased to 200 basis points in December 2007 and 335 basis points in October 2008; certificate of deposit and commercial paper spreads also rose.
  - Model interpretation: market discipline through threat of early withdrawal induces traditional banks to commit to a minimum recovery rate on deposits (“no-withdrawal constraint”) and re-allocate toward safe assets; fire-sales reduce capacity to re-allocate and create a wedge between solvency and liquidity, potentially generating self-fulfilling bank-runs when liquidity risk is introduced.

### Overview of the simple model (no liquidity risk)
- Economy populated by four agents: households, banks, entrepreneurs, and outside investors.
- Time structure: three periods (see Figure 5 timeline).
  - Period 1: banks collect deposits from households and invest in cash (M, safe and liquid) and assets (I, risky and illiquid) originated by entrepreneurs.
  - Period 2: public news signal observed. With probability q, the signal is “bad news” leading to a downward revision in expected asset returns; after observing the signal, households choose whether to withdraw early and banks trade assets with outside investors in a secondary market.
  - Period 3: assets yield payoff σ_h when fundamentals are strong and σ_l when fundamentals are weak; following bad news, fundamentals are weak with (conditional) probability p. If banks cannot pay promised returns, they become insolvent and a haircut proportional to the funding shortfall is imposed on deposits.
- Key friction: costly commitment τ > 0. Banks cannot credibly commit to a safe investment strategy unless they pay τ.
  - Endogenous heterogeneity: share γ ∈ [0,1] of banks do not pay τ and follow a shadow banking strategy (high risk-taking); remaining banks pay τ and follow a traditional banking strategy (safe enough to prevent early withdrawal).
  - Early withdrawals are optimal responses to revision in solvency prospects (not Diamond-Dybvig strategic bank-runs).

### Core probabilistic and normalization relations (as stated)
- With probability q the public news signal is bad.
- Following bad news, fundamentals are weak with (conditional) probability p.
- Normalizations and expectation:
  - Risk-free rate normalized: R* = 1.
  - Unconditional expectation of asset payoffs normalized to unity such that
    (1−qp)σ_h + qpσ_l = 1

### Entrepreneurs, production, and asset supply
- Entrepreneurs use capital K to produce period-1 assets I1 with Cobb-Douglas technology:
  I1 = A K^α
  where A > 0, α ∈ (0,1).
- Representative entrepreneur’s problem (first period): max_{I1,K} P1 I1 − K subject to production (2), where P1 is the asset price.
- First-order condition interpreted as an upward-sloping asset supply schedule (as stated in text):
  P1 =
  1
  αA
  1
  α
  I1−α
  α
  1
- Due to relationship lending frictions, banks may only purchase assets from entrepreneurs in their own island; this grants banks market power and allows extraction of a mark-up with
  ∂P1
  ∂I1
  I1
  P1
  = μ > 0
- Mark-up restriction for tractability:
  μ ≤ ̄μ ≡
  (1−q) (σ_h − 1)
  1 − (1−q) σ_h

### Notation summary (variables and parameters, as listed)
- Variables (labels exactly as in source):
  - D Deposits (per bank)
  - d Deposits (per household)
  - I Assets
  - M Cash
  - R Interest rate on deposits
  - K Capital
  - P1, P2 Asset prices
  - ̃I Excess supply in secondary market
  - ̃K Outside project
  - φ Fire-sale discount
  - θ Liquidation value
  - V Recovery rate
  - Π Bank profits
  - γ Share of shadow banks
  - C Consumption
- Parameters (labels and descriptions exactly as in source):
  - q Prob. of bad news
  - p Prob. of weak fundamentals
  - σ_h, σ_l Asset payoffs
  - A Productivity
  - α Cobb-Douglas elasticity
  - R* Risk-free rate
  - E Household endowment
  - ̃E Outside investor endowment
  - τ Commitment cost

*Italic source: wp17285 - Section 3 describes the core mechanisms of the model in a simple framework without liquidity*

### Appendix D.2 that it can be relaxed without altering the properties of the equilibrium solution.

### Appendix D.2 that it can be relaxed without altering the properties of the equilibrium solution.

### Secondary market and outside investors
- After good news:
  - Assets yield a high payoff σh with certainty; no early withdrawals.
  - Secondary markets clear at price σh; trade is inconsequential to equilibrium allocation.
- After bad news:
  - Excess supply of assets:
    - Ĩ = γ I1SB + (1−γ) (I1TB − I2TB) > 0, where γ is share of shadow banks, I1SB is shadow bank assets.
  - Secondary market price:
    - P2 = φ[(1−p)σh + pσl], where φ ∈ [0,1] is the fire-sale discount.
  - Outside investors (Stein (2012) setup):
    - Begin with endowment Ẽ and allocate between secondary-market assets and an outside project K̃ that yields safe payoff g(K̃) with g′(.)>0, g′′(.)<0.
    - First-order condition gives implicit fire-sale discount:
      - φ = 1 / g′(Ẽ − φ[(1−p)σh + pσl] Ĩ)  (equation (5)).
    - Write φ = f(Ĩ) where f(.) is continuous and decreasing.
- Restrictions on φ:
  - φ is restricted to φ ∈ (φ_, φ̄] by imposing on f(.) the restriction:
    - φ < f((Aα/α)^{1/(1−α)} ((1−q) σh/(1+μ) + q σl)^{α/(1−α)}) < φ̄  (equation (6))
  - Definitions:
    - φ ≡ σl / ((1−p)σh + pσl) ≥ 0
    - φ̄ ≡ min[1, ((1−q) σh/(1+μ) + q σl)/((1−p)σh + pσl)] > φ
  - Upper bound ensures fire-sale deepens as shadow banking grows and is necessary for interior equilibrium coexistence.
  - Lower bound simplifies exposition by eliminating another interior equilibrium case; consequences of relaxing it are discussed in Appendix E.

### Households
- Preferences and allocations:
  - Unit continuum of risk neutral households maximize expected consumption:
    - E[C] = (1−q) Cgh + q(1−p) Cbh + q p Cbl  (equation (7)).
  - First-period allocation: deposits in traditional and shadow banks (dTB, dSB) and cash M1; budget:
    - dSB + dTB + M1 = E  (equation (8)).
- Consumption profiles:
  - Good news third-period consumption:
    - Cgh = M1 + dTB RTB + dSB RSB  (equation (9)), where (RTB, RSB) are deposit interest rates.
  - Under bad news, second-period budget with only shadow-bank early withdrawal:
    - M2 = M1 + θSB dSB  (equation (10)), where θSB is liquidation value of shadow bank portfolios.
  - Third-period consumption contingent on asset payoffs:
    - Cbh = M2 + dTB RTB  (equation (11))
    - Cbl = M2 + V dTB RTB  (equation (12)), where V is recovery rate of traditional bank deposits.
- First-order conditions yield deposit interest rates:
  - RTB = 1 + qp(1−V) / (1−qp(1−V))  (equation (13))
  - RSB = 1 + q/(1−q) (1−θSB)  (equation (14))
  - Comparative statics:
    - RTB decreases with higher V.
    - RSB decreases with higher θSB.
    - RTB = 1 when V = 1; RSB > 1 unless θSB = 1.
- Early withdrawal decision and market discipline:
  - Proposition 1:
    - For all φ > φ_, households optimally withdraw deposits from shadow banks after bad news.
    - Traditional banks may avoid early withdrawal by committing to minimum recovery rate:
      - V̄ ≡ 1/p (1/RTB − (1−p))  (equation (15)).
    - In equilibrium:
      - V̄ = RTB = 1  (equation (16)).
  - Intuition:
    - No-withdrawal constraint (15) enforces market discipline on traditional banks, eliminating insolvency risk and reducing funding costs to risk-free rate.
    - Virtuous cycle: guaranteeing minimum recovery rate reduces deposit compensation demand, lowering RTB, which further strengthens withdrawal threat until solvency risk eliminated.
  - Caveat:
    - Result contingent on lack of liquidity risk; even small risk of bank run reverses the virtuous cycle (Section 4).

### Banks
- Setup:
  - Unit continuum of ex-ante identical, risk neutral banks collect deposits D and purchase assets I1 at price P1 and hold M1:
    - P1 I1 + M1 = D  (equation (17)).
  - Third-period profit after good news:
    - Πgh = σh I1 + M1 − D R  (unnumbered).
- Bad news:
  - Liquidation value for banks subject to early withdrawal:
    - θ = min{1, (P2 I2 + M2)/D}  (equation (18)).
  - Banks without early withdrawal face second-period budget:
    - P2 I2 + M2 = P2 I1 + M1  (equation (19)).
  - Profit outcomes:
    - If third-period payoff σh: Πbh = σh I2 + M2 − D R.
    - If low payoff σl, limited liability binds and deposits pay recovery rate:
      - V = min{1, (σl I2 + M2)/(D R)}  (equation (20)).
- Banking strategies and commitment cost:
  - Commitment cost τ > 0 creates discrete choice:
    - Shadow banking (SB): do not pay τ, anticipate early withdrawal after bad news.
    - Traditional banking (TB): pay τ, commit to no-withdrawal constraint (15).
  - No-withdrawal constraint in choice variables:
    - σl I2TB + M2TB ≥ V̄ D TB RTB  (equation (21)).
  - Banks choose strategy with higher expected payoff:
    - SB preferred when E[ΠSB] ≥ E[ΠTB] − τ  (equation (22)).
- Shadow banking (Lemma 1):
  - Maximize E[ΠSB] = (1−q) (σh I1SB + M1SB − DSB RSB)  (equation (23)), subject to (17).
  - Results (combining (35),(36) with household FOC (14)):
    - θSB = (1 + μ) P2 / ((1−q) σh + q(1 + μ) P2) < 1 ∀ φ ∈ [0,1]
    - RSB = σh / ((1−q) σh + q(1 + μ) P2) > 1 ∀ φ ∈ [0,1]
    - E[ΠSB] = (1−q) μ/(1 + μ) σh I1SB
    - I1SB = (Aα/α)^{1/(1−α)} ((1−q) σh/(1 + μ) + q P2)^{α/(1−α)}
- Traditional banking (Lemma 2):
  - Maximize E[ΠTB] = (1−q) (σh I1TB + M1SB) + q(1−p) (σh I2TB + M2TB) − (1−qp) D TB R TB, subject to (17),(19),(21).
  - Under restrictions μ ≤ μ̄, φ > φ_:
    - I1TB = (Aα/α)^{1/(1−α)} P2^{α/(1−α)}  (equation (24))
    - I2TB = 0
    - M2TB = P2 I1TB + M1TB  (equation (25))
    - E[ΠTB] = (1−q) (σh − P2) I1TB
    - M1TB ≥ 0 is indeterminate.
    - (RTB, V̄) given by (16).
    - ∂E[ΠTB]/∂P2 < 0 ∀ P2 > α σh.
  - Intuition: Traditional banks liquidate risky assets after bad news, re-allocate to cash to satisfy no-withdrawal constraint; lower P2 tightens constraint and reduces first-period risky asset purchases I1.

### Equilibrium
- Market clearing and rational expectations:
  - γ DSB = dSB
  - (1−γ) DTB = dTB
  - φ = f(γ I1SB + (1−γ) (I1TB − I2TB))
  - γ ∈ [0,1]; interior equilibrium when 0 < γ < 1 and banks indifferent:
    - E[ΠSB] = E[ΠTB] − τ  (equation (26)) (free entry condition determines γ).
- Fire-sales and bank strategies (interaction):
  - Entry into shadow banking increases excess supply in secondary market, exacerbating fire-sales (φ falls).
  - Fire-sales reduce expected payoff from shadow banking relative to traditional banking, deterring entry.
  - Mechanism illustrated: fall in P2 reduces θSB → raises shadow-bank funding costs → reduces E[ΠSB]; fall in P2 tightens no-withdrawal constraint → traditional banks reduce I1 → with RTB = 1 this increases excess return and E[ΠTB].
  - These generate fire-sale externality and strategic substitutability: as shadow banking grows, φ falls, reducing SB payoff until indifference.
- Interior equilibrium conditions (Proposition 2):
  - Interior equilibrium exists under parameter restrictions μ ≤ μ̄, φ ∈ (φ_, φ̄), α < 0.5, p < 1−α, and τ ∈ (τ_, τ̄), where explicit τ_ and τ̄ are given:
    - τ_ ≡ { piecewise expressions involving κ, σh, μ, q, p, α, P2-related terms } with κ ≡ (1−q) (Aα/α)^{1/(1−α)}, and τ̄ similarly defined; τ̄ > τ_ > 0.
  - Free entry condition pins down γ and φ through interactions above.
- Comparative statics (numerical example summary):
  - Rise in commitment cost τ:
    - Downward shift in E[ΠTB], makes SB more profitable → entry into shadow banking increases γ → exacerbates fire-sale (φ falls).
  - Increase in liquidity available in secondary market (rise in Ẽ):
    - Reduces fire-sale discount (φ rises) at given γ → E[ΠTB] falls and E[ΠSB] rises → entry into shadow banking until new intersection at larger γ.
  - Key property: fire-sale discount returns to initial value at new equilibrium; thicker secondary market increases size of shadow banking sector but does not alleviate the fire-sale; applies also to policy interventions that increase excess supply (asset purchases, bailouts).

### A model with liquidity risk
- Extensions introduced in Section 4:
  - Introduce liquidity risk via possibility of bank-runs (Diamond-Dybvig style), distinct from optimal early withdrawals and driven by self-fulfilling expectations when θ < 1.
  - Expand asset space to three assets: liquid (λ), safe (s), and risky (r). Risky asset as before; safe and liquid yield unit payoff with certainty but safe matures in period 3 while liquid pays in period 2.
- Purposes of richer asset space:
  - Determine conditions under which safe assets are endogenously liquid due to secondary markets (safe asset may or may not be subject to fire-sale depending on offsetting purchases by traditional banks).
  - Study interplay between fire-sales, market discipline, and liquidity risk:
    - Without fire-sale on safe assets: traditional banks’ portfolio re-allocation toward safe assets increases their liquidation values (θTB = 1) and reduces bank-run vulnerability.
    - With fire-sale on safe assets: liquidity risk raises funding costs and undermines market discipline; traditional banks may take on solvency risk.
- Section organization:
  - Section 4.1: bank-runs details.
  - Section 4.2: secondary markets and outside investors extended to multiple assets.
  - Section 4.3: analytical results on relationship between fire-sale on safe assets, market discipline, and liquidity risk.

*Source: wp17285 - Appendix D.2 that it can be relaxed without altering the properties of the equilibrium solution.*

### 4.1    Bank-runs

### wp17285 - 4.1    Bank-runs

### Bank-run mechanism and probability
- When a bank has a liquidity shortfall θ < 1, sequential service in withdrawals can generate a bank-run equilibrium where households optimally withdraw if they expect others to withdraw.
- Sunspot device:
  - Let εξ be a random variable drawn from a uniform distribution on the unit interval at the beginning of period 2 and ̄ξ ∈ [0,1] a constant threshold.
  - When εξ ≤ ̄ξ, household sentiments coordinate on a bank-run. Provided θ < 1, these sentiments become self-fulfilling and the bank suffers from a run.
  - Since εξ is uniform on [0,1], the probability of a bank run ξ is:
    - ξ = 0 for θ = 1
    - ξ = ̄ξ for θ < 1
- Only traditional banks are affected by bank-runs as there is no liquidity shortfall after good news and shadow banks face an early withdrawal after bad news.
- The sunspot realization is assumed idiosyncratic to each bank so that ξ may be interpreted as the share of traditional banks liquidated in a bank-run.
- Appendix J: ̄ξ can be depicted as a negative function of the liquidation value θ; qualitative results remain the same.

### Secondary market: multi-asset fire-sale and equilibrium price discount
- In period 2 only safe and risky assets are traded. Excess supplies for i ∈ {s, r} are:
  - ̃I(i) = γ I_SB_1(i) + (1−γ)(I_TB_1(i) − (1−ξ) I_TB_2(i)) ≥ 0 ∀ i ∈ {s, r}
- Outside investors allocate endowment ̃E between purchases of safe and risky assets and an outside project ̃K with safe payoff g(̃K) where g′(.) > 0, g′′(.) < 0.
- First-order conditions yield a common fire-sale discount φ ∈ [0,1] for the two assets, defined implicitly by:
  - φ = 1 / g′(̃E − φ[(1−p)σ_h + pσ_l] ̃I(r) − φ ̃I(s))  (equation (27))
  - Interpretation: a rise in total excess supply of safe and risky assets leads to a decrease in φ.
- Asset-specific price effects:
  - An asset type is only subject to the fire-sale discount when it is in excess supply such that its marginal buyer is an outside investor.
  - Secondary market prices depend on an asset’s own excess supply:
    - P2(s) = 1 for ̃I(s) = 0; P2(s) = φ otherwise.
    - P2(r) = (1−p)σ_h + pσ_l for ̃I(r) ≤ 0; P2(r) = φ[(1−p)σ_h + pσ_l] otherwise.

### Analytical results: Proposition 3 and implications
- Proposition 3 (equilibrium commitments and deposit rate):
  - Traditional banks commit to a minimum recovery rate:
    - ̄V = 1 − q/(p ξ) (1−θ_TB) / [1−q(1−ξ(1−θ_TB))]  (equation (28))
  - Interest rate on traditional bank deposits:
    - R_TB = 1 + q/(1−q) ξ (1−θ_TB)  (equation (29))
- Special cases and implications:
  - With P2(s) = 1, traditional banks have θ_TB = 1 → ξ = 0 and (28), (29) yield ̄V = R_TB = 1.
  - With sufficiently low P2(s) < 1, traditional banks have θ_TB < 1 → ξ > 0. Then (28), (29) yield ̄V < 1 < R_TB.
- Mechanism summary:
  - No fire-sale on safe assets (P2(s) = 1) implies no liquidity shortfall (θ_TB = 1), ξ = 0, ̄V = 1, and traditional banks collect deposits at the risk-free rate.
  - Large fire-sale on safe assets (P2(s) < 1) can produce θ_TB < 1 even if no-withdrawal constraint is satisfied, generating vulnerability to bank-runs, rising deposit rates R_TB, declining minimum recovery rate ̄V, solvency risk, and a vicious cycle between deposit rates and solvency.

### Numerical results: calibration highlights and key parameters
- Targeting the United States over the recent financial crisis; calibration details:
  - σ_l = 0.21 (Low payoff from risky assets)
  - q = 0.41 (Prob. of bad news)
  - p = 0.22 (Prob. of weak fundamentals)
  - α = 0.33 (Cobb-Douglas parameter)
  - A = 1.00 (Productivity)
  - μ = 0.18 (Mark-up)
  - ̃E = 2.80 (Outside investor endowment)
  - z = 10.0 (Outside investment parameter)
  - ̄ξ = 0.15 (Bank-run probability)
  - τ = 0.11 (Commitment cost)
- Calibration notes:
  - σ_h is backed out using normalization of expected payoffs.
  - (q, p) calibrated to NBER frequency of recessions and deep recessions.
  - ̃E and z calibrated to match percentage decline in bond prices ∆ˆP/ˆP from peak spread in November 2008 relative to pre-crisis average.
  - ̄ξ calibrated to match R_TB_2 to average TED spread in last quarter of 2008 via:
    - R_TB_2 = 1 / (1 − ̄ξ (1 − ̄ξ θ_TB^{1−p} (1−̄V)))  (equation (32) as given)
  - τ targets pre-crisis ratio of shadow bank assets to total bank assets ≈ 63%; τ = 0.11 amounts to just below 7% of value of a traditional bank’s assets in period 1.

### Numerical equilibrium behavior (qualitative findings)
- There exists a unique and globally stable interior equilibrium γ∗ where shadow and traditional banks coexist.
- Shadow banking expansion:
  - Entry into shadow banking exacerbates the fire-sale after bad news, increasing shadow bank deposit interest rates and reducing expected shadow-bank payoff.
  - Traditional bank profits rise as the fire-sale increases the excess return from the binding no-withdrawal constraint, inducing shadow bank entry until indifference at γ∗.
- Thresholds and equilibrium sector sizes:
  - ̄γ denotes threshold sector size above which safe assets suffer from a fire-sale.
  - Under calibration, high τ leads to γ∗ > ̄γ so a fire-sale on safe assets exists in equilibrium.
  - Fire-sale reduces θ_TB (liquidation value), raising R_TB and undermining market discipline; ̄V declines below full repayment for γ > ̄γ.
  - With lower τ, possible γ∗ ≤ ̄γ implying no fire-sale, no liquidity shortfall, R_TB = 1 and ̄V = 1.
- Portfolio and balance-sheet responses:
  - As γ rises and fire-sale deepens: shadow banks reduce deposits; traditional banks expand balance sheets due to relaxation of no-withdrawal constraint.
  - Traditional banks re-allocate away from risky assets up to 18% when shadow banking is small; in equilibrium with large shadow banking, net re-allocation is ≈ 4% away from risky assets.
  - Shadow banks skew investment toward risky assets and do not change asset composition in response to fire-sales due to limited liability and taking R_SB as given.

### Policy analysis: interventions and outcomes
- Three interventions considered: asset purchases in secondary market, interventions to secure traditional banks (liquidity support, deposit insurance and regulation), and a tax on shadow bank profits.

- Asset purchases (government purchases of safe assets):
  - Excess supply of safe assets with asset purchases I_AP > 0:
    - ̃I(s) = γ I_SB_1(s) + (1−γ)(I_TB_1(s) − (1−ξ) I_TB_2(s)) − I_AP
  - An intervention that absorbs excess safe assets at initial equilibrium γ∗ eliminates the fire-sale at γ∗ (shifts ̄γ to γ∗).
  - However, alleviating the fire-sale reduces shadow bank funding costs and increases expected shadow-bank payoff → entry into shadow banking until new equilibrium sector size γ∗_AP > γ∗.
  - Key insight: policy interventions in secondary markets are ineffective when there is free entry into shadow banking because entry offsets the intended reduction in fire-sales. A surprise (ex-post) asset purchase could be effective but creates potential time inconsistency.

- Interventions to secure traditional banks: collateralized liquidity provision and deposit insurance with regulation
  - Liquidity provision: permit traditional banks to borrow L from central bank at risk-free rate in period 2.
    - θ_TB = min[1, P2(r) I_TB_1(r) + P2(s) I_TB_1(s) + I_1(λ) + L] / D_TB
  - Conditions to eliminate solvency and liquidity risk:
    - L ≥ L ≡ D_TB − P2(r) I_TB_1(r) − P2(s) I_TB_1(s) − I_1(λ)  (offset liquidity shortfalls)
    - Liquidity must be fully collateralized (rules out default on central bank and makes depositors junior to central bank).
  - With collateralized liquidity provision:
    - Bank-runs eliminated (ξ = 0); no-withdrawal constraint becomes σ_l I_TB_2(r) + I_TB_2(s) ≥ L + ̄V D_TB / R_TB, so market discipline is not adversely affected.
    - By Proposition 3, absence of liquidity risk implies ̄V = 1 and R_TB = 1.
  - Numerical outcome under liquidity provision:
    - Liquidity provision ringfences traditional banking and eliminates the fire-sale on safe assets (P2(s) = 1) via arbitrage: borrowing at risk-free to buy discounted safe assets until no excess supply remains.
    - This increases portfolio reallocation toward safe and liquid assets for traditional banks.
    - Alleviating fire-sale reduces shadow bank funding costs → increases expected payoff from shadow banking → increases equilibrium shadow banking size. Despite larger shadow banking sector, traditional banks remain insulated and overall financial stability improves.
    - Liquidity provision also increases household consumption under weak fundamentals and in unconditional period-1 expectation (Panels E and F), given the model’s structure (no full welfare analysis performed).
  - Deposit insurance alternative:
    - Combining deposit insurance guarantees with bank regulation imposing ̄V = 1 achieves the same outcome for eliminating liquidity and solvency risk, but differs in role of market discipline:
      - Liquidity provision leverages market discipline while securing traditional banks.
      - Deposit insurance removes no-withdrawal constraint (depositors never lose) and requires bank regulation to replace market discipline to prevent risk-shifting.

- Tax on shadow bank profits (briefly referenced)
  - A tax on shadow bank profits can reduce shadow bank entry at a given sector size, deepening the fire-sale at a given γ, but free entry/exit can offset intended effects—qualitatively similar entry-offset logic applies as with asset purchases.

*Source: wp17285 - 4.1 Bank-runs (IMF working paper excerpt).*

### 6.3    Tax on shadow bank profits

### 6.3    Tax on shadow bank profits

### Purpose and mechanism
- The tax is designed to deter entry into the shadow banking sector by internalizing a negative externality that entry imposes on the remainder of the financial sector through its contribution to fire-sales.
- The tax can be considered a Pigouvian tax. A constant tax T is introduced so that the free entry condition becomes:
  - E[Π_SB] − T = E[Π_TB] − τ

### Equilibrium effects at 0 < T < τ
- A tax level satisfying 0 < T < τ reduces the equilibrium sector size of shadow banks to the fire-sale threshold ̄γ.
- The tax shifts down the expected payoff schedule for shadow banking and can produce an equilibrium without a fire-sale on safe assets.
- Because this is a tax on profits, the equilibrium allocation at any given sector size γ is identical to the numerical results in Figure 10.
- Reducing the equilibrium sector size to ̄γ:
  - Eliminates liquidity risk in the traditional banking sector.
  - Strengthens market discipline such that there is also no solvency risk.

### Equivalence and alternatives
- The effect of a tax on shadow banks is equivalent to a decrease in the commitment cost τ.
- When taxing shadow banks is not feasible, the same outcome can be achieved with a transfer to traditional banks.
- Shadow bank profits may be reduced indirectly with a tax on their liabilities D_SB. In a less stylized set up, this corresponds to a tax on funding instruments most widely used by shadow banks such as ABCPs.

### Policy implications and connection to broader conclusions
- Taxation of shadow bank profits (or equivalently a transfer to traditional banks) can reduce the size of the shadow banking sector to a level compatible with financial stability.
- This policy complements other findings in the paper:
  - Secondary market interventions alleviate fire-sales ex-post but can fuel shadow banking growth ex-ante.
  - Policies securing the traditional banking sector, when anticipated, can expand shadow banking but may still have a net positive impact on financial stability.
- Quantitative indication (from related discussion in the source): the necessary transfer amounts to 9% of commitment costs, which is equivalent to approximately 0.6% of the value of a traditional bank’s portfolio in period 1.

*Source: wp17285 - 6.3    Tax on shadow bank profits*

### References

### wp17285 - References

### Key literature cited
- Acharya, V. V., Schnabl, P., and Suarez, G. (2013). Securitization without risk transfer. Journal of Financial Economics, 107(3):515–536.
- Adrian, T. and Ashcraft, A. B. (2012). Shadow Banking: A Review of the Literature. The New Palgrave Dictionary of Economics. Palgrave Macmillan.
- Allen, F. and Gale, D. (1994). Limited market participation and volatility of asset prices. The American Economic Review, 84(4):933–955.
- Brunnermeier, M. K. (2009). Deciphering the liquidity and credit crunch 2007–2008. The Journal of economic perspectives, 23(1):77–100.
- Claessens, S., Zoltan, P., Ratnovski, L., and Singh, M. (2012). Shadow banking: Economics and policy. Staff discussion note, IMF.
- Gorton, G. and Metrick, A. (2012). Securitized banking and the run on repo. Journal of Financial Economics, 104(3):425–451.
- Financial Stability Board (2017). Global Shadow Banking Monitoring Report 2016.
- Pozsar, Z., Adrian, T., Ashcraft, A. B., and Boesky, H. (2013). Shadow banking. Economic Policy Review, 19(Dec):1–16.
- Shleifer, A. and Vishny, R. (2011). Fire Sales in Finance and Macroeconomics. Journal of Economic Perspectives, 25(1):29–48.
(Reference list continues in source; above preserves representative citations as provided.)

### Appendix A — Proof of Proposition 1: withdrawal incentives and no-withdrawal constraint
- Expected consumption without early withdrawal:
  - (1−P)c_bh + p c_bl = M 1+θ_SB d_SB + (1−p(1− ̄V)) d_TB R_TB
- Expected consumption with early withdrawal:
  - c_wb = M 1+θ_SB d_SB + d_TB
- No-withdrawal condition (optimal not to withdraw):
  - (1−p)c_bh + p c_bl ≥ c_wb ⇒ V ≥ ̄V ≡ 1 p (1 R_TB − (1−p) )
- Joint solution with (13) yields:
  - ̄V = 1, R_TB = 1.
- Early withdrawal from shadow banks is optimal when:
  - (1−p)R_SB + p V_SB R_SB < 1
- Shadow-bank liquidation value (using (14), (35), (36)):
  - V_SB = (1 + μ) σ_l / σ_h R_SB = σ_h (1−q)σ_h + q(1 + μ) P^2 / 2
- Sufficient condition when p = q: satisfied for any P^2 > σ_l (consistent with (6)).
- Restrictions on p depend on mark-up μ: when σ_h < (1 + μ) σ_l need p ≥ q and vice versa.

### Appendix B — Limited liability implications
- When no-withdrawal constraint binds in low-payoff state b_l, limited liability implies:
  - ̄V ≤ 1 and limited liability must bind in b_l.
- Under slack no-withdrawal and slack limited liability, banks borrow at risk-free rate R = 1 and profits in b_l:
  - Π_bl = σ_l I_2 + M_2 − D
- Bank optimization reduces to first order conditions:
  - I_1: P_1 = 1 (1 + μ) (equation (33))
  - M_1: φ = 1 (equation (34), inequality due to no-short-sale possibility)
- Limited liability binds under conditions:
  - σ_l − 1/(1 + μ) < 0
  - σ_l (1−p)σ_h + pσ_l − 1 < 0
- These are satisfied under condition (4) in the source; therefore limited liability cannot be slack when assets yield a low payoff.

### Appendix C — Proof of Lemma 1: shadow-bank liquidation and rates
- Shadow banks hold no cash in period 1 when R_SB > 1:
  - M_SB1 = 0
- Liquidation value:
  - θ_SB = P_2 / P_SB1 (equation (35))
- First-order condition for risky asset price:
  - P_SB1 = σ_h (1 + μ) / R_SB (equation (36))
- Combined expressions:
  - P_SB1 = (1−q) σ_h /(1 + μ) + q P_2 (equation (37))
  - θ_SB = (1 + μ) P_2 / [(1−q) σ_h + q (1 + μ) P_2] (equation (38))
- Liquidity shortfall when θ_SB < 1. Sufficient condition via φ = 1 gives:
  - (1 + μ) [(1−p) σ_h + p σ_l] < σ_h
- Interest rate expression:
  - R_SB = 1 / [1−q + q P_2 / (σ_h (1 + μ))]
  - R_SB > 1 follows from θ_SB < 1.
- Expected payoff:
  - E[Π_SB] = (1−q) μ/(1 + μ) σ_h I_SB1 where I_SB1 derived from asset supply (3).
- I_SB1, P_SB1, and E[Π_SB] are all positively related to P_2.

### Appendix D — Proof of Lemma 2: traditional-bank cases and equilibrium structure
- Traditional bank problem (after substitutions) written as:
  - Π = (1−q) (σ_h I_1 + M_1) + q(1−p) (σ_h I_2 + M_2) − (1−q p) D (equation (39))
  - Constraints: P_1 I_1 + M_1 = D, P_2 I_2 + M_2 = P_2 I_1 + M_1, σ_l I_2 + M_2 ≥ D (equations (40), (41))
- Three exhaustive cases:
  - Case 1: No-withdrawal constraint (41) and no-short-sale constraint on I_2 bind.
    - Implies P_1 = P_2 and I_1 = (A α/α)^(1/(1−α)) P_2^{α/(1−α)} (from supply (3)).
    - Expected payoff: E[Π_TB] = (1−q) (σ_h − P_2) I_1, derivative ∂E[Π_TB]/∂P_2 < 0 ∀ P_2 > α σ_h.
  - Case 2: No-withdrawal binds but no-short-sale slack.
    - I_2 expression: I_2 = (P_2 − P_1)/(P_2 − σ_l) I_1 (equation (42))
    - First-order condition for I_1 yields P_1 per (43).
    - Expected payoff expression (44). Case 2 eliminated when mark-up restriction μ < ̄μ holds (eliminates Case 2 via (45)).
  - Case 3: No-withdrawal slack. Leads to banks holding infinite M_1 and converting after bad news; contradiction with no-withdrawal constraint, so Case 3 eliminated.
- Conclusion: equilibrium conforms to Case 1 under given restrictions μ ≤ ̄μ, φ > φ.

### Appendix E — Solution under φ = φ̄ (fire-sale discount threshold)
- If φ < φ̄ and P_2 < σ_l, traditional banks buy risky assets until P_2 returns to σ_l (i.e. φ = φ̄).
- Define ̃I_2 implicitly by:
  - σ_l (1−p) σ_h + p σ_l = f(γ I_SB1 + (1−γ) (I_1 − ̃I_2))
- When P_2 achieves σ_l, no-withdrawal binds and:
  - P_1 = P_2 = σ_l, M_2 = σ_l (I_1 − ̃I_2), M_1 set to 0 by indeterminacy.
- Expected profits:
  - E[Π] = (σ_h − σ_l) [(1−q) I_1 + q(1−p) ̃I_2] = (σ_h − σ_l) [(1−q) A^{1/(1−α)} (α σ_l)^{α/(1−α)} + q(1−p) ̃I_2]
- In limiting case γ → 1 (only shadow banks), both safe asset holdings and traditional banks’ profits approach infinity:
  - lim_{γ→1} E[Π_TB] = lim_{γ→1} ̃M_1 = ∞
- This guarantees an inferior equilibrium for sufficiently high commitment cost τ > ̄τ.

### Appendix F — Proof of Proposition 2: existence of interior equilibrium and τ bounds
- Define π_SB(φ) and π_TB(φ) mapping fire-sale discount φ to expected payoffs:
  - π_SB(φ) = (1−q) μ/(1 + μ) σ_h (A α/α)^{1/(1−α)} ((1−q) σ_h /(1 + μ) + q φ[(1−p) σ_h + p σ_l])^{α/(1−α)}
  - π_TB(φ) = (1−q) (σ_h − φ[(1−p) σ_h + p σ_l]) (A α/α)^{1/(1−α)} (φ[(1−p) σ_h + p σ_l])^{α/(1−α)}
- Sufficient conditions for interior equilibrium:
  - π_SB(φ) > π_TB(φ) − τ (46)
  - π_SB(φ) > π_TB(φ) − τ (47) [duplicate in source; both imply lower/upper bounds considered]
  - ∂π_SB(φ)/∂φ > ∂π_TB(φ)/∂φ ∀ φ ∈ (φ^∗, 1) (48)
  - ∂φ/∂γ < 0 ∀ γ ∈ [0,1] (49)
- Lower bound τ ≥ τ_ with explicit expression (preserved from source) and τ_ > 0 shown under conditions:
  - α < 1/2 and μ ≤ ̄μ.
- Upper bound τ ≤ ̄τ with cases depending on:
  - Condition (51): σ_h ≥ 1/(1 − q p (1 − α)).
  - If (51) holds, ̄τ expression involves peak φ̃ = α σ_h /( (1−p) σ_h + p σ_l ).
  - If (51) not satisfied, ̄τ expression uses φ.
- Non-emptiness of interval (τ_, ̄τ) shown; sufficient condition in one subcase:
  - α < 1 − p (for example α < 0.5, p ≤ 0.5).

### Appendix G — Proof of Proposition 3: liquidity risk, households, bank-run and R_TB
- Household FOC for deposits in traditional banks yields R_TB:
  - R_TB = 1 + q ξ (1 − θ_TB)/(1 − q (ξ + (1−ξ) p (1 − ̄V) ) )  [equation (52) as in source; preserved form]
- Minimum repayment threshold remains:
  - V ≥ ̄V ≡ 1 p (1 R_TB − (1 − p) ) (equation (53)) — same as simple model.
- Combining (52) and (53) gives:
  - ̄V = 1 − q/p ξ (1 − θ_TB)/ (1 − q (1 − ξ (1 − θ_TB) ) )
  - R_TB = 1 + q/(1 − q) ξ (1 − θ_TB)
- With ξ = 0: ̄V = R_TB = 1. Increase in ξ leads to ̄V < 1 < R_TB.
- Fire-sale on safe assets pivotal for liquidity shortfall. Liquidation value θ_TB = 1 when:
  - P_2(r) I_TB2(r) + P_2(s) I_TB2(s) ≥ D_TB
- Without fire-sale on safe assets (P_2(s) = 1), this holds if P_2(r) ≥ σ_l.
- With fire-sale on safe assets (P_2(s) = φ), liquidity shortfall condition becomes:
  - φ (I_TB2(r) + I_TB2(s)) ≥ D_TB (equation (54))
- For sufficiently low φ, (54) fails and θ_TB < 1.

### Appendix H — Example fire-sale function
- Outside investor payoff parameterization:
  - g( ̃K ) = κ^{−1} ln( ̃K ), κ > 0
- Fire-sale function:
  - f( ̃I ) = κ ̃E /(1 + κ [(1 − p) σ_h + p σ_l] ̃I)
- Lower-bound calibration requirement: set ̃E so that f(I_SB1) = φ at γ = 0:
  - ̃E = φ [1/κ + (A α/α)^{1/(1−α)} ((1−q) σ_h /(1 + μ) + (1 − (1 − q p) σ_h)/p )^{α/(1−α)} ]
- Upper bound φ̄ approached as κ rises.

### Appendix I — Model extension with liquidity risk: entrepreneurs and banks
- Entrepreneurs produce three asset types i ∈ {λ, s, r} with production:
  - I_1(i) = A K(i)^{α} ∀ i ∈ {λ, s, r}
  - First-order conditions:
    - P_1(i) = 1/(α A^{1/α} I_1(i)^{1−α/α} ) — preserved as equation (55) style in source.
- Banks’ first-period budget:
  - Σ_{i∈{λ,s,r}} P_1(i) I_1(i) = D
- Third-period profit when high payoff:
  - Π_gh = σ_h I_1(r) + I_1(s) + I_1(λ) − D R (equation (56))
- Second-period reallocation and liquidation constraint (after bad news):
  - Σ_{i∈{s,r}} (P_2(i) − P_1(i)) I_1(i) = I_1(λ) (equation (57))
  - θ = min{1, [P_2(r) I_2(r) + P_2(s) I_2(s)] / D} (equation (58))
- Recovery rate under limited liability:
  - V = min{1, [σ_l I_2(r) + I_2(s)]/(D R)}
- Shadow-bank and traditional-bank first-order conditions extend accordingly:
  - Shadow-bank: P_1(r) = σ_h /(1 + μ) R_SB ; P_1(s) = P_1(λ) = 1 /(1 + μ) R_SB
  - Traditional-bank FOCs incorporate terms ̄Z_1, ̄Z_2 (defined in-source) reflecting liquidity shortfall and bank-run probabilities.

### Appendix J — Alternative bank-run specification (dependence on liquidity)
- Bank-run probability ξ modeled as decreasing function ζ(θ) of liquidation value θ:
  - ξ = ζ(θ), ζ′(.) ≤ 0, ζ(θ) ∈ [0,1] ∀ θ, and ζ(1) = 0.
- Parameterization used:
  - ζ(θ) = max{0, min{1, ̃ξ (1 − θ)} } with ̃ξ = 1.64 (calibrated per Section 5.1).
- Numerical results (Figure 14 in source) show:
  - Alternative bank-run specification yields the same equilibrium outcome as baseline.
  - For γ > γ^∗, alternative specification implies further increases in ξ with decreasing liquidity, leading to lower traditional bank profits and sharper decline in minimum recovery rate ̄V compared to baseline.

*Source: wp17285 - References (appendices and references as provided in the source PDF).*

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_Source: https://www.imf.org/-/media/files/publications/wp/2017/wp17285.pdf_