## wpiea2019092 — 1. Introduction; 1.1 Related Literature; 2.2 Fragility and fire sale; 2.3 Welfare Analysis

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---

### Introduction — motivation, question, and model overview
- Motivation and empirical facts:
  - Anecdote: "In the aftermath of the nancial crisis, a $60 million slice of subprime mortgage bonds from 2007 traded hands for as little as two cents on the dollar. Now, they’re higher than 90 cents..." (Bloomberg - October 1, 2018).
  - Three broad facts during the crisis:
    - "The cumulative losses as of 2013 on all subprime AAA-rated MBS has been estimated to be less than half of a percentage point."
    - Collapse of asset prices accompanied by lower trade volume, often akin to a market freeze.
    - Some agents with available resources (e.g., commercial banks) bought a significant portion of these securities between 2007-2009.
- Central research question:
  - Can coordination failure among buyers (strategic waiting) in decentralized, endogenous-participation markets generate fire-sale prices and lower trade volume even in the absence of asymmetric information?
- Main contribution:
  - Demonstrates that decentralized trading with endogenous participation can produce strategic waiting that causes collapse in asset price and trade volume.
  - Identifies four critical elements for fragility:
    - Decentralized competitive asset market with endogenous entry timing.
    - Liquidity shocks at current and future dates increasing over time.
    - A medium degree of imbalance between potential supply and demand.
    - A medium degree of market liquidity.
  - Equilibrium uniqueness holds in the absence of any one of these elements.
- Model setup (overview):
  - Agents: two types (sellers and buyers); sellers hold one indivisible asset initially; buyers can buy at most one unit.
  - Three periods t=0,1,2; asset pays dividend d_2>0 at t=2.
  - Preferences: U_S = E_0[δ_0 C_0 + δ_1 C_1 + C_2]; U_B = E_0[C_0 + C_1 + C_2]; Assumption 1: δ_0>1 and δ_1>1.
  - Participation is endogenous and costless; buyers post prices in submarkets; sellers choose submarket/price.
  - Matching function: M(s,b)=γ s^{1−α} b^{α} with 0<α<1 and 0<γ<1.
  - Limited buyer capacity interpreted as tightening borrowing constraints during distress.

- Numerical illustration of waiting-complementarity:
  - Initially 100 buyers and 40 sellers; if trade volume is 20, future sellers-to-buyers ratio = (40−20)/(100−20) = 1/4.
  - If 10 buyers withdraw and trade volume drops to 10, future sellers-to-buyers ratio = (40−10)/(100−10) = 1/3.
  - A buyer’s decision to wait raises other buyers’ incentive to wait via higher reservation utilities—generating multiple equilibria.

### Equilibrium types, fragility, and empirical implications
- Two equilibrium types:
  - Delayed equilibrium: some buyers and sellers indifferent and wait until t=1 (participation postponed).
  - Run equilibrium: all buyers and sellers try to trade in t=0.
- Fragile market: coexistence of run and delayed equilibria; fire-sale equilibrium is the lower-price equilibrium (can be delayed or run type).
  - If buyers > sellers, fire sale manifests as a delayed equilibrium (buyers wait).
  - If buyers < sellers, fire sale manifests as a run equilibrium (sellers rush).
- Implications for trade volume and price:
  - When multiple equilibria exist:
    - Fire sales coincide with lower trade volume when there are fewer sellers than buyers (delayed equilibrium).
    - Fire sales coincide with higher trade volume when there are more sellers than buyers (run equilibrium).
  - Price impact of fire sale defined as the percentage decline in asset price relative to high-price equilibrium; when fire sale is delayed, price impact is larger when markets are initially more liquid (higher initial asset price leads to bigger fall).

### Policy implications (overview)
- Policies lowering opportunity cost of participation at t=0 (e.g., accommodative monetary policy) can make delayed fire-sale less likely.
- Asset purchases by the government or credible commitment to support prices can eliminate fire-sale equilibria by keeping prices above threshold.
- Policies changing market structure or trading platform (standardization, CCP, transparency) have ambiguous effects: net effect depends on quantitative impact on market liquidity γ and demand m.

---

### Related literature — positioning and contrasts
- Classical fire-sale narratives:
  - Shleifer and Vishny (1992, 1997): experts/liquidity-constrained specialized investors cause fire sales; limitations for financial assets where non-specialized investors with resources existed.
- Information frictions / adverse selection accounts:
  - Dow and Han (2018), Guerrieri and Shimer (2014), Chang (2018): adverse selection or private information can produce fire sales or illiquidity, but mechanisms and volume implications differ from the coordination-waiting mechanism here.
- Alternative mechanisms:
  - Diamond and Rajan (2011): debt overhang causing market freezes.
- Multiplicity and search externalities:
  - Vayanos and Weill (2008): self-fulfilling equilibria via search externalities; analogous externality here is intertemporal—participation at t=0 affects future outside options and probabilities of trade.
- Empirical findings supportive of buyer-strike or market-freeze narratives:
  - Hanson and Sunderam (2013): trading for nontraditional securitizations declined dramatically 2007-2009.
  - Boudoukh et al. (2016): newly issued sovereign bonds become cheaper in crises; consistent with larger price impact and trade-volume drop for more liquid assets when fire sale is delayed.

---

### Fragility and fire sale — characterization, conditions, and comparative statics
- Key equilibrium at t=1 (unique):
  - With q_B1(σ_1)=γ σ_1^{1−α} and q_S1(σ_1)=γ σ_1^{−α}, the unique equilibrium price:
    - p^*_1 = [1 + (1−α)(δ_1−1)]/δ_1 d_2
    - with d_2/δ_1 < p^*_1 < d_2, implying full participation at t=1.
  - Continuation utilities at t=1:
    - ̄U_S1 = V_S1 = (1 + (1−α)(δ_1−1) γ σ_1^{*−α}) d_2
    - ̄U_B1 = V_B1 = [α(δ_1−1)/δ_1] γ σ_1^{*1−α} d_2
- Indifference and delayed equilibrium conditions at t=0:
  - ̄U_S0 − ̄U_S1 = 0
  - d_2 − 1/δ_0 ̄U_S1 − ̄U_B0 = 0
  - Define f(σ): f(σ) ≡ 1/δ_0 + (1−α)(δ_1−1)/δ_0 γ σ^{−α} + α(δ_1−1)/δ_1 γ σ^{1−α}
  - Equation determining σ_1^{*} in delayed/run indifference equilibrium:
    - 1/δ_0 + (1−α)(δ_1−1)/δ_0 γ σ_1^{*−α} + α(δ_1−1)/δ_1 γ σ_1^{*1−α} = 1
  - Price in delayed equilibrium at t=0:
    - p^{**}_0 = (1/δ_0) ̄U_S1 = [1/δ_0 + (1−α)(δ_1−1)/δ_0 γ σ_1^{*−α}] d_2

- Lemma 1 (preserved statement):
  - Whenever market fragility exists:
    - p^D_0 < p^R_0 ⇐⇒ m<1
  - Interpretation: fire sale occurs in a run equilibrium when m>1 or in a delayed equilibrium when m<1; ν_0 (volume at t=0) is higher when price is higher iff m<1.

- Fire sale in delayed equilibrium (m<1):
  - When m<1 and markets fragile, fire sale in delayed equilibrium coincides with collapse in trade volume (ν^D_0 < ν^R_0).
  - Selected equalities:
    - p_1 = 1+(1−α)(δ_1−1) δ_1 d_2
    - p^D_0 − p_1 = α(δ_1−1) δ_1 n[ 1−γ σ^{*1−α}_1 ] o d_2
    - Since γ σ^{*1−α}_1 <1, p_1 < p^D_0 < p^R_0 (Lemma 2).

- Role of decentralization and complementarity:
  - Centralized market benchmark (Lemma 3): equilibrium prices unique (except knife-edge m=1); fragility not present in centralized trading.
  - In decentralized market multiplicity arises because intertemporal externalities (participation at t=0 affecting σ_1 and others’ reservation utilities) are not fully priced.
  - Complementarity:
    - For m<1: a buyer waiting at t=0 increases σ_1, raising other buyers’ reservation utilities and encouraging further waiting.
    - For m>1: symmetric role reversal between buyers and sellers.

- Conditions for fragility (Proposition 1, Proposition 2, Lemma 5 — preserved statements and inequalities):
  - Fragility requires δ_1 > δ_0 and non-extreme values of δ_1/δ_0:
    - f( δ_1 δ_0 )<1 ⇐⇒ γ[ δ_1 δ_0 ]^{−α} < δ_0 −1 δ_1 −1
    - Implies 1 < δ_1 δ_0 < γ^{−1/(1−α)}
  - There must be a minimum imbalance between supply and demand (m sufficiently far from 1) for multiplicity; as m→1, participation decisions at t=0 have negligible impact on σ_1.
  - Necessary conditions for multiple equilibria (γ bounds):
    - m^{α} (δ_0 −1)δ_1 ((1−α)δ_1 + α m δ_0)(δ_1 −1) ≤ γ ≤ (δ_0 −1)δ_1 ((1−α)δ_1 + α δ_0)(δ_1 −1)

- Price impact of fire sale (Definition 4 and Proposition 3):
  - ∆p_0 defined as percentage decline in asset price when agents switch from high-price to fire-sale equilibrium; precise formula depends on whether fire sale is run or delayed.
  - Comparative statics:
    - d∆p_0/dm >0 if fire sale is a run (m>1).
    - d∆p_0/dm <0 if fire sale is delayed (m<1).
    - Implies d∆p_0/d|m−1| >0: price impact increases with imbalance |m−1|.
    - When fire sale is delayed (m<1), d∆p_0/dγ >0 (more liquid markets can have larger price impacts in delayed fire sale).
    - For run equilibrium, a sufficient (but complex) condition involving m, γ, δ_0, δ_1 leads to d∆p_0/dγ >0 (text provides full inequality).

- Funding liquidity extension (pledgeability θ_0) and Lemma 6:
  - Buyers must borrow entire price subject to pledge constraint p_0 ≤ θ_0 d_2.
  - For m<1 and parameters satisfying:
    - 1 +(1−α)(δ_1 −1)γm^{−α} δ_0 < 1 +(1−α)(δ_1 −1) δ_1
  - There exists a range of θ_0 such that an equilibrium exists with p^D_0 < p_1, all buyers participate at t=0 while some sellers wait for t=1.
  - Within that θ_0 range:
    - ∂p^D_0 / ∂θ_0 >0
    - ∂ν_0 / ∂θ_0 >0
  - Interpretation: limited funding liquidity (low θ_0) can exacerbate fire-sale price declines and reduce trade volume; θ_0 may depend on p_0, potentially reinforcing multiplicity.

---

### Welfare analysis, ranking of equilibria, and policy effects
- Planner objective (time-zero consumption equivalent):
  - W ≡ Ū_B0 + m δ_0 Ū_S0  (equation 25).
- Run equilibrium welfare representation (equation 27):
  - W = [ Ū_B1 + Ū_S1/δ_0 m ] + γ m^{1−α} [ d_2 − 1/δ_0 Ū_S1 − Ū_B1 ]
    - First term: sum of reservation utilities.
    - Second term: number of transactions at t=0 times extra trade surplus per match.
- Welfare ranking results (Proposition 4 and Corollary 2):
  - For market-fragile parameters {γ, δ_0, δ_1}:
    - If m>1 ⇒ W_D(m,γ,δ_0,δ_1) > W_R(m,γ,δ_0,δ_1).
    - There exists 0<ξ<1 such that for ξ ≤ m <1 ⇒ W_D(m,γ,δ_0,δ_1) < W_R(m,γ,δ_0,δ_1).
  - Corollary 2: when market is fragile, the equilibrium with the fire sale has the lower total welfare for all m>ξ.
- Comparative statics of welfare in σ_1^{*}:
  - W is strictly decreasing for σ_1^{*} < (δ_1/δ_0) (m − γ m^{1−α})/(1 − γ m^{1−α}) and strictly increasing beyond that threshold; minimum at σ_1^{*}_{min} = (δ_1/δ_0) (m − γ m^{1−α})/(1 − γ m^{1−α}).
  - For m>1: σ^{*D}_1 < σ^{*R}_1 = (m − γ m^{1−α})/(1 − γ m^{1−α}) < σ^{*}_{1min} ⇒ W_D > W_R.
  - For m<1: σ^{*R}_1 = (m − γ m^{1−α})/(1 − γ m^{1−α}) < σ^{*D}_1 < m; if m ≤ σ^{*}_{1min} (i.e., m ≥ ξ), then W_R > W_D.

- Sources of inefficiency:
  - Non-priced intertemporal externality: agents do not internalize how participation at t=0 affects future probability of trade and others’ reservation utilities.
  - Buyers set t=0 prices considering contemporaneous market tightness but not their effect on future tightness and reservation values.
  - Inefficiency and multiplicity arise in random-search decentralized markets and do not rely on additional non-priced search externalities.

- Policy implications and scenarios (section 3 recap):
  - Monetary policy:
    - If initial equilibrium is delayed and potential fire sale would be a run, accommodative monetary policy can raise probability of run fire sale by lowering opportunity cost of trading at t=0.
    - If initial equilibrium is a run, accommodative policy reinforces trading at t=0 and can prevent a delayed fire sale.
  - Asset purchases / price floor:
    - Purchasing asset at minimum price p^{min}_0 can implement the higher-price equilibrium when market fragile; credible commitment suffices—actual purchases may not be needed in equilibrium.
    - Quantitative Easing (QE) is a relevant example.
  - Market-structure reforms:
    - Standardization (increasing γ) and broadening buyer base (increasing m) can eliminate fire sale if large enough.
    - If impact on liquidity is modest and market liquidity initially very low, increasing liquidity can make the market fragile and subject to fire sale (non-monotone effects).
    - Higher transparency and centralized clearing have ambiguous effects via their influence on γ and m.

- Conclusion — key takeaways:
  - Decentralized asset markets can be fragile and prone to fire sales when future liquidity shocks exceed current ones (δ_1 > δ_0), market liquidity and imbalance are in intermediate ranges, and participation is endogenous.
  - Fire sales may be inefficient; a social planner can implement Pareto-superior allocations via transfers or credible price-support policies.
  - Extensions (balance-sheet effects, heterogeneous asset liquidity) may deepen discounts, exacerbate welfare losses, or change fragility boundaries.

*Source: wpiea2019092 (IMF working paper sections 1, 1.1, 2.2, 2.3).*

### 1. Introduction ........................................................................................................

### wpiea2019092 - 1. Introduction

### Motivation and key empirical facts
- Anecdote: "In the aftermath of the nancial crisis, a $60 million slice of subprime mortgage bonds from 2007 traded hands for as little as two cents on the dollar. Now, they’re higher than 90 cents..." (Bloomberg - October 1, 2018).
- Three broad facts about asset markets during the crisis:
  - The initial shock to asset values has been surprisingly small: "The cumulative losses as of 2013 on all subprime AAA-rated MBS has been estimated to be less than half of a percentage point."
  - Collapse of asset prices was accompanied by lower trade volume, often akin to a market freeze.
  - Some agents with available resources (e.g., commercial banks) bought a significant portion of these securities between 2007-2009.
- Additional evidence and anecdotes:
  - WSJ (June 23, 2008) quote: buyers wary of buying because of "so much concern about further supply out there...that people are just wary of buying anything."
  - Mention of funds (e.g., Marathon Asset Management) formed to exploit potential future price drops; Marathon required that investors leave money "for longer than a year."

### Central research question and contribution
- Key question: Can coordination failure among buyers (strategic waiting) in decentralized, endogenous-participation markets generate fire-sale prices and lower trade volume even in the absence of asymmetric information?
- Main contribution:
  - Shows that when trade is decentralized and participation is endogenous, buyers’ strategic waiting can cause collapse in asset price and trade volume.
  - Identifies four critical elements for market fragility in the model:
    - Decentralized (yet competitive) asset market with endogenous entry timing.
    - Liquidity shocks at current and future dates which are increasing in magnitude over time.
    - A medium degree of imbalance between potential supply and demand.
    - A medium degree of market liquidity.
  - Equilibrium uniqueness holds in the absence of any one of these elements.

### Model setup (overview)
- Agents:
  - Two types: sellers and buyers.
  - Both consume over three periods; sellers have higher propensity to consume in the first two periods due to liquidity shocks.
  - Sellers hold one unit of an indivisible asset initially; buyers can buy at most one unit.
- Participation is endogenous and costless: agents can trade at initial date or postpone to second period.
- Buyers post prices in submarkets; sellers choose submarket and corresponding price.
- Market capacity: "The number of buyers and sellers should be seen as proxies for the potential demand and supply in the market."
- Limited buyer capacity interpreted as tightening borrowing constraints during financial distress.

### Equilibria, fragility, and the waiting-complementarity mechanism
- Two equilibrium types:
  - Delayed equilibrium: some buyers and sellers are indifferent and wait until the second period (participation postponed).
  - Run equilibrium: all buyers and sellers try to trade in the first period.
- Market fragility defined as coexistence of run and delayed equilibria.
- Fire-sale equilibrium: the equilibrium with the lower asset price (can be either delayed or run type).
  - If buyers > sellers, fire sale manifests as a delayed equilibrium (buyers wait).
  - If buyers < sellers, fire sale manifests as a run equilibrium (sellers rush to sell).
- Mechanism of strategic complementarity among buyers:
  - A buyer’s decision to wait reduces current and future market tightness (ratio of buyers to sellers), increasing other buyers’ reservation utility to wait.
  - Numerical illustration from text:
    - Initially 100 buyers and 40 sellers; if trade volume is 20, sellers-to-buyers ratio in future = (40−20)/(100−20) = 1/4.
    - If 10 buyers withdraw and trade volume drops to 10, future sellers-to-buyers ratio = (40−10)/(100−10) = 1/3.
  - Complementarity can generate multiple equilibria: waiting by some raises incentives for others to wait, producing delayed trade and low prices.

### Welfare analysis and efficiency
- Fire-sale equilibrium can be dominated (inefficient) relative to the high-price equilibrium:
  - When buyers outnumber sellers and multiple equilibria exist, the run (high-price) equilibrium dominates the delayed (fire-sale) equilibrium as long as sellers-to-buyers ratio is not too low.
  - A social planner can implement lump-sum transfers to achieve a Pareto-superior allocation relative to the fire-sale equilibrium.
  - When buyers are fewer than sellers, the high-price equilibrium always dominates the fire-sale equilibrium in welfare terms.
- Comparison to other explanations:
  - Unlike adverse selection narratives, the model can generate fire sales despite small actual losses on some assets (e.g., small cumulative losses on AAA-rated RMBS).
  - Distinct from models where fire sales are efficient (e.g., Davila and Korinek (2018)): here inefficiency arises from agents not internalizing their participation effects on future probability of trade.

### Implications for price and trade volume
- When multiple equilibria exist:
  - Fire sales coincide with lower trade volume when there are fewer sellers than buyers (delayed equilibrium).
  - Fire sales coincide with higher trade volume when there are more sellers than buyers (run equilibrium).
  - Thus empirical prediction: direction of trade-volume change during fire sales depends on relative potential demand versus supply.
- Price impact of fire sale:
  - Defined as the percentage decline in asset price relative to high-price equilibrium.
  - When fire sale is a delayed equilibrium, the price impact is larger when markets are initially more liquid (i.e., higher initial asset price leads to a bigger fall in fire sale).

### Policy implications
- Policies lowering opportunity cost of participation at initial date (e.g., accommodative monetary policy) can make delayed fire-sale less likely.
- Asset purchases by the government or credible commitment to support prices can eliminate fire-sale equilibria by keeping asset prices above threshold.
- Policies that change market structure or trading platform (e.g., standardization of securities, introduction of Central Counterparty Clearing House (CCP), higher transparency) have ambiguous effects on fragility:
  - Net effect depends on their quantitative impact on market liquidity and demand.

*Source: wpiea2019092 (https://www.imf.org/-/media/files/publications/wp/2019/wpiea2019092.pdf).*

### 1.1  Related Literature

### 1.1  Related Literature

### Classical and foundational fire-sale explanations
- Shleifer and Vishny (1992): fire sales happen because agents who are experts in using the asset are liquidity constrained.
- Shleifer and Vishny (1997): emphasizes limited arbitrage capital by specialized investors who understand the asset.
- Limitations noted:
  - Liquidity-constrained expert explanation is less applicable to financial assets; not all specialized investors (e.g., banks) were liquidity constrained during the crisis (He, Khang, and Krishnamurthy (2010)).
  - Limited arbitrage capital explanation is questioned because non-specialized investors with abundant resources existed and it is unclear why they did not step in.

### Information frictions and adverse selection narratives
- Dow and Han (2018): model shows asymmetric information problems may increase during a crisis, reducing demand by less liquidity-constrained investors and depressing prices.
  - Mechanism: when specialized investors become liquidity constrained, market price becomes less informative about fundamentals, exacerbating adverse selection by decreasing supply of high-quality assets and lowering valuations by nonspecialized investors.
- Guerrieri and Shimer (2014): model of fire sale in decentralized markets with competitive search and private information about asset quality.
  - Equilibrium: unique; sellers signal quality by waiting longer; fire sale occurs when distribution of asset quality worsens; fire sales accompanied by illiquidity.
- Chang (2018): two-dimensional private information (asset quality and degree of distress).
  - Implication: fire sale happens only when trading volume is high and distressed sellers accept steep discounts in a semi-pooling equilibrium.
  - Distinction: adverse selection plays no role in Chang (2018); model yields equilibria with both liquid/high-volume and illiquid/low-volume markets depending on ratio of potential sellers to buyers.

### Alternative mechanisms: market freezes and debt overhang
- Diamond and Rajan (2011): market freezes due to a debt overhang problem.
  - Financial intermediaries may refuse asset sales that could raise survival chances because much of survival benefits do not accrue to shareholders.

### Multiplicity, search externalities, and specialness/liquidity
- Vayanos and Weill (2008): decentralized asset markets with multiple equilibria; newly issued treasury bonds sold at premium because short sellers borrowing newly issued bonds make them more liquid and special.
  - Self-fulfilling equilibria arise from search externalities: each agent’s decision to borrow enhances liquidity/specialness, encouraging others to borrow.
- Guerrieri (2010): competitive search in labor market with private information and limited commitment; equilibrium inefficient outside steady state due to firms not internalizing externality on workers’ outside option.
  - Analogous externality in current model: intertemporal externality from past actions on future probability of trade.

### Empirical findings on demand and trading volumes during the crisis
- Hanson and Sunderam (2013): drivers of investors’ demand for mortgage-backed securities before and during subprime crisis.
  - Findings: trading for nontraditional securitizations declined significantly through the boom 2003-2007 and was extremely low during 2007-2009 bust — consistent with buyer strike narrative (market freezes due to adverse selection or other frictions).
  - Also find investors sold more liquid securities (e.g., government-guaranteed MBS) during the crisis.
- Boudoukh et al. (2016): puzzling behavior of newly issued sovereign bond spreads during financial distress.
  - Finding: newly issued sovereign bonds (more liquid and more expensive in normal times) become cheaper in crises, especially for low-quality sovereigns.
  - In this paper: price impact of fire sale and drop in trade volume are larger for more liquid assets when fire sale happens in a delayed equilibrium.
  - Consistency note: findings align if trade volume in newly issued sovereigns declines more significantly than in old bonds.

### Related literature on runs, currency attacks, and stock-market runs
- Currency-market multiplicity: models of multiple equilibria in currency markets (Obstfeld (1996) review).
  - Contrast: currency-attack models feature centralized markets and a strategic agent (government) that can generate bad equilibria with large devaluations.
- Bernardo and Welch (2004): fire sale and run in a stock market.
  - Mechanism: risk-neutral investors fear forced liquidation after a run and may sell today, potentially causing the run; future liquidity shocks are key in causing a run today.

### Connection to the model presented (overview of model features and equilibria)
- Model setup (brief):
  - Three periods t=0,1,2; two agent types: measure 1 of buyers and m>0 of sellers; indivisible asset pays dividend d_2>0 at t=2.
  - Preferences:
    - U_S = E_0[δ_0 C_0 + δ_1 C_1 + C_2]
    - U_B = E_0[C_0 + C_1 + C_2]
  - Assumption 1: δ_0>1 and δ_1>1 (sellers face liquidity shocks / higher marginal utility of consumption in t=0,1).
  - Buyers may buy at most one unit in either t=0,1; buyers have big enough endowments in t=0,1.
- Market microstructure:
  - Decentralized competitive search with random matching; buyers post prices and form submarkets; sellers choose submarket/price.
  - Matching function (Assumption/Specification):
    - M(s,b)=γ s^{1−α} b^{α} with 0<α<1 and 0<γ<1.
  - Assumption 2 (parameter restriction):
    - γ^{1/α} < (m−γ m^{1−α})/(1−γ m^{1−α}) < γ^{−1/(1−α)}.
- Key equilibrium results and conditions:
  - At t=1: with q_B1(σ_1)=γ σ_1^{1−α} and q_S1(σ_1)=γ σ_1^{−α}, the unique equilibrium price:
    - p^*_1 = [1 + (1−α)(δ_1−1)]/δ_1 d_2
    - with d_2/δ_1 < p^*_1 < d_2, implying full participation at t=1 (V_S1>R_S1 and V_B1>R_B1).
  - Continuation utilities at t=1:
    - ̄U_S1 = V_S1 = (1 + (1−α)(δ_1−1) γ σ_1^{*−α}) d_2
    - ̄U_B1 = V_B1 = [α(δ_1−1)/δ_1] γ σ_1^{*1−α} d_2
  - At t=0: participation constraint and buyer maximization yield interior FOC for full participation:
    - σ_0^{*−α} = [̄U_S0 − ̄U_S1/δ_0] / [γ(1−α)(d_2 − 1/δ_0 ̄U_S1 − ̄U_B0)]
    - Full participation requires d_2 − 1/δ_0 ̄U_S1 − ̄U_B0 >0 and ̄U_S0 − ̄U_S1 >0.
    - Price at t=0 under full participation:
      - p^*_0 = (1−α)(d_2 − ̄U_B1) + α (1/δ_0) ̄U_S1
  - Multiple equilibria (fragility) and delayed equilibrium:
    - Indifference conditions (agents indifferent between participating or staying out at t=0) occur when:
      - ̄U_S0 − ̄U_S1 = 0
      - d_2 − 1/δ_0 ̄U_S1 − ̄U_B0 = 0
    - Define function f(σ):
      - f(σ) ≡ 1/δ_0 + (1−α)(δ_1−1)/δ_0 γ σ^{−α} + α(δ_1−1)/δ_1 γ σ^{1−α}
    - Equation determining market tightness inverse at t=1 in delayed/run indifference equilibrium:
      - 1/δ_0 + (1−α)(δ_1−1)/δ_0 γ σ_1^{*−α} + α(δ_1−1)/δ_1 γ σ_1^{*1−α} = 1
    - Price in the delayed equilibrium at t=0 (p^{**}_0):
      - p^{**}_0 = (1/δ_0) ̄U_S1 = [1/δ_0 + (1−α)(δ_1−1)/δ_0 γ σ_1^{*−α}] d_2
- Definitions:
  - Fragile market: both full participation and limited participation equilibria exist.
  - Run and delayed equilibrium: names for full and limited participation equilibria, respectively (superscripts R and D).
  - Fire sale price: the strictly lower price at t=0 between the run and delayed equilibria (p^D_0 or p^R_0).

*Italic source attribution: wpiea2019092 - 1.1  Related Literature (wpiea2019092 - 1.1  Related Literature)*

### 2.2  Fragility and re sale

### 2.2  Fragility and fire sale

### Characterization of equilibrium price and comparative statics
- Equilibrium price at t=0 for the run equilibrium:
  - p∗0 =(1−α)(d2 − ̄UB1)+α 1 δ0 ̄US1 ⇒ p∗0(σ∗1)= n(1−α) [ 1− α(δ1−1) δ1 γ σ∗1−α1 ] + α δ0 [ 1 +(1−α)(δ1−1)γ σ∗−α1 ] o d2 (19)
- For the delayed equilibrium the equation collapses to 18 as we have d2 − ̄UB1 = 1 δ0 ̄US1.
- Derivative of the equilibrium price at t=0 with respect to the (inverse of) market tightness:
  - dp∗0(σ∗1) dσ∗1 = [ − α δ0 σ∗−(1+α)1 − 1−α δ1 σ∗−α1 ] d2 <0 (20)
  - Interpretation: p∗0 is decreasing in σ∗1 because higher σ∗1 makes finding buyers next period less likely, reducing sellers’ continuation value and leading sellers to accept lower prices at t=0.
- Relation between σ∗1 and volume of trade at t=0:
  - σ∗1 = m−ν0 1−ν0 (21)
  - dσ∗1 dν >0 iff m>1.
- Lemma 1 (preserved statement):
  - Whenever we have market fragility, the following hold for the delayed and run equilibria:
    - p∗D0 < p∗R0 ⇐⇒ m<1
  - Interpretation: fire sale occurs either in a run equilibrium where m>1 or in a delayed equilibrium where m<1. Moreover, ν0 (volume of trade at t=0) is higher when the price is higher iff m<1.

### Fire sale in delayed equilibrium (m<1 case)
- When m<1 and markets are fragile, fire sale in a delayed equilibrium coincides with a collapse in the volume of trade.
- Using 8,16 and 19 we have (selected equalities from the source):
  - p1 = 1+(1−α)(δ1−1) δ1 d2
  - pD0 = n(1−α)[ 1− α(δ1−1) δ1 γ σ∗1−α1 ] + α δ0 [ 1 +(1−α)(δ1−1)γ σ∗−α1 ] o d2 1 δ0 + (1−α)(δ1−1) δ0 γ σ∗−α1 + α(δ1−1) δ1 γ σ∗1−α1 = 1
  - From elimination:
    - pD0 = n[ 1− α(δ1−1) δ1 γ σ∗1−α1 ] o d2
    - pD0 − p1 = α(δ1−1) δ1 n[ 1−γ σ∗1−α1 ] o d2
- Since γ σ∗1−α1 <1, we have pD0 > p1. Therefore:
  - Lemma 2 (preserved statement):
    - When m<1 and markets are fragile, fire sale happens in the delayed equilibrium. Moreover we have:
      - p1 < pD0 < pR0
- Figure 2 interpretation (textual): Two equilibrium prices pD0 < pR0 at t=0 due to coordination failure among buyers when m<1; trade volume declines simultaneously with asset price, νD0 < νR0. This cannot happen in centralized markets.

### Conditions for fragility; role of decentralization and complementarity
- Centralized market benchmark (Lemma 3, preserved statement summary):
  - In a centralized competitive asset market (except knife-edge m=1), equilibrium prices at which trade takes place are unique. Cases depend on comparisons of δ0 and δ1 and on m>1 or m<1; central feature: fragility is not a feature of centralized trading because no externalities/coordination failure.
- Multiplicity in decentralized market (Lemma 4, preserved structure):
  - Two types of equilibria with strictly positive volume at t=0:
    1. Run (full participation) equilibrium characterized by (equations 8,9,12,13 and additional conditions):
       - VS0 > RS0, VB0 > RB0, μ∗S0 = m, μ∗B0 = 1,
       - μ∗S1 = m−γm 1−α, μ∗B1 = 1−γm 1−α
       - σ∗1 = m−γm 1−α 1−γm 1−α , σ∗0 = m
    2. Delayed (limited participation) equilibrium where sellers and buyers are indifferent about participating at t=0, with equilibrium pinned down using 9,8,15,16,18 and additional conditions:
       - γ μ∗S0 1−α μ∗B0 α = m−σ∗1 1−σ∗1 , σ∗0 = μ∗S0 μ∗B0 μ∗S1 = m−γ μ∗S0 1−α μ∗B0 α , μ∗B1 = 1−γ μ∗S0 1−α μ∗B0 α
- Source of multiplicity: intratemporal externalities are fully priced, but intertemporal externalities (an agent’s participation at t=0 affecting others’ probability of trade at t=1 and thus reservation utilities at t=0) are not fully captured by price at t=0. Complementarity among agents’ actions is necessary for multiplicity:
  - When m<1, a buyer’s decision to wait at t=0 reduces market tightness at t=1 (increases σ∗1), increasing reservation utility of other buyers and encouraging them to wait — complementarity.
  - When m>1, symmetric statements apply with roles of buyers and sellers swapped.
- Assumption 3 (preserved):
  - We have f(γ −1 1−α )>1, f(γ 1 α )>1.
- Proposition 1 (preserved statement):
  - Given 3, necessary and sufficient conditions in terms of liquidity shocks δ0 and δ1 that ensure existence of fragility in the asset market at least for some values of m is:
    - f( δ1 δ0 )<1 ⇐⇒ γ[ δ1 δ0 ] −α < δ0 −1 δ1 −1 ,
  - The above implies:
    - 1 < δ1 δ0 < γ −1 1−α ,
  - Moreover there is no fragility for m<1, iff f(1)>1.
  - Interpretation: fragility requires δ1 > δ0 (future liquidity shock bigger than current), so waiting can be valuable; non-extreme values of δ1/δ0 are needed.
- Proposition 2 (preserved statement summary):
  - Assume f(1)<1. Given {γ,δ0,δ1}, there exist quadruple {m, m, m, m} such that:
    - γ 1 α < m < m < 1,
    - 1 < m < m < γ −1 1−α
    - Markets are fragile iff m satisfies one of:
      - m<1, m < m < m,
      - m>1, m < m < m
  - Interpretation: there must be a minimum imbalance between supply and demand (m sufficiently far from 1) for fragility; when m→1, participation decisions at t=0 have negligible impact on σ∗1.
- Liquidity (γ) conditions for multiplicity (Lemma 5, preserved inequalities):
  - For given {α,δ0,δ1,m}, necessary conditions for multiple equilibria:
    - m α (δ0 −1)δ1 ((1−α)δ1 + α m δ0)(δ1 −1) ≤ γ ≤ (δ0 −1)δ1 ((1−α)δ1 + α δ0)(δ1 −1)

### Price impact of fire sale and comparative statics
- Definition 4 (preserved):
  - When markets are fragile, the price impact of fire sale is the percentage decline in asset price when agents switch from the high asset price to the fire sale equilibrium.
  - More precisely, ∆p0 ≡ |p∗D0 − p∗R0| p∗D0 when fire sale happens in a run, and ∆p0 ≡ |p∗D0 − p∗R0| p∗R0 when it happens in a delayed equilibrium.
- Proposition 3 (preserved statements and implications):
  - Assume markets are fragile for parameters {m,γ,δ0,δ1,α}. Then:
    - d∆p0 dm >0 if fire sale is a run (if m>1)
    - d∆p0 dm <0 if fire sale is a delayed equilibrium (if m<1)
  - Implication:
    - d∆p0 d|m−1| >0
    - Effect of γ:
      - When fire sale happens in a delayed equilibrium (m<1), d∆p0 dγ >0.
      - When fire sale happens in a run equilibrium, if m> δ1 δ0 , a sufficient condition to have d∆p0 dγ >0 is:
        - ( (m−γm 1−α)(1−γm 1−α) γm 1−α (m−1) − α )( 1 δ0 − m−γm 1−α (1−γm 1−α)δ1 ) <1
  - Interpretation: price impact ∆p0 is unambiguously higher when imbalance |m−1| is larger. Changes in ∆p0 are driven by what happens to p∗R0 (p∗D0 does not depend on m by 17).
- Corollary 1 (preserved statement):
  - Let ν∗D0 and ν∗R0 denote volume of trade at t=0 for delayed and run equilibria respectively. When m<1 and market is fragile:
    - ν∗D0 is decreasing and ν∗R0 is increasing in γ.
    - The impact on trade volume (ν∗R0 − ν∗D0)/ν∗R0 is increasing in γ.
  - Interpretation: when sellers are the short side (m<1), more liquid markets experience higher prices and trade volume in normal times and lower prices and trade volume during fire sales.

### Funding liquidity, price and volume (buyers borrow; pledgeability θ0)
- Extension: buyers have no own resources and must borrow entire price; lending subject to pledge constraint:
  - p0 ≤ θ0 d2
  - If borrower defaults at t=1, lender seizes asset; fraction 1−θ0 of dividend lost to bankruptcy costs; only θ0 d2 can be pledged at t=0.
- Lemma 6 (preserved statement):
  - Suppose m<1 and model parameters satisfy:
    - 1 +(1−α)(δ1 −1)γm −α δ0 < 1 +(1−α)(δ1 −1) δ1
  - Then there exists a range of θ0 for which there exists an equilibrium such that pD0 < p1, all buyers participate at t=0 while some sellers participate and some wait to trade at t=1.
  - Moreover within that range of θ0:
    - ∂pD0 ∂θ0 >0,
    - ∂ν0 ∂θ0 >0
    - where ν0 is the volume of trade at t=0.
  - Interpretation: limited funding liquidity (low θ0) can exacerbate fire-sale price declines and reduce trade volume; θ0 may itself depend on market conditions (e.g., θ0 = θ0(p0)), potentially reinforcing multiplicity if agents do not internalize their impact on θ0.

_Italic: Source: wpiea2019092 - 2.2 Fragility and fire sale (IMF PDF chapter/section)._

### 2.3  Welfare Analysis

### 2.3 Welfare Analysis

### Welfare measure and ranking of equilibria
- Planner objective (time zero consumption equivalent of utilities of all agents): W ≡ ŪB0 + m δ0 ŪS0 (equation 25).
- Lemma 7: If Wi and Wj are total welfare for two different equilibria i and j where Wj > Wi, the planner can design transfers in j to achieve an allocation j′ that is Pareto superior to i.
- The total welfare in a run equilibrium is derived from the FOC and objective (equations 12 and 5). Key expressions (from equation 26):
  - ŪB0 and 1 δ0 ŪS0 written as functions of γ, σ∗0, α, d2, 1 δ0 ŪS1 − ŪB1, and ŪB1, ŪS1.
- Simplified welfare in a run equilibrium (equation 27):
  - W = [ ŪB1 + ŪS1/δ0 m ] + γ m1−α [ d2 − 1/δ0 ŪS1 − ŪB1 ]
  - Interpretation:
    - First term: sum of (time zero consumption equivalent) reservation utilities of sellers and buyers.
    - Second term: product of total number of transactions at t=0, γ m1−α, and the extra trade surplus per match, d2 − 1/δ0 ŪS1 − ŪB1.
- In a delayed equilibrium the extra surplus d2 − 1/δ0 ŪS1 − ŪB1 equals zero; total welfare equals the sum of reservation utilities.

### Main welfare ranking results
- Proposition 4 (market fragile parameters {γ, δ0, δ1}):
  - When m>1:
    - m>1 ⇒ W_D(m, γ, δ0, δ1) > W_R(m, γ, δ0, δ1)
  - There exists 0<ξ<1 such that for m<1:
    - ξ ≤ m < 1 ⇒ W_D(m, γ, δ0, δ1) < W_R(m, γ, δ0, δ1)
  - W_R and W_D denote total welfare for the run and delayed equilibria respectively.
- Corollary 2:
  - When market is fragile, the equilibrium with fire sale has the lower total welfare for all m>ξ where ξ<1 is defined above.
- Comparative statics:
  - Welfare W as a function of σ∗1 (inverse market tightness at t=1) is:
    - W = [ γ m1−α + (m − γ m1−α)/δ0 + (1−α)(m − γ m1−α)(δ1 − 1)/δ0 ] γ σ∗−α1 + α(δ1 − 1)(1 − γ m1−α)/δ1 γ σ∗1−α1 d2 (as rearranged using equations 9 and 27).
  - dW/dσ∗1 sign:
    - W is strictly decreasing for σ∗1 < (δ1/δ0) (m − γ m1−α)/(1 − γ m1−α) and strictly increasing for σ∗1 > that threshold; minimum at σ∗1min = (δ1/δ0) (m − γ m1−α)/(1 − γ m1−α).
  - For m>1: σ∗D1 < σ∗R1 = (m − γ m1−α)/(1 − γ m1−α) < σ∗1min ⇒ W_D > W_R.
  - For m<1: σ∗R1 = (m − γ m1−α)/(1 − γ m1−α) < σ∗D1 < m. If m ≤ σ∗1min (i.e., m ≥ ξ defined implicitly), then W_R > W_D.

### Sources of inefficiency and mechanism
- Inefficiency arises from non-priced externalities:
  - Agents do not internalize how participation decisions affect future probability of trade and others’ reservation utilities at t=0.
  - Buyers set price at time zero considering contemporaneous market tightness but fail to account for the effect on future market tightness and reservation values.
- Multiplicity, complementarity among participation decisions, and resulting inefficiency persist in an environment with random search and do not depend on inherent non-priced externalities in random search.

### Policy implications and scenarios (section 3)
- Conventional monetary policy:
  - If initial equilibrium is delayed and the resulting fire sale would be a run, an accommodative monetary policy (lowering the real interest rate between t=0 and t=1) reinforces incentives to trade at t=0 and may raise the probability of a run fire sale.
  - If initial equilibrium is a run, accommodative policy reinforces incentives to trade at t=0 and can prevent a fire sale in the form of a delayed equilibrium.
- Asset purchase (price floor) policy:
  - A policy to purchase the asset at a minimum price pmin0 can implement the equilibrium with the higher asset price when market is fragile.
  - This policy eliminates the possibility of fire sale as long as pL0 < pmin0 < pH0, where pH0 and pL0 are time zero prices in the two equilibria. A credible commitment is sufficient; no purchases need occur in equilibrium.
  - Asset purchases implemented as part of quantitative easing (QE) are relevant examples.
- Market-structure policies:
  - Standardization can increase liquidity via higher γ and broaden buyer base via higher m; if strong enough, this can eliminate the fire sale equilibrium.
  - If impact on liquidity is modest and market liquidity is initially very low, increasing liquidity may make the market fragile and subject to fire sale (see Proposition 5 referenced in the text).
  - Higher transparency in OTC markets and introduction of centralized clearing platforms may have ambiguous effects on fragility via liquidity changes.

### Conclusion (key takeaways)
- Decentralized asset markets can be fragile and prone to fire sales when liquidity shocks increase over time.
- Conditions for fragility include:
  - Liquidity shocks that are increasing in magnitude over time (δ1 > δ0 in the model).
  - A medium degree of liquidity (parameters γ, m interact nontrivially).
  - Imbalance between sellers and buyers in the market.
- Fire sales may be inefficient, producing suboptimally low aggregate trade surplus relative to the equilibrium with the higher asset price.
- Extensions (not modeled here) could include balance sheet effects of fire sales and portfolios with assets of different liquidity; such extensions may deepen discounts and exacerbate welfare losses or alter fragility boundaries.

*Source: wpiea2019092 - 2.3 Welfare Analysis*

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_Source: https://www.imf.org/-/media/files/publications/wp/2019/wpiea2019092.pdf_