Financial Stability with Fire Sale Externalities

Layer 1: Overview

Research question and motivation: Asset fire sales were a defining feature of the 2007-08 crisis, and post-crisis reforms (Basel III liquidity requirements, Money Market Mutual Fund reforms) were introduced to mitigate fire sale externalities by reducing distressed debt obligations and forcing larger liquidity buffers. The paper asks whether policies that successfully mitigate fire sale externalities actually improve financial stability, since it is not obvious how banks re-optimize in response.

Model setup (no empirical data — this is a theoretical paper): The authors build a three-period (t = 0,1,2) Diamond-Dybvig (1983) model of financial intermediation augmented with (i) cash-in-the-market pricing in a financial market as in Allen and Gale (1998), and (ii) limited commitment as in Ennis and Keister (2009), following Li (2017). A unit continuum of ex ante identical depositors have CRRA preferences with relative risk aversion γ > 1. Each depositor is impatient with known probability π. There are two assets: a short-term storage asset (1 unit yields 1 next period) and a long-term asset (1 unit at t=0 yields R > 1 at t=2). The bank invests fraction x in the long-term asset and 1−x short. Long-term assets can be sold at t=1 at an endogenous price p to risk-neutral investors who receive endowment ws (market liquidity) and have outside return R* > 0. Runs are introduced via a sunspot s ∈ {α, β} with run probability q; runs are partial (stop after fraction π is served), following Ennis and Keister. The authors assume R* = R, which implies p ≤ 1 in equilibrium. Financial fragility is measured by q-bar, the maximum run probability q for which the run strategy is an equilibrium (run condition c1 ≥ c2β).

Main analytical findings: (1) Without intervention, banks over-invest in long-term assets relative to the socially efficient level because each competitive bank takes p as given and does not internalize that selling long-term assets in a run depresses p (the fire sale externality); the equilibrium price is inefficiently low. (2) The bank’s best response is in Case I (no excess liquidity, fire sale occurs) when 0 < q < q_l, and Case II (excess liquidity held) when q_l ≤ q < 1 (Lemma 1). There is a unique q_c at which the market-clearing price p* turns from decreasing to increasing in q (Lemma 3). (3) Comparative statics on market liquidity ws (Proposition 1): when the relevant q-bar lies in Case II (low ws), q-bar is strictly increasing in ws, so a small rise in market liquidity raises fragility; when q-bar lies in Case I (high ws), q-bar is strictly decreasing in ws. The mechanism (Lemmas 4-5) is that a higher p* raises c1 via intertemporal substitution; the c2α/c2β effect is always dominant, flipping the sign of dq-bar/dws between cases. (4) The intervention: a regulator controls (x, c1), internalizing the effect on p, while the bank still chooses (c2α, c1β, c2β) taking p as given. The regulator chooses lower x and higher c1 than the bank in Case I (Lemma 6: c1 ≤ c1R, x ≥ xR), raising the market-clearing price (Proposition 2: p* ≤ pR* in Case I). (5) Key result (Proposition 3): q-bar_R ≥ q-bar when both solutions are in Case I (intervention always raises fragility); ambiguous otherwise. When ws (or R) is high, intervention raises fragility (q-bar_R > q-bar); when ws or R is low, intervention involves excess liquidity and lowers fragility (q-bar_R < q-bar). Proposition 4 gives a sufficient condition for q-bar_R > q-bar via four thresholds ws1≤ws≤ws2 and ws3<ws<ws4. When ws is sufficiently high, p = pR = 1, the externality vanishes, and q-bar = q-bar_R. (6) Welfare (Proposition 5): WR(q-bar) ≤ W(q-bar) when both in Case I, and for some parameter values otherwise — intervention does not always improve welfare and can worsen it when market liquidity is large.

Policy implication: Mitigating fire sale externalities does not necessarily increase stability. Because the regulator takes q as given, it ignores that its own intervention can raise q-bar. Policymakers must internalize the fragility effect and balance externality mitigation against increased fragility, especially when market liquidity is high.

Layer 2: Deep Dive

Is there an identification strategy or empirical data? What are the threats?

No. This is a purely theoretical paper with no data, sample period, or estimation. The quantitative content consists of analytical comparative-statics results (Lemmas 1-6, Propositions 1-5) and numerical illustrations rendered as figures (Figures 4-9) for specific parameter combinations of (ws, R, q, γ, π). There is no econometric identification; the analog of robustness is the set of modeling assumptions and the parameter regions over which results hold.

What is the core economic mechanism, and how does intervention raise fragility?

The regulator internalizes the fire sale externality by reducing the bank’s long-term holdings x and holding more short-term assets, which reduces asset supply in a crisis and raises the market value p of each long-term asset (this mitigates the externality and is the intended benefit). But two competing effects act on long-term payments c2β: the higher price raises the value of remaining long-term assets, while there are fewer long-term assets left for c2β (whose period-2 return R is fixed, so the price increase does not help c2β as it does c1β). The net effect on c2β is ambiguous. Simultaneously, reducing x lowers the relative cost of t=1 consumption, optimally pushing the regulator to raise short-term payment c1. Since the run condition is c1 ≥ c2β, raising c1 while c2β may fall makes early withdrawal more attractive, raising q-bar. When market liquidity is high, the net effect always increases fragility.

What is the role of ’excess liquidity’ and how does it reverse the result at low market liquidity?

Excess liquidity (Case II: πc1 < 1−x, holding more short-term assets than needed for the first π payments) is the bank’s/regulator’s hedge against runs. When ws is low, the anticipated fire sale price is low, so the regulator chooses to hold more excess liquidity than the bank. Excess liquidity supplies additional resources to pay c1β and further reduces asset supply (raising p), leaving more resources for c2β. This makes the net effect on c2β favorable enough that q-bar falls. Thus at low market liquidity the regulator can simultaneously mitigate the externality and reduce fragility; at high market liquidity, excess liquidity is small or zero and the fragility-increasing channel dominates.

What heterogeneity / regime dependence is documented?

Results depend critically on the regime (Case I = no excess liquidity / fire sale; Case II = excess liquidity; Case III = excess liquidity, no fire sale, which never arises in equilibrium). The sign of dq-bar/dws flips between Case I (decreasing) and Case II (increasing). The intervention’s effect on fragility flips with market liquidity ws and long-term return R: low ws or low R → intervention reduces fragility; high ws or high R → intervention raises fragility; very high ws → externality vanishes (p = pR = 1) and intervention is neutral (q-bar = q-bar_R). The switch from Case I to Case II is governed by thresholds q_l (bank) and q_l,R (regulator), with q_l,R < q_l because the regulator internalizes the price and is more inclined to hold excess liquidity.

What robustness / generality checks are discussed?

Several modeling-assumption relaxations are argued not to change results qualitatively: (i) the assumption R* = R (giving p ≤ 1) can be generalized to allow p > 1, which does not undermine findings in the p < 1 range; (ii) partial runs can be generalized to multiple waves via a richer sunspot space without changing mechanisms; (iii) depositors not observing the bank’s portfolio can be replaced by observing it only after the withdrawal decision, with identical results; (iv) the simultaneous-move game is shown equivalent to a dynamic game in which the regulator moves first, as long as depositors cannot observe regulator choices; (v) the assumption that interventions convey no information to depositors can be relaxed (justified by the complexity of post-crisis regulation, e.g., the 848-page Dodd-Frank Act) without undermining the structure.

How does this paper relate to and differ from prior work?

It builds on the fire sale externality literature (Lorenzoni 2008; Gale and Gottardi 2015; He and Kondor 2016; Davila and Korinek 2018 on over/under-investment; Acharya et al. 2011 and Gale and Yorulmazer 2020 on distorted portfolios; Perotti and Suarez 2011, Walther 2016, Kara and Ozsoy 2019 on optimal capital/liquidity regulation). It also builds on the bank-run literature (Bryant 1980; Diamond-Dybvig 1983) and on general-equilibrium / endogenous-portfolio extensions (Allen-Gale 2004; Farhi et al. 2009; Eisenbach-Phelan 2021; Cooper-Ross 1998; Ennis-Keister 2006; Li 2017). The stated novel contribution is being the first to show that policies designed to correct fire sale externalities can worsen financial fragility, achieved by jointly endogenizing the portfolio choice, the general-equilibrium asset price, and the equilibrium probability of a run.

What are the policy implications and their scope conditions?

Macroprudential interventions that regulate short-term liabilities and portfolio choice to curb fire sale externalities can increase the equilibrium probability of runs. The scope condition is market liquidity: the harmful trade-off (mitigate externality but raise fragility, and sometimes lower welfare) arises specifically when market liquidity ws is high (and/or R high); when ws is low, the regulator’s optimal excess-liquidity holding lets intervention both mitigate the externality and reduce fragility. A central caveat is that the regulator takes q as given and so does not perceive that its policy raises q-bar; the prescriptive takeaway is that policymakers must internalize q-bar (the endogenous run probability) when designing such policies, balancing externality mitigation against fragility.

Are the quantitative results exact magnitudes or signs?

The paper’s results are predominantly signs and ordinal comparisons (e.g., x ≥ xR, p* ≤ pR*, q-bar_R ≥ q-bar, monotonicity in ws and p) plus closed-form threshold expressions (q_l, p_l, p_u, the four ws thresholds in Proposition 4) given in the text and appendices. Specific numeric magnitudes appear only as illustrative figure values (e.g., the example in Figure 9 where intervention raises fragility when ws is near 0.2); the paper does not report calibrated point estimates beyond such illustrative figures.

Key Concepts

Fire sale externality: In this model, the inefficiency arising because each competitive bank takes the t=1 asset price p as given and does not internalize that its long-term holdings and crisis-time asset sales depress p, harming other banks. It leads banks to over-invest in long-term assets and sell more than the efficient amount, pushing the equilibrium price below its efficient level.

Cash-in-the-market pricing: The price of long-term assets at t=1 is set by the limited cash (endowment ws) that risk-neutral investors bring to the market rather than by fundamental value; when banks must sell, scarce market liquidity forces the price down (p ≤ 1 under the R*=R assumption).

Financial fragility (q-bar): Measured as q-bar, the maximum run probability q for which the partial-run strategy profile is part of an equilibrium, i.e., the largest q satisfying the run condition c1 ≥ c2β. Higher q-bar means the banking system is more fragile.

Excess liquidity: Short-term asset holdings beyond what is needed to pay the first π withdrawals (πc1 < 1−x; Case II). It is a precautionary buffer that supplies resources for crisis payments c1β, reduces asset supply, and raises the fire sale price; the regulator holds more of it than the bank when market liquidity is low.

Case I vs Case II vs Case III: Regimes of the bank’s best response: Case I = no excess liquidity, fire sale occurs (small q, high ws); Case II = excess liquidity held with fire sale (large q, low ws); Case III = excess liquidity so large that no fire sale occurs — shown never to be an equilibrium because it implies c2β > c2α > c1 (no run condition).

Regulator/intervention: A planner that chooses (x, c1) internalizing the effect of these choices on the asset price p, while the bank still chooses (c2α, c1β, c2β) taking p as given and the regulator cannot direct depositors’ withdrawal decisions; it represents the two policy instruments of regulating short-term liabilities and portfolio choice.