Liquidity Crises and the Market-Maker of Last Resort
What this paper finds — and why it matters
Layer 1: Overview
This paper develops a theoretical model to explain why financial markets can suffer self-fulfilling liquidity crises and how a central bank acting as a “market-maker of last resort” (MMLR) can mitigate them. The motivation is policy-driven: during the 2008-09 crisis and the COVID-19 pandemic, the Fed, ECB, and other central banks purchased assets at above-market prices (e.g., Maiden Lane I/II/III and the TALF) to support markets, a function distinct from the traditional lender-of-last-resort (LLR) role. The authors note that formal theoretical analysis of MMLR remains sparse (citing Buiter et al. 2023) and aim to fill that gap.
Model setup: It is an overlapping-generations (OLG) model with two-period-lived agents and fully rational expectations. There are two assets: a risk-free storage technology with gross return 1-δ (0<δ<1, a negative net return capturing the cost of self-insurance) and a non-depreciating Lucas tree in unit measure paying a constant dividend r (0<r<1). Young agents receive a unit endowment and save (natural buyers); old agents sell their tree to finance consumption (natural sellers). The tree price p_t is set by decentralized Nash bargaining with β denoting the seller’s (old agent’s) bargaining power. Old agents face an i.i.d. idiosyncratic liquidity shock γ∈{0,1} with probability q; if hit (γ=1) they must pay one unit of the good or suffer a utility penalty ω times the shortfall, with ω>1 (focus on large ω). A key parameter restriction is 0<r<δ<1, which rules out a trivial case where liquidity crises could never occur.
Main results: Because trading is by bilateral bargaining (not Walrasian), the model has multiple Pareto-rankable stationary rational-expectations equilibria, each sustained by self-fulfilling beliefs about future prices; lower-price equilibria are Pareto-inferior, more pessimistic, and entail lower consumption. Three benchmark equilibria are derived: (1) an efficient stationary equilibrium with p_t=1 (zero storage), which exists for large ω if seller bargaining power β exceeds a threshold β̃=(1-δ)(1-r)/[δ+(1-δ)(1-r)]; (2) an inefficient stationary equilibrium at p_t=p*=1-r/δ, which exists for any β∈(0,1) and large ω; and (3) a nonstationary equilibrium where prices asymptotically approach p* via p_{t+i}=p*-(1-δ)^i(p*-p_t), requiring β below a threshold β*. The authors introduce a nonfundamental “sunspot” shock that occurs each period with small probability π, inducing pessimistic beliefs that lower the price below the continuation path (to C(p_{t-1})) and leave old agents illiquid (W<1) — a liquidity crisis with flight-to-quality (increased costly storage), run-like behavior, and fire-sale-like price collapse. Crucially, along non-crisis recovery paths all later generations remain liquid, and the increased output loss from storage is exactly offset by greater price appreciation (the wealth difference across adjacent non-crisis periods nets to zero).
Policy: An “aggressive” MMLR — government issuing bonds to young agents and buying trees via Nash bargaining with a positively sloped excess-utility function — can support the unique first-best (p=1) allocation, but the authors argue this is likely politically infeasible (looks like a Wall Street bailout) and fragile (requires persistent intervention if β<β̃). A “conservative” MMLR embedding a “no-bailout” constraint (buy low / sell high) can support p=p*, eliminating utility-cost (crisis) inefficiency but leaving storage-cost inefficiency. Finally, replacing bilateral bargaining with a centralized Walrasian auction yields a unique, efficient equilibrium (p_t=1) with no storage and no liquidity crises, motivating regulatory pushes toward centralized/transparent trading (e.g., Dodd-Frank swap execution facilities, Treasury central clearing proposals). The model abstracts from moral hazard and from distinguishing fundamental vs. nonfundamental price declines.
Layer 2: Deep Dive
What is the core mechanism that generates multiple equilibria and liquidity crises?
The combination of (a) decentralized Nash bargaining as the trading mechanism and (b) the concavity of the indirect utility function when ω>1. With ω>1, the liquidity penalty makes storage relatively more valuable to a poorer young agent, so an equal fall in the tree price today and tomorrow reduces young agents’ wealth and shifts demand from the tree toward storage. This makes pessimistic beliefs self-fulfilling: a fall in p_t justified by expected low p_{t+1} is itself an equilibrium. With ω=1 (no liquidity penalty) Proposition 1 shows there is a single stationary equilibrium and no nonstationary equilibria.
How exactly is a liquidity crisis defined in the model?
An old agent is ’liquid’ if end-of-trading wealth W(p_t,p_{t-1})≥1, which is enough to fund a unit liquidity shock. A liquidity crisis is a state where W<1, so an old agent hit by γ=1 cannot fund the shock and incurs the utility penalty. The crisis is triggered by a nonfundamental sunspot that makes the young pessimistic, pushing the price to a crisis-deviation value C(p_{t-1}) satisfying p_underbar < C(p_{t-1}) < κ^o(p_{t-1}), which renders the date-of-crisis old agents illiquid.
What are the three benchmark equilibria and their existence conditions?
(1) Efficient stationary p_t=1 ∀t: exists for large ω if β>β̃=(1-δ)(1-r)/[δ+(1-δ)(1-r)]; under the tighter condition β>1-δ it exists for all ω>1; not an equilibrium if β<β̃ for large ω. (2) Inefficient stationary p_t=p*=1-r/δ: exists for any β∈(0,1) and large ω; here κ^o(p*)=κ^y(p*)=p* so all agents are liquid. (3) Nonstationary equilibrium p_{t+i}=p*-(1-δ)^i(p*-p_t) approaching p*: requires β<β*=(1-δ)p*/[δ+(1-δ)p*] and appropriate starting prices; along this path W=1 for all i≥1 so all agents are liquid.
Why does the recovery after a crisis leave subsequent generations liquid even though prices recover only gradually?
Although a crisis raises costly storage (flight to quality) and prices recover only asymptotically, the authors decompose wealth in adjacent non-crisis periods and show the reduction in output from increased storage is exactly offset by a greater rate of price appreciation: W_{t’+i}-W_{t’+i-1}=(p_{t’+i-2}-p_{t’+i-1})(1-δ) + (p_{t’+i-1}-p_{t’+i-2})(1-δ) = 0. So later generations remain liquid (W=1) until the next crisis hits.
What distinguishes the ‘aggressive’ from the ‘conservative’ MMLR policy?
Aggressive MMLR (Proposition 6): government traders act with an excess-utility function having strictly positive slope in p_t (prefer buying at higher prices), which can enforce p=1 and support the first-best. The authors deem it politically infeasible (appears to subsidize/bailout Wall Street) and fragile (if β<β̃, sustaining p=1 requires persistent intervention). Conservative MMLR (Proposition 7): government adopts a ’no-bailout’ excess-utility function strictly decreasing in p_t and increasing in expected future price (buy low, sell high), supporting p=p* and ruling out p=1 as an equilibrium. It eliminates utility-cost (crisis) inefficiency but not storage-cost inefficiency, and p* remains a natural equilibrium even if political support wavers (absent a current crisis).
What role does the Walrasian alternative play?
Proposition 8 shows that if trading occurs via a centralized Walrasian auction rather than bilateral bargaining, there is a unique equilibrium with p_t=1 ∀t, no storage, and no liquidity crises. The multiplicity arises in the bargaining model precisely because there is no market to sell storage and buy more trees, permitting interior solutions p_t∈(0,1). This yields the normative implication that regulators should favor centralized, transparent trading venues (cited examples: national bid/offer dissemination for stocks, Dodd-Frank swap execution facilities, proposals for Treasury central clearing).
How is bargaining power β interpreted, and what is its normative significance?
β∈[0,1] is the old agent’s (seller’s) bargaining power, taken as a primitive standing in for unmodeled market characteristics (e.g., the seller of an MBS may have superior information, or fire-sale conditions may disadvantage sellers). Bargaining power inheres in the role (seller vs. buyer), not the individual; the same agent has power β when old/selling and 1-β when young/buying. High β supports the efficient p=1 equilibrium; low β makes the economy prone to crises. The authors note the Hosios-type efficiency condition on β from labor-search models is not relevant here.
How does the paper relate to and differ from the closest prior work, Choi and Yorulmazer (2023, ‘CY’)?
Both study multiple equilibria in financial markets and the MMLR’s role in removing multiplicity. Differences: CY’s model is fundamentally static, whereas this is a dynamic stochastic equilibrium model used to generate periodic crises from exogenous bouts of pessimism. Price determination differs: CY uses the cash-in-the-market paradigm (Allen and Gale 1994), whereas this paper uses decentralized Nash bargaining, in which the Walrasian equilibrium is unique and efficient but many Pareto-inferior bargaining equilibria coexist, letting the authors ask whether MMLR can eliminate some or all inferior equilibria. The paper also relates to Holmström-Tirole (self-insurance via low-yield assets is suboptimal; government has a role), but there the friction is a pledgeability/principal-agent problem, whereas here suboptimality comes from a small-probability inferior equilibrium.
Is the Nash bargaining assumption robust to an alternative bargaining solution?
The authors check Kalai (1977) proportional bargaining. Holding the Kalai weight ν constant, there exist two values of ν supporting the efficient and inefficient equilibria of Propositions 2 and 3 (with parameters r=0.2, δ=0.25, ω=200, q=0.1, the young’s proportional weight is 0.243 in the efficient equilibrium and 0.555 in the inefficient one). For the nonstationary equilibrium of Proposition 4, the ratio of old-to-young excess utility changes over time, so no single constant ν supports it; the Nash solution, by contrast, holds over a range of weights. Inefficient equilibria are supported under both Nash and Kalai.
What are the main caveats and scope conditions on the policy conclusions?
The model is highly stylized: two-period OLG rules out LLR analysis (old agents do not live long enough to repay loans). In practice policymakers must distinguish price declines due to equilibrium shifts from those due to changing fundamentals (the authors say both were likely active in 2007-08), and must determine the ‘correct’ equilibrium price, which is nontrivial. The model abstracts entirely from moral hazard in public backstopping (citing Farhi-Tirole 2012, Gradstein 2022). The aggressive policy supporting p=1 is fragile and politically vulnerable; the conservative no-bailout policy only removes crisis (utility-cost) inefficiency, leaving storage-cost (flight-to-quality) inefficiency intact.
What real-world MMLR interventions does the paper map its model to?
Maiden Lane LLC (March 2008, Bear Stearns mortgage assets to facilitate the J.P. Morgan merger), Maiden Lane II and III (October 2008, addressing AIG’s exposure to RMBS and CDOs), and the TALF (supporting certain asset-backed securities). It also cites Buiter et al. (2023) documenting extensive MMLR use by the Fed, ECB, Sveriges Riksbank, Bank of Japan, and Bank of Canada during COVID-19 (repo participation, corporate bond and commercial paper purchases, restarting TALF).