When Disaster Wipes Out Income, Can Collateral Markets Still Reach Equilibrium?

A disaster can leave someone with no new income or resources. For economists studying markets under uncertainty, that possibility creates a mathematical question: can a market still reach an equilibrium when some participants have nothing coming in after uncertainty resolves?

In a theoretical preprint, Jaime Orrillo examines an economy in which people trade securities backed by collateral. The paper's result is an existence theorem: under its assumptions, prices and individual plans can be found that satisfy both people's choices and market clearing, even if future endowments—the resources agents receive in a given future state—are zero for some or all agents in any or all states. The result does not say that everyone will have positive consumption after a disaster.

A market with many possible futures

The model has two dates, a finite number of agents and securities, and one good at each date. At the first date, each agent must have a positive endowment of the good. At the second, uncertainty resolves into a state drawn from a continuum of possibilities, represented by points on an interval. Future endowments can vary across states and can be zero, although they must be nonnegative, measurable and bounded in the sense specified by the paper.

Agents can consume the good initially, hold it for the future, and buy or sell securities. A held unit may produce a state-dependent quantity of the same good later; that quantity may also be zero. Each security promises a state-dependent delivery, and a seller must hold a specified amount of the good as collateral.

The securities are non-recourse: the seller's delivery is limited to the smaller of the promised amount and the value of the collateral in that state. A buyer therefore may receive less than promised. But the seller cannot be required to give up more than the collateral's return.

The paper defines equilibrium as a set of security prices and plans at which each agent chooses the best affordable plan, purchases and sales of each security balance, and the good available at each date is allocated across agents.

Why collateral matters

The central observation is a solvency argument. Because the seller holds enough collateral and deliveries cannot exceed its return, paying securities buyers cannot reduce the seller's future resources below the agent's own future endowment. This holds for every portfolio satisfying the initial budget and collateral constraints, in every state. Since endowments are nonnegative, the resulting resources are nonnegative too.

That does not ensure a comfortable outcome. If an agent's endowment and remaining resources are zero in a state, consumption can be zero. If everyone has zero future endowments in a state, the only aggregate resources available then come from whatever the good held from the first date yields; securities redistribute resources rather than create them.

The paper contrasts its approach with earlier work that imposed stronger future-resource conditions, including a uniform positive lower bound on endowments in a previous collateral model. The new result removes that lower bound for the one-good setting while retaining positive first-date endowments.

From finite scenarios to a continuum

The proof starts with approximations containing finitely many future states. It divides the interval of possible states into pieces, averages the relevant data within each piece and adds a small positive amount to future endowments. An existing finite-state result supplies an equilibrium for each approximation.

The author then bounds prices and individual holdings so that a sequence of these equilibria has a convergent subsequence. As the partitions become finer and the added endowment tends to zero, the approximations converge to the original economy. The solvency result keeps future resources nonnegative along the way. With only one good, optimal future consumption can be taken to equal the resources available in each state, simplifying the passage to the limit.

The proof also uses positive first-date endowments to show that suitably scaled alternative plans remain affordable at nearby prices. This helps establish that the limiting plans are optimal, not merely that markets balance.

The result is mathematical rather than an empirical test of lending or disaster losses. It does not establish equilibrium for loans with recourse beyond collateral, or for economies with multiple goods; the paper identifies those as different or open questions. Its conclusion also depends on the stated assumptions about preferences, endowments and securities. Within those boundaries, it shows that a positive income floor in every future state is not necessary for equilibrium.