Markets recursive · not sovereign

The Bank Run

A bank takes money you can ask for at any moment and puts it into things that take years to mature. Everyone is better off for it, and the arrangement that makes everyone better off is the same arrangement that can be destroyed by the belief that it will be.

Diamond and Dybvig proved in 1983 that a bank of this kind has two equilibria at exactly the same fundamentals. In one, the people who need their money early take it and everyone else waits and is paid more. In the other, everyone queues at once, the bank liquidates everything at a loss, and the people who waited get nothing. Neither equilibrium is a mistake. Both are what a rational depositor should do given what they expect everyone else to do, and nothing in the bank’s books decides between them. What decides is what people think is about to happen.

Two Hundred Depositors and One Rumour

R = 2 held to maturity, 1 if broken early · a quarter need their money at date 1

promised to an early withdrawer,

promised to everyone else,

to make running worth it,

everyone is better off by,

smallest rumour that becomes a run,

outcome,

starting

Both curves at the bottom are derived, not drawn. The contract that maximises welfare comes out of the first-order condition U′(c₁) = R U′(c₂), which for this utility gives c₂ = c₁√R and a closed form for c₁. The run threshold comes out of asking when a patient depositor would rather not be patient. The panel computes both at every point on the axis, and finds the smallest rumour that becomes a run by bisecting on the dynamics rather than by looking up the threshold.

Why a Bank Helps at All

the part that is not about panic

Put your money in the ground and it grows. Dig it up early and you get back what you put in, with nothing for the time. You do not know yet whether you will need it early.

A bank knows that a predictable fraction of its depositors will need money early even though no individual knows it about themselves. So it can pay early withdrawers more than they could have got alone, funded by the ones who wait. That is insurance against a risk nobody can insure alone, and it is the entire economic reason banks exist.

With a quarter of depositors needing money early and a long asset that doubles, the best contract is:

early withdrawal c₁ = 1.28151 patient withdrawal c₂ = 1.81233 = c₁√R everyone better off by 1.61% against keeping it yourself

Those are computed on this page from the first-order condition, not copied. The published exposition of the model quotes 1.28 and 1.813.

The Second Equilibrium

the same contract, read differently

Now suppose you are patient, and you believe a fraction f of depositors is about to withdraw early. The bank has to liquidate f × c₁ of its assets at par to pay them, and whatever survives compounds for the rest:

c₂(f) = (1 − f c₁) R / (1 − f)

That falls as f rises, and at some point it falls below c₁. Past that point you are better off in the queue than waiting, and so is everybody else who is patient, which makes the belief correct. The threshold is:

f̂ = (R − c₁) / ( c₁ (R − 1) ) = 56.1% at the best contract

Nothing about the bank has changed between the two paragraphs above. Same assets, same contract, same depositors. The only difference is what people expect, and the expectation makes itself true either way.

The Trade-off With No Fix

the plot across the bottom of the panel

Drag the contract slider and watch the two curves. They point in opposite directions, and where they cross each other is the whole problem.

promised early	patient gets	everyone better off by	run needs
1.000	2.000	0.00%	no run possible
1.100	1.933	+0.98%	81.8%
1.200	1.867	+1.49%	66.7%
1.282	1.812	+1.61%	56.1%
1.450	1.700	+1.14%	37.9%
1.600	1.600	0.00%	25.0%

The safe bank is the useless one. At c₁ = 1 the bank offers nothing you could not do yourself, and no rumour of any size can hurt it, because there is nothing to gain by being first in the queue. Every unit of insurance it adds past that point buys a smaller and smaller improvement in welfare and a larger and larger reduction in how many people have to panic.

The contract that makes everyone best off is already 56% of the way to being run on, and it is chosen freely, by a bank doing nothing wrong. The exposure is not a defect in the design. It is the design.

At the far end, c₁ = 1.6, the two curves meet the floor together. The patient depositor gets exactly what the early one gets, so the bank is providing no insurance again, and the run threshold has fallen to 25%, which is the fraction who withdraw early in the normal course of things. That bank is permanently one step from a run and is worth nothing to anybody.

The Basin

how big a rumour has to be

Set the contract to its best value and move the rumour slider. Below about 53% the rumour dies within two rounds and the bank is fine. Above it the thing feeds itself and everyone is at the counter within three.

rumour 50% -> 50.0 43.5 25.0 25.0 25.0 the rumour dies rumour 55% -> 55.0 59.2 72.2 100 100 the bank fails

That boundary is the basin edge, and the panel finds it by bisection on the dynamics rather than by using the formula. It is the same object as the critical kick on the blocking page: two states available at one setting of the forcing, and a measurable distance between where you are and the edge of the other one.

Why This Is Recursive and Not Sovereign

the tag on the card

The run supplies its own cause. Each person withdrawing reduces what is left for the people who wait, which is the fact that makes withdrawing correct for the next person. Nothing outside the depositors produces that. It is a closed loop in the same sense as the storm cluster building the cold pool that triggers the next storm.

It is not sovereign, for the plainest possible reason: the whole structure runs on an asset somebody else grows. Take away the return R and there is no contract, no insurance and no run. The bank transforms a supply it does not make.

What This Model Is Not

the limits

Diamond and Dybvig’s game is static. It has two equilibria and says nothing about how one of them gets picked. The rounds you watch on this page are an addition: depositors best-respond to what happened last round, and they differ in how nervous they are, so a depositor runs when the expected withdrawal fraction exceeds f̂ minus their own nervousness, drawn uniformly over a spread of 0.12. That device is this page’s, not theirs. As the spread goes to zero the basin edge it produces converges on f̂ to ten decimal places, which is the check that the addition did not change the model underneath it.

No deposit insurance and no suspension. Both are in the original paper and both remove the run equilibrium, which is why the paper is a foundation of deposit-insurance policy. The panel leaves them out because it is about the fragility, not the fix.

The bank is not a firm. It has no capital, no equity holders, no other creditors, and no ability to borrow against its assets. Every one of those changes the picture in a real bank.

Sequential service is assumed away. In the original, depositors arrive in a queue and are paid until the money runs out, which is what makes being early worth anything. Here the fraction f is treated as known within a round.

Diamond, D. W. & Dybvig, P. H. (1983). Bank runs, deposit insurance, and liquidity. Journal of Political Economy, 91(3), 401–419.
Ennis, H. M. & Keister, T. (2007) and related expositions in the Federal Reserve Bank of Richmond Economic Quarterly. Fetched and read 2026-09-13; source of the worked parameterisation used here (R = 2, t = 1/4, U(c) = 1 − 1/c) and of the published figures c₁ = 1.28, c₂ = 1.813 and f̂ = 0.5625 that this panel reproduces.

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