ECON30005 · Money and Banking
The Diamond-Dybvig Model of Bank Runs
Week 6 works through the Diamond-Dybvig model: patient and impatient savers face liquidity shocks, and a bank pools funds to offer a demand-deposit contract (c1, c2) that insures against them, characterised by the first-order condition u′(c1) = βR·u′(c2) subject to the resource constraint. The same contract admits two equilibria — a good liquidity-insurance one and a self-fulfilling bank-run one under sequential service — and deposit insurance removes the bad one. This is a prime Section 3 worked model and a rich source of true/false statements.
What this chapter covers
- 01The environment: N savers, a fraction θ hit by an impatience (liquidity) shock, a project returning 1 if liquidated early and R > 1 if held to maturity, with βR > 1
- 02Autarky: impatient consume 1, patient consume R, each bearing liquidity risk alone
- 03The demand-deposit contract (c1, c2): withdraw c1 in period 1 or c2 in period 2
- 04The resource constraint (1−θ)c2 = (1 − θc1)R and the first-order condition u′(c1) = βR·u′(c2)
- 05The optimal allocation satisfies 1 < c1 < c2 < R — risk sharing smooths consumption across types
- 06Liquidity creation and liquidity insurance: impatient get more, patient get less than autarky, everyone better off ex ante
- 07Multiple equilibria: a good equilibrium (only impatient withdraw) and a self-fulfilling bank-run equilibrium under sequential (first-come-first-served) service
- 08Deposit insurance removes the run equilibrium by making waiting safe regardless of others' beliefs
Solving the Diamond-Dybvig demand-deposit contract
- +1Autarky benchmark. Without a bank an impatient saver liquidates early and consumes c1 = 1; a patient saver waits and consumes c2 = R = 2.25. Each bears their own liquidity risk; there is no sharing.
- +1Bank's problem. The bank chooses (c1, c2) to maximise θ·u(c1) + (1−θ)·u(c2) subject to the resource constraint (1−θ)c2 = (1 − θc1)R, giving the first-order condition u′(c1) = βR·u′(c2).
- +1Apply the utility function. For u(c) = 1 − 1/c, u′(c) = 1/c². The FOC becomes 1/c1² = βR/c2² = 2.25/c2², so c2² = 2.25·c1² and c2 = √2.25·c1 = 1.5·c1.
- +1Solve with the resource constraint. Substitute c2 = 1.5·c1: (1−0.2)(1.5·c1) = (1 − 0.2·c1)·2.25 → 1.2·c1 = 2.25 − 0.45·c1 → 1.65·c1 = 2.25 → c1 ≈ 1.36, and c2 = 1.5·c1 ≈ 2.05. Check the ordering: 1 < 1.36 < 2.05 < 2.25 ✓.
- +1Interpret. Risk sharing: impatient depositors get more than autarky (1.36 > 1) and patient depositors less (2.05 < 2.25), and because they are risk-averse everyone is better off ex ante. But the contract is fragile: under sequential (first-come-first-served) service, if patient depositors expect everyone to withdraw early, promised withdrawals N·c1 exceed the N units available from liquidating all projects, so running becomes each patient depositor's best response — a self-fulfilling bank run. Full deposit insurance removes this bad equilibrium.
Key terms
- Liquidity shock
- The random event, occurring at date 1, that makes a saver impatient (wanting to consume early) with probability θ or patient (willing to wait) with probability 1−θ. Types are private information, revealed only after funds are invested.
- Demand-deposit contract
- The contract (c1, c2) a bank offers: withdraw c1 in period 1 or wait and receive c2 in period 2. Chosen to maximise ex-ante expected utility subject to the resource constraint, it provides liquidity insurance against the shock.
- Resource constraint
- (1−θ)c2 = (1 − θc1)R: total period-2 consumption of patient depositors equals the return on the projects not liquidated early. It links the two payments the bank can promise.
- Liquidity insurance
- The risk sharing a bank provides by paying impatient depositors more than the early-liquidation value and patient depositors less than the full maturity value, smoothing consumption across states so that risk-averse depositors are better off ex ante.
- Sequential service
- The first-come-first-served rule by which a bank pays withdrawals until it runs out of funds. It is what makes waiting risky when others withdraw, creating the strategic complementarity behind a bank run.
- Bank-run equilibrium
- The self-fulfilling bad equilibrium in which patient depositors, fearing others will withdraw early, also withdraw; total demand exceeds the bank's liquidation value and the bank fails. It depends on beliefs, not fundamentals, and deposit insurance removes it.
The Diamond-Dybvig Model of Bank Runs FAQ
Why does the bank contract make everyone better off than going it alone?
Because savers are risk-averse and do not know in advance whether they will be impatient or patient. Under autarky an impatient saver is stuck with the low early-liquidation value 1 and a patient saver gets the full R, a risky spread. The bank pools funds and offers (c1, c2) with 1 < c1 < c2 < R, raising the low outcome and lowering the high one. Since the depositor values the smoother allocation, expected utility is higher ex ante — that is liquidity insurance.
If the bank contract is optimal, why can a bank run still happen?
Because the same contract has two equilibria. In the good equilibrium only the impatient withdraw early and the patient wait, which is self-consistent. But under sequential service, if a patient depositor believes everyone else will withdraw early, the bank will have to liquidate all its projects and cannot pay c1 to everyone, so waiting risks getting nothing — the best response is to run too. That belief is also self-consistent, so a run is a genuine equilibrium driven by expectations, not by any deterioration in the bank's assets.
How does deposit insurance stop runs?
By making waiting safe regardless of what others do. If the government fully guarantees deposits, a patient depositor has no reason to withdraw early out of fear, because their period-2 payment is assured. That removes the strategic complementarity that generated the run equilibrium, so only the good equilibrium survives. This is the theoretical case for deposit insurance and the lender of last resort as backstops for fractional-reserve banking.
Can AI help me with the Diamond-Dybvig model in ECON30005?
Yes. This is one of the models students most often bring to Sia. It can set up the resource constraint and the first-order condition, solve for (c1, c2) with your utility function, verify the ordering, and explain the two equilibria one step at a time. Use it to rehearse the derivation, not to answer a graded quiz, and confirm assessment details on Canvas.
Exam move
Diamond-Dybvig is a top candidate for the 10-mark worked model, so drill the full solve from a blank page: autarky benchmark, the bank's objective and resource constraint, the first-order condition u′(c1) = βR·u′(c2), the utility-specific step (for u = 1 − 1/c you get c2 = √(βR)·c1), and the final (c1, c2) with the ordering check 1 < c1 < c2 < R. Then rehearse the words that earn the conceptual marks: liquidity insurance (impatient gain, patient lose, everyone better off ex ante), the two equilibria under sequential service, and how deposit insurance removes the run. A classic true/false is that runs are driven by beliefs rather than fundamentals — be ready to defend it. Keep β, R and θ symbolic so you can re-solve with any numbers under exam pressure. Confirm the exam structure on Canvas.
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