Diamond–Dybvig Model

Diamond–Dybvig model is the foundational mathematical framework in business and economics theory for analyzing financial intermediation, maturity transformation, and systemic banking crises. Published by American economists Douglas W. Diamond and Philip H. Dybvig in their landmark 1983 Journal of Political Economy paper "Bank Runs, Deposit Insurance, and Liquidity"—work recognized with the 2022 Nobel Memorial Prize in Economic Sciences alongside Ben S. Bernanke—the model demonstrates that commercial banks perform a vital social function by transforming illiquid, high-yield long-term productive investments into liquid demand deposits that insure households against idiosyncratic consumption-timing shocks. However, this very structure of financing illiquid assets with short-term demandable liabilities creates a strategic coordination problem in game theory with multiple Nash equilibria: a welfare-maximizing risk-sharing equilibrium in which only depositors with genuine liquidity needs withdraw early, and a Pareto-inferior bank run equilibrium in which mutual fear of premature asset liquidation drives all depositors to withdraw simultaneously, toppling a fundamentally solvent institution.

Timeline and Production Technology

The Diamond–Dybvig economy unfolds over three discrete dates—t = 0 (contracting and investment), t = 1 (interim liquidity shock and early withdrawal), and t = 2 (long-term project maturity)—populated by a continuum of ex-ante identical depositors of measure 1, each endowed with 1 unit of a homogeneous consumption good at t = 0.

The economy possesses a constant-returns-to-scale productive investment technology that converts 1 unit of input at t = 0 into:

• Early Liquidation (t = 1): Yielding exactly 1 unit of consumption good if interrupted prematurely at t = 1 (zero net return, representing costly physical interruption or salvage liquidation).
• Full Maturity (t = 2): Yielding R > 1 units of consumption good if left undisturbed until t = 2.

In addition, goods can be stored costlessly between t = 1 and t = 2 at a gross return of 1. Thus, productive investment is illiquid: it earns a high long-run yield R > 1 at t = 2, but yields no gain over simple storage if liquidated early at t = 1.

Depositor Preferences and Idiosyncratic Liquidity Types

At t = 0, all consumers are identical and uncertain about when they will need to consume. At the beginning of t = 1, each consumer privately learns their personal liquidity type, modeled as an independent realization involving information asymmetry because a household's sudden medical, business, or family emergency cannot be verified by third-party insurers:

1. Impatient Consumers (Fraction λ ∈ (0, 1)): Care only about consumption at t = 1 (c₁) and derive zero utility from consumption at t = 2 (c₂), with utility u(c₁).
2. Patient Consumers (Fraction 1 − λ): Value consumption in both periods equally (or can store goods from t = 1 to t = 2), with utility u(c₁ + c₂) (or ρ · u(c₁ + c₂) with discount factor ρ ≤ 1 such that ρR > 1).

Consequently, ex-ante expected utility at t = 0 is:

where the Period-1 and Period-2 consumption utility function u(c) is twice continuously differentiable, increasing (u′(c) > 0), concave (u″(c) < 0), and exhibits a coefficient of relative risk aversion greater than unity:

This risk-aversion condition implies that marginal utility rises steeply when consumption is low, so risk-averse households strongly desire insurance that raises early consumption c₁ above 1 in the event they turn out to be impatient, in exchange for accepting a slightly lower late payout c₂ < R when they turn out to be patient.

Financial Autarky Benchmark

If households invest on their own in autarky without banks (or trade in competitive spot bond markets at t = 1), each household invests its 1 unit in the productive technology at t = 0. At t = 1, an impatient household is forced to liquidate its project early, receiving c₁^aut = 1, whereas a patient household holds its project to maturity at t = 2, receiving c₂^aut = R. Notice that autarky provides zero liquidity insurance: the gap between c₁^aut = 1 and c₂^aut = R leaves impatient households fully exposed to their consumption-timing shock.

Social Planner's First-Best Liquidity Insurance

By the law of large numbers, although each individual faces idiosyncratic uncertainty, the aggregate fraction of impatient consumers across the population is deterministically known to be λ. A mutual bank or social planner pooling all depositors' endowments at t = 0 chooses early consumption c₁^* for the λ impatient depositors and late consumption c₂^* for the (1 − λ) patient depositors to maximize ex-ante expected utility subject to the resource constraint that liquidating λc₁ units at t = 1 leaves (1 − λc₁) units to compound at rate R until t = 2:

The first-order condition equates the marginal utility of early consumption to R times the marginal utility of late consumption:

Because R > 1 and the coefficient of relative risk aversion exceeds 1 (−*cu*″(c)/u′(c) > 1 implies that c · u′(c) is a decreasing function of c), the Pareto-efficient risk-sharing allocation satisfies the fundamental Diamond–Dybvig inequality:

Because c₁^* < c₂^*, this allocation is incentive-compatible even when liquidity types are private information: patient consumers prefer waiting until t = 2 to receive c₂^* rather than misrepresenting themselves as impatient at t = 1 to receive the smaller payout c₁^*, provided they are confident the bank will remain solvent at t = 2.

Sequential Service Constraint and Payoff Structure

Commercial banks implement the optimal allocation (c₁^*, c₂^*) by offering demand deposit contracts subject to a sequential service constraint (first-come, first-served): any depositor who demands withdrawal at t = 1 is paid a fixed face value r₁ = c₁^* > 1 so long as the bank still has assets left to liquidate, and whatever assets remain at t = 2 are divided pro rata among depositors who waited.

Let f ∈ [λ, 1] denote the total fraction of depositors who demand withdrawal at t = 1. Because each early withdrawal requires liquidating c₁^* > 1 units of the bank's productive assets, the payout r₂(f) per remaining depositor at t = 2 is a decreasing function of f:

Notice the fragility embedded in maturity transformation: because c₁^* > 1, if everyone demands withdrawal at t = 1 (f = 1), the bank's entire asset pool is exhausted after serving only a fraction 1 / c₁^* < 1 of the queue (f · c₁^* = 1), leaving zero assets (r₂ = 0) for latecomers and anyone who waits until t = 2.

Two Pure-Strategy Nash Equilibria

For a patient depositor deciding at t = 1 whether to keep funds in the bank or withdraw early and store the cash until t = 2, the optimal best response depends on their belief about the aggregate withdrawal fraction f:

1. The "Good" Equilibrium (f = λ): If all other patient depositors keep their deposits in the bank, only the impatient fraction f = λ withdraws at t = 1. A patient depositor who waits receives r₂(λ) = c₂^* > c₁^*, whereas withdrawing early yields only c₁^*. Waiting is therefore a dominant best response, confirming f = λ as a Pareto-optimal Nash equilibrium.
2. The "Bad" Bank Run Equilibrium (f = 1): Suppose f exceeds the critical tipping threshold f̂ = (R − c₁^*) / [c₁^*(R − 1)], where r₂(f̂) = c₁^*. Whenever a patient depositor believes that f > f̂ (for instance, triggered by a sunspot panic, rumor, or macroeconomic shock), waiting until t = 2 yields r₂(f) < c₁^* (and r₂(1) = 0 in a full run), whereas joining the queue at t = 1 yields an expected value of (1 / c₁^*) · c₁^* = 1 > 0. Thus, running on the bank becomes the unique individual best response for every patient depositor, destroying real productive projects (R → 1) and leaving society worse off than under financial autarky.

Diamond and Dybvig evaluated three institutional mechanisms for eliminating the bank run equilibrium:

1. Suspension of Convertibility: When the aggregate fraction of early withdrawals λ is deterministic and known at t = 0, the bank can write a contract stipulating that the teller window will close at t = 1 as soon as f = λ withdrawals have been paid out. Because every patient depositor is then guaranteed that at least (1 − λc₁^*) of the bank's productive assets will survive until t = 2 to pay c₂^* > c₁^*, no patient depositor has any incentive to run, and the good equilibrium becomes unique. However, when λ is itself a stochastic random variable (aggregate liquidity uncertainty), a rigid suspension threshold cannot achieve the first-best outcome because halting withdrawals too early starves genuinely impatient households of consumption.
2. Government Deposit Insurance: A government deposit insurance agency (such as the U.S. Federal Deposit Insurance Corporation, established in 1933) or a central bank acting as lender of last resort can eliminate bank runs even under stochastic λ. Because the state possesses the sovereign power to levy proportional taxes on deposits at t = 1 (or issue central bank liquidity via monetary policy), it can guarantee that any patient depositor who waits until t = 2 receives c₂^* regardless of how many others withdraw at t = 1. Crucially, because this guarantee removes any strategic incentive for patient depositors to run, the bank run never occurs in equilibrium, and the insurance fund never actually has to pay out a bailout along the equilibrium path—though in real-world banking with risky loan portfolios, deposit insurance introduces moral hazard and principal-agent problems that require prudential capital regulation.

Why can a completely solvent bank fail in the Diamond–Dybvig model?

In the baseline Diamond–Dybvig model, there is zero credit risk or asset-quality uncertainty: every long-term project is guaranteed to yield R > 1 if held until t = 2. A bank fails in a run not because its assets are bad ex ante, but because its assets are illiquid (yielding only 1 instead of R when liquidated early at t = 1) while its liabilities are liquid demand deposits paying c₁^* > 1. When panic induces patient depositors to withdraw early, premature liquidation destroys the long-run yield R, turning an illiquidity panic into actual insolvency.

Why must relative risk aversion be greater than 1 in the Diamond–Dybvig model?

If consumers had a coefficient of relative risk aversion less than or equal to 1 (such as logarithmic utility u(c) = ln c, where relative risk aversion equals 1), solving u′(c₁^*) = R · u′(c₂^*) would yield c₁^* ≤ 1 and c₂^* ≥ R. Only when relative risk aversion exceeds 1 do households value consumption smoothing strongly enough to want c₁^* > 1 at t = 1, which is the condition that makes bank demand deposits superior to autarky while simultaneously creating the vulnerability to bank runs.

How did Carlsson and van Damme's global games refinement extend the Diamond–Dybvig model?

In the original 1983 Diamond–Dybvig model, the selection between the good equilibrium and the bank run equilibrium is indeterminate (driven by arbitrary "sunspot" beliefs). In 1998 and 2004, Stephen Morris, Hyun Song Shin, and Itay Goldstein and Ady Pauzner applied global games—where each depositor observes a slightly noisy private signal of the bank's underlying asset return R—to pin down a unique threshold equilibrium R^* below which a self-fulfilling run occurs with certainty and above which the bank remains stable.

1. Diamond, D. W., & Dybvig, P. H. (1983). "Bank Runs, Deposit Insurance, and Liquidity." Journal of Political Economy, 91(3), 401–419. https://doi.org/10.1086/261155
2. Bryant, J. (1980). "A Model of Reserves, Bank Runs, and Deposit Insurance." Journal of Banking & Finance, 4(4), 335–344. https://doi.org/10.1016/0378-4266(80)90012-690012-6)
3. Jacklin, C. J. (1987). "Demand Deposits, Trading Restrictions, and Risk Sharing." In E. C. Prescott & N. Wallace (Eds.), Contractual Arrangements for Intertemporal Trade (pp. 26–47). University of Minnesota Press. https://www.jstor.org/stable/10.5749/j.ctttt86n.5
4. Peck, J., & Shell, K. (2003). "Equilibrium Bank Runs." Journal of Political Economy, 111(1), 103–123. https://doi.org/10.1086/344803
5. Goldstein, I., & Pauzner, A. (2005). "Demand-Deposit Contracts and the Probability of Bank Runs." The Journal of Finance, 60(3), 1293–1327. https://doi.org/10.1111/j.1540-6261.2005.00762.x