Investors face fundamental uncertainty about when they will need to consume. Some will need funds soon; others can wait. The problem is that the most productive investments are illiquid — they lose value when liquidated early.
NoteDiamond’s Framing
As Diamond (2007) puts it: “An illiquid asset is one where proceeds from liquidation are less than the present value of future payoffs.” The gap between what an asset is worth if held to maturity versus what you can recover if forced to sell early — that gap is the essence of illiquidity.
Why does this create a demand for liquidity?
Many financial institutions intermediate between savers and borrowers:
Investment banks: Underwrite securities
Mutual funds: Pool investments
Pension funds: Manage retirement savings
What makes banks special?
Banks combine:
Illiquid, long-term assets (loans)
Liquid, short-term liabilities (deposits)
This combination is essential to their function, not accidental. The reason lies in the consumer’s problem: investors don’t know when they will need to consume, so they value the ability to access funds early without sacrificing too much return. This is the demand for liquidity, and it is what banks exist to serve.
2.1.2 The Model Setup
2.1.2.1 Timeline
Three periods: \(t = 0, 1, 2\)
2.1.2.2 Investment Technology
Two investment technologies available:
Short-term (liquid) investment
Invest 1 at \(t=0\)
Returns \(r_1 = 1\) at \(t=1\)
Can be rolled over: 1 at \(t=1\) → \(r_2\) at \(t=2\)
Assumption: \(r_1 = 1\) (normalization)
Long-term (illiquid) investment
Invest 1 at \(t=0\)
Returns \(R > 1\) at \(t=2\)
Returns 0 at \(t=1\) (not productive if liquidated early)
Alternatively: Can liquidate at \(t=1\) for \(\ell < 1\)
Key relationship: Long-term investment dominates for patient investors: \[
R > r_1 \cdot r_2 = r_2
\]
NoteIntuition
If you know you won’t need money at \(t=1\): Invest in long-term (get \(R > r_2\))
If you might need money at \(t=1\): Face a dilemma
Short-term: Safe but low return
Long-term: High return but can’t access at \(t=1\)
2.1.2.3 Consumer Preferences
Ex ante identical consumers
Each consumer has initial wealth = 1 at \(t=0\)
Uncertainty about consumption timing:
At \(t=1\), consumers learn their type:
NoteConsumer Types
Impatient (probability \(\lambda\)): Want to consume at \(t=1\)
Utility: \(u(c_1)\)
Patient (probability \(1-\lambda\)): Want to consume at \(t=2\)
Now consider a more liquid asset that offers \(r_1 = 1.28\) at \(t=1\) and \(r_2 = 1.813\) at \(t=2\). Expected utility: \[
EU_{\text{liquid}} = \frac{1}{4}\, u(1.28) + \frac{3}{4}\, u(1.813) = \frac{1}{4}\left(1 - \frac{1}{1.28}\right) + \frac{3}{4}\left(1 - \frac{1}{1.813}\right)
\]\[
= \frac{1}{4}(0.21875) + \frac{3}{4}(0.44845) = 0.0547 + 0.3363 = 0.391
\]
ImportantRisk Aversion Creates the Demand for Liquidity
The more liquid asset gives higher expected utility (\(0.391 > 0.375\)) even though a risk-neutral investor would prefer the illiquid asset:
Expected payoff of the liquid asset: \(\frac{1}{4}(1.28) + \frac{3}{4}(1.813) = 1.68\)
Expected payoff of the illiquid asset: \(\frac{1}{4}(1) + \frac{3}{4}(2) = 1.75\)
Since \(1.68 < 1.75\), a risk-neutral investor would choose the illiquid asset. It is risk aversion — the concavity of \(u\) — that makes the liquid asset more attractive. The smoother payoff profile is worth the sacrifice in expected return.
This is the key insight: risk aversion creates the demand for liquidity.
2.1.4 Benchmark: Autarky
Without a bank, each investor holds assets directly. This is the autarky benchmark.
Each consumer individually chooses investment portfolio: \((x, y)\)
\(x\): Amount in short-term (liquid) asset
\(y\): Amount in long-term (illiquid) asset
Budget constraint: \(x + y = 1\)
Payoffs:
If impatient: Consume \(c_1 = x \cdot r_1 = x\) at \(t=1\)
If patient: Consume \(c_2 = x \cdot r_2 + y \cdot R\) at \(t=2\)
Consumer’s problem at \(t=0\): \[
\max_{x, y} \quad \lambda u(x) + (1-\lambda) u(x r_2 + y R)
\]\[
\text{s.t.} \quad x + y = 1
\]
where \(c_1^A = x\) and \(c_2^A = (1-x)R + x r_2\)
2.1.4.1 Autarky Solution: Interpretation
The autarky first-order condition: \[
\frac{u'(c_1^A)}{u'(c_2^A)} = \frac{(1-\lambda) R}{\lambda}
\]
Interpretation:
Left side: Marginal rate of substitution
Willingness to trade \(t=1\) consumption for \(t=2\) consumption
High \(c_1\) → low \(u'(c_1)\) → low MRS
Right side: Marginal rate of transformation
Technology for transforming \(t=1\) consumption into \(t=2\) consumption
Giving up 1 unit at \(t=1\) allows investing in long-term asset
Returns \(R\) at \(t=2\) with probability \((1-\lambda)\)
ImportantThe Key Limitation of Autarky
In autarky, each consumer must hold both assets:
Liquid asset: Insurance against being impatient
Illiquid asset: To get high returns if patient
This is inefficient because impatient consumers waste the high returns from illiquid assets. Crucially, an individual investor needs all or none of his liquidity — he does not know in advance whether he will be impatient, so he cannot fine-tune his portfolio to match his realized type. The bank, by contrast, knows that exactly fraction \(\lambda\) will need liquidity at date 1.
2.2 How Banks Create Liquidity
2.2.1 The Banking Contract
A bank offers a deposit contract\((c_1, c_2)\) that specifies:
\(c_1\): Consumption for depositors who withdraw at \(t=1\)
\(c_2\): Consumption for depositors who withdraw at \(t=2\)
Key features:
Depositors make deposits at \(t=0\): Each deposits 1 unit
Bank chooses investment portfolio\((x, y)\):
\(x\): Liquid short-term investment
\(y\): Illiquid long-term investment
Budget: \(x + y = 1\)
At \(t=1\):
Fraction \(\lambda\) reveal they are impatient, withdraw and get \(c_1\)
Fraction \((1-\lambda)\) wait
At \(t=2\):
Remaining fraction \((1-\lambda)\) withdraw and get \(c_2\)
2.2.1.1 Diamond’s 100-Investor Example
Following Diamond (2007), consider a bank with 100 depositors, \(\lambda = 1/4\), and \(R = 2\). Suppose the bank offers \(c_1 = 1.28\):
At \(t=1\): 25 depositors withdraw, each getting 1.28
Total paid out: \(25 \times 1.28 = 32\)
The bank must liquidate 32% of its portfolio (since the illiquid asset returns 1 per unit if liquidated at \(t=1\))
Remaining assets: \(100 - 32 = 68\) units invested in the long-term asset
At \(t=2\): These 68 units mature, each worth \(R = 2\), yielding \(68 \times 2 = 136\)
Per remaining depositor: \(136 / 75 = 1.813\)
So the bank delivers \((c_1, c_2) = (1.28, 1.813)\) — exactly the “more liquid” payoff profile from our earlier comparison. This is liquidity creation: the bank offers depositors a more liquid claim than the underlying illiquid asset.
2.2.1.2 The Budget Constraint
For the contract \((c_1, c_2)\) to be feasible, the bank must be able to meet withdrawals:
Date 1 constraint (liquidity constraint):
Bank must have enough liquid funds to pay impatient depositors: \[
\lambda c_1 \leq x \cdot r_1 = x
\]
Impatient depositors (fraction \(\lambda\)) each get \(c_1\), must come from liquid assets.
Date 2 constraint (solvency constraint):
After paying impatient depositors, remaining funds must cover patient depositors: \[
(1-\lambda) c_2 \leq (x - \lambda c_1) \cdot r_2 + y \cdot R
\]
Combined with budget constraint\(x + y = 1\), when both constraints bind and \(r_1 = r_2 = 1\), the budget constraint reduces to:
This is the bank’s feasibility frontier: every dollar of \(c_1\) promised to impatient depositors reduces the resources available for patient depositors, amplified by the forgone long-term return \(R\).
2.2.2 The Optimal Amount of Liquidity
The bank chooses \((c_1, c_2, x)\) to maximize expected consumer welfare:
Taking the ratio:\[
\frac{u'(c_1)}{u'(c_2)} = R
\]
ImportantProposition: Optimal Deposit Contract
The optimal banking contract satisfies: \[
u'(c_1^*) = R \cdot u'(c_2^*)
\]
Interpretation (Diamond, 2007): The marginal utility of consumption at date 1 must be in line with the marginal cost of providing liquidity. Since each unit of \(c_1\) costs the bank \(R\) units of forgone \(t=2\) output (because the liquid asset earns 1 while the long-term asset earns \(R\)), the optimum equates the marginal benefit of insurance with this marginal cost.
2.2.2.3 Diamond’s Closed-Form Solution
For the utility function \(u(c) = 1 - 1/c\) (CRRA with \(\gamma = 2\)), we have \(u'(c) = 1/c^2\). The FOC becomes: \[
\frac{1}{c_1^2} = R \cdot \frac{1}{c_2^2} \implies \frac{c_2}{c_1} = \sqrt{R}
\]
Substituting \(c_2 = c_1 \sqrt{R}\) into the budget constraint \(c_2 = \frac{(1 - \lambda c_1)R}{1-\lambda}\):
What does this mean? Autarky has too large a gap in marginal utility between the two types. Impatient depositors consume little (\(c_1^A\) low, marginal utility high), while patient depositors consume a lot (\(c_2^A\) high, marginal utility low). This means there are unexploited gains from redistribution — a dollar moved from patient to impatient types would increase total expected welfare.
The bank exploits exactly this gap. It raises \(c_1\) and lowers \(c_2\), narrowing the marginal utility ratio until it falls to \(R\) — the point where the insurance benefit (utility gained by impatient types) exactly equals the opportunity cost (forgone return \(R\) from diverting resources away from the long-term asset). At that point, no further redistribution can improve welfare.
ImportantDiamond’s Key Insight
“An investor’s opportunity set without the bank is worse because an investor needs all or none of his liquidity, while the bank knows that a fraction \(\lambda\) will need liquidity at date 1.”
— Diamond (2007, p. 196)
The individual faces binary uncertainty (impatient or patient), but the bank faces only the known aggregate (exactly fraction \(\lambda\)). This is what allows the cross-subsidy: the bank can promise more than any individual could achieve alone.
Shape determined by \(u(\cdot)\) and probabilities \(\lambda, 1-\lambda\)
Optimum: Highest indifference curve tangent to budget constraint
Banking expands the budget constraint by efficiently allocating the long-term investment.
2.2.4 Asset Management of Liquidity
Diamond (2007) introduces an important extension that further clarifies why banks dominate individual investors. Suppose that liquidating the long-term asset early is costly but not completely worthless: at \(t=1\), each unit of the long-term asset can be liquidated for \((1 - \tau)\), where \(\tau > 0\) represents the liquidation cost.
2.2.4.1 The Bank’s Strategy
The bank can hold a fraction \(\lambda c_1\) in short-term assets (just enough to pay impatient depositors) and invest the rest, \(1 - \lambda c_1\), in the long-term asset. Since the bank knows exactly fraction \(\lambda\) will withdraw early, the bank never needs to liquidate the long-term asset. It achieves the same payoffs as before:
The liquidation cost \(\tau\) is irrelevant to the bank because it manages its portfolio so that early withdrawals are funded entirely from short-term assets.
2.2.4.2 The Individual Investor’s Problem
An individual investor, by contrast, does not know whether she will be impatient or patient. If she invests fraction \(\alpha\) in the short-term asset and \((1-\alpha)\) in the long-term asset:
If impatient (realized at \(t=1\)): she gets \(c_1 = \alpha + (1-\alpha)(1-\tau)\)
She liquidates her short-term holdings plus her long-term holdings at the penalty \((1-\tau)\)
If patient (realized at \(t=2\)): she gets \(c_2 = \alpha + (1-\alpha)R\)
She rolls over her short-term asset and collects the long-term return
Eliminating \(\alpha\) from these two equations gives the individual’s opportunity set: \[
c_2 = 1 + (1 - c_1)\frac{R - 1}{\tau}
\]
This is a linear tradeoff: for every unit increase in \(c_1\), the investor sacrifices \((R-1)/\tau\) units of \(c_2\).
2.2.4.3 Diamond’s Numerical Example
With \(\lambda = 1/4\), \(\tau = 1/2\), \(R = 2\):
Bank: offers \(c_1^* = 1.28\) and \(c_2^* = 1.813\) (same as the \(\tau = 0\) case, since the bank avoids liquidation entirely).
If she wants \(c_1 = 1\): \(c_2 = 1 + 0 = 1\) (everything in short-term).
If she wants \(c_1 = 0.5\) (all in long-term): \(c_2 = 1 + 0.5 \times 2 = 2\), but \(c_1 = 0.5\) is very low.
ImportantWhy the Bank Dominates the Individual
The individual’s opportunity set is strictly dominated by the bank’s. The bank offers \((1.28, 1.813)\) while an individual choosing \(c_1 = 0.9\) can only achieve \(c_2 = 1.2\).
The reason is fundamental: the individual must choose a portfolio before knowing her type. If she turns out to be impatient, she must liquidate long-term assets at a loss. The bank, by contrast, knows the aggregate fraction \(\lambda\) and can separate its short-term and long-term holdings perfectly.
This is what Diamond calls asset management of liquidity: the bank manages its asset portfolio to provide liquidity without incurring the liquidation penalty that an individual investor would face.
NoteRelated Literature
The concept of asset management of liquidity connects to several important papers:
Bryant (1980): First showed that banks can provide liquidity insurance by pooling deposits
Jacklin (1987): Analyzed the role of trading restrictions in sustaining the banking contract
Haubrich and King (1990): Explored asset management of liquidity in banking models with more general asset structures
2.2.5 What Drives the Insurance Motive?
Before proceeding, let’s clarify what kind of “risk” we’re insuring against and what role risk aversion plays.
NoteThe Nature of Uncertainty in DD
The Diamond-Dybvig model features timing uncertainty, not wealth uncertainty:
Agents don’t know WHEN they’ll need to consume (early at \(t=1\) vs. late at \(t=2\))
This is different from traditional risk aversion over wealth lotteries
Ex ante (at \(t=0\)), each agent faces probability \(\lambda\) of being impatient, probability \(1-\lambda\) of being patient
This is closer to “state-contingent preferences” than classical risk aversion
Key question: Why does this timing uncertainty create a demand for insurance?
Answer: Because of diminishing marginal utility (concave \(u\))
2.2.5.1 The Role of Concavity
The concavity of utility (\(u'' < 0\)) drives the desire for consumption smoothing across types.
With concave utility:
High consumption — low marginal utility
Low consumption — high marginal utility
Moving consumption from high-marginal-utility state to low-marginal-utility state raises expected utility
In the DD context:
Starting from autarky where impatient types consume less: \[
c_1^A < c_2^A \implies u'(c_1^A) > u'(c_2^A)
\]
Insurance improves welfare by:
Transferring resources from patient types (who have low marginal utility) to impatient types (who have high marginal utility)
This is what the DD contract does: \(c_1^* > c_1^A\) and \(c_2^* < c_2^A\)
The transfer continues until marginal utilities are equalized (weighted by \(R\))
ImportantKey Insight
Risk aversion (concave utility) is what creates the gains from insurance:
With risk-neutral utility (\(u(c) = c\)), autarky would be optimal
Higher risk aversion — greater desire for consumption smoothing
The degree of risk aversion determines how much insurance is optimal
2.2.5.2 Relative Risk Aversion and the Optimal Contract
Relative risk aversion (RRA) measures the curvature of utility: \[
\text{RRA}(c) = -\frac{c \cdot u''(c)}{u'(c)}
\]
Higher RRA (\(\gamma\)): - More desire for consumption smoothing - Optimal contract provides more insurance - Consumption levels \(c_1^*\) and \(c_2^*\) become more similar - Front-loading of consumption increases
Lower RRA: - Less desire for smoothing - Closer to autarky allocation - Greater consumption inequality across types
Important clarifications:
Risk aversion drives insurance demand: Without concave utility, no gains from banking
RRA determines insurance intensity: Higher RRA — more front-loading of consumption
But the mechanism is the same: All cases satisfy \(\frac{u'(c_1^*)}{u'(c_2^*)} = R\)
Timing vs. wealth uncertainty: This is about smoothing consumption across different timing needs, not insuring against wealth shocks
2.2.5.3 DD Requires RRA > 1
ImportantCritical Requirement
For banking to improve on autarky (i.e., \(c_1^* > 1\)), we need RRA \(> 1\).
Why \(c_1^* > 1\) is the benchmark:
In autarky, each depositor has endowment of 1 and invests individually
An impatient depositor can get at most\(c_1^A = 1\) (by putting everything in the liquid asset)
But doing so sacrifices all long-term returns (\(c_2^A = r_2\), not \(R\))
In practice, autarky agents invest some in illiquid assets, so \(c_1^A < 1\)
If the bank offers \(c_1^* > 1\), impatient depositors get more than their entire endowment — something impossible in autarky. The bank achieves this by pooling: it knows exactly fraction \(\lambda\) will be impatient, so it can cross-subsidize from patient depositors (who accept \(c_2^* < R\)) to impatient ones.
Why does transferring resources from patient to impatient depositors improve welfare?
In autarky, patient depositors consume \(R > 1\) while impatient depositors consume at most 1. Because utility is concave (diminishing marginal utility), an extra dollar is worth less to someone already consuming \(R\) than to someone consuming only 1. The patient depositor’s marginal utility is low; the impatient depositor’s marginal utility is high.
So when the bank takes a small amount from each patient depositor (reducing \(c_2\) slightly below \(R\)) and gives it to impatient depositors (raising \(c_1\) above 1), the utility gained by impatient types exceeds the utility lost by patient types. This is exactly what insurance does — it transfers resources from states where marginal utility is low to states where marginal utility is high.
But how can impatient depositors get \(c_1^* > 1\) when each only deposited 1?
This is the power of pooling. An individual acting alone can never consume more than their endowment at \(t=1\). But the bank collects from everyone. With \(\lambda = 0.3\) and 100 depositors, the bank knows exactly 30 will be impatient. It can pay each of them \(c_1^* = 1.2\) (costing \(36\) total) because the remaining 70 patient depositors don’t need funds until \(t=2\). The extra \(0.2\) per impatient depositor comes from the patient depositors’ share — they each accept slightly less than \(R\).
An individual faces binary uncertainty (impatient or patient), but the bank faces only the known aggregate (exactly fraction \(\lambda\)). This is what allows the cross-subsidy: the bank can promise more than any individual could achieve alone.
But this only works when the gap in marginal utility is large enough to justify the cost. The long-term asset earns \(R > 1\), so diverting resources to early consumption has a real opportunity cost. When RRA \(> 1\), the curvature of utility is steep enough that the insurance benefit outweighs this cost. When RRA \(\leq 1\), marginal utility doesn’t fall off fast enough — the gains from smoothing are too small relative to the forgone returns.
Proof (optional). With CRRA utility \(u(c) = \frac{c^{1-\gamma}}{1-\gamma}\), the optimality condition \(u'(c_1^*) = R \cdot u'(c_2^*)\) gives: \[
(c_1^*)^{-\gamma} = R \cdot (c_2^*)^{-\gamma} \implies \frac{c_2^*}{c_1^*} = R^{1/\gamma}
\]
Substituting into the budget constraint \((1-\lambda)c_2 = (1-\lambda c_1)R\): \[
c_1^* = \frac{R}{(1-\lambda) R^{1/\gamma} + \lambda R}
\]
For \(c_1^* > 1\) (bank beats autarky), we need: \[
R > (1-\lambda) R^{1/\gamma} + \lambda R
\]
The Diamond-Dybvig model provides a theoretical foundation for understanding liquidity creation, but how do we measure it empirically? Berger and Bouwman (2009) develop the first comprehensive measures of bank liquidity creation.
2.2.6.1 The Challenge
Although modern financial intermediation theory portrays liquidity creation as an essential role of banks, comprehensive empirical measures did not exist before this paper. The challenge is to quantify how banks transform illiquid assets into liquid liabilities across their entire balance sheet.
2.2.6.2 What They Do
Berger and Bouwman (2009) construct four measures of bank liquidity creation and apply them to data on virtually all U.S. banks from 1993 to 2003.
NoteResearch Question
How much liquidity do banks actually create? How does liquidity creation vary across bank characteristics (size, ownership structure, type)? What is the relationship between bank capital and liquidity creation?
2.2.6.3 How They Measure Liquidity Creation
The methodology follows a three-step process:
Step 1: Classify each balance sheet item as liquid, semi-liquid, or illiquid
For assets: Classification based on the ease, cost, and time for banks to dispose of (sell or securitize) the asset
For liabilities and equity: Classification based on the ease, cost, and time for customers to obtain liquid funds from the bank
Off-balance sheet items: Also classified by their liquidity characteristics
Step 2: Assign weights to each category
The weighting scheme captures how each item contributes to liquidity creation:
Category
Assets
Liabilities & Equity
Off-Balance Sheet
Illiquid
+0.5
-0.5
+0.5 (illiquid guarantees)
Semi-liquid
0
0
0
Liquid
-0.5
+0.5
-0.5 (liquid guarantees)
Intuition: - Banks create liquidity by transforming illiquid assets (+0.5) into liquid liabilities (+0.5) - Holding liquid assets (-0.5) or issuing illiquid liabilities (-0.5) destroys liquidity - This captures the Diamond-Dybvig insight: liquidity creation comes from maturity transformation
Step 3: Aggregate across all items
Total liquidity creation = Sum of (weight × dollar amount) across all balance sheet and off-balance sheet items
2.2.6.4 The Four Measures
Berger and Bouwman construct four measures:
CAT FAT (Category, includes off-balance sheet):
Classifies items by category (not maturity)
Includes off-balance sheet activities
Preferred measure — captures both on- and off-balance sheet liquidity creation
Classifies items by maturity (short-term vs. long-term)
Includes off-balance sheet activities
MAT NONFAT (Maturity, excludes off-balance sheet):
Classifies items by maturity
Excludes off-balance sheet activities
TipWhy Category over Maturity?
Berger and Bouwman argue that the category-based measure is preferred because for liquidity creation, banks’ ability to securitize or sell loans is more important than loan maturity. A 30-year mortgage may be illiquid by maturity but quite liquid if it can be easily securitized and sold.
2.2.6.5 What They Find
1. Liquidity creation is substantial and growing
Bank liquidity creation exceeded $2.8 trillion in 2003
Liquidity creation increased every year from 1993 to 2003
This confirms that liquidity creation is a quantitatively important activity
2. Large banks dominate liquidity creation
Large banks create the most liquidity in absolute terms
Banks that are members of multibank holding companies create more liquidity
Retail banks create more liquidity than wholesale banks
Recently merged banks create more liquidity
3. Capital and liquidity creation: Size matters
The relationship between bank capital and liquidity creation depends on bank size:
Large banks: Positive relationship — more capital → more liquidity creation
Small banks: Negative relationship — more capital → less liquidity creation
ImportantKey Insight
This differential effect by size suggests that capital requirements may have different impacts on liquidity creation for large vs. small banks. For large banks, capital provides a cushion that allows them to take on more liquidity risk. For small banks, capital crowds out liquid liabilities (deposits), reducing their ability to create liquidity.
2.2.6.6 Connecting Theory to Evidence
The Berger-Bouwman measures operationalize the Diamond-Dybvig theory:
Theory: Banks create liquidity by offering liquid deposits backed by illiquid loans
Measurement: Berger-Bouwman quantify this transformation with their weighting scheme
Evidence: U.S. banks created $2.8 trillion of liquidity by 2003, confirming the empirical importance of the theoretical mechanism
This measurement framework has become the standard in the literature and is used in hundreds of subsequent studies of bank liquidity creation, financial crises, monetary policy, and regulation.
2.2.7 The Role of the Term Structure
The Diamond-Dybvig model also provides insights into why we see different maturities of securities.
Key observation: The model generates a natural term structure
Short-term deposits: Liquid, redeemable at \(t=1\), pay \(c_1\)
Long-term deposits: Illiquid, mature at \(t=2\), pay \(c_2\)
Returns:
Return on short-term: \(c_1 / 1 = c_1\)
Return on long-term: \(c_2 / 1 = c_2\)
Since \(\frac{u'(c_1^*)}{u'(c_2^*)} = R\) and \(u\) is concave:
If \(R > 1\), then \(c_2^* > c_1^*\)
NoteUpward-Sloping Term Structure
Long-term deposits pay higher returns because:
They are less liquid (can’t withdraw early)
They fund the higher-return long-term investments
Patient depositors are compensated for giving up liquidity
2.2.7.1 Implications for Bank Liabilities
The model justifies the bank’s liability structure:
Demand deposits:
Short-term, liquid
Payoff: \(c_1\) if withdrawn at \(t=1\)
Serves impatient depositors
Time deposits / CDs:
Longer maturity
Payoff: \(c_2\) at \(t=2\)
Serves patient depositors
Higher return compensates for illiquidity
The mix of maturities:
Fraction \(\lambda\) of depositors optimally choose demand deposits
Fraction \(1-\lambda\) optimally choose time deposits
Bank structures liabilities to match depositor preferences
TipModern Banking
In practice, banks offer a menu of deposit products with different:
Liquidity (instant withdrawal vs. penalty for early withdrawal)
Returns (lower for more liquid, higher for less liquid)
This menu allows depositors to self-select based on their liquidity needs.
2.3 Can Markets Replicate Banking?
2.3.1 Jacklin (1987): Trading and Restrictions
NoteJacklin (1987)
Jacklin, Charles J. (1987). “Demand Deposits, Trading Restrictions, and Risk Sharing.” In Contractual Arrangements for Intertemporal Trade, ed. E.C. Prescott and N. Wallace, University of Minnesota Press.
Main question: What happens if agents can trade securities at \(t=1\), after types are revealed? Can markets replicate the DD banking contract?
So far we’ve compared two extremes:
Autarky
No coordination
Each agent self-insures
Inefficient allocation
DD Banking Contract
Perfect coordination
Optimal insurance
First-best allocation
Jacklin asks: what about an intermediate arrangement — markets open at \(t=1\) where agents can trade claims? Where does this fall relative to autarky and the DD optimum?
Preview: Trading provides partial insurance — better than autarky but worse than the DD contract. And crucially, if trading is allowed alongside the DD contract, it unravels the bank’s insurance.
2.3.2 Market Equilibrium with Trading
Timeline with trading:
What can be traded?
Claims to consumption at \(t=2\)
Impatient types want to sell their \(t=2\) claims
Patient types want to buy additional \(t=2\) claims
All trades must satisfy no-arbitrage pricing
At \(t=1\), let \(p\) be the price of consumption at \(t=2\) in terms of consumption at \(t=1\).
No-arbitrage bounds:
ImportantArbitrage-Free Pricing
The price \(p\) must satisfy: \[
\frac{1}{R} \leq p \leq 1
\]
Recall: \(p\) is the price you pay at \(t=1\) for a claim that pays 1 unit at \(t=2\).
Upper bound: \(p \leq 1\)
The liquid asset acts as a storage technology: 1 unit at \(t=1\) becomes 1 unit at \(t=2\) (it doesn’t grow, but it doesn’t shrink either). So why would you pay more than 1 today for a claim that delivers 1 tomorrow, when you can just hold onto your unit in the liquid asset and get the same 1 for free? No one would buy at \(p > 1\). So \(p \leq 1\).
Lower bound: \(p \geq \frac{1}{R}\)
If the price is too low, the implied return on buying a claim (\(\frac{1}{p}\)) exceeds what the real economy can produce (\(R\)). That can’t persist in equilibrium. So \(p \geq \frac{1}{R}\).
Equilibrium determination:
Impatient agents supply \(t=2\) claims (want to consume at \(t=1\))
Patient agents demand \(t=2\) claims (want to consume at \(t=2\))
Price \(p\) clears the market
2.3.2.1 Deriving the Market Allocation
Suppose each agent starts with the symmetric portfolio: - \(x\) in liquid assets - \(y = 1-x\) in illiquid assets
At \(t=1\):
Each agent has: - Liquid resources: \(x\) units - Illiquid claims: \((1-x) \cdot R\) units at \(t=2\)
Trading behavior:
Impatient types (fraction \(\lambda\)):
Want to consume everything at \(t=1\)
Sell all \(t=2\) claims at price \(p\)
Total consumption: \(c_1^M = x + p(1-x)R\)
Patient types (fraction \(1-\lambda\)):
Want to consume everything at \(t=2\)
Buy \(t=2\) claims from impatient types
Total consumption: \(c_2^M = \frac{x}{p} + (1-x)R\)
2.3.2.2 Market Clearing Condition
Supply of \(t=2\) claims (from impatient types): \[
\lambda \cdot (1-x) R
\]
Demand for \(t=2\) claims (from patient types):
Patient types have \(x\) units at \(t=1\) to spend on buying claims: \[
(1-\lambda) \cdot \frac{x}{p}
\]
Market clearing:\[
\lambda (1-x) R = (1-\lambda) \frac{x}{p}
\]
Solving for \(p\): \[
p = \frac{(1-\lambda) x}{\lambda (1-x) R}
\]
Anticipating the market equilibrium, agents at \(t=0\) choose \(x\) to maximize: \[
\max_{x} \quad \lambda u(c_1^M) + (1-\lambda) u(c_2^M) = \lambda u\left(\frac{x}{\lambda}\right) + (1-\lambda) u\left(\frac{(1-x)R}{1-\lambda}\right)
\]
First-order condition:\[
u'\left(\frac{x}{\lambda}\right) = R \cdot u'\left(\frac{(1-x)R}{1-\lambda}\right)
\]
Which simplifies to: \[
\frac{u'(c_1^M)}{u'(c_2^M)} = R
\]
This is the same condition as the DD optimum! So why isn’t the market allocation identical to DD?
2.3.3 The Incentive Compatibility Constraint
The market imposes an additional constraint that the DD bank does not face.
In DD without markets, a patient depositor has no reason to withdraw early — they don’t want to consume at \(t=1\), and there’s nothing else they can do with the money. So they wait and collect \(c_2\).
With open markets, a patient depositor now has a profitable alternative:
Pretend to be impatient — withdraw \(c_1\) from the bank at \(t=1\)
Take that \(c_1\) and buy \(t=2\) claims on the market at price \(p\)
Receive\(\frac{c_1}{p}\) units of consumption at \(t=2\)
If \(\frac{c_1}{p} > c_2\), the patient depositor is better off pretending to be impatient. For the contract to survive, patient types must not gain from this deviation: \[
c_2 \geq \frac{c_1}{p} \implies c_1 \leq p \cdot c_2
\]
With \(p = \frac{1}{R}\), this becomes \(c_1 \leq \frac{c_2}{R}\).
But the DD optimum has \(c_1^* > \frac{c_2^*}{R}\) (front-loaded). This violates the constraint — if markets are open, patient depositors would withdraw early and reinvest, unraveling the DD contract.
The market is therefore constrained to allocations satisfying \(c_1 = \frac{c_2}{R}\). Combined with the budget constraint, this pins down the equilibrium:
ImportantMarket Equilibrium
The IC constraint binds and pins down the allocation:
In autarky, each agent hedges individually by splitting between liquid and illiquid. Impatient types get only \(c_1^A = x < 1\) (their liquid holdings), while their illiquid investment is wasted. Patient types get \(c_2^A = x + (1-x)R\).
With market trading, impatient types can now sell their illiquid claims at \(t=1\). This raises their consumption from \(c_1^A < 1\) all the way to \(c_1^M = 1\) — a real improvement.
But the market cannot push \(c_1\) above 1, because that would require \(c_1 > c_2/R\) (front-loading), which violates no-arbitrage. Patient depositors would mimic impatient ones and exploit the difference.
The DD bank breaks through this barrier by restricting trading: it offers \(c_1^* > 1\), cross-subsidizing from patient to impatient depositors. This is full insurance.
In summary: \(c_1^A < 1 = c_1^M < c_1^*\). Markets get you to the no-arbitrage boundary; only the bank can go beyond it.
Autarky — Markets: Trading lets impatient types sell illiquid claims, raising \(c_1\) from below 1 to exactly 1
Markets — DD: The bank breaks the no-arbitrage barrier, pushing \(c_1\) above 1 through cross-subsidization
But: DD requires trading restrictions — otherwise patient types exploit the front-loading and the contract unravels
Modern relevance:
Shadow banking: When deposits become tradable securities, we move from DD toward the market equilibrium
Money market funds: Redeemable shares are closer to the market solution than traditional banking
Narrow banking proposals: Holding only liquid assets gives market-like outcomes but loses the insurance benefit
2.3.5 Diamond’s Response
Diamond, Douglas W. (1987). “Discussion of Jacklin’s Paper.” In Contractual Arrangements for Intertemporal Trade, ed. E.C. Prescott and N. Wallace, University of Minnesota Press.
Diamond acknowledges Jacklin’s point but argues that trading restrictions are not an artificial limitation — they are exactly what makes banking valuable:
The demand deposit contract is deliberately designed to be illiquid
This illiquidity is what enables the insurance: it prevents patient depositors from exploiting the front-loading
Without this restriction, the bank collapses to the market outcome and loses its purpose
2.3.6 How Banks Restrict Trading in Practice
Banks use several mechanisms to implement the DD contract:
1. Non-Transferable Deposits
Traditional bank deposits:
Deposits are in your name only
Cannot sell to others
Can only withdraw, not transfer
Prevents secondary markets
Why this works:
No market price for deposits
Patient types cannot profitably withdraw and resell
The illiquidity is a feature, not a bug
2. Penalties for Early Withdrawal
Certificates of Deposit (CDs) impose penalties that eliminate the arbitrage incentive.
Recall the problem: if a secondary market exists, a patient depositor can withdraw \(c_1^*\) at \(t=1\) and buy \(t=2\) claims at price \(p = \frac{1}{R}\), receiving \(\frac{c_1^*}{p} = c_1^* \times R > c_2^*\) at \(t=2\) — strictly better than waiting. (Note: the return \(R\) comes from buying market claims, not from direct reinvestment — at \(t=1\), the only available asset is storage at \(r_2\).)
The bank sets an early withdrawal penalty equal to the arbitrage gain: \[
\text{Penalty} = c_1^* \times R - c_2^*
\]
Now a patient depositor who withdraws early gets \(c_1^*\), reinvests to earn \(c_1^* \times R\), but pays the penalty — netting exactly \(c_2^*\). Since this is the same as waiting, there is no incentive to deviate. Only truly impatient depositors — who need to consume at \(t=1\) and cannot reinvest — withdraw early.
3. Monitoring and Verification
Partial verification of liquidity needs:
Medical emergencies, job loss — allowed
“Frivolous” withdrawals — discouraged
Not fully enforceable (private information) but helps
TipThe Consensus
Jacklin is right about the mechanism:
Trading restrictions are necessary for optimal insurance
Tradable deposits lead to market equilibrium, not DD optimum
But this creates a new problem:
The same restrictions that enable insurance also make banks fragile
If depositors lose confidence, coordination failures can occur
This leads to bank runs (next chapter)
2.3.7 Application: The 2008 Financial Crisis
The Jacklin critique became reality in 2008:
Traditional banking: - Non-tradable deposits - Penalties for early withdrawal - FDIC insurance - Relatively stable
Shadow banking (money market funds, repo): - Tradable securities (can sell to others) - No penalties for redemption - No deposit insurance - Highly vulnerable
What happened:
September 2008: Reserve Primary Fund “broke the buck” - Held Lehman Brothers debt - Net asset value fell below $1 - Massive withdrawals (run!) - Spread to other money market funds
The problem: Money market funds offered “bank-like” liquidity but with tradable shares
When investors got nervous, they could sell shares to others
No penalties for redemption — everyone can exit simultaneously
This validated Jacklin’s concern about tradable deposits
Government had to step in with emergency guarantees
ImportantThe Lesson
The 2008 crisis validated Jacklin’s insight:
Bank-like claims + Tradability = Fragility
If you want deposit-like insurance, you need deposit-like restrictions: - Non-tradability, OR - Penalties for early redemption, OR - Insurance/guarantees (covered in the next chapter)
You can’t have full tradability and full insurance.
2.3.8 Modern Debate: Narrow Banking
Narrow banking proposal: Banks hold only liquid, safe assets (Treasury bills)
Proponents argue: - Eliminates bank runs (assets perfectly liquid) - No need for deposit insurance - No systemic risk
Critics respond (Diamond & Dybvig view):
What we lose:
Liquidity transformation
No maturity mismatch — no liquidity creation
Back to market equilibrium
Credit creation
Banks can’t lend to businesses
Monitoring function lost
Insurance value
With only safe assets, no need for insurance
But also no value creation!
The tradeoff:
Narrow banking is safe but: - Equivalent to everyone holding T-bills individually - No better than autarky with safe assets - Loses the whole point of banking!
Bottom line: The Jacklin debate shows why traditional banking (with restrictions) dominates both: - Pure markets (insufficient insurance) - Narrow banking (no liquidity creation)
NoteKey Takeaway
The banking structure we observe — illiquid deposits funding illiquid loans — is not an accident:
It’s necessary for optimal insurance (DD result)
It requires trading restrictions (Jacklin insight)
It creates fragility that needs policy support (Diamond-Dybvig)
But alternatives (pure markets, narrow banking) are worse!
2.4 Preview and Summary
2.4.1 Preview: The Dark Side of Banking
We have shown that banks create value by offering liquid deposits backed by illiquid assets. But the very trading restrictions that make this possible also create a vulnerability.
WarningBank Runs: Multiple Equilibria
The Diamond-Dybvig model has two equilibria:
Good equilibrium: Only impatient depositors withdraw at \(t=1\). The bank operates as designed, providing optimal insurance.
Bad equilibrium: All depositors attempt to withdraw at \(t=1\). The bank must liquidate long-term assets at a loss, and late withdrawers receive nothing. This is a self-fulfilling panic — it is rational to run if you believe others will run.
As Diamond (2007) notes, the bad equilibrium arises because of the sequential service constraint: the bank pays depositors in the order they arrive, and it cannot distinguish impatient from patient depositors.
Possible solutions (to be studied in detail in the next chapter):
Suspension of convertibility: The bank stops paying after a certain number of withdrawals, eliminating the incentive to run
Deposit insurance: A government guarantee removes the fear of being a late withdrawer
These solutions preserve the insurance benefits of banking while eliminating (or reducing) the fragility. We will analyze them formally in the next chapter.
2.4.2 Key Takeaways
ImportantMain Insights
Banks as liquidity providers: Banks exist to insure depositors against idiosyncratic liquidity shocks
Risk aversion creates the demand for liquidity: Risk-neutral investors would prefer illiquid assets; it is the concavity of utility that makes liquidity valuable (Diamond’s numerical example: EU = 0.391 > 0.375)
Maturity transformation: Banks optimally engage in maturity mismatch because:
Depositors face uncertain timing of consumption needs
Law of large numbers makes aggregate withdrawals predictable
This allows efficient investment in high-return illiquid assets
The optimal deposit contract: Satisfies \(u'(c_1^*) = R \cdot u'(c_2^*)\) — marginal utility of early consumption equals marginal cost of liquidity
Asset management of liquidity: The bank’s ability to manage its portfolio (holding short-term assets for predictable withdrawals, long-term assets for the rest) strictly dominates what any individual can achieve
Trading restrictions are essential: Without them, markets unravel the banking contract (Jacklin, 1987)
Term structure: Different deposit maturities emerge naturally to serve depositors with different liquidity needs
What we haven’t covered yet:
What happens if depositors panic? — Bank runs (next chapter)
What if aggregate liquidity demand is uncertain? — Aggregate liquidity shocks
How do banks monitor borrowers? — Asset side of banking
What regulations are needed? — Banking regulation and policy
2.4.3 References
2.4.3.1 Primary Sources
Diamond, D. W., & Dybvig, P. H. (1983). Bank runs, deposit insurance, and liquidity. Journal of Political Economy, 91(3), 401-419.
Diamond, D. W. (2007). Banks and liquidity creation: A simple exposition of the Diamond-Dybvig model. Federal Reserve Bank of Richmond Economic Quarterly, 93(2), 189-200.
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.
Diamond, D. W. (1987). Discussion of Jacklin’s paper. In E. C. Prescott & N. Wallace (Eds.), Contractual Arrangements for Intertemporal Trade. University of Minnesota Press.
2.4.3.2 Related Papers
Berger, A. N., & Bouwman, C. H. S. (2009). Bank liquidity creation. Review of Financial Studies, 22(9), 3779-3837.
Bryant, J. (1980). A model of reserves, bank runs, and deposit insurance. Journal of Banking & Finance, 4(4), 335-344.
Diamond, D. W. (1984). Financial intermediation and delegated monitoring. Review of Economic Studies, 51(3), 393-414.
Haubrich, J. G., & King, R. G. (1990). Banking and insurance. Journal of Monetary Economics, 26(3), 361-386.
2.4.3.3 Textbooks
Tirole, J. (2006). The Theory of Corporate Finance. Princeton University Press. Chapter 12.
Freixas, X., & Rochet, J. C. (2008). Microeconomics of Banking (2nd ed.). MIT Press.