Diamond–Rajan Liquidity Creation and Bank Fragility Model

The Diamond–Rajan Liquidity Creation and Bank Fragility Model is a landmark incomplete-contracts theory of financial intermediation, bank run fragility, and bank capital structure in business and economics theory, developed by economists Douglas W. Diamond and Raghuram G. Rajan in their 2001 Journal of Political Economy article "Liquidity Risk, Liquidity Creation, and Financial Fragility: A Theory of Banking" and their companion 2000 Journal of Finance paper "A Theory of Bank Capital." Building on Oliver Hart and John Moore's (1994) theory of incomplete contracts and debt renegotiation under the inalienability of human capital, Diamond and Rajan asked why commercial banks simultaneously hold illiquid relationship loans on the asset side and issue fragile, run-prone demand deposits on the liability side. In their model, a relationship lender acquires specific human capital during the course of lending that allows the lender to extract a higher repayment c₁ from an entrepreneurial borrower than any arm's-length market buyer could recover (c₂ < c₁). Because the relationship lender cannot commit ex ante not to withhold this specialized loan-collection skill at an interim date, financing the loan with outside equity or non-run-prone debt allows the lender to threaten a holdup and bargain outside investors down to a fraction of the loan's value, rendering the loan illiquid. Issuing demandable deposits subject to a sequential service constraint eliminates the banker's holdup power: if the banker attempts to renegotiate deposit face value d₀ downward, depositors stage a collective bank run that forces early liquidation of the loan and disintermediates the banker, committing the banker to pass 100% of relationship collection value c₁ through to outside depositors.

Why Financial Assets Become Illiquid Under Hart–Moore Incomplete Contracts

In classical general equilibrium theory, any asset generating a known future cash flow is liquid: its holder can sell the asset or borrow against 100% of its discounted present value at any interim date. Why, then, are commercial loans, entrepreneurial projects, and relationship credits illiquid in practice—selling only at fire-sale discounts if liquidated early?

Following Hart and Moore (1994), Diamond and Rajan traced asset illiquidity to the inalienability of specialized human capital:

1. Entrepreneurial Human Capital (First Layer of Inalienability): An entrepreneur requires 1 unit of capital at date t = 0 to build a firm producing cash flow C₁ at date t = 1 using the entrepreneur's specialized managerial knowledge. Because courts cannot compel an entrepreneur to supply specialized human capital, the entrepreneur can threaten to walk away before date t = 1 unless the financier renegotiates loan repayment downward.
2. Relationship Lender's Collection Skill (Second Layer of Inalienability): During loan origination and monitoring, the original Relationship Lender (Banker) learns how to redeploy the borrower's physical collateral in its best alternative use, recovering a relationship liquidation value c₁ < C₁. By contrast, any arm's-length third-party buyer lacking this relationship-specific knowledge can liquidate the assets only for a lower market fire-sale value c₂ < c₁ < C₁:

Because the entrepreneur knows that the relationship lender can seize and redeploy the assets for c₁ (whereas an outside buyer can recover only c₂), the entrepreneur pays c₁ to the relationship lender at date t = 1, but at most c₂ to an unskilled third-party creditor.

The Relationship Lender's Bilateral Holdup Problem

Suppose that at an interim date t = ½, the relationship lender needs to raise cash from outside investors (or raises funds at t = 0 from passive outside equity holders or a single long-term bondholder):

• Direct Sale of the Loan Yielding Only c₂: Selling the loan outright to an unskilled buyer yields at most c₂ < c₁, losing the relationship surplus c₁ − c₂ > 0.
• Financing via Outside Equity or a Single Renegotiable Lender: If the relationship lender retains management and issues outside equity or debt to a single outside investor, the lender's collection skill is inalienable: just before date t = 1, the lender can threaten to walk away from collecting the loan unless paid a collection fee, leaving the investor with c₂.
• Nash Bargaining Over the Relationship Surplus (c₁ − c₂): Nash bargaining (with investor bargaining weight θ ∈ [0, 1]) splits the relationship quasi-rent c₁ − c₂ between the relationship lender and the outside investor:

Because the relationship lender cannot commit ex ante not to hold up the outside investor at date t = 1, the loan is illiquid: the lender can pledge or raise at most ½(c₁ + c₂) < c₁ from non-bank investors.

Collective Action Problem and the Sequential Service Constraint

Diamond and Rajan (2001) proved that a commercial bank resolves this two-sided illiquidity problem by financing relationship loans with dispersed demand deposits (n ≫ 1 small depositors) governed by a sequential service constraint (first-come, first-served par redemption, as in the Diamond–Dybvig model and the Calomiris–Kahn model).

Suppose the relationship lender (organized as a Bank) issues demand deposits with an aggregate promised face value d₀ = c₁, giving each of n depositors a claim to d₀/n redeemable on demand at any interim date t ∈ [½, 1]:

1. Right to Trigger Asset Seizure: If a depositor demands redemption at date t < 1 and the bank lacks sufficient cash on hand, the bank must sell or liquidate loan assets at the market value c₂ on a first-come, first-served basis until the bank's assets are exhausted.
2. Partial Recovery in a Run (c₂/c₁ < 1): If all depositors run simultaneously before date t = 1, the bank is forced to liquidate the loan early, disintermediating the banker and reducing total recoverable value from c₁ to the market value c₂ < d₀ = c₁. Consequently, only the first c₂/d₀ = c₂/c₁ < 1 fraction of depositors in the withdrawal queue receive full par d₀/n, while the remaining 1 − c₂/c₁ depositors at the back of the line receive zero.

Why the Banker Cannot Renegotiate Demand Deposits Downward

Now trace what happens if the banker attempts at date t = ½ to hold up the depositors by offering to write down the aggregate deposit claim from d₀ = c₁ to ½(c₁ + c₂). Consider the strategic decision of an arbitrary depositor i ∈ {1, ..., n} when confronted with the banker's renegotiation threat:

• If depositor i believes that other depositors will accept the write-down or wait, depositor i can unilaterally go to the teller window and demand full par redemption d₀/n = c₁/n, forcing the bank to sell enough loan assets at market value to pay depositor i in full (100 cents on the dollar > ½(c₁ + c₂)/c₁).
• If depositor i believes that other depositors are going to run to the teller window, waiting yields zero (since total run liquidation value c₂ < d₀), whereas joining the run gives depositor i a positive probability c₂/c₁ > 0 of standing near the front of the queue and recovering d₀/n.

Thus, whenever the banker attempts to renegotiate the deposit face value below d₀ = c₁, demanding withdrawal is a dominant strategy for every individual depositor. A collective bank run ensues, stripping the banker of control over the loan, forcing early liquidation at c₂, and driving the banker's own collection rent to zero:

Knowing that any attempt to hold up depositors for a collection rent triggers a bank run that reduces the banker's rent to zero, the banker never attempts to renegotiate in equilibrium. Instead, the banker collects the full relationship value c₁ from the entrepreneur at date t = 1 and pays out d₀ = c₁ to the depositors.

The Diamond–Rajan Fragility Theorem (2001): Financial fragility (vulnerability to a collective, value-destroying bank run under sequential service) is essential for liquidity creation. Because a bank run disintermediates the banker and destroys the banker's relationship rent, issuing run-prone demand deposits (d₀ = c₁) acts as a hard commitment device that eliminates the banker's ex-post holdup power and increases the pledgeable value of illiquid relationship loans from ½(c₁ + c₂) to 100% of c₁.

A distinctive strength of the Diamond–Rajan framework (complementing the liquidity-creation metrics of Berger & Bouwman, 2009, and Kashyap, Rajan, & Stein, 2002) is that it explains why commercial banking combines deposit-taking and relationship lending inside the same institution, creating liquidity on both sides of the balance sheet:

1. Liquidity Creation for Depositors (Liability Side)

Suppose the initial depositors at date t = 0 face a probability α > 0 of suffering an idiosyncratic liquidity shock at date t = ½. By organizing as a bank financed with demand deposits d₀ = c₁, when initial depositors withdraw at date t = ½ for liquidity needs, the bank can issue fresh demand deposits of face value d₀ = c₁ to new incoming depositors at date t = ½ without liquidating the underlying relationship loan or losing value to banker holdup. The bank transforms an illiquid relationship claim into a liquid medium of exchange.

2. Liquidity Creation for Entrepreneurial Borrowers (Asset Side)

Because a relationship bank funded by demand deposits can refinance withdrawals from the broader pool of depositors up to the full pledgeable value of its relationship collection skill, the bank can extend long-term credit and committed lines of credit to entrepreneurs without forcing premature liquidation of viable long-term projects.

When Uncertainty Over Loan Returns (c̃₁) Triggers Fundamental Runs

In "A Theory of Bank Capital" (Journal of Finance, 2000), Diamond and Rajan introduced aggregate uncertainty over the relationship value of bank loans: suppose that at date t = 1, the relationship collection value of the bank's loan portfolio is a random variable c̃₁ ∈ {c₁^L, c₁^H}, where the high state c₁^H occurs with probability p and the low recession state c₁^L < c₁^H (with market liquidation value c₂^L < c₁^L) occurs with probability 1 − p.

A stochastic loan value c̃₁ creates a trade-off for bank leverage d₀:

1. High-Leverage All-Deposit Bank (d₀ = c₁^H): Setting demand deposits equal to the high-state relationship value d₀ = c₁^H leaves the banker with zero holdup rent in state H. However, if the low state c₁^L < d₀ is realized, the bank cannot pay d₀ even when the banker exerts full collection effort. Because demand deposits cannot be renegotiated downward without triggering a sequential-service panic, the realization of c₁^L < d₀ forces an equilibrium bank run, losing the low-state relationship surplus c₁^L − c₂^L > 0.
2. Low-Leverage Safe Bank (d₀ = c₁^L) Financed Partly by Bank Capital (Equity): Suppose instead the bank limits demand deposits to the recession-proof level d₀ = c₁^L and finances the remainder of the balance sheet with outside bank equity (capital). Now the bank never suffers a run in state L (preserving c₁^L − c₂^L). However, in the boom state H where the loan is worth c₁^H > d₀, once the banker pays off d₀ = c₁^L to the depositors, the banker faces soft, renegotiable outside equity holders over the remaining surplus c₁^H − c₁^L and extracts half of the equity residual:

The Fundamental Trade-Off of Bank Capital Regulation

This yields the Diamond–Rajan Bank Capital Trade-Off:

• Higher Bank Capital (Lower Deposit Leverage d₀) acts as a buffer that reduces the probability of costly fundamental bank runs in downturns (saving (1 − p)(c₁^L − c₂^L));
• Lower Bank Capital (Higher Deposit Leverage d₀) increases discipline on the banker in normal times and increases liquidity creation and total pledgeable loan value (raising state-H pledgeability ½(c₁^H + d₀) by ½ per dollar of deposits d₀).

Why can't a relationship banker sign a binding contract promising not to hold up outside investors?

Under the Hart–Moore incomplete-contracts framework (1994), specialized human capital—such as a relationship banker's specific skill in monitoring a borrower and redeploying its physical collateral for c₁ > c₂—is inalienable. Because courts cannot compel a human being to supply specialized effort or prevent two consenting parties (the banker and a single lender or equity board) from mutually renegotiating a contract when the alternative is a lower payoff c₂, any bilateral contract is vulnerable to ex-post renegotiation and holdup unless structured so that a renegotiation attempt triggers a collective bank run.

Why does a collective bank run occur whenever the banker tries to renegotiate demand deposits?

Because demand deposits are held by dispersed depositors and paid on a first-come, first-served basis (sequential service), if the banker offers to pay less than the promised par value d₀, each individual depositor has a unilateral incentive to refuse the write-down, walk up to the teller window, and demand 100% par value d₀/n by forcing the bank to sell loan assets at market value c₂. Because c₂ < d₀, the bank cannot pay everyone in liquidation, so every depositor rushes to the front of the line simultaneously, disintermediating the banker and driving the banker's collection rent to zero.

How does the Diamond–Rajan (2001) model differ from the Diamond–Dybvig (1983) model?

In the Diamond–Dybvig model (1983), bank assets are technological constants, there is no banker agency or holdup problem, and bank fragility is an undesirable side effect of providing liquidity insurance to households. By contrast, in the Diamond–Rajan model (2001), bank loans are illiquid due to the inalienability of the banker's collection skill, and fragility is necessary to create liquidity: if a bank were prohibited from failing via unconditional suspension of convertibility, the banker would hold up depositors and liquidity creation would collapse.

Why does bank equity capital reduce liquidity creation in Diamond and Rajan (2000)?

Unlike demand depositors—who are induced by the sequential service constraint to run if offered less than d₀—outside equity holders have no right to demand fixed par redemption and cannot trigger a sequential-service run. Therefore, over any cash flow above the deposit threshold d₀, the relationship banker can threaten to withhold collection effort and bargain outside equity holders down to 50% of the residual surplus ½(c₁ − d₀). Replacing demand deposits with outside bank equity therefore lowers the fraction of relationship loan value c₁ that can be pledged to outside investors.

1. Diamond, D. W., & Rajan, R. G. (2001). "Liquidity Risk, Liquidity Creation, and Financial Fragility: A Theory of Banking." Journal of Political Economy, 109(2), 287–327. https://doi.org/10.1086/319552
2. Diamond, D. W., & Rajan, R. G. (2000). "A Theory of Bank Capital." The Journal of Finance, 55(6), 2431–2465. https://doi.org/10.1111/0022-1082.00296
3. Hart, O., & Moore, J. (1994). "A Theory of Debt Based on the Inalienability of Human Capital." The Quarterly Journal of Economics, 109(4), 841–879. https://doi.org/10.2307/2118350
4. Kashyap, A. K., Rajan, R., & Stein, J. C. (2002). "Banks as Liquidity Providers: An Explanation for the Coexistence of Lending and Deposit-Taking." The Journal of Finance, 57(1), 33–73. https://doi.org/10.1111/1540-6261.00415
5. Berger, A. N., & Bouwman, C. H. S. (2009). "Bank Liquidity Creation." The Review of Financial Studies, 22(9), 3779–3837. https://doi.org/10.1093/rfs/hhn104