The demand deposit is contracted the following way: Each demand deposit contract has a face value of one unit of fiat money. Depositors are free to choose whether to demand early withdrawal in period 1 (early withdrawers), or wait until period 2 (late withdraw- ers). Bankers promise to payR1∗per dollar deposited to early withdrawers, andR∗2 to late withdrawers, whereR2∗ > R∗1 > 1. Specifically, the payment consists of two parts: inter- est payment (short-termr∗1 and long-termr∗2) and principal (par value of the deposit, one unit of fiat money). By definition,R∗1 = 1+r∗1 and R2∗ = 1+r2∗. By settingR∗1 = c1∗ and
R∗2 =c∗2, the demand deposit delivers the first-best allocation in an incentive-compatible way.
But due to maturity mismatch and liquidation loss, bankers cannot fully honor their promises should all depositors demand early withdrawal. The actual payoffs, ˜R1and ˜R2, respectively, are dependent onωthe number of early withdrawers in period 1, andωithe
the number of early withdrawers in line before early withdraweriin period 1.
assets to meet the withdrawal need. A depositoriwill get ˜ R(ω,ωi) = R∗1 (early withdrawal) (1−ωR∗1)Ys 1−ω (late withdrawal)
If ω ≥ 1/R∗1, the status quo cannot be maintained since the withdrawal demand exceeds the market value of bank assets in period 1. A depositoriwill get
˜ R(ω,ωi) = R1∗ (early withdrawal, ωi<1/R∗1) 0 (withdrawal, ωi ≥1/R∗1)
The payoff structure characterizes the sequential service constraint on the financiers’ side. The fulfillment of withdrawal requests is on a first-come-first-served basis15.
A threshold ˆω is the number of early withdrawers which will make a patient depos- itor indifferent between demanding early withdrawal and waiting:
(1−ωRˆ ∗1)Ys 1−ωˆ = R ∗ 1, ωˆ = Ys−R∗1 (Ys−1)R∗1 <1
As analyzed in Diamond and Dybvig(1983), the good equilibrium can be achieved by setting R∗1 = c∗1 and R2∗ = (1−λR1∗)Ys
1−λ = c∗2. In the socially optimal equilibrium, only impatient depositors will withdraw money in period 1, and all patient depositors will wait until period 2.
Yet there is a strategic complementarity in demanding early withdrawals. The strate- gic complementarity leads to multiple equilibria, including a bad equilibrium, namely a self-fulfilling bank run. Expecting other people’s early withdrawal will induce a de- 15A variation incorporates suspension of convertibility and liquidation, where the depositors will line up before the bank opens and the financier will count the number first. It then will decide whether to open business (whenω < yj/r∗1) or to declare bankruptcy, liquidate its assets, and allocate the proceeds pro rata to all depositors (whenω ≥ yj/r∗1). This is more like the case of MMF, where same-day redemption requests are treated equally together after the market closes.
positor to do the same thing. All depositors will withdraw money in period 1 for fear that other people will do the same and nothing would be left in period 2. Even without the inefficiency of excessive risk taking, traditional banks are plagued by runs. Keeping excessive but still fractional reserves to satisfy liquidity needs will not work in times of crisis, and is extremely costly in good times.
In a world with safe projects, only self-fulfilling runs exist. Deposit insurance, which guarantees a paymentc1∗to each early withdrawer, can remedy the case, without having to pay out anything or incurring moral hazard problem. The insured demand deposit contract achieves first-best allocation.
The timeline is the following: In period 0, firms acquire funding from financial inter- mediaries to pay for capital and labor provided by households, who demand fiat money (“hoarder"), bank deposits (“depositor"), or shadow bank shares (“investor"). Fiat money is used as unit of account of consumption goods.
In period 1, households find out their preference types. Impatient households will withdraw deposits, redeem shares or use fiat money to buy consumption goods right away, while patient ones will decide whether to do the same or wait until next period. The financiers will decide how many projects to liquidate in order to satisfy withdrawal or redemption requests and whether the status quo (full debt repayment or stable NAV) can be maintained. If so, enter into period 2. If not, then the financiers will have to declare bankruptcy/breaking the buck, and liquidate all assets and pay back to the de- positors/investors in a pro rata manner.
In period 2, the returns of projects that haven’t been liquidated (if any) are realized, and the financiers will pay residual value to depositors/investors who choose to wait until this period.
3.3.3.2 The Equivalence
The capital structure of financial institution doesn’t matter in terms of creating liquidity or achieving socially optimal allocation as long as the following conditions are satisfied: (1) Households are identical ex ante, (2) No aggregate uncertainty, (3) The fraction of impatient households are known so optimal dividend policy can be devised, (4) Demand deposits are protected by insurance, (5) Bankruptcy and liquidation are resolved in a costless, timely manner, (6) The secondary market is frictionless. Bank debt contract and fund equity contract are equivalent and trading in the secondary market or directly with the fund also do not matter.
Suppose a financial intermediary issues both demand deposit contracts and equity shares contracts. In period 1, if a household is revealed to be impatient, s/he can either withdraw deposits and receive R∗1 = c∗1, or request redemption/trade the ex-dividend share in the secondary market and get backd1+ϕor d1+√. All are equal toc∗1. If the household is patient, then s/he will either wait until period 2 and receiveR∗2 =c∗2, or use the dividend to purchase more shares directly with the fund or in the secondary market in period 1 and redeem shares in period 2, still getting(1+d1/ϕ)d2 = (1+d1/p)d2 =c2∗. Since the payoff structures are the same, households in period 0 would like to pay identical prices for these contracts. The no-arbitrage condition excludes different pricing of the contracts.