23. Diamond and Dybvig's Model (1983)
本组导读：银行挤兑（Bank Run） 美国大萧条期间（1929–1933），累计实际 GDP 下降 34%、价格水平下降 24%、银行存款下降 48.2%。美联储当时（1910 年代）已成立，但尚无存款保险。
在引入存款保险机制之前，每当危机逼近就会出现银行账户的资金外逃，因为人们只想持有现金、以免银行倒闭时蒙受损失。后来引入存款保险，银行账户被视为安全资产，几乎不再发生挤兑。但类似问题仍在重演：2008 金融危机中，美联储未对货币市场账户提供保险，纽约联储当时单方面宣布对货币账户提供保险。
于是值得关注的问题是：为何会有银行挤兑？有无办法有效避免它？下面这个简单模型给出对这两个问题的解答。
23. Diamond 与 Dybvig 模型（1983，Journal of Political Economy）
23.1 设定
模型的主要特征：
- 共三期：\(T=0\)、\(T=1\)、\(T=2\)。
- 只有一种技术，家庭、银行、保险公司都相同且可用：
- 在 \(T=0\) 每存入 1 单位，若在 \(T=1\) 取出则偿付 1 单位，若在 \(T=2\) 取出则偿付 \(R\) 单位，其中 \(R>1\)。技术现金流如下表：
| 技术现金流 | \(T=0\) | \(T=1\) | \(T=2\) |
|---|---|---|---|
| 若 \(T=1\) 取出 | $-1$ | $+1$ | \(0\) |
| 若 \(T=2\) 取出 | $-1$ | \(0\) | \(R>1\) |
- 银行：
- 在 \(T=0\) 吸收家庭存款，投资于技术（长期、即两期内有生产力）；
- 提供期限转换（maturity transformation）：把资金贷给长期活动，同时吸收面临短期流动性冲击的家庭存款（流动性冲击指使家庭想在短期即 \(T=1\) 消费的冲击）。
- 家庭：
- $[0,1]$ 上的连续统家庭；
- 每个家庭在 \(T=0\) 获得禀赋 \(1\)，并在 \(T=0\) 决定投向：
- \(T=0\) 自己储藏：只在 \(T=1\) 或 \(T=2\) 获得恰如技术所定的偿付（该技术对家庭、银行、保险公司皆相同；后文给出最优选择）；
- \(T=0\) 购买保险保单：在 \(T=1\)、\(T=2\) 获得 \(T=0\) 保单所定的偿付；
- \(T=0\) 存入银行：在 \(T=1\)、\(T=2\) 获得 \(T=0\) 银行公告所定的偿付；
- 从 \(T=1\) 持币至 \(T=2\)：无利息、无折旧。
- 面临 i.i.d. 偏好冲击 \(\theta=1,2\)：
- 若 \(\theta=1\)（概率 \(\alpha\)），家庭想在 \(T=1\) 消费；
- 若 \(\theta=2\)（概率 \(1-\alpha\)），家庭想在 \(T=2\) 消费；
- 时序：冲击 \(\theta\) 在 \(T=1\) 之初揭示。
- 效用函数为
$$ u(c_1,c_2)=\begin{cases}u(c_1) & \text{if }\theta=1\\ \rho u(c_1+c_2) & \text{if }\theta=2\end{cases} $$
其中 \(c_1,c_2\) 分别是 \(T=1\)、\(T=2\) 的消费，\(\rho\) 是效用贴现因子；假设 \(\rho<1\) 且 \(R\rho>1\)。
为何 \(\theta=2\) 的效用 \(\rho u(c_1+c_2)\) 同时含 \(c_1\) 与 \(c_2\)？ 因为我们假设家庭可在 \(T=1\) 储藏商品、留待 \(T=2\) 消费而不赚利息（此时家庭不使用技术）。但实际上，由于冲击 \(\theta\) 在 \(T=1\) 之初揭示，\(\theta=2\) 的理性家庭不会在 \(T=1\) 从银行取款，除非发生银行挤兑。如此构造效用，是想让 \(\theta=2\) 的家庭能从 \(T=1\) 无息储藏至 \(T=2\)，从而在他们觉得资产在银行不安全时有动机在 \(T=1\) 挤兑。
23.2 两种可能均衡
- 好均衡：家庭不挤兑，只遵循其冲击，即仅当 \(\theta=1\) 时在 \(T=1\) 取款、\(\theta=2\) 时等到 \(T=2\)。
- 坏均衡：人人担心拿不回钱，故所有家庭（无论冲击 \(\theta=1\) 还是 \(\theta=2\)）一起在 \(T=1\) 去银行取款。于是银行破产，只向一部分家庭支付全额，其余家庭蒙受全部损失。注意这未必是银行之过，可能只是全体储户集体决定挤兑的集体厄运。
23.3 求解家庭的最优选择
23.3.1 家庭的自给（autarky）问题
先考虑自给情形。家庭只能在"\(T=1\) 取出并消费 1 单位"与"\(T=2\) 取出并消费 \(R\) 单位"之间选择，即
$$ u(c_1,c_2)=\begin{cases}u(1) & \text{if }\theta=1\\ \rho u(R) & \text{if }\theta=2\end{cases} $$
故期望效用为
$$ \mathbb{E}[u(c_1,c_2)]=\alpha u(1)+(1-\alpha)\rho u(R) $$
这是家庭只能被动接受的偿付——作为个体，它无法对该期望偏好作任何改变，也无法对冲冲击。
23.3.2 最佳保险保单
考虑由拥有同种技术的保险公司提供的完美保险保单。
- 保险公司有连续统的家庭客户，由大数定律可完全对冲冲击。
- 假设经济竞争，保险公司愿提供家庭所能得到的最优可行保单；否则因竞争性，不这样做的公司会失去所有客户、退出市场。
- 故完美保险保单求解如下效用最大化问题：
$$ \max_{c_1,c_2}\big[\alpha u(c_1)+(1-\alpha)\rho u(c_2)\big] $$
$$ \text{s.t.}\quad \alpha c_1+(1-\alpha)\left(\frac{c_2}{R}\right)=1 \tag{23.1} $$
其中 (23.1) 是使保险保单零利润的盈亏平衡条件。构造拉格朗日函数
$$ \mathcal{L}=\alpha u(c_1)+(1-\alpha)\rho u(c_2)+\lambda\left[1-\alpha c_1-(1-\alpha)\left(\frac{c_2}{R}\right)\right] $$
一阶条件分别为 \(\alpha u'(c_1^*)=\lambda\alpha\) 与 \((1-\alpha)\rho u'(c_2^*)=\lambda(1-\alpha)\frac{1}{R}\)，合并得
$$ \frac{(1-\alpha)\rho u'(c_2^*)}{\alpha u'(c_1^*)}=\frac{(1-\alpha)\frac{1}{R}}{\alpha}\ \Rightarrow\ u'(c_1^*)=\rho R\,u'(c_2^*) \tag{23.2} $$
由假设 \(R\rho>1\)，(23.2) 蕴含
$$
c_1^* 为进一步分析，聚焦于常相对风险厌恶（CRRA）效用 \(u(c)=\dfrac{c^{1-\sigma}}{1-\sigma}\)，\(\sigma>0\)。则 (23.2) 蕴含 $$
(c_1^*)^{-\sigma}=\rho R(c_2^*)^{-\sigma}\ \Rightarrow\ \left(\frac{c_2^*}{c_1^*}\right)^{\sigma}=\rho R\ \Rightarrow\ c_2^*=c_1^*\underbrace{(\rho R)^{\frac{1}{\sigma}}}_{>1}>c_1^*
$$ 为得到支持挤兑动机的结果，我们希望 \(c_1^*>1\)： $$
\begin{aligned}
\alpha\cdot1+(1-\alpha)\left(\frac{(\rho R)^{\frac{1}{\sigma}}}{R}\right)&<1\\
\Rightarrow\ (1-\alpha)\left(\frac{(\rho R)^{\frac{1}{\sigma}}}{R}\right)&<1-\alpha\\
\Rightarrow\ \frac{(\rho R)^{\frac{1}{\sigma}}}{R}&<1\\
\Rightarrow\ \rho^{\frac{1}{\sigma}}& $$
1 这对银行挤兑分析至关重要。 最后考虑银行。因这是竞争市场，银行若想拥有任何客户，就必须模仿最佳保险保单、向家庭提供同样的存款偿付。这里假设同样的特定 CRRA 效用与附加约束 (23.4)。故由 (23.5)，银行的报价为： 由上述讨论，我们在 (23.5) 得到 \(1 设 \(T=1\) 时一些人开始担心其资金安全，无论流动性冲击 \(\theta\) 为何都去取款。设人人如此，则因 \(c_1^*>1\)，人人都索取严格大于 \(1\) 的取款，这使银行无法支付太多。于是只有排在队首 \(f\) 比例的人（\(f=\dfrac{1}{c_1^*}<1\)）能拿到全额 \(c_1^*\)，其余人最终一无所获，这称为银行的序贯服务约束（sequential service constraint）。 而当其他人观察到这种行为，他们知道银行的序贯服务约束，故其最优行动是无论 \(\theta\) 如何都尽快去排队——毕竟，拿到一些总好过一无所获。 于是银行挤兑成为自我实现的预言（self-fulfilling prophecy），即上文讨论的坏均衡。 那么，我们能做什么来排除这个坏均衡？至少有两种办法： 所以，两种办法都能让好均衡实现。
23.3.3 银行的报价
23.4 银行挤兑问题
Group overview: Bank Run During the Great Depression in the U.S., cumulatively in 1929–1933, real GDP dropped 34%, price level dropped 24%, and bank deposits dropped 48.2%. The Fed had already been established (in 1910s), but there was no deposit insurance at that time.
Before the introduction of deposit insurance mechanism, there were flights out of bank account when crisis was approaching because people simply wanted to hold the money in their own hands to avoid loss when the bank failed. Later, when the deposit insurance was introduced, banking account is considered as safe asset, so there was almost no run on banks. But, similar problem is still repeating itself. During the 2008 financial crisis, there had been no insurance offered by the Fed on money market accounts. The New York Fed unilaterally announced insurance on money account at that time.
So, the question of interest is that, why do we have bank run? Is there any way to effectively avoid it? See the following simple model for answers to these two questions.
23. Diamond and Dybvig's Model (1983, Journal of Political Economy)
23.1 Set-up
The main feature of this model is:
- Three periods in total in this model: \(T=0\), \(T=1\), and \(T=2\).
- Only one technology in the economy, which is the same and available to household, bank and insurance company:
- For each 1 unit deposited at \(T=0\), the technology pays back 1 unit if taken out at \(T=1\), and pays back \(R\) units if taken out at \(T=2\) where \(R>1\). The technology cash flow is in the table below:
| Technology Cash Flow | \(T=0\) | \(T=1\) | \(T=2\) |
|---|---|---|---|
| if taken out at \(T=1\) | $-1$ | $+1$ | \(0\) |
| if taken out at \(T=2\) | $-1$ | \(0\) | \(R>1\) |
- The bank:
- takes deposits from the household at \(T=0\), and invest in the technology (which is productive in the long run, i.e. in 2 periods);
- provides the maturity transformation: it lends to long term activities, and takes deposit from household who is subject to short term liquidity shock (liquidity shock here means the shock that makes the household want to consume in short run, i.e. in period \(T=1\)).
- Household:
- There is a continuum of households on $[0,1]$.
- Each household gets endowment \(1\) at \(T=0\), and he can decide where to invest at \(T=0\):
- If he chooses to store it by himself at \(T=0\), then he can get payoff only at \(T=1\) or at \(T=2\) exactly as specified by the technology (common to every household, bank and insurance company; optimal shown later);
- If he chooses to invest in an insurance policy at \(T=0\), he will get payoffs at \(T=1\) and \(T=2\) as specified by the insurance policy at \(T=0\);
- If he chooses to deposit in a bank account at \(T=0\), he will get payoffs at \(T=1\) and \(T=2\) as specified by the bank announcement at \(T=0\);
- If he chooses to hold the money from \(T=1\) to \(T=2\), there will be no interest earned and no depreciation.
- Households are subject to i.i.d. preference shocks \(\theta=1,2\):
- if \(\theta=1\) (with probability \(\alpha\)), household wants to consume at \(T=1\);
- if \(\theta=2\) (with probability \(1-\alpha\)), household wants to consume at \(T=2\);
- timing: the shock \(\theta\) is revealed at the beginning of period \(T=1\).
- Then, the utility function is
$$ u(c_1,c_2)=\begin{cases}u(c_1) & \text{if }\theta=1\\ \rho u(c_1+c_2) & \text{if }\theta=2\end{cases} $$
where \(c_1,c_2\) are consumption at \(T=1\) and \(T=2\) respectively and \(\rho\) is the utility discount factor; we assume \(\rho<1\) and \(R\rho>1\).
Why does the \(\theta=2\) utility \(\rho u(c_1+c_2)\) include both \(c_1\) and \(c_2\)? Because we assume that the household can store the good at \(T=1\) to be consumed in \(T=2\) without earning interest (since household doesn't use the technology in that case). But actually, since the shock \(\theta\) is revealed at the beginning of period \(T=1\), a rational household whose \(\theta=2\) would not withdraw from the bank at \(T=1\) unless there is a bank run happening. So we constructed the utility in such way because we want to make \(\theta=2\) household be able to store from \(T=1\) to \(T=2\) without interest so that they have incentive to run on a bank at \(T=1\) if they feel the assets are not safe in bank.
23.2 Two possible equilibria
- The good equilibrium: household would not run on the bank, and they simply follow their shocks, i.e. withdraw at \(T=1\) only when \(\theta=1\), and wait until \(T=2\) if \(\theta=2\).
- The bad equilibrium: everyone is worried about not being able to get their money back, so all of the household, whether the shock is \(\theta=1\) or \(\theta=2\), go to the bank together to withdraw at \(T=1\). Then, the bank would go bankrupt, and only pays a fraction of household full amount while other household would suffer total loss. Note that this may or may not be the fault of the bank. It could be just the collective bad luck of all depositors as they collectively decide to run on the bank.
23.3 Solving for the optimal choice of the household
23.3.1 The household's autarky problem
First, consider the autarky case. He can only choose between withdraw and consume 1 unit at \(T=1\) and withdraw and consume \(R\) units at \(T=2\), i.e.
$$ u(c_1,c_2)=\begin{cases}u(1) & \text{if }\theta=1\\ \rho u(R) & \text{if }\theta=2\end{cases} $$
So, his expected utility is
$$ \mathbb{E}[u(c_1,c_2)]=\alpha u(1)+(1-\alpha)\rho u(R) $$
This is the payoff that the household take as given, and he cannot make any changes to this expected payoff because he is just an individual, so he cannot insure himself against the shocks.
23.3.2 The best insurance policy
Then, let's consider the perfect insurance policy that could be offered by a insurance company which has access to the same technology.
- Since the insurance company has a continuum of household customers, it can completely insure itself against shocks by Law of Large Numbers.
- We assume the economy is competitive, which makes the insurance companies want to offer the best feasible policy that the household could ever have. Otherwise, by competitiveness, the company that doesn't do this would lose all of its customer and leave the market.
- So, we assume that the perfect insurance policy solves the following utility maximizing problem for the household:
$$ \max_{c_1,c_2}\big[\alpha u(c_1)+(1-\alpha)\rho u(c_2)\big] $$
$$ \text{s.t.}\quad \alpha c_1+(1-\alpha)\left(\frac{c_2}{R}\right)=1 \tag{23.1} $$
where (23.1) is the break even condition for the insurance policy that makes zero profit. Construct the Lagrangian
$$ \mathcal{L}=\alpha u(c_1)+(1-\alpha)\rho u(c_2)+\lambda\left[1-\alpha c_1-(1-\alpha)\left(\frac{c_2}{R}\right)\right] $$
The f.o.c. are \(\alpha u'(c_1^*)=\lambda\alpha\) and \((1-\alpha)\rho u'(c_2^*)=\lambda(1-\alpha)\frac{1}{R}\), and combined,
$$ \frac{(1-\alpha)\rho u'(c_2^*)}{\alpha u'(c_1^*)}=\frac{(1-\alpha)\frac{1}{R}}{\alpha}\ \Rightarrow\ u'(c_1^*)=\rho R\,u'(c_2^*) \tag{23.2} $$
Then, since we assumed \(R\rho>1\), (23.2) implies
$$
c_1^* In order to make further analysis, we will focus on a specific case with Constant Relative Risk Aversion (CRRA) utility function \(u(c)=\dfrac{c^{1-\sigma}}{1-\sigma}\) with \(\sigma>0\). Then, (23.2) implies $$
(c_1^*)^{-\sigma}=\rho R(c_2^*)^{-\sigma}\ \Rightarrow\ \left(\frac{c_2^*}{c_1^*}\right)^{\sigma}=\rho R\ \Rightarrow\ c_2^*=c_1^*\underbrace{(\rho R)^{\frac{1}{\sigma}}}_{>1}>c_1^*
$$ To have a result that supports incentive for bank run, we hope that \(c_1^*>1\): $$
\begin{aligned}
\alpha\cdot1+(1-\alpha)\left(\frac{(\rho R)^{\frac{1}{\sigma}}}{R}\right)&<1\\
\Rightarrow\ (1-\alpha)\left(\frac{(\rho R)^{\frac{1}{\sigma}}}{R}\right)&<1-\alpha\\
\Rightarrow\ \frac{(\rho R)^{\frac{1}{\sigma}}}{R}&<1\\
\Rightarrow\ \rho^{\frac{1}{\sigma}}& $$
1 which is crucial for our bank run analysis. Finally, let's consider a bank. Since this is a competitive market, if the bank wants to have any single customer, it must mimic the best insurance policy and offer the household the same deposit payoff. Here we assume the same specific CRRA utility with additional constraint (23.4). So, by (23.5), the bank's offer is: By the above discussion, we achieved the conclusion in (23.5) that \(1 Suppose at \(T=1\) some people feel worried about the safety of their money and go to withdraw regardless of his liquidity shock \(\theta\). Suppose everyone does the same thing, then since \(c_1^*>1\), everyone is claiming a withdraw strictly greater than 1, which makes it impossible for the bank to pay much. So, only the first \(f\) fraction in the line (where \(f=\dfrac{1}{c_1^*}<1\)) can get the full repayment \(c_1^*\) and others get zero in the end, which is called the bank's sequential service constraint. Then, when others observe such behavior, they know bank's sequential service constraint, so their optimal action is to stand in the line as quick as possible regardless of their \(\theta\). After all, getting something is better than nothing. So, the bank run becomes a self-fulfilling prophecy, which becomes the bad equilibrium discussed above. Then, what can we do to rule out this bad equilibrium? There are at least two ways: So, both ways make the good equilibrium happen.
23.3.3 The bank's offer
23.4 The bank run problem