2. The model

Time goes on for three periods, $t=0,1,2$ . There is an infinitely divisible good used for consumption and investment. There are three types of agents: retail depositors, wholesale depositors, and bankers.

A measure $\sigma \in [0,1]$ of retail depositors is born at date 0 with an endowment of one unit of good each. Additionally, a measure $\lambda$ of wholesale depositors is born with an endowment of $\Lambda$ units of good each, where $\Lambda$ should be thought of as being much larger than 1. For convenience, depositors’ aggregate endowment is normalized to one, that is, $\sigma + \lambda \Lambda = 1$ , so that $\sigma$ equals the share of the aggregate endowment held by retail depositors. Retail and wholesale depositors are indexed by $i_{\sigma } \in [0,\sigma ]$ and $i_{\lambda } \in [0,\lambda ]$ , respectively. The payoff of both retail and wholesale depositors is given by $u(c_1+c_2)$ , where $c_t$ denotes consumption at date $t$ and $u(\cdot)$ is a continuous, strictly increasing function. The only possibility to store the endowment from date 0 to the future is through a constant-return-to-scale investment technology that returns 1 unit of good at date 2 per unit of good invested at date 0. For reasons that lie outside of the model, depositors access the investment technology only through demand deposits issued by bankers.

There is a finite number of bankers indexed by $j=1,2,..,\mathcal{J}$ , where $\mathcal{J}$ is some large, finite number and denotes both the number of bank(er)s as well as the set of bank(er)s. Bankers are born at date 0 without endowment and they maximize $U(c_0)$ , where $U(\cdot)$ is a strictly increasing function. At date 0, bankers sell demand deposits to depositors in return for depositors’ endowment. For simplicity, I assume each depositor can only invest in one bank. Banks invest part of the collected endowment in the investment technology and keep the remaining part as profits. I denote by $r(j)$ the gross return on deposits offered by banker $j$ . It may be helpful to anticipate here that, as a result of Bertrand competition among banks, we will have $r(j)=1$ for all active banks in equilibrium, so that banks make no profits.

Deposits can be redeemed at par either at date 1 or at date 2. If depositors withdraw at date 1, banks need to sell assets (claims to investment return) in order to pay out redeeming depositors. Denote by $p$ the date 1 price of an asset paying 1 unit at date 2, and denote by $\mu$ the total fundamental value (i.e., the total date 2 payout) of all assets sold by banks at date 1. Suppose the liquidation price follows

(1) \begin{equation} p(\mu ) = \begin{cases} 1 & \text{if} \, \, \mu \leqslant \overline{\mu } \\[5pt] k(\mu ) & \text{if} \, \, \mu \geqslant \overline{\mu } \end{cases}, \quad \text{with} \quad 0 \lt \overline{\mu } \lt 1, \end{equation}

where $k(\mu )$ is continuous and satisfies $k(\overline{\mu })=1, k(\mu )\gt 0$ and $k'(\mu )\lt 0$ for $\mu \geqslant \overline{\mu }$ . In words, assets can be sold at fundamental value as long as the aggregate amount of assets sold is below some threshold $\overline{\mu }$ ; once the amount of assets sold exceeds this threshold, the fire sale price is strictly decreasing in the amount of assets sold.Footnote ⁸ Both depositors and banks act as price-takers with regard to the liquidation price. For future reference, I define $p^m \equiv p(1)$ as the lowest liquidation price that may possibly occur.

The final ingredient of the model is an exogenous deposit insurance scheme that insures the deposits of retail but not wholesale depositors.Footnote ⁹ Banks cannot discriminate at date 0 between retail and wholesale depositors, that is, any depositor who wishes to open an account at any given bank at date 0 can do so. Whenever a bank is not able to pay out an amount of good corresponding to the face value of its retail depositors at date 2, deposit insurance steps in and repays retail depositors. Deposit insurance payments are financed by taxes which are not modeled explicitly; they are made at the end of date 2 and the deposit insurance agency remaining passive up to that point. Notably, if banks have both retail and wholesale deposits outstanding at date 2, wholesale depositors can redeem first, even if the bank incurred losses at date 1.Footnote ¹⁰

3. Runs

In this section, I focus on depositors’ decision whether or not to withdraw their deposits at date 1, taking their investment choices at date 0 as given. Since retail depositors never have an incentive to withdraw at date 1, we can limit attention to the behavior of wholesale depositors.

A bank pays out depositors who redeem at date 1 in full as long as the bank can raise enough funds to do so. If requests for redemptions exceed the liquidation value of the bank’s portfolio then, following Allen and Gale (Reference Allen and Gale>1998) and others, the bank will liquidate its entire portfolio at the going market price and distribute the proceeds equally among those who redeem. In this situation, wholesale depositors who do not redeem receive nothing. It follows that a bank will be susceptible to runs at date 1 whenever the amount of good it can raise by liquidating its entire portfolio at the going market price is insufficient to pay out all its wholesale depositors in full.

Whether a bank $j$ is susceptible to runs depends both on the share of the bank’s outstanding deposits held by retail depositors, denoted $\vartheta (j)$ , as well as on the liquidation price, $p$ . To illustrate this, consider a bank that has a measure $D$ of outstanding deposits at date 1, which also equals the fundamental value of assets held by the bank. Suppose half the deposits are held by retail depositors, that is, $\vartheta (j)=0.5$ . Suppose further the liquidation price of assets at date 1 equals $p\lt 0.5$ . Repaying all its wholesale depositors at date 1 would require to raise $0.5 \cdot D$ units of good at date 1. However, by liquidating its entire asset portfolio, the bank can raise only $p \cdot D \lt 0.5$ units of good. The bank is then susceptible to a run at date 1 since nothing will be left in the bank for wholesale depositors who do not redeem in case all (other) wholesale depositors run. Consider now a different example where, as before, 50% of the bank’s deposits are held by retail depositors but the liquidation price equals $p=0.8$ . The bank could then repay all its wholesale depositors at date 1 by selling a fraction $\dfrac{0.5}{0.8} = 0.625$ of its portfolio at the prevailing market price of $p=0.8$ . No matter how many wholesale depositors redeem at date 1, the bank has always enough funds left at date 2 to repay wholesale depositors who did not redeem at date 1. It follows that the bank is not susceptible to runs at date 1. In general, by the same reasoning, a bank $j$ is susceptible to runs at date 1 if and only if

(2) \begin{equation} 1-\vartheta (j) \gt p, \end{equation}

That is, if and only if the share of deposits held by wholesale depositors exceeds the liquidation price.

When hit by a run, banks liquidate their entire asset portfolio which, by (1), depresses the secondary market price $p$ . This introduces a systemic element to runs: the more banks are hit by a run, the lower $p$ and hence the more banks are susceptible to runs. Note further that not redeeming deposits is always the best response if no other depositor at the same bank redeems at date 1. This implies that wholesale depositors at a given bank can always coordinate on not running on their bank, independent of the liquidation price $p$ . In principle, it is thus possible that depositors coordinate on running on some subset of banks (which is large enough to cause the liquidation price $p$ to drop below 1) and not on the remaining banks.Footnote ¹¹ The economy may thus exhibit many different run equilibria at date 1 in which different subsets of banks are hit by a run. In the remainder of the paper, I will limit attention to two date 1 equilibria: (i) the one where nobody runs and (ii) the largest run that is possible, denoted the systemic run.Footnote ¹² To make this precise, it is useful to introduce the following definitions:

It is not hard to show that the systemic run equilibrium contains all other run equilibria in the sense that the set of banks hit in the systemic run equilibrium is a strict superset of the set of banks hit in all other run equilibria. Notice that the economy may not exhibit any run equilibria, in which case the notion of systemic run is obsolete. I assume that whenever the economy does exhibit multiple date 1 equilibria, depositors select the systemic run equilibrium with probability $\pi ^{sr} \in (0,1)$ and the no-run equilibrium with probability $1-\pi ^{sr}$ . Finally, it will be useful to introduce the following labels for banks with and without retail depositors, respectively:

4. The optimal structure of the banking system

In this section, I study how a social planner aiming to minimize the magnitude of systemic runs would distribute insured and uninsured deposits across banks.Footnote ¹³ In Section A.2. of the online appendix, I show that minimizing the magnitude of systemic runs is equivalent to maximizing the expected aggregate payout to depositors net of deposit insurance payments.

4.1. An illustrative example

We start with an example that illustrates why the magnitude of systemic runs depends on the distribution of insured (retail) and uninsured (wholesale) deposits across banks. Consider a banking system with $\sigma =0.5$ , such that 50% of all deposits are held by retail depositors. Suppose further the liquidation price (1) follows $k(\mu ) = \overline{\mu }/ \mu$ , with $\overline{\mu }=0.25$ . Figure 1 shows three different ways how retail and wholesale deposits could be distributed across banks, with retail deposits being depicted in gray and wholesale deposits in white.

Figure 1.

Different distributions of retail and wholesale deposits across banks.

The left-hand side of Figure 1 shows a banking system in which retail and wholesale deposits are distributed uniformly across banks, that is, there is one representative bank with 50% retail and 50% wholesale deposits. Consider a scenario where all banks liquidate their portfolios, in which case the liquidation price falls to $p(1)=0.25$ . Since the share of wholesale depositors is higher than 25% at all banks, by condition (2), all banks are susceptible to runs at a liquidation price of $p=0.25$ . It follows that systemic runs in this economy encompass all banks.

Consider next the middle part of Figure 1, which shows a banking system with two types of banks: some with only retail depositors (“commercial banks”) and some with only wholesale depositors (“shadow banks”). Consider a scenario where all shadow banks liquidate their portfolios, in which case the liquidation price falls to $p(0.5)=0.5$ . By condition (2), banks with 100% wholesale depositors are susceptible to runs at this liquidation price. The largest possible run in this banking system (the systemic run) is thus one that affects all shadow banks. Compared to the uniform distribution of deposits across banks depicted on the left-hand side of Figure 1, systemic runs encompass only half the banking system.

The extreme distribution of retail and wholesale deposits across banks depicted in the middle of Figure 1 does not in general minimize the magnitude of systemic runs. To see this, suppose that starting from the situation depicted in the middle of Figure 1, some wholesale deposits are moved from shadow banks into commercial banks, as depicted on the right-hand side of Figure 1. In the example on the right-hand side of Figure 1, the share of wholesale deposits at commercial banks equals $\displaystyle \frac{0.1}{0.6}\lt 0.25$ . Since there is no scenario in which the liquidation price falls below $p(1)=0.25$ , systemic runs still do not encompass commercial banks. As before, systemic runs affect the entire shadow banking sector; however, since the size of the shadow banking sector is smaller, the magnitude of systemic runs is smaller.

4.2. Optimal distribution of deposits across banks

In this subsection, I formally derive the structure of the banking system that minimizes the magnitude of systemic runs. To start, let $D(\mathcal{J'})$ denote the total face value of deposits held at some subset of banks $\mathcal{J'} \subseteq \mathcal{J}$ , which is the same as the fundamental value of assets held by these banks. For each bank, the planner chooses the total amount of deposits held at the bank, $D(j)$ , as well as the share of retail deposits at the bank, $\vartheta (j)$ .

In a scenario where all banks in $\mathcal{J'}$ are hit by a run, they all liquidate their assets, so that the liquidation price falls to $p(D(\mathcal{J'})) \leqslant 1$ . From the discussion in Section 3, it follows that the economy exhibits a run equilibrium encompassing all banks in $\mathcal{J'}$ iff the share of wholesale deposits at each bank in $\mathcal{J'}$ is higher than the liquidation price in a scenario where all banks in $\mathcal{J'}$ liquidate their portfolios. The condition for a run equilibrium encompassing all banks $j \in \mathcal{J'}$ to exist is thus

(3) \begin{equation} 1-\vartheta (j) \gt p(D(\mathcal{J'})) \hspace{0.75cm} \text{for all banks} \hspace{0.4cm} j \in \mathcal{J'}. \end{equation}

The following result shows that the systemic run encompasses all banks whose share of retail deposits is below some threshold $\vartheta ^{sr}$ :

Lemma 4.1. For any given structure of the banking system $\{D(j),\vartheta (j)\}_{j \in \mathcal{J}}$ , there exists a threshold $\vartheta ^{sr} \in [0,1]$ such that a systemic run encompasses all banks $j \in \mathcal{J}$ with $\vartheta (j) \lt \vartheta ^{sr}$ and does not encompass banks with $\vartheta (j) \geqslant \vartheta ^{sr}$ .

The proof of Lemma 4.1 is an application of Tarski’s fixed point theorem and is given in Appendix A.1. Using the result of Lemma 4.1, we can formulate the planner’s problem as:

(4) \begin{equation} \begin{aligned} & \min \limits _{\{{D(j), \vartheta (j)\}}_{j \in \mathcal{J}}} \, \, D \left (\{j \, \in \mathcal{J} \, | \, \vartheta (j)\lt \vartheta ^{sr} \} \right ) \\ & \text{s.t.} \, \, \sum _{j \in \mathcal{J}} D(j) = 1 \hspace{0.25cm} \text{and} \hspace{0.25cm} \sum _{j \in \mathcal{J}} \vartheta (j) D(j) = \sigma \end{aligned} \end{equation}

We then get the following result:

Proposition 4.1. Minimizing the magnitude of systemic runs requires $D(\{j \in \mathcal{J} \, | \, \vartheta (j)=0\}) + D(\{j \in \mathcal{J} \, | \, \vartheta (j) \geqslant \vartheta ^{sr}\})=1$ .

Stated verbally, Proposition 4.1 says that minimizing the magnitude of system runs requires that there be only two types of banks: banks without retail deposits and banks with a high enough share of retail deposits to prevent them from being susceptible to runs. The formal proof of Proposition 4.1 is given in Appendix A.2. Figure 2 provides a graphical illustration of the proof. Suppose there are banks with $\vartheta \in (0, \vartheta ^{sr})$ that issue some amount $D\gt 0$ of deposits. The fundamental value of assets liquidated by these banks in a systemic run equals $D$ . Suppose we reallocate an amount $D'$ of wholesale deposits away from these banks into newly created banks, where $D'$ is such that the share of retail deposits at the original banks reaches $\vartheta ^{sr}$ . An amount $D-D'$ of deposits is now held at banks that are run-proof, which decreases the amount of assets liquidated in a systemic run. Therefore, the initial structure of the banking system cannot minimize the magnitude of systemic runs.

Figure 2.

Illustration of the proof of Proposition 4.1.

In the following, without loss of generality, I assume the share of retail deposits at all commercial banks is set to the same level $\vartheta _{CB}$ .Footnote ¹⁴ From Proposition 4.1, we know that minimizing the magnitude of systemic runs requires $\vartheta _{CB} \geqslant \vartheta ^{sr}$ . We continue with the following result, whose proof is given in Appendix A.3:

Lemma 4.2. Suppose $\vartheta (j) \in \{0, \vartheta _{CB} \}$ for all banks $j \in \mathcal{J}$ . Then $\vartheta _{CB} \geqslant \vartheta ^{sr}$ is equivalent to $\vartheta _{CB} \geqslant 1- p^m$ .

To streamline notation, denote $D_{CB} \equiv D(\{j \in \mathcal{J} \, | \, \vartheta (j)\gt 0\})$ as the (relative) size of the commercial banking sector and $D_{SB} \equiv D(\{j \in \mathcal{J} \, | \, \vartheta (j)=0\})$ as the (relative) size of the shadow banking sector. Since the aggregate share of deposits held by retail depositors equals $\sigma$ , we have $D_{CB} \, \vartheta _{CB} = \sigma$ . Combining this with $\vartheta _{CB} \geqslant 1-p^m$ (from Lemma 4.2) and substituting $D_{SB}=1-D_{CB}$ yields the following condition on the size of the shadow banking sector:

(5) \begin{equation} D_{SB} \geqslant \max \left \{ 1- \displaystyle \frac{\sigma }{1-p^m}, 0 \right \} \equiv D_{SB}^{min}(\sigma ) \end{equation}

In words, $D_{SB}^{min}(\sigma )$ is the minimum size of the shadow banking sector required to absorb enough wholesale deposits from the commercial banking sector such as to keep commercial banks shielded from runs. Notice that $D_{SB}^{min}(\sigma )$ is decreasing in $\sigma$ ; if $\sigma$ is high enough, we have $D_{SB}^{min}(\sigma )=0$ , that is, commercial banks are not susceptible to runs even if all wholesale deposits are held at commercial banks.

So far we have seen that if $\sigma$ is relatively low, then shadow banks need to have some minimum size in order to keep commercial banks stable. But at which point do shadow banks themselves become susceptible to runs? By condition (2), shadow banks are susceptible to runs whenever the liquidation price $p$ drops below one. Given that commercial banks are never hit by a run, this implies that shadow banks are susceptible to systemic runs iff the liquidation price drops below one in a situation where all shadow banks are hit by a run, which, by (1), is the case iff $D_{SB} \gt \overline{\mu }$ . We can therefore denote $D_{SB}^{max} \equiv \min \{1-\sigma, \overline{\mu }\}$ as the maximum size of the shadow banking sector for which shadow banks are not susceptible to systemic runs, given that commercial banks are not susceptible to runs.Footnote ¹⁵ The discussion in the previous paragraphs leads us to the following proposition, whose proof is given in Appendix A.4:

Proposition 4.2. Denote $D_{SB}^{opt}$ as the (relative) size of the shadow banking sector for which the magnitude of systemic runs is minimized. $D_{SB}^{opt}$ depends on $\sigma$ as follows:

1. (i) If $\sigma \in [(1-p^m),1]$ , then $D_{SB}^{opt} \in [0,D_{SB}^{max}].$

2. (ii) If $\sigma \in [(1-\overline{\mu })(1-p^m), (1-p^m)]$ , then $D_{SB}^{opt} \in [D_{SB}^{min}(\sigma ), \, \, D_{SB}^{max} ].$

3. (iii) If $\sigma \in [0,(1-\overline{\mu })(1-p^m)]$ , then $D_{SB}^{opt}=D_{SB}^{min}(\sigma ).$

According to Proposition 4.2, $\sigma$ can be divided into three regions. In region (i), we get from condition (5) that $D_{SB}^{min}(\sigma )=0$ , meaning that commercial banks are not susceptible to systemic runs even if all wholesale deposits are held in the commercial banking sector. Systemic runs then do not occur in the economy as long as the size of the shadow banking sector is below $D_{SB}^{max}$ . Next, in region (ii), we have $D_{SB}^{min}(\sigma ) \gt 0$ , that is, the relative size of the shadow banking sector must be larger than zero in order to absorb enough wholesale deposits from the commercial banking sector to keep commercial banks stable. Since $D_{SB}^{min}(\sigma ) \leqslant D_{SB}^{max}$ , systemic runs can be avoided by setting the relative size of the shadow banking sector within $[D_{SB}^{min}(\sigma ), D_{SB}^{max}]$ . Finally, in region (iii), we have $D_{SB}^{min}(\sigma )\gt D_{SB}^{max}$ , that is, the minimum size of the shadow banking sector required to keep commercial banks stable is such that shadow banks are susceptible to systemic runs. Preventing runs entirely is then not feasible, and the magnitude of systemic runs is minimized by setting the shadow banking sector to the smallest size necessary to keep commercial banks stable.

We can also derive from Proposition 4.2 what fraction of the banking system can be shielded from systemic runs for given $\sigma$ . For $ \sigma \geqslant (1-\overline{\mu })(1-p^m)$ , systemic runs can be prevented entirely by distributing retail and wholesale deposits optimally across banks. If $\sigma \lt (1-\overline{\mu })(1-p^m)$ , the maximum fraction of the banking system that can be kept run-proof corresponds to the relative size of the (stable) commercial banking sector in the optimal structure of the banking system, which equals $1-\sigma _{SB}^{min}(\sigma ) = \dfrac{\sigma }{1-p^m}$ . This leads to the following result:

Corollary 4.1. Denote $S(\sigma )$ as the maximum share of the banking system that can be shielded from systemic runs for given $\sigma$ . Then:

\begin{equation*} S(\sigma ) = \begin {cases} 1 & \text {for} \, \, \sigma \in \big [(1-\overline {\mu })(1-p^m), 1 \big ] \\ \dfrac {\sigma }{1-p^m} & \text {for} \, \, \sigma \in \big [0, (1-\overline {\mu })(1-p^m) \big ) \end {cases} \end{equation*}

5. Date-0 equilibrium

In this section, I examine what structure of the banking system results in a decentralized equilibrium at date 0 where banks offer deposit contracts and depositors choose in which bank to invest. I then compare the structure of the banking system resulting in a decentralized date 0 equilibrium to the optimal structure derived in Section 4.

In a decentralized allocation, each bank $j$ sets the gross return $r(j) \in [0,1]$ of the deposit contract it offers. The bank chosen by a given retail depositor $i_{\sigma }$ is denoted by $\alpha _{\sigma }(i_{\sigma }) \in \mathcal{J}$ , and $\alpha _{\lambda }(i_{\lambda }) \in \mathcal{J}$ denotes the same for a given wholesale depositor $i_{\lambda }$ . The equilibrium definition is standard: A date-0 equilibrium is a set of deposit contracts $\{r(j)\}_{j \in \mathcal{J}}$ together with a set of investment choices $\{\alpha _{\sigma }(i_{\sigma })\}_{i_{\sigma } \in [0,\sigma ]}$ and $\{\alpha _{\lambda }(i_{\lambda })\}_{i_{\lambda } \in [0,\lambda ]}$ such that (i) each depositor chooses the bank that gives her the highest expected payoff and (ii) no bank can strictly increase profits by changing the return paid on deposits.

Equilibrium behavior of banks and retail depositors is straightforward. Since retail depositors are guaranteed to receive the offered return $r(j)$ , they always invest in the bank offering the highest return, and they are indifferent between all banks that offer the same return. Furthermore, it is not hard to see that, by the usual argument of Bertrand, all banks that are active in equilibrium offer a return $r(j)=1$ , that is, banks pay a return equal to the return of the investment technology and make zero profits in equilibrium. To streamline the exposition, I will take this as given from now on.

What makes the decentralized equilibrium interesting are the investment choices of wholesale depositors. In order to write down expected payoffs of wholesale depositors, we need to introduce some additional notation. First, we denote the liquidation price in a systemic run by

(6) \begin{equation} p^{sr} \equiv p \!\left [ D \big (\{j \, \in \mathcal{J} \, | \, \vartheta (j)\lt \vartheta ^{sr} \} \big ) \right ]. \end{equation}

Next, we denote by $\varphi (\vartheta,p)$ the fraction of wholesale deposits (in terms of their face value) a bank can pay out when hit by a run. The share of wholesale deposits a bank can repay in a run equals the amount good the bank can raise by liquidating its entire asset portfolio at price $p$ , divided by the face value of wholesale deposits held at the bank. We thus have

(7) \begin{equation} \varphi (\vartheta, p) = \min \left \{ \frac{p}{1-\vartheta }, 1 \right \}. \end{equation}

When choosing which bank to invest in, an individual wholesale depositor takes the distribution of retail and wholesale deposits across banks as well as the liquidation price in a systemic run as given. Denote $\tilde{r}(j)$ as the effective return on wholesale deposits at bank $j$ . The no-run equilibrium is selected with probability $1-\pi ^{sr}$ , in which case wholesale deposits at all banks yield a gross return of one. With probability $\pi ^{sr}$ , a systemic run occurs, in which case the effective gross return on wholesale deposits held at bank $j$ equals $\varphi (\vartheta (j), p^{sr})$ . We thus have

(8) \begin{equation} \tilde{r}(j) = (1-\pi ^{sr})u(1) + \pi ^{sr} u\left (\varphi (\vartheta (j), p^{sr})\right ). \end{equation}

If systemic runs do not occur in the economy, then we have $\varphi (\cdot)=1$ for all banks, which implies $\tilde{r}(j)=1$ for all banks. If systemic runs do occur, then $\tilde{r}(j)$ is strictly higher for banks with a higher share of retail deposits, which means that wholesale depositors will invest in the banks with the highest share of retail deposits. Before we proceed, it is useful to introduce the following definitions:

In words, a type A equilibrium is an equilibrium in which systemic runs do not occur, and a type B equilibrium is an equilibrium in which systemic runs affect all banks. We continue with the following result:

The proof of Proposition 5.1 is relatively straightforward, which is why a formal proof is omitted. First, note that if the equilibrium is such that systemic runs do occur, then systemic runs must affect all banks. Otherwise, wholesale depositors would move away from the banks that are susceptible to runs to those that are not. Further, if all banks are susceptible to systemic runs, the share of retail deposits must be the same across all banks. Otherwise, wholesale depositors would move from the banks with a lower share of retail deposits to those with a higher share. Finally, if all banks have the same share of retail deposits, the share of retail deposits at any given bank must equal the aggregate share $\sigma$ .

We continue with the following result, whose proof is given in Appendix A.5:

Consider first item (i) in Proposition 5.2. We know from Section 4 that preventing runs entirely is not feasible in this region of $\sigma$ and that runs affect only part of the banking system if deposits are distributed optimally. In a decentralized date-0 equilibrium, runs affect the entire banking system; the optimal structure of the banking system in which fragile shadow banks coexist with stable commercial banks does not constitute a date 0 equilibrium since wholesale depositors have a private incentive to move from fragile shadow banks into stable commercial banks, causing the latter to become fragile as well.

Consider next region (ii) in which multiple date 0 equilibria exist. As seen in Section 4, systemic runs can be avoided in this region of $\sigma$ if the shadow banking sector has the right size. In a situation where neither commercial nor shadow banks are susceptible to runs, wholesale depositors are indifferent between the two, so that this constitutes a date 0 equilibrium. However, once we get to a situation where systemic runs do occur, wholesale depositors will flock to the banks with the highest shares of retail deposits, and we end up in a date 0 equilibrium where all banks have the same share of retail deposits. Note that in the “good” type A equilibrium in region (ii), the fact that commercial banks are not susceptible to runs sustains the liquidation price of assets. This in turn implies that shadow banks are not susceptible to runs either, since they can sell assets on the secondary market without incurring fire sale losses.Footnote ¹⁶ Lastly, item (iii) in Proposition 5.2 represents the case where commercial banks are not susceptible to runs even if all wholesale deposits are held at commercial banks and need no further elaboration.

Even though the transition from one equilibrium to another is not explicitly modeled, we can discuss informally the stability of the type A equilibrium in region (ii). According to Proposition 4.2, systemic runs do not occur in this region of $\sigma$ iff the relative size of the shadow banking sector is within $D_{SB} \in \![D_{SB}^{min}(\sigma ), D_{SB}^{max}]$ . This implies that for any $D_{SB} \in (D_{SB}^{min}(\sigma ), D_{SB}^{max})$ , the type A equilibrium is stable in the sense that small changes in the relative size of the shadow banking sector will not cause the economy to move to a type B equilibrium. If either $D_{SB} \gt D_{SB}^{max}$ or $D_{SB}\lt D_{SB}^{min}(\sigma )$ , then systemic runs occur. In the first case ( $D_{SB}\gt D_{SB}^{max}$ ) systemic runs only affect shadow banks. Depositors would then move away from fragile shadow banks into stable commercial banks, which would cause $D_{SB}$ to fall below $D_{SB}^{max}$ again, so that the economy would move back to the type A equilibrium. In the second case ( $D_{SB}\lt D_{SB}^{min}$ ) both commercial and shadow banks are fragile. Since losses caused by runs are smaller at commercial banks, depositors would move away from shadow into commercial banks, causing the economy to transition to a type B equilibrium in which only fragile commercial banks exist.