Central bank screening, moral hazard, and the lender of last resort policy

Abstract

This paper constructs a theoretical model to examine the LOLR policy when a central bank can imperfectly screen insolvent from solvent banks. We find that: (1) Central bank screening produces a “positive” stigma associated with central bank borrowing by punishing insolvent banks. (2) With central bank screening, the LOLR policy in fact reduces moral hazard rather than inducing it. (3) If the central bank can better identify solvent and insolvent banks when they apply for central bank loans, it will improve social welfare first by forcing the insolvent banks to efficiently liquidate their assets and second by deterring banks from choosing the risky assets to start with.

Appendices

Appendix 1: Conditions for the three assumptions to hold

First, we assume that

$$\begin{aligned} p R_H +(1-p) R_L< \gamma \end{aligned}$$

(20)

which implies that it is socially efficient for an L-type bank to liquidate its asset at date 1 rather than carry the asset to date 2. Further, we assume that

$$\begin{aligned} \gamma A <D \end{aligned}$$

(21)

which means that for both types of banks, the date 1 liquidation value of their assets is not enough to repay their debts. This condition can be satisfied when $e_0$ is sufficiently low such that the asset–debt ratio of $\frac{A}{D}$ is high or when $\gamma$ is sufficiently low. Note that by combining conditions (20) and (21), we derive

$$\begin{aligned} p R_H +(1-p) R_L<\gamma <\frac{D}{A} \end{aligned}$$

or

$$\begin{aligned} p AR_H +(1-p)A R_L<D \end{aligned}$$

(22)

Since $AR_H>D$ (because $A>D$ and $R_H>1$), it is straightforward to see that $A R_L<D$, implying that an L-type bank cannot repay its debts in the down state and will default.

To simplify the math derivation, we assume that

$$\begin{aligned} p\frac{R_H}{\gamma }+(1-p)\frac{AR_L}{D} <1 \end{aligned}$$

(23)

This condition guarantees that a creditor will never roll over his debts to an L-type bank. To see this, let $r_M$ denote the market rate that creditors offer to banks. We know that a bank will never borrow in the market if $1+r_M>\frac{R_H}{\gamma }$. This is because with such a high market rate, the bank is always better off by liquidating its own assets (see "Appendix 3" for a rigorous proof). As a result, the highest possible return rate that a creditor can gain from lending to an L-type bank is $1+r_M=\frac{R_H}{\gamma }$, and the highest possible expected return that a creditor can gain from lending to an L-type bank is $p\frac{R_H}{\gamma }+(1-p)\frac{AR_L}{D}$. When $p\frac{R_H}{\gamma }+(1-p)\frac{AR_L}{D}<1$, a creditor will never lend to an L-type bank because his expected return rate from lending is less than 1. $\square$

Appendix 2: Proof that no separating equilibrium exists

Proof

Consider the equilibrium where only L-type banks apply for CBLs. L-type banks will always deviate in this equilibrium, because they can mimic H-type banks by not applying for CBLs without incurring any cost. By doing so, an L-type bank will be identified as H-type and receive the lowest possible borrowing rate in the market. Consider the equilibrium where only H-type banks apply for CBLs. If an L-type bank follows the equilibrium strategy and does not apply, it will be identified as L-type by the market and will not be able to borrow in the market. As a result, it will be forced to liquidate all of its asset and go bankrupt at date 1. If it mimics an H-type bank and borrows from the central bank, then at least with probability $1-\phi$, it will successfully get loans and gain positive equity in the up state. As a result, L-type banks will always deviate, and such an equilibrium cannot exist either. $\square$

Appendix 3: Proof of Proposition 1

Suppose that an H-type bank chooses to liquidate l of its asset, where $0\le l \le A$. When it does not apply for CBLs, its net asset value is

$$\begin{aligned} NV_H=(A-l)R_H-(D-\gamma l)(1+r_M) \end{aligned}$$

(24)

When it applies for CBLs, its net asset value is

$$\begin{aligned} NV_H &= (A-l)R_H-(D-\gamma l-L_\mathrm{CB})(1+r_M) \nonumber \\&\quad - \, L_\mathrm{CB}(1+r_\mathrm{CB}) \end{aligned}$$

(25)

In both cases, the first-order derivative of $NV_H$ with respect to l is given by $\frac{\partial NV_H}{\partial l}=-R_H+\gamma (1+r_M)$. Thus, when $1+r_M<\frac{R_H}{\gamma }$, $\frac{\partial NV_H}{\partial l}<0$, implying that $l^*=0$. That is, an H-type bank will never liquidate its asset when $1+r_M<\frac{R_H}{\gamma }$. When $1+r_M>\frac{R_H}{\gamma }$, $\frac{\partial NV_H}{\partial l}>0$, implying that $l^*=A$. That is, an H-type bank will liquidate all of its asset when $1+r_M>\frac{R_H}{\gamma }$. Note that the bank will have to liquidate all of its asset and go bankrupt in this case because we assume $\gamma A<D$.

For an L-type bank, first note that its net asset value is always negative such that its equity value is always zero in the down state. This is because no matter an L-type bank applies for CBLs or not, its maximum asset value in the down state is $A\gamma$, when it liquidates all the asset at date 1, because $\gamma >R_L$ by assumption. No matter an L-type bank applies for CBLs or not, its minimum liability is D, when it is charged a zero interest rate. Thus, the maximum net asset value for an L-type bank is $A\gamma -D$ in the down state. Because by assumption, $A\gamma <D$, an L-type bank’s net asset value in the down state is always negative, resulting in zero equity. It implies that an L-type bank aims only at maximizing its equity value in the up state. An L-type bank’s net asset value in the up state is the same as that of an H-type bank. Thus, we prove the results. $\square$

Appendix 4: An explanation about the condition to ensure $r_M^\mathrm{PBB}$ and ${\hat{r}}_{M,{\hat{\lambda }}}$ exist

An equilibrium market rate exists only when: (1) it will give creditors an expected riskless rate of zero; (2) both types of banks are willing to borrow at this rate; and (3) it can actually be paid by H-type banks and L-type banks in the up state. The equilibrium rate derived from criterion 1 might be too high due to a low g or ${\hat{\lambda }}$ such that criteria 2 and 3 are not satisfied. In this case, a market freeze occurs and no equilibrium rate exists.

Based on criterion 1, an equilibrium market rate $r_M^\mathrm{PBB}$ should satisfy:

$$\begin{aligned} 1 &= g(1+r_M^\mathrm{PBB})+(1-g) \nonumber \\&\quad \times \, \left( p(1+r_M^\mathrm{PBB})+(1-p)\frac{AR_L}{D}\right) \end{aligned}$$

(26)

This is because creditors’ expected return rate from lending to an H-type bank is $1+r_M^\mathrm{PBB}$. Similarly, their expected return rate from an L-type bank in the up state is also $1+r_M^\mathrm{PBB}$. Creditors’ expected return rate from an L-type bank in the down state is $\frac{AR_L}{D}$. This is because, in the down state, an L-type bank goes bankrupt. As a result, its asset is allocated between the central bank and creditors.

We impose condition (2) such that criteria 2 and 3 hold. For criterion 2 to hold, an equilibrium market rate must be lower than $\frac{R_H}{\gamma }-1$ as explained following Proposition 1. For criterion 3 to hold, an equilibrium rate cannot exceed $\frac{AR_H-L_\mathrm{CB}(1+r_\mathrm{CB})}{D-L_\mathrm{CB}}-1$, which is the maximum return rate an H-type bank or an L-type bank in the up state can afford.

Similarly, for ${\hat{r}}_{M,{\hat{\lambda }}}$ to exists, criterion 1 implies

$$\begin{aligned} 1 &= {\hat{\lambda }}(1+{\hat{r}}_{M,{\hat{\lambda }}})+(1-{\hat{\lambda }}) \\&\quad \times \, \left( p(1+{\hat{r}}_{M,{\hat{\lambda }}})+(1-p)\frac{AR_L}{D}\right) \end{aligned}$$

For criterion 2 to hold, an equilibrium market rate must be lower than $\frac{R_H}{\gamma }-1$ as explained in Proposition 1. For criterion 3 to hold, an equilibrium rate cannot exceed $\frac{AR_H}{D}-1$, which is the maximum return rate an H-type bank or an L-type bank in the up state can afford. Note that by assumption $A\gamma <D$, implying $\frac{AR_H}{D}-1<\frac{R_H}{\gamma }-1$. Thus, for criteria 2 and 3 to hold, we must have the equilibrium market rate lower than $\frac{AR_H}{D}-1$.

Appendix 5: Derivation of $r_M^\mathrm{PBB}$ and ${\hat{r}}_{M,{\hat{\lambda }}}$

Because creditors are risk neutral and aim at an expected rate of zero, the equilibrium market rate, $r_M^\mathrm{PBB}$, is given by Eq. (26), or equivalently Eq. (4). Similarly, we find that ${\hat{r}}_{M,{\hat{\lambda }}}$ is identical to $r_M^\mathrm{PBB}$ except that g is replaced by ${\hat{\lambda }}$.

Appendix 6: Proof of Proposition 2

When an H-type bank follows the equilibrium strategy, its equity value is given by

$$\begin{aligned} e_H &= AR_H-L_\mathrm{CB}(1+r_\mathrm{CB})-(D-L_\mathrm{CB})(1+r_M^\mathrm{PBB}) \nonumber \\ &= AR_H-D(1+r_M^\mathrm{PBB})+L_\mathrm{CB}(r_M^\mathrm{PBB}-r_\mathrm{CB}) \end{aligned}$$

(27)

where $r_M^\mathrm{PBB}$ is given by Eq. (4).

When an L-type bank follows the equilibrium strategy, it will be rejected with a probability of $\phi$ and will be accepted with a probability of $1-\phi$. Using subscripts A and R to denote the cases where the loan application is accepted and rejected, respectively, we have

$$\begin{aligned} Ee_L=p(1-\phi ) e^u_{L,A}+(1-p)(1-\phi ) e^d_{L,A}+\phi e_{L,R} \end{aligned}$$

(28)

where

$$\begin{aligned} e^u_{L,A} &= AR_H-L_\mathrm{CB}(1+r_\mathrm{CB}) \nonumber \\&\quad -\, (D-L_\mathrm{CB})(1+r_M^\mathrm{PBB}) \end{aligned}$$

(29)

$$\begin{aligned} e^d_{L,A} &= 0 \end{aligned}$$

(30)

$$\begin{aligned} e_{L,R} &= 0 \end{aligned}$$

(31)

The equity value of an individual bank deviating to not applying for CBLs is as follows. An H-type bank will borrow a debt of D at the market rate of ${\hat{r}}_{M,{\hat{\lambda }}}$. Thus, its equity value is:

$$\begin{aligned} {\hat{e}}_{H,\mathrm{NCB}}=AR_H-D(1+{\hat{r}}_{M,{\hat{\lambda }}}) \end{aligned}$$

(32)

An L-type bank will also borrow a debt of D at the market rate of ${\hat{r}}_{M,{\hat{\lambda }}}$, and its expected date 2 equity will be

$$\begin{aligned} E{\hat{e}}_{L,\mathrm{NCB}}=p{\hat{e}}_{L,\mathrm{NCB}}^u+(1-p){\hat{e}}_{L,\mathrm{NCB}}^d \end{aligned}$$

(33)

where

$$\begin{aligned}&{\hat{e}}_{L,\mathrm{NCB}}^u = AR_H - D(1+{\hat{r}}_{M,{\hat{\lambda }}})= {\hat{e}}_{H,\mathrm{NCB}} \end{aligned}$$

(34)

$$\begin{aligned}&{\hat{e}}_{L,\mathrm{NCB}}^d =0 \end{aligned}$$

(35)

An H-type bank’s no-deviation condition is $e_H > {\hat{e}}_{H,\mathrm{NCB}}$. We have

$$\begin{aligned}&e_H-{\hat{e}}_{H,\mathrm{NCB}} \\&\quad = [AR_H-D(1+r_M^\mathrm{PBB})+L_\mathrm{CB}(r_M^\mathrm{PBB}-r_\mathrm{CB})] \\&\qquad -\, [AR_H - D(1+{\hat{r}}_{M,{\hat{\lambda }}})] \\&\quad = L_\mathrm{CB}(r_M^\mathrm{PBB}-r_\mathrm{CB})-D(r_M^\mathrm{PBB}-{\hat{r}}_{M,{\hat{\lambda }}})>0 \end{aligned}$$

An L-type bank’s no-deviation condition is

$$\begin{aligned} Ee_L-E{\hat{e}}_{L,\mathrm{NCB}}=(1-\phi ) p e^u_{L,A}- p {\hat{e}}_{L,\mathrm{NCB}}^u>0 \end{aligned}$$

(36)

or $(1-\phi )e^u_{L,A}> {\hat{e}}_{L,\mathrm{NCB}}^u$. We have

$$\begin{aligned}&(1-\phi )e^u_{L,A}- {\hat{e}}_{L,\mathrm{NCB}}^u \\&\quad = (1-\phi )\left[ AR_H-D(1+r_M^\mathrm{PBB})+L_\mathrm{CB}(r_M^\mathrm{PBB}-r_\mathrm{CB})\right] \\&\qquad -\, [AR_H- D(1+{\hat{r}}_{M,{\hat{\lambda }}})]>0 \end{aligned}$$

which is condition (5). Comparing these two conditions, we find that $e_H-{\hat{e}}_{H,\mathrm{NCB}}>(1-\phi )e^u_{L,A}- {\hat{e}}_{L,\mathrm{NCB}}^u$ such that an H-type bank’s no-deviation condition is easier to hold. Thus, the essential no-deviation condition is L-type banks’ no-deviation condition, condition (5). To see why, note that $e_H=e^u_{L,A}>(1-\phi )e^u_{L,A}$, while ${\hat{e}}_{H,\mathrm{NCB}}={\hat{e}}_{L,\mathrm{NCB}}^u$. The intuition here is that an H-type bank gains a higher equity value from applying for CBLs than an L-type bank in the up state, because an L-type bank may not survive central bank screening. On the other hand, both H-type banks and L-type banks in the up state share the same equity value when deviating to not applying CBLs. Thus, an H-type bank is less likely to deviate. $\square$

Appendix 7: Proof of Proposition 3

When banks follow the equilibrium strategy and borrow only in the market, two types of banks’ payoffs are given by

$$\begin{aligned}&e_{H,\mathrm{NCB}}=AR_H-D(1+r_M^\mathrm{PNB}) \\&e_{L,\mathrm{NCB}}^u = AR_H-D(1+r_M^\mathrm{PNB}) \\&e_{L,\mathrm{NCB}}^d= 0 \end{aligned}$$

Now consider banks’ payoffs when they deviate. If an H-type bank deviates, its date 2 equity will be

$$\begin{aligned} {\tilde{e}}_{H} &= AR_H-L_\mathrm{CB}(1+r_\mathrm{CB})-(D-L_\mathrm{CB})(1+{\tilde{r}}_{M,{\tilde{g}}}) \nonumber \\ &= AR_H-D(1+{\tilde{r}}_{M,{\tilde{g}}})+L_\mathrm{CB}({\tilde{r}}_{M,{\tilde{g}}}-r_\mathrm{CB}) \end{aligned}$$

(37)

If an L-type bank deviates, it will be identified as L-type and gain zero equity with probability $\phi$. With probability $1-\phi$, it can successfully get CBLs and then borrow in the market at the rate of ${\tilde{r}}_{M,{\tilde{g}}}$. Its equity values in different cases are specified as follows.

$$\begin{aligned} {\tilde{e}}_{L,\mathrm{Rej}} &= 0 \end{aligned}$$

(38)

$$\begin{aligned} {\tilde{e}}^u_{L,\mathrm{Acc}} &= AR_H-L_\mathrm{CB}(1+r_\mathrm{CB})-(D-L_\mathrm{CB})(1+{\tilde{r}}_{M,{\tilde{g}}}) \nonumber \\ &= {\tilde{e}}_{H} \end{aligned}$$

(39)

$$\begin{aligned} {\tilde{e}}^d_{L,\mathrm{Acc}} &= 0 \end{aligned}$$

(40)

An H-type bank’s no-deviation condition is $e_{H, \mathrm{NCB}}> {\tilde{e}}_H$, or

$$\begin{aligned}&[AR_H-D(1+r_M^\mathrm{PNB})] \nonumber \\&\qquad -\, [AR_H-D(1+{\tilde{r}}_{M,{\tilde{g}}})+L_\mathrm{CB}({\tilde{r}}_{M,{\tilde{g}}}-r_\mathrm{CB})] \nonumber \\&\quad = D({\tilde{r}}_{M,{\tilde{g}}}-r_M^\mathrm{PNB})-L_\mathrm{CB}\left( {\tilde{r}}_{M,{\tilde{g}}}-r_\mathrm{CB}\right) >0 \end{aligned}$$

(41)

which is condition (10).

An L-type bank’s no-deviation condition is $p e^u_{L,\mathrm{NCB}}> (1-\phi ) p {\tilde{e}}^u_{L,\mathrm{Acc}}$, or

$$\begin{aligned}&[AR_H-D(1+r_M^\mathrm{PNB})]-(1-\phi ) \nonumber \\&\quad \times \, [AR_H-D(1+{\tilde{r}}_{M,{\tilde{g}}})+L_\mathrm{CB}({\tilde{r}}_{M,{\tilde{g}}}-r_\mathrm{CB}) ]>0 \end{aligned}$$

(42)

It is obvious to see that an L-type bank in the up state has the same equilibrium payoff as an H-type bank, but has a lower payoff under the deviating strategy due to central bank screening. As a result, an H-type bank has a stronger incentive to deviate than an L-type bank. Thus, this equilibrium exists as long as H-type banks’ no-deviation condition, condition (10), holds.

Appendix 8: Proof of Proposition 4

Proof

We first prove result (1). First, we prove that a higher $\phi$ makes PNB less likely. Recall that the no-deviation condition for PNB is given by condition (10). Note that the market rate under the deviating strategy, ${\tilde{r}}_{M,{\tilde{g}}}$, decreases with $\phi$, because creditors are more optimistic about bank quality and charge a lower market rate with a higher $\phi$. As a result, the LHS of condition (10) decreases with $\phi$, implying that the no-deviation condition is easier to be violated.

On the other hand, we find that a higher $\phi$ does not necessarily make PBB less likely. Recall that the no-deviation condition for PBB is given by condition (5). We find that a higher $\phi$ has two opposite effects on the LHS of condition (5): (1) It lowers $1-\phi$, inducing a lower first term of the LHS, $(1-\phi )\left[ AR_H-D(1+r_M^\mathrm{PBB})+L_\mathrm{CB}(r_M^\mathrm{PBB}-r_\mathrm{CB})\right]$. (2) It increases $\left[ AR_H-D(1+r_M^\mathrm{PBB})+L_\mathrm{CB}(r_M^\mathrm{PBB}-r_\mathrm{CB})\right] =AR_H-(D-L_\mathrm{CB})(1+r_M^\mathrm{PBB})-L_\mathrm{CB}(1+r_\mathrm{CB})$, inducing a higher first term of the LHS. To see this, note that $r_M^\mathrm{PBB}$ is lower with a higher $\phi$ due to more optimistic beliefs of creditors about bank quality. Depending on which effect is dominant, a higher $\phi$ could make PBB more or less likely.

Next, we prove result (2). First, we prove that more generous LOLR policy with a higher $L_\mathrm{CB}$ and lower $r_\mathrm{CB}$ makes PBB more likely. From the no-deviation condition for PBB, condition (5), we can tell that the RHS of condition (5) is increasing in $L_\mathrm{CB}$ and decreasing in $r_\mathrm{CB}$, given our assumption that $r_\mathrm{CB}<r_M^\mathrm{PBB}$. As a result, a higher $L_\mathrm{CB}$ and lower $r_\mathrm{CB}$ makes the no-deviation condition less tight. Next, we prove that more generous LOLR policy with a higher $L_\mathrm{CB}$ and lower $r_\mathrm{CB}$ makes PNB less likely. From the no-deviation condition for PNB, condition (10), we find that this condition holds if and only if ${\tilde{r}}_{M,{\tilde{g}}}>r_M^\mathrm{PNB}$. Given our assumption that $r_\mathrm{CB}<r_M^\mathrm{PNB}$, we find that $r_\mathrm{CB}<{\tilde{r}}_{M,{\tilde{g}}}$, implying that the RHS of condition (10) is decreasing in $L_\mathrm{CB}$ and increasing in $r_\mathrm{CB}$. As a result, a higher $L_\mathrm{CB}$ and lower $r_\mathrm{CB}$ makes the no-deviation condition tighter.

Finally, we prove result (3). The intuitive criterion [6] is related to the equilibrium-dominated strategies. In a perfect Bayesian equilibrium in a signaling game, a message $m_j$ is equilibrium-dominated for type $t_i$ if $t_i$’s equilibrium payoff is greater than $t_i$’s highest possible payoff from $m_j$. This approach requires that if the information set following $m_j$ is off the equilibrium path and $m_j$ is equilibrium-dominated for type $t_i$, but is not equilibrium-dominated for other types, then the receiver’s belief should place zero probability on type $t_i$.

Using the intuitive criterion, we find that as long as $\phi$ is sufficiently high, PNB cannot exist. We prove it as follows: For H-type banks, their equilibrium payoff in PNB is $AR_H-D(1+r_M^\mathrm{PNB})$. Their maximum possible payoff under the deviating strategy of borrowing CBLs is $AR_H-L_\mathrm{CB}(1+r_\mathrm{CB})-(D-L_\mathrm{CB})=AR_H-D-L_\mathrm{CB}r_\mathrm{CB}$. Thus, borrowing CBL is an equilibrium-dominated strategy if and only if $D r_M^\mathrm{PNB}<L_\mathrm{CB}r_\mathrm{CB}$, which is impossible if we assume $r_\mathrm{CB}<r_M^\mathrm{PNB}$. Thus, borrowing CBLs is not an equilibrium-dominated strategy for H-type banks.

For L-type banks, their equilibrium payoff in PNB is $p[AR_H-D(1+r_M^\mathrm{PNB})]$. Their maximum possible payoff under the deviating strategy of borrowing CBLs is $p(1-\phi )[AR_H-L_\mathrm{CB}(1+r_\mathrm{CB})-(D-L_\mathrm{CB})]$. Thus, borrowing CBL is an equilibrium-dominated strategy if and only if

$$\begin{aligned}&p[AR_H-D(1+r_M^\mathrm{PNB})] \nonumber \\&\quad > p(1-\phi )[AR_H-L_\mathrm{CB}(1+r_\mathrm{CB})-(D-L_\mathrm{CB})] \end{aligned}$$

(43)

or equivalently condition (11).

Provided that this condition holds, according to the intuitive criterion, creditors should assign ${\tilde{\lambda }}=1$ for a bank deviating to borrowing CBLs. H-type banks’ no-deviation condition thus becomes $AR_H-D(1+r_M^\mathrm{PNB})>AR_H-D-L_\mathrm{CB}r_\mathrm{CB}$, which is always violated as long as $r_\mathrm{CB}<r_M^\mathrm{PNB}$. Thus, PNB is eliminated.

We can prove that ${\bar{\phi }}$ is decreasing in $L_\mathrm{CB}r_\mathrm{CB}$ as follows:

$$\begin{aligned}&\frac{\partial {\bar{\phi }}}{\partial L_\mathrm{CB}r_\mathrm{CB}} \nonumber \\&\quad =\frac{-[AR_H-D-L_\mathrm{CB}r_\mathrm{CB}]+[D r_M^\mathrm{PNB}-L_\mathrm{CB}r_\mathrm{CB}]}{[AR_H-D-L_\mathrm{CB}r_\mathrm{CB}]^2} \nonumber \\&\quad =\frac{D r_M^\mathrm{PNB}-(AR_H-D)}{[AR_H-D-L_\mathrm{CB}r_\mathrm{CB}]^2}<0 \end{aligned}$$

(44)

because $AR_H-D>D r_M^\mathrm{PNB}$. This is because for an equilibrium market rate to exist, $1+r_M^\mathrm{PNB}$ must be lower than $\frac{AR_H}{D}$. $\square$

Appendix 9: An explanation about baseline parameter value choices

We set the capital adequacy ratio at 10%, which matches the data for US banks before the global financial crisis. We set $R_H$ at 0.01343 to match the average ROE of $13.43\%$ of US banks over the five-year period of 2002–2006 before the global financial crisis. Note that because $A/e_0=10$, a ROE of $13.43\%$ implies that $R_H$ is 1.01343. We set $R_L$ at 0.65 to match the average recovery rate of failed US banks, according to Bennett and Unal [4]. We set $\lambda$ at 0.85 to match the data in the US banking sector during the global financial crisis. More detailed explanations for these parameter value choices are available in Li et al. [15].

Appendix 10: Proof of Proposition 5

First, we find the threshold level of $h_i$, ${\bar{h}}^\mathrm{PBB}$. A bank with a private benefit of $B({\bar{h}}^\mathrm{PBB})$ must be indifferent between investing in safe and risky assets. Its expected date 2 equity value from investing in the safe asset is given by:

$$\begin{aligned} Ee^{s}=\pi e^s_\mathrm{no\, shock}+(1-\pi )e^s_\mathrm{shock} \end{aligned}$$

(45)

Here $e^s_\mathrm{no\, shock}=AR_H-D$ denotes the bank’s equity value at date 2 in the absence of a crisis. In this case, all the banks will be solvent and roll over their debts at the riskless rate of zero. $e^s_\mathrm{shock}$ denotes the bank’s equity value at date 2 when a crisis occurs. In this case, a proportion $\lambda ={\bar{h}}^\mathrm{PBB}$ of banks will be H-type, and a proportion $1-\lambda =1-{\bar{h}}^\mathrm{PBB}$ of banks will be L-type. The equity value at date 2 for the bank choosing the safe asset is given by Eq. (27), which we replicate here:

$$\begin{aligned} e^s_\mathrm{shock} &= AR_H-L_\mathrm{CB}(1+r_\mathrm{CB})-(D-L_\mathrm{CB})(1+r_{M,{\bar{h}}}^\mathrm{PBB}) \\ &= AR_H-D(1+r_{M,{\bar{h}}}^\mathrm{PBB})+L_\mathrm{CB}(r_{M,{\bar{h}}}^\mathrm{PBB}-r_\mathrm{CB}) \end{aligned}$$

where $r_{M,{\bar{h}}}^\mathrm{PBB}$ is given by Eq. (13), which we replicate here:

$$\begin{aligned} r_{M,{\bar{h}}}^\mathrm{PBB} = \left( 1-\frac{AR_L}{D}\right) \left( \frac{1}{{\bar{g}}+(1-{\bar{g}})p}-1\right) \end{aligned}$$

Here ${\bar{g}}=\frac{{\bar{h}}^\mathrm{PBB}}{{\bar{h}}^\mathrm{PBB} +(1-{\bar{h}}^\mathrm{PBB})(1-\phi )}$. Note that here we focus on the case with no market freeze. As a result, the bank’s expected equity from choosing the safe asset is

$$\begin{aligned} Ee^{s} &= \pi (AR_H-D) + (1-\pi ) \nonumber \\&\quad \times \, [AR_H-D(1+r_{M,{\bar{h}}}^\mathrm{PBB})+L_\mathrm{CB}(r_{M,{\bar{h}}}^\mathrm{PBB}-r_\mathrm{CB})] \end{aligned}$$

(46)

If a market freeze occurs, which is possible when ${\bar{h}}^\mathrm{PBB}$ is sufficiently low, we have

$$\begin{aligned}&e^s_\mathrm{shock,\, freeze} \\&\quad = \max \left\{ 0, \left( A-\frac{D-L_\mathrm{CB}}{\gamma }\right) R_H - L_\mathrm{CB}(1+r_\mathrm{CB})\right\} \end{aligned}$$

The bank’s expected equity from choosing the safe asset is

$$\begin{aligned}&Ee^{s}_\mathrm{freeze} \nonumber \\&\quad = \pi (AR_H-D) + (1-\pi ) \nonumber \\&\qquad \times \, \max \left\{ 0, \left( A-\frac{D-L_\mathrm{CB}}{\gamma }\right) R_H - L_\mathrm{CB}(1+r_\mathrm{CB})\right\} \end{aligned}$$

(47)

If the bank chooses the risky asset, its expected equity value at date 2 is given by:

$$\begin{aligned} Ee^{r}=\pi e^r_\mathrm{no\, shock}+(1-\pi )Ee^r_\mathrm{shock} \end{aligned}$$

(48)

Here $e^r_\mathrm{no\, shock}=e^s_\mathrm{no\, shock}=AR_H-D$ denotes the bank’s equity value at date 2 in the absence of a crisis. $Ee^r_\mathrm{shock}$ denotes the bank’s expected equity value at date 2 when a crisis occurs. Without a market freeze, we have

$$\begin{aligned} Ee^r_\mathrm{shock}=(1-p)\times 0+p\phi \times 0+p(1-\phi )e_{L,A}^{u} \end{aligned}$$

(49)

where $e_{L,A}^{u}=e^s_\mathrm{shock}$ is given by Eq. (27). We have the above equation because when a crisis occurs, with a probability of $1-p$, the down state is realized and the bank’s equity is zero. With a probability of $p\phi$, the up state is realized and the bank’s application is rejected by the central bank. In this case, the bank’s equity is zero too. With a probability of $p(1-\phi )$, the up state is realized and the bank’s application is accepted by the central bank. In this case, the bank’s equity is given by $e_{L,A}^{u}$.

If a market freeze occurs, $e_{L,A}^{u}$ will be replaced by $e^s_\mathrm{shock,\, freeze}$.

The bank with $h_i={\bar{h}}^\mathrm{PBB}$ must be indifferent between the two choices. Thus, we have

$$\begin{aligned} Ee^s = Ee^r+(a_0+a_1 {\bar{h}}^\mathrm{PBB}) \end{aligned}$$

(50)

As a result, without a market freeze we have

$$\begin{aligned}&a_0+a_1{\bar{h}}^\mathrm{PBB} \nonumber \\&\quad = (1-\pi )\left[ 1-p(1-\phi )\right] \nonumber \\&\qquad \times \, [AR_H-D(1+r_{M,{\bar{h}}}^\mathrm{PBB})+L_\mathrm{CB}(r_{M,{\bar{h}}}^\mathrm{PBB}-r_\mathrm{CB})] \end{aligned}$$

(51)

We can prove that the RHS of Eq. (51) is strictly increasing and concave in h as follows.

$$\begin{aligned} \frac{\partial \mathrm{RHS}}{\partial h}=\frac{\partial \mathrm{RHS}}{\partial r_{M,h}^\mathrm{PBB}}\frac{\partial r_{M,h}^\mathrm{PBB}}{\partial h} \end{aligned}$$

(52)

Note that

$$\begin{aligned} \frac{\partial \mathrm{RHS}}{\partial r_{M,h}^\mathrm{PBB}}=-(1-\pi )[1-p(1-\phi )](D-L_\mathrm{CB})<0 \end{aligned}$$

(53)

Note that

$$\begin{aligned} \frac{\partial r_{M,h}^\mathrm{PBB}}{\partial h}=\frac{\partial r_{M,h}^\mathrm{PBB}}{\partial {\bar{g}}}\frac{\partial {\bar{g}}}{\partial h}<0 \end{aligned}$$

(54)

because

$$\begin{aligned} \frac{\partial r_{M,h}^\mathrm{PBB}}{\partial {\bar{g}}} &= \left( 1-\frac{AR_L}{D}\right) \frac{-(1-p)}{[{\bar{g}}+(1-{\bar{g}})p]^2}<0 \end{aligned}$$

(55)

$$\begin{aligned} \frac{\partial {\bar{g}}}{\partial h} &= \frac{1-\phi }{[h+(1-h)(1-\phi )]^2}>0 \end{aligned}$$

(56)

Thus, we prove that the payoff gap between safe and risky assets without a market freeze is increasing in h. Intuitively, this is because a higher h induces a lower market rate, $r_{M,{\bar{h}}}^\mathrm{PBB}$. Because safe assets are more likely to survive and benefit from a lower market rate than risky assets, a lower market rate makes safe assets more attractive.

We can also prove that the RHS of Eq. (51) is concave in h. Note that

$$\begin{aligned}&\frac{\partial \mathrm{RHS}}{\partial h} \nonumber \\&\quad =\frac{(1-\pi )[1-p(1-\phi )](D-L_\mathrm{CB})(1-\phi )(1-p)\left( 1-\frac{AR_L}{D}\right) }{[{\bar{g}}+(1-{\bar{g}})p]^2[h+(1-h)(1-\phi )]^2} \end{aligned}$$

(57)

Let $C\equiv (1-\pi )[1-p(1-\phi )](D-L_\mathrm{CB})(1-\phi )(1-p)\left( 1-\frac{AR_L}{D}\right)$, which is independent of h. We have

$$\begin{aligned}&\frac{\partial ^2 \mathrm{RHS}}{\partial h^2} \nonumber \\&\quad = C\frac{-2\{[{\bar{g}}+(1-{\bar{g}})p]^2[h+(1-h)(1-\phi )]\phi +[{\bar{g}}+(1-{\bar{g}})p](1-p)(1-\phi )\}}{[{\bar{g}}+(1-{\bar{g}})p]^4[h+(1-h)(1-\phi )]^4}\nonumber \\&\quad <0 \end{aligned}$$

(58)

With a market freeze, which happens when h is sufficiently low, $Ee^s-Ee^r$ is independent of h. There are two possible cases for a market freeze to occur. First, $\frac{AR_H-L_\mathrm{CB}(1+r_\mathrm{CB})}{D-L_\mathrm{CB}}>\frac{R_H}{\gamma }$. Second, $\frac{AR_H-L_\mathrm{CB}(1+r_\mathrm{CB})}{D-L_\mathrm{CB}}<\frac{R_H}{\gamma }$. In the first case, the market freezes once $r_{M,{\bar{h}}}^\mathrm{PBB}$ reaches $\frac{R_H}{\gamma }$. Let $h^*$ be the threshold level of h below which a market freeze occurs. We can prove that $Ee^s-Ee^r$ is continuous at $h^*$, that is, at $h^*$, $Ee^s-Ee^r$ is the same with and without a market freeze. To see this, note that when $r_{M,{\bar{h}}}^\mathrm{PBB}=\frac{R_H}{\gamma }$, without a market freeze, $e^s_\mathrm{shock}=\max \left\{ 0, \left( A-\frac{D-L_\mathrm{CB}}{\gamma }\right) R_H - L_\mathrm{CB}(1+r_\mathrm{CB})\right\}$, which is identical to $e^s_\mathrm{shock,\, freeze}$.

In the second case, the market freezes once $r_{M,{\bar{h}}}^\mathrm{PBB}$ reaches $\frac{AR_H-L_\mathrm{CB}(1+r_\mathrm{CB})}{D-L_\mathrm{CB}}$. In this case, at $h^*$, without a market freeze, $e^s_\mathrm{shock}=0$, because the banks will earn zero equity after paying the maximum borrowing rate they can afford. With a market freeze, the banks will liquidate assets to repay a debt of $D-L_\mathrm{CB}$, implying that their financing cost is even higher than $\frac{AR_H-L_\mathrm{CB}(1+r_\mathrm{CB})}{D-L_\mathrm{CB}}$. Thus, again they will earn zero equity. That is, $e^s_\mathrm{shock,\, freeze}=0$. Hence, $Ee^s-Ee^r$ is continuous at $h^*$ in both cases.

Thus, we prove that when $h<h^*$, $Ee^s-Ee^r$ is independent of ${\bar{h}}^\mathrm{PBB}$. When $h>h^*$, $Ee^s-Ee^r$ is increasing and concave in h. Further, $Ee^s-Ee^r$ is continuous at $h^*$.

To ensure a unique stable interior solution for ${\bar{h}}^\mathrm{PBB}\in (h^*,1)$, we impose the following conditions: First, at $h^*$, RHS > LHS in Eq. (51). Second, at $h=1$, RHS < LHS in Eq. (51). Because the RHS is concave with a decreasing slope, and the LHS is linear with a constant slope, the imposed conditions ensure that the LHS will cross the RHS from below only once and ${\bar{h}}^\mathrm{PBB}\in (h^*,1)$ at the intersection. Additionally, around ${\bar{h}}^\mathrm{PBB}$ given by the intersection, we must have the slope of the RHS lower than the slope of the LHS. That is, at ${\bar{h}}^\mathrm{PBB}$, RHS−LHS is decreasing in h. Further, note that this equilibrium is stable in the sense that whenever $h<{\bar{h}}^\mathrm{PBB}$, RHS > LHS, implying that more banks will choose safe assets and h will increase until it reaches ${\bar{h}}^\mathrm{PBB}$. On the other hand, whenever $h>{\bar{h}}^\mathrm{PBB}$, RHS < LHS, implying that more banks will choose risky assets and h will decrease until it reaches ${\bar{h}}^\mathrm{PBB}$.

Next we prove that this switching strategy ${\bar{h}}^\mathrm{PBB}$ is optimal for each bank. It is straightforward to see that given that each bank follows this switching strategy, any bank with $h_i>{\bar{h}}^\mathrm{PBB}$ will have the LHS of the above equation higher, and the RHS unchanged. Thus, it is indeed optimal for it to choose the risky asset. On the other hand, any bank with $h_i<{\bar{h}}^\mathrm{PBB}$ will have the LHS lower and the RHS unchanged. Thus, it is indeed optimal for it to choose the safe asset.

Note that the RHS−LHS of Eq. (51) is increasing in $L_\mathrm{CB}$ and $\phi$ and decreasing in $r_\mathrm{CB}$. Because we have proved that at ${\bar{h}}^\mathrm{PBB}$, RHS−LHS is decreasing in h, according to the implicit function theorem, we find that ${\bar{h}}^\mathrm{PBB}$ is increasing in $L_\mathrm{CB}$ and $\phi$, and is decreasing in $r_\mathrm{CB}$. $\square$

Appendix 11: Proof of Proposition 6

First, we find the threshold level of $h_i$, ${\bar{h}}^\mathrm{PNB}$. A bank with a private benefit of $B({\bar{h}}^\mathrm{PNB})$ must be indifferent between investing in safe and risky assets. Without a market freeze, its expected date 2 equity value from investing in the safe asset is given by:

$$\begin{aligned} Ee^{s,\mathrm{PNB}}=\pi e^{s,\mathrm{PNB}}_\mathrm{no\,shock}+(1-\pi )e^{s,\mathrm{PNB}}_\mathrm{shock} \end{aligned}$$

(59)

where

$$\begin{aligned} e^{s,\mathrm{PNB}}_\mathrm{no\,shock}=AR_H-D \end{aligned}$$

(60)

and

$$\begin{aligned} e^{s,\mathrm{PNB}}_\mathrm{shock}=AR_H-D(1+r_{M,{\bar{h}}}^\mathrm{PNB}) \end{aligned}$$

(61)

Here $r_{M,{\bar{h}}}^\mathrm{PNB}$ is given by

$$\begin{aligned} r_{M,{\bar{h}}}^\mathrm{PNB}= \left( 1-\frac{AR_L}{D}\right) \left( \frac{1}{{\bar{h}}^\mathrm{PNB}+(1-{\bar{h}}^\mathrm{PNB})p}-1\right) \end{aligned}$$

(62)

For a bank with a private benefit of $B({\bar{h}}^\mathrm{PNB})$, its payoff of choosing the risky asset without a market freeze is given by

$$\begin{aligned} Ee^{r,\mathrm{PNB}}=\pi e^{r,\mathrm{PNB}}_\mathrm{no\,shock}+(1-\pi )e^{r,\mathrm{PNB}}_\mathrm{shock} \end{aligned}$$

(63)

where

$$\begin{aligned} e^{r,\mathrm{PNB}}_\mathrm{no\,shock}=AR_H-D \end{aligned}$$

(64)

and

$$\begin{aligned} e^{r,\mathrm{PNB}}_\mathrm{shock}=pe^{s,\mathrm{PNB}}_\mathrm{shock}=p[AR_H-D(1+r_{M,{\bar{h}}}^\mathrm{PNB})] \end{aligned}$$

(65)

Thus, without a market freeze, ${\bar{h}}^\mathrm{PNB}$ is determined by

$$\begin{aligned} a_0+a_1{\bar{h}}^\mathrm{PNB} &= (1-\pi )(1-p) \nonumber \\&[AR_H-D(1+r_{M,{\bar{h}}}^\mathrm{PNB})] \end{aligned}$$

(66)

which is Eq. (15).

Let the RHS of Eq. (15) be denoted by $\mathrm{RHS}^\mathrm{PNB}$. We prove that it is increasing and concave in h as follows:

$$\begin{aligned} \frac{\partial \mathrm{RHS}^\mathrm{PNB}}{\partial h} &= (1-\pi )(1-p)\times (-D) \nonumber \\&\times \frac{\partial r_{M,h}^\mathrm{PNB} }{\partial h}>0 \end{aligned}$$

(67)

because

$$\begin{aligned} \frac{\partial r_{M,h}^\mathrm{PNB} }{\partial h}= \left( 1-\frac{AR_L}{D}\right) \frac{-(1-p)}{[h+(1-h)p]^2}<0 \end{aligned}$$

(68)

Thus, we prove that $\mathrm{RHS}^\mathrm{PNB}$ is increasing in h.

Additionally, note that

$$\begin{aligned} \frac{\partial ^2 \mathrm{RHS}^\mathrm{PNB}}{\partial h^2} &= (1-\pi )(1-p)^2D\left( 1-\frac{AR_L}{D}\right) \nonumber \\&\quad \times \, \frac{-2[h+(1-h)p](1-p)}{[h+(1-h)p]^4}<0 \end{aligned}$$

(69)

Thus, we prove that $\mathrm{RHS}^\mathrm{PNB}$ is concave in h.

Now we consider the case of a market freeze. Similar to our analysis for PBB, with a market freeze, which happens when h is sufficiently low, $\mathrm{RHS}^\mathrm{PNB}$ is independent of h. When the market freezes, $r_{M,h}^\mathrm{PNB}$ exceeds $\frac{AR_H}{D}-1$. We have $\mathrm{RHS}^\mathrm{PNB,\,freeze}=0$ because $e^{s,\mathrm{PNB}}_\mathrm{shock}=0$ when the market freezes. Similar to the analysis for PBB, let $h^{\mathrm{PNB}*}$ be the threshold level below which the market freezes. We find that $\mathrm{RHS}^\mathrm{PNB}$ is continuous at $h^{\mathrm{PNB}*}$. That is, $\mathrm{RHS}^\mathrm{PNB}$ is the same with and without a market freeze at $h^{\mathrm{PNB}*}$. To see this, note that without a market freeze, $e^{s,\mathrm{PNB}}_\mathrm{shock}=0$ at the equilibrium market rate of $\frac{AR_H}{D}$, which is the same as when the market freezes.

Thus, we prove that when $h<h^{\mathrm{PNB}*}$, $\mathrm{RHS}^\mathrm{PNB}$ is independent of h. When $h>h^{\mathrm{PNB}*}$, $\mathrm{RHS}^\mathrm{PNB}$ is increasing and concave in h. Further, $\mathrm{RHS}^\mathrm{PNB}$ is continuous at $h^{\mathrm{PNB}*}$.

To ensure a unique stable interior solution for $h\in (h^{\mathrm{PNB}*},1)$, we impose the following conditions: First, at $h^{\mathrm{PNB}*}$, $\mathrm{RHS}^\mathrm{PNB}>\mathrm{LHS}^\mathrm{PNB}=a_0+a_1h$. Second, at $h=1$, $\mathrm{RHS}^\mathrm{PNB}< \mathrm{LHS}^\mathrm{PNB}$. Because the RHS is concave with a decreasing slope, and the LHS is linear with a constant slope, the imposed conditions ensure that the LHS will cross the RHS from below only once and ${\bar{h}}^\mathrm{PNB}\in (h^{\mathrm{PNB}*},1)$ at the intersection. Additionally, around ${\bar{h}}^\mathrm{PNB}$ given by the intersection, we must have the slope of the RHS lower than the slope of the LHS. That is, at ${\bar{h}}^\mathrm{PNB}$, $\mathrm{RHS}^\mathrm{PNB}-\mathrm{LHS}^\mathrm{PNB}$ is decreasing in h. Further, note that this equilibrium is stable in the sense that whenever $h<{\bar{h}}^\mathrm{PNB}$, $\mathrm{RHS}^\mathrm{PNB}>\mathrm{LHS}^\mathrm{PNB}$, implying that more banks will choose safe assets and h will increase until it reaches ${\bar{h}}^\mathrm{PNB}$. On the other hand, whenever $h>{\bar{h}}^\mathrm{PNB}$, $\mathrm{RHS}^\mathrm{PNB}<\mathrm{LHS}^\mathrm{PNB}$, implying that more banks will choose risky assets and h will decrease until it reaches ${\bar{h}}^\mathrm{PNB}$.

Appendix 12: Moral hazard comparison between PBB and PNB

Recall that in PBB and PNB, ${\bar{h}}^\mathrm{PBB}$ and ${\bar{h}}^\mathrm{PNB}$ are determined by Eqs. (12) and (15), respectively. Now we compare the RHS of the two equations at the same level of h. Let them denoted by $\mathrm{RHS}^\mathrm{PBB}$ and $\mathrm{RHS}^\mathrm{PNB}$, respectively. We find that $\mathrm{RHS}^\mathrm{PNB}<\mathrm{RHS}^\mathrm{PBB}$ as long as

$$\begin{aligned}&(1-\pi )(1-p)[AR_H-D(1+r_{M,h}^\mathrm{PNB})] \nonumber \\&\quad < (1-\pi )\left[ 1-p(1-\phi )\right] \nonumber \\&\qquad \times \, [AR_H-D(1+r_{M,h}^\mathrm{PBB})+L_\mathrm{CB}(r_{M,h}^\mathrm{PBB}-r_\mathrm{CB})] \end{aligned}$$

(70)

or

$$\begin{aligned}&AR_H-D(1+r_{M,h}^\mathrm{PNB}) \nonumber \\&\quad < AR_H-D(1+r_{M,h}^\mathrm{PBB})+L_\mathrm{CB}(r_{M,h}^\mathrm{PBB}-r_\mathrm{CB}) \end{aligned}$$

(71)

because $1-p<1-p(1-\phi )$. A simple transformation of the above condition leads to

$$\begin{aligned} \left( 1-\frac{L_\mathrm{CB}}{D}\right) r^\mathrm{PBB}_{M,h}+\frac{L_\mathrm{CB}}{D}r_\mathrm{CB}<r^\mathrm{PNB}_{M,h} \end{aligned}$$

(72)

Note that $r^\mathrm{PBB}_{M,h}$ is always lower than $r^\mathrm{PNB}_{M,h}$ at the same level of h as long as $\phi >0$. With the assumption of $r_\mathrm{CB}$ is always below the market rate, $r^\mathrm{PBB}_{M,h}$, the above condition always holds. Thus, we prove that $\mathrm{RHS}^\mathrm{PNB}<\mathrm{RHS}^\mathrm{PBB}$ at the same level of h.

We have proved that both $\mathrm{RHS}^\mathrm{PBB}$ and $\mathrm{RHS}^\mathrm{PNB}$ are increasing and concave in h without a market freeze. With our imposed conditions, we ensure that in both PBB and PNB, there is a unique solution to $h<1$ that is sufficiently high such that there is no market freeze in equilibrium. We now prove that with the imposed conditions, ${\bar{h}}^\mathrm{PBB}$, is always higher than ${\bar{h}}^\mathrm{PNB}$. That is, PBB always induces less severe moral hazard than PNB.

Because $\mathrm{RHS}^\mathrm{PNB}$ is always lower than $\mathrm{RHS}^\mathrm{PBB}$ at the same level of h, we find that $\mathrm{RHS}^\mathrm{PBB}({\bar{h}}^\mathrm{PNB})> \mathrm{RHS}^\mathrm{PNB}({\bar{h}}^\mathrm{PNB})=a_0+a_1{\bar{h}}^\mathrm{PNB}$. That is, $\mathrm{RHS}^\mathrm{PBB}$ is above $\mathrm{LHS}^\mathrm{PBB}$ at ${\bar{h}}^\mathrm{PNB}$. Recall that we have proved that in PBB, $\mathrm{RHS}^\mathrm{PBB}$ crosses $\mathrm{LHS}^\mathrm{PBB}$ only once from above. With the imposed constraint that $\mathrm{RHS}^\mathrm{PBB}<\mathrm{LHS}^\mathrm{PBB}$ at $h=1$, we conclude that ${\bar{h}}^\mathrm{PBB}$ must be greater than ${\bar{h}}^\mathrm{PNB}$ and less than 1. $\square$

Appendix 13: Proof that CBL losses decrease with $\phi$

Proof

From Eq. (19), we have

$$\begin{aligned} CBLL &= L_\mathrm{CB}(1-{\bar{h}}^\mathrm{PBB})(1-\phi ) \nonumber \\&\quad \times \, \left[ (1-p)\left( 1-\frac{AR_L}{D}\right) -pr_\mathrm{CB}\right] \nonumber \\&\quad -\, {\bar{h}}^\mathrm{PBB}L_\mathrm{CB}r_\mathrm{CB} \end{aligned}$$

(73)

Note that

$$\begin{aligned} \left[ (1-p)\left( 1-\frac{AR_L}{D}\right) -pr_\mathrm{CB}\right] >0 \end{aligned}$$

(74)

This is because

$$\begin{aligned} 1 &= g(1+r_M^\mathrm{PBB})+(1-g) \nonumber \\&\quad \times \, \left( p(1+r_M^\mathrm{PBB})+(1-p)\frac{AR_L}{D}\right) \end{aligned}$$

(75)

Because $g>p$ and $\frac{AR_L}{D}<1+r_M^\mathrm{PBB}$, we have

$$\begin{aligned} p(1+r_M^\mathrm{PBB})+(1-p)\frac{AR_L}{D}<1 \end{aligned}$$

(76)

Because by assumption, $r_\mathrm{CB}<r_M^\mathrm{PBB}$, we have

$$\begin{aligned} p(1+r_\mathrm{CB})+(1-p)\frac{AR_L}{D}<1 \end{aligned}$$

(77)

or equivalently

$$\begin{aligned} pr_\mathrm{CB}<(1-p)\left( 1-\frac{AR_L}{D}\right) \end{aligned}$$

(78)

Thus, at a given ${\bar{h}}^\mathrm{PBB}$, $L_\mathrm{CB}(1-{\bar{h}}^\mathrm{PBB})(1-\phi )\left[ (1-p)\left( 1-\frac{AR_L}{D}\right) -pr_\mathrm{CB}\right]$ decreases with $\phi$. Additionally, we have proved that ${\bar{h}}^\mathrm{PBB}$ increases with $\phi$. It is straightforward to see that CBLL decreases with ${\bar{h}}^\mathrm{PBB}$. Thus, a higher $\phi$ further decreases CBLL by increasing ${\bar{h}}^\mathrm{PBB}$. In sum, we find that CBLL decreases with $\phi$. $\square$