Optimal design of bank regulation under aggregate risk

Abstract

We develop a tractable general equilibrium model of banking under aggregate risk. Our novel framework includes the main tools of banking regulation—capital and liquidity requirements, deposit insurance and transfers—and shows how they interact and jointly emerge as an optimal response to the inefficiency of the equilibrium of the unregulated economy. This enables us to contribute to several ongoing debates between proponents and opponents of stricter bank regulation. We show that, for low levels of aggregate risk, the general efficient allocation of the economy can be implemented via deposit insurance and transfers alone with no need for capital or liquidity requirements. When aggregate risk exceeds a threshold, however, all four regulatory instruments are essential for efficiency. Capital and liquidity requirements become stricter as aggregate risk increases. Our results, therefore, support the views of opponents of stricter bank regulation when aggregate risk is moderate, but validate the views of proponents when aggregate risk is high. In contrast with recent proposals to reduce or eliminate depositor subsidies to constrain bank leverage, our analysis also demonstrates that depositor subsidies are necessary for efficiency under aggregate risk as they achieve optimal risk-sharing among agents and mitigate underinvestment.

Appendices

Appendix A: Proofs

1.1 Proof of Lemma 1

We organize the proof into several steps for clarity.

• 1. Since the firms’ production function, $\Lambda (.),$ satisfies the Inada condition, $\Lambda '(0)=\infty ,$ all firms demand a non-zero amount of capital in equilibrium. By (5), the expected marginal productivity of each private firm $n\in [0,{\mathcal {F}}]$ must equal the expected marginal cost of bank loan financing, $q{\mathscr {R}}_{L,n}^{*}+(1-q)\Lambda _{liq}$, in equilibrium. Similarly, by (7), the expected marginal productivity of each public firm $n\in [0,{\mathcal {G}}]$ equals the expected marginal cost of equity financing, $q{\mathscr {R}}_{E,n}^{*}+(1-q)\Lambda _{liq}$. Moreover, in any equilibrium, $q{\mathscr {R}}_{L,n}^{*}+(1-q)\Lambda _{liq}\ge 1.$ To the contrary, if $q{\mathscr {R}}_{L,n}^{*}+(1-q)\Lambda _{liq}<1,$ then banks prefer the safe asset to investing in the firm. In this case, however, the firm does not raise any capital, which is a contradiction. Similarly, $q{\mathscr {R}}_{E,n}^{*}+(1-q)\Lambda _{liq}\ge 1$ in any equilibrium.

• 2. As argued in the paragraph following the statement of the lemma, to ensure that bank loan markets clear, all private firms must receive the same capital investment in equilibrium. If not, private firms that raise less capital would have lower expected marginal returns from production and, therefore, lower expected returns from bank loans. Such firms would then receive no capital at all from banks. Similarly, to ensure that public firm equity markets clear, all public firms must also receive the same capital investment in equilibrium. It follows that the expected returns on bank loans for all private firms are equal, and the espected equity returns of all public firms are equal. By similar arguments, all banks must offer the same expected equity return in equilibrium.

• 3. In any equilibrium, the expected equity return of a public firm must equal the expected equity return of a bank. If not, investors would strictly prefer to invest either in public firms or in banks, which cannot be the case in equilibrium.

• 4. For each bank, the expected return on bank loans, $\sup _{n\in [0,{\mathcal {F}}]}q{\mathscr {R}}_{L,n}^{*}+(1-q)\Lambda _{liq},$equals its expected marginal cost of financing. Consider any bank $m\in [0,1]$. As the bank’s objective function, (10) is linear, its expected asset return, ${\mathbb {E}}[{\widetilde{A}}_{m}^{*}]$ in equilibrium equals its expected marginal cost of financing, $\frac{{\mathbb {E}}\left[ D_{m}^{*}{\widetilde{R}}_{D,m}^{*}+E_{m}^{*}{\widetilde{R}}_{E,m}^{*}\right] }{D_{m}+E_{m}}$. If $\sup _{n\in [0,{\mathcal {F}}]}q{\mathscr {R}}_{L,n}^{*}+(1-q)\Lambda _{liq}>1,$ then the bank optimally invests all its capital in private firms and its expected asset return equals $\sup _{n\in [0,{\mathcal {F}}]}q{\mathscr {R}}_{L,n}^{*}+(1-q)\Lambda _{liq}.$ If $\sup _{n\in [0,{\mathcal {F}}]}q{\mathscr {R}}_{L,n}^{*}+(1-q)\Lambda _{liq}=1,$ then banks are indifferent between investing in private firms and the safe asset, and ${\mathbb {E}}[{\widetilde{A}}_{m}^{*}]=\sup _{n\in [0,{\mathcal {F}}]}q{\mathscr {R}}_{L,n}^{*}+(1-q)\Lambda _{liq}=1.$

• 5. For any bank $m\in [0,1],$ its expected equity and deposit returns, ${\mathbb {E}}[{\widetilde{R}}_{E,m}^{*}],{\mathbb {E}}[{\widetilde{R}}_{D,m}^{*}]$ are greater than or equal to the expected loan return, $\sup _{n\in [0,{\mathcal {F}}]}q{\mathscr {R}}_{L,n}^{*}+(1-q)\Lambda _{liq}.$ If the bank raises nonzero equity (deposit) capital, then its expected equity (deposit) return, ${\mathbb {E}}[{\widetilde{R}}_{E,m}^{*}]$ $({\mathbb {E}}[{\widetilde{R}}_{D,m}^{*}])$ equals the expected loan return, $\sup _{n\in [0,{\mathcal {F}}]}q{\mathscr {R}}_{L,n}^{*}+(1-q)\Lambda _{liq}$ in equilibrium. Suppose that ${\mathbb {E}}[{\widetilde{R}}_{E,m}^{*}]<\sup _{n\in [0,{\mathcal {F}}]}q{\mathscr {R}}_{L,n}^{*}+(1-q)\Lambda _{liq}.$ Then the bank has a profitable deviation where it can raise additional equity capital and invest it in loans. A similar argument holds for the deposit capital. Hence, ${\mathbb {E}}[{\widetilde{R}}_{E,m}^{*}],{\mathbb {E}}[{\widetilde{R}}_{D,m}^{*}]\ge \sup _{n\in [0,{\mathcal {F}}]}q{\mathscr {R}}_{L,n}^{*}+(1-q)\Lambda _{liq}$ in equilibrium. Suppose that $D_{m}^{*},E_{m}^{*}>0.$ If ${\mathbb {E}}[{\widetilde{R}}_{E,m}^{*}]\ne {\mathbb {E}}[{\widetilde{R}}_{D,m}^{*}]$. The bank then has a profitable deviation where it either lowers or raises its deposit capital relative to its equity capital. It then follows from step 2 that ${\mathbb {E}}[{\widetilde{R}}_{E,m}^{*}]={\mathbb {E}}[{\widetilde{R}}_{D,m}^{*}]=\sup _{n\in [0,{\mathcal {F}}]}q{\mathscr {R}}_{L,n}^{*}+(1-q)\Lambda _{liq}.$ If $D_{m}^{*}=0,E_{m}^{*}>0,$ then ${\mathbb {E}}[{\widetilde{R}}_{E,m}^{*}]=\sup _{n\in [0,{\mathcal {F}}]}q{\mathscr {R}}_{L,n}^{*}+(1-q)\Lambda _{liq}$ by step 2. If $D_{m}^{*}>0,E_{m}^{*}=0$, then ${\mathbb {E}}[{\widetilde{R}}_{D,m}^{*}]=\sup _{n\in [0,{\mathcal {F}}]}q{\mathscr {R}}_{L,n}^{*}+(1-q)\Lambda _{liq}$ by step 2.

• 6. If some banks raise nonzero equity (deposit) capital in equilibrium, then all banks raise nonzero equity (deposit) capital. Consider a bank $m\in [0,1]$ that does not raise equity. Suppose that ${\mathbb {E}}\left[ {\widetilde{R}}_{E,m}^{*}\right] >\sup _{n\in [0,{\mathcal {F}}]}q{\mathscr {R}}_{L,n}^{*}+(1-q)\Lambda _{liq}$. By Step 4, the expected equity return of any bank that raises nonzero equity capital equals the expected bank loan return, $\sup _{n\in [0,{\mathcal {F}}]}q{\mathscr {R}}_{L,n}^{*}+(1-q)\Lambda _{liq},$ which is the same for all banks. Hence, bank m’s expected equity return exceeds the expected equity return of banks that raise nonzero equity capital. In this case, however, investors would strictly prefer this bank to other banks, and the equity market would not clear. Now consider a bank $m\in [0,1]$ that does not raise deposit capital and suppose that ${\mathbb {E}}\left[ {\widetilde{R}}_{D,m}^{*}\right] >\sup _{n\in [0,{\mathcal {F}}]}q{\mathscr {R}}_{L,n}^{*}+(1-q)\Lambda _{liq}$. By the result of part 1, the expected deposit return of the bank exceeds that of banks which raise nonzero deposit capital. In this case, however, it is optimal for depositors to increase their supply of capital to bank m,  which implies that the bank must raise nonzero deposit capital in equilibrium.

• 7. All firms—public and private—raise the same amount of capital. The expected return on bank loans to a private firm equals the expected equity return of a public firm. We first argue that, for any bank $m\in [0,1],$ ${\mathbb {E}}[{\widetilde{R}}_{E,m}^{*}]=\sup _{n\in [0,{\mathcal {F}}]}q{\mathscr {R}}_{L,n}^{*}+(1-q)\Lambda _{liq}$ in equilibrium. Indeed, if the bank raises nonzero equity capital, the assertion follows from step 5. Suppose that $E_{m}^{*}=0.$ By step 6, all banks must then raise zero equity capital. Suppose that ${\mathbb {E}}[{\widetilde{R}}_{E,m}^{*}]>\sup _{n\in [0,{\mathcal {F}}]}q{\mathscr {R}}_{L,n}^{*}+(1-q)\Lambda _{liq}$. Investors would then strictly prefer to invest in bank m relative to firms. Hence, we must have ${\mathbb {E}}[{\widetilde{R}}_{E,m}^{*}]=\sup _{n\in [0,{\mathcal {F}}]}q{\mathscr {R}}_{L,n}^{*}+(1-q)\Lambda _{liq}$ for any bank $m\in [0,1]$ in equilibrium. It then follows from Steps 2 and 3 that the expected equity bank loan returns of private firms equal the expected equity returns of public firms. By Step 1, the expected marginal productivities of all firms must then be equal from which it follows that all firms raise the same amount of capital.

• 8. In any equilibrium, banks raise nonzero deposit capital. By Step 4, if one bank raises no deposit capital, then all banks raise no deposit capital. By the first inequality in (16) and the strict concavity of firms’ production function, $\Lambda (.),$ the expected marginal return from each firm’s technology exceeds one. Hence, by steps 1 and 3, $\sup _{n\in [0,{\mathcal {F}}]}q{\mathscr {R}}_{L,n}^{*}+(1-q)\Lambda _{liq}>1$. By step 5, therefore, ${\mathbb {E}}[{\widetilde{R}}_{E,m}^{*}]>1$ as well. Suppose that ${\mathbb {E}}[{\widetilde{R}}_{D,m}^{*}]<{\mathbb {E}}[{\widetilde{R}}_{E,m}^{*}]$. In this case, bank m has a profitable deviation where it lowers its equity capital and replaces it with deposit capital. Hence, we must have ${\mathbb {E}}[{\widetilde{R}}_{D,m}^{*}]\ge {\mathbb {E}}[{\widetilde{R}}_{E,m}^{*}]>1$. In this case, however, depositors strictly prefer investing some capital in bank m relative to the safe asset. Hence, all banks must raise nonzero deposit capital. Steps 1 to 8 together imply that, in equilibrium, all banks raise nonzero deposit capital. Further, in equilibrium, the expected marginal return from firm production equals the expected public firm equity, bank equity and bank deposit returns.

• 9. Equilibria are either default equilibria where all banks default with positive probability or no-default equilibria where all banks remain solvent. By Steps 6 and 8, all banks raise nonzero deposit capital and ${\mathbb {E}}[{\widetilde{R}}_{D,m}^{*}]=\sup _{n\in [0,{\mathcal {F}}]}q{\mathscr {R}}_{L,n}^{*}+(1-q)\Lambda _{liq}$. Hence, the expected deposit return for all banks is the same. In this case, however, risk-averse depositors strictly prefer the banks that do not default so deposit markets do not clear. Hence, if any bank defaults with nonzero probability, all banks must default with nonzero probability.

• 10. In a risky equilibrium, the (common) expected return on bank deposits, bank equity, bank loans,public firm equity as well as the expected marginal return on production for any firm exceeds the risk-free return. Because a bank’s deposits are risky, its risk-averse depositors must receive a risk premium for investing in the bank so the expected deposit return must exceed the risk-free return. The remaining assertions then follow from the arguments above.

• 11. In a no-default equilibrium, the (common) expected return on bank deposits, bank equity, bank loans, firm equity as well as the expected marginal return on investment for any firm equal the risk-free return. The common return cannot be less than the risk-free return by step 1. Suppose the common return exceeds the risk-free return. It would then be optimal for depositors to invest all their capital in banks, and banks and investors to invest all their capital in firms. However, in this case, the second condition in (16) implies that the expected marginal return on firm investment is less than the risk-free return, which is a contradiction.

• 12. In a risky equilibrium, all banks offer identical deposit and equity contracts, and raise the same amounts of deposit and equity capital. Depositors invest their capital in a single bank that offers the maximum expected utility. If multiple banks offer the same maximum expected utility, depositors randomly choose a single bank to invest in with uniform probability. Any bank that does not offer the maximum expected utility will receive no capital at all so that its deposit market will not clear. Hence, all banks must offer the same maximum expected utility and raise the same amount of deposit capital. Similarly, all banks must offer the same expected equity return and raise the same amount of equity capital. It then follows that banks must offer identical deposit and equity contracts in equilibrium. Q.E.D.

1.1.1 Proof of Proposition 1

Consider the set, $[0,{\mathcal {F}}],$ of private firms endowed with the Borel $\sigma$-algebra. Let $\mathcal {\widetilde{{\mathscr {R}}}}_{L,n}^{^ {}*}$ denote the equilibrium loan contract of firm $n\in [0,F].$ Suppose that the representative bank invests proportion, $d\zeta (n)$, of its capital in firm n where $d\zeta (.)$ is a Borel measure on $[0,{\mathcal {F}}]$ with $\int _{0}^{{\mathcal {F}}}d\zeta (n)=1$. The representative bank’s portfolio return is given by

$$\begin{aligned} {\widetilde{Q}}=\int _{0}^{{\mathcal {F}}}{\widetilde{{\mathscr {R}}}}_{L,n}^{^ {}*}d\zeta (n) \end{aligned}$$

(A1)

Define $\psi ({\widetilde{Q}})=((D^{*}+E^{*}){\widetilde{Q}}-D^{*}{\widetilde{R}}_{D}^{* })^{+}.$ $\psi (.)$ is a convex function. The expected payoff of the bank’s shareholders is

$$\begin{aligned} {\mathbb {E}}\left[ \psi ({\widetilde{Q}})\right]&={\mathbb {E}}\left[ \psi \left( \int _{0}^{{\mathcal {F}}}{\widetilde{{\mathscr {R}}}}_{L,n}^{^ {}*}d\zeta (n)\right) \right] \\&\le {\mathbb {E}}\left[ \left( \int _{0}^{{\mathcal {F}}}\psi \left( {\widetilde{{\mathscr {R}}}}_{L,n}^{^ {}*}\right) d\zeta (n)\right) \right] \\&=\left( \int _{0}^{{\mathcal {F}}}{\mathbb {E}}\left[ \psi \left( {\widetilde{{\mathscr {R}}}}_{L,n}^{^ {}*}\right) \right] d\zeta (n)\right) \\&={\mathbb {E}}\left[ \psi \left( {\widetilde{{\mathscr {R}}}}_{L}^{^ {}*}\right) \right] . \end{aligned}$$

The weak inequality in the second line above follows from Jensen’s inequality because $\psi$ is convex. The equality in the third line follows from Fubini’s theorem. The last equality follows because $\int _{0}^{{\mathcal {F}}}d\zeta (n)=1,$ and the bank loan returns are identically distributed across firms by Lemma 1. In the R.H.S. of the last equality, ${\widetilde{{\mathscr {R}}}}_{L}^{^ {}*}$ is the loan return of the representative firm. The weak inequality in the second line above is, in general, strict if the bank invests nonzero proportions of its capital in firms across different sectors because their returns may be imperfectly correlated. Note that the bank does not know the correlation between different sectoral shocks ex ante. Because the equilibrium loan returns are identically distributed across firms, and are perfectly correlated for firms in the same sector, the weak inequality becomes an equality when the bank invests its capital in any portfolio of firms in the same sector, that is, $\zeta (n)>0$ for any subset of firms in the same sector and $\zeta (n)=0$ otherwise. It follows that the expected payoff of the bank’s shareholders is maximized when the bank invests in any portfolio of firms within the same sector. Q.E.D.

1.1.2 Proof of Proposition 2

Consider a candidate risky equilibrium where banks’ equity capital $E^{*}\in [0,{\mathcal {E}}].$ Let $D^{*}$ be the total deposit capital raised by banks in the equilibrium. As discussed in Sect. 4.1.5, the deposit capital, $D^{* }$and the loan rate, ${\mathscr {R}}_{L}^{*},$ satisfy conditions (17) and (27) with the deposit returns upon bank failure and success, respectively, given by (24) and (25).

Since $\Lambda '(\frac{\mathcal{E}+D^{* }}{{\mathcal {F}}+{\mathcal {G}}})={\mathscr {R}}_{L}^{*}$ and $\Lambda$ is strictly concave and twice continuously differentiable, ${\mathscr {R}}_{L}^{*}$ is a strictly decreasing and continuous function of $D^{*}$. Further, by (16), $q{\mathscr {R}}_{L}^{*}+(1-q)\Lambda _{liq}>1$ for $D^{*}=0$ and $q{\mathscr {R}}_{L}^{*}+(1-q)\Lambda _{liq}<1$ for $D^{*}={\mathcal {D}}.$

We now show that, for a given loan rate, ${\mathscr {R}}_{L}^{*},$ with $q{\mathscr {R}}_{L}^{*}+(1-q)\Lambda _{liq}>1,$ there is a unique deposit level $D^{*}$ that satisfies (27) and, moreover, $D^{*}$ increases with ${\mathscr {R}}_{L}^{*}$. By (27), (24) and (25), we have

$$\begin{aligned} D^{*}&={\mathcal {D}}d(R_{D}^{s*},R_{D}^{f*})={\mathcal {D}}d \Bigg (\frac{q{\mathscr {R}}_{L}^{*}+(1-q)\Lambda _{liq}}{q}\nonumber \\&\quad -\frac{1-q}{q}\left( 1+E^{*}/D^{*}\right) \Lambda _{liq},\left( 1+E^{*}/D^{*}\right) \Lambda _{liq}\Bigg ). \end{aligned}$$

(A2)

By Assumption 2 (ii), when ${\mathbb {E}}[{\widetilde{R}}_{D}^{*}]>1$ is fixed, the demand function $d(R_{D}^{s*},R_{D}^{f*})$ is continuously decreasing in $R_{D}^{s*}-R_{D}^{f*}$. It follows that the last expression in (A2) is continuously decreasing in $D^{*}.$ Further, it is strictly positive for $D^{*}=0$ and is less than or equal to ${\mathcal {D}}$ for $D^{*}={\mathcal {D}}$. Hence, there is a unique solution of (A2), which determines the deposit capital corresponding to the loan rate, ${\mathscr {R}}_{L}^{*},$ as implied by (27). Moreover, by Assumption 2 (i), the deposit demand function, $d(R_{D}^{s*},R_{D}^{f*})$, increases with ${\mathscr {R}}_{L}^{*}$ so that the solution, $D^{*}$, to (A2) also increases with ${{{\mathscr {R}}_{L}^{*}}}.$ When $q{\mathscr {R}}_{L}^{*}+(1-q)\Lambda _{liq}\rightarrow 1,$ $D^{*}\rightarrow 0$ because risk-averse depositors must obtain a risk premium to supply nonzero capital.

It follows from the above that the production technology and deposit demand curves defined by (17) and (27) have a unique intersection, which determine the deposit capital, $D^{*},$ and the loan rate, ${\mathscr {R}}_{L}^{*}$ in the equilibrium. Moreover, $q{\mathscr {R}}_{L}^{*}+(1-q)\Lambda _{liq}>1.$ The corresponding equilibrium equity capital, $E^{*},$of the representative bank is then given by (20).By Lemma 1, a necessary and sufficient condition for an unregulated risky equilibrium is that the expected marginal return from investment, $q\Lambda '(\frac{\mathcal{E}+D^{*}}{{\mathcal {F}}+{\mathcal {G}}})+(1-q)\Lambda _{liq}$ is strictly greater than one. Q.E.D.

1.1.3 Proof of Proposition 3

The social planner’s objective is

$$\begin{aligned}&\max _{X_{F},X_{S},P_{D}^{l},P_{D}^{h},P_{E}^{l},P_{E}^{h},P_{F}^{l},P_{F}^{h}\ge 0}{\mathcal {D}}{\mathbb {E}}\left[ u\left( \frac{{\widetilde{P}}_{D}}{{\mathcal {D}}}\right) \right] \text{ subject } \text{ to } \end{aligned}$$

(A3)

$$\begin{aligned}&X_{S}+X_{F}\le {\mathcal {D}}+\mathcal {E\text {, }} \end{aligned}$$

(A4)

$$\begin{aligned}&P_{D}^{h}+P_{E}^{h}+P_{F}^{h}\le \omega ^{h}({\mathcal {F}}+{\mathcal {G}})\Lambda \left( \frac{X_{F}}{{\mathcal {F}}+{\mathcal {G}}}\right) +(1-\omega ^{h})X_{F}\Lambda _{liq}+X_{S}, \end{aligned}$$

(A5)

$$\begin{aligned}&P_{D}^{l}+P_{E}^{l}+P_{F}^{l}\le \omega ^{l}({\mathcal {F}}+{\mathcal {G}})\Lambda \left( \frac{X_{F}}{{{\mathcal {F}}}+{\mathcal {G}}}\right) +(1-\omega ^{l})X_{F}\Lambda _{liq}+X_{S}, \end{aligned}$$

(A6)

$$\begin{aligned}&{\mathbb {E}}[{\widetilde{P}}_{E}]\ge \Delta _{E};{\mathbb {E}}[{\widetilde{P}}_{F}]\ge \Delta _{F}. \end{aligned}$$

(A7)

Equation (A4) is the aggregate resource constraint. Inequalities (A5) and (A6) mean that the total capital allocated to all agents does not exceed the total available capital in the low and high states, respectively. Because a proportion $\omega ^{h}$ of firms succeeds in the high aggregate state, the total revenue of all firms in the high state is $\omega ^{h}\left( {\mathcal {F}}+{\mathcal {G}}\right) \Lambda (\frac{X_{F}}{{{{\mathcal {F}}}{+{\mathcal {G}}}}})+(1-\omega ^{h})X_{F}\Lambda _{liq}.$ Similarly, the total revenue of all firms in the low state is $\omega ^{l}({\mathcal {F}}+{\mathcal {G}})\Lambda (\frac{X_{F}}{{\mathcal {F}}+{\mathcal {G}}})+(1-\omega ^{l})X_{F}\Lambda _{liq}.$ The constraints, (A7) specify that the total allocated expected payoffs to investors and firms is no less than the specified reservation payoff levels that determine the efficient allocation.

Set

$$\begin{aligned} \Delta _{E}+\Delta _{F}&=\Delta ,\\ P_{E}^{\omega }+P_{F}^{\omega }&=P_{N}^{\omega };\omega \in \left\{ l,h\right\} . \end{aligned}$$

Consider the following planning problem:

$$\begin{aligned}&\max _{X_{F},X_{S},P_{D}^{l},P_{D}^{h},P_{N}^{l},P_{N}^{h}\ge 0}{\mathcal {D}}{\mathbb {E}}\left[ u\left( \frac{{\widetilde{P}}_{D}}{{\mathcal {D}}}\right) \right] \text{ subject } \text{ to } \end{aligned}$$

(A8)

$$\begin{aligned}&X_{S}+X_{F}\le {\mathcal {D}}+{\mathcal {E}}, \end{aligned}$$

(A9)

$$\begin{aligned}&P_{D}^{h}+P_{N}^{h}\le \omega ^{h}({\mathcal {F}}+{\mathcal {G}})\Lambda \left( \frac{X_{F}}{{\mathcal {F}}+{\mathcal {G}}}\right) +(1-\omega ^{h})X_{F}\Lambda _{liq}+X_{S}, \end{aligned}$$

(A10)

$$\begin{aligned}&P_{D}^{l}+P_{N}^{l}\le \omega ^{l}({\mathcal {F}}+{\mathcal {G}})\Lambda \left( \frac{X_{F}}{{\mathcal {F}}+{\mathcal {G}}}\right) +(1-\omega ^{l})X_{F}\Lambda _{liq}+X_{S}, \end{aligned}$$

(A11)

$$\begin{aligned}&{\mathbb {E}}[{\widetilde{P}}_{N}]\ge \Delta . \end{aligned}$$

(A12)

Given a solution to (A8), we can construct a solution to (A3) by using a transfer between firms and investors. Limited liability and an increasing depositor utility function together imply that we can replace (A8) with the following problem:

$$\begin{aligned}&\max _{X_{F},X_{S},P_{D}^{L},P_{D}^{H},P_{N}^{L},P_{N}^{H}\ge 0}{\mathcal {D}}{\mathbb {E}}\left[ u\left( \frac{{\widetilde{P}}_{D}}{{\mathcal {D}}}\right) \right] \text{ subject } \text{ to }\nonumber \\&\quad P_{D}^{l}\le \omega ^{l}({\mathcal {F}}+{\mathcal {G}})\Lambda \left( \frac{X_{F}}{{\mathcal {F}}+{\mathcal {G}}}\right) +(1-\omega ^{l})X_{F}\Lambda _{liq}+X_{S}=A_{1}, \end{aligned}$$

(A13)

$$\begin{aligned}&P_{D}^{h}\le \omega ^{h}({\mathcal {F}}+{\mathcal {G}})\Lambda \left( \frac{X_{F}}{{\mathcal {F}}+{\mathcal {G}}}\right) +(1-\omega ^{h})X_{F}\Lambda _{liq}+X_{S}=A_{2}, \end{aligned}$$

(A14)

$$\begin{aligned}&{\mathbb {E}}[{\widetilde{P}}_{D}]\le pA_{1}+(1-p)A_{2}-\Delta \nonumber \\&\quad =q({\mathcal {F}}+{\mathcal {G}})\Lambda \left( \frac{X_{F}}{{\mathcal {F}}+{\mathcal {G}}}\right) +(1-q)X_{F}\Lambda _{liq}+X_{S}-\Delta =A_{0}, \end{aligned}$$

(A15)

where the first equality in the last expression above follows from (1). Now we show that, in the optimum, the equation (A15) is binding. If not, since

$$\begin{aligned} A_{0}<pA_{1}+(1-p)A_{2}. \end{aligned}$$

we can increase the payout to depositors in at least one state of the world such that (A13), (A14), and (A15) are satisfied which contradicts optimality. Hence there are two cases:

• Case 1: Inequality (A15) is binding, but inequality (A13) is not binding. In this case, since depositors are risk averse, we have

  $$\begin{aligned} P_{D}^{h,{\textit{eff}}}= & {} P_{D}^{l,{\textit{eff}}}=\max _{X_{F},X_{S}}q({\mathcal {F}}+{\mathcal {G}})\Lambda \left( \frac{X_{F}}{{\mathcal {F}}+{\mathcal {G}}}\right) \nonumber \\{} & {} +(1-q)X_{F}\Lambda _{liq}+X_{S}-\Delta =A_{0}. \end{aligned}$$

  (A16)

  If not, we can transfer capital from high state to low state such that equation ${\mathbb {E}}[P_{D}^{\omega }]=A_{0}$ is satisfied, which would increase depositor’s expected utility. Now recall that since the central planner optimally invests all the resources in either the safe asset or firms, we have $X_{S}={\mathcal {D}}+{\mathcal {E}}-X_{F}.$ The first condition for problem (A16) gives us the following equation:

  $$\begin{aligned}&q\Lambda '\left( \frac{X_{F}^{{\textit{eff}}}}{{\mathcal {F}}+{\mathcal {G}}}\right) +(1-q)\Lambda _{liq}=1. \end{aligned}$$

  (A17)

This can only happen when there is enough capital to pay depositors $A_{0}$ in the low state. Therefore,

$$\begin{aligned} A_{0}\le A_{2}, \end{aligned}$$

which, by (1) and (2) is equivalent to

$$\begin{aligned} \tau \le \min \left( \frac{\Delta }{[q({\mathcal {F}}+{\mathcal {G}})\Lambda (\frac{X_{F}^{\text {e}{\textit{ff}}}}{{\mathcal {F}}+{\mathcal {G}}})+(1-q)X_{F}^ {\text {e}{\textit{ff}}}\Lambda _{liq}](\omega ^{h}-\omega ^{l})},1\right) ={\bar{\tau }}(\Delta ) \end{aligned}$$

(A18)

• Case 2: Inequality (A15) and (A13) are both binding. In this case we have:

$$\begin{aligned} P_{D}^{h,{\textit{eff}}}=\frac{A_{0}}{p}-\frac{(1-p)P_{D}^{l,{\textit{eff}}}}{p}. \end{aligned}$$

Because (A13) is binding, depositors receive the entire economic output in the low aggregate state. The central planner problem reduces to:

$$\begin{aligned} \max _{X_{F}}{\mathcal {D}}\left[ pu\left( \frac{\frac{A_{0}}{p}-\frac{(1-p)P_{D}^{l,{\textit{eff}}}}{p}}{{\mathcal {D}}}\right) +(1-p)u\left( \frac{P_{D}^{l,{\textit{eff}}}}{{\mathcal {D}}}\right) \right] . \end{aligned}$$

The first order condition for the problem is

$$\begin{aligned}&\frac{\partial }{\partial X_{F}}{\mathcal {D}}[pu\left( \frac{\omega ^{h}({\mathcal {F}}+{\mathcal {G}})\Lambda (\frac{X_{F}^{{\textit{eff}}}}{{\mathcal {F}}+{\mathcal {G}}})+(1-\omega ^{h})X_{F}^{{{\textit{eff}}}}\Lambda _{liq}+{\mathcal {D}}+{\mathcal {E}} -X_{F}-\frac{\Delta _{E}+\Delta _{F}}{p}}{{\mathcal {D}}}\right) \nonumber \\&\quad +(1-p)u\left( \frac{\omega ^{l}({\mathcal {F}}+{\mathcal {G}})\Lambda (\frac{X_{F}^{{\textit{eff}}}}{{\mathcal {F}}+{\mathcal {G}}})+(1-\omega ^{l})X_{F}^{{\textit{eff}}}\Lambda _{liq}+{\mathcal {D}}+{\mathcal {E}}-X_{F}^{{\textit{eff}}}}{{\mathcal {D}}}\right) ] =0. \end{aligned}$$

(A19)

The above reduces to

$$\begin{aligned}&p\left( \omega ^{h}\Lambda '(\frac{X_{F}^{{\textit{eff}}}}{{\mathcal {F}}+{\mathcal {G}}})+(1-\omega ^{h})\Lambda _{liq}-1\right) u' \left( \frac{P_{D}^{h,{\textit{eff}}}}{{\mathcal {D}}}\right) \nonumber \\&\quad +(1-p)\left( \omega ^{l}\Lambda '(\frac{X_{F}^{{\textit{eff}}}}{{\mathcal {F}}+{\mathcal {G}}})+(1-\omega ^{l})\Lambda _{liq}-1\right) u'\left( \frac{P_{D}^{l,{\textit{eff}}}}{{\mathcal {D}}}\right) =0. \end{aligned}$$

(A20)

Since $u'\left( \frac{P_{D}^{h,{\textit{eff}}}}{{\mathcal {D}}}\right) ,u'\left( \frac{P_{D}^{l,{\textit{eff}}}}{{\mathcal {D}}}\right) >0,$ and $\omega ^{h}\Lambda '(\frac{X_{F}^{{\textit{eff}}}}{{\mathcal {F}}+{\mathcal {G}}})+(1-\omega ^{h})\Lambda _{liq}>\omega ^{l}\Lambda '(\frac{X_{F}^{{\textit{eff}}}}{{\mathcal {F}}+{\mathcal {G}}})+(1-\omega ^{l})\Lambda _{liq},$ we must have

$$\begin{aligned} \omega ^{h}\Lambda '\left( \frac{X_{F}^{{\textit{eff}}}}{{\mathcal {F}}+{\mathcal {G}}}\right) +(1-\omega ^{h})\Lambda _{liq}-1&>0,\nonumber \\ \omega ^{l}\Lambda '\left( \frac{X_{F}^{{\textit{eff}}}}{{\mathcal {F}}+{\mathcal {G}}}\right) +(1-\omega ^{l})\Lambda _{liq}-1&<0. \end{aligned}$$

(A21)

By (1), and rearranging expressions, we can rewrite (A20) as

$$\begin{aligned}&\left( q\Lambda '\left( \frac{X_{F}^{{\textit{eff}}}}{{\mathcal {F}}+{\mathcal {G}}}\right) +(1-q)\Lambda _{liq}-1\right) u'\left( \frac{P_{D}^{h,{\textit{eff}}}}{{\mathcal {D}}}\right) \\&\quad +\ (1-p)\left( \omega ^{l}\Lambda '\left( \frac{X_{F}^{{\textit{eff}}}}{{\mathcal {F}}+{\mathcal {G}}}\right) +(1-\omega ^{l})\Lambda _{liq}-1\right) \left( u'\left( \frac{P_{D}^{l,{\textit{eff}}}}{{\mathcal {D}}}\right) -u'\left( \frac{P_{D}^{h,{\textit{eff}}}}{{\mathcal {D}}}\right) \right) =0. \end{aligned}$$

Because u is strictly concave, and $P_{D}^{l ,{\textit{eff}} }<P_{D}^{h,{\textit{eff}}}$, we see that $u'\left( \frac{P_{D}^{l,{\textit{eff}}}}{{\mathcal {D}}}\right) -u'\left( \frac{P_{D}^{h,{\textit{eff}}}}{{\mathcal {D}}}\right) >0$. It then follows from (A21) that the second expression on the L.H.S. above is strictly negative so that the first expression must be strictly positive. Hence,

$$\begin{aligned} q\Lambda '\left( \frac{X_{F}^{{\textit{eff}}}}{{\mathcal {F}}+{\mathcal {G}}}\right) +(1-q)\Lambda _{liq}>1, \end{aligned}$$

which implies that the expected marginal return on investment in the representative firm exceeds the risk-free return. Q.E.D.

1.1.4 Proof of Proposition 4

• (i) For $\tau <{\bar{\tau }}(\Delta )$, as defined in (A18), $X_{F}^{\text {e}{\textit{ff}}}$ is constant. For $\tau >{\bar{\tau }}(\Delta )$, in order to prove $X_{F}^{\text {e}{\textit{ff}}}$ is decreasing, it is enough to show the cross derivative of the objective function is negative. The cross derivative is:

  $$\begin{aligned}{} & {} \frac{\partial ^{2}}{\partial X_{F}\partial \tau }{\mathcal {D}}\left[ pu\left( \frac{P_{D}^{h,{\textit{eff}}}}{{\mathcal {D}}}\right) +(1-p)u\left( \frac{P_{D}^{l,{\textit{eff}}}}{{\mathcal {D}}}\right) \right] \\{} & {} \quad = p(1-p)(\omega ^{h}-\omega ^{l})\Lambda '\left( \frac{X_{F}^{{\textit{eff}}}}{{\mathcal {F}}+{\mathcal {G}}}\right) \left( u'\left( \frac{P_{D}^{h,{\textit{eff}}}}{{\mathcal {D}}}\right) -u'\left( \frac{P_{D}^{l,{\textit{eff}}}}{{\mathcal {D}}}\right) \right) \\{} & {} \qquad + p(1-p)(\omega ^{h}-\omega ^{l})[{\mathcal {F}}\Lambda \left( \frac{X_{F}^{{\textit{eff}}}}{{\mathcal {F}}+{\mathcal {G}}}\right) \frac{1}{{\mathcal {D}}}\\{} & {} \qquad \times \left[ (\omega ^{h}\Lambda '\left( \frac{X_{F}^{{\textit{eff}}}}{{\mathcal {F}}+{\mathcal {G}}}\right) +(1-\omega ^{h})\Lambda _{liq}-1)u'' \left( \frac{P_{D}^{h,{\textit{eff}}}}{{\mathcal {D}}}\right) \right. \\{} & {} \left. \qquad -(\omega ^{l}\Lambda '(\frac{X_{F}^{{\textit{eff}}}}{{\mathcal {F}}+ {\mathcal {G}}})+(1-\omega ^{l})\Lambda _{liq}-1)u''\left( \frac{P_{D}^{l,{\textit{eff}}}}{{\mathcal {D}}}\right) \right] . \end{aligned}$$

  Concavity of the utility function u implies that the first term is negative since $u'\left( \frac{P_{D}^{h,{\textit{eff}}}}{{\mathcal {D}}}\right) <u'\left( \frac{P_{D}^{l,{\textit{eff}}}}{{\mathcal {D}}}\right)$. Hence, it is enough to show the last term is negative as well. Since $\omega ^{h}-\omega ^{l}>0,$ It suffices to show the following:

  $$\begin{aligned}&(\omega ^{h}\Lambda '\left( \frac{X_{F}^{{\textit{eff}}}}{{\mathcal {F}}+{\mathcal {G}}}\right) +(1-\omega ^{h})\Lambda _{liq}-1)u'' \left( \frac{P_{D}^{h,{\textit{eff}}}}{{\mathcal {D}}}\right) \\&\quad -(\omega ^{l}\Lambda '\left( \frac{X_{F}^{{\textit{eff}}}}{{\mathcal {F}} +{\mathcal {G}}}\right) +(1-\omega ^{l})\Lambda _{liq}-1)u''\left( \frac{P_{D}^{l,{\textit{eff}}}}{{\mathcal {D}}}\right) <0. \end{aligned}$$

  Since $u''<0,$ it is enough to show that

  $$\begin{aligned} A^{h}&=\omega ^{h}\Lambda '\left( \frac{X_{F}^{{\textit{eff}}}}{{\mathcal {F}}+{\mathcal {G}}}\right) +(1-\omega ^{h})\Lambda _{liq}-1\ge 0,\\ A^{l}&=\omega ^{l}\Lambda '\left( \frac{X_{F}^{{\textit{eff}}}}{{\mathcal {F}}+{\mathcal {G}}}\right) +(1-\omega ^{l})\Lambda _{liq}-1\le 0. \end{aligned}$$

  But we have already established the above in (A21).

• (ii) If $p=q$ and $\tau =1$, it follows from (1) and (2) that $\omega ^{h}=1,\omega ^{l}=0$ and all firm shocks are perfectly correlated. In this case, the lack of diversification of depositors is costless so that the competitive equilibrium of the unregulated economy is Pareto efficient. Indeed, the equilibrium condition (24) along with the perfect correlation of firm shocks implies that depositors receive the entire output of the economy in the low aggregate state as in the efficient allocation, that is, constraint (A13) in the optimization program that determines the efficient allocation is binding. Constraint (A15) is also satisfied with equality by the unregulated equilibrium allocation. Finally, when $\tau =1,$ the unregulated equilibrium allocation maximizes depositors’ expected utility subject to the above constraints and, therefore, coincides with the efficient allocation obtained by setting the reservation payoffs of investors, banks and firms equal to their respective expected payoffs in the unregulated equilibrium. In particular, therefore, the efficient level of investment coincides with the investment level in the unregulated equilibrium. Because the efficient investment level is weakly decreasing with $\tau$ by the preceding analysis, and is strictly greater than the equilibrium level for $\tau <{\bar{\tau }}(\Delta ),$ it follows that the basic economy underinvests in firms relative to the efficient allocation for $\tau <1$ with the investment levels being equal for $\tau =1.$ Q.E.D.

1.1.5 Proof of Lemma 2

• (i) If the liquidity requirement is not binding, then the arguments in the discussion following the statement of the lemma show that (40) holds.

  Suppose that the liquidity requirement is binding in the regulated equilibrium. If $x=e^{reg}+d^{reg}$ is the total capital raised by a bank, and $\alpha ^{reg}$ is the proportion of equity capital to total capital in the regulated equilibrium, we have:

  $$\begin{aligned} e^{reg}&=\alpha ^{reg}x,\; d^{reg}=(1-\alpha ^{reg})x,\\ s^{reg}&=\beta ^{reg}(1-\alpha ^{reg})x,\; l^{reg}=(1-\beta ^{reg}+\beta ^{reg}\alpha ^{reg})x. \end{aligned}$$

  Hence the marginal cost of capital for the representative bank is equal to

  $$\begin{aligned}&\alpha ^{reg}{\mathbb {E}}[{\tilde{C}}_{E}^{reg}]+(1-\alpha ^{reg}){\mathbb {E}}[{\tilde{C}}_{D}^{reg}], \end{aligned}$$

  and the expected marginal revenue is equal to:

  $$\begin{aligned}&\beta ^{reg}(1-\alpha ^{reg})+(1-\beta ^{reg}+\beta ^{reg}\alpha ^{reg})(q{\mathscr {R}}_{L}^{reg}+(1-q)\Lambda _{liq}). \end{aligned}$$

  As the bank must make zero expected profit in any regulated equilibrium, (40) is satisfied.

• (ii) By arguments identical to those used in the proof of Lemma 1, the expected return on bank loans in a competitive equilibrium must equal the expected marginal productivity of each private firm. This, in turn, coincides with the expected marginal productivity of each firm, ${\mathbb {E}}[{\widetilde{R}}^{\text {e}{\textit{ff}}}],$ in the efficient allocation as the regulated equilibrium implements the efficient allocation by hypothesis. In equilibrium, the expected return on bank loans must equal the expected marginal cost of equity financing for each public firm as both coincide with the expected marginal productivity of each firm in the efficient allocation. Further, to ensure that bank equity and public firm equity markets both clear, investors must be indifferent between investing in bank equity and public firm equity. It follows that the expected equity returns of banks and public firms must be equal. This, in turn, implies that the expected marginal costs of equity financing for banks and public firms should also be equal. Hence, the expected return on bank loans equals the expected marginal cost of equity financing for banks so that (41) is satisfied. Finally, (42) follows directly from (41) and (40). Q.E.D.

1.1.6 Proofs of Proposition 5

Assuming that the equilibrium that implements the efficient allocation exists, we first derive the policy tools and allocations. We begin by assuming that, in the candidate regulated equilibrium, depositors supply all their capital to banks. We later establish that this is, indeed, the case.

Case 1 $\tau \le {\overline{\tau }}(\Delta )$

By Proposition 3 (i), the total capital, $X_{F}^{{\textit{eff}}},$ invested in all firms—public and private—in the efficient allocation is such that the expected marginal productivity of each firm equals the risk-free return, that is,

$$\begin{aligned} {\mathbb {E}}[{\widetilde{R}}^{\text {e}{\textit{ff}}}]=1. \end{aligned}$$

(A22)

Let $X_{S}^{{\textit{eff}}}$ denote the total investment in the safe asset in the efficient allocation.

Consider the representative bank. The bank pays a deposit insurance premium, $\pi ^{reg}{\mathcal {D}},$ from the total deposit capital that it raises, which goes into the deposit inurance fund that is invested in the safe asset. Let $L^{reg}$ and $S^{reg}$ be the capital invested by the representative bank in private firms and the safe asset, respectively in a candidate regulated equilibrium that implements the efficient allocation. $L^{reg}$ is uniquely pinned down by the condition that the expected marginal productivity of each private firm that the bank invests in must equal ${\mathbb {E}}[{\widetilde{R}}^{\text {e}{\textit{ff}}}],$that is,

$$\begin{aligned} q\Lambda ^{'}\left( \frac{L^{reg}}{{\mathcal {F}}}\right) +(1-q)\Lambda _{liq}={\mathbb {E}}[{\widetilde{R}}^{\text {e}{\textit{ff}}}]. \end{aligned}$$

(A23)

In the L.H.S. above, we incorporate the fact that there is a mass one of banks so that the total capital invested by all banks is $L^{reg},$ and this capital is invested uniformly in the mass ${\mathcal {F}}$ of private firms. The total investment by investors in public firms is then given uniquely by $X_{F}^{{\textit{eff}}}-L^{reg}.$ Further, we know from the proof of Proposition 3 that the expected marginal productivity of private and public firms must be equal in the efficient allocation, that is,

$$\begin{aligned} q\Lambda ^{'}\left( \frac{L^{reg}}{{\mathcal {F}}}\right) +(1-q)\Lambda _{liq}=q\Lambda ^{'}\left( \frac{X_{F}^{{\textit{eff}}}-L^{reg}}{{\mathcal {G}}}\right) +(1-q)\Lambda _{liq}={\mathbb {E}}[{\widetilde{R}}^{\text {e}{\textit{ff}}}]. \end{aligned}$$

By Lemma 2, the expected marginal costs of equity and deposit financing for banks, the expected bank loan return, and the expected marginal cost of equity financing for public firms all equal the risk-free return. Hence, banks are indifferent between investing in private firms and the safe asset. As we discuss below, we can design the taxes on equity payouts and returns from safe asset investments by investors so that they are indifferent between investing in public firm equity, bank equity and the safe asset. Any allocation in which the total investments by banks, investors and the deposit insurance fund in the safe asset equals the efficient safe asset investment, $X_{S}^{{\textit{eff}}}$, then constitutes an equilibrium. The equilibria differ only in the relative investments by banks and investors in the safe asset as long as the total safe asset investment is $X_{S}^{{\textit{eff}}}$.

We can implement a unique regulated equilibrium in which banks invest a specific capital level, $S^{reg}$, in the safe asset by specifying a liquidity requirement, $\beta ^{reg}$, on banks such that $\beta ^{reg}{\mathcal {D}}=S^{reg}$. This pins down the safe asset investment by banks. Hence, the safe asset investment by investors must equal $X_{S}^{{\textit{eff}}}-S^{reg}-\pi ^{reg}{\mathcal {D}}$. Alternatively, we can implement a unique regulated equilibrium by specifying a capital requirement, $\theta ^{reg}$, on banks such that each bank’s total deposit and equity capital equals $L^{reg}+S^{reg}$, thereby ensuring that its safe asset investment is $S^{reg}.$ In other words, we determine the capital requirement, $\theta ^{reg},$ by

$$\begin{aligned} {\mathcal {D}}+\theta ^{reg}L^{reg}-\pi ^{reg}{\mathcal {D}}&=L^{reg}+S^{reg},\text {or}\nonumber \\ \theta ^{reg}&=\frac{L^{reg}+S^{reg}-(1-\pi ^{reg}){\mathcal {D}}}{L^{reg}}. \end{aligned}$$

(A24)

Let us now determine the allocations to depositors and investors in the candidate equilibrium. If the representative bank fails, its asset payoff is

$$\begin{aligned} \Lambda _{liq}L^{reg}+S^{reg}-\pi ^{reg}D. \end{aligned}$$

(A25)

In this scenario, its entire payoff goes to depositors. Therefore,

$$\begin{aligned} \mathcal{D}C_{D}^{f,,reg}=\Lambda _{liq}L^{reg}+S^{reg}-\pi \mathcal{D}, \end{aligned}$$

(A26)

Hence, $C_{D}^{f,reg}$ is given by

$$\begin{aligned} C_{D}^{f,reg}&=\frac{\Lambda _{liq}L^{reg}+S^{reg}-\pi \mathcal{D}}{\mathcal{D}}. \end{aligned}$$

(A27)

Lemma 2 and (A22 together imply that:

$$\begin{aligned}&{\mathbb {E}}[{\tilde{C}}_{D}^{reg}]={\mathbb {E}}[{\widetilde{R}}^{\text {e}{\textit{ff}}}]-\frac{S^{reg}-\pi ^{reg}\mathcal{D}}{\mathcal{D}}({\mathbb {E}}[{\widetilde{R}}^{\text {e}{\textit{ff}}}]-1)=1. \end{aligned}$$

(A28)

The marginal cost of deposits when the representative bank succeeds, $C_{D}^{s,reg}$, is then uniquely determined by (A27) and (A28). Since the equilibrium implements the efficient allocation, we must have

$$\begin{aligned} R_{D}^{j,reg}=\frac{P_{D}^{j,\text {e}{\textit{ff}}}}{\mathcal{D}};j\in \left\{ h,l\right\} \end{aligned}$$

(A29)

We set the deposit insurance premium so that there is no subsidy in the high aggregate state, that is, any additional payment to depositors is via deposit insurance. By (38), therefore, we have

$$\begin{aligned} R_{D}^{h,reg}&=C_{D}^{s,reg}-t_{D}^{h,reg}-\pi ^{reg}=C_{D}^{f,reg}-\pi ^{reg}+\mu ^{h,reg},\nonumber \\ R_{D}^{l,reg}&=C_{D}^{s,reg}-t_{D}^{l,reg}-\pi ^{reg}=C_{D}^{f,reg}-\pi ^{reg}+\mu ^{l,reg}+\phi ^{l,reg}. \end{aligned}$$

(A30)

The above equations along with (31) uniquely identify $\mu ^{h,reg},\mu ^{l,reg},\phi ^{l,reg}$,$t_{D}^{h,reg},t_{D}^{l,reg}$ and $\pi ^{reg}.$

Let us now turn to investors. Because banks and public firms default when their assets fail, the marginal costs of equity for banks and public firms, and the corresponding returns to investors are equal to zero, that is,

$$\begin{aligned} R_{E}^{f,h,reg}=R_{E}^{f,l,reg}=C_{E}^{f,reg}=0. \end{aligned}$$

(A31)

Hence, the marginal tax on equity payouts to investors by banks and public firms when they fail is also zero. In the above, we we use the same notation to refer to marginal costs of equity for banks and public firms as well as the corresponding returns to investors to avoid complicating the notation unnecessarily. Relation (42) determines the expected marginal cost of equity to banks and public firms. This, in turn, pins down the marginal cost of equity, $C_{E}^{s,reg}$, upon bank/public firm success. Since the regulated equilibrium implements the efficient allocation, the total expected payoff to investors must be $\Delta _{E}.$ Hence,

$$\begin{aligned} {\mathbb {E}}\left[ {\widetilde{R}}_{E}^{reg}\right] =\frac{\Delta _{E}}{{\mathcal {E}}}. \end{aligned}$$

(A32)

We set $R_{E}^{s,h,reg}=R_{E}^{s,l,reg}$, that is, investors receive the same return upon bank/public firm success in both aggregate states. From the above, we therefore have

$$\begin{aligned} qR_{E}^{s,h,reg}=qR_{E}^{s,l,reg}={\mathbb {E}}\left[ {\widetilde{R}}_{E}^{reg}\right] =\frac{\Delta _{E}}{{\mathcal {E}}}. \end{aligned}$$

(A33)

Relations (39 then determine the marginal taxes on equity payouts upon bank/public firm success. We set the marginal tax on payouts from the safe asset so that the expected returns to investors from investing in bank equity, public firm equity, and the safe asset are equal. Because marginal taxes are imposed only on payouts by successful firms, and the marginal taxes are the same in both aggregate states, this is guaranteed by setting the marginal tax rate on safe asset investments to $qt_{E}^{s,h,reg}=qt_{E}^{s,l,reg}.$

Let us turn now to firms. Recall that they receive a positive payoff only upon firm success. We set the payoffs to be equal in the high and low aggregate states. Hence, we have

$$\begin{aligned} qP_{F}^{s,h,reg}=qP_{F}^{s,l,reg}=\frac{\Delta _{F}}{{\mathcal {F}}+{\mathcal {G}}}, \end{aligned}$$

(A34)

as the regulated equilibrium implements the efficient allocation. As $X_{F}^{{{\textit{eff}}}}$ is the total capital invested in firms in the efficient allocation, and $C_{E}^{s,reg}$ is the marginal cost of bank loan (equity) financing for a private (public) firm when it succeeds, we must have

$$\begin{aligned} \Lambda \left( \frac{X_{F}^{{\textit{eff}}}}{{\mathcal {F}}+{\mathcal {G}}}\right) -\frac{X_{F}^{{\textit{eff}}}}{{\mathcal {F}}+{\mathcal {G}}}C_{E}^{s,reg}-T_{F}^{s,j,reg}=P_{F}^{s,j,reg};j\in \left\{ h,l\right\} \end{aligned}$$

(A35)

The above determines the lump sum taxes on firms upon firm success.

It is straightforward to verify that the above, indeed, determine a unique equilibrium. By (36), the risk-free return to depositors strictly exceeds the safe asset return. Hence, depositors supply all their capital in banks as hypothesized. By construction, banks and investors are indifferent between investing in firms and the safe asset. Given the marginal costs of bank loan financing and equity financing, each private or public firm demands capital, $\frac{X_{F}^{eff}}{{\mathcal {F}}+{\mathcal {G}}},$ by construction. Because banks and investors are indifferent between investing in firms and the safe asset, the capital demand by each private or public firm is met by capital supplied by banks and investors, respectively. Similarly, banks’ demand for deposits and equity are met by capital supply so that all the market clearing conditions are satisfied.

Case 2 $\tau >{\overline{\tau }}(\Delta )$

As in Case 1 where $\tau \le {\overline{\tau }}(\Delta ),$ the investment by the representative bank in bank loans, $L^{reg}$, is determined by (A23), and the total investment by investors in public firms is given uniquely by $X_{F}^{{\textit{eff}}}-L^{reg}.$ In this case, however, ${\mathbb {E}}[{\widetilde{R}}^{\text {e}{\textit{ff}}}]>1.$ Hence, the expected return on bank loans, which equals the expected marginal cost of bank equity financing, is greater than one so that banks do not voluntarily invest any capital in the safe asset. Similarly, investors too do not voluntarily invest any capital in the safe asset. Since a regulator cannot monitor investor portfolios, it must force banks to invest capital in the safe asset that matches the efficient allocation. The total deposit insurance premia, $\pi ^{reg}{\mathcal {D}}$, in the deposit insurance fund is invested in the safe asset. Hence, the investment by the representative bank, $S^{reg}$, in the safe asset must be such that the total investment in the safe asset is $X_{S}^{{\textit{eff}}.}$ so that $S^{reg}=X_{S}^{{\textit{eff}}}-\pi ^{reg}{\mathcal {D}}$. The binding liquidity requirement, $\beta ^{reg},$ on banks is, therefore, given by $\beta ^{reg}=\frac{S^{reg}}{{\mathcal {D}}}$.

By (42), the expected marginal cost of deposits is strictly less than the expected marginal cost of equity for banks. Therefore, banks do not voluntarily raise equity capital unless compelled to do so. The regulator must, therefore, impose a binding capital requirement, $\theta ^{reg},$that satisfies (A24).

The marginal cost of bank deposits, the return to depositors, the marginal taxes on deposit payments, the deposit insurance indemnity payments, and the depositor subsidy in the low aggregate state are determined exactly as in Case 1, that is, by equations (A26) - (A30).

Let us now turn to investors. Recall from section 5 that, when $\tau >{\bar{\tau }}(\Delta )$, investors do not receive any payoff in the efficient allocation when the aggregate shock is low. Therefore, $R_{E}^{s,l,reg}=0$ so that $t_{E}^{l,reg}=C_{E}^{s,reg}.$ As the total expected payoff to investors must be $\Delta _{E},$we must have

$$\begin{aligned} p\omega ^{h}R_{E}^{s,h,reg}&=\Delta _{E},\\ t_{E}^{h,reg}&=C_{E}^{s,reg}-R_{E}^{s,h,reg}. \end{aligned}$$

Finally, firms too receive no payoff in the low aggregate state in the efficient allocation. Hence, $P_{F}^{s,l,reg}=0$ so that

$$\begin{aligned} \Lambda \left( \frac{X_{F}^{{\textit{eff}}}}{{\mathcal {F}}+{\mathcal {G}}}\right) -\frac{X_{F}^{{\textit{eff}}}}{{\mathcal {F}}+{\mathcal {G}}}C_{E}^{s,reg}=T_{F}^{l,reg}. \end{aligned}$$

The lump sum tax on firms in the high aggregate state is then pinned down by the fact that the total expected payoff to firms must be $\Delta _{F},$that is,

$$\begin{aligned} p\omega ^{h}P_{F}^{s,h,reg}&=\Delta _{F},\\ T_{F}^{h,reg}&=\Lambda \left( \frac{X_{F}^{{\textit{eff}}}}{{\mathcal {F}}+{\mathcal {G}}}\right) -\frac{X_{F}^{{\textit{eff}}}}{{\mathcal {F}}+{\mathcal {G}}}C_{E}^{s,reg}-P_{F}^{s,h,reg}. \end{aligned}$$

By the above, the payoffs to agents match their payoffs in the efficient allocation. Hence, depositors’ payoffs solve the planning problem (A3). Because the representative depositor’s expected utility is maximized in (A3), it is, in fact, optimal for depositors to supply all their capital to banks as hypothesized. The arguments that the above constitute an equilibrium are exactly as in Case 1. Q.E.D.

1.1.7 Proof of Proposition 6

• 1. Most of the analysis has been carried out in the proof of Proposition 5. We summarize the results here. For the first part note that we have

  $$\begin{aligned} R_{D}^{h,reg}=R_{D}^{l,reg}={\mathbb {E}}\left[ {\widetilde{R}}_{D}^{reg}\right] =\frac{P_{D}^{{ef \text {f}}}}{{\mathcal {D}}}>1, \end{aligned}$$

  where the last inequality follows from (36). Hence, the expected return on deposits is at least equal to the safe return of one. Recall that, when $\tau \le {\bar{\tau }}(\Delta )$, ${\mathbb {E}}[{\widetilde{R}}^{\text {e}{\textit{ff}}}]-1=0$. Hence, (A28) implies that ${\mathbb {E}}[{\tilde{C}}_{D}^{reg}]={\mathbb {E}}[{\tilde{C}}_{E}^{reg}]=q{\mathscr {R}}_{L}^{reg}+(1-q)\Lambda _{liq}={\mathbb {E}}[{\widetilde{R}}^{\text {eff}}]=1$. Finally, as we saw in the construction of the equilibrium, since banks are indifferent between investing in firms and safe asset, neither a liquidity nor a capital requirement is needed to implement the regulated equilibrium. Imposing either a liquidity requirement or a capital requirement, however, ensures a unique regulated equilibrium.

• 2. Similar to the proof of part 1, we gather the required equations from the proof of Proposition 5 to show our claims here. For the first claim, note that, if the expected return on deposits is not greater than one, then because deposits are risky and depositors are risk-averse, they supply no capital which contradicts $D^{reg}={\mathcal {D}}$ in the equilibrium. The expected return to investors must also exceed the safe return by (37). By (41), the expected marginal cost of equity for banks and public firms equals the expected marginal productivity of each firm, that is,

  $$\begin{aligned} {\mathbb {E}}[{\tilde{C}}_{E}^{reg}]={\mathbb {E}}[{\widetilde{R}}^{\textit{eff}}]. \end{aligned}$$

  To compare the marginal costs of deposit and equity, Lemma 2 implies that

  $$\begin{aligned}&{\mathbb {E}}[{\tilde{C}}_{D}^{reg}]={\mathbb {E}}[{\widetilde{R}}^{\text {e}{\textit{ff}}}]-\frac{X_{S}^{\text {e}{\textit{ff}}}-\pi ^{reg}\mathcal{D}}{\mathcal{D}}({\mathbb {E}}[{\widetilde{R}}^{\text {e}{\textit{ff}}}]-1)<{\mathbb {E}}[{\tilde{C}}_{E}^{reg}]. \end{aligned}$$

  Q.E.D.

1.1.8 Proof of Proposition 7

If $\tau <{\bar{\tau }}(\Delta )$, the efficient allocation does not vary with $\tau$ by Proposition 3 (i). Hence, the regulated equlibrium and optimal regulatory tools do not vary with $\tau$.

If $\tau >{\bar{\tau }}(\Delta ),$ Proposition 4 shows that the efficient level of investment in the safe asset increases with $\tau .$ Hence, $\frac{X_{S}^{\text {e}{\textit{ff}}}}{{\mathcal {D}}}$ increases with $\tau$ so that the optimal liquidity requirement, $\beta ^{reg}$, also increases. Again, by Proposition 4, the levels of investment in public and private firms both decrease with $\tau$ in the efficient allocation. Since investors’ supply of capital to public firms must decrease, the capital that they provide to banks as bank equity capital should increase. At the same time, the capital that banks supply as loans to private firms must decrease with $\tau$ in an efficient regulated equilibrium. It then follows immediately from (34) that the capital requirement, $\theta ^{reg},$ must also increase with $\tau .$ Q.E.D.

1.1.9 Proof of Proposition 8

We will show that the structure of the tax and bailout policies is as follows.

• 1. If $0<\tau <{\overline{\tau }}(\Delta )$,

  $$\begin{aligned} t_{D}^{h.reg}&=t_{D}^{l.reg}<0, {\mathbb {E}}[{\widetilde{t}}_{D}^{reg}]<0,\text { }\phi ^{l,reg}>\phi ^{h,reg}, \end{aligned}$$

  (A36)
  $$\begin{aligned} {\mathbb {E}}[{\widetilde{t}}_{E}^{reg}&]<0,\text { }{\mathbb {E}}[{\widetilde{T}}_{F}^{reg}]>0. \end{aligned}$$

  (A37)

• 2. If ${\tau >{\overline{\tau }}(\Delta )},$ then

  $$\begin{aligned} t_{D}^{h.reg}&<t_{D}^{l.reg};\text { }{\mathbb {E}}[{\widetilde{t}}_{D}^{reg}]\le 0 ,\text { }\phi ^{l,reg}>\phi ^{h,reg}, \end{aligned}$$

  (A38)
  $$\begin{aligned} {\mathbb {E}}[{\widetilde{t}}_{E}^{reg}]&\le 0, {\mathbb {E}}[{\widetilde{T}}_{F}^{reg}]\ge 0. \end{aligned}$$

  (A39)

If $0<\tau <{\overline{\tau }}(\Delta )$ then depositors are fully insured, therefore, $P_{D}^{h,reg}=P_{D}^{l,reg}$. Equation (38) implies that $t_{D}^{h,reg}=t_{D}^{l,reg}$ and $\mu ^{h,reg}+\phi ^{h,reg}=\mu ^{l,reg}+\phi ^{l,reg}$. Equation (31) implies that $\mu ^{h,reg}>\mu ^{l,reg}$, therefore $\phi ^{h,reg}<\phi ^{l,reg}$.

If $\tau >{\overline{\tau }}(\Delta )$, then $P_{D}^{h,reg}>P_{D}^{l,reg}$. Equation (38) implies that $t_{D}^{h,reg}<t_{D}^{l,reg}$ and $\mu ^{h,reg}+\phi ^{h,reg}<\mu ^{l,reg}+\phi ^{l,reg}$. Since $\mu ^{h,reg}>\mu ^{l,reg}$, $\phi ^{h,reg}<\phi ^{l,reg}$.

Note that the representative investor and firm’s respective expected payoffs in the unregulated equilibrium are, respectively,

$$\begin{aligned} {\mathbb {E}}[{\tilde{R}}^{*}]= & {} q\Lambda '\left( \frac{\mathcal{E}+D^{* }}{{\mathcal {F}}+{\mathcal {G}}}\right) +(1-q)\Lambda _{liq}>1,\\{} & {} q\Lambda \left( \frac{\mathcal{E}+D^{* }}{{\mathcal {F}}+{\mathcal {G}}}\right) +(1-q)\Lambda _{liq}-{\mathbb {E}}[{\tilde{R}}^{*}]\left( \frac{\mathcal{E}+D^{* }}{{\mathcal {F}}+{\mathcal {G}}}\right) . \end{aligned}$$

Let

$$\begin{aligned} X_{F}^{*}=\mathcal{E}+D^{* }. \end{aligned}$$

Since the function $\Lambda (.)$ is strictly concave, the representative investor’s expected payoff decreases with $X_{F}^{*}$. The envelope theorem implies that the derivative of the firm’s expected payoff with respect to $X_{F}^{*}$ is $-\frac{\partial {\mathbb {E}}[{\tilde{R}}^{*}]}{\partial X_{F}}(\frac{X_{F}^{*}}{{\mathcal {F}}+{\mathcal {G}}})$, which is positive. Hence the representative firm’s expected payoff increases with $X_{F}^{*}$. Note that the total expected payoff of the firms and the investors is equal to:

$$\begin{aligned}&\Psi (X_{F}^{*})=q(\mathcal{F}+{\mathcal {G}})\Lambda \left( \frac{X_{F}^{*}}{{\mathcal {F}}+{\mathcal {G}}}\right) +(1-q)X_{F}^{*}\Lambda _{liq}-{\mathbb {E}}[{\widetilde{R}}^{*}]X_{F}^{*}+{\mathbb {E}}[{\widetilde{R}}^{*}]\mathcal{E}. \end{aligned}$$

Note that

$$\begin{aligned}&\Psi '(X_{F}^{*})=\frac{\partial {\mathbb {E}}[{\tilde{R}}]}{\partial X_{F}^{*}}(-X_{F}^{*}+\mathcal{E}). \end{aligned}$$

Since the investors’ entire capital is directly or indirectly invested in firms $\Psi '(X_{F}^{*})>0$. The rest of the proof follows from the discussion after the proposition. Q.E.D.

Appendix B: Extensions

1.1 Distinct production functions and output distributions for private and public firms

In this section, we briefly present a generalization of the basic model to the scenario where private and public firms may have different production functions and output probability distributions. Our main results are also robust to this extension.

Suppose that private firms have the production function and output distribution described in Sect. 3.1.1, but public firms have a production function, $\Sigma ,$ that is strictly increasing and concave (satisfying the Inada conditions), and a success probability r. The payoff upon failure is $\Sigma _{liq}.k$. As in Sect. 3.1.1, the range of capital values is such that $\Sigma _{liq}.k<\Sigma (k)$ so that the payoff upon failure is less than the payoff upon success. Sectoral and aggregate shocks are as modeled in Sect. 3.1.2. However, the conditional probabilities of the success of any firm’s project in the high and low aggregate states are $\sigma ^{h},\sigma ^{l},$ respectively. To ensure that the unconditional success probability of a firm is r, we must have

$$\begin{aligned} p\sigma ^{h}+(1-p)\sigma ^{l}=r. \end{aligned}$$

(B1)

Analogous to the baseline model, we define aggregate risk, $\tau ,$ as the difference between the proportions of firms that succeed in the high and low aggregate states, that is,

$$\begin{aligned} \tau =\frac{(\omega ^{h}-\omega ^{l}){\mathcal {F}}+(\sigma ^{h}-\sigma ^{l}){\mathcal {G}}}{{\mathcal {F}}+{\mathcal {G}}} \end{aligned}$$

(B2)

The rest of the model is as described in Sect. 3.

1.1.1 Unregulated equilibrium

Define $K_{1}$ as the unique capital level that solves the following equation:

$$\begin{aligned} q\Lambda '\left( \frac{\mathcal{E}-K_{1}}{\mathcal{F}}\right) +(1-q)\Lambda _{liq}=r\Sigma '\left( \frac{K_{1}}{\mathcal{G}}\right) +(1-r)\Sigma _{liq} \end{aligned}$$

(B3)

A unique $K_{1}$ exists because $\Lambda$ and $\Sigma$ are both strictly concave and satisfy the Inada conditions. The above condition implies that, if only investor capital, ${\mathcal {E}}$, is invested in (public and private) firms with no depositor capital being invested, then $K_{1}$ is the capital invested in public firms such that the expected marginal productivities of public and private firms are equalized.

Similarly, define $K_{2}$ as as the unique capital level that solves the following equation:

$$\begin{aligned} q\Lambda '\left( \frac{\mathcal{E}+{\mathcal {D}}-K_{2}}{\mathcal{F}}\right) +(1-q)\Lambda _{liq}=r\Sigma '\left( \frac{K_{2}}{\mathcal{G}}\right) +(1-r)\Sigma _{liq} \end{aligned}$$

(B4)

$K_{2}$ has a similar interpretation to $K_{1}$ if, instead, all capital in the economy, ${\mathcal {E}}+{\mathcal {D}}$, is invested in public and private firms.

We make the following assumption that is analogous to Assumption 1 with very similar intuition.

Assumption 3

$$\begin{aligned} q\Lambda '\left( \frac{\mathcal{E}-K_{1}}{\mathcal{F}}\right) +(1-q)\Lambda _{liq}&=r\Sigma '\left( \frac{K_{1}}{\mathcal{G}}\right) +(1-r)\Sigma _{liq}>1,\nonumber \\ q\Lambda '\left( \frac{\mathcal{E}+{\mathcal {D}}-K_{2}}{\mathcal{F}}\right) +(1-q)\Lambda _{liq}&=r\Sigma '\left( \frac{K_{2}}{\mathcal{G}}\right) +(1-r)\Sigma _{liq}<1. \end{aligned}$$

(B5)

Lemma 1 holds exactly as stated in this more general setting with identical intuition. Importantly, in any equilibrium, the expected marginal productivities of public and private firms must be equal. Further, the expected returns on bank loans, bank equity and public firm equity must all be equal to the expected marginal productivity of each (public or private) firm. The analysis of the unregulated equilibrium now proceeds as in Sect. 4 culminating in Proposition 2.

1.1.2 Efficient allocations

The characterization of Pareto efficient allocations proceeds as in Sect. 5. However, because public and private firms have different production functions, the capital invested in each public firm in the efficient allocation differs in general from the capital invested in each private firm. More precisely, let $X_{F1}^{\textit{eff}}(\Delta ),X_{F2}^{\textit{eff}}(\Delta )$ be the total capital invested in private and public firms, respectively, in the efficient allocation when the total expected reservation payoff of investors and firms is $\Delta .$ We must then have

$$\begin{aligned} q\Lambda '\left( \frac{X_{F1}^{\textit{eff}}(\Delta )}{\mathcal{F}}\right) +(1-q)\Lambda _{liq}=r\Sigma '\left( \frac{X_{F2}^{\textit{eff}}(\Delta )}{\mathcal{G}}\right) +(1-r)\Sigma _{liq}. \end{aligned}$$

In other words, the expected marginal productivities of private and public firms must be equal in an efficient allocation,.

Proceeding along the lines of the proof of Proposition 3 in “Appendix A”, let $X_{F1}^{*\textit{eff}}(\Delta )$ and $X_{F2}^{*\textit{eff}}(\Delta )$ solve

$$\begin{aligned} q\Lambda '\left( \frac{X_{F1}^{*\textit{eff}}(\Delta )}{\mathcal{F}}\right) +(1-q)\Lambda _{liq}=r\Sigma '\left( \frac{X_{F2}^{*\textit{eff}}(\Delta ,\tau )}{\mathcal{G}}\right) +(1-r)\Sigma _{liq}=1 \end{aligned}$$

(B6)

We can use arguments identical to those in the proof of Proposition 3 to show that $X_{F1}^{*\textit{eff}}(\Delta )$ and $X_{F2}^{*\textit{eff}}(\Delta )$ are the efficient levels of investment in private and public firms, respectively, when $\tau <{\overline{\tau }}(\Delta ),$ where

$$\begin{aligned} {\overline{\tau }}(\Delta )= \min \left( \frac{\Delta }{[q{\mathcal {F}}\Lambda (\frac{X_{F1}^{*\textit{eff}}(\Delta )}{{\mathcal {F}}})+(1-q)X_{F1}^{*\textit{eff}}(\Delta )\Lambda _{liq}](\omega ^{h}-\omega ^{l})+[r{\mathcal {G}}\Sigma (\frac{X_{F2}^{*\textit{eff}}(\Delta )}{{\mathcal {G}}})+(1-r)X_{F2}^ {*\textit{eff}}(\Delta )\Sigma _{liq}](\sigma ^{h}-\sigma ^{l})},1\right) . \end{aligned}$$

The intuition for the threshold is identical to the intuition in the case of the basic model. It is the level of aggregate risk such that the total capital held by investors and firms in the low aggregate state is just sufficient to fully insure depositors without violating investors’ and firms’ limited liability constraints, while ensuring that the total expected reservation payoff of investors and firms is $\Delta$.

Proposition 3 then holds as stated except that we now distinguish between the levels of investment in private and public firms, respectively. Proposition 4 also directly extends to this setting with the levels of investment in private and public firms declining with $\tau$ for $\tau >{\overline{\tau }}(\Delta ).$

1.1.3 Regulation

The analysis of regulation proceeds as in Sect. 6 with the only modification being the threshold aggregate risk level and the expressions for the transfers, capital and liquidity requirements. In particular, Lemma 2 and Propositions 5–8 extend directly to this setting.

1.2 Public firm financing

The baseline model assumes that private firms obtain financing via bank loans, while public firms obtain financing from investors via publicly traded securities. In this section, we extend the model to allow for public firms to also obtain financing via bank loans. The description of the model proceeds as in Sect. 3 except that a public firm $n\in [0,{\mathcal {G}}]$ can obtain financing either via bank loans or via public securities that we again assume to be equity to simplify the exposition. The firm provides a return of ${\widetilde{{\mathscr {R}}}}_{L,n}\equiv \left( {\mathscr {R}}_{L,n}^{s},{\mathscr {R}}_{L,n}^{f}\right)$ if it obtains capital via bank loans and a return of ${\widetilde{{\mathscr {R}}}}_{E,n}\equiv \left( {\mathscr {R}}_{E,n}^{s},{\mathscr {R}}_{E,n}^{f}\right)$ on equity financing. As in the analysis of the baseline model, ${\mathscr {R}}_{L,n}^{f}={\mathscr {R}}_{E,n}^{f}=\Lambda _{liq}.$

1.2.1 Unregulated equilibrium

We impose Assumption 1 as stated with identical motivation. Lemma 1 holds as stated with very similar intuition. Moving to the analysis of the risky equilibrium conditions, the analysis in Sect. 4.1.1 holds as stated. The investigation of investors’ decisions in Sect. 4.1.2, however, changes because banks can now invest a portion of the capital they raise in public firms. Let $E^{*}$ be the total equity capital that banks raise from investors in any equilibrium, and let $E_{1}^{*}$ be the portion of this capital that they allocate to private firms. As in the analysis of the unregulated equilibrium, Lemma 1 implies that each private or public firm has the same expected marginal productivity and raises the same amount of capital in equilibrium. Hence, we must have

$$\begin{aligned} \frac{D^{*}+E_{1}^{*}}{{\mathcal {F}}}=\frac{{\mathcal {E}}-E_{1}^{*}}{{\mathcal {G}}}=\frac{{\mathcal {E}}+D^{*}}{{\mathcal {F}}+{\mathcal {G}}}. \end{aligned}$$

(B7)

The above equations pin down the amount of equity capital that banks invest in private firms. However, we now have multiple unregulated equilibria that differ only in the amount of equity capital that banks raise from investors with a portion $E_{1}^{*}$ of this capital invested in private firms and the remaining invested in public firms. As banks must raise at least equity capital, $E_{1}^{*},$the equity capitalizations of banks in these multiple equilibria lie in the interval, $[E_{1}^{*},{\mathcal {E}}].$ The equilibrium with bank equity capitalization, $E_{1}^{*},$ corresponds to the case where banks raise just enough equity capital to invest in private firms. The equilibrium with bank equity capitalization, ${\mathcal {E}}$, corresponds to the case where investors provide all their capital to banks, and banks invest in private and public firms. However, these multiple unregulated equilibria deliver identical expected utilities to depositors, investors and firms as they differ only in the relative allocations of capital to public firms by banks and investors with the total capital allocated being constant.

1.2.2 Efficient allocations

Any Pareto efficient allocation maximizes the expected utility of depositors while ensuring that investors and firms receive pre-specified expected reservation payoffs. Consequently, the analysis in Sect. 5 is unaffected by how banks allocate their capital between private and public firms as long as the capital allocated to each private or public firm is such that the expected marginal productivities of firms are equalized. Hence, Propositions 3 and 4 hold exactly as stated.

1.2.3 Regulation

Proceeding to the analysis of regulation, the regulatory instruments are as described in Sect. 6.1, and Lemma 2 holds as stated. Propositions 5, 6 and 7 also hold as stated with identical intuition. Regarding 8, where the expected reservation payoffs of investors and firms are set to their expected reservation payoffs in an unregulated equilibrium, we note that the multiple unregulated equilibria deliver identical expected utilities to depositors, investors and firms. Hence, Proposition 8 also holds as stated.

1.3 General capital structures for public firms

In this section, we further extend the model in Sect. B.2 to allow for more general capital structures for public firms that comprise both bank loans and public equity. As we discuss below, bank loan contracts may now be distinct for private and public firms in contrast with the baseline model. For convenience, let us use the index $m\in [0,{\mathcal {F}}]$ for a private firm and the index $n\in [0,{\mathcal {G}}]$ for a public firm. A bank loan contract for private firm $m\in [0,{\mathcal {F}}]$ is denoted by ${\widetilde{{\mathscr {R}}}}_{L,m}\equiv \left( {\mathscr {R}}_{L,m}^{s},{\mathscr {R}}_{L,m}^{f}\right)$, while a bank loan contract for a public firm $n\in [0,{\mathcal {G}}]$ is denoted ${\widetilde{{\mathscr {R}}}}_{L,n}\equiv \left( {\mathscr {R}}_{L,n}^{s},{\mathscr {R}}_{L,n}^{f}\right)$. The other notation is as in the baseline model.

As in the analysis of the baseline model, ${\mathscr {R}}_{L,m}^{f}=\Lambda _{liq}.$ However, the bank loan return upon the failure of a public firm, ${\mathscr {R}}_{L,n}^{f}$, need not be equal to $\Lambda _{liq}$ in contrast with the model of Sect. B.2 because a public firm may choose a combination of bank loan and public equity financing, and bank loans have absolute priority upon bankruptcy. As bank loans have absolute priority, we must have ${\mathscr {R}}_{L,n}^{f}\ge \Lambda _{liq}.$ If ${\mathscr {R}}_{L,n}^{f}=\Lambda _{liq}$, then the firm clearly chooses bank loan financing exclusively. The analysis is then identical to that in Sect. B.2, where each public firm either chooses bank loan financing or public equity financing. Hence, it suffices to consider the case where ${\mathscr {R}}_{L,n}^{f}>\Lambda _{liq}.$

1.3.1 Unregulated equilibrium

We impose Assumption 1 as stated with identical motivation. Lemma 1 holds as stated with very similar intuition. In particular, in any equilibrium in which public firms choose a combination of bank loan and public equity financing, they must be marginally indifferent between them, that is, the expected marginal costs of bank loans and equity are equal. In summary, the expected returns on bank loans to public and private firms, the expected returns on bank and public firm equity, the expected returns on bank deposits, and the expected marginal productivities of public and private firms must all be equal in equilibrium.

Let us now move to the analysis of the risky equilibrium conditions. If $E_{1}^{*}$ is the capital allocated by banks to private firms, then $E_{1}^{*}$ must satisfy (B7) because the expected marginal productivities of all firms—public and private—are equal in equilibrium. The modification relative to Sect. B.2 is in the capital structures of public firms. Consider a public firm $n\in [0,{\mathcal {G}}]$. Let $K_{L,n}^{*},K_{E,n}^{*}$ be the equilibrium capital levels raised via bank loans and public equity, respectively, by the firm. As in the baseline model, the following condition must hold.

$$\begin{aligned} {\mathbb {E}}[\widetilde{{\mathscr {R}}}_{L,n}^{*}]={\mathbb {E}}[\widetilde{{\mathscr {R}}}_{E,n}^{*}]=q\Lambda '\left( \frac{\mathcal{E}+D^{* }}{{\mathcal {F}}+{\mathcal {G}}}\right) +(1-q)\Lambda _{liq}>1. \end{aligned}$$

In contrast with the baseline model, however, a bank loan to a public firm may be risky or risk-free in equilibrium, and we must distinguish between the two cases. Note, however, that because the expected bank deposit return exceeds the safe return, bank deposits are still risky as must be the case in a risky equilibrium.

If the bank loan is risky, then we must have $\widetilde{{\mathscr {R}}}_{L,n}^{f*}<{{\widetilde{{\mathscr {R}}}_{L,n}^{s*}}}$ and

$$\begin{aligned} \widetilde{{\mathscr {R}}}_{L,n}^{f*}=\frac{K_{L,n}^{*}+K_{E,n}^{*}}{K_{L,n}^{*}};\widetilde{{\mathscr {R}}}_{E,n}^{f*}=0. \end{aligned}$$

The above expresses the fact that the bank receives the firm’s asset payoff when it fails and is unable to make the entire bank loan payment, while equityholders get nothing.

If the bank loan is risk-free, then $\widetilde{{\mathscr {R}}}_{L,n}^{f*}={{\widetilde{{\mathscr {R}}}_{L,n}^{s*}}{=}{{\mathscr {R}}_{L}^{*}}}$ where

$$\begin{aligned} {{\mathscr {R}}_{L}^{*}}&=q\Lambda '\left( \frac{\mathcal{E}+D^{* }}{{\mathcal {F}}+{\mathcal {G}}}\right) +(1-q)\Lambda _{liq}\\ \widetilde{{\mathscr {R}}}_{E,n}^{f*}&=\frac{(K_{L,n}^{*}+K_{E,n}^{*})\Lambda _{liq}-K_{L,n}^{*}{\mathscr {R}}_{L,n}^{*}}{K_{E,n}^{*}}. \end{aligned}$$

The above conditions express the fact that the bank loan rate, which the bank receives if the firm succeeds or fails as the bank loan is risk-free, must necessarily equal the expected bank loan return that is given by the R.H.S. of the first equation above.

The rest of the analysis of the unregulated equilibrium proceeds as in Sect. 4. There can, however, be multiple unregulated equilibria. First, the equilibria derived in Sect. B.2 remain equilibria in this setting, that is, public firms choose either bank loan financing or equity financing. These equilibria differ only in the amount of equity capital that banks raise from investors with a portion $E_{1}^{*}$ of this capital invested in private firms and the remaining invested in public firms. In addition to these equilibria, we could also have multiple equilibria where public firms choose a combination of bank loan and equity financing, that is, they have interior capital structures. These equilibria differ in whether bank loans to public firms are risky or risk-free. In each set of equilibria—risky or risk-free—all public firms have the same capital structure that is pinned down by the conditions derived above. The equilibria, however, differ in the amount of equity capital that banks raise from investors.

1.3.2 Efficient allocations and regulation

As in the baseline model and in the extension of Sect. B.2, any Pareto efficient allocation maximizes the expected utility of depositors while ensuring that investors and firms receive pre-specified expected reservation payoffs. The analysis in Sect. 5 is unaffected by how banks allocate their capital between private and public firms, as well as public firms’ capital structures, as long as the capital allocated to each private or public firm is such that the expected marginal productivities of firms are equalized. Hence, Propositions 3 and 4 hold exactly as stated. The analysis of regulation too is qualitatively unaffected since it does not hinge on the capital structures of public firms.