Welfare and bank risk-taking

Abstract

Our study investigates a model of general equilibrium banking that incorporates moral hazard and incentive mechanisms for bank risk-taking, with a particular focus on deposit market competition. Our findings reveal that when banks compete perfectly in the deposit market, it leads to maximal welfare and an optimal level of bank failure risk. This outcome remains valid even if the risk of failure for competitive banks is higher than that of banks with monopoly rents, and it is not affected by social costs associated with bank failures. Our model suggests that there is no trade-off between bank competition and financial stability. Our results support the empirical findings of Carlson, Correia, and Luck (J Polit Econ 130(2): 462–520, 2022).

Appendix

Lemma 3

The risk of failure of the monopolistic bank declines monotonically with deposit insurance coverage.

Proof

Differentiating Eq. (14) with respect to $g$ we get:

$$ \frac{{dR_{M}^{*} }}{dg} = \frac{1}{2}\left( {1 - \frac{1}{2}\left( {X^{2} - 4\alpha^{ - 1} \rho + g\left( {g - 2X + 4\alpha^{ - 1} } \right)} \right)^{ - 1/2} \left( {2g - 2X + 4\alpha^{ - 1} } \right)} \right) $$

Therefore, $sign\left\{ {\frac{{dR_{M}^{*} }}{{dg}}} \right\} = sign\left\{ 1 - \frac{1}{2}\left( {X^{2} - 4\alpha ^{{ - 1}} \rho + g\left( {g - 2X + 4\alpha ^{{ - 1}} } \right)} \right)^{{ - 1/2}} \right.\break\left. \left( {2g - 2X + 4\alpha ^{{ - 1}} } \right) \right\}.$

$\frac{{dR_{M}^{*} }}{dg} < 0$ is equivalent to the following inequalities:

$$ \begin{aligned} 1 < & \frac{1}{2}\left( {X^{2} - 4\alpha^{ - 1} \rho + g\left( {g - 2X + 4\alpha^{ - 1} } \right)} \right)^{ - 1/2} \left( {2g - 2X + 4\alpha^{ - 1} } \right) \\ & \Leftrightarrow 2\left( {X^{2} - 4\alpha^{ - 1} \rho + g\left( {g - 2X + 4\alpha^{ - 1} } \right)} \right)^{1/2} < 2g - 2X + 4\alpha^{ - 1} \\ & \Leftrightarrow 4\left( {X^{2} - 4\alpha^{ - 1} \rho + g\left( {g - 2X + 4\alpha^{ - 1} } \right)} \right) < \left( {2\left( {g - X} \right) + 4\alpha^{ - 1} } \right)^{2} \\ & \Leftrightarrow 4X^{2} - 16\alpha^{ - 1} \rho + 4g\left( {g - 2X + 4\alpha^{ - 1} } \right) < 4\left( {g - X} \right)^{2} + 16\alpha^{ - 2} + 16\left( {g - X} \right)\alpha^{ - 1} \\ & \Leftrightarrow 4X^{2} - 16\alpha^{ - 1} \rho + 4g^{2} - 8gX + 16g\alpha^{ - 1} < 4g^{2} + 4X^{2} - 8gX + 16\alpha^{ - 2} + 16\alpha^{ - 1} g - 16X\alpha^{ - 1} \\ & \Leftrightarrow - 16\alpha^{ - 1} \rho < 16\alpha^{ - 2} - 16X\alpha^{ - 1} \Leftrightarrow \\ & - \rho < \alpha^{ - 1} - X \\ \end{aligned} $$

By (A1), $\alpha^{ - 1} - X \ge 0$. Therefore, $\frac{{dR_{M}^{*} }}{dg} < 0$, which implies $\frac{{dP_{M}^{*} }}{dg} > 0$ by Eq. (15). $\hfill\square$

Lemma 4

For all $g \in [0,1]$, $\frac{\partial Z}{{\partial \sigma }} > 0$.

Proof

$$ \begin{aligned} \frac{\partial Z}{{\partial \sigma }} = & \frac{2A}{{(.)^{2} }}\left( {\frac{\partial r}{{\partial \sigma }}\rho \left[ {2r\left( {\sigma ,g} \right)\left( {\rho + g} \right) + \left( {\pi_{C}^{{}} \sigma + \pi_{M}^{{}} \left( {1 - \sigma } \right)} \right)\rho } \right] - r\left( {\sigma ,g} \right)\rho \left[ {2\frac{\partial r}{{\partial \sigma }}\left( {\rho + g} \right) + \rho \left( {\pi_{C}^{{}} - \pi_{M}^{{}} } \right)} \right]} \right) \\ & \Leftrightarrow \frac{2A\rho }{{(.)^{2} }}\left( {\frac{\partial r}{{\partial \sigma }}\left( {\pi_{C}^{{}} \sigma + \pi_{M}^{{}} \left( {1 - \sigma } \right)} \right) - r\left( {\sigma ,g} \right)\rho \left( {\pi_{C}^{{}} - \pi_{M}^{{}} } \right)} \right) \\ \end{aligned} $$

The term $\frac{\partial r}{{\partial \sigma }}\left( {\pi_{C}^{{}} \sigma + \pi_{M}^{{}} \left( {1 - \sigma } \right)} \right) - r\left( {\sigma ,g} \right)\rho \left( {\pi_{C}^{{}} - \pi_{M}^{{}} } \right)$ is strictly positive for all $g \in \left[ {0,1} \right]$, since $\frac{\partial r}{{\partial \sigma }} = P_{C} R_{C} - \rho + g\left( {P_{M} - P_{C} } \right) > 0$ and $\pi_{C}^{{}} < \pi_{M}^{{}}$. Thus, $\frac{\partial Z}{{\partial \sigma }} > 0$. $\hfill\square$

Proposition 1

For all $g \in \left[ {0,1} \right]$, $\frac{\partial Y}{{\partial \sigma }} > 0$: perfect competition ($\sigma = 1$) maximizes welfare.

Proof

Using the bank profit functions in the two sectors, we can write:

$$ P_{C} X - \frac{1}{2\alpha }P_{C}^{2} = \pi^{C} + P_{C} R_{C} $$

(a)

$$ P_{M}^{{}} X - \frac{1}{2\alpha }P_{M}^{2} = \pi^{M} + \rho $$

(b)

Hence, expected output net of monitoring and production costs in the two sectors are:

$$ P_{C} X - \frac{1}{2\alpha }P_{C}^{2} - \frac{1}{2}\pi_{C} = P_{C} R_{C} + \frac{1}{2}\pi_{C} $$

(c)

$$ P_{M}^{{}} X - \frac{1}{2\alpha }P_{M}^{2} - \frac{1}{2}\pi_{M} = \pi^{M} + \rho - \frac{1}{2}\pi_{M} = \rho + \frac{1}{2}\pi_{M} $$

(d)

Substituting (c) and (d) in (32), and using (30), we can write:

$$ \begin{aligned} Y(\sigma ,g) &\equiv \left[ {\left( {P_{C} X - \frac{1}{2\alpha }P_{C}^{2} - \frac{1}{2}\pi_{C} } \right)\sigma + \left( {P_{M}^{{}} X - \frac{1}{2\alpha }P_{M}^{2} - \frac{1}{2}\pi_{M} } \right)\left( {1 - \sigma } \right) + g} \right]Z\left( {\sigma ,g} \right) \\ &= \frac{{\left[ {2\left( {P_{C} R_{C} \sigma + \left( {1 - \sigma } \right)\rho } \right) + \pi_{C} \sigma + \pi_{M} \left( {1 - \sigma } \right) + g} \right]r\left( {\sigma ,g} \right)\rho }}{{2r\left( {\sigma ,g} \right)\left( {\rho + g} \right) + \left( {\pi_{C}^{{}} \sigma + \pi_{M}^{{}} \left( {1 - \sigma } \right)} \right)\rho }}A \\ \end{aligned} $$

Let $g = 0$. Then

$$ Y(\sigma ,0) = \frac{{\left[ {2\left( {P_{C} R_{C} \sigma + \left( {1 - \sigma } \right)\rho } \right) + \pi_{C} \sigma + \pi_{M} \left( {1 - \sigma } \right)} \right]r\left( {\sigma ,0} \right)}}{{2r\left( {\sigma ,0} \right) + \left( {\pi_{C}^{{}} \sigma + \pi_{M}^{{}} \left( {1 - \sigma } \right)} \right)}}A $$

Since $r(\sigma ,0) = \sigma P_{C} R_{C} + (1 - \sigma )\rho$, $Y(\sigma ,0)$ can be written as:

$$ Y(\sigma ,0) = \frac{{\left[ {2\left( {\sigma P_{C} R_{C} + (1 - \sigma )\rho } \right) + \pi_{C} \sigma + \pi_{M} )(1 - \sigma )} \right]r\left( {\sigma ,0} \right)}}{{2\left( {\sigma P_{C} R_{C} + (1 - \sigma )\rho } \right) + \left( {\pi_{C}^{{}} \sigma + \pi_{M}^{{}} (1 - \sigma )} \right)}}A = r(\sigma ,0)A $$

Thus, $\frac{\partial Y}{{\partial \sigma }} = \frac{\partial r}{{\partial \sigma }}\left( {\sigma ,0} \right)A = \left( {P_{C} R_{C} - \rho } \right)A > 0$, since $P_{C} R_{C} > \rho$.

Let $g \in (0,1]$ and re-write:

$$ Y(\sigma ,g) = h(\sigma )f(\sigma )A, $$

where

$$ f(\sigma ) \equiv \frac{r(\sigma ,g)\rho }{{2r(\sigma ,g)(\rho + g) + \left( {\pi_{C}^{{}} \sigma + \pi_{M}^{{}} \left( {1 - \sigma } \right)} \right)\rho }} $$

$$ h(\sigma ) \equiv 2\left( {P_{C} R_{C} \sigma + (1 - \sigma )\rho } \right) + \pi_{C} \sigma + \pi_{M} (1 - \sigma ) + g $$

Next, we show that both functions $f(\sigma )$ and $h(\sigma )$ are monotonically increasing in $\sigma .$

Consider

$$ \begin{aligned} f^{\prime}(\sigma ) & = \frac{1}{{\left( {2r(\sigma ,g)(\rho + g) + \left( {\pi_{C}^{{}} \sigma + \pi_{M}^{{}} (1 - \sigma )} \right)\rho } \right)^{2} }}x \\ & \quad \frac{\partial r}{{\partial \sigma }}\rho \left( {2r(\sigma ,g)(\rho + g) + \frac{\partial r}{{\partial \sigma }}\rho \left( {\pi_{C}^{{}} \sigma + \pi_{M}^{{}} (1 - \sigma )} \right)\rho } \right) \\ &\quad - r(\sigma ,g)\rho 2r^{\prime}(\sigma ,g)(\rho + g) - r(\sigma ,g)\rho \left( {\pi_{C}^{{}} - \pi_{M}^{{}} } \right)\rho \\ & = \frac{{\left( {\frac{\partial r}{{\partial \sigma }}\rho \left( {\pi_{C}^{{}} \sigma + \pi_{M}^{{}} (1 - \sigma )} \right)\rho } \right) - r(\sigma ,g)\rho \left( {\pi_{C}^{{}} - \pi_{M}^{{}} } \right)\rho }}{{\left( {2r(\sigma ,g)(\rho + g) + \left( {\pi_{C}^{{}} \sigma + \pi_{M}^{{}} (1 - \sigma )} \right)\rho } \right)^{2} }} \\ \end{aligned} $$

By Lemma 3, $\frac{\partial r}{{\partial \sigma }}\rho \left( {\left( {\pi_{C}^{{}} \sigma + \pi_{M}^{{}} (1 - \sigma )} \right)\rho ) - r(\sigma ,g)\rho \left( {\pi_{C}^{{}} - \pi_{M}^{{}} } \right)\rho } \right) > 0$, hence $f^{\prime}(\sigma ) > 0$.

Now consider:

$$ h^{\prime}(\sigma ) \equiv 2\left( {P_{C} R_{C} - \rho } \right) + \pi_{C} - \pi_{M} $$

Using equilibrium values, this derivative can be written as:

$$ \begin{aligned} h^{\prime}(\sigma ) &= 2\left( {P_{C} R_{C} - \rho } \right) + \pi_{C} - \pi_{M} \\ &= 2\alpha \left( {\frac{X - g}{2}} \right)\frac{X + g}{2} + \left( {\alpha \frac{{\left( {X - g} \right)^{2} }}{8} - \alpha \frac{{\left( {X - g + \sqrt {X^{2} - 4\alpha^{ - 1} \rho + g\left( {g - 2X + 4\alpha^{ - 1} } \right)} } \right)^{2} }}{8}} \right) \\ \end{aligned} $$

Therefore:

$$ \begin{array}{*{20}l} {h^{\prime}(\sigma ) \Leftrightarrow 2\alpha \left( {\frac{X - g}{2}} \right)\frac{X + g}{2}} \hfill \\ \quad { > \alpha \left( {\frac{{X^{2} - 4\alpha^{ - 1} \rho + g\left( {g - 2X + 4\alpha^{ - 1} } \right) + 2\left( {X - g} \right)\sqrt {X^{2} - 4\alpha^{ - 1} \rho + g\left( {g - 2X + 4\alpha^{ - 1} } \right)} }}{8}} \right) + 2\rho } \hfill \\ \end{array} \left( * \right) $$

By (A1):

$$ \begin{aligned} & \sqrt {X^{2} - 4\alpha^{ - 1} \rho + g\left( {g - 2X + 4\alpha^{ - 1} } \right)} < X + g \hfill \\ &\quad \Leftrightarrow X^{2} - 4\alpha^{ - 1} \rho + g\left( {g - 2X + 4\alpha^{ - 1} } \right) < \left( {X + g} \right)^{2} \hfill \\ \end{aligned} $$

Therefore:

$$ \begin{aligned} & \alpha \frac{{X^{2} - 4\alpha^{ - 1} \rho + g\left( {g - 2X + 4\alpha^{ - 1} } \right) + 2(X - g)\sqrt {X^{2} - 4\alpha^{ - 1} \rho + g\left( {g - 2X + 4\alpha^{ - 1} } \right)} }}{8} \hfill \\ &\quad < \frac{{X^{2} - 4\alpha^{ - 1} \rho + g\left( {g - 2X + 4\alpha^{ - 1} } \right) + 2(X - g)(X + g)}}{8} \hfill \\ \end{aligned} $$

Hence, inequality (*) is satisfied if:

$$ 2\alpha \left( {\frac{X - g}{2}} \right)\frac{X + g}{2} > \left( {\frac{{X^{2} - 4\alpha^{ - 1} \rho + g\left( {g - 2X + 4\alpha^{ - 1} } \right) + 2\left( {X - g} \right)\left( {X + g} \right)}}{8}} \right) + 2\rho \;\;\;\;\left( {**} \right) $$

Note that if (**) holds, we can write it as:

$$ \begin{aligned} 2\alpha \left( {\frac{X - g}{2}} \right)\frac{X + g}{2} & > \left. {\frac{{X^{2} - 4\alpha^{ - 1} \rho + g\left( {g - 2X + 4\alpha^{ - 1} } \right) + 2(X - g)(X + g)}}{8}} \right) \\ & \Leftrightarrow 4\left( {X^{2} - g^{2} } \right) > X^{2} - 4\alpha^{ - 1} \rho + g\left( {g - 2X + 4\alpha^{ - 1} } \right) + 2\left( {X^{2} - g^{2} } \right) + 16\rho \\ & \Leftrightarrow 2X^{2} - 2g^{2} > X^{2} - 4\alpha^{ - 1} \rho + g\left( {g - 2X + 4\alpha^{ - 1} } \right) + 16\rho \\ & \Leftrightarrow X^{2} > - 4\alpha^{ - 1} \rho + 2g^{2} + g\left( {g - 2X + 4\alpha^{ - 1} } \right) + 16\rho \\ & \Leftrightarrow X^{2} + 4\alpha^{ - 1} (\rho - g) + g\left( {2X - g} \right) > 2g^{2} + 16\rho \\ &\quad X^{2} + 4\alpha^{ - 1} (\rho - g) + g2X > 3g^{2} + 16\rho \\ \end{aligned} $$

By (A4) and (A1), $g2X > 3g^{2}$, since $X > 4\rho > \frac{3}{2}g$, and by (A3)$X^{2} > 16\rho$, which implies that inequality (**) holds, since $X^{2} + 4\alpha^{ - 1} (\rho - g) + g2X > 3g^{2} + 16\rho$. Hence, $h^{\prime}(\sigma ) > 0.$

In conclusion, $Y(\sigma ,g) = h(\sigma )f(\sigma )A$ is strictly increasing in $\sigma$ since both component functions are increasing in $\sigma .$ $\hfill\square$

Proposition 2

For any admissible social cost function that is increasing and convex in investment (deposits), for all $g \in [0,1]$, $\frac{\partial W}{{\partial \sigma }} > 0$: perfect competition ($\sigma = 1$) maximizes welfare.

Proof

The welfare function (35) can be written as:

$$ \begin{aligned} W(\sigma ,g) = & Z(\sigma ,g)\left[ {\left( {P_{C} R_{C} \sigma + \rho (1 - \sigma )} \right) + \frac{1}{2}\pi_{C} \sigma + \frac{1}{2}\pi_{M} (1 - \sigma ) + g} \right] + \\ & - C\left[ {(1 - P_{M} )(1 - \sigma )^{\gamma } + (1 - P_{C} )\sigma^{\gamma } } \right]Z(\sigma ,g)^{\gamma } \\ \end{aligned} $$

The upper bound defined by inequality (36) for all $\sigma \in [0,1]$ and $g \in [0,1]$ implies:

$$ \begin{aligned} W(\sigma ,g) = & Z(\sigma ,g)\left[ {\left( {P_{C} R_{C} \sigma + \rho (1 - \sigma )} \right) + \frac{1}{2}\pi_{C} \sigma + \frac{1}{2}\pi_{M} (1 - \sigma ) + g} \right] + \\ & - C\left[ {(1 - P_{M} )(1 - \sigma )^{\gamma } + (1 - P_{C} )\sigma^{\gamma } } \right]Z(\sigma ,g)^{\gamma } \ge \rho A \Rightarrow \\ & C \le \frac{{Z(\sigma ,g)\left[ {\left( {P_{C} R_{C} \sigma + \rho (1 - \sigma )} \right) + \frac{1}{2}\pi_{C} \sigma + \frac{1}{2}\pi_{M} (1 - \sigma ) + g} \right] - \rho A}}{{\left[ {(1 - P_{M} )(1 - \sigma )^{\gamma } + (1 - P_{C} )\sigma^{\gamma } } \right]Z(\sigma ,g)^{\gamma } }} \\ & \equiv \overline{C}(\sigma ,g) \\ \end{aligned} $$

Function $\overline{C}(\sigma ,g)$ is the highest level of social costs consistent with the existence of essential intermediation. Thus, a lower bound to any welfare function can be defined as:

$$ \begin{aligned} \underline {W} (\sigma ,g) = & Z(\sigma ,g)\left[ {\left( {P_{C} R_{C} \sigma + \rho (1 - \sigma )} \right) + \frac{1}{2}\pi_{C} \sigma + \frac{1}{2}\pi_{M} (1 - \sigma ) + g} \right] + \\ & - \overline{C}(\sigma ,g)\left[ {(1 - P_{M} )(1 - \sigma )^{\gamma } + (1 - P_{C} )\sigma^{\gamma } } \right]Z(\sigma ,g)^{\gamma } = Z(\sigma ,g)g + \rho A \\ \end{aligned} $$

If $g > 0$, then $\frac{{\partial \underline {W} }}{\partial \sigma } > 0$ by Lemma 4. If $g = 0$, then $\frac{{\partial \underline {W} }}{\partial \sigma } = 0$ since $\underline {W} (\sigma ,g) = \rho A$. But in this case investing all resources in the safe asset would be best, which would make bank intermediation inessential. Thus, $\frac{\partial W}{{\partial \sigma }} > 0$ for any admissible social cost function. Therefore, perfect bank competition $\left( {\sigma = 1} \right)$ maximizes welfare. $\hfill\square$