Banking competition and welfare

Abstract

We develop a simple general equilibrium model in which investment in a risky technology is subject to moral hazard and banks can extract market power rents. We show that more bank competition results in lower economy-wide risk, higher social welfare, lower bank capital ratios, more efficient production plans and Pareto-ranked real allocations. Perfect competition supports a second best allocation and optimal levels of bank risk and capitalization. These results are at variance with those obtained by a large literature that has studied a similar environment in partial equilibrium, they are empirically relevant, and carry significant implications for policy guidance.

Appendix

Proposition 1

In the moral hazard economy with banks and depositors an equilibrium with intermediation exists and it is unique for every $R\in (\frac{Xk}{2W},X]$.

Proof

Rearranging (10) and rewriting (11), we obtain:

$$\begin{aligned} I_1 (q)\equiv & {} I=\frac{2RWq-Xk}{X-R}; \end{aligned}$$

(41)

and

$$\begin{aligned} I_2 (q)\equiv I=\frac{W}{2}(1-q^{2}). \end{aligned}$$

(42)

An equilibrium is a value of $q\in (0,1)$ that satisfies (41) and (42). Observe that $I_2 (0)=\frac{W}{2}>-\frac{Xk}{X-R}=I_1 (0),$ while $I_2 (1)=0<\frac{2RW-Xk}{X-R}=I_1 (1)$ if $R\ge \frac{Xk}{2W}$, which is necessary for $I_1$ to be non-negative for every value of $q\in (0,1)$, otherwise no equilibrium would exist. Let $F(q)\equiv I_2 (q)-I_1 (q)=\frac{W}{2}(1-q^{2})-\frac{2RWq-Xk}{X-R}$. Clearly, ${F}'(q)<0$ holds. Therefore, there exists a unique value of $q^{*}\in (0,1)$ such that $F(q^{*})=0$. $I^{*}$ is found using either (41) or (42). Using (8), $p^{*}=\alpha ^{-1}\left[ {(X-R)I^{*}+Xk} \right]$. $\square$

Proposition 2

In the moral hazard economy with banks and depositors:

$$\begin{aligned} I_R^*>0, \quad \hat{{q}}_R^*<0, \quad p_R^*>0, \quad K_R^*<0 ~{ and}~ p_k^*>0. \end{aligned}$$

Proof

Differentiating totally (10) and (11) with respect to I, q, R and k, we get:

$$\begin{aligned}&\displaystyle \frac{X-R}{2}dI-RWdq=qWdR-\frac{X}{2}dk; \end{aligned}$$

(43)

$$\begin{aligned}&\displaystyle dI+qWdq=0. \end{aligned}$$

(44)

The determinant of the system (43)–(44) is $\Delta =\frac{X-R}{2}qW+RW>0$.

Setting $dk=0$, by Cramer’s rule,

$$\begin{aligned} I_R =\frac{dI}{dR}=\frac{1}{\Delta }(qW)^{2}>0, \end{aligned}$$

(45)

and

$$\begin{aligned} q_R \equiv \frac{dq}{dR}=-\frac{1}{\Delta }qW<0. \end{aligned}$$

(46)

Substituting (41) in (8), we get:

$$\begin{aligned} p=\frac{2RWq}{\alpha } \end{aligned}$$

(47)

Differentiating (47) with respect to R, we get:

$$\begin{aligned} p_R \equiv \frac{dp}{dR}=\frac{2Wq}{\alpha }+\frac{2RW}{\alpha }q_R =\frac{2Wq}{\alpha }\left( {1-\frac{RW}{\Delta }} \right) >0, \end{aligned}$$

(48)

where the second equality was obtained using (46), and the last inequality derives from $\Delta >RW$.

Finally, by (45) and (46)

$$\begin{aligned} K_R =\frac{k}{(I+qk)^{2}}(q_R I-qI_R )<0 \end{aligned}$$

(49)

Setting $dR=0$ and following the same procedure above, we obtain:

$$\begin{aligned} q_k \equiv \frac{dq}{dk}=\frac{X}{2\Delta }>0. \end{aligned}$$

(50)

Therefore,

$$\begin{aligned} p_k \equiv \frac{dp}{dk}=\frac{2RW}{\alpha }q_k >0 \end{aligned}$$

(51)

$\square$

Proposition 3

In the no-moral hazard economy with banks and depositors an equilibrium with intermediation exists and it is unique for every $R\in (\frac{X^{2}k^{2}}{2\alpha W},\frac{X^{2}(W+2k)^{2}}{4\alpha W})$.

Proof

Rearranging (17) and (18), we obtain:

$$\begin{aligned} q_1 (I)\equiv q=\frac{X^{2}(I+k)^{2}}{2\alpha RW}-\frac{I}{W}; \end{aligned}$$

(52)

and

$$\begin{aligned} q_2 (I)\equiv q=\sqrt{1-\frac{2I}{W}}. \end{aligned}$$

(53)

An equilibrium is a value of $I\in (0,\frac{W}{2}]$ that satisfies (41) and (42).

Observe that $q_1 (0)=\frac{X^{2}k^{2}}{2\alpha RW}$, $q_1 (I)$ is strictly concave in I and strictly decreasing for all $I>I_{\max }$, where $I>I_{\max }$ is the maximum of $q_1 (I)$. Thus, there exists a value $\hat{{I}}>I_{\max }$ such that $q_1 (\hat{{I}})=0$. On the other hand, $q_2 (0)=1$, $q_2 (I)$ is strictly decreasing, and $q_2 (\frac{W}{2})=0$.

Thus, a unique intersection of $q_1 (I)$ and $q_2 (I)$—that is, a value of $I\in (0,\frac{W}{2}]$ that satisfies (41) and (42)—will occur for some $I\in (0,\frac{W}{2}]$ only if $q_2 (0)=1\ge \frac{X^{2}k^{2}}{2\alpha RW}=q_1 (0)$ (51), and $q_2 (\frac{W}{2})=0\le \frac{X^{2}(W+2k)^{2}}{8\alpha RW}-\frac{1}{2}=q_1 (\frac{W}{2})$ (A12). Inequality (51) implies $R\ge \frac{X^{2}k^{2}}{2\alpha W}$, while inequality (A12) implies $R\le \frac{X^{2}(W+2k)^{2}}{4\alpha W}$. Thus, for every $R\in (\frac{X^{2}k^{2}}{2\alpha W},\frac{X^{2}(W+2k)^{2}}{4\alpha W})$, an equilibrium exists and it is unique. Using (16), $p^{*}=\alpha ^{-1}[X(I^{*}+k)]$. $\square$

Proposition 4

In the no-moral hazard economy with banks and depositors:

$$\begin{aligned} I_R^*>0, \quad \hat{{q}}_R^*<0, \quad p_R^*>0, \quad K_R^*<0 ~{ and}~p_k^*>0. \end{aligned}$$

Proof

Differentiating totally (17) and (18) with respect to I, q, R and k, we obtain:

$$\begin{aligned}&\displaystyle \left( {\frac{X^{2}(I+k)}{\alpha }-R} \right) dI-RWdq=(qW+I)dR-\frac{X^{2}(I+k)}{\alpha }dk; \end{aligned}$$

(54)

$$\begin{aligned}&\displaystyle dI+qWdq=0. \end{aligned}$$

(55)

The determinant of the system (54)–(55) is $\Delta =\frac{X^{2}(I+k)}{\alpha }qW+RW(1-q)>0$.

Setting $dk=0$, by Cramer’s rule,

$$\begin{aligned} I_R =\frac{dI}{dR}=\frac{1}{\Delta }(qW+I)qW>0, \end{aligned}$$

(56)

and

$$\begin{aligned} q_R \equiv \frac{dq}{dR}=-\frac{1}{\Delta }(qW+I)<0. \end{aligned}$$

(57)

Differentiating (16) with respect to R, we get:

$$\begin{aligned} p_R \equiv \frac{dp}{dR}=\frac{XI}{\alpha }I_R >0, \end{aligned}$$

(58)

where we have used (56).

Finally, by (56) and (57)

$$\begin{aligned} K_R =\frac{k}{(I+qk)^{2}}(q_R I-qI_R )<0 \end{aligned}$$

(59)

Setting $dR=0$, and following the same procedure above, we obtain:

$$\begin{aligned} I_k =\frac{dI}{dk}=-\frac{1}{\Delta }qW\frac{X^{2}(I+k)}{\alpha }<0, \end{aligned}$$

(60)

Hence,

$$\begin{aligned} p_R \equiv \frac{dp}{dR}=\frac{X}{\alpha }(I_k +1)=\frac{X}{\alpha }\left( 1-\frac{qWX^{2}(I+k)\alpha ^{-1}}{qWX^{2}(I+k)\alpha ^{-1}+RW(1-q)}\right) >0,\qquad \end{aligned}$$

(61)

where we have used (60) and $\Delta$. $\square$

Proposition 5

In the moral hazard economy with banks and depositors, there is no equilibrium with intermediation that supports the Pareto optimal allocation.

Proof

Inserting (11) in (8), replacing (p, q) with $(p^{O},q^{O})$ in (8) and (10), and solving for R in (8) and (10) respectively, we obtain

$$\begin{aligned} R_1\equiv & {} R=X+\frac{2Xk-2\alpha p^{O}}{W(1-q^{O 2})} \end{aligned}$$

(62)

$$\begin{aligned} R_2\equiv & {} R=\frac{XW(1-q^{O 2})+2Xk}{W(1-q^{O 2})+4q^{O}W} \end{aligned}$$

(63)

Using (25) in (62) and (63) yields $R_1 =0\ne \frac{4XW}{4W-2k+20W\sqrt{\frac{2k+W}{5W}}}=R_2$.

Thus, there does not exist a value of which yields an equilibrium with $(p,q)=(p^{O},q^{O})$. Thus, there is no equilibrium that supports the Pareto optimal allocation. $\square$

Proposition 6

In the moral hazard economy with banks and depositors, $R^{*}$ is an equilibrium with intermediation that supports the second best allocation.

Proof

A solution $R^{*}$ to problem (26) subject to (27) exists, since the continuous function V(.) in (26) is maximized over the compact set (27). Since function V(.) is strictly concave in (p, q), there exist a unique pair $(\hat{{p}}^{*},\hat{{q}}^{*})$ that solves (26) subject to (27). Since $\hat{{p}}(R)$ and $\hat{{q}}(R)$ are uniquely determined by the equilibrium conditions (8)-(11), there exists a unique $R^{*}$ such that $\hat{{p}}^{*}=\hat{{p}}(R^{*})$ and $\hat{{q}}^{*}=\hat{{q}}(R^{*})$. $\square$

Proposition 7

In the no-moral hazard economy with banks and depositors, there exist a unique $R^{0}$ such that the corresponding equilibrium with intermediation supports the (first best) Pareto optimal allocation.

Proof

Using (18) and $(p^{O},q^{O})$ in (16) and (17), we obtain:

$$\begin{aligned}&\displaystyle 2\alpha p^{O}=X\left( {W(1-q^{O 2})+2k} \right) \end{aligned}$$

(64)

$$\begin{aligned}&\displaystyle X^{2}\left( {\frac{W}{2}(1-q^{O 2})+k} \right) ^{2}-\alpha RW(1-q^{O 2})=2\alpha Rq^{O}W \end{aligned}$$

(65)

Equation (64) is independent of R. Solving (65) for R, we obtain:

$$\begin{aligned} R^{*}=\frac{\left( {2k+W(1-q^{O 2})} \right) ^{2}X^{2}}{4\alpha W(1+2q^{O}-q^{O 2})}. \end{aligned}$$

(66)

Lastly, we need to verify that $R^{*}$ is an equilibrium. By proposition 3, this amounts to verify that $R^{*}\in (\frac{X^{2}k^{2}}{2\alpha W},\frac{X^{2}(W+2k)^{2}}{4\alpha W})$. Thus, $R^{*}$ is an equilibrium if

$$\begin{aligned} \frac{X^{2}k^{2}}{2\alpha W}<\frac{\left( {2k+W(1-q^{O 2})} \right) ^{2}X^{2}}{4\alpha W(1+2q^{O}-q^{O 2})} \end{aligned}$$

(67)

and

$$\begin{aligned} \frac{\left( {2k+W(1-q^{O 2})} \right) ^{2}X^{2}}{4\alpha W(1+2q^{O}-q^{O 2})}<\frac{X^{2}(W+2k)^{2}}{4\alpha W}. \end{aligned}$$

(68)

Rearranging (67), we obtain

$$\begin{aligned} 4k^{2}+W^{2}(1-q^{O 2})^{2}+4kW(1-q^{O 2})>2k^{2}(1+2q^{O}-q^{O 2}). \end{aligned}$$

(69)

By assumption, $k<1$, $W>1$ and $0\le q\le 1$. Therefore, $2k^{2}(1+2q^{O}-q^{O 2})\le 4k^{2}$ and $W^{2}(1-q^{O 2})^{2}+4kW(1-q^{O 2})>0$. Thus, inequality (69) holds, hence, inequality (67) is verified.

(68) is equivalent to

$$\begin{aligned} 2k+W(1-q^{O 2})<(2k+W)\sqrt{1+2q^{O}-q^{O 2}} \end{aligned}$$

(70)

Since $1\le 1+2q^{O}-q^{O 2}\le 2$ and$1-q^{O 2}\le 1$, (70) holds, thus (68) is verified. $\square$

Proposition 8

In both the moral-hazard and no-moral hazard economies with banks and depositors, the equilibrium level of risk $\hat{{p}}$ converges to the optimal level of risk $p^{*}$ from below, to the optimal level of bank capitalization $K^{*}$ from above, and to a best allocation as market power rents vanish, i.e. $\hat{{p}}\uparrow p^{*}$, and $\hat{{K}}\downarrow K^{*}$ as $\rho \rightarrow 0$.

Proof

By propositions 2 and 4, $\hat{{p}}$ increases monotonically in R. By propositions 6 and 7, there exists a unique optimal $R^{*}$ that supports the optimal level of risk $p^{*}$. Therefore $\hat{{p}}(R)\uparrow \hat{{p}}(R^{*})=p^{*}$ and $\hat{{K}}(R)\downarrow \hat{{K}}(R^{*})=K^{*}$ as $\rho =R^{*}-R\rightarrow 0$. $\square$