Banking competition, production externalities, and the effects of monetary policy

Abstract

Since the global financial crisis, there has been a significant amount of concern about the presence of large-scale financial intermediaries which affects the competitive landscape of the banking sector in advanced economies. In light of this issue, this paper develops a framework to demonstrate how the degree of concentration impacts economic activity. As is standard in the growth literature, we incorporate production externalities from the aggregate capital stock which promote economic development. In this setting, we show that monetary policy may need to accommodate departures from perfect competition by setting a higher rate of money growth. In fact, in the presence of large capital externalities, neither low inflation nor perfect competition may be optimal. That is, in environments where capital accumulation would be expected to be inefficiently low, the optimal rate of money growth is higher than the Friedman rule in order to encourage investment—yet, the optimal competitive structure favors increased concentration to foster a large seigniorage tax base that also adds to the capital stock.

Technical appendix

1. A typical bank’s problem when all seigniorage revenues are rebated to young agents:

First, consider the case where the incentive compatibility constraint is non-binding. As discussed in the text, each intermediary makes its portfolio choice to maximize (5) subject to (6)–(8). Upon substituting (7) and (9) into objective function, (5), the problem can be written as:

$$\begin{aligned} \underset{m_{t},k_{t+1}}{\mathrm{Max}}\ \frac{\pi \left[ \frac{N}{\pi }m_{t}\frac{ P_{t}}{P_{t+1}}\right] ^{1-\theta }+\left( 1-\pi \right) \left[ \frac{N}{ 1-\pi }r\left( k_{t+1},K_{t+1}^{N-1}\right) k_{t+1}\right] ^{1-\theta }}{ 1-\theta } \end{aligned}$$

(44)

Next, using (6), the bank’s problem is reduced to a choice of $k_{t+1}:$

$$\begin{aligned} \underset{k_{t+1}}{\mathrm{Max}}\ \frac{\pi \left[ \frac{N}{\pi }\frac{P_{t}}{P_{t+1}} \left( \frac{1}{N}\left( w_{t}\!+\!\tau _{t}\right) -k_{t+1}\right) \right] ^{1-\theta }+\left( 1-\pi \right) \left[ \frac{N}{1-\pi }r\left( k_{t+1},K_{t+1}^{N-1}\right) k_{t+1}\right] ^{1-\theta }}{1-\theta } \end{aligned}$$

The bank’s choice of capital investment is such that:

$$\begin{aligned}&\pi ^{\theta }\left( N\frac{P_{t}}{P_{t+1}}\right) ^{1-\theta }\left( \frac{1 }{N}\left( w_{t}+\tau _{t}\right) -k_{t+1}\right) ^{-\theta }\\&\quad =\left( 1-\pi \right) ^{\theta }N^{1-\theta }\left[ r\left( k_{t+1},K_{t+1}^{N-1}\right) k_{t+1}\right] ^{-\theta }\\&\qquad *\left[ r\left( k_{t+1},K_{t+1}^{N-1}\right) +\frac{\partial r\left( k_{t+1},K_{t+1}^{N-1}\right) }{\partial K_{t+1}}\frac{\partial K_{t+1}}{ \partial k_{t+1}}k_{t+1}\right] \end{aligned}$$

where $r\left( k_{t+1},K_{t+1}^{N-1}\right) =\alpha A \left( K_{t+1}^{N-1}+k_{t+1}\right) ^{\rho +\alpha -1}=\alpha AK_{t+1}^{\rho +\alpha -1}$ and $K_{t+1}=K_{t+1}^{N-1}+k_{t+1}$. Therefore:

$$\begin{aligned} \frac{\partial r\left( k_{t+1},K_{t+1}^{N-1}\right) }{\partial K_{t+1}} =-\left( 1-\rho -\alpha \right) r\left( k_{t+1},K_{t+1}^{N-1}\right) K_{t+1}^{-1} \end{aligned}$$

and $\frac{\partial K_{t+1}}{\partial k_{t+1}}=1$. Upon substituting this information into the choice of $k_{t+1}$, we obtain (12):

$$\begin{aligned}&\pi ^{\theta }\left( N\frac{P_{t}}{P_{t+1}}\right) ^{1-\theta }\left( \frac{1 }{N}\left( w_{t}+\tau _{t}\right) -k_{t+1}\right) ^{-\theta } \\&\quad = \left( 1-\pi \right) ^{\theta }N^{1-\theta }\left[ r\left( k_{t+1},K_{t+1}^{N-1}\right) k_{t+1}\right] ^{-\theta } \\&\qquad *\left[ 1-\left( 1-\rho -\alpha \right) \frac{k_{t+1}}{ k_{t+1}+K_{t+1}^{N-1}}\right] r\left( k_{t+1},K_{t+1}^{N-1}\right) \end{aligned}$$

Next, from a bank’s balance sheet condition, (6), we have: $m_{t}=\frac{1}{N} \left( w_{t}+\tau _{t}\right) -k_{t+1}$ . Plugging this information into (12):

$$\begin{aligned} \pi ^{\theta }\left( N\frac{P_{t}}{P_{t+1}}\right) ^{1-\theta }m_{t}^{-\theta }= & {} \left( 1-\pi \right) ^{\theta }N^{1-\theta }\left[ r\left( k_{t+1},K_{t+1}^{N-1}\right) k_{t+1}\right] ^{-\theta } \nonumber \\&*\left[ 1-\left( 1-\rho -\alpha \right) \frac{k_{t+1}}{ k_{t+1}+K_{t+1}^{N-1}}\right] r \nonumber \\&\left( k_{t+1},K_{t+1}^{N-1}\right) \end{aligned}$$

(45)

With some simplification, it is easily verified that (45) can be written as:

$$\begin{aligned} \begin{aligned}&m_{t}\left( K_{t+1}^{N-1},\frac{P_{t+1}}{P_{t}}\right) \\&\quad =\frac{k_{t+1}}{\frac{1-\pi }{\pi } \left[ 1-\left( 1-\rho -\alpha \right) \frac{k_{t+1}}{k_{t+1}+K_{t+1}^{N-1}} \right] ^{\frac{1}{\theta }}\left( r\left( k_{t+1},K_{t+1}^{N-1}\right) \frac{P_{t+1}}{P_{t}}\right) ^{^{\frac{1-\theta }{\theta }}}} \end{aligned} \end{aligned}$$

(46)

Next, balance sheet condition, (6), implies that $k_{t+1}=\frac{1}{N}\left( w_{t}+\tau _{t}\right) -m_{t}$. Using this information:

$$\begin{aligned} \begin{aligned}&m_{t}\left( K_{t+1}^{N-1},\frac{P_{t+1}}{P_{t}}\right) \\&\quad =\frac{\frac{w_{t}+\tau _{t}}{N}}{ 1+\frac{1-\pi }{\pi }\left[ 1-\left( 1-\rho -\alpha \right) \frac{k_{t+1}}{ k_{t+1}+K_{t+1}^{N-1}}\right] ^{\frac{1}{\theta }}\left( r\left( k_{t+1},K_{t+1}^{N-1}\right) \frac{P_{t+1}}{P_{t}}\right) ^{^{\frac{1-\theta }{\theta }}}} \end{aligned} \end{aligned}$$

(47)

Recall that in a symmetric Nash equilibrium, $K_{t+1}^{*} =Nk_{t+1}^{*}$. Plugging the definition of $I_{t}^{*}$, where $I_{t}^{*}\equiv I_{t}\left( k_{t+1}^{*},\frac{P_{t+1}}{P_{t}};N\right) =r\left( k_{t+1}^{*};N\right) \frac{P_{t+1}}{P_{t}}$ , into (47):

$$\begin{aligned} m_{t}\left( k_{t+1}^{*},\frac{P_{t+1}}{P_{t}};N\right) =\frac{\frac{ w_{t}+\tau _{t}}{N}}{1+\frac{1-\pi }{\pi }\left[ 1-\frac{\left( 1-\rho -\alpha \right) }{N}\right] ^{\frac{1}{\theta }}\left( I_{t}\left( k_{t+1}^{*},\frac{P_{t+1}}{P_{t}};N\right) \right) ^{^{\frac{1-\theta }{ \theta }}}} \end{aligned}$$

which is the expression obtained in the text.

Using (7) and (9), the relative return to depositors is such that:

$$\begin{aligned} \frac{r_{t}^{n^{*}}}{r_{t}^{m^{*}}}=\frac{\pi Nr\left( k_{t+1}^{*};N\right) k_{t+1}^{*}}{\left( 1-\pi \right) Nm_{t}\left( k_{t+1}^{*},\frac{P_{t+1}}{P_{t}};N\right) \frac{P_{t}}{P_{t+1}}} \end{aligned}$$

(48)

Substituting for $m_{t}\left( k_{t+1}^{*}, \frac{P_{t+1}}{P_{t}};N\right)$ from (46) yields:

$$\begin{aligned} \frac{r_{t}^{n^{*}}}{r_{t}^{m^{*}}}= & {} \left[ 1-\left( 1-\rho -\alpha \right) \frac{k_{t+1}^{*}}{K_{t+1}^{*}}\right] ^{\frac{1}{ \theta }}\left( r\left( k_{t+1}^{*};N\right) \frac{P_{t+1}}{P_{t}} \right) ^{\frac{1}{\theta }} \\= & {} \left[ 1-\left( 1-\rho -\alpha \right) \frac{1}{N}\right] ^{\frac{1}{ \theta }}\left( r\left( k_{t+1}^{*};N\right) \frac{P_{t+1}}{P_{t}} \right) ^{\frac{1}{\theta }} \end{aligned}$$

Next, suppose the incentive compatibility constraint is binding above the Friedman rule level. From (48):

$$\begin{aligned} \frac{r_{t}^{n^{*}}}{r_{t}^{m^{*}}}=\frac{\pi Nr\left( k_{t+1}^{*};N\right) k_{t+1}^{*}}{\left( 1-\pi \right) Nm_{t}\left( k_{t+1}^{*},\frac{P_{t+1}}{P_{t}};N\right) \frac{P_{t}}{P_{t+1}}}=1 \end{aligned}$$

which implies that:

$$\begin{aligned} m_{t}\left( k_{t+1}^{*},\frac{P_{t+1}}{P_{t}};N\right) =\frac{\pi }{ 1-\pi }r\left( k_{t+1}^{*};N\right) \frac{P_{t+1}}{P_{t}}k_{t+1}^{*} \end{aligned}$$

(49)

In addition, as $m_{t}^{*}=\frac{1}{N}\left( w_{t}+\tau _{t}\right) -k_{t+1}^{*}$:

$$\begin{aligned} \frac{1}{N}\left( w_{t}+\tau _{t}\right) -k_{t+1}^{*}=\frac{\pi }{1-\pi } r\left( k_{t+1}^{*};N\right) \frac{P_{t+1}}{P_{t}}k_{t+1}^{*} \end{aligned}$$

or equivalently:

$$\begin{aligned} k_{t+1}^{*}=\frac{\frac{1}{N}\left( w_{t}+\tau _{t}\right) }{\left( 1+ \frac{\pi }{1-\pi }r\left( k_{t+1}^{*};N\right) \frac{P_{t+1}}{P_{t}} \right) } \end{aligned}$$

which is Eq. (17).

2. Proof of Proposition 1. The incentive compatibility constraint does not bind if $I^{*}>\underline{I}$. Given our characterization of (29), $I^{*}>\underline{I}$ if $\Gamma \left( \underline{I}\right) >1$. Using the expression for $\Gamma \left( \underline{I}\right)$, this condition holds if:

$$\begin{aligned} \left( \sigma \left[ 1-\frac{\left( 1-\alpha -\rho \right) }{N}\right] + \frac{\pi }{1-\pi }\right) \frac{\alpha }{\left( 1-\alpha \right) }>1 \end{aligned}$$

With some simple algebra, the condition can be written as $\sigma >\frac{ \frac{1}{\alpha }-\frac{1}{1-\pi }}{1-\frac{\left( 1-\alpha -\rho \right) }{N }}=\underline{\sigma }_{2}$. Therefore, for all $\sigma >\underline{\sigma } _{2}$, we have $I^{*}>\underline{I}$ and $\frac{r^{n^{*}}}{ r^{m^{*}}}>1$. Furthermore, the incentive compatibility constraint binds if $I^{*}\in \left[ 1,\underline{I}\right]$. When $\sigma =\underline{ \sigma }_{2}$, $I^{*}=\underline{I}=\left[ 1-\frac{\left( 1-\rho -\alpha \right) }{N}\right] ^{-1}$ and $\frac{r^{n^{*}}}{r^{m^{*}}}=1$.

As discussed in the text, when the incentive compatibility constraint binds above the Friedman rule level, the solution to the problem yields $I^{*}= \frac{\sigma }{\frac{1}{\alpha }-\frac{1}{1-\pi }}$. Therefore, $I^{*}\ge 1$ if $\frac{\sigma }{\frac{1}{\alpha }-\frac{1}{1-\pi }}\ge 1$, which can be rewritten as: $\sigma \ge \frac{1}{\alpha }-\frac{1}{1-\pi }= \underline{\sigma }_{1}$. In this manner, $\frac{r^{n^{*}}}{r^{m^{*}} }=1$ and $I^{*}=\frac{\sigma }{\frac{1}{\alpha }-\frac{1}{1-\pi }}>1$ for all $\sigma \in \left( \underline{\sigma }_{1},\underline{\sigma }_{2} \right]$. Finally, $\frac{r^{n^{*}}}{r^{m^{*}}}=1$ and $I^{*}=1$ for all $\sigma \in \left( 0,\underline{\sigma }_{1}\right]$. In sum, the unique solution to polynomial, (29), $I^{*}$ is such that: $I^{*}\in \left[ 1,\underline{I}\right]$ and $\frac{r^{n^{*}}}{r^{m^{*}}}=1$ if $\sigma \in (0,\underline{\sigma } _{2}]$, whereas $I^{*}>\underline{I}$ with $\frac{r^{n^{*}}}{ r^{m^{*}}}>1$ if $\sigma >\underline{\sigma }_{2}$. This completes the proof of Proposition 1.

3. Proof of Proposition 2. In the text, we discuss how bank activities are independent of the degree of concentration when the incentive compatibility constraint is binding, $\sigma \in \left( 0,\underline{\sigma } _{2}\right]$. Therefore, we focus in the proof on case ii, where $\sigma >\underline{\sigma }_{2}$. To begin, we use some simple algebra to rewrite the capital market clearing condition in the following manner. First, from (6) and (19):

$$\begin{aligned} m(k^{*};\sigma ,N)=\frac{k^{*}}{\frac{1-\pi }{\pi }\left[ 1-\frac{ \left( 1-\alpha -\rho \right) }{N}\right] ^{\frac{1}{\theta }}(I(k^{*};\sigma ,N))^{\frac{1-\theta }{\theta }}} \end{aligned}$$

(50)

Moreover, using (22), (23), the definition of $I^{*}$, and the fact that $K^{*}=Nk^{*}$ in (50), we get:

$$\begin{aligned} m(k^{*};\sigma ,N)= & {} \frac{k^{*}I(k^{*};N,\sigma )}{\frac{1-\pi }{ \pi }\left[ 1-\frac{\left( 1-\alpha -\rho \right) }{N}\right] ^{\frac{1}{ \theta }}(I(k^{*};\sigma ,N))^{\frac{1}{\theta }}} \\ m(k^{*};\sigma ,N)= & {} \frac{\frac{K^{*}}{N}\sigma \frac{\left( 1-\alpha \right) }{1-\alpha }\alpha AK^{*\alpha +\rho -1}}{\frac{1-\pi }{\pi } \left[ 1-\frac{\left( 1-\alpha -\rho \right) }{N}\right] ^{\frac{1}{\theta } }(I(k^{*};\sigma ,N))^{\frac{1}{\theta }}} \end{aligned}$$

Since $w(k^{*};N)=\left( 1-\alpha \right) A(Nk^{*})^{\rho +\alpha }$:

$$\begin{aligned} Nm(k^{*};\sigma ,N)= & {} \frac{\sigma \frac{\alpha }{1-\alpha }w(k^{*};N) }{\frac{1-\pi }{\pi }\left[ 1-\frac{\left( 1-\alpha -\rho \right) }{N}\right] ^{\frac{1}{\theta }}(I(k^{*};\sigma ,N))^{\frac{1}{\theta }}} \nonumber \\ \frac{Nm(k^{*};\sigma ,N)}{\sigma w(k^{*};N)}= & {} \frac{1}{\frac{ 1-\alpha }{\alpha }\frac{1-\pi }{\pi }\left[ 1-\frac{\left( 1-\alpha -\rho \right) }{N}\right] ^{\frac{1}{\theta }}(I(k^{*};\sigma ,N))^{\frac{1}{ \theta }}} \end{aligned}$$

(51)

After substituting for the amount of transfers in equilibrium from (2), bank’s balance sheet constraint, (6), can be written as:

$$\begin{aligned} w(k^{*};N)-\left( \frac{1}{\sigma }\right) Nm(k^{*};\sigma ,N)=K^{*} \end{aligned}$$

In turn:

$$\begin{aligned} \frac{Nm(k^{*};\sigma ,N)}{\sigma w(k^{*};N)}=1-\frac{K^{*}}{ w(k^{*};N)} \end{aligned}$$

As a result, the aggregate supply of capital by the banking sector is such that:

$$\begin{aligned} \frac{K^{*1-\alpha -\rho }}{\left( 1-\alpha \right) A}=1-\frac{\pi }{ 1-\pi }\frac{\alpha }{1-\alpha }\frac{1}{\left[ 1-\frac{\left( 1-\alpha -\rho \right) }{N}\right] ^{\frac{1}{\theta }}(I(k^{*};\sigma ,N))^{ \frac{1}{\theta }}} \end{aligned}$$

(52)

where $I^{*}=\sigma \alpha AK^{*\alpha +\rho -1}$. Thus, the aggregate stock of capital is a solution to the following polynomial:

$$\begin{aligned} \frac{K^{*1-\alpha -\rho }}{\left( 1-\alpha \right) A}=1-\frac{1}{\frac{ 1-\alpha }{\alpha }\frac{1-\pi }{\pi }\left[ 1-\frac{\left( 1-\alpha -\rho \right) }{N}\right] ^{\frac{1}{\theta }}\left( \alpha A\right) ^{\frac{1}{ \theta }}}\frac{K^{^{*}\frac{1-\rho -\alpha }{\theta }}}{\sigma ^{\frac{1 }{\theta }}} \end{aligned}$$

(53)

With some algebra, total differentiation of (53) with respect to N yields:

$$\begin{aligned} \frac{N}{K^{*}}\frac{\mathrm{d}K^{*}}{\mathrm{d}N}=\frac{\frac{\frac{\left( 1-\alpha -\rho \right) }{N}}{1-\frac{\left( 1-\alpha -\rho \right) }{N}}}{\frac{ \left( 1-\alpha -\rho \right) \theta K^{*1-\alpha -\rho }}{\left( 1-\alpha \right) A\left( 1-\frac{K^{*1-\alpha -\rho }}{\left( 1-\alpha \right) A}\right) }+\left( 1-\rho -\alpha \right) }>0 \end{aligned}$$

(54)

Since $r^{*}(k^{*};N)=\alpha A\left( K^{*}\right) ^{\rho +\alpha -1\text { }}$and $\frac{\mathrm{d}K^{*}}{\mathrm{d}N}>0$, $\frac{\mathrm{d}r^{*}}{\mathrm{d}N}<0$ and $\frac{\mathrm{d}I^{*}}{\mathrm{d}N}<0$. We proceed to show the effect of N on $\gamma ^{*}$ and $\frac{r^{n^{*}}}{r^{m^{*}}}$. By definition of $\gamma ^{*}$ and from bank’s balance sheet, (6):

$$\begin{aligned} \gamma ^{*}= & {} \frac{m(k^{*};N,\sigma )}{\frac{w(k^{*},N)+\tau (k^{*};N,\sigma )}{N}}=1-\frac{k^{*}}{\frac{w(k^{*};N)+\tau (k^{*};\sigma ,N)}{N}} \nonumber \\ \gamma ^{*}= & {} \frac{1-\frac{K^{*}}{w(k^{*};N)}+\frac{\sigma -1}{ \sigma }\frac{Nm(k^{*};\sigma ,N)}{w(k^{*};N)}}{1+\frac{\sigma -1}{ \sigma }\frac{Nm^{*}}{w^{*}}} \nonumber \\ \gamma ^{*}= & {} \frac{1}{\frac{1}{\sigma }\frac{1}{1-\frac{K^{*}}{ w\left( k^{*};N\right) }}+\frac{\sigma -1}{\sigma }} \end{aligned}$$

(55)

This suggests that each bank allocates smaller resources toward cash reserves when the banking sector is more competitive as $\frac{K^{*}}{ w\left( k^{*};N\right) }=\frac{K^{*1-\alpha -\rho }}{\left( 1-\alpha \right) A}$ is increasing in N. As for the effects on risk-sharing, using (6) and (51) in (21), the relative return to depositors can be expressed as:

$$\begin{aligned} \frac{r^{n^{*}}}{r^{m^{*}}}=\frac{\pi }{1-\pi }\frac{\alpha }{ 1-\alpha }\frac{1}{\left( 1-\frac{K^{*}}{w\left( k^{*};N\right) } \right) } \end{aligned}$$

(56)

Therefore, higher N leads to higher $K^{*}$ and less risk-sharing. This completes the proof of Proposition 2.

4. Proof of Proposition 3. The effect of a change in the rate of money growth on capital formation under case iii can be examined by differentiating (53) with respect to $\sigma$ and some simplification:

$$\begin{aligned} \frac{\sigma }{K^{*}}\frac{\mathrm{d}K^{*}}{\mathrm{d}\sigma }=\frac{1}{\theta \left( 1-\alpha -\rho \right) }\frac{1}{\frac{1}{1-\frac{K^{*1-\alpha -\rho }}{ \left( 1-\alpha \right) A}}+\frac{1-\theta }{\theta }}>0 \end{aligned}$$

(57)

Moreover, from the polynomial for $I^{*}$, (29), it is trivial to show that $\frac{\mathrm{d}I^{*}}{\mathrm{d}\sigma }>0$. The impact of $\sigma$ on $\gamma ^{*}$ and $\frac{r^{n^{*}}}{ r^{m^{*}}}$ directly follows. Finally, the effects of monetary policy when the incentive compatibility constraint is binding can be directly inferred from the closed-form solutions for $K^{*}$ and $I^{*}$ in the text. This completes the proof of Proposition 3.

5. Proof of Proposition 4. From (57), define $z\left( K^{*}\right) =\frac{1}{1-\frac{K^{*1-\alpha -\rho }}{\left( 1-\alpha \right) A}}$ and differentiate (57) with respect to N:

$$\begin{aligned} \theta \left( 1-\alpha -\rho \right) \sigma \frac{d^{2}K^{*}}{\mathrm{d}\sigma \mathrm{d}N} =\frac{\left[ \left[ z\left( K^{*}\right) +\frac{1-\theta }{\theta } \right] -z^{\prime }\left( K^{*}\right) K^{*}\right] \frac{\mathrm{d}K^{*} }{\mathrm{d}N}}{\left[ z\left( K^{*}\right) +\frac{1-\theta }{\theta }\right] ^{2} } \end{aligned}$$

Given that $\frac{\mathrm{d}K^{*}}{\mathrm{d}N}>0$, then $\frac{\mathrm{d}^{2}K^{*}}{\mathrm{d}\sigma \mathrm{d}N} \ge 0$ if $\left[ \left[ z\left( K^{*}\right) +\frac{1-\theta }{\theta } \right] -z^{\prime }\left( K^{*}\right) K^{*}\right] \ge 0$. Alternatively, $\frac{d^{2}K^{*}}{\mathrm{d}\sigma \mathrm{d}N}<0$ if $\left[ \left[ z\left( K^{*}\right) +\frac{1-\theta }{\theta }\right] -z^{\prime }\left( K^{*}\right) K^{*}\right] <0$. Using the definition of $z\left( K^{*}\right)$, $\frac{d^{2}K^{*}}{\mathrm{d}\sigma \mathrm{d}N}\ge 0$ if:

$$\begin{aligned} 1+\frac{1-\theta }{\theta }\left( 1-\frac{K^{*1-\alpha -\rho }}{\left( 1-\alpha \right) A}\right) \ge \frac{\frac{K^{*1-\alpha -\rho }}{A}}{1- \frac{K^{*1-\alpha -\rho }}{\left( 1-\alpha \right) A}} \end{aligned}$$

(58)

Define $\chi \left( K^{*}\left( N\right) \right) =1+\frac{1-\theta }{ \theta }\left( 1-\frac{K^{*1-\alpha -\rho }}{\left( 1-\alpha \right) A} \right)$ and $\Phi \left( K^{*}\left( N\right) \right) =\frac{\frac{ K^{*1-\alpha -\rho }}{A}}{1-\frac{K^{*1-\alpha -\rho }}{\left( 1-\alpha \right) A}}$. It is easily verified that $\frac{\mathrm{d}\chi }{\mathrm{d}N}<0$ and $\frac{\mathrm{d}\Phi }{\mathrm{d}N}>0$. The incentive compatibility constraint is non-binding if $\sigma >\sigma _{2}$. Using the definition of $\sigma _{2}$, this condition can be written as one on N, $N>\frac{\left( 1-\alpha -\rho \right) }{1-\frac{\frac{1}{\alpha }-\frac{1}{1-\pi }}{\sigma }}=\underline{N}$, where at $\underline{N}$, $\frac{r^{n^{*}}}{r^{m^{*}}}=1$.

From the expression for the relative return to depositors, (21), and the expression for the supply of capital, (52), (52) can be expressed as:

$$\begin{aligned} \frac{K^{*1-\alpha -\rho }}{\left( 1-\alpha \right) A}=1-\frac{\pi }{ 1-\pi }\frac{\alpha }{1-\alpha }\frac{1}{\frac{r^{n^{*}}}{r^{m^{*}}}} \end{aligned}$$

(59)

Equilibrium condition, (59) at $\underline{N }$, implies that:

$$\begin{aligned} \frac{K^{*}(\underline{N})^{1-\alpha -\rho }}{\left( 1-\alpha \right) A} =1-\frac{\pi }{1-\pi }\frac{\alpha }{1-\alpha } \end{aligned}$$

Using this information and some algebra, $\chi \left( K^{*}\left( \underline{N}\right) \right) \ge \Phi \left( K^{*}\left( \underline{N} \right) \right)$ if:

$$\begin{aligned} \theta \le \frac{1}{\frac{1-\pi }{\pi }\frac{1-\alpha }{\alpha }\left[ \left( 1-\alpha \right) \frac{1-\pi }{\pi }\frac{1-\alpha }{\alpha }-\left( 2-\alpha \right) \right] +1}=\tilde{\theta } \end{aligned}$$

(60)

Next, evaluate at the upper bound on N. It is easy to see that $\underset{ N\rightarrow \infty }{lim}K^{*}(N)\rightarrow \widetilde{K}^{*}$ as $\widetilde{K}^{*}$ is simply the equilibrium capital stock under perfect competition. In order for a finite value of N such that $\chi \left( K^{*}\left( N\right) \right) =\Phi \left( K^{*}\left( N\right) \right)$ to exist, we must show that:

$$\begin{aligned} \chi \left( \widetilde{K}^{*}\right) <\Phi \left( \widetilde{K}^{*}\right) \end{aligned}$$

or equivalently:

$$\begin{aligned} 1+\frac{1-\theta }{\theta }\left( 1-\frac{\widetilde{K}^{*1-\alpha -\rho }}{\left( 1-\alpha \right) A}\right) <\frac{\frac{\widetilde{K}^{*1-\alpha -\rho }}{A}}{1-\frac{\widetilde{K}^{*1-\alpha -\rho }}{\left( 1-\alpha \right) A}} \end{aligned}$$

(61)

By the Tobin effect, $\chi \left( \widetilde{K}^{*}\right)$ is decreasing in $\sigma$, while $\Phi \left( \widetilde{K}^{*}\right)$ is increasing in $\sigma$. We proceed to show that there exists a rate of money growth, $\hat{\sigma }_{0}$ such that (61) holds.

Next, recall that:

$$\begin{aligned} \frac{r^{n^{*}}}{r^{m^{*}}}=\left[ 1-\frac{\left( 1-\rho -\alpha \right) }{N}\right] ^{\frac{1}{\theta }}\left[ I^{*}\right] ^{^{\frac{1}{ \theta }}} \end{aligned}$$

From (29), it can be easily shown that $I^{*} \rightarrow \infty$ as $\sigma \rightarrow \infty$. Therefore, $\frac{ r^{n^{*}}}{r^{m^{*}}}\rightarrow \infty$ as $\sigma \rightarrow \infty$. Consequently, this implies that $\widetilde{K}^{*}$ converges to $\widetilde{K}_{1}^{*}$ as $\sigma \rightarrow \infty$. From (59), $\lim \limits _{\sigma \rightarrow \infty } \frac{\widetilde{K}^{*}}{w(k^{*};N)}=\frac{\widetilde{K}_{1}^{*}{}^{1-\alpha -\rho }}{\left( 1-\alpha \right) A}\rightarrow 1$ as $\frac{ r^{n^{*}}}{r^{m^{*}}}\rightarrow \infty$.

From here, we can pin down $\widetilde{K}_{1}^{*}:$$\widetilde{K} _{1}^{*}=\left[ \left( 1-\alpha \right) A\right] ^{\frac{1}{1-\alpha -\rho }}$. Thus, $\chi \left( \widetilde{K}^{*}\right) <\Phi \left( \widetilde{K}^{*}\right)$ as $\sigma \rightarrow \infty$. In addition, as $N\rightarrow \infty$, $\underline{\sigma }_{1}=\underline{\sigma }_{2}= \frac{1}{\alpha }-\frac{1}{1-\pi }$. As a result, at $\underline{\sigma } _{1}$:

$$\begin{aligned} \frac{\widetilde{K}^{*}{}^{1-\alpha -\rho }}{\left( 1-\alpha \right) A} =1-\frac{\pi }{1-\pi }\frac{\alpha }{1-\alpha } \end{aligned}$$

Therefore, $\chi \left( \widetilde{K}^{*}\right) |_{\sigma =\underline{ \sigma }_{1}}>\Phi \left( \widetilde{K}^{*}\right) |_{\sigma =\underline{ \sigma }_{1}}$ if (60) holds. In this manner, if (60) holds, a $\hat{\sigma }_{0}$ such that $\chi \left( \widetilde{K}^{*}\right) =\Phi \left( \widetilde{K}^{*}\right)$ exists. In sum, suppose $\theta <\tilde{\theta }$. Under this condition, $\frac{d^{2}K^{*}}{\mathrm{d}\sigma \mathrm{d}N}>0$ if $\sigma <\hat{\sigma } _{0}$ as $\chi \left( K^{*}\left( N\right) \right)>$$\Phi \left( K^{*}\left( N\right) \right)$$\forall$$N>\underline{N}$. By strict monotonicity of $\chi \left( K^{*}\left( N\right) \right)$ and $\Phi \left( K^{*}\left( N\right) \right)$ with respect to both $\sigma$ and N, if $\sigma >\hat{\sigma }_{0}$, $\chi \left( K^{*}\left( N\right) \right)$ and $\Phi \left( K^{*}\left( N\right) \right)$ intersect at a finite value of N, $\hat{N}$$>\underline{N}$. This implies that $\frac{ d^{2}K^{*}}{\mathrm{d}\sigma \mathrm{d}N}\ge \left( <\right) 0$ if $N\le \left( >\right) \hat{N}$. This completes the proof of Proposition 4.

6. Proof of Proposition 5. We begin by differentiating (29) with respect to $\rho$ and some simplification to get:

$$\begin{aligned} \frac{\mathrm{d}I^{*}}{\mathrm{d}\rho }=\frac{-\frac{1}{\theta }\frac{1}{N}\left[ 1-\frac{ \left( 1-\alpha -\rho \right) }{N}\right] ^{-1}}{\left( \frac{1-\theta }{ \theta }I^{*-1}+\frac{\left( 1-\alpha \right) }{\alpha }\frac{1}{\frac{ \left( 1-\alpha \right) I^{*}}{\alpha }-\sigma }\right) }<0 \end{aligned}$$

(62)

Next, we know that $I^{*}=\sigma \alpha AK^{*\alpha +\rho -1}$. Taking the log and differentiating with respect to $\rho$:

$$\begin{aligned} \ln K^{*}-\frac{1}{I^{*}}\frac{\mathrm{d}I^{*}}{\mathrm{d}\rho }=\left( 1-\rho -\alpha \right) \frac{1}{K^{*}}\frac{\mathrm{d}K^{*}}{\mathrm{d}\rho } \end{aligned}$$

(63)

As $\frac{\mathrm{d}I^{*}}{\mathrm{d}\rho }<0$, if $K^{*}>1$, then $\ln K^{*}>0$ and $\frac{\mathrm{d}K^{*}}{\mathrm{d}\rho }>0$. However, if $K^{*}<1$, $\ln K^{*}<0$ and $\frac{\mathrm{d}K^{*}}{\mathrm{d}\rho }$ can be negative. For this comparative static, we choose to focus on cases where $K^{*}>1$. From the polynomial yielding $K^{*}$, (53), the term on the left-hand side (LHS) is increasing in K, while that on the right-hand side (RHS) is decreasing in K. Moreover, for a given K, as A rises, LHS falls, while RHS rises. Overall, $\frac{\mathrm{d}K^{*}}{\mathrm{d}A}>0$. $K^{*}>1$ if at $K=1$, $LHS<RHS$. This condition can be written as:

$$\begin{aligned} \frac{1}{\left( 1-\alpha \right) A}+\frac{1}{\frac{1-\alpha }{\alpha }\frac{ 1-\pi }{\pi }\left[ 1-\frac{\left( 1-\alpha -\rho \right) }{N}\right] ^{ \frac{1}{\theta }}\left( \alpha A\right) ^{\frac{1}{\theta }}}\frac{1}{ \sigma ^{\frac{1}{\theta }}}<1 \end{aligned}$$

Therefore, there exists an $A_{0}$ such that this condition holds with equality. For $A>A_{0}$, $K^{*}>1$ and $\frac{\mathrm{d}K^{*}}{\mathrm{d}\rho }>0$.

We proceed by examining the effects of $\rho$ on risk-sharing. Taking the natural log of (21) and differentiating with respect to $\rho$, we get:

$$\begin{aligned} \frac{1}{\frac{r^{n^{*}}}{r^{m^{*}}}}\frac{\mathrm{d}\frac{r^{n^{*}}}{ r^{m^{*}}}}{\mathrm{d}\rho }=\frac{1}{\theta }\frac{1}{\left[ N-\left( 1-\alpha -\rho \right) \right] }+\frac{1}{\theta }\frac{1}{I^{*}}\frac{\mathrm{d}I^{*} }{\mathrm{d}\rho } \end{aligned}$$

Therefore, $\frac{\mathrm{d}\frac{r^{n^{*}}}{r^{m^{*}}}}{\mathrm{d}\rho }>0$ if:

$$\begin{aligned} \frac{\rho }{\left[ N-\left( 1-\alpha -\rho \right) \right] }>-\frac{\rho }{ I^{*}}\frac{\mathrm{d}I^{*}}{\mathrm{d}\rho } \end{aligned}$$

(64)

Substituting (62) into (64) along with some simplification, $\frac{\mathrm{d}\frac{ r^{n^{*}}}{r^{m^{*}}}}{\mathrm{d}\rho }>0$ if:

$$\begin{aligned} -\sigma <0 \end{aligned}$$

The condition always holds given that $\sigma >0$. This completes the proof of Proposition 5.

7. Proof of Proposition 6. We begin by deriving an expression for welfare over the range, $\sigma \in \left( 0,\underline{\sigma }_{1}\right]$ . First, from (14) and (15), complete risk-sharing in a symmetric Nash equilibrium implies that:

$$\begin{aligned} \begin{aligned}&\frac{\delta _{t}(k_{t+1}^{*},\frac{P_{t+1}}{P_{t}};N)Nm_{t}(k_{t+1}^{ *},\frac{P_{t+1}}{P_{t}};N)}{\pi }\frac{P_{t}}{P_{t+1}} \\&\quad =\frac{N}{1-\pi } \left[ \begin{array}{c} r\left( k_{t+1}^{*};N\right) k_{t+1}^{*}+\left( 1-\delta _{t}(k_{t+1}^{*},\frac{P_{t+1}}{P_{t}};N)\right) \\ *m_{t}(k_{t+1}^{*},\frac{P_{t+1}}{P_{t}};N)\frac{P_{t}}{P_{t+1}} \end{array} \right] \end{aligned} \end{aligned}$$

(65)

Imposing that $r\left( k_{t+1}^{*};N\right) =\frac{P_{t}}{P_{t+1}}$ at the Friedman rule:

$$\begin{aligned} \begin{aligned}&\frac{\delta _{t}\left( k_{t+1}^{*},\frac{P_{t+1}}{P_{t}};N\right) m_{t}\left( k_{t+1}^{ *},\frac{P_{t+1}}{P_{t}};N\right) }{\pi } \\&\quad =\frac{1}{1-\pi }\left[ k_{t+1}^{*}+\left( 1-\delta _{t}\left( k_{t+1}^{*},\frac{P_{t+1}}{P_{t}};N\right) \right) m_{t}\left( k_{t+1}^{*},\frac{P_{t+1}}{P_{t}};N\right) \right] \\&\qquad \delta _{t}\left( k_{t+1}^{*},\frac{P_{t+1}}{P_{t}};N\right) =\pi \left( \frac{ K_{t+1}^{*}}{Nm_{t}\left( k_{t+1}^{*},\frac{P_{t+1}}{P_{t}};N\right) }+1\right) \end{aligned} \end{aligned}$$

(66)

In the steady state:

$$\begin{aligned} \delta (k^{*};\sigma ,N)=\pi \left( \frac{K^{*}}{Nm(k^{*};\sigma ,N)}+1\right) \end{aligned}$$

(67)

where from (2) and (6) in the steady state we have:

$$\begin{aligned} \sigma \left( w\left( k^{*};N\right) -K^{*}\right) =Nm(k^{*};\sigma ,N) \end{aligned}$$

Upon using this information in (67):

$$\begin{aligned} \delta (k^{*};\sigma ,N)=\pi \left( \frac{\frac{K^{*}}{w\left( k^{*};N\right) }}{\sigma \left( 1-\frac{K^{*}}{w\left( k^{*};N\right) }\right) }+1\right) \end{aligned}$$

(68)

From (14), the consumption of a depositor in the steady state is such that:

$$\begin{aligned} r^{m^{*}}\left( w\left( k^{*};N\right) +\tau (k^{*};\sigma ,N)\right) =\frac{\delta ^{*}Nm(k^{*};\sigma ,N)}{\sigma \pi } \end{aligned}$$

Upon substituting for the expression of $Nm(k^{*};\sigma ,N)$ and $\delta (k^{*};\sigma ,N)$, it can be verified that:

$$\begin{aligned} r^{m^{*}}\left( w\left( k^{*};N\right) +\tau (k^{*};\sigma ,N)\right) =\left( \frac{1}{\sigma }-1\right) K^{*}+w\left( k^{*};N\right) \end{aligned}$$

(69)

Furthermore, using (22) and the fact that $K^{*} =\left( \alpha A\sigma \right) ^{\frac{1}{1-\alpha -\rho }}$ in (69):

$$\begin{aligned} c^{m}(k^{*};\sigma ,N)=r^{m^{*}}\left( w\left( k^{*};N\right) +\tau (k^{*};\sigma ,N)\right) =\alpha ^{\frac{1}{1-\alpha -\rho }}A^{ \frac{1}{1-\alpha -\rho }}\left( \frac{1}{\alpha }-\sigma \right) \sigma ^{ \frac{\alpha +\rho }{1-\alpha -\rho }} \end{aligned}$$

(70)

Given that $c^{m}(k^{*};\sigma ,N)=c^{n}(k^{*};\sigma ,N)=c(k^{*};\sigma ,N)$, the expected utility of a depositor over this parameter space is:

$$\begin{aligned} U^{*}=\frac{1}{1-\theta }c(k^{*};\sigma ,N)^{1-\theta } \end{aligned}$$

(71)

which is identical to the one presented in the text upon substituting for the expression for $c(k^{*},\sigma ;N)$ above.

Next, differentiate (71) with respect to $\sigma$ to obtain:

$$\begin{aligned} \frac{\mathrm{d}U^{*}}{\mathrm{d}\sigma }=c(k^{*};\sigma ,N)^{-\theta }\alpha ^{\frac{1 }{1-\alpha -\rho }}A^{\frac{1}{1-\alpha -\rho }}\left[ -1+\frac{\alpha +\rho }{1-\alpha -\rho }\sigma ^{-1}\left( \frac{1}{\alpha }-\sigma \right) \right] \sigma ^{\frac{\alpha +\rho }{1-\alpha -\rho }} \end{aligned}$$

(72)

Since $c(k^{*};\sigma ,N)$ and $\sigma$ are positive, $\frac{\mathrm{d}U^{*} }{\mathrm{d}\sigma }>0$ if the term in solid brackets is positive. Simple algebra implies that $\frac{\mathrm{d}U^{*}}{\mathrm{d}\sigma }\ge 0$ if $\sigma \le \frac{ \alpha +\rho }{\alpha }$. Upon using the definition of $\underline{\sigma } _{1}$, $\underline{\sigma }_{1}\ge \frac{\alpha +\rho }{\alpha }$ if:

$$\begin{aligned} \frac{1}{\alpha }-\frac{1}{1-\pi }\ge \frac{\alpha +\rho }{\alpha } \end{aligned}$$

With some simplification, the condition above can be written as:

$$\begin{aligned} \rho \le 1-\left( \frac{2-\pi }{1-\pi }\right) \alpha =\hat{\rho } \end{aligned}$$

Therefore, under this condition, $\frac{\mathrm{d}U^{*}}{\mathrm{d}\sigma }\ge \left( <\right) 0$ if $\sigma \le \left( >\right) \frac{\alpha +\rho }{\alpha }$ and a local optimum is present. By comparison, if $\rho >\hat{\rho }$, $\frac{ \mathrm{d}U^{*}}{\mathrm{d}\sigma }>0 \sigma \in \left( 0,\underline{\sigma }_{1}\right)$.

It is important to highlight from (70) that consumption is positive if $\sigma <\frac{1}{\alpha }$. It suffices to show that: $\underline{\sigma }_{1}<\frac{1}{\alpha }$. Upon using the definition of $\underline{\sigma }_{1}$, $\underline{\sigma }_{1}<\frac{1}{ \alpha }$ if $\frac{1}{\alpha }-\frac{1}{1-\pi }<\frac{1}{\alpha }$. This condition can be written as: $-\frac{\pi }{1-\pi }<0$, which always holds. Also notice that we need $\delta ^{*}<1$. Using (68) and some algebra, this condition can be written as:

$$\begin{aligned} 1<\frac{1-\pi }{\pi }\sigma \left( \frac{w\left( k^{*};N\right) }{ K^{*}}-1\right) \end{aligned}$$

where $\frac{K^{*}}{w\left( k^{*};N\right) }=\frac{K^{*1-\alpha -\rho }}{\left( 1-\alpha \right) A}=\frac{\alpha }{\left( 1-\alpha \right) r^{*}}=\frac{\alpha \sigma }{\left( 1-\alpha \right) I}=\frac{\alpha \sigma }{\left( 1-\alpha \right) }$ as $I=1$ over this parameter space. Upon using this information, $\delta ^{*}<1$ if:

$$\begin{aligned} \sigma <\frac{1}{\alpha }-\frac{1}{1-\pi }=\underline{\sigma }_{1} \end{aligned}$$

Next, from our work in the text, welfare does not vary with $\sigma$ if $\sigma \in (\underline{\sigma }_{1},\underline{\sigma }_{2})$. We proceed by deriving the expression for welfare, (30), under $\sigma \ge \underline{\sigma }_{2}$. From a bank’s problem, we substitute resource constraints, (7)–(9), and the expression for rental rate, (23), into expected utility function, (5), to get:

$$\begin{aligned} U^{*}=\frac{\pi \left[ \frac{N}{\pi }\frac{1}{\sigma }m^{*}\right] ^{1-\theta }+\left( 1-\pi \right) \left[ \frac{1}{1-\pi }\alpha AK^{*\rho +\alpha }\right] ^{1-\theta }}{1-\theta } \end{aligned}$$

(73)

Subsequently, using the expression for $\frac{Nm(k^{*};N,\sigma )}{ w\left( k^{*};N\right) }\frac{1}{\sigma }$ from (51) and the fact that $I^{*}=\sigma \alpha AK^{*\rho +\alpha -1}$ we obtain the expression for welfare in the text, (30), for all under $\sigma > \underline{\sigma }_{2}$.

We proceed to differentiate (30) with respect to $\sigma$ to get:

$$\begin{aligned} \frac{\mathrm{d}U^{*}}{\mathrm{d}\sigma }=\frac{\frac{\pi \left( \frac{\left[ 1-\left( \rho +\alpha \right) \left( 1-\theta \right) \right] \left( 1-\theta \right) }{\theta }\frac{1}{K^{*}}\frac{\mathrm{d}K^{*}}{\mathrm{d}\sigma }-\frac{1-\theta }{ \theta }\frac{1}{\sigma }\right) }{\left[ 1-\frac{\left( 1-\alpha -\rho \right) }{N}\right] ^{\frac{1-\theta }{\theta }}\left( \alpha A\right) ^{ \frac{1-\theta }{\theta }}\sigma ^{\frac{1-\theta }{\theta }}}K^{*\frac{ \left[ 1-\left( \rho +\alpha \right) \left( 1-\theta \right) \right] \left( 1-\theta \right) }{\theta }} +\left( \rho +\alpha \right) \left( 1-\theta \right) \left( 1-\pi \right) (K^{*})^{\left( \rho +\alpha \right) \left( 1-\theta \right) -1}\frac{\mathrm{d}K^{*}}{\mathrm{d}\sigma }}{\frac{\left( 1-\theta \right) \left( 1-\pi \right) ^{1-\theta }}{\left[ \alpha A\right] ^{1-\theta }}} \end{aligned}$$

With some simplifying algebra, $\frac{\mathrm{d}U^{*}}{\mathrm{d}\sigma }\ge 0$ if:

$$\begin{aligned} \frac{\sigma }{K^{*}}\frac{\mathrm{d}K^{*}}{\mathrm{d}\sigma }\ge \frac{1}{\left[ 1-\left( \rho +\alpha \right) \left( 1-\theta \right) \right] +\left( \rho +\alpha \right) \theta \frac{1-\pi }{\pi }\left( \sigma ^{\frac{1}{\theta }} \left[ 1-\frac{\left( 1-\alpha -\rho \right) }{N}\right] ^{\frac{1}{\theta } }\left( \alpha A\right) ^{\frac{1}{\theta }}K^{*\left[ \frac{-1+\left( \rho +\alpha \right) }{\theta }\right] }\right) ^{1-\theta }} \end{aligned}$$

(74)

From the polynomial yielding $K^{*}$, (53), we have:

$$\begin{aligned} \sigma ^{\frac{1}{\theta }}\left[ 1-\frac{\left( 1-\alpha -\rho \right) }{N} \right] ^{\frac{1}{\theta }}\left( \alpha A\right) ^{\frac{1}{\theta }}= \frac{1}{\frac{1-\alpha }{\alpha }\frac{1-\pi }{\pi }}\frac{K^{*\frac{ 1-\rho -\alpha }{\theta }}}{1-\frac{K^{*1-\alpha -\rho }}{\left( 1-\alpha \right) A}} \end{aligned}$$

Substituting into (74) along with some simplification, the condition becomes:

$$\begin{aligned} \frac{\sigma }{K^{*}}\frac{\mathrm{d}K^{*}}{\mathrm{d}\sigma }\ge \frac{1}{\left[ 1-\left( \rho +\alpha \right) \left( 1-\theta \right) \right] +\left( \rho +\alpha \right) \theta \left( \frac{1-\pi }{\pi }\right) ^{\theta }\left( \frac{\alpha }{1-\alpha }\frac{1}{1-\frac{K^{*1-\alpha -\rho }}{\left( 1-\alpha \right) A}}\right) ^{1-\theta }} \end{aligned}$$

Finally, using the expression for $\frac{\sigma }{K^{*}} \frac{\mathrm{d}K^{*} }{\mathrm{d}\sigma }$ from (57), $\frac{\mathrm{d}U^{*}}{\mathrm{d}\sigma }\ge 0$ if:

$$\begin{aligned} \begin{aligned} \psi \left( \sigma \right)&\equiv \left( 1-\frac{K^{*1-\alpha -\rho }}{ \left( 1-\alpha \right) A}\right) +\left( \rho +\alpha \right) \left( \frac{ 1-\pi }{\pi }\right) ^{\theta }\left( \frac{\alpha }{1-\alpha }\right) ^{1-\theta } \\&\quad \left( 1-\frac{K^{*1-\alpha -\rho }}{\left( 1-\alpha \right) A}\right) ^{\theta }\ge \left( 1-\alpha -\rho \right) \end{aligned} \end{aligned}$$

(75)

Given that $\frac{\mathrm{d}K^{*}}{\mathrm{d}\sigma }>0$, $\psi ^{\prime }\left( \sigma \right) <0$. Furthermore, from (59), $\lim \limits _{\sigma \rightarrow \infty }\frac{K^{*}}{w\left( k^{*};N\right) }\rightarrow 1$ as $\frac{r^{n^{*}}}{r^{m^{*}}} \rightarrow \infty$, where $\frac{K^{*}}{w\left( k^{*};N\right) }= \frac{K^{*1-\alpha -\rho }}{\left( 1-\alpha \right) A}$. Therefore, $\lim \limits _{\sigma \rightarrow \infty } \psi \rightarrow 0$. In addition, $1-\frac{K^{*1-\alpha -\rho }}{\left( 1-\alpha \right) A}= \frac{\alpha }{1-\alpha }\frac{\pi }{1-\pi }$ when $\sigma =\underline{ \sigma }_{2}$. Upon using this information, $\psi \left( \underline{\sigma } _{2}\right) =\left( \frac{\pi }{1-\pi }+\left( \rho +\alpha \right) \right) \frac{\alpha }{1-\alpha }$.

From our characterization of $\psi$, $\psi \left( \sigma \right) \le \left( 1-\alpha -\rho \right)$ for all $\sigma \ge \underline{\sigma }_{2}$ if $\psi \left( \underline{\sigma }_{2}\right) \le \left( 1-\alpha -\rho \right)$. Using some algebra, this condition can be written as: $\rho \le 1-\alpha \left( \frac{2-\pi }{1-\pi }\right) =\hat{\rho }$. In this manner, if $0<\rho \le \hat{\rho }$, $\frac{\mathrm{d}U^{*}}{\mathrm{d}\sigma }\le 0$ for all $\sigma \ge \underline{\sigma }_{2}$. Given our characterization of the welfare function over the ranges of $\sigma \in \left( 0,\underline{\sigma } _{2}\right)$, $\sigma ^{*}=\frac{\alpha +\rho }{\alpha }$ is a global optimum.

Next, suppose $\rho>\hat{\rho }\Longleftrightarrow \psi \left( \underline{ \sigma }_{2}\right) >\left( 1-\alpha -\rho \right)$. Under this condition there exists an (interior) value of $\sigma > \underline{\sigma }_{2}$, $\sigma ^{*}:\psi \left( \sigma \right) =\left( 1-\alpha -\rho \right)$. For all $\sigma > \sigma ^{*}$, $\frac{\mathrm{d}U^{*}}{\mathrm{d}\sigma }<0$ and for all $\underline{\sigma }_{2}\le \sigma \le \sigma ^{*}$, $\frac{ \mathrm{d}U^{*}}{\mathrm{d}\sigma }\ge 0$. Since $\frac{\mathrm{d}U^{*}}{\mathrm{d}\sigma }>0$ if $\sigma \in (0,\underline{\sigma }_{1})$ and $\frac{\mathrm{d}U^{*}}{\mathrm{d}\sigma }=0$ if $\sigma \in [\underline{\sigma }_{1},\underline{\sigma }_{2})$, the optimal value of $\sigma$ is such that $\sigma ^{*}>\underline{ \sigma }_{2}$. Additionally, for a given value of $\sigma \ge \underline{ \sigma }_{2}$, $\frac{\partial \psi }{\partial N}<0$ (since $\frac{\mathrm{d}K^{*} }{\mathrm{d}N}>0$ over that range). That is, for a given $\sigma$, the locus shifts down. As a result $\frac{\mathrm{d}\sigma ^{*}}{\mathrm{d}N}<0$. This completes the proof of Proposition 6.

8. Proof of Proposition 7. Suppose $\sigma > \underline{\sigma }_{2}$. Differentiating welfare function (30) with respect to N and some simplification yields:

$$\begin{aligned} \frac{\mathrm{d}U^{*}}{\mathrm{d}N}=\frac{ \begin{array}{c} \frac{\pi \left( \frac{\left[ 1-\left( \rho +\alpha \right) \left( 1-\theta \right) \right] }{\theta }\frac{\mathrm{d}K^{*}(N)}{\mathrm{d}N}-\frac{\left( 1-\alpha -\rho \right) }{N}\frac{1}{\theta }\left[ 1-\frac{\left( 1-\alpha -\rho \right) }{N}\right] ^{-1}\right) K^{*}(N)^{\frac{\left[ 1-\left( \rho +\alpha \right) \left( 1-\theta \right) \right] \left( 1-\theta \right) }{ \theta }}}{\left( \alpha A\right) ^{\frac{1-\theta }{\theta }}\sigma ^{\frac{ 1-\theta }{\theta }}\left[ 1-\frac{\left( 1-\alpha -\rho \right) }{N}\right] ^{\frac{1-\theta }{\theta }}}+ \\ \left( \rho +\alpha \right) \left( 1-\pi \right) K^{*}(N)^{\left( \rho +\alpha \right) \left( 1-\theta \right) -1}\frac{\mathrm{d}K^{*}(N)}{\mathrm{d}N} \end{array} }{\left( \frac{1-\pi }{\alpha A}\right) ^{1-\theta }} \end{aligned}$$

Next, with some algebra, $\frac{\mathrm{d}U^{*}}{\mathrm{d}N}\ge 0$ if:

$$\begin{aligned} \begin{aligned}&\frac{N}{K^{*}(N)}\frac{\mathrm{d}K^{*}(N)}{\mathrm{d}N} \\&\quad \ge \frac{\frac{\frac{\left( 1-\alpha -\rho \right) }{N}}{1-\frac{\left( 1-\alpha -\rho \right) }{N}}}{ \begin{array}{c} \left[ 1-\left( \rho +\alpha \right) \left( 1-\theta \right) \right] + \\ \left( \rho +\alpha \right) \left( \alpha A\right) ^{\frac{1-\theta }{\theta }}\sigma ^{\frac{1-\theta }{\theta }}\theta \left( \frac{1-\pi }{\pi } \right) \left[ 1-\frac{\left( 1-\alpha -\rho \right) }{N}\right] ^{\frac{ 1-\theta }{\theta }}(K^{*}(N))^{-\left[ \frac{1-\left( \rho +\alpha \right) }{\theta }\right] \left( 1-\theta \right) } \end{array}} \end{aligned} \end{aligned}$$

(76)

Upon using the polynomial yielding $K^{*}(N)$, (53), condition (76) can be written as:

$$\begin{aligned} \frac{N}{K^{*}(N)}\frac{\mathrm{d}K^{*}(N)}{\mathrm{d}N}\ge \frac{\frac{\frac{\left( 1-\alpha -\rho \right) }{N}}{1-\frac{\left( 1-\alpha -\rho \right) }{N}}}{ \left[ 1-\left( \rho +\alpha \right) \left( 1-\theta \right) \right] +\left( \rho +\alpha \right) \theta \left( \frac{1-\pi }{\pi }\right) ^{\theta } \frac{\left( \frac{\alpha }{1-\alpha }\right) ^{1-\theta }}{\left( 1-\frac{ K^{*}(N)^{1-\alpha -\rho }}{\left( 1-\alpha \right) A}\right) ^{1-\theta }}} \end{aligned}$$

Furthermore, substituting for $\frac{N}{K^{*}}\frac{\mathrm{d}K^{*} (N)}{\mathrm{d}N}$ from (54), $\frac{\mathrm{d}U^{*}}{\mathrm{d}N}\ge 0$ if:

$$\begin{aligned} \begin{aligned} \Phi \left( N\right)&\equiv \left[ 1+\left( \frac{1-\pi }{\pi }\right) ^{\theta }\left( \frac{\alpha }{1-\alpha }\right) ^{1-\theta }\left( 1-\frac{ K^{*}(N)^{1-\alpha -\rho }}{\left( 1-\alpha \right) A}\right) ^{\theta } \right] \frac{1}{K^{*}(N)^{1-\alpha -\rho }} \\&\ge \frac{1}{\left( \rho +\alpha \right) \left( 1-\alpha \right) A} \end{aligned} \end{aligned}$$

(77)

Given that $\frac{\mathrm{d}K^{*}(N)}{\mathrm{d}N}>0$, $\Phi ^{\prime }\left( N\right) <0$ . The incentive compatibility constraint is non-binding when $\sigma > \underline{\sigma }_{2}=\frac{\frac{1}{\alpha }-\frac{1}{1-\pi }}{1-\frac{ \left( 1-\alpha -\rho \right) }{N}}$, which can also be written as a condition on N. In particular, the incentive compatibility constraint does not bind if $N>\frac{\left( 1-\alpha -\rho \right) }{1-\frac{\frac{1}{\alpha }-\frac{1}{1-\pi }}{\sigma }}=\underline{N}$. We proceed to evaluate $\Phi$ at the lower bound on N, $\Phi \left( \underline{N}\right)$. From equilibrium condition, (59) at $\underline{N}$, we have:

$$\begin{aligned} \frac{K^{*}(\underline{N})^{1-\alpha -\rho }}{\left( 1-\alpha \right) A} =1-\frac{\pi }{1-\pi }\frac{\alpha }{1-\alpha } \end{aligned}$$

Upon substituting into the expression for $\Phi$ along with some simplifying steps:

$$\begin{aligned} \Phi \left( \underline{N}\right) =\frac{1}{1-\alpha }\frac{1}{\left( 1-\frac{ \pi }{1-\pi }\frac{\alpha }{1-\alpha }\right) \left( 1-\alpha \right) A} \end{aligned}$$

It is easily verified that $\Phi \left( \underline{N}\right) \le \frac{1}{ \left( \rho +\alpha \right) \left( 1-\alpha \right) A}$ if $\rho \le 1-\alpha \left( \frac{2-\pi }{1-\pi }\right) =\hat{\rho }$. As $\Phi ^{\prime }\left( N\right) <0$, $\Phi \left( N\right)$ lies below the $\frac{1}{\left[ \left( \rho +\alpha \right) \left( 1-\alpha \right) A\right] }$ line and $\frac{\mathrm{d}U^{*}}{\mathrm{d}N}\le 0$ for all $N\ge \underline{N}$. Given that $\frac{\mathrm{d}U^{*}}{\mathrm{d}N}=0$ for $N<\underline{N}$, optimal $N:N^{*}\in \left( 1,\underline{N}\right]$.

By comparison, $\Phi \left( \underline{N}\right) >\frac{1}{\left( \rho +\alpha \right) \left( 1-\alpha \right) A}$ if $\rho > \hat{\rho }$. This implies that if the external effects from capital formation are sufficiently strong, the optimal number of banks exceeds $\underline{N}$. In fact, we next seek to find conditions in which perfect competition is optimal. To do so, we evaluate $\Phi$ under perfect competition, which we denote as $\underline{\Phi }$. Recall from the proof of Proposition 4 that $\widetilde{K }^{*}$ is the equilibrium value of K as $N\rightarrow \infty$. From the polynomial yielding $K^{*}$, (53), $\widetilde{K}^{*}$ is such that:

$$\begin{aligned} \left( 1-\frac{\widetilde{K}^{*1-\alpha -\rho }}{\left( 1-\alpha \right) A}\right) ^{\theta }=\frac{\widetilde{K}^{*1-\alpha -\rho }}{\left( \frac{1-\alpha }{\alpha }\frac{1-\pi }{\pi }\right) ^{\theta }\sigma \alpha A } \end{aligned}$$

Upon substituting into $\Phi$ and some simplification, we get:

$$\begin{aligned} \underset{N\rightarrow \infty }{\lim }\Phi =\left[ \frac{1}{\widetilde{K}^{*1-\alpha -\rho }}+\frac{\alpha }{1-\alpha }\frac{1}{\sigma \alpha A} \right] \equiv \underline{\Phi } \end{aligned}$$

Next, we evaluate $\underline{\Phi }$ at the lower bound on $\sigma$. As $N\rightarrow \infty$, $\underline{\sigma }_{1}=\underline{\sigma }_{2}= \frac{1}{\alpha }-\frac{1}{1-\pi }$, which corresponds to $\frac{r^{n^{*}}}{r^{m^{*}}}=1$. Substituting this information into (59),

$$\begin{aligned} \frac{1}{\widetilde{K}^{*1-\alpha -\rho }}=\frac{\frac{1-\alpha }{\alpha }\frac{1-\pi }{\pi }}{\frac{1-\alpha }{\alpha }\frac{1-\pi }{\pi }-1}\frac{1 }{\left( 1-\alpha \right) A} \end{aligned}$$

Upon substituting into $\underline{\Phi }$ and some simplification, we obtain:

$$\begin{aligned} \underline{\Phi }|_{\sigma =\underline{\sigma }_{1}}=\frac{1}{\alpha }\frac{1 }{\frac{1}{\alpha }-\frac{1}{1-\pi }}\frac{1}{\left( 1-\alpha \right) A} \end{aligned}$$

It can be easily verified that $\underline{\Phi }|_{\sigma =\underline{ \sigma }_{1}}>\frac{1}{\left( \rho +\alpha \right) \left( 1-\alpha \right) A}$ if $\rho >\hat{\rho }$. Further, note that $\Phi$ is decreasing in $K^{*}(N)$. As shown in Proposition 3, money growth exhibits a Tobin effect over the range from 0 to $\underline{\sigma }_{1}$. As a result, for rates of money growth below $\underline{\sigma }_{1}$, $\Phi$ exceeds $\underline{\Phi }|_{\sigma =\underline{\sigma }_{1}}$. This establishes that perfect competition is optimal if the rate of money growth is sufficiently low, $\sigma \le \underline{\sigma }_{1}=\underline{\sigma }_{2}$.

We finish by finding conditions in which perfect competition does not maximize welfare. For an interior solution, $\tilde{N}$ to exist, $\Phi \left( N\right)$ must lie below the $\frac{1}{\left[ \left( \rho +\alpha \right) \left( 1-\alpha \right) A\right] }$ line over a certain range of N . We define $\tilde{N}$ to be the number of banks such that $\Phi \left( \tilde{N}\right) =\frac{1}{\left( \rho +\alpha \right) \left( 1-\alpha \right) A}$. In other words, $\tilde{N}$ is the number of intermediaries which provides an interior solution to the first-order condition for the social welfare function in (73). In order to prove that an $\tilde{N}$ can exist, we need to find conditions where $\underline{\Phi }< \frac{1}{\left( \rho +\alpha \right) \left( 1-\alpha \right) A}$. By the Tobin effect, $\frac{\mathrm{d}\underline{\Phi }}{\mathrm{d}\sigma }<0$. We proceed to show that there exists a rate of money growth, $\hat{\sigma }$, beyond which, $\underline{\Phi } <\frac{1}{\left( \rho +\alpha \right) \left( 1-\alpha \right) A}$.

First, with some simple algebra, we rewrite $\underline{\Phi }$ as:

$$\begin{aligned} \underline{\Phi }=\frac{1}{\left( 1-\alpha \right) A}\left[ \frac{\widetilde{ K}^{*}}{w(\widetilde{k}^{*};N)}+\frac{1}{\sigma A}\right] \end{aligned}$$

From (59), $\lim \limits _{\sigma \rightarrow \infty } \frac{\widetilde{K}^{*}}{w(\widetilde{k}^{*};N)}\rightarrow 1$ as $\frac{r^{n^{*}}}{r^{m^{*}}}\rightarrow \infty$. Therefore, $\frac{\widetilde{K}^{*}}{w(\widetilde{k}^{*};N)}\rightarrow 1$ as $\sigma \rightarrow \infty$ and $\lim \limits _{\sigma \rightarrow \infty } \underline{\Phi }\rightarrow \frac{1}{\left( 1-\alpha \right) A}$, which is clearly less than $\frac{1}{\left( \rho +\alpha \right) \left( 1-\alpha \right) A}$.

In sum, when $\rho >\hat{\rho }$ and $\sigma \le \underline{\sigma }_{1}= \underline{\sigma }_{2}$, $\underline{\Phi }|_{\sigma =\underline{\sigma } _{1}}>\frac{1}{\left( \rho +\alpha \right) \left( 1-\alpha \right) A}$ which implies that perfect competition is optimal. However, there also exists a $\hat{\sigma }>\underline{\sigma }_{1}$, such that for all $\sigma > \hat{\sigma }$, $\underline{\Phi }<\frac{1}{\left( \rho +\alpha \right) \left( 1-\alpha \right) A}$. Therefore, for $\sigma >\hat{\sigma }$, (77) has an interior solution, $\tilde{N}>\underline{N}$, where $\tilde{N}:\Phi \left( \tilde{N}\right) =\frac{1}{\left( \rho +\alpha \right) \left( 1-\alpha \right) A}$. Consequently, for $\sigma \le \hat{\sigma }$, perfect competition is optimal. For all $N\in [\underline{N},\tilde{N})$, $\frac{\mathrm{d}U^{*}}{\mathrm{d}N}>0$ and $\frac{\mathrm{d}U^{*}}{\mathrm{d}N}<0$ for $N>\tilde{N}$. In this manner, optimal N is such that $N^{*}=\tilde{N}$. Moreover, $\frac{\mathrm{d}\tilde{N}}{\mathrm{d}\sigma }<0$ as $\Phi$ shifts downward under higher values of $\sigma$. The result in the Proposition directly follows. This completes the proof of Proposition 7.

9. A typical bank’s problem when seigniorage revenues are rebated to both young and old agents:

We begin by solving the problem when the self-selection constraint is non-binding. Upon substitution for payments to movers and non-movers from (34)–(36) into objective function, (33), the problem is reduced to a choice of $k_{t+1}:$

$$\begin{aligned} \underset{k_{t+1}}{\mathrm{Max}}\ \frac{\pi \left[ \frac{N}{\pi }\frac{P_{t}}{P_{t+1}} \left( \frac{1}{N}\left( w_{t}+\lambda \tau _{t}\right) -k_{t+1}\right) +\left( 1-\lambda \right) \tau _{t+1}\right] ^{1-\theta }+\left( 1-\pi \right) \left[ \frac{Nr\left( k_{t+1},K_{t+1}^{N-1}\right) k_{t+1}}{1-\pi } +\left( 1-\lambda \right) \tau _{t+1}\right] ^{1-\theta }}{1-\theta } \end{aligned}$$

The first-order condition with respect to $k_{t+1}$ is such that:

$$\begin{aligned} \begin{aligned}&-\pi \left( c_{t}^{m}\right) ^{-\theta }\frac{N}{\pi }\frac{P_{t}}{P_{t+1}} +\left( 1-\pi \right) \left( c_{t}^{n}\right) ^{-\theta }\frac{N}{1-\pi } \\&\quad \left[ r\left( k_{t+1}\right) +\frac{\partial r\left( k_{t+1},K_{t+1}^{N-1}\right) }{\partial K_{t+1}}\frac{\partial K_{t+1}}{\partial k_{t+1}}k_{t+1}\right] =0 \end{aligned} \end{aligned}$$

(78)

where $c_{t}^{m}=\frac{N}{\pi }\frac{P_{t}}{P_{t+1}}\left( \frac{1}{N}\left( w_{t}+\lambda \tau _{t}\right) -k_{t+1}\right) +\left( 1-\lambda \right) \tau _{t+1}$ and $c_{t}^{n}=\frac{Nr\left( k_{t+1},K_{t+1}^{N-1}\right) k_{t+1}}{1-\pi }+\left( 1-\lambda \right) \tau _{t+1}$. From our derivations of the benchmark model, $\frac{\partial r\left( k_{t+1},K_{t+1}^{N-1}\right) }{\partial K_{t+1}}=-\left( 1-\rho -\alpha \right) r\left( k_{t+1},K_{t+1}^{N-1}\right) K_{t+1}^{-1}$. Upon using this information in (78):

$$\begin{aligned} -\left( c_{t}^{m}\right) ^{-\theta }\frac{P_{t}}{P_{t+1}}+\left( c_{t}^{n}\right) ^{-\theta }\left[ 1-\left( 1-\rho -\alpha \right) K_{t+1}^{-1}k_{t+1}\right] r\left( k_{t+1},K_{t+1}^{N-1}\right) =0 \end{aligned}$$

Imposing symmetry, it is easy to verify that:

$$\begin{aligned} \frac{c_{t}^{n^{*}}}{c_{t}^{m^{*}}}=\left( \left[ 1-\frac{\left( 1-\rho -\alpha \right) }{N}\right] r\left( k_{t+1}^{*};N\right) \frac{ P_{t+1}}{P_{t}}\right) ^{\frac{1}{\theta }} \end{aligned}$$

(79)

which is identical to (39) in the steady state.

We proceed to get an expression for the amount of money balances. From the expressions for consumption of each type of depositor, we have:

$$\begin{aligned} c_{t}^{m^{*}}=\frac{N}{\pi }m_{t}^{*}\frac{P_{t}}{P_{t+1}}+\left( 1-\lambda \right) \tau _{t+1} \end{aligned}$$

and

$$\begin{aligned} c_{t}^{n^{*}}=\frac{N}{1-\pi }r\left( k_{t+1}^{*};N\right) k_{t+1}^{*}+\left( 1-\lambda \right) \tau _{t+1} \end{aligned}$$

Substituting from (79):

$$\begin{aligned} \begin{aligned} \frac{N}{1-\pi }r\left( k_{t+1}^{*};N\right) k_{t+1}^{*}+\left( 1-\lambda \right) \tau _{t+1}&=\left[ \frac{N}{\pi }m_{t}^{*}\frac{P_{t}}{ P_{t+1}}+\left( 1-\lambda \right) \tau _{t+1}\right] \\&\quad \left( \left[ 1-\frac{ \left( 1-\rho -\alpha \right) }{N}\right] r\left( k_{t+1}^{*};N\right) \frac{P_{t+1}}{P_{t}}\right) ^{\frac{1}{\theta }} \end{aligned} \end{aligned}$$

(80)

Imposing steady state on (80) and using the fact that $\tau ^{*}=\frac{\sigma -1}{\sigma } Nm^{*}$, simple algebra yields:

$$\begin{aligned} Nm^{*}=\frac{1}{1-\pi }\frac{K^{*}}{\left[ \frac{1}{\pi }+\left( 1-\lambda \right) \left( \sigma -1\right) \right] \left[ 1-\frac{\left( 1-\rho -\alpha \right) }{N}\right] ^{\frac{1}{\theta }}I^{*\frac{ 1-\theta }{\theta }}-\frac{\left( 1-\lambda \right) \left( \sigma -1\right) }{I^{*}}} \end{aligned}$$

(81)

Finally, from the balance sheet condition in the steady state we have: $\frac{1}{N}\left( w\left( k_{t+1}^{*};N\right) +\lambda \tau ^{*}\right) =m^{*}+k^{*}$. From (81):

$$\begin{aligned} m^{*}=\frac{\frac{1}{N}\left( w\left( k_{t+1}^{*};N\right) +\lambda \tau ^{*}\right) }{1+\left( 1-\pi \right) \left[ \frac{1}{\pi }+\left( 1-\lambda \right) \left( \sigma -1\right) \right] \left[ 1-\frac{\left( 1-\rho -\alpha \right) }{N}\right] ^{\frac{1}{\theta }}I^{*\frac{ 1-\theta }{\theta }}-\frac{\left( 1-\pi \right) \left( 1-\lambda \right) \left( \sigma -1\right) }{I^{*}}} \end{aligned}$$

which is identical to that in (37) when the self-selection constraint does not bind.

Next, using (34), (81), and the expression for transfers:

$$\begin{aligned} \frac{K^{*}}{w\left( k^{*};N\right) }=\frac{1}{1+\left[ 1-\lambda \frac{\sigma -1}{\sigma }\right] \frac{1}{\left( 1-\pi \right) \left[ \frac{1 }{\pi }+\left( 1-\lambda \right) \left( \sigma -1\right) \right] \left[ 1- \frac{\left( 1-\rho -\alpha \right) }{N}\right] ^{\frac{1}{\theta }}I^{*\frac{1-\theta }{\theta }}-\frac{\left( 1-\pi \right) \left( 1-\lambda \right) \left( \sigma -1\right) }{I^{*}}}} \end{aligned}$$

where $\frac{K^{*}}{w\left( k^{*};N\right) }=\frac{K^{*1-\alpha -\rho }}{\left( 1-\alpha \right) A}$. Furthermore, from (35) and (36), the relative return to depositors is such that:

$$\begin{aligned} \frac{r_{t}^{n^{*}}}{r_{t}^{m^{*}}}=\frac{\frac{N}{1-\pi }r\left( k_{t+1}^{*};N\right) k_{t+1}^{*}}{\frac{N}{\pi }m_{t}^{*}\frac{ P_{t}}{P_{t+1}}} \end{aligned}$$

Imposing steady state:

$$\begin{aligned} \frac{r^{n^{*}}}{r^{m^{*}}}=\frac{I^{*}K^{*}}{\frac{1-\pi }{ \pi }Nm^{*}} \end{aligned}$$

As a final step, we substitute for $Nm^{*}$ from (81):

$$\begin{aligned} \frac{r^{n^{*}}}{r^{m^{*}}}=\left[ 1+\pi \left( 1-\lambda \right) \left( \sigma -1\right) \right] \left[ 1-\frac{\left( 1-\rho -\alpha \right) }{N}\right] ^{\frac{1}{\theta }}I^{*\frac{1}{\theta }}-\pi \left( 1-\lambda \right) \left( \sigma -1\right) \end{aligned}$$

which is the same expression obtained in (38). Simple examination of (38) indicates that $\frac{ r^{n^{*}}}{r^{m^{*}}}>1$ if $I^{*}>\left[ 1-\frac{\left( 1-\rho -\alpha \right) }{N}\right] ^{-1}=\underline{I}$.

We proceed to find the expression for the expected utility of depositors when the incentive compatibility constraint is non-binding, $U^{*}$. It is easy to verify that (33) in the steady state can be written as:

$$\begin{aligned} U^{*}=\frac{\pi +\left( 1-\pi \right) \left[ \frac{c^{n^{*}}}{ c^{m^{*}}}\right] ^{1-\theta }}{1-\theta }\left[ c^{m^{*}}\right] ^{1-\theta } \end{aligned}$$

(82)

Substituting for $\frac{c^{n^{*}}}{c^{m^{*}}}$ from (39):

$$\begin{aligned} U^{*}=\frac{\pi +\left( 1-\pi \right) \left[ \left[ 1-\frac{\left( 1-\rho -\alpha \right) }{N}\right] I^{*}\right] ^{\frac{1-\theta }{ \theta }}}{1-\theta }\left[ c^{m^{*}}\right] ^{1-\theta } \end{aligned}$$

(83)

where $c^{m^{*}}=\frac{N}{\pi }m^{*}\frac{1}{\sigma }+\left( 1-\lambda \right) \tau$. Upon using the definition for $\tau$,

$$\begin{aligned} c^{m^{*}}=\left( \frac{1}{\pi }+\left( 1-\lambda \right) \left( \sigma -1\right) \right) \frac{Nm^{*}}{\sigma } \end{aligned}$$

As a final step, using (81), $c^{m^{*}}$ is such that:

$$\begin{aligned} c^{m^{*}}=\frac{\left( \frac{1}{\pi }+\left( 1-\lambda \right) \left( \sigma -1\right) \right) K^{*}}{\left( 1-\pi \right) \left[ \frac{1}{\pi }+\left( 1-\lambda \right) \left( \sigma -1\right) \right] \left[ 1-\frac{ \left( 1-\rho -\alpha \right) }{N}\right] ^{\frac{1}{\theta }}I^{*\frac{ 1-\theta }{\theta }}-\frac{\left( 1-\pi \right) \left( 1-\lambda \right) \left( \sigma -1\right) }{I^{*}}}\frac{1}{\sigma } \end{aligned}$$

(84)

Direct substitution of (84) into (83) yields (43) in the text.

Now suppose the incentive compatibility constraint binds above the Friedman rule level. That is, $1<I^{*}\le \underline{I}$. From (35) and (36), in the steady state, the relative return to depositors is such that:

$$\begin{aligned} \frac{r^{n^{*}}}{r^{m^{*}}}=\frac{I^{*}K^{*}}{\frac{1-\pi }{ \pi }Nm^{*}}=1 \end{aligned}$$

which implies that: $K^{*}=\frac{1-\pi }{\pi }\frac{1}{I^{*}} Nm^{*}$. Using bank’s balance sheet condition, (34):

$$\begin{aligned} K^{*}=\left( w\left( k^{*};N\right) +\lambda \tau ^{*}\right) -Nm^{*}=\frac{1-\pi }{\pi }\frac{1}{I^{*}}Nm^{*} \end{aligned}$$

Simple algebra implies that:

$$\begin{aligned} m^{_{*}}=\frac{\frac{1}{N}\left( w\left( k^{*};N\right) +\lambda \tau ^{*}\right) }{\left( 1+\frac{1-\pi }{\pi }\frac{1}{I^{*}} \right) } \end{aligned}$$

which is the expression obtained in (37).

Using the expression for transfers in the equation for money balances:

$$\begin{aligned} Nm^{*}=\frac{w\left( k^{*};N\right) }{1+\frac{1-\pi }{\pi }\frac{1}{ I^{*}}-\lambda \frac{\sigma -1}{\sigma }} \end{aligned}$$

(85)

Finally, from (2) and (34) we have:

$$\begin{aligned} Nm^{*}=\frac{w\left( k^{*};N\right) -K^{*}}{1-\lambda \frac{ \sigma -1}{\sigma }} \end{aligned}$$

(86)

From (85) and (86), we obtain:

$$\begin{aligned} \frac{K^{*}}{w\left( k^{*};N\right) }=\frac{\frac{1-\pi }{\pi }\frac{ 1}{I^{*}}}{1+\frac{1-\pi }{\pi }\frac{1}{I^{*}}-\lambda \frac{\sigma -1}{\sigma }} \end{aligned}$$

(87)

which is identical to (40) in the text (after some simplification).

Plugging the definitions of $w\left( k^{*};N\right)$ and $I^{*}$, where $I^{*}=\frac{\sigma \alpha A}{K^{*1-\alpha -\rho }}$ into (87), it is easy to verify that:

$$\begin{aligned} K^{*1-\alpha -\rho }=\left( 1-\alpha \right) A-\frac{\pi \alpha A}{1-\pi }\left[ \sigma \left( 1-\lambda \right) +\lambda \right] \end{aligned}$$

and

$$\begin{aligned} I^{*}=\frac{\sigma }{\frac{1-\alpha }{\alpha }-\frac{\pi }{1-\pi }\left[ \sigma \left( 1-\lambda \right) +\lambda \right] } \end{aligned}$$

which are, respectively, Eqs. (41) and (42) in the text. From the expression for I, it can be established that $I^{*}>1$ if $\sigma >\frac{\frac{1-\alpha }{\alpha }-\lambda \frac{\pi }{1-\pi }}{1+\frac{ \pi }{1-\pi }\left( 1-\lambda \right) }=\underline{\sigma }_{1}^{\prime }$.

In addition, $I^{*}\ge \underline{I}$ if

$$\begin{aligned} \frac{\sigma }{\frac{1-\alpha }{\alpha }-\frac{\pi }{1-\pi }\left[ \sigma \left( 1-\lambda \right) +\lambda \right] }\ge \frac{1}{\left[ 1-\frac{ \left( 1-\rho -\alpha \right) }{N}\right] } \end{aligned}$$

which can be written as:

$$\begin{aligned} \sigma \ge \frac{\frac{1-\pi }{\pi }\frac{1-\alpha }{\alpha }-\lambda }{ 1-\lambda +\frac{1-\pi }{\pi }\left[ 1-\frac{\left( 1-\alpha -\rho \right) }{ N}\right] }=\underline{\sigma }_{2}^{\prime } \end{aligned}$$

In this manner, $I^{*}>\underline{I}$ and $\frac{r^{n^{*}}}{ r^{m^{*}}}>1$ if $\sigma >\underline{\sigma }_{2}^{\prime }$. Moreover, $I^{*}\in \left[ 1,\underline{I}\right]$ and $\frac{r^{n^{*}}}{ r^{m^{*}}}=1$ if $\sigma \in \left[ \underline{\sigma }_{1}^{\prime }, \underline{\sigma }_{2}^{\prime } \right]$. Finally, $I^{*}=1$ for all $\sigma \le \underline{\sigma }_{1}^{\prime }$.

We proceed to derive an expression for the welfare of depositors, (43). As agents receive complete risk-sharing, $r^{n^{*}}=r^{m^{*}}$ and the same amount of transfers when old, it directly follows that $c^{m^{*}}=c^{n^{*}}=c^{*}$. Substituting this information into (82):

$$\begin{aligned} U^{*}=\frac{1}{1-\theta }\left[ c^{m^{*}}\right] ^{1-\theta } \end{aligned}$$

where from the work above:

$$\begin{aligned} c^{m^{*}}=\left( \frac{1}{\pi }+\left( 1-\lambda \right) \left( \sigma -1\right) \right) \frac{Nm^{*}}{\sigma } \end{aligned}$$

and $m^{*}$ is given by (85). Using (3), (41), and (42) in (85):

$$\begin{aligned} \frac{Nm^{*}}{\sigma }=\frac{\pi \alpha A}{1-\pi }\left[ \left( 1-\alpha \right) A-\frac{\pi \alpha A}{1-\pi }\left[ \sigma \left( 1-\lambda \right) +\lambda \right] \right] ^{\frac{\alpha +\rho }{1-\alpha -\rho }} \end{aligned}$$

Upon substituting into $c^{m^{*}}$ and $U^{*}$, we get (43) in the text.

Now suppose $\sigma \le \underline{\sigma }_{1}^{\prime }$, where $I^{*}=1$. Using the definition of $I^{*}$, in the steady state we have: $K^{*}=\left( \alpha A\sigma \right) ^{\frac{1}{1-\alpha -\rho }}$. As under the benchmark model, the constraints on payments to movers and non-movers are such that:

$$\begin{aligned} \pi \frac{1}{N}r_{t}^{m}\left( w_{t}+\lambda \tau _{t}\right) =\delta _{t}m_{t}\frac{P_{t}}{P_{t+1}} \end{aligned}$$

(88)

and

$$\begin{aligned} \frac{1-\pi }{N}r_{t}^{n}\left( w_{t}+\lambda \tau _{t}\right) =r\left( k_{t+1},K_{t+1}^{N-1}\right) k_{t+1}+\left( 1-\delta _{t}\right) m_{t}\frac{ P_{t}}{P_{t+1}} \end{aligned}$$

(89)

Imposing steady state and symmetry on (88) and (89), complete risk-sharing implies that:

$$\begin{aligned} \delta ^{*}=\pi \left( \frac{K^{*}}{Nm^{*}}+1\right) \end{aligned}$$

Substituting (86) into the expression for $\delta ^{*}$, it is easily verified that:

$$\begin{aligned} \delta ^{*}=\pi \left( \frac{1-\lambda \frac{\sigma -1}{\sigma }\frac{ K^{*}}{w\left( k^{*};N\right) }}{1-\frac{K^{*}}{w\left( k^{*};N\right) }}\right) \end{aligned}$$

(90)

where $\delta ^{*}<1$ if:

$$\begin{aligned} \pi \left( \frac{1-\lambda \frac{\sigma -1}{\sigma }\frac{K^{*}}{w\left( k^{*};N\right) }}{1-\frac{K^{*}}{w\left( k^{*};N\right) }} \right) <1 \end{aligned}$$

where from the work above, at the Friedman rule we have:$\frac{K^{*}}{ w\left( k^{*};N\right) }=\frac{K^{*1-\alpha -\rho }}{\left( 1-\alpha \right) A}=\frac{\alpha }{\left( 1-\alpha \right) r}=\frac{\alpha \sigma }{ \left( 1-\alpha \right) I}=\frac{\alpha \sigma }{\left( 1-\alpha \right) }$. Using this information, it is easy to verify that $\delta ^{*}<1$ if: $\sigma <\frac{\frac{1-\alpha }{\alpha }-\lambda \frac{\pi }{1-\pi }}{1+\frac{ \pi }{1-\pi }\left( 1-\lambda \right) }=\underline{\sigma }_{1}^{\prime }$.

Next, from (86) and (90) we have:

$$\begin{aligned} \delta ^{*}Nm^{*}=\pi \left( \frac{1-\lambda \frac{\sigma -1}{\sigma }\frac{K^{*}}{w\left( k^{*};N\right) }}{\left( 1-\lambda \frac{ \sigma -1}{\sigma }\right) }\right) w\left( k^{*};N\right) \end{aligned}$$

(91)

Using the definitions of $I^{*}$ and $w\left( k^{*};N\right)$, along with the expression for $K^{*}=\left( \alpha A\sigma \right) ^{ \frac{1}{1-\alpha -\rho }}$ into (91) to get:

$$\begin{aligned} \delta ^{*}Nm^{*}=\pi \left( \frac{1-\lambda \frac{\sigma -1}{\sigma }\frac{\alpha \sigma }{\left( 1-\alpha \right) }}{\left( 1-\lambda \frac{ \sigma -1}{\sigma }\right) }\right) \left( 1-\alpha \right) A\left( \alpha A\sigma \right) ^{\frac{\alpha +\rho }{1-\alpha -\rho }} \end{aligned}$$

(92)

In order to derive an expression for the welfare of depositors, we solve for the consumption of a typical depositor which is expressed as:

$$\begin{aligned} c^{*}=r^{m}\left( w\left( k^{*};N\right) +\lambda \tau ^{*}\right) +\left( 1-\lambda \right) \frac{\sigma -1}{\sigma }Nm^{*} \end{aligned}$$

where

$$\begin{aligned} r^{m^{*}}\left( w\left( k^{*};N\right) +\lambda \tau ^{*}\right)= & {} \frac{\delta ^{*}Nm^{*}}{\pi }\frac{1}{\sigma } \\= & {} \left( \frac{ 1-\lambda \frac{\sigma -1}{\sigma }\frac{\alpha \sigma }{\left( 1-\alpha \right) }}{\left( 1-\lambda \frac{\sigma -1}{\sigma }\right) }\right) \left( 1-\alpha \right) A\left( \alpha A\sigma \right) ^{\frac{\alpha +\rho }{ 1-\alpha -\rho }}\frac{1}{\sigma } \end{aligned}$$

and from (86):

$$\begin{aligned} Nm^{*}=\frac{w\left( k^{*};N\right) -K^{*}}{\left( 1-\lambda \frac{\sigma -1}{\sigma }\right) } \end{aligned}$$

Using this information,

$$\begin{aligned} \begin{aligned} c^{*}&=\left( 1-\alpha \right) A\left( \alpha A\right) ^{\frac{\alpha +\rho }{1-\alpha -\rho }} \\&\quad \left[ \left( 1-\lambda \left( \sigma -1\right) \frac{\alpha }{\left( 1-\alpha \right) }\right) \frac{1}{\sigma }+\left( 1-\lambda \right) \frac{\left( \sigma -1\right) }{\sigma }\left( 1-\frac{ \alpha \sigma }{\left( 1-\alpha \right) }\right) \right] \\&\quad \frac{\sigma ^{ \frac{\alpha +\rho }{1-\alpha -\rho }}}{\left( 1-\lambda \frac{\sigma -1}{ \sigma }\right) } \end{aligned} \end{aligned}$$

which yields (43) when substituting into $U^{*}=\frac{1}{1-\theta }\left[ c^{m^{*}}\right] ^{1-\theta }$. This completes the derivation.

10. Proof of Proposition 8. From our derivations above, it is clear that a steady-state equilibrium where both money and capital are held exists when $\sigma \in \left( 0,\underline{\sigma }_{2}^{\prime }\right]$. We proceed to show that a unique solution exists if $\sigma >\underline{\sigma }_{2}^{\prime }$. That is, we proceed to show that (27) and (40) intersect once at $I^{*}>\underline{I}$. We begin by rewriting the expression for the supply of capital over that range of $I^{*}$ as:

$$\begin{aligned} \Omega \left( K^{*}\right) =\frac{1}{1+\left[ 1-\lambda \frac{\sigma -1}{ \sigma }\right] \frac{1}{\left( 1-\pi \right) \left[ \frac{1}{\pi }+\left( 1-\lambda \right) \left( \sigma -1\right) \right] \left[ 1-\frac{\left( 1-\rho -\alpha \right) }{N}\right] ^{\frac{1}{\theta }}I^{*\frac{ 1-\theta }{\theta }}-\frac{\left( 1-\pi \right) \left( 1-\lambda \right) \left( \sigma -1\right) }{I^{*}}}} \end{aligned}$$

(93)

where $\Omega \left( K^{*}\right) =\frac{K^{*}}{w\left( k^{*};N\right) }=\frac{K^{*1-\alpha -\rho }}{\left( 1-\alpha \right) A}$. In addition, recalling that : $I^{*}=\sigma \alpha AK^{*\rho +\alpha -1}$ from (27), $\Omega \left( K^{*}\right)$ can be rewritten as:

$$\begin{aligned} \Omega \left( K^{*}\right) =\sigma \frac{\alpha }{\left( 1-\alpha \right) }\frac{1}{I^{*}} \end{aligned}$$

(94)

which behaves as described in the text.

Using (27) and (40), the following polynomial yields the solution for $I^{*}$

$$\begin{aligned} \begin{aligned}&\frac{1}{1+\left[ 1-\lambda \frac{\sigma -1}{\sigma }\right] \frac{1}{\left( 1-\pi \right) \left[ \frac{1}{\pi }+\left( 1-\lambda \right) \left( \sigma -1\right) \right] \left[ 1-\frac{\left( 1-\rho -\alpha \right) }{N}\right] ^{ \frac{1}{\theta }}I^{*\frac{1-\theta }{\theta }}-\frac{\left( 1-\pi \right) \left( 1-\lambda \right) \left( \sigma -1\right) }{I^{*}}}} \\&\quad =\sigma \frac{\alpha }{\left( 1-\alpha \right) }\frac{1}{I^{*}} \end{aligned} \end{aligned}$$

which can be rewritten as:

$$\begin{aligned} \begin{aligned} \zeta \left( I^{*}\right)&=\frac{\alpha }{\left( 1-\alpha \right) }\frac{ 1}{I^{*}} \\&\quad \left[ \sigma +\frac{1}{\left( \frac{1}{\left[ \sigma \left( 1-\lambda \right) +\lambda \right] }\frac{1}{\pi }+\frac{\left( 1-\lambda \right) \left( \sigma -1\right) }{\left[ \sigma \left( 1-\lambda \right) +\lambda \right] }\right) \left( 1-\pi \right) \left[ 1-\frac{\left( 1-\rho -\alpha \right) }{N}\right] ^{\frac{1}{\theta }}I^{*\frac{1-\theta }{ \theta }}-\frac{\left( 1-\pi \right) \left( 1-\lambda \right) \left( \sigma -1\right) }{I^{*}\left[ \sigma \left( 1-\lambda \right) +\lambda \right] }}\right] \\&=1 \end{aligned} \end{aligned}$$

(95)

It is trivial to show that $\zeta ^{\prime }\left( I^{*}\right) <0$ and $\lim \limits _{I^{*}\rightarrow \infty } \zeta \rightarrow 0$. In this manner, a unique solution exists if $\zeta \left( \underline{I}\right) \ge 1$. Upon using the definition of $\underline{I}$ and some algebra, $\zeta \left( \underline{I}\right) \ge 1$ if $\sigma \ge \frac{\frac{1-\pi }{\pi } \frac{1-\alpha }{\alpha }-\lambda }{1-\lambda +\frac{1-\pi }{\pi }\left[ 1- \frac{\left( 1-\alpha -\rho \right) }{N}\right] }=\underline{\sigma } _{2}^{\prime }$. From the solution to the problem above, a unique steady-state equilibrium where both money and capital are held exists for all $\sigma >0$. This completes the proof of Proposition 8.

11. Proof of Proposition 9. To begin, suppose $\sigma \in \left( 0, \underline{\sigma }_{1}^{\prime }\right]$, where $\frac{c^{n^{*}}}{ c^{m^{*}}}=1$ and $I^{*}=1$. Over this range of money growth, $K^{*}=\left( \alpha A\sigma \right) ^{\frac{1}{1-\alpha -\rho }}$, which is independent of $\lambda$. Since $I^{*}=\sigma \alpha AK^{*\rho +\alpha -1}$, then we also have $\frac{\mathrm{d}I^{*}}{\mathrm{d}\lambda }=0$. Clearly, $\frac{\mathrm{d}\frac{c^{n^{*}}}{c^{m^{*}}}}{\mathrm{d}\lambda }=0$. Now, suppose $\sigma \in \left( \underline{\sigma }_{1}^{\prime },\underline{\sigma } _{2}^{\prime }\right]$, where $\frac{c^{n^{*}}}{c^{m^{*}}}=1$ and $I^{*}>1$. From the work above, the capital is stock is given by (41) which can be rewritten as:

$$\begin{aligned} K^{*1-\alpha -\rho }=\left( 1-\alpha \right) A-\frac{\pi \alpha A}{1-\pi }\left[ \sigma -\lambda \left( \sigma -1\right) \right] \end{aligned}$$

It is clear that $\frac{\mathrm{d}K^{*}}{\mathrm{d}\lambda }\ge \left( <\right) 0$ if $\sigma \ge \left( <\right) 1$. The impact of a change in $\lambda$ on $I^{*}$ directly follows. Moreover, $\frac{\mathrm{d}\frac{c^{n^{*}}}{ c^{m^{*}}}}{\mathrm{d}\lambda }=0$ as $\frac{c^{n^{*}}}{c^{m^{*}}}=1$ over this range of $\sigma$.

Now suppose $\sigma >\underline{\sigma }_{2}^{\prime }$. The equilibrium values of $K^{*}$ and $I^{*}$ are obtained from system (27) and (40). First, capital demand function, (27), is as described in the text. In particular, $\frac{\mathrm{d}I^{*}}{\mathrm{d}K^{*}}<0$ and $\lim \limits _{I\rightarrow \infty } K\rightarrow 0$. On the supply side, as $\theta <1$, it is clear that (93) is such that: $\lim \limits _{I^{*}\rightarrow \infty }$$K^{*}\rightarrow \Omega ^{-1}\left( 1\right)$ and since $\Omega ^{\prime }\left( K^{*}\right) >0$, $\frac{\mathrm{d}I^{*}}{\mathrm{d}K^{*}}>0$ as long as $\sigma \ge 1$. We proceed to find a sufficient condition on $\sigma :\frac{\mathrm{d}I^{*}}{\mathrm{d}K^{*}}>0$ for $\sigma <1$.

Define $\Delta$:

$$\begin{aligned} \Delta \left( I^{*}\right) =\frac{\left[ 1-\lambda \frac{\sigma -1}{ \sigma }\right] }{\left( 1-\pi \right) \left[ \frac{1}{\pi }+\left( 1-\lambda \right) \left( \sigma -1\right) \right] \left[ 1-\frac{\left( 1-\rho -\alpha \right) }{N}\right] ^{\frac{1}{\theta }}I^{*\frac{ 1-\theta }{\theta }}-\frac{\left( 1-\pi \right) \left( 1-\lambda \right) \left( \sigma -1\right) }{I^{*}}} \end{aligned}$$

where $\left[ 1-\lambda \frac{\sigma -1}{\sigma }\right] >0$ for all $\sigma >0$ and $\lambda \in \left[ 0,1\right]$. Capital supply function, (93), can be written as:

$$\begin{aligned} \Omega \left( K^{*}\right) =\frac{1}{1+\Delta \left( I^{*}\right) } \end{aligned}$$

Clearly, $\frac{\mathrm{d}I^{*}}{\mathrm{d}K^{*}}>0$ if $\Delta ^{\prime }\left( I^{*}\right) <0$.

$$\begin{aligned} \begin{aligned}&\frac{1}{\left[ 1-\lambda \frac{\sigma -1}{\sigma }\right] }\Delta ^{\prime }\left( I^{*}\right) \\&\quad =-\frac{\frac{1-\theta }{\theta }\left( 1-\pi \right) \left[ \frac{1}{\pi }+\left( 1-\lambda \right) \left( \sigma -1\right) \right] \left[ 1-\frac{\left( 1-\rho -\alpha \right) }{N}\right] ^{ \frac{1}{\theta }}I^{*\frac{1-\theta }{\theta }-1}+\frac{\left( 1-\pi \right) \left( 1-\lambda \right) \left( \sigma -1\right) }{I^{*2}}}{ \left[ \left( 1-\pi \right) \left[ \frac{1}{\pi }+\left( 1-\lambda \right) \left( \sigma -1\right) \right] \left[ 1-\frac{\left( 1-\rho -\alpha \right) }{N}\right] ^{\frac{1}{\theta }}I^{*\frac{1-\theta }{\theta }}-\frac{ \left( 1-\pi \right) \left( 1-\lambda \right) \left( \sigma -1\right) }{ I^{*}}\right] ^{2}} \end{aligned} \end{aligned}$$

In this manner, $\Delta ^{\prime }\left( I^{*}\right) <0$ if the term in the numerator is positive. With some algebra, $\Delta ^{\prime }\left( I^{*}\right) <0$ if:

$$\begin{aligned} \frac{1-\theta }{\theta }\left[ \frac{1}{\pi }+\left( 1-\lambda \right) \left( \sigma -1\right) \right] \left[ 1-\frac{\left( 1-\rho -\alpha \right) }{N}\right] ^{\frac{1}{\theta }}I^{*\frac{1}{\theta }}+\left( 1-\lambda \right) \left( \sigma -1\right) >0 \end{aligned}$$

which always holds if $\sigma \ge 1$ since $N>1$. Now suppose $\sigma <1$, rewriting the inequality as:

$$\begin{aligned} \left[ 1-\frac{\left( 1-\rho -\alpha \right) }{N}\right] ^{\frac{1}{\theta } }I^{*\frac{1}{\theta }}>\frac{\left( 1-\lambda \right) \left( 1-\sigma \right) }{\left( \frac{1}{\pi }-\left( 1-\lambda \right) \left( 1-\sigma \right) \right) }\frac{\theta }{1-\theta } \end{aligned}$$

Since $\left[ 1-\frac{\left( 1-\rho -\alpha \right) }{N}\right] ^{\frac{1}{ \theta }}I^{*\frac{1}{\theta }}>1$ when the incentive compatibility constraint is non-binding, it is sufficient to find a condition such that: $\frac{\left( 1-\lambda \right) \left( 1-\sigma \right) }{\left( \frac{1}{\pi }-\left( 1-\lambda \right) \left( 1-\sigma \right) \right) }\frac{\theta }{ 1-\theta }<1$$\Longleftrightarrow \sigma >1-\frac{\left( 1-\theta \right) }{ \left( 1-\lambda \right) }\frac{1}{\pi }=\underline{\sigma }_{0}^{\prime }$. Under this condition, $\Delta ^{\prime }\left( I^{*}\right) <0$ and (93): $\frac{\mathrm{d}I^{*}}{\mathrm{d}K^{*}}>0$. Notice that $\underline{\sigma }_{2}^{\prime }>\underline{\sigma }_{0}^{\prime }$ if:

$$\begin{aligned} \frac{\frac{1-\pi }{\pi }\frac{1-\alpha }{\alpha }-\lambda }{1-\lambda + \frac{1-\pi }{\pi }\left[ 1-\frac{\left( 1-\alpha -\rho \right) }{N}\right] } >1-\frac{\left( 1-\theta \right) }{\left( 1-\lambda \right) }\frac{1}{\pi } \end{aligned}$$

which can be written as:

$$\begin{aligned} \theta <1-\pi \left( 1-\lambda \right) \left( \frac{1+\frac{1-\pi }{\pi } \left[ 1-\frac{\left( 1-\alpha -\rho \right) }{N}\right] -\frac{1-\pi }{\pi } \frac{1-\alpha }{\alpha }}{1-\lambda +\frac{1-\pi }{\pi }\left[ 1-\frac{ \left( 1-\alpha -\rho \right) }{N}\right] }\right) =\tilde{\theta }_{1} \end{aligned}$$

Now suppose $\sigma >\max \left( \underline{\sigma }_{0}^{\prime }, \underline{\sigma }_{2}^{\prime }\right)$, over which the capital supply function is upward sloping. The supply of capital is given by (93):

$$\begin{aligned} \Omega \left( K^{*}\right) =\frac{1}{1+\Delta \left( I^{*},\lambda \right) } \end{aligned}$$

where $\Delta \left( I^{*},\lambda \right)$ is defined above and can be rewritten as:

$$\begin{aligned} \begin{aligned}&\Delta \left( I^{*},\lambda \right) \\&\quad =\frac{\left[ 1-\lambda \frac{\sigma -1}{\sigma }\right] }{\frac{\left( 1-\pi \right) }{\pi }\left[ 1-\frac{ \left( 1-\rho -\alpha \right) }{N}\right] ^{\frac{1}{\theta }}I^{*\frac{ 1-\theta }{\theta }}+\left( \left[ 1-\frac{\left( 1-\rho -\alpha \right) }{N} \right] ^{\frac{1}{\theta }}I^{*\frac{1}{\theta }}-1\right) \frac{\left( 1-\pi \right) \left( 1-\lambda \right) \left( \sigma -1\right) }{I^{*}}} \end{aligned} \end{aligned}$$

For a given $I^{*}$,

$$\begin{aligned} \frac{\partial \Delta }{\partial \lambda }=\frac{ \begin{array}{c} -\frac{\sigma -1}{\sigma }\left\{ \frac{\left( 1-\pi \right) }{\pi }\left[ 1-\frac{\left( 1-\rho -\alpha \right) }{N}\right] ^{\frac{1}{\theta } }I^{*\frac{1-\theta }{\theta }}+\frac{\left( 1-\pi \right) \left( 1-\lambda \right) \left( \sigma -1\right) }{I^{*}}\left( \left[ 1-\frac{ \left( 1-\rho -\alpha \right) }{N}\right] ^{\frac{1}{\theta }}I^{*\frac{1 }{\theta }}-1\right) \right\} + \\ \frac{\left( 1-\pi \right) \left( \sigma -1\right) }{I^{*}}\left( \left[ 1-\frac{\left( 1-\rho -\alpha \right) }{N}\right] ^{\frac{1}{\theta } }I^{*\frac{1}{\theta }}-1\right) \left[ 1-\lambda \frac{\sigma -1}{ \sigma }\right] \end{array} }{\left[ \frac{\left( 1-\pi \right) }{\pi }\left[ 1-\frac{\left( 1-\rho -\alpha \right) }{N}\right] ^{\frac{1}{\theta }}I^{*\frac{1-\theta }{ \theta }}+\frac{\left( 1-\pi \right) \left( 1-\lambda \right) \left( \sigma -1\right) }{I^{*}}\left( \left[ 1-\frac{\left( 1-\rho -\alpha \right) }{N }\right] ^{\frac{1}{\theta }}I^{*\frac{1}{\theta }}-1\right) \right] ^{2} } \end{aligned}$$

In an (I, K) plane, capital supply locus, (93), shifts to the right for a given I if $\frac{\partial \Delta }{\partial \lambda }<0$. Given that the capital supply function is upward sloping, the equilibrium capital stock is higher if $\frac{\partial \Delta }{\partial \lambda }<0$. Suppose $\sigma >1$. With a few lines of algebra, $\frac{ \partial \Delta }{\partial \lambda }<0$ if:

$$\begin{aligned} 0<\frac{1-\pi }{\pi }\left[ 1-\frac{\left( 1-\rho -\alpha \right) }{N}\right] ^{\frac{1}{\theta }}I^{*\frac{1-\theta }{\theta }}+1 \end{aligned}$$

which always holds. Now, suppose $\sigma <1$. It is easy to verify that $\frac{\partial \Delta }{\partial \lambda }<0$ if $0>\frac{1-\pi }{\pi }\left[ 1-\frac{\left( 1-\rho -\alpha \right) }{N}\right] ^{\frac{1}{\theta } }I^{*\frac{1}{\theta }}+1$, which never holds. Therefore, $\frac{ \partial \Delta }{\partial \lambda }>0$ if $\sigma <1$. The result in the Proposition directly follows. This completes the proof of Proposition 9.

12. Proof of Proposition 10. The effect of monetary policy on capital formation when the incentive compatibility constraint is binding is easily observed from the explicit solutions for $K^{*}$ and $I^{*}$. We proceed to show that $\frac{\mathrm{d}K^{*}}{\mathrm{d}\sigma }>0$ if $\sigma \ge \max \left( \underline{\sigma }_{0}^{\prime },\underline{\sigma }_{2}^{\prime }\right)$.

To begin, it is clear that the demand for capital is higher under higher values of $\sigma$. Therefore, in an I, K plane, locus (27) shifts to the right. On the supply side, from (93), it is trivial to show that $\frac{\partial \Delta }{\partial \sigma }<0$ for a given $I^{*}$. Therefore, (93) locus shifts to the right. In this manner, unambiguously, $\frac{ \mathrm{d}K^{*}}{\mathrm{d}\sigma }>0$. The impact of $\sigma$ on $I^{*}$ is less straightforward. We proceed to show that $\frac{\mathrm{d}I^{*}}{\mathrm{d}\sigma }>0$.

Recall from our characterization of the polynomial yielding the solution for $I^{*}$, (95)

$$\begin{aligned} \begin{aligned} \zeta \left( I^{*}\right)&=\frac{\alpha }{\left( 1-\alpha \right) }\frac{ 1}{I^{*}}\\&\quad \left[ \sigma +\frac{1}{\left( \frac{1}{\left[ \sigma \left( 1-\lambda \right) +\lambda \right] }\frac{1}{\pi }+\frac{\left( 1-\lambda \right) \left( \sigma -1\right) }{\left[ \sigma \left( 1-\lambda \right) +\lambda \right] }\right) \left( 1-\pi \right) \left[ 1-\frac{\left( 1-\rho -\alpha \right) }{N}\right] ^{\frac{1}{\theta }}I^{*\frac{1-\theta }{ \theta }}-\frac{\left( 1-\pi \right) \left( 1-\lambda \right) \left( \sigma -1\right) }{I^{*}\left[ \sigma \left( 1-\lambda \right) +\lambda \right] }}\right] \\&=1 \end{aligned} \end{aligned}$$

(96)

We have $\zeta ^{\prime }\left( I^{*}\right) <0$. In this manner, $\frac{ \mathrm{d}I^{*}}{\mathrm{d}\sigma }>0$ if $\frac{\partial \zeta }{\partial \sigma }>0$. Define $Z_{1}=\frac{1}{\left[ \sigma \left( 1-\lambda \right) +\lambda \right] }\frac{1}{\pi }+\frac{\left( 1-\lambda \right) \left( \sigma -1\right) }{\left[ \sigma \left( 1-\lambda \right) +\lambda \right] }$ and $Z_{2}=\frac{\left( 1-\pi \right) \left( 1-\lambda \right) \left( \sigma -1\right) }{I^{*}\left[ \sigma \left( 1-\lambda \right) +\lambda \right] }$, where

$$\begin{aligned} \frac{\partial Z_{1}}{\partial \sigma }=-\frac{\left( 1-\lambda \right) }{ \left[ \sigma \left( 1-\lambda \right) +\lambda \right] ^{2}}\frac{1-\pi }{ \pi } \end{aligned}$$

and

$$\begin{aligned} \frac{\partial Z_{2}}{\partial \sigma }=\frac{\left( 1-\pi \right) \left( 1-\lambda \right) }{I^{*}}\frac{1}{\left[ \sigma \left( 1-\lambda \right) +\lambda \right] ^{2}} \end{aligned}$$

Obviously, $\frac{\partial Z_{1}}{\partial \sigma }<0$ if $\lambda \in [0,1)$. Similarly, $\frac{\partial Z_{2}}{\partial \sigma }>0$. This clearly shows that $\frac{\partial \zeta }{\partial \sigma }>0$ and $\frac{ \mathrm{d}I^{*}}{\mathrm{d}\sigma }>0$ if $\lambda \in [0,1)$. It is also obvious that $\frac{\mathrm{d}I^{*}}{\mathrm{d}\sigma }>0$ if $\lambda =1$ by visual inspection of Eq. (96). This completes the proof of Proposition 10.