\section{Introduction}

After the Great Recession, many economists have been trying to better understand the substantial changes in the conduct of monetary policy. Most of the literature focuses on the independence of the quantity of reserves from interest rate management in an ample-reserves regime (e.g., 
\citealp*{keister2008divorcing};
\citealp*{curdia2011central};
\citealp*{bech2011mechanics};
\citealp*{kashyap2012optimal}; \citealp*{cochrane2014monetary};    \citealp*{ennis2018simple}).\footnote{Many works on the central banks' large-scale asset purchases also could be included in this. Since the central banks purchase securities from financial intermediaries by paying with reserves, the independence of the quantity of reserve from interest rate management implies the independence of the amount of asset purchases from interest rate management and vice versa.} The rationale is that the interest rate paid on reserves provides a floor for the short-term interest rate that the central bank seeks to control. Then, as the central bank's target reaches the interest rate on reserves, paying interest on reserves “divorces” money from interest rate management, and the central bank can determine the amount of reserves independently of the interest rate. This implies market rates are equal to or higher than the interest rate on reserves.

\begin{figure}[b!]
\centering
\includegraphics[width=11cm,height=4.8cm]{figures_1/target_ranges.pdf}
\caption{US short-term Interest rates and the target range by the Federal Reserve}
\label{fig:multip2}
\end{figure}

However, as Figure \ref{fig:multip2} shows, since the Federal Reserve introduced the target interest rate range in December 2008 the interest on reserves has been higher than the federal funds rate and has been equal to the upper limit of the target range. Also, the short-term interest rate is closely correlated to the amount of reserves rather than independent. The left panel of Figure \ref{fig:motive4} plots the excess reserves ratio against the spread between the federal funds rate and the interest on reserves. It shows that the excess reserves ratio and the interest rate spread move in opposite direction. It suggests that, contrary to the previous work, the quantity of reserves is not independent of the central bank's interest rate management even in the  post-2008 period with ample reserves. In addition to that, the right panel of Figure \ref{fig:motive4} shows that the money multiplier moves together with the interest rate spread, which is just opposite to the excess reserves ratio. The time-series variation in the excess reserve ratio is directly reflected in the M1 money multiplier. This implies that the monetary policy in the post-2008 period is closely related to the banks' money creation activity,\footnote{As mentioned above, the central bank buys assets by paying with reserves. In the ample-reserves regime, the central bank's purchases are directly reflected in the central bank's liability with the same amount of excess reserves.} whereas many of monetary models did not pay attention to the money multiplier during last decades. These discrepancies call for a model that better captures for the transmission of monetary policy through the banking system. 



\begin{figure}[tp!]
\centering
\includegraphics[width=7cm,height=5.7cm]{figures_1/ior_excess.pdf}
\includegraphics[width=7cm,height=5.7cm]{figures_1/ior_multi.pdf}
\caption{US Excess reserves and M1 multiplier in the post-2008 period}
\label{fig:motive4}
\end{figure}
 
This paper revisits the issue of money creation to understand the role of banking in conducting monetary policy during the pre-2008 period as well as the post-2008 period. To guide the modeling, Section \ref{sec:motive} presents a series of empirical facts on money creation and money demand. In contrast to the textbook explanation of the money multiplier, there is no negative relationship between the required reserve ratio and the M1 money multiplier during the period of zero excess reserves. I identify two structural breaks in the money creation process since 1960: (1) one associated with consumer credit, and (2) one associated with interest on reserves. The first structural break in 1992 coincides with a structural break of the deposit component of money demand. A stable long-run money demand relationship, however, is recovered if one accounts for the impact of unsecured credit. The second structural break in 2008 coincides with the Fed's introduction of the interest on reserves. Banks have been holding excess reserves since then. 

These findings suggest that if one wants to study monetary transmission, a desirable monetary model should have the following properties. First, the model should feature the distinct role of the interest on reserves and the nominal interest rate.  Second, the model should be able to answer why banks are holding excess reserves now whereas they did not before 2008. Lastly, the model needs to capture the interaction between money and credit.

This paper builds a model based on \cite{lagos2005unified} and \cite{berentsen2007money} of money and banking to understand the monetary transmission under different regimes (e.g., ample-reserves regime, scarce-reserves regime) and identify the factors that changed the regimes from one to another. The bank's demand for reserves and inside money creation are endogenously determined.\footnote{See \cite{sargent1982real}, \cite{freeman1987reserve},
\cite{freeman1991inside},
\cite{haslag1998money}, and
\cite{freeman2000monetary} for the previous works on inside money creation. Recent works based on a search-theoretic framework include \cite{gu2013banking} and  \cite{andolfatto2020money}.} The model includes the explicit structure of monetary exchange and the role of financial intermediation. Agents can trade by using cash, claims on deposits (e.g., check or debit card), banknotes, and unsecured credit (e.g., credit card).\footnote{By modeling unsecured credit with exogenous credit limit, this paper follows \cite{gu2016money}. For other approaches to introducing credit to the monetary economy, see \cite{sanches2010money},  \cite{lotz2016money} and  \cite{williamson2016scarce}.}  The bank creates banknotes by making loans, and its lending is constrained by the reserve requirement. The equilibrium falls into one of the following three cases: a scarce-reserves equilibrium, an ample-reserves equilibrium, and a no-banking equilibrium.

In the scarce-reserves equilibrium, the nominal interest rate is sufficiently high. The bank's lending limit binds. If the central bank lowers the nominal interest rate, the reserve balance increases. The bank creates loans proportional to reserves, which implies the reserve requirement affects the money multiplier. If the central bank pays interest on reserves and sets the nominal interest rate at some moderate level, the bank holds excess reserves. We call this an ample-reserves equilibrium. In the ample-reserves equilibrium, the bank's lending limit does not bind. The reserve requirement does not change the money multiplier. Instead, the money multiplier is determined by the nominal interest rate and the interest on reserves. Lowering the nominal interest rate increases reserves, but the banks do not create banknotes proportionally, which lowers the money multiplier. A higher interest on reserves decreases the money multiplier since the bank has more incentive to hold reserve and less incentive to create banknotes. The interest on reserves and the nominal interest rate play distinct roles and they jointly determine the quantity of reserves. These are new findings compared with the literature. In both cases, better credit conditions reduce the real balance of inside money and reserves, but not cash, which results in a lower money multiplier. When the nominal interest rate is low enough, there is sufficient outside money to facilitate trade so buyers do not need to deposit their balances in the bank. The bank can not create inside money because it does not hold any reserves. This is called a no-banking equilibrium.



Next I quantify the model to determine the impact of the monetary policy and the introduction of consumer credit on reserves and the money multiplier. The model is parameterized to match pre-2008 US data. Quantitatively, the calibrated model can account for the behavior of money creation before and after 2008. The model-generated series can mimic the historical behavior of the M1 money multiplier, the excess reserves to deposit ratio, and the currency deposit ratio. The welfare analysis shows that lowering the reserve requirement or paying interest on reserves can reduce the welfare cost of inflation. Also, the quantitative analysis identifies the source of changes in the money multiplier and means of payment. The counterfactual analysis shows that the pre-2008 trend of a decreasing money multiplier is driven by an increase in unsecured credit, whereas the post-2008 trend of a decreasing money multiplier is not attributed to the increase in unsecured credit. From the model and data, I provide evidence that suggests the dramatic changes in the money multiplier after 2008 are mainly driven by the Federal Reserve's monetary policy: the introduction of the interest on reserves with low nominal interest rate.   \\

This paper is organized as follows. Section \ref{sec:motive} provides motivating evidence. Section \ref{sec:model} constructs the search-theoretic monetary model of money creation. Section \ref{sec:quant} calibrates the model to quantify the theory. Section \ref{sec:conclusion} concludes. 




\section{Motivating Evidence}
\label{sec:motive}

\begin{figure}[t]
\centering
\includegraphics[width=7cm,height=5.4cm]{figures_1/m1multi.pdf}
\includegraphics[width=7cm,height=5.4cm]{figures_1/cdr1.pdf}
\includegraphics[width=7cm,height=5.4cm]{figures_1/multireq.pdf}
\includegraphics[width=7cm,height=5.4cm]{figures_1/multicdr.pdf}
\caption{Money multiplier, currency/deposit ratio and required reserve ratio}
\label{fig:multip1}
\medskip
\begin{minipage}{0.97\textwidth}
{\footnotesize Chow tests for structural breaks are implemented. The bottom-left panel reports a test statistic with the null hypothesis of no structural breaks in 1992Q2 and 2008Q4 and the bottom-right panel reports a test statistic with the null hypothesis of no structural break in 2008Q4. Sample periods are 1959Q1-2015Q4. Appendix \ref{sec:append_1} contains details of the Chow tests. \par}
\end{minipage}
\end{figure}

Whereas the money multiplier decreased drastically since 2008, the decrease in the money multiplier itself is not is a recent phenomenon. The top-left panel of Figure \ref{fig:multip1} shows that the US M1 money multiplier has been decreasing since 1992. However, the declining trends before and after 2008 are different. As the top-right panel of Figure \ref{fig:multip1} shows, the decline during 1992-2007 is accompanied by a huge increase in the ratio of currency to deposit, whereas the decline after 2008 is accompanied by a huge drop in the ratio of currency to deposit. The bottom-left panel of Figure \ref{fig:multip1} reports two structural breaks in the relationship between M1 multiplier and the required reserve ratio: 1992Q3 and 2008Q4.\footnote{The required reserve ratio presented in Figure \ref{fig:multip1} is computed by (Required Reserves)/(Total Checkable Deposits). The legal reserve requirement for net transaction accounts was 10\% from April 2, 1992, to March 25, 2020, but some banks are imposed upon by lower requirements or exempt depending on the size of their liabilities. These criteria changed 27 times from the 1st quarter of 1992  to the last quarter of 2019. From March 2020, all the required reserve ratios have become zero. See \cite{feinman1993reserve} and \href{https://www.federalreserve.gov/monetarypolicy/reservereq.htm}{https://www.federalreserve.gov/monetarypolicy/reservereq.htm} for more details on the historical evolution of the reserve requirement policy of the United States.}$^{,}$\footnote{Appendix \ref{sec:append_2} contains more detail on the Chow tests reported in Figure \ref{fig:multip1}.}  It also shows that there is no negative relationship between M1 money multiplier and the required reserve ratio from 1992Q2 to 2008Q3; the excess reserve ratio had been zero until 2008, which suggests that the required reserve ratio does not drive the money multiplier.

\begin{figure}[tp!]
\centering
\includegraphics[width=5.05cm,height=4.5cm]{figures_1/m1y.pdf}
\includegraphics[width=5.05cm,height=4.5cm]{figures_1/cdy.pdf}
\includegraphics[width=5.05cm,height=4.5cm]{figures_1/cy.pdf}
\caption{US Money demand for M1 and its components}
\label{fig:motive3}
\end{figure}

To offer a better picture of what had happened at the structural break of 1992, Figure \ref{fig:motive3} plots the ratio of M1 to GDP and the ratio of M1's components to GDP against the 3 month Treasury Bill rate. There is a breakdown in M1 in 1992 that coincides with the structural break in Figure \ref{fig:multip1}. As pointed out by \cite{lucas2015stability}, this is due to the breakdown of the deposit component, not the currency component. My hypothesis is that the increased availability of consumer credit crowded out deposit but not cash, which implies that once one accounts for the substitution effect of the newly available consumer credit, there should still be a negative relationship between the real money balance and the interest rate.\footnote{One may think this is due to the relaxation of bank deposit regulation in the 1980s and 1990s that stimulated financial innovations such as money market deposit accounts (MMDAs) in the 1980s or retail sweep accounts in the 1990s (e.g., \citealp*{vanhoose2001sweep}, \citealp*{teles2005stable}, \citealp*{lucas2015stability},
\citealp*{berentsen2015financial}). However, Appendix \ref{sec:md2_unstable} shows there is a breakdown in M2 in 1992 as well and MMDAs and retail sweeps are part of M2.} 

Following \cite{cagan1956monetary}, and \cite{ireland2009welfare}, I relate the natural logarithm of $m$, the ratio of money balances to income, to the short-term nominal interest rate, denoted by $r$. I also regress $r$ on the natural logarithm of $d$, the ratio of deposit balances to income.
$$ln(m_t)=\beta_0+\beta_1 r_t+\epsilon_t, \quad ln(d_t)=\beta_0+\beta_1 r_t+\epsilon_t $$
In addition to the above specifications, to capture the impact of the improved availability of consumer credit that can substitute the deposit, I add a logarithm of $uc$, the ratio of unsecured credit to income as another regressor as follows.\footnote{Following to \cite{krueger2006does}, I use revolving consumer credit.}
$$ln(m_t)=\beta_0+\beta_1 r_t+\beta_2 ln(uc_t)+\epsilon_t, \quad ln(d_t)=\beta_0+\beta_1 r_t+\beta_2 ln(uc_t)+\epsilon_t
$$
\begin{table}[tp!]

\caption{Cointegration regressions and tests}
\label{tab:motive_reg}
\footnotesize
\centering
\begin{threeparttable}
\begin{tabular}{lcccc}
\toprule \hline
Dependent Variable:  & 
\multicolumn{2}{c}{$ln(m_t)$}&\multicolumn{2}{c}{$ln(d_t)$} \\
&\multicolumn{1}{c}{OLS}&\multicolumn{1}{c}{CCR}  & \multicolumn{1}{c}{OLS}&  \multicolumn{1}{c}{CCR} \\ 
&\multicolumn{1}{c}{(1)}&\multicolumn{1}{c}{(2)}  & \multicolumn{1}{c}{(3)}       & \multicolumn{1}{c}{(4)}    \\ \hline
$r_t$      & $0.016^{***}$ & $-0.027^{***}$  & $0.049^{***}$ & $-0.053^{***}$  \\ 
           &  (0.004)    &  (0.004)      & (0.009)     &  (0.009)        \\
$ln(uc_t)$ &             &  $-0.279^{***}$ &             & $-0.574^{***}$   \\
           &             &   (0.033)     &             &  (0.040)     \\ \hline
$adj R^2$  &  0.109    &  0.970       & 0.229        &  0.962       \\  
$N$  & \multicolumn{1}{c}{112}    & \multicolumn{1}{c}{112}    &  \multicolumn{1}{c}{112}   &  \multicolumn{1}{c}{112}    \\  \hline
Johansen $r=0$              & \textbf{15.004} & 41.744        & \textbf{14.934} & 49.174    \\
5\% Critial Value for $r=0$ & 15.41       & 29.68        & 15.41  &  29.68    \\
1\% Critial Value for $r=0$ & 20.04       & 35.65        & 20.04  &  35.65  \\
Johansen $r=1$              & 0.027        & \textbf{12.163}  & 0.26   &  \textbf{14.319}     \\
5\% Critial Value for $r=1$ & 3.76        &  15.41       & 3.76  &  15.41     \\
1\% Critial Value for $r=1$ & 6.65        & 20.04        & 6.65  &  20.04  \\
\hline \bottomrule 
\end{tabular}
\end{threeparttable}
\begin{minipage}{0.90\textwidth}
{\scriptsize  Notes: Columns (1) and (3) report OLS estimates and columns (2) and (4) report the canonical cointegrating regression (CCR) estimates.  First-stage long-run variance estimation for CCR is based on Bartlett kernel and lag 1.  For (1) and (2) Newey-West standard errors with lag 1 are reported in parentheses. Intercepts are included but not reported. ***, **, and * denote significance at the 1, 5, and 10 percent levels, respectively. Johansen cointegration test results are reported in columns (1)-(4). Appendix \ref{sec:append_2} contains unit root tests for each series. The data are quarterly from 1980Q1 to 2007Q4.  \par}
\end{minipage}
\label{tab:reg1}
\end{table}
I focus on the post-1980 period, until the arrival of the Great Recession. In Table \ref{tab:reg1},  columns (1) and (3) report the estimates without unsecured credit, and columns (2) and (4) report the estimates with unsecured credit. The Johansen tests in columns (1) and (3) fail to reject the null hypothesis of no cointegration, which confirms the apparent breakdowns from Figure \ref{fig:motive3}, and ordinary least squares (OLS) estimates from columns (1) and (3) both report positive coefficients on $r_t$ that contradict the conventional notion of money demand: the stable downward-sloping relationship
between real balances and interest rates. In columns (2) and (4), however, the Johansen tests reject their null of no cointegration at a 99 percent confidence level, suggesting there exists a stable relationship between real money balances, interest rates, and real balances of unsecured credit. To estimate the cointegration relationship, I implement the canonical cointegrating regression, proposed by \cite{park1992canonical}, in columns (2) and (4). The estimated coefficients on $r_t$ and $ln(uc_t)$ both are negative and significantly different from zero. Thus, using the cointegrating regressions and tests, I document the evidence that once one accounts for the substitution effect of consumer credit, there still exists a stable negative relationship between real money balances and the interest rates. This substitution effect is a potential explanation for the decline of the money multiplier during 1992Q2-2008Q3. 

\begin{figure}[tp!]
\centering
\includegraphics[width=11cm,height=4.7cm]{figures_1/exratios.pdf}
\caption{Excess reserves ratio}
\end{figure}





The second structural break at 2008Q3, which can be detected from the bottom-right panel of Figure \ref{fig:multip1}  as well as the bottom-left panel, coincides with the Fed's introduction of the interest on reserves. As \cite{nakamura2018unconven} points out, the amount of reserves had skyrocketed before interest rates hit zero and its dramatic increase was simultaneous with the Fed's introduction of the interest on reserves.

To interpret the observed patterns from data, in the following section I develop a model that can incorporate the evolution of the money creation process.





\section{Model}
\label{sec:model}
The model constructed here extends the standard monetary search model \citep{lagos2005unified} by introducing fractional reserve banking and unsecured credit. Time is discrete and two markets convene sequentially in each time period: (1) a frictionless centralized market (CM, hereafter), where agents work, consume, and adjust their balances, following after (2) a decentralized market (DM, hereafter), where buyers and sellers meet and trade bilaterally. The DM trade features imperfect record-keeping and limited commitment. Due to these two frictions, some means of payment are needed in DM trades. Below I describe the economic agents in this economy and the different types of DM meetings.   \\




\noindent\textbf{Buyers and Sellers } The economy consists of a unit mass of  buyers and a unit mass of sellers who discount their utility each period by $\beta$. The preferences of buyers and sellers  for each period are
$$\mathcal{U}^b= U(X)-H+u(q) \quad\text{ and }\quad \mathcal{U}^s=U(X)-H-c(q), $$
where $X$ is the CM consumption, $H$ is the CM disutility from production, and $q$ is the DM consumption. As standard, assume $U'$, $u'$, $c'>0$, $U''$, $u''<0$, $c''\geq0$, and $u(0) = c(0)=0$. Consumption goods are perishable. One unit
of $H$ produces one unit of $X$ in the CM. The efficient consumption in the CM and DM is denoted by $X^*$ and $q^*$, respectively, which solve $U'(X^*)=1$ and $u'(q^*)=c'(q^*)$, respectively. \\

\noindent\textbf{The Bank } There are measure $n$ of active banks that is endogenously determined by the free entry condition in the equilibrium. In the CM, the banks accept deposit and decide how much to deposit at the central bank as reserves and how many banknotes to issue. Managing a deposit payment facility incurs a cost. The cost is represented by a cost function $\gamma(\tilde{d})$, where $\tilde{d}$ is the amount of deposit in real terms, $\gamma',\gamma''>0$, and $\gamma(0)=\gamma'(0)=0$. The reserve earns a nominal interest rate of $i_r$. The bank extends loans by issuing banknotes. The loans are paid back with interest rate $i_\ell$. Enforcing repayment is costly. The cost function is described by $\eta(\tilde{\ell})$, where $\tilde{\ell}$ is the loan in real terms, $\eta',\eta''>0$, and $\eta(0)=\eta'(0)=0$. The bank's lending is constrained by the reserve requirement,  i.e., the bank cannot lend more than $(1-\chi)\tilde{r}/\chi$, where $\chi$ is the reserve requirement and $\tilde{r}$ is the real balance of reserves. \\


\noindent\textbf{Types of DM meetings } There are three types of DM meetings. In DM1, there is no record-keeping device, and the seller can only recognize cash. In DM2, the seller can recognize cash, the claims on bank accounts, and private banknotes. So she accepts cash, deposit receipts and banknotes. In DM3, in addition to the means of payment accepted in DM2, the buyer can trade using unsecured credit with credit limit $\bar{\delta}$ as the trading is monitored imperfectly.\footnote{The acceptance of different means of payment can be endogenized as in \cite{lester2012information} or \cite{lotz2016money} but here we assume the types of meetings are exogenously given. \cite{lester2012information} endogenize the meeting types by allowing sellers' costly \textit{ex ante} choice to acquire the technology for recognizing certain type of assets. Similarly, \cite{lotz2016money} study the environment with costly record-keeping technology where sellers must invest in a record-keeping technology to accept credit.}  The probability of joining a type $j$ meeting is $\sigma_j$. The agents get to know which type of meeting they will be going to in the preceding CM. \\

\noindent\textbf{The Central Bank } The central bank controls the base money supply $M$ in the CM. Let $\mu$ denote the base money growth rate. Then the changes in the real balance of base money can be written as 
$$\mu M=M^+-M,$$
where $x^{+}$ is the value of (any variable) $x$ in the next period. The base money is held in two ways: (1) $C$ as currency in circulation, i.e., outside money held by agents; (2) $R$ as reserves held by a representative bank. Thus,
$$ M=C+ R. $$
The central bank can control the base money supply in two ways. First, it can conduct a lump-sum transfer or collect a lump-sum tax in the CM. Second, it can increase the money supply by paying interest on reserves, $i_r$. Let $T$ represents a lump-sum transfer (or tax if it is negative). The central bank's constraint is
$$\mu \phi M=\phi (M^+-M)=T+  i_r\phi R, $$ where $\phi$ is the price of money in terms of the CM consumption good.  




\subsection{The CM Problem}

\noindent\textbf{Buyers' Decisions } At the beginning of the CM, each buyer's subsequent DM meeting type is realized. Therefore, the buyers' CM problem depends on their DM meeting type. Let $W^B_j(m,d,b,\ell,\delta)$ denote the CM value function where $j$ is the type of the following DM meeting, $m$ is the cash holding, $d$ is the deposit balance, $b$ is the private banknote holding, $\ell$ is the loan borrowed from the bank during the last CM period, and $\delta$ is the unsecured debt owed to the seller from the previous DM. 
All the state variables are in unit of the current CM consumption good. Let $\tau$ denote the lump-sum transfer (or tax if it is negative) to the buyer in the CM. Now, consider the value of the CM. For an agent who is going to a type $j$ DM meeting, the CM problem is 
\begin{equation}
\begin{split}
W^B_j(m,d,b,\ell,\delta) & = \max_{\substack{ X,H, \hat{m}_j, \hat{d}_j, \hat{b}_j, \hat{\ell}_j   } } U(X)-H+\beta V^B_j(\hat{m}_j,\hat{d}_j,\hat{b}_j,\hat{\ell}_j)  \\
\text{  s.t.  }  (1+\pi)\hat{m}_j+(1+\pi)\hat{d}_j+X & = m+(1+i_d)d+b-(1+i_\ell)\ell-\delta+H+\tau \\
\hat{b}_j&=\hat{\ell}_j,
\end{split}
\end{equation}
where $\hat{m}_j$, $\hat{d}_j$, $\hat{b}_j$ and $\hat{\ell}_j$ are the cash holding, deposit balance, private banknote balance, and  debt balance, respectively, carried to the next DM. The first-order conditions (FOCs) are $U'(X)=1$ and 
\begin{align} 
-(1+\pi)+\beta \partial V^B_j(\hat{m}_j,\hat{d}_j,\hat{b}_j,\hat{\ell}_j) / \partial\hat{m}_j \leq 0, &= \text{ if } \hat{m}_j>0 \label{eq:gcmfoc1} \\ 
-(1+\pi)+\beta \partial V^B_j(\hat{m}_j,\hat{d}_j,\hat{b}_j,\hat{\ell}_j) / \partial\hat{d}_j \leq 0, &= \text{ if } \hat{d}_j>0  \label{eq:gcmfoc2} \\ 
\beta \partial V^B_j(\hat{m}_j,\hat{d}_j,\hat{b}_j,\hat{\ell}_j)/ \partial\hat{b}_j+\beta \partial V^B_j(\hat{m}_j,\hat{d}_j,\hat{b}_j,\hat{\ell}_j)/ \partial\hat{\ell}_j \leq 0, &= \text{ if } \hat{\ell}_j>0.
\label{eq:gcmfoc3}
\end{align}
The first term on the left-hand side (LHS) of equation (\ref{eq:gcmfoc1}) is the marginal cost of acquiring cash. The second term is the discounted marginal value of carrying cash to the following DM. Therefore, the choice of $\hat{m}_j>0$ equates the marginal cost and the marginal return on cash. A similar interpretation applies to equation (\ref{eq:gcmfoc2}) for the decision on deposit. For equation (\ref{eq:gcmfoc3}), the first term on the LHS captures the discounted marginal value of carrying privately issued banknotes from the CM to the following DM, and the second term captures the discounted marginal cost of  getting a bank loan.
The envelope conditions for $W^B_j(m,d,b,\ell,\delta)$ are 
$$\frac{\partial W^B_j}{\partial m}=1, \quad \frac{\partial W^B_j}{\partial d}=1+i_d, \quad \frac{\partial W^B_j}{\partial b}=1, \quad \frac{\partial W^B_j}{\partial \ell}=-(1+i_\ell), \quad \frac{\partial W^B_j}{\partial \delta}=1$$
for all $j=1,2,3$, which implies $W^B_j(m,d,b,\ell,\delta)$ is linear. This linearity allows us to write  $$W^B_j(m,d,b,\ell,\delta)=m + (1+i_d)d+b-\delta-(1+i_\ell)\ell+W^B_j(0,0,0,0,0).$$ Let $W^B(m,d,b,\ell,\delta)$ be the buyers'  expected value function before the CM at period $t$ opens, i.e., before their subsequent DM meeting type is realized. Then one can write the buyer's expected value function in the CM as $
W^B(m,d,b,\ell,\delta)=\sum\sigma_j W^B_j(m,d,b,\ell,\delta)$. \\

\noindent\textbf{Sellers' Decisions }  A seller enters the CM with cash, $m$, deposits, $d$, private banknotes, $b$, and unsecured credit $\delta$ that a buyer owes to the seller from the previous DM. The seller does not borrow from the bank as long as $i_\ell>0$. Let $W^{S}_j(m,d,b,0,\delta)$  be the sellers' value function in the CM at period $t$. It can be written as follows: 
\begin{equation}
\begin{split}
W^S_j(m,d,b,0,\delta) & = \max_{X,H,\hat{m}_j,\hat{d}_j,\hat{b}_j,\hat{\ell}_j} U(X)-H+\beta V^S_j(\hat{m}_j,\hat{d}_j,\hat{b}_j,\hat{\ell}_j)  \\
\text{  s.t.  } (1+\pi)\hat{m}_j+(1+\pi)\hat{d}_j+X & =  m+(1+i_d)d+b+\delta+H+\tau  \\
\hat{b}_j&=\hat{\ell}_j
\end{split}
\end{equation}
As we will see below, the DM terms of trade does not depend on the seller's portfolio, there is no incentive for the sellers to carry any liquidity to the next DM as the cost of holding liquidity is positive. The envelope conditions are
$$\frac{\partial W^S_j}{\partial m}=1, \quad \frac{\partial W^S_j}{\partial d}=1+i_d, \quad \frac{\partial W^S_j}{\partial b}=1, \quad \frac{\partial W^B_j}{\partial \delta}=1$$
for all $j\in\{1,2,3\}$, which implies $W^S_j(m,d,b,0,\delta)$ is linear. By linearity, the CM value function can be written as  $$W^S_j(m,d,b,0,\delta)= m +(1+i_d)+b+\delta+W^S_j(0,0,0,0,0).$$

\subsection{The DM Problem}
In the DM, the buyer and seller trade bilaterally. Let $q_j$ and  $p_j$  be the DM consumption and payment in a type-$j$ DM meeting. The bilateral trade is characterized by $(p_j,q_j)$. This trade is subject to $p_j \leq z_j$ where $z_j$ is the total liquidity of the buyer in a type-$j$ meeting. The liquidity position for each type of buyer is 
\begin{align}
z_1&=m_1  \\
z_2&=m_2+ d_2(1+i_d)+b_2   \\
z_3&= m_3+d_3(1+i_d)+b_3+\bar{\delta}.
\end{align}

The DM terms of trade is determined by Kalai (1977) proportional bargaining. Kalai bargaining solves the following problem:
$$\max_{} u(q)-p \quad \textit{s.t} \quad u(q)-p=\theta\left[ u(q)-c(q)\right]$$
where $\theta \in [0,1]$ denotes the buyers' bargaining power. The payment, $p$, is a function of DM consumption, $q$. This can be expressed as $p=v(q)=(1-\theta)u(q)+\theta c(q)$.
Define \textit{liquidity premium}, $\lambda(q)$, as follows:
\begin{equation*}\lambda(q)=\frac{u'(q)}{v'(q)}-1=\frac{\theta [u'(q)-c'(q)]}{(1-\theta) u'(q)+\theta c'(q)}
\end{equation*} 
where $\lambda(q)>0$ for $q<q^*$ and $\lambda(q^*)=0$ with $\lambda'(q)<0$ for $q\in[0,q^*)$. When $z_j\geq p^*$, the buyer has sufficient liquidity to purchase efficient DM output $q^*$. In this case, the payment to the seller is $p^*=v(q^*)$. 

By the linearity of $W^B_j$, we can write a DM value function for a buyer in a type-$j$ meeting as follows:
\begin{align}
V^B_j(m_j,d_j,b_j,\ell_j)= u(q_j)-p_j+W^B(m_j,d_j,b_j,\ell_j,0) \quad 
\end{align} 
where $p_j \leq z_j$. The third term on the right-hand side (RHS) is the continuation value when there is no trade. The rest of the RHS is the surplus from the DM trade. DM payments are constrained by $p_j\leq z_j$. With $v(q_j)=p_j$ and $z_j\leq p^*$, differentiating $V^B_j$ and substituting its derivatives into the FOCs from the CM problem yields
\begin{align}
-(1+\pi) +\beta\big[1+\lambda(q_1)\big] & = 0  \label{eq:agent_ps} \\ 
-(1+\pi) +(1+i_d)\beta &\leq 0, = \text{ if } d_1>0  \\
-i_\ell &\leq 0, = \text{ if } \ell_1>0  \\
-(1+\pi) +\beta\big[1+\lambda(q_i)\big] &\leq 0, = \text{ if } m_j>0 \text{ for } j=2,3\\ 
-(1+\pi) +(1+i_d)\beta\big[1+\lambda(q_i)\big] &\leq 0, = \text{ if } d_j>0 \text{ for } j=2,3\\ 
-i_\ell+\lambda(q_i)  &\leq 0, = \text{ if } l_j>0 \text{ for } j=2,3,
\label{eq:agent_pe}
\end{align}
where $q_j=\min\{q^*,v^{-1}(z_j)\}$ and  $\lambda(q^*)=0$. 

Similarly, the sellers' DM value function is
$$V^S_j(m_j,d_j,b_j,\ell_j)= p_j-c(q_j)+W^S_j(m_j,d_j,b_j,\ell_j,0).$$ 

\subsection{The Bank's Problem}
A bank maximizes its profit subject to the lending constraint. 
\begin{align}
\max_{\Tilde{r},\Tilde{d},\Tilde{\ell}} \quad & i_r \Tilde{r}-i_d \Tilde{d}-\gamma(\Tilde{d}) +i_\ell \Tilde{\ell}-\eta(\Tilde{\ell})  \\
\textit{s.t.  } & \Tilde{\ell} \leq \bar{\ell} \equiv\frac{1-\chi}{\chi} \Tilde{r} 
\label{eq:constraint} \\
& \Tilde{r}\leq \Tilde{d}
\end{align}
Let $\lambda_L$ denote the Lagrange multiplier for the lending constraint. It is straightforward to show $\tilde{r}=\tilde{d}$. The FOCs for the bank's problem can be written as 

\begin{align}
&0=  i_r-i_d-\gamma'(\Tilde{r})+\lambda_L \left(\frac{1-\chi}{\chi}\right) \label{eq:bank_foc1} \\
&0= i_\ell-\eta'(\Tilde{\ell})-\lambda_L. \label{eq:bank_foc2} 
\end{align}
The bank's \textit{ex post} profit equals to the entry cost, $k$
\begin{equation}
\label{eq:bank_entry} 
(i_r -i_d)\tilde{r}+i_\ell\tilde{\ell}-\gamma\left(\tilde{r}\right)-\eta(\tilde{\ell})=k.   
\end{equation
Suppose there are active banks. Consider two cases. In the first case, the bank's lending constraint is binding, i.e., $\lambda_L>0$. In the second case, the bank's lending constraint is loose, i.e., $\lambda_L=0$.  We call the first case a \say{scarce-reserves case,} and the second an \say{ample-reserves case.}  \\

\noindent\textbf{The Scarce-Reserves Case } The bank does not have enough reserves. It needs to acquire reserves to make more loans, which implies a binding constraint. With $\lambda_L>0$, the bank's FOCs (\ref{eq:bank_foc1}) and (\ref{eq:bank_foc2}) give
\begin{align}
0=  i_r-i_d-\gamma'(\tilde{r})+\Bigg[i_\ell-\eta'(\tilde{\ell})\Bigg] \frac{1-\chi}{\chi}.
\label{eq:bank_foc_b}
\end{align}

\noindent\textbf{The Ample-Reserves Case } The bank has sufficient reserves. Its lending constraint does not bind. Then the two FOCs for the bank's problem are
\begin{align}
&0=  i_r-i_d-\gamma'(\tilde{r}) \label{eq:bankfoc_amp1} \\
&0= i_\ell-\eta'(\tilde{\ell}). \label{eq:bankfoc_amp2}
\end{align}
The bank's unconstrained optimal lending, $\ell^*$, satisfies
$$i_\ell=\eta'(\ell^*) $$
and increases as $i_\ell$ rises.

\subsection{Stationary Equilibrium}
\label{sec:st_eq}
I focus on a symmetric stationary monetary equilibrium in which the same type of agents make the same decisions and the real balances are constant over time.  Given that $\phi/\phi^{+}=M^{+}/M=C^{+}/C=1+\mu$, the net inflation rate, $\pi$, is equal to the currency growth rate, $\mu$, in the stationary monetary equilibrium.  By the Fisher equation, $1+i=(1+\mu)/\beta$. The market clearing conditions are
\begin{align}
 \sigma_2 \ell_2+\sigma_3 \ell_3=n\Tilde{\ell}&=\ell \label{eq:clear21} \\
\sigma_2 d_2+\sigma_3 d_3=n\Tilde{r}= r  &=\phi R  \\
 \sigma_1 m_1+\sigma_2 m_2+\sigma_3 m_3= m&=  \phi C,\label{eq:clear22}
\end{align}
where $M=C+R$. Note that $i\geq0$, i.e., $\mu\geq \beta-1$ is necessary for the existence of equilibrium.\footnote{Whereas the lower bound of the nominal interest rate is zero in this setting, one can relax this by introducing liquid assets or threats of theft. See \cite{rocheteau2018open}, \cite{lee2016money} and \cite{williamson2019central} for details.}  Define the stationary equilibrium as follows:

\begin{definition}[\textbf{Stationary Equilibrium}]\label{def:ste}
Given monetary policy, $i$, $i_r$, and $\chi$ and credit limit $\bar{\delta}$, a stationary monetary equilibrium consists of real balances $( m_j, d_j,\ell_j)_{j=1}^3,$ allocation $(q_1,q_2,q_3,X),$ the measure of banks $n$, and prices $(i_\ell,i_d)$,  such that
\begin{enumerate}[label=(\roman*)]
\setlength\itemsep{0.0em}
\item $(i_d, i_\ell,q_1,q_2,q_3)$ solves $(\ref{eq:agent_ps})$-$(\ref{eq:agent_pe})$ and $(\ref{eq:bank_foc1})$-$(\ref{eq:bank_entry})$ with $q_j=\min\{q^*,v^{-1}(z_j)\}$, where $z_1= m_1$, $z_2= m_2+ (1+i_d)d_2+\ell_2$, and $z_3= m_3+ (1+i_d)d_3+\ell_3+\bar{\delta}$.
\item The bank lending $\Tilde{\ell}=\min(\bar{\ell},\ell^*)$, where $\bar{\ell}=(1-\chi)\Tilde{r}/\chi$ and $\ell^*$ solves $i_\ell=\eta'(\ell^*)$
\item  Asset markets clear $(\ref{eq:clear21})$-$(\ref{eq:clear22})$.
\end{enumerate}
\end{definition}
Given Definition \ref{def:ste}, there are three types of equilibria, which are defined as follows:
\begin{definition}\label{def:type} 
In a no-banking  equilibrium, $\bar{\ell}=r=n=0$. In an ample-reserves equilibrium, $\bar{\ell}> \ell^*\geq 0$. In a scarce-reserves equilibrium, $\ell^*\geq\bar{\ell}>0$. 
\end{definition}



In the no-banking equilibrium, the deposit interest rate is zero, $i_d=0$ and there is no active bank. Because the return to holding deposits is dominated or equal to the return to holding currency, agents do not have any incentive to deposit their balances. With zero reserve, the lending limit is zero. Therefore, in this equilibrium, agents only use cash for DM trading. All agents hold the same balance of cash and consume the same amount of consumption goods.  
\begin{align}
i=L( m_j ) \text{ for } j=1,2
\end{align}
and $m_3=\max\{0,L^{-1}(i)-\bar\delta\}$. In this equilibrium, it is straightforward to see that DM consumption is efficient when $i=0$, i.e. the Friedman rule applies.  In the ample-reserves equilibrium, each bank holds  sufficient reserves to lend $\ell^*$, i.e., $\ell^*<\bar\ell=(1-\chi)\tilde{r}/\chi$, where $\tilde{r}$ represents the equilibrium reserve balance of each bank. Thus, the unconstrained optimal lending is less than the lending limit. Given $i$ and $i_r$, $(\tilde{r},\tilde{\ell})$ solves
$$\left\{1+\eta'(\tilde{\ell})\right\}[1+i_r-\gamma'(\tilde{r})]=1+i,  \quad
\gamma'(\tilde{r})\tilde{r}-\gamma(\tilde{r})+\eta'(\tilde{\ell})\tilde{\ell}-\eta(\tilde{\ell})=k.$$In the scarce-reserves equilibrium, however, the lending limit is lower than the bank's unconstrained optimal lending. Therefore, the lending constraint (\ref{eq:constraint}) binds, i.e., $\ell^*>\tilde{\ell}=\bar\ell=(1-\chi)\tilde{r}/\chi$. Given $i_\ell$ and $i_d$, the equilibrium real balance of reserves satisfies 

\begin{equation}\label{eq:scarce_reserved}
r=\begin{dcases} 
\frac{(\sigma_2+\sigma_3)\chi}{1+i_d\chi}L^{-1}(i_\ell)-\frac{\sigma_3\bar\delta\chi}{1+i_d\chi}    & \text{if } \hat\delta >  \bar\delta \\ 
\frac{\sigma_2\chi}{1+i_d\chi}L^{-1}(i_\ell)    & \text{if } \hat\delta \leq  \bar\delta \end{dcases}\end{equation}
where $\hat\delta=L^{-1}(i_\ell)$.


For each type of equilibrium, the following results are proved in Appendix \ref{sec:proof}. 

\begin{proposition}
\label{prop:compastatics1} (i) In the ample-reserves and scarce-reserves equilibrium, $\partial i_d/\partial i>0$, $\partial i_\ell/\partial i>0$, $\partial i_d/\partial i_r>0$, and $\partial i_\ell/\partial i_r<0$. (ii) In the no-banking equilibrium, $\partial i_d/\partial i=\partial i_d/\partial i_r=\partial i_\ell/\partial i=\partial i_\ell/\partial i_r=0.$
\end{proposition}

Proposition \ref{prop:compastatics1} tells us that as long as the measure of banking is positive, the monetary policy rates pass through the deposit rate and lending rate. The deposit rate is  strictly increasing in the nominal interest rate and the interests on reserves. The lending rate is  strictly increasing in the nominal interest rate but is strictly decreasing in interest on reserves. I also establish conditions for the determination of equilibrium types. 
\begin{figure}[tp!]
\centering
\includegraphics[width=7cm,height=5cm]{figures_2/fig_entry.pdf}
\caption{Monetary equilibrium regions in $(i,i_r)$ space}
\label{fig:equ}
\end{figure} 
\begin{proposition}\label{prop:threshold1}
Given $(i_r, \chi)$: (i)
$\exists !$  ample-reserves equilibrium if and only if $i_r>\gamma'(\hat{r})$ and $i\in (\underline{i},\bar{i})$;
(ii) $\exists !$ scarce-reserves equilibrium if and only if either $i>\hat{i}$ and $i_r<\gamma'(\hat{r})$ or $i>\bar{i}$ and $i_r>\gamma'(\hat{r})$;    (iii) $\exists !$ no banking equilibrium  if and only if either $i\in [0,\hat{i})$ and $i_r <\gamma'(\hat{r})$, or $i \in [0,\underline{i})$ and $i_r >\gamma'(\underline{r})$; and the thresholds   satisfy 
$$ \hat{i}=\frac{\chi}{1-\chi}[\gamma'(\hat{r})-i_r]+\eta'\left(\frac{1-\chi}{\chi}\hat{r}\right), \quad \bar{i}=[1+i_r-\gamma'(\hat{r})]\left[1+\eta'\left(\frac{1-\chi}{\chi}\hat{r}\right)\right]-1$$
and $\underline{i}=i_r-\gamma'(\underline{r})$, where  $\hat{r}$ solves $\gamma'(\hat{r})\hat{r}-\gamma(\hat{r})+\eta'\left(\frac{1-\chi}{\chi}\hat{r}\right)\frac{1-\chi}{\chi}\hat{r}-\eta\left(\frac{1-\chi}{\chi}\hat{r}\right)=k$ and  $\underline{r}$ solves $\gamma'(\underline{r})\underline{r}-\gamma(\underline{r})=k$.
\end{proposition}

Banks hold excess reserves when the central bank pays sufficiently high interest on reserves with the nominal interest rate at some moderate level. To see the intuition consider the case in which the bank holds reserves. The reserve requirement and the reserve balances determine the lending limit. Due to monotone pass-through from the nominal interest rate to the bank's lending rate, the bank's unconstrained optimal lending $\ell^*$ is increasing in the nominal interest rate. There exists a threshold of the nominal interest rate below which the lending limit is lower than the bank's unconstrained lending (scarce-reserves),  and above which the lending limit is higher than the bank's unconstrained lending (ample-reserves). In other words, there is a critical value $\bar{i}$ that satisfies $\ell^*=\bar{\ell}=(1-\chi)\Tilde{r}/\chi$.



However, when the equilibrium deposit rate
is zero, agents have no incentive to deposit their balance in the bank, implying the no-banking equilibrium. As shown in Proposition \ref{prop:compastatics1}, the deposit rate is monotone in the nominal interest rate and the deposit rate could be zero given some nominal interest rate. There exists a threshold $\hat{i}$, below which the deposit rate is zero and above which the deposit rate is positive. The bank's constraint plays a crucial role in determining the equilibrium type. Lowering the nominal interest rate or increasing interest on reserves loosens the bank's lending constraint. These lead to the following results:
\begin{corollary}\label{cor:threshold11}
The thresholds $\bar{i}$ and $\underline{i}$ are increasing in $i_r$ and $\hat{i}$ is decreasing in $i_r$.
\end{corollary}
As Figure \ref{fig:equ} illustrates, the equilibrium type is determined by $(i,i_r)$. The bank holds excess reserves when the interest on reserves $i_r$ is high enough and the nominal interest rate is not too high. 

In the ample-reserves equilibrium, $q_2=q^*$ if $i_r-i=\gamma'(\underline{r})$. Therefore, the DM2 meeting consumption can be efficient even though the economy is not under the Friedman rule. This result can be formally summarized in the following proposition.
\begin{proposition}\label{prop:uncon}
In the ample-reserves equilibrium with $i>0$, DM2 consumption is efficient if $i_r=i+\gamma'(\underline{r})$. 
\end{proposition}
The intuition behind the efficient DM2 consumption is straightforward. In many monetary models, a high inflation or interest rate increases the opportunity cost of holding money. In the environment where money is valued as a medium of exchange, having less liquidity in the economy because of an opportunity cost of holding money is inefficient. However, the interest on reserves provides a proportional return. If this return is properly distributed across agents, it eliminates the inefficiency that arises from the opportunity cost of holding money, which results in efficient DM2 consumption.

In the scarce-reserves equilibrium, it is easy to show that the equilibrium reserve balance is decreasing in $i$  
$$\frac{\partial r}{\partial  i} =\begin{dcases} 
   \frac{(\sigma_2+\sigma_3)\chi L'^{-1}(i_\ell)}{1+i_d\chi}\frac{\partial i_\ell}{\partial  i}-
   \frac{\chi[(\sigma_2+\sigma_3)L'^{-1}(i_\ell)-\sigma_3\bar\delta]}{(1+i_d\chi)}\frac{\partial i_d}{\partial i}
   <0    & \text{if } \hat\delta > \bar\delta  \\
   \frac{\sigma_2\chi L'^{-1}(i_\ell)}{1+i_d\chi}\frac{\partial i_\ell}{\partial  i}-\frac{\sigma_2\chi L^{-1}(i_\ell)}{(1+i_d\chi)^2}\frac{\partial i_d}{\partial  i}<0    & \text{if } \hat\delta \leq \bar\delta  
  \end{dcases}$$
because $\partial i_\ell/\partial i<0$ and $\partial i_d/\partial i>0$, as shown in Proposition \ref{prop:compastatics1}. 





\begin{figure}[tp!]
\centering
\includegraphics[width=7cm,height=5.5cm]{figures_2/md_ex1.pdf}
\includegraphics[width=7cm,height=5.5cm]{figures_2/md_ex2.pdf}
\caption{Demand for reserves and the monetary aggregate}
\label{fig:example1}
\end{figure} 
Whereas Proposition \ref{prop:compastatics1} shows that how the deposit rate can change as the central bank sets the nominal interest rate, the pass-through of nominal interest rate to deposit rate depends on other policy variables and the equilibrium type. 
\begin{proposition}\label{prop:pass}
Higher reserve requirement weakens the pass-through from the nominal interest rate to the deposit rate in the scarce-reserve equilibrium, i.e., 
$$\frac{\partial^2 i_d}{\partial\chi\partial i}<0 \quad \text{ if } \bar\ell<\ell^*.$$
Higher interest on the reserve raises the pass-through from the nominal interest rate to the deposit rate in the ample-reserves equilibrium, i.e., 
$$\frac{\partial^2 i_d}{\partial i_r\partial i}>0 \quad \text{ if } \bar\ell\geq\ell^*.$$
\end{proposition}
Regardless of the equilibrium type, there exists a downward-sloping demand curve for total liquidity. To illustrate this, I define the monetary aggregate as a sum of cash holdings, reserves, and banknotes in the economy.  The right (left) panel of Figure \ref{fig:example1} shows that there exist stable downward-sloping demand curves for monetary aggregates (reserves) with different interest on reserves. An increase in $i_r$ shifts the money demand to the right, both in the scarce-reserves equilibrium and in the ample-reserves equilibrium. In the scarce-reserves case, higher $i_r$ raises $\tilde{r}$ and $\tilde{\ell}$ with a looser lending constraint and shifts the money demand to the right. However, in the ample-reserves equilibrium, higher $i_r$ raises reserves $\tilde{r}$ but decreases $\tilde{\ell}$. Last, a rise in $i_r$ increases $\bar{i}$, which allows the monetary authority to induce the ample-reserves equilibrium with higher nominal interest rates.  As the central bank pays interest on reserves, the economy can shift to the ample-reserves equilibrium, which has a flatter demand curve for reserves. This is consistent with the observation from \cite{nakamura2018unconven} that the quantity of reserves skyrocketed before interest rates hit zero, but its dramatic increase was simultaneous with the Fed's introduction of the interest on reserves.  \\




\noindent\textbf{The Role of Access to Unsecured Credit } One also can check the effect of changes in the access to unsecured credit, $\sigma_3$, or changes in the credit limit, $\bar\delta$. An increase in $\sigma_3$ implies that more buyers can use unsecured credit in the DM. Some DM2 buyers become DM3 buyers and they hold less deposit than they used to hold. With higher $\bar\delta$, although the measure of DM3 buyers stays same, DM3 buyers can use more unsecured credit for the DM trade. As long as there are positive amount of reserve balances, $r>0$, an increase in $\sigma_3$ or $\bar\delta$ lowers $r$. Appendix \ref{sec:proof} verifies the following: 




\begin{figure}[tp!]
\centering
\includegraphics[width=7cm,height=5.5cm]{figures_2/md_ex3.pdf}
\includegraphics[width=7cm,height=5.5cm]{figures_2/md_ex4.pdf}
\caption{Demand for reserves and the monetary aggregate with different credit limits}
\label{fig:example2}
\end{figure} 


\begin{proposition}\label{prop:credit}
Let $\hat\delta >\bar\delta$. In scarce-reserve and ample-reserves equilibria, better credit condition and more credit access decrease the real balance of reserves, i.e., 
$$\frac{\partial r}{\partial \bar\delta}<0, \quad \frac{\partial r}{\partial \sigma_3}<0. $$
\end{proposition}

This result is consistent with the finding in Section \ref{sec:motive}. As more unsecured credit becomes available, real balances of inside money decrease. By summing up the results, we now establish the results on the money multiplier. Define money multiplier $\zeta\equiv(m+r+\ell)/(m+r)$, then we have following results:

\begin{proposition}\label{prop:multi}
In ample-reserves equilibrium, for small $m$, we have
$$\frac{\partial\zeta}{\partial i}>0, \quad \frac{\partial\zeta}{\partial i_r}<0.$$
Let $\hat\delta >\bar\delta$. In  the  ample-reserves  and  scarce-reserves  equilibria, a better credit condition lowers the money multiplier as long as  $m>0$ and  $\chi<1$, i.e.,
$$\frac{\partial \zeta}{\partial\bar\delta}<0 \quad \text{ if }  m>0 \text{ }\& \text{ } \chi<1. $$
\end{proposition}
Thus, the model can successfully address the mechanism illustrated in Section \ref{sec:motive}. We can interpret the decline in the money multiplier in the pre-2008 economy as a result of improved availability of consumer credit under the scarce-reserve equilibrium. For the post-2008 period, after the Fed started paying interest on reserves the economy moved to the ample-reserves equilibrium. The model suggests that the changes in the money multiplier and the excess reserve ratio are the results of the Fed's management of two interest rates, the nominal interest rate and the interest on reserves.





\section{Quantitative Analysis}
\label{sec:quant}
To evaluate the theory quantitatively, I calibrate the model to match several targets using pre-2008 data. Using calibrated parameters, I compare the model predictions with the data of the pre- and post-2008 periods. Given the parameters, the stationary equilibrium is characterized by $(i,i_r,\chi,\bar\delta)$. The required reserves ratio is computed by dividing the required reserves by total checkable deposits. Whereas the first three series are easy to obtain, it is hard to get the unsecured credit limit, $\bar\delta$, from either macro or micro data. Since the use of unsecured credit corresponds to the credit limit in the model, the unsecured credit limit is computed using the unsecured credit to output ratio, instead of using explicit credit limit data. In the model, the unsecured credit to output ratio is given by $\sigma_3\bar\delta/(B+\sum_j^3 \sigma_j z_j)$, so we can compute $\bar\delta$ using the model with the given policies $(i,i_r,\chi)$  and other parameters. 
Following \cite{krueger2006does}, the revolving consumer credit is used as the unsecured credit. For this exercise, I generate simulated data by using 4 series: (i) nominal interest rates\footnote{I use 3-month Treasury-bill rates as standard. Whereas I use 3-month Treasury-bill rates instead of the federal funds rates because the model does not include interbank lending, Section \ref{sec:robust_} and Appendix \ref{sec:quant_robustness1} check robustness by using different measures of monetary policy. The sensitivity analysis suggests that the main results are not overly sensitive to the choice for the measure of monetary policy.}; (ii) the interest on reserves; (iii) the required reserve ratio; and  (iv) the unsecured credit to GDP ratio.


\begin{table}[tp!]
\footnotesize
\caption{Model parametrization}
\vspace{-0.2cm}
\centering
\begin{threeparttable}
\begin{tabular}{llcD{.}{.}{3}D{.}{.}{3}}
\toprule \hline
Parameter                        &    Value  & Target/source  & \multicolumn{1}{r}{Data} & \multicolumn{1}{r}{Model}  \\ \hline
\multicolumn{5}{c}{\textbf{External Parameters}}       \\ 
DM3 matching prob, $\sigma_3$       & 0.69 & SCF 1970-2007&        &       \\
\multicolumn{5}{c}{\textbf{Internal Parameters}}      \\
Bargaining power, $\theta$           & 0.454 & avg. retail markup & 1.384 & 1.384   \\
Enforcement cost level, $E$          & 0.001 & avg. $UC/DM$       & 0.387 & 0.370  \\
Deposit operating cost level, $A$    & 0.0017 & avg. $R/Y$        & 0.014 & 0.017   \\ 
Entry cost, $k$                      & 0.0011 & avg. $\Pi/Y$      & 0.0016 & 0.0011  \\
DM1 matching prob, $\sigma_1$        & 0.187 & avg. $C/D$         & 0.529 & 0.523  \\
DM utility level, $B$                & 0.825 & avg. $C/Y$         & 0.044 & 0.044    \\
DM utility curvature, $b$& 0.398 & semi-elasticity of $C/Y$ to $i$& -3.713& -3.712   \\
\hline \bottomrule
\end{tabular}
\begin{tablenotes}
\item Note: $C$, $R$, $DM$, $D$, $UC$, and $Y$ denote currency in circulation, reserves, DM transactions, deposit, unsecured credit, and nominal output, respectively. $\Pi$ denotes the net income of banks.  
\end{tablenotes}
\end{threeparttable}
\label{tab:param}
\end{table}
\begin{figure}[tp!]
\centering
\includegraphics[width=10cm,height=6.6cm]{figures_3/cd.pdf}\caption{Money demand for currency}
\label{fig:cd}
\end{figure}

\subsection{Calibration}

The utility functions for the DM and the CM are $u(q)=Bq^{1-b}/(1-b)$ and $U(X)=\log(X)$ implying $X^*=1$ (a normalization). The cost function for the DM is $c(q)=q$. The enforcement cost for lending is assumed to be quadratic, $\eta(\tilde{\ell})=E\tilde{\ell}^2$, and the management cost for deposit facility takes the form, $\gamma(\tilde{d})=A\tilde{d}^{a}$.\footnote{To rule out the no-banking equilibrium from the exercise, I set the curvature parameter of $\gamma(\cdot)$, $a$, to be 1.2 so that $\gamma(\cdot)$ will be sufficiently less convex than $\eta(\cdot)$. Appendix \ref{sec:quant_robustness2} includes a sensitivity analysis for the curvature parameter of $\gamma(\cdot)$.} The fraction of buyers who can use unsecured credit is set to $\sigma_3=0.69$.\footnote{The Survey of Consumer Finances provides triennial series for the percentage of U.S. households holding at least one credit card from 1970 to 2007. The average percentage during 1970-2007 is 69\%.} 
The remaining 7 parameters $(\theta,A,B,b,k,E,\sigma_1)$ are set to match the following eight targets: (i) the average retail market markup; (ii) the average credit share of the DM transactions, $\sigma_3\delta/DM$; (iii) the average currency to deposit ratio, $C/D$; (iv) the average reserves to output ratio, $R/Y$; (v) the average currency to output ratio, $C/Y$; (vi) the semi-elasticity of $C/Y$ to $i$, where $i$ denotes the nominal interest rate; and (vii) the average net income of banks to output ratio;\footnote{I use the net attributable income of FDIC-insured commercial banks and savings institutions as the net income of banks.} (viii) the average net income of banks to deposit ratio. The targets are computed on the basis of 1968-2007 data, except for the markup, which uses the average from 1993 to 2007, and the net income of banks, which uses the average from 1984 to 2007. 

The bargaining power $\theta$ is set to match the DM markup to the retail markup.\footnote{In the Annual Retail Trade Survey, the average ratio
of gross margins to sales from 1993-2007 is 0.2776, implying the average markup is 1.3844.}  Set $(B,b)$ to match the currency to output ratio and the semi-elasticity of $C/Y$ with respect to $i$. The costs of operating deposit services and issuing loans from the bank, captured by $A$ and $E$, are set to match the reserves to output ratio,  the unsecured credit to DM transaction ratio and the  banking industry profit to deposit ratio. The entry cost $k$ is set to match banking industry profit to output ratio. Lastly, I set $\sigma_1$ to match the currency-deposit ratio. The calibrated parameters and the targets are summarized in Table \ref{tab:param}, and the calibrated money demand of currency is shown in Figure \ref{fig:cd}. 








\subsection{Results and the Model Fit}

\begin{figure}[tp!]
\centering
\includegraphics[width=7.5cm,height=5cm]{figures_3/m1multifit1.pdf}
\includegraphics[width=7.5cm,height=5cm]{figures_3/exr1.pdf}
\includegraphics[width=7.5cm,height=5cm]{figures_3/cd_ratio1.pdf}
\includegraphics[width=7.5cm,height=5cm]{figures_3/resy1.pdf}
\caption{In-sample fit: 1968-2007} 
\label{fig:model_fit1}
\end{figure}
\begin{figure}[tp!]
\centering
\includegraphics[width=7.5cm,height=5cm]{figures_3/m1multifit2.pdf}
\includegraphics[width=7.5cm,height=5cm]{figures_3/exr2.pdf}
\includegraphics[width=7.5cm,height=5cm]{figures_3/cd_ratio2.pdf}
\includegraphics[width=7.5cm,height=5cm]{figures_3/resy2.pdf}
\caption{Out-of-sample fit: 2008-2017}
\label{fig:model_fit2}
\end{figure}

Figure \ref{fig:model_fit1} compares the model and data for the sample period, 1968 to 2007. The top-left panel of Figure \ref{fig:model_fit1} shows the M1 money multiplier from 1968 to 2007. The model  generated decreases in the M1 multiplier during 1987-2007, while the peak was in 1987 in the data and 1984 in the model. During this period, the excess reserves to deposit ratio had been almost zero both in the data and in the model, suggesting the US economy had been in the scarce-reserves equilibrium. In the model, the declining trend of the M1 multiplier in the pre-2008 period is driven by the increase in unsecured credit, which crowds out the inside money (private banknotes and reserves) but not currency. This induces increases in the currency to deposit ratio, as shown in the  bottom-left panel of Figure \ref{fig:model_fit1}. The bottom-right panel of Figure \ref{fig:model_fit1} compares how unsecured credit crowds out reserves in the model and the data. 


The next step is to evaluate model projections by comparing them with the data after 2007. Overall, the model can match the patterns in the data. Timeplots of Figure \ref{fig:model_fit2} compares the model projections for the M1 money multiplier, the excess reserves to deposit ratio, the currency to deposit ratio, and the reserves to output ratio with data from 2008 to 2017. The model-implied series shows similar patterns to the actual data series. The model can generate the change in the equilibrium type, from scarce-reserves to ample-reserves, and a similar pattern of excess reserves to deposit ratio. This change in the equilibrium type is represented by a huge drop in the money multiplier in the top-left panel and a huge increase in the excess reserves to deposit ratio in the top-right panel.

Regression estimates shown in Table \ref{tab:reg} illustrate the main mechanism of the model. Columns (1) and (2) show the regression coefficient estimates using the following equation for 1968-2007.
$$\text{Reserves}_t/\text{GDP}_t=\beta_0+\beta_1\text{UnsecuredCredit}_t/\text{GDP}_t+\beta_2\texttt{Tbill3}_t+\epsilon_t $$
Since all three series have a unit root and are cointegrated, both in the data and in the model-generated series, the coefficients are estimated using the canonical cointegrating regression.\footnote{Unit root and cointegration test results are reported in  Appendix \ref{sec:append_2}.}
The estimated negative coefficient on the 3-month T-bill rate suggests a downward sloping demand for reserves with respect to the interest rate; but other coefficients on unsecured credit suggest that this demand for reserves can shift as the credit condition changes, as shown in Figure \ref{fig:example2}. This is consistent with Proposition \ref{prop:credit}, and the model-implied regression produces similar results.

\begin{table}[tp!]
\caption{Model-implied regression coefficients, model vs. data}
\footnotesize
\vspace{-0.2cm}
\centering
\begin{threeparttable}
\begin{tabular}{lD{.}{.}{4}D{.}{.}{4}D{.}{.}{4}D{.}{.}{4}D{.}{.}{4}D{.}{.}{4}}
\toprule \hline
Dependent Variable:  & 
\multicolumn{2}{c}{Reserves/GDP}&\multicolumn{2}{c}{M1 Money Multiplier}&\multicolumn{2}{c}{Excess Reserve/Deposit}  \\
                     & 
\multicolumn{2}{c}{(1968-2007)}   & \multicolumn{2}{c}{(2009-2017)} & \multicolumn{2}{c}{(2009-2017)} \\ \hline
&\multicolumn{1}{c}{Data}&\multicolumn{1}{c}{Model}  & \multicolumn{1}{c}{Data} & \multicolumn{1}{c}{Model} & \multicolumn{1}{c}{Data} & \multicolumn{1}{c}{Model} \\ 
&\multicolumn{1}{c}{(1)}&\multicolumn{1}{c}{(2)}  & \multicolumn{1}{c}{(3)}       & \multicolumn{1}{c}{(4)} & \multicolumn{1}{c}{(5)}       & \multicolumn{1}{c}{(6)}  \\ \hline
Unsecured Credit/GDP &-0.125^{***}&-0.200 &            &        &            &   \\ 
                     &(0.010)     &       &            &        &            &          \\
3 Month T-bill Rate   &-0.001^{***}&-0.001&0.686^{***} & 0.864  &-1.561^{***}& -1.697 \\
                     &(0.000)     &       &(0.184)     &        &(0.498)     &          \\
Interest on Reserves &            &       &-0.648^{***}& -0.848 &1.461^{**} & 1.688 \\
                     &            &       &(0.210)     &        &(0.567)     &        \\ \hline
$adj R^2$     &  0.830    & 0.814         & 0.656       &  0.918  &  0.577     & 0.997         \\
\hline \bottomrule 
\end{tabular}
\end{threeparttable}
\begin{tablenotes}
\item {\footnotesize Notes: Columns (1)-(2) report the canonical cointegrating regression (CCR) estimates. First stage long-run variance estimation for CCR is based on Bartlett kernel and lag 1. Columns (3)-(6) report OLS estimates. For (3) and (5) Newey-West standard errors with lag 1 are reported in parentheses. ***, **, and * denote significance at the 1, 5, and 10 percent levels, respectively. Intercepts are included but not reported.}
\end{tablenotes}
\label{tab:reg}
\end{table}

Columns (3) and (4) regress the M1 multiplier on the 3 month T-bill rate and the interest on reserves, and Columns (5) and (6) regress the excess reserves ratio on the same variables. Because the number of observations in the data is too small, Columns (3) and (5) use data from the 1st quarter of 2009 to the 4th quarter of 2017. Regressions using the data and the model-implied series provide similar results. Based on the regression results in (3)-(6), for a given interest on reserves, raising the 3-month T-bill rates increases the M1 multiplier while it decreases excess reserves. At the same time, for a given 3-month treasury rate, lowering the interest on reserves decreases the M1 multiplier while it increases excess reserves. In the model, when the bank faces higher interests on reserves, it holds more reserves and does not lend as much as before. This is because interest on reserves yields profits to the bank with low cost, but lending is associated with the enforcement cost. This increases the excess reserve ratio and lowers the money multiplier. The model also provides the composition of the monetary base over time.  Figure \ref{fig:portf} compares the composition of the monetary base from the data and the model. The model successfully generates the changes for each component of the monetary base - currency, required reserves, and excess reserves - both before and after 2008.   \\

\begin{figure}[tp!]
\centering
\includegraphics[width=7.3cm,height=5.1cm]{figures_3/portiond.pdf}
\includegraphics[width=7.3cm,height=5.1cm]{figures_3/portionm.pdf}
\caption{Composition of monetary base: data vs. model}
\label{fig:portf}
\end{figure} 

\noindent\textbf{A Digression on Model Fit  } For the post-2007 period, although the model projections can match the patterns in the data well, they do not fit very well in levels. This discrepancy is from the fact that the theoretical lower bound for the money multiplier is 1 in the model. In reality, however, the U.S. economy has experienced M1 multipliers lower than 1.\footnote{The M1 multiplier of the U.S. was lower than 1 from December 2008 (0.975) until June 2018 (0.991).} There are two potential explanations for this.  

One possible reason is that monetary policy can be conducted in different ways than the lump-sum transfer in the model. In the model, all the base money is distributed to agents through the lump-sum transfer, and they keep some in their bank accounts.  Reserves are in the bank deposits in this setup, and this implies the money multiplier can not be lower than 1. In contrast to most of the monetary models that assume money is injected as a lump-sum transfer across the agents (buyers and sellers, in this model), much money injection is made to the banking system directly in the real economy. For example, in the quantitative easing program, the Fed purchased large amounts of financial assets from financial intermediaries and gave them the same amount in reserves. These reserves are directly injected into the banking system and this is different from lump-sum transfers. In this case, reserves can only be held by banks, not by the public, and banks do not lend out reserves. One may need to consider a more explicit mechanism for monetary policy implementation.\footnote{Previous works on the explicit model of the interbank market with monetary policy implementation include \cite{armenter2017excess},  \cite{afonso2019model}, \cite{bianchi2014banks}, and  \cite{chiu2020relationships}. Those models explicitly describe search frictions and the market structure of the interbank market for reserves, whereas this paper assumes a centralized market for reserves. Noting that the Fed controls the effective federal funds rates, which are interbank rates, introducing the interbank market can allow more realistic monetary transmission.}

Another possible reason is that reserves could be kept in saving accounts or time deposits, which is in M2 but not in M1.  Even though one assumes that the monetary base is distributed through a lump-sum transfer, it does not have to be kept in a checkable account. In this case, there is no discrepancy between the data and the theoretical lower bound for the money multiplier because the M2 money multiplier has never been lower than 1.\footnote{The lowest M2 money multiplier during 1959-2019 was 2.812 at August 2014.} From a balance sheet point of view, reserves are recorded as a cash asset on the commercial bank's balance sheet because reserves are held as an account for the commercial banks at the Federal Reserve Bank, but deposits are liabilities. In this case, reserves cannot exceed total liabilities (total deposits), but they can exceed checkable deposits. 
\newpage

\subsection{Welfare}

\begin{figure}[t]
\centering
\includegraphics[width=7cm,height=5cm]{figures_4/welfare1.pdf}
\includegraphics[width=7cm,height=5cm]{figures_4/welfare2.pdf}
\includegraphics[width=7cm,height=5cm]{figures_4/welfare3.pdf}
\includegraphics[width=7cm,height=5cm]{figures_4/welfare4.pdf}
\caption{Welfare}
\label{fig:welfare}
\end{figure}

So far, I have shown that different tools of monetary policy have distinct roles in monetary transmission. This section focuses on the impacts of these different tools of monetary policy in terms of welfare. I measure the welfare of the seller in type $j$ meeting using her DM trade surplus.
$$J^S_j(i,\chi,i_r)=(1-\theta)[u(q_j)-c(q_j)],$$
and the welfare of the buyer who trades in the $j$ type DM meeting is a DM trade surplus with the cost for acquiring the cash and reserves. 
$$J^B_j(i,\chi,i_r)=-im_j(i,\chi,i_r)-(i-i_d)r_j(i,\chi,i_r)+(1-\theta)[u(q_j)-c(q_j)]$$
I define the total welfare as a weighted sum of each agent's welfare. 
$$J(i,\chi,i_r)=\sum_{j=1}^3 \sigma_j[J^B_j(i,\chi,i_r)+J^S_j(i,\chi,i_r)]$$ 
The top-left panel of Figure \ref{fig:welfare} illustrates the effects of monetary policy, $i$, on total welfare $J$ ranging from 0\% to 16\% and how its impact can change depending on different reserve requirements and different interest rates on reserves. Each curve denotes the welfare under the different reserve requirements and interest rates on reserves. The welfare is monotonically decreasing in $i$, and each curve can be shifted up by paying interest on reserves or lowering the reserve requirement. These results are well expected. The higher opportunity cost of holding money lowers welfare. Paying interest on reserves, however, is welfare improving because it compensates the opportunity cost of holding money. Lowering reserve requirement also improves welfare since it provides more liquidity to the economy. The difference becomes smaller as the economy faces lower $i$, and the Friedman rule gives the optimal level of welfare.

In addition to total welfare, one can examine the distributional effect of monetary policy. The top-right panel of Figure \ref{fig:welfare} plots the standard deviation of each agent's welfare, $J^k_j$ under different policies. Clearly, the effects of the monetary policy are different across the agents. The bottom-left (bottom-right) panels of Figure \ref{fig:welfare} plots the buyer's (seller's) welfare in different DM meetings depending on different policies. Among buyers,  DM1 buyer's welfare is lower than others. Whereas DM2 and DM3 buyers consume the same amount in the DM, given that the access to unsecured credit allows the DM3 buyer carry less money than the DM2 buyer, the DM3 buyer's welfare is higher than the DM2 buyer's. For the sellers, because the consumption in DM2 and DM3 are equal, they enjoy the same welfare. As in the buyers' case, DM1 seller's welfare is the lowest among all the sellers. Lowering reserve requirement increases the welfare of buyers and sellers in DM2 and DM3. However, DM1 agents' welfare is independent of reserves requirement, access to credit, and the interest on reserves but only depends on the nominal interest rate. 


\subsection{Counterfactual Analysis}
In this section, I use calibrated parameters to assess how the money multiplier and currency deposit ratio would be changed by setting different reserve requirements. I also demonstrate that it is important to distinguish the effect of the reserve requirement from the effect of credit.

\begin{figure}[tp!]
\centering
\includegraphics[width=7cm,height=4.8cm]{figures_4/cf11.pdf}
\includegraphics[width=7cm,height=4.8cm]{figures_4/cf12.pdf}
\includegraphics[width=7cm,height=4.8cm]{figures_4/cf21.pdf}
\includegraphics[width=7cm,height=4.8cm]{figures_4/cf22.pdf}
\caption{Counterfactual analysis}
\label{fig:cf1}
\end{figure}


The top panel of Figure \ref{fig:cf1} shows the counterfactual under different reserve requirements while keeping $(i,i_r,\bar\delta)$ the same as in the benchmark case. 
With a lower reserve requirement, the money multiplier increases. However, we see a trend of decreasing multipliers regardless of reserve requirement. As illustrated in Table \ref{tab:reg}, this gradual decrease in the money multiplier since the late 1980s is driven by an increase in unsecured credit in the model. The currency deposit ratio increases with a higher reserve requirement and all the cases show the similar trend. 

The bottom panel of Figure \ref{fig:cf1} shows the counterfactual under a different reserve requirement with $\bar\delta=0$, and $(i,i_r)$ set to the data. In the bottom-right panel, we see almost no changes in the currency deposit ratio. Since the money demand for currency and inside money is stable, if unsecured credit did not crowd out inside money, there would not be a substantial increase in the currency deposit ratio, as the U.S. economy witnessed. In the case of the money multiplier, whereas it shows a stationary pattern before 2009, it drops drastically once the Federal Reserve started paying interest on reserves and lowered the nominal interest rate. This suggests that the gradual decline of the money multiplier from the late 1980s to 2007 can be attributed to an increase in unsecured credit, whereas the dramatic decline in the money multiplier since 2008 can be explained by the monetary policies of the Federal Reserve. 

\subsection{Robustness}
\label{sec:robust_}
This section briefly summaries a few results from Appendix \ref{sec:quant_robustness}. Appendix \ref{sec:quant_robustness1} examines the sensitivity of the results using different measures of the monetary policy target: the federal funds rate, and the commercial paper rate. Using different measures does not change the main results.  In Appendix \ref{sec:quant_robustness2}, to check whether the curvature parameter 1.2 for the deposit operating cost is sensitive, I change the benchmark parameter from 1.2 to 1.15 or 1.25. Changing these parameters does not have a significant impact on the results. 



\section{Concluding Remarks}
\label{sec:conclusion}

This paper develops a monetary-search model with fractional banking and unsecured credit and studies the money creation process. In the fractional reserve system, the money is created when banks make loans. The bank's lending, however, can be constrained by the reserve requirement and the reserves. 

Banks hold excess reserves when the central bank pays sufficiently high interest on reserves with the nominal interest rate at some moderate level. In this case, the money multiplier and the quantity of the reserve depend on the nominal interest rate and the interest on reserves rather than the reserve requirement. In contrast to the previous works, these two interest rates play distinct roles. Whether the banks hold excess reserves or not, there exists a downward-sloping demand curve for reserves, and the Friedman rule is optimal. Paying interest on reserves with low nominal interest rates can move the economy from the scarce reserves regime to the ample reserves regime, which is consistent with what we have seen in the US economy. The quantitative analysis can generate simulated data that resemble the actual data. This paper provides evidence from the model and the data that suggests that the dramatic changes in the money multiplier after 2008 are mainly driven by the introduction of the interest on reserves with the low nominal interest rate.

This work can be extended in various ways. Although I focus on the centralized market for the reserves with homogeneous banks, in reality, the market for reserves is a decentralized interbank market and banks have different portfolios. Therefore, one can further investigate how much the market structure and heterogeneity matter for the transmission of monetary policy (e.g., \citealp*{afonso2015trade}; \citealp*{armenter2017excess}; \citealp*{afonso2019model}). Second, it would be worthwhile to study how inside creation via loan extension is related to investment and firms' dynamics (e.g., \citealp*{ennis2018simple}; \citealp*{bianchi2014banks}; \citealp*{altermatt2019bank}). This will allow us to understand the investment channels of monetary policy more explicitly.  Moreover, I assume that bank assets are composed of loans and reserves. But commercial banks' assets are mainly composed of securities, loans, and reserves. Extending the model to incorporate banks' portfolio choices and analyzing the role of investment, financial regulation, and monetary policy can open up other
research avenues. (e.g., \citealp*{rocheteau2018corporate}).




\newpage
\bibliographystyle{aer}
