%Paper: ewp-mac/9802010
%From: renea-jay@uiowa.edu
%Date: Thu, 5 Feb 98 11:29:13 CST


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%TCIDATA{Created=Mon Aug 11 11:43:39 1997}
%TCIDATA{LastRevised=Thu Jan 22 16:35:52 1998}
%TCIDATA{Language=American English}

\input tcilatex
\begin{document}


\begin{center}
PRIVATE MONEY AND RESERVE MANAGEMENT

IN A RANDOM MATCHING MODEL\footnote{%
We wish to thank, without implicating in any way, Narayana Kocherlakota,
Neil Wallace, Warren Weber, and Ruilin Zhu for helpful comments. All errors
are ours. Part of this paper was completed while the third author held a
position as an economist at the Federal Reserve Bank of Philadelphia. The
views expressed here are those of the authors and do not necessarily reflect
those of the Federal Reserve Bank of Philadelphia or the Federal Reserve
System.}

\smallskip

\smallskip\ 

\smallskip\ 

Ricardo de O. Cavalcanti

Department of Economics, University of Miami

P.O. Box 248126, Coral Gables, FL 33124

\smallskip \ E-mail: ricardoc@walleye.econ.umn.edu

\smallskip

\bigskip

Andres Erosa

Department of Economics, University of Western Ontario

London, Ontario, N6A 5C2 Canada

\smallskip \ E-mail: EROSA@sscl.uwo.ca

\smallskip

\bigskip

Ted Temzelides

Department of Economics, University of Iowa

Iowa City, IA 52242

\smallskip E-mail: tedt@blue.weeg.uiowa.edu

\smallskip\ 

First version: September 1996; This version: September 1997.

\newpage\ 

\ {\bf Abstract}
\end{center}

\noindent We introduce an element of centralization in a random matching
model of money that allows for private liabilities to circulate as media of
exchange. Some agents, which we identify as banks, are endowed with the
technology to issue notes and to record-keep reserves with a central
clearinghouse, which we call the treasury. The liabilities are redeemed
according to a stochastic process that depends on the endogenous trades. The
treasury removes the banking technology from banks that are not able to meet
the redemptions in a given period. This, together with the market
incompleteness, gives rise to a reserve management problem for the issuing
banks. We demonstrate that ``sufficiently patient'' banks will concentrate
on improving their reserve position instead of pursuing additional issue.
The model provides a first attempt to reconcile limited note issue with
optimizing behavior by banks during the National Banking Era.

\newpage\ 

\section{Introduction}

While there are several interesting questions concerning private money, a
useful framework for studying the operation of private monetary systems and
the implications of interventions into such systems has not yet been
developed.\footnote{%
See, for example, Fischer (1986) and King (1983) for a discussion of some of
the issues and challenges in modelling private money.} Hayek has argued that
private money would have a positive effect against sustained inflation by
subjecting the government to the discipline of competition. However, other
economists are skeptical about the stability of a competitive monetary
system: How can a stable real-valued currency emerge if, having established
a currency, the supplier can produce more at zero cost? A satisfactory
answer to this question has not been offered. For this reason, the theory on
the workings of a private monetary system is commonly seen as inherently
difficult.

In this paper, we introduce a model of private money and show its potential
usefulness by demonstrating that a stable monetary system can emerge in a
way that resembles the conservative note issue by banks during the National
Banking Era in the United States (1863-1913). This period provides a
challenge for any model of private money. Standard theory suggests that if
banks can issue their own currency, and if the public treats the private
currency as a perfect substitute to lawful money, then banks will overissue
notes unless they are obliged, by law, to back them by 100\% reserves in
lawful money. According to this argument, if the public does not distinguish
between private bank notes and fiat money, a bank should always be able to
exchange any amount of its redeemed notes with fiat money from the
indifferent public, keeping them, in effect, outstanding. In other words,
even if banks have to provide an amount of lawful money equal to the amount
of their notes redeemed in any given period, this will not cause a problem
for them as long as the amount of lawful money in the economy exceeds the
amount of notes redeemed in any given period.

One problem with this conclusion is that, apparently, it is inconsistent
with at least one important historical episode. When banks faced this
opportunity during the National Banking Era, they did not issue as many
notes as the collateral and other restrictions in place during that period
would have permitted.\footnote{%
This has been identified as a puzzle. See, for example, Friedman and
Schwartz (1963).} Champ, Wallace, and Weber (1994) carried out one of the
recent studies on the question of underissue during this period and
concluded that the note-issuing banks were concerned about the
demand-liability feature of their outstanding notes, which points to a
reserve management problem faced by banks. Under a complete markets
assumption, theory predicts overissue since banks can then always keep any
amount of their notes outstanding. Thus, these models are not able to
reconcile the public acceptance of the private notes with the facts about
underissue during this period.

In this paper, we build a model of private money in the decentralized
markets framework introduced by Kiyotaki and Wright (1989). Indeed, little
needs to be added to the standard random matching model of money in order
for inside money in the form of private bank notes to circulate as media of
exchange and for a reserve management problem to arise for the note-issuing
banks. Our model contains the following two features. First, the
no-note-issuing public treats all notes and fiat money as perfect
substitutes. Second, banks cannot keep the entire amount of their notes
outstanding, since notes are redeemed as a result of the random and
endogenous trades. The same market incompleteness that gives rise to money
as a medium of exchange thus creates a reserve management problem for banks.
In other words, banks are subject to frictions when raising the funds
required to meet the random redemptions of their notes. We demonstrate that
solving this problem makes it possible to reconcile optimizing behavior with
limited note issue.

In our model, a subset of the agents is endowed with the ability to Fssue
liabilities in order to finance consumption during encounters with producers
of their preferred good. We will identify these agents with banks and the
liabilities with bank notes. Banks also have the capability of recording
with a clearinghouse, the treasury, their earnings from money accepted in
decentralized trades. This technology enables them to build reserves by
making deposits with the treasury, but it also creates a redemption process,
since circulating notes might get redeemed in any given period if they are
deposited as reserves of other banks. We assume that the treasury removes
the banking technology from banks that hold an amount of reserves smaller
than the amount of notes redeemed in a given period. Only a fraction of the
outstanding notes of a bank is redeemed in any given period, so this rule is
much weaker than, say, 100\% reserve requirements. Since banks do not have
the opportunity to always keep their issued notes outstanding, they must be
concerned with the amount of reserves they hold, keeping in mind the
probability of losing their note-issuing privilege if caught with negative
reserves.

We demonstrate the existence of a monetary steady state equilibrium where
liabilities circulate as private media of exchange, as a perfect substitute
for fiat money. We characterize the optimal policy rules for banks and find
that, for high discount factors, banks will limit their note issue. Indeed,
concerned about the amount of their notes redeemed, banks in some states
forgo consumption and do not issue new notes while, at the same time, they
suffer the disutility of production in order to improve their reserve
position. Finally, we demonstrate that if banks discount the future at a
sufficiently low rate, they will never issue notes unless they can back them
in reserves. Thus, 100\% reserves may arise endogenously as a special case
in our model. We also find that for low discount factors, most banks will be
``illiquid'' and may even display ``wildcat behavior,'' i.e., they might
find it optimal to concentrate on the short-run benefits of note issue and
eventually exit the banking sector by failing to honor redemptions. Perhaps
not surprisingly, our results on a stable monetary system depend on
conditions guaranteeing the long-run profitability of banks. More precisely,
three features of our model guarantee that the frequency of consumption for
banks is higher than that of non-banks. First, there is limited entry in the
banking sector. Second, since we do not impose 100\% reserve requirements,
banks have access to a form of borrowing through the ``floating'' of notes,
which is not available to non-banks. Finally, the banking technology allows
for the accumulation of reserves (in the form of record-keeping), while
non-banks can hold at most one unit of money. Although the upper bound on
money holdings greatly simplifies the analysis, we conjecture that it is not
essential for generating a profitable bank sector, especially in the
presence of the generous reserve requirement assumed here.

If we interpret private note issue as a form of credit, our model makes the
methodological contribution of introducing credit in a
lack-of-double-coincidence model of money. In the Kiyotaki-Wright model,
market incompleteness, as modelled by a random matching technology, together
with the assumption that agents' histories are not part of a public record,
creates a role for money as a medium of exchange. However, in a world with
fully decentralized markets and with agents' histories being private
information, arrangements such as credit cannot exist. Diamond (1990), Shi
(1996), and Aiyagari and Williamson (1997) provide alternative ways of
introducing credit in a search framework, but in none of these models do
liabilities circulate as private media of exchange, a necessary condition
for them to be interpreted as private money.

The important question of whether this arrangement is part of an optimal one
is left for future research. A standard Kiyotaki-Wright type model follows
as a special case of ours. This leads us to the conjecture that the
introduction of private liabilities might, at least in the cases where
outside money is scarce, lead to improved welfare. We further speculate on
this issue in our Conclusions section. The paper proceeds as follows:
Section II describes the economic environment. Section III contains the
steady state value functions. Section IV characterizes the optimal policy
rules and deals with the existence of a steady state monetary equilibrium.
Section V concludes the paper. The appendix contains some of the proofs.

\section{The Economic Environment}

Time is discrete, $t,$ measured over the positive integers. There is a $[0,%
\frac{1}{k}]$ continuum of each of $k$ types of infinitely lived agents, and
there are $k\geq 3$ indivisible perishable goods. The total measure of
agents in the economy is 1. Agents are specialized in production and
consumption of goods. Agents of type $i$ consume good $i$ only and produce
good $i+1$ only ($\func{mod}$ $k$). All agents are expected utility
maximizers, and the ``discount factor'' is $\tilde{\beta}$, a positive
number. Agents are randomly matched pairwise, once in every period. As is
common in this literature, the assumptions on specialization rule out double
coincidence meetings. The only storable assets are indivisible money objects
that can be either government or private money. Each person has a storage
capacity of one unit of money, government or private, and can produce at
most one unit of good. We let $m_{fiat}$ denote the fraction of each
consumption type that is holding government money. Consumption of one unit
of good gives utility $u,$ and production of one unit gives disutility $e$.
We assume that $u>e.$

An uncountable subset of the $[0,\frac{1}{k}]$ continuum for each type is
endowed with the ability to costlessly issue one unit of liability per
period in exchange for purchases of goods. We will identify these agents
with {\it banks} and will refer to these liabilities as {\it bank notes}. We
assume that private notes are treated as a perfect substitute for each other
and for outside money in all trades and demonstrate that this assumption is
consistent with a steady state equilibrium. While there might exist other
monetary equilibria where some or all notes are not accepted as a substitute
of outside money, we thus concentrate on the steady state equilibrium that
is consistent with historical observations from the National Banking Era.

Banks also have a capacity to hold, at most, one unit of outside money but
have the ability to record-keep any notes or fiat money they earned from
production with a central location. We will identify this storage as {\it %
reserves deposited with} {\it the} {\it treasury.} Given the production
technology for goods, these assumptions imply that one unit of money (fiat
or notes) is exchanged for one unit of goods in all decentralized trades,
i.e., prices are exogenous. We let $d$ denote the (integer) amount of
reserves for a bank agent. Except for their ability to issue notes and
record reserves, banks in this model are identical to non-banks. Notes
issued circulate as private media of exchange until they are deposited by
other banks. We let $m$ be the (integer) amount of its own notes in
circulation for a bank. We assume that the treasury acts according to the
following exogenous rule: whenever a note issued by bank $i$ is deposited by
bank $j,$ the account of the depositing bank $j$ is credited and the account
of the issuing bank $i$ is debited by one. We identify this process with a 
{\it redemption process} {\it for the notes}. This note subsequently becomes
a mere record-keeping device.\footnote{%
Clearly, there are many ways to model the treasury's information in this
model. We keep this information minimal, assuming that the treasury knows
banks' reserves but not their number of notes in circulation.} If a bank has
a negative balance in a given period, i.e., if the amount of its own notes
deposited by other banks with the treasury in that period exceeds its
reserves, the treasury adopts the exogenous policy of depriving it of the
banking technology. In this case, the bank becomes a non-bank in the model
and is given one unit of fiat money upon exit.\footnote{%
This is only a simplifying assumption. It guarantees that banks always
deposit every unit of fiat money or notes they earn so that we need to keep
track of one less state variable.} We let $q\in [0,1]$ be the fraction of
banks in the population and in each consumption type that exits the banking
sector because of a negative balance in each period. Notes issued by banks
that exit the industry might circulate in periods after the exit. We assume
that the treasury and, therefore, all other agents in the model will honor
such notes. Even though we do not model collateral explicitly in the model,
this assumption captures the feature that bank notes during the National
Banking Era had an implicit government guarantee, so that note holders were
not facing any substantial risk from failures of note-issuing banks.%
\footnote{%
During the National Banking Era, notes were fully backed by purchases of
U.S. government bonds. The bonds were paying interest that, absent any
hidden costs, made note issue a most profitable investment for banks. One
question is why the entire amount of eligible bonds was not used in order to
support note issue during most of that period (see Friedman and Schwartz
(1963)). Here, we explore the implications of having the market for putting
redeemed notes back in circulation {\it immediately} being ``missing.''}

Since exit from the banking sector is possible, for a steady state where the
size of this sector is constant, we will also need entry as well as exit
from the non-banking sector. We assume that with probability $\delta \in
(0,1),$ each agent in the model dies and is replaced by a newborn. In the
case where the exiting agent holds a note, the treasury redeems that note
and adjusts the reserve balances.\footnote{%
Our results would go through if we assumed that agents' money holdings
``disappear'' after their deaths. We, however, find our current assumption
more convenient to work with.} We let $\mu _{1}\in (0,1)$ be the fraction of
newborns that are banks. Banks enter the economy with $(d,m)=(0,0).$ We also
assume that a fraction, $\mu _{2},$ of the non-bank newborns enters with one
unit of money holdings. We let $\mu =\mu _{1}+\mu _{2}.$ We let $\beta
=(1-\delta )\tilde{\beta}\in (0,1)$ be the effective discount factor.%
\footnote{%
Notice that this specification does not require that $\tilde{\beta}$ be less
than 1.} Banks and non-banks alike trade after the period starts. Each
agent's type, including whether it is a bank or not, is private information.
If a bank knew that it is dealing with another bank, then it might want to
avoid issuing a note because this note would be redeemed in the current
period if issued. With individual banking privileges being private
information, we need to consider one fewer decision variable. We will
concentrate on steady state equilibrium outcomes where all private notes are
accepted as perfect substitutes. As mentioned above, the upper bound and
indivisibility assumptions make prices exogenous, i.e., one unit of a good
is always exchanged for one unit of fiat money or a note. For simplicity, we
will rule out money-for-money trades.\footnote{%
Clearly, a bank has an incentive to exchange one of its own notes for a note
issued by another bank. However, since the non-bank is indifferent between
the two, the case where all non-banks refuse such trades is consistent with
an equilibrium. Including such money-for-money trades will not change the
results, provided that the same upper bound conditions hold.} After trades
occur, and since we assume that banks that exit because of a negative
balance are given one unit of fiat money, banks will always deposit their
earnings with the treasury. The reserve balances are then adjusted. Finally,
right before the new period starts, each bank is informed of its new reserve
balance, from which each can infer the amount of its notes remaining in
circulation. The timing of actions within a period is as follows:

\begin{center}
\begin{equation}
t:\mapsto [v_{d,m}]\mapsto trade\mapsto deposits\mapsto deaths\mapsto
[w_{d,m}]\mapsto balance\text{ }adjustment\mapsto :t+1,
\end{equation}
\end{center}

\smallskip

\noindent where $v_{d,m}$ is the steady state value function of a bank of
type $(d,m)$, and $w_{d,m}$ is the steady state value function after a bank
has deposited its earnings for the period, but before the redemption process
and the balance readjustment occur. For non-banks, the state is $m\in
\{0,1\},$ and in each period, they choose the probability $\alpha \in [0,1]$
of accepting money (fiat or notes) in exchange for producing. For banks, the
state is $(d,m)\in {\Bbb N}\times {\Bbb N},$ and in each period, they choose 
$(\gamma _{d,m},\phi _{d,m})\in [0,1]\times [0,1],$ where $\gamma _{d,m}$ is
the probability of accepting money in exchange for producing and $\phi
_{d,m} $ is the probability of issuing money in exchange for consuming, at
state $(d,m)$.\footnote{%
Recall that, being consistent with observations from the National Banking
Era, we concentrate on arrangements where all notes are treated
symmetrically in decentralized trades. Thus, we do not index the agents'
choice variables by the type of notes they are offered.} We let $x_{d,m}$ be
the measure of banks of type $(d,m)$ within each consumption type and,
therefore, in the population. We define $\gamma x=\sum_{d,m}\gamma
_{d,m}x_{d,m}$ and $\phi x=\sum_{d,m}\phi _{d,m}x_{d,m}$ to be the fraction
of banks that are willing to increase their reserves by producing and the
fraction of banks that are willing to issue a new note in exchange for
consumption, respectively, within a given period. Similarly, $c_{i}$ stands
for the measure of non-banks with 0 or 1 unit of money, respectively.
Feasibility requires that $c_{0}+c_{1}+\sum_{d,m}x_{d,m}=1.$ In order to
calculate the stationary measure of banks, observe that the fraction of
banks tomorrow equals the fraction of banks today minus the exogenous
fraction of banks that die, minus the endogenous fraction of banks that exit
because of a negative balance, plus the fraction of newborns that are banks.
In other words, $\Sigma x_{d,m}^{^{\prime }}=(1-\delta )(\Sigma
x_{d,m})-q+\delta \mu _{1}.$ Thus, provided that $q\leq \frac{\delta \mu _{1}%
}{1-\delta },$ the steady state measure of banks is $\mu _{1}-\frac{1-\delta 
}{\delta }q.$

Since a bank will always find it optimal to deposit with the treasury every
unit of money it earns, a note that is already in circulation is redeemed if
it is earned by another bank in exchange for production or if it is in the
hands of a non-bank that dies.\footnote{%
Since there is a continuum of banks, we will ignore the case where a bank is
presented with one of its own notes.} Therefore, $\pi ,$ the probability
that any note is redeemed, is given by: $\pi =\delta +(1-\delta )\frac{%
\gamma x}{k}.$ Hence $\pi $\ is bounded below by $\delta .$\ \ Because there
is a constant inflow of non-banks with and without money in the economy, $%
c_{0}$ and $c_{1}$ are also bounded away from zero. In what follows, we will
concentrate on equilibria that are symmetric across agent types.

\smallskip\ 

\section{Steady State Value Functions}

In this section, we describe the steady state value functions. We let $s_i$
be the value of a non-bank with $i\in \{0,1\}$ units of money. First, for a
non-bank with one unit of money holdings we have

\begin{center}
\begin{equation}
\ s_1=\frac 1k\left( \alpha c_0+\gamma x\right) (u+\beta s_0)+\left[ 1-\frac
1k\left( \alpha c_0+\gamma x\right) \right] \beta s_1.
\end{equation}
\end{center}

\noindent The first part of the above equation gives the expected payoff
from a trade as a consumer in a single-coincidence meeting with a producer.
The second part describes the expected payoff if no such trade occurs. For a
non-bank with zero units of money holdings we have

\begin{center}
\begin{equation}
\ s_0=\frac 1k\left( c_1+\phi x\right) \max_\alpha \left[ \alpha (-e+\beta
s_1)+(1-\alpha )\beta s_0\right] +\left[ 1-\frac 1k\left( c_1+\phi x\right)
\right] \beta s_0.
\end{equation}
\end{center}

\noindent The first part of the above equation gives the expected payoff of
a non-bank from a trade as a producer in a single-coincidence meeting with a
consumer. The second part gives the expected payoff if no such trade occurs.
Next, the symmetric steady state value function for a bank of type $(d,m)$
is given by

\begin{center}
\[
v_{d,m}=\frac 1k\left( c_1+\phi x\right) \max_{\gamma _{d,m}}\left[ \gamma
_{d,m}(-e+\beta w_{d+1,m}^0)+(1-\gamma _{d,m})\beta w_{d,m}^0\right] 
\]

\[
\ +\frac 1k(\alpha c_0+\gamma x)\max_{\phi _{d,m}}\left\{ \phi _{d,m}\left[
u+p\beta w_{d-1,m}^0+\ (1-p)\beta w_{d,m}^1\right] +(1-\phi _{d,m})\beta
w_{d,m}^0\right\} 
\]

\begin{equation}
+\left[ 1-\frac 1k\left( c_1+\phi x\right) -\frac 1k(\alpha c_0+\gamma
x)\right] \beta w_{d,m}^0.
\end{equation}
\end{center}

\smallskip\ 

\noindent The probability of redemption for a note in a given period depends
on whether the note is issued in that period or whether it was already in
circulation. A newly issued note is redeemed if it is accepted by a bank or
if it is held by a non-bank that dies. We denote the probability of this
event by $p,$ where $p=\ \frac{\gamma x}{\gamma x+\alpha c_{0}}+\frac{\delta
\alpha c_{0}}{\gamma x+\alpha c_{0}}.$ For a note already in circulation the
probability of redemption is $\pi ,$ as defined earlier. The first part of
the value function describes the expected payoff of a bank of type $(d,m)$
from a trade as a producer in a single coincidence meeting with a consumer.
The second part of the equation gives the expected payoff of a bank of type $%
(d,m)$ from a trade as a consumer in a single-coincidence meeting with a
producer. The last part describes the expected payoff if no such trades
occur. We let $w_{d,m}$ be the value function after the bank has deposited
the earnings for the period with the treasury, but before the redemption
process and the balance readjustment occur. The index $j$ takes value 1 if a
bank issues a note in a given period, and value 0 otherwise. We then have:

\begin{equation}
w_{d,m}^{j}=\sum_{0\leq i\leq \min \left\{ d,m\right\} }p(i,m)v_{d-i,m+j-i}\
+\sum_{d<i\leq m}\text{ }I_{d,m}p(i,m)s_{1},
\end{equation}

\smallskip \noindent \noindent where $p(i,m)=\binom{m}{i}\pi ^{i}(1-\pi
)^{m-i},$ $j=1$ if a new note enters circulation, while $j=0$ otherwise, and 
$I_{d,m}=\left\{ 
\begin{array}{lll}
1, & if & m>d; \\ 
0, & if & m\leq d.
\end{array}
\right. $ In the expression above, the first sum corresponds to the
possibility that the notes redeemed do not exceed the bank's reserves and,
therefore, the bank remains in business. The second sum corresponds to the
possibility that the notes redeemed from the treasury in that period exceed
the bank's recorded reserves and, therefore, the bank will exit the sector
in the next period. The number of notes redeemed within a period is a random
variable following a binomial distribution, and the bank takes the
redemption probability, $\pi $, as given.\ In order to define a steady state
equilibrium for this economy, we first need to consider the law of motion of 
$x_{d,m}$. In the appendix, we demonstrate how the optimal decision rules
define an operator, $Q$, mapping the distribution of banks across states at
the beginning of the period to distributions of banks across states at the
end of the period. As we mentioned before, we assume that those banks that
end the period with a negative balance will exit the industry. To capture
this fact, we also define an operator $T$\ mapping the distribution of bank
types at the beginning of a period to the distribution at the beginning of
the next period. At a steady state equilibrium we have that $x=Tx,$\ and the
measure of banks that exit the industry is given by $q=\sum_{\{d<0,\text{ }%
m\geq 0\}}Qx_{(d,m)}.$ We have the following:

\bigskip

\bigskip

\smallskip

\noindent {\bf Definition:}{\it \ A symmetric steady state equilibrium is a
set of value functions} $\left\{ s,v\right\} ${\it \ with} $%
s:\{0,1\}\rightarrow {\Bbb R}$ {\it and} $v:{\Bbb N}\times {\Bbb N}%
\rightarrow {\Bbb R},$ {\it together with a set of policy functions,} $%
\{\alpha ,\gamma ,\phi \},$ {\it a distribution over non-banks and banks,} $%
\{x,c_{0},c_{1}\},$ {\it where} $x:{\Bbb N}\times {\Bbb N}\rightarrow [0,1]$ 
{\it and }$c_{0},c_{1}\in [0,1],$ {\it a measure of fiat money in
circulation, }$m_{fiat},${\it \ and a probability of a note being redeemed,} 
$\pi \in [0,1],$ {\it such that}

$(1)$ $s_{0},s_{1}$ {\it are the solutions to the functional equation for
non-banks when} $\alpha ${\it \ is optimal.}

$(2)$ $v${\it \ is the solution to the functional equation for banks when} $%
\gamma $ {\it and} $\phi $ {\it are optimal.}

$(3)$ $x=Tx.$

$(4)$ $c_{1}=(1-\delta )\left[ \frac{c_{1}+\phi x}{k}\alpha c_{0}+\left( 1-%
\frac{\gamma x+\alpha c_{0}}{k}\right) c_{1}\right] +q+\delta \mu _{2}.$

$(5)$ $c_{0}=(1-\delta )\left[ \left( 1-\frac{c_{1}+\phi x}{k}\right) c_{0}+%
\frac{\gamma x+\alpha c_{0}}{k}c_{1}\right] +\delta (1-\mu ).$

$(6)$ $\pi =\delta +(1-\delta )\frac{\gamma x}k,$ $q=\sum_{\{(m,d):d<0,\text{
}m\geq 0\}}Qx_{(d,m)}.$

$(7)$ $q+\delta \mu _{2}=\delta m_{fiat}+(1-\delta )m_{fiat}\frac{\gamma x}{k%
}.$

$(8)$ $c_0+c_1+\sum_{d,m}x_{d,m}=1,$ $\sum_{d,m}mx_{d,m}+m_{fiat}=c_1.$

\smallskip\ 

Conditions (1) and (2) above are self-explanatory. Conditions (3),(4) and
(5) require that the agents' behavior is consistent with the steady state
distribution over banks and non-banks. Condition (6) requires that the
redemption probability and the measure of banks that exit are consistent
with the agents' optimal behavior. Condition (7) equates the inflow to the
outflow of fiat (outside) money in the economy. Finally, condition (8)
describes aggregate feasibility. Next we move to the question of
characterizing the banks' optimal policies at steady state.

\smallskip\ 

\section{Characterizing Policy Rules}

In this section, we characterize the optimal policy rules for the agents in
the economy and demonstrate the existence of a monetary steady state
equilibrium. The first Lemma gives a sufficient condition for non-banks to
accept money (fiat and private) in exchange for service in a symmetric
steady state equilibrium. This will be helpful in showing that a monetary
equilibrium exists.

\smallskip\ \ 

\noindent {\bf Lemma 1:} {\it Suppose that} $\frac{1}{k}\delta (1-\mu )\geq
\left( \frac{1}{\beta }-1\right) \frac{1}{\frac{u}{e}-1}$. {\it Then,} {\it %
non-banks accept money if other non-banks do; so}{\sl \ }$\alpha =1.$

\smallskip\ 

Recall that $\mu =\mu _{1}+\mu _{2}.$ Note that the condition of Lemma 1
will tend to be satisfied if the discount factor is large and if the
fraction of newborns that are banks is small. If the fraction of agents that
can issue notes is too high, a steady state equilibrium where notes are
valued might not exist. It is also intuitive that non-banks will tend to
accept money when the ratio $u/e$ is sufficiently high. The next Lemma
establishes the monotonicity properties of the value function for banks. It
asserts that the value function is increasing in the amount of reserves and
decreasing in the amount of notes in circulation.

\smallskip\ \ 

\noindent {\bf Lemma 2:} {\it The value function }$v_{d,m},${\it \ is (a)
weakly increasing in} $d$ {\it and (b)} {\it weakly decreasing in} $m$.

\smallskip\ 

The next Lemma says that the value of a bank is always greater than the
value of a non-bank with one unit of money holdings which, in turn, is
greater than the value of a non-bank with no money holdings. Banks in our
model can do everything that non-banks can, and, in addition, they can
record earnings in reserves and ``borrow'' by costlessly issuing new
liabilities. This additional value of remaining in business will make the
reserve management problem meaningful.

\bigskip

\bigskip

\smallskip

\noindent {\bf Lemma 3:} $s_{0}<s_{1}<v_{d,m}$ for all $d$ and all $m$.

\smallskip

Our goal in this section is to characterize the behavior of banks at a
monetary steady state. We build toward the characterization through a
sequence of claims describing the optimal policy rules for banks throughout
the state space. A summary of this exercise is presented in Figure 1. First,
we find it convenient to prove a Lemma for the following special case.

\smallskip\ 

\noindent {\bf Lemma 4:} {\it If} $\beta $ {\it is sufficiently large, then} 
$\phi _{0,0}=0${\it .}

\smallskip\ 

The above Lemma provides a sufficient condition for a bank at state $(0,0)$\
to choose not to issue a note in exchange for consumption if this note is
not backed, in order to avoid a positive probability of having to exit.
Recall that exit will occur if the producer in this meeting is another bank
and thus the note gets redeemed. Although Lemma 4 is about a special case,
we will find that its proof can be used in order to prove more general
propositions later. Here is some intuition for the proof. By issuing a note
at state{\bf \ }$(0,0),$ a bank faces the possibility of having negative
reserves and being forced to exit. The immediate gain of issuing a note is
equal to the utility of consumption, $u.$ As agents become more patient, the
difference in utility between being a bank or a non-bank grows unboundedly.
Thus, by issuing a note a bank faces an arbitrarily large utility loss with
positive probability. This cost outweighs the short-run gain of printing a
note today. A higher $\beta $\ can, therefore, reduce liquidity in the
economy. Furthermore, as the discount factor approaches 1, a self-imposed
100\% reserve requirement becomes the optimal policy for most banks in a
steady state. The proof of Lemma 4 explores the fact that any small entry of
non-banks with money in every period $(\mu _{2}>0)$ suffices to provide a
positive lower bound on the liquidity in the economy and, as a result, on
the probability that a newborn bank will accumulate enough reserves to enjoy
consumption.

{\bf \ }This behavior originates from assumptions guaranteeing that, on
average, banks have a higher frequency of consumption than non-banks. First,
there is limited entry in the banking sector. Second, since we do not impose
100\% reserve requirements, banks have access to a form of borrowing through
the floating of notes, which is not available to non-banks. Third, the
banking technology allows for the accumulation of reserves (in the form of
record-keeping), while non-banks can hold at most one unit of money. The
next proposition provides a benchmark case. It says that if banks discount
the future at a sufficiently high rate, they will always choose to borrow
and consume today. The proof is trivial and, thus, omitted.

\smallskip\ 

\noindent {\bf Proposition 1:}{\it \ If} $\beta ${\it \ is sufficiently
small, then }$\phi _{d,m}=1$ {\it for all} $d$ {\it and all} $m$.

\smallskip\ 

The next proposition characterizes the optimal behavior of banks around the
45$^{0}$ line in the $(d,m)$ space for different values of the underlying
parameters (see Figure 1). States above this line correspond to ``illiquid''
banks in the sense that for banks in this region the amount of notes
outstanding is greater than the amount of reserves. States below this line
have the opposite implication. Generally, a bank will choose to cross into
the illiquid zone provided that it is large enough, since, in that case, the
probability of a resulting negative balance in this region is small. At the
same time, if the discount factor is high enough, banks around the 45$^{0}$
line will also choose to produce in order to improve their reserve position.

\smallskip\ 

\noindent {\bf Proposition 2:}{\it \ (a) Fix} $d$.{\bf \ }{\it Then as }$%
\beta \rightarrow 1${\it , }$\phi _{d,d}=0.$

\noindent {\it (b) Fix }$\beta \in (0,1).$ {\it There exists }$D$ {\it such
that} $\phi _{d,d}=1$ {\it if} $d\geq D$.

\noindent {\it (c) Fix }$(d,m)${\it \ such that }$m<d.$ {\it Then }$\phi
_{d,m}=1.$

\smallskip\ 

The proof of part (a) follows from Lemma 1. For fixed $d$, a bank issuing a
note at state $(d,d)$ faces a probability of being caught with negative
reserves that is bounded away from zero. Thus, for beta large enough, $\phi
_{d,d}=0.$ The proof of part (b) relies on two facts. First, for $\beta $
less than one, the cost of exiting the industry is finite. Second, as $d$
grows, a bank issuing a note at state $(d,d)$ faces a probability of being
caught with negative reserves that is close to zero. Part (c) is established
by comparing the immediate gain of issuing an additional note (i.e., the
utility of consumption, $u$) with the loss from having one extra note in
circulation. It is shown that when $m$ is less than $d,$ this loss is less
than $u$ in present value terms. This is true since, when $m$ is less than $%
d,$ the value function satisfies the condition that $v_{d,m}-v_{d,m+1}<u/%
\beta .$

The next proposition characterizes the behavior of banks that are ``too
liquid'' or ``too illiquid'' in the sense that either $d-m$ or $m-d$ are
positive and large, respectively. Banks that have too many reserves compared
with the amount of their notes outstanding will concentrate on issuing more
notes instead of building additional reserves. Perhaps not surprisingly,
banks that have too few reserves compared with the amount of their notes
outstanding will do the same, since they expect to exit the banking sector
with probability one in the near future.

\smallskip\ \ 

\noindent {\bf Proposition 3:}{\it \ (a) Fix} $m$. {\it There exists a }$%
D_{m}$ {\it such that for all} $d\geq D_{m},$ $\phi _{d,m}=1.$

\noindent {\it (b) Fix }$m$. {\it There exists a }$D_{m}${\it \ such that
for all }$d\geq D_{m},$ $\gamma _{d,m}=0.$

\noindent {\it (c) Fix }$d$. {\it There exists an} $M_{d}$ {\it such that
for all }$m\geq M_{d},$ $\phi _{d,m}=1.$

\noindent {\it (d) Fix }$d$. {\it There exists an} $M_{d}${\it \ such that
for all }$m\geq M_{d},$ $\gamma _{d,m}=0.$

\smallskip\ 

The next proposition asserts that if $\beta $ is high enough and $d<m,$
banks will always work toward improving their reserve position.{\bf \ }The
proof follows from a similar argument to that in the proofs of Propositions
2 and 3. An increase in the reserve position implies a discrete reduction in
the probability of being caught with negative reserves. Since, as $\beta $
approaches 1, the loss of exiting the banking industry grows unboundedly, it
is worth it for a bank to suffer the disutility of production ($e$) and to
forfeit the utility of consumption ($u$) in order to improve its reserve
balance.

\smallskip\ \ 

\noindent {\bf Proposition 4:} {\it Fix} $(d,m)${\it , such that }$d<m.${\it %
\ As} $\beta $ {\it approaches 1, we have}

\noindent {\it (a) }$\gamma _{d,m}=1$.

\noindent {\it (b)} $\phi _{d,m}=0$.

\smallskip\ 

The statements in the above propositions are summarized in Figure 1. As the
values of $d$ and $m$ vary, the optimal policy rules give rise to four
regions in the $(d,m)$ space. In region I, the bank's reserves are high
compared to the number of its notes in circulation. In that case, a bank
will find it optimal to issue a note when faced with the opportunity, thus
increasing the number of its notes in circulation. At the same time, such a
bank will reject opportunities to increase its reserves. Banks in region II
will still find it optimal to put a new note in circulation, given the
opportunity, but being less liquid, they will now also accept trades that
increase their reserves. Banks in region III are becoming alarmingly
illiquid and will find it optimal to both improve their reserves and stop
issuing new notes. In other words, concerned about the possibility of having
to give up their note issuing privilege, these banks will concentrate on
improving their reserve position. Finally, banks in region IV have too few
reserves compared to the number of their notes in circulation and will thus
have to exit the banking sector with high probability in the near future.
These banks would not benefit from increasing their reserves since
redemptions will arrive at a faster rate, making them even less liquid than
before. They will, therefore, exhibit ``wildcat banking'' behavior and will
only issue new notes until they are forced to exit due to a negative
balance. Notice that in all four cases, the redemption process reduces both
reserves and the notes in circulation by the same number and thus always
moves a bank southwest, along a 45$^{0}$ line. While banks in our model will
not enter region IV voluntarily, we cannot rule out the possibility that the
redemption process might bring them there from another region. In that case,
they will never exit this region before they exit the banking sector.

Given our assumptions on the technology and the treasury policy, and given
the optimal policy rules for banks, the next Lemma asserts that at each
point in time, all but an arbitrarily small number of banks will have a
bounded number of notes in circulation and, therefore, a bounded number of
notes in reserves. Recall that a bank can issue at most, one note per
period, and the probability of a note's being redeemed is bounded away from
zero. As the number of notes in circulation becomes large, the fraction of
these notes redeemed approaches a constant, and the number of notes redeemed
becomes greater than 1 for all but an arbitrarily small fraction of banks,
which we will ignore. Let $\chi $ be an{\it \ }invariant distribution of
banks across states{\it \ }$(d,m).$

\smallskip\ 

\noindent {\bf Lemma 5:}{\it \ For all }$\epsilon >0,$ {\it there exists an }%
$M${\it \ large enough such that for all }$d,${\it \ }$\sum_{d,m\geq M}\chi
_{d,m}<\epsilon $ {\it and for all }$m,${\it \ }$\sum_{d\geq M,m}\chi
_{d,m}<\epsilon .$

\smallskip\ \ 

Next, Proposition 5 says that there exists a unique invariant distribution
on the state space, associated with a symmetric monetary equilibrium. An
arbitrarily small probability of deaths guarantees that transition
probabilities to any state are bounded away from zero. This, in turn,
implies that the mapping from the distribution of banks across $(d,m)$ types
today to the distribution tomorrow satisfies a contraction property. For the
proof of this proposition we impose an upper bound on the state space for
banks. The previous Lemma suggests that this bound, if large enough, will
not bind but for an arbitrarily small number of banks. In addition,
Proposition 5 asserts that in a monetary equilibrium, a positive fraction of
banks will both issue and accept private liabilities.

\smallskip\ 

\noindent {\bf Proposition 5:}{\it \ Suppose that a monetary equilibrium
exists. This equilibrium is characterized by a unique invariant
distribution, }$\chi ,${\it \ of banks across states }$(d,m).$ {\it %
Furthermore,} {\it if }$\beta ${\it \ is sufficiently large and }$e${\it \
is sufficiently small, }$\gamma x>0$ {\it and }$\phi x>0$ {\it .}

\smallskip \ $\ $

Proposition 6 asserts the existence of a steady state monetary equilibrium
with a nontrivial distribution over bank states. The proof is similar to the
one in Aiyagari and Wallace (1991) and, thus, omitted.

\pagebreak

\smallskip\ 

\noindent {\bf Proposition 6: }{\it For the parameters satisfying the
condition of Lemma 1, there exists a monetary steady state equilibrium with
trade and a non-degenerate asset distribution for the banks.}

\smallskip\ 

Let $NB=\sum_{\{(d,m):d<m\}}(m-d)x_{d,m}$ be the total amount of circulating
notes that are not backed by reserves in the economy. The following
proposition suggests that for low enough discount factors, banks will issue
notes not backed in reserves, while if the discount factor is sufficiently
high, they will remain completely liquid, in which case there will be no
exit from the sector because of negative reserves. It is worth mentioning
that the last case is consistent with 100\% reserves arising endogenously as
part of an equilibrium outcome.

\smallskip\ \ \ 

\noindent {\bf Corollary:}{\it \ (a) If} $\beta ${\it \ is high enough,
there exists a steady state equilibrium with }$NB=0.$

\noindent {\it (b) If} $\beta ${\it \ and }$e$ are {\it low enough, there
exists a steady state equilibrium with} $NB>0.$\ 

\smallskip\ \ 

Figure 1 provides the intuition for part (a). Proposition 2(a) implies that
for high enough discount factors, banks will never cross the 45$^{0}$ line
in the $(d,m)$ space (with $d$ and $m$ less than a fixed upper bound $M).$
In this case, there are no illiquid banks in equilibrium. To see why the
claim in part (b) holds, consider the limit case where $e=0.$ Notice that
the condition in Lemma 1 is satisfied for all $\beta \in (0,1).$ Then
Proposition 6 guarantees the existence of a monetary steady state. Using our
characterization of the decision rules, we can conclude that, for $\beta $
sufficiently low, there is a positive probability that a newborn bank will
become illiquid in some finite time. Since there is a positive measure of
banks at state $(0,0),$ we can then conclude that there will be a positive
number of illiquid banks. By a continuity argument, we can then argue that
there exists an equilibrium with illiquid banks for $e$ close to zero.

\section{Conclusions}

We studied private money issue and redemption in a version of the framework
of the Kiyotaki-Wright model. In our model, a departure from the extreme
decentralized markets allows for private liabilities to circulate as media
of exchange in a symmetric steady state equilibrium. The optimal policy
rules, as summarized in Figure 1, suggest that liability issuers in certain
states forgo consumption and suffer the disutility of production in order to
build a better reserve position. The market incompleteness leads
liability-issuing banks to be concerned about the amount of their
outstanding notes, even though the public views all notes and currency as
perfect substitutes. This suggests that rules less restrictive than 100\%
reserve requirements can be consistent with a money-issuing banking system
that is stable. Admittedly, the model is extreme in many respects. For
example, the market frictions of the matching model that we studied rule out
the possibility of redeemed notes being put back in circulation immediately,
or that borrowing is used in order to prevent a negative balance. Certainly,
the (limited) ability of banks to borrow or to keep issued notes outstanding
is not described by either extreme. However, as long as banks are not able
to keep the entire amount of their notes in circulation, they will be
concerned about their reserve position even when the public views all notes
and currency as perfect substitutes. In such cases, the reserve management
problem will be relevant.

Extensions of the basic model studied here could include studying the case
where all agents can have access to the note-issuing technology, as well as
the case where agents' types are publicly known in decentralized trades. In
the latter case, a bank might refuse to trade with another bank, knowing
that in this case redemption will occur instantly and, instead, might wait
for a meeting with a non-bank producer in order to consume. By introducing
divisible production, we could study endogenous price formation via a
bargaining protocol. This would complicate the model significantly, but it
should not affect any of our results. Another extension would be to allow
for the case where notes issued by banks that exit the industry cannot be
used as reserves and, therefore, there is some risk to the note holders. We
believe that most of our results will also hold true in that case.

One question is that of a welfare comparison between our model and, say, a
standard Kiyotaki-Wright model with the same steady state amount of outside
money. Although we do not explicitly study this question in this paper, it
is worth speculating. If the stock of outside money in the economy is small,
i.e., if outside money is scarce, the ability to issue inside money will
almost certainly enhance trade and, therefore, improve welfare.
Interestingly, this is consistent with historical observations about
scarcity of money and the introduction of private money in colonial America.%
\footnote{%
See, for example, Hanson (1979).}

Potentially important issues to study are the optimality properties of our
steady state equilibrium and alternative policy rules for the treasury. For
example, this framework seems suitable for a welfare comparison between
inelastic and elastic currency regimes in the presence of periodic cycles in
the demand for currency. We leave such questions for future research.

\smallskip\ 

\section{References}

\noindent Aiyagari R., Wallace, N. and Wright R. (1996): Coexistence of
Money and Interest-Bearing Securities. {\it Journal of Monetary Economics} 
{\bf 37}.

\noindent Aiyagari R. and S. Williamson (1997): Credit in a Random Matching
Model with Private Information. University of Iowa Working Paper \#97-03.

\noindent Aiyagari R. and N. Wallace (1991): Existence of Steady States with
Positive Consumption in the Kiyotaki-Wright Model. {\it Review of Economic
Studies }{\bf 58,} 901-916.

\noindent Champ B., N. Wallace, and W. Weber (1994): Interest Rates Under
the U.S. National Banking System. {\it Journal of Monetary Economics }{\bf 34%
}, 101-115.

\noindent Diamond P. (1990): Pairwise Credit in a Search Equilibrium. {\it %
Quarterly Journal of Economics} {\bf 105}, 285-320.

\noindent Freeman S. (1996): Clearinghouse Banks and Banknote Over-issue. 
{\it Journal of Monetary Economics} {\bf 38}, 101-115.

\noindent Fischer S. (1986): Friedman Versus Hayek on Private Money. Review
Essay. {\it Journal of Monetary Economics }{\bf 17}, 433-439.

\noindent Friedman M. and A. Schwartz (1963): A Monetary History of the
United States 1867-1960. Princeton, N.J.: Princeton University Press.

\noindent Green E. (1996): Money and Debt in the Structure of Payments.
Manuscript. Federal Reserve Bank of Minneapolis.

\noindent Hanson J.R. (1979): Money in the Colonial American Economy: An
Extension. {\it Economic Inquiry} {\bf XVII}, 281-286.

\noindent Hayek F.A. (1990): Denationalization of Money - The Argument
Refined. Third Edition. Institute for Economic Affairs, London.

\noindent King R.G. (1983): On the Economics of Private Money. {\it Journal
of Monetary Economics} {\bf 12}, 127-158.

\noindent Kiyotaki N. and R. Wright (1989): On Money as a Medium of
Exchange. {\it Journal of Political Economy }{\bf 97(4),} 927-954.

\noindent Salin P. (Ed.) (1984): Currency Competition and Monetary Union.
Financial and Monetary Policy Studies 8. Martinus Nijhoff Publishers.

\noindent Shi S. (1996): Credit and Money in a Search Model with Divisible
Commodities. {\it Review of Economic Studies }{\bf 63,} 627-652.

\noindent Taub Bart (1985): Private Money with Many Suppliers. {\it Journal
of Monetary Economics} {\bf 16}, 195-208.

\newpage

\section{Appendix}

\noindent {\bf The law of motion of }$x_{d,m}${\bf :} The optimal decision
rules define an operator, $Q$, mapping the distribution of banks across
states at the beginning of the period to distributions of banks across
states at the end of the period. Formally, let $G=\left\{ f:{\Bbb N}\times 
{\Bbb N}\rightarrow {\Bbb R}\right\} $ and $G^{^{\prime }}=\left\{ g:{\Bbb Z}%
\times {\Bbb N}\rightarrow {\Bbb R}\right\} $. Then we define the operator $%
Q:G\rightarrow G^{^{\prime }}$ by

\begin{center}
\[
Qx_{d^{^{\prime }},m^{^{\prime }}}=\sum_{d,m}(1-\delta )\frac{c_1+\phi x}%
k\left[ \gamma _{d,m}P_{d+1,m}^0(d^{^{\prime }},m^{^{\prime }})+(1-\gamma
_{d,m})P_{d,m}^0(d^{^{\prime }},m^{^{\prime }})\right] x_{d,m}\ 
\]

\[
\ +\sum_{d,m}(1-\delta )\frac{\gamma x+\alpha c_0}k\left\{ \phi _{d,m}\left[
pP_{d,m}^0(d^{^{\prime }},m^{^{\prime }})+(1-p)P_{d,m}^1(d^{^{\prime
}},m^{^{\prime }})\right] +(1-\phi _{d,m})\ P_{d,m}^0(d^{^{\prime
}},m^{^{\prime }})\right\} x_{d,m} 
\]

\[
+\sum_{d,m}(1-\delta )\left[ 1-\frac{c_1+\phi x}k-\frac{\gamma x+\alpha c_0}%
k\right] P_{d,m}^0(d^{^{\prime }},m^{^{\prime }})x_{d,m} 
\]

\begin{equation}
+\delta \mu x_{0,0}I_{\{(d^{^{\prime }},m^{^{\prime }})=(0,0)\}}.
\end{equation}
\end{center}

\smallskip\ 

The first sum in the above expression describes the transition to state $%
(d^{^{\prime }},m^{^{\prime }})$, provided the bank trades as a producer.
The second and third sums describe the transition to state $(d^{^{\prime
}},m^{^{\prime }}),$ provided the bank trades as a consumer or does not
trade at all, respectively. Finally, the last expression refers to the entry
to state $(0,0)$ by a newborn bank. More precisely, for the transition
probabilities we have

\begin{center}
\begin{equation}
P_{d,m}^{j}(d^{^{\prime }},m^{^{\prime }})=\left\{ 
\begin{array}{cc}
p(i,m), & if\text{ }(i=m-m^{^{\prime }}\geq 0,\text{ }j=0)\text{ }or\text{ }%
(i=m+1-m^{^{\prime }}\geq 0,\text{ }j=1);\text{ } \\ 
0, & otherwise.
\end{array}
\right.
\end{equation}
\end{center}

The above expression describes the respective steady state transition
probabilities, $P_{d,m}^{j}(d^{^{\prime }},m^{^{\prime }}),$ from any state $%
\left( d,m\right) $ at the beginning of the period, to state $(d^{^{\prime
}},m^{^{\prime }}),$ after the balance adjustments take place. The index $j$
reflects the fact that the transition probabilities depend on whether a new
note was issued in the current period. The number of notes redeemed within a
period is a random variable following a binomial distribution, and the bank
takes the redemption probability, $\pi $, as given.

As we mentioned before, we assume that those banks that end the period with
a negative balance will exit the industry. To capture this fact we define
the operator $T:$ $G\rightarrow G$, mapping the distribution of bank agent
types at the beginning of a period to the distribution at the beginning of
the next period. Formally, for all $(d^{^{\prime }},m^{^{\prime }})$, $T$ is
defined as the restriction of $Q$ to the set of banks that stay in business: 
$Tx_{d^{^{\prime }},m^{^{\prime }}}=Qx_{d^{^{\prime }},m^{^{\prime }}}.$

\bigskip

\bigskip

\smallskip

\noindent {\bf Proof of Lemma 1: }We have that $\alpha =1$ if $%
s_{1}-s_{0}\geq \frac{e}{\beta }$ or, by substituting, if $[\frac{1}{k}%
(c_{0}+\gamma x)u+\frac{1}{k}\left( c_{1}+\phi x\right) e][1-\beta (1-$ $%
\frac{1}{k}(c_{0}+\gamma x)-\frac{1}{k}\left( c_{1}+\phi x\right) ]^{-1}\geq 
\frac{e}{\beta }$ or, by rearranging, if $\frac{1}{k}(c_{0}+\gamma x)\geq
\left( \frac{1}{\beta }-1\right) \frac{1}{\frac{u}{e}-1}.$ Since $c_{0}\geq
\delta (1-\mu )$, the above condition will be satisfied if $\frac{1}{k}%
\delta (1-\mu )\geq \left( \frac{1}{\beta }-1\right) \frac{1}{\frac{u}{e}-1}%
. $ This condition holds if either $u-e$ or $\beta $ is high enough.%
%TCIMACRO{\TeXButton{End Proof}{\endproof}}
%BeginExpansion
\endproof%
%EndExpansion

\smallskip\ 

\noindent {\bf Proof of Lemma 2: }(a) We first demonstrate that $v$ is
weakly increasing in $d$. Let $G=\left\{ f:{\Bbb N}\times {\Bbb N}%
\rightarrow {\Bbb R}\right\} .$ Define the operator $T:G\rightarrow G$ by $%
Tv=$ $\{${\it left hand side of the functional equation for a bank}$\}$. We
know that $T$ has a unique fixed point, $v$, in the space of bounded
continuous functions. If $T$ also maps the space of weakly increasing
continuous functions into itself, and since this space is complete, this
fixed point will also be a weakly increasing continuous function. We need to
show that $T$ preserves monotonicity, i.e., for any fixed $m$, $d_{1}\leq
d_{2}$ implies that $Tv(d_{1},m)\leq Tv(d_{2},m)$. We first show that for
all $m$ and any $j$, $w_{d_{1},m}^{j}\leq $ $w_{d_{2},m}^{j}.$ First,
consider the case where $d_{1}\geq m.$ Then $w_{d_{h},m}^{j}=\sum_{0\leq
i\leq m}\binom{m}{i}\pi ^{i}(1-\pi )^{m-i}v_{d_{h}-i,m+j-i},$ where $h=1,2.$
Notice that the terms multiplying $v$ are the same for $h=1,2.$ Since $v$ is
monotone, it follows that $w_{d,m}^{j}$ is monotone. Next, consider the case
where $d_{2}<m.$ Then

\begin{center}
$w_{d_h,m}^j=\sum_{0\leq i\leq m}\binom mi\pi ^i(1-\pi
)^{m-i}v_{d_h-i,m+j-i}+\sum_{d_h\leq i\leq m}\binom mi\pi ^i(1-\pi
)^{m-i}s_1.$
\end{center}

\noindent Note that the first $d_{1}$ terms in the first sum are weakly
greater when $d_{h}=d_{2}$ than when $d_{h}=d_{1}$, given that $v$ is
monotonically increasing in $d$. The result then follows, since, for all $%
(d,m),$ $v_{d,m}>s_{1}.$ By the same reasoning, the desired inequality
follows for the case where $d_{1}<m<d_{2}$, since, once again, $v$ is
monotonically increasing in $d$ and, for all $(d,m),$ $v_{d,m}>s_{1}.$

\smallskip\ 

\noindent (b) Now we show that $v$ is weakly decreasing in $m$. Using the
argument in (a), we need to show that for any fixed $d$, $m_{1}\leq m_{2}$
implies that $Tv(d,m_{1})\geq Tv(d,m_{2})$. As before, we first show that
for all $d$ and any $j$, $w_{d,m_{1}}^{j}\geq w_{d,m_{2}.}^{j}$ We consider
the case where $m_{2}<d$ first. In that case, $w_{d,m_{h}}^{j}=\sum_{0\leq
i\leq m_{h}}\binom{m_{h}}{i}\pi ^{i}(1-\pi )^{m_{h}-i}v_{d-i,m_{h}+j-i}.$
Define $p(i,m)$ by $p(i,m)=\binom{m}{i}\pi ^{i}(1-\pi )^{m-i}.$ We then have 
$\frac{p(i,m)}{p(i,m+1)}=\frac{\frac{m!}{(m-i)!i!}}{\frac{(m+1)!}{(m+1-i)!i!}%
}\frac{\pi ^{i}}{\pi ^{i}}\frac{(1-\pi )^{m-i}}{(1-\pi )^{m+1-i}}=\frac{m+1-i%
}{m+1}\frac{1}{1-\pi }.$ This expression is less than 1 if and only if $%
i>\pi (m+1).$ Therefore, $p(i,m)$ is greater than $p(i,m+1)$ for low values
of $i$ and is lower for high values of $i$. The result then follows, since $%
w_{d,m}^{j}$ is a convex combination of decreasing functions of $m$.
Therefore, $w_{d,m_{1}}^{j}\geq w_{d,m_{2}}^{j}.$ Since $v_{d,m}>s_{1},$ for
all $(d,m),$ the same argument provides the result for the cases where $%
d<m_{1}$ and the case where $m_{1}<d<m_{2}.$%
%TCIMACRO{\TeXButton{End Proof}{\endproof}}
%BeginExpansion
\endproof%
%EndExpansion

\bigskip

\bigskip

\smallskip

\noindent {\bf Proof of Lemma 3: }For the first inequality, notice that $%
\alpha >0$ implies that $-e+\beta s_{1}\geq \beta s_{0},$ which, in turn,
implies that $s_{1}\geq s_{0}+\frac{e}{\beta }$ and, therefore, $%
s_{1}>s_{0}. $ Since $v$ is increasing in $d$ and decreasing in $m,$ for the
second inequality it is sufficient to show that $v_{0,m}>s_{1},$ for $m$
large. In this case, the bank will have a negative balance with probability
1 at the end of the period and, therefore, will exit the sector. The bank
can issue one more note this period, and in the next period it will still
have the same value function as a non-bank with one unit of money, $s_{1}.$
So $v_{0,m}>s_{1},$ even for an arbitrarily high $m$.%
%TCIMACRO{\TeXButton{End Proof}{\endproof}}
%BeginExpansion
\endproof%
%EndExpansion

\bigskip

\bigskip

\smallskip

\noindent {\bf Proof of Lemma 4: }We have that $\phi _{0,0}=0$ if$\ u<p\beta
(v_{0,0}-s_{1})+(1-p)\beta (v_{0,0}-v_{0,1}),$ where $p$ is the probability
that a newly issued note is redeemed. Since $v_{0,0}>v_{0,1},$ it is
sufficient to show that $v_{0,0}-s_{1}>\frac{u}{p\beta }.$ Consider a
voluntary 100\%-reserves rule for a bank. According to this arbitrary
decision rule, banks issue notes only if they can fully back them in
reserves. We shall deal with a lower bound for the bank's value attained by
such a rule under a worst-case scenario: that each bank faces a redemption
probability equal to $1,$ so that notes never stay in circulation. Let $%
\tilde{v}$ denote the expected discounted utility for banks under this
scenario. Notice that $\tilde{v}$ depends only on available reserves, $d.$
To complete the description of $\tilde{v},$ we choose the production
decision rule for banks as follows. Banks with $d=0$ and $d=1$ always
produce to acquire reserves. Banks with $d\geq 2$ produce if and only if \ $%
\beta (\tilde{v}_{d+1}-\tilde{v}_{d})\geq e$. Given these restrictions, $%
\tilde{v}$ is a lower bound on the optimal value function $v.$ Therefore, it
will be sufficient to show that $\tilde{v}_{0}-s_{1}>\frac{u}{p\beta },$ for 
$\beta $ large enough. Let $q_{1}=\frac{1}{k}(c_{0}+\gamma x)$ and $q_{2}=%
\frac{1}{k}(c_{1}+\phi x)$ be the probabilities of a single coincidence
meeting as a consumer and as a producer respectively for a bank. Since both $%
\mu _{1}$ and $\mu _{2}$ are strictly positive, $c_{0}$ and $c_{1}$ are
bounded away from zero, and so are $q_{1}$ and $q_{2}.$ For $d=0,1,$ the
value function $\tilde{v}$ satisfies: 
\begin{equation}
\tilde{v}_{0}=\beta \tilde{v}_{0}+q_{2}[-e+\beta (\tilde{v}_{1}-\tilde{v}%
_{0})]  \label{v0}
\end{equation}
and 
\begin{equation}
\tilde{v}_{1}=\beta \tilde{v}_{1}+q_{1}[u+\beta (\tilde{v}_{0}-\tilde{v}%
_{1})]+q_{2}[-e+\beta (\tilde{v}_{2}-\tilde{v}_{1})].  \label{v1}
\end{equation}
Regarding non-banks, we shall work with their optimal values which satisfy $%
s_{0}=\beta s_{0}+q_{2}[-e+\beta (s_{1}-s_{0})]$ and $s_{1}=\beta
s_{1}+q_{1}[u+\beta (s_{0}-s_{1})].$ Given that $q_{2}$ is bounded away from
zero, a straightforward calculation reveals that $\tilde{v}_{0}\rightarrow 
\tilde{v}_{1}-e$ as $\beta \rightarrow 1.$ Since $p$ is bounded away from
zero, because $\delta $ is strictly positive, the assertion that $\tilde{v}%
_{0}-s_{1}>\frac{u}{p\beta }$ holds for $\beta $ sufficiently high, now
follows from showing that $\tilde{v}_{1}-s_{1}\rightarrow +\infty $ as $%
\beta \rightarrow 1.$ This limit is computed as follows. Given the
production decision rule attaining $\tilde{v},$ 
\[
(1-\beta )\tilde{v}_{2}=q_{1}[u+\beta (\tilde{v}_{1}-\tilde{v}%
_{2})]+q_{2}\max \{0,-e+\beta (\tilde{v}_{3}-\tilde{v}_{2})\}\geq
q_{1}[u+\beta (\tilde{v}_{1}-\tilde{v}_{2})]. 
\]
Since $(1-\beta )\tilde{v}_{1}=q_{1}[u+\beta (\tilde{v}_{0}-\tilde{v}%
_{1})]+q_{2}[-e+\beta (\tilde{v}_{2}-\tilde{v}_{1})],$ we can work with $%
\tilde{v}_{2}-\tilde{v}_{1}$ in order to obtain the inequality $[1-\beta
(1-q_{1}-q_{2})](\tilde{v}_{2}-\tilde{v}_{1})\geq q_{2}e+q_{1}\beta (\tilde{v%
}_{1}-\tilde{v}_{0}).$ Now (\ref{v1}) implies 
\begin{eqnarray*}
\lbrack 1-\beta (1-q_{1}-q_{2})](1-\beta )\tilde{v}_{1} &\geq &(1-\beta
(1-q_{1}-q_{2}))q_{1}u-(1-\beta (1-q_{1}))q_{2}e \\
&&\ +(1-\beta (1-q_{1}))q_{1}\beta (\tilde{v}_{0}-\tilde{v}_{1}),
\end{eqnarray*}
or, rearranging terms, 
\begin{eqnarray}
\lbrack 1-\beta (1-q_{1}-q_{2})](1-\beta )\tilde{v}_{1} &\geq &(1-\beta
)(q_{1}u-q_{2}e)+q_{1}^{2}\beta u+  \label{vv1} \\
&&q_{1}q_{2}\beta (u-e)+(1-\beta (1-q_{1}))q_{1}\beta (\tilde{v}_{0}-\tilde{v%
}_{1}).  \nonumber
\end{eqnarray}
Similarly, expressions for $s_{0},$ $s_{1}$ and $s_{1}-s_{0}$ promptly imply 
\begin{equation}
\lbrack 1-\beta (1-q_{1}-q_{2})](1-\beta )s_{1}=(1-\beta
)q_{1}u+q_{1}q_{2}\beta (u-e).  \label{ss1}
\end{equation}
Because $\tilde{v}_{1}-\tilde{v}_{0}\rightarrow $ $e$ as $\beta \rightarrow
1,$ we can use (\ref{vv1}) and (\ref{ss1}) to notice that 
\[
\lim_{\beta \rightarrow 1}\{[1-\beta (1-q_{1}-q_{2})](1-\beta )(\tilde{v}%
_{1}-s_{1})\}\geq \lim_{\beta \rightarrow 1}\{q_{1}^{2}\beta (u-e)-(1-\beta
)(q_{1}+q_{2})e\}. 
\]
Since $u>e$ and $q_{1}$ is bounded away from zero, we conclude that $\tilde{v%
}_{1}-s_{1}\rightarrow +\infty $ as desired.%
%TCIMACRO{\TeXButton{End Proof}{\endproof}}
%BeginExpansion
\endproof%
%EndExpansion

\smallskip\ \ \ 

\noindent {\bf Proof of Proposition 2:} (a) $\phi _{d,d}=0$ if and only if $%
u+p\beta w_{d-1,d}^{0}+(1-p)\beta w_{d,d}^{1}<\beta w_{d,d}^{0},$ where $p$
is the probability that the note is redeemed instantly. This, in turn, is
true if $u<p\beta (w_{d,d}^{0}-w_{d-1,d}^{0})+(1-p)\beta
(w_{d,d}^{0}-w_{d,d}^{1}).$ Since $w_{d,d}^{0}\geq w_{d,d}^{1},$ the last
inequality follows if $u<p\beta (w_{d,d}^{0}-w_{d-1,d}^{0}).$ Note that $%
w_{d,d}^{0}-w_{d-1,d}^{0}=\sum
{}_{i=0}^{d-1}p(i,d)(v_{d-i,d-i}-v_{d-1-i,d-i})+p(d,d)(v_{0,0}-s_{1})\geq
p(d,d)(v_{0,0}-s_{1}).$ Therefore, it suffices to show that for $\beta $
close to 1, $u<p\beta p(d,d)(v_{0,0}-s_{1}).$ This follows since $%
v_{0,0}-s_{1}\rightarrow \infty $ as $\beta \rightarrow 1,$ and since $p\geq
\mu _{1}\delta >0$ and $p(d,d)>\delta ^{d}>0.$%
%TCIMACRO{\TeXButton{End Proof}{\endproof}}
%BeginExpansion
\endproof%
%EndExpansion

\smallskip\ 

\noindent (b) Consider the case where $m=d$ and $d$ is large. We have that $%
\phi _{d,d}=1$ if and only if $u+p\beta w_{d-1,d}^{0}+(1-p)\beta
w_{d,d}^{1}>\beta w_{d,d}^{0}.$ This is true if and only if $u>p\beta
(w_{d,d}^{0}-\beta w_{d-1,d}^{0})+(1-p)\beta (w_{d,d}^{0}-\beta
w_{d,d}^{1}). $ Note that $w_{d,d}^{0}$ is bounded, since it belongs to the
interval $(0,\frac{u}{1-\beta }),$ and an increasing function of $d$.
Therefore, $\lim_{d\rightarrow \infty }w_{d,d}^{0}=\lim_{d\rightarrow \infty
}w_{d-1,d}^{0}=K,$ for some finite constant $K.$ Also, $%
w_{d-1,d}^{0}<w_{d,d}^{1}<w_{d,d+1}^{1},$ which implies that $%
\lim_{d\rightarrow \infty }w_{d,d}^{0}=\lim_{d\rightarrow \infty
}w_{d,d}^{1}=K.$ Therefore, the above inequality holds for $d$ large.

\smallskip\ 

\noindent (c) Fix $(d,m)$ with $d>m.$ For a bank that can issue a note we
have that $\phi _{d,m}=1$ if and only if $u+p\beta w_{d-1,m}^{0}+(1-p)\beta
w_{d,m}^{1}\geq \beta w_{d,m}^{0}.$ We know that $w_{d-1,m}^{0}\leq
w_{d,m}^{1}.$ Thus, it is sufficient to show that $u+\beta w_{d-1,m}^{0}\geq
\beta w_{d,m}^{0}.$ This, in turn, is true if $u+\beta \sum
{}_{i=0}^{m}p(m,i)v_{d-1-i,m-i}\geq \beta \sum {}_{i=0}^{m}p(m,i)v_{d-i,m-i}$
which holds if $v_{d-1-i,m-i}+\frac{u}{\beta }\geq v_{d-i,m-i},$ for all $%
0\leq i\leq m.$ The proof then reduces to showing that for all $(d,m)$ such
that $d>m$, $v_{d,m}-v_{d-1,m}\leq \frac{u}{\beta }.$ Let $C$ be the set of
functions $f:[0,\bar{M}]^{2}\rightarrow [0,\frac{u}{\frac{1}{\beta }-1}]$
that satisfy this property, where $\bar{M}$ is an upper bound on the state
space (see Lemma 5 below). For all $(d,m)$ in $[0,\bar{M}]^{2},$ define an
operator $T:C\rightarrow C$ by

\begin{center}
$T\bar{v}_{d,m}=\frac 1k\left( c_1+\phi x\right) \max_{\gamma _{d,m}}[\gamma
_{d,m}(-e+\beta \bar{w}_{d+1,m}^0)+(1-\gamma _{d,m})\beta \bar{w}_{d,m}^0]$

$+\frac 1k(\alpha c_0+\gamma x)\max_{\phi _{d,m}}\left\{ \phi
_{d,m}[u+p\beta \bar{w}_{d-1,m}^0+\ (1-p)\beta \bar{w}_{d,m}^1]+(1-\phi
_{d,m})\beta \bar{w}_{d,m}^0\right\} $

$+[1-\frac{1}{k}\left( c_{1}+\phi x\right) -\frac{1}{k}(\alpha c_{0}+\gamma
x)]\beta \bar{w}_{d,m}^{0},$
\end{center}

\noindent where

\begin{center}
$\bar{w}_{d,m}^j=\sum_{0\leq i\leq \min \left\{ d,m\right\} }\binom mi\pi
^i(1-\pi )^{m-i}\bar{v}_{d-i,m+j-i}\ +\sum_{d<i\leq m}$ $I_{d,m}\binom mi\pi
^i(1-\pi )^{m-i}s_1.$
\end{center}

\noindent We know that $T$ has a unique fixed point, $v$, in the space of
bounded continuous functions. If $T$ also maps $C$ into itself, and since
this space is complete, the unique fixed point will also satisfy the
desirable property. Therefore, we need to show that if $\bar{v}_{d,m}$
satisfies $\bar{v}_{d,m}-\bar{v}_{d-1,m}\leq \frac u\beta $ for all $(d,m)$
with $d<m$, then so does $T\bar{v}_{d,m}.$ We have that $T\bar{v}_{d,m}-T%
\bar{v}_{d-1,m}=p_1q_1+p_2q_2+p_3q_3,$ where $p_1=\frac 1k\left( c_1+\phi
x\right) ,$ $p_2=\frac 1k(\alpha c_0+\gamma x),$ $p_3=1-p_1-p_2,$ and

\begin{center}
$q_1=\max_{\gamma _{d,m}}[\gamma _{d,m}(-e+\beta \bar{w}_{d+1,m}^0)+(1-%
\gamma _{d,m})\beta \bar{w}_{d,m}^0]$

$-\max_{\gamma _{d,m}}[\gamma _{d,m}(-e+\beta \bar{w}_{d,m}^0)+(1-\gamma
_{d,m})\beta \bar{w}_{d-1,m}^0],$

$q_2=\max_{\phi _{d,m}}\left\{ \phi _{d,m}[u+p\beta \bar{w}_{d-1,m}^0+\
(1-p)\beta \bar{w}_{d,m}^1]+(1-\phi _{d,m})\beta \bar{w}_{d,m}^0\right\} $

$-\max_{\phi _{d,m}}\left\{ \phi _{d,m}[u+p\beta \bar{w}_{d-2,m}^0+\
(1-p)\beta \bar{w}_{d-1,m}^1]+(1-\phi _{d,m})\beta \bar{w}_{d-1,m}^0\right\}
,$

$q_3=\beta (\bar{w}_{d,m}^0-\bar{w}_{d-1,m}^0).$
\end{center}

\noindent It is then sufficient to show that $\max \{q_1,q_2,q_3\}\leq \frac
u\beta .$

\noindent {\it Step 1:} $q_{3}\leq \frac{u}{\beta }.$

\noindent We have that $q_3=\beta (\bar{w}_{d,m}^0-\bar{w}_{d-1,m}^0).$ By
the definition of $w$, and since $d<m,$ this expression equals $\beta \sum
{}_{i=0}^mp(m,i)[\bar{v}_{d-i,m-i}-\bar{v}_{d-1-i,m-i}].$ Since $\bar{v}$
satisfies the desirable property, this expression is less than or equal to $%
\beta \frac u\beta =u<\frac u\beta .$

\noindent {\it Step 2:} $q_{2}\leq \frac{u}{\beta }.$

\noindent By the same argument as in step 1, we have that $\bar{w}_{d-1,m}^0-%
\bar{w}_{d-2,m}^0,$ $\bar{w}_{d,m}^1-\bar{w}_{d-1,m}^1,$ and $\bar{w}%
_{d,m}^0-\bar{w}_{d-1,m}^0$ are all less than $\frac u\beta .$ It is then
straightforward to show that $q_2\leq \frac u\beta ,$ for all possible
combinations of $\phi _{d,m}$ and $\phi _{d-1,m}.$ For example, suppose that 
$\phi _{d,m}=1$ and $\phi _{d-1,m}=0.$ Then

\begin{center}
$q_3=u+p\beta \bar{w}_{d-1,m}^0+\ (1-p)\beta \bar{w}_{d,m}^1-\beta \bar{w}%
_{d-1,m}^0$

$\leq p\beta (\bar{w}_{d-1,m}^0-\bar{w}_{d-2,m}^0)+(1-p)\beta (\bar{w}%
_{d,m}^1-\bar{w}_{d-1,m}^1)$

$\leq p\beta \frac u\beta +(1-p)\beta \frac u\beta =u\leq \frac u\beta .$
\end{center}

\noindent This implies that $T\bar{v}_{d,m}$ satisfies the desirable
property and the proof is complete.%
%TCIMACRO{\TeXButton{End Proof}{\endproof} }
%BeginExpansion
\endproof%
%EndExpansion
\ 

\smallskip\ 

\noindent {\bf Proof of Proposition 3:} (a) A bank with an opportunity to
consume faces

\begin{center}
\[
\max_{\phi _{d,m}}\left\{ \phi _{d,m}[u+p\beta w_{d-1,m}^0+(1-p)\beta
w_{d,m}^1]+(1-\phi _{d,m})\beta w_{d,m}^0\right\} . 
\]
\end{center}

\noindent We have that $\phi _{d,m}=1$ if and only if $u+p\beta
w_{d-1,m}^{0}+(1-p)\beta w_{d,m}^{1}>\beta w_{d,m}^{0}.$ This is true if and
only if $u>p\beta (w_{d,m}^{0}-w_{d-1,m}^{0})+(1-p)\beta
(w_{d,m}^{0}-w_{d,m}^{1}).$ Since $w_{d,m}^{0}\geq w_{d,m}^{1}$, it is
sufficient to show that $u>p\beta (w_{d,m}^{0}-w_{d-1,m}^{0}).$ Note that $%
\lim_{d\rightarrow \infty }w_{d,m}^{0}=\lim_{d\rightarrow \infty }\sum
{}_{i=0}^{m}p(m,i)v_{d-i,m-i}=\sum {}_{i=0}^{m}\lim_{d\rightarrow \infty
}\{p(m,i)v_{d-i,m-i}\}.$ Also, $v$ is increasing in $d$ and bounded. Thus,
the above limit exists and equals a constant, i.e., $0<\lim_{d\rightarrow
\infty }v_{d-i,m-i}=K_{m-i}<\frac{u}{1-\beta }.$ Therefore, $%
\lim_{d\rightarrow \infty }w_{d,m}^{0}=\lim_{d\rightarrow \infty
}w_{d-1,m}^{0}=\sum {}_{i=0}^{m}p(m,i)K_{m-i}=\bar{K}_{m},$ a constant,
thus, $\phi _{d,m}=1$, for $d$ large.

\smallskip\ 

\noindent (b) Consider a bank facing an opportunity to increase reserves. We
have that $\gamma _{d,m}=0$ if and only if $-e+\beta w_{d+1,m}^{0}\leq \beta
w_{d,m}^{0}.$ We know that $\lim_{d\rightarrow \infty
}w_{d,m}^{0}=\lim_{d\rightarrow \infty }w_{d-1,m}^{0}=\bar{K}_{m}.$ The
result then follows for $d$ large.

\smallskip\ 

\noindent (c) A bank that is given the opportunity to issue a note faces the
following problem:

\begin{center}
\[
\max_{\phi _{d,m}}\left\{ \phi _{d,m}[u+p\beta w_{d-1,m}^0+(1-p)\beta
w_{d,m}^1]+(1-\phi _{d,m})\beta w_{d,m}^0\right\} . 
\]
\end{center}

\noindent We have that for any fixed $d$, $\lim_{m\rightarrow \infty
}w_{d,m}^{j}=\lim_{m\rightarrow \infty }\left\{ \sum
{}_{i=0}^{d}p(m,i)v_{d-i,m+j-i}+\sum {}_{i=d+1}^{m}p(m,i)s_{1}\right\} .$
The first sum in this expression is finite, so we have $\lim_{m\rightarrow
\infty }p(m,i)=\lim_{m\rightarrow \infty }\left\{ \frac{m!}{(m-i)!i!}\pi
^{i}(1-\pi )^{m-i}\right\} =$ $\lim_{m\rightarrow \infty }\pi ^{i}(1-\pi
)^{m-i}=0.$ Thus, $\lim_{m\rightarrow \infty }w_{d,m}^{j}=\lim_{m\rightarrow
\infty }\left\{ \ 0+s_{1}\sum {}_{i=d+1}^{m}p(m,i)\right\} =s_{1}$, and, for
any fixed $d$, there exists an $M_{d}$ large enough such that $%
u+ps_{1}+(1-p)\beta s_{1}>\beta s_{1}$ and, therefore, $\phi _{d,m}=1.$

\smallskip\ 

\noindent (d) We know that $\gamma _{d,m}=0$ if and only if $-e+\beta
w_{d+1,m}^{0}\leq \beta w_{d,m}^{0}.$ Fix $d\geq 0.$ We have $%
\lim_{m\rightarrow \infty }w_{d+1,m}^{j}=\lim_{m\rightarrow \infty
}w_{d,m}^{j}=s_{1}.$ So given $d$, there exists a large enough $M_{d}$ such
that $-e+\beta w_{d+1,m}^{0}=-e+\beta s_{1}<\beta s_{1}=\beta w_{d,m}^{0}$,
for $m\geq M_{d}.$ Therefore, $\gamma _{d,m}=0$ for all $m\geq M_{d}.$%
%TCIMACRO{\TeXButton{End Proof}{\endproof}}
%BeginExpansion
\endproof%
%EndExpansion

\smallskip\ 

\noindent {\bf Proof of Proposition 4:}\ (a) We will prove the claim for the
case where $m=d+1.$ The proof can be generalized for any state $(d,m)$, such
that $d<m.$ Fix $d$. Then $\phi _{d,d+1}=0$ if and only if $u+p\beta
w_{d-1,d+1}^{0}+(1-p)\beta w_{d,d=1}^{1}<p\beta w_{d,d+1}^{0}+\beta
pw_{d,d+1}^{0}.$ Since $w_{d,d+1}^{0}\geq w_{d,d+1}^{1}$, it is sufficient
to show that $u+p\beta w_{d-1,d+1}^{0}<\beta pw_{d,d+1}^{0}.$ We have $%
w_{d-1,d+1}^{0}=\sum
{}_{i=0}^{d-1}p(i,d+1)v_{d-1-i,d+1-i}+[p(d,d+1)+p(d+1,d+1)]s_{1}$ and $%
w_{d,d+1}^{0}=\sum {}_{i=0}^{d}p(i,d+1)v_{d-i,d+1-i}+p(d+1,d+1)s_{1}.$
Therefore, $w_{d,d+1}^{0}-w_{d-1,d+1}^{0}=\sum
{}_{i=0}^{d-1}p(i,d+1)(v_{d-i,d+1-i}-v_{d-1-i,d+1-i})+p(d,d+1)(v_{0,1}-s_{1}). 
$ The result then follows, since $w_{d,d+1}^{0}-w_{d-1,d+1}^{0}\rightarrow
\infty $ as $\beta \rightarrow 1$. To see why this is true, notice that $%
v_{0,1}=\frac{1}{k}(c_{1}+\phi x)\beta w_{1,1}+A$ and $s_{1}=\frac{1}{k}%
(c_{1}+\phi x)\beta s_{1}+B$, where for the constant terms we have that $%
A\geq B$. In addition, $v_{d-i,d+1-i}-v_{d-1-i,d+1-i}\geq 0,$ $p(d,d+1)>0$,
and $w_{1,1}\geq v_{0,0}.$ The implication then follows, since $%
v_{0,0}-s_{1}\rightarrow \infty $ as $\beta \rightarrow 1$.

\smallskip\ 

\noindent (b) We have that $\gamma _{d,m}=1$ if and only if $-e+\beta
w_{d+1,d+1}^{0}\geq \beta w_{d,d+1}^{0}.$ In addition, $%
w_{d+1,d+1}^{0}-w_{d,d+1}^{0}=%
\sum_{i=0}^{d}p(i,d+1)(v_{d+1-i,d+1-i}-v_{d-i,d+1-i})+p(d+1,d+1)(v_{0,0}-s_{1}) 
$. Again, $v_{d+1-i,d+1-i}-v_{d-i,d+1-i}\geq 0,$ and, since $\pi >0,$ $%
p(d+1,d+1)>0$. Therefore, $v_{0,0}-s_{1}\rightarrow \infty $ as $\beta
\rightarrow 1$. Thus, the result follows.%
%TCIMACRO{\TeXButton{End Proof}{\endproof}}
%BeginExpansion
\endproof%
%EndExpansion

\smallskip\ \ \ \ 

\noindent {\bf Proof of Lemma 5: }Since at a monetary equilibrium we have
that $\pi \geq \delta >0,$ there exists a large enough value of $m$, say $M,$
such that the number of notes redeemed is greater than 1, with probability
arbitrarily close to 1. Given that banks can issue, at most, one unit of
money per period, we conclude that $m\leq M$ for all banks, i.e., for any $%
\epsilon >0,$ $M$ can be chosen such that $\sum_{m>M}x_{d,m}<\epsilon $.%
%TCIMACRO{\TeXButton{End Proof}{\endproof}}
%BeginExpansion
\endproof%
%EndExpansion

\smallskip\ \ \ 

\noindent {\bf Proof of Proposition 5: }For mathematical convenience, we
impose an exogenous upper bound, $M,$ on $m.$ Lemma 5 shows that if this
bound is large enough, it will not bind with probability 1. This also
implies that $d$ is bounded above by $M.$ We then have that $d\in
\{0,1,...,M\}\equiv Z_{D}$ and $m\in \{0,1,...,M\}\equiv Z_{M}.$ Let $%
Z=Z_{D}\times Z_{M}.$ The agents' optimal policies together with the
matching technology define a Markov chain on the finite state space: $Z\cup
\{0,1\},$ where $\{0,1\}$ represents the two possible states for the
non-banks. Let $\Pi $ denote the Markov matrix associated with the Markov
chain. Let $l$ denote the cardinality of $Z\cup \{0,1\}.$ Define a mapping $%
T:S^{l}\rightarrow S^{l}$ by $Ts=s\Pi ,$ for all $s\in S^{l},$ the $l$%
-dimensional unit simplex. Consider the following labeling of the state
space: Let 0 denote the state of a non-bank with no money, 1 be the state of
a non-bank with one unit of money, 2 be the state of a bank with $%
(d,m)=(0,0),$ 3 be the state of a bank with $(d,m)=(1,0),$... . For any such
state, $j$, we have the following lower bounds on the transition
probabilities: $\pi _{j,0}\geq \delta (1-\mu _{1}),$ $\pi _{j,1}\geq q,$ $%
\pi _{j,2}\geq \delta \mu _{1},$ $\pi _{j,3}\geq \delta \mu _{1}\frac{1}{k}%
(c_{1}+\phi x),$ ... . For $j=0,...,l,$ let $\epsilon _{j}=\min_{i}\pi
_{i,j}.$ Then $\sum {}_{j=0}^{l}\epsilon _{j}\geq \delta (1-\mu
_{1})+q+\delta \mu _{1}+\delta \mu _{1}\frac{1}{k}(c_{1}+\phi x)+...\geq
\delta +q>0.$ Therefore, $T$ is a contraction of modulus at least $1-\delta
-q.$ By the contraction mapping theorem, $T$ has a unique fixed point. In
addition, from any initial distribution across states, the process converges
to the invariant distribution at a geometric rate and there are no
cyclically moving subsets. To show that $\gamma x>0,$ and since $x_{0,0}>0,$
it is sufficient to show that $\gamma _{0,0}=1.$ This is true if $-e+\beta
v_{1,0}>\beta v_{0,0.}$ Using $\phi _{1,0}=1$ (see proposition 2c), we have $%
v_{1,0}=A+\rho \{u+p\beta v_{0,0}+(1-p)\beta w_{1,0}^{1}\}$ and $%
v_{0,0}=B+\rho [\max \{u+p\beta s_{1}+(1-p)\beta w_{0,0}^{1},\beta
w_{0,0}^{0}\}],$ where $A$ and $B$ represent the payoffs when there is no
opportunity to consume and $\rho $ is the probability of facing the
opportunity to issue a note in exchange for consumption. Note that $u+p\beta
v_{0,0}+(1-p)\beta w_{1,0}^{1}>u+p\beta s_{1}+(1-p)\beta w_{0,0}^{1}$ and,
since $\phi _{1,0}=1$, we also have that $u+p\beta v_{0,0}+(1-p)\beta
w_{1,0}^{1}>u+p\beta s_{1}+(1-p)\beta w_{0,0}^{1}>\beta w_{1,0}^{0}\geq
\beta w_{0,0}^{0}.$ We conclude that $v_{1,0}>v_{0,0}$ and, therefore, that $%
\gamma _{0,0}=1,$ for $e$ small enough. This, in turn, implies that $\gamma
x>0.$ To show that $\phi x>0,$ notice that $\gamma _{0,0}>0$ implies that $%
x_{1,0}>0.$ But since $\phi _{1,0}=1,$ this implies that $\phi x>0.$ 
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\bigskip

\bigskip

\noindent {\bf Proof of Corollary: }(a) By Lemma 5, $\sum_{m>M}x_{d,m}=0$.
By proposition 3(b), there exists a $D$ such that $\gamma _{d,m}=0,$ for all 
$d\geq D.$ Then, in a steady state equilibrium, we have: $\sum_{\{(d,m):m>%
\bar{M}\text{ or }d>\bar{M}\}}=0,$ where $\bar{M}=\max \{D,M\}.$ By
proposition 2(a), as $\beta \rightarrow 1,$ we have that $\phi _{d,d}=0$ for
all $d\in [0,\bar{M}].$ This, in turn, implies that $NB=0$.

\smallskip\ 

\noindent (b) Follows from Proposition 2(b).%
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