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%From: Ted Temzelides <tedt@blue.weeg.uiowa.edu>
%Date: Thu, 06 May 1999 11:06:52 -0500


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\begin{center}
ON THE LOCAL INTERACTION OF MONEY AND CREDIT\footnote{%
We thank Narayana Kocherlakota, B. Ravikumar, Deirdre McCloskey, and Steve
Williamson for comments and discussions.}

\vspace{1.6cm}

Yi Jin

Department of Economics, University of Iowa

Iowa City, IA 52242

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

\bigskip

\bigskip

Ted Temzelides

Department of Economics, University of Iowa

Iowa City, IA 52242

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

\bigskip

October 1998

This draft: April 22, 1999

JEL: E4, E5

\vspace{2cm}

\ \textbf{Abstract}
\end{center}

\noindent We study the emergence and coexistence of monetary and credit
transactions in a model where exchange is decentralized. Agents belong to
different villages which are informationally separated. The frequency of
meetings between any two different villages decreases as their respective
geographic distance from one another increases. The equilibrium mix of
monetary and credit transactions is characterized as a function of the
frequency of meetings among agents from different villages. Our economy may
be interpreted as a medieval economy. Trade takes place only among a small
set of nearby villages via the use of credit. Monetary trades emerge only
after interactions with faraway villages become sufficiently frequent. Even
in that case, trades among nearby villages remain non-monetized.

\newpage

\section{ Introduction}

Transactions between people that meet often might involve neither a double
coincidence of wants nor the use of money. For example, one person borrows
some sugar from his next-door neighbor and, instead of offering money, she
implicitly agrees to help out a neighbor on a similar occasion in the
future. On the other hand, this behavior might not be extended to people
that we meet only once. In this case, some form of payment, say money, is
offered on the spot in exchange for the good or service provided. Credit
transactions among non-strangers have been prevalent throughout history. Of
course, people that we meet often do not have to be located literally close
to us. For example, economists might meet more often with other economists,
say in conferences, regardless of their own geographic location.

In this paper, we try to capture the emergence and coexistence of monetary
and credit transactions in a model where trade is decentralized. Our
analysis is motivated by observations from transactions in medieval village
economies. In some cases these villages were largely closed, non-market
economies.\footnote{%
See, for example, Townsend (1993) for a detailed analysis of the medieval
village economy.}\footnote{%
For example, there seems to have been little commercial exchange and no
markets in Wales before 1066. See Thirsk (1991), page 332.} In his study of
markets in England and Wales between the years 1200-1500, David L. Farmer
refers to ``... a network of lending and borrowing among acquaintances,
mainly for small amounts, [which] could well strengthen the social fabric of
a village with a fairly static population. It could not serve in the same
way an international fair, a port, or even a major market town, where
litigants could not wait for the next regular borough court to press their
pleas.\footnote{%
See Thirsk (1991), page 423.}'' However, as mobility increased, so did the
frequency of interactions with outsiders, and monetary transactions became
commonplace. Still, villages kept their integrity and, at least in some
cases, trades among people from the same or neighboring villages remained
largely non-monetized for a long time.\footnote{%
Non-monetized transactions include gifts from the lord to tenants and credit
transactions between the manor and villagers. There is also evidence that
other trades, for example in wool, were mainly among members of the village
and family groups. Virtually all trade was reported to be local, mostly
within the radius of six or seven miles, or about two hours walking distance.%
}

The setup we use is based on the work of Kiyotaki and Wright (K-W 1989). We
think that the decentralized way in which goods and information are
exchanged in this model makes it especially appropriate for the study of
issues concerning early economies in which centralized markets were scarce
or non-existent. In order to study the emergence and coexistence issues, we
add two features to the standard setup. First, random matching is not
uniform but local. That is, people from the same (or neighboring) villages
are more likely to meet and trade with each other than with people from
faraway places. Second, public record-keeping of agents' trading histories
is available only within their village. As we shall see, these frictions are
necessary and sufficient for the coexistence of monetary and credit
transactions in our setup.

First, we study conditions that guarantee that trade take place only among
nearby villages and only via the use of credit. In this case, money has no
possible welfare-improving role. Monetary trades will emerge only when,
perhaps as a result of improvements in transportation, interactions with
faraway villages become sufficiently frequent. In that case, both means of
payment coexist. Credit is used in trades across neighbors, whereas money is
used in trades among ``strangers.''\footnote{%
Our analysis is mainly positive in the sense that we will build a model that
delivers this coexistence as an equilibrium outcome, and we shall largely
abstract from normative issues.} Finally, we discuss the implications of
improvements in record-keeping technology that lead to information on past
credit histories being shared across locations on the equilibrium mix of
monetary and credit transactions.

The idea that credit-like instruments are used in trades among agents with
known histories while currency is used in trades among relative strangers is
not new. Townsend (1989), for example, studies a model in which both types
of transactions coexist and are essential to support efficient outcomes. In
his model, as in ours, money and credit differ in their communication and
record keeping aspects. In earlier work Lucas (1980) suggested that a
friction involved in establishing one's creditworthiness might lead to the
coexistence of money and credit transactions since, in that case, money will
economize on record-keeping costs. This distinction is also present in
Prescott (1985), who studied a model with transaction costs in which a form
of bank drafts is used for large transactions while currency is used for
small ones. In his model, the use of monetary transactions can be thought of
as economizing on very costly information collection on agents' trading
histories. In our model, money is shown to be essential for facilitating
trade among agents that meet infrequently and do not have access to
information about each other's trading histories. Thus, one contribution of
our paper is to characterize conditions under which the random matching
model of money is consistent with the above observations.

It is worth mentioning that while here we concentrate on the implications of
a local matching rule and of a locally available record keeping technology
for monetary theory, our setup might be useful in the study of other issues
that involve local interactions. For example, search in labor markets often
has a local feature, and reputation of certain goods and services is in many
cases available only locally. Similarly, separate information networks to
which different groups of agents belong constitute an example of an
incomplete record-keeping technology.

\subsection{An Informal Description of the Model}

In the standard random matching model, credit cannot exist since with
probability 1, there are no two agents that meet repeatedly, and there is no
public record-keeping technology. We amend the uniform random matching
technology by assuming that agents meet different sets of other agents with
different probabilities and by introducing record-keeping of past actions at
the local level. Although other interpretations are possible, as mentioned
above, for reasons of concreteness we will interpret these different meeting
probabilities as being the result of the agents' respective physical
distance from one another. The unit circle is assumed to be divided into a
large number of arcs of equal length. Each arc is interpreted as a location
or village. Villages are symmetric, each populated by a continuum of agents.
Agents are specialized in production and consumption of goods. For
simplicity, we assume that each agent likes only one type of good and can
produce, by suffering some disutility, at most one unit of an indivisible
good that gives him no utility. Agents are assumed to be randomly matched in
pairs in each period. Trade is thus possible in the case where a potential
producer meets with an agent that likes his production good (hence called a
single-coincidence meeting). The distribution of all agents' characteristics
is the same across villages. We assume that, with high probability, each
agent meets with someone from his own village, and that the probability of
meeting another agent decreases as the distance from the village that the
agent belongs to increases. In addition, information about agents' trading
histories is assumed to be publicly available only locally. More precisely,
we will assume that each individual agent's trading history is public only
within the village where they belong, and only in regard to their meetings
with agents from that village.

What are the likely methods of payment if trade occurs in this environment?
One possibility corresponds to a regime under which producers offer to
produce without payment in all single-coincidence meetings regardless of the
location of the consumer. In the context of our model, we will identify such
a non-monetized type of exchange with a credit transaction. Suppose that if
one agent deviates from this implicit agreement, say by not producing for
free in a single-coincidence meeting as a producer with someone from village 
$j,$ then this deviation triggers a collective punishment to permanent
no-trade between the two villages by all agents in the village of the agent
that was deviated on.\footnote{%
In the next sections we will discuss the sensitivity of our results to the
exogenous specification of the punishment.} Since the credit regime implies
a higher frequency of consumption for the representative agent, one may
think that the above threat might induce a credit equilibrium in all
meetings in the economy. However, suppose that an agent has a
single-coincidence meeting as a producer with someone from a very
``faraway'' village. In this case, the potential producer has a very high
incentive to deviate from producing. The disutility of producing is suffered
now, while the punishment from not producing will be borne sometime in the
distant future.\footnote{%
This requires that the close neighbors of the deviating producer remain
unaware of his deviation and, thus, are not punishing him. The next section
deals with assumptions that guarantee that this will be true in our setup.}
For sufficiently low discount factors, it will be best for the producer to
deviate and refuse to produce in such a meeting. Our main result establishes
conditions for the existence of a critical distance (frequency of meetings)
such that in a stationary equilibrium only credit transactions will take
place among close neighbors (people whose frequency of meetings is higher
than a critical value) while only monetary transactions will take place in
meetings between people from faraway places (people whose frequency of
meetings is lower than a critical value). In addition, monetary transactions
emerge as mobility improves while credit transactions become more prevalent
as the record-keeping technology improves.

Our work is related to a number of papers that study credit arrangements in
a search-based setup. Diamond (1990) introduced credit in a non-monetary
search economy. Shi (1996) and Corbae and Ritter (1997) also study money and
credit in a search setup. Unlike them, we do not allow pairs of agents to
form ongoing relationships by staying together for more than one period.
Kocherlakota and Wallace (1998) build on Kocherlakota (1998) and study the
coexistence of money and credit in an environment where public
record-keeping is incomplete because it is updated with a lag. Public
record-keeping is also incomplete in our model, but in the sense that it is
available only locally. This, together with the local matching rule,
generates an endogenous lag that allows us to differentiate between
``frequent'' transactions that use a form of credit, and ``rare''
transactions that use money. Finally, Shi (1997) uses a search model in
which each household consists of a continuum of agents in order to eliminate
uncertainty at the household level. The modelling of local interactions via
a random matching technology has been applied in evolutionary game theory in
order to study the emergence of conventions (see Ellison 1993), but there it
is almost always assumed that agents are boundedly rational (see also Wright
1995). In contrast, in our model agents are assumed to be fully rational.

The paper proceeds as follows. Section 2 describes the environment. In
Section 3, we present a benchmark case of the model under a uniform matching
rule. In Section 4 we discuss the main results. Section 5 offers some
preliminary conclusions. The Appendix gives the value functions.

\section{The Environment}

Time is discrete and the horizon is infinite. There is a large number,
formally a continuum\textbf{,} of agents in the economy. Agents are matched
bilaterally in every period. There is a very large but finite number of
villages denoted by $j=1,2,...,J,$ forming a partition of the unit circle.
There is a continuum of agents in each village. We normalize the population
size of the entire economy to be of measure one. There is a finite number, $%
k\geq 3$, of types of consumption goods. All consumption goods are
perishable and indivisible. Agents specialize in consumption and production
of goods in a symmetric fashion. At the beginning of each period, there is a 
$[0,\frac{1}{k}]$ continuum of each type. Individuals of type $i$ can only
consume good $i$ and produce good $i+1$ (modulo $k$). The instantaneous
utility derived from consumption is $u>0.$ We normalize the instantaneous
utility of not consuming to be $0$. Production requires an effort that gives
disutility $e>0$, where $u>e.$ Agents maximize expected discounted utility,
and the time discount factor is $\beta \in (0,1).$ The solution concept we
employ is stationary perfect equilibrium. Because $k\geq 3,$ there is no
double coincidence of wants. We assume that a fraction $M$ of agents each
starts with one unit of indivisible, storable, and intrinsically useless
fiat money. Current period money holdings are observable within a match. We
impose an upper bound on individual holdings, i.e., individuals can hold at
most one unit of currency or one unit of a good. This implies that in
meetings where there is monetary trade, one unit of money is exchanged for
one unit of good. Agents are assumed to be assigned to villages at the
beginning of time. We assume that the initial distribution of all agents'
characteristics, including money holdings, is symmetric across all
consumption types and villages. We will concentrate on equilibrium outcomes
that respect this symmetry.

One difference between the environment of our model and the standard setup
in existing search models is in the matching technology. Matching here is
not uniform but local. Each agent in any given village is matched with an
agent from the same village with probability $p_{0};$ he is matched with an
agent from either of the two immediately neighboring villages with
probability $p_{1}$ where $p_{0}>p_{1}>p_{2}>\ldots $ etc. This assumption
captures the feature that people in the economy meet more frequently with
some people and less frequently with others. By symmetry, there is no loss
of generality in concentrating on a generic village, say $j$. We will denote
by $i$ the distance between our generic village and any other village, $%
j^{\prime },$ i.e., $i=\left| j-j^{\prime }\right| .$ Hence, $p_{i}$ stands
for the probability that an agent from a given village is matched with an
agent from an $i-th$ order neighboring village, where $\sum {}_{i}p_{i}=1$.
In order for an actual match to take place between any two villages, it will
have to be the case that the random matching technology assigns both
villages to each other.\footnote{%
Consider, for example, the case of four villages, $A,B,C$, and $D,$ located
in that order around the circle. Let $\widetilde{p}_{0}$ be the probability
that, for example, $A$ is assigned to itself, $\widetilde{p}_{1}$ the
probability that $A$ is assigned to $B$ or $D$ and $\widetilde{p}_{2}$ be
the probability that $A$ is assigned to $C.$ These probabilities are
symmetric across villages. Then $p_{2}=\widetilde{p}_{2}^{2}$, $p_{1}=2\frac{%
\widetilde{p}_{1}}{2}\frac{\widetilde{p}_{1}}{2}=\frac{\widetilde{p}_{1}^{2}%
}{2},$ and $p_{0}=1-p_{1}-p_{2}.$ In the remaining of the paper, we deal
with the $p_{i}^{\prime }s$ directly and will require that $%
p_{0}>p_{1}>p_{2}.$} Thus, the matching probabilities are described by a
step function. Agents return to their village at the end of each period. We
will find it convenient to assume that the realizations of the matching
probabilities are \textit{perfectly correlated} across agents that live in
the same village. In other words, if the random matching technology assigns
one agent from village $j$ to some agent from village $j^{^{\prime }}$ in a
given period, then each agent from village $j$ is assigned to an agent from
village $j^{^{\prime }}$ in that period.\footnote{%
If the matching probabilities were $iid$ across agents in the same village,
and since there is a continuum of agents in each village and a positive
probability of visiting any other, the logic of the law of large numbers
implies that a positive measure of agents from any village visit any other
village in every period. In this case, if one village collectively triggers
the punishment strategy in a given period, then this may trigger the
punishment strategy for the entire economy in the next period.}

When two agents have a single-coincidence meeting, after they observe each
other's type and village of origin, they simultaneously make a trading
proposal. The consumer can announce the request of a unit of a good offering
nothing in exchange, or the offer of one unit of money in exchange for a
unit of his consumption good, or propose no trade. The producer can announce
the offer of a unit of the good in exchange for nothing (i.e., extend
credit), or the request of one unit of money in exchange for producing
(i.e., offer credit), or propose no trade. If the two proposals match in the
obvious way, the proposals are realized. If they are different, then a
punishment is triggered.

An essential assumption for what follows is that information about agents'
histories of actions does not travel across villages. This captures the
feature that, especially in early village economies, it was costly to have
access to credit histories, and these costs increase as a function of
distance.\footnote{%
One could argue that these costs are relevant even with today's technology.
For example, it is easier for a bank to have access to the credit history of
someone who lives in the same country than of someone that lives overseas.}
Thus, we do not allow agents from different villages to exchange messages
regarding histories. If an agent from village $j$ refuses to produce in
exchange for nothing during a single-coincidence meeting with an agent from
village $j^{^{\prime }},$ then this deviation is communicated to all agents
in village $j^{^{\prime }}$ by the agent that experienced the deviation and
triggers a punishment to permanent autarky by all agents in village $%
j^{^{\prime }}$ in all future meetings between agents from these two
villages. Of course, the deviator has no incentive to reveal to agents in
his village that he deviated. We will assume that in the future, when other
agents from village $j$ experience the punishment by agents in village $%
j^{^{\prime }},$ being unaware of what triggered this behavior, they will
treat it as a new deviation, leading to a collective punishment of village $%
j^{^{\prime }}$ by agents in village $j$.\footnote{%
In this discussion, $j$ and $j^{^{\prime }}$ could be the same location.} In
addition, we assume that individual actions during meetings with agents from
other villages are not publicly observable in the village that an agent
belongs to. This assumption implies that agents within, say, village $j$ do
not know whether an agent from their own village have deviated in a past
meeting with village $j^{^{\prime }}$; they will only observe that agents
from village $j^{^{\prime }}$ deviate from the credit regime in the future.%
\footnote{%
Following common practice, we assume that any deviation triggers a
punishment to the worst ``reasonable'' equilibrium. The results would still
hold if we had assumed that only the deviator is punished. One issue is
whether this punishment is renegotiation-proof. We believe that, while this
will not be the case in general, this punishment is renegotiation-proof for
some parametrizations. Nevertheless, we will concentrate on unilateral
deviations henceforth.}

Fix a single-coincidence meeting between an agent from a given village, $j,$
and an agent from a village that is $i$ steps away. Let $D_{j}$ denote the
subset of villages that, as a result of past deviations, have switched to
permanent no trade in meetings with village $j$. We denote by $\alpha
^{iD_{j}}$ the probability with which an agent accepts money in exchange for
producing, and by $m^{iD_{j}}$ the probability that an agent offers money in
order to consume. In addition, we denote by $g_{M_{p},M_{c}}^{iD_{j}}$ and
by $h_{M_{p},M_{c}}^{iD_{j}}$ the respective probabilities that a producer
extends credit and that a consumer requests credit, when the producer's and
the consumer's money holdings are given by $M_{p}$ and $M_{c},$
respectively. The remaining probability is assigned to the event that no
trade proposal is made.

The punishment to permanent no-trade with the village of the deviator by the
village that experienced the deviation is arbitrary, and one could think of
several alternatives. The harshest possible punishment that a deviated-upon
village could adopt is to revert to permanent autarky during all future
meetings with all villages.\footnote{%
This punishment defines one end of the spectrum, but it is unattractive. For
example, agents in the deviated-upon village, $j$, know the location of the
deviator. It does not seem reasonable to punish themselves by adopting
autarky in meetings within the village, or in meetings between village $j$
and its close neighbors, especially if the deviating village is ``very far
away.''} Even if this punishment is adopted, the long lag before it will
reach the village of the deviator, provided that this village is
sufficiently far away in terms of the frequency of meetings, will guarantee
that the qualitative properties of our result still hold. Of course, more
credit can be supported in this case.

\section{A Benchmark Model - Uniform Matching}

Our main focus is the study of a model where both record-keeping and
matching are local. In order to separate the effects of these two
assumptions, in this section we consider a benchmark case where matching is
uniform, while information about histories is available only at the local
level. Suppose there are $n$ villages. Since the probabilities of meeting
with an agent from any given village are assumed to be perfectly correlated,
we can think of the random matching technology as operating in two steps.
First, a village is matched to any other, including itself, with probability 
$p=\frac{1}{n}$. The second step of the random matching technology
determines the individual agent matches. The following proposition asserts
that in this environment, money and credit payments cannot coexist in a
non-trivial way. More precisely, regardless of which villages are involved,
credit exchange across all meetings can be supported as an equilibrium
outcome for low values of $n$ and credit exchange cannot be supported as
part of an equilibrium outcome for high values of $n.$

\bigskip

\begin{proposition}
Fix $\beta $ such that $0\ll \beta \ll 1$. There exists a unique $n^{*}$
such that if $\ p\in (\frac{1}{n^{*}},1),$ there exists a steady state
equilibrium in which all transactions involve credit. If $p\in (0,\frac{1}{%
n^{*}}),$ there exists no equilibrium in which at least some transactions
involve credit.
\end{proposition}

\bigskip

The above Proposition follows as a Corollary of the main Proposition in the
next section, and we will omit the proof. The intuition behind it is the
following. By our assumption on the exogenous punishment, once an agent
deviates from offering credit, this triggers permanent no-trade between his
village and the village of the deviated-upon agent. Since everything is
symmetric, each agent's decision whether to offer credit or not is
independent of his village and of that of his partner. Producing implies a
certain amount of disutility that the producer suffers today. On the other
hand, the potential cost from the autarky punishment depends on the
frequency with which the no trade penalty is experienced by the deviator in
the future. Thus, if the probability of meeting the same village in the
future is high enough ($n$ is small), offering credit will be the best
response, while if the probability of meeting the same village in the future
is low enough ($n$ is large), deviating will be the best response. Next, we
consider the case where matching is not uniform but local.

\section{Local Matching and the Main Result}

In this section we state our main proposition establishing conditions that
guarantee the coexistence of monetary and credit transactions in a symmetric
stationary equilibrium for our environment. We first prove some preliminary
results. The first Lemma asserts that the search economy in Kocherlakota
(1998) follows as a special case of our setup if we shut down all meetings
across different villages $(p_{0}=1)$. In that case, each village is an
isolated economy in which uniform matching prevails. A high enough discount
factor guarantees the existence of a credit equilibrium in that case. In
addition, there is no way for a monetary arrangement to provide a higher
frequency of consumption for the representative agent than the credit
equilibrium. This is because in a credit equilibrium trade takes place in
each single-coincidence meeting while, in an equilibrium where money is
essential, it is also needed that in at least some single-coincidence
meetings the consumer has one unit of money holdings while the producer has
none. While we will concentrate on the Pareto superior credit regime
whenever such a regime exists between two locations, it is worth stating
that, given our trading mechanism, for high enough discount factors both a
pure credit and a pure monetary regime might exist.

\bigskip

\begin{lemma}
Set $p_{0}=1.$ Then: (a) there exists unique $\beta ^{C}$ and $\beta ^{M}$
with $0<\beta ^{C}<\beta ^{M}<1$ such that for all $\beta \in (\beta ^{C},1)$%
, a pure credit equilibrium exists, and for all $\beta \in (\beta ^{M},1),$
a monetary equilibrium exists, and (b) welfare in the credit equilibrium is
strictly higher than in any alternative arrangement where money is essential
in facilitating exchange.
\end{lemma}

\medskip

\noindent \textbf{Proof:} \textit{(a) }In a credit equilibrium where money
is not used, the value function of a representative agent is given by

\[
V^C=\frac{u-e}{k\left( 1-\beta \right) }>0. 
\]

\noindent In order for $g^{iD_{j}}=1$ to be the best response for a producer
in a single coincidence meeting where $h^{iD_{j^{^{\prime }}}}=1,$ we need
that $-e+\beta V^{C}\geq 0,$ which implies that 
\[
\beta \geq \frac{ek}{ek+u-e}\equiv \beta ^{C}. 
\]
On the other hand, since $u+\beta V^{C}>0,$ $h^{iD_{j}}=1$ is the best
response given $g^{iD_{j^{^{\prime }}}}=1.$ That is, the credit equilibrium
exists provided that $\beta \in \left( \beta ^{C},1\right) .$ In a monetary
equilibrium we have 
\begin{eqnarray*}
V_{1}^{M} &=&\frac{1}{k}\left( 1-M\right) \left( u+\beta V_{0}^{M}\right)
+\left[ 1-\frac{1}{k}\left( 1-M\right) \right] \beta V_{1}^{M} \\
V_{0}^{M} &=&\frac{1}{k}M\left( -e+\beta V_{1}^{M}\right) +\left( 1-\frac{1}{%
k}M\right) \beta V_{0}^{M}.
\end{eqnarray*}

\noindent Solving for $V_0^M$ and $V_1^M,$ we get 
\begin{eqnarray*}
V_0^M &=&\frac{M\left[ \beta \left( 1-M\right) \left( u-e\right) -ke\left(
1-\beta \right) \right] }{\left( 1-\beta \right) \left[ k\left( 1-\beta
\right) +\beta \right] } \\
V_1^M &=&\frac{\left( 1-M\right) \left[ ku\left( 1-\beta \right) +\beta
M\left( u-e\right) \right] }{\left( 1-\beta \right) \left[ k\left( 1-\beta
\right) +\beta \right] }.
\end{eqnarray*}

\noindent For a monetary equilibrium to exist, we need that $-e+\beta
V_1^M\geq \beta V_0^M$, which is satisfied if 
\[
\beta \geq \frac{ek}{ek+\left( u-e\right) \left( 1-M\right) }\equiv \beta
^M. 
\]

\bigskip

\noindent \textit{(b) }Frequency of consumption in the pure credit
equilibrium is greater since an agent always consumes whenever he has a
single-coincidence meeting as a consumer. For consumption to take place in a
monetary regime, it is also required that the consumer has one unit of
money, and the producer has no money. More explicitly, welfare in the
monetary equilibrium is given by 
\[
W^{M}=\left( 1-M\right) V_{0}^{M}+MV_{1}^{M}. 
\]
It can be shown that $W^{M}$ reaches the maximum, $\overline{W}^{M},$ when $%
M=\frac{1}{2}$. In that case, 
\[
\overline{W}^{M}=\frac{u-e}{4k\left( 1-\beta \right) }. 
\]

\noindent Welfare in the credit equilibrium is given by 
\[
W^{C}=\frac{u-e}{k\left( 1-\beta \right) }, 
\]
which is clearly greater than the best monetary outcome.%
%TCIMACRO{\TeXButton{End Proof}{\endproof}}
%BeginExpansion
\endproof%
%EndExpansion

\bigskip

The next Lemma describes a sufficient condition for the cost of breaking the
credit regime between two villages to increase in the expected frequency of
meetings between these two villages. Autarky implies a lower expected
frequency of consumption for the representative agent than a credit regime.
The more probable the meetings between villages $j$ and $j^{^{\prime }},$
the bigger the costs to reverting to no trade in future meetings between
them. Since the punishment becomes more painful, it becomes easier to
support the credit regime.

\medskip

\begin{lemma}
Fix $\beta $ such that $0<\beta ^{C}<\beta \ll 1$. Consider a village $%
j^{^{\prime }}$ that is matched to village $j$ with probability $p$. Then:
(a) the difference between the value functions of a representative agent
under a credit regime and under autarky between villages $j$ and $%
j^{^{\prime }}$ increases in $p$, and (b) let $\overline{p}=\frac{ek\left(
1-\beta \right) }{\beta \left( u-e\right) }$. If $p\in (\overline{p},1),$
all transactions between the two villages use credit. If $p\in (0,\overline{p%
}),$ there is no equilibrium where credit transactions between the two
villages take place.
\end{lemma}

\medskip

\noindent \textbf{Proof:} \textit{(a)} Let $V_{j,j^{\prime }}^{C}$ and $%
V_{j,j^{\prime }}^{A}$ stand for the value functions under a credit regime
and under an autarky regime between villages $j$ and $j^{^{\prime }},$
respectively. In order to support credit transactions between villages $j$
and $j^{^{\prime }},$ we need that $-e+\beta V_{j,j^{\prime }}^{C}\geq \beta
V_{j,j^{\prime }}^{A}$ or $\beta \left( V_{j,j^{\prime }}^{C}-V_{j,j^{\prime
}}^{A}\right) \geq e.$ It can be shown that the difference $V_{j,j^{\prime
}}^{C}-V_{j,j^{\prime }}^{A}$ is given by $\frac{p\left( u-e\right) }{%
k\left( 1-\beta \right) },$ which is clearly increasing in $p.$

\medskip

\noindent \textit{(b)} The expression for $V_{j,j^{\prime
}}^{C}-V_{j,j^{\prime }}^{A}$ is independent of how agents in village $j$
trade in meetings with agents from villages other than $j^{\prime }.$ Using
part \textit{(a) }and substituting $\frac{p\left( u-e\right) }{k\left(
1-\beta \right) }$ for the difference $V_{j,j^{\prime }}^{C}-V_{j,j^{\prime
}}^{A}$ it can be shown that the condition that $-e+\beta V_{j,j^{\prime
}}^{C}\geq \beta V_{j,j^{\prime }}^{A}$ is satisfied if 
\[
p\geq \frac{ek\left( 1-\beta \right) }{\beta \left( u-e\right) }\equiv 
\overline{p}. 
\]
Notice that $\overline{p}$ is decreasing in $\beta .$ If $\beta =1,$ then $%
\overline{p}=0,$ and if $\overline{p}=1,$ then $\beta =\beta ^{C}.$%
%TCIMACRO{\TeXButton{End Proof}{\endproof}}
%BeginExpansion
\endproof%
%EndExpansion

\bigskip

The previous Lemma does not automatically imply the existence of monetary
trades. Producers in such meetings, however, will produce for money provided
that it sufficiently increases their probability of consuming in the future.
This requires that a meeting with an arbitrary ``faraway'' village is very
likely in the near future although the probability of meeting any specific
village from that set is arbitrarily small.

The next Lemma offers sufficient conditions for all transactions in meetings
between agents from two given villages to be monetary. First, credit will
not be used in meetings between agents from villages $j$ and $j^{^{\prime }}$
if these two villages will not be matched with each other again with high
enough probability in the future. In that case, the expected large delay
before the punishment from a deviation is felt makes deviating the best
choice for producers. Let $B$ be the set of villages that are each matched
to village $j$ with probability less than $\overline{p},$ so that a credit
arrangement could not be implemented in meetings between village $j$ and any
of the villages in $B$. Let $q$ be the probability that village $j$ will be
matched to a village in $B.$ That is, $q=\sum_{i}p_{i}$, where $p_{i}<%
\overline{p}.$

\bigskip

\begin{lemma}
Fix a meeting between villages $j$ and $j^{^{\prime }}$ such that the
probability of a future meeting between these two villages is lower than $%
\overline{p}=\frac{ek\left( 1-\beta \right) }{\beta \left( u-e\right) }$.
Provided that the probability of a meeting with a village in $B$ is greater
than $\overline{q}=\frac{ek\left( 1-\beta \right) }{\beta \left( 1-M\right)
\left( u-e\right) },$ there exists an equilibrium where all transactions in
meetings between villages $j$ and $j^{^{\prime }}$ are monetary.
\end{lemma}

\medskip

\noindent \textbf{Proof: }From Lemma 3, we know that if $\beta $ $>\beta
^{C},$ then a credit equilibrium can be supported in meetings between
village $j$ and any village in the complement of the set $B.$ The value
functions for an agent in village $j$ are given by 
\begin{eqnarray*}
V_{1} &=&(1-q)[\beta V_{1}+\frac{1}{k}\left( u-e\right) ]+q\{\frac{1}{k}%
\left( 1-M\right) \left( u+\beta V_{0}\right) +[1-\frac{1}{k}\left(
1-M\right) ]\beta V_{1}\} \\
V_{0} &=&(1-q)[\beta V_{0}+\frac{1}{k}\left( u-e\right) ]+q[\frac{1}{k}%
M\left( -e+\beta V_{1}\right) +(1-\frac{1}{k}M)\beta V_{0}].
\end{eqnarray*}
Solving these two equations we get

\begin{eqnarray*}
V_{0} &=&\frac{\left( u-e\right) \left[ \left( 1-\beta \right) \left(
k+q\beta -kq\right) +q^{2}\beta M\left( 1-M\right) \right] -\left( 1-\beta
\right) kqme}{k\left( 1-\beta \right) \left[ k\left( 1-\beta \right) +q\beta
\right] }, \\
V_{1} &=&\frac{\left( u-e\right) \left[ \left( 1-\beta \right) \left(
k+q\beta \right) +q^{2}\beta M\left( 1-M\right) \right] +\left( 1-\beta
\right) kq\left( e-mu\right) }{k\left( 1-\beta \right) \left[ k\left(
1-\beta \right) +q\beta \right] }.
\end{eqnarray*}

For a monetary equilibrium to exist in a meeting between village $j$ and a
village $j^{^{\prime }}\in B$ we need that $\alpha =1$, or $-e+\beta
V_{1}>0+\beta V_{0}.$ This requires that

\[
q\geq \frac{ek\left( 1-\beta \right) }{\beta \left( 1-M\right) \left(
u-e\right) }\equiv \overline{q}.%
%TCIMACRO{\TeXButton{End Proof}{\endproof}}
%BeginExpansion
\endproof%
%EndExpansion
\]

\bigskip

The above Lemmata lead to our main Proposition. If the probability of a
meeting between two villages in the future is less than $\overline{p}$, the
credit regime between these two villages cannot prevail. If the probability
of a meeting between two villages in the future is greater than $\overline{p}
$, the credit regime between the two villages can be implemented. On the
other hand, in the case where credit will not prevail, monetary trades may
be implemented if money sufficiently increases the probability of consuming
in the future, i.e., if a meeting with some other ``faraway'' village in the
near future is very likely. To better understand the economic principle at
work, consider a community of people that trade almost exclusively with
people from inside the community, offering credit whenever there is a
single-coincidence meeting with a member of the community. Suppose that, in
a very unlikely event, one of the people in that community meets someone
from far away who likes his good. It will certainly not be optimal to offer
to produce for free for the stranger, but what if the stranger offers money
in exchange for production? Should the potential producer accept the money
and produce? If, with very high probability, future meetings are expected to
be only with members in the community, the offered money will not be used
for a very long time and, therefore, will be rejected. On the other hand, if
meetings with different strangers are frequent enough, with high probability
the producer can use the money to finance consumption in the near future,
and so he will accept it. These observations define the existence of a
critical probability that separates the two types of transactions.

\bigskip

\begin{proposition}
Fix $\beta $ such that $0<\beta ^{C}<\beta \ll 1$. Let $\overline{p}=\frac{%
ek\left( 1-\beta \right) }{\beta \left( u-e\right) }.$ Transactions between
village $j$ and villages that are matched to $j$ with probability $p\in (%
\overline{p},1)$ use credit and, provided that the probability of a meeting
between village $j$ and a village from $B$ is greater than $\overline{q}=%
\frac{ek\left( 1-\beta \right) }{\beta \left( 1-M\right) \left( u-e\right) }%
, $ transactions between village $j$ and villages that are matched to $j$
with probability $p\in (0,\overline{p})$ are monetary.
\end{proposition}

\medskip

\noindent \textbf{Proof: }The sufficient conditions of $\beta $ to support
the credit regime and monetary regime are given by Lemma 2.\textbf{\ }The
existence of a credit regime when $p_{i}\in (\overline{p},1)$ and the
non-existence of a credit regime when $p_{i}\in (0,\overline{p})$ follows
from Lemma 3. The existence of a monetary regime when $p_{i}\in (0,\overline{%
p})$ and $q\in (\overline{q},1)$ follow from Lemma 4. 
%TCIMACRO{\TeXButton{End Proof}{\endproof}}
%BeginExpansion
\endproof%
%EndExpansion

\bigskip

To demonstrate this proposition, we constructed a simple example. Let $u=8,$ 
$e=2,$ $k=3,$ and $M=0.5.$ Then we can solve for $\overline{p}=\frac{1-\beta 
}{\beta }$ and $\overline{q}=\frac{2\left( 1-\beta \right) }{\beta }.$ These
two functions are plotted in Figure 1 in the Appendix. Figure 1 also
demonstrates the region of values that $p,q$ and $\beta $ can take, and the
possible equilibria that might be supported in these regions. We know that $%
\overline{p}$ $=1$ implies that $\beta =\beta ^{C},$ and that $\overline{q}%
=1 $ implies that $\beta =\beta ^{M}.$ The two vertical lines divide the
whole area into three regions. Consider the generic village $j$ matched with
another village $j^{\prime }$ with probability $p_{i}$. In region I, the
only equilibrium is autarky since the discount factor $\beta $ is too low to
support either credit or monetary transactions between the two villages. In
region II, if in addition $p_{i}$ lies above the $\overline{p}$ line, all
transactions between $j\,$ and $j^{\prime }$ use credit. If $p_{i}$ lies
below that line, no trade will take place. Monetary transactions cannot be
supported in this region since $q$ is always less than $\overline{q}$ ($%
\beta $ is always less than $\beta ^{M}$). In region III, if in addition $%
p_{i}$ lies above the $\overline{p}$ line, all transactions between $j\,$
and $j^{\prime }$ use credit. If $p_{i}$ lies below that line, then there
are two possibilities: autarky or monetary exchange depending on how many
other villages with which village $j$ can be matched lie below the $%
\overline{p}$ line. If $q,$ the sum of the probabilities that are less than $%
\overline{p},$ is above the $\overline{q}$ line, the transactions between
village $j$ and $j^{\prime }$ are monetary; otherwise autarky prevails.

\section{Conclusions}

We studied conditions for the coexistence of monetary and credit
transactions by using a random matching model of money. These conditions are
that agents meet each other with frequencies that are inversely related to
their geographic distance, and that they populate regions that are
informationally separated. In our model, the cost of setting up a local
record-keeping system is assumed to be zero, while the cost of setting up a
record-keeping system across villages is infinite. In a steady-state
equilibrium, credit is used among agents that meet each other frequently,
while money is used in transactions among those that meet only infrequently.
A necessary condition for money to be accepted in trade between agents that
are not likely to meet again in the near future is that a meeting with some
other ``faraway'' village in the near future is likely.

It is hard to know the exact form of non-monetized transactions in medieval
villages, but observations of contemporary village economies provide some
clues. In their study of the financial structure of three villages in India,
Lim and Townsend (1998) found evidence consistent with credit functioning
well within villages while, at the same time, external credit markets do not
function as well, perhaps due to a disadvantage in information sharing.

As shown earlier, the local matching rule is necessary for our result. If we
consider our environment under uniform probabilities of matching across all
villages, the only possible outcomes would involve either no trade or only
credit transactions across all villages. Perhaps the most appropriate
interpretation of our model is as a parable of a medieval village economy.
Our main proposition makes predictions about the \textit{emergence} of money
as an equilibrium outcome when meetings between people from faraway villages
become frequent enough, perhaps as a result of increased mobility.

In future work we would like to introduce multiple locally-issued fiat
objects and study their circulation and redemption properties as a function
of the local matching rule. It could also be interesting to study monetary
injections and price dispersion in the context of a local interaction model
with divisible goods. In such a model, newly injected money is likely to
stay within a small set of villages in the short run. In addition, locally
issued money could circulate globally but at discount that is proportional
to the respective distance from the village of issuance.

\newpage

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\section{Appendix - Value Functions}

Here we describe parts of the value functions for the agents in our model
economy. We concentrate on the symmetric steady state value function of a
representative agent from a generic village $j$. We let $V_{0}^{D_{j}}$ and $%
V_{1}^{D_{j}}$ denote the value function of an agent with no money holdings
and with one unit of money holdings respectively. There are four
possibilities regarding the money holdings of two agents that are in a
single coincidence meeting. For the case of an agent with no money holdings
we have:

$V_0^{D_j}=\sum_{i\in I}p_i$%
\[
\left\{ 
\begin{array}{c}
\left( 1-\frac 2k\right) \beta V_0^{D_j} \\ 
+\frac 1k\left( 1-M\right) \\ 
\left\{ 
\begin{array}{c}
\max_{g_{00}^{iD_j}}\left[ g_{00}^{iD_j}h_{00}^{iD_{j^{\prime }}}\left(
-e+\beta V_0^{D_j}\right) +\left( 1-g_{00}^{iD_j}h_{00}^{iD_{j^{\prime
}}}\right) \beta V_0^{D_j+j^{\prime }}\right] \\ 
+\max_{h_{00}^{iD_j}}\left[ g_{00}^{iD_{j^{\prime }}}h_{00}^{iD_j}\left(
u+\beta V_0^{D_j}\right) +\left( 1-g_{00}^{iD_{j^{\prime
}}}h_{00}^{iD_j}\right) \beta V_0^{D_j+j^{\prime }}\right]
\end{array}
\right\} \\ 
+\frac 1kM \\ 
\left\{ 
\begin{array}{c}
\left( 1-m^{iD_{j^{\prime }}}\right) \left\{ \max_{g_{01}^{iD_j}}\left[
g_{01}^{iD_j}h_{01}^{iD_{j^{\prime }}}\left( -e+\beta V_0^{D_j}\right)
+\left( 1-g_{01}^{iD_j}h_{01}^{iD_{j^{\prime }}}\right) \beta
V_0^{D_j+j^{\prime }}\right] \right\} \\ 
+m^{iD_{j^{\prime }}}\left\{ \max_{\alpha ^{iD_j}}\left[ \alpha
^{iD_j}\left( -e+\beta V_1^{D_j}\right) +\left( 1-\alpha ^{iD_j}\right)
\beta V_0^{D_j+j^{\prime }}\right] \right\} \\ 
+\max_{h_{10}^{iD_j}}\left[ g_{10}^{iD_{j^{\prime }}}h_{10}^{iD_j}\left(
u+\beta V_0^{D_j}\right) +\left( 1-g_{10}^{iD_{j^{\prime
}}}h_{10}^{iD_j}\right) \beta V_0^{D_j+j^{\prime }}\right]
\end{array}
\right\}
\end{array}
\right\} . 
\]

The first part of the value function describes the case of a meeting in
which there is no coincidence of wants. The second part describes the case
of a single coincidence meeting as a producer and as a consumer,
respectively, with an agent that has no money holdings. In that case, our
distinguished agent chooses the probability of offering a gift and the
probability of receiving a gift, respectively. Finally, the third part
describes the case of a single coincidence meeting as a producer and as a
consumer, respectively, with an agent that has one unit of money holdings.
In that case, our distinguished agent chooses the probability of offering a
gift or offering to produce in exchange for his partner's money holdings,
etc. Similarly, for the case of an agent with one unit of money holdings we
have:

$V_1^{D_j}=\sum_{i\in I}p_i$%
\[
\left\{ 
\begin{array}{c}
\left( 1-\frac 2k\right) \beta V_1^{D_j} \\ 
+\frac 1kM \\ 
\left\{ 
\begin{array}{c}
\max_{g_{11}^{iD_j}}\left[ g_{11}^{iD_j}h_{11}^{iD_{j^{\prime }}}\left(
-e+\beta V_1^{D_j}\right) +\left( 1-g_{11}^{iD_j}h_{11}^{iD_{j^{\prime
}}}\right) \beta V_1^{D_j+j^{\prime }}\right] \\ 
+\max_{h_{11}^{iD_j}}\left[ g_{11}^{iD_{j^{\prime }}}h_{11}^{iD_j}\left(
u+\beta V_1^{D_j}\right) +\left( 1-g_{11}^{iD_{j^{\prime
}}}h_{11}^{iD_j}\right) \beta V_1^{D_j+j^{\prime }}\right]
\end{array}
\right\} \\ 
+\frac 1k(1-M) \\ 
\left\{ 
\begin{array}{c}
\max_{g_{01}^i}\left[ g_{01}^{iD_j}h_{01}^{iD_{j^{\prime }}}\left( -e+\beta
V_1^{D_j}\right) +\left( 1-g_{01}^{iD_j}h_{01}^{iD_{j^{\prime }}}\right)
\beta V_1^{D_j}\right] \\ 
+\max_{m^{iD_j}}\left\{ 
\begin{array}{c}
\left( 1-m^{iD_j}\right) \left\{ \max_{h_{01}^{iD_j}}\left[ 
\begin{array}{c}
g_{01}^{iD_{j^{\prime }}}h_{01}^{iD_j}\left( u+\beta V_1^{D_j}\right) \\ 
+\left( 1-g_{01}^{iD_{j^{\prime }}}h_{01}^{iD_j}\right) \beta
V_1^{D_j+j^{\prime }}
\end{array}
\right] \right\} \\ 
+m^{iD_j}\left[ \alpha ^{iD_{j^{\prime }}}\left( u+\beta V_0^{D_j}\right)
+\left( 1-\alpha ^{iD_{j^{\prime }}}\right) \beta V_1^{D_j+j^{\prime
}}\right]
\end{array}
\right\}
\end{array}
\right\}
\end{array}
\right\} . 
\]

\end{document}


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