%Paper: ewp-ge/9606001
%From: ejg@sup.mpls.frb.fed.us (Edward J. Green)
%Date: Sat, 1 Jun 1996 09:11:39 -0500
%Date (revised): Thu, 25 Jul 1996 23:01:53 -0500
%Date (revised): Mon, 23 Dec 1996 21:39:09 -0600

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\begin{document}

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\title{A Rudimentary Random-Matching Model with \\ Divisible Money and
Prices \thanks{This paper was written while Ed Green was BP Visiting
Professor at the London School of Economics. We thank Dean Corbae,
Juan Renero, Helene Rey and Warren Weber for helpful comments on an
earlier draft. Zhou's research received support from the Research
Foundation and the Institute of Economic Research of the University of
Pennsylvania. Green's research received support from the U.S.~National
Science Foundation. The authors can be contacted at
``ejg@res.mpls.frb.fed.us'' and ``ruilin@ysidro.sas.upenn.edu''. The
views expressed in this paper are those of the authors, and do not
necessarily reflect those of the Federal Reserve Bank of Minneapolis
or the Federal Reserve System.}} This paper is available as
ewp-ge/9606001 from the Washington University archive
(http://econwpa.wustl.edu).}}

\author{{\ } \\
Edward J.~Green \\
Research Department \\
Federal Reserve Bank of Minneapolis \\
250 Marquette Avenue \\
Minneapolis, MN 55401-2171 \\
{\ } \and {\ } \\
Ruilin Zhou \\
Department of Economics\\
University of Pennsylvania and\\
Federal Reserve Bank of Minneapolis \\
Philadelphia, PA 19104-6297\\
{\ }}
\date{First Draft: May, 1995\\ Current Draft: October, 1996}
\maketitle

\vskip 1.8in
\begin{abstract} 
We consider a version of Kiyotaki and Wright's monetary search model
in which agents can hold arbitrary amounts of divisible money. A
continuum of stationary equilibria, indexed by the aggregate
real-money stock, exist with all trading occurring at a single price.
There is always a maximum level of the real money stock consistent
with existence of such an equilibrium. In the limit as trading becomes
faster relative to discounting, any real money stock becomes feasible
in such an equilibrium. In contrast to the original Kiyotaki-Wright
model, higher equilibrium real money stocks unambiguously correspond
to higher welfare in this costless-production environment.

\bigskip

J.E.L Classification: D51, E40
\end{abstract}

\newpage
\pagestyle{plain}
\setcounter{page}{1}
\Section a{Introduction}

Monetary theory deals with a feature of the economy from which
Walrasian general equilibrium theory deliberately abstracts: the
inability of all traders to meet concurrently to transact mutually
beneficial trades. Models of economies with random pairwise meetings,
but without double coincidence of wants, are suitable to study this
problem. Kiyotaki and Wright (1989) provide a prototypical formal
model of this sort.

The advantage of taking this approach is the ability to proceed with
tractable, fully consistent models that deepen our understanding of
fundamental issues. At present, though, these highly schematic models
are still long way from being straightforwardly applicable to many
policy questions. In this paper, we formulate and analyze a
random-matching model that is less schematic in one important respect
than its predecessors. Specifically, fiat money is modeled as being
perfectly divisible and not subject to inventory
constraints.\footnote{Diamond and Yellin (1990) study a search model
in which both money and goods are divisible, but they assume an
exogenous cash-in-advance constraint for one class of agents rather
than deriving the monetary nature of trade as an equilibrium
phenomenon. Shi ({\sl Econometrica}, forthcoming) studies a search
model in which money is assumed to be divisible from the viewpoint of
the household (which consists of infinitely many individual traders)
but in which each transaction involves an indivisible unit of money.
Molico (1996) computationally studies a model in which both goods and
money are divisible.}

Our model is closely similar to the Kiyotaki-Wright (1989) model in
other respects, and it also broadly resembles more recent models by
Shi (1995) and Trejos and Wright (1995). In all of those models, money
is assumed to be indivisible and a trader is assumed not to be able to
hold more than a single unit of it.  This pair of assumptions has two
restrictive consequences. It trivializes the ``law of one price,'' 
and it leads to difficulty in welfare analysis.

One of the goals of studying models of decentralized trade is to
understand how approximately uniform terms of trade become established
throughout an entire economy where agents do not deal with one another
directly. If there is no auctioneer with whom all of the agents in the
economy can and must deal simultaneously, and if moreover the
decentralization of exchange amplifies the heterogeneity among agents by
generating dispersion in money holdings, then is it consistent to assume
that all transactions occur at identical terms? If so, does the process of
exchange move agents systematically toward such uniformity?  These are
questions that can be addressed using a search model with divisible money,
but that cannot even be posed in the context of models that incorporate a
one-unit inventory constraint on the holding of an indivisible money
object.\footnote{Camera and Corbae (1996), Hendry (1993) and Wallace
(1996) relax the constraint, specifying an arbitrary finite upper bound,
which is sufficient to produce heterogeneity among trading pairs. Such
heterogeneity would also be consistent with the inventory constraint if,
for example, utility or production opportunities were not time separable.
However, time separability is another assumption that Kiyotaki and Wright
and subsequent researchers have adopted for analytical tractability.} The
only trading pairs that can make mutually beneficial trades in those
models (even if fiat money is valued) are those in which the buyer has
exactly one unit of money, and this assumption plus the assumed symmetry
of traders' preferences and technologies imply uniformity of the terms of
trade. That is, the ``law of one price'' holds trivially. In contrast,
because agents with different money holdings have different willingness to
pay for consumption in the equilibrium of our model, the existence of a
single-price equilibrium is nontrivial here.

The imposition of a one-unit inventory constraint limits the
usefulness of currently available models for policy analysis. The most
obvious instance is that an increase in the nominal money stock can be
Pareto worsening in these models because the agents who hold the
newly-minted money would be unable to gain from engaging in
production.  The possibility that an increase in nominal money can
decrease welfare deserves serious consideration, but the crude
mechanism by which occurs is incredible. Aiyagari, Wallace and Wright
(1995) discuss further difficulties in this same vein, in the context
of a model that builds on Trejos and Wright's (1995) model. Our
present model avoids the problematic assumptions.

The spirit of our present investigation is to relax a particularly
stringent assumption of the Kiyotaki-Wright (1989) model, and to
examine the consequences. Three consequences are particularly
striking. First, there can be equilibria in which the ``law of one
price'' holds exactly, rather than only approximately as would be
anticipated. Second, there exists a continuum of such steady-state
equilibria that correspond to distinct real allocations. Third,
although money is neutral (in the sense that the equilibrium
conditions can be defined with respect to real money balances alone) and
the underlying technology of pairwise meetings is exogenous, the
equilibrium price level and velocity of money are indeterminate. The
last two results, especially, pose challenges for the applicability of
random-matching models to policy analysis. Deeper knowledge about the
source of these results, and about whether or not they are robust,
will clarify how serious these challenges are. 


\Section b{The Environment}

The set of agents is a nonatomic mass of measure 1. There are $k \ge
3$ types of agent. Each type $i \in \{1, \ldots k\}$ has mass $1/k$ in
the population. Time is continuous, and agents are infinite lived. 
There are $k + 1$ goods. Of these goods, $k$ (which we index by 1
through $k$) are indivisible, immediately perishable goods that are
produced by the agents. The remaining good is a divisible, perfectly
durable, fiat-money object. The total nominal stock of this fiat money
is a constant $M$.

An agent of type $i$ can produce one unit of good $i+1$ (mod $k$)
instantaneously and costlessly at any time.  He consumes only good
$i$, from which he derives instantaneous utility $u > 0$. Each agent
maximizes the discounted expected utility of his consumption stream,
with discount rate $\r$.

Meetings between agents are pairwise. Each agent meets other agents
randomly according to a Poisson process with parameter $\mu$. The
distribution of partners' characteristics from which an agent'
meetings are drawn matches the demographic distribution of
characteristics in the entire population of the economy. A meeting
partner has two characteristics, his type and the amount of money that
he holds. An agent's type is observable, but not his money holding.

In this economy there is no double coincidence of wants (in the sense
of trades that give strictly positive utility to both traders) between
any pair of agents. Consumption goods cannot be used as commodity
money because they are perishable. Thus trade must involve using fiat
money as a medium of exchange.  We assume that transactions occur
according to a seller-posting-price protocol. When a type-$i$
agent who possesses fiat money meets a type-$(i-1)$ trader who can
produce his desired good, the seller (the type-$(i-1)$ agent) posts an
offer that the buyer (the type-$i$ agent) must either accept or
reject. Trade occurs if and only if the offer is accepted, and in that
case the buyer pays exactly the seller's offer price. This specific
assumption about the trading protocol is crucial to the results which
follow. 

\Section c{Definition of Stationary Equilibrium}

We are going to consider stationary equilibrium in the trading
environment just described. Moreover we restrict attention to
equilibria in which all agents with identical characteristics act
alike, and in which all of the $k$ types are symmetric. (Hereafter,
all of our discussion will be in terms of a generic type $i$.) A
stationary equilibrium can be characterized in terms of six
theoretical constructs: agents' offer strategy and reservation-price
strategy, the stationary measure on traders' money holdings, the
stationary distributions of offers and reservation prices, and the
value function for money holdings. 

The domain of possible money holdings is $\Re_+$, the set of nonnegative
real numbers. A type-$i$ agent's trading strategy is a pair of real-valued
functions on $\Re_+$, $\o(\h)$ that specifies the offer that he will make
as a seller when his current money holding is $\h$ and he meets a
type-($i+1$) agent, and $\b(\h)$ that specifies the maximum willingness to
pay as a buyer when his current money holding is $\h$ and he meets a
type-($i-1$) agent. Note that $\b$ is a reduced-form description of
the actual decisions made by a buyer. The buyer with a particular
money holding accepts certain offers and rejects others. It is obvious
that {\it optimal} decisions will involve a threshold offer level, below
which offers are accepted and above which they are rejected. $\b(\h)$
specifies this threshold for a buyer with money holding $\h$.

A buyer must always be able to pay his reservation price, so we impose
the feasibility constraint that  
%
\display i{\b(\h) \le \h.}
%

The stationary distribution of money holdings is given by a measure $H$
defined on $\Re_+$. 

A strategy (or, more precisely, a symmetric strategy profile) and a
stationary distribution of money holdings imply stationary distributions
of offers and reservation prices. Note that a buyer's willingness to pay
depends on his current money holding, so a trader's reservation price as a
function of his money holding is the solution of an optimization problem.
Thus we will often refer to a trader's optimal reservation price.

Define the stationary distribution of offers by 
%
\display a{\O(x)=H\{\h\,|\,\o(\h)\le x\}} 
%\O(x) = H(\o^{-1}([0, x])}
%
and the stationary distribution of reservation prices by
%
\display b{\B(x)=H\{\h\,|\,\b(\h)< x\}.} 
%\B(x) = H(\b^{-1}([0, x)).}
%
(Note that, for convenience, we are defining $\B$ to be continuous
from the left, rather than from the right as would be conventional.)

The value function $\V \colon \Re_+ \to \Re_+$ of money holdings
specifies the expected discounted utility that an agent will receive,
given his current money holding, if he adopts an optimal trading
strategy. 

The value function is studied in terms of its Bellman equation.
Intuitively the Bellman equation states that $\V(\h)$ is the discounted
expected value of $W + \V(\h')$, where $\h'$ is the agent's money
holding immediately after his next meeting with a potential trading
partner, and $W = 0$ if that transaction will not result in a purchase
but $W = u$ if it will result in a purchase (and hence will be
accompanied by consumption). By a potential trading partner of an
agent of type $i$, we mean either an agent of type $i-1$ from whom the
agent might make a purchase, or else an agent of type $i+1$ to whom he
might make a sale. Since the mass of those two types together in the
population is $2/k$, the Poisson parameter for the frequency of
meetings with them is $2\mu /k$. Therefore the value function with
appropriate discount rate is given by 
%
\display c{\V(\h) = \int_0^\infty e^{-\r t} \mean [W + \V(\h') | \eta] 
{2\mu \over k} e^{-(2\mu /k)t} \; dt}
%
where the first exponential expression inside the integrand is the
discount rate, and the second is the exponential waiting time
implied by the Poisson process. Note that the expectation does not
have to be conditioned on $t$ because, since we are assuming
stationarity, the expectation does not depend on the time at which it
is taken. Evaluation of the integral yields
%
\display d{\V(\h) = {2\mu \over k\r + 2\mu} \mean [W + \V(\h') | \eta].}
%

Since agents of types $i-1$ and $i+1$ are equally numerous in the
population, the probability that the first potential trading partner
will be a seller is $1/2$. If the type-$i$ trader's reservation price
is $r$, then the conditional expectation of $W + \V(\h')$ in that event is 
%
\display e{\int_0^r [u + \V(\h - x)] \; d \O(x) + (1 - \O(r)) \V(\h).}
%
In the complementary event that the first potential trading partner
will be a buyer, if the type $i$ trader makes offer $o$, then the
conditional expectation of $W + \V(\h')$ is
%
\display f{\B(o) \V(\h) + (1 - \B(o)) \V(\h + o).}
%
Substitution of the equally weighted average of the optimized values
of \eqn e and \eqn f for the expectation in \eqn d yields
%
\disparray g{
\V(\h) & = & {\mu \over k\r + 2\mu} \Biggl[ \max_{r \in [0, \h]}
\biggl[ \int_0^r [u + \V(\h - x)] \; d \O(x) + (1 - \O(r)) \V(\h)
\biggr] \nonumber \\
& &  \hbox{\hskip .6in}
+ \max_{o \in \Rel_+} \biggl[ \B(o) \V(\h) + (1 - \B(o)) \V(\h + o) \biggr]
\Biggr] .}
%

Equation \eqn g is the Bellman equation for $\V$. Standard arguments
establish that it has a unique solution in the space of bounded
measurable functions, and that this solution does indeed specify the
optimal expected discounted value of each possible level of money
holding. 

At this point we need to consider an implication of the way in which
we have collapsed buyers' decisions into the reduced-form
representation $\b$. Literally, in the seller-posting-price protocol,
for any possible offer $o$, the buyer must issue either an acceptance
or a rejection. The criterion for making this decision optimally is to
accept $o$ if and only if the offer is below the buyer's full
valuation of the good, that is, 
%
\display T{\forall o \quad \bigl[ \b(\h) \ge o \iff u + V(\h - o) \ge
V(\h) \bigr].} 
%
Perfectness of equilibrium requires that this condition be satisfied
even for offers that are never made in equilibrium. 

The state of the environment is summarized by the distribution of money
holdings. Implicitly, the money holding of each agent is a continuous-time,
pure-jump Markov process on the state space $\Re_+$. The transition
probabilities are the probabilities of transactions occurring, induced by
the optimal strategies $(\o, \b)$. The environment is stationary if
the measure $H$ is a stationary initial distribution of this process. 

The equilibrium concept that we will adopt is stationary perfect
Bayesian Nash equilibrium. We will refer to this simply as stationary
equilibrium. 

\medskip
{\sc Definition.} A {\it stationary equilibrium} consists of a
time-invariant profile $\langle H$, $R$, $\O$, $\o$, $\b,$ $\V\rangle$
that satisfies  
\newcounter{def}
\begin{list}{(\Roman{def})}{\usecounter{def}
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\setlength{\leftmargin}{1.5cm} \setlength{\topsep}{0.15cm} 
\setlength{\parsep}{0pt} \setlength{\itemsep}{0cm}} 
\item $H$ is stationary under trading strategies $\o$ and $\b$, and the
reservation-price distribution $R$ and the offer distribution $\O$ are
derived from $H$, $\o$ and $\b$ according to \eqn a and \eqn b. 
\item Given the distributions for money-holdings $H$, reservation-price $R$
and offer $\O$, the reservation-price strategy $\b$ satisfies feasibility
condition \eqn i and perfectness condition \eqn T, and trading
strategies $(\o,\,\b)$ and value function $\V$ solve Bellman equation
\eqn g. That is, 
\disparray h{
\V(\h) & = & {\mu \over k\r + 2\mu} \Biggl[ \biggl[\int_0^{\b(\h)} \bigl(u
+ \V(\h - x)\bigr) \; d \O(x) + (1 - \O(\b(\h))) \V(\h) \biggr] \nonumber
\\ & &  \hbox{\hskip .45in}
+ \biggl[ \B(\o(\h)) \V(\h) + (1 - \B(\o(\h))) \bigl(\V(\h + \o(\h)) 
\bigr) \biggr]\Biggr]. 
}
\end{list}

\medskip

In the equilibria that we are going to study in this paper, the
support of $H$ will be the discrete set $\{0, p, 2p, 3p, \ldots
\}$. Giving an exact statement of the stationarity condition is much
easier in this special case than in general, so we will state the
formal condition of stationarity in the context of this class of
equilibria.

\Section d{Single-Price Equilibrium}

In this section we are going to characterize a sufficient condition
for a single-price equilibrium to exist.  We are going to begin by
supposing that all trades occur at a single price $p$, and that all
agents' money holdings are integer multiples of $p$. Also we assume that
agents always offer to sell at $p$, and that every agent who
holds money is willing to purchase at $p$. We characterize the
stationary measure on traders' money holdings under these assumptions.
Then we find the corresponding solution for the value function and use
it to calculate the optimal reservation-price function. Finally we
find a sufficient condition such that the optimal offer function is
constant at price $p$. Thus, under this condition, the offer function,
reservation-price function, and stationary measure that we have found
are an equilibrium. 

\Subsection e{Stationary Measure on Traders' Money Holdings}

Consider the formulation of equilibrium just given, in the special
case that all trades occur at a single price $p$, and that the
support of the population measure $H$ of money holdings is on the
discrete set of points $p^\Natl = \{0, p, 2p, 3p, \ldots \}$.\note{This is
evidently the simplest stationary equilibrium. We do not know whether
there are other single-price equilibria in which the support of $H$
is not $p^\Natl$.} In this case, define 
%
\display j{h(n) = H(\{np\}).}
%
That is, $h(n)$ is the measure of the set of agents who hold precisely
$np$ units of fiat money. Now, instead of working with the measure $H$
on $\Re_+$, we can work with the equivalent measure $h$ on $\Nat$.
We will say that a trader is in state $n$ when his money holding is
$np$. The proportion of agents who hold positive money holdings is
defined to be 
%
\display k{m \equiv \sum_{n=1}^\infty h(n).}
%
Note that 
%
\display m{h(0) = 1-m.}
%

% In a single-price equilibrium, an agent only moves into the set of
% states in which his money holding is at most $np$ by making a purchase
% when his money holding is initially $(n+1)p$, and he only moves out of
% this set of states by making a sale when his money holding is
% initially $np$. We can use this fact, and the fact that a stationary
% population measure must leave the size of the population having money
% holdings at most $np$ unchanged over time, to characterize
% stationarity. Clearly an agent of type $i$ will make a sale whenever he 
% meets an agent of type $i+1$ whose money holding is positive, and he
% will make a purchase whenever he meets an agent of type $i-1$ if his
% own money holding is positive. The aggregate ...
% Therefore a necessary and sufficient
% condition for stationarity of $h$ is that 
% %
% \display l{\forall n \in \Nat \quad \mu h(n+1) - \mu m h(n) = 0.}
% %

In a single-price equilibrium, an agent only moves into state $n$ (the
state of having money holding $np$) by either making a sale from state
$n-1$ or making a purchase from state $n+1$. He moves out of state $n$
by either making a purchase or a sale. Clearly an agent of type $i$
will make a sale whenever he meets an agent of type $i+1$ whose money
holding is positive, and he will make a purchase whenever he meets an
agent of type $i-1$ if his own money holding is positive. Stationarity
requires that the sum of time rates of inflow to state $n$ from all
other states must equal the time rate of outflow from state $n$. The
time rate of a population flow is the instantaneous transition
probability for an individual multiplied by the population of the
state from which the transition occurs. The time derivative of $h(n)$,
for all $n > 0$ is thus 
%
\display p{\dot h(n) = \mu h(n+1) + \mu m h(n-1) - \mu
(m+1) h(n).} 
%
The time derivative of $h(0)$ is 
%
\display q{\dot h(0) = \mu h(1) - \mu m h(0).}
%
Setting these two derivatives equal to zero and arguing recursively,
it is seen that the only candidates for stationary measures are of the
form 
%
\display n{\forall n \in \Nat \quad h(n) = m^n (1-m)}
%
for some $m \in (0, 1)$. 
Given this geometric functional form specified by \eqn n, the
quantities $p$, $m$, and $M$ are related by the equation    
%
\display o{M = p \sum_{n=1}^\infty n h(n) = {m \over 1-m} \, p.}
%
This characterization of stationarity by equations \eqn n and \eqn o
is the remaining equilibrium condition that was postponed from the end
of the preceding section. 

\Subsection f{Equilibrium Value Function}

Now we solve equation \eqn h for the equilibrium value function. To
begin, recall that the presumed optimal strategy in single-price
equilibrium is that agents are always willing to sell at $p$, and that
every agent who holds money is willing to purchase at $p$. Formally
this assumption means that 
%
\display r{\O(p) = 1 \mand \forall x \! < \! p \enspace \O(x) = 0}
%
and
%
\display s{\B(p) = 1 - m.}
%
It will be convenient to define $V(n) = \V(np)$. Then, using \eqn r
and \eqn s, \eqn h simplifies for $n = 0$ to 
%
\display u{V(0) = {\mu \over k\r + 2\mu} [V(0) + (1-m) V(0) + mV(1)],}
%
which yields
%
\display E{V(0) = {\mu m \over k\r + \mu m} V(1).}
%
For all positive $n$, \eqn h simplifies to
%
\display t{V(n) = {\mu \over k\r + 2\mu} \bigl[ 
[u + V(n-1)] + [(1-m) V(n) + mV(n+1)] \bigr].}
%
The system of equations \eqn E and \eqn t defines the value function
implicitly in terms of five parameters: $\mu$, $k$, $\r$, $m$, and
$u$. Note that $k\r/\mu$ actually functions as a single
parameter. To simplify further computations, define
%
\display I{\phi = {k\r \over \mu}.}
%
Equation \eqn t can be rewritten in matrix form as 
%
\display v{
\left( \begin{array}[]{c} V(n+1) \\ V(n) \\ u \end{array} \right)
=
\left( \begin{array}[]{c c c} 
\left[ {\phi \over m} +{1 \over m} +1 \right] & 
{-1 \over {}\; m} & {-1 \over {}\; m} \\ 1 & 0 & 0 \\ 0 & 0 & 1
\end{array} \right)
\left( \begin{array}[]{c} V(n) \\ V(n-1) \\ u \end{array} \right)
}
%
Equation \eqn v is an inhomogeneous second-order linear difference
equation. Its family of solutions is given in terms of eigenvectors of
the matrix 
%
\display l{\D = \left( \begin{array}[]{c c c} 
\left[ {\phi \over m} +{1 \over m} +1 \right] & 
{-1 \over {}\; m} & {-1 \over {}\; m} \\ 1 & 0 & 0 \\ 0 & 0 & 1
\end{array} \right)}
%
in \eqn v, and the correct solution is determined by means of two
endpoint conditions.\note{See Lefschetz (1977, Chapter III) for a
discussion of the continuous-time theory, which is completely
analogous.} One of these endpoint conditions is equation \eqn E. The
other condition  is that $V$ is bounded. It is bounded below
because it is nonnegative and above because, even if a trader's rate
of consumption were not constrained by his need to pay for the goods
that he acquires in trade, he would still have only discrete
consumption opportunities that would occur at times separated by $\mu$
on average, and the utility of which would thus be discounted. 

The matrix $\D$ has three distinct eigenvectors, all of which have
real eigenvalues. The solution of equation \eqn v is therefore
determined by a linear combination of the eigenvectors for which
(because $V$ is bounded) the eigenvalue has absolute value at most 1. 
Two of the eigenvectors satisfies this criterion. They can be 
expressed as  
%
\display x{\D \Xone = \Xone \mand \D \Xtwo = \l \, \Xtwo\;.}
%
where
%
\display y{\l = {1 \over 2} \left( {\phi \over m} + {1 \over m} + 1 -
\sqrt{ \left( {\phi \over m} + {1 \over m} + 1 \right)^2 - {4 \over m}}
\; \right) \in (0, 1).}
%

Now, by Lefschetz (1977), there are two coefficients $\theta_1$ and
$\theta_2$ such that 
%
\display z{\vector{V(1)}{V(0)}{u} = \theta_1 \Xone + \theta_2
\Xtwo.}
%
It follows from \eqn E and \eqn z that 
%
\display A{\theta_1 = u \,; \qquad \theta_2 = V(0) - {u \over \phi} \,; \qquad
V(0) = {(1-\l) m \over  \phi + (1-\l) m} \, {u\over \phi}.}
%
Moreover, by induction, for all $n \ge 0$   
%
\disparray w{\vector{V(n+1)}{V(n)}{u} &
= & \D^n \vector{V(1)}{V(0)}{u} \nonumber \\ 
& & \rule{0pt}{8pt} \nonumber \\ 
& =  & u \Xone + \l^n \left( V(0) - u/\phi \right) \Xtwo .}
%
In particular, the second row states that for all $n$
%
\display B{V(n) = \l^n V(0) + (1-\l^n) u/\phi.}
%
Equations \eqn A and \eqn B imply the following lemma.

\lemma a{$V$ is increasing and satisfies the concavity condition that,
for all  $j > 0$, $V(n+j) - V(n)$ is a decreasing function of $n$.}

Although we are characterizing a class of equilibria in which each
trader's money holding is always an integer multiple of $p$, the value
function is defined for non-integer multiples as well. Given the
presumed optimal trading strategy, the value function is evidently a
step function. Specifically, if $[x]$ denotes the integer part of $x$
(that is, $x = [x] + \epsilon$ for some $\epsilon \in [0,1)\,{}$)
then
%
\display D{\V(\h) = V([\h/p]).}
%
This completes the derivation of the value function.

\Subsection g{Equilibrium Strategy}

Suppose that an agent $a$ of type $i$ with money holding $\h$ meets
a trading partner of type $(i-1)$ with money holding $\h'$. Each
observes the other's type but neither observes the other's money
holding. Independently of one another, $a$ chooses a reservation price $r$
and the partner posts an offer $o$. If $r \ge o$, then the partner
supplies a unit of good $i$ to $a$ in exchange for amount $o$ of money,
and otherwise no transaction takes place. Agent $a$ should choose an
optimal reservation price that solves the maximization problem with
respect to $r$ that occurs on the right hand side of the Bellman equation
\eqn g. The solution to this maximization problem may not be unique.
The solution that satisfies the perfectness condition \eqn T can be
written as 
%
\display C{\b(\h) = \max \{r \in [0, \h] | u + \V(\h - r) \ge
\V(\h)\}.}
%
This reservation price is $a\/$'s full value for a unit of good $(i+1)$. 
(Analogously, in Vickrey's (1960) analysis of a second-price auction, to
bid this full value is the buyer's weakly dominant action.) An alternative
assumption, that a buyer only accepts an offer when he gains strictly from
it, would not change our general conclusions. The following lemma provides
further information about the reservation-price function $\b$ that will be
used below.  
 
\lemma f{The reservation-price function $\b$ specified by equation \eqn C 
satisfies the conditions that $\b(p) = p$ and that, for every positive
integer multiple $np$ of $p$, $\b(np)$ is an integer multiple of $p$. 
It also satisfies the condition that $[\b(\h)/p]$ is a nondecreasing
function of $\h$.}

\proof Equations \eqn A and \eqn C imply that $\b(p) = p$. By \eqn D
and \eqn C, $\V(\h) = V([\h/p])$ and $\h - \b(\h) = jp$ for some $j
\in \Nat$. To prove that $[\b/p]$ is nondecreasing, suppose that $\h <
\h'$. By \eqn C, $\V(\h) - \V(\h - \b(\h)) \le u$. By \eqn
D, then, $V([\h/p]) - V((\h - \b(\h))/p) = V([\h/p]) - V([\h/p] -
[\b(\h)/p]) \le u$. By the concavity of $V$ established in \lem a,
$V([\h'/p]) - V([\h'/p] - [\b(\h)/p]) \le u$. That
is, this concavity condition implies that $\V(\h') - \V(\h' - p
[\b(\h)/p]) \le u$.  Thus by \eqn C and the increasingness 
assertion in \lem a, $\b(\h') \ge p [\b(\h)/p]$.
Therefore $[\b(\h)/p] \le [\b(\h')/p]$. \endproof 

\medskip

\uppercase \lem f has the following, intuitively obvious, consequence.

\lemma c{For an agent whose money holding is at least $p$, it is optimal
to accept an offer of $p$ if all sellers' offers are almost
surely at price $p$.}

\uppercase \lem c establishes one of the two presumptions about the
equilibrium strategy, stated at the beginning of \sec f, that we have
used to derive the value function. The other presumption is that all
offers are made at price $p$. The following lemma establishes a
sufficient condition for this latter presumption to characterize agents'
optimizing behavior in an equilibrium.

\lemma b{If all agents' reservation prices are integer multiple of $p$,
then the optimal offer $\o(\h)$ is an integer multiple of $p$ for every
$\h$. If the proportion of agents with positive money holdings in a
stationary measure of form \eqn n is $m \le 1/2$, and if all agents with
positive money holdings have reservation price at least $p$, then it is
optimal for an agent always to offer to sell at $p$.}

\proof The first assertion is obvious. To prove the second
assertion, define the set of money holdings $\h$ at which an offer at
price $o$ would be accepted by $A(o) = \b^{-1}([o, \infty)) \subseteq
\Re$, and define $a(o) = \min \{n | np \in A(o) \}$. Note that $a(o)
\ge [o/p]$ because an agent's reservation price cannot exceed his money
holding. By \lem a and \eqn C, $\{n | np \in A(o) \} = \{ a(o), a(o)+1,
\ldots \}$. Thus, by \eqn j and \eqn n, $H(A(o)) = m^{a(o)} \le
m^{[o/p]}$. If the seller's money holding is $\h$, then his expected value
of offering $o$ is 
%
\disparray J{W(\h, o) & =  & H(A(o)) \V(\h + o) + (1 - H(A(o))) \V(\h)
\nonumber \\ & \le & m^{[o/p]} \V(\h + o) + (1-m^{[o/p]}) \V(\h).}
%
By the first assertion of this lemma, there must be an optimal offer
of the form $o = jp$, so we can restrict attention to this case and
also assume that $\h = np$, and simplify the upper bound on the
expected value of offering $o$ to $W(\h,o) \le m^j V(n+j) + (1-m^j)
V(n)$. By the concavity of $V$ established in \lem a, $V(n+j) < V(n) +
j(V(n+1) - V(n))$. Therefore $W(\h, o) \le V(n) + j m^j (V(n+1) -
V(n))$. If $m \le 1/2$, then $j m^j \le m$ for all $j > 1$, so $W(\h,
o) \le V(n) + m (V(n+1) - V(n)) = H(A(p)) \V(\h + p) + (1 - H(A(p)))
\V(\h) = W(\h, p)$. \endproof 


\Subsection h{A Continuum of Equilibria}

We began \sec d by making two assumptions about the form of a
possible stationary equilibrium. We assumed that all offers are made
at a single price $p$ and that reservation prices of all agents
with positive money holdings are at least $p$. Under these assumptions, 
we have shown in \sec e that every geometrically-distributed 
measure on money holdings that are nonnegative integer multiples of
$p$ is stationary.  In \sec f, we have characterized the
equilibrium value function. Finally, in \sec g, we have shown that the
characterizations of stationarity and optimality imply that our
assumptions regarding reservation prices and offers are implied by agents' 
optimizing decisions if $m \le 1/2$. Equation \eqn o establishes that
$m \le 1/2$ if $p \ge M$. Thus we have proved the following theorem.
%
\theorem d{In every trading environment described in \sec b, for every
price $p \ge M$, there is a stationary equilibrium in which all
transactions occur at price $p$ and all traders' money holdings are integer
multiples of $p$. The proportion of agents with positive money
holdings is an increasing function of $M/p$, so there is a continuum of
distinct stationary-equilibrium allocations.}

\uppercase \thm d states that if the price level $p$ is not below 
$M\/$ (that is, essentially, if the per capita real money stock is not
greater than 1), then a stationary single-price equilibrium exists.  This
is a sufficient condition, not a necessary condition for existence.
The question remains, then, whether there is any maximum real money
stock that is consistent with equilibrium. In fact, for any $\phi$,
such a maximum real money stock does exist. We prove this via two
lemmas.

\lemma i{Suppose that the following condition holds.
%
\display P{m^{a(2)} > {1 \over 1 + \l}.}
%
Then there exists an $n$ such that $\o(np) \ge 2p$.}

\proof Using notation developed in the proof of \lem b, a seller with
money holding $np$ will offer at least $2p$ if $W(np, 2p) > W(np, p)$.
Substituting \eqn J into this inequality yields $m^{a(2)} [V(n+2) -
V(n)] > m^{a(2)+1} [V(n+2) - V(n)] > m [V(n+1) - V(n)].$ Thus the
inequality between the second and third terms is a  sufficient
condition for an offer of at least $2p$. Applying \eqn B yields the
equivalent condition \eqn P. \endproof 

\lemma j{For any given $\phi$, there exists some $J \in \Nat$ such
that,  for all $m$, $a(2) \le J$.} 

\proof By equation \eqn C, an agent with money holding $np$ will have
reservation price at least $2p$ if $u + V(n-2) \ge V(n)$. That is, 
%
\display Q{u \ge {1 \over \phi + (1-\l) m} u (\l^{n-2} -
\l^n).}
%
For this inequality to hold, it is sufficient that $u \ge u \l^{n-2}/\phi$, or 
%
\display R{n \ge 2 + {\ln(\phi) \over \ln(\l)}.}
%
Noting that $\l$ is a function of $m$ and that the right hand of \eqn
R reaches a finite maximum at $m = 1$, $J$ can be taken to be the
first natural number greater than that maximum. Since $a(2)$ is the
smallest number satisfying \eqn Q and $J$ satisfies \eqn Q, $a(2) \le
J$. \endproof

\theorem h{For any given $\phi$, there is some $m^* <1$ such that, 
for any $m > m^*$, a stationary single-price equilibrium does not exist.}

\proof Since $a(2) \le J$ (where $J$ is described in \lem j),
$m^{a(2)} \ge m^J$. Combining this inequality with \eqn P, 
%
\display S{m^J > {1 \over 1 + \l}}
%
is a sufficient condition for there to exist an $n$ such that $\o(np)
\ge 2p$. Both sides of \eqn S are continuous functions of $m$, and
condition \eqn S is not satisfied at $m = 1/2$ but it is satisfied at
$m = 1$. These two facts, combined with the fact that the set of
values $m$ at which \eqn S is not satisfied is closed, imply that the
set has a maximum $m^*$ which is strictly less than 1.
\endproof

\Section i{The Limiting Case}

\uppercase \thm h shows that, for any given $\phi$ and for sufficiently
high $m$ (or equivalently, for sufficiently high $M/p$), a stationary
single-price equilibrium cannot exist. However, in an economy where the
parameter $\phi$ is small, reflecting high frequency of meetings or
insignificance of discounting, the minimum price level for such an
equilibrium to exist is actually arbitrarily low.

The intuition for this result is that, if a buyer is confident that he
will almost immediately meet another seller whose offer is very close to
the minimum offer in the market, then he should be unwilling presently to
accept a high offer unless his money holding is huge. The key to
the formal derivation is a closer examination of the optimal reservation
price, characterized in \eqn C by 
%
\[ \b(\h) = \max \{r \in [0, \h] | u + \V(\h - r) \ge
\V(\h)\}. \]
%
It is clear from this formula and the formula \eqn D for $\V$ that, if
$\h$ is an integer multiple of $p$, then $\b(\h)$ must also be an
integer multiple of $p$. Define, for the optimal reservation-price
function $\b$ in an economy with parameters $\mu$, $k$, and $\r$
satisfying \eqn I and with proportion $m$ of agents having positive money
holdings, 
%
\display F{\b(np) = r(n, \phi)p}
%
We want to study what happens in the limit as $\phi$ approaches
zero. As in the preceding section, we assume that all offers are
at price $p$ and then verify that (asymptotically, in this section)
such offers are indeed optimal given agents' optimal reservation-price
functions. 

\lemma e{If all offers are at price $p$, then for every natural number
$n \ge 3$, there exists a $\phi_n \in \Re$ such that
%
\display G{\forall \phi \le \phi_n \quad \forall j \in \{1, \ldots,
n-1 \} \quad r(j, \phi) \le 1.}}
%
\proof By \lem f, it is sufficient to show that $\forall \phi \! \le
\! \phi_n \enspace r(n-1, \phi) \le 1.$
Equation \eqn C implies that, if $r(n-1, \phi) > 1$ (that is, the
optimal reservation price is at least 2), then $V(n-1) - V(n-3) \le
u$. Note that, by equations \eqn A and \eqn B, 
%
\disparray H{V(n-1) - V(n-3) & = & [u/\phi - V(0)]
(\l^{n-3} - \l^{n-1}) \nonumber \\ & = & {u (1-\l^2) \l^{n-3} \over
\phi + (1-\l) m} \;.} 
%
Substitution of the value of $\l$ defined in \eqn y into \eqn H, and
application of l'H\^ opital's rule, yield $\lim_{\phi \to 0}
(V(n-1) - V(n-3)) = 2u$. Thus the lemma follows from \lem f and \eqn
C, since $\lim_{\phi \to 0} (V(n-1) - V(n-3)) > u$ implies that
the trader's unique optimal reservation price is 1 for sufficiently small $\phi$.
\endproof   

Now we use \lem e to show that, for sufficiently small values of
$\phi$, the optimum offer for all sellers is $p$. Recall that $W(\h,
o)$ was defined in \eqn J to be the expected value, to a seller
of type $i$ holding quantity $\h$ of money, of making an offer of $o$
to a buyer of type $(i+1)$.

\theorem g{For every $m \in (0, 1)$ there exists a $\phi^*_m \in \Re$
such that  
%
\display K{\forall \phi \le \phi^*_m \quad \forall \h \in \Re_+ \quad 
W(\h, p) = \max_{o \in \Rel_+} W(\h, o).}
%
}

\proof Note first that $\lim_{x \to \infty} x m^x = 0$. Thus we can
choose $n \in \Nat$ such that $\max_{n \le x} x m^x < m$. Let
$\phi^*_m$ be the value of $\phi_n$ satisfying \eqn G. As in the
proof of \thm d, we can restrict attention to the case where $\h$ and
$o$ are both integer multiples of $p$. Specifically let $\h = ip$ and
$o = jp$. Then 
%
\def\case#1#2{#1 & #2}
\display L{W(\h, o) = \left\{\begin{array}{l@{\quad :\quad }l}
\case{V(i)}{j = 0;}\\
\case{V(i) + m (V(i+1) - V(i))}{j = 1;}\\
\case{V(i) + m^{a(o)} (V(i+j) - V(i))}{j > 1;}
\end{array} \right. }
%
where $a(o)$, which has been defined formally in the proof of \lem b,
corresponds to the least level of money holding at which 
an offer of $o$ will be less than or equal to the optimal reservation
price. 
If $1 < j \le n,$ $a(2p)\ge n$ by \lem e. This implies that for all $o\ge
2p,\; a(o) \ge n.$ Therefore, $m^{a(o)}\le m^n.$ Also since $V$ is concave, 
\display M{V(i) + m^{a(o)} (V(i+j) - V(i)) \le V(i) + n m^n (V(i+1) -
V(i)).}
%
If $j>n,$ the following bound can be derived as in the proof of \lem b,
\display Z{V(i) + m^{a(o)} (V(i+j) - V(i)) \le V(i) + j m^j (V(i+1) -
V(i)).}
%
With these bounds, $p$ is seen to be the optimal reservation price by
choice of $n$. 
\endproof

\Section j{Welfare}

This version of the Kiyotaki-Wright model with divisible money, and
without inventory constraints on the holding of it, provides a
more natural environment to study welfare questions. In contrast to
the original version, here stationary equilibria with higher real
money stocks always provide higher levels of welfare. Intuitively, the
fewer agents there are without money, the fewer trading opportunities will
be foregone, and therefore the higher welfare will be.

To show this formally, we consider the standard welfare measure of summing
agents' utility levels. That is, our welfare measure is 
%
\display N{U(m, \phi, u) = \sum_{n=0}^\infty h(n) V(n) 
= (1-m) \sum_{n=0}^\infty m^n V(n).}
%
Substituting the values of $V(0)$ and $V(n)$ given in \eqn A and \eqn
B into equation \eqn N yields
%
\display O{U(m, \phi, u) = {m u\over \phi}.}
%
Given this equation and the fact that $\phi$ and $u$ are parameters
of the model, welfare would be maximized by selecting the highest
level of $m$ consistent with existence of equilibrium. By equation
\eqn o, welfare is maximized by minimizing the price level given a
fixed nominal money stock $M$ (that is, by maximizing the real money
stock in economy). 

Our strong conclusion that a higher real money stock unambiguously
corresponds to higher steady-state welfare arguably results from our
having abstracted from production costs. Zhou (1996) models production
costs, and shows that welfare need not be monotonic in the aggregate
real-money stock. The reason is that traders do not produce when they
are holding sufficiently high real money balances. An intuition for
this result is that the cost of production must be borne immediately,
while the benefit from acquiring additional money would be discounted
to the time of its expenditure. Since an increase in the economy's
real money stock can lead to an increase in the number of traders
holding high real money balances, such an increase might cause
aggregate production to decrease. In contrast to the technological
incompatibility between money holding and production in models
incorporating the Kiyotaki-Wright inventory constraint, such an
equilibrium incentive effect of the real money stock on production
would presumably disappear in the limit as $\phi$ approaches zero.

\Section k{Discussion}

We have studied a very preliminary, schematic version of a
search-equilibrium model with divisible money. We show that there
always exists a continuum of stationary monetary equilibria where all
transactions occur at a single price. Agents in this model economy set
their prices strategically rather than taking prices to be parametric
as in the Walrasian model. The prevalence of a single price results
from self-fulfilling beliefs of the agents. Besides the single-price
equilibria of the model, we conjecture that there may be other
stationary equilibria in which prices are dispersed. Nonstationary
equilibria may also exist.

The existence of a single-price equilibrium might be viewed in either
a positive or a negative light. On the positive side, the model
provides support for conclusions drawn from the Kiyotaki-Wright
analysis in which the parity of exchange between money and goods is
exogenous. On the negative side, the existence of an exact
single-price equilibrium seems to be a consequence of extreme features
of the model such as the seller-posting-price protocol (rather than a
trading protocol that splits the difference between seller's and
buyer's reported valuations) and the indivisibility of the consumption
good. While privacy of traders' information about their money holdings
also play a role, we regard this as being a more appealing assumption
than the others.

We conjecture that, even if our model were amended in ways that would
eliminate exact single-price equilibrium from occurring, equilibrium
price dispersion would vanish in the limit as $\phi$ is taken to
zero. If a buyer is confident that he will almost immediately meet
a seller whose offer is very close to the minimum offer in the market,
then he should be unwilling to post a high reservation price and trade
with any sellers who would force him to make a higher payment, unless his
money holding is huge. This is the same insight on which our present
analysis of frequent meetings is based. 

As in all search-equilibrium models without production costs, there is
a non-monetary equilibrium in this model. Each agent simply gives his
good away for free to anyone he meets who wants it. It is noteworthy that
this equilibrium has a greater amount of trade (and hence provides a
higher level of welfare) than does any monetary equilibrium. However, this
is a suspect equilibrium, because its existence is directly traceable to
our schematic assumption that production of goods is completely costless.
Introduction of any production cost, however small, removes it from the
equilibrium set. 

Monetary equilibrium in a random-matching environment with costly
production is studied by Zhou (1996). The extension to costly
production requires more careful attention to the equilibrium concept,
including use of a perfectness-type refinement. Despite this
refinement, the existence of a continuum of single-price equilibria
persists. The concluding section of Zhou (1996) discusses this, as
well as other issues regarding the economic interpretation of our
model. 

\newpage
\bigskip\bigskip\bigskip
\section*{\large\bf References}

\def\Aer{{\it American Economic Review,\/ }}
\def\Econ{{\it Econometrica,\/ }}
\def\Jet{{\it Journal of Economic Theory,\/ }}
\def\Jme{{\it Journal of Mathematical Economics,\/ }}
\def\Jpe{{\it Journal of Political Economy,\/ }}
\def\Res{{\it Review of Economic Studies,\/ }}

\parindent=0pt
\medskip \hangindent=30pt  
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\medskip \hangindent=30pt  
G.~Camera and D.~Corbae, ``Monetary patterns of exchange with search,''
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\medskip \hangindent=30pt  
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\medskip \hangindent=30pt
S.~Hendry, ``Search Models of Money,'' Ph.D. Dissertation, University of
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\medskip \hangindent=30pt
N.~Kiyotaki and R.~Wright, ``On money as a medium of exchange,'' \Jpe 97
(1989): 927-954.

\medskip \hangindent=30pt  
S.~Lefschetz, {\it Differential Equations: Geometric Theory}.
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\medskip \hangindent=30pt  
M.~Molico, ``The distribution of money and prices in a search
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\medskip \hangindent=30pt  
S.~Shi, ``A simple divisible search model of fiat money,''
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\medskip \hangindent=30pt  
S.~Shi, ``Money and prices: a model of search and bargaining,'' \Jet 67
(1995): 467-496.

\medskip \hangindent=30pt  
A.~Trejos and R.~Wright, ``Search, bargaining, money and prices,'' \Jpe
103 (1995): 118-141. 

\medskip \hangindent=30pt  
W.~Vickrey, ``Counterspeculation, auctions, and competitive
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\medskip \hangindent=30pt  
N.~Wallace, ``Questions concerning rate-of-return dominance and
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\medskip \hangindent=30pt  
R.~Zhou, ``Individual and aggregate real balances in a random
matching model,'' manuscript, Federal Reserve Bank of Minneapolis, 1996.

\end{document}


%%%

