%Paper: ewp-ge/9612001
%From: ruilin@ysidro.sas.upenn.edu (Ruilin Zhou)
%Date: Mon, 23 Dec 1996 17:47:13 -0500 (EST)
%Date (revised): Mon, 23 Dec 1996 18:04:55 -0500 (EST)

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{{\LARGE Individual and Aggregate Real Balances \par
\smallskip
in a Random Matching Model}\footnote{I would like to thank Harold Cole,
Dean Corbae, Ed Green, Neil Wallace and Randall Wright for helpful
conversations. The views expressed in this paper are those of the author
and not necessarily those of the Federal Reserve Bank of Minneapolis or
the Federal Reserve System.} \par} 

\bigskip\bigskip

{\large Ruilin Zhou \par
 University of Pennsylvania \par
\smallskip
and \par
\smallskip
Federal Reserve Bank of Minneapolis \par
250 Marquette Avenue \par
Minneapolis, MN 55401-2171 \par
ruilin@ysidro.sas.upenn.edu \par

\bigskip
\bigskip
November, 1996 \par}

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\vskip 1.8in
\begin{abstract} 
This paper investigates the existence and properties of stationary
single-price equilibrium in a monetary random matching model where agents
can hold an arbitrary amount of divisible money, and where production is
costly. For some parameter values of the model, there exists a continuum of 
single-price equilibria indexed by the aggregate real-money balance. 
At such an equilibrium, an agent accumulates money only up to a certain
point where his marginal value of holding money drops below the cost of
production. Different upper bounds on money holdings imply different
distributions of money holdings. The coexistence of multiple equilibria
with distinct upper bounds but identical aggregate real-money balances
suggests the importance of money distribution in determining trade
velocity, production and welfare.

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

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

Despite the success of research based on Kiyotaki and Wright (1989) in
endogenizing monetary trade as a means of overcoming market frictions,
there are apparent limitations with respect to questions such as how price
is formed. Two key simplifying assumptions are the main restrictions:
indivisibility of traded objects (including both money and goods) and
unit-inventory constraint on the holdings of objects. Together, these
assumptions imply that the only transaction pattern is one-for-one swapping
of goods and/or money. Shi (1995) and Trejos and Wright (1995) relax the
indivisibility constraint on goods by endowing each agent with the
capacity to produce any amount of goods instantaneously with cost, while
keeping both constraints on money. The equilibrium price is the outcome of
bargaining between an agent holding one unit of money and an agent with
production capacity. I will study a model that is dual to these models in
the sense that both indivisibility and unit-inventory constraints are kept
on goods, but money will be divisible and agents can carry any amount of
money. This approach is adopted for a better understanding of the crucial
aspects of the random-matching model, and for its potential in addressing
issues such as the uniformity of terms of trade and indeterminacy and/or
multiplicity of monetary equilibrium. Addressing these issues is a
necessary step toward applying these models to a broad range of policy
questions.

Green and Zhou (1995) study a version of Kiyotaki and Wright's monetary
search model in which agents can hold an arbitrary amount of divisible
money.\footnote{Some parallel work has been done by Camera and Corbae
(1996), Molico (1996) and Wallace (1996). They endow agents ability to
produce perfectly divisible good instantaneously, and adopt a
full-information bargaining mechanism in pairwise trading.  Camera and
Corbae (1996) and Wallace (1996) allow agents to hold multiple units of
money up to a finite upper bound. Molico (1996) computationally studies a
model without upper bound on money holdings.} The main result of the paper
is the existence of a continuum of stationary single-price equilibria in a
costless-production environment.  These equilibria are indexed by the
aggregate real-money balance, and they are also welfare-ranked. The main
problem of the paper is that money is not necessary to facilitate trades
given that production is costless. In fact, a gift-giving non-monetary
equilibrium always exists and Pareto-dominates any monetary equilibrium.
The question is then whether the assumption of costless production is
essential to the existence of a continuum of monetary equilibria.  

To answer this question, I study the stationary single-price equilibrium in
a costly-production environment.\footnote{I am indebted to Dean Corbae for
correcting my initial erroneous conjecture that a stationary distribution
of single-price equilibrium would not exist in a costly-production
environment.} The main results of the paper are as follows. First, at any
single-price equilibrium, agents' money holdings are uniformly bounded.
That is, there exists an endogenous upper bound on money holdings, at
which the cost of production exceeds agents' marginal value of holding
money, therefore ``rich'' people stop selling and accumulating money.
Second, for some parameter values of the model, a single-price equilibrium
exists. Generically, if one such equilibrium exists where the price is
$p$, for the same set of parameters, the equilibrium also exists in the
neighborhood of $p$. In other words, the existence of a continuum of
single-price equilibria indexed by price or aggregate real-money balance
survives the introduction of costly production. Third, there is another
dimension of multiplicity of equilibrium in this model. For a given
real-money balance (i.e., nominal money balance and price), single-price
equilibria with distinct upper bounds on money holdings may coexist. They
differ in the distribution of money holdings, velocity of trading, and
therefore welfare. This result suggests that the distribution of money may
be an important contributing factor to the non-neutrality of money.
Fourth, the equilibrium set with endogenous upper bound on money holdings
in general is different from those with exogenous upper bound. 
 
\Section b{The Model}

There is a continuum of infinitely-lived agents with mass of measure 1.
The population is equally divided into $k$ types, $k \ge 3.$ Time is
continuous. There are $k+1$ goods. Of these goods, $k$ (indexed
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, with a constant total nominal stock $M$.
A type $i$ agent can produce one unit of good $i+1$ (mod $k$)
instantaneously and at a utility cost $c > 0 $ any time. He consumes
only good $i$, from which he derives an instantaneous utility $u>c>0$. Each
agent maximizes the discounted expected utility of his consumption stream,
with discount rate $\r$. Note that two key assumptions of Kiyotaki and
Wright on fiat money are relaxed here: money is divisible and agents can
hold any amount of money at any time, even while they are producers. But
both indivisibility and inventory constraints on consumption goods are
maintained.

Agents meet pairwise randomly according to a Poisson process with
parameter $\mu$. The specialization in preference and production ensures a
complete lack of double coincidence of wants 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. An agent is characterized by his type and the amount of money he
holds, which varies across traders. Agents' types are known, but their
money holdings and trading histories are private information.  Pairwise
trading is conducted through a seller-posting-price mechanism.  When a
type $i$ agent who has fiat money (buyer) meets a type $i-1$ agent who can
produce his desired good (seller), the seller posts an offer at which he is
willing to sell that the buyer must either choose to 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.  

I will be focusing on stationary symmetric equilibrium where all agents
of the same type with the identical money holdings act alike, all of
the $k$ types are symmetric, and agents' trading strategies are
time-invariant. 

Let $H$ denote the measure of stationary distribution of money holdings in
the economy, with domain $\Re_+$. A generic type $i$ agent's trading
strategy consists of a pair of real-valued functions defined on $\Re_+$. An
offer strategy specifies the offer that an agent will make as a seller
when his current money holding is $\h$ and he meets a type $i+1$ agent,
denoted by $\o(\h)$. A reservation-price strategy specifies the maximum
willingness to pay as a buyer when his current money holding is $\h$ and
he meets a type $i-1$ agent, denoted by $\b(\h)$. A buyer with
money-holding $\h$ accepts an offer if it does not exceed his
reservation-price $\b(\h)$. A buyer cannot accept an offer that exceeds
his money holding, which is the feasibility constraint, 
\begin{equation}
\b(\h) \le \h.
\label{eqn:a}
\end{equation}

A symmetric strategy profile and a stationary distribution of money holdings
imply stationary distributions of offers, called $\O$,
\begin{equation}
\O(x) = H\{\h\, |\, \o(\h)\le x\},
\label{eqn:b1}
\end{equation}
and of reservation prices, called $\B$,
\begin{equation}
\B(x) = H\{\h \,|\,\b(\h)<x\}.
\label{eqn:b2}
\end{equation}
Note that, for convenience, $\B$ is defined to be continuous from the
left, rather than from the right as is conventional.

The value function $\V \colon \Re_+ \to \Re_+$ specifies the expected
discounted utility that an agent will receive as a function of his current
money holding, if he adopts an optimal trading strategy. Consider an agent
of type $i$ with money holding $\h$. The value of his money holding
$\V(\h)$ depends on its usage in potential trade meetings. With
probability $1/k$, the agent meets a producer of his consumption good,
from whom he might be able to make a purchase, pay an offer price $x$ which
decreases his money holding to $\h-x$, and enjoy a utility gain of $u$.
With equal probability $1/k$, the agent meets a consumer of his
production good to whom he might be able to make a sale, receive a payment
$o$ of his offer price which increases his money holding to $\h+o$, but
suffer a disutility cost of production $c$. Formally, given the discount
rate $\r$ and the random-matching frequency $\mu$, the value function
can be written as follows (for detailed derivation, see Green and Zhou
(1995)), 
\begin{eqnarray}
\V(\h) & = & {\mu \over k\r + 2\mu} \Biggl[ \max_{r \in [0, \h]}
\biggl[ \int_0^r \bigl(u + \V(\h - x)\bigr) \; 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)) \bigl(\V(\h + o) - c
\bigr) \biggr]\Biggr] .
\label{eqn:f}
\end{eqnarray} 
Equation \eqn f is the Bellman equation for $\V$. Standard arguments
establish that there exists a unique solution in the space of bounded
measurable functions, and that this solution specifies the optimal
expected discounted value of each possible level of money holding.  

A potential trade meeting between a buyer and a seller entails a
sequential bargaining game (the seller posts an offer, and then the buyer
accepts or rejects it). Extending the reduced-form representation of the
buyers' strategy --- the reservation price --- back to the sequential
setting, a buyer accepts an offer if and only if it does not exceed his
full valuation of a unit of consumption good. That is, his reservation
price is determined by his willingness to pay, 
\begin{equation}
\forall o \in \Re_+ \enspace \bigl[ \b(\h) \ge o \iff u +
\V(\h - o) \ge \V(\h) \bigr]. 
\label{eqn:f1}
\end{equation}
Perfectness of the equilibrium requires that this condition be satisfied
at any offer. This condition is particularly important because 
the equilibrium studied in this paper does not have full support.   

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 adopted here is stationary perfect
Bayesian equilibrium. I 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}
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\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 {b1} and \eqn {b2}. 
\item Given the distributions for money-holdings $H$, reservation-price $R$
and offer $\O$, the reservation-price strategy $\b$ satisfies feasibility
condition \eqn a and perfectness condition \eqn {f1}, and trading
strategies $(\o,\,\b)$ and value function $\V$ solve Bellman equation
\eqn f. That is, 
\begin{eqnarray}
\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)) - c
\bigr) \biggr]\Biggr].
\label{eqn:g}
\end{eqnarray}
\end{list}
\medskip

%Applying the definition of stationary perfect Bayesian equilibrium of
%Green and Zhou (1995) in above trading environment, an equilibrium
%consists of a distribution of money holding $H$, offer strategy $\o$, 
%reservation-price strategy $\b$, and value function $\V$ that jointly
%satisfy conditions of feasibility \eqn a, perfectness \eqn {f1},
%optimization and stationarity. The optimization condition 
%requires that given the distributions for money-holdings $H$,
%reservation-price $R$ and offer $\O$, the trading strategies $(\o,\,\b)$
%and value function $\V$ solve Bellman equation \eqn f. That is, 
%\begin{eqnarray}
%\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)) - c
%\bigr) \biggr]\Biggr],
%\label{eqn:g}
%\end{eqnarray}
%The stationarity condition is that under the transition probabilities
%induced by the optimal trading strategies $(\o, \b)$, the distribution $H$
%is a stationary initial distribution of the continuous-time markov process
%that governs the random matching and transaction.

\Section c{The Characterization of Single-Price Equilibrium}

In this section, I will study the existence of single-price equilibria.
Consider a potential single-price equilibrium at which all trades occur at
a single price $p$. I will first describe the set of trading strategies
for a single-price equilibrium. Assume that such strategies are adopted, I
will characterize the stationary money-holding distribution, and solve the
corresponding value function. Given the money-holding distribution and
value function, I will then verify that the presumed trading strategies
are indeed optimal. I will show that for any strictly positive production
cost $c$, every single-price equilibrium is associated with an endogenous
upper bound on agents' money holdings, above which no one will produce at
the prevailing price $p$. I will give a set of sufficient conditions under
which the single-price equilibrium exists, and discuss the choice of
equilibrium among an infinite set with equilibrium refinement. 

Consider the formulation of equilibrium above in the special case that all
trades occur at price $p$. By definition, for any pair of trade meeting at
equilibrium, either the buyer is not willing to pay the seller's offer
price, or they trade at exactly price $p$. Formally, let ${\cal H}$ denote
the product measure on $\Re \times \Re$, with $H$ in each coordinate. Then 

\medskip
{\sc Definition.} A stationary single-price equilibrium at which all trades
occur at price $p$ is a stationary equilibrium that satisfies an
additional condition. For meetings between a buyer and a seller
with money holdings $(\h_b,\,\h_s) \in \Re \times \Re$, the following two
events occur with probability $1$ (measurable with respect
to ${\cal H}$): either $\b(\h_b)<\o(\h_s)$ thus no trade occurs, or
$\o(\h_s)=p \le \b(\h_b)$ so trade occurs at price $p$. 
\medskip
   
If all trades occur at price $p$, the effective money holdings for an
agent are those that are integer multiples of $p$. I will consider
money-holding distributions only of this kind. That is, the support of the
distribution $H$ is a subset of the discrete set $p^\Natl\equiv \{0, p, 2p,
\ldots \}$.  For simplicity, define 
\begin{equation}
h(n)\equiv H(\{np\}).
\label{eqn:h}
\end{equation}
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 distribution $h$ on $\Nat$.
I will say that an agent is in state $n$ if his money holding is $np$.

We are looking for an equilibrium where money has value and where $p$
units of money can be exchanged for one unit of consumption goods. An
agent can only obtain $p$ units of money by incurring production cost $c$
and selling his good. Presumably, the agent will be unwilling to incur the
cost in exchange for the same amount of money if he has a lot of money in
his pocket.  That is, he will charge a higher price, at which he will not
make any sale. This intuition suggests a potential upper bound on an
agent's money holding. Let $\Np$ ($\N$ units of $p$) denote the maximum
quantity of money that an agent will hold in equilibrium, which can be
finite or infinite. If $\N$ is infinity, as in the case of Green and Zhou
(1995) with zero production cost, an agent may hold any amount of money.
Denote the effective support of money-holding distribution $H$, $\{0,p,2p,
\ldots,\Np\}$, by $\Lambda(p,\N)$. I am going to show later that the above
intuition can be justified: with positive production cost, there is an
endogenously determined finite $\N$ for every single-price equilibrium.  

Given the upper bound $\Np$ on an agent's money holdings, the set of 
optimal trading strategies for single-price equilibrium that are
consistent with the above definition are as follows. Sellers who hold less
than $\Np$ units of money choose to post price $p$ to sell their goods,
and sellers with money balance $\Np$ demand a higher price than $p$. But
with probability $1$, any price higher than $p$ is never accepted by any
buyers. In other words, if there is any seller posting higher than $p$
price (in particular, sellers with money balance $\Np$), there is no buyer
with money holding in the support $\Lambda(p,\N)$ who is willing to pay
that price. Therefore if an agent's money balance reaches $\Np$, he will
wait until he succeeds in spending $p$ on his consumption good before he
will produce again. As an outcome, no agents' money balance exceeds the 
upper bound $\Np$. Furthermore, any buyer holding at least $p$ units of
money will be willing to buy at price $p$, since there is no lower price
and not accepting $p$ would only delay consumption which is costly. To
summarize, the characteristics of the conjectured equilibrium strategies
are as follows. 
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\item All buyers with money holdings at least $p$ accept offer $p$:\par
$\qquad\qquad\qquad \forall n=1,2,\ldots,\; \b(np) \ge p$; 
\item All sellers with money holdings less than $\Np$ offer price $p$: \par
$\qquad\qquad\qquad \forall n=0,1,2,\ldots,\N-1 \quad\o(np)=p;$ 
\item Agents with money holdings greater or equal to $\Np$ offer above
$p$: \par
$\qquad\qquad\qquad \forall n\ge \N,\;\o(np)>p;$
\item Offers made by agents with money holdings $\Np$ or more are
not acceptable for any buyer: \par
$\qquad\qquad\qquad \forall n\ge \N,\;\o(np)>\max_{k\le \N}\b(kp).$
\end{list} 

Note that agents with money holdings not in the support of $H$, i.e., more
than $\Np$, may be willing to accept a higher price than $p$. But since
this group of agents has measure $0$, such probability $0$ event should not
affect trading strategies (i)---(iv). However, the price that sellers in
state $\N$ offer will affect the values of these states off the support.
There are potentially many equilibria that are outcome-equivalent --- all
trades occur at price $p$ with probability $1$ --- that have sellers in
state $\N$ posting different prices. The choice among these equilibria
will be made later with a specific equilibrium refinement. 

For now, suppose that sellers in state $\N$ offer to sell at price $\Jp$.
By (iv), no agents with money holding in the support will be willing to
pay $\Jp$. But agents off the support, with money holdings more than some
$\Kp$ ($\K>\N$) might be willing to pay $\Jp$. That is, in addition to 
(i)---(iv), the following characteristics of the strategies are relevant, 

\begin{list}{(\roman{fig})}{\usecounter{fig}
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\item Sellers in state $\N$ sell at price $\Jp$: $\;\o(\Np)=\Jp$;  
\item There exists a least money balance $\Kp$ ($\K \ge \N$), buyers with
money holdings above which are willing to pay $\Jp$: $\;\b(\Kp)<\Jp$, and 
$\forall n > \K,\;\b(np)\ge \Jp.$ 
\end{list}
 
\noindent Obviously, $\K$ is determined by $\J$.  The offer strategy
implies the two types of positive-probability trading opportunities for
agents holding more than $\Np$: buy consumption goods at price $p$, and
buy consumption goods at price $\Jp$ which they will engage only if their
money holding is more than $\Kp$.  From now on, I will refer to a
single-price equilibrium at which all trades occur at price $p$, the
support of money-holding distribution is $\Lambda(p,\N)$, and all agents
adopt trading strategies (i)---(vi), a single-price-$\N$-$\J$-$p$
equilibrium.  

\Subsection e{Stationary Money-Holding Distribution}

Assume that agents adopt the above set of trading strategies. Let the
proportion of agents who hold positive money holdings to be $m$, 
\begin{equation}
m \equiv \sum_{n=1}^\n h(n).
\label{eqn:j}
\end{equation}
Note that 
\begin{equation}
h(0)=1-m. 
\label{eqn:k}
\end{equation}

Since all transactions are made at price $p$, an agent moves into state
$n$ (the state of having money holding $np$) only 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 a type $i$ agent
with a money balance less than $\Np$ will make a sale whenever he meets 
an agent of type $i+1$ who has money, the Poisson frequency of such
trading is $\mu m/k$. A type $i$ agent with money will make a purchase
whenever he meets an agent of type $i-1$ who is willing to sell at price
$p$ (everybody except agents in state $\N$), and the Poisson
frequency of such trading is $\mu (1-h(\N))/k$. The trading flows and
frequencies are represented in the following graph. Stationarity requires
that the time rate of inflow to any state $n$ equals the time rate of
outflow from state $n$. 
%
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\put(6.32,0.75){${\mu (1-h({\n})) \over k}$}
\put(8.3,1.5){\oval(1.0,0.6)} \put(8.25,1.4){$n$}
\put(9.0,1.45){$\cdots\cdots$}
\put(10.6,1.5){\oval(1.0,0.6)} \put(10.15,1.35){$\N-1$}
\put(11.6,2.0){${\mu m \over k}$}
\put(11.12,1.66){\vector(1,0){1.46}}
\put(12.58,1.34){\vector(-1,0){1.46}}
\put(11.12,0.75){${\mu (1-h({\n})) \over k}$}
\put(13.1,1.5){\oval(1.0,0.6)} \put(13.0,1.35){$\N$}
\end{picture}
\end{center}
%
Or equivalently, as indicated by the graph, the
time rate of inflow into all the states above an arbitrary $n$ (state
$n$, $n+1$ up to $\N$) should be equal to the time rate of outflow from
these states. That is, for any $n=1,2,\ldots,\N,$
\begin{equation}
{\mu m \over k} h(n-1)= {\mu (1-h(\N)) \over k} h(n).
\label{eqn:l}
\end{equation}
Combining \eqn j, \eqn k and \eqn l, we can express the size of
population in state $n$ as a function of the proportion of agents having
money $m$ and the upper bound $\N$, for any $n=0,1,\ldots,\N,$
\begin{equation}
h(n)=\Bigl( {m \over 1-h(\N)}\Bigr)^n (1-m).
\label{eqn:m}
\end{equation} 
Note that equation \eqn m is consistent with the costless production case
of Green and Zhou (1995) where $\N=\infty$, the money-holding
distribution is geometric and $h(\N)=0$. The fact that \eqn m holds for
$n=\N$ implies 
\begin{equation}
h(\N) \bigl(1-h(\N)\bigr)^\n=(1-m) m^\n.
\label{eqn:n}
\end{equation}

Given the functional form of money-holding distribution \eqn m, the
nominal money stock $M$ is the sum of money held by all agents. That is, 
\begin{equation}
M= \sum_{n=1}^\n np h(n) = p (1-m) \sum_{n=1}^\n n \Bigl( {m \over
1-h(\N)}\Bigr)^n.
\label{eqn:o}
\end{equation}
Rearranging equation \eqn o, we can solve $h(\N)$ as a function of
real-money balance ${M \over p}$, 
\begin{equation}
h(\N)={m - (1-m){M\over p} \over m (\N+1) - {M\over p}}.
\label{eqn:p}
\end{equation} 
For a given $\N$, real-money balance ${M\over p}$ and the proportion of
people holding money $m$ have to jointly satisfy \eqn n--\eqn p, as
well as satisfy $0\le m\le 1$ and $0\le h(\N)\le 1$.  We can show that
the solution exists, which together with \eqn m defines the distribution
completely. Equations \eqn m--\eqn p are the equilibrium
stationarity condition. More specifically, the distribution has the
following features.

\medskip
\lemma z{Conditional on $0\le m\le 1$ and $0\le h(\N)\le 1$, the following
relationship holds among $\N$, ${M\over p}$, $m$ and the distribution $h$.
\newcounter{lemz}
\begin{list}{(\alph{lemz})}{\usecounter{lemz}
\setlength{\labelwidth}{1cm} \setlength{\labelsep}{0.3cm}
\setlength{\leftmargin}{1.5cm} \setlength{\topsep}{0.15cm} 
\setlength{\parsep}{0pt} \setlength{\itemsep}{0cm}}
\item $\N > {M\over p}$. 
\item Given $\N$, $m$ is uniquely determined by ${M\over p} < \N$ 
and vice versa. 
\item The distribution $h$ satisfies the following conditions: 
if ${M\over p}<{\N \over 2}$, then $m<{\N\over \N+1}$, $1-h(\N)>m$
and the distribution $\{h(n)\}_{n=0}^\n$ is a decreasing sequence of $n$;
if ${M\over p}={\N \over 2}$, then $m={\N\over \N+1}$, and $h$ is 
a uniform distribution, i.e., $\forall n=0,1,\ldots,\N$, $h(n)={1\over
\N+1}$;
if ${M\over p}>{\N \over 2}$, then $m>{\N\over \N+1}$, $1-h(\N)<m$
and the distribution $\{h(n)\}_{n=0}^\n$ is an increasing sequence of $n$.
\end{list} }

\medskip

\uppercase \lem z asserts that for a given $\N$, there is a one-to-one
mapping between $m$ and ${M\over p}$, therefore either one uniquely
determines the stationary distribution. Note that in case (c), the
money-holding distribution is censored-geometric:  the proportion of
people holding a higher quantity of money declines geometrically before it
hits the upper bound. The proof of \lem z is given in appendix A.  

\Subsection f{Equilibrium Value Function}

Now we solve equation \eqn g for the equilibrium value function. The
presumed optimal trading strategies (i)---(vi) postulate that sellers in
states in the support except those in state $\N$ will sell at price $p$
and no lower than price $p$, 
\begin{equation}
\O(p)=1-h(\N); \qquad \forall x<p \quad \O(x)=0; 
\label{eqn:q}
\end{equation}
that agents in state $\N$ will sell at price $\Jp$,
\begin{equation}
\forall p<x<\Jp, \quad \O(x)=\O(p);\qquad \O(\Jp)-\O(p)=h(\N); \qquad
\forall x>\Jp \quad \O(x)=1; 
\label{eqn:r}
\end{equation}
and that all buyers with money will accept offer $p$, but not any higher
offer made by agents with money holding $\Np$,
\begin{equation}
1-\B(p)=m,\qquad R(\Jp)=1.
\label{eqn:r1}
\end{equation}
For convenience, define $V(n)=\V(np)$. Given \eqn q---\eqn {r1},
the presumed trading strategies (i)---(iv) imply that the value function
\eqn g can be written as the following set of equations: 
\begin{equation}
\label{eqn:s1}
V(0) = {\mu \over k\r+2\mu}\Bigl(V(0) + (1-m)V(0) + m(V(1)-c)\Bigr), 
\end{equation}
for all $n=1,2,\ldots,\N-1$,
\begin{equation}
\label{eqn:s2}
V(n) = {\mu \over k\r+2\mu}\Bigl((1-h(\N))(u+V(n-1))+h(\N)V(n) + (1-m)V(n)
+m(V(n+1)-c)\Bigr),
\end{equation}
and
\begin{equation}
\label{eqn:s3}
V(\N) = {\mu \over k\r+2\mu}\Bigl((1-h(\N))(u+V(\N-1))+h(\N)V(\N) +
V(\N)\Bigr). 
\end{equation}
The value function for states off the support, $n>\Np$, can be
defined similarly given (i)---(vi).
For all $n=\N+1,\ldots,\K$, 
\begin{equation}
\label{eqn:s4}
V(n) = {\mu \over k\r+2\mu}\Bigl((1-h(\N))(u+V(n-1))+h(\N)V(n) +
V(n)\Bigr),
\end{equation}
for $n\ge\K+1$,
\begin{equation}
\label{eqn:s5}
V(n) = {\mu \over k\r+2\mu}\Bigl((1-h(\N))(u+V(n-1)) +
h(\N)(u+V(n-\J)) + V(n)\Bigr).
\end{equation}
Note that the parameters $\mu$, $\r$ and $k$ function as a single
parameter in the form of $k\r/\mu$. For simplicity, define
\begin{equation}
\phi={k\r\over \mu}.
\label{eqn:t}
\end{equation}

The system of equations \eqn {s2} can be rewritten in the matrix form,
\begin{equation}
\left( \begin{array}{c} V(n+1) \\ V(n) \\ B \end{array} \right)
= \D \;
\left( \begin{array}{c} V(n) \\ V(n-1) \\ B \end{array} \right)
\label{eqn:u}
\end{equation}
where $B=(1-h(\N))u-mc$, and 
\begin{equation}
\D = \left( \begin{array}{c c c} 
\left[ {\phi \over m} +{1-h(\N) \over m} +1 \right] & 
-{1-h(\N) \over {}\; m} & -{1 \over {}\; m} \\ 1 & 0 & 0 \\ 0 & 0 & 1
\end{array} \right).
\end{equation}
Equation \eqn u is a second-order linear difference equation. Its family
of solutions is given in terms of eigenvectors of the matrix $\D$, which
have three real eigenvalues, $\l_0(=1)$, $\l_1\in (0,1)$ and $\l_2>1$. The
precise solution is determined by the two endpoint equations \eqn {s1} and
\eqn {s3}.  Appendix B solves the difference equation in detail. To summarize
the result, the value function $V$ takes the following form: for all
$n=0,1,\ldots,\N$, 
\begin{equation}
V(n)=\t_0 /\phi + \t_1 \l_1^n + \t_2 \l_2^n
\label{eqn:w}
\end{equation}
where $\t_0$, $\t_1$ and $\t_2$ are functions of $\N$, $m$, $h(\N)$, and the
parameters of the model $u$, $c$ and $\phi$. The explicit functional forms
for $\l_1$, $\l_2$, $\t_0$, $\t_1$ and $\t_2$ are given in appendix B. 

By \eqn {s4} and \eqn {s5}, the value function for $n>\N$ can be solved
recursively through the following, for $n=\N+1,\ldots,\K$, 
\begin{equation}
V(n)={1-h(\N)\over \phi +1-h(\N)}(u+V(n-1)),
\label{eqn:x}
\end{equation}
for $n\ge\K+1$, 
\begin{equation}
V(n)={1\over \phi +1}\Bigl((1-h(\N))(u+V(n-1)) + h(\N)(u+V(n-\J))\Bigr).
\label{eqn:x1}
\end{equation}
When we analyze the explicit solution of the value function \eqn w, \eqn x
and \eqn {x1}, it is easy to show the following.

\medskip
\lemma a{The value function $V$ is increasing in $n$.} 
\medskip

Although we characterize 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
\begin{equation}
\V(\h) = V([\h/p]).
\label{eqn:z}
\end{equation}
This completes the derivation of the value function.

\Subsection g{Equilibrium Strategy}

Given the characterization of the distribution of money holdings and value
function, I will first show the existence of the finite upper bound $\N$
on money holdings in a single-price equilibrium.

\medskip
\theorem f{For any given $\phi$ and $c>0$, there is no single-price
equilibrium at which all agents' money holdings are unbounded.} 
\medskip

\proof Suppose to the contrary, there is a single-price equilibrium where
all transactions occur at price $p$ and all agents' money holdings are
unbounded, i.e., $\N=\infty$. Since the only way for an agent to acquire
money is by selling his product, the supposition implies that the agent
must be willing to sell at price $p$ regardless of his money holding. That
is, for any $n\in \Nat,$ $V(n+1)-V(n)\ge c$. As $\N\to\infty$, $h(\N)\to
0$ by \eqn p. For a given $\phi$, $\l_1 \in (0,1)$ and $\l_2 >1$. By \eqn
w, for any $n=0,1,\ldots,\N$, $V(n+1)-V(n)=\t_1 \l_1^n (\l_1-1) + \t_2
\l_2^n (\l_2-1)$, therefore it holds for $n=[\N/2].$ It is easy to show
that as $\N\to \infty$, $\t_1$ is finite but $\l_1^{[\n/2]} \to 0$ since
$\l_1<1$, and $\t_2 \l_2^{[\n/2]} \to 0$, thus, $V([\N/2]+1)-V([\N/2]) = 
\t_1 \l_1^{[\n/2]} (\l_1-1) + \t_2 \l_2^{[\n/2]} (\l_2-1) \to 0.$ But this
contradicts the implication of the supposition that $V(n+1)-V(n)$ is at
least $c>0$ for any $n$. Therefore, there exists some finite $\N$ such
that for $n \ge \N$, an agent with money holding $np$ will not be willing
to sell at $p$. Whatever his posting price will be, trade will not take
place at equilibrium because such trade conflicts with the definition of
{\it single-price} equilibrium. That is, an agent will not be able to sell
while holding $\Np$ units of money. He can succeed only to purchase his
consumption good at $p$ and therefore reduces his money balance. Such a
trading pattern implies that no agent's money holding will exceed $\N$.  
\endproof

\medskip
Now we can study the set of single-price equilibrium strategies in the
context of a finite money-holding upper bound $\N$. I will characterize
the set of sufficient conditions under which the presumed optimal trading
strategies (i)---(vi) of the single-price-$\N$-$\J$-$p$ equilibrium are
optimal relative to the value function derived above for an agent,
given the money-holding distribution $H$, the reservation-price
distribution $R$ and the offer distribution $\O$. The choice of $\J$ will
be discussed in the next subsection. 

Suppose that a buyer of type $i$ with money holding $\h$ meets a seller of
type $i-1$ who produces the buyer's consumption good. Without the
knowledge of each other's money holding, the seller makes an offer $o$ to
sell his product, and the buyer chooses a reservation price $r$. If $r\ge  
o$, the buyer accepts the offer and pays $o$ units of money in exchange
for one unit of his consumption good. Otherwise, the offer is rejected and
the buyer and the seller part ways without trade. The buyer should
choose an optimal reservation price, given the sellers' offer distribution
$\O$, that solves the maximization problem with respect to $r$ of the
Bellman equation \eqn f, subject to the perfectness constraint \eqn {f1}.
The solution is,
\begin{equation}
\b(\h) = \max \{r \in [0, \h] | u + \V(\h - r) \ge \V(\h)\}.
\label{eqn:A}
\end{equation} 
This reservation price reflects the agent's full value for a unit of his
consumption good. The seller also chooses an optimal offer strategy, given
the buyers' reservation-price distribution $R$, that solves the
maximization problem with respect to $o$ of the Bellman equation \eqn f, 
\begin{eqnarray}
\o(\h)&=&\hbox{arg}\max_{o \in \Rel_+} \biggl[ \B(o) \V(\h) + \Bigl(1 -
\B(o)\Bigr) \Bigl(\V(\h + o) - c \Bigr) \biggr] \nonumber \\
&=&\hbox{arg}\max_{o \in \Rel_+} \biggl[ \Bigl(1 - \B(o)\Bigr)\Bigl(\V(\h +
o) - c-\V(\h) \Bigr) \biggr].
\label{eqn:E1}
\end{eqnarray}

Since the value function is a step function defined by \eqn z, we can
eliminate some obvious weakly dominated trading strategies with \eqn A and
\eqn {E1}.

\medskip
\lemma y{Given the support of money-holding distribution $\Lambda(p,\N)$, the
equilibrium offer and reservation-price strategies involve only integer
multiples of $p$. More precisely, for every positive integer multiple $np$
of $p$, the reservation price $\b(np)$ is an integer multiple of $p$; 
and given all agents' reservation prices are integer multiples of $p$, the
optimal offer $\o(\h)$ is an integer multiple of $p$ for any $\h$.}
\medskip

\noindent
See Green and Zhou (1995), lemma 2 and lemma 4 for proof. We can now focus
on the set of offer and reservation-price strategies that are integer
multiples of $p$. 

If a single-price equilibrium exists, where agents' money holdings are
bounded by $\Np$, the sellers' offer strategy must change distinctly at
money holding $\Np$. In particular, a seller offers to sell at price $p$
in state $\N-1$, and he is not willing to offer at $p$ in state $\N$.
Given the reservation-price distribution \eqn {r1}, these offer strategies
imply 
\begin{eqnarray}
\label{eqn:y}
&&V(\N)-c \ge V(\N-1), \\
\label{eqn:D}
&&V(\N+1)-c<V(\N).
\end{eqnarray}
The conditions can be expressed as functions of the parameters of the
model by applying the solutions for the value function \eqn w, \eqn x and
\eqn {x1}. This pair of inequalities defines precisely the upper bound
$\N$ on money holdings as the incremental value of $p$ units money drops
below the cost of production $c$. The necessity of these two conditions in
defining the upper bound $\N$ implies that they are necessary conditions
for the existence of the single-price equilibrium with that upper bound.
Assuming these two conditions hold, let's examine the rest of
equilibrium trading strategies. 

First, condition \eqn y implies the following characteristics of the value
function.  

\medskip
\lemma b{At equilibrium, the value function $V$ is concave in $n$ up to
$\K$. That is, for all $1\le n \le \K$, $V(n)-V(n-1)$ is decreasing in $n$.} 
\medskip

\noindent 
\uppercase \lem b is proved in appendix C. The value function may not be
concave above $\K$. Remember that $\K$ is the state above which buyers 
accept offer $\Jp$ made by sellers in state $\N$, $\K\ge \N$. Given \lem b,
it is easy to show the existence of $\K$.

\medskip
\lemma x{For any finite $\J>1$, at equilibrium there exists a finite
$\K$ such that buyers in state $\K$ reject offer $\Jp$ and buyers in
state $\K+1$ accept offer $\Jp$.} 
\medskip

\proof Suppose $\K$ is infinite, that is, no money holding is high enough
for agents to accept $\Jp$. By \lem b, the value function is concave
in $n$ throughout. By \eqn x, it is easy to show that for a finite $\J$,
$V(n)-V(n-\J) \to 0$, as $n \to \infty$. Therefore there exists an $n^*$
such that $V(n^*)-V(n^*-\J)> u$ but $V(n^*+1)-V(n^*+1-\J)\le u$. By \eqn A,
then, $\b((n^*+1)p)\ge \Jp$, contradicts to the supposition. The threshold
$n^*$ is the $\K$ we are looking for. That is,
\begin{equation}
\K=\max\{n\,|\,V(n)-V(n-\J) >u \} 
\label{eqn:O}
\end{equation}
which is finite and unique for a given $\J$.
\endproof

\medskip
Now we are ready to verify the presumed equilibrium reservation-price
strategy (i). 

\medskip
\lemma c{If condition \eqn y holds, it is optimal for agents with money
holdings at least $p$ to accept offer $p$.}
\medskip

\proof By \lem b, condition \eqn y guarantees that the value function is
concave up to $\K$. Consider buyers in state $n$. If $n=1$, subtracting
\eqn {s1} from \eqn {s2} for $n=1$ and rearranging the equation, we have 
\begin{equation}
(1-h(\N))(u+V(0)-V(1))=\phi(V(1)-V(0)) + m\Bigl((V(1)-V(0)) -
(V(2)-V(1)) \Bigr).
\label{eqn:B}
\end{equation}
The right hand side of \eqn B is positive because $V$ is increasing
by \lem a and concave in $n$.\footnote{In the special case where
$\N=\K=1$, \eqn B is replaced by $(1-h(\N))(u+V(0)-V(1))=\phi V(1)
>0.$ No concavity condition is needed here.} Therefore $u+V(0)>V(1)$. By
the reservation-price function \eqn A, $\b(p)=p$. 

If $2\le n\le\K$, again since the value function $V$ is concave in
$n$, $V(n)-V(n-1)<V(1)-V(0)<u.\,$ Therefore, $u+V(n-1)>V(n)$. 

If $n\ge \K+1$, from \eqn {x1}, 
\begin{equation}
u+V(n-1)-V(n)=\phi V(n)+h(\N)(V(n)-V(n-\J)).
\label{eqn:B1}
\end{equation}
The right hand side of \eqn {B1} is positive since $V$ is increasing. So
$u+V(n-1)>V(n)$.

Combining all three cases, by \eqn A, we find for all $n\ge 1$, $\b(np)
\ge p$.  That is, it is optimal that buyers with money holding at least
$p$ set their reservation prices no lower than $p$. 
\endproof

\medskip
Next, I examine the validity of sellers' offer strategy profile
(ii)---(iv) as equilibrium strategy profile.  

\medskip
\lemma d{Given that conditions \eqn y and \eqn D hold, if 
\begin{equation}
V(\N) - V(\N-2)>u,
\label{eqn:C}
\end{equation} 
then it is optimal for buyers with money holdings in the support pay no
more than $p$, for sellers with money holdings less than $\Np$ to offer
$p$, and for sellers with money holdings at least $\Np$ but less than
$\Kp$ to offer higher than $p$.} 
\medskip

\proof By \lem b, condition \eqn y guarantees the concavity of the value
function up to $\K$. If $V(\N)-V(\N-2)>u$ holds, by \eqn A agents
with money holdings $\Np$ do not pay $2p$ for their consumption goods. By
the concavity of the value function, $\forall \;n=2,3,\ldots,\N$,
$V(n)-V(n-2) > u$. Also by the increasingness of $V$, for any
$l=2,3,\ldots,\N$ and $n=l,\ldots,\N$, $V(n)-V(n-l) >u$. Thus no agents
will be willing to pay higher than $p$ price. By \lem c, all agents with
money holdings at least $p$ accept $p$ for trade, therefore, for all
$n=1,2,\ldots,\N$, $\b(np)=p$, or $R(p)=1-m$ and for any $l>1$,
$1-R(lp)=H\{\h\,|\,\b(\h)\ge lp\}=0$. 

Consider a seller with money holding $np$, denote his expected
value of offering $lp$ by $W(np,lp)$. For $l=0,1,\ldots,\N$,
\begin{eqnarray}
W(np,lp)&=&R(lp)V(n)+(1-R(lp))(V(n+l)-c) \nonumber\\
&=&V(n)+(1-R(lp))(V(n+l)-c-V(n)).
\label{eqn:E}
\end{eqnarray}
Since $V(\N)-V(\N-1)\ge c$, by concavity $V(n)-V(n-1)>c$
for all $n=1,2,\ldots,\N$. Together with the result regarding the
reservation distribution $R$ above, for any $n<\N$ and $l>1$,
$W(np,p)=V(n)+m(V(n+1)-c-V(n))>W(np,lp)=V(n)>W(np,0)$. Thus for all
$n<\N$, $\o(np)=p$. That is, all agents holding less than $\Np$ units of
money should offer exactly $p$. 

Given $V(\N+1)-V(\N)<c$, the concavity of $V$ up to $\K$ implies
that for all $\N \le n< \K$, $V(n+1)-V(n)<c$.  Applying these inequalities
to \eqn E, we have for all $\N\le n <\K$ and any $l>1$, $W(np,0) < W(np,p)=
V(n)+m(V(n+1)-c-V(n))< W(np,lp)=V(n)$. Therefore, the optimal offer
should be higher than $p$, for $\N\le n <\K$, $\o(np)>p$. 
\endproof

\medskip

Condition \eqn C ($V(\N)-V(\N-2)>u$) is imposed as a sufficient condition
for (iv). It eliminates all higher value (than $p$) of reservation price,
and therefore ensures $R(x)=1$ for any $x>p$ which is stronger than (iv)
requires. It is conceivable that there is a single-price equilibrium that
does not satisfy \eqn C. For example, there may be agents with reservation
price $2p$, but the optimal offer is $3p$ or higher for agents holding
$\Np$. In this case, (iv) is satisfied, but \eqn C is not. On the other
hand, if we impose $V(\N)-V(\N-\J)>u$, i.e., no agents are willing to pay
$\Jp$, it may not be enough to guarantee that there is no trade conducted
at a price other than $p$ with positive probability. In case there are
agents willing to pay some $lp$ which is above $p$ but below $\Jp$, it may
be the best response for agents holding $\Np$ units of money to offer
$lp$.  
 
Since the imposition of \eqn C implies that for any $l>1$, $1-R(lp)=0$,
agents with money holding $\Np$ are in fact indifferent among posting any
offer $\Jp$ for $\J>1$. To simplify the analysis temporarily, let's assume
that a seller can post `infinity' as a price, and so make it impossible
for a buyer with any finite money holding to purchase his consumption
good, i.e., $\K$ is infinity as well. This is the simplest single-price
equilibrium possible. The value function in the states above $\N$ is given
by \eqn x, and is concave in $n$ throughout. In this case, \lem y---\lem d
completely characterize the set of equilibrium strategies, with conditions
\eqn y,
\eqn D, \eqn C as the set of sufficient conditions for single-price
equilibrium of this kind to exist. In fact, when the instantaneous utility
of consumption $u$ and production cost $c$ are such that $u \le 2c$ ($u>c$
is assumed in the beginning),
\eqn y and \eqn D become the necessary and sufficient conditions for the
existence of the single-price-$\N$-$\infty$-$p$ equilibrium (\eqn y
implies condition \eqn C). To summarize,

\medskip
\theorem e{For a given set of the parameters $M,\,u,\,c,\,\phi$
in the above environment, 
\newcounter{thme}
\begin{list}{(\alph{thme})}{\usecounter{thme}
\setlength{\labelwidth}{1cm} \setlength{\labelsep}{0.3cm}
\setlength{\topsep}{0.1cm} \setlength{\leftmargin}{1.5cm}
\setlength{\parsep}{0pt} \setlength{\itemsep}{0cm}}
\item For any $\J>1$, if a single-price-$\N$-$\J$-$p$ equilibrium exists,
conditions \eqn y and \eqn D hold.
\item A single-price-$\N$-$\infty$-$p$ equilibrium exists if conditions
\eqn y, \eqn D and \eqn C are satisfied.  
\item When $u \le 2c$, a single-price-$\N$-$\infty$-$p$ equilibrium exists
if and only if \eqn y and \eqn D hold.
\end{list}}
\medskip

Readers who are not interested in equilibrium refinement can skip the next
subsection.

\Subsection j{Equilibrium Refinement}

There are many equilibria at which all offer prices are finite. We can
shrink the equilibrium set a bit by imposing ex-post individual
rationality for sellers. An offer $o$ is ex-post individually rational
if it does not make the seller worse off in case a trade does occur at the
offer price, or 
\begin{equation}
\V(\h + o) - c \ge \V(\h). 
\label{eqn:L}
\end{equation}
But the set of single-price-$\N$-$\J$-$p$ equilibria (indexed by $\J$),
with the offer $\Jp$ in state $\N$ satisfying \eqn L, i.e., $V(\N+\J)-c
\ge V(\N)$, is still infinite. One can stop here, pick a $\J$ 
at random, and study that particular single-price-$\N$-$\J$-$p$
equilibrium.  Here, I will attempt to use an equilibrium refinement
concept, Myerson's {\it properness} (1991) in essence, heuristically to
select a particular equilibrium, or more specifically, select an offer
strategy among the set that agents with money holding $\Np$ are
indifferent.
 
Under condition \eqn C, the optimal reservation price is $p$ for
all agents with money holdings in the support. Suppose instead of assuming
that agents accept only offer $p$ with probability $1$, we assume that
they might make mistakes, and that the chance of a small mistake (small
loss of utility) is bigger than the chance of a big mistake (big loss of
utility). More precisely, denote the net gain of a buyer with money
holding $\h$ who purchases at an offer $o$ ($o\le\h$) to be $\tau(\h,o)$,
$\tau(\h,o)=u+\V(\h-o)-\V(\h)$. The net gain can be positive or negative.
Suppose that buyers adopt the following ``tremble'' strategy, for some
$\varepsilon\in (0,1)$, for any money holding $\h$ and any offer $o\le\h$,

\bigskip
\begin{center}
\begin{tabular}{llcc llcc} 
if $o\le p$ \quad & accept $o$&wp&$1-\varepsilon^{1+\tau(\h,o)}$ &
\qquad if $o> p$ \quad & accept $o$&wp&$\varepsilon^{1-\tau(\h,o)}$\\ 
& reject $o$&wp&$\varepsilon^{1+\tau(\h,o)}$ &
& reject $o$&wp&$1-\varepsilon^{1-\tau(\h,o)}$.\\ 
\end{tabular}
\end{center}
\bigskip

\noindent 
That is, when $o\le p$, the net gain for the buyer is positive, and he
should accept it. He rejects it only mistakenly with a tremble probability
$\varepsilon^{1+\tau(\h,o)}$, which is smaller as the offer $o$ becomes lower
and the net gain becomes higher.\footnote{The $1$ is added to ensure the
probability of rejection when $o\le p$ and the probability of acceptance
when $o>p$ are not 1 in case the net gain is zero.} When $o > p$, the net
gain of the deal for the buyer is negative and he should reject it. He
accepts it only mistakenly with a tremble probability
$\varepsilon^{1-\tau(\h,o)}$, which is smaller as the offer $o$ becomes
higher and the net loss becomes higher.  

Given this tremble strategy of the buyers and given that the value
function is a step function defined by \eqn z, first of all, all
non-integer multiples of $p$ are weakly dominanted by the integer
multiples of $p$ as offers. Then, among the set of offers that are integer
multiples of $p$, one can choose $\varepsilon$ small enough such that for
sellers holding $\Np$, if selling at $lp$ and $(l+1)p$ both are ex-post
individually rational, posting the lower offer $lp$ yields a higher
expected return, therefore dominates posting the higher offer $(l+1)p$.
That is, 
\begin{equation}
\o(\Np)=p\min\{l\,|\, V(\N+l)-c\ge V(\N)\,\}.
\label{eqn:K}
\end{equation}
Taking $\varepsilon$ to $0$, the tremble strategy converges to the original
buyers optimal strategy, represented by its reduced-reform
reservation-price strategy \eqn A, and $\o(\Np)$ remains the offer
strategy for agents in state $\N$. That is, \eqn K in addition to \eqn A
and \eqn {E1} are the set of equilibrium strategies that survive this
equilibrium refinement process. To distinguish single-price-$\N$-$\J$-$p$
equilibrium with $\J=\o(\Np)$ as in refinement condition \eqn K, I
will call it single-price-$\N$-$\J$-$p$ {\it proper} equilibrium.

To illustrate the workings of the single-price-$\N$-$\J$-$p$ proper
equilibrium, let's consider the simple case of $\J=2$. Given \eqn D,
$V(\N+1)-c<V(\N)$, \eqn K requires 
\begin{equation}
V(\N+2)-c \ge V(\N).
\label{eqn:N}
\end{equation} 
Given agents in state $\N$ posting offer $2p$, condition \eqn C,
which guarantees that no agents are willing to pay $2p$, becomes a 
necessary condition for no trades occur at a price other than $p$.   
Now we are ready to verify the rest of the presumed optimal
strategies (iii) and (vi) for agents off the support with money holdings
more than $\Kp$ for the case of $\J=2$. We have already shown that it
is optimal for agents to accept an offer of $p$ in states higher than $\K$
by \lem c, and to accept an offer of $2p$ in state $\K+1$ by \lem x.  

\medskip
\lemma f{In addition to condition \eqn y, \eqn D, \eqn C and \eqn N, if
\begin{equation}
V(\N+2)-c < V(\N+1),
\label{eqn:M}
\end{equation} 
then it is optimal for agents holding at least $\Kp$ units of money to
offer a price above $p$, and for agents holding more than $\Kp$ units of
money to accept an offer of $2p$.} 

\medskip 
The proof is in appendix D. Condition \eqn M is imposed to ensure that
agents with money holding more than $\Kp$ do not offer to sell at $p$ when
$\K=\N$ or $\K=\N+1$. In case of $\K \ge \N+2$, \eqn M is always
satisfied. Note that depending on the value of $\K$, $V(\N+1)$ and $V(\N+2)$
may be different. If $\K=\N$, both $V(\N+1)$ and $V(\N+2)$ are computed
through \eqn {x1}. If $\K=\N+1$, $V(\N+1)$ is given by \eqn x and
$V(\N+2)$ by \eqn {x1}. If $\K\ge \N+2$, both $V(\N+1)$ and $V(\N+2)$ are
given by \eqn x. 

For the cases of $\J>2$, the analogous of conditions \eqn N and \eqn M can
be constructed accordingly. Combining the previous results with \lem f, we
have 

\medskip
\theorem g{For a given set of the parameters $M,\,u,\,c,\,\phi$
in the above trading environment, a single-price-$\N$-$2$-$p$ proper
equilibrium exists if conditions \eqn y, \eqn D, \eqn C, \eqn N and \eqn M
are satisfied.} 
\medskip

We can compare the sets of sufficient conditions for the existence of
single-price-$\N$-$\J$-$p$ proper equilibrium and that of the
single-price-$\N$-$\infty$-$p$ equilibrium discussed in the end of \sec g.

\medskip
\corollary h{For any $\J>1$, the existence region of the 
single-price-$\N$-$\J$-$p$ proper equilibrium is a subset of the existence 
region of the corresponding single-price-$\N$-$\infty$-$p$ equilibrium.}
\medskip

Proof. By \thm e, the sufficient conditions for the existence of
single-price-$\N$-$\infty$-$p$ equilibrium are \eqn y, \eqn D, and \eqn C,
which are a subset of the equilibrium conditions for
single-price-$\N$-$\J$-$p$ proper equilibrium. Among this subset of
conditions, \eqn y and \eqn C involve value function only in the support,
therefore are common to both equilibria. Condition \eqn D,
$V(\N+1)-c<V(\N)$, differs in two equilibria only if $V(\N+1)$ is 
computed differently. This can happen only if $\K=\N$ in the proper
equilibrium, in which case $V(\N+1)$ is computed through \eqn {x1},
denoted by $V^{\j}(\N+1)$, but it is computed through \eqn x
in the $\N$-$\infty$-$p$ equilibrium for $\K=\infty$, denoted by
$V^{\infty}(\N+1)$. By \eqn O, $\K=\N$ requires $u+V(\N+1-\J)\ge
V^{\j}(\N+1)$, which also implies $u+V(\N+1-\J)\ge V^{\infty}(\N+1)$. 
Under these conditions, it is easy to show that $V^{\j}(\N+1) \ge
V^{\infty}(\N+1)$. That is, \eqn D is a tighter constraint for the
$\N$-$\J$-$p$ proper equilibrium. If $\K>\N$, \eqn D is the same for both
equilibria. Therefore, in general, if a single-price-$\N$-$\J$-$p$ proper
equilibrium exists, a single-price-$\N$-$\infty$-$p$ equilibrium exists as
well. \endproof 

\medskip

Note that if a single-price equilibrium (with or without proper refinement)
exists, generically, all conditions hold with strict inequality. Then for
the same $\N$, these conditions will hold for prices in a neighborhood of
$p$ as well. That is, single-price equilibria with the same money-holding
upper bound $\N$, but different prices in the neighborhood of $p$,
coexist. Thus, if one single-price equilibrium exists, a continuum of
similar single-price equilibria with different prices exist in general. 
 
\Section h{Existence of Continuum of Single-Price Equilibria}

We have discussed the set of sufficient conditions for the existence of a
single-price equilibrium. In this section, I will explore some features of
the single-price equilibrium with a special case of $\N=1$ and some
numerical examples for higher $\N$. These features include the existence of
a continuum of single-price equilibria, the implication of endogenous
upper bound on money holdings at equilibrium, and the coexistence of
single-price equilibria with different money-holding distributions. 

\Subsection k{Single-Price-$1$-$\infty$-$p$ Equilibrium}

The reason to study a single-price equilibrium with money-holding upper
bound $\N=1$ is that it corresponds, in outcome, to the monetary
equilibrium with exogenous unit-inventory constraint that has been widely
studied. The choice of $\J=\infty$, thus equilibrium without proper
refinement, is made because this equilibrium has the largest existence
region (by \cor h) among all single-price-$\N$-$\J$-$p$ equilibria. If this
equilibrium does not exist, there is no other single-price equilibrium with
price $p$ and money-holding bounded at $1\cdot p$. 

Consider an environment defined by an arbitrary set of parameters $M$,
$\phi$, $u$ and $c$. At a single-price-$1$-$\infty$-$p$ equilibrium, all
trades occur at price $p$, and all traders hold either no money (in which
case they can not buy) or $p$ units of money (in which case they choose
not to sell). The stationary money-holding distribution is simple: 
\begin{equation}
h(1)=m={M\over p}, \quad \hbox{and} \quad h(0)=1-{M\over p}. 
\label{eqn:F}
\end{equation}
The value function can be solved from \eqn w and \eqn x.
%\begin{eqnarray}
%&&V(0)={m\Bigl((1-m)u-(\phi+1-m)c\Bigr) \over \phi
%(1+\phi)}, \qquad
%V(1)={(1-m)\Bigl((m+\phi)u-mc\Bigr) \over \phi
%(1+\phi)} \nonumber\\
%&&V(2)={(1-m)\Bigl((\phi(1+\phi)+(1-m)(m+
%\phi))u - m(1-m)c\Bigr) \over (\phi+1-m)\phi
%(1+\phi)}
%\label{eqn:G}
%\end{eqnarray}

Since at equilibrium, all agents hold at most $p$ units of money, no
agent with money holding in the support can afford to pay more than $p$
units of money. Therefore, condition \eqn C which ensures that all agents
pay no more than $p$ is not needed. By \thm e, then, conditions \eqn y and 
\eqn D fully characterize the set of single-price-$1$-$\infty$-$p$
equilibria. With the solution of the value function, conditions \eqn y
($V(1)-V(0)\ge c$) and \eqn D ($V(2)-V(1)<c$) are reduced to 
\begin{equation}
\Psi_2 <{M\over p} \le \Psi_1
\label{eqn:I}
\end{equation}
where
$$\Psi_1=1-{\phi \over ({u\over c}-1)}, \qquad 
\Psi_2={1\over 2({u\over c}-1)}\Bigl( 2({u\over c}-1)- \phi -
\sqrt{(4({u\over c}-1)+1)\phi^2 + 4\phi ({u\over c}-1)}
\Bigr) $$
and $\Psi_1>\Psi_2$ always holds. One can show that $\Psi_1\le 1$ and
$\Psi_2\le 1$, but they may not always be positive. If $\phi < {u\over c}-1$,
$\Psi_1>0$. Let $\Psi_0\equiv \max\{\Psi_2,0\}$. By \thm e,

\medskip
\corollary i{If $\phi < {u\over c}-1$, a single-price-$1$-$\infty$-$p$
equilibrium exists if and only if the price $p$ is such that
the real-money balance ${M\over p} \in (\Psi_0, \Psi_1]$.}
\medskip

Note that under the condition $\phi < {u\over c}-1$, the interval
$(\Psi_0, \Psi_1]$ is not empty. Given that both the money-holding upper
bound $\N$ and price $p$ are equilibrium choice variables, \cor i states
the existence condition such that one single-price equilibrium exists. 

This particular equilibrium demonstrates several interesting points. One
is that the money-holding upper bound $1\cdot p$ (i.e., $\N=1$) is
endogenous here.  This upper bound results from the fact that it is not
optimal to sell a unit of goods as one holds $p$ units of money, since the
value of an additional $p$ units of money $V(2)-V(1)$ is lower than the
cost of producing the good $c$. Had an exogenous upper bound on an agent's
ability to hold money been imposed, namely, an agent {\it can} hold at
most one unit of goods {\it or} $p$ units of money, a monetary equilibrium
exists if ${M\over p} \le \Psi_1$. Condition $\Psi_2<{M \over p}$ when
$\Psi_2>0$, which corresponds to \eqn D, is not needed since nobody is
able to carry more money anyway. Therefore, as one eliminates the
inventory holding constraint and endows each agent with the ability to
hold any amount of money, the equilibrium region will shrink. If the proper
equilibrium concept is adopted, the equilibrium existence region for
${M\over p}$ is a subset of $(\Psi_0, \Psi_1]$ (or empty if the
equilibrium does not exist).  Thus, the conclusion that the equilibrium
existence region might be too large with exogenous upper bound continues
to hold.  

\uppercase \cor i shows that for a set of parameters $M$, $\phi$, $u$
and $c$ where the single-price-$1$-$\infty$-$p$ equilibrium exists, there
is a continuum of such equilibria indexed by the real-money balance
${M\over p}$, ${M\over p}\in (\Psi_0, \Psi_1]$. This answers the question
posted in the beginning of the paper. Namely, the existence of a continuum
of single-price equilibria is robust to the introduction of costly
production. But not all results in Green and Zhou (1995) are robust to
such change. An important difference is in welfare implications of the
model. For this equilibrium, the average welfare can be computed as  
\begin{equation}
\hbox{Welfare }= h(0) V(0) + h(1) V(1) = {1\over \phi} {M\over p}
(1-{M\over p})(u-c).
\label{eqn:J}
\end{equation} 
Therefore, among the continuum of equilibria, the closer an
equilibrium's real-money balance ${M\over p}$ or the proportion of people
holding money $m$ to one half, the higher its average welfare. This is
essentially the same result as many papers with indivisible goods and
money and exogenous unit upper bound on inventory holdings have gotten
(e.g., Kiyotaki and Wright (1991) and (1993)). But it differs from the
costless production case of Green and Zhou (1995), in which higher
real-money balance gives higher welfare. Here the welfare loss from higher
real-money balance stems from the slowing down of trade by the fact that
agents with money holdings $1\cdot p$ do not produce. The cost of
production (obtaining money) induces agents to economize on the amount of
money to hold, which simultaneously shrinks the productive population as
well.  

\Subsection l{Coexistence of Single-Price-$\N$-$2$-$p$ Proper Equilibria}

In this section, I will return to the proper equilibrium concept studied
in \sec j and look at examples of single-price-$\N$-$2$-$p$ proper
equilibria. 

Similar results regarding the existence of a continuum of single-price
equilibria hold here for $N\ge 2$. The explicit conditions such as those
in \cor i on parameters of the model are more complicated for higher
values of $\N$, but the equilibrium is always well defined by \thm g where
the conditions are implicit functions of the parameters of the model.
Figure 1 shows the equilibrium existence region in terms of the proportion
of people holding money $m$ for some numerical examples. In all the
examples, $u=1$ and $c=0.5$. The first panel represents
single-price-$\N$-$2$-$p$ proper equilibrium with $\N=1,\,2,\,3$ for
$\phi=0.2$. The second panel shows the equilibrium with
$\N=1,\,2,\,3,\,4,\,5,\,6,\,7$ for $\phi=0.1$. Since the equilibrium
conditions in \thm g are only sufficient, the equilibrium range may be
larger than what is shown on the graph. Each thin line represents the
value that $m$ can take, namely the unit interval. The heavy bar on top of
each thin line indicates the single-price equilibrium range of $m$ for
each case. For example, the bottom line of the first panel shows that the
single-price-$1$-$2$-$p$ proper equilibrium exists for $m\in (0.4, 0.8)$
(the heavy bar runs from $0.4$ to $0.8$).  Since there is a one-to-one
mapping between $m$ and ${M\over p}$ by \lem z, the equilibrium existence
region in $m$ can be easily translated into that of ${M\over p}$. 

\bigskip\medskip

\centerline{Figure 1. Equilibrium Existence Range for $m$}
\centerline{($u=1,\,c=0.5$)}

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\put(10,38){$\N$$=$3}
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\put(60,-3){$\phi=0.2$}

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}

\medskip\medskip

These examples confirm once again the previous result. In each case, the
equilibrium existence region is an interval rather than a single point.
That is, for the fixed environment represented by parameters $M$, $\phi$,
$u$ and $c$, for a given $\N$, there is a continuum of
single-price-$\N$-$2$-$p$ proper equilibria with different prices $p$
(uniquely determined by $m$). 

Another observation from figure 1 is that equilibria with different
money-holding upper bound $\N$ but the same $m$ may coexist, although the
price $p$ and money-holding distribution are different. A more interesting
comparison among coexisting equilibria with different money-holding upper
bound $\N$ can be made for those with the same real-money balance ${M\over
p}$ (as well as other parameters $u$, $c$ and $\phi$). Figure 2 shows the
distributions of money holdings $h$ of four coexisting single-price
equilibria for $\N=2,\,3,\,4,\,5\,$ where $u=1,\,c=0.5,\,\phi=1/14\,$ and
${M\over p}=1.5$.  These examples confirm the result of \lem z: the
money-holding distribution is increasing in $n$ for $\N=2$, uniform for
$\N=3=2 {M\over p}$, and decreasing in $n$ for $\N=4,\,5.$ 

\bigskip
\bigskip
\centerline{Figure 2. Distribution $h$ of Coexisting Equilibria}
\centerline{($u=1,\,c=0.5,\,\phi=1/14,\,{M\over p}=1.5$)}
\nopagebreak

\medskip
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\medskip\medskip

In these particular examples, the welfare is increasing in money-holding
upper bound $\N$, but this is not true in general. There are cases where
welfare is higher with a lower value of $\N$ even though ${M\over p}$
remains constant. By increasing the money-holding upper bound $\N$, it has
the welfare enhancing effect of reducing the size of un-productive
population $h(\N)$, simultaneously it has the welfare worsening effect of
enlarging the group of people who do not have money $h(0)$. It is possible
for either effect to dominate.  

The coexistence of equilibrium allocations of consumption with different
welfare levels in an environment with all identical exogenous variables,
and also with identical real-money balances strongly suggests the
non-neutrality of money. The channel for such non-neutrality is the
distribution of money holdings. Different distributions generate different
velocities of trading which dictate the speed of the production and
consumption, therefore the level of welfare.  

\Section i{Concluding Remarks}

I have studied a random matching model with divisible money and costly
production. The model confirms the main findings of Green and Zhou (1995) 
regarding the existence of a continuum of single-price equilibria. 
I show that the phenomenon is robust to the introduction of costly
production. 

The mechanism through which the single-price equilibrium is reached in
this costly production environment is somewhat different from that of the
costless production model. In Green and Zhou (1995), agents prefer posting
the prevaling price given the nondegenerate reservation-price distribution
because a higher offer price reduces potential trading opportunities too
much relative to the gain of getting more money. Here, the cost of
production induces agents to ask for more when their money holdings reach
a level at which the prevailing price is not enough to compensate for the
cost of production.  To maintain common price for all trades, the
reservation price function has to be flat and equal to the posting price
of agents with money holdings in the support.  The nontrivial decision
here for an agent is when to ask for more. Although in both environments,
the self-fulfilling belief of the prevailing price plays the crucial role
for the existence of single-price equilibria. 

In addition to the continuum of single-price equilibria with different
aggregate real-money balances, there exist multiple equilibria with
identical aggregate real-money balances but different money-holding
distributions. My conjecture for this multiplicity of equilibria is that
it is a result of decentralized trade. Several distinct distributions can
reinforce themselves through different trading patterns provided by the
decentralized trading environment. The result shows that it is the
distribution of money holdings, not the aggregate real-money balance,
which provides the summary statistics of trading opportunities. The result
suggests that the velocity of money is endogenously determined rather than
an exogenous feature of the economy. Furthermore, for a given aggregate
real money balance, the equilibrium velocity may not be unique. It is
through the velocity that an important aspect of the environment, the
distribution of money, plays out its role on shaping economic activities.  

The existence of a continuum of equilibria is a lot more difficult to
explain. It might be a robust phenomenon, or it might be caused by the
modelling assumptions, e.g., the indivisibility and the unit-inventory
constraints of goods.  An important point made by Wallace (1996) should be
emphasized here. What I have shown here is the existence of a continuum of
{\it stationary} equilibria, not equilibrium indeterminacy in the sense
that all of these equilibria can be reached from the same initial
distribution of money holdings. It is possible that the set of equilibrium
paths from any given initial state is finite.  Without the ability to
characterize the full equilibrium path, moreover, without understanding
the fundamental source of the problem, one cannot assess this class of
models' applicability in substantial policy analysis.  

\newpage

\begin{appendix}

\begin{center}
{\Large\bf Appendix}
\end{center}

\bigskip

\addtocounter{section}{1}
\noindent{\large {\bf \thesection. The Proof of \uppercase \lem z}}
\medskip

This lemma states the relationship among $\N$, $m$ and ${M\over p}$, three
variables determining the stationary distribution of money holdings $h$,
that satisfies \eqn n--\eqn p, $0\le m\le 1$ and $0\le h(\N)\le 1$.  First
note that for any given $\N$, $(m, h(\N),{M\over p})=(0,1,0)$ and
$(m,h(\N),{M\over p})=({\N\over \N+1},{1\over \N+1},{\N\over 2})$ are
solutions to \eqn n--\eqn p. While the former is a distribution for a
non-monetary equilibrium, the latter corresponds to a uniform distribution
of money holdings, i.e., $\forall n=0,1,\ldots,\N$, $h(n)={1\over \N+1}$.
We are looking for other nontrival stationary distributions for potential
monetary equilibrium. For a given $\N$, I will show that $m$ uniquely
determines ${M\over p}$ and vice versa. 

Take a constant $m$ as given. Rewrite $h(\N)$ as $h(\N;{M\over
p})$ to index the dependence of $h(\N)$ on ${M\over p}$,
 
\parbox{5.5in}{$$
\begin{array}{ll}
h(\N;{M\over p})={m-(1-m){M\over p} \over m(\N+1)-{M\over p}}, &\qquad 
1-h(\N;{M\over p})={m(N-{M\over p}) \over m(\N+1)-{M\over p}}.
\end{array}$$} \hfill
\parbox{1.25cm}{$$\begin{array}{r}(A.1) \end{array}$$}
 
\noindent Define 

\parbox{5.5in}{$$
\begin{array}{ll}
T({M\over p})\equiv h(\N;{M\over p})\Bigr(1- h(\N;{M\over p})\Bigr)^\n,
&\qquad S(x)\equiv (1-x) x^\n.
\end{array}$$} \hfill
\parbox{1.25cm}{$$\begin{array}{r}(A.2) \end{array}$$}
 
\noindent 
$S(m)$ is a positive constant given $m$. Instead of finding solution
${M\over p}$ to \eqn n, we will look for solutions for $T({M\over
p})=S(m)$. This can be done in the following two cases, depending on the
values of $\N$ and $m$.

\medskip

\noindent {\it Case 1.} $m(\N+1)<\N$.

Divide $\Re_+$ into three segments: $I_1=(0,m(\N+1)]$,
$I_2=(m(\N+1),\N)$, $I_3=[\N,\infty)$. 

If ${M \over p} \in I_3$, it is easy to show that $T({M\over p})$ is
increasing on $I_3,\,$ $T(\N)=0,\,$ and $T({M\over p})\to S(m)\,$ as
${M\over p}\to \infty.\,$ That is, $\forall {M\over p} \in I_3\,$
$T({M\over p})<S(m),\,$ therefore there is no solution on $I_3$.  

If ${M \over p} \in I_2,\,$ by $(A.1),\,$ $1-h(\N)<0.\,$ Thus,
there is no solution on $I_2$. 

If ${M \over p} \in I_1,\,$ we can show that $T({M\over p})$ is
decreasing on $I_1,\,$ $T(0)={\N^\n\over (\N+1)^{\n+1}}>0,\,$ and
$T({M\over p})\to S(m)\,$ as ${M\over p}\to m(\N+1).\,$ Since $S(x)$ is a
increasing function when $x<{\N\over \N+1}\,$ and by assumption
$m<{\N\over \N+1},\,$ $S(m)<S({\N\over \N+1})=T(0).\,$ Thus, there exists
a ${M\over p}^* \in (0,m(\N+1))\,$ such that $T({M\over p}^*)=S(m)$. 

\medskip

\noindent {\it Case 2.} $m(\N+1)> \N$.

Similarly, divide $\Re_+$ into three intervals: $I_1=(0,\N]$,
$I_2=(\N,m(\N+1))$, $I_3=[m(\N+1),\infty)$. 

If ${M \over p} \in I_3$, again we can show that $T({M\over p})$ is
increasing on $I_3,\,$ $T({M\over p})\to -\infty\,$ as ${M\over p}\to
m(\N+1),\,$ and $T({M\over p})\to S(m)\,$ as ${M\over p}\to \infty.\,$ That
is, $\forall {M\over p} \in I_3\,$ $T({M\over p})<S(m),\,$ therefore there
is no solution on $I_3$.  

If ${M \over p} \in I_2,\,$ by $(A.1),\,$ $1-h(\N)< 0.\,$ Thus,
there is no solution on $I_2$. 

If ${M \over p} \in I_1,\,$ we can show that $T({M\over p})$ is
decreasing on $I_1,\,$ $T(0)={\N^\n\over (\N+1)^{\n+1}}>0,\,$ and
$T(\N)=0< S(m).\,$ Since $S(x)$ is a decreasing function when $x>{\N\over
\N+1}\,$ and by assumption $m\ge {\N\over \N+1},\,$ $S(m)<S({\N\over
\N+1})=T(0).\,$ Thus, there exists a ${M\over p}^* \in (0,\N)\,$ such that
$T({M\over p}^*)=S(m)$.  

\medskip
The case of $m(\N+1)=\N$ corresponds to the special uniform distribution
above. To summarize, for a given $\N$ and $m$, there exists a unique 
${M\over p}^* <\N,\,$ such that $T({M\over p}^*)=S(m)$. 

\bigskip
Now let's consider the reverse case of finding $m$ for a given ${M\over
p}$. Substituting $h(\N)$ of \eqn p into \eqn n, we have  

\parbox{5.5in}{$$
\begin{array}{l}
{\Bigl(m({M\over p}+1)-{M\over p}\Bigr) m^\n (\N-{M\over p})^\n \over
\Bigl(m(\N+1)-{M\over p}\Bigr)^{\n+1}}=(1-m)m^\n.
\end{array}$$} \hfill
\parbox{1.25cm}{$$\begin{array}{r}(A.3) \end{array}$$}
 
\noindent Define

\parbox{5.5in}{$$
\begin{array}{ll}
O(m)\equiv {(1-m)\Bigl(m(\N+1)-{M\over p}\Bigr)^{\n+1} \over
m({M\over p}+1)-{M\over p}},&\qquad 
Q\equiv (\N-{M\over p})^\n.
\end{array}$$} \hfill
\parbox{1.25cm}{$$\begin{array}{r}(A.4) \end{array}$$}

\noindent Then $(A.3)$ can be written as 

\parbox{5.5in}{$$
\begin{array}{l}
O(m)=Q.
\end{array}$$} \hfill
\parbox{1.25cm}{$$\begin{array}{r}(A.5) \end{array}$$}
 
\noindent So instead of looking for solutions for \eqn n, we will solve
for $m$ for $(A.5)$. Given the above result ($({M\over p})^*<\N$ for a
given $m$), we only need to study the case of ${M\over p}<\N$. 

Given ${M\over p}<\N$, we have $0<{M\over p}{1\over \N+1} < {M\over
p}{1\over {M\over p} +1} < {\N\over \N+1}.\,$ If $m\in (0,{M\over p}{1\over
\N+1}),\,$ $1-h(\N)<0$. If $m\in [{M\over p}{1\over \N+1}, {M\over
p}{1\over {M\over p} +1}),\,$ $h(\N)<0$. So the only possible solution of
$m$ is in $[{M\over p}{1\over {M\over p} +1},1].\,$ On this interval,
$O(m)\to +\infty\,$ as $m\to {M\over p}{1\over {M\over p} +1},\,$ and
$O(1)=0<Q.\,$ Also

\parbox{5.5in}{$$
\begin{array}{l}
{\partial O\over \partial m}={\Bigl(m(\N+1)-{M\over p}\Bigr)^\n \over
\Bigl(m({M\over p}+1)-{M\over p}\Bigr)^2}(m(\N+1)-\N)\Bigl({M\over
p}(\N+2)-m(\N+1)({M\over p}+1)\Bigr).
\end{array}$$} \hfill
\parbox{1.25cm}{$$\begin{array}{r}(A.6) \end{array}$$}
 
\noindent The solutions for ${\partial O\over \partial m}=0$ are 

\parbox{5.5in}{$$
\begin{array}{ll}
m_1={\N\over \N+1}, &\qquad m_2={{M\over p}(\N+2)\over ({M\over
p}+1)(\N+1)},
\end{array}$$} \hfill
\parbox{1.25cm}{$$\begin{array}{r}(A.7) \end{array}$$}
 
\noindent and $O(m_1)=(\N-{M\over p})^\n=Q.\,$ Note that $m_1$ gives the
special uniform distribution discussed above, which requires ${M\over p} =
{\N\over 2}.\,$ In this case $m_1=m_2\,$ is saddle point of $O$ which 
is the only solution of $(A.5)$. Let's consider the cases where ${M\over
p} \not ={\N\over 2}.\,$

If ${M\over p} < {\N\over 2},\,$ $m_1>m_2.\,$ Since $O$ is a continuous
function of $m$, there exists a $m^*\in ({M\over p}{1\over {M\over p}
+1},m_2)$ such that $O(m^*)=(\N-{M\over p})^\n=Q$. That is,
$m^*<m_2<m_1,\,$ and by \eqn p, $1-h(\N)>m.\,$ Then by \eqn m,
$\{h(n)\}_{n=0}^\n$ is a decreasing sequence of $n$.

If ${M\over p} > {\N\over 2},\,$ $m_1<m_2.\,$ Again since $O$ is a
continuous function of $m$, there exists a $m^*\in (m_2,1)$, such that
$O(m^*)=(\N-{M\over p})^\n=Q$. That is, $m^*>m_2>m_1,\,$ and by \eqn p,
$1-h(\N)<m.\,$ Then by \eqn m, $\{h(n)\}_{n=0}^\n$ is an increasing
sequence of $n$. 

This concludes the second part of the proof: for a given ${M\over p}$,
there is a unique $m^*$ that satisfies $(A.5)$. \endproof

\bigskip\bigskip

\addtocounter{section}{1}
\noindent{\large {\bf \thesection. The Derivation of the Value Function}}
\medskip

The solution to the linear difference equation \eqn u can be represented
as a linear combination of the eigenvectors of matrix $\D$. The matrix
$\D$ has three distinct eigenvectors, all of which have real eigenvalues.
The eigenvalues $\l_0$, $\l_1$ and $\l_2$ are

\parbox{5.5in}
{$$\begin{array}{l}
\l_0=1 \\
\l_1={1 \over 2} \left( {\phi \over m} + {1-h(\N) \over m} + 1 -
\sqrt{ \left( {\phi \over m} + {1-h(\N) \over m} + 1 \right)^2 - {4
(1-h(\N)) \over m}} \; \right) \in (0, 1) \\
\l_2={1 \over 2} \left( {\phi \over m} + {1-h(\N) \over m} + 1 +
\sqrt{ \left( {\phi \over m} + {1-h(\N) \over m} + 1 \right)^2 - {4
(1-h(\N)) \over m}} \; \right) > 1. 
\end{array}$$} \hfill
\parbox{1.25cm}{$$\begin{array}{r} (B.1) \end{array}$$}

\noindent The corresponding eigenvectors $\xi_0$, $\xi_1$ and $\xi_2$ can
be expressed as 

\medskip
\parbox{5.5in}{$$
\begin{array}{lll}
\xi_0=\left( \begin{array}{c} 1/\phi \\ 1/\phi \\ 1 \end{array} \right), &
\qquad 
\xi_1=\left( \begin{array}{c} \l_1 \\ 1 \\ 0 \end{array} \right), & \qquad
\xi_2=\left( \begin{array}{c} \l_2 \\ 1 \\ 0 \end{array} \right) 
\end{array}$$} \hfill
\parbox{1.25cm}{$$\begin{array}{r} (B.2) \end{array}$$}

\noindent that satisfy the following relations,

\parbox{5.5in}{$$
\begin{array}{lll}
\D \xi_0=\xi_0, &\qquad \D \xi_1= \l_1 \xi_1, &\qquad \D \xi_2=\l_2 \xi_2.
\end{array}$$} \hfill
\parbox{1.25cm}{$$\begin{array}{r}(B.3) \end{array}$$}
 
\noindent Define

\parbox{5.5in}{$$
\begin{array}{l}
w^{n+1} \equiv \left( \begin{array}{c} V(n+1) \\ V(n) \\ B \end{array}
\right). 
\end{array}$$} \hfill
\parbox{1.25cm}{$$\begin{array}{r} (B.4) \end{array}$$}

\noindent Let $\t_0$, $\t_1$ and $\t_2$ be the parameters satisfies

\parbox{5.5in}{$$
\begin{array}{l}
w^1=\t_0 \xi_0 + \t_1 \xi_1 + \t_2 \xi_2
\end{array}$$} \hfill
\parbox{1.25cm}{$$\begin{array}{r}(B.5) \end{array}$$}

\noindent  or

\parbox{5.5in}
{$$\begin{array}{rcl}
({\phi \over m}+1) V(0) + c &=& \t_0 / \phi + \t_1 \l_1 + \t_2 \l_2 \\
V(0) &=& \t_0/ \phi + \t_1 + \t_2  \\ 
B &=& \t_0
\end{array}$$} \hfill
\parbox{1.25cm}{$$\begin{array}{r} (B.6)\\(B.7)\\(B.8) \end{array}$$}

\noindent where $(B.6)$ is derived by substituting $V(1)$ solved from 
\eqn {s1}. By $(B.3)$ and $(B.5)$, induction on $w^{n+1}$ for
$n=0,1,\ldots,\N-1$ implies,

\medskip
\parbox{5.5in}{$$
\begin{array}{l}
w^{n+1}=\t_0 \xi_0+\t_1 \l_1^n \xi_1 + \t_2 \l_2^n \xi_2 = \t_0\left(
\begin{array}{c} 1/\phi \\ 1/\phi \\ 1 \end{array} \right) + \t_1
\l_1^n\left( \begin{array}{c} \l_1 \\ 1 \\ 0 \end{array} \right) + \t_2
\l_2^n \left( \begin{array}{c} \l_2 \\ 1 \\ 0 \end{array} \right).
\end{array}$$} \hfill
\parbox{1.25cm}{$$\begin{array}{r} (B.9) \end{array}$$}

\noindent Take the second row of $(B.9)$ we have for $n=1,2,\ldots,\N$,

\parbox{5.5in}{$$
\begin{array}{l}
V(n) =\t_0 /\phi + \t_1 \l_1^n + \t_2 \l_2^n
\end{array}$$} \hfill
\parbox{1.25cm}{$$\begin{array}{r}(B.10) \end{array}$$}

\noindent which is the general representation of value function \eqn w. 
Rewrite equation \eqn {s3},

\parbox{5.5in}{$$
\begin{array}{l}
(\phi+1-h(\N)) V(\N) = (1-h(\N))(u+V(\N-1))
\end{array}$$} \hfill
\parbox{1.25cm}{$$\begin{array}{r}(B.11) \end{array}$$}

\noindent
and substitute $(B.10)$ for $n=\N-1$ and $n=\N$ into $(B.11)$, we have

\parbox{5.5in}{$$
\begin{array}{l}
(\phi+1-h(\N)) \Bigl(\t_0 /\phi + \t_1 \l_1^\n + \t_2
\l_2^\n\Bigr) =(1-h(\N))\Bigl(u+\t_0 /\phi + \t_1 \l_1^{\N-1} + \t_2
\l_2^{\N-1}\Bigr). \quad \end{array}$$} \hfill
\parbox{1.25cm}{$$\begin{array}{r}(B.12) \end{array}$$}

\noindent Equations $(B.6)$, $(B.7)$, $(B.8)$ and $(B.12)$ are a
four-equation linear system with four unknowns $\t_0$, $\t_1$, $\t_2$ and
$V(0)$. Solving this linear system, we have

\medskip
\parbox{5.5in}{$$
\begin{array}{rcl}
\t_0&=&(1-h(\N))u-mc\\
\t_1&=&{1\over G}\Bigl((\l_2-1-{\phi \over m}) mc-({1\over
m}\t_0+c) F\Bigr) \\
\t_2&=&{1\over G}\Bigl((1+{\phi \over m}-\l_1) mc+({1\over
m}\t_0+c) E\Bigr) \\
V(0)&=&{1\over G}\Bigl((\l_2-\l_1)(\t_0 F/\phi +mc)-(c+(\l_2-1)
\t_0/\phi)(F-E)\Bigr) 
\end{array}$$} \hfill
\parbox{1.25cm}{$$\begin{array}{r} \\(B.13)\\ \end{array}$$}

\noindent where

\parbox{5.5in}{$$
\begin{array}{lll}
& &E=\phi\l_1^\n - \l_1^{\N-1}(1-h(\N))(1-\l_1) \\
& &F=\phi\l_2^\n + \l_2^{\N-1}(1-h(\N))(\l_2-1) \\
& &G=(1-\l_1+{\phi \over m})F + (\l_2-1-{\phi \over m}) E.
\end{array}$$}

\noindent
Equation $(B.10)$ and $(B.13)$ fully characterize the value function in
terms of the parameters of the model $u$, $c$ and $\phi$, the upper bound
on money holdings $\N$, and the distribution parameter $m$ and $h(\N)$.

\bigskip\bigskip

\addtocounter{section}{1}
\noindent{\large {\bf \thesection. The Proof of \uppercase \lem b}}
\medskip

The value function is concave in $n$ for $\N\le n\le \K$ can be directly
shown by equation \eqn {s3} and \eqn {s4}, that is, for $k=0,1,2,\ldots,\K$ 

\parbox{5.5in}{$$
\begin{array}{l}
V(\N+k)-V(\N+k-1) > V(\N+k+1)-V(\N+k). 
\end{array}$$} 
\hfill \parbox{1.25cm}{$$\begin{array}{r} (C.1)\end{array}$$}

\noindent I will focus on the case $n <\N.$ From \eqn {s2} and \eqn {s3},

\parbox{5.5in}{$$
\begin{array}{rcl}
(\phi+m+1-h(\N))V(\N-1)&=&(1-h(\N))(u+ V(\N-2))+m(V(\N)-c), \\
(\phi+1-h(\N))V(\N)&=&(1-h(\N))(u+ V(\N-1)).
\end{array}$$}
\hfill \parbox{1.25cm}{$$\begin{array}{r} (C.2)\\(C.3)\end{array}$$}

\noindent Subtracting $(C.3)$ from $(C.2)$ and rearranging the equation,
we have

\parbox{5.5in}{$$
\begin{array}{l}
(1-h(\N))\Bigl((V(\N-1)-V(\N-2)-(V(\N)-V(\N-1))\Bigr)= \qquad\qquad\\
\qquad\qquad\qquad\qquad\qquad\qquad\qquad \phi(V(\N)-V(\N-1)) +
m(V(\N)-c-V(\N-1)). 
\end{array}$$} 
\hfill \parbox{1.25cm}{$$\begin{array}{r} (C.4)\end{array}$$}

\noindent The right hand side of $(C.4)$ is positive since $V(\N)>V(\N-1)$
by \lem a, and $V(\N)-c\ge V(\N-1)$ by \eqn D. Therefore,

\parbox{5.5in}{$$
\begin{array}{l}
V(\N-1)-V(\N-2)>V(\N)-V(\N-1).
\end{array}$$} 
\hfill \parbox{1.25cm}{$$\begin{array}{r} (C.5)\end{array}$$}

\noindent Similarly, subtracting equation \eqn {s2} for $n$ from that of
$n-1$ and rearranging the difference, we have

\parbox{5.5in}{$$
\begin{array}{l}
(1-h(\N))\Bigl((V(n-1)-V(n-2)-(V(n)-V(n-1))\Bigr) = 
\phi(V(n)-V(n-1))\\ \qquad\qquad\qquad\qquad\qquad\qquad\qquad
 + m\Bigl((V(n)-V(n-1))-(V(n+1)-v(n))\Bigr). 
\end{array}$$} 
\hfill \parbox{1.25cm}{$$\begin{array}{r} (C.6)\end{array}$$}

\noindent Again the right hand side of (C.6) is positive for $n=\N-1$ by
\lem a and by inequality $(C.5)$. So $V(\N-2)-V(\N-3)>V(\N-1)-V(\N-2).$
Induction on $n$ backward using $(C.6)$, we have for $n=1,2,\ldots,\N-1,$

\parbox{5.5in}{$$
\begin{array}{l}
V(n)-V(n-1) > V(n+1)-V(n). 
\end{array}$$} 
\hfill \parbox{1.25cm}{$$\begin{array}{r} (C.7)\end{array}$$}

\noindent Inequalities $(C.1)$ and $(C.7)$ give the statement of \lem b.
\endproof 

\bigskip\bigskip

\addtocounter{section}{1}
\noindent{\large {\bf \thesection. The Proof of \uppercase \lem f}}
\medskip

The proof of this lemma involves three cases depending on the value of
$\K$. Compute the value of $V(\N+1)$ by \eqn x. If $V(\N+1)-V(\N-1)\le u$,
by \eqn A, agents in state $\N+1$ are willing to accept an offer of $2p$.
This is the first case, $\K=\N$. If $V(\N+1)-V(\N-1) > u$, compute next
the value $V(\N+2)$ by \eqn x. If $V(\N+2)-V(\N)\le u$, then agents in state
$\N+2$ are willing to accept an offer of $2p$. Then, $\K=\N+1$. Otherwise,
$\K\ge \N+2$. I will consider each case separately.

\bigskip
\noindent {\it Case 1.} $\K=\N$, $V(\N+1)-V(\N-1)\le u$.

{}From \eqn {s3} and \eqn {s5}, we have 

\parbox{5.5in}{$$
\begin{array}{l}
(\phi+1)\Big(V(\N+2)-V(\N)\Bigr)=(1-h(\N))\Bigl(V(\N+1)-V(\N-1)\Bigr)+h(\N)u
\le u
\end{array}$$} 
\hfill \parbox{1.25cm}{$$\begin{array}{r} (D.1)\end{array}$$}

\noindent
because $V(\N+1)-V(\N-1)\le u$. Therefore $V(\N+2)-V(\N)<u$. That is,
agents in state $\N+2$ are willing to accept an offer of $2p$. Similarly,
for $n > \N+2$, from \eqn {s5} we can derive the following,

\parbox{5.5in}{$$
\begin{array}{l}
(\phi+1)\Bigl(V(n)-V(n-2)\Bigr)=(1-h(\N))\Bigl(V(n-1)-V(n-3)
\Bigr)+\qquad\qquad\\ 
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
h(\N)\Bigl(V(n-2)-V(n-4)\Bigr). 
\end{array}$$} 
\hfill \parbox{1.25cm}{$$\begin{array}{r} (D.2)\end{array}$$}

\noindent
Since $V(\N+1)-V(\N-1)\le u$ and $V(\N+2)-V(\N)<u$, by $(D.2)$,
$V(\N+3)-V(\N+1)< u$. Applying (D.2) recursively, we have $V(n)-V(n-2)< u$
for any $n>\N+2$. Combining all cases, we find that it is optimal for
agents with money holding more than $\Kp(=\Np)$ accept an offer of $2p$.

In regard to sellers, by \eqn D, $V(\N+1)-c<V(\N)$, and by \eqn M,
$V(\N+2)-c<V(\N+1)$. For $n> \N+2$, from \eqn {s5}

\parbox{5.5in}{$$
\begin{array}{l}
(\phi+1)\Bigl(V(n)-V(n-1)\Bigr)=(1-h(\N))\Bigl(V(n-1)-V(n-2)\Bigr)
+\qquad\qquad\\
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
h(\N)\Bigl(V(n-2)-V(n-3)\Bigr).
\end{array}$$} 
\hfill \parbox{1.25cm}{$$\begin{array}{r} (D.3)\end{array}$$}

\noindent
Again by applying $(D.3)$ recursively, we have $V(n)-c < V(n-1)$ for all
$n>\N+1$. Then given \eqn C, the expected value of making an offer, $W(np,
\cdot\;)$ for all $n> \K$, satisfies $W(np,0)<W(np,p)=V(n)+m(V(n+1)-c-V(n))
<W(np,lp)=V(n)$ for any $l>1$. Therefore, it is optimal for agents with
money holding at least $\Kp$ offer higher than $p$. 
 
\bigskip
\noindent {\it Case 2.} $\K=\N+1$, $V(\N+2)-V(\N)\le u$.

In this case, given $V(\N+2)-V(\N)\le u$, we can show that
$V(\N+3)-V(\N+1)< u$ by way of showing $V(\N+2)-V(\N)< u$ as in case 1.
Then applying $(D.1)$ recursively for $n=\N+4, \N+5, \ldots$, we have
$V(n)-V(n-2)< u$. That is, it is optimal for agents with money holding more
than $\Kp$ accept an offer of $2p$.

For offer strategy of the sellers with money holding at least $\Kp$, the
same argument as in case 1 applies here.
 
\bigskip
\noindent {\it Case 3.} $\K\ge \N+2$, $V(\N+2)-V(\N) > u$.

The proof for the optimality of buyers with money holding more than $\Kp$
accepting an offer of $2p$ is similar as in the previous two cases.

For the sellers, by the definition of $\K$ in \eqn O, 
$V(\K+1)-V(\K-1) \le u < V(\K)-V(\K-2)$. Rewrite this inequality,

\parbox{5.5in}{$$
\begin{array}{l}
V(\K+1)-V(\K) < V(\K-1)-V(\K-2).
\end{array}$$} 
\hfill \parbox{1.25cm}{$$\begin{array}{r} (D.4)\end{array}$$}

\noindent
By \eqn D, $V(\N+1)-V(\N)<c$. Given $\K\ge \N+2$, and the value function is
concave up to $\K$, $V(\K-1)-V(\K-2)<c$. Then by $(D.4)$,
$V(\K+1)-V(\K)<c$. We also know $V(\K)-V(\K-1)<c$ from the concavity of $V$.
Starting with the last two inequalities, applying $(D.3)$ recursively, we
can show $V(n)-c < V(n-1)$ for all $n\ge \K+1$. Following the rest of the
proof as in case 1, it is optimal for agents with money holding at least
$\Kp$ offer higher than $p$.

Note that condition \eqn M is not needed in the last case. The concavity
of the value function below $\K$ is sufficient to show that no agent is
willing to offer $p$. But \eqn M is needed for the other two cases.
\endproof 

\end{appendix}
\def\Aer{{\it American Economic Review,\/ }}
\def\Econ{{\it Econometrica,\/ }}
\def\Jet{{\it Journal of Economic Theory,\/ }}
\def\Jedc{{\it Journal of Economic Dynamics and Control,\/ }}
\def\Jme{{\it Journal of Mathematical Economics,\/ }}
\def\Jpe{{\it Journal of Political Economy,\/ }}
\def\Res{{\it Review of Economic Studies,\/ }}
\def\mimeo{{\it mimeographed.\/ }}

\newpage
\begin{center}
{\large\bf References}
\end{center}
\bigskip

\parindent=0pt
\medskip \hangindent=30pt  
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\medskip \hangindent=30pt
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\medskip \hangindent=30pt
N.~Kiyotaki and R.~Wright, ``On money as a medium of exchange,'' \Jpe 97
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\medskip \hangindent=30pt  
N.~Kiyotaki and R.~Wright, ``A contribution to the pure theory of
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\medskip \hangindent=30pt  
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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  
R.~B.~Myerson, {\it Game Theory}, Cambridge: Harvard University
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\medskip \hangindent=30pt  
S.~Shi, ``Money and prices: a model of search and bargaining,'' \Jet 67
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\medskip \hangindent=30pt  
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\medskip \hangindent=30pt  
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\end{document}

