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Federal Reserve Bank of Chicago
Price Level Uniformity in a Random
Matching Model with Perfectly Patient
Traders
Edward J. Green and Ruilin Zhou
WP 2001-17
Price Level Uniformity in a Random Matching Model
with Perfectly Patient Tradersy
Edward J. Green
and
Ruilin Zhou
Federal Reserve Bank of Chicago
December, 2001
Abstract
This paper shows that one of the de ning features of Walrasian equilibrium|law
of one price|characterizes equilibrium in a non-Walrasian environment of (1) random
trade matching without double coincidence of wants, and (2) strategic, price-setting
conduct. Money is modeled as perfectly divisible and there is no constraint on agents'
money inventories. In such an environment with discounting, the endogenous heterogeneity of money balances among agents implies di erences in marginal valuation
of money between distinct pairs of traders, which raises the question whether decentralized trade would typically involve price dispersion. We investigate the limiting
case in which agents are patient, in the sense that they have overtaking-criterion
preferences over random expected-utility streams. We show that in this case the \law
of one price" holds exactly. That is, in a stationary Markov monetary equilibrium,
all transactions endogenously must occur at a single price despite the decentralized
organization of exchange. The result is in the same spirit as the work of Gale (1986a,
b) on bargaining and competition, although the model di ers from Gale's in some
signi cant respects.
J.E.L. Classi cation: C78, D51, E40
Ruilin
Zhou thanks the National Science Foundation for nancial support. The views expressed herein
are those of the authors and not necessarily those of the Federal Reserve Bank of Chicago or the Federal
Reserve System.
y Corresponding author: Ruilin Zhou, Research Department, Federal Reserve Bank of Chicago, P.O.
Box 834, Chicago, IL 60690-0834. Email: rzhou@frbchi.org.
1. Introduction
In a decentralized-trading environment, di erence among traders creates a real possibility that decentralized trade could typically involve price dispersion. For example, it is
plausible that traders with low money balances would be more desperate to acquire money
than those with high balances, and would therefore o er a lower price to increase the likelihood of the o er being accepted. Indeed, Camera and Corbae (1999), Aiyagari, Wallace
and Wright (1996), and Kamiya and Sato (2001) calculate examples of price-dispersion
equilibrium in such environments.
On the other hand, there is a presumption that the \law of one price" should hold in a
market environment where trade is decentralized but nevertheless competitive. One view
of the counterexamples just cited is that, in a random-matching environment where traders
maximize discounted expected utility, trade cannot be competitive. That is, if traders i
and j are currently matched and i desires to consume the good that j can produce, then j
is the only trader in the entire economy who can enable i to consume that good without
discounting the utility derived from it, because time must elapse before i can meet another
potential seller. That is, j has some monopoly power with respect to his current trading
partner, i. Equilibrium price dispersion can be viewed as a consequence of such monopoly
power.
One might conjecture, then, that price dispersion would be negligible in a pairwisematching economy where sellers have little monopoly power, either because buyers' discount factor is very close to unity or because successive matches are very closely spaced
in time. Camera and Corbae (1999) have proved a limit theorem to the e ect that a
special class of monetary equilibrium in a somewhat di erent random-matching environment must converge to single-price equilibrium as the discount factor approaches unity.
This paper concerns another conjecture in the same spirit: that there should be no price
dispersion|that is, that the \law of one price" should hold exactly|in a random-matching
environment where traders have overtaking-criterion preferences. Restricting attention to
stationary Markov equilibria, we prove this fact in this paper.
Our result is in the same spirit as the work of Gale (1986a, b) on bargaining and
competition. There is a number of technical di erences between Gale's model and ours,
of which two are noteworthy. Gale models sequential exchange of claims on consumption
goods, but consumption does not actually occur until after a trader has left the market;
we model a trader as consuming immediately when a trade has been transacted, and then
remaining in the market to transact further trades. Gale assumes that each trader has
positive marginal utility for all goods, which implies that a double coincidence of wants
1
exists in each trade meeting and money would be inessential; our speci cation rules out
double coincidence of wants entirely to focus on the role of money.
1
2. The Environment
Economic activity occurs at dates 0; 1; 2; : : :. Agents are in nitely lived, and they
are nonatomic. For convenience, we assume that the measure of the set of all agents
is one. Each agent has a type in (0; 1]. The mapping from the agents to their types
is a uniformly distributed random variable, independent of all other random variables in
the model. Similarly, there is a continuum of di erentiated goods, each indexed by a
number j 2 (0; 1]. These goods are perfectly divisible but nonstorable. Each agent of
type i receives an endowment of one unit of \brand" i good in each period. Each agent
consumes his own endowment and half of the brands in the economy; agent i consumes
goods j 2 [i; i + ](mod 1) (for example, agent 0:3 consumes goods j 2 [0:3; 0:8], and agent
0:7 consumes goods j 2 [0:7; 1] [ (0; 0:2]). He prefers other goods in his consumption range
to his own endowment good; while consumption of his endowment yields utility c per unit,
consumption of any other good in his feasible range yields utility u per unit, and u > c > 0.
In addition to the consumption goods, there is a perfectly divisible and durable at-money
object. An agent can costlessly hold any quantity of money. The total nominal stock of
money remains constant at M units per capita. We assume that agents do not discount
future utility. Their preferences are characterized by an overtaking criterion with respect
to expected utility, which will be formalized below.
Agents randomly meet pairwise each period. By the assumed pattern of specialization
in production and consumption, there is no double coincidence of wants in any pairwise
meeting. Each agent meets an owner of one of his consumption goods with probability
one-half, and a consumer of his endowment good with probability one-half. So, every
meeting is between a potential buyer and seller.
Consumption goods cannot be used as a commodity money because they are nonstorable, so money is the only medium of exchange available. An agent is characterized by
his type and the amount of money he holds. Within a pairwise meeting, each agent observes
the other's type, but not the trading partner's money holdings and trading history. They
cannot communicate about this information either. However, the economy-wide moneyholdings distribution is common knowledge. For simplicity, we assume that each transaction
1
2
2
1 Gale
points out that his parametrization can be relaxed, but not suÆciently broadly to cover our case.
speaking, there is a double coincidence of wants only when types i and j are matched, with
i j + 1=2 (mod 1). Such a match occurs with probability zero. Hence, we ignore this possibility.
2 Strictly
2
occurs according to the following simultaneous-move game. The potential buyer submits a
bid specifying a maximum price and also a quantity that he is willing to buy at any price
weakly below that maximum price. And the potential seller submit an o er specifying the
price at which she is willing to sell and the quantity she will sell at that price. Trade occurs
if and only if the bid price is at least as high as the o er price. In that case, the buyer pays
with money at the seller's o er price, and gets the smaller of the bid and o er quantities
of the seller's endowment good.
3. The De nition of Equilibrium
The domain of agents' money holdings is R . Let be the space of countably additive
probability measures on R . Let t 2 denote the money-holdings distribution of the
environment at date t.
At each date, the set of agents is randomly partitioned into pairs. Within each pair,
one of the agents desires to consume the other's endowment. Thus, a bid and o er are
associated with each pair.
Now we provide an intuitive discussion of the distributions of bids and o ers, and we
state some formal assumptions about those distributions. Our assumptions are in the
spirit if a \continuum law of large numbers." For each random partition of the agents
into buyer-seller pairs at date t, there is a sample distribution Bt of bids and a sample
distribution Ot of o ers. We assume that these sample distributions do not depend on the
partition. That is, these are bid and o er distributions Bt and Ot such that for all partition
; Bt = Bt and Ot = Ot . Moreover, because each agent has a trading partner assigned
at random, the probability distribution of the trading partner's bid and o er should be
identical to the sample distribution. That is, Bt and Ot are the probability distributions
of bid and o er respectively that are received at date t by each individual agent, as well as
being the sample distribution in each random pairing of the population of agents.
Now let the probability space ( ; B; P ) represent the stochastic process of encounters
faced by a generic agent. This agent faces a sequence ! of random encounters, one at each
date. His date-t encounter, with some agent of type j , is characterized by her trading type
(buyer or seller) in the meeting and her bid/o er price and quantity. Denote the trading
partner's characteristics by !t = (!t ; !t ; !t ),
If the trading partner is a buyer, !t = b; !t = bid price, !t = bid quantity
If the trading partner is a seller, !t = s; !t = o er price, !t = o er quantity.
+
+
3
1
2
3
1
2
3
1
2
3
3 That
is, we believe that they are logically consistent with the results from probability theory that
we will apply in our analysis, although they cannot be derived from those results. See Green (1994) and
Gilboa and Matsui (1992) for further discussion.
3
The encounters f!tg1t ! are independent across time. is the set of all possible
sequences of encounters that an arbitrary agent in the economy faces.
At each date t, pairwise meetings are independent across the population. That is, for
each agent, !t follows a Bernoulli distribution, a potential buyer's bid (!t ; !t ) is drawn
from the bid distribution Bt , and a potential seller's o er (!t ; !t ) is drawn from the o er
distribution Ot. For t 1, let Bt be the smallest -algebra on that makes the rst
t 1 coordinates, ! t
= (! ; ! ; : : : ; !t ), measurable, and B = f ; g. Let Pt be the
probability measure de ned on Bt . Then, for all t 0, x 2 R , and y 2 [0; 1],
1
(1)
Pt f!t = bg = Pt f!t = sg =
2
n
o
Pt !t x; !t y j !t = b = Bt (x; y )
(2)
n
o
Pt !t x; !t y j !t = s = Ot (x; y ):
(3)
De ne B = B1 and P = P1.
We are going to focus on symmetric equilibrium at which agents are anonymous, and an
agent's strategy is only a function of his own trading history and initial money holdings. In
particular, the trading strategy does not depend on an agent's type. Let be the trading
strategy of a generic agent of type i with initial money holdings . His date-t strategy
t (t ; t ; t ; t ) speci es his bid and o er as a function of his initial money holdings
and his encounter history !. The strategy t is measurable with respect to Bt . The bid
(t ; t ) is the maximum price t at which the agent is willing to buy and the quantity
t that he is willing to purchase (at price no higher than t ) if he is paired with a seller of
his consumption goods. The o er (t ; t ) represents the price t at which he is willing
to sell and the maximum quantity t that he is willing to sell at price t if he meets a
consumer of his endowment good. Because of the restriction on endowment, t 1: As a
buyer, the agent has to be able to pay his bid. Let t denote the agent's money holdings
at the beginning of date t by adopting strategy . (Note that t is a function of and !,
and that it is Bt -measurable in !.) Then
t ( ; ! ) t ( ; ! ) t ( ; ! ):
(4)
Given the agent's initial money holdings , encounter history !, and strategy =
ft g1t , his money holdings evolves recursively as follows: ( ; !) = and, for t 0,
8
if !t = b; t ( ; !) !t
< t ( ; ! ) + t ( ; ! ) minft ( ; ! ); !t g
t ( ; ! ) !t minft ( ; ! ); !t g
if !t = s; t ( ; !) !t
t ( ; ! ) =
:
t ( ; ! )
otherwise
(5)
=0
1
2
2
1
0
1
1
3
3
0
+
1
1
2
3
1
2
3
1
0
1
1
2
3
4
2
1
2
1
3
4
3
4
3
4
0
1
0
2
0
0
0
=0
+1
0
0
0
3
0
2
0
4
2
0
0
3
3
0
4
0
0
1
3
0
2
1
1
0
2
Let vt ( ; !) denote the agent's utility achieved at date t adopting strategy relative to
no trade that date. Then
8
c minft ( ; ! ); !t g
if !t = b; t ( ; !) !t
<
vt ( ; ! ) =
u minft ( ; ! ); !t g
if !t = s; t ( ; !) !t
(6)
:
0
otherwise
Then, strategy overtakes another strategy ^ if for all 2 R ,
0
4
0
2
0
3
0
3
1
3
0
2
1
1
0
2
0
t
hX
lim
inf E
t!1
=0
(
v 0 ; !
t
X
)
=0
(
+
v ^ 0 ; !
i
)
>
0
(7)
where E is the expectation operator with respect to the probability measure P .
At the beginning of date t, given all agents' trading strategy t and the initial moneyholdings distribution , rational expectation requires that agents' belief regarding the bid
distribution Bt and the o er distribution Ot that prevail during date-t trading con rm with
the actual distributions implied by the strategy. That is, for all x; y 2 R ,
0
+
( )=
Bt x; y
( )=
1
Z
0
Z
Ot x; y
1
0
n
j (
) x; t (z; !) y
o
j (
) x; t (z; !) y
o
Pt ! t1 z; !
n
Pt ! t3 z; !
2
4
()
(8)
()
(9)
d 0 z
d 0 z :
The o er and bid distribution each extend uniquely to a corresponding measure on R ,
which will also be denoted by O and B respectively. (No confusion will result, since the
argument of the c.d.f. is an ordered pair of numbers while the argument of the measure
is a subset of the nonnegative orthant of the plane.) That is, for all p 0 and q 0,
O ([0; p] [0; q ]) = O (p; q ) and B ([0; p] [0; q ]) = B (p; q ).
Similarly, the money holdings distribution at the beginning of the of date t is de ned
as follows, for any set A 2 Bt ,
2
+
( )=
t A
1
Z
0
n
j (
Pt ! t z; !
o
)2A
()
(10)
d 0 z
The equilibrium concept we adopt is Bayesian Nash equilibrium with overtaking criterion.
.
Definition 1
A Bayesian Nash equilibrium is a four-tuple h; ; fBtg1t ; fOtg1t i
0
that satis es
(i) is the initial money-holdings distribution in the environment.
0
5
=0
=0
(ii) Given the bid distributions fBt g1t and the o er distributions fOtg1t , and given
that all other agents adopt strategy , it is optimal for an arbitrary agent to adopt
strategy , that is, there is no strategy that overtakes strategy .
(iii) For each t 0, Bt and Ot satisfy equations (8) and (9). That is, these distributions
re ect the adoption of strategy by all agents.
=0
=0
In this paper, we are interested in stationary equilibria where all the distributions
(money-holdings t, bid Bt, and o er Ot) are time-invariant, and agents' trading strategy
is stationary Markov. A strategy is stationary Markov if there is a function : R ! R
such that for all t 0, 2 R , and ! 2 ,
t ( ; ! ) = (t ( ; ! ))
(11)
that is, the strategy is only a function of an agent's current money holdings.
+
0
4
+
+
0
0
A stationary Markov monetary equilibrium (SMME) is a Bayesian Nash
equilibrium h; ; fBtg1t ; fOtg1t i such that
(i) There exist measures O, B and such that, for all t 0, Bt = B , Ot = O and
t = .
(ii) A generic agent's random process of bid prices converges almost surely to the marketwide bid distribution in the sense that, for every open subset A of R ,
X
lim 1 ( ( ; !); ( ; !)) ! B (A) ; a.s. :
.
Definition 2
=0
=0
2
+
n!1 n
t<n
A
t1
t2
0
0
(iii) A generic agent's sample path of o ers converges almost surely to the market-wide
o er distribution in the sense that (with A denoting the characteristic function of a
set A), for every open subset A of R ,
X
lim 1 ( ( ; !); ( ; !)) ! O(A) ; a.s. :
2
+
n!1 n
t<n
A
t3
t4
0
0
(iv) is a stationary Markov strategy.
(v) monetary trade is strictly better than autarky in the sense that
lim
inf E
t!1
t
X
=0
6
(
) 0:
v 0 ; ! >
Items (i){(iv) of the de nition specify various aspects of stationarity, with (ii) and (iii)
suggesting that agents' strategic behavior is ergodic as would be expected in a stationary
equilibrium. Item (v) rules out autarkic equilibrium. Note that, since an agent can choose
autarky unilaterally, no equilibrium can be worse than autarky.
Note that this equilibrium concept is a specialization of Bayesian Nash equilibrium in
the space of all strategies de ned above. That is, a SMME must be impervious to deviation
to an arbitrary strategy, whether or not the deviation strategy is stationary Markov.
One particular type of SMME is the single-price equilibrium, which is de ned as an
equilibrium where all trades occur at the same price, say p, almost surely. That is, all
traders bid to buy one unit or as much as they can a ord of their desired consumption
goods at price p, and o er to sell one unit of their endowment goods at price p. From the
result of Green and Zhou (2000), we can conclude that a stationary single-price-p equilibria
exists if the initial money holdings is distributed geometrically on the lattice de ned by
p, that is, on f0; p; 2p; : : :g. Another type of potential equilibrium is the price-dispersion
equilibrium, at which trades occur at di erent prices across di erent pairs of traders who
have di erent money holdings and trading histories. In the next section, we are going
to show that at a SMME, price dispersion can not occur. In other words, single-price
equilibrium is the only kind of stationary Markov equilibrium that exists. Note that an
autarkic equilibrium is always an single-price equilibrium.
0
4. Nonexistence of SMME with Price-Dispersion
Suppose that h; ; B; Oi is a SMME de ned as above. We want to show that at this
equilibrium, all trades occur at the same price.
Recall that, for a probability measure P on a separable topological space T , the support
supp(P ) of the measure is the intersection of all closed subsets H T such that P (H ) = 1,
and that P (supp(P )) = 1.
Let o be the in mum of the support of the marginal distribution of o er prices at
which positive quantities are o ered. That is,
o = supfx j O ([0; x) (0; 1)) = 0g:
(12)
(Note that o ers to sell a zero quantity are excluded.)
We show that at equilibrium, all trades occur at price o almost surely. We prove
this via several lemmas. Note that by the de nition of the equilibrium, the bid and o er
distributions B and O are generated by the equilibrium strategy and the money-holdings
distribution through equations (8) and (9).
7
. Given that h; ; B; Oi is a SMME, the in mum
o er price distribution is positive.
Lemma 1
o
of the support of the
Proof.
First observe that O(f0g (0; 1)) = 0, since otherwise an agent would be giving
away for free some quantity of endowment that he would have obtained utility from having
consumed instead.
Using this fact, we show that for some p > 0 and q > 0, (p; q) 2 supp(B ). Suppose
that, to the contrary, every bid in the supp(B ) is either (p; 0) or (0; q). A bid of (p; 0)
results in no trade since the size of the transaction is no greater than the bid quantity, and
a bid of (0; q) results in no trade since O(f0g (0; 1)) = 0. Since SMME is not autarkic
by de nition 2(v), bids of these two forms cannot constitute all of supp(B ). That is, for
some p > 0 and q > 0, (p; q) 2 supp(B ).
Let p > 0 and q > 0, suppose that (p; q) 2 supp(B ), and consider o er (p=2; q=2). This
o er results in a sale of q=2 units for revenue p q=4 with some positive probability , since
(p; q) 2 supp(B ). Therefore the expected revenue from making this o er is at least p q=4.
The ratio of expected revenue to expected amount of endowment sold is p.
If every o er in an open set U of the topological space R (with its Euclidean topology
relative to R ) provides expected revenue less than what (p=2; q=2) provides and also yields
a lower ratio of expected revenue to expected amount of endowment sold than (p=2; q=2)
yields, then an argument involving the Strong Law of Large Numbers and Fatou's Lemma
shows that U \ supp(O) = ;. In particular, U = [0; p q=4) (0; 1) satis es this condition,
so U \ supp(O) = ;. That is, o p q=4 > 0.
The next lemma is a technical result that will be used in the proof of the subsequent
lemma.
2
+
2
4
. For a given subset Q of , for any > 0, there exist T 0 and Q 2 BT
Lemma 2
such that
( \ Q) (1
P Q
) ( )
and
P Q
( n Q) < P (Q):
P Q
(13)
Lemma 2 says that for a given set Q, there exist a date T and a set Q, measurable
with respect to BT , that is an arbitrarily close approximation of Q. The proof of Lemma
2 is in the appendix.
4 Fatou's lemma (c.f.
Royden, 1988) justi es converting an expectation of limits to a limit of expectations,
as the overtaking criterion requires. See the complete proof of Lemma 3 in the appendix for a closely related
argument.
8
Consider an agent with an initial money holdings who adopts the equilibrium strategy
. Given that the environment is stationary, the agent's money holdings t is markovian
with full support of economy-wide distribution . The next lemma states that the agent's
money holdings drops below any arbitrary level in nitely often.
0
. If h; ; B; Oi is a SMME, then for any
in nitely often almost surely.
Lemma 3
t <
>
0, an agent's money holdings
Proof.
A detailed proof is given in appendix. The intuitive argument is as follows.
Suppose that, with positive probability, an agent's money holdings falls below > 0
only nitely often. Then, by countable additivity, there must be a date T such that
P (f! j8 t T t g > 0. De ne Q = f! j8 t T t g. By Lemma 2, for every > 0
there is an event Q such that P (Q \ Q) (1 )P (Q) and P (Q n Q) < P (Q) with
Q being BT -measurable, where without loss of generality T T .
This situation is proved to lead to a contradiction by specifying an overtaking-criterionpreferred deviation ^ from with the following intuitive form. The agent follows until
T , and after T if ! 2
= Q .
Choose a number o > o . What ^ prescribes for ! 2 Q is that beginning at T ,
the agent should abstain from selling (that is, set t = 0) whenever would have set
the o er price no higher than o . This abstention should continue until a date t when
either a cumulative value of in sales has been forgone, or else t would have been less
than . If ! 2 Q, then eventually the cumulative value of forgone sales will equal
almost surely (conditional on Q \ Q ). Because the price at which these sales are forgone
does not exceed o , forgoing the sales provides at least =o of the agent's endowment
for consumption. Asymptotically, then, this consumption adds c(=o )P (Q \ Q ) to the
sum of expected utilities by which overtaking is calculated. Note that, by choice of Q ,
c(=o )P (Q \ Q ) c(=o )(1 )P (Q).
If ! 2= Q, then the agent will learn this at the rst time t when t < . (Note that the
agent has the information required to know what t would have been, although he does
not play strategy .) When this happens (that is, in the event Q n Q), ^ prescribes that
the agent should abstain from purchasing until has cumulatively been saved. (Limited
purchasing, rather than complete abstention, is required in some circumstances to prevent
cumulatively saving more than .) Because the forgone purchases would have been made
at prices of at least o , no more than =o consumption will be forgone. That amount
+
4
+
5
+
+
+
+
+
5A
positive, but reduced, value of t4 may have to be set in some circumstances, in order to reduce sales
below what would have entailed but ensure that the cumulative value of forgone sales does not exceed .
9
of consumption, in the event Q n Q, asymptotically subtracts at most u(=o )P (Q n Q)
from the sum of expected utilities by which overtaking is calculated. Note that, by choice
of Q, u(=o )P (Q n Q) u(=o )P (Q).
Except for these changes just speci ed, ^ prescribes the same bids and o ers as does .
Asymptotically, the net bene t of deviating to ^ is at least [c(=o )(1 ) u(=o )]P (Q).
A positive value of can be chosen that is small enough so that this asymptotic net bene t
is positive if P (Q) > 0. This argument by contradiction shows that P (Q) = 0, that is, that
t < in nitely often, almost surely.
Now we are ready to prove the main result of the paper.
If h; ; B; Oi is a SMME, then it is a single-price equilibrium. In other
words, given o is the in mum of the support of the o er price distribution O , all trades
occur at o almost surely.
Proof. We rst show that for any arbitrary Æ > 0, all trades occur at price no higher
than o + Æ almost surely. This claim holds trivially if no trader o ers to sell at price
higher than o + Æ, i.e., supp(O) \ (o + Æ; 1) (0; 1) = ;, since trade occurs at the
o er price. The more complicated case is when o ers higher than o + Æ are made with
positive probability, i.e., supp(O) \ (o + Æ; 1) (0; 1) 6= ;. In this case, we show that
supp(B ) \ (o + Æ; 1) (0; 1) = ;. This would contradict trade occurring at prices higher
than o + Æ with positive probability.
Suppose to the contrary, supp(B ) \ (o + Æ; 1) (0; 1) 6= ;. Consider a generic trader
whose strategy is , and whose random trading partners are described by . For each
! 2 , let (! ) denote the set of dates at which the agent bids to buy at price above
o + Æ,
(! ) = ft j t ( ; ! ) > o + Æ g:
By assumption and De nition 2 (ii), (!) is an in nite set almost surely. Let " be a small
positive number, which value will be chosen later. For any n 1, let '"n(!) denote the set
of dates at which the agent's money holdings are below "n,
'"n (! ) = ft j t ( ; ! ) < "n g:
For each n 1, since "n > 0, by Lemma 3, '"n(!) is an in nite set almost surely.
We construct a strategy ^ as follows. For all ! 2 and t 0, the o er (^t ; ^t )( ; !)
is identical to (t ; t )( ; !). The bid (^t ; ^t ), however, di ers from (t ; t ) at the
following dates.
+
Proposition.
1
0
0
3
3
4
0
1
10
2
1
4
2
0
1. Change the bid price from above o + Æ to o + Æ through the rst date at which
a trade that would have occurred under fails to occur because the changed bid is
below the o er. That is, let
t = minft j t 2 (! ); !t = s; t !t > o + Æ; t > 0; !t > 0g
and for all t 2 (!) and t t, set ^t ( ; !) = o + Æ. Let X be the quantity that
would have traded on date t under , X minft ; !t g, and let S be the money
saved on date t by not buying at price above o + Æ, that is,PS = !t X > (o + Æ)X .
Then, at the beginning of date t + 1, t = t + S , t (v v ) = uX >
uS=(o + Æ ).
2. Spend the money saved at date t , S , at a sequence of dates after t when the agent's
money holdings are below "n (i.e. dates in '"n(!)), consecutively, for n = 1; 2; : : :.
Speci cally, let l = t, and set n=1.
(a) For all t > ln such that t 2 '"n(!), set
1
1
1
2
2
3
0
2
^
+1
2
2
+1
=0
^
0
1
^ (
) = o + Æ=2;
o
n
) = min 1; o +S Æ=2
until a trade is accomplished with the modi ed strategy at date ln. That is,
ln = minft j t > ln ; t 2 '"n (! ); !t = s;
^t ( ; !) !t ; !t > 0g:
Such an occasion exists because '"n(!) is an in nite set, o is the in mum of the
support of the o er price distribution, hence there exist ps 2 [o ; o + Æ=2] and
qs > 0 such that (ps ; qs ) 2 supp(O ).
(b) Let S = S !ln minf^ln ( ; !); !ln g. If S > 0, set n = n + 1 and return to
step (a). Otherwise, all money saved at date t is spent, and the agent resume
strategy ( ; !).
This process stops in nite time almost surely given that S is nite, and transaction
sizes are determined by f!ln gn , which is i.i.d. and f^ln gn which is constructed
to be a non-binding constraint as long as S remains positive.
Now, let us examine the utility lost and gained by adopting the modi ed strategy. In
step 2, the utility gained is from consuming goods purchased by the S units of saved money
at price no higher than o + Æ=2, hence it is no less than uS=(o + Æ=2). However, since we
modify the bid price for t 2 '"n(!), if the original bid price is higher than o + Æ=2, some
t1 0 ; !
^ (
t2 0 ; !
1
2
1
2
0
1
0
2
3
0
3
1
2
11
1
3
trades that would have occurred under strategy may fail to occur under strategy ^ . In
each of such instances (i.e., for all n 1), the maximum amount of goods the agent would
have bought under strategy is "n=o . Hence, the total utility loss due to this modi cation
is less than
1
X
"
u
:
u
"n =o =
o 1 "
n
Combining all the gains and losses in steps 1 and 2, given that the rest of strategy is
unchanged,
=1
1
X
=0
(
v ^
Let
v
) =
"
By (14), for any " < ",
1
X
Hence,
=0
(v
^
t
X
)
=0
v
)+
1
X
=t +1
S
(v
v
^
+ Æ + u o + Æ=2
o
2(o + Æ)(o + Æ=2)
o
>
u
=
u
(
v ^
S
o
u
1
SÆ
)
1
1
"
"
"
"
:
(14)
SÆo + 2(o SÆo
+ Æ)(o + Æ=2) :
v > u
SÆ
2(o + Æ)(o + Æ=2)
lim
inf E
t!1
t
hX
=0
v ^
t
X
=0
i
1
o
1
"
"
0
> :
0
v > :
By (7), strategy ^ overtakes strategy , which contradicts the assumption that is an
equilibrium strategy. Therefore, no price higher than o + Æ is in the support of the bid
price distribution.
We have shown that for any Æ > 0 such that any price higher than o + Æ is in the
support of the o er price distribution, it is not in the support of the bid price distribution.
Thus, no trade occurs at price above o + Æ with positive probability. Taking Æ ! 0, all
trades occurs at price no higher than o almost surely. Since o is the in mum of the
support of the o er price distribution, all trades occur at o almost surely.
The intuition for the above result is quite simple. If with strictly positive probability
that o ers are made at some o + Æ > o , then the optimal strategy has to be such that
with probability one no such o er is accepted, since otherwise, a perfectly patient agent
12
can improve the strategy by switching purchases at price o + Æ to other dates at some
lower price without any loss. The lowest price possible is o . Hence, no trades will occur
at price other than o .
To summarize, despite the decentralized trading arrangement, a SMME is always a
single-price equilibrium, where all trades take place at the same price among all pairwise
meetings, almost surely.
5. Conclusion
In this paper, we have shown that any stationary Markov monetary equilibrium in a
random-matching economy with perfectly patient agents must be a single-price equilibrium.
Our demonstration of this fact does not depend on showing that equilibrium is walrasian.
In this respect, the demonstration is very di erent from the type of theorem proved by
Douglas Gale (1986a, b) for non-monetary economies. We are con dent that our theorem
in the limit regarding perfectly patient agents can be complemented with a limit theorem,
which will state that equilibrium is approximately single-price, in a suitable sense, in an
economy where agents' discount factor is almost equal to unity. We note that Camera
and Corbae (1999) have proved such a limit theorem regarding a special class of monetary
equilibrium in a somewhat di erent random-matching environment.
13
Appendix
The Proof of Lemma 2.
For all t 0, de ne a measure t on Bt such that for any C 2 Bt , t(C ) = P (Q \ C ).
By this de nition, forR any C 2 Bt , t (C ) P (C ). Therefore, there exists a density ft such
that for all C 2 Bt , C ft dP = t (C ). In fact, ft = E[ Q j Bt] where Q(!) = 1 if ! 2 Q
and Q(!) = 0 if ! 62 Q, since for any C 2 Bt ,
Z
Z
Z
Z
Q dP +
Q dP = P (C \ Q) = t (C ) =
ft dP:
E[ Q j Bt ] dP =
C \Q
C
C nQ
C
Therefore, fft g1t is a martingale. By Martingale convergence theorem, ft ! Q a.s..
Furthermore, E[ft Q] ! 0.
For any > 0, for all t 0, de ne Ct f! j ft(!) 1 g. Then,
Z
Z
Z
1
1
ft dP =
(
ft Q ) dP +
Q dP
P (Ct n Q)
1 Ct nQ
1 Z CtnQ
Ct nQ
1
= 1
(ft Q) dP
(15)
Ct nQ
Furthermore, since for ! 62 Ct, ft (!) < 1 ,
Z
Z
Z
P (Q n Ct ) =
Q dP =
( Q ft ) dP +
ft dP
=0
QnCt
Z
<
which implies that
therefore,
QnCt
QnCt
Z
1
P (Q n Ct ) <
( Q
QnCt
QnCt
)
+ (1
ft dP
( Q
)
ft dP
Z
1
P (Ct \ Q) = P (Q) P (Q n Ct ) > P (Q)
Then, since E[ft
Q
] ! 0, there is T 0 such that
Z
(ft Q) dP minf(1
CT nQ
) ( n Ct)
P Q
QnCt
( Q
)
ft dP:
) g ( )
; 2 P Q :
By (15) and (16), (17) implies that
P (CT \ Q) (1 )P (Q) and
P (CT n Q) < P (Q):
Let Q CT 2 BT , (18) is the result we set out to proof. .
14
(16)
(17)
(18)
The Proof of Lemma 3.
Suppose that, with positive probability, an agent's money holdings falls below > 0
only nitely often. Then, by countable additivity, there must be a " > 0 and a date T such
that P (f!j8 t T t g > ". De ne Q = f!j8 t T t g. By Lemma 2, for every
> 0 there exist T > T and an event Q 2 BT such that
P (Q \ Q) (1 )P (Q) and
P (Q n Q) < P (Q):
(19)
We show that a deviation strategy ^ can overtake strategy , hence contradict that
is an equilibrium strategy. Strategy ^ is de ned as follows. The agent follows until T ,
and after T if ! 2= Q .
Choose a number o o such that there exist p o , qb > 0 and qs > 0 such that
(o ; qb ) 2 supp(B ) and (p; qs) 2 supp(O). Such an o exists (possibly equals to o ) because
at equilibrium, trade occurs at some price in nitely often. By assumed trading rule, trading
price is the o er price when bid price is higher than o er price. Then by De nition 2 (iii),
the agent o er to sell at price no higher than o in nitely often almost surely. For ! 2 Q ,
strategy ^ speci es that beginning at T , the agent should abstain from selling (that is,
set t = 0) whenever would have set the o er price no higher than o . This abstention
should continue until a date when either a cumulative value of in sales has been forgone,
or else t would have been less than .
Formally, for all ! 2 , let (!) denote the set of dates that the agent's o er price t
is below o ,
(! ) = ft j t ( ; ! ) < o g:
The set (!) is in nite almost surely. Set S = , l = T , and n = 1. For each ! 2 Q ,
strategy is modi ed as follows.
1. First consider the o er strategy. For each t > ln , if t 2 (!), set ^t = t and
^t = maxf0; t S=t g, until the agent succeed selling ^t instead of t at date
ln ,
n
o
ln = min t j t > ln ; t 2 (! ); !t = b; !t t ; minf!t ; t g >
^t :
Such a date exists because ^t < t , and because by de nition 2 (ii), the agent runs
into buyers with bid as high as o in nitely often. If t 62 (!), the agent behaves as if
( S ) has not been spent at l ; : : : ; ln ; (^t ; ^t )( ( S ); !) = (t ; t )( ; !):
For the bid strategy, for any t > ln , there are two possibilities:
+
+
+
+
+
+
4
3
+
3
+
0
0
1
4
4
3
3
4
1
1
4
2
3
3
4
3
4
4
4
+
1
1
1
15
3
4
0
3
4
0
At date t, the agent has enough money to purchase as he originally planned;
i.e., t = t ( S ) t t . In this case, again behave as if ( S ) has not
^
1
2
been spent at l ; : : : ; ln ; (^t ; ^t )( ( S ); !) = (t ; t )( ; !):
It is discovered that ! 62 Q. That is, after selling S value less at l ; : : : ; ln ,
the agent does not have enough money to buy at t that he would have had he
followed strategy , i.e., t < t t ,. In such a case, set t = ln and go to
step (3) directly.
For ln < t < ln, if the agent has enough money, trade is carried out as if he has not
adopt the modi ed strategy ^. Hence vt = vt . At date ln, the agent sells ^t units of
his endowment instead of t , and consume the remaining portion of the endowment.
Let yn denote the reduction of the agent's sale at ln,
yn minf!ln ; ln g minf!ln ;
^ln g:
The utility gain is vln vln = c yn, and the corresponding money loss is ln yn
(o + Æ)yn:
2. Let S = S ln yn. If S > 0, set n = n + 1, return to step (1). Otherwise, selling
-value less of endowment has been accomplished at time lN ! where N (! ) = n.
N (! ) is nite because is nite, and because trading quantities are determined by
i.i.d. bid quantities and o er quantities which are non-binding constraint on reducing
to zero. Set t = ln , and go to step (3).
3. For all t > t , the agent resume strategy as if ( S ) has not been spent previously,
i.e., ^t ( ( S ); !) = t ( ; !). When this is possible, vt = vt . If the agent nds
that he does not have enough money to buy at t what he would have had he followed
strategy , then t < , or equivalently, ! 62 Q. In such a case, ^ prescribes that the
agent buys what he can on dates when he can not a ord to make the original bid.
The number of such occasions is limited by the amount S overspent previously.
Formally, for all t > t , if t < t t , set ^t = t , but ^t = t =t . Let R = S
and k = 1. The following process records the utility loss as the agent buys less than
under strategy .
(a) Let tk be the k-th time that the agent is unable to purchase what he originally
planed under .
n
o
tk = min t j t > tk ; !t = s; !t t ; and t < t t
1
1
1
2
0
1
2
0
1
^
1
2
0
1
1
1
^
4
4
3
4
3
4
^
3
3
( )
0
0
0
^
0
0
^
1
2
1
1
1
16
1
2
^
2
2
^
1
1
2
Let zk denote the reduction of the agent's purchase at tk ,
zk minf!tk ; tk g minf!tk ;
^tk g:
Then the utility loss on date tk is vtk vtk = u zk , and the amount of money did
not spend is !tk zk .
(b) Let R = R !tk zk . If R > 0, set k = k + 1, and return to step (a). Otherwise,
the money balance has been returned to what it would have been in the original
strategy pro le. The agent resumes strategy . Let K (!) = k.
As previously argued, this process ends in nite time.
To compare strategies ^ and , let D( ; !) denotePthe di erence in utility generated
by ^ and for any ! 2 and the given , D( ; !) = 1 v ( ; !) v ( ; !) . Then
3
2
3
2
^
1
1
0
0
(
8 ! 2 Q \ Q
D 0 ; !
(
8 ! 2 Q n Q
D 0 ; !
0
^
=0
)=c
)=c
N
(! )
X
n=1
N
(! )
X
n=1
yn
0
0
co
+
yn
u
K
(! )
X
k=1
By (19) and (20) and Fatou's Lemma (Royden[10]),
zk
u
o
(20)
:
6
lim
inf E
t!1
t
hX
=0
(
v ^ 0 ; !
(
i
)
t
hX
=0
+
Z
lim ( (
t!1
)
v ^ 0 ; !
( \ Q)
o
c
(1 )P (Q)
o
), since P (Q) ", (21) implies
Take < c o =2(c o + u o
lim
inf E
t!1
)
v 0 ; !
(
v ^ 0 ; !
)
(
v 0 ; !
)
i
c
+
o
+
=
c
o+
c
o+
c
(1
)
co
)
( )
dP !
( n Q)
u
P (Q):
o
u
P Q
(
v 0 ; !
P Q
u
o
+uo
( )
P Q
+
o o+
(21)
"
2 o " > 0:
By (7), strategy ^ overtakes strategy , which contradicts the assumption that is an
equilibrium strategy.
>
6 The
+
Fatou's Lemma stated on Royden[10], p86, requires the sequence that converges to be nonnegative.
In fact, the lemma holds for sequence bounded from below, which is the case here.
17
References
1. S. R. Aiyagari, N. Wallace and R. Wright (1996). \Coexistence of money and interestbearing securities," Journal of Monetary Economics , 397-419.
2. G. Camera and D. Corbae (1999), \Money and Price Dispersion," International Economic Review, 40, 985{1008.
3. D. Gale (1986a). \Bargaining and competition part I: characterization," Econometrica , 785-806.
4. D. Gale (1986b). \Bargaining and competition part II: existence," Econometrica ,
807-818.
5. I. Gilboa, A. Matsui (1992). \A model of random matching," Journal of Mathematical
Economics , 185-97.
6. E. J. Green (1994). \Individual Level Randomness in a Nonatomic Population,"
ewp-ge/9402001, econwpa.wustl.edu.
7. E. J. Green and R. Zhou (2000). \Dynamic monetary equilibrium in a random matching economy," forthcoming, Econometrica .
8. H. L. Royden (1988). Real Analysis. MacMillan Publishing Company, New York.
9. K. Kamiya and T. Sato (2001). \Equilibrium Price Dispersion in a Matching Model
with Divisible Money," manuscript, University of Tokyo.
37
54
54
21
18
@FedFRASER