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Federal Reserve Bank of Chicago
Dynamic Monetary Equilibrium in a
Random-Matching Economy
Edward Green and Ruilin Zhou
WP 2000-01
Dynamic Monetary Equilibrium in a
Random Matching Economyy
Edward J. Green
Federal Reserve Bank of Chicago
and
Ruilin Zhou
Federal Reserve Bank of Chicago
First version: February, 2000
Current version: April, 2001
Abstract
This article concerns an in nite horizon economy where trade must occur pairwise, using
a double auction mechanism, and where at money overcomes lack of double coincidence of
wants. Traders are anonymous and lack market power. Goods are divisible and perishable,
and are consumed at every date. Preferences are de ned by utility-stream overtaking. Money
is divisible and not subject to inventory constraints. The evolution of individual and economywide money holdings distributions is characterized. There is a welfare-ordered continuum of
single price equilibria, re ecting indeterminacy of the price level rather than of relative prices.
J.E.L. Classi cation: C78, D51, E40
Ruilin Zhou thanks the National Science Foundation for nancial support. Part of this research was done
while both authors held appointments at the Federal Reserve Bank of Minneapolis. The views expressed herein are
those of the authors and not necessarily those of the Federal Reserve Bank of Chicago, Federal Reserve Bank of
Minneapolis 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
This article contributes to a research program, both classical and contemporary, concerning
the relationship between decentralized trading and allocative eÆciency. Classical economists
observed that chains of transactions link traders who do not deal directly with one another or
even have speci c knowledge of one another's existence. They understood that such a pattern of
sequential trade within overlapping coalitions re ects environmental constraints (e.g., geographic
separation) that rule out having an economywide market (as envisioned by Walras' parable of an
auction) in which all traders would participate directly. Those classical economists, then, regarded
fragmentation of trading coalitions as an important fact for economic theory to take explicitly
into account. In fact, their view about the eÆciency of competitive trade was based on their
assessment that traders largely succeed in overcoming this obstacle, at least insofar as achieving
economywide uniformity of relative prices is concerned.
A closely similar perspective, that a credible welfare characterization of market equilibrium
needs to come to terms with how traders interact strategically in an environment that enforces
fragmented market participation, motivates the model of pairwise trading to be studied here. The
main results establish the existence of equilibrium, display indeterminacy by looking at single
price equilibrium, and characterize the long run behavior of trade and money holdings in these
equilibria.
Analysis of this model supports the classical economists' conjecture that, despite the absence
of a central market, it is possible for equilibrium relative prices to remain constant across trading
pairs and throughout time. This is a nontrivial nding about the model economy because exchange
and consumption occur in real time and result in endogenous heterogeneity of wealth among both
buyers and sellers. Such heterogeneity might have been suspected to induce disparity of terms of
trade across trading pairs, but is shown not necessarily to do so.
While equilibrium in the model economy is consistent with economywide uniformity of relative
prices, nevertheless the equilibrium price level is indeterminate. Given a xed, nominal stock of
money, which is taken to be a parameter of the economy, a higher price level implies a stochastically
dominated distribution of real balances among traders in the economy. The lower level of real
balances results in a lower incidence of completed transactions among trading pairs, hence in
economic ineÆciency. The model economy possesses a continuum of distinct, Pareto-ranked,
equilibrium allocations. It remains an open question whether this indeterminacy of equilibrium
re ects a fundamental fact about decentralized competition or whether it may be an accidental
consequence of speci c, fragile, modeling assumptions. This question is discussed further in the
1
conclusion.
An informal description of some main features of the model economy, and a brief comparison
with formulation of three prior models of decentralized exchange may be a helpful prelude to
technical exposition. The model economy comprises a continuum of in nitely lived traders who
populate a discrete time environment. At each date, every trader receives an endowment of a
perishable, divisible, di erentiated good and enjoys consumption of his own endowment good
and of the endowment goods of some, but not all, other agents. Agents' preferences between
random consumption streams are determined according to a von Neumann-Morgenstern utility
function for \temporary utility" at each date, and by an overtaking criterion to compare in nite,
expected temporary utility sequences. Indeed, there are gains to economywide trade because each
trader receives higher marginal utility from consumption of others' endowment goods than from
consumption of the good with which he is endowed.
At each date, the population of agents is randomly partitioned into pairs who are able to trade
with one another. All trading pairs satisfy the condition that exactly one agent can obtain utility
from consumption of the other's endowment good. Trading is anonymous, in the sense that no
pair meets more than once and also that each agent knows the variety of good with which his
partner is endowed but nothing else about the partner. A particular double auction mechanism
governs trade within all pairs. The equilibrium concept for the economy is a version of Bayesian
Nash equilibrium.
The model just described most closely resembles that of Green and Zhou (1998). The present
model di ers from its predecessor in three respects. Equilibrium is de ned in terms of the evolution
of the economy from an initial state, rather than the analysis being concerned exclusively with
steady state analysis, in order to investigate indeterminacy of equilibrium in its strict sense. The
utility-overtaking criterion is adopted, rather than a discounted utility formulation, in order to
facilitate analysis of dynamic equilibrium without sacri cing essentiality of money.1 Goods, as
well as money, are modeled as being divisible, in order to remove indivisibility and nonconvexity
as possible causes of price level indeterminacy.
Further perspective on the model is obtained by comparing its formulation with those of Gale
(1986a,b) and of Shi (1995) and Trejos and Wright (1995). As in Gale's model, traders transfer
endowments of divisible goods in random, pairwise meetings that take place in discrete time,
and assumptions of anonymity and absence of time preference prevent monopoly power from
1 Money is inessential in the model of Green and Zhou (1998) because agents in that model are unable to consume
their own endowments. Using a combination of analytical and numerical approaches, Zhou (1999a) studies steady
state equilibrium of a model economy in which money is essential and in which agents maximize expected discounted
utility.
2
being exercised in these meetings. As in the Shi-Trejos-Wright model, economic activity (i.e.,
the process of receiving endowments, trading, and consumption) takes place repeatedly through
time, rather than each trader consuming only once and then leaving the market as in Gale's
model. Also as Shi-Trejos-Wright, the insuÆciency of trading pairs to garner directly the full
gain to economywide trade takes the particularly stark form of a complete absence of double
coincidence of wants. This assumption gives money an essential role as medium of exchange, as
Gale emphasized would be desirable in an extension of his model. In particular, the presence
of money (which is absent in Gale's model economy) enables the price level, as well as relative
prices of various goods, to be considered explicitly. However, unlike the Shi-Trejos-Wright model,
money here is modeled as being divisible and not subject to inventory constraints.2 Finally, as in
both the Gale and Shi-Trejos-Wright models, exchange within each trade meeting is assumed to
be governed by a strategic form mechanism broadly resembling an auction protocol.
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. An agent can consume his own endowment and half of the other brands in the
economy; agent i consumes goods j 2 [i; i + 21 ](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 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 > 0.3 In addition to the consumption goods, there is a at money.4 Money is perfectly
2 The Shi-Trejos-Wright model amends the Kiyotaki-Wright (1989) model by making goods divisible, while
retaining assumptions of money indivisibility and a one-unit inventory constraint on money holdings, to model
non-par exchange of money for goods. See Green and Zhou (1998) for a discussion of why the indivisibility and
inventory-constraint assumptions are undesirable. Models along the lines of Shi-Trejos-Wright that partially relax
those ad hoc constraints (by posting a nite bound, greater than 1, on the amount of money carried into a trade
meeting) include Camera and Corbae (1999), Hendry (1993), Molico (1997), and Wallace (1996).
3 In principle, a consumption bundle could be de ned to be a nite measure on [0,1) and the utility of to an
agent i could be de ned to be c(fig) + u((i; i + 12 ](mod 1)). In practice, at any date an agent can only consume
his and his trading partner's endowment goods.
4 Logically, at money is an economywide accounting system that satis es restrictions such as we now describe.
It is customary in the money/search literature, but not logically necessary, to interpret at money as some physical
object.
3
divisible, and an agent can costlessly hold any quantity of it. 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 endowments and
consumption sets, there is no double coincidence of wants in any pairwise meeting.5 Each agent
meets a partner endowed with one of his consumption goods with probability one half, and a
partner who can consume his endowment good with probability one half. So, in every meeting,
one partner is a potential buyer and the other is a potential seller.
Consumption goods cannot be used as 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. Each agent has an initial money holdings, which, like the agent's
type, is exogenously and deterministically given. Within the population, types and initial money
holdings are independently distributed. The economywide initial money holdings distribution is
common knowledge.
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.
For simplicity, we assume that each transaction 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 maximum 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, but the bid quantity is no higher than the o er quantity. In that case, the seller transfers
the quantity of his endowment good prescribed by the bid to the buyer, and the buyer pays with
money at the seller's o er price.
This particular double auction mechanism is closely related to a family of such mechanisms in
which trade occurs if and only if the bid price is at least as high as the o er price, the quantity
transferred from the seller to the buyer is then the minimum of the bid and the o er, and the
transaction price is weakly between the o er price and the bid price.6 The mechanism de ned here
fails to belong to the family only because trade is speci ed not to take place if the o er quantity
5 Strictly 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.
6 Examples of such mechanisms are (1) a short side mechanism in which trade occurs at the o er price if the bid
quantity is less than the o er quantity and at the bid price otherwise, and (2) a mechanism in which trade occurs
at the midpoint of the bid and o er prices. The latter mechanism was studied by Chatterjee and Samuelson (1983)
and was shown to be eÆcient for sale of an indivisible good in a static setting by Myerson and Satterthwaite (1983).
4
is smaller than the bid quantity|a situation that intuitively would not occur in an equilibrium of
an auction type mechanism in this economy because the seller (who has linear temporary utility)
should be willing to sell his entire endowment if he is willing to sell any of it. By enforcing the bid
quantity to be no greater than the o er quantity when trade occurs, some algebraic expressions
in the de nitions and proofs below are made simpler than if explicit reference to the minimum
of the bid and o er quantities were necessary. However the proofs remain sound, with inessential
modi cations, for any mechanism in the related family just characterized.
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+ . Suppose that the initial money holdings distribution is given by 0 .
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 of a \continuum
law of large numbers."7 For each random partition of the agents into 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, there are bid and o er distributions
Bt and Ot such that for all partitions , 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. Agent
i's date-t encounter, with some agent of type j , is characterized by agent j 's trading type (buyer
or seller) in the meeting and her bid/o er price and quantity. Denote the trading partner's type
by !t = (!t1 ; !t2 ; !t3 ), which is interpreted as follows.
If the trading partner is a buyer, !t1 = b; !t2 is the bid price; !t3 is the bid quantity
If the trading partner is a seller, !t1 = s; !t2 is the o er price; !t3 is the o er quantity:
7 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.
5
The encounters f!t g1
t=0 ! are independent across time. is the set of all possible sequences of
encounters that a generic agent in the economy faces.
At each date t, pairwise meetings are independent across the population. That is, for each
agent, !t1 follows a Bernoulli distribution, a potential buyer's bid (!t2 ; !t3 ) is drawn from the
bid distribution having c.d.f. Bt , and a potential seller's o er (!t2 ; !t3 ) is drawn from the o er
distribution having c.d.f. Ot . For t 1, let Bt be the smallest -algebra on that makes the
vector of the rst t coordinates, !t = (!0 ; !1 ; : : : ; !t 1 ), measurable, and B0 = 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],
Ptf!t1 = bg = Pt f!t1 = sg = 21
n
o
Pt !t2 x; !t3 y j !t1 = b = Bt (x; y)
n
o
Pt !t2 x; !t3 y j !t1 = s = Ot (x; y):
(1)
(2)
(3)
De ne B = B1 and P = P1 .
We focus on symmetric equilibrium, that is, equilibrium in which agents are anonymous, an
agent's strategy is a function of only his own trading history and initial money holdings, and
strategy is symmetric with respect to agents' types. Let be the trading strategy of a generic
agent with initial money holdings 0 . His date-t strategy t (t1 ; t2 ; t3 ; t4 ) speci es his
bid and o er as a function of his initial money holdings and his encounter history !, and it is
measurable with respect to Bt . The bid (t1 ; t2 ) is the maximum price t1 at which the agent is
willing to buy and the quantity t2 that he is willing to purchase (at price no higher than t1 ) if
he is paired with a seller of his consumption goods. The o er (t3 ; t4 ) represents the price t3 at
which he is willing to sell and the maximum quantity t4 that he is willing to sell at price t3 if
he meets a consumer of his endowment good. Because of the restriction on endowment, t4 1:
As a buyer, the agent has to be able to pay his bid. Let t denote the agent's money holdings at
the beg inning of date t by adopting strategy . Then
t1 (0 ; !) t2 (0 ; !) t (0 ; !):
(4)
Given the agent's initial money holdings 0 , encounter history !, and strategy = ft g1
t=0 ,
his money holdings evolves recursively as follows: 0 (0 ; !) = 0 and, for t 0,
if !t1 = b; t3 (0 ; !) !t2 ; t4 (0 ; !) !t3
if !t1 = s; t1 (0 ; !) !t2 ; t2 (0 ; !) !t3
otherwise
8
<
t (0 ; !) + t3 (0 ; !)!t3
t+1 (0 ; !) = t (0 ; !) !t2 t2 (0 ; !)
:
t (0 ; !)
6
(5)
Let vt (0 ; !) denote the agent's utility achieved at date t by adopting strategy . Then
8
<
vt (0 ; !) =
:
if !t1 = b; t3 (0 ; !) !t2 ; t4 (0 ; !) !t3
if !t1 = s; t1 (0 ; !) !t2 ; t2 (0 ; !) !t3
otherwise
c !t3
u t2 (0 ; !)
0
(6)
Then, strategy overtakes another strategy ^ if for all 0 2 R+ ,
t
hX
lim
inf E
v (0 ; !)
t!1
=0
t
X
=0
i
v ^ (0 ; !) > 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 money holdings
distribution 0 , rational expectation requires that agents' beliefs regarding the c.d.f. of the bid
distribution Bt and the c.d.f. of the o er distribution Ot that prevail during date-t trading coincide
with the actual distributions implied by the strategy. That is, for all x; y 2 R+ ,
Z 1
n
o
Bt (x; y) =
Pt ! j t1 (z; !) x; t2 (z; !) y d 0(z)
(8)
0
Z 1
n
o
Ot (x; y) =
Pt ! j t3 (z; !) x; t4 (z; !) y d 0 (z):
(9)
0
Similarly, the money holdings distribution at the beginning of the of date t is de ned as follows,
for any set A 2 Bt ,
Z 1
n
o
t (A) =
Pt ! j t (z; !) 2 A d 0 (z)
(10)
0
The equilibrium concept we adopt is Bayesian Nash equilibrium with respect to the overtaking
criterion.
1 ; fOt g1 i that satisDefinition. A Bayesian Nash equilibrium is a four tuple h; 0 ; fBt g
t=0
es
t=0
(i) 0 is the initial money holdings distribution in the environment.
1
(ii) No strategy overtakes , given that fBt g1
t=0 and fOt gt=0 characterize trading partners'
decisions.
(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.
We are going to study one particular example of equilibrium, single price equilibrium. In
such an equilibrium, all trades occur at the same price, say p, at all dates, p > 0. 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. We call this a price-p
7
equilibrium. Price-p equilibrium is markovian in the sense that the dependence of agents' strategy
on time and trading history is only through their own current money holdings, despite the dynamic
environment. Formally, de ne the strategy ~ p as follows, for all 0 2 R+ , encounter history ! 2 ,
and t 0,
~tp1 (0 ; !) = p; ~tp2 (0 ; !) = minf~t (0 ; !)=p; 1g
~tp3 (0 ; !) = p; ~tp4 (0 ; !) = 1
(11)
(12)
where ~t (0 ; !) = t~p (0 ; !): Let ~t 2 denote the money holdings distribution at the beginning
of date t induced by strategy ~ p. The bid distribution implied by strategy ~ p are as follows: for
any x; y 2 R+ ,
8
if x < p
< 0
B~t (x; y) = ~t ([0; py])
if x p and y < 1
(13)
:
1
if x p and y 1
p
The o er distribution implied by ~ is stationary and degenerate with mass at price p and quantity
1. That is, for any x; y 2 R+ ,
O~t (x; y) =
8
<
:
if x p
if x p and y < 1
if x p and y 1
0
0
1
(14)
The evolution of the money holdings distribution ~t is speci ed in the next section.
~ 1
In order to show that h~ p ; 0 ; fB~t g1
t=0 ; fOt gt=0 i is an equilibrium, we rst investigate the
properties of the dynamic path of the two distributions relevant to the equilibrium at hand:
the economywide money holdings distribution which determines the bid distribution, and the
distribution of an arbitrary individual's money holdings which helps to de ne the optimality of
the strategy. In the next section, we show both distributions converge asymptotically under a mild
condition on the distribution of initial money holdings. In section 5, we show that the strategy
~ p is optimal.
4. The Convergence of Money Holdings Distributions at Price-p Equilibrium
Given that there is a continuum of nonatomic agents, if all agents adopt strategy ~ p , the
convergence path of the economywide money holdings distribution over time is deterministic.
However, the trading path of a single agent in the economy is random. The probability structure
introduced in section 3 is de ned in terms of the stochastic process of encounters faced by such an
agent. For a generic agent with initial money holdings 0 , the distribution for his possible money
holdings at date t is not necessarily given by the economywide money holdings distribution ~t ,
8
which is the money holdings distribution of a potential trading partner at date t. In order to study
the optimality of strategy ~ p , we need to know the evolution of the money holdings distribution
for a single agent with arbitrary initial money holdings 0 .
In this section, we show that if all agents adopt the strategy ~ p , and if the initial money
holdings distribution 0 satis es a certain condition, then the economywide money holdings distribution converges weakly to a unique geometric distribution at which the economy is stationary.
Furthermore, we show that given the economywide money holdings distribution converges, the
distribution of a generic agent with an arbitrary initial money holdings converges to the aggregate
limit distribution, and the mean of his money holdings converges to the per capita money holdings
in the economy M .
To show the convergence of the economywide money holdings distribution, as a technical intermediate step, we rst show the convergence of the economywide distribution and the distribution
of an individual agent if money is indivisible with unit p. That is, all agents' money holdings
are in multiple units of p. We then use the result to prove the convergence of money holdings
distributions when money is divisible.
We now introduce a notation that will be used throughout the paper. In the trading environment introduced above, the aggregate nominal quantity of money does not change over time.
That is, in an economy with aggregate nominal quantity of money M , the probability measure
that represents the economywide money holdings distribution satis es the aggregation condition,
R1
M
restricted probability space where the aggregation condition holds.
0 d = M . Let be the
R1
That is, M = f j 2 ; 0 d = M g.
4.1. The Indivisible Money Case
Consider an economy where the aggregate nominal quantity of money is M , M > 0, and where
money is indivisible with unit p. In this economy, the support of money holdings distribution
is in the lattice pN of multiples of p. (pN f0; p; 2p; : : :g.) Denote the set of these probability
measures with support in pN by pN , and the subset of probability measures in pN with mean
M to be M
~ p , and given the initial distribution 0 , the
pN . Given that all agents adopt strategy
economywide money holdings distribution evolves deterministically as follows: for any t 0,
t+1 (f0g) = (1
m(t )
1
)t (f0g) + t (fpg)
2
2
(15)
and for all k 1,
t+1 (fkpg) = (1
m(t )
2
1
1
m(t )
)t (fkpg) + t (f(k + 1)pg) +
(f(k
2
2
2 t
9
1)pg)
(16)
P
1
where m(t ) = 1
k=1 t (fkpg) is the measure of agents who have money. The sequence ft gt=0
of money holdings distributions can be obtained by applying (15) and (16) recursively. It is easy
M
to check that if 0 2 M
pN , then at any point of time t 1, t 2 pN .
For technical convenience, we work with a transformation of the probability measure instead
of itself. De ne a mapping L: pN ! [0; 1]1 as follows,
8 2 pN 8k 2 N
Lk () = ([kp; 1)) =
X
j k
(fjpg):
(17)
Obviously, L0 () = 1, Lk () 2 [0; 1] and Lk () Lk+1() for all k 2 N . Let = L(pN ). Then,
for any x 2 , x satis es that x0 = 1, xk 2 [0; 1] and xk xk+1 for all k 2 N . By de nition, L
is a one-to-one aÆne mapping from pN to . The aggregate
real money
balance corresponding
P1
P1
to the distribution 2 M
pN can be written as k=1 k Lk () Lk+1 () = k=1 Lk () = M=p.
De ne S = L(M
pN ). Clearly S is a subset of .
n
S = xjx 2 ;
1
X
k=1
o
xk = M=p :
(18)
The set S is the space we are going to work with primarily in the rst half of this section. It is
easy to show the following (the proof is omitted).
. Both S and
Lemma 1
8 2 [0; 1], x + (1
are convex. That is, for X = S or X =
)y 2 X .
, 8x; y
2X
and
By equations (15) and (16), the law of motion of the transformation of money holdings distribution L() is a mapping T : S ! S such that for all x 2 S ,
8 k 1 Tk (x) = 1 2 x1 xk + 21 xk+1 + x21 xk 1:
(19)
0
t
It is easy to show that T0 (x) = 1 and T (x) 2 S . For any given 0 2 M
pN , x = L(0 ), x =
T (xt 1 ) = L(t ) for all t 1. The following lemma states that the mapping T has a unique xed
point.
. The mapping T has a unique xed point x 2 S such that x = T (
x):
Lemma 2
8k 2 N xk = m k ;
where m
=
10
M=p
:
1 + M=p
(20)
Proof. For all x 2 S , by equation (19), T (x) = x requires that for all k 1,
1 x1
1
x
)xk + xk+1 + 1 xk 1
2
2
2
and x0 = 1. This system of equations has a unique solution x that satis es, for all k 2 N ; xk =
P
P
(x1 )k . Since x 2 S , 1
k = 1
x1 )k = M=p, which implies that x1 = 1+M=p
.
k=1 x
k=1 (
M=p = m
xk = (
The unique xed point x of T given in Lemma 2 corresponds to a geometric money holdings
distribution with parameter m
: = L 1 (x) 2 M
pN . In particular, for all k 2 N , (fkpg) =
k
(1 m
)m
. We want to show that starting from a given initial state x0 , the economy as a
dynamic system evolving according to mapping T converges asymptotically to the steady state
characterized by x. Toward this objective, we construct a Liapunov function that is a function
of the state of the dynamic system. We show that the Liapunov function decreases over time
and asymptotically approaches its minimum. Therefore, by a standard argument of dynamical
systems theory, the economy asymptotically approaches a steady state, which is represented by
the unique xed point of T , x, which does not depend on the initial state.
The Liapunov function we choose to use can be interpreted as the expected hazard rate for
the corresponding distribution. De ne Z : ! R+ , for all x 2 ,
1 (x x )2
X
k
k+1
:
(21)
Z (x) =
x
k
k=0
For technical reasons, we de ne the function Z on the larger space instead of on S . For Z to
be a Liapunov function, it should be continuous in some metric, it should be decreasing along
the trajectory of the system de ned by T , and it should have a unique minimum on S where it is
applied. We show that Z has these properties one by one. Many of the technical proofs are given
in the appendix.
. The function Z is strictly convex on .
Lemma 3
Using Lemma 3, the following lemma shows that the function Z is strictly decreasing along
the trajectory de ned by T .
. For all x 2 S , Z (T (x)) < Z (x), unless x = T (x).
Lemma 4
Proof. De ne mappings : S
k (x) =
! and : S ! as follows: for all x 2 S , k 2 N ,
xk+1
;
x1
0 (x) = 1; k+1 (x) = x1 xk :
11
The measure is a normalized left shift of x, and is a normalized right shift of x. It is easy to
check that (x) 2 and (x) 2 , but neither is necessarily an element of S . Then by (19), T (x)
can be rewritten as a convex combination of x, (x) and (x),
x
1
1 x1
T (x) = 1 (x) + (x) +
x:
2
2
2
Since Z is strictly convex on
by Lemma 3, unless (x) = (x) = x,
x1
1
1 x1
Z ((x)) + Z ((x)) +
Z (x)
2
2
2
1
1 x1
x 1
Z (x) (1 x1 )2 + (1 x1 )2 + x1 Z (x) +
Z (x)
= 1
2 x1
2
2
= Z (x):
Z (T (x)) <
It is easy to verify that (x) = x if and only if x = T (x). Therefore, unless x = T (x), we have
Z (T (x)) < Z (x).
P
Because of the aggregation condition ( 1
k=1 xk = M=p), S is a subset of the complete metric
P1
space (X ; d), where X = fx 2 [0; 1]1 j
xk < 1g and d is the usual `1 -metric associated
with X , for any x; y 2 X ,
k=0
d(x; y) =
1
X
k=0
jxk yk j:
(22)
By standard argument, (S; d) is a complete subspace of (X ; d). The following two lemmas state
that both Z and T are continuous mappings in metric d.
. The function Z is continuous on S .
Lemma 5
. The mapping T is continuous on S .
Lemma 6
The set S we have been working with is unfortunately not compact. To insure the convergence of
the system from some initial state, we introduce a subset of S that is compact and closed under
mapping T .
De ne an ordering relation between two vectors x and y: y dominates x (denoted by x d y)
if and only if for all k 2 N ; xk yk . For a given 2 X , let S be the set of vectors in S that are
dominated by ,
S = fx 2 S j x d g:
(23)
. For every vector 2 X , the set S is compact.
Lemma 7
12
The vector can be any element of X . In particular, let denote the geometric vector de ned
by some 2 (0; 1): for all k 2 N ,
k = k :
(24)
The vector as de ned above is an element of X as well as . Also, it is a xed point of T , i.e.,
T ( ) = . The following lemma states that for , S is closed under T .
. For any x 2 S and any 2 (0; 1), if x d , then T (x) d .
Lemma 8
By Lemma 8, if state x0 satis es the following condition on the initial distribution of money
holdings,
(ID1 ) there exist 2 (0; 1) and t 0 such that T t (x0 ) d
then all the subsequent states of the dynamic system T n (x0 ), for all n t; are dominated by
as well, hence, they are elements of S .
. Consider an environment where money is indivisible with unit p. Suppose
that the initial money holdings distribution 0 is such that x0 = L(0 ) satis es condition ID1 . As
a dynamic system evolving from x0 according to mapping T , this economy converges asymptotically
to the steady state characterized by distribution x, which uniquely satis es T (
x) = x and d(x; 0) =
d(x0 ; 0).
Proposition 1
Proof. Suppose that condition ID1 holds, that is, there exist 2 (0; 1) and t 0 such that
T t (x0 ) d . Then T n (x0 ) 2 S for all n t. By Lemma 7, S is a compact set, and by
Lemma 5, the function Z is continuous on S , hence on S , so Z achieves its minimum on S .
Furthermore, by Lemma 3, Z is strictly convex on S , hence on S , Z has a unique minimum on
S . Last, Z is strictly decreasing along the trajectory of the system de ned by T by Lemma 4.
Therefore, Z is a Liapunov function. With this Liapunov function, we show next the convergence
of the system from the initial state x0 .
n
n 0
From the given x0 , construct a sequence fxn g1
n=1 by applying T recursively, x = T (x ).
Consider the sequence excluding the rst t elements, fxn g1
n=t , which has just been shown to lie
within S . By Lemma 4, the corresponding sequence fZ (xn )g1
n=t is monotonically decreasing.
n
Since S is compact, there exist a subsequence fx k g that converges to some x^ 2 S . Suppose
that x^ is not a xed point of T . Then by Lemma 4, Z (T (^x)) < Z (^x). Since Z is continuous and T
is continuous, there exists Æ > 0 such that for all y satisfying d(^x; y) < Æ, Z (T (y)) < Z (^x). Since
13
fxnk g converges to x^, there exists K such that for all k K , d(^x; xnk ) < Æ, hence, Z (T (xnk )) <
Z (^x), or,
Z (xnk +1 ) < Z (^x):
(25)
But since fZ (xn )g1
^ is the limit of xnk ,
n=t is monotonically decreasing, and since x
Z (xnk +1 ) Z (^x)
(26)
which contradicts (25). Therefore, the limit x^ has to be a xed point of T . Since T has a unique
xed point x in S by Lemma 2, x = x^ 2 S . Hence, for the given initial state x0 , T n(x0 ) ! x as
n ! 1. This strengthened statement, that the entire sequence (rather than only the subsequence
selected above) converges to x, follows from a standard argument involving the Liapunov function
Z.
The convergence of T t (x0 ) to x as t ! 1 in `1 -metric implies that for each k, Tkt (x0 ) ! xk as
t ! 1, which by de nition, implies weak convergence of the corresponding sequence of probability
measures t ) as t ! 1, where is the probability measure corresponding to the xed point
x.
. Consider an environment where money is indivisible with unit p. Suppose
that the initial money holdings distribution 0 is such that x0 = L(0 ) satis es condition ID1 ,
and that all agents adopt strategy ~ p . Then the economywide money holdings distribution ft g1
t=0
converges weakly to the unique geometric distribution .
Corollary 1.1
Next we show that the money holdings distribution of a generic agent with an arbitrary initial
money holdings converges, given that the economywide money holdings distribution converges.
Suppose that all agents adopt strategy ~ p , and that the initial state x0 = L(0 ) satis es
condition ID1 , that is, the economywide money holdings distribution converges to a geometric
distribution de ned on pN . Consider an agent with initial money holdings 0 = lp, l 2 N . Let
'lt 2 pN represent the probability distribution of the agent's date-t money holdings ~t (lp; !),
that is, for any set D 2 Bt , 'lt (D) = Pt f! j ~t (lp; !) 2 D g. Then, 'l0 (flpg) = 1 and 'l0 (fkpg) = 0
for all k 6= l. For any 'lt 2 pN , de ne ylt = L('lt ), hence, ylt 2 . Obviously, 'lt and ylt
uniquely determine each other. As xt represents the date-t aggregate state of the economy, ylt
represents the distribution of the date-t personal state for the agent with initial money holdings
lp. Given that the distribution of money holdings in the population follows the path of fxt g1
t=0 ,
the distribution of the agent's personal state from one date to the next is a mapping U : !
14
such that for any arbitrary individual state y 2 , for any aggregate state x 2 , and for all k 1,
1 x1
1
x
Uk (y; x) =
yk + yk+1 + 1 yk 1
(27)
2
2
2
and U0 (y; x) = 1, where x1 is the measure of agents holding money at the date (hence able to
purchase) as de ned above, which is taken as given by each agent. That is, if ylt represents
the distribution of the agent's date-t state and xt represents the aggregate date-t state, then the
distribution of his date-(t + 1) state is given by yl(t+1) = U (ylt ; xt ). The following proposition
states that each agent's money holdings probability distribution converges to the same geometric
distribution as the economywide money holdings sample distribution does. The initial money
holdings of an agent do not matter in the limit.
. Consider an environment where money is indivisible with unit p. Suppose
that the initial money holdings distribution 0 is such that x0 = L(0 ) satis es condition ID1 , and
that all agents adopt strategy ~ p . Then the money holdings distribution of a generic agent with
initial money holdings 0 = lp, f'lt g1
t=0 , converges weakly to the same aggregate limit regardless
of 0 .
Proposition 2
Proof. Consider a trader with initial money holdings 0 = lp, l
~ p.
2 N who, as everyone else in
the economy, adopts strategy
To prove that the trader's money holdings ~t (0 ; !) converges
weakly to the aggregate limit, we need to show that for all k, jyklt xk j ! 0 as t ! 1. Given
that the aggregate state xt converges to x, i.e., for any k 1, xtk ! xk as t ! 1, it is suÆcient
to show that for all k, jyklt xtk j ! 0 as t ! 1. We show this by induction on k.
For any t 0, given that the date-t aggregate state xt and the distribution of the agent's
personal state ylt , by equations (19) and (27), the corresponding date-(t + 1) states are de ned
as follows, for any k 1,
xt
1 xt1 t 1 t
xk + xk+1 + 1 xtk 1
xtk+1 =
2
2
2
1 xt1 lt 1 lt
xt1 lt
l(t+1)
yk
=
y + y + y :
2 k 2 k+1 2 k 1
The di erence of the above two equations is, for any k 1,
1 xt1 lt t
1
xt
ykl(t+1) xtk+1 =
(yk xk ) + (yklt+1 xtk+1 ) + 1 (yklt 1 xtk 1 ):
(28)
2
2
2
Then, given that y0lt = xt0 = 1,
1
1
1
X
X
1 X
l(t+1)
t
+1
l
(
t
+1)
t
+1
lt
t
d(y
; x ) = jyk
xk j
j
yk xk j + jyklt xtk j
2 k=1
k=1
k=2
15
or
1 lt t
jy x1 j:
(29)
2 1
That is, fd(ylt ; xt )g1
t=0 is a weakly decreasing sequence, and since it is bounded below by zero, it
has a limit . Then, equation (29) implies that jy1lt xt1 j ! 0 as t ! 1.
Now suppose that for all j k, jyjlt xtj j ! 0. We want to show that jyklt+1 xtk+1j ! 0.
P
P
lt xt j ! , we have
By induction hypothesis, kj=1 jyjlt xtj j ! 0. Since d(ylt ; xt ) = 1
j
j =1 jyj
P1
lt
t
xj j ! . Then,
j =k+1 jyj
1
1
X
X
jyjl(t+1) xtj+1 j
jyjlt xtj j ! 0 as t ! 1:
(30)
d(yl(t+1) ; xt+1 ) d(ylt ; xt )
j =k+1
By equation (28),
1
X
jyjl(t+1)
j =k+1
j =k+1
xtj+1 j
or
1 lt
jy
2 k+1
xtk+1 j
1
X
j =k+1
1
X
j =k+1
t
jyjlt xtj j 21 jyklt+1 xtk+1j + x21 jyklt xtk j
jyjl(t+1) xtj+1 j
1
X
j =k+1
jyjlt xtj j + jyklt xtk j:
(31)
Applying (30) and induction hypothesis to (31), we have jyklt+1 xtk+1 j ! 0 as t ! 1. Hence, by
induction, for all k 1, jyklt xtk j ! 0 as t ! 1.
We have studied an economy with the following features: money is indivisible with unit p,
the aggregate nominal quantity of money is M , the initial money holdings distribution 0 is
such that x0 = L(0 ) satis es condition ID1 , and all agents adopt strategy ~ p . We have shown
that, in such an environment, both the economywide money holdings sample distribution and
the money holdings probability distribution of a generic agent with an arbitrary initial money
holdings converge weakly to the unique geometric distribution de ned on support pN with mean
M.
4.2. The Divisible Money Case
Now, we return to the environment introduced in section 2 where aggregate nominal quantity
of money is M , and where money is perfectly divisible. We rst de ne the law of motion of the
economywide money holdings distribution given that all agents adopt strategy ~ p .
Suppose that the economywide money holdings distribution is at the beginning of a date.
Parallel to the L transformation made in the indivisible money case, we need to know only the
16
evolution of the money holdings distribution de ned on sets of the form [y; 1) rather than on any
arbitrary set D R+ . Let ()[y; 1) for any y 2 R+ be the measure of agents whose after-trade
money holdings are at least y conditional on being a seller. To characterize ()[y; 1) for any
y p, note that a seller's higher-than-y money holdings after trade can result from either of
two types of transaction. The seller can begin with a before-trade money holdings no lower than
y b , and acquire b 2 [0; p) units from a buyer with only b units of money by selling b =p units
of his endowment. He can also start with at least y p units of money before trade, and acquire
p units from a buyer with at least p units of money by selling 1 unit of his endowment. That is,
8y p ()[y; 1) =
Z
[0;p)
[y b ; 1)d(b ) + [p; 1)[y p; 1):
(32)
If y < p, then a seller may have more than y units of money after trade from either of two kinds of
transaction: starting with at least y b units before trade, and acquiring b 2 [0; y) units from a
buyer with only b units of money; or trading with a buyer with at least y units of money. That
is,
Z
8y 2 (0; p) ()[y; 1) =
[y b ; 1)d(b ) + [y; 1):
(33)
[0;y)
Similarly, for any y 2 R+ , let ()[y; 1) be the measure of agents whose after-trade money
holdings are at least y conditional on being a buyer. In particular, a buyer's after-trade money
holdings is always nonnegative, that is,
()[0; 1) = 1:
(34)
For any y > 0, a buyer can only have a higher-than-y money holdings after trade if he has more
than y + p units of money before trade and spends p units to buy consumption goods. If he has
less than p units, he will spend all of it and reduce his money holdings to 0. That is,
8y > 0 ()[y; 1) = [y + p; 1):
(35)
Then, since half of the agents are buyers and half are sellers, the evolution of the money
holdings distribution, T : ! , is given by the following, for all 2 ,
Z
i
1h
[y b ; 1)d(b ) + [p; 1)[y p; 1) + [y + p; 1) (36)
2 [0;p)
Z
i
1h
8y 2 (0; p) T ()[y; 1) = 2
[y b ; 1)d(b ) + [y; 1) + [y + p; 1) :
(37)
[0;y)
8y p
T ()[y; 1) =
Obviously, T ()[0; 1) = 1. It is easy to check that if 2 M ; then T () 2 M : Equations (36)
and (37) together de ne the evolution of the economywide money holdings distribution, which
17
is t = T t (0 ) at date t. Note that when the support of is the p-lattice pN , for any k 2 N ,
T ()[kp; 1) de ned by (36) and (37) coincides with Tk (L()) de ned by (19) in the indivisible
money case.
The dominance relationship d de ned on in last subsection is the restriction to pN of the
relationship of stochastic dominance de ned on : for any 1 ; 2 2 , 1 2 if and only if for
any y 2 R+ , 1 [y; 1) 2 [y; 1). The following lemma states that the mapping T preserves the
relationship of stochastic dominance.
. For any 1 ; 2 2 , if 1 2 , then T (1 ) T (2 ).
Lemma 9
Divide the agents according to their money holdings into two groups: those whose money
holdings are integer multiples of p (on-lattice) and those whose money holdings are between
integer multiples of p (o -lattice). It can be shown that over the dynamic path of economy, the
measure of the on-lattice group is increasing in time. Intuitively, in most pairwise trades, buyers
and sellers either keep their status (on- or o -lattice) or exchange their status. These trades do not
a ect the size of either group. But when a buyer and a seller, both with money holdings strictly
between 0 and p (o -lattice), meet, the buyer spends all his money on the preferred consumption
good, and hence reduces his after-trade money holdings to 0. In other words, trading among this
group of agents results half of the group moving from o -lattice status to on-lattice status. The
following lemma summarizes this intuition and states a quantitative implication.
. For any 2 , fT n ()(pN )g1
0 is a nondecreasing sequence, and satis es the
following relationship: for all n 0,
Lemma 10
h
i2
T n+1 ()(pN ) (pN ) + 2 (2n+1) (np; (n + 1)p) :
(38)
Proof. Consider all the possible pairwise meetings between a buyer with b units of money and
a seller with s units of money where trade occurs. Let b0 and s0 denote the corresponding
after-trade money holdings.
(i) If b 2 pN and s 2 pN , then, b0 2 pN and s0 2 pN .
(ii) If b p, then z0 2 pN i z 2 pN where z = b; s.
(iii) If b 2 (0; p) and s 2 pN , then, b0 = 0 2 pN and s0 62 pN .
18
(iv) If b 2 (0; p) and s 62 pN , then, b0 = 0 2 pN and s0 62 pN almost surely.
Among these four types of active trade, the rst three do not change the measure of agents with
money holdings on the lattice pN . The last type of trade moves the buyer's money holdings onto
the lattice. A subset of the last type of trades occur among agents (both buyers and sellers) with
money holdings in (0; p). Therefore,
i2
1h
T ()(pN ) (pN ) (0; p) 0
(39)
2
that is, T ()(pN ) (pN ). Hence, for any 2 , fT n ()(pN )g1
0 is nondecreasing in n.
To prove the second part of the lemma, we rst prove the following claim: For any 2 ,
n 1 and n,
T () (n )p; (n + 1)p
2 (np; (n + 1)p):
(40)
We prove the claim by induction on . When = 0, (40) holds with equality. Suppose the
claim holds for = k < n, consider the case when = k + 1. Given the matching technology
and the strategy, half of the agents with money holdings in ((n k)p; (n k + 1)p) are buyers
(note that n k 1) whose after-trade money holdings will be in ((n k 1)p; (n k)p) =
((n (k + 1))p; (n (k + 1) + 1)p). Hence,
i
1h
T k+1 () (n (k + 1))p; (n (k + 1) + 1)p T k () (n k)p; (n k + 1)p
2
i
1h k
2 2 (np; (n + 1)p) = 2 (k+1) (np; (n + 1)p)
where the second inequality is due to the induction hypothesis. That is, (40) holds for = k + 1.
By induction, (40) holds for all n.
By the above claim, for all n 0,
T n ()(0; p) 2 n (np; (n + 1)p):
(41)
By (39), (41) and the fact that T n ()(pN ) is nondecreasing in n, for all n 0, for all 2 ,
i2
h
i2
1h
T n+1 ()(pN ) T n ()(pN ) + T n ()(0; p) (pN ) + 2 (2n+1) (np; (n + 1)p) :
2
That is, (38) holds.
Applying Lemma 10, we show an even stronger result: asymptotically, all agents' money
holdings will be integer multiples of p.
. For any 2 M , T n ()(pN )
Lemma 11
! 1 as n ! 1.
19
For an arbitrary probability measure 2 , de ne + be the probability measure resulting
from right shifting all the probability mass in interval ((n 1)p; np) to the point np on the p-lattice
for all n 1, that is,
+ f0g = f0g;
i
8n 1 +fnpg = (n 1)p; np :
(42)
Similarly, let be the probability measure resulting from left shifting all the probability mass
in interval (np; (n + 1)p) to the point np on the p-lattice for all n 0, that is,
8n 0
h
fnpg = np; (n + 1)p :
The supports of + and are within pN , and + ;
(43)
2 . By de nition,
+:
(44)
Let g denote the probability measure de ned on R+ but with g (pN ) = 1 that corresponds to
the geometric vector for some 2 (0; 1) introduced in last subsection, that is, for any k 1
and x 2 [0; p),
g [kp x; 1) = g [kp; 1) = k
g 2 . It is easy to check that g is a xed point of T , that is, T (g ) = g . If + is stochastically
dominated by some g where 2 (0; 1), given that the mapping T preserves stochastic dominance
(Lemma 9), we have
T ( ) T () T (+ ) g :
(45)
. For an arbitrary 2 , if there exists 2 (0; 1) such that + g , then for any
n 0, [T n ()]+ g .
Lemma 12
Now we are ready to show the convergence of the economywide money holdings distribution.
The condition on the initial money holdings distribution 0 that guarantees the convergence of
the distribution is formally very similar to condition ID1 .8
(ID2 ) There exist 2 (0; 1) and t > 0 such that [T t (0 )]+ g .
The following proposition states that, under this condition, the limit distribution is the probability
measure corresponding to the geometric distribution with mass on the p-lattice pN and mean M .
That is, the limit distribution is gm , where m = 1+MM=p
=p .
8 However the mapping T is de ned in terms of p, which is an exogenous parameter of the indivisible money
economy but an endogenous price in the divisible money economy. Thus, despite their formal similarity, condition
ID2 does not relate the initial money holdings distribution to exogenous parameters of the economy as ID1 does.
A suÆcient condition ID3 for ID2 , to be formulated below, will avoid reference to the price or other endogenous
quantities.
20
. Suppose that the initial money holdings distribution 0 2 M satis es
condition ID2 . Then the sequence of money holdings distributions evolving from 0 according to
mapping T converges weakly to the steady state characterized by geometric distribution gm : That
is, T n (0 ) ) gm as n ! 1.
Proposition 3
Proof. Take an arbitrary initial distribution 0 2 M such that there exist 2 (0; 1) and
t 0, satisfying [T t (0 )]+ g . Without loss of generality, assume that +0 g . Let l1 = 0 and
n
1
n1 = 0. We construct a subsequence fT lk +nk (0 )g1
n=0 of the sequence fT (0 )gn=0 by repeating
the following, three-step procedure for all k 2.
(i) By Lemma 11, there exists nk nk 1 + lk 1 such that
1
T nk (0 )(pN ) > 1
:
kp
(ii) Let +k [T nk (0 )]+ , and k
Hence,
[T nk (0)] . By Lemma 12, given that +0 g , +k g :
k
T nk (0 ) +k g :
(46)
R
R
Furthermore, de ne Mk+ = 01 d+k , and Mk = 01 dk . Since T nk (0 )(pN ) < kp1 (pN
R
denotes the complement of pN ), and 01 dT nk (0 ) = M , we have
Mk+ < M + 1=k and Mk > M
De ne
1=k:
(47)
Mk+ =p
Mk =p
and mk
:
+
1 + Mk =p
1 + Mk =p
Consider +k and k as the initial distributions of an economy where money is indivisible
with unit p. Then, given that +k g and k g , both L(+k ) and L(k ) satisfy condition
+
ID1 . By Corollary 1.1, T n (+k ) weakly converges to gmk and T n(k ) weakly converges to
gmk as n ! 1. Hence, for = 6=(kp), for all j 0,
m+k
9 lkj+ 8 n lkj+ T n(+k )[jp; 1) [m+k ]j < =6 = 1=(kp):
Since the mean of T n(+k ) is Mk+ and m+k < 1, there exists J such that for all j
T n (+k )[jp; 1) < =12 and (m+k )j < =12. Take lk+ = maxflk0 ; : : : ; lkJ g. Then,
8 n lk+ 8 j 0 T n(+k )[jp; 1) [m+k ]j < =6 = 1=(kp):
21
J,
By the same argument, there exists lk such that
8 n lk 8 j 0 T n(k )[jp; 1) [mk ]j < =6 = 1=(kp):
Take lk = maxflk+ ; lk g. Then, by (46),
T lk (k ) T lk +nk (0 ) T lk (+k )
and for all j 0,
T lk (+k )[jp; 1) [m+k ]j < 1=(kp) and T lk (k )[jp; 1) [mk ]j < 1=(kp):
(48)
(49)
(iii) Increase k by 1. Go back to step (i).
n
1
Now we have a subsequence fT lk +nk (0 )g1
n=0 of the sequence fT (0 )gn=0 that satis es (48)
and (49). We want to show that this subsequence weakly converges to gm . By (48), for all
y 2 R+ ,
T lk (k )[y; 1) T lk +nk (0 )[y; 1) T lk (+k )[y; 1)
(50)
By (47), it is easy to check that for any j 0,
(m+k )j
(m )j < 1=(kp) and
(mk )j
(m )j < 1=(kp):
(51)
Therefore, for any " > 0, k 6=("p), for any y > 0, write y = jp x with j 1, and x 2 [0; p),
T lk +nk (0 )[y; 1) gm [y; 1) = T lk +nk (0 )[y; 1) (m )j
T lk +nk (0 )[y; 1) T lk (+k )[y; 1) + T lk (+k )[y; 1) (m )j
T lk (k )[y; 1) T lk (+k )[y; 1) + T lk (+k )[y; 1) (m+k )j + (m+k )j
T lk (k )[y; 1) (mk )j + 2 T lk (+k )[y; 1) (m+k )j + (mk )j
6=(kp) < ":
(m )j
(m )j + 2 (m+k )j
(m )j
The second inequality is due to (50) and the last is due to (49) and (51) given that T lk (k )[y; 1) =
T lk (k )[jp; 1) and T lk (+k )[y; 1) = T lk (+k )[jp; 1). That is, the subsequence fT lk +nk (0 )g1
n=0
m
weakly converges to g .
The foregoing argument can be generalized as follows. Consider an arbitrary number sequence
i0 < i1 < i2 < : : : and consider the sequence fT ij (0 )g1
j =0 . This sequence has a subsequence
m
that converges weakly to g . Since every subsequence of fT n (0 )g1
n=0 has a subsequence that
converges weakly to gm , the entire sequence must converge weakly to gm .
The requirement that the initial distribution 0 satis es condition ID2 is fairly weak. The
following lemma gives a class of initial distributions that satisfy the condition.
22
. For any given equilibrium price p > 0, if there exist Jp > 0 and p
that for all j > Jp , 0 (jp; 1) jp+1 , then 0 satis es condition ID2 .
Lemma 13
Proof. To prove condition ID2 holds, we need to show that there exist
such that for all j > 0, T t (0 )(jp; 1) j +1.
2 (0; 1) such
2 (0; 1) and t > 0
Let K = 0 if 0 f0g > 0, and K = minfj j 0 ((j 1)p; jp ] 6= 0g otherwise. Then, 0 (jp; 1) = 1
for all j < K . We rst show by induction on K a claim: that T K (0 )f0g > 0. For K=0, the
claim holds automatically. Suppose that it holds for K = n, that is, 0 ((n 1)p; np ] 6= 0 and
0 (jp; 1) = 1 for all j < n implies that T n (0 )f0g > 0. Consider the case when K = n + 1.
Given that 0 (np; (n + 1)p ] 6= 0 and 0 (jp; 1) = 1 for all j < n + 1, by the de nition of T given
in equations (36) and (37),
i
i
1
T (0 ) (n 1)p; np = 0 np; (n + 1)p > 0:
2
Applying the induction hypothesis, we have T n+1 (0 )f0g > 0. That is, the claim holds.
Given that 0 (jp; 1) jp+1 for all j > Jp , applying equations (36) recursively, we have
T K (0 )(jp; 1) < jp+1 for all 8 j > Jp + K .
By the above claim, T K (0 )f0g > 0, hence T K (0 )(0; 1) < 1. Then, there exists 2 [ p ; 1)
such that T K (0 )(0; 1) Jp+K +1. For j Jp + K , T K (0 )(jp; 1) T K (0 )(0; 1)
Jp+K +1 j +1. For j > Jp + K , T K (0 )(jp; 1) jp+1 j +1 since p . That is, for all
j 2 N , T K (0 )(jp; 1) j +1. Therefore, [T K (0 )]+ g , or equivalently, 0 satis es condition
ID2 .
Condition in Lemma 13 is more transparent than ID2 given that it is expressed directly in
terms of the initial distribution 0 . However, it is still cumbersome since it depends on equilibrium
price p, which is endogenous. In fact, if the condition holds for one particular price, it holds for
any other price. The following proposition exploits this feature and gives a suÆcient condition
for ID2 that depends only on exogenous parameters.
. If the initial money holdings distribution 0 has a tail that is stochastically
dominated by the tail of a geometric distribution, that is, if
Proposition 4
(ID3 ) there exist J > 0 and
2 (0; 1) such that for all j > J; 0 (j; 1)
then 0 satis es condition ID2 .
23
j +1
Proof. We need to show only that if 0 satis es condition ID3 , then for any p > 0, there exist
Jp > 0 and p 2 (0; 1) such that for all j > Jp , 0 (jp; 1) jp+1 . Then, by Lemma 13, condition
ID2 holds.
Take an arbitrary p > 0. First, note that if p 1, take Jp = J and p = . Then by ID3 , for
all j > Jp , 0 (jp; 1) 0 (j; 1) j +1 = jp+1 .
Next, consider the nontrivial case 0 < p < 1. De ne Jp to be any integer such that
Jp
J +1
and
p
p
J +1
= 1=x where x = p :
pJp
Obviously, p 2 ( ; 1) since x > 1. For any j > Jp , take kj to the integer such that kj =
maxfk j jp k g. That is, kj jp < kj + 1. The rst inequality implies that 0 (jp; 1)
0 (kj ; 1): The second inequality leads to
kj + 1 > jp > Jp p
J +1
p = J +1
p
that is, kj > J . Then, by condition ID3 ,
0 (kj ; 1)
Since j > Jp ,
Jp +1
Jp
kj +1
=(
x )kj +1
p
(
> j +1
j . Therefore, given that
0 (jp; 1) 0 (kj ; 1)
p
x )jp
p
+1 jp
Jp
pJp
=
p
=
+1 j
Jp
Jp
p
:
< 1,
+1 j
Jp
Jp
p
That is, the claim holds: for all j > Jp , 0 (jp; 1)
<
j
p
+1 j
j
=
j +1 :
p
j +1 .
p
As a practical matter, economists are not likely to nd condition ID3 restrictive. There are
at least two classes of initial money holdings distributions 0 satisfy the condition. One class
of distributions consists of those with nite support (that is, there is a Y > 0 such that for
all y Y , 0 [y; 1) = 0). Probability measures with nite support are dense in the space of
probability simplex . The other class of distributions includes those that are results of injecting
money into a steady state economy with a geometric distribution of money holdings by giving
uniformly bounded amount to agents whose money holdings are below some particular level (i.e.,
\poor" people).
We can conclude now that if the initial money holdings distribution satis es condition ID3
and if all agents adopt strategy ~ p, then by Proposition 3, the economywide money holdings
distribution converges weakly to a unique geometric distribution at which the environment is
stationary.
24
Given that the economywide money holdings sample distribution converges, in what follows,
we apply the result of the indivisible money case and show that the probability distribution of
money holdings of a generic agent with an arbitrary initial money holdings converges to the
aggregate limit distribution gm .
Consider an agent with an initial money holdings 0 . Let 't 0 2 denote the probability
distribution of the agent's date-t money holdings ~t (0 ; !), that is, for any set D 2 Bt , 't 0 (D) =
Ptf! j ~t (0 ; !) 2 D g. Then '00 is degenerate and satis es that for all y 0 , '00 [y; 1) = 1, and
for all y > 0 , '00 [y; 1) = 0. Similar to the evolution of the economywide distribution (which
follows the path of ft g1
t=0 given by (36) and (37)), the evolution of the individual agent's money
holdings distribution from one date to the next is a mapping U : ! such that for any
individual distribution ' 2 and any economywide distribution 2 , for all y p,
Z
i
1h
U ('; )[y; 1) =
'[y b ; 1)d(b ) + [p; 1)'[y p; 1) + '[y + p; 1)
(52)
2 [0;p)
and for all y 2 (0; p),
Z
1h
U ('; )[y; 1) =
'[y
2 [0;y)
i
b ; 1)d(b ) + [y; 1) + '[y + p; 1) :
(53)
Note that if the support of is the p-lattice pN , then for any k 2 N , U ('; )[kp; 1) de ned in
(52) and (53) coincides with Uk (L('); L()) de ned by (27) in the indivisible money case. Then,
given initial individual money holdings distribution '00 and initial economywide distribution 0 ,
0
0
t
the sequence f't 0 g1
t=0 is recursively de ned: for all t 0, 't+1 = U ('t ; t ) where t = T (0 )
given by (36) and (37).
The following proposition shows that in a divisible money environment, an individual agent's
money holdings distribution, regardless of his initial money holdings, converges to the same geometric distribution as the economywide money holdings distribution does. Similar to the proof
of Proposition 3, we bound the money holdings distribution by two distributions in indivisible
money environments, and then applying Proposition 2.
. Consider any p > 0. Suppose that the economywide initial money holdings
distribution 0 2 M satis es condition ID3 , and that all agents adopt strategy ~ p . Then the
money holdings distribution of a generic agent with initial money holdings 0 , f't 0 g1
t=0 , converges
weakly to the same aggregate limit distribution gm regardless of his initial money holdings 0 .
Proposition 5
Proof. Consider an agent with an arbitrary initial money holdings 0 . By Proposition 3, the
economywide money holdings distribution converges weakly to the geometric distribution gm .
25
Let +n = [T n (0 )]+ , n = [T n (0 )] , and let Mn+ , Mn , m+n and mn be de ned the same way as
in the proof of Proposition 3. Then for any " > 0, there exists t 0 such that jm+t m j < "=6
and jmt m j < "=6, which implies that for any j 1,
j(m+t )j (m)j j < "=6 and j(mt )j (m)j j < "=6:
(54)
By the evolution of the individual's money holdings distribution (52) and (53), it is easy to check
the following claim holds: For any '1 ; '2 2 and 1 ; 2 2 , if '1 '2 and 1 2 , then for
all n 0, U n ('1 ; 1 ) U n ('2 ; 2 ). For the t chosen above, we have
['t 0 ]
't 0 ['t 0 ]+ and t t +t
where [ ]+ and [ ] are de ned as in (42) and (43). Applying the above claim, for all n 0,
U n(['t 0 ] ; t ) U n+t ('00 ; 0 ) U n(['t 0 ]+ ; +t )
which implies that for all y 2 R+ ,
U n (['t 0 ] ; t )[y; 1) U n+t ('00 ; 0 )[y; 1) U n (['t 0 ]+ ; +t )[y; 1)
(55)
As noted earlier, both U (['t 0 ] ; t ) and U (['t 0 ]+ ; +t ) can be interpreted as the law of motion of the individual agent's money holdings distribution given by (27) in an indivisible money
environment, where the individual agent's initial money holdings distribution is given by ['t 0 ]
and ['t 0 ]+ respectively, and the economywide initial distribution is given by t and +t respectively. It is easy to verify the hypothesis that the economywide initial money holdings distribution satis es condition ID3 , hence it satis es condition ID2 , implies that xt = L(t ) and
x+t = L(+t ) satisfy condition ID1 . Then, by applying Proposition 2, U n (['t 0 ] ; t ) ) gmt and
+
U n (['t 0 ]+ ; +t ) ) gmt as n ! 1. (Note that, the proof of Proposition 2 does not require the
individual agent's initial distribution to be degenerate). With the same argument as in the proof
of Proposition 3, then, there exists n^ 0 such that for all n n^ , for all y = jp x with j 1
and x 2 [0; p),
U n (['t 0 ] ; t )[y; 1) (mt )j < "=6 and
U n (['t 0 ]+ ; +t )[y; 1) (m+t )j < "=6:
(56)
This is because U n(['t 0 ] ; t )[y; 1) = U n (['t 0 ] ; t )[jp; 1) and U n (['t 0 ]+ ; +t )[y; 1) =
U n (['t 0 ]+ ; +t )[jp; 1). By (54) and (56), for all n n^ ,
U n (['t 0 ] ; t )[y; 1) (m )j < "=3 and
26
U n (['t 0 ]+ ; +t )[y; 1) (m )j < "=3
(57)
Therefore, for any n n^ , for y = jp x with j 1 and x 2 [0; p),
U n+t ('00 ; 0 )[y; 1) gm [y; 1) = U n+t ('00 ; 0 )[y; 1) (m )j
U n+t('00 ; 0)[y; 1) U n(['t 0 ]+; +t )[y; 1) + U n(['t 0 ]+; +t )[y; 1) (m)j
U n(['t 0 ] ; t )[y; 1) U n(['t 0 ]+; +t )[y; 1) + U n(['t 0 ]+ ; +t )[y; 1) (m )j
U n(['t 0 ] ; t )[y; 1) (m )j + 2 U n(['t 0 ]+; +t )[y; 1) (m )j
< ":
The second inequality is due to (55) and the last one is due to (57). Hence, regardless of the
agent's initial money holdings 0 , U n ('00 ; 0 )[y; 1) ) gm as n ! 1.
The weak convergence of the random variable ~t (0 ; !) established in Proposition 5 is now
shown to imply the convergence of E[~t (0 ; !)] to the aggregate mean money holdings M .
. Consider any p > 0. Suppose that the economywide initial money holdings
distribution 0 2 M satis es condition ID3 , and that all agents adopt strategy ~ p . Then the
expected money holdings of a generic agent adopting strategy ~ p converges to the per capita money
holdings M regardless of his initial money holdings 0 . That is, for any 0 2 N ,
Proposition 6
lim E[~t (0 ; !)] = M :
t!1
(58)
Proof. Consider a generic agent with initial money holdings 0 = lp + x, l 2 N and x 2 [0; p), who,
as everyone else in the economy, adopts strategy ~ p . Suppose the economywide initial distribution
0 satis es condition ID3 , hence it satis es condition ID2 . We rst show that there exists a date tl
such that the sequence of distributions of the trader's money holdings from tl on, f't 0 (0 ; !)g1
t=tl ,
is dominated by a geometric distribution. Note that since condition ID2 is satis ed, by Lemma
0
12, there exist t0 and 0 2 (0; 1) such that for all t t0 , [T t (0 )]+ g .
Given that the agent's initial money holdings is 0 , for all y 0 , '00 [y; 1) = 1, and for
all y > 0 , '00 [y; 1) = 0. By the law of motion (52) and (53), after l + 1 repeated operations
0 f0g = U l+1 ('0 ; )f0g > 0, and for all y > (2l + 1)p + x, '0 [y; 1) = 0.
of U on '00 , 'l+1
0
0
l+1
0
0
t
Then, for all t > l + 1, 't f0g = U ('0 ; 0 )f0g > 0, and for all y > (l + t)p + x, 't 0 [y; 1) =
U t ('00 ; 0 )[y; 1) = 0. Or equivalently, ['t 0 ]+ [p; 1) < 1, and for all j > l +1+ t, ['t 0 ]+ [jp; 1) =
0. Take tl = maxfl + 1; t0 g. Choose l 2 [0; 1) such that ['t 0 ]+ [p; 1) < (l )l+1+tl . Since
['t 0 ]+ [jp; 1) is decreasing in j , for j l +1+ tl , ['t 0 ]+ [jp; 1) ['t 0 ]+ [p; 1) (l )l+1+tl (l )j .
27
For j > l + 1 + tl , ['t 0 ]+ [jp; 1) = 0 (l )j . Therefore, ['t 0 ]+ gl . Given that 0 l and
tl t0 , [tl ]+ = [T tl (0 )]+ g0 gl . That is, both [tl ]+ and ['tl0 ]+ are dominated by gl .
Then, by a similar argument as in the proof of Proposition 5, for all t tl ,
't 0 = U t ('00 ; 0 ) = U t tl ('tl0 ; tl ) U t tl (['tl0 ]+ ; [tl ]+ ) gl :
A random variable with geometric distribution corresponding to gl is integrable. Because the
distributions of the sequence of trader's money holdings from tl on, f~t (0 ; !)g1
t=tl , is dominated
by the same geometric distribution, ~t (0 ; !) is uniformly integrable for t tl . Then, by Theorem
25.12 (Billingsley 1995), weak convergence of 't 0 to the aggregate limit distribution which is
geometric with mean M by Proposition 5, implies that the expectation of ~t (0 ; !) converges to
the same mean M .
This concludes our investigation of the distributions in the economy, given the all agents adopt
the presumed optimal strategy ~ p . To summarize, if the economywide initial money holdings
distribution 0 satis es condition ID3 , then the economywide money holdings sample distribution
as well as each individual agent's money holdings probability distribution converges. Furthermore,
the mean money holdings of each individual agent converges to the per capita money holdings
M in the economy.
5. The Existence of Price-p Equilibrium
In this section, we show that if the initial distribution 0 satis es condition ID3 , the price-p
equilibrium de ned in Section 3 is a Bayesian Nash equilibrium. In particular, we show that for
an arbitrary agent, given that all other agents in the economy adopt the strategy ~ p de ned in
(11) and (12),9 it is optimal for the agent to adopt strategy ~ p as well. That is, no strategy
overtakes ~ p .
Consider a generic agent of any type. Suppose that the agent's initial money holdings is
0 . Since 0 is xed and is taken as given when we compare di erent strategies, for notational
convenience, we will suppress 0 as an argument of all functions such as and t in the rest of
the section, and write them as functions of ! alone. Also note that given all the other agents
adopt strategy ~ p and the agent in question has measure 0, although his trading history will be
determined by his strategy , his encounter history ! is independent of the strategy he adopts.
Let t (!) denote the agent's money holdings at the beginning of date t with encounter history
! if he adopts strategy , 0 (!) = 0 . De ne the agent's achievement function at the beginning
9 Hence, the bid and o er distributions are given by fB~t g1
~ 1
t=0 and fOt gt=0 de ned in (13) and (14).
28
of date t, At : ! R+ , to be the sum of his total utility up to date t and the future utility that
will be bought by the money accumulated up to date t, t , given that the agent buys his future
consumption goods at price p. That is, for any encounter history ! 2 ,
At (!) =
1
t
X
=0
v (!) +
t (!)
u
p
(59)
where v (!) is de ned in (6), and t (!) is de ned recursively by (5). For notational convenience,
de ne for all t 0,
p
p
p
A~t = At~ ;
v~t = vt~ ;
~t = t~ :
(60)
By the de nition of the overtaking criterion (7), given that all other agents adopt strategy
~ p , any strategy that speci es at any point of time to o er at a price lower than p is obviously
overtaken by the corresponding strategy which replaces the lower o er price by p. This is because
any transaction that would occur at price lower than p would have occurred at price p. Hence, the
seller in transaction would have been better o by obtaining more money while su ering the same
endowment loss or by obtaining the same amount of money while su ering the less endowment
loss. In the rest of the paper when we compare strategies with ~ p , we exclude those strategies
with o er price lower than p at any point of time. The following lemma shows that strategy ~ p
is associated with the highest achievement function of any strategy.
. Consider any p > 0. If all other agents adopt strategy ~ p, then for an arbitrary
agent facing any encounter history ! 2 , adopting a strategy , for all t 0, At (!) A~t (!):
Lemma 14
2 .
~
Obviously, 0 = A0 (!) = 0 u=p. Assume that the lemma holds up to date t, we compare an
arbitrary strategy with ~ p at the beginning of date t + 1, t 0.
Proof. We prove the lemma by induction. Consider an agent of type i with a history !
A (!)
(i) !t1 = s; !t2 = p; and !t3 = 1. In this case, regardless of the agent's strategy (including ~ p ),
At+1 (!) = At (!).
(ii) !t1 = b; !t2 = p; and !t3 = 0: This is a case that the buyer encountered has no money. So
regardless of the strategy (including ~ p ), no trade can take place. Consequently At+1 (!) =
At (!).
(iii) !t1 = b; !t2 = p; and !t3 = y 2 (0; 1]. Adopting strategy ~ p implies A~t+1 (!)
y(u c) > 0: Depending on the strategy , there are three subcases.
29
A~t (!) =
(a) t3 (!) > p. Since the seller's o er price is higher than the buyer's bid price p, no trade
will take place if strategy is adopted. Hence, At+1 (!) At (!) = 0 < A~t+1 (!) A~t (!).
(b) t4 (!) < y. In this case, the seller's o er quantity is smaller than the amount buyer
wants to buy, no trade will take place if strategy is adopted. Hence, At+1 (!)
At (!) = 0 < A~t+1 (!) A~t (!).
(c) t3 (!) = p and t4 (!) y. In this case, trade yields the same net gain as by adopting
strategy ~ p , At+1 (!) At (!) = y(u c) = A~t+1 (!) A~t (!).
In all three cases, At+1 (!) A~t+1 (!) At (!)
by the induction hypothesis.
A~t (!) 0: The last inequality is implied
Therefore, we conclude that for any strategy , for all history ! 2 , A0 (!) A~0 (!) = 0, and for
all t 0, At+1 (!) A~t+1 (!) 0: By induction, for all t 0, At (!) A~t (!).
Proposition 6 states that if all other agents adopt strategy ~ p , and if the economywide initial
distribution 0 satis es condition ID3 , the expected money holdings of an agent adopting strategy
~ p converges to M . The next lemma shows that if an agent adopts some other strategy , then
in the limit, he may end up with more money on average.
. Consider any p > 0. Suppose that the economywide initial money holdings
distribution 0 satis es condition ID3 , and that an arbitrary agent adopts strategy while all other
agents adopt strategy ~ p . If E[At (!) A~t (!)] 6! 1 as t ! 1, then lim inf t!1 E[t (!)] M .
Lemma 15
Proof. For strategy ; for all !
2 , de ne Æ (!) to be the set of dates at which the agent
who adopts strategy meets a buyer with money, and either his o er price is above p or his o er
quantity is below the buyer's bid quantity,10
n
o
Æ (!) = t j !t1 = b and !t3 > 0 and (t3 (!) > p or (t3 (!) = p and t4 (!) < !t3 )) :
On these occasions, trade will not occur which would have occurred had the agent adopted strategy
~ p .
t2Æ !t3 < 1 a.s.
10 These are the relevant deviations of o er strategy, since they block trade that would have occurred otherwise.
As stated earlier, we do not explicitly consider the deviation of o ering at price below p since it is obviously
suboptimal. Also, we need not consider the case in which an agent deviates by o ering to sell at price p and
quantity below 1, but the quantity happens to be above the buyer's bid quantity. In that case, trade will occur at
the buyer's bid quantity.
Claim 1. If E[At (!)
A~t (!)] 6!
1 as t ! 1, then
30
P
To prove this, consider an arbitrary encounter sequence ! 2 . For any date t 2 Æ (!), given that
the agent adopts strategy , no trade takes place, hence, At+1 (!) At (!) = 0. On the other hand,
if the agent adopts strategy ~ p , then trade takes place at price p, A~t+1 (!) A~t (!) = !t3 (u c) > 0.
Therefore,
8t 2 Æ (!) At+1 (!) A~t+1 (!) = At (!) A~t (!) !t3(u c):
(61)
For t 62 Æ (!), it is easy to check, as for all the cases considered in the proof of Lemma 14,
that At+1 (!) A~t+1 (!) = At (!) A~t (!). Hence, by (61) and the fact that A0 (!) = A~0 (!), if
P
t2Æ (!) !t3 = 1, then
lim [A (!) A~t (!)] = tlim
t!1 t
!1
X
t2 (!)
Æ
!t3 (u c) =
1:
P
Therefore, if Pf! j t2Æ (!) !t3 = 1g > 0, then limt!1 E[At (!) A~t (!)] = 1, which contradicts to the assumption. Thus, the claim holds.
P
Given claim 1, for any " > 0, there exists t" > 0 such that Pf! j t2Æ (!); t>t" !t3 < "g >
1 "=2. Recall that for all t 2 Æ (!), ~tp2 (!) = p, ~tp3 (!) = 1. De ne " (!) min ft j t
t"; ~t (!) = 0g.
Claim 2. " < 1 a.s.
Suppose to the contrary. Let A = f! j 8t ~t (!) > 0 g and suppose that P (A) > 0. Take an
S
arbitrary date t. For all n 2 N , de ne Dn f! j ~t (!) 2 [np; (n +1)p) g. Then A = n2N (A \ Dn ).
Since P (A) > 0, there exists j 2 N such that P (A \ Dj ) > 0. Given that all other agents play
strategy ~ p and the random matching at each date is independent of those at other dates, the
probability that an agent with money holdings ~t (!) 2 Dj spends all of his money in next j
consecutive dates is (1=2)j . That is,
P ! j ~t+j (!) = 0 and ! 2 A \ Dj = 21 P (A \ Dj ) > 0
n
o
j
which contradicts the de nition of the set A.
Claim 3. For all ! 2 , for all t "(!), ~t (!) t (!) + p
P
2Æ (!); t" < <t ! 3 .
This claim can be proved by induction. For t = " (!), the claim holds automatically since
P
~t (!) = 0 t (!) + p 2Æ (!); t" < <t ! 3 . Suppose that it holds for some t " (!), consider the
date-(t + 1) transaction.
(i) If !t1 = b; !t2 = p; !t3 2 [0; 1]; then ~t+1 (!) = ~t (!) + p !t3 .
31
(a) If t 2 Æ (!), then trade does not occur if the agent adopts strategy , so t (!) =
t+1 (!). By induction hypothesis,
~t+1 (!) t (!) + p
X
2 (!); t" < <t
Æ
X
! 3 + p !t3 = t+1 (!) + p
2 (!); t" < <t+1
Æ
! 3 :
(b) If t 62 Æ (!), trade occurs at price p and quantity !t3 with both strategies and ~ ,
then, by induction hypothesis,
~t+1 (!) t (!) + p
X
2 (!); t" < <t
Æ
X
! 3 + p !t3 = t+1 (!) + p
2 (!); t" < <t+1
Æ
! 3 :
(ii) If !t1 = s; !t2 = p; !t3 = 1; then ~t+1 (!) = ~t (!) p, and t 62 Æ (!).
(a) If ~t (!) < p, then, ~t+1 (!) = 0 t+1 (!) + p
P
2Æ (!); t" < <t+1 ! 3 :
(b) If ~t (!) p, then, given that t3 (!) 1 and the induction hypothesis,
~t+1 (!) t (!) + p
X
2 (!); t" < <t
Æ
! 3 p t3 (!) = t+1 (!) + p
X
2 (!); t" < <t+1
Æ
! 3 :
That is, ~t+1 (!) t+1 (!) + p 2Æ (!); t" < <t+1 ! 3 . Hence, the claim holds for all t "(!).
P
By Claim 3, for all ! 2 such that t2Æ (!); t>t" !t3 < ", for all t " (!),
P
t (!) ~t (!) p ":
(62)
Since " < 1 a.s., for the " chosen above, there exists " > 0 such that Pf! j " (!) "g > 1 "=2.
De ne
n
o
X
!t3 < " and " (!) "
1 (") = !
t2Æ (!); t>t"
n
2 (") = !
o
X
t2Æ (!); t>t"
!t3 " or "(!) > " :
Then = 1 (") [ 2 ("), P ( 1 (")) > 1 " and P ( 2 (")) < ". Take " = 1=n2 . For ! 2 1 (1=n2 ),
t1=n2 1=n2 (!) 1=n2 . For a xed n, consider the sequence ft (!) ~t (!)g for t 1=n2 . Let
1n 1 (1=n2 ) and 2n 2 (1=n2 ).
lim
inf E[t (!)
t!1
~t (!)]
lim
inf
t!1
Z
+ lim
inf
t!1
(t (!) ~t (!))dP (!)
1n
Z
2n
t (!)dP (!)
Z
2n
~t (!)dP (!) :
(63)
The rst term on the right hand side of (63) is greater than ( p=n2 )P ( 1n ) p=n2 because
(62) holds for all ! 2 1n . We will now prepare to consider the second term on the right side of
(63). Note that by Proposition 6, the limit of E[~t (!)] exists and equals M .
32
Claim 4. For any n 1, limt!1
R
2n ~t (! )dP (! ) = M
To prove this claim, for any t 0, de ne
P ( 2 n ).
n
o
t ! j 9t^[t1=n2 < t^ t and t^ 2 Æ (!)] or 1=n2 (!) > 1=n2 :
Then, for all t 0, t t+1 , [1
t=0 t = 2n , and P (t ) ! P ( 2n ) as t ! 1. Hence, for any
" > 0, there exists t1 0 such that for all t t1 ,
jP (t ) P ( 2n )j < 3M" :
(64)
Let be the parameter such that the geometric distribution g dominates both the aggregate
money holdings distribution as well as the agent's money holdings distribution from some date
onward. (See the proof of Proposition 6 for the construction of .) For the " above, choose k > 0
such that
X
j (1 )j < "=3:
j k
By Proposition 5, the distribution of the agent's money holdings converges weakly to the aggregate
money holdings distribution gm . That is, for the k chosen above, for all 2 (0; (m )k ), there
exists t2 t1 such that for all t t2 , jPf~t (!) kpg (m )k j < , which implies 0 < (m )k <
Pf~t (!) kpg. Since f 2n nt g1
t=0 is decreasing and P ( 2n nt ) ! 0 as t ! 1, there exists
t3 t2 such that for all t t3 , P ( 2n nt ) (m )k < Pf~t (!) kpg. Then, for all t t3 ,
Z
2n nt
~t (!)dP (!)
Z
f!: ~t (!)kpg
~t (!)dP (!):
(65)
Furthermore, given that the agent's money holdings distribution is dominated by g from some
date on, there exists t4 t3 such that for all t t4 , Pf~t (!) kpg k . Then, for all t t4 ,
Z
f!: ~t (!)kpg
~t (!)dP (!)
By inequalities (65) and (66), for all t t4 ,
Z
2n nt
X
j k
j (1 )j < "=3:
~t (!)dP (!) "=3:
(66)
(67)
For any t 1=n2 , t 2 Bt . For any > t, ! is independent of Bt , and in particular,
the distribution of the trading partner's money holdings conditional on Bt is given by , and
the conditional probability that the trading partner is a potential seller is one half. Therefore,
analogously to Proposition 6, lim !1 E[~ (!)jt ] = M . Then, for all t t4 , there is a t such
that for all t ,
Z
~ (!)dP (!) M P (t ) < "=3:
(68)
t
33
Combining the results given by (64), (67) and (68), for some t t4 , for all
Z
Z
2n
~ (!)dP (!) M P ( 2n )
M P (t ) + M P (t )
~ (!)dP (!)
t
t, we have
P(
2n ) +
Z
< "=3 + M ("=3M ) + "=3 = ":
2n nt
~t (!)dP (!)
That is, the claim holds.
By Claim 4, the second lim inf on the right hand side of (63) can be broken down to two terms,
Z
lim
inf
t!1
2n
Z
t (!)dP (!)
~t (!)dP (!) = lim
inf
t!1
2n
Z
2n
t (!)dP (!) M P ( 2n ):
(69)
The rst term of the right hand side of (69) is nonnegative. Therefore,
p + M
:
(70)
n2
Take limits as n ! 1 in inequality (70). The left hand side is unrelated to n, hence constant,
and the right hand side goes to 0. Therefore, lim inf t!1 E[t (!)] limt!1 E[~t (!)] = M by
Proposition 6.
M P ( 2n )
lim
inf E[t (!) ~t (!)] p=n2
t!1
Now, we are ready to show the last step for the existence of the price-p equilibrium.
. Consider any p > 0. If the economywide initial money holdings distribution
0 2 satis es condition ID3 , and if all other agents adopt strategy ~ p , then it is optimal for
an arbitrary agent to take strategy ~ p as well. That is, there is no strategy that overtakes ~ p .
Proposition 7
M
Proof. For an arbitrary strategy , consider the following two cases.
Case 1. lim supt!1 E[t (!)] M . Then, for any " > 0, there exists an in nite set
G" = ft j E[t (!)] M "=2g. Since limt!1 E[~t (!)] = M by Proposition 6, the set J" =
ft j E[~t (!)] < M + "=2g is also in nite. For all t 2 G" \ J" , by Lemma 14,
0 E[At
1
t
X
A~t ] = E[
=0
v
1
t
X
=0
v~ ] + E[
t 1
X
~t
]u E[ v
p
=0
t
p
Since " can be arbitrarily small, the above inequality implies that
1
t
X
1
t
X
lim
inf E[
v~ ] 0:
t!1
=0
=0
By de nition of overtaking criterion (7), strategy does not overtake ~ p .
v
34
1
t
X
=0
v~ ]
u
"
p
Case 2. lim supt!1 E[t (!)] < M . By the proof of Lemma 14, for all ! 2 , fAt (!)
~
A~t (!)g1
t=0 is a weakly decreasing sequence. If E[At (! ) At (! )] 6! 1 as t ! 1, by Lemma 15,
lim inf t!1 E[t (!)] M , which contradicts to the assumption. If E[At A~t ] ! 1 as t ! 1,
and since
t 1
t 1
X
X
~t
~
v~ ] + E[ t
]u
E[At At ] = E[ v
p p
=0
=0
then we have
1
t
X
1
t
X
u
A~t ] + M
p
lim
inf E[
v~ ] lim
inf
t!1
t!1
=0
=0
Again by the de nition (7), strategy does not overtake ~ p .
v
E[At
lim
inf
t!1
E[t ]
=
1:
By Proposition 7, strategy ~ p is a Bayesian Nash strategy according to the overtaking criterion.
This proves (ii) of the equilibrium de nition, and (iii) is evidently satis ed. Hence, the price-p
equilibrium always exists.
6. Conclusion
This article has provided an analysis of equilibrium in an in nite horizon economy where
trade must occur pairwise rather than in a central market, and where the exchange of at money
for goods overcomes a lack of double coincidence of wants. Although an agent can bargain
with only one trading partner at a time, trade has the characteristics of anonymity and absence
of market power. These characteristics are ensured by assumptions about the random matching
process for pairwise trade, agents' lack of information about trading partners' money holdings and
histories, and absence of time preference. (This last assumption deprives an agent's current trading
partner of monopoly power because the agent considers consumption in the future to be a perfect
substitute for current consumption.) Goods in the economy are divisible and perishable, with new
endowments being received and consumption occurring at every date. Money is also divisible,
and is not subject to inventory constraints. Exchange within each trading pair is governed by a
double auction mechanism.
Two main results have been obtained. The rst is a characterization of the evolution of
individual agents' money holdings and of the economywide distribution of these holdings when
trading occurs as extensively as possible at a speci ed, economywide price. This characterization
is instrumental to deriving the second result: that for any price, and for any initial distribution of
money holdings that is dominated in the tail by a geometric distribution, there is an equilibrium
in which all trades occur at the speci ed price. This second result shows the existence and also
35
the indeterminacy of single price equilibrium, a special class of Bayesian Nash equilibrium of the
economy.
That single price equilibrium exists, despite the fragmentation of trading activity, con rms
an expectation that economists have long held.11 While not a foregone conclusion, this result
is consistent with numerous earlier ndings such as those cited in the introduction. Camera
and Corbae (1999) show that price dispersion characterizes equilibrium in a model economy
that resembles the present one in many respects, but trading in that economy is not completely
anonymous because agents observe their trading partners' money holdings. As Rubinstein and
Wolinsky (1990) emphasize, lack of anonymity is conducive to the existence of many equilibria in
which agents can be treated disparately.
The indeterminacy of equilibrium in the model economy is a surprising result. The nature of
this indeterminacy is rather di erent from that which is familiar from Walrasian general equilibrium models such as overlapping generations models with at money.12 In those models, there
is an indeterminacy of relative prices between various dated, location speci c, state contingent
commodities. In the present model, all such commodities would trade at par because they all
have the same nominal price. It is the nominal price level, rather than the relative prices of
various goods, that is indeterminate. This price level indeterminacy has a real e ect through its
in uence on the distribution of money holdings, with agents not holding money being unable to
participate in bene cial transactions. This e ect is a further example of the existence of Pareto
dominated equilibria in coordination game models, such as macroeconomic models due to Cooper
and John (1988) and Diamond (1984), where an ineÆciently low level of economywide search effort or other complementary investment results from agents' failure to take proper account of the
positive external e ects of their actions. The trading mechanism in the present model does not
provide a coordination device to take advantage of the positive externality that sellers collectively
could provide by o ering their endowment at a lower nominal price. If the market price could be
lowered in that way, then the aggregate real money balances M =p in the economy would increase
and welfare would improve.
There are a number of alternative model economies that might be examined to investigate
whether the indeterminacy of equilibrium is robust to re-speci cation. Among them would be
to substitute Stahl-Rubinstein strategic bargaining for a double auction mechanism as a representation of strategic price/quantity determination for transactions; to impute discounted utility
11 Indeed, we strongly suspect that all stationary Bayesian Nash equilibria in this model economy are single price
equilibria. Green and Zhou (2001) prove this result for a virtually identical model.
12 Cass (1992) and Werner (1990) show that indeterminacy of equilibrium is a generic feature of Walrasian
economies with incomplete markets or other forms of restricted market participation.
36
preferences, rather than preferences characterized by an overtaking criterion, to agents; and to
introduce a durable good alongside the perishable commodities currently traded. We consider the
rst of these alternatives, Stahl-Rubinstein bargaining, to be the one most likely to overturn the
indeterminacy result by itself.
Indeterminacy of the steady state money holdings distribution, which is the key to the equilibrium indeterminacy result in the present model, has been exhibited in closely related model
economies with discounted utility preferences and a durable, tradable good.13 We conjecture that
these economies exhibit indeterminacy of dynamic equilibrium from an initial state (under some
assumptions such as that the initial money holdings distribution is nice and that the discount
factor is close to 1), as well as indeterminacy of steady state distributions.
Regarding the issue of Stahl-Rubinstein bargaining versus double auction mechanisms, and
more generally regarding a research program of investigating the robustness of the equilibrium indeterminacy result derived here, we would o er three methodological observations. First, neither
of the two types of representation of strategic transactions is a literal model of actual economic
activity or is the uniquely privileged representation in any other way, especially since a wide
spectrum of mechanisms are actually used to conduct various transactions. At the very least, the
result proved here warrants the interpretation that some transaction mechanisms may be susceptible to indeterminacy of equilibrium if they are predominantly used in an economy. We hope
that the present result may stimulate the development of further models that pay closer attention
to the micro-structure of transactions and that might provide deeper understanding of what is
required for equilibrium to be determinate. Second, while indeterminacy of equilibrium may be a
symptom that a model is incompletely speci ed (in a sense resembling the idea that when there
are fewer equations than unknowns, a well motivated equation may have been ignored), indeterminacy does not necessarily show that a model is misspeci ed or implausible. While indeterminacy
of equilibrium would be an inconvenient situation for policy analysis, the possibility that actual
economies may exhibit this feature cannot be ruled out. Third, assumptions or features of specication that make the di erence between a model economy having determinate or indeterminate
equilibrium should be regarded as economically crucial. Once discovered to have such an e ect,
an assumption should not be regarded as merely a convenient formal simpli cation|even if that
is why it was originally introduced. In view of the indeterminacy of equilibrium that results when
indivisibility and nite-inventory-constraint assumptions about money are avoided in a random
matching model, such determinacy-inducing assumptions should not casually be adopted.
13 Cf. Green and Zhou (1998), Zhou (1999a,b).
37
References
Billingsley, P. (1995): Probability and Measure. New York: John Wiley and Sons.
Camera, G.,
D. Corbae (1999), \Money and Price Dispersion," International Economic
Review, 40, 985{1008.
and
Chatterjee, K.,
W. Samuelson (1983), \Bargaining under Incomplete Information," Operations Research, September-October, pp. 835{851. Reprinted in Bargaining under Incomplete Information, P. Linhart, R. Radner and M. Satterthwaite (edited), Academic Press
1992.
and
Cass, D. (1992): \Incomplete Financial Markets and Indeterminacy of Competitive Equilibrium,"
in Advances in Economic Theory: Sixth World Congress, Vol. 2, ed. by Jean-Jacques
La ont, 263{288. Cambridge: Cambridge University Press.
Cooper, R.
A. John (1988): \Coordinating Coordination Failures in Keynesian Models,"
Quarterly Journal of Economics, 103, 441{463.
and
Diamond, P. (1984): A Search-Equilibrium Approach to the Micro Foundations of Macroeconomics
(The Wicksell Lectures, 1982). Cambridge, The MIT Press.
Gale, D. (1986a): \Bargaining and Competition Part I: Characterization," Econometrica, 54,
785{806.
|| (1986b): \Bargaining and Competition Part II: Existence," Econometrica, 54, 807{818.
Gilboa, I.,
A. Matsui (1992): \A Model of Random Matching," Journal of Mathematical
Economics, 21, 185{97.
and
Green, E. J. (1994): \Individual Level Randomness in a Nonatomic Population," ewp-ge/9402001,
http://econwpa.wustl.edu.
Green, E. J., and R. Zhou (1998): \A Rudimentary Random-Matching Model with Divisible
Money and Prices," Journal of Economic Theory, 81, 252{271.
|| (2001): \Money and the Law of One Price: the Case Without Discounting," manuscript,
Federal Reserve Bank of Chicago.
Hendry, S. (1993): \Endogenous Money and Goods Production in a Search Model," University
of Western Ontario, Department of Economics Research Report, 9316.
Kiyotaki, N., and R. Wright (1989): \On Money as a Medium of Exchange," Journal of Political
Economy, 97, 927{953.
38
Myerson, R., and M. Satterthwaite (1983): \EÆcient Mechanisms for Bilateral Trading," Journal
of Economic Theory, 29, 265{281.
Molico, M. (1997): \The Distribution of Money and Prices in a Search Equilibrium," Ph.D.
Dissertation, University of Pennsylvania.
Rubinstein, A., and A. Wolinsky (1990): \Decentralized Trading, Strategic Behavior and the
Walrasian Outcome," Review of Economic Studies, 57, 63{78.
Shi, S. (1995): \Money and Prices: A Model of Search and Bargaining," Journal of Economic
Theory, 67, 467-496.
Trejos, A. and R. Wright (1995): \Search, Bargaining, Money and Prices," Journal of Political
Economy, 103, 118-141.
Wallace, N. (1996): \Questions Concerning Rate-of-Return Dominance and Indeterminacy in
Absence-of-Double-Coincidence Models of Money," manuscript, Pennsylvania State University.
Werner, J. (1990): \Structure of Financial Markets and Real Indeterminacy of Equilibria," Journal of Mathematical Economics, 19, 217{232.
Zhou, R. (1999a): \Individual and Aggregate Real Balances in a Random Matching Model,"
International Economic Review, 40, 1009{1038.
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39
Appendix
The Proof of Lemma 3: The function Z is strictly convex on .
Take arbitrary x; y 2 , x 6= y, and 2 (0; 1). Let w( ) = (1
)x + y = x + (y
The set is convex, hence w( ) 2 . For all k 2 N , de ne Æk yk xk and
zk (w( ))
(wk ( ) wk+1 ( ))2 (xk
=
wk ( )
Direct computation reveals that
1 d2 z (w( )) X
1
d2 Z (w( )) X
2
k
=
=
Æk
2
d 2
d
x
+
Æ
k
k
k=0
k=0
x).
xk 1 + (Æk Æk+1 ))2
:
xk + Æk
Æk+1
Æk (xk
xk+1 + (Æk
xk + Æk
Æk+1 ) 2
0:
2
2
Moreover d Zd(w2( )) = 0 if an only if 8k 2 N d zkd(w2( )) = 0, which is equivalent to 8k 2 N yk+1xk =
2
yk xk+1 , that is (since x0 = y0 = 1), 8k 2 N xk = yk . Given that x 6= y, d Zd(w2( )) > 0. Hence, Z
is strictly convex on .
The Proof of Lemma 5: The function Z is continuous on S .
We need to show that for any given " > 0, for any x 2 S , there exists a Æ-neighborhood of x
such that for all y satisfying d(x; y) < Æ, jZ (x) Z (y)j < ".
P
Fix an arbitrary " > 0, and an arbitrary x 2 S . Since xk is decreasing in k and 1
k=1 xk = M ,
there exists I 1 such that
xI < "=8:
(71)
Let J = max fj j j I; xj > 0g. So xJ > 0. Without loss of generality, assume J I 1. Let
Æ = ("=8)xJ > 0. (Otherwise xJ +1 = 0, so J + 1 satis es xJ +1 < "=8.) Then for any y such that
d(x; y) < Æ,
yI jyI xI j + xI d(x; y) + xI < ("=8)xJ + "=8 "=4:
(72)
For all k 2 N , de ne k (x)
k I 1, xk xJ ,
(xk xk+1)=xk 1, and k (y) (yk yk+1)=yk 1. Then, for
jk (x) k (y)j x1 jxk+1 yk+1j + jxk yk j yyk+1 < x2 8" xJ = 4" :
J
k
k
Now, applying (71){(73), we have
jZ (x) Z (y)j
40
(73)
=
1
I
X
(xk
k=0
I 1
X
k=0
I 1
X
(xk
xk+1 )k (x) (yk
yk+1)k (y) +
xk+1 )jk (x) k (y)j + jxk
X
kI
(xk
xk+1)k (x) +
yk jk (y) + jxk+1
X
k I
(yk
yk+1)k (y)
yk+1jk (y) + xI + yI
I 1
I 1
X
" X
xk+1) + jxk yk j + jxk+1 yk+1j + "=8 + "=4
4 k=0
k=0
k=0
< "=4 + "=8 + "=8 + "=8 + "=4 < ":
<
(xk
We have shown that for any given " > 0, for any x 2 S , there is Æ > 0 such that for all y satisfying
d(x; y) < Æ, jZ (y) Z (x)j < ". Hence, Z is continuous on S .
The Proof of Lemma 6: The mapping T is continuous on S .
We need to show that for any given " > 0 and x 2 S , there is a Æ > 0 such that for all y
satisfying d(x; y) < Æ, d(T (x); T (y)) < ".
Fix an arbitrary " > 0 and an arbitrary x 2 S . By (19), for all y 2 S , for all k 1,
Tk (y) =
1 y1
1
y
yk + yk+1 + 1 yk 1 :
2
2
2
Take Æ = "=3 > 0, and let y be such that d(y; x) < Æ. Then,
1
X
d(T (x); T (y)) =
jTk (y) Tk (x)j
=
<
1 1
X
k=1
jxk yk j + jxk+1 yk+1j + x1jxk yk j + x1jxk+1 yk+1j + (yk + yk+1)jx1 y1j
2
k=1
1
"=3 + "=3 + "=3 + "=3 + "=3 + "=3 = "
2
Therefore, T is continuous on S .
The Proof of Lemma 7: For every vector 2 X , the set S is compact.
To prove S is compact, we need to show that S is complete and totally bounded subset of
X . The completeness of S is trivial given that S is complete, and the proof is omitted here. To
show that S is totally bounded, we need to show that there exist a nite "-net for S in X for
any " > 0.
41
P
Fix an arbitrary " > 0. Since 2 X , 1
k=0 k < 1. Hence, there exists I > 0 such that
P
^ be the vector of x truncated at I , x^ = (x0 ; x1 ; : : : ; xI ; 0; 0; : : :).
k>I k < "=2. For any x 2 S , let x
P
P
Then d(x; x^) = k>I xk k>I k < "=2. Let S^ be the set of x^ associated with x 2 S . The
set S^ is a totally bounded in I -dimensional Euclidean space (with the usual metric). Let A be
a nite "=2-net for S^ . Then A is a nite "-net for S .
The Proof of Lemma 8: For any x 2 S and any 2 (0; 1), if x d , then T (x) d .
Suppose that there exists 2 (0; 1) such that x d . By de nition, x d implies that
xk k for all k 2 N . By equation (19), for all k 1,
Tk (x) =
1
1
(1 x1 )xk + xk+1 + x1 xk 1 (1 x1 )k + k+1 + x1 k 1 :
2
2
Since the expression in the right hand side of the above inequality is an increasing function of x1
and by assumption, x1 ,
Tk (x)
1
(1 )k + k+1 + k 1 = k :
2
By de nition, T0 (x) = 1 = 0 . Therefore, T (x) d .
The Proof of Lemma 9: For any 1 ; 2 2 , if 1 2 , then T (1 ) T (2 ).
For any 1 ; 2
have14
2 such that 1 2 , for any nondecreasing function f : R+ ! [0; 1], we
Z
1
0
For any y p, de ne
f (x) =
Then, by (32) and (74),
(1 )[y; 1) =
=
1
Z
0
fd1
[0;p)
Z
[0;p)
1
0
1 [y x; 1)
1 [y p; 1)
f (x)d1 (x)
Z
Z
Z
1
0
fd2 :
if x < p
if x p
f (x)d2 (x)
1 [y x; 1)d2 (x) + 2 [p; 1)1 [y p; 1)
2 [y x; 1)d2 (x) + 2 [p; 1)2 [y p; 1) = (2 )[y; 1)
14 This is a standard result about stochastic dominance.
42
(74)
The last inequality holds because 1 2 . For any y < p, de ne
<y
f (x) = 1 1 [y x; 1) ifif xx
y
Then, by (33) and (74),
(1 )[y; 1) =
=
1
Z
0
Z
f (x)d1 (x)
[0;y)
Z
[0;y)
1
Z
0
f (x)d2 (x)
1 [y x; 1)d2 (x) + 2 [y; 1)
2 [y x; 1)d2 (x) + 2 [y; 1) = (2 )[y; 1)
That is, (1 ) (2 ). Also, by equations (34) and (35), (1 )[0; 1) = (2 )[0; 1) = 1, and for
any y > 0,
(1 )[y; 1) = 1 [y + p; 1) 2 [y + p; 1) = (2 )[y; 1)
we have (1 ) (2 ): Hence, by (36) and (37), 1 2 implies T (1 ) T (2 ).
The Proof of Lemma 11: For any 2 M , T n ()(pN )
! 1 as n ! 1.
By Lemma 10, for a given 2 M , T n()(pN ) is nondecreasing in n. Since T n ()(pN ) is
bounded by 1, it has a limit 1. We want to show that = 1. Suppose to the contrary, < 1.
R
Let n minfn j np > 2M =(1 ) g. Since for any n 0, T n () 2 M , or 01 dT n () = M ,
we have T n ()[n p; 1) < (1 )=2, which implies
1
X
1
T n ()(lp; (l + 1)p) <
:
2
l=n
Correspondingly,
1
nX
T n ()(lp; (l + 1)p) >
1
:
(75)
2
l=0
Therefore, there exists l < n such that for an in nite sequence n0 ; n1 ; : : :, for any k 0,
1
T nk ()(l p; (l + 1)p) >
(76)
2n
(otherwise, a contradiction of (75)). Without loss of generality, assume that for any k
nk+1 nk + l + 1. Then, by Lemma 10 and (76), for any k 0,
T nk+1 ()(pN )
T nk +l+1()(pN )
h
i2
T nk ()(pN ) + 2 (2l +1) T nk ()(np; (n + 1)p)
T nk ()(pN ) + 2
43
h
(2l +1) 1
2n
2
i
0,
The assumption < 1 then implies that limk!1 T nk+1 ()(pN ) = 1, which contradicts the fact
that for any k 1, T nk+1 ()(pN ) 1. Hence, = 1.
The Proof of Lemma 12: For an arbitrary
then for any n 0, [T n ()]+ g .
2 , if there exists 2 (0; 1) such that + g ,
Take an arbitrary 2 and some 2 (0; 1) such that + g . We rst show that
[T ()]+ T (+ ):
(77)
Since both [T ()]+ and T (+ ) have all their probability mass concentrated on pN , we need to
check (77) only on pN . For any k 1, the only transactions that would result di erence between
[T ()]+ fkpg and T (+ )fkpg are those trades between buyers with money holdings b 2 (0; p)
and sellers with money holdings s 2 ((k 1)p; kp) such that b + s < kp. Such a trade
will add the seller to the measure [T ()]+ fkpg, but for T (+ ), the same trade occurs as if it
is between a buyer with b = p units of money and a seller with s = kp units of money,
which increases the seller's money holdings to (k + 1)p rather than kp. That is, for all k 1,
[T ()]+ [(k + 1)p; 1) T (+ )[(k + 1)p; 1). It is easy to check that [T ()]+ fpg = T (+ )fpg.
Therefore, [T ()]+ T (+ ). Repeatedly applying (77), we have for any n 0,
[T n()]+ T ([T n 1 ()]+ ) T 2 ([T n 2 ()]+ ) : : : T n (+ ) g :
The last inequality is because + g .
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@FedFRASER