HIAS-E-84

Market Structure and Indeterminacy of Stationary Equilibria ina Decentralized Monetary Economy
So Kubota
Hitotsubashi Institute for Advanced Study, Hitotsubashi University

March, 2019

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Market structure and indeterminacy of stationary equilibria in adecentralized monetary economy∗
So Kubota†
March 10, 2019

Abstract
This study investigates which market structure gives rise to indeterminacy of stationary equilibria in adecentralized economy with non-degenerate distributions of money holdings. I develop a price-posting modelwith divisible money and then, examine two alternative markets: a pairwise random matching market and amany-to-many exchange. Importantly, the former market balances the number of matched buyers and sellers
by definition. As a result, indeterminacy arises under the pairwise matching while a unique equilibrium exists
in the many-to-many market. This balancing assumption also leads to the indeterminacy in a Walrasianmarket.
Keywords: search theory, money, indeterminacy
Journal of Economic Literature Classification Number: D31, D51, D83, E41

1Introduction

This study analyzes which type of market structure of decentralized monetary trade that gives
rise to the real indeterminacy of stationary equilibria initially found by Green and Zhou (1998).
Notably, the indeterminacy of equilibrium distribution is of non-degenerate money holdings,
which is of growing concern in monetary economics. Theoretically, such indeterminacy is potentially a straightforward consequence of decentralized trades with divisible money. Moreover,recent empirical studies emphasize the influence of heterogeneity of household balance sheetson the eﬀects of inflation (e.g., Doepke and Schneider (2006) and Auclert (2017)). Thus, thepresent study aims to provide a foundation for one aspect of the non-degenerate money holding
distributions, namely, indeterminacy.
A series of works following Green and Zhou (1998) study the circulation of divisible money
using a variety of search models and find a specific type of indeterminacy. There is a continuumof stationary equilibria where each equilibrium diﬀers not only in nominal prices but also realterms, such as consumption and production. In addition, the distributions of money holdings
and the number of matchings are uncertain. As a result, economic welfare is unpredictable.
∗ The previous title was ”Indeterminacy in Search Theory of Money: Bilateral vs. Multilateral Trades.” I am grateful to Edward
Green, Kazuya Kamiya, Nobu Kiyotaki, Guido Menzio, Ezra Oberfield, Yohei Sekiguchi, Takashi Shimizu, and Chang Sun for theirvaluable comments. I thank the participants of The Search Theory Workshop (Osaka) and Summer Workshop on Money, Banking,
Payments, and Finance (Chicago). I also acknowledge financial support from The Nakajima Foundation.
† Hitotsubashi Institute for Advanced Studies (E-mail: gkubotaso@gmail.com)

1

I develop a model of price-posting whereby sellers post unit prices of a consumption good andform submarkets. Then, each buyer visits a submarket and quotes a purchase quantity. Finally,a market structure determines the allocation in every submarket. This price-posting market
represents daily shopping, for example, a person decides how much meat to buy after looking
at its price by weight in a supermarket. On this model, I examine two alternative market
structures: bilateral trade with random pairwise matching and a two-sided many-to-manyexchange. The random matching one is a variant of the standard competitive search markets(e.g., Moen (1997); and Rocheteau and Wright (2005)) and the many-to-many matching is a
type of market game (Shapley and Shubik (1977)). As a result, the bilateral trade market gives
rise to the indeterminacy of the stationary equilibrium distributions of money holdings, but the
many-to-many exchange market holds a unique equilibrium.
Pairwise matching equalizes the number of buyers and sellers in both on and oﬀ paths.
This balanced trade assumption creates one more identity in the system of equations that
determine the distribution of the stationary equilibrium money holdings; it makes a continuumof solutions. On the other hand, the many-to-many exchange assumption excludes this identity.
In addition to the decentralized price-posting model, the present study shows that a similarassumption of balancing buyers and sellers may give rise to indeterminacy in a centralized
market. I consider a Walrasian market with buyer and seller sides. In the standard case,the Walrasian market holds a unique equilibrium because it allows a many-to-many exchange.
However, under a limited participation restriction by which only the same population of buyers
and sellers enters the market, the model also leads to indeterminacy.
This study builds on the literature on decentralized trade models with divisible money andindeterminacy. Green and Zhou (1998); Zhou (1999); Green and Zhou (2002); Kamiya andSato (2004); Kamiya and Shimizu (2006, 2007, 2011); Ishihara (2010) examine many typesof money search models with non-degenerated distributions of money holdings and show thesimilar indeterminacy. It also arises in other models such as market place choice by Matsui andShimizu (2005), auction market by Kamiya and Shimizu (2013), New Monetarist model withindivisible good by Jean et al. (2010), and Walrasian market with indivisible good by Kamiya
et al. (2017).
Wallace (1998) conjectures that the nominal nature of money is the cause of indeterminacy.
However, Zhou (2003) questions this finding by showing indeterminacy in a model with commodity money. Another conjecture is that the indivisibility of goods assumed in Green andZhou (1998) is the source of indeterminacy. On the contrary, Ishihara (2010) and Kamiya andShimizu (2011) find indeterminacy with divisible goods. This study’s price-posting model also
reveals indeterminacy with divisible consumption goods. Kamiya and Shimizu (2006) show
the kind of system of equations about stationary distributions of money holdings that support
indeterminacy but a question remains about the underlying market structure causing inde-2

terminacy1 . Kamiya and Shimizu (2013) study a centralized auction market and conjecturethat the non-Walrasian price determination causes the indeterminacy. However, the present
study derives indeterminacy in a Walrasian market too. In a diﬀerent context, another stationary equilibrium indeterminacy is found in economic growth models with knowledge diﬀusion(e.g., Luttmer (2007)). Their indeterminacy is associated with fat-tail distributions of firm
productivity. Nevertheless, indeterminacy in decentralized monetary models possibly emerges
on bounded distributions.
This study is further related to tractable search models of non-degenerate distributions ofmoney holdings. Berentsen et al. (2005) and Rocheteau et al. (2018) study delayed meanreverting adjustment processes of money holdings using the new-monetarist model and obtain
non-degenerate distributions. Menzio et al. (2013) provide a competitive search model withblock recursive structure, which results in non-degenerate distribution associated with sorting into diﬀerent submarkets. While these models incorporate Walrasian markets, the present
study analyzes entirely decentralized markets2 . Interestingly, this study derives similar distributions which hold discrete masses. Other notable approaches to generating non-degenerate
distributions in money search models include a countable number of money holdings (Camera
and Corbae (1999)) and numerical methods (Molico (2006); Chiu and Molico (2010, 2011)).
The present study’s many-to-many exchange trade is a variant of Shapley and Shubik(1977)’s market game. There are other approaches to incorporate the many-to-many exchange
in decentralized monetary models: Corbae et al. (2003) consider a type of coalition formation
game with indivisible money, Howitt (2005) studies the appearance of trading places, and Julien
et al. (2008), Galenianos and Kircher (2008), and Kamiya and Shimizu (2013) analyze auction
markets.
The structure of the rest of the paper is as follows. In Section 2, I propose a simple example
and provide the intuition for the process by which a bilateral trade assumption makes the equilibrium indeterminate. In Section 3, the environment of the economy and equilibrium concept
are introduced. Section 4 introduces a pairwise matching market and obtains indeterminacy.
In Section 5, a many-to-many exchange market attains the unique equilibrium. Section 6 considers a Walrasian model with and without the limited participation of buyers and sellers, andstudies the emergence of indeterminacy. Section 7 presents the conclusion.

3

population

population

60

60

40

40

$0
$50
Pairwise matching

money

$0
$50 $75
Many-to-many trademoney

Figure 1: Distribution of money holdings

2

A simple example

This section proposes a simple example and provides the intuition for the main result. The
example is presented in a reduced-form version of the models in the later sections. There is amarket in which buyers and sellers exchange money and goods every period. The population is100. The money supply is fixed at $3000. Disregarding consumption, production, and incentivesof agents, I assume only a money transfer rule: an agent holding positive units of money enters
the buyer side and spends all the money holding. An agent holding no cash enters the seller
side and receives money from the buyer side.
I propose two alternative market structures.
1. Pairwise matching: a random draw decides the matchings of one buyer and one seller.
Each buyer gives the whole amount of money to the partner. The number of matchings isthe smaller of the population of either the buyer or the seller side,
2. Many-to-many exchange: all agents on the buyer side pay all money holdings. The money
is distributed equally to agents on the seller side.
Pairwise matching represents a search theory of money with the most eﬃcient matching function, that is, there are no unmatched agents on the smaller population side. Many-to-manyexchange is interpreted as Shapley and Shubik (1977)’s market game or Walrasian market.
Suppose there is a money holding distribution in which 60 agents hold $50 each and 40
people hold no cash3 . Under the pairwise trade assumption, the number of matchings is 40 =1 Jean et al. (2010) point out the importance of strategic coordination issues. It is required for multiple equilibria in general.
Since indeterminacy following Green and Zhou (1998) arises with a continuum of equilibria, additional conditions provided by
Kamiya and Shimizu (2006) are necessary.
2 These studies do not report indeterminacy, possibly because of Walrasian markets, which indeed allow many-to-many matching.
Another diﬀerence is that these studies consider only equilibria with continuous and diﬀerentiable value functions; indeterminacy
often arises with step value functions in the literature whereas the value functions in this study have kinks.
3 This example considers only distribution with two masses.

4min{60, 40}. All 40 sellers hold $50 in the next period. There are 40 among 60 buyers who
exhaust their cash. As in the left diagram of Figure 1, the money holding distribution is
stationary. In general, all the distributions in which x people hold $3000/x and 1 − x holdnothing can be steady-state equilibria. The pairwise trade market exchanges the same numberof buyers and sellers and keeps the distribution unchanged.
However, in the many-to-many trade case, all the buyers spend their money. In the next
period, 60 agents hold no cash, and 40 agents accumulate $75 each. Therefore, the distribution
is not stationary as shown in the right diagram of Figure 1. The unique stationary distribution
is that 50 agents hold $60 each, and 50 agents hold no cash4 . The many-to-many exchange
procedure matches diﬀerent populations of buyers and sellers. It diﬀerentiates the inflow andoutflow of each point on the distribution and violates the stationarity.

3

Environment and Equilibrium ConceptThis section introduces a baseline structure of the price-posting model and the equilibrium
concept. The solution is provided with the pairwise trade procedure in Section 4, and with
many-to-many matching in Section 5.
3.1

Environment

Time is discrete and infinite. There are two types of goods: a consumption good and money.
Both goods are divisible. Money is storable while the consumption good is perishable. Moneysupply is constant at M through all periods.
There is a unit measure of homogeneous agents. They are anonymous, that is, the historyof actions and money holdings are unobserved by other agents5 . Each agent has a linear utility
function uq where q is the unit of the consumption. Each agent also has a production technology that converts her labor input to the consumption good. I assume capacity-constrained
production, following the industrial organization literature since Levitan and Shubik (1972).
Production of q units yields the following labor disutility:⎧⎪
if q = 0⎨0
C(q) = c
if q ∈ (0, 1] .
⎪⎩+∞if q > 1
In other words, each seller produces at most one unit of the consumption good with constantdisutility c. The function induces a strong incentive for sellers to lower prices because of the

4 Note that this study considers a distribution of money holdings to be stationary if it is unchanged every period. Cyclicalequilibria may also exist, for example, 60 agents hold $50 in even periods and 40 agents hold $75 in odd periods. This study focuses
only on the strict concept of stationary equilibrium, because it is the main issue in the existing literature. Cyclical equilibrium isanother worthwhile topic to study, for example, Arbuzov et al. (2019)
5 These assumptions preclude non-monetary equilibrium by keeping history, as in Kocherlakota (1998) and contagion equilibrium,
as in Araujo (2004).

5

zero marginal costs. This ensures a kind of Bertrand–Edgeworth competition among sellers
which makes the equilibrium simple. Each agent can either only consume or only produce ineach period. The utilities are discounted by a factor β.
Each period consists of four stages.
• Stage 1: each agent decides whether to become a seller or a buyer.
• Stage 2: each seller posts a unit price of the consumption good p in terms of money. The
sellers posting the same price p forms submarket p.
• Stage 3: each buyer observes the distribution of the posted prices and chooses one submarket.
• Stage 4: each buyer in each submarket p quotes demand q. In submarket p, given theparticipating buyers and sellers, either one-to-one matching or multilateral exchange procedure decides the allocations of money and the consumption good.
This study assumes that the consumption good is general, that is, each buyer demands goods
produced by any other sellers. The division of buyers and sellers is enough to generate a single
coincidence of wants and circulate money, as in Rocheteau and Wright (2005)6 .
3.2

Strategy and Equilibrium

A probability measure λ represents the distribution of money holdings. It satisfies the total pop%∞%∞
ulation 1 = 0 dλ(m) and the money supply M = 0 mdλ(m). I assume a symmetric strategy
in which all agents holding m units of money choose the same strategy in the equilibrium. The
strategy allows the use of money holding m as the index of agents7 .
Definition 1. A strategy of an agent is a sequence of actions over the four stages x(m) =&'xk (m) k=1,2,3,4 contingent on her money supply m at the beginning of stage 1.

• The action in stage 1, x1 (m), is buying or selling. Let S be the set of sellers and B be thatof buyers.
• In stage 2, each seller observes the set of buyers and sellers, and then posts a price p. No
buyer is active.
(p > 0 if x1 (m) ∈ S
x2 (m|x1 , B, S) =∅otherwise.
Let Sp be the set of sellers in submarket p. The set of all the submarkets is P = {p|Sp ̸= ∅}.

6 Another possible assumption is that there are K ≥ 3 types of agents and the population of each type is 1/K. A type k agent

consumes the good produced only by type k − 1. Since this economy deals with a price-posting procedure, a type k buyer can go
to a type k − 1 seller’s submarket. Hence, the types can be ignored without loss of generality.
7 In other words, each agent chooses a pure strategy. The assumption does not lose generality, because only a measure-zero agent
at a margin may choose a mixed strategy. Thus, to consider a deviation from an equilibrium, I assume a unilateral deviation. Eachagent’s state variable can be represented only by m instead of λ(m) because λ(m) is stationary.

6• In stage 3, each buyer enters one submarket. Each seller is inactive.
(&'p ∈ P if x1 (m) ∈ B
x3 m|x1 , B, (Sp )p∈P =∅otherwise.
Let Bp be the set of buyers in submarket p.

• In stage 4, each buyer posts a non-negative demand q given her budget constraint. Eachseller is inactive.
(&'q ∈ [0, m/p] if x1 (m) ∈ B and p = x3 (m)x4 m|x1 , x3 , (Sp )p∈P , (Bp )p∈P =∅if x1 (m) ∈ S.
The definition of equilibrium consists of three functions of money holding m at the beginningof the period: strategy x(m), the discounted sum of utilities V (m) and the distribution of money
holdings λ(m).
Definition 2. The equilibrium consists of stationary x(m), λ(m), and V (m) satisfying thefollowing conditions: (i) λ(m) is consistent with x(m), (ii) V (m) is induced by x(m) andλ(m), (iii) given λ(m) and V (m), x(m) is a subgame perfect equilibrium from stage 1 to stage
4, and (iv) V (0) > 0.
This concept is called stationary monetary equilibrium in the literature on the search theoryof money. While most studies8 consider a one-shot game every period, the present study derives
the subgame perfect equilibrium given the dynamic game in each period. Given stationary
λ(m), the discounted value depends only on m. Each agent keeps track of only m, because
the history of other agents is anonymous and the agent is atomic. Condition (iv) means thatengaging in monetary trade is more attractive than autarky. Then, there exist some buyers
and sellers exchange money and goods in the stationary monetary equilibrium. The equilibrium
necessary holds the following submarket:
Definition 3. Submarket p is active if λ(Sp ) > 0 and λ(Bp ) > 0.
Meanwhile, a submarket may exist that has measure zero agents. It is crucial to consider
each seller’s incentive because one seller’s deviation from the equilibrium makes such a measurezero submarket. To decide each buyer’s incentive in stage 3, the model needs to specify whatkind of matching and trades appear in such a submarket. This study follows the literature of
the large economy with atomless agents (e.g., Dubey et al. (1982)) and assumes an approximate
equilibrium. The equilibrium in a measure zero submarket is solved by assuming a small andpositive measure of sellers. Then, the equilibrium is obtained when the measure of market
participants converges to zero, which this study calls ε-submarket.
8 The models in the literature since Green and Zhou (1998) use a take-it-or-leave-it game, which is generally a two-shot game ingeneral because an oﬀerer takes into account the receiver’s response. However, all the models assume unobservable money holding of
the trade partner. The oﬀerer cares only about the average actions of all possible receivers. Then, the dynamic decision disappears.

7Definition 4. Suppose that there exists p ∈ P such that Sp ̸= ∅ and λ(Sp ) = 0. This is called
ε-submarket. The equilibrium is obtained by solving the buyers’ actions in stage 3 and 4 giventhat there is a suﬃciently small ε > 0 measure of sellers, and then taking ε → 0.
The following two lemmas are straightforward implications of the definition.
Lemma 1. V (m) is weakly increasing in the equilibrium.
Proof. Suppose x = {xt , xt+1 , · · · } is the optimal actions of agent m from time t to the future,
which induces V (m). Consider agent m′ > m. Indeed, agent m′ can also takes x and acquire
the same discounted utility V (m). By the optimality of the strategy of agent m′ , V (m′ ) ≥V (m).
Lemma 2. In the equilibrium, each agent becomes a buyer and spends money in finite periods
ahead.
Proof. By Lemma 1, V (m) ≥ V (0) > 0 for all m > 0 in the equilibrium. In the model, theonly way to acquire positive utility is by being a buyer and spending money. If agent m neverpurchases consumption goods, the discounted sum of utility is never positive.

4

Pairwise Trade with Random MatchingThis section builds a variant of the competitive search market, that is, stage 4 incorporates
a pairwise trade with random matching. I derive a set of equilibria diﬀering in both the
distribution of money holdings and real allocation.
In submarket p, the number of matchings is determined by the measure of sellers λ(Sp ) andthat of buyers λ(Bp ). I assume that the number of matchings is min{λ(Sp ), λ(Bp )} without
loss of generality. Acemoglu and Shimer (1999) call this matching function frictionless. This
assumption is for an explicit comparison of the pairwise trade and the many-to-many matchingmarkets. Indeterminacy can be obtained under any constant returns to scale matching function.
In a matching, buyer m optimally decides a purchase quantity qm under production limitqm ≤ 1 and budget constraint pqm ≤ m.
4.1

Equilibrium

There is a continuum of stationary equilibria, represented by Proposition 1. In each equilibrium,
all sellers post the same price p∗ . All buyers participate in submarket p∗ and spend all theircash. The single price p∗ is indeterminate. Each p∗ has an associated two-point equilibrium
distribution of money holdings.
There are two main properties that make the equilibrium tractable. The first is a buyer’s
pay-all strategy in stage 4. Given posted price p in stage 4, each buyer optimally chooses

8quantity q:
V4B (m|p) = max uq + βV (m − pq), s.t. pq ≤ m, q ≤ 1.
q(1)This equation defines a buyer’s discounted sum of utilities evaluated at stage 4 given a price p
as V4B (m|p). In the equilibrium, p = p∗ . The marginal gain from an increase in q is u. However,
it yields disutility βp∗ V ′ (m − p∗ q). The stationarity implies that the current period marginal
gain is strictly larger than future one. Saving one unit of money enables the purchase of 1/p∗
units of good in the future, because p∗ is unique and stationary. The utility will increase byu/p∗ . By the discount factor β < 1,βp∗ V ′ (m − pq ∗ ) ≤ βp∗ (u/p∗ ) = βu < u.
Equation (1) has a corner solution in which either p∗ q = m or q = 1 holds.
The second property making the equilibrium tractable is the single-price equilibrium instages 2 and 3. Suppose that all the buyers hold m = p∗ . Then, each seller posts the same
price p∗ and each buyer necessarily chooses submarket p∗ . This is a Nash equilibrium. By thecorner solution in Equation (1), each buyer chooses q = 1 and p∗ q = m = p∗ . No seller has anincentive to oﬀer a diﬀerent price. An oﬀer p > p∗ does not generate profit because each buyer
pays at most p∗ . A lower price p < p∗ is worse, because the sales amount never increases owing
to the capacity limit q ≤ 1.
Given these two properties, indeterminate distributions of money holdings appear. In the
two-point distribution9 , both the single price and the buyer’s money holdings are p∗ . The total
number of matchings is λ(0) = min{λ(0), λ(p∗ )}. By the first property, all matched buyers
spend p∗ , and the same number of sellers receive p∗ . The equilibrium exchanges the same
population of agents between two points, and hence, the distribution is stationary. In the caseof a slightly higher price p̃∗ > p∗ , there are fewer buyers. The second property ensures that the
new price p̃∗ is an equilibrium in stages 2 and 3. Then, the market again exchanges the same
number of buyers and sellers. Another equilibrium with a diﬀerent distribution arises.
Indeterminacy is real because the amount of production is λ(0) in each period. Eﬃciency
and the welfare change according to the single price p∗ . The result is unsurprising. While thismarket is a variant of the competitive search market, the main feature is similar to the random
search theories.
Proposition 1. Suppose βu − c > 0. For any p∗ such that)*
(1 − β)c∗M ≤p <M+1 ,β(u − c)an equilibrium exists in which

9 Some models, such as those of Green and Zhou (1998) and Zhou (1999), derive single-price equilibria with more than two points
distributions. Some agents accumulate money several times owing to random matching. However, such a distribution disappears
in the present model by the buyer/seller choice in stage 1. Interestingly, the many-to-many matching model presented in the next
section causes diﬀerent incentives for agents and gives rise to more than two point distributions.

91. λ(p∗ ) = M/p∗ and λ(0) = 1 − λ(p∗ ), otherwise λ(m) = 0 where λ(0) < λ(p∗ ),2. The strategy is x(m) = (S, p∗ , ∅, ∅) if m ≤ m̄ and x(m) = (B, ∅, p̃, max{1, m/p̃}) otherwise,
where the equilibrium matching probability of a buyer is λ̃ ≡ λ(0)/λ(p∗ ), the buyer’s choiceof the submarket isp̃ = arg max λ̃V4B (m|p),p

and the cutoﬀ amount of money holding in stage 1 ism̄ =p∗ [β λ̃u − (1 − β + β λ̃)c],λ̃u(1 + β λ̃)

3. The value function is defined by the following equations:
V (0) =β λ̃u − (1 − β + β λ̃)c,(1 − β)(1 + β λ̃)(2)V (m) = V (0) if m ≤ m̄+,m
V (m) = λ̃ ∗ u + βV (0) + β(1 − λ̃)V (m) if m̄ < m ≤ p∗p
V (m) = λ̃ [u + βV (m − p∗ )] + β(1 − λ̃)V (m) if p∗ < m

(3)
(4)
(5)

The strategy x(m) is defined as history dependent in one period. In the equilibrium, buyers
have only one option p∗ . The strategy does not depend on the history before the current period,
because each agent is anonymous and atomic. Equation (3) represents the discounted valuethat an agent with a low money holding becomes a seller in the current period. She holdsp∗ + m units of money in the next period. However, given the unique posted price p∗ and thecapacity constraint q ≤ 1, she can pay only p∗ in each matching. This means that m > 0 unitsof money are never used. Equation (4) describes an agent who becomes a buyer in the current
period. She meets a seller with probability λ̃ and spends all her cash. Finally, Equation (5)shows that, if an agent holds m > p∗ amount of money, she becomes a buyer and spends onlyp∗ units of production capacity.
Proof. This proof constructs a set of equilibria by the guess and verify method. I first assume
a set of V (m) and λ(m), which is characterized by the indeterminate single price p∗ . Then,
given assumption V (m) and λ(m), the subgame perfect equilibrium strategy x(m) is derived.
Each step is summarized as a lemma.
The best response actions are derived backward from stages 4 to 1.
stage 410

Given an oﬀered unit price p, agent m’s problem ismax uq + βV (m − pq), s.t. pq ≤ m, q ≤ 1.
q∈R+

Lemma 3. If p ≤-1−β(1−λ̃)
λ̃

.

(6)

p∗ , the buyer chooses q = 1.

λ̃uProof. By Equation (3), (4) and (5), the slope of V (m) is less than or equal to p∗ [1−β(1−;λ̃)]
hence, q = 1 maximizes the utility.

If p = p∗ , a buyer chooses q = 1 because [1 − β(1 − λ̃)]/λ̃ ≥ 1.
stage 2 and 3
I consider stages 2 and 3 one at a time and show the single price equilibrium.
Lemma 4. In stages 2 and 3, an equilibrium exists at which all sellers post p∗ .
Proof. Consider a seller’s strategy given that all other sellers post p∗ . If the seller posts p = p∗ ,she succeeds in matching with probability 1 and, by Lemma 3, acquires p∗ units of money. If
the seller posts p ̸= p∗ , an ε-market opens. The matching probability does not increase morethan 1. A lower price p < p∗ is not profitable. Moreover, a higher price p > p∗ does notraise revenue because all buyers spend p∗ units of money given the assumed money holding
distribution.

stage 1
Here, I calculate the discounted sum of the utilities of being a seller and a buyer depending
on m, and then derive the cutoﬀ m̄. By Lemma 4, each seller m posts p∗ and matches withprobability one. The discounted value is−c + βV (m + p∗ ).

(7)

Consider a case in which the agent chooses to be a buyer. If m < p∗ , she meets a seller withprobability λ̃ and spends all her money holding. Hence, the discounted utility is Equation (4).
If m ≥ p∗ , she can spend at most p∗ units of money, which means (5).
The following proposition shows the existence of m̄ which equates Equations (3) and (4).
The former is larger if m ≤ m̄ and smaller if m̄ < m ≤ p∗ . Then, Equation (5) gives a wealthy
agent’s incentive to choose the buyer side.
Lemma 5. There exists m̄ ≤ p∗ such that agent m becomes a seller if m ≤ m̄ and a buyer
otherwise.
11

Proof. In Appendix.
The following two lemmas show that x(m) induces λ(m) and V (m). The distribution λ(m)is clear because λ(0) sellers and buyers match and exchange p∗ money and goods. Indeed thisflow makes λ(m) stationary.
Lemma 6. The strategy x(m) makes λ(m) stationary.
Proof. The λ(0) measure of agents choose the seller side and oﬀer p∗ because 0 < m̄. By
p∗ > m̄, the λ(p∗ ) become buyers. The number of matching is λ(0) = min{λ(p∗ ), λ(0)}. In
each matching, the buyer quotes q = 1 and spends p∗ by Lemma 3. Therefore, all the m = 0
agents will accumulate p∗ and the λ(0) among λ(p∗ ) buyers exhaust their cash in the next
period. Therefore, λ is stationary under x.
Lemma 7. Strategy x(m) is consistent with V (m).
Proof. In Appendix.
Finally, the equilibrium requires V (0) > 0 to enable sellers to participate in the market. By
Equation (2), the condition is rewritten asλ̃ >(1 − β)c.
β(u − c)Since M < p∗ ≤ 2M , 0 < λ̃ ≤ 1. Such λ̃ exists if βu > c. The right-hand side also
- yields the.
(1−β)clower bound of p∗ . The two-point distribution implies p∗ = M (λ̃+1), then p∗ > M β(u−c)
+1 .
!

5

Many-to-many matching market

This section proposes another market to that in Section 4, a many-to-many matching, and%obtains the unique equilibrium. Suppose that there are QSp = m∈Sp dλ(m) measure of sellers insubmarket p. Since each seller produces at most one unit, QSp is interpreted as the aggregate sup%ply. An agent m posts a quote q(m) ∈ [0, m/p]. The total amount is QB
p = m∈Bp q(m)dλ(m),which is interpreted as the aggregate demand.
Definition 5. In submarket p, the many-to-many matching procedure yields an allocation suchthat every seller exchanges qpS consumption good for pqpS money and each buyer m trades pqpB (m)money for qpB (m) good defined as
(
/
(
/
B
S
Q
Q
p
p
qpS = min 1, S , qpB (m) = q(m) · min 1, B .
Qp
Qp

12

S
B
Call submarket p excess supply if QSp > QB
p , balanced if Qp = Qp , and excess demand otherwise.
In an excess supply submarket, each buyer acquires the exact posted quantity q(m). Every seller
produces less than 1 amount to meet the demand. In an excess demand submarket, each buyer
obtains the posted good reduced by the same proportion QSp /QB
p . In contrast to the pairwise
matching market, the many-to-many market allows all agents to trade. Instead of the rationing,each quantity of trade is adjusted.
This procedure is related to the market game of Shapley and Shubik (1977). In the market
S
game, the nominal price per unit is determined by the demand/supply ratio QB
p /Qp . By
contrast, in the present study, the price per unit is predetermined by the sellers’ postings. The
S
10demand/supply ratio QB
p /Qp determines only the allocation of the consumption good .
This amount of trade is defined only if QSp > 0. Therefore, the ε-submarket assumption in
Definition 4 is crucial. To consider one seller’s deviation to p, the consequence depends on how
many buyers enter submarket p and how much they quote11 .

5.1

Equilibrium

The unique equilibrium is characterized by the single price p∗ . Given a set of parameters, p∗and λ(m) are uniquely determined. The main diﬀerence between the pairwise matching andthe many-to-many markets is that the latter possibly generates an equilibrium with more than
two masses. Agents hold 0, p∗ , 2p∗ , · · · , N p∗ money. Note that this is not an indeterminacy: a
set of parameters uniquely pins down both p∗ and N .
To provide intuition, consider the case of N = 1, which gives rise to a similar two-point
distribution. This case arises under some parameters. In the equilibrium, each agent with 0
units of money becomes a seller and earns p∗ units of money. In the next period, the agent
becomes a buyer and spends p∗ . Because of no rationing, the equilibrium distribution of money
holdings is unique. The measure of no-cash agents is λ(0) and they hold p∗ in the next period.
Therefore the distribution is stationary only if λ(0) = λ(p∗ ).
The equilibrium is characterized by a similar strategy as the one-to-one matching case. In
stage 4, each buyer pays all money holdings. A buyer’s problem is written as(/
QSp
V4B (m|p) = max uqB + βV (m − pqB ), s.t. pq ≤ m, qB = q · min 1, B ,(8)qQpwhich is slightly diﬀerent from the pairwise matching market. The consumption amount isadjusted by the seller/buyer ratio, and the capacity constraint q ≤ 1 does not bind. Nonetheless,
pq = m holds at the optimum because of the stationary price p∗ and the discount factor β.
10 In a market game, a good is never allocated to a person who derives no utility from it, such as a seller in my model. However,
my model allows the possibility that there remains a good left unsold.
11 Note that a submarket with a positive measure of sellers and zero-measure buyers is straightforward: it is an excess supply
submarket with qpS = 0 and qpB (m) = q(m)13The single price equilibrium with the unique balanced submarket holds in stages 2 and3. In the pairwise trade market, Bertrand–Edgeworth competition gives sellers lower prices
and increases the probability of matching. In the many-to-many market, a cheaper oﬀer may
attract more buyers and raise revenue. Sellers in an excess supply submarket have this incentiveof deviation, because they produce less than 1 unit each. Then, excess supply submarkets
disappear. Excess demand submarkets also do not exist. In such a submarket, buyers purchaseless amount than they desire. Hence, some sellers may post slightly higher prices. Some buyers
may also deviate and choose the new submarkets to purchase more.
In an equilibrium with N > 1, the distribution satisfies λ(0) = λ(p∗ ) = λ(2p∗ ) = · · · =λ(N p∗ ). Similarly, there is a unique balanced submarket p∗ . An agent m = 0 becomes a seller
first and earns p∗ . Then, she chooses a seller again and saves 2p∗ . She selects the seller sidein a row until m = N p∗ . After that, she becomes a buyer and spends N p∗ . In other words,
the cutoﬀ level of money holding in stage 1 is between (N − 1)p∗ and N p∗ . This one-way
process is similar to Menzio et al. (2013). The submarket is balanced, because, on the sellerside, N/(N + 1) measure of sellers supplies goods. In the buyer side, 1/(N + 1) population ofbuyers pay N p∗ amount of money. Hence, the total demand is also N/(N + 1).
This N > 1 distribution emerges because each buyer is able to purchase more than one unit ofgood in one period. Consider discounted values from the current period to three periods ahead
without matching friction. In the pairwise matching market, a seller obtains −c+βu−β 2 c+β 3 u.
If she accumulates N = 2, then it would be −c−βc+β 2 u+β 3 u, which is strictly worse, owing to
β < 1. However, in the many-to-many matching market, she may accumulate N > 1, because
she is able to spend all the amount in one period. A discounted value −c − βc − β 2 c + β 3 3u is
possible. It may be better if u is suﬃciently larger than c and that β is close enough to 1.
This uniqueness result can be interpreted in a general framework provided by Kamiya andShimizu (2006), which mainly considers N -point distribution with the single a price p∗ . Stationarity holds by equalities of inflow and outflow at m = 0, p∗ , 2p∗ , . . . , N p∗ . They are written
as a system of equations to determine the distribution. Kamiya and Shimizu (2006) show theexistence of a hidden identity in the system. It is derived by an assumption that, besides the
population flow, the amounts of money transferred among diﬀerent points are balanced bothon and oﬀ paths. The framework of Kamiya and Shimizu (2006) does not explicitly depends on
search markets; however, this identity still relies on a kind of one-to-one matching assumption.
The many-to-many trade market does not share the identity.
Proposition 2. If an integer N satisfies the following condition:[(1 − β N +1 ) − (1 − β)N ]u > (1 − β)c ≥ [(1 − β N +2 ) − (1 − β)(N + 1)]u,
then there exists the unique equilibrium where
1. The distribution of money holdings is λ(m) = 1/(N + 1) if m = 0, p∗ , 2p∗ , . . . , N p∗ withp∗ = 2M/N , otherwise λ(m) = 0.
14

2. The value function V (m) is recursively defined as- N.
β N N u − 1−βc1−βV (0) =,
1 − β N +1
V (m) = −c + βV (m + p∗ ) if m < m̄,um
V (m) = ∗ + βV (0) if m ≥ m̄,
p
where
+
,
∗ βu − cm̄ = p− V (0) ,1−β

3. The strategy is x(m) = (S, p∗ , ∅, ∅) if m < m̄ and x(m) = (B, ∅, p̃, m/p̃}) if m ≥ m̄ wherep̃ = arg maxp λ̃V4B (m|p).
The proof consists of several necessary conditions of the equilibrium, which are summarized
as lemmas. First, the equilibrium value function V (m) is strictly increasing (Lemma 8). An
agent accumulating an additional small amount of money has a chance to spend it and yield
more utility. It is because the capacity constraint does not bind. The equilibrium does notcontain any excess supply submarket (Lemma 9). A seller’s deviation to oﬀer a lower pricebenefits buyers as well as the seller, because lower supply/demand ratio is generated. Similarly,
any excess demand submarket disappears because a higher price brings benefits to a seller andimproves the quantity adjustment ratio for a buyer (Lemma 11). Therefore, only balanced
submarkets may exist in the equilibrium. The balanced submarket must be unique; otherwise,
each buyer chooses a lower price submarket (Lemma 10). By the linear utility function, eachbuyer is incentivized to spend all their money holdings given the unique posted price p∗ (Lemma
12). This makes the value function continuous (Lemma 13). No jump on V (m) makes a unique
cutoﬀ action in stage 1 such that becoming a buyer if m ≥ m̄ and a seller otherwise (Lemma
14). Finally, the condition that the supply and demand are balanced in the unique submarketp∗ makes the unique distribution of money holdings.
Proof. Lemma 8 to Lemma 15 obtain necessary conditions of the equilibrium. According tothese lemmas, if the equilibrium exists, it is unique and holds the single balanced submarketp∗ . Lemma 16 assures the existence of this equilibrium.
Lemma 8. V (m) is strictly increasing in the equilibrium.
Proof. Suppose x = {xt , xt+1 , · · · } is the optimal actions of agent m from the current period t,which induces V (m). By Lemma 2, an agent becomes a buyer and uses money in finite periods
ahead, that is, there exists T ≥ t, p > 0, and q > 0, such that xT = {B, ∅, p, q}. Consider asequence of actions of agent holding m + σ where σ > 0. Take x̃ = {x̃t , x̃t+1 , · · · } such that
15

0
1
S
xτ = x̃τ for all τ ̸= T and xT = {B, ∅, p, q̃}, where q̃ = (m + σ)/p · min{1, (QBp /Qp )}. Let Ṽbe the associated discounted sum of utility. x̃ is a feasible action of agent m + σ. By q̃ > q,
Ṽ > V (m). By the optimality, V (m + σ) ≥ Ṽ . Therefore, V (m + σ) > V (m).
Lemma 9. There is no excess supply submarket in the equilibrium.
Proof. Suppose there is an excess supply submarket p. This lemma considers a seller’s incentive
to post p − σ where σ > 0 is suﬃciently small. By Definition 4, QSp−σ = ε < QSp .
In submarket p, there is a buyer m who consumes q and saves m′ = m−pq. If submarket p−σ
is excess supply or balanced, while keeping m′ , the buyer can consume (m′ − m)/(p − σ) > q.
Every buyer in submarket p deviates while submarket p − σ is excess supply or balanced. In
the equilibrium, submarket p − σ is excess demand under ε → 0.
Consider a seller in submarket p. Her revenue is pQSp /QB
p < p. If she deviates and postsp − σ, submarket p − σ is excess demand. Then, she earns p − σ, which is higher than pQSp /QB
pfor suﬃciently small σ. Since V (m) is strictly increasing, posting p is not optimal.
Lemma 10. The submarket is unique in the equilibrium.
Proof. By Lemma 9, only excess demand and/or balanced submarkets exist in the equilibrium.
Suppose there are two submarkets p < p′ . Each seller provides 1 unit in both submarkets. The
revenue is higher in p′ ; hence, given strictly increasing V (m), every seller in submarket p moves
to p′ . This does not satisfy the equilibrium.
Lemma 11. The unique submarket is balanced in the equilibrium.
Proof. By Lemma 9, there is no excess supply submarket. Suppose that the unique submarketp∗ is excess demand. Suppose a seller posts p∗ (1 + σ) where σ > 0 is suﬃciently small12 . If
submarket p∗ (1 + σ) is also excess demand, the seller has a strict incentive of deviation. Thisis because that she sells 1 unit of goods in both submarkets and V (m) is strictly increasing.
Hereafter, I show that submarket p∗ (1+σ) is excess demand. Suppose x(m) = {xt , xt+1 , · · · }is the optimal actions of a buyer m from the current period t, which induces V (m). Suppose
the buyer chooses qt in period t. By Lemma 2, there is a period T > t when this agent becomesa buyer and spends a positive amount of money again. Let xT = {B, ∅, p∗ , qT }. This buyer’s∗
SB∗
posts qt and qT satisfy the constraints: qt ≤ (QSp∗ /QBp∗ )(m/p ) and qT ≤ (Qp∗ /Qp∗ )(mT /p ),where mT is her money holding at the beginning of period T .
On the contrary, assume that submarket p∗ (1 + σ) is excess supply or balanced. Consider aseries of actions x̃ = {x̃t , x̃t+1 , · · · }, where xτ = x̃τ for all τ ̸∈ {t, T }. Assume x̃t = {B, ∅, p∗ (1 +σ), q(1 + δ)}, where
2
3
QSp∗
1
δ=
−
σ.
1+σ
QB
p∗
12 I use p∗ (1 + σ) instead of p∗ + σ for simplifying the equations.

16

QS∗

This implies δ > 0 and δ → 0 as σ → 0. Since QpB +σ > 1, there exist σ > 0 which satisfiesp∗ +σ

the budget constraint, that is, p∗ qt (1 + σ)(1 + δ) ≤ m. Then, the agent’s money holding at the
beginning of period T is mT − p∗ qt (σ + δ + σδ). Let x̃T = {B, ∅, p∗ , q̃T } whereq̃T = qT −QSp∗ p∗ qt (σ + δ + σδ)
,
p∗
QB
p∗

which satisfies the budget constraint.
The discounted utility of periods t and T under x(m) is uqt + β T −t uqT . That for x̃ is
uqt (1 + σ) + β T −t uq̃T . Therefore, the agent has a strict incentive to move to p∗ (1 + σ) if
uqt σ > β T −t u(qT − q̃T ) = β T −t uqt (QSp∗ /QBp∗ )[σ + δ + δσ]
By the definition of δ, the inequality is 1 > β T −t (1 + σ). This is satisfied for suﬃciently small
σ > 0.
Lemma 12. Every buyer in the unique submarket exhausts cash, that is, p∗ q = m.
Proof. Suppose, on the contrary, that buyer m decides mt+1 = m − p∗ q > 0 in period t andchooses a series of actions x(m) thereafter. By Lemma 2, there exists period T in which
agent m becomes a buyer and mT − mT +1 > 0. Consider x̃(m) where m̃t+1 = mt+1 + σ
and m̃T − m̃T +1 = mT − mT +1 − σ for suﬃciently small σ > 0. Indeed it satisfies the budget
constraint. The period t and T actions of x(m) yields u(mT − mT +1 )/p∗ + β T −t (m̃T − m̃T +1 )/p∗ ,while x̃(m) leads to u(mT − mT +1 + σ)/p∗ + β T −t (m̃T − m̃T +1 − σ)/p∗ . x̃(m) yields strictly
higher utility than x(m).
Lemma 13. The value function V (m) is continuous. For all m′ > m ≥ 0, V (m′ ) − V (m) ≤u(m′ − m)/p∗ .
Proof. Consider cases in which agent m becomes a buyer for the first time in n period ahead,since period t for n = 0, 1, 2, · · · . By Lemma 2, there exists an upper bound N such thatn ≤ N . Agent m chooses mt+n+1 = 0 by Lemma 12. Define Ṽn (m) as
Ṽ0 (m) =um+ βV (0)p∗andṼn (m) = −cn4

β i−1 + β n u(m + np∗ ) + β n+1 V (0)i=1

for n = 0, 1, 2, · · · , N . Then, V (m) = max{Ṽ0 (m), Ṽ1 (m), . . . , ṼN (m)}. Since Ṽn (m) for all n =0, . . . , N and max function are continuous, V (m) is also continuous. For all n = 0, . . . , N andm′ > m ≥ 0, Ṽn (m′ )−Ṽn (m) ≤ (u/p∗ )(m′ −m). Therefore, V (m′ )−V (m) ≤ (u/p∗ )(m′ −m).
17

Lemma 14. There exists m̄ such that an agent becomes a buyer if m > m̄ and a seller otherwise.
Proof. Let VS (m) = −c + βV (m + p∗ ) and VB (m) = um/p∗ + βV (0). Then, V (m) =max{VS (m), VB (m)}. By V (0) > 0, V (0) = VS (0) > VB (0). Let m̄ be the minimum money
holding of being a buyer, that is, VS (m) > VB (m) for all m < m̄. By the continuity ofV (m), VS (m̄) = VB (m̄). Take m > m̄. VB (m) − VB (m̄) = u(m − m̄)/p∗ . By Lemma 13,
VS (m) − VS (m̄) = β[V (m + p∗ ) − V (m)] < u(m − m̄)/p∗ . Therefore, VB (m) > VS (m) for allm > m̄.
Lemma 15. Suppose the unique equilibrium posted price p∗ satisfies (N −1)p∗ < m̄ ≤ N p∗ witha positive integer N . Then, the equilibrium money holding distribution is λ(m) = 1/(N + 1) if
m = 0, p∗ , 2p∗ , . . . , N p∗ and λ(m) = 0 otherwise.
Proof. Given Lemmas 11 and 14, the unique submarket has price p∗ , and N ∈ N exists suchthat (N − 1)p∗ < m̄ ≤ N p∗ . Let If (M) and Of (M) denote the inflow to and outflow fromthe set of money holdings M ⊂ R in terms of the measure of agents. The money holding
distribution λ is stationary if If (M) = Of (M) = λ(M) for all M. By Lemma 14 and Lemma
12, If (0) = Of ({m|m > m̄}). Each agent m ≤ m̄ become a seller and earns p∗ . Hence,
If ({m|np∗ < m < (n + 1)p∗ for n = 0, 1, 2, . . . , N − 1}) = 0. Then, λ(0) = λ(p∗ ) = λ(2p∗ ) =&'&'· · · = λ (N −1)p∗ = λ {m|m ≥ m̄} . By the single price p∗ again, If ({m|m ≥ m̄}) = If (N p∗ ).
The stationary distribution satisfies λ(0) = λ(p∗ ) = · · · = λ(p∗ N ) = 1/(N + 1). The single
price satisfies p∗ = 2M/N , because the total money supply is M .
The last lemma shows the existence of the unique equilibrium.
Lemma 16. If a positive integer N satisfies [(1 − β N +1 ) − (1 − β)N ]u > (1 − β)c ≥ [(1 −β N +2 ) − (1 − β)(N + 1)], then a unique equilibrium with N -point distribution exists.
Proof. Suppose that the equilibrium has N masses as in Lemma 15. Let VN (m) be the discounted utilities in the equilibrium with a positive integer N , VNS (m) = −c + βVN (m + p∗ ) be
the value of selling goods, and VNB (m) = um/p∗ + βVN (0) be the value when purchasing goods
and spending all money holding. On the equilibrium path, agent m = 0 sells goods N times,
being a buyer, and spends all money holding. Therefore,
- N.
N
−1
β N N u − 1−βc41−βVN (0) = −cβ i + β N N u + β N +1 VN (0) ⇔ VN (0) =.
(9)1 − β N +1
i=0
The equilibrium exists if VNS (p∗ n) > VNB (p∗ n) for all n = 0, 1, . . . , N − 1 and VNS (p∗ N ) ≤VNB (p∗ N ). The incentive condition for n ≤ N − 1 isVN (np∗ ) = VNS (np∗ ) > VNB (np∗ ).

(10)

18

By Lemma 13,
& '
&
'
& '
&
'
& '
&
'
VNS np∗ − VNS (n − 1)p∗ = VN np∗ − VN (n − 1)p∗ ≤ u = VNB np∗ − VNB (n − 1)p∗&'&'for n = 1, . . . , N − 1. Therefore, it is suﬃcient to show VNS (N − 1)p∗ > VNB (N − 1)p∗ forEquation (10). This is rewritten as−c + βV (N p∗ ) > (N − 1)u + βV (0).
By Equation (9), the condition is rewritten by f1 (N ) asf1 (N ) ≡ [(1 − β N +1 ) − (1 − β)N ]u > (1 − β)c.
Similarly, the incentive condition for m = N is&'&'
VNS N p∗ ≤ VNB N p∗ .
It is characterized by f2 (N ) asf2 (N ) ≡ [β(1 − β N +1 ) − (1 − β)N ]u ≤ (1 − β)c.
Then,f2 (N ) = β(1−β N +1 )−(1−β)N = 1−β N +2 −1+β−(1−β)N = 1−β N +2 −(1−β)(N +1) = f1 (N +1)
The sequence of f1 (N ) and f2 (N ) satisfies
f1 (1) > f2 (1) = f1 (2) > f2 (2) = f1 (3) > f2 (3) = · · · .
Therefore, if f1 (N ) > (1 − β)c ≥ f2 (N ) is satisfied, a unique equilibrium with N -point mass
distribution exists.
The proof of Lemma 16 defines the cutoﬀ amount m̄ asum̄
βum̄+ V (0) = −c + ∗ + βV (0) = V S (m̄)
p∗p+,βu − c⇔ m̄ = p∗− V (0)1−β
V B (m̄) =It also solves the value function V (m) recursively as in this proposition.
!

6

Walrasian market

In this section, I show a similar result of the indeterminacy in a Walrasian market. Consider
a unit measure of a continuum of agents who have the same preference as in the price-posting19

model. Similarly, there are consumption good and fixed amount of money M . There is one
competitive market which has two sides: the seller and the buyer. As the price taking casein Rocheteau and Wright (2005), money is still essential. Each agent decides which side toenter at the beginning of each period. I consider two cases about the centralized market: norestriction and limited participation.
6.1

No Restriction

I derive an equilibrium in which the unit price p is stationary. Each agent solves
V (m) = max{VS (m), VB (m)},VS (m) = max −c + βV (m + pq),q∈[0,1]uq+ βV (m − pq),q∈[0,m/p] p

VB (m) = max

where V (m) is the ex-ante discounted utility, VS (m) is the ex-post value of a seller, and VB (m)is that of a buyer. Since the utility function is linear, no buyer has an incentive to save money;
hence, VB (m) = um/p + βV (0). The fixed costs of production allows the seller to produce
q = 1. Therefore,
56um
V (m) = max −c + βV (m + p),+ βV (0) .
p
The solution is( βu−cV (m) =

1−β 2
.
βu−c
um
β
+
p
1−β 2

if m ≤ m̄,if m > m̄,

where m̄ = p(βu−c). The optimal behavior is that of being a seller if m ≤ m̄ and of being au(1+β)
buyer otherwise. This satisfies m̄ < p.
Let the money holding distribution be 0 < λ(p) < 1 and λ(0) = 1 − λ(p). The transition ofmoney is similar: an agent m = 0 becomes a seller and earns p, and then, becomes a buyer andexhausts her cash.
The aggregate good demand is λ(p) · p/p = λ(p), while the supply is λ(0). The equilibrium
price p∗ solves λ(p∗ ) = λ(0) = 1/2. This is the unique distribution.
6.2

Limited participation

Next, the Walrasian market incorporates a type of pairwise matching assumption: the measuresof participating buyers and sellers must be equal. Let λ(B) and λ(S) be the measures of agents
who are willing to enter the buyer side and seller side, respectively. The market randomly
chooses min{h(S), h(B)} measure of buyers and sellers to enter and make the others autarky.
20

Consider a case in which λ(B) ≥ λ(S) and define λ̃ = λ(S)/λ(B). Given a stationary price p,
the values of a potential seller and buyer are
VS (m) = max −c + βV (m + pq),q∈[0,1]

VB (m) = λ̃ · max

q∈[0,m/p]+,uq+ βV (m − pq) + (1 − λ̃)βV (m).
pGiven the similar actions of each buyer and seller, the ex-ante value function is
5
+
,
6
um
V (m) = max −c + βV (m + p), λ̃+ βV (0) + (1 − λ̃)βV (m) .
pThen, the solution is⎧⎨ β λ̃u−[1−β(1−λ̃)]c
1−β(1−λ̃+β λ̃) .
V (m) =β λ̃u−[1−β(1−λ̃)]c⎩ λ̃um+βp[1−β(1−λ̃)]
1−β(1−λ̃+β λ̃)

if m ≤ m̄,if m > m̄,

where m̄ = p(βu−c).
u(1+β)
Assume a similar distribution of money holdings with two masses at 0 and p. The optimal
choice is the same as in the no-restriction case. However, the measure of participating buyers
in this market is λ(0) = min{λ(0), λ(p)} instead of λ(p) in the no-restriction case. Then,the aggregate demand is λ(0) · p/p = λ(0). Indeed the aggregate supply is λ(0); hence, theequilibrium holds with a continuum of stationary prices p∗ .
It is worthwhile to compare the result with Kamiya and Shimizu (2013), who show the indeterminacy in an auction market and uniqueness in a Walrasian market in a similar environment.
My result indicates that the uniqueness might not depend on Walrasian competitive price determination. From the viewpoint of pairwise matching, their indeterminacy in the auction market
seems to depend on the indivisible good assumption, that is, the measure of successful sellers
is always the same as that of the buyers. They also conjecture that a uniqueness result mighthold in an all-pay auction market, which is one example of many-to-many matching.

7

Conclusion

This study analyzes the indeterminacy of stationary distributions of money holdings in a decentralized economy. A price-posting model with two alternative market structures is considered.
The equilibrium is indeterminate with the pairwise trade market but unique in the many-tomany matching. A Walrasian market model also gives arise to a similar result under a similar
participation restriction.
The market structures in this study are extremes in opposite directions. On the one hand, thepairwise trade assumption completely excludes agents who fail to make matches in the market.
21

On the other hand, the many-to-many matching market allows all agents to be involved. Aninteresting extension would be a mixed case in which some agents match bilaterally, some havemultilateral negotiations, and some are excluded. I conjecture that the unique equilibrium may
still arise because only the exact equality of the measures of buyer and seller make indeterminate
distributions. The mixed case might be useful as a method of equilibrium selection in an
indeterminate model, such as introducing the small possibility of many-to-many matching in asearch market.

22

Appendix

Proof of Lemma 5
I check the incentives of three cases: m < m̄, m̄ ≤ m ≤ p∗ , and m > p∗ . Since the last caseis independent from m̄, I consider this one first and show that agent m becomes a buyer. Thisis equivalent to Equation (5) and (7) satisfying−c + βV (m + p∗ ) ≤ λ̃ [u + βV (m − p∗ )] + (1 − λ̃)βV (m),0101 λ̃u + c
⇔ λ̃ V (m + p∗ ) − V (m − p∗ ) + (1 − λ̃) V (m + p∗ ) − V (m) ≤.
βFor later use, by Equation 4,
V (p∗ ) − V (0) =hu − c.
1 + βh(A.1)

The equations of V (m) implies V (m+p∗ )−V (m) ≤ V (p∗ )−V (0) and V (m+p∗ )−V (m − p∗ ) ≤V (2p∗ ) − V (0) ≤ 2[V (p∗ ) − V (0)] for all m ≥ 0. Hence, it is suﬃcient to show
(1 + λ̃)[V (p∗ ) − V (0)] ≤λ̃u + c.
β

By Equation (A.1),⇔(1 + λ̃)(λ̃u + c)
λ̃u + c≤.
β
1 + β λ̃

Since β(1 + λ̃) ≤ 1 + β λ̃, the condition is satisfied.
The next case is m < m̄, in which each agent becomes a seller. First, suppose m < m̄. If
agent m chooses to be a seller, the discounted value, by Equation (4), isβV (m + p∗ ) = λ̃ [u + βV (m)] + β(1 − λ̃)V (m + p∗ ).
By Equation (5),βV (p∗ ) = λ̃ [u + βV (0)] + β(1 − λ̃)V (p∗ ).
Since V (m) = V (0), V (m + p∗ ) = V (p∗ ). Then,
7
8
λ̃u
+
c
−c + βV (p∗ ) = −c + β+ V (0)1 + β λ̃=β(λ̃u + c) − c(1 + β λ̃)
β λ̃u − c(1 − β + β λ̃)+ βV (0) =+ βV (0).
1 + β λ̃1 + β λ̃23

If the agent selects the buyer side, the discounted value is
)
*
mu
λ̃mu
λ̃
+
βV
(0)
+ β(1 − λ̃)V (m) = ∗ + βV (0).
p∗pTherefore, I need to show
β λ̃u − c(1 − β + β λ̃)
λ̃mu
> ∗ .
p
1 + β λ̃This can be rewritten asp∗ [β λ̃u − c(1 − β + β λ̃)]> m,
λ̃u(1 + β λ̃)whose left-hand side is m̄.
Finally, I consider m̄ ≤ m < p∗ . Equation (4) is rewritten as+,m[1 − β(1 − λ̃)]V (m) = λ̃ ∗ u + βV (0) .
pThis is satisfied if m = m̄. Then,⇒ [1 − β(1 − λ̃)][V (m) − V (0)] = [1 − β(1 − λ̃)][V (m) − V (m̄)] =Similarly, Equation (5) implies[1 − β(1 − λ̃)]V (m + p∗ ) = λ̃ [u + βV (m)] ,which also holds if m = 0. Then,[1 − β(1 − λ̃)][V (m + p∗ ) − V (p∗ )] =⇔ V (m + p∗ ) = V (p∗ ) +β λ̃2 u(m − m̄),[1 − β(1 − λ̃)]p∗

β λ̃2 u(m − m̄).
[1 − β(1 − λ̃)]2 p∗

If agent m̄ ≤ m < p∗ chooses to be a seller, the expected value is−c + βV (m + p∗ ) = −c + βV (p∗ ) +2

β 2 λ̃2 u(m − m̄)[1 − β(1 − λ̃)]2 p∗

3
λ̃u + cβ 2 λ̃2 u(m − m̄)= −c + β+ V (0) +1 + β λ̃p∗ [1 − β(1 − λ̃)]2=β λ̃u − (1 − β + β λ̃)cβ 2 λ̃2 u(m − m̄)++ βV (0)1 + β λ̃p∗ [1 − β(1 − λ̃)]2
24

'
λ̃u &
m
−
m̄
.
p∗

If she decides to be a buyer, it is
)
*
mu
λ̃
+ βV (0) + β(1 − λ̃)V (m)
p∗
2
3
)
*
mu
λ̃u(m − m̄)= λ̃+ βV (0) + β(1 − λ̃)+ V (0)p∗p∗ [1 − β(1 − λ̃)]=λ̃mu
λ̃u(m − m̄)+ β(1 − λ̃)+ βV (0).
∗∗pp [1 − β(1 − λ̃)]I need to show∗

−c + βV (m + p ) ≤ λ̃)*mu
+ βV (0) + β(1 − λ̃)V (m),p∗which is rewritten as
β λ̃u − (1 − β + β λ̃)cβ 2 λ̃2 u(m − m̄)λ̃muλ̃u(m − m̄)+≤ ∗ + β(1 − λ̃)∗2∗p
1 + β λ̃p [1 − β(1 − λ̃)]p [1 − β(1 − λ̃)]23β λ̃u − (1 − β + β λ̃)cβ λ̃u(m − m̄)
β λ̃
λ̃mu
⇔
+
− (1 − λ̃) ≤ ∗∗p
1 + β λ̃p [1 − β(1 − λ̃)] 1 − β(1 − λ̃)23p∗ [β λ̃u − (1 − β + β λ̃)c]β(m − m̄) (1 − λ̃)(1 − β) − λ̃2 β⇔+≤m
λ̃u(1 + β λ̃)
1 − β(1 − λ̃)
1 − β(1 − λ̃)
23β(m − m̄) (1 − λ̃)(1 − β) − λ̃2 β⇔ m̄ +≤m
1 − β(1 − λ̃)
1 − β(1 − λ̃)2323β(1 − λ̃)(1 − β) − λ̃2 β 2β(1 − λ̃)(1 − β) − λ̃2 β 2⇔ m̄ 1 −≤m 1−[1 − β(1 − λ̃)]2[1 − β(1 − λ̃)]2⇔ m̄ ≤ m.

!

Proof of Lemma 7
This proof is divided into several cases. First, suppose an agent holds m < m̄. Given thestrategy x, she becomes a seller. Then, the Bellman equation satisfies
9:
V (m) = −c + βV (m + p∗ ) = −c + β λ̃ [u + βV (m)] + (1 − λ̃)V (m + p∗ )Given λ̃ = [1−λ(p∗ )]/λ(p∗ ) and λ(p∗ ) = M/p∗ , this equation is independent from m. Therefore,
V (0) = V (m) for m < m̄.
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Then, the expected return is−c + βV (m + p∗ ) = −c + βV (p∗ )
7
8
β λ̃u
β 2 λ̃
= −c ++V (0)1 − β(1 − λ̃)
1 − β(1 − λ̃)
7
8
β λ̃u
β 2 λ̃
β λ̃u − (1 − β + β λ̃)c
= −c ++
1 − β(1 − λ̃)
1 − β(1 − λ̃)(1 − β)(1 + β λ̃)
7788(1 − β)(1 + β λ̃) + β 2 λ̃
β 2 λ̃
=β λ̃u −+1 c
[1 − β(1 − λ̃)](1 − β)(1 + β λ̃)(1 − β)(1 + β λ̃)=

β λ̃u − (1 − β + β λ̃)c,(1 − β)(1 + β λ̃)(A.2)

which yields (3).
Second, if p∗ > m ≥ m̄, the agent becomes a buyer. Given x, the agent trades withprobability λ̃; then the discounted utility is)*mλ̃ ∗ u + βV (0) + β(1 − λ̃)V (m),(A.3)
pwhich is (4).
Finally, suppose m ≥ p∗ . The agent becomes a buyer and the expected utility isλ̃[u + βV (m − p∗ )] + β(1 − λ̃)V (m).

(A.4)

It is Equation (5).
!

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