DP2019-21

Efficiency of Monetary Exchange with Divisible
Fiat Money: An Experimental Approach
Kazuya KAMIYA
Hajime KOBAYASHI
Tatsuhiro SHICHIJO
Takashi SHIMIZU

October 28, 2019

Eﬃciency of Monetary Exchange with Divisible Fiat
Money: An Experimental Approach∗Kazuya Kamiya†, Hajime Kobayashi‡, Tatsuhiro Shichijo§, and Takashi Shimizu¶
October 28, 2019

Abstract
In this paper, we investigate a search model with divisible ﬁat money in a laboratory
setting where transaction prices are endogenously determined. In the model, there exist
welfare-ranked multiple stationary monetary equilibria and gift-giving equilibria. We ﬁndthat endogenizing transaction prices enhanced the coordination of subjects through monetary
exchange and deteriorated it through gift-giving. In other words, the subjects endogenously
reduced the trade friction of monetary exchanges. We also compare our experimental resultswith those in search models with exogenously given transaction prices.
Keywords: Real Indeterminacy, Random Matching, Divisible Money, Experiment, Equilibrium Selection, Gift-giving
JEL Classiﬁcation Number: C91, C92, D51, D83, E40∗

We thank Maria Bigoni, Marco Casari, John Duﬀy, Hirokazu Ishise, Masahiko Shibamoto and Takayuki
Tsuruga for their helpful comments. This research is partly conducted when the second author was visiting atUniversity of Bologna. He appreciates their hospitality. This research is ﬁnancially supported by the Nomura
Foundation for Social Science, a grant-in-aid from Zengin Foundation for Studies on Economics and Finance,
Grant-in-Aid for Scientiﬁc Research (B) from JSPS (16H03596) and Grant-in-Aid for Scientiﬁc Research (C) from
JSPS (24530207 for the second author, 26380242 for the third author, and 26380252 for the last author). Of
course, any remaining errors are our own.
†
Research Institute for Economics and Business Administration, Kobe University, 2-1 Rokkodai-cho Nada-ku,
Kobe, 657-8501 JAPAN (e-mail: kkamiya@rieb.kobe-u.ac.jp)‡
Faculty of Economics, Kansai University, 3-3-35 Yamate-cho, Suita 564-8680 JAPAN (e-mail: khajime@kansaiu.ac.jp)§
School of Economics, Osaka Prefecture University, 1-1 Gakuen-cho, Naka-ku, Sakai 599-8531 JAPAN (e-mail:shichijo@eco.osakafu-u.ac.jp)¶
Graduate School of Economics, Kobe University, 2-1 Rokkodai-cho Nada-ku, Kobe, 657-8501 JAPAN (e-mail:shimizu@econ.kobe-u.ac.jp)

1

Introduction

In the real-world economy, monetary exchange prevails, and gift-giving plays only an auxiliaryrole. In this paper, we study a search model with divisible ﬁat money in a laboratory setting
and investigate whether the coordination through monetary exchanges is higher than it is in
the case of indivisible money.1 In so-called ﬁrst-generation monetary search models, such as
Kiyotaki and Wright’s [11] model, both money and goods are indivisible, and the transaction
price is exogenously given. That is, one unit of goods is traded for one unit of money. If money
is divisible and agents can hold any amount of money, then a buyer and a seller can use any
amount of money in their transactions, and thus, a transaction price is endogenously determined.
The purpose of this paper is to show through experiments that endogenizing transaction prices
enhances the coordination of subjects through monetary exchange and deteriorate it throughgift-giving. We compare our experimental results with those in search models with exogenously
given transaction prices, such as Camera and Casari [2], Camera, Casari, and Bigoni [3], andDuﬀy and Ochs [5]. We also compare our results with Duﬀy and Puzzero [6], where both money
and goods are divisible and the goods are traded in search markets and in Walrasian markets.
The divisibility of money allows agents to trade using any amount of money, and thus,transaction prices are endogenized and opportunities for monetary exchange increase. Suppose
there are 6 agents and 3 units of money in an economy. If money is indivisible and each agent canhold at most one unit of money, then the money holdings distribution is uniquely determined,i.e., 3 agents have one unit of money and the other agents do not have money. On the other hand,
if money is divisible and agents can hold any amount of money, then a variety of money holdings
distribution can be chosen through monetary transactions. Indeed, Kamiya and Shimizu [9] [10]show that in monetary search models with divisible money, there generically exists a continuumof stationary equilibria. Therefore, a money holdings distribution with small trade friction canbe endogenously chosen. For example, suppose agents have 0, p, 2p, or 3p amounts of money andthe transaction price is p. Then agents with p, 2p and 3p (0, p, and 2p) can buy (sell). That is,monetary exchange opportunities increase compared with the case in which money is indivisible
and an agent can have at most one unit of money, where only an agent with money (without
money) can buy (sell). Of course, a very ineﬃcient allocation can also be endogenously chosen.
For example, suppose that one agent has all money while the other agents do not; then, in
most of the matched pairs, monetary exchange is infeasible. Therefore, we cannot theoretically
predict which allocation agents choose through monetary exchanges. It is worth noting that in
our experiments, such an ineﬃcient distribution was not chosen.
Our baseline model is a variant of Zhou’s [14] model with divisible ﬁat money. Agents are
randomly matched pairwise, and each matched pair trades indivisible goods of which utility isu and production cost is c, where u > c > 0. In each matched pair, the roles of buyer and seller
are randomly assigned. Bargaining over a price proceeds with the seller’s take-it-or-leave-it oﬀer.
We investigate the stationary equilibria, called a single price equilibrium (SPE), in which every
good is traded with a common price, say p, and the support of the stationary money holdings
1

In our experiments, the total amount of money is 600, and a subject can use an integer amount of money intransactions.; 600 considered to be suﬃciently large and money can be considered to be almost divisible.

2

distribution is {0, p}. Section 2 theoretically shows that there is a region of parameters in which
multiple SPEs exist. It is worth noting that in an SPE, agents play the game as if money is
indivisible, and they can hold at most one unit of money, i.e., the p amount of money is one
unit, and it is traded for one unit of goods.
We conducted experiments using the above model. We ﬁrst investigate whether gift-giving
occurred. To see this, we count the number of transactions with zero price and ﬁnd that it wasclose to zero, i.e., gift-giving rarely occurred. Next, we investigate monetary exchanges. We
calculate successful trade rates of monetary exchanges and ﬁnd that it was more than or close
to that of the most eﬃcient SPE (MSPE). More precisely, there were four treatments, and thesuccessful trade rate was more than that of the MSPE in one treatment, while it was close tothat of the MSPE in the other treatments. For the money holdings, a signiﬁcant number ofsubjects had more than or equal to twice the transaction price in all treatments. Therefore,
we can conclude that the agents did not play any SPEs. We also investigate whether an initialmoney holding aﬀects the eﬃciency of trades and ﬁnd that successful trade rates increased as
the initial money holding became equal.
Some experiments using monetary search models are closely related to ours. Camera andCasari [2] investigated an equilibrium selection problem in an economy with indivisible ﬁatmoney. In their model, the agents can hold at most two units of money and can use only
one unit in transactions; that is, a transaction price is exogenously given. They found thatthe subjects used both monetary transactions and gift-giving, and the former was 61.4 % whenmonetary exchange was feasible and the latter was 12.5% when it was infeasible. Camera, Casari,and Bigoni [3] investigated the same game with various numbers of subjects. They observed thatthe successful trade rate of monetary transaction increased and that of gift-giving decreased as
the number of subjects increased. Duﬀy and Puzzello [6] investigated an equilibrium selection
problem in an economy with divisible ﬁat money. Their model is a variant of Lagos and Wright’s[12] model, and there are unique stationary monetary equilibrium and multiple non-monetarygift-giving equilibria. In their model, there are two types of markets: a search market and a
Walrasian market, and both money and goods are divisible. One of their main concerns waswhether subjects chose an eﬃcient non-monetary gift-giving equilibrium. They found that themonetary equilibrium was selected in almost all cases because more than 95% of the acceptedoﬀers included positive token (money) quantities. We will more precisely discuss the above
results in Section 5.
The plan of this paper is as follows. In Section 2, we present the variant of Zhou’s [14] money
search model used in the experiments. In Section 3, we report the experimental design, including
our parameterization and the hypotheses. We report the results of the experiments in Section
4. In Section 5, we discuss the ﬁndings of the experiment and conclude the paper.

32

Theoretical Considerations

2.1

Environment

Below, we present a monetary model with an inﬁnite number of agents and a model with a ﬁnite
number of agents. Both models are variants of Zhou’s [14] model with ﬁat money. We ﬁrst
present the common environment of the models.
Time is discrete, and the time horizon is inﬁnite, as denoted by t = 1, 2, . . .. Each agent
can produce one unit of indivisible goods in each period, and he or she cannot consume his or
her production good, while he or she can consume the other agents’ production good. Money isperfectly divisible, and the agents can hold any amount of money. The total amount of moneyin the economy is denoted by M > 0.
The timeline in each period is as follows. In the beginning of each period, agents observe thecurrent economy-wide money holdings distribution. Then, pairwise random matching occurs.
In each matching, each agent cannot observe his or her partner’s money holding, and one agent
becomes a seller with probability 12 , while the other becomes a buyer. Then, bargaining overthe price proceeds with the seller’s take-it-leave-it-oﬀer. That is, the seller posts a price p ≥ 0,
and if the buyer accepts it, then the trade occurs, i.e., the seller produces one unit of goods,
and the buyer pays p and consumes one unit of goods; otherwise, there is no trade. Finally, thematching resolves and the economy ends with probability 1 − δ and moves to the next periodwith probability δ. If the economy ends, the utilities of the agents are zero thereafter.
u is the utility of consumption and c is the cost of production. We assume u > c > 0. The
agents do not discount future payoﬀs, and they maximize time-additive expected utility. Clearly,
δ plays a role as a discount factor because a future payoﬀ is multiplied by a probability. Below,
we investigate the Nash equilibria with Markov strategies; that is, depending only on each agent’s
money holding and an economy-wide money holdings distribution, each seller chooses an oﬀer
price and each buyer chooses a reservation price, i.e., he or she accepts any oﬀer price less thanor equal to the reservation price.

2.2

The Model with a Continuum of Agents

We ﬁrst consider a model with a continuum of agents and a measure is normalized to 1.
We consider a benchmark equilibrium called SPE. We compare it with the experimental
results in the following section. An SPE is deﬁned as a stationary (time-invariant) equilibriumwith the following features:• there exists a common price p > 0,
• a stationary money holdings distribution is deﬁned by h = (h0 , h1 ), where hn is the measure, i.e., the proportion of agents holding np amount of money, n = 0, 1,• p satisﬁes M = ph1 ,• when an agent is a seller, the agent oﬀers price p if his or her current money holding η isless than p, and otherwise, he or she oﬀers a price that is never accepted, e.g., 2p, and4• when an agent is a buyer, the agent accepts a price oﬀer p if he or she holds η ≥ p.
Note that, for simplicity, the strategy is not completely speciﬁed in the above. For example,
strictly speaking, a buyer has a reservation price strategy depending on his or her money holding
η, denoted by R(η), such that p ≤ R(η) for η ≥ p.
In the SPEs, the agents play the game as if money is indivisible and they can hold at mostone unit of money, i.e., the p amount of money is one unit and it is traded for one unit of goods.
Note that any h satisfying h0 + h1 = 1 is stationary, since before trades the potential sellers
are agents without money, and the potential buyers are agents with p amount of money; after
trades, the former have p amount of money and the latter have no money. That is, the outﬂow
of agents from 0 (the number of sellers) is equal to the inﬂow of agents to 0 (the number ofbuyers), and a similar argument applies to the outﬂow from p and inﬂow to p.
Next, we investigate the condition for the existence of SPE. The value function for agents
holding np amount of money is deﬁned as:()
 h1 {−c + δV (p)} + 1 − h1 δV (0)if n = 0,2(2)
V (np) = h
 0 {u + δV ((n − 1)p)} + 1 − h0 δV (np)if n ≥ 1.
2
2
Solving the above, we obtainh0u−
V (np) =2(1 − δ){δh02(1 − δ) + δh0}n {2(1 − δ) + δh02(1 − δ)(2 − δ)}(h0 u + h1 c).

Based on this, we consider the incentive condition for the prescribed strategy. First, thefollowing condition must hold:−c + δV (1) ≥ δV (0),This means that if an agent without money becomes a seller, then the agent wants to sell his or
her production good with the price p instead of oﬀering a price that cannot be accepted by any
buyer. This condition is equivalent tou2(1 − δ) + δh0
≥
.
c
δh0

(1)

Second, the following condition must hold:δV (np) ≥ −c + δV ((n + 1)p)∀n ≥ 1.

This condition means that if an agent with np amount of money becomes a seller, then he or she
wants to oﬀer a price that cannot be accepted rather than selling his or her production good at
the price p. By the concavity of the value function, this expression is reduced toδV (p) ≥ −c + δV (2p),5which is equivalent tou2(1 − δ)(2 − δ) + 2δ(1 − δ)h0 + δ 2 h20≤.
cδ 2 h20(2)

It is veriﬁed that (1) and (2) are suﬃcient for an SPE, i.e., all the other conditions of thedeﬁnition of an SPE can be veriﬁed.2
Zhou [14] also analyzes another type of single price equilibrium in which the equilibriummoney holdings distribution is on {0, p, 2p, . . . , N p}, where N ≥ 2. The features of an equilibrium
are as follows:• there exists a common price p > 0,
• a stationary money holdings distribution is deﬁned by h = (h0 , h1 , . . . , hN ), where hn isthe proportion of agents holding np amount of money,∑• p satisﬁes M = p N
n=1 nhn ,• when an agent is a seller, the agent oﬀers price p if his or her current money holding η isless than N p; otherwise, he or she oﬀers a price that is never accepted, e.g., (N + 1)p, and• when an agent is a buyer, the agent accepts a price oﬀer p if he or she holds η ≥ p.
In the case of a continuum of agents, Kamiya, Morishita, and Shimizu [8] show that there
exists a stationary money holdings distribution consistent with the above features. However,in the case of a ﬁnite number of agents, it can be shown that a money holding distribution
with N ≥ 2 cannot be stationary. Let I ≥ 4 be the total number of agents. To guaranteethat all agents match pair-wise, we assume that I is an even number. Suppose N = 2 andh = (h0 , h1 , h2 ),∑hn > 0, n = 0, 1, 2, is a stationary distribution. Then, hn can be expressed
as In /I, where 3n=0 In = I. Suppose I1 > 0 is an even number. With a positive probability,each agent with p meet another agent with p, and in the other matches monetary exchange is
not feasible. Then agents with p trade and the other agents do not trade, i.e., after trades,
the buyers with p have zero, the sellers with p have 2p, and the money holdings of the otheragents remain the same.3 Then, I1 becomes zero in the next period and thus h is not stationary.
Suppose I1 > 0 is an odd number larger than one. With a positive probability, all agents with p
except one agent meet each other, one agent with p meets an agent without money, and in theother matches monetary exchange is not feasible. Then all agents with p and one agent without
money trade, and the other agents do not trade. Then, after trades, h0 and h2 increase and h1
decreases, i.e., h is not stationary. Suppose I1 = 1 and I0 ≥ 2. Then, with a positive probability,
the agent with p and at least one agent with 2p buy goods. Then, h2 decreases, and thus, h is notstationary. Suppose I1 = 1 and I2 ≥ 2. Then with a positive probability, the agent with p and2

In almost all analyses of models with a continuum of agents, the optimality oﬀ the equilibrium distribution is
not required as an equilibrium condition because it cannot be reached by the deviation of a single agent.
3
In the case of a continuum of agents, this probability is zero due to the law of large number.

6

at least one agent without money sell goods. Then, h0 decreases, and thus h is not stationary. A
similar argument applies to all N ≥ 2. Therefore, in a stationary distribution, N must be one.
In the following sections, we show that the subjects were playing the above strategy withN ≥ 3, although the money holdings distributions were not stationary. It is worth noting thatthe total welfare can be considered to be an increasing function of N . Indeed, among all pair-wisematchings, only in the case that a buyer does not have money or a seller has N p, a trade does
not occur. That is, the buyer does not have money to accept p and the seller does not have an
incentive to trade because he or she has a suﬃcient amount of money. It can be considered thatthe number of agents with N p and without money decreases as N increases. Therefore, tradeopportunity is increasing in N .4

2.3

The Model with a Finite Number of Agents

We consider a model with I agents, where I is a ﬁnite even integer larger than one. Because only
SPEs can have a stationary distribution in the case with a ﬁnite number of agents, we focus on aclass of SPEs, in which some agents have p ≥ 0 amount of money, the other agents do not havemoney, and the equilibrium price of goods is p. Similar to the case of a continuum of agents, pmust satisfy
M = pI1 ,(3)

where In , n = 0, 1, is the number of agents holding np amount of money.
In the case of ﬁnite agents, the value function is deﬁned as follows:{()
H1{−c + δV (p)} + 1 − H21 δV (0)if n = 0,
2()
V (np) = H0
H0
if n ≥ 1,
2 {u + δV ((n − 1)p)} + 1 − 2 δV (np)In
where Hn = I−1. Solving this, we obtain

H0
V (np) =u−2(1 − δ){δH02(1 − δ) + δH0}n {2(1 − δ) + δH02(1 − δ) (2 − δ(2 − H0 − H1 ))}(H0 u + H1 c).

Then, the equilibrium conditions in the case of a continuum of agents, (1) and (2), are respectively
written as follows:u2(1 − δ) + δH0
,
≥
c
δH0
u
(2(1 − δ) + δH0 ) (2(1 − δ) + δ(H0 + H1 )) − δ 2 H0 H1≤.
cδ 2 (H0 )2

4

(4)
(5)

For a given N , there exist multiple stationary equilibrium money holdings distributions. Therefore, strictly
speaking, we cannot conclude that trade opportunity is increasing in N . However, if we compare the most eﬃcient
stationary equilibrium money holdings distribution for each N , we can prove that it is increasing in N .

7

3

Experimental Design and Hypothesis

The experiments were conducted at Kansai University on January, February, and December in2015 and June and July in 2016. In total, 180 subjects voluntarily participated at The Center
for Experimental Economics, Kansai University. The subjects were all undergraduate students
at Kansai University with no prior experience in the game we conducted.
We considered the case of I = 6. In each session, 24 or 18 subjects were randomly divided
into 4 or 3 groups, respectively, consisting of 6 members and interacted through z-Tree software(Fichbacher [7].) In the beginning of each session, subjects were given written instructions on
the game they were about to play.5 After the written instructions were read aloud, subjects had
to correctly answer a number of questions about the rules of the game.
Speciﬁcally, in our experiment, we used the term “substitute currency” (“daiyo-kahei” inJapanese) as the equivalent of ﬁat money in our model. The instruction clearly stated that thesubstitute currency can not be exchanged for rewards in the experiments. Furthermore, thecomprehensibility test conﬁrmed this fact.
Each session consisted of several cycles. A cycle consisted of an indeﬁnite number of periodsof a stage game. In each stage game, members of each group were randomly matched in pairs.
Each pair bargained over a price according to the process we described in Section 2. At the endof each stage game the cycle continued with probability δ = 0.9. If a cycle ended and less than
one hour had passed, since the instructions were given, the current group was dissolved and newgroups comprised of 6 subjects were randomly formed.
We choose the following parameter values across all treatments:• the production cost of the seller is c = 10, and• the total amount of money in the economy is M = 600.
As for the information structure, we set up the environment in which each subject is informed
about the prices of the group with which trades were successfully made.
We focused on two treatment variables: the levels of utility from trade u and the initial
distributions of money. First, we selected two diﬀerent values of u: u = 14 and u = 20. In thecase of u = 14, from (4) and (5), it is veriﬁed that there exist the following 3 types of SPE:• the price is 200, 3 agents hold 200 units of money, and the other agents hold no money ina stationary money holdings distribution,• the price is 300, 2 agents hold 300 units of money, and the other agents hold no money ina stationary money holdings distribution, and• the price is 600, 1 agent holds 600 units of money, and the other agents hold no money ina stationary money holdings distribution.
In the case of u = 20, from (4) and (5), it is veriﬁed that there exist the following 2 types ofsingle price equilibria:
5

Instructions are given in the Appendix.

8• the price is 150, 4 agents hold 150 units of money, and the other agents hold no money ina stationary money holdings distribution, and• the price is 200, and 3 agents hold 200 units of money and the other agents hold no money
in a stationary money holdings distribution.
Then we can make the following hypothesis under the assumption that the subjects are playing
an SPE.
Hypothesis 1 A transaction price in u = 20 is less than or equal to that in u = 14.
The intuitive explanation is as follows. As u increases, the incentive condition for the seller
without money becomes relaxed. Because the right-hand sides of (4) decrease in the numberof agents without money, h0 , then the condition still holds even if h0 decreases. Decreasing
h0 means increasing h1 so that the number of agents with money increases. Because the total
amount of money M is constant and M = ph1 , then transaction price must decrease.
Of course, the above argument can be applied under the assumption that the subjects play
an SPE. If the above hypothesis is not veriﬁed, it could be an indirect evidence that the subjects
were not playing any SPE. In the following section, we show that the hypothesis cannot be
veriﬁed, and directly ﬁnd that the subjects were not playing any SPE.
Any SPE mentioned above cannot attain the full eﬃciency because there always exists a
positive probability that a mismatch between the seller and buyer occurs. Theoretically, we cancompute the successful trade rate for each SPE as in Table 1. 6 Table 1 also shows that the SPEwith p = 200 is the most eﬃcient in both u = 14 and u = 20. That is, a half of the agents havemoney, and the probability of a trade is the maximum.
I1
1
2
3
4

Theoretical Prediction
5/30 ≈ 0.166
8/30 ≈ 0.266
9/30 = 0.3
8/30 ≈ 0.266

u = 14◦◦◦-

u = 20◦◦

Table 1: Theoretical Successful Trade Rates
However, we cannot determine which level of utility increases successful trade more by using
only an equilibrium analysis. For example, Table 1 shows that if the equilibrium with I1 = 3 isrealized in u = 14 and the equilibrium with I1 = 4 is realized in u = 20, successful trade occurs
more in u = 14. However, if the equilibrium with I1 = 2 is realized in u = 14 and the equilibrium
6

The idea of the computation is as follows. A subject becomes a money holder with probability n/I. Then,with probability 1/2, he or she becomes a buyer, and with probability (I − n)/(I − 1), he or she meets a sellern(I−n)
without money. Therefore, he or she can successfully trade with probability 2I(I−1). Additionally, a subject
becomes a non-money holder with probability (I − n)/I. Then, he or she becomes a seller with probability 1/2,
and with probability n/(I − 1), he or she meets a buyer with money. Therefore, he or she can successfully traden(I−n)
with prob. 2I(I−1). Hence, he or she can successfully trade with prob. n(I−n)
in total.
I(I−1)

9

with I1 = 3 is realized in u = 20, successful trade occurs more frequently in u = 20. Therefore,
we apply the arguments in Blonski et al. [1] to our environment. Based on their theory, therange of discount factors required for equilibrium aﬀects the successful monetary transactions
rate. 7 Because the ranges of the discount factors are wider in the case of u = 20 than in thecase of u = 14, successful trades might occur more in u = 20. This argument can be applied tothe case that the subjects are not playing an SPE because the incentive to sell a good is always
larger in the case of u = 20, and the ranges of the discount factors are considered to be wider inthe case of u = 20. Therefore, we can always make the following hypothesis.
Hypothesis 2 Successful trades occur more in u = 20 than in u = 14.
Second, we choose two diﬀerent initial distributions of money holdings:• dis = 200: 3 agents hold 200 units of money and the other agents have no money,• dis = 100: all agents hold 100 units of money.
Note that dis = 200 constitutes SPE both in the cases of u = 14 and u = 20. Note also that
dis = 100 does not constitute SPE either in the cases of u = 14 or u = 20. Therefore, we canmake the following hypothesis.
Hypothesis 3 Successful trades occur more in dis = 200 than in dis = 100.
Of course, the above hypothesis can be applied under the assumption that the subjects play
an SPE. If the hypothesis is not veriﬁed, it could be an indirect evidence that the subjects are
not playing any SPE.
Although we have focused on equilibria using positive amount of money, there exists anothertype of equilibrium in our game, called a gift-giving equilibrium. Let us consider the strategy
with two phases: C and D. In phase C,• a seller oﬀers p = 0, and• a buyer accepts the seller’s oﬀer if and only if he or she has enough money to accept theoﬀer.
In phase D,• a seller oﬀers a price that exceeds the maximum amount of money in that period, and• a buyer accepts the seller’s oﬀer if and only if he or she has enough money to accept the
oﬀer.

7

Because our model is diﬀerent from inﬁnitely repeated prisoner’s dilemma, there is no negative payoﬀ associated with “strategic uncertainty.” Therefore, we do not use here risk dominance as an index for the occurrence of
successful trades.

10

The transition rule is as follows. The strategy starts in phase C. If players do not observe any
sign of deviations, i.e., they have observed that all the three pairs successfully made transactions
with price oﬀer p = 0, the phase remains at C. Otherwise, the phase moves to phase D. Once
players move to D, they remains at D forever. It is easy to see that this strategy can be astationary equilibrium if δ is suﬃciently close to one (for details, see the Appendix).
However, previous experimental studies found that this type of gift-giving equilibrium rarely
occurs in experiments on monetary search models. Camera and Casari [2] found in Result 2 of
their paper that successful trade decreased when money transfer was unfeasible. Furthermore,
Camera, Casari and Bigoni [3] veriﬁed that the successful trade rate was much higher in monetary transactions than in gift-giving environments, when the population of the economy is 6.
Therefore, we have the following hypothesis.
Hypothesis 4 A gift-giving equilibrium rarely occurs.
Each subject was initially endowed with 300 points as an initial endowment. Total points
consisted of the initial 300 points plus the cumulative points he or she acquired in all the cycles.
Total points were converted into cash at the end of the session at an exchange rate of 1 point =10JPY. The summary of the experiments is described as in Table 2.
Session No.: Treatment
1: u = 14, dis = 200
2: u = 14, dis = 100
3: u = 14, dis = 100
4: u = 14, dis = 200
5: u = 20, dis = 200
6: u = 20, dis = 200
7: u = 20, dis = 100
8: u = 20, dis = 100

Subjects
6×4
6×4
6×4
6×4
6×4
6×3
6×4
6×3

Total Periods
49
53
57
49
44
64
46
48

Cycles
7
4
6
6
4
8
2
4

Ave. Duration
7
13.25
9.5
8.2
11
8
23
12

Table 2: Summary of Experimental Sessions

4

Experimental Results

4.1

No Gift-Giving

As a starting point of our experimental results, we show that gift-giving equilibria rarely occurredin our experiment. For that purpose, Figure 1 depicts the frequencies of the oﬀer prices. The
percentage of p = 0 is below 1% and thus we can conclude that gift-giving equilibria rarely
occurred.
Result 1 Gift-giving equilibria rarely occurred.
This ﬁnding supports Hypothesis 4. Therefore, we hereafter suppose that our experimental
data are generated from some behavior using a positive amount of money.
11

100

200

0.6

39.5%
14

37.3%

0.4

22.0%

0.2

25.0%

24.5%
18.6%

Frequency

11.9%
0.0

3.0%

0.8%

7.5% 8.2%
1.7%

64.9%
0.6

48.9%
20

0.4

26.3%

21.7%

0.2

13.4%

10.0%
0.0

1.1% 2.0%0.2%
0(0,100] (.,200](.,300](.,400](400,.]

3.3%

0.1%
0

(0,100] (.,200](.,300](.,400]8.1%(400,.]

category of price

Figure 1: The Distributions of Oﬀer Prices
In this ﬁgure, Category 0 means that the oﬀer price level is equal to 0. Category (0, 100] means that the oﬀer
price level is within the interval (0, 100]. Category (·, 200] means that the oﬀer price level is within the interval(100, 200]. Category (·, 300] means that the oﬀer price level is within the interval (200, 300]. Category (·, 400]
means that the oﬀer price level is within the interval (300, 400]. Category (400, ·] means that the oﬀer price levelis higher than 400.

12

4.2

Successful Trade Rate

We ﬁrst investigate how successful trade rates diﬀer according to treatments. Figure 2 reports
successful trade rates when averaging across all periods in all cycles for each treatment.

Figure 2: Successful Trade Rate
In the treatment of u = 20, the successful trade rate in dis = 200 is almost 30%, which corresponds to the most eﬃcient theoretical successful trade rate, and the one in dis = 100 is morethan 40%, which is beyond the most eﬃcient theoretical successful trade. On the other hand,the successful trade rates in u = 14 are less than 30%, although they are not much smaller than
30%.
First, the subjects were not playing any SPEs in all treatments. Figure 3 reports the average
ratios of subjects over all periods who had more than or equal to twice the transaction price,where the transaction price in each period is the average transaction price up to the period.
More precisely, let the average transaction price up to period t be p̄t and the ratio of subjects
who have more than or equal to twice of p̄t at period t be rt .8 (Note that, in Subsection 4.5,
we statistically verify that the subjects recognized that p̄t is a fair transaction price.) Figure 3
reports the average rt over all periods.9 The ratios were between 0.284 and 0.530. That is, asigniﬁcant number of subjects had more than or equal to twice the transaction price, and thus,
the opportunities for monetary exchange were larger than those in the SPEs. For example, when
the role of a buyer was assigned to a subject with more than or equal to 2p̄ twice in a row,
he or she could buy goods twice in a row, where p̄ is the average transaction price up to the∑p̄t = 1t ts=1 ps , where ps is the average transaction price in period s.
∑
9The average rt is T1 Ts=1 rt , where T is the last period.

8

13

current period.10 Indeed, in the treatment of u = 20 and dis = 100, the successful trade rate
0.418 was larger than that of the most eﬃcient SPE. In the other treatments, the successful
trade rates were almost equal to or slightly smaller than those of the most eﬃcient SPE, becausesome subjects refused monetary exchange even when it was feasible. (See Figure 4.) In Section
4.5, we report individual behaviors that support the above results.
100

200

53.0%

0.4

35.0%
Ratio of 2p

29.6%

28.4%

u
14
20

0.2

0.0
14

20

14

20

u

Figure 3: Subjects with Money More Than Twice the Average Price
Second, the diﬀerences in the eﬃciency of the treatments can also be checked in the statisticalanalysis. Table 3 reports the results of the random eﬀect probit estimation whose dependent
variable is binary: trade is successful or not. That is, if a trade is successful, the variable takes avalue of 1 and zero otherwise. In Table 3, u20 and dis200 are the treatment variables. Speciﬁcally,
the variable u20 (dis200) is a dummy variable that is equal to 1 if subject i is assigned to thetreatment of u = 20 (dis = 200) and equal to zero otherwise. Additionally, the variables Cycle
and PeriodInCycle are control variables. Cycle stands for the number of cycles in which subject
i is involved, and PeriodInCycle speciﬁes the period in a cycle that subject i faces. By looking
at Table 3, we can verify that u20 is signiﬁcantly positive, dis200 is signiﬁcantly negative, andthe cross-term of both variables is insigniﬁcant.
In summary, we obtain the following results.
Result 2 (i) The subjects are not playing any SPEs, and in some treatment the allocation ismore eﬃcient than those of the SPEs.
10

For simplicity, we denote p̄ instead of p̄t .

14

Figure 4: Average Rate of Buyers who Accept an Oﬀer

Figure 5: Successful Trade Rate over Cycles
15

VARIABLES

Successful Trade

PeriodInCycle-0.0599***(0.00464)-0.102***(0.0275)0.450**(0.216)-0.515***(0.173)-0.147(0.271)1.512***(0.219)Cycle
u20
dis200
u20 × dis200
Constant

Observations
1,528
Session-clustered robust standard errors in parentheses.
*** p<0.01, ** p<0.05, * p<0.1
Table 3: Random Eﬀects Probit Estimation of Successful Trades(ii) The successful trade rate is enhanced as the gain from trade grows, irrespective of the initialmoney distributions.
(iii) When the initial money holding distribution is dis=200, the successful trade rate is signiﬁcantly lower than dis=100.
Note that (iii) contradicts Hypothesis 3, which could be an indirect evidence that the subjects
are not playing any SPE.

4.3

Transaction Prices

Next, we explore how transaction prices diﬀer according to the treatments. Figure 6 reports
transaction prices in each treatment when averaging across all periods in all cycles. In Figure
6, we ﬁnd that the average transaction prices in all treatments were less than the theoreticalequilibrium prices in any SPE. We also examine transaction prices in each cycle and ﬁnd thatthey were less than any SPE prices except the last cycles in u = 14. Figure 7 shows the results.
Based on the above observation, we implement the random eﬀects linear regression estimationin order to investigate treatment eﬀect on transaction prices. The ﬁrst column of Table 4 reports
the estimation result of the basic model. It does not show any signiﬁcant treatment eﬀects on
the transaction prices, while transaction prices signiﬁcantly increase over cycles. However, if we
add the treatment eﬀects at each cycle to the basic model, we can verify the following results
as in the second column of Table 4. Both u20 and dis200 have a signiﬁcantly positive eﬀect on
16

Figure 6: Transaction Price

Figure 7: Transaction Prices over Cycles

17

VARIABLES
PeriodInCycle
Cycle
u20
dis200
u20 × dis200(1)
Transaction Price(2)
Transaction Price

2.184***(0.398)13.67***(2.546)-11.86(13.51)
8.012(11.98)31.51(19.27)1.894***(0.308)25.81***(0.869)32.98***(10.49)47.91***(11.17)2.678(13.82)-13.25***(0.538)-11.72***(1.267)8.099***(1.367)-24.30***(8.851)

u20 × Cycle
dis200 × Cycle
u20 × dis200 ×Cycle
Constant

15.27(13.09)Observations
1,406
1,406
Session-clustered robust standard errors in parentheses*** p<0.01, ** p<0.05, * p<0.1
Table 4: Random Eﬀect Linear Regression on Transaction Pricethe transaction price at the initial point, and their marginal eﬀects decreases over cycles. In
summary, we obtain the following results.
Result 3 (i) The levels of transaction prices in each treatment are less than those of any SPE.
(ii) Transaction prices increase over cycles in all treatments(iii) Both u20 and dis200 have a signiﬁcantly positive eﬀect on the transaction price at the initialpoint, and the marginal increase in the transaction price over the cycle is smaller in u20 thanin u14.
There are two reasons why transaction prices in each treatment are lower than those in any
SPE. First, as shown in the previous subsection, a signiﬁcant number of subjects have morethan or equal to 2p̄. Thus, for a given amount of total money supply, the transaction price mustbe smaller than those in the case of the SPEs. Second, some amount of money is not used in18

monetary exchanges. Let Mbt be a level of money holdings of buyer b at the beginning of periodt and pbt be her transaction price at period t. Then, Mbt may not be an integer multiple ofpbt and the remainder of the division, called the remaining amount, is typically positive in our
experiments. Because the total money supply is given, transaction prices must be smaller thanthe case without a remaining amount.
Note that the ﬁrst part of Result 3(iii) contradicts Hypothesis 1. Recall that the hypothesis
is supposed to hold under the assumption that the subjects are playing an SPE. As shown inResult 2 (i), they were not playing any SPE, and the assumption does not hold.

4.4

Money Holdings Distributions

As shown in Subsection 4.2, the trade rates were relatively large because the money holdings
distributions in our experiments were very diﬀerent from those of the SPEs. In this subsection,
we more precisely investigate the money holdings distributions in our experiments, and ﬁnd a
cause of the eﬃciency found in Subsection 4.2.
First, Figure 3 shows that the ratio of subjects having more than or equal to 2p̄ is very large
between 28.4% and 53.0%, where p̄ is an average transaction price up to the current period. Notethat the ratio must be zero in the SPEs. Moreover, Figure 8 shows that the ratio of subjects
having more than or equal to 3p̄ is also large, between 12.5% and 32.0%. Note that the ratio
must be zero in Camera and Casari [2], where a subject can hold at most twice of the given
transaction price.
100

200

32.0%
0.3

Ratio of 3p

0.2

18.9%

18.5%

u
14

12.5%

20

0.1

0.0
14

20

14

20

u

Figure 8: Subjects with More Money Than Three Times the Average Price

19

Result 4 (i) A large proportion of the subjects have more than or equal to 2p̄, and thus, themoney holdings distributions are diﬀerent from those of the SPEs.
(ii) A large proportion of the subjects have more than or equal to 3p̄, and thus, the money holdings
distributions are diﬀerent from those of Camera and Casari [2].
In the following subsection, we report that buyers’ acceptance rates of oﬀer prices were very
high both in the cases with having more than or equal to 2p̄ and 3p̄. Together with the above
results, we can conclude that buyers often bought goods more than or equal to twice in a row
when the role of a buyer was assigned to his or her more than or equal to twice in a row. Thatis, the subjects endogenously chose the eﬃcient allocations.

4.5

Individual Behaviors

In this subsection, we examine the mechanism by which the results in Sections 4.2, 4.3, and 4.4
are presented. Then we turn to the estimation of individual behaviors. Because the roles ofseller and buyer are randomly determined in our experiment, we investigate each role separately.
4.5.1

Acceptance Decisions by Buyers

In Result 2 (i), we report that the subjects were not playing any SPE, and in a treatment, theallocation was more eﬃcient than those of the SPEs. That is, buyers often bought goods morethan or equal to twice in a row. Behind the result, a large proportion of subjects must havemore than or equal to 2p and the acceptance rates of buyers must be high. The former has beenshown in the previous subsection. Below, we show the latter.
As shown in Figure 4, the average acceptance rates are between 69.9% and 77.7%. Moreover,in the cases of money holding η satisfying η < 2p̄, 2p̄ ≤ η < 3p̄, and 3p̄ ≤ η are between
75.6% and 84.4%, 67.7% and 79.1%, and 67.5% and 79.0%, respectively. (See Figures 9 to 11.)
Therefore, we can conclude that subjects often bought goods twice or more in a row.
We deﬁned p̄ as the average transaction price up to the current period. Below, we showthat it is the correct choice of a transaction price. First, Figure 12 shows that acceptance rates
drastically decreased around p̄, i.e., many subjects recognized that p̄ was a fair price. Second,
we also statistically verify that acceptance rates depend on whether an oﬀer price is less than orequal to p̄. That is, applying the strategy frequency estimation method by Dal Bo and Frechette[4], we tested the consistency of the strategy of buyers that they only accept an oﬀer price lessthan or equal to p̄. As in Table 5, the estimated rate of inconsistency is 0.247 that is signiﬁcant
at 1%. Therefore, we can conclude that almost 75% of subjects follow the above type of buyer’s
strategy.
Finally, we statistically investigate how the buyer’s behavior diﬀers across treatments. Thatis, we estimate the buyer’s accept/reject pattern using the random eﬀect probit model. Table 6
reports the estimation results. In Table 6, Price, Before money, and Period are controls. Here,the variable Before money stands for subject i’s level of money holdings in the beginning of
the period. The explanatory variable TheoAc is a dummy variable that a subject follows theacceptance strategy with p̄, and it is signiﬁcantly positive. This fact supports the conjecture
20

100

200

84.4%

81.6%
0.8

77.4%

75.6%

Acceptance Rate

0.6

u
14

0.4

20

0.2

0.0
14

20

14

20

u

Figure 9: Buyers’ Response When η < 2p̄
100
0.8

200

79.1%
71.3%

72.2%

67.7%

Acceptance Rate

0.6

u
0.4

14
20

0.2

0.0
14

20

14

20

u

Figure 10: Buyers’ Response When 2p̄ ≤ η < 3p̄
21

100
0.8

200

79.0%
74.3%
67.9%

67.5%

Acceptance Rate

0.6

u
0.4

14
20

0.2

0.0
14

20

14

20

u

Figure 11: Buyers’ Response When η ≥ 3p̄
100

200

1.00

69.9%

64.7%

58.5%

60.2%

0.25

u
0.00
14
1.00

91.6%

86.3%

93.5%

93.2%

14

20

20
Less or equal to average

Acceptance Rate

0.50

Larger than average

0.75

0.75

0.50

0.25

0.00
14

20

u

Figure 12: Buyers’ Response to Average Oﬀered Price
22

Estimate
Inconsistency Rate
0.247***(0.026)
Log Likelihood-2563.364
Session-clustered robust standard errors in parentheses*** p<0.01, ** p<0.05, * p<0.1
Table 5: Strategy Frequency Estimationthat buyers follow the above described strategy. Moreover, the cross terms of TheoAc and thetreatment variables u20, and dis200 are all insigniﬁcant at the 5% level. Therefore, we can statethat the above type of buyer’s behavior is common across treatments.
In summary, we obtain the following results.
Result 5 (i) The buyer’s acceptance rates depend on whether an oﬀer price is less than or equal
to p̄.
(ii) The buyer’s acceptance rates depend on his or her money holdings and are between 67.5%
and 84.4%.
(iii) The gain from trade has a signiﬁcant but small eﬀect on the buyer’s acceptance decision.
At the end of Subsection 2.2, we present a strategy with N ≥ 2, where acceptance rates ofbuyers with money holdings η ≥ p are 100%. In the experiments, they are high but less than
100%. The reasons are (i) there are price oﬀers larger than p̄ and (ii) some subjects are inactive.
4.5.2

Price Oﬀers by Sellers

Next, we investigate seller behavior. To estimate the seller’s behavior, we must take into accountthe following problem associated with the seller’s decision making. Although sellers only oﬀer
prices in our experiment, they might decide the following two things:
1. whether to oﬀer an acceptable price or not, and
2. if a seller decides to oﬀer an acceptable price, then he or she decides the price’s level.
Thus, the seller’s problem possibly contains an endogenous sample selection problem.
Before moving on to the estimation, we brieﬂy show how unacceptable price oﬀers diﬀer
according to the treatments. Here, we deﬁne unacceptable price oﬀer to be a price levels thateven the subject with the most money in the period cannot aﬀord. It can be seen in Figure 13
that in both dis = 100 and dis = 200, the subjects with u = 20 oﬀer an unacceptable price lessoften than those with u = 14. This ﬁnding means that, at least from the seller’s point of view,
the subjects with u = 20 did not oﬀer unacceptable prices more often than those with u = 14.
Therefore, this behavior contributes to a higher transaction success rate in the case of u = 20.
23

VARIABLES
Price
Before money
PeriodTheoAcu20
dis200
u20 × dis200(1)
Accept / Reject(2)
Accept / Reject(3)
Accept / Reject-0.0168***(0.00229)
0.0116***(0.00287)
0.0354***(0.00758)
0.428***(0.0966)-0.469**(0.195)-0.590*(0.340)0.725*(0.374)-0.0169***(0.00231)
0.0116***(0.00288)
0.0355***(0.00768)
0.289***(0.109)-0.547**(0.250)-0.576*(0.341)0.710*(0.378)0.266(0.207)0.174(0.327)0.218(0.329)-0.0170***(0.00235)
0.0116***(0.00290)
0.0359***(0.00773)
0.285***(0.109)-0.578**(0.258)-0.577*(0.343)0.784*(0.431)0.363*(0.206)-0.233(0.313)0.219(0.331)

TheoAc × u20
TheoAc × u20 × dis200
Constant

Observations
1,721
1,721
Session-clustered robust standard errors in parentheses*** p<0.01, ** p<0.05, * p<0.1
Table 6: Buyer’s Acceptance Decision

24

1,721

Figure 13: Unacceptable Oﬀer Price Rate

Figure 14: Acceptable Oﬀer Price Levels

25

VARIABLES
Before money
Periodu20
dis200
u20 × dis200
Constant

Two Part (1st: Probit)
SeriousOﬀer

Two Part (2nd: OLS)
Price

-0.00218***(0.000410)-0.0319***(0.00647)
0.439*(0.249)0.0895(0.441)0.0268(0.574)1.480***(0.296)-0.0455(0.0458)
2.228***(0.404)-39.54*(18.43)1.026(19.71)
38.72(25.51)70.58**(20.91)

Observations
4,584
3,309
R-squared
0.202
Session-clustered robust standard errors in parentheses*** p<0.01, ** p<0.05, * p<0.1
Table 7: Estimation of the Seller’s Behavior Using Two-Part Estimation

26

Furthermore, in our model, it is diﬃcult to build up the statistical model of seller’s behavior
that satisﬁes the exclusion restriction. Puhani [13] suggests that the two part model is less susceptible to bias than other models, such as a maximum likelihood estimation of the simultaneous
equations. We implement the estimation of the seller’s decision by following the suggestion.
The ﬁrst column of Table 7 gives the estimation results for the ﬁrst part of the two-partmodel by the probit model whose dependent variable is whether a seller seriously oﬀers a price.
Formally, we consider that a seller seriously oﬀers a price if the seller oﬀers an acceptable price.
By looking at the estimation result, we ﬁnd that u20 is signiﬁcantly positive, although dis200
and the cross term of u20 and dis200 are insigniﬁcant. Furthermore, we ﬁnd that sellers are lesslikely to seriously oﬀer a price as the level of money holdings increases.
The second column of Table 7 gives the estimation results of the second part of the two-partmodel with the linear regression on the level of a serious oﬀer price. We ﬁnd that the treatment
variable u20 is signiﬁcantly negative but dis200 and that the cross term of u20 and dis200 are
insigniﬁcant.
Result 6 (i) The possibility of a serious oﬀer increases as the gain from trade increases.
(ii) The possibility of a serious oﬀer increases as the level of money holdings increases.
(iii) The level of a serious price oﬀer decreases as the gain from trade increases.
(iv) The level of a serious price oﬀer increases over periods.

5

Discussion and Conclusion

In this paper, we conducted experiments using a monetary search model with multiple stationary
SPE and gift-giving equilibria. We observed that gift-giving rarely occurred, the subjects did
not play any SPE, and in a treatment, they chose a more eﬃcient allocation than in any SPE.
That is, the subjects endogenously reduced the friction of monetary exchanges.
In the case of divisible money and endogenously determined prices, any money holdings
distribution can be endogenously chosen through monetary exchanges, while it is limited in thecase of indivisible money. Therefore, the opportunity for monetary exchanges grows; that is,
the set of feasible allocations is larger than that in the case of indivisible money. Of course, avery ineﬃcient allocation, which is more ineﬃcient than the case of indivisible money, can beendogenously chosen. In our experiments, such an ineﬃcient money holdings distribution was
not chosen, and a large proportion of the subjects had money equal to or more than three timesof the average transaction price. Therefore, the transaction prices were much less than those in
the SPEs. Note that the average transaction prices were between 50.7 and 86.9, while the oﬀer
prices were between 74.8 and 126.1 and larger than the transaction prices because relatively highprice oﬀers were often rejected. Indeed, as shown in the previous section, buyers recognized thatp̄ was a fair price and often rejected a price larger than p̄.
Some experiments using monetary search models are closely related to ours. Camera andCasari [2] (hereafter, referred to as CC) investigated an equilibrium selection problem in an27

economy with indivisible ﬁat money.11 In their model, the agents can hold at most two unitsof money and can use only one unit in monetary transactions; that is, a transaction priceis exogenously given. They ﬁnd that the subjects used both monetary transactions and giftgiving. The former was 61.4% when monetary exchange was feasible, and the latter was 12.5%
when it was infeasible. There are some diﬀerences between their model and ours in addition
to the divisibility of money. The main diﬀerences are (1) in their model, the buyer and seller
know whether monetary exchange is feasible, while in our model, they know the money holdings
distribution in the current period, and (2) in their model, the bargaining protocol is simultaneousoﬀers of a buyer and a seller, while in our model, it is a take-it-or-leave-it oﬀer from a seller.
Below, we compare the rates of monetary transactions and of gift-giving with the counterpartsin our experiments.
The acceptance rate of the buyers, when they have a suﬃcient amount of money to accept
an oﬀer price, can be considered as a counterpart of the rate of monetary exchange when it isfeasible in CC because both are the rates of monetary exchange when it is feasible. As shownin Figure 4, the acceptance rates of the buyers are between 69.9% and 77.7% and are larger
than the rate of monetary exchange in CC. In their experiments, the diﬀerence in payoﬀs of abuyer and a seller in the case of cooperation is 18, and it is larger than u − c = 10 or 4 in our
experiments. Therefore, from the viewpoint of payoﬀ, it can be considered that the subjects arelikely to cooperate through monetary exchange in CC. However, the cooperation rate is higher
in our case, which is a reason why, in our model, subjects focus more on monetary exchanges
than on gift-giving. Indeed, in our model, a seller chooses gift-giving by oﬀering a price of zero,
while in CC, he or she can directly choose ‘cooperation’.
In our experiments, the rates of a zero price oﬀer were between 0.091% and 1.7%. (SeeFigure 1.) On the other hand, CC reported that the rate of gift-giving was 12.5% when monetary exchange was infeasible. It is not possible to directly compare these results, because theinformation on the subjects is diﬀerent. That is, in their model, in each matching, the buyer andseller know whether monetary exchange is feasible. However, in our case, they do not have thisinformation, but they know the money holding distribution in each period. That is, the devices
for coordinating on gift-exchange equilibria are diﬀerent. Therefore, there are several ways tocompare these two results. First, the rate of cooperation, which means that a seller gives a goodno matter what the buyer’s choice is, may be a counterpart to a zero price oﬀer. This rate is
12.6% and is much larger than our case. The second way is to compare the rates of successful
gift-giving: the rate of an accepted zero price oﬀer in our case and the rate of gift-giving whenmonetary exchange was infeasible in CC. The former was between 0% and 1.62%, and the latter
was 12.5%. (See Figure 15.) In our experiments, subjects had positive amounts of money withhigh ratios between 68.5% and 94.2%, and thus a seller might think that a buyer almost always
had a positive amount of money. Therefore, the third way is to compare the rate of acceptedzero price oﬀers in our case and the rate of gift-giving when monetary exchange was feasible in
11

Duﬀy and Ochs [5] also conducted experiments using a model with indivisible ﬁat money, and reported thatﬁat money often served as a medium of exchange. However, their model is quite diﬀerent from CC and our model.
For example, in their model gift-giving is not allowed and goods can be circulated as media of exchange. Therefore,
it is very diﬃcult to compare their results with ours.

28CC. The former was between 0% and 1.62%, as in the above, and the latter was 0.1%. However,this might not be the correct way to compare the results because the money holdings for a buyer
could be much smaller than p̄ with some probability, and thus, a seller might think that a buyer
does not have a suﬃcient amount of money to accept p̄.
100

200

1.6%

Rate of Accepted Zero Price Offer

0.015

0.010

u
14

0.8%

20

0.005

0.2%
0.0%

0.000
14

20

14

20

u

Figure 15: Rate of Accepted Zero Price Oﬀer
Duﬀy and Puzzello [6] also investigate an equilibrium selection problem in an economy withdivisible ﬁat money. Their model is a variant of Lagos and Wright’s [12] model, where goods aretraded in a search market and a Walrasian market, and there are a unique stationary monetary
equilibrium and multiple non-monetary gift-exchange equilibria. One of their main concerns iswhether subjects choose an eﬃcient non-monetary gift-giving equilibrium. They ﬁnd that themonetary equilibrium is selected in almost all cases, because more than 95% of the acceptedoﬀers include positive token (money) quantities. Their model is clearly very diﬀerent from ours.
First of all, the agents rebalance their money holdings in the Walrasian market, i.e., their money
holdings become the same after trading in the market, and thus, their model is not an appropriate
model for investigating the eﬃciency of money holdings distributions. In addition to the market
institutions, the main diﬀerences are as follows. Goods are divisible and buyers have all thebargaining power in their case, while goods are indivisible and sellers have all the bargaining
power in our case. Therefore, it is almost impossible to directly compare the results. However,slightly less than 5% of oﬀers in their experiments have a zero price, which is much larger thanour case. On the other hand, monetary oﬀer acceptance rates are between 34.2% and 59.4%,
and they are smaller than our case. Therefore, in Duﬀy and Puzzero [6], gift-giving occurred
more frequently and monetary exchange occurred less frequently than those in our experiments.
In conclusion, the divisibility of money enhanced monetary exchanges and relatively eﬃcient29

allocations were chosen through experiments, although ineﬃcient allocation could be chosen.
Moreover, gift-giving very rarely occurred. Finally, we compared the diﬀerence between our
results and those in the literature.

References[1] Matthias Blonski, Peter Ockenfels, and Giancarlo Spagnolo. Equilibrium selection in therepeated prisoner’s dilemma: Axiomatic approach and experimental evidence. American
Economic Journal: Microeconomics, 3(3):164–192, 2011.
[2] Gabriele Camera and Marco Casari. The coordination value of monetary exchange: Experimental evidence. American Economic Journal: Microeconomics, 6(1):290–314, 2014.
[3] Gabriele Camera, Marco Casari, and Maria Bigoni. Money and trust among strangers.
Proceedings of the National Academy of Sciences, 110(37):14889–14893, 2013.
[4] Pedro Dal Bo and Guillaume Frechette. The evolution of cooperation in inﬁnitely repeatedgames: Experimental evidence. American Economic Review, 101(1):411–429, 2011.
[5] John Duﬀy and Jack Ochs. Intrinsically worthless objects as media of exchange: Experimental evidence. International Economic Review, 43(3):637–673, 2002.
[6] John Duﬀy and Daniela Puzzello. Gift exchange versus monetary exchange: Theory andevidence. American Economic Review, 104(6):1735–1776, 2014.
[7] Urs Fischbacher. Z-tree: Zurich toolbox for ready-made economic experiments. Experimental Economics, 10(2):171–178, 2007.
[8] Kazuya Kamiya, Noritsugu Morishita, and Takashi Shimizu. The existence of single-priceequilibria in a matching model with divisible money and production cost. International
Journal of Economic Theory, 1(3):219–231, 2005.
[9] Kazuya Kamiya and Takashi Shimizu. Real indeterminacy of stationary equilibria in matching models with divisible money. Journal of Mathematical Economics, 42(4):594–617, 2006.
[10] Kazuya Kamiya and Takashi Shimizu. On the role of the tax-subsidy scheme in money
search models. International Economic Review, 48(2):575–606, 2007.
[11] Nobuhiro Kiyotaki and Randall Wright. A search-theoretic approach to monetary economics. American Economic Review, 83(1):63–77, 1993.
[12] Ricardo Lagos and Randall Wright. A uniﬁed framework for monetary theory and policy
analysis. Journal of Political Economy, 113(3):463–484, 2005.
[13] Patrick Puhani. The Heckman correction for sample selection and its critique. Journal of
Economic Surveys, 14(1):53–68, 2000.
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[14] Ruilin Zhou. Individual and aggregate real balances in a random-matching model. International Economic Review, 40(4):1009–1038, 1999.

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Appendix: Existence of Gift-Giving Equilibria
To analyze the strategy for the gift-giving equilibrium described in this paper, let us consider the
incentive conditions for the seller and buyer in each phase. Suppose that the phase is C. First,
consider the seller’s incentives in Phase C. If the seller follows the prescription of the strategy,
that is, the seller oﬀers a zero price, then he or she obtains:
VsC = (1 − δ)(−c) + δ[(1/2)VsC + (1/2)VbC ].

(6)

If the seller deviates to oﬀer a positive price, then he or she obtains 0 in this period and thephases from the next periods on are D so that he or she obtains 0 forever. Therefore, we canverify that the seller has an incentive to follow the strategy in this situation if δ is suﬃciently
close to 1.
Second, consider the buyer’s incentive in phase C. Suppose the situation in which the selleroﬀers a zero price in phase C. If the buyer follows the prescription of the strategy, he or she
obtains:
VbC = (1 − δ)u + δ[(1/2)VsC + (1/2)VbC ].
(7)
If the buyer deviates and rejects the oﬀer a zero price, then he or she obtains 0 in this period
and the phases from the next periods on are D so that he or she obtains 0 forever. Because thecondition (6) ≥ 0 implies (7) ≥ 0, the existence of δ that supports (6) ≥ 0 guarantees the buyer’s
incentive in this situation.
On the other hand, suppose that the seller oﬀers a positive price in phase C. If the buyer
has enough money to accept the oﬀer and follows the prescription of the strategy, then the buyer
obtains u in this period and obtains 0 from the next periods on. Therefore, the buyer obtains(1 − δ)u + δ · 0 = (1 − δ)u If the buyer has enough money to accept the oﬀer and deviates fromthe prescription of the strategy, the buyer obtains 0 in this period and obtains 0 from the next
periods on. By the comparing payoﬀs, we can verify that buyers have an incentive to follow thestrategy in this situation.
Next suppose that the phase is D. First, consider the seller’s incentives in Phase D. If theseller follows the prescription of the strategy, he or she obtains 0 forever. Suppose that the seller
deviates to oﬀer a positive price that is less than the maximum amount of money in that period.
If the buyer has enough money to accept the oﬀer, then the seller pays the cost c in this period
and obtains 0 from the next periods on. Therefore, the seller does not have incentives to deviate.
On the other hand, if the buyer does not have enough money to accept the oﬀer from the seller,
then the seller obtains 0 forever. Therefore, the seller does not also have an incentive to deviate
in this situation.
Second, consider the buyer’s incentive in phase D. Suppose the situation in which the selleroﬀers a price that is more than the maximum amount of money in that period. Because thebuyer can not accept the oﬀer, there is no incentive problem in this situation.
On the other hand, suppose that the seller oﬀers a positive price in phase D. If the buyer
has enough money to accept the oﬀer and follows the prescription of the strategy, then the buyer
obtains u in this period and obtains 0 from the next periods on. Therefore, the buyer obtains(1 − δ)u + δ · 0 = (1 − δ)u. If the buyer has enough money to accept the oﬀer and deviates from32

the prescription of the strategy, the buyer obtains 0 in this period and obtains 0 from the next
periods on. By the comparing payoﬀs, we can verify that buyers have an incentive to follow thestrategy in this situation.

Appendix: Sample Instructions
The instructions were originally written in Japanese. Here, we provide a translated sample copyof the experimental instructions used in our treatment “dis = 200 & u = 20 .”12 The instructions
for the other treatments were adapted accordingly.

Instructions
This is an experiment in economic decision-making. If you understand these instructions well,and make the appropriate decisions, you can earn a reasonable amount of money. The points
that are acquired during the course of this experiment will be converted into monetary funds andpaid out in cash once the experiment is over. At the start of the experiment, you will be given
300 points, and this amount will ﬂuctuate in accordance with your decision-making thereafter.
In this experiment, 1 point will be converted to 10 yen.
1. In this experiment, you will engage in multiple periods of decision-making. You are to
randomly gather with other subjects in one group consisting of six people, and makemultiple decisions. The sequence of periods in which the same six people form the groupwill be called a “cycle.”
2. The length of the cycle will be randomly determined. After each decision-making in aperiod, there is a 90% probability that the cycle will continue. In other words, there is a
10% probability that the cycle will end.
3. Once the cycle ends, six new people will form a group at random, and a new cycle will
begin.
4. During each cycle, six people form one group and engage in decision-making. These six
people will not change (groups) until that cycle is over.
5. In this experiment, besides points, a substitute currency will be used. At the start of eachcycle, three members from each group selected at random will be assigned 0 units of thesubstitute currency, while the remaining three members will be assigned 200 units of thesubstitute currency.
Figure 1 shows a summary of the above ﬂow.
Each period will proceed according to the following format.
12

In the main part of this paper, we use the term “cycle” to refer to a cycle of the stage games until the period
when play is randomly terminated. However, we used the term “cycle” to refer the same concept in our instruction.
This is because the term “cycle” better ﬁts this context in Japanese.

33

1. In each period, among the six people who created the same group at the beginning of thecycle, you will randomly form a group of two with subjects other than you and engage indecision-making.
2. Although you are allowed to know the distribution of the substitute currencies among thesix members of your group, you will not know the amount held by your paired subject.
3. At random, one of the people in the pair will be assigned the role of the “buyer,” while
the other will be assigned the role of the “seller”.
4. The seller will always have the opportunity to sell one unit of goods to the buyer. When
doing so, the seller will present the amount of substitute currency requested to the buyer
in exchange for the goods.
5. Upon examining the amount of substitute currency that has been requested, the buyer
decides whether to “make a deal” or to “not make a deal.” However, when the amountof substitute currency requested by the seller is more than the amount held by the buyer,“do not make a deal” is automatically selected.
6. If the buyer chooses to “make a deal” and a deal is established, the buyer acquires 20
points; at the same time, the buyer loses the requested amount of substitute currency.
On the other hand, when a deal is established, the seller loses 10 points to represent thecost of producing the goods; at the same time, the seller acquires the requested substitutecurrency.
7. If the buyer does not make a trade, both the buyer and seller will earn 0 points. In otherwords, there will be no increase or decrease in terms of points. Furthermore, there will be
no increase or decrease in the substitute currency.
8. Here, it is worth noting that the buyer will lose the requested amount of substitute currency
instead of being able to earn points by trading and acquiring goods, and the seller will lose
points because the deal is made and the goods are handed over to the buyer. In particular,
points are not acquired by holding a substitute currency.
9. The following three pieces of information about the group will be displayed at the end ofeach period: (1) Success or failure in regard to deals made among the three pairs in thegroup, (2) the amount of substitute currency requested by the seller for groups in which
deals are made, (3) the distribution of the substitute currency held by the six members of
the same group after these deals have been made.
Figure 2 shows a summary of the aforementioned ﬂow.
Notes:

34

(1) At the beginning of the cycle, the assignment of the substitute currency is determined
randomly, regardless of the amount of substitute currency held in the prior cycle. In otherwords, at the end of the cycle, all substitute money held will be set to zero, and will be
newly assigned randomly at the start of the next cycle.
(2) One hour after the start of the experiment, the experiment will end at the completion of
the current cycle.

Figure 1. Relationship Between Each Period and Cycle

35

Figure 2. Decision-Making Flow for Each Period

36Appendix: Screen shots
The screen shots of the program are illustrated in Figures 16-19.

Figure 16: Seller’s Decisions Screen

37Figure 17: Buyer’s Decisions Screen: Acceptable Oﬀer

38

Figure 18: Buyer’s Decisions Screen: Non-Acceptable Oﬀer

39

Figure 19: Transaction Result Screen

40