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Is Money Essential?
An Experimental Approach
WP 21-12
Janet Hua Jiang
Bank of Canada
Peter Norman
University of North Carolina
Daniela Puzzello
Indiana University
Bruno Sultanum
Federal Reserve Bank of Richmond
Randall Wright
Zhejiang University, University of
Wisconsin - Madison, and Federal
Reserve Bank of Minneapolis
Is Money Essential?
An Experimental Approach∗
Janet Hua Jiang
Peter Norman
Daniela Puzzello
Bank of Canada
University of North Carolina
Indiana University
Bruno Sultanum
FRB Richmond
Randall Wright
Zhejiang University, University of Wisconsin - Madison & FRB Minneapolis
June 30, 2021
Abstract
Monetary theorists say money is essential if more desirable outcomes are incentive feasible when money is available. We develop two models: one where
frictions make money essential; one where they do not. Then we study them
experimentally. Unlike past work, money can be valued with finite horizons,
crucial because that is necessary in the lab. Also different from past experiments, we make suggestions about strategies — e.g., “accept money” —
that subjects may follow, or not, especially if they are incentive incompatible. Results are largely consistent with theory, with some anomalies that we
investigate using measures of social preferences and exit surveys.
Keywords: Money, mechanism design, experimental economics
JEL Classification numbers: E4, E5
Wright acknowledges research support from the Ray Zemon Chair in Liquid Assets
and the Kenneth Burdett Professorship in Search Theory and Applications at the
University of Wisconsin. The views expressed in this paper are those of the authors and
not necessarily those of the Federal Reserve Bank of Richmond, the Federal Reserve
Bank of Minneapolis, or the Federal Reserve System.
1
Introduction
A central issue in monetary economics is to understand what features of an economy
make money a socially useful institution. Based on Hahn (1973), money is said to
be essential if more desirable outcomes are incentive feasible with it than without it.
While money has no such role in traditional, frictionless, general equilibrium theory,
there are by now various models that endeavor to take frictions seriously, and in
those environments money can be essential (see Lagos et al. 2017 or Rocheteau and
Nosal 2017 for surveys). While the formulations differ in detail, it seems clear that
three ingredients are needed for essentiality: a double coincidence problem; limited
commitment; and imperfect information.
A double coincidence problem means that there are gains from trade that cannot
be fully exploited by direct barter. As in Jevons (1875), suppose agents meet bilaterally at random, and are restricted for now to quid pro quo exchange. It may be
hard (a coincidence) to meet someone who has what you like, and harder (a double
coincidence) to meet someone who has what you like and likes what you have. In
many environments, ex ante payoffs are higher if everyone simply produces when
asked. If agents can commit, they would agree to always produce when asked, which
typically delivers constrained efficiency. Hence, money is not essential.
So, suppose there is no commitment. Then agents may be tempted to renege
when asked to produce, rendering the commitment agreement inconsistent with dynamic incentives. Yet that is not enough to make money essential if trading histories
are monitored, since then desirable outcomes can potentially be supported as in the
repeated game literature: agents who do not produce when asked are punished by
having others not produce for them going forward. This is sometimes described as
a credit arrangement, with the punishment interpreted as denying future credit to
those who fail to honor current obligations.
As Kocherlakota (1998) emphasizes, such punishments must be precluded for
money to be essential. Here is a robust result: if it is incentive feasible to implement
1
monetary exchange, and histories are public, it is also incentive feasible to implement
the credit arrangement described above, and the latter is always at least as good if
not better. Therefore essentiality requires information frictions. There are different
ways to capture this (see Gu et al. 2016 and references therein), but a common
thread is that it must be hard for society to monitor, communicate or record what
happens in pairwise meetings, making it hard to punish bad behavior.
Pursuing the view that essentiality and frictions are of paramount importance,
Wallace (2001, 2011) refers to this as the “mechanism design approach” to monetary
economics. This is because mechanism design provides a clear distinction between
the underlying environment and the rules of the game that map actions into outcomes. Then, once one specifies the set of feasible mechanisms, it is possible to
decide whether money is essential in a given environment. Much of the literature
on the microfoundations of monetary economics follows this approach.
What may not have been anticipated is that this leads to models of monetary
exchange that are in many ways ideally suited for experimental economics. Namely,
the theories are tractable enough that their properties are very well understood,
and transparent enough that subjects in a lab can comprehend the environment,
yet the outcomes can be complex and interesting. In particular, there are generally
multiple equilibria, with different properties, due to the self-referential nature of
liquidity (what you accept in payment depends on what others accept). A body of
work has developed analyzing these models in the lab, although no one has directly
investigated essentiality the way we do.1
There is another, rather crucial, difference here from virtually all related work:
the theory on which we build uses a finite horizon. Why is this crucial? Previous
papers generally work with infinite horizon models to avoid the following problem:
1
There are too many papers for an exhaustive literature review, but a few that use models
similar in spirit to the ones presented below are: Brown (1996), Duffy and Ochs (1999, 2002) and
Duffy (2001), who experiment with the commodity money model in Kiyotaki and Wright (1989);
Rietz (2019), Jiang and Zhang (2018) and Ding and Puzzello (2017) who use the fiat money model
in Kiyotaki and Wright (1993) or Matsuyama et al. (1993); and Duffy and Puzzello (2014a,b), who
use Lagos and Wright (2005).
2
if there is a terminal time at which the game ends, no one at should sacrifice
anything to get money; but then no one at − 1 should; and so on. Thus, backward
induction implies money is never valued, at least not fiat money, defined as a medium
of exchange that is “intrinsically worthless and inconvertible” (Wallace 1980). In
experiments, this means subjects get no payment for any they have at the end — the
“play” money in the lab has no redemption value in “real” dollars.
Now, there is nothing wrong with an infinite horizon in theory, but in practice
there is always a terminal time at which, by rule of law, experiments must end
with probability 1. Hence, the games being played in the lab do not actually have
monetary equilibria. Experimentalists are aware of this, and there are various attempts to emulate an infinite horizon in the lab. One popular idea is to assume the
game ends after each round with some probability, but given the experiments all
have a hard stop at time , that does nothing to get around the induction argument.
This point applies not only to monetary economics, but many other games. As
Selten et al. (1997) put it, “Infinite supergames cannot be played in the laboratory.
Attempts to approximate the strategic situation of an infinite game by the device of
a supposedly fixed stopping probability are unsatisfactory since play cannot continue
beyond the maximum time available.” Repeated game experiments try to address
this in various ways (see Cooper and Kuhn 2014, Fréchette and Yuksel 2017, on
general games, and Jiang et al. 2021 on monetary games).
Another attempt to reconcile the infinite horizon in theory and the finite horizon
in the lab in monetary experiments is to assign value to money at based on what
payoffs would be if the game were to continue, which is interesting, but treads close
to giving up on the fiat nature of fiat currency. Another approach asserts that when
the chance of hitting the hard stop is small subjects somehow regard the game as
approximately infinite. Without debating the merit of this approximation, it seems
problematic that experimenters are trying to learn about monetary equilibria in
settings where, strictly speaking, they do not exist.
Given this, we follow earlier work in which some of us were involved (Davis et
3
al. 2020) by using models where monetary equilibria exist even if ∞, and, in fact,
we use models with = 2 periods. The insight that monetary equilibria are possible
in finite environments is not new: our approach is closely related to the discussion
of money in Kovenock and Vries (2002), somewhat related to the discussion of
bubbles in Allen et al. (1993) or Allen and Gorton (1993), and ultimately related to
discussions like Samuelson (1987) of how lack of common knowledge about the final
period ameliorates end-game effects. What is novel is to bring this into the lab.2
Details are below, but here is the idea. Suppose there are only two trading opportunities and agents do not know if they are in the first or second. Then agents
may accept money in the second, even though there are no future trading opportunities, because they are uncertain whether it is the first or second opportunity.
Even when all players understand the horizon is finite, this uncertainty implies that
accepting money can be consistent with equilibrium. Moreover, since the monetary equilibrium yields higher payoffs than the best outcome without money, it is
essential.
Yet questions arise. One is, do agents actually use money when there is a monetary equilibrium? Theory says they may or may not, because there always coexists
a nonmonetary equilibrium. Another is, do they accept money when there is no
monetary equilibrium? Theory says they should not, but they might. If money is
accepted when that is not an equilibrium, why? Is it due to altruism or some kind
of social preferences? Another difference from past work is that we use exist surveys
and other techniques to address some of these questions.
Yet another difference is that we sometimes offer subjects suggestions about
strategies. The idea is that the multiplicity of equilibria leads to a coordination
problem: agents may settle on the inferior nonmonetary equilibrium even if mone2
To be clear, Davis et al. (2020) take a finite horizon model into the lab, but there are many
differences: (i) We adopt a mechanism design perspective. (ii) We study whether recommendations
(see below) affect equilibrium selection. (iii) We elicit information about social preferences and ask
if anomalous behavior can be understood in terms of altruism, inequality aversion etc. (iv) We use
exit surveys to uncover subjects’ motivation. (v) We made design changes to minimize potential
repeated game effects.
4
tary equilibrium exists. Hence, we ask if a mediator can help with the coordination
problem, by making suggestions, consistent with an interpretation of equilibria going
back to Nash (1950).
Suggestions are typically frowned upon in experimental work: “the researcher
must be careful to avoid demand effects — avoid suggesting the desired results to
the subjects either explicitly or implicitly” (Croson 2002). Yet suggestions seems
appropriate and consistent with mechanism design methods. A suggestion can be
something like “accept money” (this is made precise later), but it must be emphasized that subjects need not follow it, and if monetary equilibrium does not exist
they ought not follow it. Unlike most related papers, our goal is not to see if subjects
can learn to use money over time, but to ask whether money is helpful for supporting
efficient outcomes, with recommendations used as a device for coordinating beliefs.
Our findings indicate that suggestions can help, but a suggestion to accept money
is more likely to be followed if it is consistent with equilibrium. Another finding
is that agents can be rather sophisticated, trying to make inferences about future
opportunities to spend money by the timing of offers to accept money. All of this
is fairly supportive of the idea that individuals, or at least a good number of them,
behave in ways consistent with theory, and that money is useful in supporting good
outcomes in environments with trading frictions.
2
Theory
There are three agents and two events called (pairwise) meetings. In the first meeting, one agent is the consumer and the other the producer. Consistent with the
relevant literature, we think of these two agents and their roles, consumer or producer, as determined randomly by nature. Agents know their roles as producer or
consumer in meetings. Let us label the consumer Player 1 and the consumer Player
2, and the one not in the meeting Player 3. In this meeting nature may or may
not endow Player 1 with money, an intrinsically useless token that is storable and
5
transferable but indivisible. Actions in meetings are described below. After actions
in the first meeting, there is a second meeting between Players 2 and 3, where 2 is
the consumer and 3 the producer. After actions in the second meeting the game
ends.
Possible actions in first meeting are described as follows:
• Player 1 chooses from: make no offer; ask for the good for free; ask for the
good in exchange for money if 1 has money (in principle 1 could also offer
money for nothing but we ignore that).
• If no offer is made Player 1 and 2 part ways.
• If an offer is made Player 2 chooses from: accept, at which point the trade is
executed and they part ways; reject, at which point they part ways.
After Players 1 and 2 part ways the meeting between Players 2 and 3 occurs. The
possible actions are the same, although note that in the second meeting, whether
Player 2 has money depends on what happens in the first meeting and not on nature,
and that after Players 2 and 3 part ways the game is over. In each meeting, if the
producer transfers the good to the consumer the former gets disutility − and the
latter gets utility , where 0. Given that the meeting technology is a
primitive of the environment, andt agents’ roles as Player 1, 2, or 3 are randomly
assigned by nature, the efficient arrangement is for producers to always produce,
similar to standard infinite horizon models.3
Let us first discuss what happens when nature endows Player 1 with money. As
mentioned in the introduction we consider two scenarios: in Model 1, producers in
meetings know if it is the first or second meeting; in Model 2 they do not (effectively
this means producers do not know if they are Player 2 or Player 3). Model 1 has a
unique equilibrium and it entails autarky — i.e., no trade. The argument is simple:
3
The setup has a double coincidence problem. If agents met trilaterally, they could agree on
this: let them be randomly assigned labels 1, 2 and 3, then 2 produces for 1 and 3 produces for 2.
The ex ante payoff is 23 ( − ). The problem is that they meet sequentially in pairs.
6
in the second meeting Player 3 will not bear cost to produce unless Player 2 gives
something of value in exchange, but the only thing 2 could have is money, and that
is worth nothing since the game is over after this meeting. Given this, in the first
meeting Player 2 will not produce by the same reasoning. Therefore, there is no
trade.
Some comments are in order. First, this argument does not even require subgame perfection, but holds using Nash equilibrium or iterated elimination of strictly
dominated strategies as a solution concept. Also, the result easily extends to any
finite , with many agents, whether or not we adopt random terminations at ;
here we use = 2 because it is the shortest interesting horizon, and that minimizes
effects like agents regarding big ∞ as somehow (irrationally) approximating
= ∞. Also, as should be clear, the unique equilibrium outcome is autarky in
Model 1 when there is no money.
Now consider Model 2. If there is no money the unique equilibrium is again
autarky. If Player 1 is endowed with money autarky is still an equilibrium: if others
do not produce for money it is a best response to follow suit. However, there
can be a monetary equilibrium with production in exchange for money in both
meetings. To confirm this, suppose a producer does not know if it is the first or
second meeting, and assigns the correct probability to each, 12. The the expected
payoff to producing is
1
1
(− + ) + (−)
2
2
which is strictly positive if 2. In this case producing for money is a strict best
response to others doing the same.
Therefore monetary equilibrium exists. Moreover, money is essential. Without
it, the unique equilibrium is autarky with payoff 0, while the monetary equilibrium
gives positive expected payoffs to everyone. Although the realized payoff to 3 is
− 0 after getting stuck with the money, we still say this is desirable because ex
ante payoffs are higher, or, amounting to the same thing, because average payoffs
7
are higher if the game is played multiple times.4
There is also a mixed strategy equilibrium. Let be the producer’s probability
of accepting an offer to trade for money. Now ∈ (0 1) is a best response if the
producer is indifferent,
1
2
(− + ) +
1
2
(−) = 0. Hence there is an equilibrium
where goods trade for money with probability = 2 in each meeting. Some
realizations will see production in the first meeting but not the second.5
An extension of the model may also be relevant. Although in theory Model 2
has Players 2 and 3 unable to distinguish between the first and second meeting,
if the game is played in real time, inferences can be made. In particular, agents
might use waiting time as a signal determining the likelihood of being in meeting
1, which seemed to happen in the experiments. Hence, we consider a setup where
such inference is possible and show monetary equilibrium still exists if waiting time
is a sufficiently noisy signal.
Let agents distinguish between { }, indicating early, middle and late
in the game (this can be extended to richer sets of signals at a cost in terms of
tractability). Assume the first meeting can occur at or and the second can
occur at or , creating a signal-extraction problem: agents cannot tell meeting
1 from meeting 2 when = . The probability distribution over { }
conditional of being in the first meeting is
Pr ( |meeting 1) = 1 − , Pr ( |meeting 1) = , Pr ( |meeting 1) = 0
where is an objective probability that is part of the environment. In the second
4
Here readers may notice a resemblance to work on “optimal opacity” by Andolfatto et al. (2014)
or Dang et al. (2017). At the risk of oversimplifying their contributions, the connection is the following: Suppose in our environment some authority knows which meeting is which. Then society is
better off in terms of expected or average utility when the authority does not reveal the information.
5
Shevchenko and Wright (2004) point out that there are robustness issues with the mixed strategy equilibrium: consider the deviation where the consumer offers a lottery, where the producer
gets money and the consumer gets the good with some probability. This deviation is not directly
applicable to our experiments, however, as there is no way to propose a lottery. There can also
exist sunspot equilibria: if players had access to some intrinsically irrelevant, commonly observed
random variable, called a sunspot, they could potentially condition strategies on it. Hence, trade
can occur for some realizations and not others. In the experiments this seems unlikely, however,
mainly because subjects do not see any such random variable.
8
meeting
Pr ( |meeting 2) = 0, Pr ( |meeting 2) = , Pr ( |meeting 2) = 1 −
where is also an objective probability.
If the meeting occurs early, it is known to be meeting 1. If it occurs late, it is
known to be meeting 2. However, the information content in = depends on
strategies, because if players do not accept money, then a money offer reveals that
it is meeting 1. In contrast, assuming that a monetary equilibrium exists in which
money is accepted for sure at ∈ { }, Bayes rule implies
Pr (meeting 1| ) =
+
If the player produces in the first meeting in exchange for money, the next producer
will not produce if that player can detect that it is meeting 2. The probability of
that is , and the expected payoff from accepting at = is
µ
¶
( − ) + 1 −
(−) =
−
+
+
+
Acceptance at gives − , so if players are best responding by accepting
money at they will optimally accept offers at Hence, there is a pure strategy
equilibrium where players produce in exchange for money, except when they know
it is the last meeting, provided that
− ≥ 0
+
Note that production rates will be higher at meeting 1 than in meeting 2. Also note
that = = 1 is Model 2 and = = 0 is Model 1, so this extension spans the
two baseline models. Finally, note that there is also a mixed equilibrium whenever
( + ) , where acceptances are probabilistic at
3
Experimental Design
Each experimental session was conducted as follows. First, subjects participated in
a series of rounds that parameterize the models in Section 2. Then they were given
9
an exit survey asking questions about their considerations in playing the game, plus
a few demographic questions. Finally, they participated in a series of experiments
designed to elicit information about social preferences. The reason for the last part
is this: predictions in Section 2 are based on agents that only care about their own
payoffs, which need not be a good description of our participants. Hence, they play a
generalized dictator game that provides evidence of that. The Online Appendix has
details on the experiments and surveys: https://www.sultanum.com/research.html.
In total we ran four monetary treatments and one treatment without money.
The equilibrium allocation without money is the same for the two models, so we
opted to run experiments without money in Model 2 only. For the treatments with
money, two are based on Model 1, where monetary equilibrium does not exist, and
two are based on Model 2, where monetary equilibrium exists. For each, we ran
treatments where it was left up to the subjects to find (or not) equilibrium, plus
treatments where we act as a mediator by suggesting strategies. These suggestions
were equilibrium strategies in Model 2, but not Model 1.
Part 1 of every treatment consisted of 15 rounds, where in each round a game
from Section 2 is played once. As is standard in experimental economics, participants
played the games multiple times to gain experience, learn the incentives, and possibly
coordinate beliefs. To avoid participants approaching the 15 rounds as a repeated
game, we randomly assigned subjects to groups of three in each round. While
some subjects interacted more than once, they were anonymous, and the number of
participants seems large enough that reputation-building effects should minimal.
At the beginning of the Model 1 experiments, each participant was randomly
selected into the role of Player 1, 2, or 3 with equal probability. These roles were
then fixed for all 15 rounds, implying that in each session there were players in
each role. Then, in each round groups were formed with one player randomly
selected from the sets of players 1, 2, and 3.
In each round of the monetary treatments Player 1 was endowed with a token
and could in meeting 1 offer it to Player 2 in exchange for production. Then Player
10
2 accepted or rejected. Then in meeting 2, if the token was acquired, Player 2 could
choose whether to offer it to Player 3 in exchange for production. Then Player 3
accepted or rejected. Player 3 had to surrender the token, for nothing, if it was
acquired. This completed the round, and the players were then randomly assigned
to new groups of 3, except in round 15, which ended this part of the session.
Model 2 treatments started with selecting participants to be Player 1 or not
Player 1. If Player 1, which happened with probability 1/3, the subject stayed in
that role for all 15 rounds. The other subjects were randomly assigned as Player 2
or 3 with equal probability in each round, and were not informed about that until
after production.
As in Model 1, when Player 1 is endowed with a token it could be offered it in
exchange for production. After accepting or rejecting, the recipient was informed
about their role, Player 2 or 3, in that round. If the token was accepted by a
subject in their role as Player 2, it could be used in the next meeting with Player
3. Importantly, the participant in the role of Player 3 was uninformed whether they
were Player 3 meeting 2 or Player 2 meeting 1. After accepting or rejecting, Player 3
learned their role and payoffs (points) were tallied. Then the players were randomly
assigned to new groups of 3, and those not previously assigned as Player 1 were
randomly assigned new roles as 2 or 3, except in round 15, which ended this part of
the session.
The instructions for the treatments with strategy suggestions, Model 1-S and
Model 2-S, were the same except that the instructions and decision screens included
the following language:
A suggestion: Each player in a group may consider making the following choices:
1. Whenever you have the token, transfer it to the next player (if there
is one).
2. Produce ONLY if you are offered the token.
This is simply a suggestion. Feel free to follow it or not.
11
The suggestion recommends following the monetary strategy of always offering
the token if in possession of it, and always producing in exchange for it. The treatments with this non-binding recommendation were designed with two purposes in
mind. First, we can study whether providing the suggestion facilitates coordination
on the monetary equilibrium in Model 2. Second, since the monetary strategy can
be supported as an equilibrium in Model 2 but not Model 1, introducing the suggestion in Model 1 allows us to determine the strength of so-called demand effects,
meaning that subjects follow the suggestion indiscriminately, regardless of whether
it is consistent with equilibrium.6
Model 1 and Model 2 treatments were designed to test whether fiat money is
more likely to increase production and welfare when its usage is an equilibrium
outcome (Model 2), compared to when it is not (Model 1). Everything else in the
design keeps other differences at a minimum.
In all treatments, subjects were endowed with 3 points, and could earn additional points from playing the games. Specifically, they earned = 3 points from
consumption and lost = 1 point from production. Three out of the 15 rounds
of each session were randomly selected for payment.7 Points were converted into
dollars a the rate 1 point equals 2 dollars. Tokens are not converted into dollars and
subjects were explicitly told “The token does not yield points directly and cannot
be transferred from one game to another.”
After this, subjects were asked to complete an exit survey tailored to the position
they were assigned in the experiment. This survey was designed to better understand
considerations that played a role in their decisions. After this, subjects answered
demographic questions on gender, age, English proficiency and field of study.
Then subjects moved on to the second part of the experiment, where they played
6
In each treatment, the instructions for the first part were read aloud by the experimenter and
followed by a questionnaire to test understanding of the environment, including payoff determination, position assignment, timing and grouping.
7
In each game the lowest possible score is -1, when the subject produces but does not consume.
Since three games were randomly selected for payment, the endowment of 3 points guaranteed that
no player incurred losses in this part of the experiment.
12
a series of generalized dictator games. This part of the experiment provides a measure of their social value orientation ( ) following Murphy et al. (2011). This
measure is meant to control for differences between theoretical and real-world preferences that might be characterized by altruism or inequality aversion.
We used the computerized module developed by Crosetto et al. (2019) for implementation. It consists of a series of 15 generalized dictator games where each
subject chooses as a sender how to allocate payoffs to sender and receiver (the other
subject). We used the ring matching protocol so that each subject functions as both
a sender and a receiver. From the first six (primary) games, one can derive each
subject’s index as the average allocation for the receiver over the average allocation for the sender. A high index represents a more prosocial or altruistic
orientation.
The nine secondary games can further separate efficiency motives from equality
motives (as the two motives are maximized simultaneously in the primary games).
From the secondary games one can calculate the inequality aversion score. After
every subject finished the 15 SVO games, one game of each subject is randomly
selected to determine each subject’s payoff as a sender and as a receiver. The points
in this part of the experiment were converted to cash at the rate 100pts = $3.
The experiments were conducted in 2020 and 2021 at the IELAB at Indiana University or online with remote participation. The subject pool consisted of Indiana
University students recruited via the Online Recruitment System for Economic Experiments ORSEE (Greiner 2015). The experiments conducted face-to-face in the
IELAB were programmed using zTree (Fischbacher 2007), while the online experiments were programmed using oTree (Chen et al. 2016), keeping the decision screens
and procedures as close as possible. In the online sessions, subjects were admitted
into a “zoom room” where they received links to the experiment. Subjects could
only use the private chat function to communicate with the experimenter. In faceto-face laboratory sessions, subjects could raise their hand and the experimenter
would address the subject privately.
13
We ran four sessions in each monetary treatment with the exception of Model 1,
where we conducted six sessions, and we conducted two sessions of Model 2 without
money.8 The two sessions without money were conducted to further validate the
essentiality results from other treatments.
Table 1: Experimental Design
Treatment
Suggestion Sessions Subjects
Model 1
No
6
72
Model 2
No
4
51
Model 1-S
Yes
4
48
Model 2-S
Yes
4
48
Model 2-No Money
No
2
18
We used an across-subject design and no subject participated in more than one
session. For each session 9 to 15 subjects were recruited, for a total of 237 subjects.
The number of subjects per session varied depending on the show-up rate. Subjects
were paid for the first and second part of the experiment, and earned an average of
approximately $19 for between 45 and 60 minutes. Table 1 summarizes the design
and session characteristics.
Based on theoretical predictions, we formulate the following hypotheses, which
is simple enough, but gets to the heart of the matter:9
Hypothesis 1. In Model 2, production is higher with money than without money.
Theory also predicts that autarky is the unique equilibrium in Model 1, while autarky and monetary exchange are equilibria in Model 2. Past experimental evidence
indicates that in coordination games subjects tend to gravitate toward more efficient
outcomes. Based on these observations, we formulate the following hypotheses:
8
For the record, four of the six sessions for Model 1 were conducted face-to-face in 2020 prior
to the COVID-19 pandemic, after which we moved the remaining sessions online. In the Online
Appendix, we provide a comparison between the online and in-person sessions.
9
Note that previous experiments find higher production with money even in models that do
not, stricly speaking, have monetary equilibria. Also, some experiments indicate that money can
be useful even if not essential. Duffy and Puzzello (2014) and Camera and Casari (2014) study
economies where credit (as described in the Introuction) is feasible, so money is not technically
esstential, but it still seems to help deliver superior outcomes.
14
Hypothesis 2. Production is higher in Model 2 than Model 1.
Hypothesis 3. Production is enhanced by the suggestion in Model 2, but not in
Model 1.
4
Results
Here we do several things. First we test whether money is essential in Model 2,
then we test whether production is higher in model 2 than Model 1. The we check
whether making a suggestion leads to higher production in Model 2, where it could
help coordinate on the better equilibrium, and in Model 1, where it should not.
4.1
Money
We first document that money leads to significant increases in production. As seen
in Figure 1, production in Model 2 with money converges towards approximately
60%. When there is no money it is about 20% towards the end. While this is
higher than theory predicts, note that results in Model 2 without money look very
similar to those in Model 1 with money, which is a model with the same equilibrium
allocation.
Figure 1: Production Rates with and Without Money
As another way to see that money has significant effects, consider Table 2. Effects
are statistically significant overall, in early, and later in rounds.
15
Table 2: Production Averages for Model 2 with and without Money
2
2
Difference
# of
without money with money
(t-test)
Observations
03000∗∗∗
05771∗∗∗
02771∗∗
1,140
All Rounds
(0.1222)
(0.0287)
∗∗∗
Rounds 1—3
∗∗∗
05278
06302
(0.0278)
(0.0350)
∗∗∗
Rounds 4—9
∗∗∗
02361
Rounds 10—15
05964
(0.0955)
01025∗∗
228
(0.0404)
03602∗∗∗
(0.1250)
(0.0301)
(0.0979)
02500∗∗∗
05312∗∗∗
02812∗
(0.1667)
(0.0341)
(0.1288)
456
456
N o te : S ta n d a rd e rro rs in p a renth e se s a re clu stere d a t th e sessio n le ve l.
4.2
Model 1 vs Model 2
Theory predicts that when agents know their position, as in Model 1, there is only
the autarkic equilibrium. However, if they do not, as in Model 2, then there is a
monetary equilibrium. As stated in Hypothesis 2, production should be higher in
Model 2 than Model 1. Our first finding supports this.
Finding 1 Production occurs in 24% of Model 1 meetings and 58% of Model 2
meetings.
Table 3: Production Averages by Model
All Rounds
Rounds 1—3
Rounds 4—9
1
02400∗∗∗
2
05771∗∗∗
(0.0299)
(0.0287)
∗∗∗
03750
∗∗∗
06302
Difference (t-test) # of Obs.
03371∗∗∗
2,160
(0.0402)
02552∗∗∗
(0.0578)
(0.0350)
(0.0659)
02187∗∗∗
05964∗∗∗
03776∗∗∗
(0.0350)
(0.0301)
(0.0342)
Rounds 10—15 01937∗∗∗
05312∗∗∗
03437∗∗∗
(0.0268)
(0.0341)
(0.0420)
432
864
*864
N o te : S ta n d a rd e rro rs in p a renth e se s a re clu stere d a t th e sessio n le ve l.
Figure 2 shows production by round in Model 1 and 2 over all meetings. There
is a large difference in production between models and it persists across all rounds.
16
Production tends to decrease initially before stabilizing, consistent with the idea
that subjects are learning, perhaps about how the experiment works in general,
or about what others are doing. Overall production is roughly 24% in Model 1,
but it starts close to 50% and then declines to 20%. While theory suggests that
there should be no production in these treatments, based on other experiments
we expected that some production would occur, particularly in early rounds, when
subjects are learning. Of course, altruism or plain misunderstanding could also lead
some subjects to produce. Based on experience it would have been surprising if
there were no production in Model 1.
Figure 2: Production Rate by Model
Overall production in Model 2 is higher, roughly 60%, starting at 70% and
declining slightly. Because of the multiplicity of equilibria there is no point prediction from economic theory. In fact, the rate of production is remarkably close
to the production probability in the mixed strategy equilibrium, which is 23 for
the parameters in the experiment. However, it may be difficult to imagine agents
honing in on mixed-strategy equilibrium, and very difficult to test whether that is
actually happening, so we do not push this too far.
Table 3 reports average production for Model 1 and 2 and t-test results on
differences (see Table 5 for average production by session). We look at the full
sample, plus we separate early rounds 1-3 from later rounds. As shown in Table 3,
production in Model 2 is significantly higher than Model 1, regardless of how we
17
segment the data. These results are confirmed by a two-sided Wilcoxon rank-sum
test (for details, again see the Online Appendix that was mentioned above).
The difference between production in Models 1 and 2 is consistent with theory:
production is much higher when monetary exchange is an equilibrium. Non-zero
production in Model 1 goes against the theoretical prediction, as mentioned above,
and we will discuss this more below. We will also discuss whether results from
the exit surveys provide insight as to why subjects produce when it is not a best
response, as is particularly puzzling when Player 3 produces.
4.3
Suggestions
Recommending that subjects produce for money in Model 1 should not have an
impact because it is inconsistent with equilibrium, but in Model 2 it is plausible
that recommending that subjects play in accordance with that equilibrium could
help coordination. It is true that participants may just ignore the suggestion if they
believe that others will ignore it, but should they believe others will follow it, the
following suggestion is a strict best response in Model 2.
In order to disentangle the various effects, at this point, consider running the
following regression,
= 0 + 1 × 2 + 2 × × 2 + 3 × × 1 + .
Here denotes production, 1 and 2 are dummies identifying Models 1 and 2,
is a dummy for whether we suggest that subjects produce for money, and are
controls. We control for the meeting, which is 1 or 2, and the round, which goes
from 1—15. We also restrict our sample to production decisions in meetings where a
subject was offered money, and found that the results are robust to this and other
changes in specification.10
10
It is worth mentioning that subjects almost always offer the token when they have it. In Model
1 the token offer rates are 96% and 97% in meetings 1 and 2; and in Model 2 they are 98% and
96% in meetings 1 and 2.
18
Table 4: Number of Observations of Monetary Treatments
Total
With Production
With Money Offer
Meetings Meeting 1 Meeting 2
2,160
1,080
1,080
842
574
268
1,708
1,047
661
We find evidence in support of Hypothesis 3.
Finding 2 Comparing production with and without suggestions: the suggestion significantly increases production in Model 1 only in early rounds. while it significantly increases production in Model 2 in early, middle and late rounds.
Table 5: Impact of Strategy Suggestions
All Rounds
2
03124∗∗∗
× 1
00998
× 2
00941∗∗
(0.0366)
(0.0634)
(0.0346)
∗∗∗
∗∗
Rounds 1—3
03136
(0.0826)
(0.0881)
(0.0679)
Rounds 4—9
03476∗∗∗
00786
01075∗∗∗
(0.0387)
(0.0416)
(0.0377)
00600
01067∗∗∗
(0.0738)
(0.0320)
∗∗∗
Rounds 4—15 03137
(0.0424)
02266
# of Obs.
1,708
365
00419
668
1,343
N o te : S ta n d a rd e rro rs in p a renth e se s a re clu stere d a t th e sessio n le ve l.
Figure 3: Production by Model and Suggestion
19
Table 5 and Figure 3 provide support for Finding 2 (see also the Online Appendix where we report robustness checks). Figure 3 depicts average production
by round for different treatments. The panel on the left plots production with and
without suggestion for Model 1, while the one on the right plots it for Model 2.
The overall conclusion from Figure 3 and Table 5 is that the suggestion seems to
improve coordination when consistent with equilibrium, and is largely ignored otherwise. This indicates that the increased production in Model 2 has to do with
improved coordination, as opposed to, say, a desire by the participants to please the
experimenter.
Specifically, in Model 1 the suggestion may impact production early on, but then
it converges to the same level. In Model 2 the suggestion increases production by
about 10%. These results are also confirmed by a two-sided Wilcoxon rank-sum
test where the unit of observation is average production at the session level (again
se the Online Appendix). These results are consistent with theory: subjects learn
to ignore the suggestion in Model 1, where it should not have an impact, while it
increases production in Model, where using money is an equilibrium.
4.4
Production in Meetings 1 and 2
In Model 2 there is a choice about how to perform the experiments. One possibility,
that we did not pursue, is to use have Players 2 and 3 specify contingent plans
before observing offers. The upside of this is that there would be nothing the
players could use to infer whether they are in meeting 1 or 2. However, it also
makes the experiment seem less like dynamic economic interaction. Hence we ran it
as a sequential game. The issue this creates is that sophisticated participants may
try to make inferences about which meeting they are in based on how long they wait
for an offer. As a matter of fact, in our exit surveys, some subjects said they tried
just that,
As shown in Section 2, there are still monetary equilibria when waiting time
is a noisy signal, as long as it is noisy enough. Should production in meeting 1
20
exceed meeting 2, this is evidence of sophisticated reasoning, which we find rather
intriguing. Consider the following regression,
= 0 + 1 × 2 + 2 × × 2 + × × 1 +
where the new variable identifies whether the production is taking place in meeting
1 or 2. In the regressions we tried various controls, and considered the interaction
of meeting and suggestions, but the effects are small, insignificant and not robust
to the specification; results are quantitatively and qualitatively the same with or
without these variables. Meetings without a money offer are dropped, leaving 1,708
meetings, where 1,047 are meeting 1 and 661 are meeting 2.
Finding 3 Comparing production in meetings across models: production in Model
2 is 20% lower in meeting 2; Production is 30% higher in Model 2 for meetings
1 and 2.
Table 6: The impact of Meeting on Production
All Rounds
2
02606∗∗
(0.0919)
Rounds 1—3
00637
(0.2206)
Rounds 4—15 03085∗∗∗
(0.1063)
× 1
−02459∗∗∗
(0.0378)
∗∗∗
−03383
× 2
−02099∗∗∗
(0.0326)
× 1
00779
× 2
01857∗∗
(0.1465)
(0.0703)
∗∗∗
(0.0348)
−01654
(0.1423)
(0.0942)
(0.2223)
−02240∗∗∗
−02215∗∗∗
−00027
01889∗∗
(0.0523)
(0.0140)
04039
(0.1746)
01654
# of Obs.
1,708
365
1,343
(0.0688)
N o te : S ta n d a rd e rro rs in p a renth e se s a re clu stere d a t th e sessio n le ve l.
Support for finding 3 is in Table 6 and Figure 4. Table 6 shows that, as before,
behavior in early rounds is very different from later rounds. In Model 1, subjects
tend to produce 0.25 less in meeting 2. In Model 2, there is no significant difference
between production in meetings 1 and 2. That is consistent with subjects not
knowing in which meeting they are in. However, in rounds 4—15 subjects produce
about 0.22 less in both models, and both coefficients are significant at the 1% level.
Of course, because the information is not perfect, production is still 0.30 higher in
21
Model 2 compared to Model 1, in meeting 2, because production falls by about the
same amount in both models.
Thus, it appears that subjects may be able to infer to some degree which meeting
they are in, but the inference is noisy, as otherwise we would not observe higher
production in Model 2. As detailed in Section 2 such behavior is consistent with
equilibrium play when the time of meeting is random.
Figure 4: Production by Model and Meeting
Figure 4 depicts average production by round for meetings 1 and 2. The panel on
the left shows production in both meetings for Model 1, while the one on the right
shows it for Model 2. The interpretation, again, lines up fairly well with theory.
Production is lower in meeting 2 for both Models 1 and 2. For Model 1 this should
not be the case, but it is still not a huge surprise, as Player 2 can always hope
Player 3 accepts the money, while there is no such hope for Player 3. Production
in meeting 2 is still substantially higher in Model 2, which rationalizes production
in meeting 1. To see this, note that the payoff from consuming in the game is 3,
while the cost of production is 1. From Figure 5, production occurs in about 50%
of the meetings. Then we have that 05 × 3 = 15 1, so producing in meeting 1 is
optimal given behavior in the experiment.
22
4.5
Discussion
Our results are largely consistent with theoretical predictions, but there are some
departures. The main departures include production by Player 3 in Model 1 and
production by Players 2 and 3 in Model 2 without money.11 Given that production in
Model 1 is significantly different from zero, one may wonder whether other-regarding
preferences could play a role. To test this, we re-ran the regressions including
subjects’ scores from social value orientation, , as a control,
= 0 + 1 × + 2 × 2 + 3 × × 2 + 4 × × 1 +
The measure is constructed using outcomes from generalized dictator
games which differ in how costly it is for a player making the offer to give utility to the receiver. Details are provided in the Online Appendix (see also Murphy
et al. 2011, who first suggested this measure). However, for our purposes, all that
matters is that low (high) values are interpreted as the individual caring less (more)
about others.
Table 7: The Impact of Social Value Orientation
All Rounds
2
02070∗∗∗
(0.0619)
Rounds 1—3
∗∗∗
03180
(0.1083)
Rounds 4—15
01795∗∗
(0.0707)
× 1
−00057∗∗
× 2
00008
× 1
00997∗
× 2
00923∗∗
(0.0012)
(0.0530)
(0.0364)
−00016
02251
00456
(0.0026)
−00017
∗∗
(0.0022)
(0.0869)
(0.0690)
−00069∗∗
−00015
−00617 01035∗∗∗
(0.0025)
(0.0028)
(0.0015)
(0.0609)
# of Obs.
1,708
365
1,343
(0.0340)
N o te : S ta n d a rd e rro rs in p a renth e se s a re clu stere d a t th e sessio n le ve l.
Table 7 reports on the results. While we conjectured that high production in
Model 1 would be correlated with prosocial preferences, there is no evidence for this.
On the contrary, we find that the variable has a negative impact on production
11
Production in Model 1 by Player 2 tends to be higher than by Player 3, which can be in part
rationalized by the observation that occasionally Player 3 chose to produce for the token.
23
in Model 1 and no significant effect in Model 2. Moreover, in our main specification,
and alternatives considered for robustness, the magnitude of the impact is very
small.
Deviations from theory seemingly must be explained by other factors. Consider
the exit survey, where subjects reflect and explain their decisions by answering
multiple choice and open-ended questions. In the following, we describe the answers
by those who produced for at least 5 rounds.
In Model 1 without suggestions, only 1 out of 24 subjects who were Player 3
produced in more than 5 rounds, and the answers provided by this subject suggest
confusion as much as anything else. In Model 1 with suggestions, 4 out of 16 subjects
who were Player 3 produced in more than 5 rounds. One appeared to be confused,
selecting the option “To increase the chance of trading it for the good with player
3” as the reason to produce for the token. The others selected “to help the other
player,” “to follow the suggestion,” “I made a mistake,” or “I wanted the token for
the sake of it.”
In Model 2 without money, 5 out of 12 Players 2 or 3 produced for more than
5 rounds. Two of these subjects appear confused as they selected “To increase the
chance of others producing for me in this game,” which is not possible. One player
selected “To increase the chance of others producing for me in a following game,”
suggesting that there may be some repeated game effects, as much as we tried to
eliminate these. The remaining two subjects indicated that they produced to “help
the other player,” with one of them adding “I made a mistake.”
Overall, helping the other player and confusion appear to be the dominant explanations for production choices that departed from theoretical predictions. It is
perhaps puzzling to find that “help the other player” appears as one of the main
reasons to produce when the social value orientation does not seem to explain anything. Perhaps if exit surveys are more often used in experiments, in the future, we
may gain additional insight. For now we simply report that certain subjects’ play
is hard to understand.
24
However, it is important to emphasize the following: The fact that there are
some agents in an economy that use strategies we do not fully understand is not
especially relevant for the purposes of this project. As long as there are other agents
that are sophisticated enough to grasp the environment and play best responses, it
is clear from the results that money is essential — we can achieve better outcomes
with it than without it.
It is also clear that suggestions help, which or may not be a surprise, but it is
still interesting. For society, learning to adopt the institution of monetary exchange
is a complicated enterprise, and it is useful to know that having a mediator making
recommendations can help in the selection of a better outcome. Indeed, this can
be interpreted as a primitive form of “forward guidance,” a recently popular policy
option among central bankers.
5
Conclusion
This project has developed simple models where monetary equilibria exist in finite
environments, making them better suited (for some issues) to experimental methods
than models used before. There may well be disagreement among people, even
among coauthors, as to how problematic is the tension between infinite-horizon
models and finite-horizon experiments, but it seems unambiguous that it is useful
to have alternative environments that avoid the issue. We then took the theory
to the lab to address several substantive issues. Perhaps the main question is the
simplest: Is money essential? The answer is yes, in the precise sense that we can
achieve better outcomes with money than without it. This may not seem surprising,
but it is worth mentioning that it is inconsistent with many economic models: in
standard general equilibrium theory, money cannot help; in general equilibrium
theory with cash-in-advance constraints, it only makes things worse.
Our experimental framework is similar to work on the microfoundations of monetary theory in several ways, featuring bilateral trade, random matching and in-
25
formation frictions. Other innovations here include making suggestions, not in an
effort get the desired results, but as a coordination device. It is comforting that
suggestions are more likely to have an impact when they are consistent with best responses. We also used exit surveys and other information to help understand what
subjects cared about or had in mind when they played puzzling strategies. Play
deviating from theory did not seem to be related econometrically to a standard
measure of caring for others; perhaps it is best chalked up to confusion. Having said
that, many subjects were remarkably sophisticated in their reasoning, even making
inferences based on the timing of events. All things considered, we learned a lot
from this project about both monetary economics and experimental economics.
26
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