Rational Bubbles and Middlemen∗
Yu Awaya†

Kohei Iwasaki‡

Makoto Watanabe§

October 8, 2021

Abstract
This paper develops a model of rational bubbles where trade of an asset takes
place through a chain of middlemen. We show that there exists a unique androbust equilibrium, and a bubble can occur due to information frictions in bilateral and decentralized markets. Under reasonable assumptions, the equilibrium
price is increasing and accelerating during bubbles although the fundamental value
is constant over time. Bubbles may be detrimental to the economy, but any announcement from the central bank has no effect on welfare with risk neutral agents.
Middlemen are the source of financial fragility.

Keywords: Rational bubbles; Middlemen; Higher-order uncertainty; Asymmetric
information; Flippers
JEL Classification Numbers: D82, D83, D84, G12, G14∗

We would like to thank Gadi Barlevy, Henrique de Oliveira, Antonio Doblas-Madrid, Ed Green,
Shigeki Isogai, Michihiro Kandori, Vijay Krishna, Erwan Quintin, and Marzena Rostek for their helpfulcomments. In particular, we are deeply indebted to John Conlon and Randy Wright for their detailed
comments and suggestions. Awaya acknowledges financial support from NSF Grant (SES-1626783).
†
University of Rochester. Address: 208 Harkness Hall, Rochester, NY 14627, US. Email: YuAwaya@gmail.com‡
Osaka University.
Address: 6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan.
Email:
iwasaki@iser.osaka-u.ac.jp§
Vrije Universiteit Amsterdam. Address: DeBoelelaan 1105, 1081 HV Amsterdam, The Netherlands.
Email: makoto.wtnb@gmail.com

1

1

Introduction

As was emphasized in an article “How Tales of ‘Flippers’ Led to a Housing Bubble” by
Shiller (2017), the bubble preceding the Great Recession and financial crisis of 2007 to2009 is tightly linked to flipping in certain housing markets. In fact, this applies tobubbles in many other markets as well. Flipping in decentralized markets often occurs
through chains of middlemen who are specialized in purchasing an asset at a low priceand quickly reselling it at a higher price. The objective of our paper is to explore a
tractable equilibrium framework to understand the theoretical connection between theoccurrence of bubbles and middlemen.1
In this paper, we construct a tractable finite-period model of rational bubbles caused
by a chain of middlemen. We show that
Theorem. There is a unique equilibrium. A bubble occurs with positive probability
and once it occurs, it bursts for sure. The equilibrium is robust to perturbations ofparameters.
Of course, a standard backward induction argument implies that if the value of anasset were common knowledge, then a bubble would never occur. However, we show thatwith lack of common knowledge, a bubble occurs in a unique and robust equilibrium.
In our model, middlemen contact each other individually and negotiate the terms oftrade bilaterally. Prior to trade, the final user may or may not know the consumption
value of an asset. Importantly, because of rumors about the final user, downstream
middlemen may or may not know whether the final user knows the consumption valueof the asset, and upstream middlemen may or may not know whether the downstream
middlemen know whether the final user knows the consumption value of the asset.
1

Bubbles that occur via a chain are common not only in housing markets (see Bayer et al. 2020),

but also, for example, in a market for securities, where the chain starts with an originator of loans(typically, a bank), the agents in the middle of the chain are investment and commercial banks, andthe final user is an individual investor.

2

This opens room for a bubble—an upstream middleman acquires an asset, knowing it isoverpriced, in hopes of finding a downstream middleman, who also knows it is overpriced
but believes that he can sell the asset to someone else. In other words, each middlemancares less about the fundamental value of the asset, and more about how much the otheragents value it.
The general property that a higher-order uncertainty associated with flippers causes
bubbles captures many historical episodes. Notably, Kindleberger and Aliber (2005)
says “The bubble involves the purchase of an asset, usually real estate or a security, notbecause of the rate of return on the investment but in anticipation that the asset or
security can be sold to someone else at an even higher price; the term ‘the greater fool’has been used to suggest the last buyer was always counting on finding someone else towhom the stock or the condo apartment or the baseball cards could be sold.”2
Under reasonable assumptions, we show that the equilibrium price increases during
bubbles even when the fundamental value is constant over time. This is because each
middleman always faces the risk that his downstream middleman rejects the asset, andprices are determined in such a way that the risk is compensated. The price also tends
to accelerate because the probability that a middleman can sell the asset decreases
over time. So, middlemen who trade in later periods are exposed to bigger risks. Thisequilibrium price trajectory explains, for instance, the housing bubble during 2003–
2005 where over the course of market fluctuations, the price tends to increase, possibly
accelerating, until it is interrupted by a sudden markdown or even a crash. Similar pricetrajectories are observed in many other bubbles, not only in the modern bubbles, e.g.,
the dot com bubble or the Japanese asset bubble, but also in the classical Dutch Tulip
bubble (see Figure 1), etc. We show it as a unique and robust equilibrium—we claim
not only that it can occur as an equilibrium, but also that it must be this way.
2

There is a literature of “greater fool models” initiated by Allen et al. (1993). While both theirmodels and ours share some common features—in particular, higher-order uncertainty causes bubbles—,there are important differences. We will detail them in Section 3.3.

3

Price Index
250
200
150
100
50
November 1

February 3

May 1

Time

Figure 1: Tulip prices over the 1636-1637 period. Source: Thompson (2007)

We focus attention on a bubble which occurs in a decentralized market with bilateral
trades. In particular, the market is assumed to be opaque in the sense that the prices atwhich individual trades occur are not publicly observable. This is a feature of over-thecounter markets, where middlemen contact each other individually and negotiate theterms of trade bilaterally. Indeed, this opaqueness assumption plays the key role for therobustness of our bubble equilibrium. That is, we show that a bubble can occur evenwhen prices are publicly observable but such an equilibrium is not robust to a small
perturbation of parameters (see the discussion section of our main result in Section
3.3). The opaqueness assumption echos Duffie’s (2012) sentiment that “The financial
crisis of 2007–2009 brought significant concerns and regulatory action regarding the roleof OTC markets, particularly from the viewpoint of financial instability. OTC markets for derivatives, collateralized debt obligations, and repurchase agreements playedparticularly important roles in the crisis and in subsequent legislation.”Thanks to the features of our model—rationality, common prior, uniqueness, androbustness—, we are able to offer the following characterizations of our equilibrium.

4First, like in the traditional view, bubbles facilitate trade in our model. However, weshow that bubbles are not necessarily welfare improving. We characterize exactly whena bubble is detrimental: it is welfare reducing (improving) if there are losses (gains)from trade between middlemen.3
Second, “irrational exuberance” is the phrase used by Alan Greenspan in a 1996
speech, “The Challenge of Central Banking in a Democratic Society.” The speech wasgiven during the dot-com bubble. The Greenspan’s comment was broadcasted by dedicated financial TV channels around the world live, and its meaning was widely discussed
by financial journalists. The Tokyo market moved down sharply after his speech, withother markets following. Inspired by the recent progress in the information design literature (see Kamenica and Gentzkow 2011), we analyze the welfare consequence of suchbubble-bursting policies à la Conlon (2015) as an information design problem. We assume that the information designer (the central bank), who aims at maximizing total
expected utility, is able to send out a public message about the value of the asset. Suchpublic announcements create common knowledge, and burst a bubble. The information
designer can burst the bubble even if it does not have superior information technology tothe other market participants. Surprisingly, however, given risk neutral agents, it cannot
influence ex ante welfare even if it has superior information technology. This neutrality
result breaks down if agents are risk averse, and it depends on how we introduce risk
aversion whether the bubble-bursting policy is beneficial or detrimental.
Finally, based on comparative statics exercises we analyze how market structure
influences financial fragility. We find that the wider the rumors are spread, the lesslikely a bubble occurs and the shorter the expected duration of the bubble is, but thelarger its size is. The recent regulation on Credit Default Swap (CDS) markets (e.g., theDodd-Frank Act) aims at preventing bubbles by facilitating information sharing among
investors. Our result predicts a side-effect—while a larger amount of information sharing3

Of course, even if there are losses from trade between middlemen, there are (potential) gains fromtrade between middlemen and the final user.

5

reduces the probability and length of a bubble, it may increase its size. Also, we findthat with a larger number of middlemen, the ex ante probability that a bubble occurs
and the expected duration of a bubble can be all magnified—middlemen are the sourceof financial fragility.

Related Literature
Allen et al. (1993) construct a model of bubbles with information asymmetry, relaxing the common-knowledge assumption. This line of models is often referred to the“greater fool theory” approach and substantially simplified in a sequence of papers by
Conlon (2004, 2015), Liu and Conlon (2018) and Liu, White, and Conlon (2020). Thesepapers and the current paper both point the crucial role of higher-order uncertainty
in the occurrence of bubbles. Our innovation is to establish both the robustness andthe uniqueness of equilibrium, which is obtained by explicitly highlighting the role ofmiddlemen in an opaque market. We are able to pursue relatively easily the welfare
analysis, policy analysis, and comparative statics based on standard tools. Further, we
identify an intuitive necessary and sufficient condition for which price increases, andbubbles are detrimental to welfare.
One crucial difference is that, in these greater fool models, buyers trade either with“bad” sellers—sellers who know that they are selling worthless objects—or “good” sellers. For the occurrence of bubbles, the two types of sellers must behave in the same way.
Since this occurs only at knife-edge parameter values, their bubble equilibria are notrobust to small perturbations of parameter values.4 In contrast, in our model, there are
no good or bad sellers because buyers are better informed than sellers. The exception
is the middleman next to the final user. However, since the final user cannot observe
past prices in the opaque market, the final user does not learn whether the middleman
4

See, e.g., Conlon (2015) and Liu and Conlon (2018) for a discussion of this matter. They argue

that with a continuum of states, the bubble equilibrium is robust.

6

is a good or bad seller from behavior. Thus, our bubble equilibrium is robust.5
There are papers that model bubbles as a timing game—a game where each trader
wants to sell an overpriced asset quicker than its rivals. Abreu and Brunnermeier (2003)
show that overpricing can occur in such a model along with an exogenously increasing
price path. Our modeling approach differs from them. Moreover, they do not argue
whether the equilibrium is unique or not.6
Matsushima (2013) and (2020) also model bubbles as a timing game. These papers
obtain uniqueness, but unlike the current paper, they assume existence of irrational
agents who ride a bubble no matter what. A rational agent also rides the bubble,
pretending to be irrational. In our model, uniqueness is obtained in a simpler setting without irrational agents, where only a standard backward induction argument isneeded. Further, in Matsushima (2013) and (2020), a bubble occurs in a mixed-strategy
equilibrium (a pure-strategy equilibrium does not exist in his models), while in oursimpler setting, a bubble occurs in a pure-strategy equilibrium. This is advantageous
because we obtain a sharper characterization.
Finally, we also contribute to the literature of middlemen initiated by Rubinsteinand Wolinsky (1987). We find a novel motive of middlemen—middlemen care less about
the quality of the asset but more about what others know about it once the commonknowledge assumption is relaxed. Our environment is akin to Wright and Wong (2014),
who develop a model of intermediation chains. They show that a bubble can occur with
5

Other existing models in this literature emphasize risk-sharing (Allen et al. 1993, Liu and Conlon2018) or intertemporal consumption-smoothing (Liu et al. 2020) as the motive for trade. In contrast,
the motive for trade in our model is neither risk-sharing nor consumption-smoothing, but is based on
middlemen’s incentive to flipping.
6
In a timing game model of bubble, Doblas-Madrid (2012) also generates a bubble in which a priceincreases, and his model, gains from trade are rooted in a liquidity shock. In his model, the priceincreases due to the growth in fundamentals, whereas in our model, the price increases even whenfundamentals do not grow. Moreover, like Abreu and Brunnermeier (2003), Doblas-Madrid (2012) also
does not obtain uniqueness. See also Araujo and Doblas-Madrid (2019).

7

symmetric information only when there is an infinite number of middlemen. In contrast,
we offer a finite-period model with asymmetric information. Moreover, unlike in Wrightand Wong (2014), middlemen in our model can be active even when the asset does nothave a positive fundamental value. We study the role of intermediaries for bubbles infinite intermediation chains and show that they are essential for bubbles. Though it ishighly stylized, our model delivers novel insights—the role of higher-order uncertainty
in the trade involving middlemen—into the occurrence and burst of rational bubbles.

The Structure of the Paper
The rest of the paper is organized as follows. Section 2 presents the model. Section
3 shows the existence, the uniqueness and the robustness of equilibrium. In Section 4,
we study how the equilibrium price changes over time. Section 5 derives the welfare
implications, and in Section 6, we discuss the implications of policies that affect agents’beliefs. Section 7 examines how market structure affects the occurrence and durationof a bubble. Section 8 concludes. The Appendix A explores an alternative setup whereindividual prices are publicly observable. In the Appendix B, we show that when agentsare risk-averse, policies that affect agents’ beliefs have welfare consequences. Finally in
the Appendix C we provide a parametric example.

2

The Model

In this section, we describe the economic environment in the first subsection and theknowledge of agents in the second subsection.

2.1

The Environment

There are N agents A1 , A2 , · · · , AN where 2 < N < ∞. They are spatially separated
in the following fashion: An can meet and hence, trade with An−1 and An+1 but with8

no one else. Therefore, trade between An−1 and An+1 must go through An . We assume
that trade is sequential, An and An+1 trade in period n, and An exits the economy after
trading with An+1 . Hence, time is discrete and continues for N − 1 periods.
There are two objects in this economy. One is an indivisible asset x in fixed supply,
and the other is a divisible good y that every agent can produce at unit cost. Only A1
is endowed with x, that is, A1 is the initial owner of x. He can try to trade it to A2 in
exchange for some amount of y, say y1 . We assume the consumption value of x for A1
is zero. More generally, if An acquires x from An−1 , he can try to trade it to An+1 foryn , which generates a payoff
U (yn ) = κynwhere κ > 0 is the value of An+1 ’s good to An . The consumption value of x for An is
0 for each n < N . The value of x for the final user AN is v > 0 with some probability
and 0 with the remaining probability. The key here is that the final user may have alarger consumption value than the other agents.7 For simplicity, agents do not discountutilities between any two periods. Middlemen A2 , A3 , · · · , AN −1 are a necessary part ofthe process of getting x from the initial owner A1 to the final user AN .
For our purpose of studying the occurrence of bubbles in the presence of middlemen,
we keep the determination of terms of trade as simple as possible. We therefore employ−1
a generalized Nash bargaining to determine the terms of trade (yn )N
n=1 , where agents’utilities are zero if they disagree to trade.8 In trade between An and An+1 , let θ be the7Middlemen could have a positive consumption value as well. More precisely, even if the consumption

value of x for An is f ≥ 0 for each n < N , and the value for AN is f + v with some probability andf with the remaining probability, all our results hold as long as v is sufficiently large, f is sufficiently
small, or κ is not too small. Just for simplicity, we assume f = 0.
8
Formally, we consider a game form where a fictitious third party suggests the exchange ratio
following the Nash bargaining solution, and then each agent either accepts or rejects the trade. The
trade occurs only when both accept (in order to get rid of an equilibrium where an agent rejects because
the other rejects, we assume that agents move sequentially). While the agents have private information,
the terms of trade that the third party proposes do not depend on it. Our result is, however, robust to9

bargaining power of An with 0 < θ < 1.9
The chain structure of trade can be justified by the presence of search frictions that
only allow trade between two adjacent agents.10 Alternatively, it can be motivated by asetup where good y produced by An+1 differs from good y by An , and only An can enjoy
good y produced by An+1 . This interpretation would be relevant especially when κ > 1,
in which case homogeneous-good interpretation might allow each agent to produce andconsume good y for himself as much as he wants.
We assume that the final user cannot observe trade among the other agents. The
Appendix A explores an alternative setup where individual prices are publicly observable. With a transparent market that we study in the Appendix, bubbles are shown tooccur in equilibrium in a knife-edge case. With an opaque market that we study in themain text, bubbles are shown to occur in equilibrium for any parameter values.

2.2

Knowledge

In this section, we describe the knowledge of agents. It is reminiscent of Rubinstein’s(1989) Email game and meant to capture the essence of a rational bubble in the simplest
possible way. Following the literature, we say that
Definition 1. A rational bubble occurs if all agents know that the asset x is worthless,but it is traded for a positive amount of the good y.
There are two necessary ingredients for a rational bubble. First, there should be atleast one agent who may not know the consumption value—otherwise, the value of theother bargaining protocols.
9
When θ = 0, it will turn out that the terms of trade are zero, and hence we assume θ > 0. When
θ = 1, agents are indifferent on whether they buy the asset x or not. Thus, to ensure the uniqueness ofequilibrium, we assume θ < 1. However, if we assume that agents buy whenever they are indifferent,
all the results hold even with θ = 1.
10Hollifield et al. (2017) and Li and Schürhoff (2019) document the empirical evidence of intermediation chains often observed in OTC financial markets.

10asset is commonly known and the asset can never be overpriced. Second, there should
be at least some situation where all agents, including the agent in question, know thevalue of the asset—given our definition of a rational bubble, everyone should know it isoverpriced.
To capture these ingredients, we shall assume that all parameters describing utilities,
costs, etc., are common knowledge, except for the consumption value of asset x. We
also assume that prior to trade, all agents except for AN observe the consumption valueof x for AN .11 This setup can be applied to, for example, housing or antique markets.
In such a market, middlemen are experts who know the fundamental value of the asset,
and the final user is not an expert (e.g. imagine a temporal consumer like a mover into
a new city or an amateur collector of artistic paintings).
If the value of the asset is zero, AN receives a signal with some probability. Otherwise,
AN does not receive any signals. Thus, if he receives a signal, then AN is sure that theconsumption value is zero, and in this event, every other agent also knows that x is
worthless. Moreover, if AN receives a signal, he (non-strategically) sends a signal (emailin the terminology of Rubinstein 1989) to AN −1 . The signal reaches AN −1 with someprobability but is lost with the remaining probability. Thus, if AN −1 receives a signal, heis sure that AN knows that the consumption value is zero. Similarly, if AN −1 receives asignal from AN , he (non-strategically) sends a signal to AN −2 . The signal reaches AN −2
with some probability but is lost with the remaining probability. This process continues
until a signal is lost between some two agents or the initial owner A1 receives a signal.
In words, the signal (rumor) that AN knows that x is worthless spreads from AN to A1 ,11

Even when we consider a more general setting where A1 , · · · , AN −1 do not know the consumption

value at some state, by definition of the rational bubble, we would end up considering a state where allagents know the consumption value. Hence, it is sufficient to focus on the model where A1 , · · · , AN −1
always know the consumption value. Also, the agent who may not know the consumption value does
not need to be AN , i.e., the final user, and could be one of the middlemen. Our formulation, however,is able to generate a bubble in the cleanest way in an intermediation chain setting.

11

but it is subject to loss between any two agents.12 We assume that information does
not flow from An−1 to An , and in particular, AN −1 cannot credibly reveal the true valueof the asset to AN . We finally assume all these signals occur prior to trade.
To describe the above situation formally, we introduce N + 2 states of the world.
The consumption value of x for AN is v > 0 at state ωv and 0 at the other states.
When the state is ωφ , no agent receives a signal although x is worthless. On the other
hand, for each n = 1, · · · , N , state ωn corresponds to the case where all the agents
AN , AN −1 , · · · , An receive signals, while the others do not. Hence, the set of states isΩ = {ωv , ωφ , ωN , ωN −1 , · · · , ω1 }.
Let µ be the common prior distribution over Ω. We assume it has full-support on Ω.
We represent agents’ knowledge by partitions of Ω. Agent AN ’s partition is
PN = {{ωv , ωφ }, {ωN , ωN −1 , · · · , ω1 }}.
The first element, {ωv , ωφ }, corresponds to the case where AN does not receive a signal
and hence, does not know whether x is worthless. The second element, {ωN , ωN −1 , · · · , ω1 },corresponds to the case where AN receives a signal and knows that x is worthless. For
each n < N , agent An ’s partition isPn = {{ωv }, {ωφ , ωN , ωN −1 , · · · , ωn+1 }, {ωn , ωn−1 , · · · , ω1 }}.
The first element, {ωv }, corresponds to the case where the consumption value of x for
AN is v. The second element, {ωφ , ωN , ωN −1 , · · · , ωn+1 }, corresponds to the case where
the consumption value is zero, but An does not receive a signal. The third element,
12

For simplicity, we say that An “non-strategically” sends a signal to An−1 when An receives a signal.

However, An does not actually care whether An−1 receives the signal. It will turn out that, if An−1
receives the signal, the only effect on An is that An does not receive an offer from An−1 , but it is always
optimal for An to reject the offer given that An has already received the signal. Hence, An does notcare whether he sends a signal to An−1 , and might therefore use a mixed strategy, or even just let the
information randomly leak out.

12

{ωn , ωn−1 , · · · , ω1 }, corresponds to the case where the consumption value is zero and
An receives a signal (An−1 , · · · , A1 may also receive signals). An agent can distinguish
any two states if those states belong to a different element of his partition, but cannot
otherwise.
Assume ω 6= ωv . Then, if AN does not receive a signal, the posterior probability thatthe consumption value of x for AN is v is µ(ωv )/[µ(ωv ) + µ(ωφ )], and hence the expectedvalue isve =µ(ωv )
v > 0.
µ(ωv ) + µ(ωφ )(1)

It will be useful to calculate the probability ψn that An+1 does not receive a signal
conditional on the event that An does not receive a signal. This will turn out to be theprobability that An can sell the asset x to An+1 , given that An has not received a signal.
This probability is as follows: for N − 1,
ψN −1 =µ(ωφ )µ(ωφ ) + µ(ωN )and for each of n = 1, · · · , N − 2,ψn =µ(ωφ ) + µ(ωN ) + · · · + µ(ωn+2 ).
µ(ωφ ) + µ(ωN ) + · · · + µ(ωn+2 ) + µ(ωn+1 )Note that 0 < ψn < 1 for each n = 1, · · · , N − 1 because µ(ω) > 0 for each ω ∈ Ω.

3

Equilibrium

In this section, we derive an equilibrium of the economy,13 and argue it is unique. In thefirst subsection, we display the equilibrium and in the second subsection, we provide a
proof. In the third subsection, we discuss our main result.
13

Our equilibrium concept is sequential equilibrium with the game form defined in footnote 8.

13

3.1

Life of a Bubble−1
Define a sequence (ŷn )N
n=1 as follows: for N − 1,ŷN −1 = θveand for each n = 1, · · · , N − 2,ŷn = θκψn+1 ŷn+1 .

(2)−1
Similarly, define a sequence (ŷnv )N
n=1 as follows: for N − 1,vŷN
−1 = θveand for each n = 1, · · · , N − 2,vŷnv = θκŷn+1.

Note that ŷn > 0 and ŷnv > 0 for each n = 1, · · · , N −1 because θ > 0, ve > 0, κ > 0, andψn+1 > 0 for each n = 1, · · · , N − 2. Here ŷn and ŷnv will be the prices An receives whenω ∈ {ωφ , ωN , · · · , ωn+2 } and ω = ωv , respectively, as shown in the following lemma.
Lemma 1. The following two statements hold.
1. Assume ω 6= ωv . In equilibrium, if agent An+1 receives a signal, agent An+1 does
not trade with agent An , or yn = 0; if agent An+1 does not receive a signal, agent
An+1 trades with agent An and obtains x in exchange for yn = ŷn . Moreover, thisoutcome is unique.
2. Assume ω = ωv . In equilibrium, An+1 always trades with An and obtains x in
exchange for yn = ŷnv . Moreover, this outcome is unique.
Recall that we assumed that the past prices cannot be observed by the final user.
Thus, agent AN cannot learn the state from whether the terms of trade are ŷn or ŷnv .
If ω ∈ {ω1 , ω2 }, a bubble does not occur because A2 receives a signal and does nottrade with A1 . If ω = ωφ , trade takes place, but this is simply because AN does not
14

know that x is worthless. If ω ∈ {ωN , ωN −1 , · · · , ω3 }, Lemma 1 describes the life of abubble.
To see this, suppose the economy is at ωn∗ where n∗ > 2. Then, agents AN , · · · , An∗
receive signals, while the others do not. Given this realization, every agent knows that x
is worthless, and hence the fundamental value of x is zero. Yet, the asset x is exchanged
for a positive amount of the good y for n∗ − 2 periods, until it reaches agent An∗ −1 . In
this sense, a bubble is occurring. Obviously, if the fact that the fundamental value of x
is zero were common knowledge, then x would not be traded. At period n∗ − 1, agentAn∗ refuses to buy the asset from An∗ −1 , and the bubble bursts. We summarize thisresult as follows.
Theorem 1. For any parameter values, the equilibrium exists and is unique. In theequilibrium, a bubble occurs when ω ∈ {ωN , ωN −1 , · · · , ω3 }. Moreover, the bubble bursts
for sure.
Since the equilibrium exists for any parameter values, obviously it is robust to small
perturbations of parameters. The theorem clarifies the role of middlemen for the occurrence of bubbles. To see this, consider the case without middlemen (that is, N = 2).14
The set of states in this case is Ω = {ωv , ωφ , ω2 , ω1 }. There is no state at which a bubble
occurs, since A2 only buys in {ωv , ωφ }, where he thinks the asset might be valuable.
This gives the following corollary.
Corollary 1. A bubble does not occur when there is no middlemen in our setup.
In Section 7, by means of examples, we further examine how middlemen affect theoccurrence and duration of bubbles.

3.2

Proof of Lemma 1

Proof is by backward induction. Here, we focus only on the case where ω 6= ωv , and asimilar proof holds for the other case.
14

The environment extends to this case in an obvious way.

15

Trade between AN −1 and AN :If AN receives a signal, AN knows that x is worthless, and hence AN does not tradewith AN −1 , so yN −1 = 0.
If AN does not receive a signal, his expected value of x is ve . Then, AN and AN −1
negotiate the terms of trade:max(κyN −1 )θ (ve − yN −1 )1−θyN −1

subject to incentive constraints: κyN −1 ≥ 0 and ve − yN −1 ≥ 0. Note that, if they do
not agree to trade, they do not obtain any utility. Recall that the consumption valueof x for AN −1 is commonly known to be zero, and as a result AN −1 ’s outside option in
bargaining does not depend on the state. The solution isŷN −1 = θve .
Hence, AN obtains x in exchange for ŷN −1 .
Trade between AN −2 and AN −1 :If AN −1 receives a signal, AN −1 knows that AN receives a signal. Then, as we haveshown above, AN −1 knows that AN will not trade with AN −1 , and hence AN −1 does nottrade with AN −2 , so yN −2 = 0.
If AN −1 does not receive a signal, there are exactly two possibilities:
1. both AN −1 and AN do not receive signals; and2. AN receives a signal, but AN −1 does not.
In the first case, AN trades with AN −1 . In the second case, however, AN does nottrade with AN −1 . The first case occurs with probability ψN −1 given that AN −1 does
not receive a signal, and the second case occurs with the remaining probability. Hence,16

AN −1 ’s expected utility of obtaining x is ψN −1 κŷN −1 . Then, AN −1 and AN −2 negotiate
the terms of trade:max(κyN −2 )θ (ψN −1 κŷN −1 − yN −2 )1−θyN −2

subject to incentive constraints: κyN −2 ≥ 0 and ψN −1 κŷN −1 − yN −2 ≥ 0. The solutionisŷN −2 = θκψN −1 ŷN −1 .
Therefore, AN −1 obtains x in exchange for ŷN −2 .
Induction Hypothesis:
Suppose that
1. if An+1 receives a signal, An+1 does not trade with An , so yn = 0;
2. if An+1 does not receive a signal, An+1 trades with An and obtains x in exchange
for yn = ŷn .
Then, we will show that
1. if An receives a signal, An does not trade with An−1 , or yn−1 = 0;
2. if An does not receive a signal, An trades with An−1 and obtains x in exchange foryn−1 = ŷn−1 .
Trade between An−1 and An :If An receives a signal, An knows that An+1 receives a signal. Then, by the induction
hypothesis, An knows that An+1 will not trade with An , and hence An does not tradewith An−1 , so yn−1 = 0.
If An does not receive a signal, there are exactly two possibilities:
1. both An and An+1 do not receive signals; and17

2. An+1 receives a signal, but An does not.
In the first case, by the induction hypothesis, An+1 trades with An and obtains x in
exchange for ŷn . In the second case, however, An+1 does not trade with An , again by
the induction hypothesis. The first case occurs with probability ψn , given that An does
not receive a signal, and the second case occurs with the remaining probability. Hence,
An ’s expected utility of obtaining x is ψn κŷn . Then, An−1 and An negotiate the termsof trade:max(κyn−1 )θ (ψn κŷn − yn−1 )1−θ
yn−1

subject to incentive constraints: κyn−1 ≥ 0 and ψn κŷn − yn−1 ≥ 0. The solution isŷn−1 = θκψn ŷn .
Therefore, An obtains x in exchange for ŷn−1 .
Uniqueness:
In each trade, both agents obtain positive expected utilities because 0 < θ < 1. Moreover, in those states where An+1 is hypothesized not to trade with An , An+1 has strict
incentives to refuse the trade, so chooses yn = 0. Hence, there is no indifference amongchoices of each agent, which implies the uniqueness of equilibrium.15
Individual Rationality:
From the equilibrium prices, we can see that the interim expected utilities are positive
for agents who do not receive signals and zero for the other agents. Hence, every agent
has incentives to participate in the economy at the interim stage—after the state is15

Note that conditional upon AN chooses an action, it is strictly dominant to trade (not to trade)when he does not get (gets) a signal. Given this, again conditional upon AN −1 chooses an action, it
is strictly dominant to trade (not to trade) when he does not get (gets) a signal, and this is true for
all An . Thus, in each round we are only eliminating conditionally dominated strategies (Shimoji and
Watson 1998).

18

determined but before trade takes place.16 This further implies that the ex ante—
before the state is determined—utility of each agent is positive, and thus every agent
has incentives to participate in the economy at the ex ante stage as well. Therefore,
before trade takes place, it is optimal for each agent to participate in the economy.

3.3

Discussion of Theorem 1

Opaqueness:
In our result, the opaqueness assumption—the final user cannot observe the transacted
prices of the other pairs—plays a key role. The prices at state ωv and at the other states
differ, because at state ωv middlemen know that the final user will buy the asset forsure. However, the final user cannot observe the prices and so cannot update his belief
on the state. To understand it more deeply, consider the case where prices are publicly
observable. Then it is easy to see that Lemma 1 (and hence Theorem 1) does not hold,because the final user AN can learn the state from prices—if asset x is traded at priceŷn (resp. ŷnv ), then the state must be ωφ (resp. ωv ).
Does a bubble occur with publicly observable prices in our model? For it to bepossible, we must have the same price movements in states ωv and ωφ . One way toachieve the same price movements in these states is to modify the current setup andadd new states where middlemen also do not know the true valuation. Even with thistrick, the final user might learn the difference between ωv and ωφ because in general
trades among middlemen depend on states. Hence, a bubble can occur only as a knifeedge case and we lose robustness. See the Appendix A for the details.
16

When the state is ωv , agents A1 , · · · , AN −1 know that the consumption value of the asset x forthe final user AN is v > 0, and the final user AN does not know this since he never receives a signal.
Thus, each trade takes place between each pair of two adjacent agents, and each agent enjoys positive
expected utility at the interim stage.

19

Relationship to “Greater Fool” Models of Bubbles:
Unlike in greater fool models of bubbles (Allen et al. 1993), in our model there areneither “bad” sellers—sellers who know that they are selling worthless objects—nor“good” sellers in the trade among middlemen. Related and more importantly, traders
are in fact selling to other traders who know (weakly) more than themselves. This isactually opposite to the existing greater fool models where sellers are better informed
than buyers are.17
Intermediation Chain:
Wright and Wong (2014) provide a model of intermediation bubbles and show thatbubbles occur if there are an infinite number of middlemen and the potential gains fromtrade in terms of y is strictly positive. In this vein, our innovation is not only in relaxing
the assumption of common knowledge and showing that bubbles can occur with a finite
number of middlemen, but also in showing that there is a bubble even when the potential
gains from trade in terms of y are zero or negative. In Section 5, we will see that thisleads us to different welfare implications from theirs.

4

Price Changes

We are ready to investigate the properties of the equilibrium. In this section, we study
how the equilibrium price changes over time. We show that ŷn , the price that An+1 has
to pay to obtain the asset x, is not only increasing but also accelerating in n during abubble under reasonable assumptions. We emphasize that in our model the fundamental
value is constant over time. To this end, we show the following technical lemma.
17

Strictly speaking, the information partition of agent An+1 is not finer than An . Still in terms of

relevant information, An+1 is (weakly) more informed than An .

20

Lemma 2. If µ(ω2 ) ≤ · · · ≤ µ(ωN ), the probability ψn is decreasing in n, that is,ψn+1 < ψn .
Proof. For n = 1, · · · , N − 3, letting M = µ(ωφ ) + µ(ωN ) + · · · + µ(ωn+3 ),M[µ(ωn+1 ) − µ(ωn+2 )] − [µ(ωn+2 )]2ψn+1 − ψn =<0
[M + µ(ωn+2 )][M + µ(ωn+2 ) + µ(ωn+1 )]where the last inequality holds since we have µ(ωn+1 ) ≤ µ(ωn+2 ) for each n = 1, · · · , N −3 and µ(ω) > 0 for each ω ∈ Ω by assumption. A similar argument holds for the casebetween N − 1 and N − 2.
We now obtain the following result on price changes.
Proposition 1. If θκ ≤ 1, then ŷn is increasing in n, that is,ŷn+1 − ŷn > 0.
Moreover, if µ(ω2 ) ≤ · · · ≤ µ(ωN ), it is accelerating in n, that is,ŷn+2 − ŷn+1
ŷn+1 − ŷn
>
.
ŷn+1
ŷn
Proof. To see that ŷn is increasing, we obtainŷn+1 − ŷn = ŷn+1 − θκψn+1 ŷn+1 > 0.
To see that ŷn is accelerating, we obtainŷn+2 − ŷn+1 ŷn+1 − ŷn
1
1
−
=
−
>0
ŷn+1
ŷn
θκψn+2 θκψn+1
where the inequality holds because we have ψn+2 < ψn+1 by Lemma 2.
The fact that ŷn is increasing follows because each middleman must be compensatedfor the risk that he may not be able to sell the asset x with positive probability, 1 − ψn .
Thus, during a bubble, middlemen “flip”—agent An+1 buys the asset x at the priceŷn in hopes of reselling it at the price ŷn+1 > ŷn . The fact that ŷn is accelerating
21

follows because the probability that one can resell, ψn , is decreasing over time. In otherwords, flippers who trade in later periods are exposed to a bigger risk, and the prices are
determined in such a way that they are compensated for this increasing risk. Note thatthe growth rate of the price, (ŷn+1 − ŷn )/ŷn , is increasing, not just the price differencesŷn+1 − ŷn . In other words, the price differences are growing more quickly than the priceitself is growing.
For price increase, κ ≤ 1 is sufficient. For price acceleration, we impose an additional
assumption, which is that states where more agents receive signals are realized withsmaller probabilities.
The price trajectory shown in Proposition 1 features many observed bubbles, notonly in the modern bubbles, e.g., the dot com bubble18 or the Japanese asset bubble
or the housing bubble before the last financial crisis (see for example Figure 1 in Bayeret al. (2020), page 5219), but also in the classical Dutch Tulip bubble, etc.

5WelfareNow, we derive welfare implications. Our welfare criterion is utilitarian, that is, welfare
is the sum of all agents’ utilities. Note that agents have incentives to participate in theeconomy, and so the welfare analysis is meaningful.
Observe that, in trade between An and An+1 , the gains from trade areκyn − yn = (κ − 1)yn
because we normalized the production cost of y to be 1. Thus, when the economy is at
ωn∗ with n∗ > 2, ex post welfare is(κ − 1)

∗ −2
nX

n=1
18

See the NASDAQ Composite Index.

22

ŷn .

This is because a bubble continues for n∗ − 2 periods, after which the bubble bursts
and An∗ −1 does not succeed in selling to An∗ . The terms of trade are yn = ŷn during abubble and yn = 0 after period n∗ − 2. We obtain the following result on welfare.
Proposition 2. Bubbles are detrimental to the economy if and only if κ < 1.
As in most models of bubbles, a bubble facilitates trade in our theory. However,unlike most greater-fool models,19 trade can be detrimental. Our innovation is not only
to show that bubbles are not necessarily welfare improving but also to characterize
exactly when a bubble is detrimental.
There are efficiency losses from trading the good y if and only if κ < 1. For example,
if there are some transaction costs that sellers must pay when they sell the asset, the expost welfare is negative.20
Proposition 2 should be interpreted with caution, because it is only about the welfare
implication of bubbles, and not about ex ante welfare.21 When analyzing the welfare
effect of a policy concerning bubbles (as we do in the next section), we will have toconsider not only its effect on the states of bubbles but also on the other states.

6

Bubble Bursting Policy

In his speech, “The Challenge of Central Banking in a Democratic Society,” Alan
Greenspan pointed out the possibility of overpriced assets. In this section, we examine
welfare implications of such policies—policies that (intend to) affect agents’ beliefs on
the quality of the asset by releasing information about it. The finding is somewhat
surprising. Any of these public policies has no welfare consequence.
19

Most greater fool bubbles involve sellers acting in bad faith, i.e., knowingly deceiving ex-ante

uninformed buyers who ex-post wish they had not bought. In our model, this occurs only in the tradewith the final user.
20
Grossman and Yanagawa (1993), Miao and Wang (2014) and Guerrón-Quintana et al. (2020) showthat bubbles may reduce welfare in models of endogenous growth.
PN −1
PN −1
21
The ex post welfare in ωφ is (κ − 1) n=1 ŷn and that in ωv is v + (κ − 1) n=1 ŷnv .

23

For this purpose, we employ the information design approach (Kamenica and Gentzkow
2011, henceforth KG). A public experimentation σ : Ω → ∆(M ) sends to the agentsa randomized message, where M denotes the message space. The probability that amessage m ∈ M is sent when the state is ω ∈ Ω is denoted by σω (m). Following theliterature, we assume that the information designer (central bank) chooses and commits
to the experimentation σ before the state is realized, and its choice of rules is observable to agents. Motivated by the fact that Greenspan’s comment was taken seriously
and spread instantly among all market participants, we restrict our attention to public
experimentation, where the realized message m is publicly observable.
We assume first agents receive message m, and update their belief to µm ∈ ∆(Ω).
Because message m is commonly observable, the updated belief µm is commonly known.
Next agents get private signals, as in Section 2. Thus, effectively, after message m isreceived, the situation is the same as the game described in Section 2, but with theupdated “prior” distribution µm .22
Of course Bayes plausibility (see KG) holds:
Eσ(m) [µm ] = µ(3)where µ ∈ ∆(Ω) is the prior.
One example of a public experimentation is full disclosure, where M = Ω andσωn (ωn0 ) = δn,n0 with δn,n0 is the Kronecker delta function. Another example is a bubble
bursting policy proposed by Conlon (2015) and further examined by Holt (2019).23
Example 1. [Bubble-bursting policy] The central bank announces whether the state
belongs to {ωN , · · · , ω1 } or {ωv , ωφ }. That is, M = {m0 , m+ } and m0 (resp. m+ )denotes ω ∈ {ωN , · · · , ω1 } (resp. ω ∈ {ωv , ωφ }) with probability 1. Thus the probability
22

Note that we have assumed full-support in Section 2. While this may not be the case here, it isroutine to extend the model.
23
Asako and Ueda (2014) use another approach, based on Abreu and Brunnermeier (2003), andexamine whether a public announcement can burst a bubble but without welfare analysis.

24that each message is sent is σω (m0 ) = 1 for ω ∈ {ωN , · · · , ω1 } and σω (m+ ) = 1 forω ∈ {ωv , ωφ }.
Note that messages m0 and m+ induce posteriorsµ(ωn )
if ω = ωn ∈ {ωN , · · · , ω1 }µ(ωN )+···+µ(ω1 )µ0 (ω) =0otherwiseandµ+ (ω) =





0




if ω = ωn ∈ {ωN , · · · , ω1 }µ(ωv )µ(ωv )+µ(ωφ )





 µ(ωφ )µ(ωv )+µ(ωφ )if ω = ωv
if ω = ωφrespectively. It is routine to check Bayes plausibility.
This policy is bubble bursting in the sense of Definition 1—we say that a bubble occurs if the asset x is traded despite all agents know it is worthless. At states ω1 , · · · , ωN ,asset x is not traded and hence a bubble does not occur. At state ωφ , asset x is still
overpriced, but given that agent AN does not know this, we do not call this a bubble.
At state ωv , asset x is traded but has a positive value v > 0.
This experiment is of particular interest, as it does not require the central bankto have a superior information technology to that of agents. Indeed, this policy isimplementable if the knowledge of the central bank is the same as that of agent AN .
Now we examine welfare consequences of such public announcements. Our welfare
criterion in this section is—unlike the previous section—ex ante. That is, the information
designer also takes into account the states at which a bubble does not occur.
We now prove the following neutrality result.
Proposition 3. Announcement through any public experimentation does not change theex ante payoff of any agent.
This proposition is a consequence of the fact that the ex ante expected prices are
not affected by the public experiments, which will be shown below.
25

v
Proof. First, using ŷN−1 = ŷN −1 = θve , the ex ante expected price paid by AN isvµ(ωv )ŷN−1 + µ(ωφ )ŷN −1 = [µ(ωv ) + µ(ωφ )]θve = θµ(ωv )v.

Hence, it is independent of any policy. Here, if yn is the (stochastic) price in period n,
yn = θκEn yn+1 .
Therefore, by the law of iterated expectations, announcement through any public experimentation does not change the expected prices paid by any agents.
This result means that the central bank cannot affect any agent’s ex ante welfare by
releasing information. For example, consider the full-disclosure policy where the central
bank always knows the consumption value of the asset x. When the state is ω 6= ωv ,the central bank announces that x is valueless, and thus trade does not take place.
When κ < 1 (when κ > 1, the opposite happens), trade of good y is detrimental tothe economy. In this case, the policy improves interim welfare in these states because
it prevents trade of good y. Of course this policy has a side effect. When the state is
ωv , the central bank announces the information that the consumption value of x for AN
is v > 0, and hence all agents including AN know that the consumption value is v > 0.
Thus, the trade of good y is facilitated by the policy in this state because the expectedconsumption value of x for AN increases. These effects are exactly canceled out.
The result in Section 5—that a bubble is detrimental when κ < 1—on the other
hand, focuses only on welfare in states where bubbles occur. Put differently, when
κ < 1 and ω ∈ {ωN , · · · , ω3 }, a benevolent information designer is happy with hisannouncement policy. In contrast, when ω = ωv , by the policy, the final user learns thatthe consumption value of the asset x is v > 0 and as a result the amounts of good ytraded increase. In this state, the benevolent policy maker suffers from ex post regrets
that it should not have implemented such a policy.
The result depends on the assumption that the utility function of y is linear, or agentsare risk neutral. However, it may serve as a benchmark result in information design for26

models of bubbles, as the revenue equivalence theorem does in auction theory. In the
Appendix B, we will provide two examples in which agents are risk averse and publicannouncements are not neutral—in one example welfare is enhanced and in the other it
is hurt.24

7

Market Structure

In this section, we study the effect of market structure on bubbles. In particular, weinvestigate the effect of the number of middlemen and the extent to which rumors are
spread on the probability, duration and size of bubbles. Below, we specify a parametricform of the prior distribution but ensure the assumptions that we have imposed in theprevious sections.
Towards this purpose, we specify the prior µ as follows.25 Consider a situation where
the signal to AN is lost with probability εF , and moreover, between any two adjacentagents, each signal is lost with a common probability, εM . In other words, the final user
AN knows that the consumption value of x for AN is zero with probability 1 − εF , andthe rumor that AN knows the fact reaches AN −1 with probability 1 − εM , and so on andso forth. The parameter εM can be thought as a measure of the extent to which rumors
spread—the bigger εM is, the more slowly rumors spread.
In this case, we haveµ(ωφ ) = [1 − µ(ωv )]εF
and, for each n = 2, · · · , N ,µ(ωn ) = [1 − µ(ωv )](1 − εF )(1 − εM )N −n εM
while for n = 1,µ(ω1 ) = [1 − µ(ωv )](1 − εF )(1 − εM )N −1 .
24Holt (2019) also considers risk-averse agents and shows that in his model, a bubble-bursting policy

tends to be detrimental as well.
25
In the Appendix C, we consider another specification.

27

We have µ(ωφ ) = [1 − µ(ωv )]εF because the final user AN does not receive any signals
with probability εF , given that the asset is worthless. For each n = 2, · · · , N , we haveµ(ωn ) = [1 − µ(ωv )](1 − εF )(1 − εM )N −n εM because the signal is not lost between any
two adjacent agents until An receives it, but it is lost between An and An−1 . We haveµ(ω1 ) = [1 − µ(ωv )](1 − εF )(1 − εM )N −1 because the signal is not lost between any two
adjacent agents. Assume 0 < µ(ωv ) < 1, 0 < εF < 1, and 0 < εM < 1. Then, it isobvious that µ(ω) > 0 for each ω ∈ Ω and µ(ω2 ) ≤ · · · ≤ µ(ωN ). The probability ψn isψn =εF + (1 − εF )[1 − (1 − εM )N −n−1 ]εF + (1 − εF )[1 − (1 − εM )N −n ]and the price ŷn is
ŷn =

εF
θN −n κN −n−1 veN−n−1εF + (1 − εF )[1 − (1 − εM )
]

where
ve =

µ(ωv )v.
µ(ωv ) + [1 − µ(ωv )]εFThe price ŷn is decreasing in εM . See Figure 2 for the graph of ŷn .
A bubble occurs when the state belongs to {ωN , ωN −1 , · · · , ω3 }, and hence the probability that a bubble occurs is
1 − [µ(ωv ) + µ(ωφ ) + µ(ω2 ) + µ(ω1 )] = [1 − µ(ωv )](1 − εF )[1 − (1 − εM )N −2 ].
The probability is increasing in εM . Focusing on the states where the final user knowsthat x is worthless, the expected duration of the bubble isNXµ(ωn )(n − 2)
(1 − εM )[1 − (1 − εM )N −2 ]=N −2−.
1 − [µ(ωv ) + µ(ωφ )]εM
n=3The expected duration is increasing in εM . We then define the expected duration ratio as
the expected duration divided by the number of periods. The difference in the expected
duration ratio is
N − 1 (1 − εM )[1 − (1 − εM )N −1 ]N − 2 (1 − εM )[1 − (1 − εM )N −2 ]
−
−
−
N
N εM
N −1
(N − 1)εM1={1 − (1 − εM )N −1 [1 + (N − 1)εM ]}.
N (N − 1)εM
28

ŷn
1.0
0.8

εM = 0.01
εM = 0.1

0.6
0.4
0.2
n

Figure 2: Graph of ŷn (N = 15, θ = 0.9, κ = 1, v = 2, µ(ωv ) = 1/17, and εF = 0.1).
Prices are lower in each state with a higher value of εM (a less transparent market).
It is routine to show that the right-hand side is positive for all εM ∈ (0, 1). That is, theexpected duration ratio is increasing in the number of agents.
The following two propositions summarize the above analysis.
Proposition 4. The wider the rumors are spread, the less likely a bubble occurs and theshorter the expected duration ratio of the bubble is, but the larger the size of the bubble
is.
Proposition 5. The larger the number of middlemen, the higher the ex ante probabilityof a bubble to occur, and the longer the expected duration ratio of the bubble.
The Dodd-Frank Act aims at improving the ability of market participants to monitor
and understand the risks in these markets. Hence, our prediction of Proposition 4 about
its effect is not only to reduce the probability and the length of a bubble, but also toincrease the size of it (if it occurs at all).
Proposition 5 is consistent with the empirical evidence by Bayer et al. (2020) who
report that a housing bubble is accompanied by a large increase in the number of flippers.
29

Further, it is interesting to mention that the Japanese asset/housing bubble from 1986
to 1991 was believed to be caused structurally through bank deregulation. It drastically
moved large corporate customers away from bank borrowing toward other financing and
as a result increased flipping. More generally, Kindleberger and Aliber (2005) argue thatbubbles are often triggered by financial innovations (e.g., futures, acceptance loans, or
securitization) or deregulation (opening new business opportunities), which most likely
enhances entry of middlemen.

8Conclusion

In this paper we construct a simple tractable model of a bubble. This tractability of themodel will allow one to investigate some other interesting properties of bubbles which
we have not explored.
For example, our model has an indivisible asset and a divisible good. By replacing thedivisible good with divisible money, we may be able to study the relationships between
bubbles and monetary policy more explicitly. Also it would be interesting to endogenize
middlemen’s entry. The model then may be able to generate a tighter prediction on how
regulations on entry of middlemen, which is an important and practical policy target inmany markets, can influence the occurrence and welfare consequence of bubble.
Another interesting issue is the relationship between market opaqueness and robustness—in models of bubbles where the motivation of trade is risk-sharing (Allen et al. 1993;
Liu and Conlon 2018) or intertemporal consumption-smoothing (Liu et al. 2020), would
equilibrium be robust when prices are not observable? We leave them for future research.

A

Appendix: Publicly Observable Prices

We have assumed that the final user cannot observe prices in past trades among middlemen. In this Appendix, we modify the setup in the main text and show that a bubble
30

can occur in equilibrium but within knife-edge parameters.
To have a bubble with publicly observable prices, we must have the same pricemovements in states ωv and ωφ because, otherwise, the price pattern reveals information
to the final user. One way to achieve the same price movements in these states is to add++++
new states ωN−1 , ωN −2 , · · · , ω1 . For each n = 1, · · · , N −1, at state ωn , the consumption

value of x for the final user AN is zero, and for each m ≤ n, agent Am does not knowwhether the consumption value is zero, while for each m > n, agent Am knows that theconsumption value is zero. The set of the states is++Ω+ = {ωv , ωφ , ωN , · · · , ω1 , ωN
−1 , · · · , ω1 }.

Let µ+ be the common prior distribution over Ω+ , and assume that µ+ (ω) > 0 for eachω ∈ Ω+ .
Now, agent AN ’s partition is++
PN+ = {{ωv , ωφ }, {ωN , · · · , ω1 , ωN
−1 , · · · , ω1 }}.

The first element, {ωv , ωφ }, corresponds to the case where AN does not know whether x++
is worthless. The second element, {ωN , · · · , ω1 , ωN
−1 , · · · , ω1 }, corresponds to the casewhere AN knows that x is worthless. For each n < N , agent An ’s partition is++++Pn+ = {{ωv , ωN
−1 , · · · , ωn }, {ωφ , ωN , · · · , ωn+1 }, {ωn , · · · , ω1 , ωn−1 , · · · , ω1 }}.
++The first element, {ωv , ωN
−1 , · · · , ωn }, corresponds to the case where An does not knowwhether x is worthless. The second element, {ωφ , ωN , · · · , ωn+1 }, corresponds to thecase where An knows that x is worthless, but An does not know whether An+1 knows+that ... AN knows that x is worthless. The third element, {ωn , · · · , ω1 , ωn−1
, · · · , ω1+ },corresponds to the case where An knows that x is worthless and that An+1 knows that... AN knows that x is worthless. Note that middlemen may not know the consumption
value of x for AN when it is v > 0.
Analogously to the definitions of ve and ψn in the previous sections, defineve+ =µ+ (ωv )vµ+ (ωv ) + µ+ (ωφ )31and for N − 1,
+
ψN
−1 =

µ+ (ωφ )µ+ (ωφ ) + µ+ (ωN )while for each n = 1, · · · , N − 2, define
ψn+ =µ+ (ωφ ) + µ+ (ωN ) + · · · + µ+ (ωn+2 ).
µ+ (ωφ ) + µ+ (ωN ) + · · · + µ+ (ωn+2 ) + µ+ (ωn+1 )N −1
as follows: for N − 1,
Moreover, define a sequence (ŷn+ )n=1++ŷN
−1 = θveand for each n = 1, · · · , N − 2,++ŷn+ = θκψn+1
ŷn+1.

The following lemma shows we can find parameters under which the equilibrium prices
are the same across ωv and ωφ .26
Lemma 3. Assume that for N − 1,
+
ψN
−1 =

µ+ (ωv )+µ+ (ωv ) + µ+ (ωN−1 )

and for each n = 2, · · · , N − 2,
ψn+ =

+
+
+
µ+ (ωv ) + µ+ (ωN−1 ) + · · · + µ (ωn+1 )
.
+
+
+
+
+
µ+ (ωv ) + µ+ (ωN−1 ) + · · · + µ (ωn+1 ) + µ (ωn )Then, for each ω ∈ Ω+ , the following is an equilibrium. Moreover, the outcome isunique.
1. If ω ∈ Ω+ \ {ωv , ωφ }, agent AN does not trade with agent AN −1 , so yN −1 = 0;otherwise, agent AN trades with agent AN −1 and obtains x in exchange for yN −1 =
+
ŷN
−1 .
26

For instance, a uniform distribution satisfies the requirement in Lemma 3.

32

+
+
2. If ω ∈ Ω+ \ {ωv , ωφ , ωN , · · · , ωn+2 , ωN
−1 , · · · , ωn+1 }, agent An+1 does not tradewith agent An , so yn = 0; otherwise, agent An+1 trades with agent An and obtains
x in exchange for yn = ŷn+ .
A price coincidence in each period is necessary to prevent the final user from distinguishing a “good” state (ωv ) from a “bad” state (ωφ ). Note that requiring prices tomatch in states ωv and ωφ is rather stringent and it holds true only for certain knife-edge
parameters. In other words, a bubble equilibrium is not robust to parameter changes.
Theorem 2. Consider an extended setup with additional states. For some knife-edge
parameter values, an equilibrium exists and is unique, and a bubble occurs when ω ∈{ωN , ωN −1 , · · · , ω3 }. Moreover, the bubble bursts for sure.

B

Appendix: Risk Aversion

We obtained a welfare neutrality result for bubble-bursting policies when U (y) = κy.
Here we consider two examples in which some of agents are risk averse. In both examples,
we compare the case where the information designer fully reveals the state ω versus thecase where it reveals no information. Of course, once ω is fully revealed, there is noroom for a bubble to occur, and asset x is traded only if it has a real value. As for thewelfare criterion, we again consider the sum of agents’ utilities. In one example, full
disclosure is welfare improving, while in the other it is welfare reducing.

Example 1: Full disclosure is welfare reducing
Suppose the utility function is concave in y:
U (y) = κy α
where α ∈ (0, 1] is a constant. More precisely, agent AN ’s utility is v − yN −1 if goodx has value and he produces yN −1 of good y. If good x does not have value, then it is33

−yN −1 . For n = 2, · · · , N − 1, agent An ’s utility is U (yn ) − yn−1 if he produces yn−1 andconsumes yn of good y.
First consider the case where the information designer reveals no information. Define
N −1
a sequence (ŷn )n=1
as follows (with α = 1, this corresponds to the same variable in themain text): for N − 1,ŷN −1 =θαve
1 − θ(1 − α)and for each n = 1, · · · , N − 2,
ŷn =

θα
α
κψn+1 ŷn+1.
1 − θ(1 − α)−1Similarly, define a sequence (ŷnv )N
n=1 as follows: for N − 1,
v
ŷN
−1 =

θα
ve
1 − θ(1 − α)and for each n = 1, · · · , N − 2,ŷnv =

θαv
κ(ŷn+1)α .
1 − θ(1 − α)As in Lemma 1, the characterization of equilibrium is as follows:
1. Assume ω 6= ωv . In equilibrium, if agent An+1 receives a signal, agent An+1 does
not trade with agent An , so yn = 0; if agent An+1 does not receive a signal, agent
An+1 trades with agent An and obtains x in exchange for yn = ŷn . Moreover, thisoutcome is unique.
2. Assume ω = ωv . In equilibrium, An+1 always trades with An and obtains x in
exchange for yn = ŷnv . Moreover, this outcome is unique.
When ω ∈ {ωv , ωφ }, trade takes place each period. When ω = ωn , trade takes place
until period n − 2 if n ≥ 3 and trade never takes place if n ≤ 2. When ω = ωv , the sumof utilities is
v
v
v
v
v
v − ŷN−1 + U (ŷN −1 ) − ŷN −2 + · · · − ŷ1 + U (ŷ1 ) = v +N−1X[κ(ŷnv )α − ŷnv ].

n=1

34

When ω = ωφ it is−ŷN −1 + U (ŷN −1 ) − ŷN −2 + · · · − ŷ1 + U (ŷ1 ) =N−1X(κŷnα − ŷn )n=1

and when ω = ωn∗ with n∗ ≥ 3 it is−ŷn∗ −2 + U (ŷn∗ −2 ) − ŷn∗ −3 + · · · − ŷ1 + U (ŷ1 ) =

∗ −2
nX

(κŷnα − ŷn ).

n=1

The ex ante welfare is therefore
(
)
∗ −2
N
−1
N
−1
N
nX
X
X
X
v α
v
α
W = µ(ωv ) v +[κ(ŷn ) − ŷn ] + µ(ωφ )(κŷn − ŷn ) +µ(ωn∗ )
(κŷnα − ŷn ).
n=1

n=1

n∗ =3

n=1

Next consider the full-disclosure policy. In this case, the consumption value of x
for AN is common knowledge, and trade takes place only at ωv . Asset x is traded inN −1of good y in period n (the superscript d denotes full disclosure)exchange for (ynd )n=1

where
d
yN
−1 =

θα
v
1 − θ(1 − α)and for each n = 1, · · · , N − 2,
ynd =

θα
d
κ(yn+1)α .
1 − θ(1 − α)The ex ante welfare is(
W d = µ(ωv ) v +

N
−1
X

)
[κ(ynd )α − ynd ] .

n=1

We consider the following numerical example:
Claim 1. Suppose µ is a uniform distribution with N = 3, θ = 0.9, κ = 1, v = 2, and
α = 0.9. Then, we obtain W = 0.4231 > 0.3682 = W d . That is, the full-disclosure
policy is welfare reducing.
In this case, a bubble is welfare improving because it allows risk-averse middlemento share risks—the Hirshleifer (1971) effect—and so the bubble bursting policy is detrimental. A similar result is also provided by Holt (2019) with a different setup. He
35

constructs a model similar to Conlon (2015), where the motive for trade is risk sharing,and studies bubble-bursting policies. He shows that such policies prevent risk sharing
and hence may be detrimental.

Example 2: Full disclosure is welfare improving
Next suppose agent AN ’s utility is u(v − yN −1 ) when ω = ωv and u(−yN −1 ) otherwise,
where u is strictly increasing and concave, and moreover satisfies u(0) = 0. For n =2, · · · , N − 1, agent An ’s utility is κyn − yn−1 . Agent 1’s utility is κy1 . In words, theinitial owner and middlemen are still risk neutral, but the final user is risk averse. For
simplicity, we assume κ = 1—that is, the social gains from trading good y is zero—andθ = 1 in what follows—that is, in the trade between An and An+1 , An makes a take-itor-leave-it offer—, but by continuity, the result also holds for κ 6= 1 or θ < 1 if they are
sufficiently close to 1.
First consider the case where the information is fully revealed and the consumption
value of x for AN is common knowledge. Since κ = 1, there are no gains from tradein terms of y. In trade between AN and AN −1 , the terms of trade are determined byddu(v −yN−1 ) = 0, or yN −1 = v. Hence, the ex ante welfare under the full-disclosure policyis
W d = µ(ωv )v.
Of course, with take-it-or-leave-it offer bargaining, only the initial owner of the asset
enjoys the surplus and the other agents’ ex ante welfare is 0.
Next consider the case where the information designer reveals no information. Since
AN −1 makes a take-it-or-leave-it offer to AN , in trade between AN and AN −1 , the termsof trade are determined byµ(ωv )µ(ωφ )u(v − ŷN −1 ) +u(−ŷN −1 ) = 0.
µ(ωv ) + µ(ωφ )µ(ωv ) + µ(ωφ )Let ŷN −1 be the quantity satisfying the above equation. Then, the ex ante welfare is
W = µ(ωv )[u(v − ŷN −1 ) + ŷN −1 ] + µ(ωφ )[u(−ŷN −1 ) + ŷN −1 ]36and thus,W= ŷN −1 .
µ(ωv ) + µ(ωφ )Again, only the initial owner of the asset enjoys the surplus and the other agents’ exante welfare is 0.
By definition of ve (see (1)), we haveµ(ωv )µ(ωφ )(v − ve ) +(−ve ) = 0.
µ(ωv ) + µ(ωφ )µ(ωv ) + µ(ωφ )Since u is strictly concave, this equation impliesµ(ωv )µ(ωφ )u(v − ve ) +u(−ve ) < 0.
µ(ωv ) + µ(ωφ )µ(ωv ) + µ(ωφ )Hence, we obtainŷN −1 < ve .
Therefore,
W = [µ(ωv ) + µ(ωφ )]ŷN −1 < [µ(ωv ) + µ(ωφ )]ve = W d .
Therefore, we established that
Claim 2. The full-disclosure policy is welfare improving.
Note that this time, it is the final user, not middlemen, who is risk averse. Hence, thefull-disclosure policy, which removes the risk, gives the final user the consumption valueof good x equal to v for sure whenever he decides to buy it. This is welfare improving
because without the policy, the expected consumption value for the risk-averse final user
is strictly less than v.

CAppendix: Uniform Distribution

We consider an example where the distribution µ is uniform, that is, for each ω ∈ Ω,µ(ω) =

1
.
N +2

37

ŷn
1.0
0.8
0.6
0.4
0.2
n

Figure 3: Graph of ŷn for uniform distribution (N = 15, θ = 0.9, κ = 1, and v = 2)
We derive comparative statics results for the number of middlemen similar to Proposition
5 in Section 7.
It is obvious that µ(ω) > 0 for each ω ∈ Ω and µ(ω2 ) ≤ · · · ≤ µ(ωN ). The probability
ψn is
ψn =

N −n
N −n+1and the price ŷn isŷn =1θN −n κN −n−1 ve
N −nwhere ve = v/2. See Figure 3 for the graph of ŷn .
A bubble occurs when the state belongs to {ωN , ωN −1 , · · · , ω3 }, and hence the probability that a bubble occurs is
1 − [µ(ωv ) + µ(ωφ ) + µ(ω2 ) + µ(ω1 )] = 1 −

4
.
N +2

This probability is increasing in N and converges to one as N → ∞. In words, a bubble
occurs with an arbitrarily high probability. Focusing on the states where the final user

38

knows that x is worthless, the expected duration of the bubble isNXµ(ωn )(n − 2)
(N − 2)(N − 1)
=
.
1
−
[µ(ω
)
+
µ(ω
)]
2N
v
φ
n=3

The expected duration and expected duration ratio are increasing in N . The results,
which are analogous to those in Section 7, are summarized as follows.
Proposition 6. The larger the number of middlemen, the higher the ex ante probabilityof a bubble to occur, and the longer the expected duration ratio of the bubble.

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