View original document

The full text on this page is automatically extracted from the file linked above and may contain errors and inconsistencies.

A simple model of gross worker flows across
labor market states
Marcelo Veracierto

Introduction and summary
While standard macroeconomic models of labor markets typically assume two labor market states (employment/nonemployment or employment/unemployment),
a recent literature has extended those models to incorporate three labor market states: employment, unemployment, and nonparticipation (or out of the labor
force). See, for example, Tripier (2004), Veracierto
(2008), Christiano, Trabandt, and Walentin (2010),
Haefke and Reiter (2011), and Shimer (2013). Most
of these papers have focused on modeling the number
of workers in each of these labor market states, but
not the gross flows of workers across them. A notable
exception is Krusell et al. (2012), which introduces search frictions (for the process through which
workers meet job opportunities) into a real business
cycle model with borrowing constraints in the household sector. Their model is rich enough to explicitly
determine the gross flows of workers across the three
labor market states, potentially providing a deeper
understanding of what drives unemployment and other
labor market shifts.
Interestingly, Krusell et al. (2012) find that under
certain specifications of aggregate shocks, their model
is able to broadly reproduce the cyclical behavior of
gross worker flows across labor market states in the
U.S. economy. While this is an important result, the
economic mechanisms behind it are somewhat obscured
by the real business cycle structure and the borrowing
constraints. The purpose of this article is to strip the
model in Krusell et al. (2012) down to its bare bones—
that is, to develop (and analyze) a very simple version
of it. The key difference between my model and theirs
is that instead of embodying the search frictions in a
real business cycle with borrowing constraints, I assume
that technology and workers’ preferences (for consumption and leisure) are linear. These assumptions

38

allow for an analytical characterization of the model
that makes the determination of gross worker flows
transparent. Moreover, the simple structure of the
model allows me to perfectly identify its shocks using
U.S. data.
A key ingredient of any model useful for analyzing unemployment behavior is the presence of search
frictions. While search frictions typically create bilateral
bargaining situations between workers and employers,
I (like Krusell et al., 2012) introduce them in such a
way that wages are determined in perfectly competitive
labor markets.1 Moreover, the simple structure that is

Marcelo Veracierto is a senior economist and research advisor in
the Economic Research Department at the Federal Reserve Bank
of Chicago. The author is grateful for comments from Gadi Barlevy
and Paco Buera.
© 2015 Federal Reserve Bank of Chicago
Economic Perspectives is published by the Economic Research
Department of the Federal Reserve Bank of Chicago. The views
expressed are the authors’ and do not necessarily reflect the views
of the Federal Reserve Bank of Chicago or the Federal Reserve
System.
Charles L. Evans, President; Daniel G. Sullivan, Executive Vice
President and Director of Research; David Marshall, Senior Vice
President and Associate Director of Research; Spencer Krane,
Senior Vice President and Senior Research Advisor; Daniel Aaronson,
Vice President, microeconomic policy research; Jonas D. M.
Fisher, Vice President, macroeconomic policy research; Anna L.
Paulson, Vice President, finance team; William A. Testa, Vice
President, regional programs; Lisa Barrow, Senior Economist and
Economics Editor; Helen Koshy and Han Y. Choi, Editors; Julia
Baker, Production Editor; Sheila A. Mangler, Editorial Assistant.
Economic Perspectives articles may be reproduced in whole or in
part, provided the articles are not reproduced or distributed for
commercial gain and provided the source is appropriately credited.
Prior written permission must be obtained for any other reproduction, distribution, republication, or creation of derivative works
of Economic Perspectives articles. To request permission, please
contact Helen Koshy, senior editor, at 312-322-5830 or email
Helen.Koshy@chi.frb.org.
ISSN 0164-0682

2Q/2015, Economic Perspectives

assumed allows for a single wage rate in the whole
economy. Both features are obtained by assuming that
all firms in the economy produce in a single geographical
location where workers can move from one firm to
another in a frictionless way.
Not all workers are present in the production location, though: Some of them are situated in a separate
geographical location in which they are able to enjoy
leisure. Search frictions are introduced by assuming
that it is difficult to move from the leisure location to
the production location. In particular, agents are assumed
to be able to move from the leisure location to the
production location with a fixed probability (interpreted
as a job-finding rate). Also, workers who are present
in the production location are forced to move to the
leisure location with a fixed probability (interpreted
as a job-separation rate). Aside from being subject to
job-separation shocks that force individuals to move
to the leisure location, workers always have the possibility of moving from the production location to the
leisure location whenever they wish. Because workers
are subject to idiosyncratic labor productivity shocks,
they face nontrivial labor supply decisions. Also,
because the shocks are idiosyncratic, workers end up
moving in and out of employment in an unsynchronized way, generating gross flows across the different
labor market states.
It turns out that calibrating the steady state (or longrun equilibrium) of the simple model’s economy to
average monthly U.S. gross worker flow rates requires
that I introduce errors in the classification of agents’
labor market states. Otherwise, the model would be
inconsistent with those actual gross worker flows. The
classification errors needed are somewhat more extreme
than those indicated by the empirical evidence, but they
are not vastly out of line. Once the model is calibrated,
the equations describing the gross flows of workers
are used to measure the reallocation shocks (that is, the
job-finding and job-separation rates) and the aggregate
productivity shocks that hit the economy. (In order to
match the U.S. data, I find that the classification errors
introduced into the model must also be allowed to
vary over time.) A crucial test of the model is whether
these measured aggregate shocks, classification errors,
and gross worker flows are consistent with the optimal
decisions of agents in the model. I find that they are.
Thus, the model seems to provide a reasonable labor
supply theory. However, the model does not provide
a deep theory of labor market dynamics. The reason
is that most of the success of the model at reproducing
the labor market dynamics found in the U.S. economy
relies on the realization of exogenous shocks: The optimal decisions of agents in the model economy play

Federal Reserve Bank of Chicago

a minor role in generating endogenous fluctuations in
the gross flows of workers across labor market states,
as well as in employment and unemployment.
In the next section, I describe the simple model’s
economy in greater detail. In the subsequent section, I
discuss how its classification of workers into the three
labor market states approximates that of the U.S. Bureau
of Labor Statistics (BLS). Next, I calibrate the model
to U.S. data. Then, I test how well the model does in
reproducing U.S. business cycle data. After that, I
compare these results with those of my previous work
(Veracierto, 2008) and offer goals for future research.
The model economy
The model economy is populated by an interval
[0,1] of workers, who are distributed across two islands:
a production island and a leisure island.2 The production island has a representative firm that produces
output with a linear production function that uses labor
as the only input of production. The productivity of a
worker is given by p + z, where p is an aggregate productivity level common to all workers and z is an idiosyncratic productivity level. Because the production
island is assumed to be competitive, the wage rate
that the worker receives is equal to p + z. While p is
assumed to be constant, the idiosyncratic productivity
level z evolves stochastically over time according to a
Markov process. In particular, I will assume that with
probability 1 – γ, the next period’s idiosyncratic productivity level zʹ will be equal to the current period’s
idiosyncratic productivity level z and that with probability γ, the next period’s idiosyncratic productivity
level zʹ will be drawn anew from a known distribution
function F.
On the leisure island, workers do not produce but
enjoy α units of leisure. Workers value consumption
and leisure according to the following preferences:
∞

∑β
t =0

t

ct + (1 − et ) α  ,

where et ∈{0,1} is an indicator function that is equal
to 1 when the worker is employed and 0 otherwise.
Observe that, given the linear preferences assumed,
the worker cares about the present value of his consumption but not about its timing.
There are frictions to move from the leisure island
to the production island. However, there are no frictions
to move the other way.
The timeline within each time period is as follows:
1) Workers start the period distributed in some way
across the two islands; 2) the idiosyncratic productivity
shock z of each worker is realized; 3) workers who
are initially located on the production island become

39

exogenously relocated to the leisure island with probability σ; 4) each agent who is located on the leisure
island at this point (including all those initially located
on the leisure island and all those that were exogenously
relocated from the production island) is exogenously
relocated to the production island with probability λ;
5) agents located on the production island at this point
(including all those that have just arrived from the leisure
island) must decide whether to stay on the production
island or move back to the leisure island; and 6) consumption and leisure are finally enjoyed. In what follows,
I will refer to σ as the (exogenous) job-separation rate
and to λ as the (exogenous) job-finding rate.
Next, I analyze the problem that workers face in
this economy.
Workers’ problem
Consider the decision problem of a worker situated on the production island during stage 5 of the
timeline just described (this is the only situation in
which a worker must make a decision). At this point,
the worker already knows the realization of his idiosyncratic productivity for the current period (that is,
z). Given this information, the worker must decide
whether to stay on the production island and work or
go back to the leisure island and enjoy leisure. Let his
value of staying on the production island be W(z) and
his value of moving back to the leisure island be L(z).
Then, his optimal value V(z) is given by
V(z) = max {W(z), L(z)}.
The value of being on the production island is
given by the following equation:
1)

(1 − γ ) L ( z )


W ( z ) = p + z + βσ (1 − λ ) 
+ γ ∫L ( z ′) dF ( z ′)
(1 − γ )V ( z )

.
+ β 1 − σ (1 − λ ) 
+ γ ∫V ( z ′) dF ( z ′)

This equation states that if the worker stays on
the production island, he receives the wage rate p + z
during the current period and starts the following period on the same island. Different things can happen
from that point on. With probability σ(1 – λ), the worker
gets exogenously relocated to the leisure island and is
not able to come back. In this case, with probability 1 – γ,
his idiosyncratic productivity level in the next period
does not change and he obtains the value of being on
the leisure island L(z). However, with probability γ,
he draws a new idiosyncratic productivity level and

40

obtains the expected value ∫L(zʹ) dF(zʹ). The alternatives, which happen with probability 1 – σ(1 – λ), are
either that the worker is not exogenously relocated to
the leisure island or that if he is, he is able to come
back to the production island. In either case, he starts
stage 5 in the following period on the production island.
Then, with probability 1 – γ, he obtains the value
V(z), and with probability γ, he obtains the expected
value ∫V(zʹ) dF(zʹ).
The value of being on the leisure island is given
by the following equation:
(1 − γ ) L ( z )


2) L ( z ) = α + β (1 − λ ) 
+ γ ∫L ( z ′) dF ( z ′)
+ βλ (1 − γ )V ( z ) + γ ∫V ( z ′) dF ( z ′) .
This equation states that if the worker is on the leisure
island during the current period, he receives the value
of leisure α during the current period and starts the
following period on the same island. Different things
can happen from then on. With probability 1 – λ, the
worker is not able to relocate to the production island
and gets the expected value (1 – γ) L(z) + γ ∫ L(zʹ) dF.
However, with probability λ, the worker is able to relocate to the production island and obtains the expected
value (1 – γ) V(z) + γ ∫V(zʹ) dF.
With equations 1 and 2, I can show straightforwardly that Wʹ(z) > 0 and that Lʹ(z) > 0 (that is, both
value functions are strictly increasing in z). Thus, the
optimal decision for the worker is characterized by
a threshold idiosyncratic productivity level z* that
satisfies that
W(z*) = L(z*),

		
		
		
		
		

W(z) < L(z), for z < z*,
W(z) > L(z), for z > z*.

That is, at the threshold level z*, the worker is indifferent
between staying on the production island and going
back to the leisure island. For values of z lower than the
threshold, the worker prefers to go back to the leisure
island; and for values of z higher than the threshold,
the worker prefers to stay on the production island.
Evaluating the condition W(z*) = L(z*) using
equations 1 and 2, I get that
3)

p + z* = α − βγ (1 − σ ) (1 − λ )
∞

×∫ * W ( z ′ ) − L ( z ′ )  dF ( z ′ ) .
z

2Q/2015, Economic Perspectives

FIGURE 1

Optimal threshold for idiosyncratic productivity

р–α

LHS

z*

RHS
z

Note: See the text for further details.

Hence, I get the familiar condition that the reservation
wage rate p + z* (that is, the lowest wage rate acceptable
to the worker) is less than the value of leisure α because
of the possibility that z may improve over time (the
value of this possibility is called its “option value”).
Evaluating equations 1 and 2 for z ≥ z * and
differentiating with respect to z, I get that
W ′ ( z ) − L′ ( z ) =

1
,
1 − β (1 − γ) (1 − λ ) (1 − σ )

for z ≥ z*. Integrating by parts in equation 3 thus results in the following:
4)

p − α = − z* −

βγ (1 − σ ) (1 − λ )

1 − β (1 − γ ) (1 − σ ) (1 − λ )

∞

×∫z * 1 − F ( z ′) dz ′,

which implicitly gives the idiosyncratic productivity
threshold z* as a function of the fundamental parameters of the model.
Figure 1 provides a graphic representation of the
determination of z*. It shows the left-hand side (LHS)
of equation 4 as a horizontal line, independent of z. The
right-hand side (RHS) of equation 4 is depicted as a
decreasing function.3 The intersection of both lines
determines the idiosyncratic productivity threshold z*.
When either the aggregate productivity level p
increases or the value of leisure α decreases, the
RHS of equation 4 is not affected but the LHS of
that equation shifts up, lowering the value of z*. This
is quite intuitive, since in both cases the value of
being employed increases relative to the value of enjoying leisure, inducing the worker to become less

Federal Reserve Bank of Chicago

picky and accept employment with lower values of
idiosyncratic productivity.
When either the probability of being exogenously
relocated to the leisure island σ or the probability of
being exogenously relocated to the production island
λ goes up, the LHS of equation 4 is not affected but
the RHS of that equation shifts up. Similar effects take
place when the probability of obtaining a new draw
for the idiosyncratic productivity level γ decreases. In
all three cases, the idiosyncratic productivity threshold
z* increases.
To get intuition about these effects, recall from
equation 3 that the reservation wage rate p + z* is less
than the value of leisure α because of the option value
of z improving over time. In all three cases, the option
value of waiting for an improvement in the idiosyncratic productivity level happens to decrease, making
the worker less tolerant of low realizations of the idiosyncratic productivity level. When the probability
of being exogenously relocated to the leisure island σ
increases, the option value of waiting decreases because
it is more likely that the worker will find himself on
the leisure island at the time that the idiosyncratic
productivity level z improves. When the probability of
being exogenously relocated to the production island
λ increases, the option value of waiting decreases
because it becomes easier to get back to the production
island in the future (once an improvement in z takes
place). When the probability of obtaining a new draw
for the idiosyncratic productivity level γ decreases, the
option value of waiting decreases because productivity
improvements become less frequent.
Measuring labor market states
Given the idiosyncratic productivity threshold z*
described in the previous subsection, it is straightforward to describe the evolution of aggregate employment in the economy.
Let E–1 be the total number of workers that have
been employed in the previous period. Then, the total
number of workers that are employed during the current period is given by
5)	 E = 	(1 – σ + σλ) (1 – γ)E–1 + (1 – σ + σλ)γ
		
×[1 – F(z*)]E–1 + λ(1 – γ) [1 – E–1 – F(z*)]
		
+ λγ[1 – F(z*)] (1 – E–1).
The first term is the total number of workers employed
in the previous period that continued with the same
idiosyncratic productivity level during the current
period and that either were not exogenously relocated
to the leisure island or, if they were, were able to come
back to the production island within the same period.

41

The second term is the total number of workers employed in the previous period that received a new
idiosyncratic productivity above the threshold z*and
that either were not exogenously relocated to the leisure island or, if they were, were able to come back
to the production island within the same period. The
third term is all those workers not employed in the
previous period, even though they had an idiosyncratic
productivity level higher than the threshold z*, who did
not receive a new productivity draw during the current
period and were relocated to the production island.
The last term is all those workers not employed in the
previous period who received a new idiosyncratic productivity level above the threshold z* during the current
period and were relocated to the production island.
Defining employment and nonemployment is quite
natural in this two-island model; however, dividing
nonemployed workers into the unemployed and nonparticipants is less obvious. Hereon I will follow
Krusell et al. (2012) and perform the classification by
surveying workers at the end of the period with the
following questionnaire:
1) 	Are you employed?
2) 	If you are not employed, do you wish you had
been employed?
If the worker answers yes to the first question, he is
classified as employed. If he answers no to the first
question but yes to the second question, he is classified
as unemployed. If he answers no to both the first and
second questions, he is classified as not in the labor
force (nonparticipant).
The total number of nonparticipants in the economy will then be given by
6)	 N = F(z );
*

that is, it is the total number of workers with idiosyncratic productivity levels below the threshold z *.
Irrespective of whether these agents had the opportunity of becoming employed or not, they end the period
being nonemployed and answering that they do not
wish to be employed. Observe that absent changes in
total productivity and other parameter values, the total
number of nonparticipants in the economy is constant
over time.
The total number of unemployed workers is then
given by
7)	 U = 1 – E – F(z*);

42

that is, it is the total number of nonemployed workers
that have a productivity level above the threshold z*.
Observe that U changes over time because E (the total
number of employed workers) does. In the long run,
as employment converges to a constant level, unemployment will also converge. In particular, by setting
E = E ̶ 1 in equation 5, I get that the long-run employment level is equal to
E=

λ 1 − F ( z * )

(

1 − (1 − σ ) (1 − λ ) 1 − γF ( z * )

)

.

From equation 7, I get that the long-run unemployment level is then equal to
U = 1−

λ 1 − F ( z * )

(

1 − (1 − σ ) (1 − λ ) 1 − γF ( z * )

)

− F ( z * ).

Measuring gross flows of workers across labor
market states
Given the classification of workers into employment (E), unemployment (U), or nonparticipation (N)
described in the previous subsection, I can now define
the flow rates of workers across the different labor
market states. The flow rates of workers are as follows:
8)	 fEN = γF(z*),
9)	 fUN = γF(z*),
10)	 fUE = λ(1 – fUN),
11)	 fNE = λγ[1 – F(z*)],
12)	 fEU = σ(1 – λ)(1– fEN),
13)	 fNU = (1 – λ)γ[1 – F(z*)],
where fij is the flow rate from labor market state i
(E, U, or N) to labor market state j (E, U, or N). The
flow rate fEN is given by the probability that an employed
worker receives a new idiosyncratic productivity level
times the probability that this level is below the threshold z*. The flow rate fUN is similarly given by the probability that an unemployed worker receives a new idiosyncratic productivity level times the probability that
this level is below the threshold z*. The flow rate fUE
is given by the probability that an unemployed worker
does not transition into nonparticipation times the
probability of moving to the production island. The
flow rate fNE is given by the product of the probability

2Q/2015, Economic Perspectives

that a nonparticipant draws a new idiosyncratic productivity level, the probability that this productivity
level is above the threshold z*, and the probability that
the worker moves to the production island. The flow
rate fEU is given by the product of the probability that
an employed worker does not transition into nonparticipation, the probability that he is exogenously relocated to the leisure island, and the probability that he
is not able to make it back to the production island
within the same period. The flow rate fNU is given by
the product of the probability that a nonparticipant
worker draws a new idiosyncratic productivity level,
the probability that this productivity level is above
the threshold z*, and the probability that the worker is
not able to move to the production island.
Observe that absent changes in total productivity
and other parameter values, the flow rates described
by equations 8–13 are constant over time.
Using the BLS classification of labor
market states
The U.S. Bureau of Labor Statistics classifies people
into employment, unemployment, and nonparticipation
by essentially asking the following two questions:
	 1) 	Are you employed?
	 2) 	If you are not employed, did you search for a
job in the past four weeks?
If the person answers yes to the first question, he is
classified as employed. If he answers no to the first
question but yes to the second one, he is classified as
unemployed. If he answers no to both questions, he is
classified as being a nonparticipant. In the model of
this article as well as that of Krusell et al. (2012), there
is no search activity. As a consequence, there is no distinction between unemployment and nonparticipation
in the BLS sense.
This article, as well as Krusell et al. (2012), works
around this difficulty by substituting the second BLS
question with the following one: If you are not employed, do you wish you had been employed? As shown
in the previous section, this rephrasing led to a very
clear classification between unemployment and nonparticipation in the model. But how well does the model’s
classification approximate the BLS’s classification?
Krusell et al. (2012) argue that it does this very well.
To see why this is the case, consider changing the
relocation from the leisure island to the production
island from being exogenous to being endogenous (for
stage 4 in the timeline described near the beginning
of the previous section). In particular, assume that

Federal Reserve Bank of Chicago

agents now have to pay an infinitesimal cost in order
to make such a transition with probability λ. If they
choose not to pay that infinitesimal cost, then the agents
remain on the leisure island.
Consider now the same productivity threshold z* as
in the original equilibrium, which determined who would
work and who would relocate to the leisure island (at
stage 5 of the timeline). Because idiosyncratic productivity shocks are realized at the beginning of the period
(during stage 1 of the timeline), it is clear that agents
located on the leisure island at stage 4 of the timeline with
a z < z* will not want to pay the infinitesimal search cost
(because they would not want to stay on the production
island anyway). Because these agents will end the
period not employed and will not have not searched,
they will be correctly classified as nonparticipants.
In turn, someone located on the leisure island at
stage 4 of the timeline with a z > z* will be willing to
pay the infinitesimal search cost in order to move to
the production island with probability λ. However,
some of these agents will not be lucky enough to move
to the production island. These agents will end the
period not employed but will have searched. Hence,
they will be correctly classified as unemployed.
Finally, those agents who were employed in the
previous period and who get to stage 5 of the timeline
while located on the production island with a z < z*
will decide to move to the leisure island. Because these
workers will end the period nonemployed and will not
have searched, they will be correctly classified as nonparticipants. Thus, it’s clear that if an infinitesimal search
cost is introduced into the model, the Krusell et al.
(2012) classification of labor market states, which I
use for my model, coincides with the BLS classification.
The particular timeline assumed plays a critical
role in obtaining this equivalence. To see why, consider
changing it slightly. In particular, instead of assuming
that idiosyncratic productivity shocks are realized at
the beginning of the period, assume that they are realized after the search decisions are made.4 Assuming
the infinitesimal search cost would now produce a
stark result: Nobody in the economy would be classified as a nonparticipant by the BLS. To see why, observe that all the agents would be willing to pay the
infinitesimal search cost to see if they are lucky enough
to draw an idiosyncratic productivity shock z > z*. If
they aren’t, they would move to the leisure island and
remain nonemployed. However, because they searched,
all of these agents will be classified as unemployed.
Thus, with this slight change in the timeline, one can
see that the Krusell et al. (2012) classification would
be widely different from the BLS classification.

43

TABLE 1

U.S. monthly transition rates
		To
			
From	

 1 − fUN

15) f NU = 
− 1 f NE .
 fUE


E	 U	N

E	
U	
N	

0.9559	 0.0140	0.0271
0.2503	 0.5290	0.2207
0.0472	 0.0278	0.9250

Notes: This table presents the average monthly transition rates
between the three labor market states—employment (E),
unemployment (U), and nonparticipation (N)—reported by the
BLS in the CPS between April 1992 and October 2014. See the
text for further details.
Source: Author’s calculations based on data from the U.S. Bureau
of Labor Statistics (BLS), Current Population Survey (CPS), from
Haver Analytics.

I conclude that both the infinitesimal search cost
and the particular timeline assumed are necessary for
using the model to analyze U.S. data in a meaningful way.
Calibration of the model
In principle, data on transition rates between employment, unemployment, and nonparticipation could
be used to determine the four “parameters” that appear
in equations 8–13: γ, λ, σ, and F(z*). To this end, I
present in table 1 the average monthly transition rates
between the three labor market states reported by the
BLS in its Current Population Survey (CPS) between
April 1992 and October 2014. It turns out that these
average transition rates are not in line with the simple
model considered so far because they are inconsistent
with two key testable implications of the model.
The first testable implication of the model is
obtained from equations 8 and 9:
14)	 fEN = fUN ;
that is, the transition rate from employment to nonparticipation is exactly the same as the transition rate
from unemployment to nonparticipation. However,
table 1 indicates that in the data, fUN is about eight
times larger than fEN : 22.07 percent versus 2.71 percent. Krusell et al. (2012) faced similar difficulties in
generating a large fUN relative to fEN, but here the difficulty indicated by equation 14 is even more striking
because of the particular stochastic process for idiosyncratic productivity assumed.
The second testable implication of the model is
obtained from equations 10, 11, and 13. From equations 11 and 13, I have that
f NU

44

Using equation 10, I then have that

1− λ 
=
 f NE .
 λ 

Using the values for fUN, fUE, and fNE in table 1, I get
from equation 15 an implied value for fNU of 9.97 percent. However, table 1 shows an empirical value of
2.78 percent for this transition rate.
Given that the results of the simple model do not
align with the empirical data thus far, I will follow
Krusell et al. (2012) and introduce classification error
as in Poterba and Summers (1986).5 In particular, I
will introduce two probabilities, ψUN and ψNU , which
represent the probability of classifying as a nonparticipant someone who actually is unemployed and
the probability of classifying as unemployed someone
who actually is a nonparticipant, respectively. Employment is assumed to be measured without error.
When classification error is introduced, I must
make a distinction between true unemployment and
nonparticipation, U and N, and measured unemployment and nonparticipation, U and N. (The true labor
market states do not account for any classification
error, whereas the measured labor market states do.)
In particular, by a law of large numbers, I have the
following relations:
16) U = U (1 − ψUN ) + N ψ NU ,
17) N = N (1 − ψ NU ) + U ψUN .
Equation 16 states that measured unemployment U
is constituted by a fraction 1 – ψUN of unemployed
workers U that do not get misclassified and a fraction
ψNU of nonparticipants N that get misclassified as unemployed. Equation 17 states that measured nonparticipation N is constituted by a fraction 1 – ψNU of
nonparticipants N that do not get misclassified and a
fraction ψUN of unemployed workers U that get misclassified as nonparticipants.
With classification error introduced into the model,
the measured transition rates become the following:
18) f EN = f EN (1 − ψ NU ) + f EU ψUN ,
19) f EU = f EU (1 − ψUN ) + f EN ψ NU ,
20) fUE = fUE
21) f NE = f NE

U (1 − ψUN )
U
N (1 − ψ NU )
N

+ f NE

N ψ NU
,
U

+ fUE

U ψUN
,
N

2Q/2015, Economic Perspectives

TABLE 2

Model’s true transition rates
		To
			
From	

E	 U	N

E	
U	
N	

0.9588	 0.0193	0.0219
0.2503	 0.7278	0.0219
0.0068	 0.0198	0.9734

Notes: This table presents the monthly transition rates between
the three true (without classification error) labor market states—
employment (E), unemployment (U), and nonparticipation (N)—in the
model economy. See the text for further details.

22) fUN =  fUU ψUN + fUN (1 − ψ NU )

U (1 − ψUN )

U
N ψ NU
+  f NN (1 − ψ NU ) + f NU ψUN 
,
U

23) f NU =  f NN ψ NU + f NU (1 − ψUN )

N (1 − ψ NU )

N
U ψUN
+  fUU (1 − ψUN ) + fUN ψ NU 
.
N

Equation 18 states that the probability that an employed
worker makes a transition to measured nonparticipation
f EN is given by the probability of transitioning to
nonparticipation fEN times the probability of not being
misclassified 1 – ψNU , plus the probability of transitioning to unemployment fEU times the probability of
being misclassified as a nonparticipant ψUN. Equation
19 is similar to equation 18, but it is for the transition
rate from employment to measured unemployment
f EU . Equation 20 states that the probability that a worker measured as an unemployed worker makes a transition to employment fUE is given by the probability
that the worker is truly unemployed

U (1− ψUN )
U

times the probability of transitioning into employment
fUE, plus the probability that the worker is actually
a nonparticipant

N ψ NU
times the probability of
U

transitioning into employment fNE. Equation 21 is
similar to equation 20, but it is for the transition rate
from measured nonparticipation to employment f NE .
Equation 22 states that the probability of transitioning
from measured unemployment to measured nonparticipation fUN is given by the sum of two terms. The
first term is that with probability

Federal Reserve Bank of Chicago

U (1− ψUN )
U

, the

worker is truly unemployed—in which case he makes
a transition to measured nonparticipation if he remains
unemployed and this lack of change in labor market
status is mismeasured, fUUψUN, or if he transitions to
nonparticipation and this change is not mismeasured,
fUN (1 – ψNU). The second term is that with probability
N ψ NU
, the worker is truly a nonparticipant—in
U
which case he makes a transition to measured nonparticipation if he remains a nonparticipant and this
lack of change in labor market status is not mismeasured, fNN (1 – ψNU), or if he transitions into unemployment and this change is mismeasured, fNUψUN.
Equation 23 is similar to equation 22, but it is for the
transition rate from measured nonparticipation to
measured unemployment.
Because I am interested in reproducing the transition rates in table 1, which are monthly averages
over a long time period, I will impose the following
steady-state conditions:
24)	 (fUE + fUN)U = fEU E + fNU N,
25)	 (fNE + fNU)N = fEN E + fUNU,
26)	 U + N + E = 1.
Equation 24 states that the total flows out of unemployment must be equal to the total flows into unemployment. Equation 25 states that the total flows out
of nonparticipation must be equal to the total flows
into nonparticipation. Finally, equation 26 states that
the sum of all workers across the three labor market
states must add up to the total population.
The system of nonlinear equations 14–26 could
in principle be solved for the 13 unknowns U , N, U,
N, E, fEN, fEU, fUN, fUE, fNE, fNU, ψUN, and ψNU with the
target values for f EN , f EU , fUE , f NE , fUN , and f NU
being taken from table 1. However, performing an
exhaustive computer analysis indicates that such a
solution does not exist. Instead, the best approximate
solution is obtained by setting ψNU to 0 (representing
a corner solution), ψUN to 0.2733, and the true transition
probabilities fEN, fEU, fUN, fUE, fNE, and fNU to the values
given in table 2. Based on equations 14–26, these transition probabilities and misclassification probabilities
imply the measured transition probabilities that are
given in table 3. Many of the values in table 3 do
exactly match those in table 1, and those that do not
are not that far apart. In fact, only f NE and f NU miss
their target values—and not by much. Also, while the
classification errors ψNU = 0 and ψUN = 0.2733 are more

45

Equation 4 allows for a normalization. I will therefore normalize the threshold value z* to 1 and find the
parameter value ɸ that satisfies equation 27; that is,

TABLE 3

Model’s measured transition rates
		To
			
From	

E	

E	
	
U
N
	

U	

0.9559	 0.0140	0.0271
0.2503	 0.5290	0.2207
0.0163	 0.0344	0.9493

Notes: This table presents the monthly transition rates between the
three measured (including classification error) labor market states—
employment (E ), unemployment (U ), and nonparticipation (N)—in
the model economy. See the text for further details.

extreme than those reported by Poterba and Summers
(1986) (their reported values are ψNU = 0.0064 and
ψUN = 0.1146), they are not completely out of line.
Given the transition rates estimated in table 2, I
can use equations 8 and 10–12 to back up the values
for F(z*), λ, γ, and σ consistent with them. In particular, from equation 10, I have that
λ=

F(z*) = 1 – e–ɸz* = 1 – e–ɸ = 0.4513.

N

fUE
0.2503
=
= 0.2558.
1 − fUN 1 − 0.0219

In turn, from equation 11, I have that

From this, I get ɸ = 0.6003.
Observe that

∫

f + λf EN 0.0068 + 0.2558 × 0.0219
γ = NE
=
= 0.0484.
λ
0.2558
From equation 8, I then have that
27) F ( z * ) =

f EN 0.0219
=
= 0.4513.
γ
0.0484

In addition, from equation 12, I get that
σ=
=

f EU
(1 − λ ) (1 − f EN )
0.0193
= 0.0265.
−
1
0
2558
.
(
) (1 − 0.0219 )

In order to calibrate the difference between aggregate productivity and the value of leisure (p – α) that
appears in equation 4, I must take a stance on the shape
of the distribution function of idiosyncratic productivity
levels F. For the sake of convenience, I will assume
that it is exponential:
28)	 F(z) = 1 – e–ɸz.

46

z*

∞

1 − F ( z ) dz = ∫ * e
z

∞

− φz

*

1
1
dz = − e − φz  = e − φz .
φ
 z* φ

Substituting this expression in equation 4 yields the
following equation:

29) p − α = − z * −

βγ (1 − σ ) (1 − λ )

1 − φz*
e .
1 − β (1 − γ ) (1 − σ ) (1 − λ ) φ

Using the values calibrated in this section, the normalization z* = 1, and a discount rate β = 0.9967 (which
implies an annual interest rate of 4 percent), I get a
value of p – α = –1.1021.
Observe that aggregate labor productivity is given
by the following:
∞
Y
1
= p+
zF ′ ( z ) dz.
* ∫z *
E
1− F (z )

fNE = λγ – λγF(z*) = λγ – λfEN .
Hence,

∞

That is, it is given by the aggregate labor productivity
level common to all workers p plus the average idiosyncratic productivity z of employed workers. Using
the functional form for the distribution function F in
equation 28 and integrating by parts, I find that aggregate labor productivity is given as follows:
30)

Y
1
= p + z* + .
E
φ

In order to determine the value of leisure α, I
follow Shimer (2005) and assume that it is equal to
40 percent of aggregate labor productivity Y/E. That is,

1
α = 0.40  p + z * +  .
φ

Because it has already been determined that
p – α = –1.1021, it follows that

1
p + 1.1021 = 0.40  p + z * + 
φ

1 

= 0.40  p + 1 +
,
0.6003 


2Q/2015, Economic Perspectives

where I have used the values for z* and ϕ already determined. It follows that p = –0.0596 and α = 1.0425.
The resulting value for aggregate productivity Y/E in
equation 30 is 2.6063.
Business cycles
In this section, I test how well the model does in
reproducing U.S. business cycle data. To this end, I
will follow Krusell et al. (2012) and allow the probability of being relocated to the production island λ, the
probability of being relocated to the leisure island σ,
and aggregate productivity p to fluctuate over time.
All other parameters, including those describing the
stochastic process for idiosyncratic productivity levels,
will be assumed to be constant.
The plan is to use monthly data on measured gross
flow rates to infer true gross flow rates in the model.
These true gross flow rates will then be used to construct monthly time series for the probability of being
relocated to the production island λ, the probability of
being relocated to the leisure island σ, and the idiosyncratic productivity threshold z*. Given that I have
an empirical time series for aggregate labor productivity Y/E, I will then use equation 30 to construct a
time series for the aggregate productivity p. A key test
of the model will be to compare this time series with
the time series for p that is obtained from equation 29,
using the already determined time series for z*, λ, and
σ. This is a key test because equation 29 reflects the
optimal decision of agents in the model economy.
In order to obtain empirical counterparts for true
gross flow rates, I will use equations 18, 19, 20, and 22.
The reason for using these equations is that these are
the four equations among equations 18–23 that happen
to hold with equality in the calibration performed in the
previous section. Because the classification error ψNU
that best described the data was equal to 0 (that is,
was a corner solution), I will impose that it is 0 over
the business cycle fluctuations. In contrast, I will allow
the classification error ψUN not only to be positive but
to fluctuate over the business cycle.6
Under the assumption that the classification error
ψNU is equal to 0, which implies that U = U (1 – ψUN),
equations 18, 19, 20, and 22 become the following:
f EN = f EN + f EU ψUN ,
f EU = f EU (1 − ψUN ),
fUE = fUE ,
fUN = (1 − fUE − fUN )ψUN + fUN ,
where, from equation 14,
fUN = fEN  .

Federal Reserve Bank of Chicago

This system of equations turns out to have the
following solution:
fUN − f EN
31) ψUN =
,
1 − fUE − f EN − f EU

32) f EU =

f EU
1 − ψUN

,

33) f EN = f EN + f EU − f EU ,
34) fUE = fUE ,
35)	 fUN = fEN .
With monthly CPS data for fUN , f EN , f EU , and
fUE , I can then construct monthly time series for ψUN,
fEU, fEN, fUE, and fUN using equations 31–35. Panels A–D
of figure 2 show the time series for these true transition
rates as well as the corresponding measured transition
rates (which coincide, by construction, with U.S. data).
On the one hand, one can see from panels A and B that
fEU is higher than f EU , and that fEN is lower than f EN
(which was expected from tables 2 and 3) but that the
true transition rates track the measured transition rates
quite closely. On the other hand, panel C shows that
fUE and fUE coincide (as indicated by equation 34). A
large discrepancy shows up in panel D, where one can
see that fUN is not only much smaller than fUN (as was
expected from tables 2 and 3) but that fUN hardly fluctuates at all. In fact, panel E shows that most of the
fluctuations in fUN are accounted for by fluctuations
in the classification error ψUN  .7
Given the constructed monthly time series for fUN
and fUE, I can obtain the probability of being relocated
to the production island λ from equation 10 as follows:
λ=

fUE
.
1 − fUN

In turn, I can obtain the probability of being relocated to the leisure island σ from equation 12 and the
constructed time series for fEU , fEN , and λ as follows:
σ=

f EU
.
(1 − λ ) (1 − f EN )

From the constructed monthly time series for fEN
and the calibrated value for γ in the previous section, I
can measure the time series for F(z*) from equation 8 as
F ( z* ) =

f EN
.
γ

47

FIGURE 2

True versus measured transition rates
A. Employment-to-unemployment
transition rates

B. Employment-to-nonparticipation
transition rates

rate

rate

0.04

0.04

0.02

0.02

0

0

50

100

150

200

250

0

300

0

number of months since April 1992

50

100

150

200

250

C. Unemployment-to-employment
transition rates

D. Unemployment-to-nonparticipation
transition rates

rate

rate

0.4

0.4

0.2

0.2

0

0

50

100

150

200

250

300

number of months since April 1992

0

300

number of months since April 1992

0

50

100

150

200

250

300

number of months since April 1992
True rate

Measured rate

E. Classification error and measured unemployment-to-nonparticipation transition rate
rate
0.4
0.2
0

0

50

100

150

200

250

300

number of months since April 1992
Classification error

Measured rate

Notes: Panels A–D present the monthly true and measured transition rates between the three labor market states—employment,
unemployment, and nonparticipation—in the model economy. True transition rates do not account for classification error, while
measured transition rates do. (The measured rates coincide, by construction, with U.S. data.) Panel E shows the measured
unemployment-to-nonparticipation transition rate and the unemployment-to-nonparticipation classification error. See the text for
further details on the panels.
Source: Author’s calculations based on data from the U.S. Bureau of Labor Statistics, Current Population Survey, from Haver Analytics.

Using this time series, equation 28, and the calibrated
value for ϕ from the previous section, I can then construct a monthly time series for z* from the following
equation:
1
z* = − ln 1 − F ( z* ) .
φ
Given this constructed time series for z* and an
empirical time series for aggregate labor productivity
Y/E, I can then use equation 30 to construct a time series
for p. For an empirical time series for aggregate output Y, I use the forecasting firm Macroeconomic
Advisers’ monthly real gross domestic product (GDP)
series, which is a monthly indicator of real aggregate

48

output that is conceptually consistent with GDP in the
U.S. Bureau of Economic Analysis’s national income
and product accounts (NIPAs).8 For aggregate employment E, I use employment of the civilian noninstitutional
population aged 16 years and over from the CPS provided by the BLS. The aggregate labor productivity
Y/E obtained from dividing both time series happens
to grow over time. Because aggregate labor productivity is constant in the model economy, I detrend the
data using a linear regression. The deviations from
trend thus obtained are then used to construct a time
series for Y/E with an average value of 2.6063, which
was the value for Y/E implied by the calibration of
the previous section.

2Q/2015, Economic Perspectives

TABLE 4

Stochastic properties of shocks
	
λ		p
σ
			
Standard deviation 	
Autocorrelation 	
	
λ	
σ	
p	

0.0490 	
0.97 	

0.0024 	
0.68 	

0.0914
0.28

Correlation matrix
1.00 	
0.40 	
0.26
	
1.00 	
0.34
		1.00

Note: See the text for further details.
Sources: Author’s calculations based on data from Macroeconomic
Advisers and the U.S. Bureau of Labor Statistics, Current Population
Survey, all from Haver Analytics.

Table 4 reports summary statistics for the joint
stochastic behavior of the time series for λ, σ, and p
obtained through the equations discussed in this section.
One can see that the probability of being exogenously
relocated to the production island λ is highly persistent,
the probability of being exogenously relocated to the
leisure island σ is less so, and aggregate productivity
p is much less persistent than usually assumed (see the
second row of table 4 reporting autocorrelation statistics). All three shocks are pairwise weakly positively
correlated (see the final three rows of table 4). This
contrasts with Krusell et al. (2012), who assumed
perfect correlations between the shocks. In particular,
their “good-times/bad-times” assumption implies that
ρ(λ, σ) = –1, ρ(λ, p) = 1, and ρ(σ, p) = –1.
Panel A of table 5 reports business cycle statistics
for U.S. data. The labels u and lfpr denote the unemployment rate and labor force participation rate, respectively.
All statistics correspond to monthly time series. Before
any statistics were computed, the data were logged
and applied a Hodrick–Prescott filter with smoothing
parameter of 105 in order to obtain their cyclical components (that is, their deviations from a slow-moving
trend that reflect their fluctuations at business cycle
frequencies). The labels σ(xt), ρ(xt, Yt ), and ρ(xt, xt–1)
denote the standard deviation of variable xt, the contemporaneous correlation of the variable xt with output Yt , and the serial autocorrelation of the variable xt,
respectively. I note that compared with output, employment is somewhat less variable, the unemployment rate
is much more variable, and the labor force participation rate is much less variable (see the first column of
table 5, panel A). Employment is strongly procyclical
(that is, rising when economic times are good and
falling when they are bad), the unemployment rate is
strongly countercyclical (that is, falling when economic
times are good and rising when they are bad), and the

Federal Reserve Bank of Chicago

labor force participation rate is roughly acyclical (that
is, moving independently of the overall state of the
economy) (see the second column of table 5, panel A).
All of these variables are significantly persistent, as
shown by the serial autocorrelation statistics (see the
third column of table 5, panel A). Reviewing the statistics for the transition rates, I note that all of them
are highly volatile. The fUE transition rate is strongly
procyclical and persistent, while the f EU and f NU
transition rates are countercyclical and somewhat persistent. All other transition rates display weak cyclical
patterns and have little persistence.
Panel B of table 5 reports similar statistics for
artificial data generated using the following procedure.
Given the monthly time series for z*, λ, and σ constructed earlier, equations 8–13 were used to construct monthly time series for the true transition rates
fEN, fUN, fUE, fNE, fEU, and fNU  . Given these time series,
monthly paths for true employment E, true unemployment U, and true nonparticipation N were constructed
using the following equations:
Ut = (1 – fUE,t–1 – fUN,t–1)Ut–1 + fEU,t–1Et–1 + fNU,t–1Nt–1,
Nt = (1 – fNE,t–1 – fNU,t–1)Nt–1 + fEN,t–1Et–1 + fUN,t–1Ut–1,
Et = 1 – Ut – Nt .
Given the time series for these variables and for the
classification error ψUN obtained earlier, paths for measured unemployment U and measured nonparticipation
N were obtained from equations 16 and 17. Given all
these series, paths for the measured transition rates
fUN , f NU , f NE , f EN , f EU , and fUE were constructed
using equations 18–23. In turn, given the time series
for z* and p constructed earlier, aggregate labor productivity Y/E was obtained from equation 30. Output
Y was then obtained by multiplying aggregate labor
productivity Y/E by employment E. Finally, the unemployment rate was calculated as u = U ( E + U )
and the labor force participation rate as lfpr = E + U .
I see many similarities between panels A and B of
table 5. To some extent this is not surprising because the
monthly time series for p, z*, λ, and σ were constructed
in such a way that this would be the case. Indeed, the
values of Y/E, f EU , f EN , fUE , and fUN must necessarily
be identical in both cases (because these variables have
been used as targets in the construction of the shocks).9
Interestingly, similarities are also apparent in the rest
of the variables. While the transition rate f NU and the
unemployment rate are right on target, the transition
rate f NE , output Y, employment E, and the labor force
participation rate lfpr are more volatile in the artificial

49

TABLE 5

Monthly business cycle statistics
	
			

A. U.S. data	

B. Artificial data

						
σ ( xt )
ρ ( x t , x t −1 )
σ ( xt )
ρ (x t , Yt )
ρ (x t , Yt )
ρ ( x t , x t −1 )
	(percent)	(percent)
Y	
E	
u	
lfpr 	
Y/E 	
fEU
	
fEN 	
fUE
	
fUN
	
fNE 	
fNU
	

1.12	1.00	0.88	
0.95	0.74	0.98	
10.79	
–0.84	
0.98	
0.28	0.22	0.76	
0.77	0.55	0.69	
7.68	 –0.70	
0.64	
4.30	
0.16	
0.07	
7.96	0.77	0.81	
5.75	0.53	0.52	
4.79	
0.43	
0.30	
7.65	 –0.68	
0.66	

1.47	1.00	
0.92
1.23	0.85	
0.97
10.70	
–0.75	
0.96
0.75	0.30	
0.91
0.77	0.55	
0.69
7.68	 –0.63	0.64
4.30	
0.14	
0.07
7.96	0.66	
0.81
5.75	0.46	
0.52
7.96	
0.58	
0.61
6.93	 –0.62	0.90

Notes: All statistics correspond to monthly time series and are for the period April 1992–October 2014. See the text for further details.
Sources: Author’s calculations based on data from Macroeconomic Advisers and the U.S. Bureau of Labor Statistics, Current Population Survey,
all from Haver Analytics.

data than in the U.S. economy. However, the differences are not large and the correlations with output
and serial autocorrelation statistics are quite similar
in both cases.
While this is all quite satisfactory, it does not represent a test of the model yet. The reason is that the relationship between the productivity threshold z* and
the shocks p, λ, and σ that has been used so far has
been determined by the data (and by part of the model
structure); but it is not clear that such a relationship is
completely consistent with the model economy. To
fully test the empirical plausibility of the model, I must
use equation 29, which represents the optimal decision
of agents.10 The way that I implement such a test is to
plug into equation 29 the time series for z*, λ, and σ
constructed earlier in this section and to solve for the
theoretical aggregate productivity level p implied by the
equation.11 The resulting time series is then compared
with the empirical time series for p constructed earlier
in this section. The comparison is displayed in figure 3,
panel A. The result is striking. Not only do both time
series display similar properties, but they align on top
of each other quite well. This is better seen in figure 3,
panel B, which displays a scatter plot of both the theoretical and empirical aggregate productivity levels.
While the points are not perfectly aligned along the
45-degree line (which should be the case if both time
series were identical), they are not far from it. I interpret this as a surprising success of the labor supply
theory embodied in the model.
While my analysis thus far has demonstrated that
the optimal decisions of agents (summarized by the
idiosyncratic productivity threshold z*) are consistent

50

with empirical observations, it is natural to wonder
about the role that the endogenous fluctuations in z*
play in business cycle dynamics. I evaluate this role by
comparing two scenarios, whose results are reported
in table 6. Panel A of table 6—which I label “Variable
z*”—reproduces the statistics of panel B of table 5 (the
business cycle statistics of the benchmark economy).
Panel B of table 6—which I label “Constant z*”—
reports business cycle statistics under the assumption
that the productivity threshold z* is constant at its steadystate value while all exogenous shocks (including the
classification error) remain the same as in the benchmark case. One can see that the constant z* significantly
reduces the fluctuations in f EN and increases its persistence (see the first and third columns of both panels A and B of table 6); however, the behavior of all
other gross worker flow rates is largely unchanged. The
constant z* hardly affects the behavior of employment
or unemployment: Their standard deviations and serial
autocorrelations remain largely the same (see the first
and third columns of both panels A and B of table 6).
However, fluctuations in labor force participation are
considerably dampened. This is not surprising because
equation 6 indicates that true nonparticipation N is
constant whenever z* is. What is largely affected by the
constant z* is aggregate labor productivity Y/E, which
becomes about four times more variable and completely
loses its persistence (see the first and third columns of
both panels A and B of table 6). Because aggregate
output Y is given by aggregate employment E (which
is hardly affected by the constant z*) times aggregate
labor productivity Y/E, Y also becomes more volatile
and less persistent.

2Q/2015, Economic Perspectives

FIGURE 3

Theoretical versus empirical aggregate productivity level p
A. Time series
productivity
0.2
0.1
0
–0.1
–0.2
–0.3
–0.4
–0.5

0

50

100

150
200
number of months since April 1992

250

300

Empirical p

Theoretical p
B. Scatter plot
empirical p
0.2
0.1
0
–0.1
–0.2
–0.3
–0.4
–0.5
–0.5

–0.4

–0.3

–0.2

–0.1

0

0.1

0.2

theoretical p
Notes: Panel B displays a scatter plot of both the theoretical and empirical aggregate productivity levels. See the text for further
details on how to interpret both panels.
Sources: Author’s calculations based on data from Macroeconomic Advisers and the U.S. Bureau of Labor Statistics, Current
Population Survey, all from Haver Analytics.

I conclude that while the model described in this
article captures salient features of labor market dynamics,
it does so by relying almost completely on exogenous
shocks. The only endogenous margin in the model, the
choice of the productivity threshold z*, provides little
insight into such dynamics. Conditional on observed
labor market dynamics, the main role of the endogenous
fluctuations in z* is to generate empirically relevant
fluctuations in aggregate output and labor productivity.
Discussion
The results in this article are widely different from
those in Veracierto (2008). In that paper, I studied a

Federal Reserve Bank of Chicago

business cycle model with three labor market states
but found that the model generated counterfactual
business cycle dynamics. In particular, I found that
unemployment was procyclical and that labor force
participation was highly volatile (in fact, as much as
employment). The intuition for why I got such a result
is straightforward. In that model, being out of the labor
force provided agents more leisure than being unemployed. Therefore, when a bad aggregate shock hit the
economy that made leisure more attractive than working,
workers made transitions from employment to nonparticipation instead of making transitions from employment to unemployment (as the data largely indicate).

51

TABLE 6

Variable versus constant z*
	
			

A. Variable z*	

B. Constant z*

						
σ ( xt )
ρ ( x t , x t −1 )
σ ( xt )
ρ (x t , Yt )
ρ (x t , Yt )
ρ ( x t , x t −1 )
	(percent)	(percent)
Y	
E	
u	
lfpr 	
Y/E 	
fEU
	
fEN 	
fUE 	
fUN 	
fNE 	
fNU 	

1.47	1.00	0.92	
1.23	0.85	0.97	
10.70	
–0.75	
0.96	
0.75	0.30	0.91	
0.77	0.55	0.69	
7.68	 –0.63	
0.64	
4.30	
0.14	
0.07	
7.96	0.66	0.81	
5.75	0.46	0.52	
7.96	
0.58	
0.61	
6.93	 –0.62	
0.90	

3.16	1.00	
0.09
1.29	0.30	
0.98
10.60	
–0.34	
0.96
0.23	
–0.07	
0.57
3.03	0.92	
0.01
7.67	 –0.21	0.64
1.87	
0.27	
0.34
7.98	0.28	
0.81
5.78	0.36	
0.52
7.85	
0.32	
0.69
6.02	 –0.26	0.94

Notes: All statistics correspond to monthly time series and are for the period April 1992–October 2014. See the text for further details.
Sources: Author’s calculations based on data from Macroeconomic Advisers and the U.S. Bureau of Labor Statistics, Current Population Survey,
all from Haver Analytics.

As a consequence, fluctuations in labor force participation ended up mirroring fluctuations in employment,
while unemployment became procyclical. Moreover,
when the negative shock reversed, there was a surge
of unemployment because agents needed to search in
order to become employed. This reinforced the procyclicality of unemployment.
In principle, the model in this article would be subject to the same difficulties. To see why, suppose that a
negative aggregate productivity shock hits the economy
that lowers p. Because this makes working less attractive
than enjoying leisure, the threshold productivity level
z* will increase. As a consequence, fewer people will
choose to work (if given the opportunity); and of those
not working, fewer will say that they would like to
work. Thus, when the negative aggregate productivity
shock hits the economy, employment and unemployment will decrease and nonparticipation will increase.
When the aggregate productivity shock reverses, more
of those on the production island will decide to work.
And of those not making it to the production island,
more of them will report that they would like to be employed. Thus, employment and unemployment will
increase and nonparticipation will decrease. It is clear
that with aggregate productivity shocks alone, the model
will tend to generate procyclical unemployment and
variations in labor force participation that mirror those
in employment. That is, the model would display the
same counterfactual behavior found in Veracierto (2008).
The reason why the model does not experience
these difficulties is because there are exogenous variations in the job-separation rate σ and the job-finding
rate λ. While in Veracierto (2008) these rates varied
endogenously in response to an aggregate productivity

52

shock, here they were chosen to fluctuate as much as
needed to reproduce U.S. observations. In fact, the
previous section showed that most of the success of
the model at reproducing labor market dynamics relied
on the exogenous variations in job-separation and
job-finding rates, with little role for the endogenous
decisions. The challenge for future researchers will
be to develop models that generate exactly those same
variations, but endogenously. This promises to be an
exciting area of research.
Conclusion
In this article, I develop and analyze a simple model
of the gross flows of workers across labor market states
that is based on a model by Krusell et al. (2012). The
simplicity of the model allows for analytical derivations
that make the determination of these flows transparent.
Moreover, this same simplicity allows me to perfectly
identify the shocks that drive labor market fluctuations
in the model by using U.S. data. I find that if errors in
the classification of agents’ labor market states are introduced and allowed to vary over time, the model has
the ability to generate business cycle dynamics similar
to those observed in the U.S. data. However, the labor
market dynamics generated by the model are essentially
driven by exogenous factors; the endogenous labor
supply decisions embodied in the model barely affect
them. The challenge for the future will be to develop
models that reproduce actual labor market dynamics
like my model did—but through endogenous factors.
Such models may help further our understanding of
what may be driving unemployment and other shifts
in the labor market.

2Q/2015, Economic Perspectives

NOTES

REFERENCES

A perfectly competitive labor market is a market comprising many
well-informed buyers (firms) and sellers (individuals) of labor that
take the wage rate as given. Competitive wage determination in
models with search frictions was first introduced by Phelps (1970)
and, more systematically, by Lucas and Prescott (1974).

Christiano, L. J., M. Trabandt, and K. Walentin,
2010, “Involuntary unemployment and the business
cycle,” National Bureau of Economic Research,
working paper, No. 15801, March, available at
http://www.nber.org/papers/w15801.

1

For the rest of the article, I will follow the Lucas and Prescott (1974)
tradition of referring to geographically distinct locations as “islands.”
2

The slope of the right-hand side of equation 4 is equal to

3

−1 +

βγ (1 − σ)(1 − λ )
1 − F ( z * ) < 0.

1 − β (1 − γ )(1 − λ )(1 − σ) 

This alternative timeline seems quite natural. In fact, standard
search models assume that agents search without knowing the
wage offer that they will receive (for example, McCall, 1970).

Haefke, C., and M. Reiter, 2011, “What do
participation fluctuations tell us about labor supply
elasticities?,” Institute for the Study of Labor (IZA),
discussion paper, No. 6039, October, available at
http://ftp.iza.org/dp6039.pdf.

4

Krusell et al. (2012) introduce classification error in an appendix
to show that it can improve certain failures of their benchmark
calibration.
5

Here I depart from Krusell et al. (2012) because in an appendix,
they only consider the case of constant classification error.
6

It is difficult to take a stance on the plausibility of the classification error ψUN fluctuating over the business cycle by this
magnitude.
7

In fact, the quarterly growth rate of the Macroeconomic Advisers’
GDP time series closely resembles the growth rate of real GDP in
the NIPAs.
8

In fact, only the standard deviations and serial autocorrelations of
these variables must be the same in both cases. Their correlations
with output will generally differ because output has not been used
as a target in the construction of the shocks.
9

Equation 29 has been used to calibrate the model, but it has not
been used so far to analyze business cycle fluctuations.
10

This “theoretical” aggregate productivity level should be interpreted as the aggregate productivity level p that is needed to reconcile the model with the data.
11

Krusell, P., T. Mukoyama, R. Rogerson, and
A. Şahin, 2012, “Is labor supply important for business
cycles?,” National Bureau of Economic Research,
working paper, No. 17779, January, available at
http://www.nber.org/papers/w17779.
Lucas, R. E., Jr., and E. C. Prescott, 1974, “Equilibrium
search and unemployment,” Journal of Economic
Theory, Vol. 7, No. 2, February, pp. 188–209.
McCall, J. J., 1970, “Economics of information and
job search,” Quarterly Journal of Economics, Vol. 84,
No. 1, February, pp. 113–126.
Phelps, E. S. (ed.), 1970, Microeconomic Foundations
of Employment and Inflation Theory, New York: Norton.
Poterba, J. M., and L. H. Summers, 1986, “Reporting errors and labor market dynamics,” Econometrica,
Vol. 54, No. 6, November, pp. 1319–1338.
Shimer, R., 2013, “Job search, labor-force participation, and wage rigidities,” in Advances in Economics
and Econometrics, D. Acemoglu, M. Arellano, and
E. Dekel (eds.), Vol. 2, Applied Economics, Cambridge,
UK: Cambridge University Press, pp. 197–234.
__________, 2005, “The cyclical behavior of equilibrium unemployment and vacancies,” American
Economic Review, Vol. 95, No. 1, March, pp. 25–49.
Tripier, F., 2004, “Can the labor market search model
explain the fluctuations of allocations of time?,”
Economic Modelling, Vol. 21, No. 1, January,
pp. 131–146.
Veracierto, M., 2008, “On the cyclical behavior of
employment, unemployment and labor force participation,” Journal of Monetary Economics, Vol. 55,
No. 6, September, pp. 1143–1157.

Federal Reserve Bank of Chicago

53

Bubbles and fools
Gadi Barlevy

Introduction and summary
In the wake of the financial crisis of 2007–08 and the
Great Recession precipitated by it, a growing chorus has
argued that policymakers ought to act more aggressively
to rein in asset bubbles—that is, scenarios in which
asset prices rise rapidly and then crash. Before the
crisis, conventional wisdom among policymakers
cautioned against acting on suspected bubbles. As laid
out in an influential paper by Bernanke and Gertler
(1999), there are two reasons for this. First, while asset
prices are increasing, it is difficult to gauge whether
these prices are likely to remain high or revert. Second,
many of the available tools for reining in asset prices,
such as raising nominal short-term interest rates,
tend to be blunt instruments that impact economic
activity more broadly. Bernanke and Gertler argued
that rather than responding to rising asset prices, policymakers should stand ready to address the consequences
of a collapse in asset prices if and when it happens.1
The severity of the Great Recession and the challenge
of trying to stimulate economic activity even after
lowering short-term rates to zero led many to rethink
whether policymakers should wait and see if asset
prices collapse and then deal with the aftermath. Of
course, just because the stakes are great does not
mean policymakers can or should do anything if they
are concerned about a possible bubble. To determine
whether anything can or should be done, we need to
understand when and why bubbles can emerge and what
that might mean for policy. This article considers one
explanation for bubbles known as the greater-fool
theory of bubbles, as well as its implications for policy.
As a first step, let me be clear on what I mean by
a bubble. Arguably, the distinguishing feature of the
various historical episodes that are usually described
as bubbles is that asset prices seem to rise too quickly,
culminating in an eventual collapse of asset prices

54

and a glut of assets created while prices were rising.
This suggests defining a bubble as an asset whose
price is somehow too high, which is indeed the way
economists typically define the term: A bubble is an
asset whose price deviates from its “natural” value.
But what is the natural value for an asset? When the
cash flows an asset pays out in dividends are drawn
from a known probability distribution, a natural benchmark value is the expected present discounted value
of the dividends it generates—also known as the fundamental value of the asset. Intuitively, society values
assets for the dividends they are expected to yield, so

Gadi Barlevy is a senior economist and research advisor in the
Economic Research Department at the Federal Reserve Bank of
Chicago. The author thanks Lisa Barrow, Bob Barsky, Antonio
Doblas-Madrid, and François Velde for their helpful comments.
© 2015 Federal Reserve Bank of Chicago
Economic Perspectives is published by the Economic Research
Department of the Federal Reserve Bank of Chicago. The views
expressed are the authors’ and do not necessarily reflect the views
of the Federal Reserve Bank of Chicago or the Federal Reserve
System.
Charles L. Evans, President; Daniel G. Sullivan, Executive Vice
President and Director of Research; David Marshall, Senior Vice
President and Associate Director of Research; Spencer Krane,
Senior Vice President and Senior Research Advisor; Daniel Aaronson,
Vice President, microeconomic policy research; Jonas D. M.
Fisher, Vice President, macroeconomic policy research; Anna L.
Paulson, Vice President, finance team; William A. Testa, Vice
President, regional programs; Lisa Barrow, Senior Economist and
Economics Editor; Helen Koshy and Han Y. Choi, Editors; Julia
Baker, Production Editor; Sheila A. Mangler, Editorial Assistant.
Economic Perspectives articles may be reproduced in whole or in
part, provided the articles are not reproduced or distributed for
commercial gain and provided the source is appropriately credited.
Prior written permission must be obtained for any other reproduction, distribution, republication, or creation of derivative works
of Economic Perspectives articles. To request permission, please
contact Helen Koshy, senior editor, at 312-322-5830 or email
Helen.Koshy@chi.frb.org.
ISSN 0164-0682

2Q/2015, Economic Perspectives

the fundamental value represents how much an asset
ought to be worth if it is to be supplied efficiently. In
practice, though, the distribution of dividends is typically
unknown; scenarios that matter for dividends may have
few or no historical precedents that can be used to
gauge their likelihood. If agents held different beliefs
as to the likelihood of these states, it isn’t clear what
the benchmark value of an asset should be: Whose
beliefs should we use to compute the expected present
discounted value of cash flows? In what follows, I discuss
some ways of extending the definition of a bubble to
the case where agents have different beliefs.2 Indeed,
one of the themes of this article is that some models
that purport to capture bubbles rely on a particular way
of defining the fundamental value of an asset when
traders hold different beliefs, though using alternative
definitions would imply the asset is in fact properly
priced. It is thus unclear whether these models should
be viewed as capturing bubbles in the sense that the
underlying asset is overvalued. That said, these models
unambiguously capture a separate phenomenon, speculative trading, by which I mean that agents trade assets
not because they expect mutually beneficial gains from
trading with others, but because they expect to profit
at the expense of others. Speculative trading may well
be something policymakers should be concerned about,
but the appropriate policy response to it need not be
framed in terms of driving asset prices back toward
fundamentals, as policy prescriptions for responding
to bubbles often are.
It turns out that it is not easy to construct economic
models that give rise to bubbles in the sense I have
just described. The reason is that people will naturally
be reluctant to pay more for an asset than the value
of the dividends it generates. Nevertheless, there are
settings in which this phenomenon can occur. One
explanation is that when asset prices are equal to fundamentals, there will be a shortage of assets relative
to the amount agents require for saving or liquidity or
to earn a satisfactory return. According to this view, a
shortage will lead agents to pile into whatever assets
are available. Even if assets trade above their fundamentals, agents might still be willing to buy them given
their inherent usefulness.3
A different explanation for bubbles is based on risk
shifting: If agents can buy risky assets and borrow
against them, they would be willing to pay more for
assets than their expected payoff, since they can shift
their losses on to their creditors by defaulting.4
A third explanation for why bubbles can arise—
and the one this article focuses on—is known as the
greater-fool theory of bubbles. According to this explanation, agents are willing to pay more for an asset

Federal Reserve Bank of Chicago

than they think it is worth because they anticipate they
might be able to sell it to someone else for an even
higher price. Such explanations have come to be known
as greater-fool theories because they all invariably involve speculative trading in the sense in which I defined
it earlier—that is, traders trade assets because they
expect to profit at the expense of others (who would
be the greater fools) rather than because they expect
mutual gains from trading.5 This feature distinguishes
this explanation of bubbles from some of the explanations based on asset scarcity that feature finitely lived
agents who buy infinitely lived assets. In the latter
case, agents also buy an asset intending to eventually
sell it to someone else for a higher price. But in such
a case, they do not expect to profit at the expense of
those they trade with and would have been willing to
hold on to the asset if they could.
Theories of bubbles based on asset shortages or
risk shifting are straightforward and fairly well understood. By contrast, greater-fool theories of bubbles
raise a host of complications, even though the idea they
represent is simple and resonates with many people.
For example, Edward Chancellor titled his 1999 book
on the history of speculation Devil Take the Hindmost,
alluding to the fact that whoever is the last to be stuck
with the asset ends up losing.6 Academics and nonacademics both refer to investors “riding the bubble”
to evoke the way one might ride up an air bubble in a
champagne flute, letting go right before the bubble
reaches the surface and pops. The problem is that
constructing a model where such bubbles arise can be
daunting. First and foremost, if we assume traders are
rational and understand the underlying environment
they face (as is common in most economic models),
the greater-fool theory may not hang together: The
traders one expects to profit off of would be aware
that others are trying to exploit them and might refuse
to buy the asset. Economists have figured out ways to
get around this problem. But even if we succeed in
getting rational agents to trade an asset in the hope
of profiting at the expense of other traders, it is not
clear whether this asset can be legitimately viewed
as a bubble. As noted earlier, if traders hold different
views about how valuable the asset is, it is not obvious how to define the asset’s fundamental value. Is it
the highest value of dividends any trader in the economy expects the asset to generate? As I discuss later
on, some models—which I shall call asymmetric information models of bubbles—can be understood as
proper models of bubbles. But even in these models,
it is not obvious what appropriate policy should be.
The remainder of this article explores these issues in
more detail.

55

When greater-fool theories are a fool’s errand
A natural starting point for any discussion of
greater-fool theories of bubbles is the work of Tirole
(1982).7 He derived conditions under which greaterfool theories can be definitively ruled out. Thus, any
successful greater-fool theory of bubbles must violate
at least one of the conditions he sets forth. These four
conditions are as follows:
1)	 The number of potential traders is finite.
2) 	All traders are rational, and this is common
knowledge among all traders.
3) 	Traders start out with common prior beliefs
(or “priors,” for short) about the economic
environment they face.
4) 	Resources are allocated efficiently prior to
any trading taking place.
The first condition requires little explanation or
justification. The second condition contains two assumptions. First, traders are assumed to be rational, in the
sense that they process information in accordance with
the laws of logic and probability and then act to maximize
their expected utility. Second, rationality is common
knowledge, implying that traders know that other traders
are rational and that other traders know they themselves
are rational. The third condition holds that all traders
begin with the same initial beliefs about the environment they face (for example, the attributes of the assets
they trade, of the markets they trade in, and so on).
Different traders may subsequently receive different
information that leads them to revise their initial beliefs
and deviate from what others believe. In other words,
the third condition requires only that traders share the
same initial understanding of the environment they face
before receiving any information, not that they receive
the same information or always hold the same views.
The fourth condition implies that individuals have no
reason to trade assets beyond the attempt to profit at the
expense of others, since the initial allocation is efficient
and, thus, there is no other reason to trade.
If these conditions are satisfied, attempting to construct a greater-fool theory of bubbles would be a fool’s
errand: Tirole (1982) proves that these conditions deny
the possibility of a bubble altogether. The formal
proof is contained in Tirole (1982). Here, I provide a
sketch of his argument, which will be useful later for
understanding why “greater-fool bubbles” might arise
in alternative environments. Consider a trader, whom
I will call Alice, who wants to sell her asset to a trader

56

who is not privy to the same information that Alice is
and who might therefore believe the asset is more valuable
than it truly is. Suppose there were such a trader, whom
I will call Bob. Since Bob is rational, he would realize
that the only reason Alice wants to sell him the asset
is that she received information indicating that the
asset is worth less than the price she is offering. Since
Bob knows that he and Alice started out with the same
beliefs, he realizes that if he saw the same information
as Alice did, he would also be convinced that the asset
is worth less than the price she is offering. As a result,
even without seeing the information Alice has, he knows
better than to buy the asset from her. What if Bob had
incontrovertible evidence that the asset is worth more
than the price at which he can buy it from Alice? In
that case, Alice would realize that Bob must have information that she would find compelling, and thus
seeing him eager to buy would cause her to refuse to
sell. Since Alice is rational and knows that other traders
are rational, she would realize that she will not be able
to unload an asset for a price above its true value.
Given this, she would never agree to buy an asset for
more than she believes it is worth. It follows that the
asset can never trade above its fundamental value.8
In short, developing a greater-fool theory of
bubbles requires violating one of the conditions Tirole
set forth. As I shall next discuss, the literature has pursued two approaches to modeling greater-fool theories of bubbles, each of which violates at least one of
Tirole’s conditions.
Fanfare for the uncommon prior
One modification to Tirole’s (1982) setup that has
attracted a great deal of attention is to dispense with
the assumption that traders start out with common
prior beliefs about the environment they face. Indeed,
work that assumes traders have different prior beliefs—
and therefore do not temper their beliefs when they
see others taking different trading positions from their
own—precedes Tirole’s work. Examples include
Miller (1977) and Harrison and Kreps (1978), who
both frame their results in terms of speculative trading
rather than bubbles. That is, both papers are concerned
with whether agents trade expecting to profit at the
expense of others that hold different beliefs, but neither is explicitly concerned with whether asset prices
reflect fundamentals or not.9 Harrison and Kreps do
offer a few brief comments on fundamental valuation
in the conclusion of their paper, which I discuss later.
But it is only more recently, starting with the work of
Scheinkman and Xiong (2003), that models featuring
agents with different prior beliefs have come to be
associated with bubbles.

2Q/2015, Economic Perspectives

Before I discuss whether models featuring heterogeneous prior beliefs among traders can rightly be
viewed as giving rise to bubbles, let me reflect on
why dispensing with the assumption that traders begin with the same priors about their environment may
allow us to avoid Tirole’s (1982) conclusion that rules
out bubbles. Let me trot out Alice and Bob again. If
Alice tries to sell her asset to Bob, then because Bob
is rational, he still realizes that there is no reason
Alice would want to sell him the asset other than that
she believes the asset is worth less than the price she
is offering. But since Bob does not start out with the
same beliefs as Alice, he will not necessarily be convinced by the evidence Alice sees. Indeed, suppose
nobody receives any new information to update their
priors. In that case, Bob would know that Alice is
trading on the basis of her priors, which he does not
agree with. Thus, he might believe the asset is selling
for less than its fundamental value even as Alice believes
the price is above the fundamental value. Alice and
Bob will agree to trade, each believing they are taking
advantage of the other. Note that in getting around
Tirole’s result, I am not abandoning the assumption
that agents are rational. Indeed, I invoke rationality
throughout my analysis. This is worth pointing out,
since models with uncommon priors are sometimes
described as models in which agents are irrational,
even though they need not be.10
To show how dropping the requirement of common priors can lead to scenarios that are suggestive
of greater-fool bubbles, consider the following adaptation of the Harrison and Kreps (1978) model. Suppose there is a single asset, available in fixed supply
that is normalized to 1. Let dt denote the dividend this
asset yields in period t, which corresponds to a single
day. There are two agents, Evelyn and Odelia, who
maintain different beliefs about dividends. In particular,
Evelyn believes the asset yields one unit of consumption goods in even periods and nothing in odd periods:
1 if t is even
Evelyn believes dt = 
1)
0 if t is odd.
Odelia instead believes that the asset yields one unit
of consumption goods in odd periods and nothing in
even periods, that is,
2)

0 if t is even
Odelia believes dt = 
1 if t is odd.

I assume that Evelyn and Odelia maintain these beliefs
regardless of what dividends are actually paid out.
That is, their respective theories about dividends at
different periods are logically independent, so even if
their theories about dividends at some period are ever

Federal Reserve Bank of Chicago

proven wrong—and in each period at least one of their
two theories about dividends that period will always
be proven wrong—neither trader will change or update her expectations about future dividends.11 I could
have replaced the aforementioned beliefs with beliefs
that involve nondegenerate probabilities—for example,
Evelyn believes that dt = 1 in any even period with
probability 1– ε and dt = 0 with probability ε for some
small but positive ε. This way, nobody would ever be
explicitly proven wrong, since each of their theories
would allow both realizations for dividends. But I
assume degenerate beliefs to simplify the exposition.
The essential feature of my example is that Evelyn is
more optimistic about dividends in even periods and
Odelia is more optimistic in odd periods.
I assume the asset in question cannot be sold short—
that is, a trader can sell any units of the asset she already
owns, but she cannot borrow additional units to sell.
Evelyn and Odelia take prices as given when they
trade.12 I also assume Evelyn and Odelia have ample
endowments each period that allow them to purchase
the entire fixed supply of the asset should they wish
to do so. Finally, I assume that both have a utility that
is linear in the amount of consumption goods they
eat, implying both are risk-neutral, and that both discount the future at the same rate β, where 0 < β < 1.
Trade takes place in the morning of each period,
while the dividend is paid out that evening. Consumption goods are not storable, so a trader who buys
the asset in the morning expects to consume any dividends the asset generates that same night.
Consider period 1. On the one hand, because the
date is odd, Evelyn believes that the asset will bear
no dividend today, but that it will bear a dividend one
period from now, three periods from now, five periods
from now, and so on. Hence, she would value the
present discounted dividends from the asset at
β
3)
β + β3 + β5 + … =
.
1 − β2
On the other hand, Odelia believes the asset will pay
a dividend today, two days from now, four days from
now, and so on. Hence, she would value the present
discounted dividends from the asset at
4)

1 + β2 + β4 + … =

1
.
1 − β2

Since 0 < β < 1, Odelia values the dividends paid by
the asset more than Evelyn. In even periods, the two
valuations switch, and Evelyn values the asset at
1
β
, while Odelia values it at
.
2
1− β
1 − β2

57

I will now argue that in equilibrium—when the
supply of the asset equals the demand for it—the price
of the asset at each date t, denoted pt , will exceed
1
, the most either Evelyn or Odelia thinks the
1 − β2
flow of dividends from the asset from that date on is
worth. To see this, observe that if the price of the asset
were ever below

1
at some period t, there would
1 − β2

always be someone (Evelyn if t is even or Odelia if t is
odd) who would want to buy as many units of the asset
as her endowment would allow, since she thinks she can
earn strictly positive profits from buying the asset and
holding it indefinitely. Since I assumed that both traders
have ample resources but that the supply of the asset is
fixed, demand would exceed supply. So the price cannot
1
fall below this level, that is, pt ≥
for all t.
1 − β2
Could the price be equal to

1
, the highest
1 − β2

valuation any agent assigns to a dividend in each
period? Suppose it did. In period 1 (or at any odd
date for that matter), Odelia could contemplate the
following strategy: Buy the asset, anticipate consuming its dividend that evening, and then sell the asset
for a price of at least

 1 
1 + β − β2
1+ β 
=
.
2 
1 − β2
1− β 
But since 0 < β < 1, it follows that
1 + β (1 − β )
1− β

2

>

Hence, if the price were equal to

1
.
1 − β2
1
in period 1,
1 − β2

Odelia would expect to earn strictly positive profits
from buying the asset. She would therefore want to buy
as many units of the asset as her endowment allows,
and demand would exceed supply. The fact that the
price of the asset can never fall below

1
at
1 − β2

1 + β − β2
in
1 − β2

period 1. Of course, by the same logic, it must also be
at least

1 + β − β2
in period 2, or else Evelyn could buy
1 − β2

the asset, earn the dividend in period 2, and then sell
the asset in period 3. And the same argument applies
in subsequent periods (t = 3, 4, 5, and so on), so the
price can never fall below this new level, meaning
pt ≥

1 + β − β2
for all t. Thus, I have established a new
1 − β2

lower bound on prices for all dates—which is higher
than the original bound of

1
that I started with.
1 − β2

I can now repeat the argument: Given the price is
at least

1 + β − β2
at all dates, can it ever equal this
1 − β2

bound at any date? If that were indeed the price in
period 1 (or at any odd date), Odelia could buy the asset
in period 1, consume its dividend that evening, and
then sell the asset for at least

1 + β − β2
in the following
1 − β2

period. Her payoff in that case would be at least

1
in the next period. She
1 − β2

would discount the proceeds of her sale by β, and so
the payoff to this strategy is at least

58

any date implies that it must be at least

 1 + β − β2 
1 + β − β3
1+ β 
.
 =
2
1 − β2
 1− β 
Since 0 < β < 1, it follows that
1 + β − β3
1 + β − β2
>
.
1 − β2
1 − β2
Hence, Odelia would expect to earn strictly positive
profits from this strategy, and so she should buy as many
units of the assets as her endowment allows. To ensure
supply is equal to demand, the price in period 1 must
be at least

1 + β − β3
, the lowest profit Odelia can
1 − β2

earn by holding the asset one period and then selling
it in the next period. Once again, this argument can
be applied to every period, and so I can conclude that
pt ≥

1 + β − β3
for all t, a bound that is higher than
1 − β2

in the previous round.
I can apply this argument repeatedly: Given the
price exceeds a new threshold in every period, I can

2Q/2015, Economic Perspectives

derive a now higher bound on the price for each period.
In particular, on the nth iteration, I will be able to
conclude that pt ≥

1 + β − βn
. Since this holds for
1 − β2

any n, it follows that
5)

pt ≥ lim

n →∞

1 + β − βn
1+ β
1
=
=
.
2
2
1− β
1− β
1− β

The limiting case where pt =

1
for all t turns out
1− β

to be an equilibrium price. To see this, observe that the
expected payoff for Odelia from buying the asset in
period 1, consuming its dividend that evening, and
selling at price

1
the next day is equal to
1− β
 1 
1
1+ β 
.
=
1− β  1− β

What Odelia pays for the asset is thus exactly equal
to the profit Odelia would earn from buying and selling the asset. One can show that buying and selling
after one period is the best Odelia can do—that is,
holding the asset for longer and then selling it will be
less profitable. Hence, Odelia is just indifferent between buying the asset in period 1 and not buying it
at all. Evelyn, by contrast, wants to sell the asset (and
would even sell it short if she could), since she believes the asset will yield no dividend that evening.
Hence, supply and demand for the asset can be equal.
At the equilibrium price path, Evelyn sells all her asset
holdings to Odelia in odd periods, and Odelia sells all
her asset holdings to Evelyn in even periods. I can appeal
to arguments in Harrison and Kreps (1978) to argue that
with some additional assumptions, pt =

1
for each t
1− β

is the only possible equilibrium price path for the asset.13
To recap, the equilibrium price of the asset exceeds
what either Evelyn or Odelia believes the asset can
generate in dividends. Some have argued that this implies the asset in my example should be viewed as a
bubble. Specifically, they argue that when agents have
different beliefs, a bubble should be defined as follows.
First, define a fundamental value for each individual
as what that individual expects the cash flow from the
asset to be or, alternatively, how much each individual
would value holding the asset indefinitely and consuming
its dividends. Then define an asset to be a bubble if its
price exceeds every individual’s fundamental value.
Note that when the distribution of dividends is known

Federal Reserve Bank of Chicago

so that all agents have the same beliefs, this definition
reverts to the original definition of a bubble for the
case where the distribution of dividends is known by
all agents. Hence, this definition extends the definition
of a bubble for a known distribution for dividends to
the case where the distribution of dividends is not
known and agents can hold different beliefs.
The equilibrium I have just constructed would
thus seem to provide an internally consistent model
of a greater-fool bubble. The market clearing price
for the asset is higher than anyone in the economy believes dividends are worth. Nevertheless, traders buy
the asset at this price, precisely because they expect
to sell it later to someone who values the asset even
more than they do. The fact that the equilibrium price
is constant rather than growing may make this seem
like an unusual model of a bubble, since most historical episodes suspected to be bubbles feature rapid
price appreciation. But the price is constant because
the dividends in my example are constant over
time—an assumption I imposed for convenience. The
model can be readily modified to allow for dividend
growth in a way that would introduce price booms
and busts without changing its key features.14 Still, as
I next explain, it is not obvious that this model should
be interpreted as a model of a bubble, since alternative ways of extending the definition of fundamental
value to the case where traders hold different beliefs
do not imply the asset is overvalued.
Is it a bubble?
To illustrate the complications for interpreting the
previous example as a bubble, I consider the following
related example of an economy with two types of goods—
say, apples and bananas. As before, there are two
people in the economy—I’ll again call them Evelyn
and Odelia—each of whom is endowed with an ample
amount of apples each period. Evelyn and Odelia have
the same beliefs, but now their preferences differ.
Evelyn enjoys bananas on only even days, when she
derives the same pleasure from one apple as she does
from one banana. On odd days, Evelyn derives no utility
from bananas. Odelia’s tastes are the exact opposite:
She enjoys bananas on only odd days, deriving the
same utility from a banana as from an apple. On even
days, Odelia derives no utility from bananas. There is
no uncertainty, and both Evelyn and Odelia discount
at the same rate β ∈ (0, 1).
Suppose this economy had no bananas initially,
and I contemplated introducing a banana tree that bears
one banana each day. How much would this tree be
worth in terms of apples? Consider first the perspective
of an outsider who shares the same discount rate β as

59

that of Evelyn and Odelia. The outsider could sell the
tree’s yield of one banana each day in exchange for one
apple. On even days, he would sell the banana to
Evelyn, while on odd days, he would sell it to Odelia.
Hence, the present discounted value of the tree for the
outsider as measured in apples is just
1
6) 1 + β + β2 + β3 +… =
.
1− β
The same would be true if I considered the perspective
of either Evelyn or Odelia. For example, if I asked
Odelia to value the tree, she would reason that on odd
days she could consume the banana directly, which
she values the same as an apple, while on even days
she could sell a banana to Evelyn in exchange for an
apple. Thus, she would value the tree as the present
discounted value of receiving an apple each day. The
same would be true if I asked Evelyn. Since there is
no uncertainty, the usual definition of the fundamental
value of an asset would imply the banana tree is worth
1
apples.
1− β
Now, what would happen if I precluded people
from selling bananas but still let them buy banana trees?
That is, I would shut down the market for the dividends generated by the asset, but not the market for
the asset. This restriction precludes the arrangements
that I used to argue the tree is worth

1
apples.
1− β

However, Evelyn and Odelia could still achieve the
same allocation as with a market for bananas by trading
the banana tree in such a way that ensures the person
who values bananas owns the tree when it yields fruit.
That is, Evelyn will buy the tree the morning of each
even date, consume the banana that evening, and then
sell the tree to Odelia the following morning. Since
the allocation is the same as before, the value of the
tree should be unchanged—that is, it should still be
1
. One can verify that the equilibrium price of the
1− β
1
tree would still equal
each period.
1− β
Now, suppose I asked Odelia and Evelyn how
much they would value the tree if they couldn’t sell it
and had to consume its bananas themselves. Since
Odelia enjoys bananas in only odd periods, if I asked
her valuation in period 1, she would say she values
owning the tree indefinitely at
7)

60

1 + β2 + β4 + ... =

1
.
1 − β2

Evelyn, who enjoys bananas in only even periods, would
say that owning the tree in period 1 and consuming
its bananas is worth
8)

β + β3 + β5 + ... =

β
.
1 − β2

Thus, the asset trades at a price above what Evelyn or
Odelia thinks it is worth if either had to consume its
yield on her own. Nevertheless, Evelyn or Odelia agrees
to buy the asset at this price because each agent intends
to sell the tree at a price that exceeds the value of
consuming its fruit herself.
The connection between this example and the
case with traders holding heterogeneous beliefs should
hopefully be clear. The two have the same underlying
structure: Each period, there is one person who values
the dividend of the asset at 1, while the other values
the dividend at 0. In the case where traders hold heterogeneous beliefs, this difference in valuation occurs
because one of the traders believes a dividend will be
paid out that period and the other doesn’t. In the case
where traders have heterogeneous preferences, this
difference in valuation occurs because one of the traders
values the good, while the other doesn’t. In both cases,
the trader who doesn’t value the dividend that accrues
that evening sells the asset to the trader who does. The
price of the asset is the same in both cases. Given
traders with heterogeneous preferences, it seems clear
that the asset is trading at its fundamental value, even
though it exceeds the value each trader assigns to
owning the tree forever and consuming its dividends.
Why shouldn’t one say the same when agents have
different beliefs, rather than different tastes?
Comparing the two examples reveals a shortcoming
with defining a bubble as an asset whose price exceeds
each individual’s fundamental value or, in other words,
as an asset whose price exceeds what each individual
is willing to pay to consume the asset’s dividends indefinitely. This is most readily apparent when there is
an explicit market for dividends—for example, when
individuals can sell bananas as opposed to just banana
trees. In that case, all agents agree that the value of
owning the tree indefinitely is

1
, because any
1− β

agent who owns the asset can sell its dividends to
those who value them most. In the case where traders
hold heterogeneous beliefs, I implicitly ruled out this
possibility by not allowing a market for dividends that
was analogous to a market for bananas. Without such
a market, agents are forced to trade the asset to achieve
the same outcome, making it seem as if trading the
asset makes it more valuable. But the same value can

2Q/2015, Economic Perspectives

be achieved without ever transferring ownership of
the asset. Indeed, the same outcome could be achieved
by introducing a rental market for the asset. Just as
capital equipment can be rented out to others who can
keep the cash flow they generate using it, an agent
who owns a financial asset can in principle rent it out
for a period and let whoever rents the asset accrue its
dividends. While rental markets for financial assets
may seem odd, they do have historical precedents.
Velde (2013) describes rental markets for government
lottery bonds in eighteenth-century England. Lottery
bonds were structured so that the interest payments on
any particular bond were random, and on any given
day there was some chance a particular bond would
be drawn and receive a prize interest payment. At the
time, individuals could rent a lottery bond for a day
and earn the associated payout if the bond happened
to be drawn that day. These arrangements were known
as “horses,” and their prices were published regularly.15
In my example where traders had heterogeneous
beliefs, if I introduced the possibility of renting out the
asset, both Evelyn and Odelia would value owning
1
the asset indefinitely at
. It therefore seems rea1− β
1
sonable to view
as the fundamental value of the
1− β
asset. That is, in the special case where agents agree on
the value of an asset while they hold different beliefs,
it would seem natural to define the fundamental value
of the asset as this common value—namely, what any
agent could earn from buying the asset and holding it
indefinitely, but with the option of renting it out in any
period. To see further why this is a reasonable definition
for the fundamental value of the asset, consider a benevolent social planner who contemplates creating another
asset on behalf of agents in this economy. The planner
would value the asset in terms of the total surplus that
could be created by promising its dividends at different
dates to different traders. That is, the planner could
collect

1
β
1
+
=
from the two agents to
2
2
1− β
1− β
1− β

create another asset by promising to give any future
dividends that accrue in even periods to Evelyn and any
future dividends that accrue in odd periods to Odelia.
This suggests

1
accurately reflects the value to
1− β

society from creating another asset, which is what the
notion of a fundamental value is meant to capture.
Harrison and Kreps (1978) offer a similar interpretation,
writing in the conclusion to their paper that the equilibrium

Federal Reserve Bank of Chicago

price they derive “is consistent with the fundamentalist
spirit, tempered by a subjectivist view of probability.”
Why, then, have some argued for treating the asset
as a bubble if its price exceeds how much each agent
values holding the asset indefinitely and consuming its
dividends? Undeniably, the example in which Evelyn
and Odelia hold different beliefs contains features that
make it reminiscent of a bubble. For example, Evelyn
and Odelia both agree that the asset pays dividends
only every other period, although they disagree as to
the periods in which these dividends will be paid out.
Isn’t a price that is equivalent to the asset paying out
a dividend every period too high given neither agent
believes this to be the case? This characterization of
beliefs, however, is misleading. If Evelyn and Odelia
disagreed about only when dividends are paid out,
Odelia would not expect to sell the asset to Evelyn
after consuming its dividends for a price of

1
,
1− β

since she knows Evelyn would realize she was wrong.
Rather, the price of

1
emerges because Evelyn and
1− β

Odelia continue to believe dividends will be paid out
in different periods regardless of what happened in the
past, which is perfectly rational if dividends in different
periods are determined through logically independent
processes. The price of

1
can be rationalized using
1− β

the most optimistic beliefs any trader holds about
dividends in each period. In other words, an outsider
who could rely on only Evelyn’s and Odelia’s beliefs
would be unable to rule out the possibility that dividends will actually be paid out each period, since for
each period he can find a logical theory advanced by
either Evelyn or Odelia that implies a dividend will
be paid out.
Still, the notion that the price can be supported by
always appealing to the most optimistic beliefs about
dividends may seem suspect. Isn’t it implausible that it
is always the most optimistic traders who are correct?
Depending on how agents form their beliefs, it may indeed be implausible to always rely on the most optimistic
agents to determine the fundamental price of the asset.
For example, Scheinkman and Xiong (2003) assume
agents receive signals about dividends but attribute too
much precision to their signals. This implies that the
traders who are the most optimistic tend to also be excessively optimistic. But this does not mean that the
reason assets are overvalued is because individuals have
different beliefs. Even when traders hold the same

61

beliefs, they might still be overconfident about the
signals they receive. The reason Scheinkman and Xiong
are correct to call the asset in their model a bubble is
because they drop Tirole’s second condition, which holds
that traders are rational and process information correctly,
rather than his third condition, which holds that they
have common prior beliefs. Without any information
about how traders form their beliefs, there is no reason
to dismiss the beliefs of the most optimistic agents any
more than those of other agents. Relying on the most
optimistic beliefs corresponds to the usual notion of
maximum willingness to pay that economists routinely
rely on to determine how resources should be allocated.
Finally, models in which agents hold heterogeneous
beliefs, as in the example I’ve constructed, imply traders
who buy and sell the asset expect to profit at the expense
of others they think value the asset incorrectly. This
feature makes my example a good model of a greaterfool theory, but not necessarily a model of a bubble.
In other words, this feature makes the example a good
model of speculative trading as opposed to a good model
of an asset that is overvalued. Indeed, what is striking
about the example is that even though traders disagree
about dividends, they can agree on what the asset is
worth. In particular, both traders in my example view
the asset as worth

1
if given the option to rent it out,
1− β

and would view accepting any price for the asset below
this one as a bad trade. The fact that traders believe
others are fools does not necessarily imply that they
must think the asset is overvalued.
Disagreement on valuation
In my example in which Evelyn and Odelia had
different beliefs, both valued holding the asset indefinitely
equally provided they could rent out the asset. This
equality in valuation is due to a particular feature of this
example—namely, that traders agree about the distribution of the most optimistic valuation for dividends
in every period. This feature can arise in other environments. For example, Scheinkman (2014) presents a
model in which beliefs are independent across time.
Specifically, Scheinkman assumes two types of traders.
Type A traders believe the dividend in each period is
equally likely to be 0 or 1. Type B traders, independent
of their beliefs in other periods, will with probability
1 – 2q share the same beliefs that type A traders hold;
but with probability q, type B traders believe the dividend that period will be 1, and with probability q, these
same type B traders believe the dividend will be 0. In
this case, type A and type B agents still agree about
the expected value from holding the asset indefinitely
given they both have the option to rent it out.

62

More generally, though, traders might disagree about
the distribution of the most optimistic beliefs in future
periods. In that case, they will disagree about the value
of holding the asset indefinitely. Indeed, this is true in
both the Harrison and Kreps (1978) model and the
Scheinkman and Xiong (2003) model. To illustrate this
possibility, I consider the following example. Suppose
there is a single asset that pays one dividend in period 4,
which can assume one of four values, that is,
9)	 d4 ∈ {0, 1, 2, 3}.

To motivate this example, suppose d4 represents the profits
of an agricultural company that plants three trees, each
of which can yield a harvest of at most 1. If a tree
bears fruit, it will do so in period 4. However, it will
be possible to tell whether some trees will bear fruit
before period 4. In particular, whether the first tree will
bear fruit is revealed in period 2, whether the second
tree will bear fruit is revealed in period 3, and whether
the third tree will bear fruit is revealed in period 4, at
the time of the harvest.16
There are two traders, Alice and Bob, who can
trade shares in the agricultural company as news about
the trees is revealed. Neither discount consumption
over time. Alice’s beliefs can be summarized as follows:
1) 	 Unless given evidence to the contrary, Alice
believes that with probability 0.9, all three
trees will bear fruit and that with probability
0.1, none of the trees will bear fruit.
2) 	 If just one of the first two trees bears fruit,
Alice believes that the third tree will bear fruit.
Bob’s beliefs can be summarized as follows:
1) 	 Unless given evidence to the contrary,
Bob believes that only the second tree
will bear fruit.
2) 	 If neither of the first two trees bears fruit,
Bob believes the third tree will bear fruit.
3) 	 If the first tree bears fruit, Bob believes no
other trees will bear fruit.
4) 	 If both of the first two trees bear fruit, Bob
believes that the third tree will not bear fruit.
These conditions fully describe what Alice and Bob
believe depending on what they know at the beginning
of each period. Figure 1 shows the same information

2Q/2015, Economic Perspectives

FIGURE 1

Alice’s and Bob’s beliefs given the number of good trees in each period
3

3

1
1

2

2
2

1
0.9

1

1

1

0.1

1
1

0
1

1
2

1

1
1

1
0

1
1

0

0
1
0

Number of
good trees by
period 2

Number of
good trees by
period 3

Number of
good trees by
period 4

Alice’s beliefs

graphically, with the numbers in green indicating the
probability each trader assigns to what will happen at
each node in the information tree.
Given these beliefs, consider how Alice and Bob
value owning the agricultural company in period 1,
before the status of any tree is revealed. First, consider
the value of consuming the dividend—that is, ignoring
the possibility of renting out the asset while maintaining
ownership of it. In that case, Alice values dividends
at the following units of consumption:
10)	 0.9 × 3 + 0.1 × 0 = 2.7.
Bob instead values dividends at 1 unit, fully expecting
only the second tree to bear fruit.
However, I argued earlier that the value of owning
an asset indefinitely should incorporate the value of
renting out the asset to those who have the most optimistic beliefs. In this example, the asset makes a single
dividend payment in period 4. Whoever owns the asset
in period 4 can thus rent it out before the status of the
last tree is revealed. In this case, Alice understands that
if the first two trees do not yield any fruit, Bob will still
believe that third tree will yield fruit, and she will be
able to rent her share to him for 1 unit. Hence, she
values the asset at
11)	 0.9 × 3 + 0.1 × 1 = 2.8.
Bob instead expects that only the second tree will bear
fruit, at which point Alice will still expect the third
tree to bear fruit. Hence, he can count on renting the
asset to Alice rather than consuming the fruit it yields,

Federal Reserve Bank of Chicago

0
Number of
good trees by
period 2

Number of
good trees by
period 3

Number of
good trees by
period 4

Bob’s beliefs

and so he would value owning the asset but being
able to rent it out to the highest bidder at 2 units
of consumption.
Alice and Bob now disagree as to the value of
owning the asset indefinitely, even when given the
option to rent it out. How then should the fundamental
value of the asset be defined in this case? One possibility is to define the fundamental value of the asset
as the most any agent would pay at any given date for
the right to own the asset indefinitely but still rent it
out. In this case, the value in period 1 would be 2.8,
the amount Alice thinks the asset is worth. I next show
that the equilibrium price of the asset in period 1 will
exceed this value, so according to this definition the
asset should be viewed as a bubble.
Since Alice values the asset more than Bob in
period 1, she will outbid him and own the asset at that
point in time. If the first tree turns out not to bear any
fruit, though, she could sell it to Bob. Recall that in
this case Bob will believe that the second tree will bear
fruit and that he will then be able to rent the asset to
Alice for 1 unit, so Bob would value the asset at 2
units. By selling the asset to Bob if the first tree does
not bear any fruit, Alice would guarantee herself an
expected payoff of
12)	 0.9 × 3 + 0.1 × 2 = 2.9.
Hence, if the price of the asset was only 2.8, Alice
would want to buy infinitely many units of the asset.
The only way to ensure she demands finitely many
units of the asset is if the price is 2.9. In this case, the
asset is more valuable to Alice precisely because she

63

can transfer the asset to Bob, something that cannot be
replicated by simply renting the asset to him. Essentially,
if the first tree does not bear any fruit, Alice and Bob
disagree in period 2 about what Alice will believe in
period 4; Bob expects she will think the asset is worth 2,
while Alice expects she will believe the asset is worth 1.
As a result, in period 2, Bob thinks that owning the
asset and later renting it out is more valuable than
Alice does. The only way for Alice to profit from Bob’s
beliefs is by selling him the right to rent out the asset
in the future.
Should this example be viewed as a bubble?
The price exceeds what any trader believes holding
the asset indefinitely is worth (even with the option
to rent out the asset). However, once again, care must
be taken in how the fundamental value of the asset is
defined. Suppose that agents have different initial
beliefs and that tomorrow all agents might receive
news that reveals the asset’s dividends will likely be
higher. If the news comes, since traders have different
beliefs, not all of them will process this information
in the same way. Some traders may not update their
beliefs. However, these traders should still recognize
that the asset would be more valuable now for society
as a whole, and incorporate the impact of the news
on the beliefs of others when assessing the fundamental value of the asset. Just as an agent would naturally
take into account that tomorrow he might receive news
that affects his beliefs about dividends when valuing
the asset today, he should also take into account that
tomorrow others might receive news that affects their
beliefs, even if it doesn’t affect his own beliefs, when
making his valuation today. For the example in
figure 1, to determine the value of the asset, it is
essential to know what Alice will believe about the
dividend in period 4 given she might be the one who
consumes this dividend. But agents disagree about
what Alice will believe then. Without any additional
information on how agents form their beliefs, there
is no reason to suppose Alice knows what her beliefs
will be better than others. By the same logic that I
described in the previous examples, it would seem
natural to rely on the most optimistic beliefs about
what Alice will believe, rather than on Alice’s own
beliefs about what she will believe, to determine the
value of the asset.
This logic suggests that the definition of a bubble
in environments where agents disagree about future
beliefs should not be based on using the same individual’s
beliefs at all dates—for the same reason as in my earlier example where Evelyn and Odelia held different
beliefs. In finite-horizon settings, one could determine
which agent holds the most optimistic beliefs at each

64

possible state of the world on the final date in which
the asset yields a dividend. Working backward, one
could evaluate at each prior node the agent who holds
the most optimistic beliefs when that agent’s future
beliefs are substituted with the beliefs of the trader who
was already determined to be the most optimistic at
future nodes. Because this is notationally cumbersome,
I omit the formal details. This approach is faithful to
Harrison and Kreps’s (1978) observation that the equilibrium price in a model where traders hold different
priors has a fundamentalist spirit, tempered by subjective beliefs. Moreover, when agents have the same
beliefs, this construction would revert to the usual
definition of fundamental value when the distribution
of dividends is known. The fact that agents disagree
on the value of holding the asset indefinitely, even
with the option to rent it out, doesn’t automatically
imply that the asset should be considered a bubble.
What does this all mean for policy?
So far, I have argued that models in which traders
have different prior beliefs exhibit speculative trading,
but there are good reasons not to view them necessarily
as models of bubbles in which the underlying asset is
overvalued. However, the relevant question is arguably
not whether traders holding heterogeneous beliefs give
rise to a bubble per se, but whether the possibility of
traders holding heterogeneous beliefs can somehow
justify policy intervention. That is, irrespective of
whether these models give rise to a bubble or not, do
they imply that policymakers should discourage agents
from speculative trading, meaning trading with the aim
of profiting at the expense of others rather than to
achieve mutual gains? After all, the definition of a
bubble is sufficiently difficult to apply in practice that
the more relevant question may be whether policymakers
should act to curb speculation. Unfortunately, the social
welfare analysis of models where agents hold heterogeneous beliefs turns out to also be ambiguous.
As a starting point, consider the case where the
asset is in fixed supply. This avoids the question of
whether speculation results in too high of a price that
would encourage agents to create too many units of the
asset. Instead, the relevant policy question is whether
people should be allowed to trade on the basis of different
opinions. The similarity between trade among agents
with heterogeneous beliefs and trade among agents
with heterogeneous preferences in my earlier examples
suggests traders should be allowed to enter into such
trades. In both of my examples, Evelyn and Odelia
wanted to trade. In one case, Evelyn preferred owning
the asset to her endowment of consumption goods and
Odelia preferred the opposite, while in the other case

2Q/2015, Economic Perspectives

Evelyn preferred an apple to a banana and Odelia
preferred the opposite. It is true that in the former
case, the agents’ willingness to trade is premised on
mutually inconsistent beliefs. However, when agents
have inconsistent beliefs, they are aware that their
beliefs are incompatible and still wish to trade. Absent
additional information on why they disagree, why should
they be denied their mutual desire to trade?
Several economists have argued that policymakers
should deny people the right to trade on the basis of
heterogeneous beliefs. Most of these offer variations
on a similar point: When two agents trade on the basis
of heterogeneous preferences, like trading an apple
for a banana, they will not regret their trade ex post.
By contrast, when two agents trade on the basis of
heterogeneous beliefs, at least one of them is bound
to be proven wrong and regret making the trade. For
example, in discussing trades based on differences in
beliefs, Stiglitz (1989, p. 106) argues that “impeding
trade is (Pareto) inefficient when viewed from the perspective of their ex ante expectations”17 and “impeding
trade may actually improve social welfare when viewed
from the perspective of their ex-post realizations.” He
likens this to a parent who forces his child to study in
a way the child will appreciate later. Mongin (2005)
discusses the notion of spurious unanimity in which
all individuals agree to take the same action but only
because they believe the action will result in different
outcomes that they value differently. He offers an example in which a majority of people in a country support
building a bridge to a neighboring country—some of
them because they believe it will lead to a massive
inflow of people who will revitalize the local economy
and others because they believe unwanted newcomers
will largely stay away, while locals will be able to travel
outside. It is easy to construct examples in which,
regardless of which hypothesis proves to be correct,
a majority of people will oppose building the bridge
knowing this hypothesis to be true. Trade between agents
with heterogeneous beliefs has a similar flavor: The
two parties do not receive mutual gains from trade
but are engaged in a zero-sum game where each one
is expecting to benefit at the expense of the other. Since
both cannot be correct, one of them is bound to regret
having entered into the trade.
A related but distinct argument is laid out in Blume
et al. (2014) and Brunnermeier, Simsek, and Xiong
(2014). They consider the case where agents hold
different beliefs but where trade is costly because it
introduces consumption volatility.18 To appreciate their
argument, suppose individuals are risk averse. Start
with the case where all individuals have the same
fixed endowment, so that they don’t need to face any

Federal Reserve Bank of Chicago

consumption risk. However, if they enter a bet based
on their different prior beliefs, they will each be convinced they are correct and expect to gain from the
bet. Of course, only one of them will win the bet. Thus,
they expose themselves to unnecessary consumption
volatility. If the individuals were told in advance, before they ever form beliefs, that they will be exposed
to a risk that their levels of consumption will be volatile and perfectly negatively correlated with one another,
they would prefer to enter into an ex-ante insurance
arrangement to guard against this risk. Here, the risk
involves the two of them forming different priors that
can’t both be correct. Insuring against this possibility
is equivalent to agreeing in advance not to trade on
the basis of heterogeneous beliefs. Thus, trade on the
basis of heterogeneous beliefs in this case is not a zerosum game, but a negative-sum game, and the parties
may be better off ex ante if they could commit not to
enter into such trades.
Guided by these observations, Brunnermeier,
Simsek, and Xiong (2014) propose a notion of beliefneutral Pareto improvement:19 According to their
definition, allocation A is said to improve on allocation B if, given a particular set S of probability distributions, it can be verified that no agents are worse off
under allocation A and some are strictly better off than
under allocation B when all agents’ expected utilities
are evaluated at each of the probability distributions
in the set S instead of by what individuals actually
believe. In particular, everyone must be no worse off
and some strictly better off under allocation A when
expected utility is computed using each of the agent’s
beliefs, as well as any mixture of the beliefs of the
different agents.20 According to this criterion, for riskaverse agents with equal endowments but different
beliefs, betting with each other is dominated by a
policy that precludes them from betting with each other.
This suggests policymakers might want to disallow
such trades.
Brunnermeier, Simsek, and Xiong (2014) go on
to argue that when the supply of the asset is variable
rather than fixed, there can be an additional social
cost from allowing agents to trade: If the price of the
asset exceeds its fundamental value, too many resources
will be allocated to creating this asset. This argument
of course presumes that agents with heterogeneous
beliefs who are allowed to trade lead to bubbles. As I
discussed earlier, it is not obvious whether the asset
should be viewed as a bubble if all that is known is
that agents differ in beliefs. Brunnermeier, Simsek,
and Xiong (2014) recognize this, and argue that their
analysis only applies when beliefs are distorted. In
this case, the notion that speculation encourages an

65

oversupply of bubble assets does not rely on the fact
that people have different beliefs; rather, that notion
relies on some people having distorted beliefs.
Although these arguments for how to treat trade
based on agents’ heterogeneous beliefs have their merits,
it is safe to say that their implications for policy remain
controversial. Stiglitz’s (1989) analogy to paternalism
is imperfect, since parental intervention is typically defended on the grounds that children are unable to reason
or comprehend the consequences of their actions. Far
fewer would argue that parents should continue to intervene in their children’s decisions when their children
are adults. If the agents who hold different beliefs are
rational in the sense of reasoning based on logic and
probability, a planner who argues they shouldn’t be
allowed to trade because their beliefs are incompatible
would not be telling them anything they don’t already
realize. The fact that they are nevertheless willing to
trade substantially weakens the case for intervention.
As for the argument about ex-post regret, if beliefs
correspond to nondegenerate probability distributions
about events that are rarely replicated, individuals may
never learn whether they were correct or not. Traders
may simply chalk up their losses to bad luck, in the same
way that a risk-averse agent will understand the fact
that a calamity didn’t happen does not mean it was a
mistake to buy insurance. Finally, even if some agents
come to regret entering into trades, those whose beliefs
were correct will not regret entering into the same trades.
Protecting those who will be proven wrong from trading
does not amount to making everyone at least as well
off as when they are allowed to trade, so the usual
Pareto improvement argument for policy does not apply.
Gilboa, Samuelson, and Schmeidler (2014) argue for
a compromise of sorts by introducing the notion of
no-betting Pareto improvement, a refinement on the
usual notion of Pareto improvement. Under their notion,
an allocation is viewed as superior not only if all agents
prefer it to an alternative allocation, but also if there
exist some common beliefs—which may be different
from the beliefs that any agent holds—such that if all
agents maintain these beliefs, all agents are no worse
off and some are strictly better off than under the alternative. By this logic, allowing agents to trade on the
basis of their heterogeneous beliefs will not be viewed
as a Pareto improvement, but preventing them from trading
will not be viewed as a Pareto improvement either.
Perhaps the best case for preventing agents with
different beliefs from trading is the scenario emphasized in Blume et al. (2014) and Brunnermeier, Simsek,
and Xiong (2014) in which trade is a negative-sum game.
In this case, society may be better off with restrictions
that prevent such trading from taking place before

66

knowing what beliefs any agent might have. This notion may be in line with the emerging view on bubbles
in the wake of the Great Recession: The apparent bubble
in housing might have left some better off (for example, homeowners and developers who sold houses in
the years leading up to the recession and traders such
as those profiled in Michael Lewis’s book The Big Short
who managed to short housing21) and some worse off
(for example, those who bought housing or invested in
mortgages just before the recession hit); however, on
the whole, society was worse off because of misallocated
resources (for example, excess housing and workers
whose skills were specific to housing-related activities) that might have depressed subsequent economic
activity. Posner and Weyl (2013) are the most forceful in making this case. But there are two important
caveats that make this policy prescription difficult to
implement in practice. First, the extent to which the
investment in an asset (whether it be housing in the
mid-2000s or dot-com ventures in the late 1990s or
railroads in the 1800s or tulips in the Netherlands in
the seventeenth century) is excessive ex ante, before
we know how things turn out, hinges not on agents
holding different beliefs but on them holding beliefs
we know to be distorted. Many would balk at the notion that policymakers can judge when agents hold
distorted beliefs and whether agents’ beliefs are correct.
Referring to models featuring agents with heterogeneous
beliefs as models of bubbles can be misleading in that
regard, since evidence that people hold different beliefs
does not prove that their beliefs are distorted. And yet,
distorted beliefs, rather than heterogeneous beliefs, are
what imply asset prices are too high. Second, since
agents are eager to trade, there is strong incentive for
agents to claim they are trading because of fundamental
reasons rather than because of differences in their beliefs.
Indeed, the response to financial reform in the wake of
the financial crisis suggests market participants have
actively sought to evade restrictions on when they can
trade. Cochrane (2014) makes a similar argument.
An alternative approach: Asymmetric
information
I now turn to the other approach for modeling
greater-fool theories of bubbles. For lack of a consensus term, I will refer to these as asymmetric information
models. This is because a key feature of these theories
is that agents receive private information other agents
may not be privy to. In particular, they may receive
information that all agents would agree establishes that
the asset is overvalued. However, since agents are unsure
what other agents know, they might still buy the asset
in the hope of selling it to a less informed agent. Thus,

2Q/2015, Economic Perspectives

agents engage in speculative trading because they hope
other agents are not privy to the same information as
they have, rather than because they think other agents
disagree with them. Loosely speaking, they do not expect to profit off of those with whom they trade because
their counterparties hold different views, but because
their counterparties are less informed. Of course, even
agents who begin with different priors may receive
asymmetric information. Indeed, an unfortunate source
of confusion is that some of the papers on bubbles that
feature asymmetric information assume that traders
have different prior beliefs, obscuring the differences
between these two approaches.
In discussing these asymmetric information models,
I find it once again natural to begin with the analysis
in Tirole (1982). Recall that his setup allowed individuals to obtain heterogeneous information. To allow
for the possibility of bubbles, then, one of Tirole’s four
conditions must be violated. Theories based on asymmetric information essentially drop the requirement that
resources be allocated efficiently before any trades take
place (that is, Tirole’s fourth condition). If this condition is dropped, then if some trader named Carol offers
to sell an asset to some other trader named Ted at what
seems to him like a good price, he will not be able to
conclude whether the offer has been made because
they can both gain from trade or because Carol received
information that the asset is worth less than Ted believes
it to be. For example, Carol may have immediate liquidity
needs and is willing to sell the asset at a price that Ted
thinks is a good value. Or Carol may have different
hedging needs than Ted, and so both of them will be
better off trading, since Carol can then go and purchase
another asset that better suits her needs. Ted will of
course still be cautious, knowing Carol might have received private information that the asset is not as valuable
as he believes and might now be taking advantage of
him. But because there is some possibility of gains from
the trade, he need not refuse to trade altogether. Note
that, as I have essentially already shown, one reason
agents may want to trade is that they have different prior
beliefs. Hence, one way to violate Tirole’s fourth condition (which holds that there is no reason to trade because resources are already allocated efficiently) is to
relax his third condition (which requires that agents have
common priors). The first papers to construct asymmetric
information models of bubbles did in fact just that,
since it is relatively easy to analyze models where agents
trade because they have different priors. But these
models differ in important respects from the models
that rely on different priors that I discussed earlier.
The main difference is that with asymmetric information, one can generate bubbles, rather than just

Federal Reserve Bank of Chicago

speculative trading, when all traders are rational. Recall
that speculative trading implies traders expect to profit
at the expense of others. This requires them to have
different beliefs from others—or else those whom they
expect to profit from would refuse to trade. For speculative trading to be sustained, it does not matter whether
agents hold different beliefs because they started with
different priors or because they receive different information. But the exact reason why agents hold different
beliefs does matter for whether one can view an asset
as being overvalued. Researchers initially working on
asymmetric information in asset markets ignored the
question of bubbles, focusing only on the possibility
of speculation. For example, Grossman and Stiglitz
(1980) were interested in whether agents with common
initial beliefs would ever engage in speculative trading.
As Grossman (1976) observed, rational agents will try
to infer what information others observed given the
price at which assets trade, and the market-clearing
price can reveal enough information so that all agents
are equally informed. In this case, there is no scope
for speculation. Grossman and Stiglitz (1980) instead
introduced into these models “noise traders”—traders
whose trades are motivated by some consideration
other than profit maximization, such as liquidity. As
their name suggests, a key feature of noise traders is
that their impact is random, which makes the price of
the asset a noisy signal among all the aggregate information agents receive.22 But the presence of these traders
can also be understood as a way of getting around the
Tirole (1982) results that deny the possibility of speculation. Grossman and Stiglitz (1980) did not go on
to show that this structure can also get around Tirole’s
(1982) results that deny the possibility of a bubble;
this was established only in subsequent work, which
involved an explicit dynamic setting that was missing
from previous work on speculative trading.
Bubbles and asymmetric information
The first researchers to show that asymmetric information models can give rise to asset bubbles were Allen,
Morris, and Postlewaite (1993). Their analysis was
subsequently sharpened and refined by Conlon (2004),
who developed a different setup and showed that some
of the features of their model were not essential for
their results. Although these models are too involved
to reproduce here in detail, the basic insight from
these papers can be understood as follows.
Suppose that in some state of the world there was
information that the price of the asset exceeded the true
discounted value of its dividends. For example, Allen,
Morris, and Postlewaite (1993) and Conlon (2004)
consider a situation in which there is information that

67

the asset pays no dividend, so its fundamental value
is zero, and yet the equilibrium price of the asset is
positive. Suppose that in these states of the world,
every trader receives the information that indicates
the dividend is zero.23 Although each agent knows
this information, none of the agents know what other
information the other traders have. In particular,
consider a setting where after observing the information that dividends are zero, each trader believes two
scenarios are possible:
	

1) 	 All traders know the dividend is zero, so at 		
any positive price, the asset is overvalued.

	

2) 	 There are some traders who still believe that 	
at a positive price, the asset is worth buying.

That is, the situation under consideration is one
in which reality corresponds to the first case, but no
agent is sure whether the truth is the first or the second
case. Given this uncertainty, a trader (again, call her
Carol) can take a gamble and buy the asset at a positive price with the aim of selling it after one period.
From Carol’s perspective, if it turns out the truth is the
first case, she will incur a loss, since she will be unable to sell the asset given all traders know the asset
is worthless. If it turns out the truth is the second case,
she will be able to sell the asset at a positive price to
some other trader (again, call him Ted). This trade will
only be profitable if the price rises between when Carol
buys it and when she sells it to Ted; the aforementioned
papers design an environment in which the equilibrium
price rises over time by the requisite amount.
The reason a trader would be willing to buy the
asset in the second case is precisely because of the
possibility of gains from trade related to dropping
Tirole’s fourth condition, which holds that there is no
reason for agents to trade. That is, in the state of the
world where Carol knows the asset is worthless but Ted
does not, Ted understands that Carol may be selling
him an overvalued asset. But Ted cannot distinguish
that state from other states of the world in which Carol
would offer to sell him an asset at the exact same price
but in which there are mutual gains from trade—for
example, because she has a need for liquidity and
would be willing to sell the asset for even less than
the expected value of its dividends. In short, Carol is
willing to buy the asset in period 1 at a price she knows
exceeds its fundamental value because she hopes to
sell it to Ted for an even higher price in period 2, when
he isn’t sure if it is overvalued or not. Hence, a bubble
can emerge in equilibrium. That is, the price of the
asset can be positive even when all traders are aware

68

that the asset is worthless, so long as traders don’t know
that everyone else realizes the asset is worthless. The
possibility of asymmetric information is crucial for
why a bubble can arise.
In the preceding paragraphs of this section, I have
described a coherent example in which the emergence
of a bubble is a logical possibility. However, the bubble
in the Allen, Morris, and Postlewaite (1993) model
bears no resemblance to the historical episodes people
usually have in mind when they talk about bubbles.
First, when all agents know the asset is overvalued,
its price collapses after one period: As soon as a Carol
tries to look for a Ted to sell the asset to, she will
immediately learn she cannot find one, and the overvaluation will disappear. Second, the bubble asset in
their model never actually changes hands. By contrast, the historical episodes that many have taken to
be examples of bubbles involved high trade volumes
and periods of prolonged asset price appreciation
before prices collapsed, allowing traders to “ride the
bubble,” or hold on to the asset and let its price appreciate before selling it. Conlon (2004) modifies the
model in a way that allows the bubble to be sustained
beyond one period and the asset to be traded back and
forth between two agents. Essentially, traders keep
gambling on the exact date at which it will become
common knowledge that the asset is worthless. However, the bubble he constructs remains fragile, in the
sense that small perturbations to beliefs or payoffs will
lead the bubble to disappear. More recent work has
sought to construct robust asymmetric information
models of bubbles that persist for several periods. These
are sometimes known as models of riding bubbles, since
they feature agents who hold assets while they appreciate and then sell them. I discuss some of them next.
Riding an asymmetric information bubble
Abreu and Brunnermeier (2003) were among the
first to try to model the phenomenon of riding a bubble.
In their model, agents are sequentially informed that
an asset is overvalued from some randomly chosen
date t0. However, no agent observes t0. Thus, each
agent learns that the asset is overvalued, but not how
many others know the asset is overvalued or how
long they have known this. Abreu and Brunnermeier
assume the price of the asset rises over time, just as it
does in Allen, Morris, and Postlewaite (1993) and
Conlon (2004). Hence, if a trader is among the first to
learn the asset is overvalued and the first to sell, he
will make a profit. If he is among the last to know
and among the last to sell, he will be unable to find a
buyer by the time he acts. Abreu and Brunnermeier
(2003) show that under additional assumptions, the

2Q/2015, Economic Perspectives

optimal strategy for a trader is to wait a fixed period
of time from when he learns the asset is overvalued and
then sell. Depending on the pace at which the asset
price grows, the rate at which agents discount, and the
distribution of t0, a trader may wait to sell for longer
than it takes all agents to learn the asset is overvalued.
Thus, there can be a situation where every agent knows
that the asset is overvalued, yet the asset continues to
trade at a price that exceeds its fundamental value,
just as in Allen, Morris, and Postlewaite (1993) and
in Conlon (2004).
Unfortunately, the analysis in Abreu and
Brunnermeier (2003) shows only that agents will
engage in speculative trading if a bubble exists. But
their work does not prove that a bubble can in fact
exist. However, subsequent work by Doblas-Madrid
(2012) shows that it is possible to construct an internally consistent model of a bubble that exhibits many
of the features of the Abreu and Brunnermeier model.
Doblas-Madrid’s analysis offers several insights. First,
contrary to some of the suggestions in Abreu and
Brunnermeier (2003), he shows that it is not necessary
to assume that some agents hold exotic beliefs to
sustain a bubble. This should not be surprising given
my earlier observation that a bubble may arise even
if agents are rational, as long as there is some reason
for them to trade. Indeed, Doblas-Madrid assumes
that in every period there will be some traders who require immediate liquidity, so there can be gains from
trade between agents with a pressing need for liquidity
and those willing to hold the asset. For this explanation to hang together, he needs to impose a limit on
how many units of the asset buyers can absorb. This
allows the asset price to remain below the present discounted value of earnings, so those who buy the asset
are strictly better off. In particular, Doblas-Madrid
assumes traders cannot borrow, so their demand for
the asset is constrained by their income.24 Another
issue Doblas-Madrid explores is under what condition
the bubble will persist even after the first cohort of
traders sell their asset holdings, so that the downward
pressure on prices the first cohort exert when they
sell their assets doesn’t tip off other agents that some
traders have started to sell their assets. In particular,
Doblas-Madrid shows that some source of randomness
is necessary so that prices can fall even when the first
traders to learn the asset is overvalued sell without
alerting other agents. Thus, sustaining trade in an overvalued asset requires more uncertainty than assumed
in the Abreu and Brunnermeier (2003) setup. This
feature is certainly plausible (the real world is complicated and features many sources of uncertainty),
but it suggests that asymmetric information models of

Federal Reserve Bank of Chicago

bubbles in which agents trade the asset are likely to
be fairly complicated. Finally, Doblas-Madrid (2012)
shows that to sustain a bubble, his model requires certain restrictions on the way agents can trade. Intuitively,
a greater-fool theory can only work if each trader expects he might profit from selling the asset to a greater
fool. The problem is that when agents sell, they reveal
to everyone that the asset is overvalued. If this revelation scares away buyers, it will be impossible to profit
from selling assets. Doblas-Madrid gets around this
by assuming agents must submit their orders before
they know the price of the asset. In finance parlance,
this means agents are only allowed to place market
orders, which dictate how much to buy or sell at the
market price, but they cannot place limit orders, which
restrict the range at which a trade will be executed
(for example, an order that says to only buy an asset
if its price is below some cutoff). Whether it is possible to sustain bubbles when traders are unrestricted in
the orders they can place remains an open question.25
Back to policy
To recap, economists have been able to construct
models of bubbles based on asymmetric information
in which the price of an asset exceeds what one can
objectively argue the asset is worth. Recall that the
models that feature agents with different prior beliefs
that I discussed earlier could also give rise to scenarios
that can be described as bubbles, but only if traders have
distorted beliefs so that the most optimistic beliefs
tend to be wrong. In that case, policy intervention is
justified only if policymakers know that agents’ beliefs
are erroneous. By contrast, in asymmetric information
models of bubbles, all agents know the asset is overvalued, so their beliefs are not erroneous. Instead, it is
only because traders are uncertain as to what others
know that they are willing to buy assets and gamble
that they can sell them to others who are less informed
than they are. The question is whether letting agents
gamble this way is undesirable—and, more generally,
whether the fact that asset prices can exceed the fundamental value of the asset is socially costly. Unfortunately, little work has been done to analyze these
issues in models of asymmetric information.
An important exception is Conlon (2015), who
studied the role of policy in an asymmetric information
model of bubbles along the lines of his earlier paper
(Conlon, 2004). Specifically, he assumed the policymaker also receives information that the asset is trading at
a price above its fundamental value, and can announce
this information publicly. If the policymaker were to make
such an announcement, he would eliminate the prospect
of exploiting less informed traders. To be sure, this is

69

not the policy response that advocates of more forceful action against potential bubbles have in mind. They
typically argue that a central bank should raise interest
rates to head off possible bubbles. That said, Conlon’s
thought experiment is still informative, since it reveals
the social welfare consequences of deflating a bubble
when it can be achieved costlessly.26
To understand Conlon’s (2015) results, it will be
helpful to return to the key intuition behind bubbles
in asymmetric information environments: Traders are
willing to buy an asset they know to be overvalued
because they are taking a gamble. Either they will be
able to sell it at an even higher price to another less
informed trader, or else they will find out that no other
trader is willing to buy the asset and they will incur a
loss. The reason a trader may be able to sell the asset
to a greater fool is that the buyer believes he may be
entering into a mutually beneficial trade. Thus, an inherent feature of greater-fool bubbles based on asymmetric
information is that when agents trade, sometimes it is
because assets are overvalued and sometimes it is because there are mutual gains from trade. In other words,
Carol can profit at Ted’s expense only because there
are other situations in which both Carol and Ted gain
from trading and Ted doesn’t know which state they
are in while Carol does. If a policymaker were to reveal
that the asset is overvalued, this information would
affect the price in both scenarios. In particular, it would
lead to a reduction in the price when the asset is overvalued, and it would lead to an increase in the price
when there are mutual gains from trade. The first part is
straightforward: By telling everyone the asset is overvalued, the policymaker prevents those who know the
asset is overvalued (for example, Carol) from passing
it off to less informed traders (for example, Ted), and
so the price of the asset will not exceed the fundamental
value. As for the second part, in the state of the world
where there are mutual gains from trade between Carol
and Ted, one should note that when Ted buys the asset
he remains nervous that Carol might be taking advantage of him. If this concern were mitigated, he would
be willing to pay more for the asset, and the price
would be higher.
When the asset is available in fixed supply, announcing a bubble will generally have an ambiguous effect
on social welfare. Given that the price of the asset rises
in some states of the world and falls in others, those who
sell the asset will be better off in some states but worse
off in others. We can abstract from these considerations
by assuming that the gains and losses exactly cancel
each other out. In this case, a commitment by a fully
informed policymaker to announce whenever she knows
the asset is overvalued will have no effect on welfare

70

when the asset is in fixed supply. But a commitment
by the policymaker to reveal when she knows there
is a bubble could still improve welfare, even when
these two effects exactly offset each other, if the asset
were in variable supply. This statement holds true because the way the price of the asset changes in different states of the world leads to fewer units of the asset
being created when there are no gains from trading it
and more units of the asset being created when there
are gains from trading it. This is reminiscent of the
welfare results in the case of traders with uncommon
priors: When an asset is in fixed supply, the case for
preventing agents from trading is ambiguous, but when
an asset is in variable supply, there can be welfare gains
from reducing the cost of resource misallocation due
to mispricing.
Conlon (2015) goes on to show that the case for
policy intervention against a bubble crucially hinges
on the policymaker being able to identify a bubble
whenever it arises. His argument is different from the
more conventional logic that allowing policymakers
to act against bubbles can be costly if they mistakenly
act thinking an asset might be a bubble when in fact it
is not.27 Conlon shows that even if policymakers are
conservative and only react when they are certain there
is a bubble, policy intervention may make agents worse
off. This can happen because if policymakers deflate
bubbles in some states of the world but not others, the
bubbles that remain in other states can be worse than
the ones that policymakers actually lean against. Consequently, the resulting misallocation of resources from
policy intervention can be exacerbated. The case for
intervention may therefore rest on a policymaker being
perfectly informed about bubbles, since responding
either too aggressively or too timidly may undercut
the case for intervention. That is, while asymmetric
information models of bubbles suggest intervention
can be helpful, they also highlight the difficulty of
justifying intervention in practice.
I conclude my discussion of policy implications
with one final observation. Arguably, the primary reason policymakers cite for why they are concerned
about bubbles is distinct from those I discussed in this
article. The justifications for intervention I have discussed so far involve preventing a glut of assets in cases
where the assets are overvalued. But the case for policy
intervention has tended to focus on the dire consequences
of the bubble bursting rather that resource misallocation while asset prices are too high. Here, it is worth
pointing out that the models of bubbles based on
asymmetric information I have described imply that
if a bubble arises, it will eventually burst. This is because a bubble corresponds to a scenario in which all

2Q/2015, Economic Perspectives

agents believe the asset is worth less than its price, yet
they are willing to buy it because they are unsure
whether other traders are aware of this. Eventually,
uncertainty about what other traders know is resolved,
at which point the price of the asset collapses. This is
not true for other theories of bubbles. For example, in
models where bubbles arise because of asset shortages,
bubbles can in principle persist indefinitely. In models
where bubbles arise because of risk shifting, agents
who buy the asset are gambling on a risky asset that
sometimes pays off. If that happens, the price of the
asset will rise further rather than collapse. The fact that
greater-fool theories of bubbles based on asymmetric
information imply that bubbles necessarily burst makes
them of natural interest for further study, especially to
determine whether merely avoiding an eventual asset
price collapse can justify policy intervention.
Conclusion
This article described the literature on greaterfool theories of bubbles, that is, theories in which
agents are willing to buy assets they know to be overvalued because they believe they can profit from selling the assets to others. The idea behind this theory is
intuitive and seems to capture aspects of what often
happens during real episodes that are suspected to be
bubbles. This theory can also capture the unsustainable nature of a bubble that makes asset bubbles a
concern for policymakers. And yet it turns out to be a
surprisingly difficult theory to model and analyze.
What specific lessons should be taken away from
this discussion? In this article, I highlight two distinctions that are important to keep in mind to better sort
through the various results in the existing literature.
The first is the distinction between speculation and
bubbles. Speculative trading concerns why agents trade—
namely, to profit at the expense of others as opposed
to intending to find mutually beneficial gains. Asset
bubbles concern features of an equilibrium price—
namely, whether the price faithfully represents what
the asset is fundamentally worth. The fact that agents
engage in speculative trading does not necessarily imply
that the asset must be a bubble. In line with this, some
models that try to capture greater-fool theories of
bubbles are really models of the greater-fool theory
of trading rather than models of bubbles per se.
The second distinction that this article highlights
is one between models based on uncommon priors and
those based on asymmetric information. Any greaterfool theory requires that traders hold different beliefs.
But it matters whether these different beliefs arise because traders start out with distinct priors or because
they receive different information. This difference is

Federal Reserve Bank of Chicago

reminiscent of the line from Shakespeare’s As You
Like It: “The fool doth think he is wise, but the wise man
knows himself to be a fool.”28 In models that feature
only uncommon priors, agents are willing to trade
because they are convinced their beliefs are correct,
even though they cannot all be right—they are like
the fool who thinks he and only he is wise. In models
that feature asymmetric information, traders are aware
that those they are trading with receive private information that may allow them to exploit the other traders
around them. While agents are willing to trade, they
are also cautious about being exploited—they are wise
to the fact that they might be the greater fool, although
ultimately they are willing to trade. The two types of
models are thus quite different. Without additional
restrictions on how beliefs are formed, models based
on uncommon priors arguably do not generate asset
bubbles, while models based on asymmetric information can. For the same reason, the policy implications
of the two types of models are not identical.
Although some of the work that assumes different priors does explicitly refer to asset bubbles, the
term bubble can be appropriate if beliefs are assumed
to be somehow distorted. The general insight from
these models is that asset prices are determined by
the most optimistic traders, in the same way that the
price of a good is determined by those who most prefer that good. But without a theory of how beliefs are
formed, there is nothing that tells us that the most optimistic person must be wrong or that the average belief is correct. The heterogeneity of agents’ beliefs on
its own is thus not a good basis for talking about bubbles or for arguing that policymakers should drive
asset prices to their correct values. Still, models with
uncommon priors are easy to work with, and so it may
be appealing to use them together with some restriction on how beliefs are formed that implies excessive
optimism. Asymmetric information models can generate bubbles without such restrictions, but they rely on
a lot of structure to make sure information isn’t somehow revealed through prices or actions that traders take.
Which framework is better depends on the particular
application one is interested in.
Finally, with regard to policy implications, my
discussion highlights various difficulties in using
greater-fool theories of bubbles to justify action against
potential bubbles. Although these theories can provide some justifications for why policymakers should
intervene, these rationales come with many caveats.
For example, policymakers may have to know that
traders have incorrect beliefs, even though policymakers would not necessarily be any better at forecasting future dividends than members of the private

71

sector. Other justifications for intervention require
policymakers to be perfectly attuned to when bubbles
arise—a condition that seems implausible in practice.
In fact, greater-fool theories of bubbles naturally
suggest the opposite, that is, that detecting bubbles is
likely to be difficult. Recall that in asymmetric information models, bubbles can arise only because there
is the possibility of mutual gains from trade. Thus,
there may be plausible reasons for why agents trade
assets beyond trying to benefit at the expense of others.
Finally, the social welfare implications that emerge
most clearly in these models do not seem to capture the
main issue policymakers are concerned with in regard
to bubbles. For example, those who argue for a more

72

forceful policy response to potential bubbles typically
expect this response to come from central banks. This
reflects a view that bubbles are fueled by loose credit
conditions, as well as the idea that the collapse of a
bubble causes the most harm when assets were purchased on leverage and a collapse in their price would
trigger a subsequent round of defaults. Yet in most
models of the greater-fool theory of bubbles, credit
plays only a minor role or is missing altogether. As
I discuss in Barlevy (2012), risk-shifting theories of
bubbles seem particularly well suited for exploring
these issues. However, introducing credit into models
of the greater-fool theory of bubbles, which some have
attempted to do, may help tackle these issues as well.

2Q/2015, Economic Perspectives

NOTES
Note that this logic concerns only how policymakers should respond
to evidence of a possible bubble. In principle, though, policy intervention might prevent bubbles from arising in the first place. Indeed,
some have argued that policies such as restricting how much agents
can borrow against an asset or taxing transactions to make trading less
profitable may prevent bubbles. These policies are also more targeted
than the interest rate rules Bernanke and Gertler (1999) considered.
1

Allen, Morris, and Postlewaite (1993) provide a clear discussion
of why defining a bubble when agents have different beliefs can be
difficult. Rather than attempt to provide a general definition for a
bubble, they argue for constructing specific circumstances in which
there is enough structure to argue that the price of the asset deviates
from its fundamental value.
2

There is an extensive literature on bubbles of this type. The classic
reference on bubbles that arise when agents need assets to serve as
a store of value is Tirole (1985), who builds on the original work of
Samuelson (1958). Caballero and Krishnamurthy (2006) and Farhi
and Tirole (2012) discuss the case where agents need assets to serve
a liquidity role.
3

For more on the risk-shifting theory of bubbles, see Allen and
Gorton (1993), Allen and Gale (2000), and Barlevy (2014).
4

It is not clear where the term greater fool originated, but it seems
to have been first used by market practitioners. For example, a discussion of potential broker-dealer misconduct in the Securities and
Exchange Commission’s annual report for the fiscal year ending
June 30, 1963, contains the following description: “What has been
colloquially referred to as the ‘bigger fool’ theory ... is simply the
assurance that regardless of whether the price paid for a security is
fair and/or reflective of the intrinsic value of the security or even
reflective of a rational public evaluation of the security, the security
is still a good buy because a ‘bigger fool’ will always come along
to take it off the customer’s hands at a higher price” (Securities and
Exchange Commission, 1964, p. 74).
5

Chancellor (1999).

6

Milgrom and Stokey (1982) independently established similar results to those in Tirole (1982), although they framed their findings
in terms of speculative trading rather than bubbles. I therefore refer
to Tirole’s work in my discussion.
7

at least one agent must be wrong whenever two agents hold different
beliefs, that agent must not be rational. But as Morris notes, rationality
restricts only how agents update their priors, not what their priors
can be.
For example, suppose dividends reflect the profits of a multiproduct
company that sold different products at each date. The fact that a
theory about how much profit the firm would earn selling apples in
Australia in period 1 was wrong may not lead us to revise our theories about how much profit the firm would earn selling bicycles in
Burundi in period 2.
11

The usual motivation for assuming agents are price takers is that
it can always be assumed there are many identical replicas of Evelyn
and Odelia in the market, in which case the actions of any one agent
have no influence on the price of the asset. 
12

13

suggests

In particular, one can introduce a mechanism similar to the one in
Zeira (1999). Suppose that both agents believe that positive dividends
grow at a constant rate until some random date T, where the distribution of date T is known to both parties and has unbounded support.
At date T, both traders Evelyn and Odelia agree that dividends will
cease growing thereafter, even if they disagree on when they will
be paid out. As long as dividends continue to grow, asset prices
will rise faster than the risk-free interest rate. Evelyn and Odelia
will therefore sell the asset at a higher price than they paid to buy
it. Moreover, at date T the price will crash, so the model admits both
a boom and a crash. Note that speculative trading would continue
beyond date T even when asset prices stop growing unless one assumes
differences in beliefs also disappear at date T.
14

Harrison and Kreps (1978) define speculation differently than I do,
using the term to refer to a situation in which traders assign positive
value to the right to resell an asset. The problem with their definition
is that it implies that finitely lived agents who buy infinitely lived
assets are speculators, even when they do not expect to profit at the
expense of younger cohorts that buy assets from them. This distinction was irrelevant for Harrison and Kreps, who assumed infinitely
lived agents for their model’s environment, but their definition may
not generalize well to other environments.

17

Morris (1995) discusses the common prior assumption and its
connection to rationality. As he notes, some have argued that since
10

Federal Reserve Bank of Chicago

1
behaves somewhat like a fundamental value—a
1− β

theme they pick up on in their paper.

15

9

1
+ c β−t for c > 0
1− β

can also be an equilibrium price path. If Evelyn’s and Odelia’s endowments do not grow, at some point the one who values the asset
more could not afford the asset, yet the other party would want to
sell all of her holdings. Hence, in this case, such a path cannot be
an equilibrium. Harrison and Kreps refer to cβ–t as a bubble, although
they use this term in the sense of an explosive solution of a difference
equation rather than the way I use the term. Still, their terminology

The condition that the number of traders be finite, which I have
ignored in my discussion, also plays an important role in ruling out
the possibility of a bubble. Even if each trader understands that the
trader he buys from is profiting at his expense, he might be willing
to buy the asset if he thinks there is another trader at whose expense
he can profit. If one never runs out of traders to exploit, it may be
possible to sustain such trading chains. An important step in Tirole’s
(1982) argument is to show that this is not possible when the number of potential traders is finite.
8

Harrison and Kreps (1978) show that pt =

A modern-day example is repurchase agreements (repos), under
which a security is sold with the promise that the seller will buy it
back. This can be viewed as effectively renting the asset, although
legally repos do transfer ownership of the asset.
Formally, the dividend corresponds to a binomial tree. Such a
process is often used in models with heterogeneous beliefs among
agents because of its tractability; see, for instance, the example in
section III.C of Brunnermeier, Simsek, and Xiong (2014).
16

Pareto efficiency is a standard criterion for evaluating policies in
economics. Stiglitz (1989, p. 113, note 3) offers the usual definition
for this term: “An economy is Pareto efficient if no one can be made
better off without making someone else worse off.”
Kreps (2012) offers another example where agents hold different
beliefs and trade is costly. Kreps (2012, p. 193) describes a bet
between two economists, Joe Stiglitz and Bob Wilson, over the
contents of a pillow. Each is willing to bet a small sum of money
that his belief is right. However, to prove which one is correct,
Stiglitz and Wilson must destroy the pillow and purchase a new
one. The destroyed pillow is the social cost associated with trade.
18

73

In line with the definition of Pareto efficiency described in note 17,
a Pareto improvement is a change that harms no one and helps at
least one person.
19

For example, suppose there are two individuals—one who believes
there is a probability of 0.7 that a tossed coin will come up heads
rather than tails and the other who believes there is a probability of
0.3 it will come up heads. Achieving belief-neutral Pareto improvement requires evaluating both of their utilities with the same probability p for every p between 0.3 and 0.7. If one allocation makes both
agents at least as well off as another allocation at each probability,
that allocation is said to represent a belief-neutral Pareto improvement.
20

Lewis (2010).

21

Formally, the presence of noise traders prevents the price from
being an invertible function of aggregate information.
22

Allen, Morris, and Postlewaite (1993) refer to this case as a strong
bubble. They contrast this with the case where the asset trades at a
positive price but not all agents know the dividend is zero.
23

Doblas-Madrid and Lansing (2014) consider a variation of the
Doblas-Madrid (2012) model in which agents can borrow. As long
as there are constraints on how much agents can borrow, it will still
be possible for agents to profit from buying the asset from those
with liquidity needs. The rate at which the price of the asset grows
will now be tied to the rate at which credit is growing.
24

Given my earlier remark on Zeira (1999), it is worth noting that
the Doblas-Madrid (2012) model achieves a boom and bust using a
similar structure to that of Zeira’s model. Recall that Zeira assumes
dividends grow until some random date T. At date T + 1, all agents
learn that dividends will no longer grow and the price crashes. In
Doblas-Madrid’s setup, there is a date t0 at which the fundamental
value ceases to grow, but not all agents learn this at date t0 because
they receive information sequentially. The crash that would ordinarily
happen when fundamentals cease growing is thus delayed. To avoid
having agents learn the fundamentals from dividends, Doblas-Madrid
assumes dividends are paid out far off in the future. But conceptually
the setup is similar to Zeira’s.
25

Conlon’s (2015) notion of relying on announcements also has a
historical precedent. In December 1996, Alan Greenspan in his capacity as Federal Reserve Chairman publicly expressed his concern
about “irrational exuberance” in financial markets, which some have
interpreted as an attempt to affect the beliefs of market participants
(Greenspan, 1996).
26

27

See Cogley (1999) for an example of this type of argument.

See http://www.shakespeare-online.com/plays/asu_5_1.html (act 5,
scene 1, lines 30–31).
28

REFERENCES

Abreu, Dilip, and Markus K. Brunnermeier, 2003,
“Bubbles and crashes,” Econometrica, Vol. 71, No. 1,
January, pp. 173–204.
Allen, Franklin, and Douglas Gale, 2000, “Bubbles
and crises,” Economic Journal, Vol. 110, No. 460,
January, pp. 236–255.
Allen, Franklin, and Gary Gorton, 1993, “Churning
bubbles,” Review of Economic Studies, Vol. 60, No. 4,
October, pp. 813–836.
Allen, Franklin, Stephen Morris, and Andrew
Postlewaite, 1993, “Finite bubbles with short sale
constraints and asymmetric information,” Journal of
Economic Theory, Vol. 61, No. 2, December,
pp. 206–229.
Barlevy, Gadi, 2014, “A leverage-based model of
speculative bubbles,” Journal of Economic Theory,
Vol. 153, September, pp. 459–505.
__________, 2012, “Rethinking theoretical models of
bubbles: Reflections inspired by the financial crisis
and Allen and Gorton’s paper ‘Churning bubbles,’” in
New Perspectives on Asset Price Bubbles: Theory,
Evidence, and Policy, Douglas D. Evanoff, George
G. Kaufman, and A. G. Malliaris (eds.), New York:
Oxford University Press, pp. 41–61.

74

Bernanke, Ben, and Mark Gertler, 1999, “Monetary
policy and asset price volatility,” Economic Review,
Federal Reserve Bank of Kansas City, Fourth Quarter,
pp. 17–51.
Blume, Lawrence E., Timothy Cogley, David A.
Easley, Thomas J. Sargent, and Viktor Tsyrennikov,
2014, “The case for incomplete markets,” New York
University and Cornell University, mimeo, February 27,
available at https://easley.economics.cornell.edu/
BECST-20140226.pdf .
Brunnermeier, Markus K., Alp Simsek, and Wei
Xiong, 2014, “A welfare criterion for models with
distorted beliefs,” Quarterly Journal of Economics,
Vol. 129, No. 4, November, pp. 1753–1797.
Caballero, Ricardo J., and Arvind Krishnamurthy,
2006, “Bubbles and capital flow volatility: Causes and
risk management,” Journal of Monetary Economics,
Vol. 53, No. 1, January, pp. 35–53.
Chancellor, Edward, 1999, Devil Take the Hindmost:
A History of Financial Speculation, New York: Farrar,
Straus and Giroux.
Cochrane, John H., 2014, “Challenges for cost-benefit
analysis of financial regulation,” Journal of Legal
Studies, Vol. 43, No. S2, June, pp. S63–S105.

2Q/2015, Economic Perspectives

Cogley, Timothy, 1999, “Should the Fed take deliberate steps to deflate asset price bubbles?,” Economic
Review, Federal Reserve Bank of San Francisco, No. 1,
pp. 42–52.
Conlon, John R., 2015, “Should central banks burst
bubbles? Some microeconomic issues,” Economic
Journal, Vol. 125, No. 582, February, pp. 141–161.
__________, 2004, “Simple finite horizon bubbles
robust to higher order knowledge,” Econometrica,
Vol. 72, No. 3, May, pp. 927–936.
Doblas-Madrid, Antonio, 2012, “A robust model
of bubbles with multidimensional uncertainty,”
Econometrica, Vol. 80, No. 5, September, pp. 1845–1893.
Doblas-Madrid, Antonio, and Kevin Lansing, 2014,
“Credit-fuelled bubbles,” Michigan State University
and Federal Reserve Bank of San Francisco, mimeo,
May, available at http://www.bostonfed.org/MacroFinance-2014/Doblas-Madrid-Lansing.pdf .
Farhi, Emmanuel, and Jean Tirole, 2012, “Bubbly
liquidity,” Review of Economic Studies, Vol. 79, No. 2,
April, pp. 678–706.
Gilboa, Itzhak, Larry Samuelson, and David
Schmeidler, 2014, “No-betting-Pareto dominance,”
Econometrica, Vol. 82, No. 4, July, pp. 1405–1442.

Kreps, David M., 2012, Microeconomic Foundations I:
Choice and Competitive Markets, Princeton, NJ:
Princeton University Press.
Lewis, Michael, 2010, The Big Short: Inside the
Doomsday Machine, New York: Norton.
Milgrom, Paul, and Nancy Stokey, 1982, “Information,
trade and common knowledge,” Journal of Economic
Theory, Vol. 26, No. 1, February, pp. 17–27.
Miller, Edward M., 1977, “Risk, uncertainty, and
divergence of opinion,” Journal of Finance, Vol. 32,
No. 4, September, pp. 1151–1168.
Mongin, Philippe, 2005, “Spurious unanimity and
the Pareto principle,” London School of Economics,
Centre for Philosophy of Natural and Social Science
(CPNSS), LSE Choice Group, working paper, Vol. 1,
No. 5, available at http://eprints.lse.ac.uk/23942/.
Morris, Stephen, 1995, “The common prior assumption in economic theory,” Economics and Philosophy,
Vol. 11, No. 2, October, pp. 227–253.
Posner, Eric A., and E. Glen Weyl, 2013, “An FDA
for financial innovation: Applying the insurable interest
doctrine to twenty-first-century financial markets,”
Northwestern University Law Review, Vol. 107, No. 3,
Spring, pp. 1307–1358.

Greenspan, Alan, 1996, “The challenge of central
banking in a democratic society,” remarks by Federal
Reserve Chairman at Annual Dinner and Francis Boyer
Lecture of the American Enterprise Institute for Public
Policy Research, Washington, DC, December 5,
available at http://www.federalreserve.gov/boarddocs/
speeches/1996/19961205.htm .

Samuelson, Paul A., 1958, “An exact consumptionloan model of interest with or without the social contrivance of money,” Journal of Political Economy,
Vol. 66, No. 6, December, pp. 467–482.

Grossman, Sanford, 1976, “On the efficiency of
competitive stock markets where trades have diverse
information,” Journal of Finance, Vol. 31, No. 2,
May, pp. 573–585.

Scheinkman, José A., and Wei Xiong, 2003, “Overconfidence and speculative bubbles,” Journal of Political
Economy, Vol. 111, No. 6, December, pp. 1183–1219.

Grossman, Sanford J., and Joseph E. Stiglitz, 1980,
“On the impossibility of informationally efficient
markets,” American Economic Review, Vol. 70, No. 3,
June, pp. 393–408.
Harrison, J. Michael, and David M. Kreps, 1978,
“Speculative investor behavior in a stock market with
heterogeneous expectations,” Quarterly Journal of
Economics, Vol. 92, No. 2, May, pp. 323–336.

Federal Reserve Bank of Chicago

Scheinkman, José A., 2014, Speculation, Trading,
and Bubbles, New York: Columbia University Press.

Securities and Exchange Commission, 1964,
29th Annual Report of the Securities and Exchange
Commission: For the Fiscal Year Ended June 30, 1963,
Washington, DC: U.S. Government Printing Office.
Stiglitz, Joseph E., 1989, “Using tax policy to curb
speculative short-term trading,” Journal of Financial
Services Research, Vol. 3, Nos. 2–3, December,
pp. 101–115.

75

Tirole, Jean, 1985, “Asset bubbles and overlapping
generations,” Econometrica, Vol. 53, No. 6, November,
pp. 1499–1528.

Velde, François R., 2013, “Lottery loans in the
eighteenth century,” Federal Reserve Bank of
Chicago, mimeo.

__________, 1982, “On the possibility of speculation
under rational expectations,” Econometrica, Vol. 50,
No. 5, September, pp. 1163–1181.

Zeira, Joseph, 1999, “Informational overshooting,
booms, and crashes,” Journal of Monetary Economics,
Vol. 43, No. 1, February, pp. 237–257.

76

2Q/2015, Economic Perspectives