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Federal Reserve Bank of Chicago

A Leverage-based Model of
Speculative Bubbles
Gadi Barlevy

REVISED
July 8, 2013
WP 2011-07

A Leverage-based Model of Speculative Bubbles
Gadi Barlevy
Economic Research Department
Federal Reserve Bank of Chicago
230 South LaSalle
Chicago, IL 60604
e-mail: gbarlevy@frbchi.org
July 8, 2013

Abstract
This paper examines whether theoretical models of bubbles based on the notion that the price of
an asset can deviate from its fundamental value are useful for understanding historical episodes that
are often described as bubbles, and which are distinguished by features such as asset price booms
and busts, speculative trading, and seemingly easy credit terms. In particular, I focus on risk-shifting
models similar to those developed in Allen and Gorton (1993) and Allen and Gale (2000). I show that
such models could give rise to these phenomena, and discuss under what conditions price booms and
speculative trading would emerge. In addition, I show that these models imply that speculative bubbles
can be associated with low spreads between borrowing rates and the risk free rate, in accordance with
observations on credit conditions during historical episodes often suspected to be bubbles.

This paper represents a substantial revision of Federal Reserve Bank of Chicago Working paper No. 2008-01. I am grateful
to Franklin Allen, Marios Angeletos, Bob Barsky, Marco Bassetto, Christian Hellwig, Guido Lorenzoni, Kiminori Matsuyama,
Ezra Ober…eld, Rob Shimer, Kjetil Storesletten, and Venky Venkateswaran for helpful discussions, as well as participants at
various seminars. I also wish to thank David Miller, Kenley Barrett, and Shani Shechter for their research assistance on this
or earlier versions of the paper. The views expressed here need not re‡ect those of the Federal Reserve Bank of Chicago or the
Federal Reserve System.

1

1

Introduction

The large ‡uctuations in U.S. equity and housing prices over the past decade and a half have led to renewed
interest in the phenomenon of asset bubbles. Among non-economists, the term “bubble” has come to refer
to any historical episode in which asset prices rise and fall signi…cantly over a relatively short period of
time.1 By contrast, economists usually use the term “bubble” to mean that an asset trades at a price that
di¤ers from its fundamental value, i.e. the expected discounted value of the dividends it generates. The
latter notion is meant to capture the idea that asset prices convey distorted signals as to the true value
of the underlying assets. The two notions are certainly compatible: The price of an asset can rise above
fundamentals and then collapse. But economists working on models of bubbles have largely ignored the
question of whether these models can account for the key features of the historical episodes suspected to
be bubbles, focusing instead on whether assets can trade at a price that di¤ers from fundamentals. The
distinguishing features of the historical episodes include not just a boom and bust in asset prices, but a high
incidence of speculative trading in which agents buy assets with the explicit aim of pro…ting from selling
them later on rather than accruing dividends and low spreads on loans taken out against these assets.
This paper examines whether one particular class of models that can generate a gap between the price of an
asset and its fundamental value –the risk-shifting theory of bubbles developed by Allen and Gorton (1993)
and Allen and Gale (2000) – can also generate the qualitative features common to the historical episodes
often described as bubbles. In risk-shifting models, traders purchase risky assets with funds obtained from
others. The …nancial contracts traders use to secure these funds are assumed to involve limited liability.
The latter feature implies traders would be willing to pay more for assets than the expected dividends they
yield, since traders can shift any losses they incur on to their …nanciers. In other words, agents value assets
above the dividends these assets generate because of the option to default on loans issued against the asset.
This intuition can be illustrated in a purely static model where assets trade hands exactly once. Thus,
overvaluation can occur independently of boom-bust dynamics or speculative trade.
While both Allen and Gorton (1993) and Allen and Gale (2000) consider dynamic models of risk-shifting,
the various assumptions they impose to maintain tractability make it di¢ cult to gauge whether and when
these models give rise to the key features that characterize the historical episodes often described as “bubbles.” For example, Allen and Gorton (1993) assume bilateral trades rather than a market for assets. This
implies asset prices in their model are not uniquely determined, and so their model has little to say on when
price booms and busts will arise. In addition, both papers e¤ectively require agents who buy an overvalued
asset to sell it after some exogenously determined holding period. As such, they cannot address whether
1 For example, Merriam-Webster.com de…nes a bubble as “a state of booming economic activity (as in a stock market) that
often ends in a sudden collapse,” while the New York Times Online Guide to Essential Knowledge de…nes it as a “market in
which the price of an asset continues to rise because speculators believe it will continue to rise even further, until prices reach
a level that is not sustainable; panic selling begins and the price falls precipitously.”

1

these models give rise to “speculative”bubbles in which traders buy an asset with the aim of pro…ting from
selling it, or to what Hong and Sraer (2011) dub “quiet” bubbles in which assets are overvalued but trade
only infrequently. To address these questions requires a dynamic model in which agents trade strategically.
There has also been little work on whether risk-shifting models of bubbles are consistent with the lax
credit conditions that accompany historical episodes identi…ed as bubbles. At a …rst glance, these models
seem to imply that if anything, bubbles should be associated with higher borrowing costs. In particular,
traders who buy overvalued assets are only able to raise funds by pooling with other borrowers whom
…nanciers would like to …nance. If creditors understand that some of those they fund will buy risky assets,
they would presumably charge a spread over the risk-free rate to cover the expected losses on speculators.
One of the points of this paper is to show why this intuition can be misleading, and to outline reasons why
the spread between the borrowing rate and the risk-free rate may in fact be lower in speculative episodes.
In what follows, I develop a dynamic model of risk-shifting where agents borrow to buy risky assets and
then choose if and when to sell them. If agents fear that future traders may not always buy the asset in
order to gamble at the expense of creditors, assets can exhibit rapid price appreciation as long as they
continue to trade. The intuition is related to the one Blanchard and Watson (1982) derive for rational
bubbles: If an asset might cease to become overvalued in the future, rational agents must be compensated
for holding the asset now while it is still overvalued rather than sell it. This compensation accrues as
capital gains if the asset remains overvalued. However, an important di¤erence between the Blanchard and
Watson (1982) framework and mine is that their model is silent on whether assets trade, since agents in
their model are always indi¤erent between holding and selling an asset. This is not true in my model, where
agents sometimes sell the asset and sometimes hold it to maturity. This feature of my model reveals that
the price dynamics in Blanchard and Watson (1982) should be seen as a bound on the rate of asset price
appreciation rather than the rate at which rational bubbles always grow. It also reveals when bubbles will
be “noisy” rather than quiet and feature repeated trade: When assets are more overvalued, when asset
price appreciation is high, and when assets returns are skewed towards a high upside potential relative to
the mean. An additional insight is that noisy bubbles can be associated with lower borrowing spreads than
quiet bubbles. This is because when traders sell bubble assets, the risk lenders are exposed to in lending
against assets is partly shifted to future lenders. Thus, it may be cheap to borrow against assets precisely
when assets trade hands repeatedly, a common feature of empirical episodes suspected to be bubbles.
The second modi…cation I consider is to allow lenders to design contracts optimally. This modi…cation
reveals another reason why speculation can be associated with low borrowing spreads, especially for speculators. Speci…cally, lenders will want to minimize the losses they incur from speculators, e.g. by restricting
loan size or providing only short-term …nancing. At the same time, lenders would not want to impose these
features on pro…table borrowers. To induce speculators to accept more restricted contracts, lenders must
make them more attractive on some other dimension, which can include lower rates.
2

While this paper only considers risk-shifting models of bubbles, other models have been developed in
which assets can trade above their fundamental value. One example are models in which there is a shortage
of assets that perform some essential function such as a store of value or liquidity. This scarcity can lead
people to value whatever assets are available above and beyond the dividends they yield. Examples of
such models include overlapping generations models such as Samuelson (1958), Diamond (1965), and Tirole
(1985).2 Another example are the so-called “greater-fool” models of bubbles, where agents buy assets
they view as overvalued because they expect to pro…tably resell these assets to other agents who value the
asset di¤erently. These models include Allen, Morris, and Postlewaite (1993) and Conlon (2004). Without
denying the importance of these models, there does seem to be value in studying risk-shifting models of
bubbles in particular. One reason is that in these models credit markets play a central role in allowing
bubbles to arise, which accords with the observation that in many historical episodes traders would often
borrow against the assets they purchase. The central role of credit markets also implies these models can be
used to explore one potential concern about bubbles, namely that the collapse of a bubble can be especially
consequential if it results in default by those who borrowed against these assets. Finally, risk-shifting models
of bubbles can lead to di¤erent policy implications than alternative models that give rise to bubbles. For
example, since bubbles do not arise in these models because assets play a valuable role such as a store of
value of liquidity, assets that trade above fundamentals may be ine¢ ciently oversupplied. At the same time,
unlike in greater fool models, risk-shifting models allow for it to be common knowledge that some assets
are overpriced, so a policymaker with no more knowledge than private agents might want to intervene.
The paper is structured as follows. Section 2 lays out the basic features of my model economy. Section 3
studies the implications of the model for price dynamics and speculation when the set of contracts agents
can enter is exogenously restricted to simple debt contracts. Section 4 allows for endogenous contracting,
and examines what type of contracts will be o¤ered when speculative bubbles arise. Section 5 concludes.

2

Setup

I begin by describing the key features of the environment I study. The model is meant to generalize the
Allen and Gale (2000) model to allow for strategic dynamic trades and, later on, for endogenous contracting.
To keep the analysis tractable, I restrict attention to a two-period model, where periods are indexed by
t 2 f1; 2g. I …rst describe the assets that agents can trade. I then describe the agents who populate the
2 There are also examples of such models where agents have in…nite horizons, e.g. Kocherlakota (1992) and Santos and
Woodford (1997). Both o¤er examples of bubbles in models inspired by Bewley (1980) where there are …nitely many agents
with in…nite horizons but limits on what agents can trade. Both papers go on to show that if agents face borrowing limits,
bubbles can also emerge on assets available in zero net supply. The latter seems less relevant for understanding historical
episodes suspected to be bubbles, where the relevant assets were in positive net supply. While early models focused on the
case where assets served the role as a store of value, more recent work, including Caballero and Krishnamurthy (2006), Farhi
and Tirole (2011), and Rocheteau and Wright (2011) instead focus on the case where assets provide a liquidity role.

3

economy. Finally, I describe the operation of credit markets in my economy.
As in Allen and Gale (2000), I assume a continuum of assets that are available in …xed supply and cannot
be sold short. Restricting supply in some manner is necessary for assets to trade above their fundamental
value. More generally, I could have allowed for an upward sloping supply schedule within a period, although
this would have been more cumbersome. The …xed supply of the asset can be viewed as a technological
constraint on the production of additional assets. The restrictions on short sales can be motivated by similar
informational frictions to those I consider, e.g. agents who sell short cannot be trusted to deliver the assets
they sell or to replicate its payo¤s. However, in what follows I take short sales restrictions as given rather
than derive them. Since the supply of assets is assumed to be …xed, I can normalize its mass to 1.
The payouts on assets are risky. For simplicity, suppose the assets pay a single common dividend d at
the end of date t = 2 that can take on just two values:
d=

D>0
0

with probability
with probability 1

(1)

For example, an asset can represent a claim to the pro…ts of a …rm with a patent that may or may not pan
out, and D represents the value of pro…ts to the …rm if the patent is successful.
At the beginning of date 1, agents know only that d is distributed according to (1). For reasons I explain
shortly, I allow d to be revealed with some probability between dates 1 and 2. That is, before agents trade
the asset at date 2, d will be revealed with probability q 2 [0; 1]. For example, sticking to the patent

example, a technological discovery may occur at the end of date 1 that reveals whether the patent is viable.
Formally, let It denote the information traders observe at date t concerning dividends. Then I1 = ? and
I2 =

d
?

with probability q
with probability 1

q

When 0 < q < 1, agents who buy assets at date 1 are uncertain whether these assets will remain risky at
date 2. This turns out to be important, since the riskiness of the asset a¤ects demand for it, and so demand
for the assets at date 2 is uncertain as of date 1. There are other reasons why demand for the asset might be
uncertain, e.g. the number of agents who show up at date 2 might be random, or there might be uncertainty
as to other assets agents may buy at date 2. I focus on early revelation of d only for convenience.
For simplicity, I will assume agents are all risk neutral and do not discount. As such, the expected utility
value of the dividends the asset generates for any agent is just E [djIt ]. I will refer to this expectation as
the fundamental value of the asset. It represents the value to society of creating an additional unit of the
asset at date t. That is, if additional units of the asset could be produced at some cost at date t, the price
of the asset would have to equal E [djIt ] to provide proper incentives for creating additional units.

4

I now turn to the agents that populate the economy. Since there are several types of agents in the model,
di¤ering in their endowments and opportunity sets, it will help to begin with a brief overview. At the core of
the model are two types who stand to gain by trading with each other, although this trade does not involve
the assets just described. Rather, some agents, whom I call creditors, are endowed with resources but can
earn only low returns on their savings, while other agents, whom I call entrepreneurs, lack resources but
have access to a production technology that yields a high rate of return. Creditors can thus bene…t from
lending to entrepreneurs in exchange for a higher return. The reason these two types have any bearing on
the assets just described is that there is a third group of agents, whom I call non-entrepreneurs, that lack
both resources and access to a production technology, but who can buy the aforementioned risky assets.
If creditors cannot distinguish entrepreneurs from non-entrepreneurs, in trying to trade with entrepreneurs
they may end up lending to non-entrepreneurs who buy risky assets. The desire by creditors to trade with
entrepreneurs allows resources to ‡ow into the asset market and in‡uence asset prices.
Closing the model requires two additional groups: The …rst group, whom I call original owners, is
endowed with the risky assets at the beginning of date 1. The second group, whom I call non-participants,
lacks resources, lacks access to a productive technology, and cannot trade assets. This group will only
become relevant when I consider endogenous contracts in the next section, where their presence serves
to limit the type of contracts creditors o¤er. In particular, their presence prevents creditors from paying
non-entrepreneurs not to buy assets, since this would also draw in non-participants who cannot buy assets.
Formally, the endowments and opportunity sets of the …ve types can be characterized as follows:

1. Creditors: Creditors are endowed with a large amount of resources at date 1 that they can store
until date 2 at a gross return of 1. They can also buy risky assets, although they will never strictly
prefer to do so in equilibrium. The mass of creditors is assumed to be large, more than enough to
supply the demand of potential borrowers whom I discuss next.
2. Entrepreneurs: Entrepreneurs are endowed with neither resources nor assets. However, they have
access to a productive technology that allows them to produce R > 1 units of output per unit of input
invested, up to a capacity of 1 unit of input. I assume they cannot buy risky assets. This avoids
having to verify that they prefer production over buying assets, although there are parameters that
guarantee this will be the case. Entrepreneurs arrive at exogenously set times, either at t = 1 or t = 2,
and if they want to invest they must contact creditors the period they arrive. Regardless of when they
invest, the output from their investment accrues at the end of date 2.
3. Non-Entrepreneurs: Non-entrepreneurs are also endowed with neither resources nor assets. In
contrast to entrepreneurs, they do not have access to a productive technology. But they can buy
risky assets. Like entrepreneurs, they arrive at exogenously set times, either at t = 1 or t = 2, and if

5

they wish to buy risky assets they must both contact creditors and buy assets the period they arrive.
Let nt < 1 denote the mass of non-entrepreneurs who arrive at date t, and let mt < 1 denote the

combined mass of entrepreneurs and non-entrepreneurs who arrive at date t. To cut down on the

number of parameters, I assume the ratio of non-entrepreneurs to entrepreneurs is the same in both
periods. That is, regardless of t, the number of non-entrepreneurs nt = mt for some

2 (0; 1).

4. Original Owners: Original owners are endowed with one unit of the asset each and a large amount
of resources, which they can store at a gross return of 1. For ease of exposition, I assume that unlike
creditors, they do not participate in the credit market, either as lenders or borrowers. This assumption
is not restrictive given that in equilibrium they could not pro…t from trading in the credit market.
5. Non-Participants: Non-participants are endowed with neither resources, assets, nor a productive
technology. In addition, they face a prohibitive cost of entering the market for risky assets. The mass
of such agents is large, in a sense that will be clari…ed in Section 4 when they become relevant.
Many of the assumptions above only serve to simplify the analysis, and can be considerably relaxed
without a¤ecting some of the key results. In particular, the result that assets can trade above their fundamental value E [djIt ] arises because creditors can pro…tably …nance entrepreneurs, but only on terms that
would make it pro…table for non-entrepreneurs to borrow and buy risky assets. The existence of bubbles
thus hinges on there being at least some entrepreneurs with pro…table investment opportunities but limited
resources, allowing non-entrepreneurs to borrow and then buy risky assets with little of their own at stake.
The remaining assumptions I impose – such as the exogenous arrival dates of agents, the …nite number of
agents who arrive at each period, and the …nite capacity of entrepreneurs –are unnecessary for this result.
At the same time, my assumptions do matter for price dynamics and trade volume. Speci…cally, if there
were no limit on how much agents could borrow at date 1, asset prices would be bid up immediately, assets
would trade hands only once, and asset prices would not grow. To generate price appreciation and repeated
trading, we need total borrowing at date 1 to be …nite, so the …nite capacity of entrepreneurs is important.
Similarly, assuming that agents arrive at exogenous dates is not entirely innocuous. If entrepreneurs and
non-entrepreneurs all arrived in period 1 and chose when to trade, the number of agents who trade in each
period would be endogenous. Since my results depend on the number of traders that arrive at each period,
it is not obvious that endogenous timing would accommodate all of the phenomena I emphasize. However,
as will become clear below, for certain parameter values such as a small , price appreciation and repeated
turnover would probably arise even if I allowed agents to time their actions.
To focus on the key features of the model I am after, I impose two parameter restrictions. First, I restrict
the return on production R to an intermediate range of values:
1+

1

1
(1
)
<R<
(1
)
6

(2)

It is easy to verify that this range is nonempty for

2 (0; 1) and 2 (0; 1). The reason R cannot be too low

is that the earnings of entrepreneurs must cover the expected losses creditors incur on non-entrepreneurs
who use the funds they borrow to buy risky assets. At low values of R, lending would simply shut down. But

high values of R can also be problematic. In particular, if R exceeded 1= , entrepreneurs would earn a higher
return than non-entrepreneurs could ever earn from buying risky assets. This implies non-entrepreneurs
might not be able to pro…t from buying risky assets, since borrowers may be charged a high rate that still
attracts entrepreneurs but would make it unpro…table to buy and hold risky assets. In addition, once I allow
more general contracts, if entrepreneurs earn more than non-entrepreneurs ever could they could prove their
type to creditors by showing their earnings. Creditors could then avoid lending to non-entrepreneurs.
Second, I restrict the total number of non-entrepreneurs over the two periods to not be too large:
n1 + n2 < (1

q) D=R + q D

(3)

Since each agent will be able to borrow at most one unit of resources given the …nite capacity of entrepreneurs, assumption (3) restricts the total amount of resources agents can borrow to spend on risky assets.
As we shall see below, this will impose an upper bound on the price of the asset. (3) rules out the case
where the price of the asset is high enough that non-entrepreneurs who buy risky assets earn zero expected
pro…ts, rendering them indi¤erent between buying and not buying the asset. The latter case introduces an
additional variable –the fraction of non-entrepreneurs who buy assets –and is thus more tedious to analyze.
Remark 1: Note that the second inequality in (2) implies D=R > D. Hence, the upper bound in (3) is
strictly greater than D for q < 1. In the next section I show that n1 + n2 > D is a necessary and su¢ cient
condition for a bubble. My parametric restrictions are thus compatible with the possibility of a bubble.
Finally, I need to describe the functioning of credit markets where creditors can trade with entrepreneurs,
non-entrepreneurs, and non-participants. Creditors cannot distinguish the di¤erent types that seek to
borrow, nor can they monitor what agents do with the funds they receive. To motivate this assumption, we
can think of entrepreneurs as earning the return R by purchasing some type of asset, e.g. purchasing an asset
they can manage better than its current owner, or buying an undervalued asset based on private information
as in Allen and Gorton (1993). If creditors cannot tell apart di¤erent types of assets, entrepreneurs and
non-entrepreneurs will look indistinguishable. Alternatively, creditors may not even get to observe the
underlying assets, as is sometimes the case with hedge funds that don’t divulge their trading strategies.
Credit markets are run as follows: First, creditors post contracts. Agents then arrive and choose among
contracts. Speci…cally, agents ‡ow in according to some pre-arranged order, where the fraction of nonentrepreneurs within each arriving cohort is . The reason I require sequential arrivals is that, as we shall
see below, it is possible that more than one type of contract will be o¤ered in equilibrium. In this case,
more attractive contracts must be rationed, and sequential arrival rations them to those who arrive …rst.
7

In terms of the types of contracts creditors can o¤er, for now I restrict lenders to o¤ering only a limited
set of contracts, retaining comparability with Allen and Gale (2000) who also focus on a particular set of
contracts. I will allow a more general class of contracts in Section 4. The set of contracts I initially study
are …xed-size, full recourse, simple debt contracts. Speci…cally, at each date t, lenders can o¤er to lend one
unit of resources to borrowers who show up at that date, the most creditors would ever agree to o¤er given
the …nite capacity of entrepreneurs. The borrower is required to pay back a pre-speci…ed amount 1 + rt at
the end of date 2, which is when entrepreneurs would …rst be able to make a payment. The only dimension
along which lenders can compete is the rate rt they charge borrowers. If the borrower fails to repay his
obligation in full, the lender has full recourse to go after the borrower’s remaining resources, up to the
amount of the obligation. Hence, entrepreneurs cannot escape repaying their debt, and wealthy agents will
not …nd it pro…table to borrow and buy risky assets given they will always be liable for losses they incur.3
Beyond the threat of recourse, an important reason borrowers repay their debt in the real world is that
default tends to be costly, e.g. it may be associated with a loss of access to future credit. One can crudely
capture this intuition in my model by assuming that if the borrower pays z < 1 + rt , he will incur a cost
k (1 + rt

z) proportional to his shortfall. I want to allow for this possibility, but to avoid keeping track of

another parameter I focus on the limiting case where k ! 0. All of my results extend to the case where k

is positive but small. At the same time, taking the limit as k ! 0 is not equivalent to setting k = 0. For
example, when k > 0, agents will not borrow if they expect to default with certainty, a property that will

be preserved in the limit as k ! 0. But agents would be willing to borrow and default when k = 0. Looking

at the limit as k ! 0 thus rules out equilibria that are not robust to the introduction of small default costs.

3

Equilibrium

I now proceed to analyze the equilibrium of this economy. Intuitively, an equilibrium consists of state2

2

contingent paths for the price of the asset fpt (It )gt=1 and the borrowing rate frt (It )gt=1 that ensure both
the asset market and credit markets clear. Speci…cally,

1. At each date t, for each information set It , demand for the asset by potential buyers at price pt is
equal to the amount those who already own the asset are willing to sell at price pt

2. At each date t, for each information set It , creditors earn zero expected pro…ts when they o¤er a loan
at rate rt , and creditors cannot expect to earn positive pro…ts by o¤ering an alternative contract

3 One could interpret these loans as collateralized by the assets the borrower purchases: If the borrower fails to repay, full
recourse allows the creditor to seize any dividends that accrue to the asset. However, as will become clear once I allow creditors
to design their contracts, creditors must have very limited information about the assets used as collateral, or else the contract
would naturally make use of such information in a way that would make it di¤erent from a debt contract.

8

As anticipated in the previous section, there may be situations in which condition (2) will require creditors
to o¤er more than one interest rate at date t = 1. Thus, a proper de…nition of equilibrium needs to be
modi…ed to allow for multiple interest rates. With this caveat in mind, I now proceed to characterize the
conditions that ensure the asset market clears and that creditors not expect to earn positive pro…ts.
First, though, I introduce some terminology that will help in describing equilibrium in the asset market.
I will refer to the risky asset as a bubble if at any date t, its price pt di¤ers from the fundamental value
E [djIt ]. Note that my de…nition for fundamental value only re‡ects the value of dividends and not the
option value to default on loans borrowed against an asset, even though non-entrepreneurs value the asset
in part because they can default on loans against it. The justi…cation for doing so is that society as a whole
is no better o¤ from this option, which merely redistributes resources from creditors to borrowers. Thus,
this option value should not be viewed as something intrinsic that makes the asset more valuable.
Next, following Harrison and Kreps (1978), I de…ne speculation to mean that agents assign a positive
value to the right to resell the asset when they purchase it. This de…nition is meant to capture the notion
that agents who buy an asset intend to pro…t by selling the asset rather than merely waiting to collect all
of its dividends.4 Consistent with these de…nitions, I will refer to a speculative bubble as a bubble where the
agents who buy it at date 1 would strictly prefer to sell it at date 2 in some state of the world.5 A speculative
bubble thus implies that the same assets trade hands multiple times in some states of the world. However, a
bubble asset may trade hands multiple times even if it does not meet the de…nition of a speculative bubble,
since agents who buy the asset at date 1 may in fact be indi¤erent about selling it at date 2. To distinguish
this case from the one in which the asset is a bubble but agents who buy at date 1 hold on to their assets
until d is realized, I borrow the terminology of Hong and Sraer (2011) and refer to the case where an asset
is a bubble that trades hands at most once as a quiet bubble. Likewise, I refer to a bubble that trades hands
more than once with some probability as a noisy bubble.6 A speculative bubble will always be noisy, but
not all noisy bubbles are speculative. As I show below, depending on parameter values, my model admits
4 Note that per this de…nition, …nitely-lived agents who buy in…nitely-lived assets are engaging in speculation. But this is
not because traders view selling the asset as inherently more pro…table than waiting to collect all of its dividends, which is the
notion Harrison and Kreps claimed to be after. Rather, it is because …nitely-lived agents cannot collect all dividend payments.
However, in both the Harrison and Kreps (1978) model and my model, agents live long enough to collect all dividends. In that
case, this de…nition for speculation does seem to capture the notion of trying to pro…t by selling the asset.
5 At …rst glance, the notions of a bubble and speculation may seem identical. On the one hand, if asset prices were always
equal to fundamentals, agents should be indi¤erent between selling an asset and holding it to maturity and the right to sell
the asset would be worthless. Thus, speculation would seem to imply a bubble. But this intuition breaks down if agents
value assets di¤erently, as occurs here where agents value the asset di¤erently because they borrow di¤erent amounts against
it. Although de…ning fundamentals is tricky when agents value the asset di¤erently, Barlevy and Fisher (2012) provide an
explicit example where prices can arguably be said to equal fundamentals but leveraged agents are engaged in speculation. In
the opposite direction, it might seem intuitive that assets can only trade above their fundamental value if agent expect to sell
their assets. However, the fact that bubbles can arise in static risk-shifting models where there is only one round of trading
such as Allen and Gale (2000) suggests an asset can be overvalued even without speculation.
6 Allen

and Gorton (1993) use the term “churning” to refer to the same phenomenon.

9

the possibility of no bubbles, quiet bubbles in which assets are overvalued but trade hands no more than
once, and noisy and speculative bubbles in which assets trade hands multiple times.
To solve for equilibrium, I work backwards from date 2. At this point, I2 2 f?; dg. Consider …rst the case

where I2 = d, i.e. where the dividend is revealed before agents trade. In this case, the equilibrium price

for the asset must be d. For suppose the price exceeded d. In that case, supply would be strictly positive:
Agents who own the asset earn more from selling the asset than holding it.7 But demand for the asset
would be zero, since agents with resources would prefer storage to buying the asset, while agents without
resources would have to default with certainty if they had to compensate creditors for their opportunity
cost. We can similarly rule out a price below d. In that case, demand for the asset would be strictly positive
since agents with resources could earn a higher return than from storage or from lending. At the same time,
supply would be zero, since agents would earn more holding the asset than selling it. This leaves p2 (d) = d.
Since p2 (d) is trivial to characterize, I will henceforth use p2 to refer to p2 (?), the price if d is not revealed.
The formal analysis for the case where I2 = ? is carried out in an Appendix. Here, I only sketch the

argument. Assumption (2) ensures that the return R on entrepreneurial activity is high enough to make it
pro…table to extend credit even when a fraction

of borrowers are non-entrepreneurs who are expected to

incur losses to their lenders. Thus, creditors will provide loans in equilibrium. Assumption (3) ensures that
the equilibrium price of the asset at date 2 is low enough that borrowing to buy risky assets and defaulting
if d = 0 is pro…table for non-entrepreneurs. Speci…cally, I derive the following result in the Appendix:
Lemma 1: Let pt denote the equilibrium price of the asset at state It = ?. If (2) and (3) hold, then

pt < D=R for t 2 f1; 2g.

Lemma 1 implies D=pt > R, i.e. holding the asset to maturity will yield more income if d = D than
an entrepreneur could be asked to pay. Pretending to be an entrepreneur and buying risky assets will
thus guarantee positive expected pro…ts. Since my assumptions on the various types ensure only nonentrepreneurs ever buy risky assets, computing demand is straightforward: Each of the n2 non-entrepreneurs
will borrow one unit of resources to buy assets, and so the amount of assets demanded at price p2 is n2 =p2 .
Turning to asset supply, those who own the asset at date 2 can include original owners who did not sell
their holdings at date 1 and non-entrepreneurs who showed up at date 1 and bought assets. In fact, using
assumptions (2) and (3), we can deduce that all n1 non-entrepreneurs who could have borrowed at date 1
would have done so, and hence n1 =p1 shares will be held at the beginning of date 2 by agents who bought
them at date 1, while 1

n1 =p1 shares will be held by original owners who held on to their asset at date

7 Here the fact that I focus on the limiting case where the cost of default k ! 0 is important, since it implies that even
agents who intend to default will strictly prefer to sell their asset holdings.

10

1. At date 2, the two groups have di¤erent reservation prices at which they would agree to sell the asset.
Original owners would sell the asset at any price p2 above D, their expected payo¤ from holding the asset.
Agents who borrowed one unit of resources at rate r1 at date 1 to buy assets have a di¤erent reservation
price. To see this, note that if they held on to their assets, their expected pro…ts would equal
D
p1

(1 + r1 )

(4)

If they sold their assets at date 2 instead, they would earn
p2
p1

(1 + r1 )

(5)

Comparing the two, they should be willing to sell their assets if
p2

D + (1

) (1 + r1 ) p1

(6)

Traders who bought assets at date 1 with borrowed funds have a higher reservation price than original
owners. Moreover, their reservation price is increasing in the rate r1 they are charged. Given the interest
rates charged at date 1, we can readily derive the supply schedule for the asset at date 2.
A market clearing price p2 is one at which supply and demand are equal. Figure 1 plots the supply and
demand curves assuming all borrowers are charged a single interest rate r1 at date 1. Under this assumption,
the supply schedule is a two-step function. The demand curve is a hyperbola that depends on n2 . Figure 1
shows the di¤erent ways supply and demand could intersect. The …gure suggests four cases are possible:

a. At least some original owners hold on to assets until the end of date 2. In this case, p2 = D.
b. All original owners sell by date 2, but no non-entrepreneur who bought at date 1 sells at date 2.
c. All original owners sell by date 2, and some non-entrepreneurs who bought at date 1 sell at date 2.
d. All original owners and all non-entrepreneurs who bought at date 1 sell at date 2.

Note that in cases (b)-(d), the price p2 exceeds D = E [dj?], i.e. there is a bubble. Although Figure 1
is suggestive, two important caveats are in order. First, it is only meant to illustrate the di¤erent ways in
which the demand curve could intersect a step-function. It does not correspond to the e¤ects of increasing
n2 , the number of non-entrepreneurs arriving at date 2. This is because changing n2 will in general a¤ect
the price of the asset at date 1, and will therefore a¤ect both demand and supply for the asset at date 2.
Second, the supply curve in Figure 1 is drawn assuming all borrowers at date 1 are charged the same rate
r1 in equilibrium. For cases (a), (b), and (d), this will indeed be the case. But in case (c), there will in fact
be two di¤erent rates o¤ered at date 1. To see why, suppose all borrowers were charged the same rate r1 at
11

date 1. Since in case (c) only some of the non-entrepreneurs who buy assets at date 1 sell them at date 2,
there must be some creditor who lends at date 1 and assigns probability less than 1 that a non-entrepreneur
who borrows from him would sell at date 2. Suppose this creditor charged a slightly lower rate. From (6),
we know that the reservation price is increasing in r1 . Hence, the creditor could induce a discrete jump in
the probability that a non-entrepreneur who borrows from him sells the asset. Inducing the borrower to
sell the asset and pay back his loan with certainty rather than hold on to the asset and pay back his loan if
d = D leads to a discrete rise in the creditors ex-ante expected pro…ts from the non-entrepreneur that can
more than o¤set the lower interest rate. Case (c) is thus incompatible with a single interest rate r1 .
Instead, in case (c), equilibrium requires two di¤erent interest rates at date 1, a low rate r1 and a high
rate r1 . Non-entrepreneurs who are charged r1 will sell the asset at date 2, while those charged r1 will
hold on to it. The supply curve in this case will be a three-step function, as shown in Figure 2. The number
of contracts o¤ered with each rate depends on the volume of trade between non-entrepreneurs who buy at
date 1 and non-entrepreneurs who buy at date 2. If we increase n2 , speci…cally if we increase the number of
borrowers m2 but keep the fraction of non-entrepreneurs

…xed, more non-entrepreneurs who buy at date

1 will have to sell at date 2, and so a larger fraction of the loans at date 1 will charge the lower rate r1 .
In all four cases (a)-(d), supply and demand for the asset intersect exactly once, so the market clearing
price p2 is unique. Moving back to date 1, we can similarly derive supply and demand for the asset.
Appealing to assumptions (2) and (3), all non-entrepreneurs will borrow one unit of resources and use it to
buy assets. Demand for the asset at date 1 is thus n1 =p1 . As for supply, original owners of the asset can
either sell the asset for p1 or hold the asset until date 2. Per my discussion above, original owners always
weakly prefer to sell their asset at date 2 regardless of the realization of I2 . Hence, waiting to sell the asset

would yield an expected payo¤ of q D + (1

q) p2 . This expression corresponds to the reservation price of

original owners at date 1. Given a value for p2 , there will be a unique equilibrium price p1 at date 1.
To summarize, the requirement that pt clear the market at each It yields a pair of conditions associated

with market clearing at I1 = ? and I2 = ? respectively that uniquely determine p1 and p2 . The market

clearing at date 1 implies p1 will depend on what agents believe about the price p2 , consistent with the
usual notion that asset prices are forward looking. More interestingly, market clearing at date 2 implies p2
will depend on the price p1 that prevailed at date 1. This non-traditional backward-looking aspect arises
because p1 governs the reservation price of agents who bought the assets at date 1. Since these agents are
leveraged, the value of their option to default will depend on the price of the asset at date 1.
The last step in solving for equilibrium is to use zero-pro…t conditions to pin down interest rates rt (It ).
When I2 = d, risk-shifting opportunities disappear. Loans are thus riskless, and competition will drive the
net interest rate on loans to 0, i.e. r2 (d) = 0. Once again, since interest rates are trivial to characterize in

this case, I will use r2 to refer to r2 (?). If I2 = ?, creditors who extend credit at date 2 will earn a return
12

r2 from entrepreneurs and from non-entrepreneurs if d = D, but will recoup nothing from non-entrepreneurs
if d = 0. Their expected pro…ts will equal 0 if r2 satis…es
(1

) r2 + [ (1 + r2 )

1] = 0

(7)

Note that r2 does not depend on the price of the asset. In particular, we have
r2 =
Moving to date 1, let

(1
)
(1
)

1

(8)

denote the probability that a non-entrepreneur who buys assets at date 1 will sell

them at date 2 if I2 = ?. As discussed above, equilibrium requires that

= 0 or

= 1, i.e. a creditor must

not anticipate that a non-entrepreneur who borrows from him will randomize whether he sells the asset or
not. For each , expected pro…ts must equal 0. Hence, r1 must satisfy
(1

) r1 +

[(1

q + q ) (1 + r1 )

1] + (1

) [ (1 + r1 )

1] = 0

(9)

A borrower who expects to sell his assets with certainty will be charged a lower rate, which I denote r1 ,
while a borrower who expects to hold on to his assets will be charged a higher rate, which I denote r1 .
Substituting in for

2 f0; 1g yields the following expressions
r1 =

q (1
)
;
1
q (1
)

r1 =

1

(1
)
(1
)

(10)

Remark 2: Note that r1 , r1 , and r2 are all positive. Hence, non-participants will never bene…t from
taking out a loan, which is why I can ignore them for now in analyzing the credit market.
The discussion above can be summarized as follows:
Proposition 1: Given (2), for each (n1 ; n2 ) 2 N

there exists a unique price path

2
fp (It )gt=1

(n1 ; n2 ) 2 R2++ : n1 + n2 < (1

q) D=R + q D ,

and a unique distribution of interest rates o¤ered at each It that

ensure market clearing and no positive expected pro…ts to lenders. N can thus be partitioned into regions
A, B, C, and D corresponding to the di¤erent types of asset market equilibria (a) - (d) when I2 = ?.
The regions A

D are illustrated graphically in Figure 3, and their boundaries are derived in the proof

of Proposition 1. In region A, asset prices equal fundamentals. Region B is associated with quiet bubbles
in which the asset trades above fundamentals but each asset trades hands no more than once. Region C
is associated with noisy bubbles but not with speculation, i.e. some traders who buy assets at date 1 will
turn around and sell them at date 2, but they are indi¤erent between selling at date 2 and holding on to
the asset. Lastly, region D corresponds to speculative (and thus noisy) bubbles: Traders who buy assets at
date 1 will strictly prefer to sell them if I2 = ?. The remainder of this section highlights several features
of the equilibrium and provides an economic interpretation for when the di¤erent cases arise.
13

3.1

Asset Price Levels

As evident from Figure 3, whether or not a bubble arises depends on the total number of traders n1 + n2 .
When n1 + n2 is small, speci…cally when n1 + n2 < D, the price of the asset will equal fundamentals in
both periods, i.e. pt = E [djIt ]. When n1 + n2 > D, the price of the asset can exceed its fundamental value
in date 1 and in date 2 if I2 = ?. In this case, the degree to which the asset is overvalued –i.e. the size of
the bubble component – is uniquely determined. This is in contrast to some other models of bubbles, e.g.
overlapping generations models, where the size of the bubble is indeterminate.
In particular, if a bubble exists, the size of the bubble component bt = pt

E [djIt ] depends on how

many non-entrepreneurs trade in assets markets. Here, it is important to distinguish between the absolute
number of such traders, nt , and their relative share among all borrowers that creditors …nance,
share parameter

. The

determines interest rates rt , but does not a¤ect the price of the asset directly other

than by a¤ecting rt . By contrast, the number of traders nt a¤ects asset prices, but does not a¤ect the
level of interest rates.8 Consider increasing the total number of borrowers mt while holding

…xed, i.e.

increasing both entrepreneurs and non-entrepreneurs while keeping their relative shares …xed. The proof of
Proposition 1 shows that p1 and p2 are both weakly increasing in n1 and n2 . Since E [djIt ] is constant, this
means b1 and b2 are also weakly increasing in n1 and n2 . In other words, the bubble component in asset
prices will be bigger the greater the aggregate amount that is borrowed against these assets.
The intuition behind this result is that the model gives rise to what Allen and Gale (1994) describe in
a di¤erent context as “cash-in-the-market pricing” meaning asset prices depend on the ratio of the cash
brought by asset buyers and the amount of assets up for sale.9 To better appreciate this, consider the case
where q = 0, so d will not be revealed until the end of date 2. Thus, there is no uncertainty about trade
at date 2. The only possible equilibrium is one where the asset trades at the same price in both periods,
and assets only trade hands once. This is because the price of the asset cannot rise between dates 1 and
2, or else all original owners will wait to sell at date 2, meaning no one will meet the demand for assets
from date-1 non-entrepreneurs. But the price of the asset also cannot fall, since in that case only those who
bought the asset at date 1 could sell it at date 2, yet their reservation price exceeds p1 . Hence, the unit
supply of the asset held by the original owners will be sold o¤ to non-entrepreneurs in exchange for the
resources they can bring to the asset market. Since each can borrow one unit, this means p1 = p2 = n1 + n2 .
8 The number of traders n will, however, a¤ect how many date 1 contracts are o¤ered with rates r
t
1 and r1 , respectively.
In particular, a higher n2 will imply more contracts that charge a low rate r1 . I will return to this result in Section 3.4.
9 In Allen and Gale (1994), agents choose between cash and assets in advance, and then a random number of agents are
hit with an immediate need for liquidity and must sell their assets for cash. If more agents are hit with liquidity shocks than
expected, assets trade below their fundamental value, with the price equal to the ratio of cash held by liquid agents and assets
held by illiquid agents. By contrast, here cash corresponds to the amount of resources agents can borrow and then use to buy
risky assets, and assets trade above their fundamental value.

14

This intuition for why higher n1 and n2 increase the size of the bubble continues to hold with uncertainty.

3.2

Asset Price Growth

Next, I examine the rate at which asset prices grow. Since I assume no time discounting, the risk-free rate
is zero. This implies that the price of the asset cannot rise unless the price of the asset at date 2 is itself
uncertain, or else agents could earn above the risk-free rate with no risk. Indeed, when q = 0 so the only
possible state at date 2 is I2 = ?, I argued above that p1 = p2 . But when q > 0, the price of the asset at

date 2 will depend on whether d is revealed, and if it is revealed, on the value of d. In this case, asset prices
can appreciate in some states of the world. Clearly, the price of the asset will rise if d is revealed to be D.
The more relevant question for understanding historical episodes often taken to be bubbles is the rate at
which asset prices grow if d is not revealed. That is, can assets become increasingly overvalued over time
even when there is no commensurate growth in the fundamental value of the asset?
The rate at which the asset price appreciates if d remains hidden turns out to depend on whether in
equilibrium all original owners sell their assets at date 1 or only some do. When only some of sell their
assets at t = 1, they must be indi¤erent between selling the asset and waiting to sell at date 2. This implies
p1 = q D + (1

q) p2

Let bt denote the size of the bubble at date t, i.e. bt = pt

(11)

E [djIt ]. The price of the asset at date 1 can

thus be expressed as the sum of the fundamental value and a bubble component, i.e.
p1 = D + b1

(12)

Substituting this expression for p1 into (11) and solving for p2 yields
p2 = D +
Thus, the bubble component b2 = p2

b1
1

q

(13)

D will be larger than the bubble component b1 at date 1 as long as

q < 1. That is, the asset will appreciate in price and becomes increasingly more overvalued. These dynamics
are identical to those derived by Blanchard and Watson (1982), who showed that when traders are rational,
bubbles grow at a risk-adjusted interest rate. Intuitively, the original owners of the asset require some
compensation to hold the asset at date 1, since by holding the asset they risk giving up the opportunity
to sell an overvalued asset. This compensation accrues in the form of capital gains if the bubble survives,
allowing them to earn potentially higher pro…ts if they wait. Previous risk-shifting models of bubbles such
as Allen and Gorton (1993) and Allen and Gale (2000) do not give rise to these pricing dynamics because
they require original owners to sell their assets rather than let them trade strategically.
When all original owners sell o¤ their holdings at date 1, though, the growth rate of asset prices can deviate
from the dynamics derived by Blanchard and Watson (1982). In this case, the original owners must weakly
15

prefer to sell the asset for price p1 at date 1 than to hold it and sell at date 2, and so p1
Thus, (1

q)

1

q D + (1

q) p2 .

represents an upper bound on the rate at which the bubble can grow. But there is also a

lower bound on the rate at which the bubble can grow. In particular, if all original owners sell their assets
at date 1, non-entrepreneurs who show up at date 2 will have to buy assets from non-entrepreneurs who
arrived at date 1. The latter will only agree to sell if the price is at least equal to their reservation price,
D + (1

) (1 + r1 ) p1 . Hence, the price of the asset p2 must satisfy
D + (1

) p1 (1 + r1 )

p2

D+

b1
1

q

We know from Lemma 1 that p1 (1 + r1 ) < D, and so the lower bound on p2 implies that p2 =p1

(14)
(1 + r1 ),

i.e. the price of the asset must grow faster than the rate at which non-entrepreneurs must pay to borrow
resources.10 This is because non-entrepreneurs who sell the asset at date 1 must not only earn enough to
pay their creditors, but must be compensated for the option value of default they give up and which they
could have exercised if they held on to their assets. Intuitively, price appreciation in this case is driven by
the fact that an increase in price allows for trade. In particular, if the price of the asset rises, agents who
buy the asset borrow more against each asset than those from whom they buy the assets. As a result, the
option to default is more valuable for new buyers than it is to its existing owners, meaning they will value
the asset more than the original cohort of buyers. The implied rate of price appreciation may be slower
than the growth rate due to the considerations …rst pointed out by Blanchard and Watson (1982). Since
this lower rate of price appreciation occurs when the original owners sell all of their asset holdings at date
1, asset price appreciation will generally be lower when trade volume in the bubble asset is high early on.
In sum, the rate at which the bubble component grows depends on when traders show up and is bounded
by (1

q)

1

, a bound which is increasing in the probability of the bubble bursting. This insight suggests why

the phenomena that distinguish historical episodes suspected to be bubbles, e.g. rapid price appreciation,
might be rare. When rapid appreciation is possible, i.e. when q is large, appreciation is also less likely
to materialize since the bubble is likely to burst. However, the fact that we observe rapid asset price
appreciation only infrequently does not imply that bubbles – instances in which assets are overvalued –
are also inherently rare. In particular, quiet bubbles associated with low values of q can be quite common,
although they are also harder to identify empirically since this requires estimating fundamentals.

3.3

Quiet Bubbles, Noisy Bubbles, and Speculation

I now turn to the predictions of the model for trading patterns. In particular, if the asset is overvalued, will
it be a quiet or a noisy bubble? The answer to this question can be inferred from Figure 3. The …gure shows
1 0 Since the borrowing rate r
1 exceeds the risk free rate, the asset must appreciate faster than the risk-free rate. In models
where bubbles cannot have a …nite end-date, e.g. overlapping generation models, this is problematic: It requires the economy
to grow faster than the risk-free rate to ensure agents can always a¤ord the asset. This in turns implies the economy is
dynamically ine¢ cient. Here, the bubble ends in …nite time, and so there is no need for high rates of economic growth.

16

the nature of the asset market equilibrium at date 2 for di¤erent values of n1 and n2 . Recall that region
B corresponds to quiet bubbles where assets trade hands only once, while regions C and D correspond to
noisy bubbles where at least some of the agents who buy assets at date 1 will sell them at date 2. Figure
3 is drawn as if all four types of equilibria are possible. However, this will not be true in general. In
particular, whether equilibria corresponding to regions C and D are possible depends on the location of the
outer boundary for region B relative to the outer boundary of the set N . If the outer boundary for region B
–whose location depends on the parameters q, , and

–is su¢ ciently far away from the origin, then the

only type of equilibrium bubbles that can arise for (n1 ; n2 ) 2 N are quiet bubbles. Whether speculation
and noisy bubbles can arise depends on these parameters.

The formal analysis of how these parameters a¤ect the types of equilibria is contained in the Appendix.
Here, I only review the results. Consider …rst q, which I argued in Section 3.2 governs the rate at which
the bubble can grow. As q ! 0, region B expands outwards to cover all pairs (n1 ; n2 ) 2 N for which

n1 + n2 > D. In other words, for small q, only quiet bubbles are possible. Intuitively, the smaller the

risk of the bubble collapsing, the less the bubble can grow over time without inducing agents to hold on to
the asset. But if asset prices do not grow much over time, traders who borrow to buy the asset will not
earn enough when they sell the asset to cover their interest obligation and the option value to default they
give up by selling the asset. Hence, noisy bubbles will not arise for low values of q: Can they arise for high
values? As q ! 1, region B shrinks towards the origin, meaning fewer values of (n1 ; n2 ) will be associated
with quiet bubbles. However, the set N also shrinks. In the Appendix, I show that if R is close to its upper

bound of 1= , noisy bubbles will exist for some (n1 ; n2 ) 2 N when q is close to 1. When R is close to its
lower bound in (2), noisy bubbles still exist for some (n1 ; n2 ) 2 N when q is close to 1 and either

or

are

large. Since q governs the rate at which asset prices can appreciate, it follows that noisy bubbles are more
likely to arise when price appreciation is high. As in Section 3.2, it also follows that speculative trading will
be rarely observed, since when it can take place the bubble is likely to burst before speculators can sell.
Turning to , in the limit as ! 1, any (n1 ; n2 ) 2 N for which n1 + n2 > D must fall in region B. Thus,

when

is large, if there is a bubble it must be a quiet bubble. In the opposite direction, as

long as q < 1, in the limit all (n1 ; n2 ) 2 N lie in either region C or D. That is, when

must arise and are inherently noisy.

11

! 0, then as

is small, bubbles

To understand this result, note that for small values of , holding

on to the asset is likely to be unpro…table, since d is likely to assume its low realization. Thus, a high
will encourage those who buy the asset early to turn around and sell it rather than hold on to it. This
suggests penny stocks, i.e. stocks that trade at low prices but have a skewed return distribution, should
be particularly vulnerable to speculative bubbles. Interestingly, Eraker and Ready (2013) …nd that such
1 1 Interestingly, Hong and Sraer (2011) also …nd that noisy bubbles are more likely for assets with a high upside potential,
but for di¤erent reasons. In particular, a high upside in their model allows for more potential for disagreement as to the value
of the asset, which encourages trade. This feature is absent from my setting.

17

stocks seem to be overvalued, since they tend to have a negative expected return that cannot be explained
by asset characteristics. As for , region B becomes larger as
to be associated with quiet bubbles. Intuitively, a higher

rises, i.e. higher values for

increases the cost of borrowing, which raises the

reservation price of agents who borrowed to buy the asset. High values of
While q, , and

are more likely

thus discourage turnover.

determine whether bubbles can be noisy, it is clear from Figure 3 that when noisy

bubbles are possible, they occur only when either n1 or n2 are large. Intuitively, noisy bubbles require that
either a lot of traders buy at date 1 who can later sell, or else a large number arrive and wish to buy assets
at date 2. Recall from Subsection 3.1 that other things equal, higher values of n1 and n2 imply the asset
will be more overvalued. This implies that noisy bubbles will tend to be associated with assets that are
more overvalued, while assets that are only slightly overvalued will be associated with quiet bubbles. What
is not evident from Figure 3 is that too much overvaluation discourages noisy bubbles. In particular, if
n1 + n2 > q D + (1

) D=R, we can no longer use (3) to ensure that all non-entrepreneurs will buy the

asset. Although I omit the formal analysis of this case, with a large number of traders the expected pro…ts
from buying the asset will turn to zero. This will reduce the volume of trade associated with the asset.
Although either n1 or n2 must be large for bubbles to turn noisy, the role of n1 and n2 in contributing to
noisy bubbles is not symmetric. On the one hand, holding the total number of non-entrepreneurs n1 + n2
…xed, a noisy bubble is more likely to arise if most traders arrive early rather than late, i.e. when n1 is large
rather than n2 . Intuitively, if a lot of traders show up at date 1, they will have to be those who sell the asset
at date 2 if there is additional demand. But if few traders show up at date 1, additional demand at date
2 might still be potentially met by original owners. At the same time, the features most associated with
historical episodes suspected to be bubbles arise when n2 is large rather than n1 . First, recall from Section
3.2 that asset price growth is likely to be higher when n1 is small. In addition, Figure 3 reveals that region
D where speculative bubbles arise requires that n2 be large relative to n1 . Thus, scenarios with rapid asset
price growth and traders who are keen to both buy and sell assets are more likely to be associated with a
rising trade volume rather than a falling one.

3.4

Borrowing Rates in Credit Markets

Finally, many of the historical episodes suspected to be bubbles are associated with low interest rates charged
to those who borrow against supposedly bubble assets. This may seem at odds with the risk-shifting view of
bubbles: Since risk-shifting requires creditors to charge higher spreads to cover losses on speculators, spreads
should presumably be higher with bubbles than when bubbles are absent. However, my model implies that
noisy bubbles can be associated with lower borrowing rates than quiet bubbles. Whether historical episodes
should be associated with high or low borrowing spreads depends on whether the tranquil times against
which these episodes are judged are periods in which asset prices re‡ect fundamentals or in which the assets

18

exhibit quiet bubbles. In principle, as noted by Diba and Grossman (1987), assets that can potentially
exhibit bubbles in the future must contain a bubble component in the present, even if it is arbitrarily small.
Since we argued in Section 3.3 that small overvaluation is likely to be associated with quiet bubbles, it
seems reasonable to view tranquil times as periods in which the asset is slightly overvalued.
To see that noisy bubbles can be associated with lower borrowing spreads than quiet bubbles, consider
the e¤ect of increasing the absolute number of non-entrepreneurs n2 who arrive at date 2 while holding
their share among total borrowers

…xed. Graphically, this implies moving up along Figure 3. The higher

is n2 , the larger the fraction of date 1 buyers who will have to sell their assets. Thus, as long as we are in
an equilibrium of type (c), more borrowers will be o¤ered a low rate r1 rather than a high rate r1 , until
eventually we move into an equilibrium of type (d) in which all the agents who buy at date 1 will sell at
date 2 if the payo¤ on the asset remains uncertain. The intuition for this is due to dynamic risk-shifting:
When traders turn around and sell the overvalued assets they buy, their lenders will anticipate that part
of the risk from lending against an overvalued asset will be borne by future lenders. This allows them to
charge lower rates to their borrowers. However, since noisy bubbles are more likely at high values of q, the
lower rate r1 may not be substantially lower than r1 since much of the risk is still associated with early
periods. In the next section, where I let creditors choose from a larger set on contracts, I discuss another
reason for why speculative bubbles may be associated with low spreads.

3.5

Summary

The preceding analysis shows that risk-shifting models of bubbles in which assets can trade above their
fundamental value may give rise to patterns consistent with historical episodes taken to be bubbles, e.g. rapid
price appreciation, speculative trading, and low spreads charged to those who borrow against assets used
for speculative trading. However, an overvalued asset will not always exhibit these patterns. Assets that are
only slightly overvalued will tend to trade infrequently and can exhibit low rates of price appreciation. More
overvalued assets, especially those with a high trade volume and against which a large amount is borrowed,
are those that are prone to become speculative bubbles. But since rapid appreciation and speculation can
only be high when bubbles are likely to burst early, such patterns should be observed relatively infrequently.

4

Endogenous Contracts

Up to now, I allowed creditors to o¤er only …xed-size, full-recourse, simple debt contracts. In this section,
I allow lenders to choose from a broader set of contracts. This serves two purposes. First, it o¤ers a
robustness exercise: Can speculative bubbles still emerge when lenders can choose from a broader set of
contracts? Second, it reveals what types of contracts will emerge when speculative bubbles arise. To

19

preview my results, I …nd that endogenizing contracts makes it harder but not impossible for bubbles to
occur: Although lenders would like to avoid funding agents who plan to buy overvalued assets, they will not
always be able to avoid doing so. When lenders cannot avoid taking on non-entrepreneurs, they will design
contracts to minimize losses on non-entrepreneurs, and the purchase of overvalued assets will be …nanced
using certain types of contracts. In particular, my model gives rise to Spence-Miyazaki-Wilson contracts,
along the lines of Spence (1977), Miyazaki (1977), and Wilson (1977), whereby creditors earn pro…ts on
entrepreneurs that cross-subsidize the minimal losses possible on non-entrepreneurs.
Analyzing contract choice requires being more explicit about what agents know and can condition their
contacts on. In line with my focus on debt contracts, I consider a costly state veri…cation model as in
Townsend (1979) and Gale and Hellwig (1985) that gives rise to debt-like contracts. That is, creditors
cannot immediately observe what assets borrowers purchased. However, at the end of date 2, creditors can
learn what assets the borrower purchased as well as their cash holdings by paying a cost. The auditing
cost is assumed to be a fraction

< 1 of the borrower’s cash holdings, meaning it is less costly to audit

agents with fewer resources to hide. Assuming a purely proportional cost is analytically convenient, since
it implies an agent with no cash can be audited at no cost. Adding a …xed cost of veri…cation would not
fundamentally change my results as long as the …xed cost component was small, but it would raise the issue
of stochastic auditing as a way to minimize auditing costs. I wish to avoid randomization, not just because
it is rarely employed in practice but because it raises some technical issues that I discuss below. In what
follows, I assume

is close to 1. As I discuss below, a su¢ ciently high value for

will deter lenders from

auditing borrowers who claim to be entrepreneurs unless they fail to pay the amount required of them.
Appealing to the revelation principle, we can replicate any contract with a direct mechanism in which
borrowers report their private information, and these reports trigger transfers as well as an audit strategy.
Transfers can only depend on the borrower’s actual private information if the lender audits the borrower.
Contracts can of course still depend on publicly observable information, e.g. payments can depend on I2 or
d even if creditors do not observe whether a given borrower bought risky assets. However, this conditioning
is possible only because I’ve assumed one type of risky asset. With more than one type of risk asset, such
conditioning might not be possible. In particular, suppose that we allowed for a continuum of ex-ante
identical risky assets, all of which pay a single dividend d distributed according to (1). However, ex-post,
only a fraction

pay out d = D, and for a fraction q of each type of asset the value of d is revealed before

date 2. Non-entrepreneurs who secure funds would prefer to concentrate in one asset rather than diversify,
and each period non-entrepreneurs would spread out uniformly across all assets whose dividend remains
uncertain. As long as we adjust the number of traders in period 2 to account for the fact that fewer assets
will trade at date 2 than at date 1, we can reinterpret the model in the previous section with only one type
of asset as one with di¤erent assets but no aggregate uncertainty and thus nothing for lenders to condition

20

on.12 Motivated by this interpretation, I consider contracts that do not condition on asset market variables.
This also leads to a contract that more closely resembles a standard debt contract.13
Formally, a contract amounts to a sequence of announcements by borrowers at the various stages where
they have private information, and a sequence of transfers and actions triggered by these announcements. I
use bj denote the reports at stage j = 1; 2; 3; ::: and xj (bj ) as the transfer from the borrower following each

report. Figure 4 summarizes the contract for agents who arrive at date 1 as a ‡ow chart. The contract for
agents who arrive at date 2 can be constructed similarly.

A contract begins with an agent reporting upon his arrival whether he is an entrepreneur (denoted b1 = e)
or not (b1 = n). While I could allow non-entrepreneurs and non-participants to distinguish themselves, this

turns out to be unnecessary. The contract then stipulates a transfer x1 (b1 ) from the creditor to the borrower.
Since agents own no resources, x1 (b1 ) 0.
The next stage at which the agent has private information is after he chooses what to do with the funds

he received. Assumptions (2) and (3) ensure agents will not be indi¤erent between actions, and will allocate
all of the funds they receive to a single use. This simpli…es what agents can report. An agent who reported
he was an entrepreneur can only truthfully report that he either initiated entrepreneurial activity (b2 = e)

or did nothing (b2 = ?), and so without loss of generality I restrict agents to these two reports. An agent
who reported he was not an entrepreneur can only truthfully report that he bought assets (b2 = b) or did

nothing (b2 = ?), so I restrict him to these two reports. Let x2 (b1 ; b2 ) denote the transfer from the lender
to the borrower following this report. If the agent reports that he used his funds, i.e. b2 2 fe; bg, he would

have no resources to transfer if he were truthful, so x2 (b1 ; b2 ) 0. If the agent reports he did nothing, the
constraint is x1 (b1 ) + x2 (b1 ; b2 ) 0, i.e. he cannot transfer more resources than he originally received.
For agents who arrive at date 1, the next stage at which they may receive private information is at the

beginning of date 2, after the asset market clears but before d and R are paid out. At this point, an agent
who reported buying assets, i.e. b2 = b, must report whether he sold his assets before d was revealed
(b3 = p2 ), sold them after learning d is high (b3 = D), sold them after learning d is low (b3 = 0), or held

on to them (b3 = h). An agent who reports b2 2 fe; ?g would have nothing to report at this stage if he
were truthful. For completeness, I let them report b3 = ?. Let x3 (b1 ; b2 ; b3 ) denote the net transfer from
the lender to the borrower after report b3 is submitted.
1 2 In

particular, if we want the mass of traders at date 2 to equal n2 , the number of traders who arrive must equal n2 = (1

q).

1 3 Allowing contracts to condition on the asset market would not change my qualititative results. In particular, it would
lead creditors to charge high payments in states where non-entrepreneurs earn pro…ts, e.g. if d = D, and low (even negative)
payments in other states. However, the most a contract can demand from entrepreneurs in any state is R, and assumption
(3) implies the return to buying the risk asset and holding it to maturity exceeds R. Hence, non-entreperneurs would want to
trade with creditors if contracts were contingent on asset market data.

21

At the end of date 2, an agent must issue a …nal report on his earnings. I denote this report yb. If the
initial transfer x1 (b1 ) > 0, it will be convenient to let yb denote earnings divided by x1 (b1 ). If x1 (b1 ) = 0,
the agent cannot do anything anyway, so I can restrict him to reporting yb = 0. When x1 (b1 ) > 0, …ve
reports are possible for yb: R

assets before learning d; D=p1

1, if the agent chose to produce; p2 =p1

1, if the agent bought and sold

1, if d = D and the agent did not sell the assets before d was revealed;

1, if d = 0 and the agent did not sell the asset before d was revealed; and 0, if the agent did nothing. I

let the agent report any of these yb regardless of previous reports. In particular, I need to let an agent who

bought assets but pretended to engage in entrepreneurial activity come clean if d = 0. After these reports,
the lender can choose whether to audit the agent. This means that the contract must specify an auditing
rule yb(b1 ; b2 ; b3 ; yb) equal to 1 if the agent is audited and 0 otherwise. I do not allow for stochastic audits

where 0 <

y
b

< 1. This restriction prevents a lender from o¤ering a contract with a slightly higher audit

probability and a slightly lower interest rate that would attract entrepreneurs who have no reason to fear
auditing. Allowing lenders to o¤er such contracts would imply no equilibrium exists, for the reason pointed
out in Rothschild and Stiglitz (1976): Any potential equilibrium would be vulnerable to cherry-picking of
good types. Restricting contracts to deterministic audits is one way to address this non-existence problem.
Other approaches would allow for stochastic auditing, but are considerably more involved.14
Finally, the contract stipulates a transfer xyb(b1 ; b2 ; b3 ; yb) from the lender to the borrower if there is no
audit, and a transfer xy (b1 ; b2 ; b3 ; yb; y) if there is an audit, where y denotes actual earnings.
Now that I’ve described contracts, I can proceed to characterize equilibrium contracts. Since the lender

chooses contracts optimally, I begin with some observations about optimal contracts. First, suppose a
borrower announces he has no cash at the end of date 2, i.e.
x1 (b1 ) + x2 (b1 ; b2 ) + x3 (b1 ; b2 ; b3 ) + ybx1 (b1 ) = 0

If this report is truthful, the agent will not be able to make any transfers, and so xyb(b1 ; b2 ; b3 ; yb)

(15)
0. As

such, it will be optimal for the lender to audit the borrower and grab all of his resources if he is untruthful.
The reason is that if the agent is telling the truth then the audit is costless, while if he is not telling the
truth the agent can be punished. Since

< 1, auditing and punishing an agent is never costly for the

lender. At the same time, punishing false reports relaxes incentive constraints, so the lender should do it.
A second observation is that a lender can use transfers to limit the misreports agents can make. In
particular, if x1 (b1 ) > 0, the lender can con…rm a report that the agent did nothing with his funds, i.e.
b2 = ?, by asking the agent to transfer back his resources, i.e. x2 (b1 ; b2 ) = x1 (b1 ). This would only
1 4 Wilson (1977), Miyazaki (1977), and Spence (1977) suggested dealing with this non-existence by rede…ning equilibrium so
that an equilibrium contract cannot be improved upon only by contracts that remain pro…table when unpro…table contracts
are withdrawn. Mimra and Wambach (2011) and Netzer and Scheuer (2013) provide game-theoretic foundations for this idea
by allowing lenders withdraw contracts (possibly at a small cost) after observing the contracts o¤ered by others.

22

be feasible if

2

= ?. Subsequent transfers can then be used to achieve any desired terminal payo¤ for

this type. Likewise, the lender can con…rm whether an agent who bought assets at date 1 sold them at
date 2 for a positive price, i.e. b3 2 fp2 ; Dg, by asking the agent to transfer more than x1 (b1 ) after this

announcement, i.e. x2 (b1 ; b2 ) < x1 (b1 ). Again, this would only be feasible if the agent in fact bought and

sold assets. Lastly, the lender can verify whether the agent earned D per asset by requiring an agent who

reports yb = D=p1

1 to transfer D=p1 . The upper bound on R in (2), together with (3), puts a limit on

the price p2 , implying that D=p2 > R. Thus, a non-entrepreneur who buys assets has enough funds to pass
himself o¤ as an entrepreneur, but not vice-versa. Note that con…rming an agent earned y = D=p1 1 may
require an additional transfer after xyb(b1 ; b2 ; b3 ; yb) to achieve a given payo¤ to the borrower, i.e. the agent
and lender must transfer resources back and forth. These transfers greatly limit the type of misreports by
agents and thus reduce the number of incentive constraints the contract is subject to.

The last observation is that it will be optimal for the lender to audit the agent and seize all of his cash
if the agent is ever inconsistent in his reports, e.g. announcing that he bought assets but then reporting
yb = R 1. This implies that in equilibrium, an agent who reports he chose to produce, i.e. b2 = e, will have

to pay some amount if he reports his income is R

1, and will be audited and left with no cash otherwise.
Likewise, an agent who reports he bought assets, i.e. b2 = b, will have to pay some amount that may depend
on what he says he did with the assets he bought, and will be audited and left with no cash otherwise.

We can use the observations above to determine when the equilibrium contract will call for an audit.
First, without loss of generality, an agent who announces yb =

1 without contradicting past reports will be

audited. This is because we can always take the equilibrium contract and construct an equivalent contract
with x2 (b1 ; b2 ) = x3 (b1 ; b2 ; b3 ) = 0 for reports consistent with a report of yb = 1, ensuring the agent will
have no cash at the end of date 2 if the true y =

1. This allows the lender to audit at no cost if the agent

is truthful while discouraging misreports. If we set xyb( ) to ensure the same terminal payo¤ to a truthful

agent as in the original contract, this contract will do at least as well as the original equilibrium contract.

Next, suppose the agent reports yb = 0 or yb = D=p1 1. Since the lender can directly con…rm these reports

using transfers, neither report will be audited in equilibrium. The lender cannot fully verify yb = p2 =p1

1,

since an agent who sells his assets early at price D can pass himself as this type. Since lenders prefer non-

entrepreneurs sell their assets if d is not revealed, they may want to audit such reports to reward agents.
For now, I take no stand on whether this report will be audited, although I argue below that it will not.
Finally, suppose the agent reports yb = R

borrower following this report if
this report. As

1. As anticipated above, it will be optimal not to audit the

is close to 1. For suppose the optimal contract did audit agents making

! 1, the rate that must be charged to entrepreneurs to ensure non-negative pro…ts must

tend to R, since auditing costs on these agents are equal to R while the expected pro…t from lending to
non-entrepreneurs is negative. At the same time, the …rst inequality in (2) implies there exists a rate strictly
23

below R such that charging this rate to all borrowers and collecting it in full from entrepreneurs and with
probability

from non-entrepreneurs will yield positive pro…ts. Since the latter is a lower bound on the

pro…ts of the lender, it follows that as

" 1, not auditing a report of yb = R

1 will be optimal.

It follows that equilibrium contracts are essentially variable-size debt contracts with a repayment schedule:
b
An agent who arrives at date 1 and reports b1 2 fe; ng receives x 1 resources together with a particular
1

repayment schedule. An agent who reports b1 = e will be asked to pay back (1 + r1e ) xe1 at the end of date
2 or else be audited. An agent who reports b1 = n will be asked to pay back an amount that depends on
when he pays. If he waits until the end of period 2, meaning he reported holding on to his assets (b3 = h),

he will have to pay xn1 (1 + r1n;late ) and will be audited otherwise. If he pays at the beginning of date 2,
meaning he sold assets for a positive price, he would owe xn1 (1 + r1n;D ) if he sold at price D and a potentially
di¤erent amount xn1 (1 + r1n;p2 ) if he sold at price p2 . If he pays earlier still at date 1, meaning he did nothing
(b2 = ?), he would owe xn1 (1 + r1n;? ). That is, the borrower either receives xe1 with a requirement to repay
n
o
at rate r1e at the end of date 2, or he receives xn1 with a rate schedule r1n;? ; r1n;D ; r1n;p2 ; r1n;late . This
description abstracts from the back-and-forth transfers to verify the borrower’s reports that are needed to
sustain the contract, but it correctly capture the terminal payo¤s to the lender.
By a similar argument, an agent who arrives at date 2 receives a variable-size debt contract depending
b
on his report b1 2 fe; ng that speci…es an initial transfer x21 and an amount he must repay at the end of

period 2, but which might charge a di¤erent rate if he shows he did nothing with the funds he received.

Given the optimal audit rules I derived above, agents who report b1 = n will at some point be either

audited or forced to make a transfer that reveals their identity. As such, we need not worry about whether
entrepreneurs will try to report b1 = n. Instead, the relevant incentive constraints are that non-entrepreneurs
not try to pass themselves as entrepreneurs or misreport when they sold assets or for how much. Alas for
lenders, these incentive constraints have real bite, since non-entrepreneurs can secure themselves positive

expected pro…ts from pretending to be entrepreneurs. In particular, they can buy risky assets and hold them
to the end of date 2, then pay back no more than R < D=p if d = D, all without being audited. Creditors
who want to to fund entrepreneurs thus have no choice but to also take on non-entrepreneurs and o¤er them
rents. However, it does not follow from this that once creditors …nance entrepreneurs, non-entrepreneurs
must end up buying risky assets. Rather, creditors can provide rents to non-entrepreneurs by paying them
o¤ rather than lending to them. Formally, they can o¤er r1n;? < 0, essentially paying non-entrepreneurs not
to buy assets. The next proposition establishes that if all agents without a production opportunity can buy
risky assets –meaning all can participate in the asset market –it would indeed be optimal for creditors to
bribe non-entrepreneurs not to buy the asset, and a bubble would not emerge.
Proposition 2: If there are no non-participants, allowing lenders to choose contracts optimally implies
that the equilibrium price of the asset must be p1 = p2 = D, i.e. bubbles will not emerge.
24

Intuitively, letting a non-entrepreneur buy an overvalued asset is a costly way for the creditor to provide
him with rents: When the asset is overvalued, the creditor ends up paying some resources to whoever the
non-entrepreneur buys the asset from beyond the rents he provides the non-entrepreneur. Thus, where there
is a bubble, it is more e¢ cient to pay agents directly than to reward them by having them buy assets.
Paying agents not to buy risky assets may not work well in practice, however. The reason is that it
may draw in people who have no intention to speculate. Here is where non-participants …nally play a role.
While they do nothing in equilibrium, their presence precludes creditors from paying non-entrepreneurs not
to speculate. As long as there are enough non-participants, the cost of having to pay them together with
non-entrepreneurs will be enough to make contracts with r1n;? < 0 too costly, and creditors who need to
deliver rents to non-entrepreneurs will choose to do so by letting the latter purchase risky assets.
Formally, if there is a large enough group of non-participants, any contract must satisfy the property
that if the cumulative transfers from the creditor to the borrower are positive – i.e. if the lender ends up
giving any gift to the borrower – it must be that the borrower has zero cash holdings at the end of date
2. Otherwise, agents with no intention to buy risky assets will also accept the contract. Since an agent
with zero cash holdings will be audited in equilibrium, this restriction can be formalized as follows. First,
if there is no audit, net transfers from the lender to the borrower must be non-positive:
x1 (b1 ) + x2 (b1 ; b2 ) + x3 (b1 ; b2 ; b3 ) + xyb(b1 ; b2 ; b3 ; yb)

0

(16)

Second, if there is a positive transfer of resources from the creditor to the agent with auditing, then the
terminal cash holdings of the agent must be 0, i.e.

implies

x1 (b1 ) + x2 (b1 ; b2 ) + x3 (b1 ; b2 ; b3 ) + xy (b1 ; b2 ; b3 ; y) > 0

(17)

x1 (b1 ) + x2 (b1 ; b2 ) + x3 (b1 ; b2 ; b3 ) + xy (b1 ; b2 ; b3 ; y; y) + y x1 (b1 ) = 0

(18)

Under these conditions, non-participants will never …nd it pro…table to borrow. We can therefore ignore
them for the remainder of our discussion: Creditors will never have to engage them, and their only role
comes from imposing the additional constraints (17) and (18) on contracts.
Since the …rst inequality in (2) ensures it will be pro…table to …nance entrepreneurs, we can conclude that
xe1

= xe2 = 1, i.e. entrepreneurs receive the maximum amount of funding they can employ productively.

Otherwise, a lender could simply scale xet and xnt up proportionally but charge entrepreneurs slightly more
than under the original contract, a small enough increase that they would still prefer to borrow a larger
amount at a slightly higher rate. Since entrepreneurs cannot pass themselves o¤ as non-entrepreneurs, they
will choose the contract designed for them. Non-entrepreneurs will prefer the contract o¤ered to them
given that under the original equilibrium they preferred the contract o¤ered them when entrepreneurs were
25

charged a lower rate. Since scaling up both loans and leaving interest rates …xed would leave pro…ts at zero,
charging entrepreneurs higher rates ensures positive pro…ts. Hence, xet < 1 cannot be an equilibrium.
The only remaining terms of the contracts o¤ered to entrepreneurs are the rates r1e and r2e . Since lenders
earn zero expected pro…ts in equilibrium, the amount collected from entrepreneurs must equal the expected
losses on non-entrepreneurs. This, in turn, depends on the terms o¤ered to non-entrepreneurs. To solve for
these, I begin with a preliminary result. Let Vt ( 1 ; b1 ) denote the expected utility to an agent of type 1 who
announces his type as b1 and then behaves optimally. The next lemma establishes that in equilibrium, a
non-entrepreneur will be indi¤erent about what type he announces:
Lemma 2: In equilibrium, Vt (n; n) = Vt (n; e)
Lemma 2 implies that the equilibrium contract o¤ered to non-entrepreneurs solves the problem of maximizing the expected revenue from non-entrepreneurs while leaving the non-entrepreneur with the same
utility as if he reported b1 = e. For suppose the equilibrium contract did not solve this problem, i.e. there

exists a contract x0 that o¤ers identical terms to entrepreneurs and leaves non-entrepreneurs with the same
utility Vt (n; e) but collects more revenue from them. A lender could then o¤er a contract x00 equivalent to x0

but which charges non-entrepreneurs slightly less so that their utility under x00 is still higher than Vt (n; e).
This allows the lender to also charge entrepreneurs slightly less than under the original equilibrium contract
without inducing non-entrepreneurs to report b1 = e. Since the original contract earned zero pro…ts, x0

would generate positive pro…ts if it attracted both types since it collects more from non-entrepreneurs but
the same amount from entrepreneurs. Since the pro…ts from x00 can be made arbitrarily close to the pro…ts
from x0 when it attracts both types, it follows that the original contract could not have been an equilibrium.
Essentially, the creditor would like to grab as much resources from non-entrepreneurs to make his contract
as attractive as possible to entrepreneurs.
Applying this insight, we can solve for the equilibrium contracts o¤ered to agents who show up at date 2.
In particular, the terms o¤ered to non-entrepreneurs who arrive at this date can be summarized by a loan
size xn2 and a repayment rate r2n that satisfy
max xn2 [ (1 + r2n )

n
xn
2 ;r2

1]

(19)

subject to
i.
ii.

r2n

D
p2

(1 + r2n ) xn2 =

D
p2

(1 + r2e )

0

Constraint (i) corresponds to the requirement that V2 (n; n) = V2 (n; e). Constraint (ii) follows from the
requirement that rates cannot be negative given (17) and (18). Solving (19) yields

26

Proposition 3: In an equilibrium where p2 > D, non-entrepreneurs will be o¤ered a contract in which
xn2

=

D (1+r2e )p2
D p2

< 1 and r2n = 0, while entrepreneurs will be o¤ered a contract where xe2 = 1 and r2e solves
(1

) r2e + xn2 ( r2e

(1

)) = 0:

To understand this result, observe that creditors always want to lend less to non-entrepreneurs, but they
need to charge a lower rate on such loans to ensure non-entrepreneurs choose these contracts which may
not make them so pro…table. When p2 > D, it turns out that lenders do not need to lower rates too much
to attract non-entrepreneurs to smaller loans. In particular, lenders will …nd it optimal to lower interest
rates and o¤er non-entrepreneurs smaller loans, up to the point where they hit the constraint that interest
rates must be nonnegative. This reveals a distinct reason for why speculative bubbles might be associated
with low spreads between the borrowing rate and the risk-free rate: In order to steer agents who intend to
buy risky assets towards contracts that incur smaller losses for lenders, these loans must carry low rates.
Remark 5: The fact that lenders do not need to worry about entrepreneurs pretending to be nonentrepreneurs plays an important role behind Proposition 3. If entrepreneurs could pass themselves o¤ as
non-entrepreneurs, e.g. if they could engage in hidden borrowing or if we precluded back-and-forth transfers
that would allow non-entrepreneurs to prove their types, we would need to add a constraint to (19) to insure
the entrepreneur does not prefer to take the smaller loan at a low rate, i.e.
R

(1 + r2e )

(R

(1 + r2n )) xn2

Comparing this constraint with (i) reveals that the nature of the equilibrium contract depends on how
D=p2 compares to R. When D=p2 > R, the fact that non-entrepreneurs are just indi¤erent between a
higher rate on a large loan and a zero rate on a small loan implies entrepreneurs would prefer the smaller
loan. In that case, in equilibrium both types will be o¤ered the same contract, i.e. xn2 = xe2 = 1 and
(1
)
. That is, if lenders are precluded from using transfers to restrict what borrowers
r2e = r2n =
1 (1
)
can report, the equilibrium contract terms may be the same for both types.
Finally, consider contracts o¤ered to agents who arrive in period 1. A few observations allow me to
simplify the characterization of these contracts. As in the previous section, in equilibrium lenders must
believe the non-entrepreneurs they lend to will either hold or sell their assets at date 2 with certainty if d
is not revealed. For lenders who expect a borrower to hold on to his assets, the equilibrium contract can be
described as a simple long-term debt contract fxn1 ; r1n;late g in which the agent receives xn1 upon arrival and
is asked to repay xn1 1 + r1n;late

at the end of date 2 or else be audited. To see this, denote the original

loan amount by x
bn1 . Since rates must be nonnegative, agents will prefer buying assets to doing nothing,

so we can ignore the rate for repayment at date 1. Denote the remaining rates by fb
r1n;D ; rb1n;p2 ; rb1n;late g.

Construct a new contract fxn1 ; r1n;late g where xn1 = x
bn1 and r1n;late = qb
r1n;D + (1
27

q) rb1n;late : If the borrower

holds on to his assets, he will only repay if d = D. Under the original contract, he would have expected to
pay qb
r1n;D + (1

q) rb1n;late with probability . The same is true for the new contract. If the agent prefers to

sell his assets under the new contract if d is not revealed, the original contract could not have been optimal,
since the new contract recoups the same expected amount but with a larger probability. Since all agents

are indi¤erent between the new contract and the original equilibrium contract, and the trading strategy of
non-entrepreneurs is unchanged, the contract fxn1 ; r1n;late g is equivalent to the original equilibrium contract.
Next, consider a borrower who intends to sell his assets. Here the equilibrium contract can be represented
using a simple short-term debt contract fxn1 ; r1n;early g in which the agent receives xn1 upon arrival and is
asked to repay xn1 1 + r1n;early

at the beginning of date 2 or else be audited and be stripped of all of his

cash holdings. To see this, denote the terms of the equilibrium contract by x
bn1 and fb
r1n;D ; rb1n;p2 ; rb1n;late g.
Construct a new contract fxn1 ; r1n;early g where xn1 = x
bn1 and r1n;early =

1 q
b1n;p2
q +(1 q) r

+

q
b1n;late :
q +(1 q) r

If the

borrower intends to sell his assets, this contract would ensure the same expected repayment. But a borrower
who accepts a short-term contract would necessarily sell the asset early at date 2. Hence, the short-term

contract fxn1 ; r1n;short g is equivalent to the original contract, since all agents are indi¤erent between this and

the original equilibrium contract and it has no e¤ect on the trading strategy of non-entrepreneurs.

We can therefore derive the terms for non-entrepreneurs using a problem analogous to (19) in which the
contract involves a loan size xn1 and a single interest rate, either r1n,early or r1n;late depending on what the
lender believes the borrower will do with the asset at date 2. Solving these reveals the following:
Proposition 4: Non-entrepreneurs who arrive at period 1 will receive the following contracts:
1. Non-entrepreneurs who intend to keep their assets will receive a long-term contract with xn1 < 1 and
r1n;late = 0 in any equilibrium where p1 > D. If p1 = D, the value of xn1 is not uniquely determined,
but a pooling contract where xn1 = xe1 = 1 and r1n;late = r1e is an equilibrium.
2. Non-entrepreneurs who intend to sell their assets will receive a short-term contract with xn1 < 1 and
r1n;early = 0 in any equilibrium where p1 > q D + (1

q) p2 . If p1 = q D + (1

q) p2 , the value of xn1

is not uniquely determined, but a contract where xn1 = xe1 = 1 and r1n;early = r1e is an equilibrium.

There are two di¤erences between date-1 contracts and date-2 contracts. First, there is now another
dimension that creditors can exploit to reduce their expected losses on non-entrepreneurs, namely shortening
the maturity of the loan. Requiring lenders to repay early forces them to sell the asset, which the lender
would prefer since it minimizes his exposure to losses if the purchase of the asset turns out to be unpro…table.
Second, when non-entrepreneurs intend to sell the asset at date 2 if I2 = ?, reducing the loan size may not
be pro…table if the non-entrepreneur is just indi¤erent about selling the asset. In this case, any bene…t to
the creditor from reducing the loan size xn1 is fully o¤set by the reduction in interest rate r1n .
28

Remark 6: The fact that contracts are separating even though creditors incur losses on some contracts
may seem surprising: Why don’t creditors o¤er only the contracts that entrepreneurs accept? The reason
is that from Lemma 2, non-entrepreneurs are indi¤erent between the two contracts. Thus, if a creditor
unilaterally withdrew the contract o¤ered to non-entrepreneurs, he could not be sure that he won’t end up
with a non-entrepreneur. A fully speci…ed equilibrium with contracts must include the beliefs of creditors
on whether they can avoid non-entrepreneurs by withdrawing contracts, and in equilibrium creditors cannot
believe they can dodge non-entrepreneurs. Another question is why creditors don’t exclusively o¤er nonentrepreneurs short-term contracts and force them to sell early. The answer is that not all non-entrepreneurs
can sell the asset in equilibrium, and so again, in equilibrium creditors must believe that only o¤ering short
term contracts will lead non-entrepreneurs to choose the contract o¤ered to entrepreneurs.
The analysis above only considers what type of contracts will be o¤ered in equilibrium. A full analysis
would need to solve for the equilibrium prices of the asset pt (It ). Here, there would be little innovation
over the analysis of the previous section: Depending on parameter values, there will be equilibria in which
non-entrepreneurs all sell their assets, all hold on to their assets, or some hold and some sell if d remains
uncertain at the beginning of date 2. For brevity, I skip the analysis.
To conclude, allowing lenders to design contracts reveals that this makes it more di¢ cult, but not impossible, for bubbles to arise. Creditors will try to avoid entering into …nancial arrangements with agents
who intend to buy overvalued assets, and would try to dissuade borrowers from buying overvalued assets if
possible. However, if lenders cannot just pay agents not to speculate, bubbles can emerge in equilibrium.
In these cases, lenders would want to o¤er contracts that minimize the losses they incur from speculators.
Speci…cally, the contracts that emerge involve lower rates on smaller, shorter-term loans. Barlevy and
Fisher (2012) build on this last observation and show that in a risk-shifting model adapted to features of
the housing market that my model here ignores, there is a similar tendency for speculative bubbles to be
associated with contracts that encourage early repayment. They …nd that such contracts were indeed used
heavily in cities with house price booms, but these contracts were used only sparingly elsewhere.

5

Conclusion

The distinguishing feature of equity and housing markets in recent years has been the rapid run-up in the
prices of these assets followed by an equally sharp collapse in these prices. While these episodes are often
described as “bubbles”in the popular press, economists who reserve the term “bubble”to mean a deviation
between the price of an asset and its underlying value should …nd it less obvious whether these episodes were
associated with overvaluation. In particular, an alternative potential explanation for volatile asset prices is
that the underlying fundamentals are themselves highly volatile. This is particularly true in models where
di¤erent agents value the same asset di¤erently for whatever reason. In that case, if the price of the asset
29

was equal to the value that the marginal trader assigns to the asset, shocks that change the identity of
the marginal trader can result in especially violent swings in asset prices, a theme explored in recent work
by Geanakopolos (2009), Burnside, Eichenbaum, and Rebelo (2011), and Albagli, Hellwig, and Tsyvinski
(2011). Thus, one could potentially understand these episodes without having to appeal to the notion that
asset prices are unhinged from the underlying valuation that some agent attaches to them. Yet the question
of whether asset prices re‡ect fundamentals is important. If asset prices fail to re‡ect their underlying value,
they signal to potential producers of the asset to produce a di¤erent amount of these assets than is socially
optimal, i.e. than the value of providing the marginal trader with another asset. In that case, there may be
a role for policy intervention that would not have a parallel when asset prices re‡ect volatile fundamentals.
It is therefore important to gauge whether episodes often branded are bubbles correspond to mispricing.
This paper shows that a model in which the deviation between asset prices and fundamentals is due to riskshifting concerns can generate some of the key aspects of these episodes –booms and busts, high turnover
in asset positions, and low spreads charged to those who borrow against these assets. Thus, the analysis
here suggests that overvaluation of assets remains a viable explanation for these episodes. Importantly, the
model is potentially falsi…able: These patterns do not arise in general, but only under certain conditions.
The model also implies that to the extent that a boom-bust pattern re‡ects a speculative bubble, agents
who borrow to speculate would be o¤ered certain types of contracts. Failure to observe these contracts
during speculative episodes would also provide evidence against the model.
To be sure, since the model developed here is quite stylized, the contracts it predicts will be used when
assets are overvalued may not match the type of contracts we observe in practice for reasons that need to be
incorporated into the model. However, the intuition behind the results is likely to carry over to more realistic
settings. For example, the model developed here assumes that creditors have very limited information on
what borrowers do with the funds they secure. This assumption is clearly implausible in some settings, e.g.
housing markets, where creditors can and typically do inspect the assets agents post as collateral. However,
risk-shifting can arise in settings where imperfect information pertains to variables other than the actual
assets themselves, e.g. creditors may lack information about the preferences or income ‡ows of the agents
they lend to. Barlevy and Fisher (2012) pursue this idea in a model of housing, where creditors can observe
the assets agents buy but are uncertain as to how much a given borrower values owning a home. They
…nd that the type of contracts one might expect if assets are overvalued, which are more elaborate than
the type of contracts that emerge here, empirically did seem to be used more heavily in places where house
price appreciation was unusually high. Further work is needed to relate dynamic risk-shifting models like
the ones examined here to more realistic settings.

30

Appendix
Lemma A1: Let pt denote the equilibrium price at date t if It = ?. Then pt

D.

Proof of Lemma A1: Suppose pt < D. None of the original owners would agree to sell assets. Thus,
demand for assets must be zero for the market to clear, so the expected utility to a non-entrepreneur must
be zero in equilibrium. However, suppose a lender at date t were to o¤er to lend out one unit of resources
1
at rate rt =
1. Since pt < D, it follows that
(1 + rt ) pt <

1

( D) = D

This implies a non-entrepreneur could earn positive pro…ts, and since he earns zero in equilibrium, he would
accept this contract. The expected pro…ts to the lender on this loan would equal
1

(1

)=

If entrepreneurs also chose this contract, pro…ts would be even greater, since they will repay min

1

;R

> 1.

Since no lender can expect to earn positive expected pro…ts in equilibrium, it follows that pt < D cannot
be an equilibrium.
Proof of Lemma 1: Suppose n1 + n2

D. We can verify that p1 = p2 = D is an equilibrium in

this case: Original owners are indi¤erent between holding and selling and are willing to supply any amount.
Since the second inequality in (2) implies D < D=R, non-entrepreneurs strictly prefer to buy the asset at
these prices, since they can ensure themselves positive expected pro…ts by buying and holding the asset.
Finally, since

n1
D

+

n2
D

< 1, the demand by non-entrepreneurs can be fully met by original owners selling

their asset.
At the same time, no price pt > D can be an equilibrium. For suppose this were the case. By Lemma
A1, pt

D for all t. Then either p2 > D or p2 = D and p1 > D: In either case, all original owners

would have sold their asset holdings by date 2. But since pt

D, we have that

n1
p1

+

n2
p2

<

n1
D

+

n2
D

1,

which is a contradiction.
Finally, since p1 = p2 = D < D=R, the statement of the lemma holds for these values of (n1 ; n2 )
Next, suppose D < n1 + n2 < (1
(1

q) D=R + q D. I …rst argue that p1 < n1 + n2 : Since n1 + n2 <

q) D=R + q D < D=R given (2) and (3), this implies the lemma holds at date 1. First, if p1 = D, the

statement follows. So suppose p1 > D. As argued above, in this case

n1
p1

+

n2
p2

1. There are two cases

to consider. First, if the asset is traded at date 2, then it must be the case that p2

p1 . Otherwise, all

the original owners would sell at date 1, and none of the non-entrepreneurs who bought the asset at date 1
31

would agree to sell it at date 2 if p2 < p1 since then they would have to default. But if p2
that

n1
p1

+

n2
p2

n1
p1

+

n2
p1 .

It follows that 1

n1
p1

is not traded at date 2. This requires that p2

+

n2
p1 ,

i.e. p1

p1 , it follows

n1 + n2 , as claimed. Next, suppose the asset

D=R, or else non-entrepreneurs could earn positive pro…ts

from buying the asset at date 2. But since p1 > D, and there is no trade at date 2, this requires that
all original owners sell their assets at date 1. Since demand is given by x=p1 where x

n1 is the amount

non-entrepreneurs borrow at date 1, market clearing implies x=p1 = 1, and so p1 = x

n1 < n1 + n2 , as

claimed.
The last step is to show that if n1 + n2 > D, then p2

D=R. If p2 = D, the statement follows from

(2) and (3). If p2 > D, then p1 > D, or else there would be excess demand for the asset at date 1 given
the asset is worth at least D regardless of whether d is revealed, so all agents would strictly prefer to buy
it at a price of p1 = D. Moreover, since the original owners can wait to sell, the fact that some agents will
sell at date 1 requires that p1

q D + (1
p2

q) p2 . But this implies
p1 q D
1 q
n1 + n2 q D
1 q
(1 q) D=R
D
=
1 q
R

The claim follows.

Proof of Proposition 1: I …rst argue that the restrictions (2) and (3) ensure that in equilibrium,
entrepreneurs will produce in both periods and that non-entrepreneurs in period 1 buy risky assets. I then
use these observations to set up and solve for the equilibrium for all (n1 ; n2 ) in the set
(n1 ; n2 ) 2 R2++ : n1 + n2 < (1

N

q) D=R + q D

(20)

As discussed in the text, the price of the asset if I2 = d must equal d. Hence, the only relevant prices

are at I1 = ? and I2 = ?. Denote these by p1 and p2 , respectively. Lemma A1 implies that p1

p2

D and

D.

(1
)
in both periods. In period 2, if
1
(1
)
some lender charges above r, another lender could charge r2 = r + ": For " su¢ ciently small, they would
I now argue that equilibrium interest rates are at most r =

be assured positive pro…ts. This is because when the equilibrium contract exceeds r, there exists an " that
would attract both entrepreneurs and non-entrepreneurs away from the equilibrium contract. The expected
payo¤s to the creditor are given by
(1

) r2 + [ (1 + r2 )

1] = [(1

32

)+

]" > 0

Next, at date 1, if lenders charged more than r, a lender could once again make positive expected pro…ts by
charging r1 = r +" and attracting both types. This contract would yield them a pro…t of [(1

)+

]" > 0

if non-entrepreneurs held on to their assets at date 2, and even higher pro…ts if non-entrepreneurs sold their
assets at date 2. This is because in the limiting case where k ! 0, agents will not sell their assets unless
they can repay their debt in full, so lenders recover the full debt obligation if agents sell the asset.
Assumption (2) tells us that R > 1 + r. From Lemma 1, it follows that
D=pt > R > 1 + r

1 + rt

Hence, all non-entrepreneurs will want to buy assets, since they can guarantee themselves positive expected
pro…ts by buying assets and holding them to maturity. Hence, demand for the assets at period t is equal
to nt =pt : All non-entrepreneurs would wish to buy risky assets in equilibrium, and only non-entrepreneurs
wish to buy.
Next, consider what happens when I2 = ?. Non-entrepreneurs who arrive at date 2 could potentially

earn information rents by buying the asset. They would have to buy these assets from either the original
owners, who value the assets at D, or non-entrepreneurs who bought the assets at date 1 and who value
the assets at D + (1

) (1 + r1 ) p1 . Hence, four types of equilibria are possible in this state of the world:

(a) at least some of the original owners hold on to them until maturity; (b) all original owners who own
assets at date 2 sell them, but none of those who bought them back in period 1 sell them; (c) all original
owners prefer to sell at date 2 and some but not all of those who bought the asset back in period 1 are are
willing to sell; and (d) all those who own the asset at date 2 sell it. Each of these cases yield a system of
equations that pins down p1 and p2 .
Before laying out these equations and solving for the equilibrium values of p1 and p2 , I derive some
preliminary results for case (c) in which only some of those who buy assets in period 1 sell them in period
2, and which involves some important subtleties. I …rst argue that traders who buy the risky asset at
date 1 must not randomize on whether to sell these assets at date 2. For suppose traders did randomize.
This would mean such traders are indi¤erent between holding on to the assets for an expected pro…t of
p2
D
(1 + r1 ) and selling for an expected pro…t of
(1 + r1 ). Suppose a lender in period 1 were to
p1
p1
o¤er a slightly lower interest rate than r1 to the same trader. Since the trader was just indi¤erent about
selling the asset at r1 , he would strictly prefer to sell at a lower rate. But this would allow the lender to earn
higher pro…ts, since the loss in interest rate can be in…nitesimally small while the probability of recovering
nearly 1 + r1 jumps up discretely. It follows that in equilibrium there can only be two types of traders at
date 1: Those who will de…nitely sell the asset at date 2 and those who will de…nitely hold on to it.
Next, I argue that in equilibrium the two types of traders cannot receive the same interest rate. For
suppose they did. Let

denote the fraction of agents who buy assets at date 1 that will de…nitely sell at
33

date 2. Further, let

0

and

1

denote the expected pro…ts from lending to those who will hold and those

who will sell, respectively. Expected pro…ts to the lender are given by
(1

) r1 + ((1

If the interest rate is the same for both types, then

1

)
>

0

0.

+

1)

But then a lender can earn positive expected

pro…ts by o¤ering a rate slightly below r1 : given some buyers are willing to sell the asset at date 2, at a
slightly lower rate all buyers would prefer to sell, yielding the lender an expected pro…t that is arbitrarily
close to
(1

) r1 +

1

which is strictly higher than the original equilibrium pro…ts. So borrowers cannot all be charged a common
rate in an equilibrium of type (c). Thus, in equilibrium, the two types must receive di¤erent rates. Moreover,
they have to earn zero pro…ts on each type. This implies that in equilibrium, non-entrepreneurs who sell
the asset at date 2 receive a single rate r1 that satis…es
(1

) r1 + [(1

q + q ) (1 + r1 )

1] = 0

or, rearranging,
1 + r1 =

1

1
q (1

)

Likewise, non-entrepreneurs who hold the asset at date 2 must receive a single rate r1 that satis…es
(1

) r1 + [(1

) (1 + r1 )

1] = 0

or, rearranging
1 + r1 =

1

1
(1

)

Since non-entrepreneurs would prefer the low rate contract, these have to be rationed. Given the assumption
on sequential arrivals, the borrowers who show up …rst will be those who receive these contracts.
Next, I argue that r1 must leave non-entrepreneurs just indi¤erent between holding it and selling it at
date 2, i.e.

p2 = D + (1

) (1 + r1 ) p1

For suppose that the non-entrepreneurs who sell their holdings in period 2 strictly prefer to sell at r1 . Then
a lender could earn positive expected pro…ts by o¤ering all agents who in equilibrium receive a rate above
r1 a new contract with a rate that is just a little above r1 . Since non-entrepreneurs would buy the asset
and sell it, they would act in the same way as those receiving the rate r1 . Entrepreneurs would continue
to borrow and produce. Since expected pro…ts were zero at r1 , they would be strictly positive at a higher

34

rate. The only way to ensure lenders who o¤er a high rate are unwilling to o¤er other contracts is if the
rate at which traders are just indi¤erent about selling at date 2 yields zero expected pro…ts.
I am now in a position to provide equations associated with the di¤erent cases (a) - (d) above. In case
(a), the price in period 2 must be D, or else all original owners would want to sell the asset. But then
p1 = D, or else all of the original owners would have sold at date 1. But in that case, the only agents who
can sell the asset in period 2 would have a reservation price that exceeds D, making it impossible for the
asset market to clear at p2 = D. Hence, case (a) implies
p1

=

D

p2

=

D

(21)

In case (b), all original owners must sell the asset at either date 1 or date 2, so total demand must equal
the original stock of the asset, 1. In addition, since demand is positive in both periods, the original owners
must be indi¤erent between selling at date 1 and waiting to sell at date 2. Hence, the prices p1 and p2 must
satisfy the system of equations
n2
n1
+
p1
p2
q) p2 + q D

(1

=

1

= p1

(22)

To ensure that none of those who bought the asset at date 1 wish to sell it, we must check that the solution
to (22) satis…es p2

D + (1

) (1 + r1 ) p1 , i.e. p2 does not exceed the reservation price of agents who

bought the asset at date 1. The reservation price depends on r1 since an agent who sells the asset will
receive the rate r1 .
In case (c), some but not all of those who bought the asset at date 1 sell it at date 2. From the analysis
above, we know that the agents who buy the asset at date 1 and sell at date 2 are o¤ered a rate r1 at date
1 and are indi¤erent about selling at date 2. In addition, either all of the original owners sell their assets at
date 1, implying n1 =p1 = 1, or else the original owners must be indi¤erent between the two periods. These
conditions can be summarized as follows:
p2

=

D + (1

p1

=

max f(1

) (1 + r1 ) p1
q) p2 + q D; n1 g

(23)

To ensure that not all of those who bought the assets at date 1 sell in date 2;we need to verify that at the
prices that solve the system (23), n2 =p2 < 1.
Finally, in case (d), all of those who own the asset at the beginning of date 2 sell it. In this case, buyers
who show up at date 2 must buy up all existing shares of the asset. Once again, either some of the original
35

owners sell in either period, requiring them to be indi¤erent between the two periods, or else the original
owners buy up all of the assets at date 1, implying n1 =p1 = 1. The two conditions are thus
p2

=

n2

p1

=

max fn1 ; (1

q) n2 + q Dg

(24)

The rest of the proof shows that we can partition the set N into distinct regions where the unique

equilibrium in each region corresponds to a di¤erent case. These regions are illustrated graphically in
Figure A1 and identi…ed in the analysis below.
First, case (a) is the unique equilibrium whenever n1 + n2 < D, which corresponds to region A in Figure
A1. This is because at p1 = p2 = D, the prices that must prevail in this equilibrium, all non-entrepreneurs
in both periods would buy the asset. But if n1 + n2 > D, the sum of demands in the two periods exceeds 1,
and so we cannot have an equilibrium in which some of the original owners hold on to the asset to maturity.
By contrast, it is easy to con…rm that p1 = p2 = D is an equilibrium when n1 + n2 < D. One can verify
that none of the other cases can be supported as equilibria in this region: The solution to the system of
equations (22) require that either p1 < D or p2 < D or both, which cannot occur in equilibrium, while
the systems of equations (23) and (24) yield solutions for p1 and p2 at which total demand for the asset
n2
n1
+
is less than 1, implying not all original owners sell the asset as required in
over the two periods
p1
p2
these two cases.
Next, consider case (b). We already ruled out that this can be an equilibrium in region A. The two
equilibrium conditions in (22) can be reduced to a single polynomial in p2 , speci…cally
(1

q) p22

(n1 + (1

q) n2

q D) p2

n2 q D = 0

(25)

Only one of the roots exceeds D: Denote this root by p2 (n1 ; n2 ). Hence, there is a unique price market
clearing price p2 , and thus a unique price p1 = q D + p2 , for each (n1 ; n2 ). Recall that for these prices are
consistent with an equilibrium of type (b), we need to verify that the implied p2 is not so high that agents
who bought at date 1 wish to sell their assets, i.e. p2

D +(1

) (1 + r1 ) p1 . Since p1 = (1

q) p2 +q D,

we can substitute for r1 and p1 to get
p2

D+

(1

) [(1 q) p2 + q D]
1 (1
)q

which, upon rearranging, yields
p2

1

1 + q (1
) (1
)
D
(1
) (1 q (1
))

p

n2
= 1, it is immediate that p2 (n1 ; n2 ) is strictly increasing in
p2
1 + (1
) q (1
)
both n1 and n2 . For n1 = 0, the condition p2 (0; n2 ) p implies n2
D: For
1 (1
) (1 q (1
))
Using the equation

n1
q D + (1

q) p2

+

36

1 (1
)q
D. By totally
1 (1
) (1 q (1
))
di¤erentiating (25), one can show that the boundary condition p2 (n1 ; n2 ) = p de…nes a negative linear

n2 = 0, the condition the condition p2 (n1 ; 0)

p implies n1

relationship between n1 and n2 . Hence, the region in which there exists an equilibrium consistent with case
(b) is bounded. This region is shown graphically as region B in Figure A1. Since values in the interior of
region B satisfy p2 < D + (1

) (1 + r1 ) p1 by construction, there can be no equilibria associated with

cases (c) or (d) inside region B.
Next, we move to case (c). To determine the values of (n1 ; n2 ) that allow such equilibria, it will help to
examine separately equilibria in which (1
(1

q) p2 + q D exceeds n1 and equilibria in which it does not. If

q) p2 + q D > n1 , then we have two equations and two unknowns, which have a unique solution:
p2

=

p1

=

1 + q (1
) (1
)
D
(1
) (1 q (1
))
1 (1
)q
D
1 (1
) (1 q (1
))
1

1 (1
)q
D n1 .
1 (1
) (1 q (1
))
At the same time, we need to verify that n2 < p2 , or else demand by non-entrepreneurs who arrive at
1 + q (1
) (1
)
date 2 would exceed the total supply of the asset. This implies n2 <
D
1 (1
) (1 q (1
))
n2 . Hence, an equilibrium consistent with case (c) in which n1 < (1 q) p2 + q D is con…ned to values

One restriction that needs to be satis…ed is that n1 < (1

q) p2 + q D =

f(n1 ; n2 ) : (n1 ; n2 ) 2
= A [ B, n1 < n1 and n2 < n2 g. This set is illustrated graphically as region C1 in Figure
A1. Note that the equilibrium prices p1 and p2 do not vary with n1 and n2 in this region.
Next, consider case (c) where n1 > (1

q) p2 + q D. In this case, the system of equations is given by

p2

=

D+

p1

= n1

1

(1
) n1
q (1
)

As before, there are two restrictions that need to be satis…ed to ensure these prices are indeed an equilibrium.
First, we need to con…rm that n1 > (1

q) p2 + q D. Substituting in the value of p2 yields n1 > n1 as

de…ned above. Second, we again need to make sure n2 < p2 so demand by non-entrepreneurs at date 2
would exceed the total supply of the asset. In this case, this condition implies
n2 < D +

1
q (1

1

)

n1

1
n1 depicted in Figure
1 q (1
)
A1, which is the only region in which an equilibrium that accords with case (c) arises. Note that in region

The de…nes the region C2 =

(n1 ; n2 ) 2 R2++ : n1 > n1 ; n2 < D +

C1 , the expressions for p1 and p2 are independent of n1 and n2 , while in region C2 , both are increasing in
n1 and independent of n2 .
37

Finally, we consider case (d). Observe that prices are weakly increasing in n1 and n2 in this case as well.
Again, we separately consider the case where (1

q) n2 + q D exceeds n1 and the case where the opposite

is true. If (1

q) n2 + q D > n1 , then we have p1 = (1

q) n2 + q D and p2 = n2 . We need to con…rm

that n1 < (1

q) n2 + q D. Since all the agents who buy the asset in period 1 must be willing to sell it in

period 2, we need to verify that
p2

D + (1

) (1 + r1 ) p1

Substituting in for p1 , p2 , and r1 yields
n2

D+

(1

) ((1 q) n2 + q D)
1 q (1
)

which, upon rearranging, yields
n2

1 + q (1
) (1
)
D
1 (1
) (1 q (1
))

n2

This corresponds to region D1 in Figure A1, which has no overlap with region C1 [ C2 . Thus, this type of
equilibrium cannot occur in these regions. Finally, we have case (d) where n1 > (1

q) n2 + q D. Again,

to ensure that all the agents who buy the asset in period 1 are willing to sell it in period 2 requires that
p2

D + (1

) (1 + r1 ) p1 . Substituting in for p1 , p2 , and r1 yields
n2

D+

(1
) n1
q (1
)

1

Hence, this region, denoted D2 in Figure A1, is de…ned by
(n1 ; n2 ) 2 R2++ : D +

D2

1

(1
) n1
q (1
)

n2

n1 q D
1 q

and does not overlap with C1 [ C2 . The uniqueness of equilibrium is thus established for all (n1 ; n2 ) 2 N .

Comparative Statics of Equilibrium in Proposition 1: In the text, I discuss some comparative
static results regarding the asset market equilibrium at date 2. I present the formal derivation for these
results here. Note that the outer boundary of region B is given by the line de…ned by two points,
(n1 ; n2 ) =

0;

1

1 + (1
) q (1
)
D
(1
) (1 q (1
))

and

1

1
(1

(1
) (1

)q
q (1

))

D; 0

In the limit as q ! 0, these points converge to (0; D) and (D; 0), respectively, implying the set B converges to
(n1 ; n2 ) 2 R2++ : D < n1 + n2 < D . Conversely, when q ! 1, these points converge to 0; 1+(1
1

)(1
)
(1 )

and ( D; 0), respectively.

As for the outer boundary of N , for ease of comparison we can also de…ne it by two points, namely
(n1 ; n2 ) =

0; (1

q)

D
+ D
R
38

and

(1

q)

D
+ D; 0
R

D

In the limit as q ! 0, these points converge to 0; D
R and
to (n1 ; n2 ) 2

R2++

D
R

: n1 + n2 <

D
R

. Since

D
R;0

, respectively, implying the set N converges

< D, it follows that N n (A [ B) = ?, i.e. the set N contains

only regions A and B. By continuity, the the same is true for q close to 0. Hence, for q close to 0, the only
possible outcomes are either no bubbles or quiet bubbles.
Conversely, when q ! 1, the above points converge to (0; D) and ( D; 0), respectively. In the limit when

q = 1, then N n (A [ B) = ?. However, it is not clear what happens for q close to but strictly less than 1.
De…ne

=

(1

q) D=R + D

1 (1 )q
1 (1 )(1 q(1

D

))

as the ratio of the values of n1 at the outer boundary of N and B, respectively, when n2 = 0. Di¤erentiating
with respect to q and evaluating at q = 1 reveals that
lim

R!1=

Since

d
dq

=
q=1

1
1

(1

)

<0

= 1 at q = 1, it follows that for large values of R, N n (A [ B) 6= ? in a neighborhood of q = 1.

In other words, for q su¢ ciently close to but not equal 1, noisy bubbles will be possible when n1 >
1 (1
)q
D and n2 = 0. For smaller values of R, we have
1 (1
) (1 q (1
))

lim
R!1+ 1

(1
)
(1
)

d
dq

(1
=

negative, provided both

and

or

)
1

q=1

This expression will be negative if either

2

) (1

1
(1

+

2

)

are su¢ ciently large to turn (1

)

2

1

+

2

are both strictly between 0 and 1. Hence, even for low values of R, there

exist parameters such that noisy bubbles will be possible for q su¢ ciently close to 1.
Turning to , as

! 1, the two points that de…ne the outer boundary of B converge to (0; D) and

(D; 0). By contrast, the two points that de…ne the outer boundary of N converge to 0; (1

and (1

q) D
R + qD

q) D
R + qD; 0 , respectively. Since D=R < D, it follows that N n (A [ B) = ?, i.e. the set N

contains only regions A and B. By continuity, the the same is true for

close to 1. Hence, for

close to 1,

the only possible outcomes are either no bubbles or quiet bubbles.
Conversely, in the limit as

! 0, the two points that de…ne the outer boundary of B both converge

to (0; 0). By contrast, the two points that de…ne the outer boundary of N converge to 0; (1
(1

q) D
R and

q) D
R ; 0 , respectively, which strictly exceed (0; 0) as long as q < 1. Thus, for small values of , noisy

bubbles and speculation both become possible, and in the limit these will be the only possible outcomes.
Finally, the parameter

only a¤ects the outer boundary of region B. Di¤erentiating the two values reveal

39

that both are increasing in . Speci…cally,
d
1 + (1
) q (1
)
D
d 1 (1
) (1 q (1
))

=

d
d 1

=

Hence, higher values of

1
(1

(1
) (1

)q
q (1

))

D

q (1
(1

(1
(1

(1

(1

) (1

2

)

q (1

q) q (1
) (1

2

D

2

D

)))
)

2

q (1

)))

increase the size of region B, meaning a larger set of parameters associated with

bubbles will correspond to quiet rather than noisy bubbles.

Proof of Proposition 2: The proof is by contradiction. Suppose there was an equilibrium where the
price of the asset at date 2 exceeded D, i.e. p2 > D. Let xn2 denote the amount of resources nonentrepreneurs receive upon arrival at date 2 under the equilibrium contract, and denote the amount they
would have to repay if d = D by (1 + r2n ) xn2 . The expected payo¤ to a non-entrepreneur from buying the
asset would then equal
xn2

D
p2

(1 + r2n )

while the expected payo¤ to the lender in this case is
xn2 ( (1 + r2n )

1)

Summing together yields
xn2

D
p2

1

This expression is negative when p2 > D. Hence,
xn2

D
p2

(1 + r2n )

< xn2 (1

(1 + r2n ))

(26)

Consider a creditor who o¤ered a contract x
b in which non-entrepreneurs receive nothing upon arrival, i.e.
D
n
x
b2 = 0 and then paid the agent
(1 + r2n ) xn2 at the end of date 2, which is just the expected payo¤
p2
from buying the asset under the original contract. From (26), we know that lenders are strictly better o¤
from this arrangement, since their incur smaller losses. Since (26) is a strict inequality, and since the …rst
inequality in (??) ensures entrepreneurs strictly prefer production to buying risky assets, it will be possible
to increase the payment to non-entrepreneurs a little to make them strictly better o¤ and to lower the
interest charged to entrepreneurs in a way that makes them better o¤ but still induces them to choose the
contract intended for them and to use the funds they borrow for production. Hence, if p2 > D, no agents
buy the asset at date 2. Since any original owners who still hold the asset at date 1 would prefer to sell at
a price of D than hold on to the asset, the only possible equilibrium is one in which all the original owners
sell the asset at date 1.
Next, we move to date 1. First, consider whether there can be an equilibrium in which the asset is
purchased at date 1. Suppose p1 > D. In that case, we know that (1 + r1 ) p1
40

p1 > D, so that any

agents who buy the asset at date 1 would only agree to sell the asset at date 2 for a price above D. But
we just argued that if p2 > D, there cannot be an equilibrium in which agents buy the asset at date 2 in
equilibrium. Hence, non-entrepreneurs who buy the asset at date 1 expect to hold on to the asset until d is
revealed. But by a similar argument to before, if p1 > D, then creditors can earn positive expected pro…ts
by giving non-entrepreneurs zero resources up front and then paying them after date 2 a little above their
expected pro…ts from buying risky assets. Hence, the only possible equilibrium is one where p1 = D.
However, if p1 = D, there cannot be an equilibrium where p2 > D. If there were such an equilibrium,
all original owners will hold on to their asset holds until date 2 and sell if p2 > D, i.e. if d was not revealed.
But we know that if p2 > D, there will not be any buyers at date 2 in equilibrium. Hence, the market will
not clear at date 2. It follows that not only is p1 = D in equilibrium, but p2 = D.

Proof of Lemma 2: Suppose not, i.e. a non-entrepreneur strictly prefers to announce n than to announce
e. It is easy to check that in equilibrium, rte > 0 for both t 2 f1; 2g: non-entrepreneurs will incur a cost on

creditors that must be made up to ensure creditors earn zero expected pro…ts. Consider a lender who only
o¤ers the following contract fe
xet ; rete g aimed at entrepreneurs:
x
eet

rete

= xet = 1
= rte

"

where " 2 (0; rte ) so the interest rate remains positive. Entrepreneurs will clearly prefer this contract

to the equilibrium contract. Non-entrepreneurs originally strictly preferred the contract they received in
equilibrium to the contract f1; rte g, and hence for " small enough will also prefer it to f1; rete g. Hence, this

contract will only attract entrepreneurs, and for " small enough the expected pro…ts to the creditor who
o¤ers it will be strictly positive. Hence, the original contract could not have been an equilibrium.

Proof of Proposition 3: Consider the maximization problem
max xn2 [ (1 + r2n )

n
xn
2 ;r2

1]

subject to
i.
ii.

r2n

D
p2

(1 + r2n ) xn2 =

D
p2

(1 + r2e )

0

Substituting (ii) into the objective function yields
D
p2

D
+ (1 + r2e )
p2

1 xn2

When p2 > D, this expression is decreasing in xn2 , so the optimal contract reduces xn2 . From constraint (i),
this implies r2n will be as small as possible. Since constraint (3) implies r2n
41

0, it follows that the optimal

contract involves r2n = 0. From constraint (1), this implies
xn2 =

(1 + r2e ) p2
D p2

D

The condition on r2e follows from the fact that lenders earn zero expected pro…ts in equilibrium.
Proof of Proposition 4: Consider …rst an agent who intends to hold on to his asset. In this case, the
contract is given by
xn1

max

n;l a t e
xn
1 ;r1

subject to
D
p1

i.

r1n;late

ii.

h

i
1

1 + r1n;late

1 + r1n;late

D
p1

xn1 =

(1 + r1e )

0

where we know from the proof of Proposition 5 that we can omit the constraint that the entrepreneur
must prefer to produce than buy assets. Since this problem is identical to the period 2 problem, the same
argument implies that when p > D, in equilibrium r1n;late = 0 and
xn2 =

(1 + r2e ) p2
D p2

D

When p1 = D, pro…ts are independent of x1 , and so any pair (xn1 ; r1n ) that yields expected utility V1 (n; e)
to the agent is an equilibrium.
Next, consider an agent who intends to sell his asset if d remains uncertain. In this case, the equilibrium
contract solves
max

n;e a r l y
xn
1 ;r1

subject to
i.
ii.

D
p1

q

r1n;early

1

h
xn1 (q + 1

r1n;early

+ (1

q) 1 + r1n;early

q)

p2
p1

r1n;early

1

i
1
xn1 = V1 (n; e)

0

Rearranging constraint (i) yields
q D + (1
p1

q) p2

xn1

V1 (n; e) = (q + 1

q) 1 + r1n;early xn1

So the objective function can be written as
q D + (1
p1
When p1 > q D + (1

q) p2

1 x1

V1 (n; e)

q) p2 , this expression is decreasing in xn1 , so the optimal contract reduces xn1 and

sets r1n;early = 0. Since p1

q D + (1

q) p2 or else the asset market would not clear at date 1, in the only

other case pro…ts are independent of x1 , and so any pair (xn1 ; r1n ) that yields expected utility V1 (n; e) to
the agent is an equilibrium.
42

p2

(d)
(c)

n2/p2

ЄD + (1‐Є)(1+r1)p1

(b)
(a)
ЄD

1‐n1/p1

1

Shares of the
risky asset

Figure 1: Supply and demand in the asset market at t = 2 for I2 = ∅,
assuming a single interest rate is charged to all borrowers at t = 1

p2

ЄD + (1‐Є)(1+r1*)p1
ЄD + (1‐Є)(1+r1*)p1

n2/p2

ЄD

1‐n1/p1

n2
ЄD + (1‐Є)(1+r1*)p1

1

Figure 2: Supply and demand in the asset market at t = 2 for I2 = ∅,
when some non‐entrepreneurs who buy at t = 1 sell at t = 2.

Shares of the
risky asset

n2

D/R

n2 = ЄD +

D

1‐Є
n
1‐q(1 – Є) 1

1+ (1‐)q(1‐Є)
ЄD
1‐(1‐Є)(1‐q(1‐))

C
B

ЄD

A
n1
ЄD

1‐q (1‐Є)
ЄD
1‐(1‐Є)(1‐q(1‐))

(1‐q)D/R+qЄD

Figure 3: Regions corresponding to different types of equilibrium at t = 2 for I2 = ∅

date 1

end of date 1

2 = e
production

1 = e
entrepreneur

1 = n
not an entrepreneur

2 = Ø
did nothing

2 = b
bought assets

3 œ

date 2

p2 - sold, d unknown
D - sold, d = D
0 - sold, d = 0
h - held assets

R-1

- production

p2/p1-1 - sold before knowing d
end of date 2

y œ

D/p1-1 - knows d = D
-1

- knows d = 0

0

- agent did nothing

Figure 4: Reports and Transfers Mandated by Contract

D/R
n1 ‐ qЄD
1‐q

n2 =

n2 = ЄD +

D1

1‐Є
n
1‐q(1 – Є) 1

D2

1+(1‐Є)q(1‐)
ЄD
1‐(1‐Є)(1‐q(1‐))
C1

C2

B

ЄD

A
ЄD

1‐(1‐Є)q
ЄD
1‐(1‐Є)(1‐q(1‐))

Figure A1: Equilibrium types for different pairs (n1,n2)

D/R

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45

Working Paper Series
A series of research studies on regional economic issues relating to the Seventh Federal
Reserve District, and on financial and economic topics.
A Leverage-based Model of Speculative Bubbles
Gadi Barlevy

WP-08-01

Displacement, Asymmetric Information and Heterogeneous Human Capital
Luojia Hu and Christopher Taber

WP-08-02

BankCaR (Bank Capital-at-Risk): A credit risk model for US commercial bank charge-offs
Jon Frye and Eduard Pelz

WP-08-03

Bank Lending, Financing Constraints and SME Investment
Santiago Carbó-Valverde, Francisco Rodríguez-Fernández, and Gregory F. Udell

WP-08-04

Global Inflation
Matteo Ciccarelli and Benoît Mojon

WP-08-05

Scale and the Origins of Structural Change
Francisco J. Buera and Joseph P. Kaboski

WP-08-06

Inventories, Lumpy Trade, and Large Devaluations
George Alessandria, Joseph P. Kaboski, and Virgiliu Midrigan

WP-08-07

School Vouchers and Student Achievement: Recent Evidence, Remaining Questions
Cecilia Elena Rouse and Lisa Barrow

WP-08-08

Does It Pay to Read Your Junk Mail? Evidence of the Effect of Advertising on
Home Equity Credit Choices
Sumit Agarwal and Brent W. Ambrose

WP-08-09

The Choice between Arm’s-Length and Relationship Debt: Evidence from eLoans
Sumit Agarwal and Robert Hauswald

WP-08-10

Consumer Choice and Merchant Acceptance of Payment Media
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WP-08-11

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Alejandro Justiniano, Giorgio E. Primiceri, and Andrea Tambalotti

WP-08-12

New Vehicle Characteristics and the Cost of the
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Thomas Klier and Joshua Linn

WP-08-13

Realized Volatility
Torben G. Andersen and Luca Benzoni

WP-08-14

Revenue Bubbles and Structural Deficits: What’s a state to do?
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WP-08-15

1

Working Paper Series (continued)
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WP-08-16

Bank Crises and Investor Confidence
Una Okonkwo Osili and Anna Paulson

WP-08-17

Life Expectancy and Old Age Savings
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WP-08-18

Remittance Behavior among New U.S. Immigrants
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WP-08-19

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WP-08-21

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WP-09-01

Why do the Elderly Save? The Role of Medical Expenses
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WP-09-02

Using Stock Returns to Identify Government Spending Shocks
Jonas D.M. Fisher and Ryan Peters

WP-09-03

Stochastic Volatility
Torben G. Andersen and Luca Benzoni

WP-09-04

The Effect of Disability Insurance Receipt on Labor Supply
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WP-09-05

CEO Overconfidence and Dividend Policy
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WP-09-06

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WP-09-07

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WP-09-09

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WP-09-11

2

Working Paper Series (continued)
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WP-09-12

WP-09-13

WP-09-14

The Price of Gasoline and the Demand for Fuel Economy:
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WP-09-15

Estimation of a Transformation Model with Truncation,
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Bo E. Honoré and Luojia Hu

WP-09-16

Self-Enforcing Trade Agreements: Evidence from Antidumping Policy
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WP-09-17

Too much right can make a wrong: Setting the stage for the financial crisis
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WP-09-18

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WP-09-19

Liquidity Constraints of the Middle Class
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Monetary Policy and Uncertainty in an Empirical Small Open Economy Model
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WP-09-21

Firm boundaries and buyer-supplier match in market transaction:
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WP-09-22

WP-09-23

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The Impact of Rosenwald Schools on Black Achievement
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WP-09-26

3

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WP-10-01

Macroeconomic Implications of Agglomeration
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Accounting for non-annuitization
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WP-10-03

Robustness and Macroeconomic Policy
Gadi Barlevy

WP-10-04

Benefits of Relationship Banking: Evidence from Consumer Credit Markets
Sumit Agarwal, Souphala Chomsisengphet, Chunlin Liu, and Nicholas S. Souleles

WP-10-05

The Effect of Sales Tax Holidays on Household Consumption Patterns
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Gathering Insights on the Forest from the Trees: A New Metric for Financial Conditions
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Identification of Models of the Labor Market
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Public Pensions and Labor Supply Over the Life Cycle
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Explaining Asset Pricing Puzzles Associated with the 1987 Market Crash
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WP-10-10

Prenatal Sex Selection and Girls’ Well‐Being: Evidence from India
Luojia Hu and Analía Schlosser

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Mortgage Choices and Housing Speculation
Gadi Barlevy and Jonas D.M. Fisher

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Did Adhering to the Gold Standard Reduce the Cost of Capital?
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Introduction to the Macroeconomic Dynamics:
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WP-10-14

Summer Workshop on Money, Banking, Payments and Finance: An Overview
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Complex Mortgages
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WP-10-17

4

Working Paper Series (continued)
The Role of Housing in Labor Reallocation
Morris Davis, Jonas Fisher, and Marcelo Veracierto

WP-10-18

Why Do Banks Reward their Customers to Use their Credit Cards?
Sumit Agarwal, Sujit Chakravorti, and Anna Lunn

WP-10-19

The impact of the originate-to-distribute model on banks
before and during the financial crisis
Richard J. Rosen

WP-10-20

Simple Markov-Perfect Industry Dynamics
Jaap H. Abbring, Jeffrey R. Campbell, and Nan Yang

WP-10-21

Commodity Money with Frequent Search
Ezra Oberfield and Nicholas Trachter

WP-10-22

Corporate Average Fuel Economy Standards and the Market for New Vehicles
Thomas Klier and Joshua Linn

WP-11-01

The Role of Securitization in Mortgage Renegotiation
Sumit Agarwal, Gene Amromin, Itzhak Ben-David, Souphala Chomsisengphet,
and Douglas D. Evanoff

WP-11-02

Market-Based Loss Mitigation Practices for Troubled Mortgages
Following the Financial Crisis
Sumit Agarwal, Gene Amromin, Itzhak Ben-David, Souphala Chomsisengphet,
and Douglas D. Evanoff

WP-11-03

Federal Reserve Policies and Financial Market Conditions During the Crisis
Scott A. Brave and Hesna Genay

WP-11-04

The Financial Labor Supply Accelerator
Jeffrey R. Campbell and Zvi Hercowitz

WP-11-05

Survival and long-run dynamics with heterogeneous beliefs under recursive preferences
Jaroslav Borovička

WP-11-06

A Leverage-based Model of Speculative Bubbles (Revised)
Gadi Barlevy

WP-11-07

5