Full text of Working Papers (Federal Reserve Bank of Chicago) : A Theoretical Model of Leaning Against the Wind, Working Paper 2017-16

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

Federal Reserve Bank of Chicago

A Theoretical Model of Leaning
Against the Wind
Franklin Allen, Gadi Barlevy,
and Douglas Gale

REVISED
April 20, 2018
WP 2017-16
Working papers are not edited, and all opinions and errors are the
responsibility of the author(s). The views expressed do not necessarily
reflect the views of the Federal Reserve Bank of Chicago or the Federal
Reserve System.

*

A Theoretical Model of Leaning Against the Wind
Franklin Allen
Imperial College London

Gadi Barlevy
Federal Reserve Bank of Chicago

Douglas Gale
New York University

April 20, 2018

Abstract
Policymakers have long debated whether central banks should raise interest rates if they are concerned about a potential bubble, a policy known as leaning against the wind. In this paper, we develop
a framework to study this question. We begin with the seminal work of Galí (2014) and argue that a
small change in the model rules out his …nding that raising rates ampli…es bubbles rather than dampening then. More importantly, we argue that his setup does not provide any reason to intervene against
bubbles. This leads us to consider an alternative model in which bubbles are credit-driven. In the
simplest case, raising rates dampens bubbles but exacerbates the distortions bubbles introduce. But in
this simple case, bubbles only cause harm when they arise, not when they burst. When we add default
costs, so that bubbles also cause harm when they burst, raising rates increases welfare for su¢ ciently
large default costs. Yet even when default costs are small, we …nd that a commitment to raise rates if
a bubble persists can raise welfare ex-ante even though raising rates itself reduces welfare.

The views here do not represent those of the Federal Reserve Bank of Chicago or the Federal Reserve System. The
authors thank Bob Barsky, Marco Basseto, John Conlon, Charles Evans, Jordi Galí, Alberto Martin, Jianjun Miao, Tomoyuki
Nakajima, Jaume Ventura, and Randy Wright for comments, as well as seminar participants at University of Tokyo, GRIPS,
Kyoto University, Tel Aviv University, the Interdisciplinary Center in Herzeliya, the LAEF Conference on Bubbles at UC Santa
Barbara, University of Wisconsin, the Barcelona GSE Summer Forum, the Society of Economic Dynamics, and the NBER
Summer Institute. A previous version of this paper circulated as “On Interest Rate Policy and Asset Bubbles.”

1

Introduction
Policymakers have long debated how to respond to asset bubbles. One option, advocated by Borio and Lowe
(2002) among others, involves raising interest rates to dampen the bubble, a policy that has been dubbed
leaning against the wind. An alternative approach, most associated with Bernanke and Gertler (1999), argues
for waiting to see if asset prices collapse and then cleaning up, or intervening to stimulate the economy,
if necessary. As the recent global …nancial crisis highlighted both the potential of a collapsing bubble to
trigger a …nancial crisis and the limits on central banks in responding to such a crisis, policymakers have
became more leery about the wait-and-see approach. This shifted the debate from whether to respond to
bubbles to how to best respond to them. The debate between lean vs clean essentially evolved into a debate
between lean vs screen, i.e., a debate between raising rates to dampen bubbles and using macroprudential
regulation to curb lending against bubble assets with the aim of depriving bubbles of their fuel.
More recent work has gone beyond the notion that there are preferable alternatives to interest rate hikes
and argued that leaning against the wind is in fact counterproductive. One example is the work of Galí
(2014). He introduced monetary policy and nominal price rigidity into a model of a dynamically ine¢ cient
economy in which it is well known that bubbles can arise, and found that raising rates can amplify bubbles
rather than dampen them. As another example, Svensson (2014) argues that raising rates does more harm
than good given all the implications of tighter monetary policy, citing the Swedish Riksbank attempt to
lean against a housing bubble as an example of where the policy arguably failed.
In this paper, we propose a theoretical framework to analyze whether and when leaning against the wind
is productive. Does raising rates dampen bubbles or amplify them? If it dampens them, do higher rates
increase or decrease welfare? We begin by arguing that raising rates dampens bubbles. As Galí (2014)
observes, his model features multiple equilibria. In some equilibria, higher rates amplify bubbles; in others
they dampen them. Galí argues that the equilibria in which higher rates amplify bubbles are more natural.
By contrast, Dong, Miao, and Wang (2017) and Ikeda (2017), who study models that similarly feature
multiple equilibria, focus on equilibria in which higher rates dampen bubbles and argue that those are more
natural. We consider a variation of Galí’s framework in which asset prices are uniquely determined, allowing
us to sidestep the question of which equilibrium is most natural. Once we eliminate multiple equilibria, we
…nd that raising the real interest rate unambiguously dampens bubbles.
Even after we modify Galí’s model so higher rates necessarily dampen bubbles, his setup still poses a
problem for the lean-against-the-wind view in that it implies no reason for policymakers to lean against
bubbles. This is because the friction that allows bubbles in his model is dynamic ine¢ ciency, and bubbles
ameliorate this ine¢ ciency. This observation leads us to modify Galí’s environment so that bubbles arise
not because of dynamic ine¢ ciency but because of a di¤erent friction altogether. Speci…cally, we modify the
economy he considers to be dynamically e¢ cient, but we also introduce credit together with information
frictions that interfere with lending. Bubbles arise when agents borrow and, against the interests of their
lenders who cannot monitor them, buy risky assets and bid up their price. In this setting, asset bubbles
1

reduce welfare by crowding out productive activities that could have been funded instead. This feature
implies that there may be scope for intervention against bubbles.
We …nd that in our model, even though raising rates dampens the bubble, it also exacerbates the underfunding of those productive activities crowded out by the bubble. Intuitively, contracting the economy leaves
fewer resources either to buy assets or to …nance productive activities, so both are crowded out. Hence,
there is a sense in which leaning against a bubble indeed does more harm than good. However, the typical
argument for intervening against bubbles is not to undo the distortions that arise while bubbles are present
but to mitigate the harm bubbles cause when they burst. Our model can speak to this possibility, since
in our model a fall in asset prices will drive those who borrowed against the bubble asset to default. The
concern is that the losses lenders incur as a result will make them unable or unwilling to extend subsequent
credit, leading to a fall in output. Hoggarth, Reis, and Saporta (2002) and Reinhart and Rogo¤ (2009)
estimate that …nancial crises were historically associated with a fall in GDP per capita of between 9 and
16%. Atkinson, Luttrell, and Rosenblum (2013) estimate the cumulative loss in output in the US from the
most recent …nancial crisis was even larger. We can capture these e¤ects in a reduced-form way by allowing
for default costs that are proportional to what agents borrow. This speci…cation implies that agents have
fewer resources to consume the more that speculators borrow and then default on when asset prices fall, in
line with what would happen if the collapse of the bubble resulted in a …nancial crisis. We con…rm that
raising rates can indeed make society better o¤ if default costs are su¢ ciently large: Even if raising rates
leaves fewer resources for productive activities, reducing the amount agents borrow against bubble assets
and thus the amount they would default on if the bubble burst can make society as a whole better o¤.
Finally, we …nd that even when default costs are small, it might still be possible to increase welfare by
committing to raise rates in the future if the bubble persists. Intuitively, if the bubble is likely to burst,
a threat to act if it persists is desirable: Such a threat both mitigates the distortions due to the bubble
and reduces the amount agents would default on if the bubble crashed, yet it is unlikely that the costly
intervention will actually be necessary. Our model can therefore reconcile the seemingly con‡icting views
about the lean-against-the-wind approach. Raising rates to dampen a bubble is indeed costly, as emphasized
by Svennson (2014). But dampening the bubble might mitigate the fallout if and when the bubble bursts,
and a commitment by a central bank to raise rates in the future should a bubble persist may promote
welfare ex ante even if raising rates is costly ex post.
Beyond these particular results, one of our contributions is to o¤er a simple framework that can be used to
explore a host of issues related to asset bubbles. For example, since in our model credit plays a key role and
is essential in allowing bubbles to arise, one can use our setup to compare macroprudential regulations such
as leverage restrictions to lean-against-the-wind type policies. We comment on some potential extensions
of our model in the Conclusion.
The paper is organized as follows. In Section 1, we revisit Galí’s analysis of a dynamically ine¢ cient
economy. In Section 2, we describe an environment in which bubbles arise not because of dynamic ine¢ ciency
but because of information frictions. We then conclude in Section 3.
2

1

Dynamic Ine¢ ciency, Bubbles, and Monetary Policy

We begin by reconsidering previous work on monetary policy and bubbles. We focus on a model that draws
on the seminal work of Galí (2014). He introduced price rigidity into a dynamically ine¢ cient economy
that is well known to allow bubbles, and found somewhat surprisingly that tighter monetary policy could
amplify bubbles. While this is also true in our model, we …nd that once we modify it so that asset prices are
uniquely determined, higher rates unambiguously dampen bubbles. We also show how we can capture the
e¤ects of monetary policy in a reduced-form endowment economy without explicitly modelling the monetary
transmission mechanism. Finally, we argue that if bubbles are due to dynamic ine¢ ciency as in Galí’s setup,
there will be no reason to intervene against them. This motivates our shift in the second half of the paper
to bubbles that are due to di¤erent frictions than dynamic ine¢ ciency.

1.1

Dynamic Ine¢ ciency and Bubbles

Consider an overlapping generations economy where agents live for two periods. For convenience, suppose
agents only value consumption in their second period of life. That is, agents born at date t value consumption
ct and ct+1 at dates t and t + 1 according to
u (ct ; ct+1 ) = ct+1

(1)

Agents are endowed with resources only when young. Let et > 0 denote the endowment of the agents born
at date t. Endowments grow at rate g > 0, so agents who are born later are wealthier:
t

et = (1 + g) e0

(2)

Given their preferences, agents are only concerned with converting the goods they are endowed with when
young into goods they can consume when old. We give them two options: They can either store their
endowment to consume later, or exchange their endowment for assets. We posit a …xed supply of assets,
normalized to 1, where each asset yields a constant dividend ‡ow of d 0 consumption goods per period.
Galí assumed d = 0. We assume this initially as well. All assets are initially endowed to the old at date 0.
Let st denote the amount of goods agents store at date t and xt denote the amount they spend on assets.
Then st + xt = et . Old agents will sell any assets they own and consume the goods they trade their assets
for together with any they previously stored. They will therefore consume ct+1 = st + pt+1 (xt =pt ).
1

The only market in this economy is the one for assets. An equilibrium is a path of asset prices fpt gt=0 ,
denominated in goods, which ensures the asset market clears at each date t. That is, prices must be such
that at each date, the old must be willing to sell all their assets and the young must be willing to buy them.
1
Although fpt gt=0 can in principle be stochastic, we restrict attention to deterministic price paths.
As is well known, this setup admits multiple equilibria, including equilibria in which pt > 0 for all t. Since
the asset yields no dividend, we can think of such equilibria as bubbles in the sense that the price of the
3

asset exceeds the present discounted value of its dividends. There is also an equilibrium in which pt = 0 for
all t and there is no bubble. In this equilibrium, agents must rely on storage to convert their endowment
into consumption next period. But note that storage is dynamically ine¢ cient: Since g > 0, if all agents
agreed to transfer their endowment to the previous generation instead of using storage, everyone would get
to consume more. Thus, the equilibrium without a bubble is ine¢ cient.
To characterize the set of deterministic equilibria in our model, let rt denote the rate of return that agents
who buy the asset at date t anticipate to earn from it in equilibrium. Since the asset yields no dividends,
this return is just the rate at which the price of the asset grows between dates t and t + 1:
1 + rt =

pt+1
pt

(3)

Suppose the equilibrium return rt from buying the asset at date t were positive. The cohort born at date t
would then strictly prefer the asset to storage, so st = 0. Since we normalized the supply of the asset to 1,
agents will spend all of their endowment on the asset and so
pt = et
Hence, in any date t in which the equilibrium interest rate rt > 0, the asset price and interest rate are
uniquely determined.
Next, suppose the equilibrium return rt at date t was equal to 0. In this case, the young would be
indi¤erent between storing their endowment and buying assets. While the equilibrium price pt is no longer
uniquely pinned down, we can still say something about the path of prices. Since rt = 0, the price doesn’t
grow between t and t + 1, i.e., pt+1 = pt . Given pt = et st et , then when rt = 0, the price in the next
period pt+1 will necessarily be less than the endowment of agents, as follows from the fact that
pt+1 = pt

et < et+1

Hence, st+1 = et+1 pt+1 > 0, meaning agents at date t + 1 store some of their endowment at date. But
they will only agree to do so if rt+1 = 0. Hence, a zero real interest rate is absorbing: Once the real interest
rate falls to 0, it will remain there inde…nitely. Since the price of the asset grows at the rate of interest, it
follows that the price of the asset will remain constant starting from any date t in which rt = 0.
Since a zero interest rate is absorbing while a positive interest rate is associated with a unique pt , any
deterministic equilibrium can be described in terms of a cuto¤ date t 2 f0; 1; 2; :::; 1g such that rt > 0
before date t and rt = 0 from t on. Before t , the asset price must equal et . From t on, the price cannot
grow. If we de…ne e 1 0, the price at date t can assume any value between et 1 and et . Formally,
1

Proposition 1 Suppose d = 0. A deterministic path fpt gt=0 is an equilibrium i¤ there exists a cuto¤ date
t with 0 t
1 and some value pt 2 [et 1 ; et ) such that
(
et if t < t
pt =
(4)
pt if t t
4

The proofs of this and other propositions not derived in the text are in Appendix A. Figure 1 illustrates
some sample equilibrium price paths. These equilibria can be indexed by the asymptotic price of the asset,
limt!1 pt . For any p 0, there exists a unique deterministic equilibrium for which limt!1 pt = p. When
p e0 , the threshold t is equal to 0. For p > e0 , the threshold t is the value for which et 1 < p et .
Finally, Figure 1 suggests that equilibria that feature higher price growth pt+1 =pt also feature higher price
levels. To put it another way, equilibria with higher interest rates feature larger bubbles. In the Appendix,
we prove that this is indeed true for the set of equilibria in Proposition 1. Intuitively, other things equal, a
higher interest rate would induce agents to shift from storage to buying more assets. But since the supply
of assets is …xed, this will only bid up the price of the asset.1

1.2

Monetary Policy and Nominal Price Rigidity

The previous subsection describes a standard model of bubbles due to dynamic ine¢ ciency. To study how
monetary policy a¤ects such bubbles, we need to move beyond an endowment economy so that monetary
policy can matter. This was the innovation in Galí (2014), who …rst incorporated a production economy
with rigid prices into a setting with bubbles. In Appendix B, we consider a similar model, but one where
preferences and incomes are consistent with the endowment economy from the previous subsection. In
contrast to the endowment economy above, young agents in the model described there are endowed with
productive inputs rather than goods. Their incomes depend on the output produced in equilibrium, and
can in principle vary with monetary policy. Given the incomes of agents, asset prices are determined just
as if agents were exogenously endowed with these incomes. In this sense, the endowment economy in the
previous subsection can be viewed as the reduced form of the production economy in Appendix B.
The full details of the production economy are in Appendix B. Here, we only sketch the outline of that
model. The way we capture monetary policy is by introducing a central bank that can announce a nominal
interest rate 1 + it at which it will borrow and lend money to agents. We only consider equilibria in which
agents do not trade with the central bank, meaning the in‡ation rate in equilibrium must be such that the
real return to trading with the central bank is the same as the return on the intrinsically worthless asset.
To allow for nominal rigidities, we assume sellers set the price of their goods each period before a sunspot
variable t is realized while the monetary authority sets the nominal interest rate after observing t . Thus,
price-setters cannot perfectly anticipate what nominal interest rate the central bank will set.
An equilibrium is a path of prices (goods prices, input prices, and the real interest rate) for every
1
realization of sunspots f t gt=0 and the nominal interest rates given these realizations such that markets
clear when agents behave optimally. The formal conditions that de…ne an equilibrium are in Appendix B.
1 This

result is equivalent to the result in overlapping generations monetary models that, under certain conditions, the real
value of money will be higher the lower the in‡ation rate is. This is because our asset is akin to money in that it pays no
dividends, and the return on the asset is akin to the growth in the value of money. See Blanchard and Fischer (1989, pp158-9).

5

The model admits multiple equilibria. In some of these equilibria, a higher nominal interest rate leads to
a higher real interest rate and a smaller bubble. But there are other equilibria in which a higher nominal
interest rate leads to a higher real interest rate and a larger bubble.
To elaborate, there exists an equilibrium in which a higher nominal interest rate is associated with
lower input prices, lower output, lower asset prices, and a higher real return to buying the intrinsically
worthless asset. This equilibrium corresponds to the typical way in which higher nominal interest rates are
contractionary in models with nominal price rigidity. In the analogous endowment economy, this would
correspond to changing the path of endowments in a way that makes agents poorer. As this induces agents
to save less, their demand for assets falls. Given the asset is in …xed supply, its price falls.
But the model admits another equilibrium in which a higher nominal interest rate is associated with
the same levels of output, higher asset prices, and a higher real return to buying the intrinsically worthless
asset. We can use Figure 1 to understand this equilibrium. Recall that this …gure illustrates all deterministic
1
equilibria that are possible for a given path of endowments fet gt=0 . If a higher nominal interest rate had no
e¤ect on output or what agents earn, the economy could still switch from an equilibrium in Figure 1 with a
lower path for real interest rates to one with a weakly higher path for real interest rates. As we observed in
the previous subsection, the equilibrium with higher real interest rates are associated with larger bubbles.
In this equilibrium, a higher nominal interest rate induces agents to buy assets rather than store their goods.
This increases demand for the asset. Given the asset is in …xed supply, its price rises.
In sum, monetary policy has an indeterminate e¤ect on bubbles in our model. Galí (2014) focuses on
equilibria that resemble the second equilibrium we describe, in which higher nominal interest rates raise
demand for assets.2 But in the model he studies, there similarly exist equilibria in which higher nominal
interest rates impoverish agents and decrease demand for assets. Dong, Miao, and Wang (2017) and Ikeda
(2017), who study models of bubbles that feature a similar indeterminacy, instead focus on equilibria
that resemble the …rst equilibrium we describe, in which higher nominal interest rates reduce demand for
assets.3 But there exist equilibria in their models in which higher nominal interest rates induce agents to
spend more on assets. How monetary policy operates in these models depends on the equilibrium we choose.
Our discussion suggests that the equilibria in which a higher nominal rate leads to a larger bubble rely on
the indeterminacy of asset prices, which allows agents to coordinate on an equilibrium in which they all
2 Galí focuses on an equilibrium in which the economy tends to a stable steady state in the long run and in which the
sunspot that leads the central bank to set a higher nominal interest rate also results in a higher asset price one period later.
This is because he assumes the sunspot is associated with the creation of new bubble assets with no o¤setting decline in the
value of existing assets. That requires that agents next period agree to spend more on assets in total.
3 These

papers focus on equilibria in which the economy tends to an unstable or saddle-path stable steady state. This
restriction implies asset prices cannot change in the long run in response to a shock. But if asset prices are unchanged in the
long run and the price of the asset grows at the rate of interest, higher interest rates require lower initial prices for asset prices
to remain at the same level in the long run. Note that in these two papers bubbles are due to borrowing constraints rather
dynamic ine¢ ciency, but the indeterminacy in these models is qualitatively similar.

6

spend more on assets when the nominal interest rate is high. If this were indeed the case, then eliminating
the indeterminacy of asset prices should eliminate the equilibrium in which a higher nominal interest rate
leads agents to coordinate to buy more assets. In the next subsection, we show that changing one feature
of the model yields a unique equilibrium in the asset market, and that this equilibrium still corresponds to
a bubble. We then con…rm that in this case, if a higher nominal interest rate leads to a higher real interest
rate, it must impoverish agents and dampen the bubble.

1.3

Eliminating Indeterminacy

While bubbles may seem to be inextricably linked with indeterminacy, Tirole (1985) already showed that
in some circumstances, a dynamically ine¢ cient economy could feature a bubble as the unique equilibrium
outcome. This occurs if (1) there is an asset that pays out positive dividends, and (2) without a bubble, the
real interest rate agents earn would tend to zero or to a negative value. Intuitively, in these circumstances
the present discounted value of dividends would tend to in…nity if there were no bubble, and agents would
not be able to a¤ord the asset. The only possible equilibrium, then, is one that features a bubble.
We can use this insight to modify our model to eliminate its indeterminacy. For simplicity, let us return
to our original endowment economy. Suppose we replace the intrinsically worthless asset in which d = 0
with an asset that pays a …xed positive dividend d > 0 at all dates. Note that over time, this dividend
becomes small relative to the endowment agents would like to save. This implies that without a bubble,
when returns can only be due to dividends, the return on investment that agents can expect would tend to
zero, the exact condition Tirole identi…ed with uniqueness.4
To con…rm that d > 0 implies a unique equilibrium, we …rst show that the equilibrium return to holding
the asset rt can never equal 0. For suppose rt did equal 0 at some date t. Then (3) would imply
pt+1 = pt

d < pt

Since pt
et , it again follows that pt+1 < pt
et < et+1 . But if agents store goods at date t + 1, then
rt+1 = 0. By this logic, the price would continue to decline in increments of d from date t + 1 on, until it
would eventually turn negative. But a negative price cannot be an equilibrium, since the cohort that owns
the assets would refuse to sell them. The only candidate equilibrium price path is one with strictly positive
interest rates at all dates. But in that case, storage is dominated and pt = et for all t. Formally, we have
1

Proposition 2 Suppose d > 0. Then the unique equilibrium path fpt gt=0 is given by
pt = et
4 More

(5)

generally, we can allow for an asset whose dividends dt vary over time. Our proof of Proposition 2 shows the
P
equilibrium is unique i¤ 1
t=0 dt = 1 with dt > 0 for all t. The fact that this equilibrium can only correspond to a bubble if
limt!1 dt =et = 0, or if we have an asymptotically vanishing dividend yield, was shown by Tirole (1985) and Rhee (1991).

7

and the unique equilibrium interest rate is given by
rt =

pt+1 + d
pt

1=g+

d
et

(6)

To con…rm that the equilibrium in Proposition 2 still represents a bubble, we …rst need to de…ne the
fundamental value of the asset. In particular, we need to take a stand on the rate at which to discount
dividends. If we took resources from a young agent at date t, she would demand 1 + rt units at date t + 1
to remain equally well o¤. Hence, the market interest rate rt captures the way society trades o¤ resources
between adjacent dates. This leads us to de…ne the fundamental value of the asset as
!
jY1
1
X
1
ft =
d
(7)
1 + rt+i
j=1 i=0
From Proposition 2, rt > g > 0 for all t. Hence, (7) is bounded above for all t, since
ft

1
X
j=1

1
1+g

j

d=

d
g

Furthermore, limt!1 ft = d=g < 1. The fundamental value of the asset is thus bounded. At the same
t
time, the asset price grows without bound, since limt!1 pt = limt!1 (1 + g) e0 = 1. The asset price
will eventually exceed its fundamental value, at which point it will be a bubble. But we can show that the
equilibrium price of the asset exceeds the fundamental value at all dates rather than just asymptotically.
To see this, note that the equilibrium interest rate rt in Proposition 2 implies that
pt =

d + pt+1
1 + rt

(8)

At the same time, the fundamental value ft in (7) satis…es
ft =

d + ft+1
1 + rt

(9)

Subtracting the latter expression from the former reveals that the di¤erence bt pt ft , which is just the
size of the bubble at date t, must satisfy
bt+1
bt =
(10)
1 + rt
Since bT > 0 as T ! 1, it follows that b0 > 0. Hence, the price of the asset exceeds its fundamental value
at all dates, and the asset is necessarily a bubble.

1.4

Monetary Policy Revisited

The fact that d > 0 yields a unique equilibrium extends to the production economy we lay out in Appendix
B. We can therefore use that framework to analyze the e¤ects of monetary policy in a production economy
where asset prices are uniquely determined. For simplicity, suppose that instead of a sunspot at each date
t, there is a single sunspot at date 0. In Appendix B, we show that in this case monetary policy has no
8

e¤ect on real variables beyond t = 0. Intuitively, since producers can set the price of goods each period,
they can perfectly anticipate monetary policy from date t = 1 on.
While the formal analysis of the production economy is contained in Appendix B, we can use our endowment economy to illustrate the key insights. The path of incomes in equilibrium when the nominal interest
t
rate is low at date 0 can be represented by a path of endowments et = (1 + g) e0 . Since monetary policy
has no real e¤ects beyond date 0, the incomes agents earn at dates t 1 must be the same regardless of
what nominal interest rate the central bank chose at date 0, and hence et must be the same from date t = 1.
Let denote the di¤erence in the equilibrium income of young agents at date 0 when nominal interest rates
are high and when they are low, meaning young agents earn e0
if the monetary authority sets a high
nominal interest rate at date 0. Proposition 2 tells us that when the asset pays a …xed dividend, the unique
equilibrium is one in which agents spend all of their income to buy assets. Hence, the equilibrium asset
price when nominal interest rates are high at t = 0 is given by
(
e0
if t = 0
pt =
(11)
et
if t = 1; 2; 3; :::
and the equilibrium real return on buying the asset is given by
8
e1 + d
>
>
if t = 0
<
e0
rt =
>
>
: g + d if t = 1; 2; 3; :::
et

(12)

Equation (12) shows that the real interest rate maps into a unique and equation (11) shows that the asset
price p0 maps into a unique . The only way monetary policy can increase the real interest rate rt is if
> 0, i.e., if it depresses output and the price of the asset. To con…rm that it also dampens the bubble
bt = pt ft , observe that destroying goods at date 0 has no e¤ect on either pt or rt for t
1, since
these depend on endowments from date t = 1 on. Hence, neither pt nor ft change beyond date 1, and
b1
. Since the bubble b1 is unchanged
so bt is unchanged for t
1. But we know from (10) that b0 = 1+r
0
and r0 is higher, b0 must fall. Thus, once we eliminate the indeterminacy in asset prices, a higher nominal
interest rate that increases the real interest rate must also dampen the bubble, and equilibria like the ones
we discussed earlier in which increasing the real interest rate ampli…es bubbles will no longer arise.
For the remainder of the paper, we will use an endowment economy to capture the e¤ect of tighter
monetary policy by looking at the e¤ects of destroying part of the endowment of some cohort in lieu of
explicitly modeling production and monetary policy. We should note that other policy interventions also
imply the same reduced form e¤ect of destroying part of the endowment. For example, a lump-sum tax
on the young at date 0 that is used to …nance government consumption would be similarly captured by
destroying part of the endowment of young agents. In addition, when we introduce credit in the next
section, a restriction on the amount agents can lend along the lines of how Allen and Gale (2000) model
credit tightening can also be represented as a destruction of endowments. While we interpret this thought
experiment as contractionary monetary policy, it captures other interventions as well.
9

1.5

Welfare

To recap, our analysis so far has dealt with the problem of indeterminacy in models of bubbles due to
dynamic ine¢ ciency. In the standard model, the e¤ects of monetary policy are ambiguous. Although Galí
(2014) emphasized the surprising fact that a contractionary monetary policy can amplify bubbles, it can
also dampen bubbles. We showed that if we change some of the assumptions of the model to ensure a unique
equilibrium in asset markets, a higher real rate is necessarily associated with a smaller bubble. This suggests
policymakers can dampen bubbles by increasing rates. But in the model of bubbles we have explored thus
far, there is in fact no reason for a policymaker to intervene against bubbles. This is because bubbles only
serve to alleviate the underlying dynamic ine¢ ciency that allows for bubbles in the …rst place.
To see this, suppose we destroy units of the endowment of those born at date 0. The policy is clearly
costly, with the cost fully borne by the old at date 0 who would otherwise have consumed those units
when they exchanged them for their assets. But there are no countervailing bene…ts to reducing the bubble:
All other agents are una¤ected, and each cohort born at date t 0 will continue to consume et+1 + d for all
t 0, since our preferences imply that each period all goods will go to that period’s old. More generally,
the cost of destroying goods at date 0 could be borne by young agents at date 0 as well as by later cohorts.
But such a policy could never be welfare improving. As Tirole (1985) and others have argued, asymptotic
bubbles in which limt!1 bt =et > 0 are dynamically e¢ cient in these economies. Any intervention which
changes what agents consume must make at least some agents worse o¤.
While our analysis assumed the supply of assets of assets is …xed, this insight extends to the case where
new assets can be created. Even though the price of the asset exceeds the present discounted value of
dividends it generates, there is no bene…t to raising rates in order to discourage the creation of assets. We
show this formally in Appendix C, where we allow for an endogenous supply of assets.
In sum, the problem with models of bubbles due to dynamic ine¢ ciency for the lean-against-the-wind view
is not that they imply raising rates ampli…es bubbles, but that they imply there is no reason to intervene
against such bubbles. To make a case for leaning against the wind, we need a model in which bubbles do
not alleviate the friction that allows them to arise in the …rst place. In the next section, we construct such
a model in which the friction that gives rise to bubbles is di¤erent than dynamic ine¢ ciency.

2

Credit-Driven Bubbles

In the previous section, we argued that models like Galí (2014) cannot be used to justify leaning-againstthe-wind policies, not because higher interest rates in these models amplify bubbles but because there is no
reason to dampen bubbles in these models. To the contrary, it is the equilibrium without a bubble in these
models that is generally ine¢ cient. These models also fail to capture the main concern of policymakers,
who worry that bubbles …nanced by bank lending may collapse and threaten the …nancial system. Mishkin
10

(2011) nicely summarizes this concern:
[N]ot all asset price bubbles are alike. Financial history and the …nancial crisis of 2007-2009
indicates that one type of bubble, which is best referred to as a credit-driven bubble, can be
highly dangerous. With this type of bubble, there is the following typical chain of events: Because
of either exuberant expectations about economic prospects or structural changes in …nancial
markets, a credit boom begins, increasing the demand for some assets and thereby raising their
prices... At some point, however, the bubble bursts. The collapse in asset prices then leads to
a reversal of the feedback loop in which loans go sour, lenders cut back on credit supply, the
demand for the assets declines further, and prices drop even more. The resulting loan losses
and declines in asset prices erode the balance sheets at …nancial institutions, further diminishing
credit and investment across a broad range of assets. The decline in lending depresses business
and household spending, which weakens economic activity and increases macroeconomic risk in
credit markets. In the extreme, the interaction between asset prices and the health of …nancial
institutions following the collapse of an asset price bubble can endanger the operation of the
…nancial system as a whole
In this section, we show how, starting with the framework developed in the previous section, we can
generate credit-driven bubbles along the lines in Mishkin’s quote. In particular, we eliminate dynamic
ine¢ ciency and introduce a di¤erent friction that allows bubbles to arise. To do this, we modify our model
in several ways. First, we assume the endowment no longer grows over time. This ensures the economy
is dynamically e¢ cient. Second, we add a credit market and introduce information frictions that prevent
lenders from monitoring their borrowers. For the latter friction to matter, we also assume dividends are
stochastic, although we continue to assume they are always positive. The fact that assets are risky allows
agents to gamble at the expense of creditors, which gives rise to a bubble.

2.1

An Economy with Credit

Our starting point is the endowment economy in the previous section in which agents can invest in an asset
that yields a constant dividend d > 0 each period. Our …rst modi…cation is to assume g = 0 so that all
cohorts receive the same endowment et = e. Since the proof of Proposition 2 does not rely on the value of
g, we can still use the proposition to determine the equilibrium. Hence, the unique equilibrium is given by
pt = e for all t and rt = d=e r for all t. However, setting g = 0 leads to an important di¤erence. The
value of all dividends discounted by the return on the asset 1 + r is given by
f0 =

1
X
t=0

1
1+r

t

d = d=r = e

The fundamental value of the asset is now equal to the price, in contrast to the case where g > 0. Intuitively,
without growth there is no role for intergenerational transfers that allow old agents to bene…t from the larger
11

endowment of the young generation. The economy is thus dynamically e¢ cient, and a bubble no longer
emerges. We will set g = 0 from now on, so any bubble that arises cannot be due to dynamic ine¢ ciency.
We next introduce a credit market. For agents to trade in this market requires some heterogeneity among
young agents, since they are the only ones willing to trade intertemporally. We continue to assume each
cohort includes a unit mass of savers who are born with an endowment of goods e and whose preferences
are given by (1) and whose only concern is to convert their endowment when young into consumption when
old. But we now assume that each cohort also includes an in…nitely large mass of potential entrepreneurs.
The latter are endowed with no resources, but they have access to a technology that converts goods at date
t into goods at date t + 1.
To be more precise, we assume each entrepreneur can transform up to one unit of the good at date t into
y > 1 units at date t + 1. Potential entrepreneurs di¤er in their productivity y. Let N (y) denote the mass
of entrepreneurs whose productivity is at least y. We assume that N (y) is di¤erentiable and satis…es
N 0 (y) < 0; lim N (y) = 1; lim N (y) = 0:
y!1

y#1

The assumption that N (y) is strictly decreasing implies N (y) > 0 for all y 1. Thus, for any …nite interest
rate that savers might charge, there would always be a positive mass of entrepreneurs who are su¢ ciently
productive that they would …nd it pro…table to borrow resources and produce.
Savers and entrepreneurs can trade in a centralized credit market with a single interest rate Rt . We
assume that trade takes the form of loans. Entrepreneur must repay a …xed amount 1 + Rt units of the
good at date t + 1 for each unit they borrowed at date t. An entrepreneur who cannot meet the required
payment, for whatever reason, is said to be in default and we assume a court can compel the transfer of all
of his or her resources to the creditor. Entrepreneurs who default incur a small utility cost of per unit
borrowed. In practice, we will take the limit as ! 0. As long as > 0, borrowers will never take out a
loan they expect to default on with certainty. This helps us avoid equilibria in which agents are willing to
take out loans they will certainly default on. For now, we assume there are no other costs of default. We
will consider the consequences of allowing for costs of default to lenders later on.
1

An equilibrium in this economy is a path of asset prices fpt gt=0 and a path of interest rates on loans
1
fRt gt=0 such that markets clear when agents optimize. To solve for an equilibrium, we need to describe
supply and demand for both assets and credit. The optimal behavior of agents in their second period of
life is straightforward: They collect repayment on loans they made in the past and pay back the loans they
have outstanding. If they own assets, they sell them if the price is nonnegative. As for young agents, savers
must choose between buying assets and lending. They allocate their wealth to the investments that o¤er the
highest returns. Entrepreneurs choose whether to borrow the resources they need to produce and whether to
borrow to buy assets. Given the interest rate on loans, Rt , entrepreneurs with productivity y > 1 + Rt will
borrow to produce, regardless of whether they also borrow to buy assets. Whether entrepreneurs borrow to
buy assets depends on whether that activity is pro…table after netting out interest costs.
12

For any interest rate Rt , the marginal entrepreneur has productivity ym = 1 + Rt and the demand for
loans from entrepreneurs who want to produce will be N (ym ) = N (1 + Rt ) > 0. As a result, the amount
of the endowment available to purchase the asset, either directly or with borrowed funds, is less than e.
Market clearing requires that
N (1 + Rt ) + pt = e
(13)
Our assumptions on N (y) imply there is a unique market-clearing interest rate Rt = (pt ) for any price of
the asset. Since N 0 (y) < 0, the function (pt ) is increasing: A higher pt reduces the amount of goods
available for productive investment, so the interest rate on loans Rt must rise to lower demand from
entrepreneurs who want to borrow in order to produce.
Since entrepreneurs would borrow unbounded amounts to buy the asset if they could earn positive pro…ts,
the interest rate on loans must be at least as high as what agents can earn from buying the asset. At the
same time, the interest rate on loans cannot exceed the return to buying the asset, or else no one would
buy the asset: Young savers would strictly prefer to make loans, and young entrepreneurs would refuse to
borrow to buy the asset knowing they will default. Thus, in equilibrium the interest rate on loans must
equal the return to buying the asset:
pt+1 + d = (1 + Rt ) pt = (1 + (pt )) pt
De…ne ' (p)

(1 + (p)) p

(14)

d. We can immediately see that ' (p) has the following properties:
'0 (p) > 1, for any p;
' (p) < 0, for p > 0 su¢ ciently small;
' (p) >

e, for p > 0 su¢ ciently large.

The …rst property follows from the fact that 0 (p) > 0; the second follows from the de…nition of ' (p) and
the fact that
lim (1 + (p)) p = 0;
p!0

and the third follows from the fact that

0

(p)

0.

The graph of ' is illustrated in Figure 2. The red curve is the graph of pt+1 = ' (pt ) and the black curve
is the 45o line. The two lines intersect at a unique point where p = ' (p ). For any initial condition, the
law of motion pt+1 = ' (pt ) de…nes a unique path of asset prices, but for any initial condition other than
p0 = p , the path will drift away from p until either pt < 0 or pt > e in …nite time, neither of which can
be an equilibrium. Thus, the unique equilibrium path is a steady state in which pt = p and Rt = (p )
for all t. A fact that will be useful later is that the steady state price p is increasing in the dividend d.
Setting pt = pt+1 = p in the zero-pro…t condition (14), we see that
d = (p ) p :
The right hand side is increasing because (p ) is increasing, so an increase in d must increase the equilibrium
asset price p . Graphically, a larger d will lead the red curve in Figure 2 to shift down, and so the steady
state p must rise.
13

Finally, observe that the equilibrium return on the asset rt = d=p is the same as the equilibrium interest
rate on loans, since Rt = (p ) = d=p . The value of dividends discounted at this rate of return is given by
ft =

1
X
j=1

j

1
1+R

d = d=R = p

Hence, the introduction of credit does not on its own introduce a wedge between the price of the asset and
its fundamental value. We collect these results in a summary proposition.
Proposition 3 The unique equilibrium for the economy with credit and a constant dividend d > 0 features
a constant asset price pt = p , a constant interest rate on loans Rt = (p ), and a constant fundamental
ft = p for all t, where p is the …xed point of the equation p = '(p).
The introduction of credit per se does not lead to bubbles. For that, we require some additional frictions.
In the next subsection, we assume asymmetric information in the market for credit. We also assume assets
are risky, which means that our assumption that agents must use non-contingent debt contracts represents
an additional market imperfection.

2.2

Risky Assets and Information Frictions

To allow for risky assets, we introduce a regime switching process inspired by Zeira (1999). The asset
initially pays a dividend dt = D > 0, but there is a probability > 0 in each period that the dividend falls
to a lower level 0 < d < D. Once the dividend dt falls to d, it remains at the lower level forever.
The combination of risky assets and credit contracts introduce the possibility of default. Because agents
can borrow unlimited amounts to buy assets, market clearing still requires that agents cannot expect to
earn positive pro…ts from borrowing to buy assets. That is, the interest rate must be high enough to ensure
zero pro…ts to buying the asset even when the return on the asset is highest. But in that case, entrepreneurs
will be forced to default whenever the return on the asset is below the maximum possible. Below we show
that the maximal return to buying the asset when its future dividend is uncertain occurs if the dividend
remains high, so entrepreneurs who borrow to buy the asset default if and when the dividend falls.
The second friction is that savers cannot observe an entrepreneur’s productivity and cannot monitor what
the borrower does with the funds he obtains. This information friction is the key feature that allows bubbles
to emerge, because it implies that less productive entrepreneurs will be able to borrow and speculate on
assets instead of producing. Allen and Gorton (1993), Allen and Gale (2000), and Barlevy (2014) previously
showed that such information frictions allow bubbles to arise. But these papers do not explore interest rate
policy and its consequences for welfare, as we do here.
Equilibrium is de…ned in two steps, corresponding to the two regimes in equilibrium: either the dividend
has not yet fallen and the asset is risky, or the dividend has fallen and the asset is safe. Let pD
t (respectively,
14

pdt ) denote the price of the asset at date t if dt = D (respectively, dt = d), and similarly let RtD (respectively,
Rtd ) denote the interest rate on loans at date t if dt = D (respectively, dt = d). Once the asset’s dividend
has fallen to d, the economy is identical to the one analyzed in the previous subsection. As summarized
in Proposition 3, there is a unique equilibrium with a constant asset price pdt = pd , which we previously
denoted by p , a constant interest rate Rtd = Rd which we previously denoted by (p ), and no bubble.
The equilibrium in the regime before the dividend falls is similar in some respects. Forget for a moment
that the dividend on the asset may fall, and consider what would happen if the dividend were D forever.
The interest rate on loans would have to equal the return to buying the asset:
(1 + Rt ) pt = pt+1 + D

(15)

By the same argument as before, there is a unique solution to equations (13) and (15), which takes the form
of a constant asset price pt = pD and a constant interest rate Rt = RD . As we noted above, an increase in
the dividend increases the steady state asset price, so pD > pd .
D
We now argue that setting pD
= pD ; RD as long as the dividend is high constitutes an equilibrium
t ; Rt
path for the economy where dividends are stochastic. We know by construction that this path satis…es
the equilibrium condition (13), so we only need to show that at these prices agents cannot pro…t from
borrowing to buy the asset. Since 1 + RD pD = pD + D, we need to verify that agents cannot expect
to earn positive pro…ts from borrowing to buy the asset and hoping for the low dividend regime, i.e., that
1 + R D pD
pd + d. Recall that the equilibrium interest rate Rt = (pt ) where (pt ) is an increasing
function that does not depend on the dividend. This implies

RD =

pD >

pd = R d

Since pD > pd and RD > Rd , it follows that 1 + RD pD > 1 + Rd pd = pd + d, where the last equality
follows from the de…nition of pd . So an agent who borrows to buy the asset will indeed earn zero pro…ts
given these prices.
Thus, in the economy where the dividend is stochastic, there exists an equilibrium in which during the
D
high dividend regime the price of the asset pD
t and the interest rate on loans Rt are the same as they would
D
be if the dividend remained high forever. But is this the only equilibrium? Let pD
denote any
t ; Rt
D
d
alternative equilibrium path for the high-dividend regime. If we can show that pt + D > p + d for every
t, then the zero pro…t condition that de…nes the equilibrium would correspond to (15). But under (15), if
D
pD
6= pD ; RD for some t, the price of the asset in the high dividend regime would eventually either
t ; Rt
turn negative or exceed e, neither of which can be an equilibrium. Suppose that in fact pD
pd + d
t +D
d
for some t. This would require that pD
t < p . Since (pt ) is increasing and the equilibrium interest rate on
loans Rt is always equal to (pt ), this would imply
RtD =

pD
<
t

pd = R d

But then we would have
d
1 + RtD pD
pd = pd + d:
t < 1+R

15

This means that an entrepreneur can make positive pro…ts at date t if the dividend falls next period,
d
which cannot be true in equilibrium. This contradiction implies pD
pd , and so pD
t
t + D > p + d in any
D
equilibrium. The equilibrium we constructed, in which pD
= pD ; RD for all t, is unique.
t ; Rt
Finally, we observe that when the dividend is d, savers are indi¤erent between buying the asset and
lending, while entrepreneurs are indi¤erent about borrowing to buy the asset. Hence, while the interest rate
on loans is uniquely determined, the amount of lending is not. By contrast, when the dividend is D, savers
will not be willing to invest in the asset. Let 1 + rD denote the expected return to buying the asset. Then
1 + rD =

(1

) D + pD +
pD

d + pd

If a saver were to lend in the credit market, she wouldn’t know whether she was lending to an entrepreneur
who will produce or an entrepreneur who plans to speculate on assets. In the …rst case, she would earn the
interest rate on loans 1 + RD , which from (15) we know is equal to D + pD =pD , and in the second she
would earn the expected return on the asset 1 + rD . Since we argued above that d + pd < D + pD , the
D
interest rate on loans exceeds the expected return to buying the asset, i.e., R > rD . Hence, the expected
return from lending is greater than the expected return from buying the asset. Only low productivity
entrepreneurs hold the asset. We summarize the equilibrium with the following proposition:
Proposition 4 The equilibrium for the economy with credit and stochastic dividends is unique. As
the prices and interest rates in this equilibrium are given by
(
pD ; R D
if dt = D
(pt ; Rt ) =
pd ; R d
if dt = d

! 0,

While dt = D, savers strictly prefer to lend out their savings, and low productivity entrepreneurs with
y < 1 + RD are the only ones who invest in the asset.
The fact that the price of the asset when the dividend is stochastic is identical to the price of the asset
when the dividend remains high forever suggests the asset is overpriced. This interpretation is consistent
with the fact that savers avoid buying the asset in the high dividend regime. This would suggest the asset
is a bubble during the high dividend regime. We con…rm this in the next subsection.

2.3

Credit-Driven Bubbles

To determine whether the asset is a bubble, we need to calculate the fundamental value of the asset in each
dividend regime. Once the dividend has fallen to dt = d, the economy is just the one with a safe asset. As
we have already seen, the equilibrium of this economy is stationary, and the fundamental value of the asset
is constant and equal to the asset price:
f d = pd = d=Rd

16

When the dividend is still high, so dt = D, the equilibrium is stationary conditional on the dividend being
high. In particular, the returns agents earn on their investments are constant. The fundamental value must
therefore also be constant. The question is at what rate we should discount dividends.
Up to now, the discount rate we used to de…ne the fundamental value of the asset was the compensation
we need to provide an agent at date t+1 if we took a unit of resources from her at date t. In the dynamically
ine¢ cient economy in the previous section, the necessary compensation was the return to buying the asset
1+rt or (pt+1 + d) =pt , which was e¤ectively the only option available for agents. In the economy with credit
and a safe asset, the return to lending 1 + Rt is the same as the return to the asset 1 + rt in equilibrium, so
there is again no ambiguity about the relevant discount rate. But in the economy with credit and a risky
asset, the return agents can earn varies in equilibrium both across investments (the expected return on loans
exceeds the expected return to buying the asset) and across individuals (more productive entrepreneurs can
earn higher returns). So the question of what is the appropriate discount rate becomes relevant.
It will be useful to single out three particular rates of return in our credit economy during the high
dividend regime, when the return on the asset is risky. One is the interest rate on loans, 1 + RD . Since (15)
holds in equilibrium, this interest rate is given by
1 + RD =

D + pD
pD

(16)

Another rate of return in this economy is the expected return to buying the asset. We previously denoted
this rate by 1 + rD . This expected return is given by
1 + rD =

d + pd + (1
pD

) D + pD

(17)

D

Finally, let us de…ne the expected return to lending 1 + R . Since some agents will default on their loans,
this expression will di¤er from the interest rate on loans 1 + RD . In particular, lenders collect 1 + RD from
each unit they lend out that is used in production, and expect to earn 1 + rD from each unit they lend out
that is used to buy assets. Let denote the fraction of the endowment e that is invested in the asset, so
= pD =e. Then the expected return to lending is just a weighted average of 1 + RD and 1 + rD , speci…cally
1 + RD = 1 + (1

) RD + r D

(18)

Since the price of the asset is positive, > 0. At the same time, because the mass of entrepreneurs who
borrow in equilibrium is positive given our assumptions on N (y), some of the endowment will be used for
D
production, so < 1. We can therefore rank the three returns, with rD < R < RD .
So, which of these rates, if any, should we use to discount dividends? E¤ectively, this amounts to asking
how much society would value an additional unit of resources during the high dividend regime. Under full
information, that extra unit would be allocated to the most productive entrepreneur, whose productivity
is 1 + RD . But since information frictions make it impossible to direct resources to the most productive
entrepreneur, the best we can do is lend it out to any willing borrower, and the expected return on a loan
17

D

D

is equal to 1 + R . Following this logic, we will de…ne the fundamentals using 1 + R as our discount rate.
But regardless of whether we discount using the return on a loan to a random borrower or a loan that is
directed towards more productive entrepreneurs, this return would exceed the expected return to buying
the asset 1 + rD . We now argue that as long as we discount dividends at a rate above 1 + rD , the price of
the asset will exceed the fundamental value.
Since the equilibrium during the high dividend regime is stationary, we can de…ne the fundamental value
of the asset f D recursively as
1 + RD f D =

d + pd + (1

) D + fD

(19)

D

Equation (19) incorporates 1 + R as the relevant discount rate. It also uses our result in Proposition 3
that there is no bubble in the low dividend regime, implying pd = f d . If we compare (19) to (17), we see
that since rD < RD , then
d + pd + (1
pD

) D + pD

<

d + pd + (1
fD

) D + fD

But this inequality can only be true if pD > f D . As long as we discount dividends at a rate that exceeds
the average return to buying the asset 1 + rD , the fundamental will be lower than the price. We summarize
this with a proposition.
Proposition 5 In the credit economy with stochastic dividends, if the fundamental value f D discounts
dividends at the expected return on loans Rt , then f D < pD and f d = pd . The bubble in the initial high
dividend regime, bD = pD f D , is given by
bD = (

d + pd + (1

) D)

1
+ rD

1
>0
+ RD

(20)

Proposition 5 formalizes the notion that, in this economy, the asset is overpriced during the high dividend
regime. Demand for the asset by less productive entrepreneurs pushes the price of the asset up. At this high
price, the return on the asset is low relative to alternative investments, which is why savers avoid buying it.
It is worth commenting on the di¤erences between the bubble in the dynamically ine¢ cient economy from
the previous section and the credit-driven bubble that emerges here. In the former, the price of the asset
exceeded the value of dividends discounted according to the return on the asset, 1 + rt or (pt+1 + d) =pt .
This is because in the dynamically ine¢ cient economy, the return to buying the asset consists of not only
a dividend yield but also a transfer from the subsequent generation. The fact that the return on the
asset is high reduces the fundamental value. By contrast, in the dynamically e¢ cient economy with credit,
discounting dividends according to the expected return on the asset yields a value equal to its price. The
reason the asset can still be viewed as a bubble is that the return on the asset is too low compared with
the return on other investment opportunities available to agents.

18

Another di¤erence between the two types of bubbles is that the bubble in our credit economy can grow
more slowly than the rate of interest. In particular, the bubble doesn’t grow even though agents can earn
a positive return on their savings. By contrast, in the dynamically ine¢ cient economy, (10) implies the
bubble grows at the rate of interest, since bt+1 = (1 + rt ) bt . Intuitively, in the credit-driven bubble in our
model, the agents who e¤ectively own the bubble, namely savers, do so reluctantly. Thus, they do not
require it to o¤er returns that compete with other assets they might hold, as is the case in models without
information frictions where agents know which assets they hold.
That said, credit-driven bubbles are not inherently …xed in size. The only reason the bubble bD remains
constant is that we assumed a stationary economy to simplify the analysis. If we had instead allowed the
endowment to grow at a rate g > 0 in the high dividend regime and then remain constant thereafter, the
economy would continue to be dynamically e¢ cient and a bubble would only arise because of information
frictions. But in that case, the price pt and the bubble bt would both grow over time, as would total credit.
This is consistent with Mishkin’s observation on the connection between bubbles and credit booms.
Finally, the conditions that allow a bubble to arise di¤er in the two cases. In the dynamically ine¢ cient
economy, a necessary condition for a bubble to occur is that the asymptotic growth rate of the economy is
at least as high as the interest rate, in which case intergenerational transfers can make all agents better o¤.
Credit-driven bubbles instead arise because information frictions create a situation in which entrepreneurs
who borrow to produce cross-subsidize those who borrow to buy risky assets. Less productive entrepreneurs
D
keep borrowing until they drive up both the asset price pD
t and the interest rate on loans Rt enough to
push the return to speculation to zero. A bubble thus requires both that assets are risky, so there is a risk
to shift onto creditors, and someone to cross-subsidize speculation. This explains why the bubble term in
(20) tends to zero if either ! 0 or d=D ! 1, meaning the asset isn’t risky, or the share of resources that
go to production tends to zero, meaning there are no agents to cross-subsidize speculation. All three cases
D
imply R ! rD , in which case bD would tend to 0.
Several features of our model line up nicely with historical episodes believed to be bubbles. Our model is
certainly consistent with Mishkin’s description that credit fuels demand for the asset, pushing its price up
until eventually the bubble bursts, which in our model occurs when the dividend falls. The asset price falls
when the bubble bursts, from pD to pd . The collapse of the bubble triggers default on the loans used to
…nance asset purchases. If we had assumed loans were intermediated, then the default by speculators could
have triggered additional defaults by …nancial intermediaries. Thus, our model can be extended to explain
why the collapse of asset bubbles can coincide with …nancial crises. The collapse of the bubble in our model
is also associated with a period of low interest rates, consistent with the recent experience of low real rates
in the wake of the collapse of house prices. At the same time, the cross-subsidization that is central to our
model implies that even during the bubble phase agents who borrow to buy assets are paying too low of an
interest rate. Moreover, the expected return on the asset during the bubble phase, 1 + rD , is low relative
to the return on other investments. However, even if the expected return to buying the asset is low during
the bubble phase, the realized return on the asset is high during the bubble phase. And while lenders su¤er

19

losses when the bubble bursts, they make pro…ts on loans during the bubble phase.

2.4

Monetary Policy and Welfare

Now that we have established that our model admits a bubble, we can turn to the question of whether a
policy intervention to raise the real interest rate in order to dampen the bubble is desirable. Recall from
the previous section that a reduced-form way to capture monetary policy tightening in our endowment
economy is to destroy some of the endowment. We will use the same approach in our economy with credit.
We con…rm that destroying part of the endowment at date 0 is still associated with higher interest rates,
increasing both the interest rate on loans and the expected return on the asset. It will also dampen the
bubble, at least under certain conditions. However, we …nd that, as our model is currently speci…ed, a
tighter monetary policy cannot make society better o¤ even though the bubble is distortionary. We then
argue that this is because the bubble only causes harm while it persists, not when it bursts. When we allow
for costly default in the next subsection, the size of the bubble will matter for the collapse of the bubble
a¤ects agents. In that case, tighter monetary policy could be welfare improving.
As in the previous section, we capture the contractionary e¤ects of monetary policy in an economy with
price rigidity by looking at the e¤ect of destroying the endowment at date 0. That is, we consider an
alternative path for endowments in which
(
e
if t = 0
et =
e
if t = 1; 2; 3; :::
Since the economy at dates t 1 is identical to our original economy, the intervention has no e¤ect at these
dates. We can therefore focus on what happens at date 0. The equilibrium conditions at date 0 are the
same as before, except that the endowment has been reduced. Market clearing requires that the amount
borrowed by entrepreneurs for production and speculation must equal the reduced endowment, e
, so
N 1 + R0D + pD
0 =e

(21)

D
The zero pro…t condition for speculators is the same as before, where now we use the fact that pD
t = p for
all t 1:
D
1 + R0D pD
(22)
0 =p +D

Using the second equation to express the interest rate on loans 1 + R0D at date 0 in terms of the asset price
p0 , we obtain a single equation in p0 ,
N

pD + D
p0

+ p0 = e

Di¤erentiating this with respect to , we get
pD + D
pD
0

2

N

0

pD + D
pD
0
20

!

+1

dpD
0
=
d

1

(23)

Since N 0 (y) < 0 is negative for all y
lower the price of the asset at date 0.

1, it follows that

1<

dpD
0
d

< 0. Destroying the endowment will

D
The fact that a higher lowers pD
0 implies that both the expected return on the asset 1 + r 0 at date 0
D
and the interest rate on loans 1 + R0 at date 0 increase with , since

1 + rD
0 =

pD + D + (1
pD
0

and
1 + R0D =

) pd + d

D + pD
pD
0

In both cases, the numerator is a constant while the denominator is decreasing in . By contrast, the e¤ect
D
of increasing on the expected return on loans 1 + R0 at date 0 is ambiguous. Recall that the expected
D
D
D
D
return on loans R0 is equal to the weighted average (1
). The e¤ect
0 ) R0 + 0 r 0 with 0 = p0 = (e
of increasing on 0 is ambiguous, since both the numerator and denominator are decreasing in . For 0
dpD
to increase with requires that d 0 be close to 1. From (23), this requires that N 0 1 + pDD be close to 0.
Hence, if N 0 1 + RD is su¢ ciently close to 0, increasing will raise the expected return on loans because
it increases both R0D and r0D as well as the share of loans that earn R0D . Intuitively, if there are only a
few marginal entrepreneurs who would be a¤ected by the crowding out of resources due to contractionary
monetary policy, tighter monetary policy will primarily drive out speculation and shift the composition of
borrowers towards those who wish to produce.
D

If increasing raises the expected return on loans R0 , it will also decrease the fundamental value of the
asset as we have de…ned it. To see this, note the fundamental value of the asset f0D at date 0 can still be
de…ned recursively as
d + f d + (1
) D + fD
(24)
f0D =
D
1 + R0
D

An intervention that raises R0 will thus reduce the fundamental value f0D at date 0. Hence, both the price
D
of the asset pD
on the
0 and the fundamental value f0 at date 0 decline with : To determine the e¤ect of
D
D
D
bubble term b0 = p0 f0 involves some tedious algebra we delegate to Appendix A. There, we show that
increasing dampens the bubble, as summarized in the next proposition.
D

Proposition 6 For N 0 1 + pDD su¢ ciently small, the expected return on loans R0 at date 0 is increasing
in and the bubble bD
at = 0.
0 at date 0 is decreasing in
To recap, under some additional restrictions on the distribution of productivity across entrepreneurs,
contractionary monetary policy will dampen the bubble in our economy just as in the dynamically ine¢ cient
economy. The key di¤erence is that now there may be a reason for a policymaker to intervene. This is
because the marginal entrepreneur in our economy can earn a return equal to the interest rate on loans,
RD , which exceeds the expected return on the asset, rD . Society would thus be better served by diverting
21

some of the resources used to buy the asset into funding entrepreneurial activity. Thus, there may be scope
for government intervention. But it turns out that even though tighter monetary policy can dampen the
bubble, it does not help divert resources towards entrepreneurial activity. To the contrary, it only further
crowds out such activity. Recall that we argued above that an increase in will raise the interest rate on
loans R0D at date 0. This means the marginal entrepreneur who produces must be even more productive,
meaning there must be fewer entrepreneurs producing than without the intervention.
The crowding out of entrepreneurial activity means that a monetary contraction cannot be welfare improving, even though it dampens the bubble. To see why, let us consider how each cohort fares under the
intervention. The cohort that is old at date 0 gets to consume the proceeds pD
0 from selling their asset,
which is decreasing in . The cohort born at date 0 consume the return to their investments from date 0,
namely the proceeds from entrepreneurial activity initiated at date 0 and the returns on the assets they
bought. In expectation, these are given by
Z 1
[ N 0 (y)] ydy + pd + d + (1
) pD + D
(25)
1+R0D

Since R0D is increasing in , this expression too is decreasing in . Finally, since the equilibrium is unchanged
from date t = 1 on, the consumption of all other cohorts is unchanged. The fact that both of the …rst two
cohorts consume less as increases means it will be impossible to make all cohorts better o¤: The only way
to make the old at date 0 better o¤ is if the young at date 0 give them some of their resources, but this would
leave the young with even less to consume at date 1, unless some other cohort got to consume less. Intuitively,
the presence of asymmetric information in this economy allows less productive entrepreneurs to speculate,
which raises the interest rate on loans and crowds out entrepreneurship. Increasing rates further thus fails to
correct the distortion in this economy, and only exacerbates the misallocation of resources. Dampening the
bubble by contracting economic activity does not undo the distortion associated with bubbles. In Appendix
C, we con…rm that leaning against the wind cannot make society better in the case where the quantity of
assets is endogenous as well, even though destroying the endowment reduces the quantity of bubble assets.
Increasing rates may dampen bubbles, and bubbles in our setting are distortionary, but such interventions
do more harm than good.

2.5

Costly Default and Welfare-Increasing Interventions

An important caveat to our result from the previous subsection on the counterproductive nature of raising
rates is that our setup abstracts from one of the main reasons policymakers cite for intervening against
bubbles, namely that it can help temper the negative consequences if and when a bubble collapses. For
example, one concern we already discussed is that if loans to those who speculate on the asset are intermediated, a collapse in asset prices may force …nancial intermediaries to default, which would limit their
ability to provide credit to future entrepreneurs. Yet in our model, the bubble only causes harm while it is
present, when it distorts the allocation of resources, not when it collapses.

22

In principle, we should explicitly add intermediation to our model. We follow a simpler approach by
assuming that when borrowers default, the lenders who …nanced them incur default costs that are proportional to the amount they lend. That is, in addition to the vanishing utility cost that borrowers su¤er
if they default, we now assume savers incur a cost of
> 0 per unit lent to those agents who default.
While we model this as a cost of verifying and recovering payment from borrowers who are in default, this
assumption implies that once the bubble burst, available resources will fall in proportion to the amount of
loans taken out against assets. As we point out below, this is equivalent to allowing a recession when the
bubble bursts. In line with the evidence on the costs of …nancial crises, we will think of as large.
Before turning to the e¤ects of any policy interventions, we …rst need to discuss how adding a cost of
default would a¤ect the equilibrium in our economy absent any intervention. The equilibrium conditions
(13) and (15) remain unchanged: The resources of the young in each period would still either …nance
entrepreneurs or be used to buy the asset, and the interest rate on loans must still equal the maximal return
an agent could earn from buying the asset, which is una¤ected by . Equilibrium prices are therefore the
same as before. However, savers might no longer strictly prefer lending to buying the asset themselves. The
expected return from buying an asset, 1 + rD , is unchanged, but the expected return on a loan will now
re‡ect the cost incurred when their borrower defaults. Once again, let denote the share of loans issued
that are invested in the asset. Then the expected return on loans is given by
1+R

D

= (1

) 1 + RD +

1 + rD

If only entrepreneurs bought the asset, as was the case when = 0, then would equal pD =e. But holding
D
…xed, if we increase default costs
by enough, 1 + R will fall by enough to make savers indi¤erent
between buying the asset and making loans. Since equilibrium prices are determined independently of , the
only variable that can adjust to sustain an equilibrium is . Low productivity entrepreneurs are indi¤erent
about buying the asset in equilibrium, so they would always be willing to buy fewer assets and let savers
buy them instead. Speci…cally, the share of loans used to buy the asset must be such that savers are
D
indi¤erent between buying the asset directly and making loans, meaning R = rD . The value of that
equates these two returns is given by
RD r D
=
+ RD r D
D

Recall from our previous discussion that the asset can be viewed as a bubble only if the rate 1 + R that
we use to discount dividends in de…ning the fundamental value exceeds rD . Hence, for su¢ ciently large ,
the asset would no longer be a bubble. We therefore restrict attention to the case where, given a value for
, the ratio pD =e < . That is, we assume the share of the asset in the total endowment of savers is small,
D
ensuring that expected default costs are small enough that R exceeds rD and the asset is a bubble.
Under the additional restriction that pD =e < a , destroying units of the endowment at date 0 will
now, under certain conditions, yield a Pareto improvement. Once again, consider the amount the di¤erent
cohorts can consume. The old at date 0 consume pD
0 , which is decreasing in . Those who are born at date
0 and consume at date 1 again consume the returns on their investment, although now we have to net out
23

default costs. Their expected consumption is thus equal to
Z 1
[ N 0 (y)] ydy + pd + d + (1

) pD + D

1+R0D

pD
0

(26)

All remaining cohorts born after date 0 are una¤ected. Note that we can reinterpret (26) to mean that
when dividends fall, the asset price is not pd but pd
pD
0 , as would be the case if young savers at date 1
D
spent p0 less on the asset they buy from the old because their endowment was smaller. In this sense, our
model is equivalent to one in which default causes a recession that impoverishes the young.
The two middle terms of the expression in (26) are independent of , so we are left with two terms:
Z 1
[ N 0 (y)] ydy
pD
0
1+R0D

Di¤erentiating and substituting in for 1 + R0D reveals the e¤ects of increasing :
pD + D 0
N
p20

pD + D
p0

dpD
0
d

This expression will be positive if and only if
pD + D 0
N
p20

pD + D
p0

<

Thus, if both and are su¢ ciently large, and the mass of entrepreneurs who are marginal is su¢ ciently
small, increasing would allow the cohort of agents born at date 0 to consume more. In contrast to the
case where = 0, we now have one cohort that is better o¤ and another that is worse o¤. This allows for
the possibility that all agents can be made better o¤ if we allowed for lump-sum transfers.
To compensate the old at date 0 requires a transfer of pD
pD resources from the young to the old at
0
D
date 0. If this is small, those resources would have earned an expected return of approximately 1 + R0 .
Thus, to ensure both the old and young cohorts at date 0 are no worse o¤ requires that the consumption of
D
pD
the latter cohort increase even after netting out the 1 + R0
pD required to keep the old at date
0
0 no worse o¤. This will be true if and only if
D

1 + R0

pD + D 0
N
p20

pD + D
p0

<

The threshold that ensures all agents can be made better o¤ is higher than the one that ensures the young
are better o¤. Still, there exists a value of large enough that increasing will be Pareto-improving.
To better appreciate these results, it is helpful to contrast them with those in previous work on the
welfare implications of intervening against bubbles by Grossman and Yanagawa (1993). They introduce
production externalities into an overlapping generations economy which imply that eliminating bubbles can
make future generations better o¤, just as in our analysis. But in their setup, the only way to make all those
generations better o¤ is by taking resources from the old at date 0 and deploying them to more productive
24

uses. This means that the only way to make the young better o¤ is to take away resources from the old
and leave them worse o¤. The reason the intervention we consider can make all agents better o¤ is that in
our economy we can make the young better o¤ without taking resources away from the old and deploying
them to another use. Instead, the problem is that transferring resources to the old through debt …nancing
is costly in a way that transferring them directly is not. The welfare gains we …nd come not from deploying
resources more e¢ ciently, but from discouraging borrowing and reducing the amount agents would default
on if their speculation failed.5
To recap, the argument for leaning against the wind in our model is based on the bene…ts of mitigating
the harm due to the collapse of a bubble rather than the bene…ts to undoing any distortions that bubbles
introduce when they are present. This is similar in spirit to Svensson (2014). He also argued that leaning
against the wind can be counterproductive, although he focused not on whether contractionary monetary
policy exacerbates distortions but on whether it increases agents’ debt burden. At the same time, he
recognized there may be potential bene…ts from mitigating the fallout from the collapse of a bubble. He
focused on whether tighter monetary policy would increase the likelihood of a …nancial crisis and argued the
e¤ect was small. By contrast, in our model the probability that the bubble will burst in any given period
is , which is exogenous and una¤ected by monetary policy. But even if raising rates has no e¤ect on the
probability of a crisis, it will have an e¤ect on the severity of the …nancial crisis, and that e¤ect might in
principle be quite large.6

2.6

Threats of Future Intervention

Our analysis so far suggests that leaning against the wind can be useful, but only if the costs associated
with defaults triggered by the collapse of a bubble are su¢ ciently large. In this last subsection, we argue
that even when the cost of default is small, there may still be scope for using monetary policy to make
society better o¤. In particular, suppose that rather than raising rates in the face of a bubble, the monetary
authority promises to raise rates in the future if a bubble persists. The reason this intervention can be
useful is that it can serve to dampen the bubble without immediately in‡icting the pain associated with
contractionary monetary policy. In particular, the agents in our model who borrow to buy risky assets are
motivated by the maximal pro…ts they can earn. These accrue if the bubble persists. By threatening to
intervene and drive down asset prices in that state of the world, the monetary authority can make buying
the asset today less pro…table, which would lower the price of the asset without any actual tightening. That
5 Another

way around Grossman and Yanagawa’s impossibility result is if the old at date 0 are not endowed with assets but
must produce them. As we discuss in Appendix C, in that case we can make the young better o¤ by redirecting resources to
produce fewer assets and deploying them to alternative uses even as we maintain the same consumption for the old.
6 Biswas, Hanson, and Phan (2017) o¤er a di¤erent reason for why the collapse of a bubble can be socially costly. In their
model, a bubble improves the allocation of resources and leads to higher wages. At the same time, they assume wages are
downwardly rigid. When the bubble collapses, wages are stuck at too high of a level and economic activity contracts. This
suggests an additional bene…t to dampening the bubble above and beyond costs associated with default.

25

can free up resources for entrepreneurial activity as well as reduce the expected default costs if the bubble
should collapse. If the bubble is very likely to burst, the monetary authority can achieve these bene…ts and
will likely not have to intervene.
Consider the economy from the previous subsection in which default is costly. Let us examine an intervention that involves a commitment to destroy units of the endowment at date 1 if d1 = D and the bubble
persists. Once again, the e¤ect of policy can be seen by analyzing the equilibrium for an economy with a
particular path for the endowment. The endowment path is now given by
(
e
if t = 1 and d1 = D
et =
e
otherwise
As before, the equilibrium from date t = 2 on will be una¤ected by policy. Moreover, the equilibrium at
t = 1 corresponds to the equilibrium we solved for at date 0 in the previous subsection. The only case that
still needs to be analyzed is date 0, the period before the endowment is destroyed.
Market clearing at date 0 is the same as in (13), i.e.,
N 1 + R0D + pD
0 =e
The interest rate on loans 1 + R0D at date 0 must ensure that buying the risky asset is unpro…table even in
the most favorable state. A similar logic as before implies this will be if the dividend remains high at date
1. This implies
D + pD
1
1 + R0D =
pD
0
As before, we can substitute in R0D =

pD
to rewrite the latter equation as
0
1+

Di¤erentiating with respect to

pD
0

D
pD
0 = D + p1

yields
1+

0

pD
0

pD
0 +1+

pD
0

dpD
dpD
0
= 1
d
d

dpD

We already showed that d 1 < 0, since above we argued that in the period where units are destroyed, the
price of the asset falls. This result is una¤ected by the presence of default costs, since those only a¤ect pd1
dpD
0
and not pD
< 0 as well, i.e., a threat of tighter monetary policy at date 1 if the bubble
1 . It follows that d
persists will reduce the price of the asset at date 0. Intuitively, if speculators know the monetary authority
will intervene and drive down the price of the asset if the bubble persists, they will …nd buying the asset
less attractive. This would cause the price of the asset to fall even before any intervention.
D
From the market-clearing condition (13), a lower pD
0 requires a higher value for N 1 + R0 . Since
dRD

N 0 (y) < 0 for all y 0, this implies that d 0 < 0. Hence, a threat to increase if the bubble persists will
divert resources at date 0 from those who wish to buy assets to those who intend to produce.
26

We now argue that this intervention can increase welfare. Again, consider the e¤ect on each cohort.
Agents who are old at date 0 still consume the proceeds they earn from selling the asset, pD
0 , which is lower.
Agents born at date 0 expect to consume
Z 1
[ N 0 (y)] ydy + pd + d + (1
) pD
pD
1 +D
0
1+R0D

Agents born at date 1 will consume the same amount if d1 = d. If d1 = D, they will expect to consume
Z 1
[ N 0 (y)] ydy + pd + d + (1
) pD + D
pD
1
1+R1D

All cohorts born at date 2 on will be una¤ected.
We want to argue that when is large, the above intervention can be Pareto improving. For close
dRD
to 1, the cohort born at date 0 will be better o¤: Since d 0 < 0, they will enjoy the results of greater
entrepreneurial activity at date 0 and they will su¤er smaller losses in case of default. The latter bene…t
applies even if we took the resources that went to fund entrepreneurial activity and gave them back to the
cohort that was old at date 0 directly to make sure they are no worse o¤. Since transferring resources directly
to the old does not involve debt …nancing, it still achieves a bene…t of reducing default costs. However, we
now need to compensate the cohort born at date 1, since although they will be una¤ected if the bubble
bursts at date 1, will be made worse if the bubble persists and the monetary authority intervenes. But as
! 1, this becomes an unlikely scenario. As long as the cohort born at date 0 shares any of their bene…ts
from smaller default costs with the cohort born at date 1 if the bubble collapses, the cohort born at date
1 will also be better o¤ ex-ante. Thus, as long as the bubble is very likely to collapse, a threat to lean
against the wind should a bubble persist can be used to make all agents better o¤ ex ante even though the
intervention itself makes agents worse o¤ ex post. Of course, this requires that the central bank can credibly
commit to intervene if the bubble persists. Whether such promises are credible is an interesting question,
but beyond the scope of this paper.

3

Conclusion

In this paper, we developed a framework to explore the merits of the lean-against-the-wind approach towards
bubbles. We argued that tighter monetary policy helps dampen bubbles, even though when Galí (2014)
…rst introduced nominal rigidities into models that allowed bubbles, he found that raising rates can amplify
bubbles in some circumstances. The problem with his setting for the lean-against-the-wind view is that
the bubble in his setup alleviates the friction that allows a bubble to arise in the …rst place, so there is
no reason to act against the bubble. We therefore considered a di¤erent set of frictions that give rise to
bubbles that serve no useful social role and looked at the e¤ects of monetary policy in that environment.
Our model speaks to many of the issues that have come up in debate over how policymakers ought to
respond to bubbles. For example, we …nd that raising rates is a blunt tool that, even if it dampens the
bubble, exacerbates the distortions a bubble causes. However, if the collapse of a bubble leads those who
27

borrowed against the bubble to default, which in turn incurs costs that are increasing in the amount of
borrowing against the bubble, raising rates to dampen bubbles can increase welfare. Finally, even when
raising rates is bad for welfare, a commitment in advance to raise rates if a bubble persists may be welfare
improving since it can dampen bubbles when they …rst arise without immediately in‡icting the pain that
goes with monetary tightening.
One of the themes that emerges from our analysis is the importance of the friction that allows bubbles
to emerge. The information frictions we consider imply bubbles serve no useful role, which allows scope
for intervention. Other models, such as Allen, Morris, and Postlewaite (1993), Conlon (2004), and DoblasMadrid (2012), have also generated bubbles in models with information frictions but which do not rely on
credit. In those models, bubbles do not serve a useful role either. By contrast, in models where bubbles
arise because of dynamic ine¢ ciency and serve to ameliorate that ine¢ ciency, eliminating a bubble would
be ine¢ cient, at least in the simplest versions of these models. The same is true in models where bubbles
arise because of binding borrowing constraints. Various papers have now demonstrated that when …rms
cannot borrow all the resources they could use productively, bubbles can arise as a way to transfer resources
to …rms who cannot borrow these resources. Examples include Kocherlakota (2009), Farhi and Tirole
(2012), Martin and Ventura (2012), Hirano and Yanagawa (2017), Miao and Wang (2015), and Martin and
Ventura (2016).7 Recent work by Dong, Miao, and Wang (2017) and Ikeda (2017) have looked at the e¤ect
of monetary policy in environments where bubbles arise because of borrowing constraints. And indeed,
neither paper concludes that monetary policy should aggressively …ght bubbles but manage them.
Finally, given our model is both simple and seems to connect to various issues the policymakers cite in
regard to bubbles, we view our model as a potential workhorse for exploring various policy questions. For
example, as we noted in the Introduction, the policy debate in the wake of the recent …nancial crisis has
focused on whether interest rate policy or macroprudential policy o¤er a better response to a potential
bubble. If we extended our model to allow for entrepreneurs who di¤er both in productivity and in initial
endowments, we could explore the e¤ect of policies like leverage restrictions. Here, we should emphasize
that while our analysis reveals when leaning against the wind can improve welfare, we are not arguing that
such a policy is optimal. Understanding the merits of di¤erent interventions remains an area for future
work. Another way in which we could extend our model is to more explicitly model the role of …nancial
intermediaries to have a better understanding of how to model (and calibrate) the cost of default. Still
another extension would be to incorporate our analysis into a small monetary open economy model as in
Galí and Tomacelli (2005) to explore how policy implications di¤er with the possibility of international
capital ‡ows that might ‡ow into a country that successfully increases its real interest rate. And given the
potential bene…ts we document from a commitment to raise rates if a bubble persists, understanding the
problem of time inconsistency and how to deal with it stands as another direction for future work.

7 Earlier

work by Kocherlakota (1992), Santos and Woodford (1997) and Kocherlakota (2008) argued that the existence of
borrowing constraints can give rise to bubbles in endowment economies where consumers are borrowing constrained. Rocheteau
and Wright (2013) argue that, at least in some cases, constraints on consumers and …rms are isomorphic.

28

price

et = (1+g)te0

0

1

2

3

4

5

6

7

8

9

time

Figure 1: Sample equilibrium price paths for the case where d = 0

10

pt+1

0

p*

Figure 2: Equilibrium relationship between pt and pt+1

pt

References
[1] Adam, Klaus, 2003. “Learning and Equilibrium Selection in a Monetary Overlapping Generations
Model with Sticky Prices” Review of Economic Studies, 70(4), October, p887-907.
[2] Allen, Franklin and Douglas Gale, 2000. “Bubbles and Crises” Economic Journal, 110(460), January,
p236-255.
[3] Allen, Franklin and Gary Gorton, 1993. “Churning Bubbles” Review of Economic Studies, 60(4), October, p813-836.
[4] Allen, Franklin, Stephen Morris, and Andrew Postlewaite, 1993. “Finite Bubbles with Short Sale
Constraints and Asymmetric Information” Journal of Economic Theory, 61(2), p206–29.
[5] Atkinson, Tyler, David Luttrell, and Harvey Rosenblum, 2013. “How Bad Was It? The Costs and
Consequences of the 2007–09 Financial Crisis” Federal Reserve Bank of Dallas Sta¤ Papers 20, July.
[6] Barlevy, Gadi, 2014. “A Leverage-based Model of Speculative Bubbles” Journal of Economic Theory,
153, p459-505.
[7] Bernanke, Ben and Mark Gertler, 1999. “Monetary Policy and Asset Price Volatility” Federal Reserve
Bank of Kansas City Economic Review, 4th Quarter, p17-51.
[8] Biswas, Siddhartha, Andrew Hanson, and Toan Phan, 2017. “Bubbly Recessions”Richmond Fed working paper.
[9] Blanchard, Olivier and Stanley Fischer, 1989. Lectures on Macroeconomics. Cambridge, Massachusetts:
MIT Press.
[10] Blanchard, Olivier and Mark Watson, 1982. “Bubbles, Rational Expectations, and Financial markets”
Crisis in the Economic and Financial Structure: Bubbles, Bursts, and Shocks, ed. P. Wachtel. Lexington, Massachusetts: Lexington Books, p295–315.
[11] Borio, Claudio and Philip Lowe, 2002. “Asset prices, …nancial and monetary stability: exploring the
nexus” BIS Working Paper.
[12] Conlon, John, 2004. “Simple Finite Horizon Bubbles Robust to Higher Order Knowledge” Econometrica, 72(3), p927–36.
[13] Diamond, Peter, 1965. “National Debt in a Neoclassical Growth Model” American Economic Review,
55(5), Part 1, December, p1126-50.
[14] Doblas-Madrid, Antonio, 2012. “A Robust Model of Bubbles with Multidimensional Uncertainty”
Econometrica, 80(5), p1845–93.
[15] Dong, Feng, Jianjun Miao, and Pengfei Weng, 2017. “Asset Bubbles and Monetary Policy” Boston
University working paper.
29

[16] Farhi, Emmanuel and Jean Tirole, 2012. “Bubbly Liquidity”Review of Economic Studies, 79, p678-706.
[17] Galí, Jordi, 2014. “Monetary Policy and Rational Asset Price Bubbles” American Economic Review,
104(3), March, p721-52.
[18] Galí, Jordi, and Tommaso Monacelli, 2005. “Monetary Policy and Exchange Rate Volatility in a Small
Open Economy” Review of Economic Studies, 72(3), p707-734.
[19] Grossman, Gene and Noriyuki Yanagawa, 1993. “Asset Bubbles and Endogenous Growth” Journal of
Monetary Economics, 31(1) February, p3-19.
[20] Hirano, Tomohiro and Noriyuki Yanagawa, 2017. “Asset Bubbles, Endogenous Growth, and Financial
Frictions” Review of Economic Studies, 84(1), p406-43.
[21] Hoggarth, Glenn, Ricardo Reis, and Victoria Saporta, 2002. “Costs of Banking System Instability:
Some Empirical Evidence” Journal of Banking and Finance, 26, p825–855.
[22] Ikeda, Daisuke, 2017. “Monetary Policy, In‡ation and Rational Asset Price Bubbles”Bank of England
Working Paper.
[23] Kocherlakota, Narayana, 1992. “Bubbles and Constraints on Debt Accumulation”Journal of Economic
Theory, 57 (1), p245— 56.
[24] Kocherlakota, Narayana, 2008. “Injecting Rational Bubbles” Journal of Economic Theory, 142(1),
September, p218-32.
[25] Kocherlakota, Narayana, 2009. “Bursting Bubbles: Consequences and Cures”FRB Minneapolis working paper.
[26] Martin, Alberto and Jaume Ventura, 2012. “Economic Growth with Bubbles” American Economic
Review, 102(6), p3033-58.
[27] Martin, Alberto and Jaume Ventura, 2016. “Managing Credit Bubbles” Journal of the European Economic Association, 14(3), June, p753-89.
[28] Miao, Jianjun and Pengfei Wang, 2015 “Bubbles and Credit Constraints” Boston University working
paper.
[29] Mishkin, Frederic, 2011. “Monetary Policy Strategy: Lessons from the Crisis’, in M Jarociński, F Smets
and C Thimann (eds), Approaches to Monetary Policy Revisited – Lessons from the Crisis, European
Central Bank, Frankfurt, p67–118.
[30] Reinhart, Carmen and Kenneth Rogo¤, 2009. “The Aftermath of Financial Crises”American Economic
Review, 99(2), May, p466-472.
[31] Samuelson, Paul, 1958. “An Exact Consumption-Loan Model of Interest with or without the Social
Contrivance of Money” Journal of Political Economy, 66, p467-82.

30

[32] Sargent, Thomas and Neil Wallace, 1975. “Rational Expectations, the Optimal Monetary Instrument,
and the Optimal Money Supply Rule” Journal of Political Economy, 83, p241–254.
[33] Santos, Manuel and Michael Woodford, 1997. “Rational Asset Pricing Bubbles” Econometrica, 65(1),
January, p19-57.
[34] Svensson, Lars, 2014 “Why Leaning Against the Wind is the Wrong Monetary Policy for Sweden”
working paper.
[35] Tirole, Jean, 1985. “Asset Bubbles and Overlapping Generations” Econometrica, 53(6), November,
p1499-1528.
[36] Rhee, Changyong, 1991. “Dynamic Ine¢ ciency in an Economy with Land”Review of Economic Studies,
58(4), June, p791-7.
[37] Rocheteau, Guillaume and Randall Wright, 2013. “Liquidity and Asset Market Dynamics” Journal of
Monetary Economics, 60, p275-94.
[38] Weil, Philippe, 1987. “Con…dence and the Real Value of Money in an Overlapping Generations Economy” Quarterly Journal of Economics, 102(1), February, p1-22.
[39] Zeira, Joseph, 1999. “Informational Overshooting, Booms, and Crashes” Journal of Monetary Economics, 43(1), p237-257

Appendix A: Proofs of Propositions
Proof of Proposition 1: In the text, we argued that rt = 0 is an absorbing state. That is, there exists
a t where 0 t
1 such that rt > 0 for t < t and rt = 0 for t t . For t < t , we know that rt > 0
implies storage is dominated, and so pt = et for t < t . Since rt = 0 for t t we can use (3) to conclude
that pt+1 = pt for t t , and by induction we can infer pt = pt for all t t .
The last step is to show that at date t , any 2 [et 1 ; et ] can be an equilibrium, and only these values
can be an equilibrium. Since rt 0, pt
pt 1 = et 1 . Since pt et at all dates t, this is also true for
date t . For any 2 (et 1 ; et ], the rate of return on the asset to those purchasing the asset at date t
1
will be positive, so they would buy the asset at this price at date t
1 and sell all their holdings at date
t . Since rt = 0, the young at date t are indi¤erent between storage and buying the asset, so they would
be willing to buy any amount the old sell. Hence, we can construct an equilibrium where pt = for any
2 (et 1 ; et ].
1

1

Claim: Given any two equilibrium paths fpt gt=0 and fp0t gt=0 , de…ne rt = pt+1 =pt and rt0 = p0t+1 =p0t . If
rt rt0 for all t and r0 > r00 , then p0 p00 and pt > p0t for all t 1.
31

Proof : Since r00
0, then r0 > 0. From Proposition 1, this implies p0 = e0 . At the same time, since in
equilibrium the price of the asset cannot exceed the endowment of young agents, we know p00 e0 . Hence,
p0 p00 . Next, since pt+1 = (1 + rt ) pt and p0t+1 = (1 + rt0 ) p0t , we can use the fact that r0 > r00 to show by
induction that pt > p0t for all t 1. In particular, for t = 1, we have
p1 = (1 + r0 ) p0 > (1 + r00 ) p00 = p01
Next, if pt > p0t , then since rt

rt0 , it follows that
pt+1 = (1 + rt ) pt > (1 + rt0 ) p0t = p0t+1

The claim then follows.
Proof of Proposition 2: We prove a more general result which implies the Proposition. In particular,
P1
we show that for a sequence of nonnegative dividends dt 0, there is a unique equilibrium i¤ t=0 dt = 1.
Since the case where dt = d > 0 satis…es this condition, the claim will follow.
P1
Suppose dt
0 for all t and
t=0 dt = 1. We argue that the interest rate rt must be positive at
all dates. For suppose rt = 0 for some date t. Then pt+1 = pt dt and since pt
et , it follows that
pt+1 < et < et+1 , which implies that rt+1 = 0. By this logic, the price declines by dt+i at each date t + i;
P1
Pk
for i = 1; 2; :::. Since t=0 dt = 1 implies limk!1 i=1 dt+i = 1, there must be a date t + k such that
Pk
pt+k = pt
i=1 dt+i < 0, i.e., there must be some date t + k at which the price of the asset turns negative.
But this is incompatible with equilibrium. Hence, rt > 0 at all dates. This implies storage is dominated,
and so the unique equilibrium price is pt = et for all t. The corresponding return on holding the asset is
then
dt
dt + pt+1
= (1 + g) +
1 + rt =
pt
et
P1
We can further show that the condition t=0 dt = 1 is a necessary condition for uniqueness. The proof
P1
P1
is constructive. Suppose t=0 dt < 1. Then there must be some date t such that et
t=t dt > 0. We
now propose an equilibrium
price sequenceo as follows. For t < t , set pt = et ; for t = t , set pt to be any
n
P
value such that max et 1 dt ; s t d < pt et ; and for t > t , let pt = pt
dt +1 : : : dt . The
P
price sequence is positive for every date t because pt = et > 0 for t < t and pt pt
s t ds > 0 for
t > t . The price sequence pt corresponds to an equilibrium because, for t < t , the interest rate is positive
and the entire endowment is invested in the asset and, for t t , the interest rate is zero. Thus, there are
multiple equilibria in this case.
Proof of Proposition 6: To compute the e¤ect of
D
f0 at date 0. Recursively, we know
f0D =

on bD
0 , we …rst rearrange the fundamental value

d + f d + (1

) D + fD
D

1 + R0

Next, recall that
D

1 + R0 = (1

) 1 + R0D +
32

1 + rD
0

where
= pD
0 = (e

)

Substituting in, we get
f0D =

d + f d + (1
) D + fD
(1
) 1 + R0D + 1 + rD
0

Next, we can write the interest rate on loans and the expected return on the asset as
1 + R0D

=

1 + rD
0

=

D + pD =pD
0
d + pd + (1

) D + pD

Substituting in these two expressions, we get

f0D =
and so

"

bD
0 = 1

(1

(1

d + f d + (1
) D + fD
pD
) (D + pD ) + ( (d + pd ) + (1
) (D + pD )) 0
#
d + f d + (1
) D + fD
pD
) (D + pD ) + ( (d + pd ) + (1
) (D + pD )) 0
dpD

dbD

We already established in the text that d 0 < 0. To show that d 0 < 0, it will su¢ ce to show that
the expression in the brackets is decreasing in . The only place where appears in this expression is in
pD
0
e

N

D+pD
pD
0

=
=1
. In the limit as N 0 1 + pDD tends to zero, must decrease with . It follows
e
that the expression in the brackets is decreasing in , and hence so does bD
0 .

33

Appendix B: Monetary Policy in a Production Economy
In this Appendix we consider a production economy that is analogous to the economy we examine in the
text, and then use it to study monetary policy and nominal price rigidity as in Galí (2014). As a brief
preview, young agents engage in production which nets them all of the income they earn over their lifetime.
Since they only value consumption when old, they will need to convert their income into consumption while
old. The can store the goods they earn as income, exchange these goods for the one asset in the economy,
or exchange for money issued by the central bank.
We study equilibria in which there is no trade in money. The central bank announces a nominal interest
rate, and the real return on the asset together with the in‡ation in goods prices adjust to make sure agents
do not wish to hold money. In the bulk of what follows, we will assume the yields a positive dividend d > 0,
so that agents spend all of their income on buying assets in equilibrium. The case where d = 0 introduces
additional equilibria in which agents are indi¤erent between buying the asset and storing goods. Although
in the text we begin with the case where d = 0, we defer the discussion of this case towards the end.
We begin by describing the economic environment in detail, and then de…ne and discuss equilibria for
this setup.

B.1 Endowments and Preferences
Consider an economy in which a new cohort of mass 2 is born each period and lives for two periods. Half
of each cohort is endowed with a unit of labor in their …rst period of life. We will refer to these agents
as workers. The other half is endowed in their …rst period of life with the knowledge of how to use labor
to produce. We will refer to these agents as producers. Note that this structure is more similar to the
Adam (2003) monetary OLG model, in which agents are heterogeneous and only earn income in their …rst
period of life, than Galí’s model in which agents and homogeneous and work as workers when young and
producers when old. We prefer the Adam formulation to ensure agents only earn income when young as in
our endowment economy.
Agents have the same preferences as in (1), i.e., the only value consumption when old. However, those
agents who are workers also incur a disutility from labor. In particular, a worker born at date t who supplies
w
nt units of e¤ort and who consumes cw
t and ct+1 when young and old, respectively, has utility given by
w
w
u cw
t ; ct+1 ; nt = ct+1

vt (nt )

where
vt (nt )

=

t

(1 + g) v (nt )
At v (nt )
34

(27)

for some convex di¤erentiable function v ( ). We assume limn!0 v 0 (n) = 0 and limn!1 v 0 (n) = 1 to
ensure an interior solution. Later cohorts value leisure more. Below we assume these cohorts are also more
productive, and the two will cancel out.
For those agents who are producers, utility is only de…ned only over consumption. A producer born at
date t who consumes cpt and cpt+1 when young and old, respectively, has utility given by
u cpt ; cpt+1 = cpt+1

(28)

B.2 Production and Pricing
Workers and producers combine e¤orts to produce a variety of intermediate goods which are then combined
to form …nal goods. We index the unit mass of producers by i 2 [0; 1] : Each producer knows how to produce
a di¤erent intermediate good, indexed by the same i as the producer. If producer i hires nit units of labor
at date t he can produce yit units of good i, where
t

yit = (1 + g) nit

At nit

Thus, as we anticipated above, later cohorts are more productive. The di¤erent intermediate goods can
be combined to produce …nal goods according to a Dixit-Stiglitz production function, i.e., yit of each good
i 2 [0; 1] can be combine to yield Yt of …nal goods according to
Yt =

Z

1

1

1

1
yit

0

di

(29)

Any agent can produce …nal goods this way, so the production of these is goods is perfectly competitive.
Any agent who produces …nal goods producers will choose the amount of intermediate goods yit to maximize
their pro…ts. If we let Pt denote the price of …nal goods, any such producer will solve
1
Z 1
Z 1
1
1
max Pt
Pit yit di
yit di
yit

0

0

The …rst-order condition with respect to yit implies
1

Pt
Pit

yit = Yt

(30)

To obtain the cost of producing one unit of the …nal good, we set Yt = 1 and compute the price of the
1

optimal bundle of intermediate goods given intermediate goods prices, yit =
Z

1

Pit yit di =

0

Z

0

1

1

Pit

1

Pt
Pit

:

1

Pt di

We assume the market for …nal goods is competitive. In this case, the price of …nal goods Pt must equal the
per unit cost of producing a good. The price of …nal goods Pt will thus equal the Dixit-Stiglitz aggregator
of intermediate good prices:
Z 1
1
1
Pt =
Pit di
(31)
0

35

Each intermediate goods producer chooses the price Pit to maximize expected pro…ts given demand (30)
and the wage, which we denote by Wt . We will consider cases both where price setters observe the interest
rate 1 + it that the central bank sets before choosing their prices, which we will call the ‡exible price case,
and where price setters set prices before observing the nominal interest rate, which we will call the sticky
price case. Let it denote producer i’s information set at date t when they choose their price. Then each
producer i will set Pit to solve
"
#
1
Wt
Pt
max E
Pit
Yt
it
Pit
At
Pit
The optimal price for each intermediate goods producer is then just
Pit =

(1

E [Wt Yt j t ]
) At E [Yt j

t]

(32)

Given the symmetry in production technology and information sets, all producers will charge the same
price. Hence, the amount of intermediate goods produced in equilibrium will be the same, i.e., yit is the
R1
same for all i 2 [0; 1]. All producers will therefore hire the same amount of labor, so nit = nt = 0 nit di.
Substituting in, we can express aggregate output Yt as a function of aggregate labor:
Yt = At nt
This output is divided between producers and workers, with workers receiving (Wt =Pt ) nt and producers
receiving (At Wt =P ) nt . Since both types only wish to consume when old, they will want to save all of
their income to consume next period. We now turn to the options young agents have to save their income.

B.3 Asset and Money Markets
Agents who wish to convert the income they earn from production when young into consumption when old
have several options. The …rst two are the same as in our endowment economy: Agents can store their
goods at a zero rate of return, or they can trade their goods for an asset which pays a …xed dividend of
d > 0 consumption goods each period and which available in a …xed supply we normalize to 1. In this case,
storage will be dominated, for the same arguments we used for the endowment economy. This will not be
true when d = 0. Thus, we abstract from any additional equilibria that may arise when d = 0 in which
agents are willing to store goods. We will discuss the case where d = 0 brie‡y at the end. Let pt denote the
price of the asset in terms of goods, and denote the (real) return on the asset by
1 + rt

d + pt+1
pt

In contrast to the endowment economy, we now allow agents to save using a third option: They can exchange
their endowment for cash, deposit it with the central bank, and receive cash from the central bank next
period at a nominal interest rate 1 + it announced by the central bank. We also allow people to borrow from
the central bank at this rate. Following Galí, we restrict attention to equilibria in which agents do not hold
36

cash or trade with the central bank. Hence, the real return from lending or borrowing cash to the central
bank must be the same as the return agents earn elsewhere on their savings. Thus, if we let t = Pt+1 =Pt
denote the gross in‡ation rate between dates t and t + 1, then in equilibrium
1 + it
t

=

d + pt+1
pt

(33)

When the central bank sets a di¤erent nominal interest rate it , either in‡ation t , the return to the asset
1 + rt , or both must adjust to ensure agents do not want to either borrow or lend from the central bank.
With storage dominated and agents holding no cash in equilibrium, young agents will exchange all of
their income to buy the asset from the old who wish to sell it. It follows that
pt = At nt

(34)

and the real return to buying the asset in equilibrium will equal
1 + rt =

d + At+1 nt+1
At nt

(35)

B.4 Labor Market
Finally, we consider the labor market. Demand for labor by intermediate goods producers is indirect;
producers set a price Pit for the goods they sell, which determines how much of each goods …nal goods
producers will demand. Producers then simply hire the workers they need to meet this demand. On the
supply side, workers know that if they put in nt units of labor, they will earn (Wt =Pt ) nt goods. Above we
argued they will use these goods to buy assets and earn a return of 1 + rt , where 1 + rt is given by (35).
Substituting this into the utility function for workers implies they will choose nt to solve
max (1 + rt )
nt

Wt
nt
Pt

At v (nt )

The …rst order condition for labor is then given by
At v 0 (nt ) = (1 + rt )

Wt
Pt

(36)

B.5 Equilibrium
To recap, in the production economy in this Appendix, young workers and producers join e¤orts to produce
goods which they exchange with the old for assets. When workers and producers turn old, they consume
the dividend d on the asset they own and the goods the exchange for the assets when they trade with the
next generation of young agents. Productivity growth implies later cohorts can produce more than their
predecessors. Hence, this economy will be dynamically ine¢ cient and admit bubbles. The key di¤erence is
that the income of young agents is now endogenous and can potentially be a¤ected by monetary policy.
37

1

Formally, for any path of nominal interest rates f1 + it gt=0 set by the monetary authority, an equilibrium
1
1
in this economy is a path of prices fPt ; Wt ; pt ; rt gt=0 and a path of employment fnt gt=0 such that agents
behave optimally and all markets clear at each date t. Collecting the relevant conditions from above yields
…ve equations that govern these …ve variables:
(1) Optimal pricing:

E [Wt Yt j t ]
) At E [Yt j t ]
d + pt+1
1 + rt =
pt
Wt
0
vt (nt ) = (1 + rt )
At Pt
pt = At nt
1 + it
t =
1 + rt
Pt =

(2) Asset market clearing:
(3) Labor market clearing:
(4) Goods market clearing:
(5) Money market clearing:

(1

Conditions (1), (3), and (4) are all static and only involve date t prices. Conditions (2) and (5) are dynamic
and relate prices at date t to prices at date t + 1, since condition (5) involves t = Pt+1 =Pt .

B.6 Equilibrium with Flexible Prices
We begin with the ‡exible price case in which producers can set their price for date t after observing
the nominal wage Wt . Since producers can deduce what other producers will do and how much e¤ort
workers provide, they can perfectly anticipate total output Yt . Hence, their information set will be given by
t = fWt ; Yt g. In this case, E [Wt Yt j t ] = Wt Yt and E [Yt j t ] = Yt , and so from the optimal pricing rule,
we have
Wt
Pt =
(1
) At
Substituting this and the asset market clearing condition into the labor market clearing condition yields
vt0 (nt ) =

d + At+1 nt+1
(1
At nt

)
1

Hence, nt is governed by one-dimensional di¤erence equation that is independent of f1 + it gt=0 . This
1
equation features a unique unstable steady state for each t. It follows that the equilibrium path for fnt gt=0
is unique. Thus, in a ‡exible price environment, monetary policy has no e¤ect on employment nt . For
reference, we denote the path of employment in the ‡exible price case by nt . In this case, monetary policy
also has no e¤ect on output Yt , the real interest rate rt , and the real price of assets pt , all of which are
functions of nt and exogenous variables. For reference, we will refer to these as Yt , rt , and pt .
The only objects that remain to be solved are the path of prices Pt and Wt . The money market clearing
condition yields a law of motion for Pt given the path of monetary policy, since 1 + it is chosen by the
central bank and 1 + rt is determined independently of prices. With this law of motion, we can determine
Pt for all t 1 given an initial value for P0 . Since the real wage Wt =Pt is pinned down in equilibrium by
the optimal pricing rule, this means that the equilibrium is uniquely determined up to the initial nominal

38

wage W0 . Given an initial nominal wage W0 , we can determine P0 , then determine Pt for all t
1, and
…nally, since we know Wt =Pt = At (1
), we can determine Wt for t 1. But the initial nominal wage
W0 is indeterminate. This is just an illustration of the price level indeterminacy of pure interest rate rules
established by Sargent and Wallace (1975).

B.7 Equilibria with Rigid Prices
We now consider the sticky price case. We assume producers must set the price of their intermediate good
Pit at the beginning of each period t, before the monetary authority moves or nominal wages Wt are set.
The monetary authority then sets 1 + it . After this, the timing is the same as before. Goods producers
hire workers in the labor market at a nominal wage Wt . Final goods producers buy intermediate goods at
the price Pit speci…ed in the beginning of the period. Workers and producers use the goods they earn as
income they earn to buy assets from the old who sell o¤ their assets.
If monetary policy is deterministic, the change in timing is inconsequential. Producers can perfectly
anticipate the nominal interest rate and deduce what the equilibrium nominal wage Wt will be. In that
case, t = fWt ; Yt g as before.
The same is not true if monetary policy is contingent on some random variable that is realized after
1
producers set their prices. That is, suppose it = i ( t ) where f t gt=0 is some sequence of random variables.
For simplicity, suppose t is only random at t = 0. That is, suppose
(
H w/prob
0 =
L w/prob 1
where

2 (0; 1), and
t

= ; for t = 1; 2; 3; :::

Since we assume t is realized after producers set their prices, the information available to price setters is
0 = fi : 0 ! Rg, i.e., producers know the monetary policy rule from but not the realization of 0 . At
dates t 1, monetary policy is deterministic and so t = fWt ; Yt g as we argued before. Thus, monetary
policy is irrelevant from date t = 1 on, as anything the monetary authority does is anticipated in advance.
For notational ease, let us use a superscript 2 fH; Lg to denote the value of a variable as a function of
the realization of the sunspot variable at date 0. Thus, Wt denotes the nominal wage at date t if back at date
0 the realization of the sunspot was given by 0 = . We now argue that the equilibrium is indeterminate
in up to two values: the nominal wage W0H the in‡ation rate H
0 at date 0 when 0 = H. Intuitively,
monetary policy determines the average in‡ation rate E [ 0 ], but it does not pin down what in‡ation must
be for each realization of 0 . Since the sunspot can assume two values at date 0, the equilibrium features
one additional free parameter.

39

Formally, we know that nt = nt for t
1. We now show that given W0H and H
0 , we can use the
equilibrium conditions above to pin down any remaining equilibrium objects. First, from the money market
H
H
clearing condition, given 1 + iH
0 and
0 we can deduce the real interest rate 1 + r0 when 0 = H. From
the goods market clearing and the asset market clearing conditions, we can deduce nH
0 ; this is because we
know n1 = n1 , and so
d + A1 n1
1 + r0H =
A0 nH
0
Next, we can use the labor market clearing condition to pin down the real wage W0H =P0 when

= H, since

v 0 nH
W0H
0
=
P0
1 + r0H
Note that P0 does not depend on 0 since producers set their prices before 0 is realized. Once we know the
real wage at date 0 if 0 = H, we can use the optimal pricing rule together with the various market clearing
conditions to solve for the real wage W0L =P0 and nL
0 . In particular, these two variables are determined by
the pair of conditions
P0
W0L
P0

=
=

1
(1

) A0

H
nH
0 W0 + (1
nH
0 + (1

L
) nL
0 W0
) nL
0

L
v 0 nL
0 A0 n0
d + A1 n1

L
L
Once we know nL
0 , we also have the return 1 + r0 = (d + A1 n1 ) = A0 n0 . Combining this with the other
L
L
conditions yields the in‡ation rate L
0 = 1 + i0 = 1 + r0 if 0 = L.

Finally, once we have W0H and the real price W0H =P0 , we can deduce the initial price P0 . We can then
1, i.e., the price level at each
use the money market clearing condition to deduce PtH and PtL for all t
H
1 from the fact
date as a function of the realization of 0 . We can then deduce Wt and WtL for all t
H
H
that Wt = (1
) At Pt . Hence, given W0 and 0 ;we can pin down all remaining equilibrium prices and
employment levels.
The fact that we now have two equilibrium objects that are not pinned down suggests a greater degree of
indeterminacy than under ‡exible prices. In fact, the set of equilibria in the sticky price economy includes
the ‡exible price equilibrium in which nt = nt for all dates for both realizes of 0 at date 0 as a special case.
In other words, the presence of price rigidity does not change the set of equilibrium outcomes on the real
side. Rather, it adds additional equilibria. To see this, suppose we set the in‡ation rate if 0 = H to equal
H
0

=

where
1 + r0 =

1 + iH
0
1 + r0
d + A1 n1
A0 n0

L
is the real interest rate that would prevail at date 0 in the ‡exible price world. Since nH
1 = n1 = n1 , it
L
follows that nH
0 = n0 . With a little bit of algebra, we can con…rm that n0 = n0 as well. Thus, even when

40

prices are rigid, there exists an equilibrium in which employment, output, and the real interest rate are all
una¤ected by monetary policy, even though producers set their prices before observing 0 and thus what
the monetary authority will do. But price rigidity introduces other equilibria in which the actions of the
monetary authority at date 0 a¤ect real variables at date 0.8 We now turn to this possibility.

B.8 Equilibria with Monetary Policy Tightening
The equilibria that we are particularly interested in are those where setting a higher nominal interest rate
1 + i0 at date 0 leads to a higher real interest rate 1 + r0 at date 0. We will refer to this case as tighter
L
monetary policy. Without loss of generality, suppose the monetary authority sets iH
0 > i0 . Consider an
H
equilibrium in which we set the free parameter 0 to some value where
H
0

<

1 + iH
0
1 + r0

In this case, the only way to satisfy the money market clearing condition is if r0H > r0 , i.e., if the real
interest rate if 0 = H exceeds the interest rate that would prevail at date 0 in the ‡exible price economy.
H
Since 1 + r0H = (d + A1 n1 ) = A0 nH
0 , it follows that n0 < n0 . That is, the only way the monetary authority
can achieve an equilibrium real interest rate that exceeds r0 is if it reduces employment nH
0 below n0 . In
other words, tighter monetary policy is contractionary. From the labor market clearing condition, we know
H
that for nH
0 to fall below n0 requires that the real wage W0 =P0 fall below the its level in the ‡exible price
equilibrium, ensuring workers put in less e¤ort even though they face a higher real interest rate.
We can rearrange the optimal pricing condition to emphasize that the weighted average wage across states
must be constant:
nH
0
nH
0 + (1

W0H
W0L
(1
) nL
0
+
= (1
) nL
nH
) nL
0 P0
0 + (1
0 P0

) A0

H
Since nH
0 and W0 =P0 are both lower than the ‡exible price levels, we can use this condition together with
the labor market clearing condition to con…rm that W0L =P0 and nL
0 must exceed the analogous ‡exible price
L
equilibrium values. We can then deduce that 1 + r0 < 1 + r0 , and so
L
0

>

1 + iL
0
1 + r0

Thus, with only two states, if the monetary authority manages to raise the real interest rate in some state
of the world at date 0 relative to the ‡exible price level, it must also lower the real interest rate relative to
the ‡exible price level in another state of the world.
8 The fact that nominal price rigidity includes the original ‡exible price equilibrium is due to the way this rigidity is modelled:
all producers can set their price unconstrained after one period. If producers could only adjust their prices in a staggered
manner or were constrained in terms of how much they adjust their price, the ‡exible price outcome would no longer necessarily
constitute an equilibrium with rigid pricing.

41

To the extent that a higher nominal interest rate achieves a higher real interest rate, then, it is by inducing
agents to work fewer hours when nominal interest rates are high, reducing the income they generate and
subsequently spend to buy assets. Since equilibrium output and real prices are una¤ected from date 1 on,
the agents who save at date 0 buy buying assets receive the same amount at date 1 in exchange for the
assets they bought, even though they spent less to buy these assets. Hence, when a higher nominal interest
rate contracts economic activity, it implies agents who buy assets earn a higher real return.
Note that we can capture the equilibrium in which a higher nominal interest rate corresponds to a higher
real interest rate using our endowment economy. The contractionary e¤ect of a higher nominal interest
rate can be captured in an endowment economy by having the endowment of the initial young depend
on the realization of a random variable 0 , with the endowment being lower in the state that corresponds
to a higher nominal interest rate. Endowments from date t = 1 on do not depend on the realization of
0 . This speci…cation will correctly capture what happens to output, consumption, and asset prices in
the equilibrium with tighter monetary policy in the production economy we’ve described in this Appendix.
Thus, we refer to thought experiments in which we destroy part of the endowment of the cohort born at
date 0 as a reduced-form representation of the e¤ects of monetary policy in a production economy with
nominal price rigidity.

B.9 Intrinsically Worthless Assets
Up to now, we have assumed the asset yields a positive dividend d > 0 at all dates. This implies that agents
will spend all of their income on assets in equilibrium. If d = 0 so the asset is intrinsically worthless, the
equilibria we have analyzed in which agents spend all of their income on assets remain. However, additional
equilibria arise in which agents are indi¤erent between buying the assets and storing goods and spend only
some of their income to buy assets. We will not attempt to characterize all such equilibria here. To give a
‡avor of equilibria that can arise when the asset is intrinsically worthless, we consider equilibria in which
the path of prices for each realization of 0 are deterministic. Once again, we will use a superscript to
denote the value of a variable at date t if 0 back at date 0 was equal to .
From date t = 1 on, we know the intermediate goods producers will set their prices so that regardless of
,
Pt =

Wt
(1

) At

Hence, the real wage from date t = 1 on is uniquely pinned down and will be independent of the realization
of 0 back at date 0. Substituting the real wage and the asset market clearing condition into the labor
market clearing condition yields
p
v 0 nt = t+1 (1
)
pt
Suppose pt < At nt for some date t, so at some date the agent is willing to store goods. This requires
42

1 + rt = pt+1 =pt = 1. The implication is that equilibrium employment at date t satis…es
v 0 nt

=1

Just as in Proposition 1, we argue that if rt = 0, then rt+1 = 0 as well. For suppose 1 + rt+1 > 1. In this
case, young agents born at date t + 1 would strictly prefer the asset to storage, and so pt+1 = At+1 nt+1 .
From the labor market clearing condition, we have
v 0 nt+1 = 1 + rt+1 (1
Since v 00 ( ) > 0 and v 0 nt

= (1

)>1

), it follows that nt+1 > nt . But this implies that pt+1 = At+1 nt+1 >

At nt
pt , which contradicts the fact that rt = pt+1 =pt 1 = 0. Once again, then, a zero interest rate is
absorbing. These deterministic equilibria thus have the same structure as the equilibria in Proposition 1:
There exists a cuto¤ date t such that before date t , the real interest rate 1 + rt > 0 and agents spend
all of their income on the asset and pt = At nt . From date t on, the real interest rate is 0, employment is
constant and solves v 0 nt = 1
, and the real price of the asset is constant. We can index the equilibrium
by its asymptotic limit, limt!1 pt = p . Note that the limiting value can vary by . Thus, an equilibrium
is determined up to two parameters, pH and pL . We now argue that these two parameters pin nt and rt
for all t 1.
In the case where p = 1, we have pt = At nt for all t and the analysis is the same as in the case where
d > 0. In that case, the same argument as in the case where d > 0 implies 1 + rt = 1 + g for t 1, and so
nt solves v 0 nt = (1 + g) (1
).
In the case where p < 1, there exists a cuto¤ t ( ) < 1 such that pt

For any date t

( ) 1

t ( ), the real interest rate rt = 0, employment nt solves v

0

= At nt

nt

=1

1

and pt

( )

=p .

, the real price

) At , so everything is determined up to the wage level Wt . For
pt = p , and the real wage Wt =Pt = (1
any dates 0 < t t ( ) 1, we can solve for these same values recursively. In particular, we have
v 0 nt

=

At+1 nt+1
At nt

(1

)

Using the boundary condition for nt at t = t ( ), we can solve for nt for all t between 1 and t ( )
which would yield the unique interest rate at each such date t and the unique asset price.

1,

H
As in the case where d > 0, the equilibrium at date 0 is determined up to two parameters, H
0 and W0 .
These determine the degree to which changes in the nominal interest rate a¤ect the real wage at date 0,
and the level of wages and prices given price-level indeterminacy in the model.

L
H
L
In Section 1 of the paper, we discuss a particular equilibrium in which nH
t = nt for all t but pt 6= pt .
Consider two equilibria in which rtH = rtL = 0 from date t = 1 on. This ensures employment is the same
regardless of from date t = 1 on . The initial interest rate r0H > r0L , but the initial real wage W0H =P0H is
L
lower by enough to ensure 1 + r0H W0H =P0H = 1 + r0L W0L =P0L . That ensures nH
0 = n0 .

43

Appendix C: Endogenous Asset Creation
In this Appendix, we consider the case where the supply of assets is endogenous rather than …xed exogenously. Suppose that in any period, old agents can convert consumption goods into assets. At date 0, old
agents determine the supply of assets they sell. Beyond date 0, old agents can sell assets they previously
purchased as well as create new ones. All other aspects are unchanged: At each date t a new cohort arrives
t
that is endowed with et = (1 + g) e0 consumption good who wishes to consume at date t + 1.
We look at the e¤ect of endogenous asset creation in both the dynamically ine¢ cient economy and the
economy with credit. For the dynamically ine¢ cient economy, we show that even when the supply of assets
is endogenous, there is still no reason to intervene against bubbles: The high price of the assets does not
lead to excessive asset creation. For the economy with information frictions, we show that intervening to
raise rates cannot improve welfare even when it frees up resources that would have been used to create
assets. We then show that a threat to raise rates if the bubble persists can generate a Pareto improvement
that makes all cohorts, including the initial old at date 0, better o¤. Allowing the supply of assets to be
endogenous thus allows us to get around the Grossman and Yanagawa (1993) result on the impossibility of
Pareto improvements in overlapping generation models of bubbles.

C.1 Endogenous Asset Creation in the Dynamically Ine¢ cient Economy
We begin with the dynamically ine¢ cient economy in Section 1. Instead of assuming that the supply of
assets is …xed and endowed to the old at date 0, suppose no assets exist at the beginning of date 0, but that
old agents can convert goods into assets. The technology for converting goods into assets features increasing
marginal costs. Formally, the cost of producing an additional asset given a mass q assets has already been
produced is c (q) units of the consumption good, where c ( ) is a di¤erentiable and increasing function. The
total cost for the old to produce the initial q0 assets that trade at date 0 is thus
Z q0
C (q0 )
c (j) dj
0

Since all assets trade in a decentralized market at a common price p0 , the amount of goods the old can
consume at date 0 is equal to p0 q0 C (q0 ).
At dates t = 1; 2; 3; ::: the economy inherits the qt 1 assets that the young purchased in the previous
period and are now held by the old. Old agents at date t can produce new assets in addition to the ones
they previously bought. Let xt denote the quantity of new assets produced at date t. Assets cannot be
destroyed, so xt 0. We assume the cost of producing new assets depends on the number of existing assets
qt 1 . That is, we assume marginal costs are increasing not in the amount produced in a given period but
in the total amount produced. In addition, since the economy is growing over time, we assume the cost of

44

producing also rises at the same rate. Formally, producing xt assets requires C (xt ) goods
Z qt 1 +xt
t
c (j) dj
Ct (xt ) = (1 + g)
qt

1

As we will see, this speci…cation implies there is no reason to delay creating assets. All assets will be created
at date 0.
All qt 1 + xt assets available at date t are traded in a decentralized market at a single price pt . In
equilibrium, old agents will keep creating assets until the marginal cost of the last asset created is equal to
the price at which assets trade. Thus, at date 0, we have
c (q0 ) = p0

(37)

We restrict attention to the case where the dividend d > 0, so prices are uniquely pinned down. Since young
agents at date 0 will spend all of their endowment to buy the asset, we have
p0 = e0 =q0

(38)

These two equations together determine the price p0 and quantity q0 of assets at date 0.
At dates t = 1; 2; 3; ::: it will still be the case that young agents born at date t will want to trade all of
their endowment for the asset. Hence, the price will equal pt = et =qt . Since assets cannot be destroyed,
qt q0 . From this, it follows that
pt =

et
qt

et
t e0
t
= (1 + g)
= (1 + g) p0
q0
q0

Since c (q0 ) = p0 , then the cost of producing new assets at date t
t

(1 + g) c (qt )

t

t

(1 + g) c (q0 ) = (1 + g) p0

pt

Thus, no new assets will be created after date 0, so xt = 0 for all t = 1; 2; 3; :::
Given that all assets will be created at date 0, beyond date 0 the economy will be the same as in the
case where the stock of assets is exogenously …xed. The new issue that arises when the supply of assets is
endogenous is whether the quantity of assets created in equilibrium is optimal or whether resources used to
create assets at date 0 are wasted.
Since the overlapping generations economy features multiple agents –in…nitely many, in fact –it features a
set of allocations that are Pareto optimal rather than a single allocation. We now argue that the equilibrium
with endogenous creation is Pareto optimal. Thus, it will not be possible to make agents better o¤ by
intervening to reduce the quantity of assets created at date 0.
Consider the equilibrium outcome in which the quantity of assets q0 solves p0 = e0 =q0 = c (q0 ). Suppose
we intervened and produced
fewer assets at date 0 than in the equilibrium. To ensure the old at date 0
45

are no worse o¤, we would have to give them
Z

q0

[p0

c (j)] dj

q0

Rq
This leaves us with q00 c (j) dj resources. Could we use them to make young agents at date 0 better o¤?
Since young agents only value consumption at date 1, they would have to either store these goods or use
them to buy the remaining q0
assets. But at date 1, the most old agents can consume is the endowment
e1 of the young at date 1 and the dividends from the q0
assets that were produced at date 0. In the
original equilibrium, they already consumed e1 + dq0 . This means that the young now consume d less
than that under the original equilibrium. Thus, there is no way to make all agents who are around at date 0
weakly better o¤ by producing fewer assets. A similar type of argument can be used to show that producing
more assets cannot make agents better o¤.
Intuitively, the best we could do with the resources used to create the marginal asset in equilibrium is to
invest them in the highest return option available. But in our economy, this would involve investing it in an
equivalent asset. Thus, there is nothing to gain on the margin from producing one less asset. Even though
the asset is a bubble, there is no sense in which the equilibrium quantity of assets is excessive. Asset prices
exceed the value of dividends these assets generate, but they also re‡ect the value assets serve in facilitating
the intergenerational transfers needed to overcome dynamic ine¢ ciency.

C.2 Endogenous Asset Creation in the Economy with Information Frictions
We now turn to endogenous asset creation in the economy with information frictions. We continue to assume
that at date 0, old agents can convert consumption goods into assets according to a features increasing
marginal costs such that the total cost to produce q0 assets is given by
Z q0
C (q0 )
c (j) dj
0

As in the dynamically ine¢ cient economy, old agents will keep creating assets until the marginal cost of the
last asset created is equal to the price at which assets trade. Since at date 0 the dividend is equal to D, let
us …rst derive the equilibrium conditional on the dividend remaining high. We begin with the equilibrium
when dt = D. The market clearing condition is now given by
D
D
pD
=e
t q t + N 1 + Rt

(39)

and the zero-pro…t condition for borrowers is given by
D
1 + RtD pD
t = pt+1 + D

Finally, the quantity of assets qtD satis…es
(
qtD = c 1 pD
t
D
qt =
qtD 1

D
D
max pD
if pD
t
0 ; p1 ; :::; pt 1
D D
D
if pD
t < max p0 ; p1 ; :::; pt 1

46

(40)

(41)

where c 1 ( ) is an increasing function. Suppose for a moment that pD
t
,
and
so
we
can
replace
(39)
with
Then qtD = c 1 pD
t
pD
t c

1

D
D
max pD
0 ; p1 ; :::; pt 1

for all t.

pD
+ N 1 + RtD = e
t

D
This yields RtD as an increasing function of pD
pD
t , i.e., Rt =
t , which we can substitute into (40).
D
Similarly to the case with a …xed supply of assets, this yields an equilibrium law of motion pD
t+1 = ' pt
D
D
D
which has a unique …xed point pD . If we consider an equilibrium in which pD
0 > p , then pt+1 = ' pt
D
1
D
would yield an increasing sequence of prices. In this case, qt = c
pt for all dates, so the path would
1
satisfy all three equilibrium conditions. However, as before, the total expenditures on the asset, pD
pD
t c
t ,
D
D
would eventually exceed e, which cannot be an equilibrium. What about paths in which p0 < p ? In that
D
1
case, p1 < p0 , and so q1 < c 1 pD
pD , and let us replace (39) with
1 . However, de…ne q = c
D
D
pD
=e
t q + N 1 + Rt
D
pD
This again yields RtD as an increasing function of pD
t , which we can substitute into (40).
t , i.e., Rt =
D
D
This yields a di¤erent law of motion pt+1 = ' pt that has a unique …xed point, and by construction the
D
D
D
…xed point is also pD . If we consider an equilibrium in which pD
would yield a
0 < p , then pt+1 = ' pt
D
D
D
decreasing sequence of prices. In this case, qt = qt 1 = q for all dates, so the path would satisfy all three
equilibrium conditions. However, in this case the total expenditures on the asset would eventually turn
negative, which means the price of the asset eventually turns negative. But this cannot be an equilibrium.
D
for all t. As long as dividends are high, no new assets will be produced.
Hence, pD
t =p

We can apply a similar analysis to characterize the price of the asset in the low-dividend regime pdt . The
main di¤erence is that the quantity q D which is equal to c 1 pD now strictly exceeds c 1 pd . The case
where pd0 > pd is now a bit more involved, since the argument that it will eventually exceed e is a bit more
involved, but the same argument applies. Once again, no new assets will be produced in the low dividend
regime. All assets in the economy will be produced at date 0.
Since the price of the asset in the high dividend regime pD is the same as the price of the asset paid
D forever, there is a sense in which there is excessive creation of assets in this economy. Essentially, the
expected return on the last asset produced is 1 + rD which is less than the expected return to lending out
D
resources to all willing borrowers, which is 1 + R . However, we will now show that raising rates at date
0 continues to crowd out entrepreneurial activity even though this intervention discourages the creation of
bubble assets and frees up resources to be used for entrepreneurial activity.
Consider the e¤ect of reducing the initial endowment e0 from e to e
equilibrium at date 0:
pD
0 c

1

D
pD
0 + N 1 + R0

1 + R0D pD
0
From (42), we can express R0D =
pD
0 , where
D
D
this into (43), we get p1 = ' p0 . Since

D
0

. We begin by solving for the

= e

(42)

= pD
1 +D

(43)

pD
is increasing in both and pD
0
0 . If we substitute
D
D
is increasing in , ' p
> p . This implies that
47

D
pD
0 < p , since otherwise the amount spent on the asset along the equilibrium path in the high dividend
D
regime would eventually exceed e. It follows that in equilibrium, pD
1 = p . That is, the equilibrium at date
1 is una¤ected by . If we substitute (43) into (42) and di¤erentiate, we get
!
pD + D 0 p D + D
dpD
0
1
D
D
10
D
c
p0 + p0 c
p0
N
= 1
2
D
d
p
pD
0
0

dpD
0
=
d
c

1
1

pD
0

+

10
pD
0 c

pD +D
0
2 N
(pD
0 )

pD
0

pD +D
pD
0

<0

(44)

Thus, intervening to raise rates at date 0 will depress the price of the asset. This implies q0D < q1D and
x1 = q1D q0D > 0. Thus, the intervention implies fewer assets will be created at date 0. However, since
D
R0D = p p+D
is increasing in , then N 1 + R0D is decreasing. Raising rates dampens the bubble, but it
D
0
still crowds out rather than stimulates more entrepreneurial activity.
Finally, we consider the e¤ects of a threat to destroy the endowment at date 1. The equilibrium price at
date 0 is now given by
pD
0 c

1

D
pD
0 + N 1 + R0

1+

R0D

pD
0

= e
=

pD
1

(45)
+D

(46)

The market clearing condition at date 0 is the same as before, and so we can still write R0D =
pD
0 .
D
D
D
D
Substituting in implies p1 = ' p0 where ' p = p . But we know from our analysis of what happens
D
D
D
in the period when we destroy units of the endowment that pD
1 < p . Hence, p0 < p . Thus, a
threat to destroy output at date 1 if the dividend remains high will dampen the price at date 0, and since
D
R0D =
pD
is an increasing function of pD
0
0 , it will also lower the interest rate on loans R0 . Hence, a
threat to destroy some of the endowment at date 1 if the bubble persists dampens the bubble and leads to
more entrepreneurial activity at date 0. Since the quantity of assets is increasing in the price, this means
such a threat would result in fewer assets being produced at date 0. On the margin, the last asset produced
yields an expected return of
1 + rD
0

=

d + pd + (1
pD
0

<

D + pD
1
= 1 + R0D
pD
0

) D + pD
1

Hence, the e¤ect of shifting resources to entrepreneurship increases the expected resources available at date
1: Since producers require fewer resources to compensate them than the young spend on assets, i.e., since
entrepreneurs only require pD c (q) to leave them no worse o¤, then c (q) worth of resources can be freed
up for entrepreneurial activity. Thus, even without default costs, a threat to destroy output in the future
can be used to generate a Pareto improvement in which both the old and the young at date 0 are at least
as well o¤. If we transfer some of the gains to the young at date 1 if the bubble collapses, as described in
Section 2.6, we can ensure all cohorts are at least as well o¤ under the intervention.
48