Bubbles, Banks and Financial Stability
Kosuke Aoki (University of Tokyo)
Kalin Nikolov (European Central Bank)May 2012 [First draft: November 2010]

Abstract
We build a model of rational bubbles in a limited commitment economy and show that the impact of the bubble on the real economy crucially depends on who holds the bubble. When banks are the bubbleholders, this ampli…es the output boom while the bubble survives but
also deepens the recession when the bubble bursts. In contrast, the real
impact of bubbles held by ordinary savers is more muted.

1

Introduction

The last 20 years have seen two spectacular episodes of …nancial instability.
First, stock prices experienced a truly unprecedented increase and subsequentcollapse around the turn of the millenium. Then came the dramatic rise andfall of world-wide housing prices, culminating in the …nancial crisis and ‘GreatWe are especially grateful to Jaume Ventura whose excellent comments helped us improve the paper a lot. We also thank Henrik Borchgrevink, Wouter den Haan, Piti Disyatat,
Martin Eichenbaum, Philipp Hartmann, Jonathan Heathcote, Xavier Freixas, Roman Inderst, Peter Karadi, Nobu Kiyotaki, Alberto Martin, Tomoyuki Nakajima, Helene Rey,
Victor Rios-Rull, Immo Schott, Shigenori Shiratsuka, Harald Uhlig, Christopher Waller andPhilippe Weil. The views expressed here are those of the authors and should not be interpreted as those of the European Central Bank.

1

Contraction’ of 2008-2009. These movements in …nancial prices were large
enough to reawaken interest in asset price bubbles and the way they a¤ect thereal economy. In addition, the near failure of the global banking system during
the Lehman Crisis of 2008 has increased awareness of the importance of banks
in amplifying macroeconomic ‡uctuations.
Motivation and economic questions Our paper studies the interactionof asset price bubbles, the banking system and the real economy in a generalequilibrium model with credit constraints. The current …nancial crisis is themost immediate motivation for wanting to study this interaction. Bank losses
due to subprime mortgage defaults depleted bank capital positions and led toa severe credit crunch and the deepest recession since the 1930s. This episodeis by no means unique in history. In Japan, during the 1980s, real estate
prices grew rapidly allowing property developers to borrow heavily from banks
using real estate as collateral. When property prices collapsed in the 1990s,
Japan went through a protracted banking crisis and a ’lost decade ’ of low
economic growth. A similar set of events occurred during the housing bustand …nancial crisis that hit the Scandinavian countries in the early 1990s. In
all of these historical episodes, the banking system was exposed to a bursting
housing bubble through a higher incidence of household default. As Reinhart
and Rogo¤ (2008) show, using their rich time-series and cross-country dataset, the resulting insolvency of the …nancial system tends to lead to a deep
downturn in real activity.
The one bubble episode that stands in clear contrast to all the costly boombust cycles described above is the 1998-2000 ’technology bubble’. Stock pricesrose and then collapsed dramatically without triggering a banking crisis or adeep recession. Why was this bubble episode so di¤erent from other (mainly2housing) recent bubbles? More generally, what makes some asset price busts
more costly than others? These are the main questions we focus on in thisresearch. We build a model which focuses on bank exposures to bubbles as animportant reason why some lead to crisis while others do not1 .
Main results Following the insights of the growing literature on rational
bubbles with credit constraints (Caballero and Krishnamurthy (2006), Kocherkalota(2009), Martin and Ventura (2011a), Martin and Ventura (2011b), Arce and
Lopez-Salido (2008), Farhi and Tirole (2011), Hirano and Yanagawa (2010))
we focus on an environment in which binding borrowing constraints lead to ashortage of means of saving and the use of dynamically ine¢ cient investment
technologies in equilibrium. This creates the conditions for bubbles to circulate, helping to improve the supply of liquidity but also exposing the economy
to the risks of their bursting. Our innovation relative to the rest of the rational bubbles literature is to model …nancial intermediation explicitly. In ourmodel credit ‡ows along a chain that starts with a saver, goes via a bank andends up with a …nal borrowing …rm. This realistic feature of our environment
also has the implication, overlooked by the rest of the literature, that asset
price bubbles can be held by a variety of agents with very di¤erent economicroles in equilibrium. Our paper shows that the identity of the bubble-holder
is vital for understanding the e¤ect of bubbles on the real economy. Bubbles
held by banks expand output more during the boom phase and then lead toa much more severe contraction in credit and output when they …nally burst.
1

Commercial banks in the US tend not to hold equities and their Commercial and Industrial Loans had fallen to 20% of total loans by the late 1990s. In contrast, almost 60%of their total loans and leases were secured by real estate. This is an important reason why
housing busts tend to be more costly for the banking system compared to equity busts.
Martin and Ventura (2011a) argue that a reason why the ’dot com’bust did not lead to arecession is that it was immediately followed by the housing bubble which helped to expand
the economy. This argument is plausible and we see it as complementary to our channel
which stresses the health of the banking system.

3

In contrast, bubbles held by ordinary savers have a relatively muted e¤ect on
the real economy.
The key to understanding our main result lies in the way the identity of the
bubble holder a¤ects the evolution of the wealth distribution in equilibrium.
Bubbles in our world are risky assets which deliver a higher return comparedto safe assets in order to compensate the holder for the possibility of losing
everything in the bust. This implies that the bubble holder experiences strong
growth of net worth while the bubble survives and then a sharp fall in net worth
when the bubble bursts. These movements in the wealth distribution have large
real e¤ects when the a¤ected agents are the credit constrained intermediaries.
The gains and losses they make on bubble holdings a¤ect aggregate credit
supply and therefore the production scale of the bank’s credit constrained
borrowers. During the bubble’s run up, banks expand their size, increase credit
supply and this allows productive …rms to increase output. The re-allocationof productive factors towards more e¢ cient …rms then boosts aggregate TFP.
When the bubble crashes, banks make large losses and in‡ict a credit crunch on
the rest of the economy. In contrast, ordinary savers are credit unconstrained
and their net worth does not play the same role in the credit intermediation
mechanism. When savers hold the bubble, their wealth also ‡uctuates but thishas a very limited impact on the credit constraints faced by other agents inthe economy.
In this paper, banks only undertake bubble exposures when they believethat potential losses will be underwritten by the government (moral hazard).
The promise of bailouts gives banks an advantage in holding bubbly assets,pushing the risky exposure into their hands. However, the main result of ourpaper is general: the identity of the bubble holder is key to understanding thereal e¤ects of bubbles during the boom as well as the bust. Bubbles held by4banks create a bigger boom and a bigger crash by stimulating a boom-bust
cycle in credit supply.
Related Literature Our paper contributes to a large literature that studies the way asset price bubbles a¤ect economic activity. The seminal workof Tirole (1985) showed that, in an environment without credit constraints,
bubbles have a contractionary e¤ect on economic activity because they displace ine¢ cient investments in savers’portfolios. This is the famous ’crowding out’e¤ect of bubbles. Subsequent papers (Caballero and Krishnamurthy(2006), Kocherkalota (2009), Martin and Ventura (2011a), Martin and Ventura(2011b), Farhi and Tirole (2011)) have shown that when there are credit market imperfections, bubbles may have an expansionary e¤ect through a variety
of mechanisms that help to reduce the severity of credit constraints.
In Martin and Ventura (2011b) the expansionary e¤ect of bubbles arises
because bubble creation increases both current and anticipated future pro…ts
for productive entrepreneurs who cannot borrow against their tangible assets
due to a moral hazard problem. Because the anticipated pro…ts from futurebubble sales are collateralisable, this allows the most productive entrepreneurs
to increase borrowing and production in the current period, thereby increasing aggregate TFP. Related to the results in our paper, Martin and Ventura(2011b) demonstrate that the e¤ects of bubble creation depend crucially on theidentity of the bubble creator. When credit constrained productive …rms create bubbles, the bubble is strongly expansionary. When unconstrained savers
bene…t from the creation, the bubble is contractionary as in Tirole (1985). In
our paper, we abstract from bubble creation, instead focusing on the impact of
bubble holding and bubble destruction on economic outcomes. We show that
this impact is also highly dependent on the identity of the bubble holder.

5

Similarly to Martin and Ventura (2011b) our paper also relies on a redistribution of productive resources towards more able …rms in order to generate
an increase in aggregate TFP and output as a result of the bubble. But theway in which resources are channelled into productive hands does not rely
on bubble creation. Instead, bubble holdings turn out to be expansionary or
contractionary due to the interplay of the ’liquidity e¤ect’and the ’competition e¤ect’identi…ed in Farhi and Tirole (2011) as well as the ’credit supplye¤ect’ which is unique to our paper. It is the ’credit supply e¤ect’ whichreally depends on who holds the bubble. It is at its strongest when the bubble is in the banks’ hands. There it boosts banks’ net worth and expands
credit supply. This also exerts a powerful positive externality on the net worthof borrower entrepreneurs, helping to increase their productive activities andthe economy’s aggregate e¢ ciency. When savers hold the bubble, such credit
supply externalities are relatively weak and the bubble is less expansionary.
Our paper’s …ndings are in line with several important stylised facts uncovered by the literature on the empirical regularities around …nancial crises. Our
focus on rational bubbles that can only occur in a low interest rate environment
is supported by Schularick and Taylor (2012) who report that growth adjusted
real interest rates are very low for several years in the run up to global crises.
Schularick and Taylor (2012) present evidence that credit booms are associatedwith larger cumulative increases and steeper subsequent declines in GDP. Our
theoretical results are exactly in line with such empirical regularities.
Plan of the paper Section 2 introduces the economic environment, section
3 describes the equilibrium without government sector and discusses the conditions for the existence of bubbles and bubble ownership in the absence ofgovernment bailout policy. Section 4 examines the e¤ect of the …nancial safety

6

net on the existence and ownership of bubbles, and analyses the implicationsof bubble ownership on the real e¤ects of bubbly episodes. Finally, section 5
concludes.

2

The Model

The economy is populated with three kinds of agents. There is a continuum of
in…nitely lived entrepreneurs and a continuum of in…nitely lived workers bothof measure 1. There is also a continuum of bankers who have …nite lives andcan die with probability 1

in any period, conditional on being alive in theprevious period.

2.1

Entrepreneurs

Each entrepreneur is endowed with a constant returns to scale production
function which converts labor ht into output in the next period yt+1 .
yt+1 = ait ht ;(1)

where ait is a productivity parameter which is known at time t.
In each period some …rms are productive (ait = aH ) and the others are
unproductive (ait = aL < aH ). Each entrepreneur shifts stochastically between
productive and unproductive states following a Markov process. Speci…cally, if
a productive entrepreneur in this period may become unproductive in the next
period with probability ; and an unproductive entrepreneur in this period may
become productive with probability n . This probability is independent acrossentrepreneurs and over time. This Markov process implies that the fraction of
productive entrepreneurs is stationary over time and equal to n=(1 + n), given

7that the economy starts with such population distribution. We assume thatthe probability of the productivity shifts is not too large:(2)+ n < 1:This assumption implies that the productivity of each agent is persistent.
Entrepreneurs are ex-ante identical and have log utility over consumption
streams
E

U = Et

1
X

t

(3)

ln ct

t=0

Entrepreneurs purchase consumption (ct ), bubbles (met ) at pricetandbonds bt . They also pay wages to the workers they hire wt ht in order to receive
future revenues ai ht which the government taxes at ratet after deducting debt

repayments. wt and ht denote real wage and labor respectively.
ct +wt ht +met t bt = (1t)

ai ht 1

Rti 1 bt 1 + met 1 t(1t ) zt(4)

where zt stands for entrepreneur’s net worth. Rti is the interest rate which isequal to the loan rate Rtl when the entrepreneur is a borrower and Rtd when
the household is a saver. The bubble asset in our economy is a durable but
intrinsically worthless asset which has no productive or consumption value. In
other words, its fundamental value is zero.
Due to limited commitment in the credit market, agents will only honour
their promises if it is in their interests to do so. We assume that only a fractionof the value of the …rm can be seized by creditors and tax collectors. Furthermore we assume that the tax authorities have a …rst call on the …rm’s resources
while private creditors are second in line. Hence the collateral constraint is

8

given by:
Rtl bt + Et

t+1 6

Et yt+1 ;

0<

(5)

<1

where
Et

t+1 = Et t+1

Rti bt + met t+1

ai ht

stands for expected tax payments in the following period. Entrepreneurs maximize (3) subject to (4) and (5).

2.2

Banks

We assume that only banks can enforce debt repayments in our economy.
Consequently, all borrowing and lending will be bank-intermediated. Bankers
are risk neutral and live for a stochastic length of time. Once bankers receive
an “end of life”shock, they liquidate all their asset holdings and consume their
net worth before exiting. This shock hits with probability 1:

Banks maximize the following objective function:

B

U = Et

1
X

(

)t cB
t

(6)

t=0

subject to the following constraints explained below.
In each period the bank has net worth (nt ). It collects deposits (dt ) fromthe savers. Then it lends to the borrowers (bt ), purchases bubbles ( t mbt ), or
consumes (cbt ). We assume that intermediation is entirely costless. The bank’sbalance sheet constraint is given by

cbt + bt +bt mt = nt + dt :9(7)

The evolution of net worth is given by

nt+1 = Rtl bt +b
t+1 mtRtd dt :(8)

Rtd dt :(9)when the bubble does not burst and by:

nt+1 = Rtl bt +when it does burst.

b
t+1 mtt+1 is the fraction of the banks’bubble investment whichis guarranteed by the government. In the event of a bubble collapse, thegovernment transfers these funds to the banks to compensate them for losses
made. The parametert+1

is a simple means of capturing the explicit or

implicit guarrantees given by the government to the banking system
Following Gertler and Karadi (2009), we model banks subject to limited
commitment. More speci…cally, the banker may divert 1

fraction of de-posits. Once he diverts, he consumes the funds and closes his bank, remaining
inactive until his ’death’. The savers can recover the remaining

fraction ofdeposits. Since the savers recognize the banker’s incentive to divert funds,
they will restrict the amount they deposit with the intermediary, according to
the following borrowing constraint:(1)dt 6 V (nt ) :(10)

The left hand side of equation (10) is the value when the banker diverts, while
the right hand side is the value when he did not (i.e., the continuation valueof the bank). We also assume that the bank cannot short mt . The bankmaximizes (6) subject to (7), (8), (9) and (10).

10

2.3

Workers

Unlike the entrepreneurs, the workers do not have access to the production
technology nor any collateralizable asset in order to borrow. They maximize
the following utility

U

W

= Et

1
X

t

t=0

cw
t

h1+
t
1+

(11)

subject to their ‡ow-of-funds constraintw
cw
t + mt tw
bw
t = wt ht + mt 1 t

Rtd 1 bw
t 1;

(12)

here superscript ‘w’stands for ‘workers’.

2.4

The Government

We assume that the only role for the government in this economy is to levy
taxes on entrepreneurs and bail out the banking system when it makes losses.
We assume that the government follows a balanced budget rule and does notissue government debt. Consequently taxes are only levied whenever bailout
spending is necessary. For the rest of the time, taxation is zero.

3

Equilibrium without government

We consider a stochastic bubble that persists with probability . With probability 1

the bubble bursts and its value reverts to zero. We assume thisprobability is constant over time. Also, we assume that once bubbles burst
they never arise again.
In equilibrium, due to the di¤erence in their productivity, productive entrepreneurs borrow from banks and unproductive entrepreneurs make deposits
11

to banks. We focus on equilibria in which the productive entreprenerus borrow
up to their borrowing constraint2 . In this section, we characterise the equilibrium without government. Therefore we set

t+1 = 0 and

t+1 = 0 at all

times.

3.1

Entrepreneurs’optimal behavior

The entrepreneurs’problem can be interpreted as a savings problem with uncertain returns. Since utility function is logarithmic and there is no labour
income or transfer income, entrepreneurs consume a constant fraction of net
worth (zt ).
ct = (1and save the remaining) zt :(13)

fraction3 .

The entrepreneur has several possibilities for accumulating net worth. Firstof all, an entrepreneur with productivity at = ah ; aL can undertake unleveraged
investments using his own technology. This yields a return of watt : the cost of
hiring a worker is wt and the worker produces output at in the following period.
Secondly, the entrepreneur can deposit in the banking system, earning a returnof Rtd . Thirdly, the entrepreneur can pledge up to fraction of future output tobanks and borrow at interest rate Rtl in order to invest in her own production
technology. In this case the leveraged rate of return is
at (1
)
:
wt
at =Rtl(14)

By borrowing from banks secured by fraction of output, the entrepreneur can…nance externally at =Rtl amount (equation (5)). Therefore the denominator23This happens when the borrowing constraints are tight enough. See Aoki et al. (2009).
See, for example, Sargent (1987).

12

is the required downpayment for the unit labor cost. Finally, the entrepreneur
can invest in bubbles. In this case the rate of return is given by
t+1 stands for the market value of the bubble next period.

t+1 = t , where

When it bursts,

the return is zero.
High productivity entrepreneurs enjoy better returns on production so they
are the ones who borrow in equilibrium. When their borrowing constraints
bind the rate of return on wealth (r(aH )) is given by:r(aH ) =aH (1)= Rtl :
Hwt
a =Rtl

(15)

From (5) and (4) the investment (employment) of a productive agent is givenbyht =The entrepreneur saves azt:
aH =Rtlwt(16)

fraction of wealth zt and uses her entire savings as

a downpayment for wage payments to the workers she hires.
While the productive entrepreneurs borrow, the unproductive entrepreneurs make deposits. In addition, they have two other means of savings: unleveraged production and investing in bubbles. Notice that both deposits andproduction are riskless in our environment. When

Rtd >

aL
wt

(17)

low productivity agents are inactive in production. However when the credit
constraints on banks and borrowing entrepreneurs are tight enough, the productive entrepreneurs cannot absorb all national saving. The supply of deposits
is limited and the low productivity technology may be viable in equilibrium.

13

In such case
Rtd =

aL
wt

(18)

and the saver entrepreneurs use both bank deposits and their own production
technology to accumulate wealth.
Bubbles are risky. When savers invest in bubbles as well as deposits, thearbitrage condition for bubbles is determined by the savers’state-contingent
wealth valuation
Et

1

t+1

cLt+1

= Et

t

1
cLt+1

Rtd ;

(19)

where 1=cLt+1 represents the shadow value of wealth at time t + 1 of the entrepreneur who is unproductive at time t4 , where expectation operator is takenover whether bubble survives or crashes.
b
t+1 with probability

t+1 =

(20)

0 with probability 1
wherebt+1 is the market value of the bubble on survival.

3.2

Banks’optimal behaviour

Next, we characterise the optimal behaviour of a representative bank in our
economy. The problem of the bank can be represented in recursive form byusing the bank’s value function representation as follows:

V (nt ) =4max

cbt ;dt ;bt ;mt

cbt + Et [ V (nt+1 ) + (1Namely, it is given by
1=cL
t+1 = (1H
where Zt+1
is given by equation (34)

14

L
)Zt+1

) nt+1 ](21)V (nt ) is the value of a bank with net worth nt which chooses current consumption, deposits, bubbles and loans optimally. This value is equal to current consumption and the expected future discounted value of bank net worth) nt+1 ]. This value includes the continuation value of

Et [ V (nt+1 ) + (1being a banker - this happens only if the banker survives with probability .
With probability 1, the banker receives the death shock and consumes his

entire net worth in the following period.
Because of risk neutrality, we can guess that the value of the bank is alinear function of net worth nt

V (nt ) =

(22)

t nt

When Rtl > Rtd , the credit constraint (10) binds and consumption is postponed
until death. Then, with equation (10) binding, deposits are given by

dt =

Here

t

1

(23)

nt :

) is the bank’s leverage.

t =(1Regarding the bank’s choice over bubbles and loans,

Et

1

+

t+1

Et

t+1

1

+

t+1

Rtl ;

(24)

t

where expectation operator is again taken over the bubble surviving or not.
If equation (24) holds with strict inequality, then the bank will not invest in
bubbles (mbt = 0). When the bank invests in bubbles (24) must hold withequality. By substituting (22), (23), and (24) into (21),

t =

1

Et (1

) + Et t+1 Rtl

Et (1) + Et t+1
15

t satis…es

Rtl Rtd1:(25)

This expression states that the value of a unit of net worth for a banker
is equal to the value of the returns on its loan book (the numerator), suitably
boosted by leverage (the denominator). The banker issues one unit of loans
but the downpayment he has to make is only given by the denominator of (25)because he can pledge some of the future expected excess returns from intermediation ( Et (1) + Et t+1

Rtl Rtd) to depositors who …nance a large
1part of the loan outlay. Note that the above formulas show that
with

t+1 .

t increases

This implies that the current leverage depends on the future fran-chise value of the bank which is re‡ected by the leverage next period.5 It also
shows that

3.3

lt is an increasing function of the spread Rt

Rtd :Workers’optimal behaviour

Workers are risk-neutral and consequently their consumption-savings behaviour is a knife-edge one. When the loan interest rate is lower than the rateof time preference, workers want to borrow unlimited quantities. Because theworkers cannot operate the production technology, they cannot pledge collateral to lenders. Hence workers cannot borrow.
Of course workers could save at the deposit rate but they only want to do
this when
Rtd >

1

:

(26)

If this condition is not satis…ed, workes will consume their entire net worth(…nancial wealth and labour income) and save nothing. Their labour supply
hst is given by
hst = wt :(27)

Because ours is a limited commitment economy, we guess and verify that Rtd <5

See Nikolov (2010), who considers a similar problem for …rms.

16

1

at all times along the equilibrium paths we consider. Hence our workers

are hand-to-mouth consumers at all times.

3.4

Aggregation and market clearing

Let the total supply of the bubble asset be normalized to 1. In other words,

met + mbt = 1;(28)

where met and mbt , respectively, denote the shares of the bubble held by unproductive entrepreneurs and banks.
Let ZtH and ZtL , respectively, denote aggregate wealth of the productive andunproductive entrepreneurs. Then we can characterise the aggregate equilibrium as follows. From (16) the aggregate employment of the productive
entrepreneurs is given by

HtH =

ZtH
:
aH =Rtlwt(29)

When (17) holds, the unproductive entrepreneurs are indi¤erent between making deposits and producing, thus their aggregate saving is split as follows

HtL = ZtL

Dt

met t

(30)

where Dt denotes aggregate deposit.
Let us turn to banks. Under the banks binding borrowing constraint, theaggregate deposit is given by

Dt =

t

(1

17

)

Nt :

(31)

Notice that 1

fraction of banks exits in each period by liquidating all their

net worth. Therefore the aggregate net worth of the operating banks is given
by Nt . The aggregate balance sheet of the operating banks is given by
Dt + Nt = Bt + mbt t :(32)

Let us turn to the transition of state variables. Note that the unproductive
entrepreneurs become productive in the next period with probability n andthe productive entrepreneurs continues to be productive with probability 1.

Their rates of return are given by (15) and (17). Therefore the net worth ofthe productive entrepreneurs evolves from (15) and (13) as

H
Zt+1
= (1

)

aH (1
)
ZH + nHwt
a =Rtl t

Rtd

ZtL

met t + met t+1(33)

Similarly, the aggregate net worth of the unproductive entrepreneurs evolves
as

L
Zt+1
=

aH (1
)
Z H + (1Hwt
a =Rtl tn ) Rtd

ZtL

met t + met t+1(34)

From aggregating production function, aggregate output is given by

Yt = aH HtH 1 + aL HtL 1 :(35)

Finally, aggregate bank net worth is given by

Nt+1 =

Rtl Bt + mbt t+1

Rtd Dt(36)

The markets for goods, labour, capital, loan and deposit must clear. Goods
market clearing implies that aggregate saving must equal to aggregate invest18

ment.
(ZtH + ZtL ) + Nt = w(HtH + HtL ) +

(37)

t:

From (27), labour market clearing implies

wt = HtH + HtL :(38)

Now equations (18), (19), (24), (25), (28)-(38) jointly determine 15 variables
Rtd ; Rtl , wt , HtH , HtL , Yt ;HLt , Dt , Bt , Zt+1 , Zt+1 , Nt+1 ,ebt ; mt ; mt with threestates ZtH , ZtL , Nt 6 . At t = 0, Z0H is given by (33).
De…nition 1 Competitive bubbly equilibrium without government is a sequence
1

1

H
L
of decision rules HtH ; HtL ; Yt ; Dt ; Bt ; met ; mbt t=0 , aggregate state variables Zt+1
; Zt+1; Nt+1 t=0

and a price sequence Rtd ; Rtl ; wt ;t;1
t t=0 such that: (i) entrepreneurs, banks

and workers optimally choose decision rules

1

HtH ; HtL ; Yt ; Dt ; Bt ; met ; mbt t=0

taking the evolution of aggregate states, prices and idiosyncratic productivity
opportunities as given; (ii) the price sequence Rtd ; Rtl ; wt ;t;1
t t=0 clears the

goods, labor, capital, loan, bubble and deposit markets and (iii) the equilibrium evolution of state variables

1

H
L; Nt+1 t=0 is consistent with theZt+1; Zt+1individual choices of entrepreneurs, banks and workers and with the exogenous
evolution of productive opportunities at the individual …rm level.

3.5

Calibration

We have 8 parameters; aH =aL ; ; n; ; ; ;we need to calibrate before

we proceed to examine the quantitative predictions of our model economy.
There is little consensus regarding , the Frisch elasticity of labour supply.
Micro-data evidence suggests a value close to zero based on the labour supply
6

By Warlas law one of these equations is redundant.

19behavior of primary earners. The real business cycles literature usually sets amuch higher value in the region of 3 or even higher. The di¤erences is justi…edby the presence of labour market frictions that ensure that aggregate labour
is highly elastic even though individuals are relatively unwilling to vary theirmarket hours over time. Gertler and Kiyotaki (2010) make this argument andset the Frisch elasticity to 10 in their model. We pick a value of= 5, which

is within the range set in calibrating macro models.
aH =aL is an important parameter, whose value is also highly uncertain. As
studies such as Bernard et al. (2003) and Syverson (2004) have documented,
the dispersion of plant level productivity in US manufacturing is enormous,with the most productive plants having more than 4 times more productive
compared to the least productive. But as Aoki et al. (2009) argue, it is hard
to believe that such a huge dispersion of productivity levels is entirely due tothe presence of credit constraints. More likely, inputs could be mismeasured ina number of ways. For example, intangible assets such as managerial quality
could be an important missing input which could explain some of the hugedi¤erences in measured plant level TFP. Following Aoki et al. (2009) we set
a value for aH =aL = 1:1 implying a substantial cross-sectional dispersion inplant level TFP in the model.
We calibrate the remaining 6 parameters in order to match the steady statepredictions of the model in the absence of bubbles to 7 moments in the US
data. These are (1) the real loan rate minus the growth rate of real GDP andminus intermediation costs; (2) the real deposit rate minus the growth rate ofreal GDP; (3) commercial bank leverage; (4) average corporate leverage; (5)average leverage for highly leveraged corporates; (6) the rate of return on bankequity and (7) the ratio of M2 to GDP.
Calibration targets (1) and (2) deserve further discussion. For simplicity, in20

our model we assume there is no growth and no intermediation costs. In reality
these two assumptions of course do not hold. Growth in the US economy
has averaged close to 3% per annum since the second world war. Since weare interested in the dynamic e¢ ciency of the investments of US savers andbanks, we want to know whether the real return on these investments exceeds
the economy’s growth rate. This is why we subtract the real growth rate fromthe real return on deposits and loans.
In addition, when it comes to evaluating the dynamic e¢ ciency of banks’loan investments, we need to take intermediation costs into account. FDIC
data on US commercial banks’ cash ‡ow sources reveals that there are substantial intermediation costs (80% of those are labour costs). Here we assume
that all of these costs arise due to loan issuance rather than deposit taking.
This assumption is not entirely unreasonable given the labour intensive natureof arranging loans, monitoring them and then recovering them if they become
non-performing. It does, however, err on the side of assuming that banks’realloan returns are more dynamically ine¢ cient. We subtract these loan costs
from banks’real loan returns to get the …nal numbers shown in Table 1 below.
Full details of data sources and construction are available in Appendix A.

21

Table 1: Model and data moments
Moment (Model concept)

Data

Model

Real deposit rate - real GDP growth (Rd )

0.950

0.971

Real loan rate - real GDP growth - costs/Assets (Rl ) 0.982

0.982

Ratio of M2 to GDP (D=Y )

0.500

0.464

Bank leverage (D=N )

10.00

10.00

Average corporate leverage (L=Z)

0.500

0.530

Leverage of indebted corporates (L=(sZ))(Rl Rd )Bank rate of return on equity (Rtl + t (1 t ) t )

2.000

2.000

1.100

1.103

Table 2 below presents the values of the parameters chosen to match themoments.
Table 2: Baseline calibration
Parameter Value
0.167
n

0.011

aH =aL

1.100
5.000
0.622
0.788
0.907
0.958

3.6

Bubbly equilibria without government

3.6.1

Credit frictions and existence of bubbly equilibria

Now we characterise the deposit rate Rtd and loan rate Rtl in the steady state
without bubbles and discuss when bubbles can circulate. For bubbles to cir22

culate, two conditions are needed. Firstly, bubbles should be attractive. For
savers, the opportunity cost of investing in bubbles is the deposite rate, andfor banks it is the lending rate. Secondly, bubbles should be a¤ordable. Thisimplies that the rate of return of bubbles conditional on survival is no larger
than the rate of economic growth. In our economy the gross growth rate of
the economy is unity because there is no technological growth.
As a benchmark case here we show the condition for the existence of bubbly
equilibria when= 1. Credit frictions suppress the interest rates and those1

when the credit constraints bind.7 Similarly to Farhi

and Tirole (2011), when

= 1 whether a bubbly steady state exists and who

rates are lower thanowns bubbles depend on whether the two interest rates are lower than thegrowth rate in the ‘no bubbles’steady state8 .
In our economy, the severity of credit frictions is represented by two parameters,

and : Figure 1a shows the region ofandin which the deposit

rate is less than one and low productivity agents produce in equilibrium (thered area). In this case, the savers (unproductive entrepreneurs) have incentive
to buy bubbles in order to boost the rate of return they receive on their savings. The blue parts of the graph show parts of the parameter space where theeconomy is very credit constrained. At such low values ofandlow produc-

tivity entrepreneurs are active but wages are so low that even such ine¢ cient
projects deliver a rate of return greater than unity. As a result, savers haveno incentive to hold bubbles in such economies. The white parts of the graph
7

See Aoki et al. (2009)) for the general discussion of the relationship between the interest
rate and credit frictions.
8
In Martin and Ventura (2011a) emergence of bubbles itself can creates a “pocket of dynamic ine¢ ciency" so that bubbly equilibria may exist even when the interest rate is greaterthan the growth rate in no-bubble equilibrium. This result depends on their model settingthat credit-constrained agents can create bubbles in subsequent periods. We abstracts fromnew bubble creation. Because of this assumption, the condition for existence of bubbly
equilibria reduces to whether the interest rates in no-bubble equilibrium are smaller than
the growth rate.

23

(very high values ofand ) shows parts of the parameter space where low

productivity entrepreneurs do not produce because the …nancial system is welldeveloped. Here again, the rate of return on deposits is greater than unity andsavers have no incentive to hold bubbles. So it should be clear from Figure
1b that the conditions for the existence of bubbles is satis…ed at intermediatelevels of …nancial development.
[Figure 1a here]
The red area of Figure 1b shows the region in which the loan rate is lessthan one. Then the banks have an incentive to buy bubbles. It is naturalthat the part of the parameter space where banks bubbles can exist is more
limited compared to the parts of the parameter space where saver bubbles
exist. Because banks’ borrowing constraints bind, this introduces a positive
spread between lending and deposit rates. Hence the parameter space wherebank bubbles exist is a subset of the space where savers have an incentive
to invest in bubbly assets. Since the deposit rate is always lower than theloan rate, the savers also have incentive to hold bubbles at these parametervalues. In equilibrium, it turns out that only the savers have bubbles because
their arbitrage implies that the rate of return on bubbles must be equal tothe deposit rate, which is lower than the loan rate. Therefore the banks are
crowded out from the market for the bubble.
[Figure 1b here]
3.6.2

Who holds risky bubbles: stochastic steady state without
government policy

In the previous section we show that in the deterministic steady state, bankdeposits and the bubble become perfect substitute and only the savers holdbubbles. Now we allow for risky bubbles and consider an environment in24which bubbles only survive with a certain probability ( < 1.) We focus ona stochastic steady state in which bubbles are traded at a positive value andall the endogenous variables including bubbles are constant. In such a steadystate, agents take the probability of bubble bursting into account when they
make their investment decision. In that case, bubbles, loans and deposits
are no longer perfect substitutes. Also, note that the savers and banks havedi¤erent preferences for risk. Then it is of interest to analyse who holds risky
bubbles.
Table 3 below shows the share of bubbles held by banks. The table looks atbanks’participation under di¤erent probabilities of the bubble’s survival ( ).
The dashes in the table show economies in which the bubble is too risky fora stochastic steady state to exist in which such a bubble trades at a positive
price. This occurs when the survival probability is 0.965. Such a risky bubble
would need to command a very high return conditional on survival in order
to compensate its holders for the losses when it eventually collapses. But
when this ’risk premium’is too high, the bubble follows an explosive path andviolates feasibility constraints in …nite time. Consequently such a bubble isnever valued in equilibrium.
It turns out that banks mostly stay out of the bubble market. Their holdings are zero for very high or very low values of
the total stock of bubbly assets when

= 0:985.

25

and barely reach 1.6% ofTable 3: Bank bubble holdings in the stochastic steady state= 0:965= 0:975= 0:985= 0:995

Fraction of bubbles held by banks

-

0.000

0.016

0.000

Ratio of bubbles to total wealth

-

0.118

0.317

0.403

Expected saver loss (fraction of wealth)

-

0.004

0.006

0.003

Expected bank loss (fraction of wealth)

-

0.000

0.002

0.000

s
b
t+1 = t+1

-

0.984

1.060

0.987

The table also shows the size of bubbles as percentage of total wealth. It
shows that bubbles are not large. As a result, the expected losses when bubbles
burst are also small for both savers and banks.
The reason why banks hold very few bubbles is straightforward. Whenbanks’borrowing constraints bind, Rl > Rd and banks’investment opportunities (Rl ) are superior to savers’(Rd ). It is natural therefore that savers should
be more willing to take on the risk of saving via the bubble rather than via
deposits or own production.
Here, it is worth noting that banks’linear utility does not necessarily makethem the natural risk-bearers in equilibrium. Due to binding borrowing constraints and time-varying loan spreads, banks behave as if they are risk-averse.
Similar to Gertler and Karadi (2009), periods when bank capital is low areperiods when lending spreads are high. Therefore banks’ marginal value ofan extra unit of net worth ( t ) is high when they choose to hold bubbles andbubbles burst. Indeed, Table 3 shows that, when the banks choose to holdbubbles ( = 0:9875), the marginal value when bubbles burst at time t + 1( bt+1 ), is larger than the marginal value when bubbles survive ( st+1 ). Consequently they have a strong pro…t motive to ensure some stability in their net
worth. Holding a very large quantity of the bubbly asset is not optimal since
the excess return (which is pinned down by savers’…rst order condition too)26

is not big enough to justify the risks.

4

Equilibrium with government bailouts

4.1

Modi…cation of optimal behavior

Now we characterize the economy with an active government. Introducting
government bailout policy changes agents’ behaviour in an important way.
Firstly, consider productive entrepreneurs. We continue to focus our analysis
on the case in which they borrow up to their borrowing constraint. However,their rate of return of production with maximum borrowing (after taxation) is
now given by
(1

t+1 )

aH (1 et );wt et aH =Rtl

(39)

where et is the maximum amount that can be pledged to private creditors after

taking into account expected tax payments. By borrowing from banks secured
by
et =

1

Et t+1
6
Et t+1

fraction of output, the entrepreneur can …nance externally et at =Rtl amount(equation (5)). The fraction of output that can be pledged to outside private
creditors (et ) is less than the total amount of pledgable returns . This isbecause the government is expected to call on the resources of the …rm in theform of taxation Et t+1 . Because the government holds the senior claim to the…rm’s pledgable returns, private creditors can only rely on the residual fraction
Et t+1 which is boosted to some extent by the fact that debt repayments
are shielded from taxation (hence the division by 1

Et t+1 ). Also, note thatthe rate of return on wealth is potentially subject to taxation which is why(39) contains a 1

t+1 term.

27

In order for the productive entrepreneurs to borrow up to limit, the following inequalities must hold:andaH (1 e)Et f(1
wt eaH =Rtl

t+1 )gaH (1 e)Et f(1
wt eaH =Rtl

t+1 )g1> Rtd Et (1

cH
t+1

1
cH
t+1

> Et (1

t+1 )

t+1 )

1
cH
t+1

t+1

1

t

cH
t+1

;

where 1=cH
t+1 is the shadow value of wealth at time t + 1 of the entrepreneur
who is productive at time t9 . The expectation operator is taken over whether
bubbles survive or not (and therefore taxes are levied or not).
The equilibrium portfolio choice of bubble owners also changes in an important way. Since they are subject to taxation, the arbitrage condition forthe savers is now given by

Et

1
cLt+1

(1

t+1 )

t+1

= Et

t

1
cLt+1

(1

t+1 )

Rtd

(40)

Rtl ;

(41)

The arbitrage condition for the bank is also modi…ed:

Et

1

+

~ t+1
t+1

= Et

1

+

t+1

t

where
b
t+1 with probability~ t+1 =:

(42)

t+1 t with probability 1

Notice that when the bubble bursts, banks receive a fraction
9

Namely, it is given by
1=cH
t+1 = (1H
where Zt+1
is given by equation (33)

28

H
)Zt+1

t+1 of their origi-

nal bubble investment. This is due to a bailout payment from the government.
The portfolio balance condition for entrepreneurs di¤ers in two crucial aspects from (41) above. First of all, the state income valuations di¤er due
to the di¤erent preferences of bankers (assumed to be linear in consumption)and entrepreneurs (assumed to be logarithmic). But secondly, the risks facedby the groups of potential bubble investors are very di¤erent because of theirdi¤erent access to government bailouts. Here we assume that banks may getat least partially bailed out in the event of losses on direct bubble holdings( t+1 > 0) whereas ordinary savers will not.

4.2

Aggregation and market clearing

The employment of the productive entrepreneurs ZtH is given by (29) but

isreplaced by ~. Net worth of the productive entrepreneurs is modi…ed as

H
Zt+1
= (1

Notice that

t+1 )

"

aH (1 e)ZtH + n
)
l
H
wt
a =Rt

(1

t+1 is equal toWith probability 1,

Rtdt+1 > 0.

met t + met t+1

:

#

:

(43)

b
t+1 > 0 with probability

t+1 = 0 and

ZtL#. In this caset+1 = 0.

Similarly, the aggregate net

worth of the unproductive entrepreneurs evolves as

L
Zt+1
= (1

t+1 )

"

a (1 e)ZtH + (1
l
H
wt
a =Rt
H

n ) Rtd

ZtL

met t + met t+1(44)

Aggregate bank net worth is given by

Nt+1 =

Rtl Bt + mbt ~ t+1

29

Rtd Dt(45)

Finally the government’s budget constraint implies that taxes will be leviedon entrepreneurs in order to bail out the banks’s bubble holdings in the eventof a bust. This implies that

c
t+1 =

b
t+1 mt t(46)

where mbt t is the value of the bank’s bubble purchase in period t. The tax
rate is zero whenever the bubble survives and no bailout is needed. The otherequilibrium conditions remain the same as Section 3.
Now equations (18), (25), (28)-(32), (35), (37), (38), (40)-(46) jointly determine 16 variables Rtd ; Rtl , wt , HtH , HtL , Yt ;met ; mbt

LHt , Dt , Bt , Zt+1 , Zt+1 , Nt+1 ,

t;

L
10
H
c
H
t with three states Zt , Zt , Nt . At t = 0, Z0 is given by (33).

De…nition 2 Competitive bubbly equilibrium with government is a sequence of
1

1

L
H
; Nt+1 t=0; Zt+1
decision rules HtH ; HtL ; Yt ; Dt ; Bt ; met ; mbt t=0 , aggregate state variables Zt+1

and a price sequence

Rtd ; Rtl ; wt ;

t;

1
t t=0

and government policy

c
t+1

such

that: (i) entrepreneurs, banks and workers optimally choose decision rules
1

HtH ; HtL ; Yt ; Dt ; Bt ; met ; mbt t=0 taking the evolution of aggregate states, prices,
government policy and idiosyncratic productivity opportunities as given; (ii)the price sequence Rtd ; Rtl ; wt ;t;1
t t=0 clears the goods, labor, capital, loan,

bubble and deposit markets and (iii) governmenta taxes
ernment budget constraint given exogenousctsatis…es the gov-t ; (iv) the equilibrium evolution of1

H
L
state variables Zt+1
; Zt+1; Nt+1 t=0 is consistent with the individual choicesof entrepreneurs, banks and workers and with the exogenous evolution of productive opportunities at the individual …rm level.
10

By Warlas law one of these equations is redundant.

30

4.3

Moral Hazard, Bubble Ownership and Ampli…cation

Having described the way anticipation of bailout policy modi…es the structureof our model economy, we proceed to examine its e¤ect on the economic allocation by using numerical simulations with our calibrated model. Our focus ison two main questions. What determines banks’ownership of bubbly assets inequilibrium? How does bubble ownership a¤ect the size of the bubble-driven
boom-bust cycle?
4.3.1

Moral Hazard Comparative Statics

We start by considering the e¤ect of the …nancial safety net on the bubbly
equilibrium in which bubbles are only expected to burst with probability 0.5%
per annum (i.e. bursts happen once every 200 years). The results are presented
in Table 4 below.
Table 4: Bailouts and bank risk with low probability of bursting ( = 0:995)= 0:00

= 1=3

= 2=3

Fraction of bubbles held by banks

0.000

0.000

0.000

Bubble to GDP ratio

0.674

0.674

0.674

E(Bubble Returnjbank) - Rl11

-0.004-0.002-0.001E(Bubble Returnjsaver) - Rd

0.006

0.006

0.006

Bank NW/GDP (pre-crash)

0.057

0.057

0.057

Bank Loss/GDP

0.000

0.000

0.000

% fall in bank NW12

0.000

0.000

0.000

11

This is the expected return to a bank of holding the bubble. It may di¤er from theexpected return for the saver because of bailouts.
12
The percentage fall in bank net worth is computed after the receipt of government
assistance. In other words, in many of the more extreme scenarios considered, the banking
system would have negative net worth without a government bailout.

31

The …rst row of the table shows that the percentage of bubbles held bybanks remains close to zero even for relatively generous …nancial safety nets
(high values of ). There, the presence of …nancing constraints ensures that thebank enjoys a spread of lending over deposit rates, giving rise to a ’franchisevalue’of the bank. When ordinary savers hold bubbles which are not expected
to burst (or are expected to burst with a very low probability), the return on
bubbles and deposits must be approximately equal in equilibrium. For banks,
in contrast, the indi¤erence condition implies an equality between the loan
rate and the return on bubbles. When the spread Rl

Rd , is large enough,banks do not …nd it pro…table to hold bubbles under this value of .
We also compute the e¤ects of the bubble’s collapse on banks’capital positions. The 6th and 7th rows of Table 4 shows banks’ loss of net worth aspercentage of GDP and the percentage fall in bank net worth when the bubbles
burst. Since banks are not exposed to bubbly assets, bank losses are zero. Despite the large bubble size (around 67.5% of GDP), the bursting of the bubble
does not cause a systemic banking crisis.

32

Table 5: Bailouts and bank risk with medium probability of bursting( = 0:98)= 0:00

= 1=3

= 2=3

Fraction of bubbles held by banks

0.000

0.123

0.515

Bubble to GDP ratio

0.296

0.385

0.415

E(Bubble Returnjbank) - Rl13

-0.002

0.007

0.010

E(Bubble Returnjsaver) - Rd

0.010

0.011

0.005

Bank NW/GDP (pre-crash)

0.055

0.067

0.092

Bank Loss/GDP

0.000

0.032

0.071

% fall in bank NW14

0.000

0.469

0.779

In Table 5 above we consider the e¤ect of the …nancial safety net when theprobability of the bubble bursting is equal to 2% (i.e. the bubble’s expected
life span is 50 years). Here again banks do not hold bubbles when
Expanding the …nancial safety net (increasing= 0.

towards unity) increases the

incentive for banks to hold bubbles because it shields them from an increasing
fraction of the potential losses. The share of bubbles held by banks increases
to 51.5% as

rises to 2=3 and the size of the bubble grows from 30% to 41.5%

of GDP. As the banks’bubble holdings grow, the banking sector expands toabsorb these and take advantage of the government guarantee on its risky bubble holdings. The net worth of the banking system relative to GDP increases
from 5.5% to 9.2% asincreases from 0 to 2=3.

As banks’bubble holdings expand, bank risk grows substantially. The last
two rows of Table 5 presents some statistics measuring the impact on bankbalance sheets when the bubble bursts. We see that as the …nancial safety net
13

This is the expected return to a bank of holding the bubble. It may di¤er from theexpected return for the saver because of bailouts.
14
The percentage fall in bank net worth is computed after the receipt of government
assistance. In other words, in many of the more extreme scenarios considered, the banking
system would have negative net worth without a government bailout.

33

expands, bank losses relative to GDP increase from zero ( = 0) to 7.1% of
GDP ( = 2=3). Bank capital also experiences much larger falls during thecrisis whenis high. For example, when

= 2=3, bank capital falls by 78%.

In the event that the expected government assistance does not materialise, thebanking system would be deeply insolvent.
Table 6: Bailouts and bank risk with high probability of bursting ( = 0:965)= 0:00

= 1=3

= 2=3

Fraction of bubbles held by banks

-

-

1.000

Bubble to GDP ratio

-

-

0.254

E(Bubble Returnjbank) - Rl15--

-0.001E(Bubble Returnjsaver) - Rd

-

-

0.015

Bank NW/GDP (pre-crash)

-

-

0.057

Bank Loss/GDP

-

-

0.109

% fall in bank NW16

-

-

0.774

Finally, Table 6 presents the case of a bubble with a 3.5% probability of
bursting (expected life span of about 30 years). As is explained in Table 3,such a bubble would not be valued in the absence of a …nancial safety net.
The dashes in the table show economies in which the bubble is too risky fora stochastic steady state to exist in which such a bubble trades at a positive
price.
However, once

rises to 2=3, the equilibrium changes. The insurance avail-able to banks means that these bubbles are now su¢ ciently safe for them andthey no longer require such a high risk premium in order to hold them. This
ensures that for high values ofa stochastic steady state with bubbles exists

15

This is the expected return to a bank of holding the bubble. It may di¤er from theexpected return for the saver because of bailouts.
16
The percentage fall in bank net worth is computed after the receipt of government
assistance. In other words, in many of the more extreme scenarios considered, the banking
system would have negative net worth without a government bailout.

34

even at high values of the probability of bursting ( ). The value of the bubble(entirely held by banks) rises sharply to 25% of GDP and so does bank risk.
Potential losses reach 10% of GDP and bank capital falls by more than threequarters in the event of a crisis.
Table 6 reveals an interesting and important result. The introduction ofa …nancial safety net widens the range of bubbles that exist in equilibrium.
Fragile bubbles now exist and trade at potentially large valuations amongst
banks who hold them due to the explicit or implicit guarantees given to them bygovernments and central banks. The result that bailout anticipation increases
the valuation of risky assets is of course well-known in the literature (see forexample Kareken and Wallace (1978)). The novel contribution of our paper is
to show how risk-shifting can relax the conditions for the existence of stochastic
rational bubbles.
4.3.2

Bubble ownership and ampli…cation

In the previous subsection we showed how the …nancial safety net a¤ected bankbubble ownership and bank losses in the event of the bubble collapsing. We
now examine what e¤ect moral hazard and bubble ownership have on the reale¤ect of the bubble.
Figure 3 below compares the dynamics of the economy starting at thestochastic steady state and tracking the economy’s evolution following the
bubble’s collapse. We compare two scenarios. In one, a generous …nancial
safety net ( = 2=3) ensures that the bubble asset is held only by banks. In theother, there is no …nancial safety net ( = 0) and the bubble is optimally heldby savers while banks stay out. In each case, we set the probability that the
bubble survives into the following period in order to generate a bubble whichis exactly 20% of GDP. Therefore the di¤erence between the two scenarios
35

shown in Figure 3 is due to di¤erent bubble ownership rather than di¤erent
bubble size. The vertical axis of the …gure shows the percentage deviation ofeach variable from its no-bubble steady state value.
[Figure 3 here]
The main feature of the simulations shown in the …gure is that the economy
with more bank guarantees experiences a more volatile path for output and net
worth during the bubbly episode. While the collapse of the saver-held bubble
actually generates an expansion in output, output falls under the bank bubble.
This di¤erence comes from the degree of banks’ exposure to bubbly assets.
When bubbles collapse, banks lose a large portion of net worth, causing a sharplending contraction and an increase in the lending-deposit rate spread. This
results in a credit crunch and pushes down the investment of the productive
entrepreneurs17 .
Table 7 below tries to go deeper into the underlying mechanism which
generates the large positive real e¤ects of bank bubbles during their survival.
The table decomposes the increase in output relative to the bubbleless steadystate according to the following identity

Y = aL HtL + aH HtH

1 L
a wt HtL + HtH + aHwt

aL wt HtH

(47)

which shows that output is determined by total investment wt HtL + HtH andthe investment of productive agents wt HtH as well as the impact of the cost of
employment w1t holding investment …xed18 . We can use the aggregate resouce
constraint (37) to substitute total investment wt HtL + HtH out from (47)17

Output expands after bubbles collapse in the case of saver-held bubble because the savers
increase their own ine¢ cient production. Bursting of bubbles causes shortage of means ofsavings. Then the savers increase their production because this represents another meansof savings.
18
Other things equal, higher wages reduce output and employment because entrepreneurs
can only a¤ord to operate on a smaller scale.

36and the balance sheet identity of the productive agents in order to substitutewt HtH out from (47). The resulting expression is given by (48) below. This iswhat we base our output decomposition on.

Y =

1 L
a ( Zt + Nt )wt

aL t + aH

aL ( st Zt + Lt )(48)

ZtH =Zt represents the share of wealth in the hands of productive

Here stagents. The aL ( Zt + Nt ) is the ’liquidity e¤ect ’ stressed by Farhi and
Tirole (2011).

Zt + Nt is the economy’s aggregate saving (comprising of

total entrepreneurial saving and bank saving). Bubbles help to increase wealthand (other things equal) this tends to boost investment. The negative aL t
term is the traditional ’bubble crowding out’e¤ect: when the economy (bothsavers and bankers) hold bubbles, they hold fewer real productive assets andthis (other things equal) tends to reduce output.
The aHaL

ZtH + Lt term is the investment composition e¤ect dis-cussed by Martin and Ventura (2011a). In a Kiyotaki-Moore environment like
ours, a greater share of wealth in the hands of productive agents (st ) and
improved access to bank credit (Lt ) lead to a re-allocation of resources from
unproductive to productive agents. Even holding total investment constant,this would tend to increase output because aggregate TFP rises due to the factthat factors of production are used by more productive …rms. Finally, higher
labour costs reduce output for given investment: this is captured by the w1t
term in (48).
The di¤erent columns of the table correspond to di¤erent values ofbutin each case we vary the probability of bubble survival ( ) in order to ensurethat the bubble is always equal to 20% of GDP.19 This allows us to focus on
19

When

= 0,

= 0:978; when

= 1=3,= 0:976; when37

= 2=3,= 0:963.

the real e¤ect of bubble ownership holding bubble size …xed.
Table 7: Decomposing the real e¤ects of bubbles
=0

= 1=3

= 2=3

Bank bubble holdings (fraction of tot. bubble value)

0.00

0.11

1.00

% increase in output relative to ’no bubble’SS

1.03

1.32

3.11

(1) Liquidity e¤ect, %

19.64

20.01

21.78

(2) Bubble ’crowding out’e¤ect, %- 19.54- 19.67- 19.99(3) Investment composition e¤ect, %

0.95

1.04

1.65

(3.1.) Productive net worth, %

0.31

0.34

0.54

(3.2.) Bank lending, %

0.64

0.70

1.11

- 0.02- 0.06- 0.32...of which(4) Labour costs, %Note: Rows (1)-(4) show the percentage point contributions of various channels to thetotal increase in output relative to the bubbleless steady state.

The table shows that the larger real e¤ect of bank-held bubbles is largely
due to the fact that it generates a bigger liquidity (wealth) e¤ect and due tothe fact that it stimulates bank lending to a greater extent.
Table 8: Cost of funds (percentage point deviation from bubbleless steadystate)
=0

= 1=3

= 2=3

Bank net worth (% increase)

14.43

25.03

113.7

Bank lending to entrepreneurs (% increase)

13.03

14.35

22.69

Lending-Deposit spread (percentage points)

0.02- 0.01- 0.22

Table 8 above provides some further intuition on the reasons behind the
expansion of credit in the bank bubble case. The …rst row of the table clearlyshows that the increase in bank capital relative to the bubbleless steady state38

is much larger in the case when banks hold the bubble. This is because banks’ownership of the risky bubble asset boosts the rate of return on wealth while thebubble survives. Bubbles carry a risk premium to compensate the holder forthe risk of bursting. Conditional upon no crisis occuring, this risk premium
allows the bubble holder to expand its balance sheet substantially. Whenthe bubble holder is a bank, the expansion of its balance sheet also leads togreater credit supply. This is shown in the second row of Table 8 - lending tocorporates increases by more when banks are the bubble holders (the= 2=3column) compared to the case when savers are the holders (the …rst and second
columns).
This credit expansion lowers lending-deposit spreads (the third row of thetable) helping to boost the leverage of productive entrepreneurs and allowing
them to expand the scale of their operations. Note again, that this e¤ect is only
present when banks hold the bubble (the third column of the table). Whensavers are the main (or only) bubble holders, the expansion in bank lending ismore limited and lending spreads do not change much.

5

Conclusions

In this paper we build a model with explicit …nancial intermediation in whichrational asset price bubbles arise due to credit frictions. Our aim is to model
the interaction between banks’net worth and the value of asset price bubbles
which has been a key feature in many historical episodes of boom and bust.
This interaction turns out to be very important in understanding the real
impact of asset price bubbles on the wider economy.
We show that in the baseline version of our economy, banks have better
investment opportunity than savers. As a result, savers have a stronger in-39

centive to ’search for yield’ by holding bubbles. Nevertheless, we invoke thepossibility that in some cases, banks may be the only bubble holders due totechnological or informational constraints on savers’direct participation. We
then proceed to compare the way the bubble’s impact on the real economy
depends on who the main holder is. It turns out that the identity of the holder
is very important. When banks are the bubble-holders, this ampli…es both theboom while the bubble survives and the bust when it …nally bursts. This isbecause the bubble delivers a high return conditional on not bursting in order
to compensate for the tail risk of total loss. During the boom phase, this highreturn leads to very high pro…ts for the bubble holder. When the banks are
the recipients of these pro…ts, this helps to expand credit and boost the net
worth of other credit constrained agents. When the savers are the recipientsof these pro…ts, no such relaxation of borrowing constraints occurs and theimpact of the bubble is relatively limited.
In the second part of the paper, we explore the possibility that government
guarantees a part of banks’ bubble investments against loss. This can makebanks the natural bubble holders even in an environment of full participationin the bubble market by all agents. When bailouts are expected, the banking
system expands rapidly to exploit its advantage in holding the bubble, bidding
up its price very substantially. This subsidy to bank risk-taking leads to adramatic increase in the riskiness of bank balance sheets: bank capital falls
very sharply when the bubble eventually bursts. Finally, banks’ diminishedsensitivity to risk increases the range of parameter values for which bubbly
equilibria exist. In particular, bank-held bubbles with a very high probabilityof bursting exist only when the …nancial safety net is su¢ ciently generous.

40

6

Appendix A: Data Sources

In this section we provide details of the sources of the data used for calibrating
the model. This is given in Table A1 below:
Table A1:
Theor. concept

Data concept

Real bank loan rate

Real prime loan rate-GDP FRB, Table H.15,
growth-costs

Real deposit rate

Source

FDIC, BEA

Real M2 own rate - GDP FRED
growthExpected in‡ation

Average CPI in‡ation (All FRED
Urban Consumers)

Expected real GDP growth

Average real GDP growth FRED
(chained measure)

Deposit stock

M2

FRED

Nominal GDP

Nominal GDP

FREDBank leverage

Bank Debt Liabilities/Bank FRB, Table H.8
Net WorthAverage corporate leverage

Corporate Debt/Corporate Welch (2004)
Net Worth

Leverage of indebted corpo- Debt/Corp Net Worth for Welch (2004)
rates

high leverage corporates

Bank rate of return on eq- Bank rate of return on eq- FDIC
uity

uity

41

7

Appendix B: Computational procedure

In this Appendix, we outline the computational procedure we use to solve forthe stochastic steady state in the general case of full participation in the bubble
market for all agents.
(1) Start by solving the deterministic steady state of the model in theabsence of bubbles. This is where the economy will converge to after the bubble
bursts because we do not consider any fundamental shocks in our simulations.
(2) We have one signi…cant forward looking variable in the modelt and

we need to characterise the way it evolves after the bubble bursts. This will
give us an expected value oft conditional on bursting. We start by guessing

such a path.
(3) Solve the system of equations that de…nes the stochastic steady state,taking the future assumed path oft as given. We use Matlab’s own non-linear

equation solver fsolve.m in order to do this.
Within step (3) we need to determine which regime our economy operates
in. In particular we check whether (a) only savers hold bubbles, (b) only banks
hold bubbles or (c) both hold bubbles. In addition we need to check whether
unproductive agents are active producers in equilibrium20 . We do this by trial
and error:(3.1) First we check who holds the bubble(a) First we try to solve the model under the assumption that only savers
hold bubbles. This means that savers’s …rst order condition determines theequilibrium growth rate of bubbles conditional upon survival. Once we solve
the model under this assumption we check whether banks would like to invest
in bubbly assets on the margin by examining the di¤erence in the expected
20

There are a multitude of other regimes the economy can operate in, depending onwhether the banks’ and the entrepreneurs’ borrowing constraints bind or not. We only
consider parameter values for which both of these constraints bind.

42

utility for bankers from investing in one unit of bubbles and loans. If we …ndthat banks stictly prefer to invest in loans on the margin, this con…rms ourassumption that only savers buy bubbles and we move on to step (4). If we…nd that banks strictly prefer to invest in bubbles, then not only savers holdbubbles in equilibrium and we need to try a di¤erent assumption.
(b) Only banks hold bubbles. This means that banks’s …rst order condition
determines the equilibrium growth rate of bubbles conditional upon survival.
Once we solve the model under this assumption we check whether savers would
like to invest in bubbly assets on the margin by examining the di¤erence in the
expected utility for savers from investing in one unit of bubbles and deposits.
If we …nd that savers stictly prefer to invest in deposits on the margin, thiscon…rms our assumption that only banks buy bubbles and we move on to step(4). If we …nd that savers strictly prefer to invest in bubbles, then not only
banks hold bubbles in equilibrium and we need to try a di¤erent assumption.
(c) Both hold bubbles. This means that both banks and savers hold bubbles
and each holds a strictly positive fraction of the total bubble stock.
(3.2) Check whether the low productivity agents produce in equilibrium(a) We solve the model, assuming that the deposit rate is equal to theunleveraged rate of return on the low productivity technology. We then checkif the investment of the low productivity …rms is positive. If not we reject thisequilibrium and assume that low productivity agents are inactive as producers.
(b) We solve the model, assuming that the employment (investment) of
low productivity entrepreneurs is zero. We check the resulting equilibrium
to ensure that the deposit rate is higher than the unleveraged rate of returnon the low productivity technology. If this condition holds, this con…rms our
initial assumption that saver entrepreneurs prefer not to produce.
(4) Use the output in (3) to compute the initial conditions of the econ43

omy when the bubble bursts. We then solve for a perfect foresight path thatconverges to the no-bubble deterministic steady state, following the bubble’s
collapse. We do this by iterating on thet path after the bubble’s collapse

until it converges(5) Return to step (1) and keep iterating until the entire path oft (includ-ing its stochastic steady state value) has converged to within error tolerance.

44References
Aoki, K., Benigno, G., Kiyotaki, N., 2009. Adjusting to capital liberalization.
Working Paper, LSE.
Arce, O., Lopez-Salido, D., 2008. Housing bubbles. Banco de Espania Working
Paper No. 0815.
Bernard, A., Eaton, J., Jensen, B., Kortum, S., 2003. Plants and productivity
in international trade. American Economic Review 93(4), 1268–1290.
Caballero, R., Krishnamurthy, A., 2006. Bubbles and capital ‡ow volatility:
Causes and risk management. Journal of Monetary Economics 53, 35–53.
Farhi, E., Tirole, J., 2011. Bubbly liquidity. Review of Economic Studies forthcoming.
Gertler, M., Karadi, P., 2009. A model of unconventional monetary policy.
Working Paper.
Gertler, M., Kiyotaki, N., 2010. Financial intermediation and credit policy inbusiness cycle analysis. In: Woodford, M. (Ed.), Handbook of Monetary
Economics. Elsevier.
Hirano, T., Yanagawa, N., 2010. Asset bubbles, endogenous growth, and …nancial crisis. Working Paper, University of Tokyo.
Kareken, J., Wallace, N., 1978. Deposit insurance and bank regulation: A
partial equilibrium exposition. Journal of Business 51, 412–438.
Kocherkalota, N., 2009. Bursting bubbles: Consequences and cures. Working
Paper, Federal Reserve Bank of Minneapolis.

45Martin, A., Ventura, J., 2011a. Economic growth with bubbles. American Economic Review forthcoming.
Martin, A., Ventura, J., 2011b. Theoretical notes on bubbles and the current
crisis. IMF Economic Review 59 (1), 6–40.
Nikolov, K., 2010. A model of borrower reputations as intangible collateral.
Working paper, LSE.
Reinhart, C., Rogo¤, K., 2008. This time is di¤erent: A panoramic view ofeight centuries of …nancial crisis. Working paper, University of Maryland.
Sargent, T., 1987. Dynamic Macroeconomic Theory. Harvard University Press,
Cambridge, USA.
Schularick, M., Taylor, A., 2012. Credit booms gone bust: Monetary policy,
leverage cycles and …nancial crises, 1870-2008. American Economic Review
forthcoming.
Syverson, C., 2004. Market structure and productivity: A concrete example.
Journal of Political Economy 112 (6), 1181–1222.
Tirole, J., 1985. Asset bubbles and overlapping generations. Econometrica
53 (6), 1499–1528.
Welch, I., 2004. Capital structure and stock returns. Journal of Political Economy 112 (1), 106–131.

46

0.9
0.8
0.7

λ

0.6
0.5
0.4
0.3
0.2
0.1
0.1

0.2

0.3

0.4

0.5

0.6

0.7

θ

Figure 1a. Deposit rate less than one (red area)

47

0.8

0.9

0.9
0.8
0.7

λ

0.6
0.5
0.4
0.3
0.2
0.1
0.1

0.2

0.3

0.4

0.5

0.6

0.7

θ
Figure 1b. Lending rate less than one (red area)

48

0.8

0.9

Output

Bank capital

1.035

2.4

1.03

2.2
2

1.025

1.8

1.02

1.6
1.015
1.4
1.01
1.2
1.005

1

1

0.8

0.995

0.6

0.99

0.4
1

11

21

31

Saver bubbl e

41

1

11

Bank bubbl e

21

Saver bubbl e

Loans

31

41

Bank bubbl eAggregate investment

1.25

1.03

1.025
1.2
1.02
1.15
1.015

1.1

1.01

1.005
1.05
1
1
0.995

0.95

0.99
1

11

21

Saver bubbl e

31

41

1

Bank bubbl e

11

21

Saver bubbl e

31

41

Bank bubbl eFigure 3: Comparing a bank-held (solid line) and a saver-held (dashed line) bubble

49