Bubbles, Banks, and Financial Stability
Kosuke Aoki (University of Tokyo)
Kalin Nikolov (European Central Bank)VERY PRELIMINARY
April 2011 [First version: November 2010]

Abstract
This paper asks two main questions:(1) What makes some asset price bubbles more costly for the real
economy than others?
(2) What does theory have to say about the appropriate early warning indicators of dangerous bubbles?
We construct a model of rational bubbles under credit frictions andshow that when bubbles held by banks burst this is followed by a costly…nancial crisis. In contrast, bubbles held by ordinary savers have relatively muted e¤ects. Bank intermediated bubbles also lead to a strong
expansion of money and credit, which explains why these variables areuseful though noisy early warning indicator of future crises.

1

Introduction

The last decade has seen the dramatic rise and fall of world-wide housing
prices, culminating in the …nancial crisis and ’Great Contraction ’ of 20081

2009. This brought the …nancial system to the brink of collapse and led tounprecedented o¢ cial intervention. Currently, politicians and central bankers
are busy putting the foundations of a new macro-prudential policy framework
which is meant to make the …nancial system more stable and less prone to thekind of boom-bust cycle we experienced over the last …ve years.
Motivation and economic questions As Reinhart and Rogo¤ (2008) document in detail, there are many episodes of boom-bust …nancial cycles but not
all of them result in a costly economic contraction. Some boom-bust cycles,
such as those in Japan and the Scandinavian countries in the 1990s, and thesubprime crisis of 2007-2009, led to banking crisis and a serious recession. But
on other well known occasions such as the 1987 crash or the dot-com bubbleof 1999-2000, the collapse of asset prices did not result in a banking crisis anda severe contraction of real economic activity. Figure 1 below illustrates the
puzzle. Panel A of the …gure compares the decline from peak of the S&P 500
during 2000-03 period with the fall of the value of the CDS on a AAA-rated
RMBS since 2007. Panel B compares the behaviour of GDP growth over thetwo periods. The message of the …gure should be clear. The ’dot com ’crash
was of a similar magnitude to the ’subprime crisis ’ while its output e¤ects
were small in comparison.
[Figure 1 here]
Why did the dot-com bubble not lead to a serious banking crisis while thesubprime bubble did? Should policy react to any sharp increase in asset prices
or are there occasions when the market can be left safely to its own devices
even when …nancial prices look to have departed from fundamentals? These
are the main questions we ask in our paper. Some policy makers (most notably
Mishkin (2008) and Mishkin (2009)) have argued that we should only worry

2about bubbles generated within the banking system. This view receives some
support from the literature on ’early warning indicators of crisis’(Borio andLowe (2002), Borio and Drehmann (2008), Alessi and Detken (2009) as wellas many others). The literature shows that an asset price boom is far more
likely to result in a costly output collapse when it is accompanied by a large
increase in money, credit and bank leverage. In this paper, we formalize thisidea and show that who owns bubbly assets indeed matters for …nancial andeconomic stability.
Model description In order to analyze the questions addressed above, weconstruct a model in which both banks and entrepreneurs are subject to credit
frictions. Entrepreneurs di¤er in their productivity levels, and those withhigher productivity become borrowers (and vice versa) in equilibrium. Following Kiyotaki and Moore (1997) they are subject to a collateral constraint
when they borrow. Following Gertler and Karadi (2009), the amount of deposits a bank can collect depends on its net worth. When those credit frictions
are severe enough, the interest rate is suppressed and bubbles can be traded
once expectations are coordinated. Using this model, we compare the case inwhich banks hold bubbles and the case in which savers hold bubbles directly.
While not modelled explicitly, we interpret bubbles held by banks as indivisible
large bubbly assets, such as commercial real estate bubbles. Since it is muchlarger than a typical savings of savers, individuals cannot a¤ord to buy it, butbanks can do so by pooling savings of individuals. Bubbles held by individual
savers can be interpreted, for example, bubbles attached to equities. Thoseare divisible and savers can buy them.
Results The main point of our paper is to formalise the intuition that asset
bubbles held by banks (sometimes known as ’credit bubbles’) are more dan3

gerous than bubbles held by other investors who are less central in the credit
allocation mechanism (we refer to these as ’saver-bubbles ’). When bank-held’credit bubbles’burst, banks become insolvent and need to be rescued by thegovernment. The fall in their net worth causes a severe credit crunch andoutput collapse. In contrast, the e¤ects of asset price busts on real activity are
milder when other investors/savers directly hold bubbles. When savers holdbubbles it is those savers who su¤er from capital loss. But because the net
worth of savers is not central to the e¢ ciency of …nancial intermediation, thecosts of the bubble collapse remain private rather than ’systemic ’. Borrowers
and other lenders (the banks) do not su¤er as a result of savers ’ losses. In
contrast, when banks hold bubbles, the bursting of bubbles directly hit thebanks’net worth, possibly leaving the banking system insolvent without government intervention. Because banks have a ’special ’ place in the …nancial
system, this fall in bank net worth results in a ’systemic crisis’, a credit crunch
and a sharp decrease in investment and output.
Our paper also provides a theoretical justi…cation for the usefulness ofmoney, credit and bank leverage in predicting future crises. The emergence ofall bubbles leads to a big boost in corporate net worth and demand for credit.
As a result, the economy experiences a dramatic expansion of credit and bankleverage, consistent with the empirical evidence. Nevertheless, we also showthat within our model it is di¢ cult for a policy-maker to tell with con…dence
whether a bubble will result in a …nancial crisis or not. This is because money
and credit expand more rapidly during the build up phase of a bank-held (asopposed to a saver-held) bubble but the di¤erences are not large enough toallow for accurate forecasting. Our model implies that crisis prediction using
aggregate data will remain an uncertain exercise.
In the …nal part of the paper, we explore the link between …nancial liberal4isation and the tendency of banks to invest in bubbly assets. History contains
many examples in which deregulation has lad to the growth of non-bank …nancial intermediation and a decline in the pro…tability of traditional lending anddeposit taking activities. Very often banks have reacted to such developmentsby trying to branch out into alternative lines of business with disastrous consequences. To model such a situation, we extend our framework by allowing
direct intermediation via a ’corporate bond market ’ and examine banks ’reaction to the growth of non-bank lending. We …nd that banks are much
more likely to invest in bubbly assets following such a ’…nancial liberalisation’. This explains why …nancial innovations that rapidly expand the total supplyof credit can eventually lead to a systemic …nancial crisis.
Literature review Motivated by the recent global economic stagnation following the subprime crisis of the US, there is growing literature on models of
economic ‡uctuations that emphasize the role of credits. Much literature incorporates various forms of credit frictions and how those frictions amplify the effects of technology and/or …nancial shocks (Kiyotaki and Moore (2009), Christiano et al. (2010), Gertler and Karadi (2009), Gertler and Kiyotaki (2010)).
The literature …nds the importance of credit shocks (shocks to net worth ofborrowers or banks) but it is not easy to identify what they are in reality. An
example of shocks that change the value of …rms and bank net worth used in
the literature is a shock that makes …rms capital obsolete (Gertler and Karadi(2009), Gertler and Kiyotaki (2010)). However, it is not very obvious how
such shocks indeed occured during every boom-bust cycles. Instead, following
Martin and Ventura (2010), our explanation of the crisis is based on changes
in investor expectations rather than changes in technology and/or …nancial
shocks. Collapse of bubbles in our model serves as credit shocks.

5Our paper contributes to the recent growing literature on rational bubbles
under credit frictions, pioneered by Ventura (2010) and subsequent work includes Caballero and Krishnamurthy (2006), Kocherlakota (2008), Arce and
Lopez-Salido (2008), Martin and Ventura (2010), Farhi and Tirole (2010), Hirano and Yanagawa (2010). Ever since the seminal work of Tirole (1985), the’rational bubbles’literature has been very interested in the question of whether
bubbles are expansionary for aggregate economic activity or not. The traditional view was that bubbles replace excessive investment and therefore havea contractionary impact on total output. Subsequent papers have shown thatwhen there are credit market imperfections, bubbles may have an expansionarye¤ect through a variety of mechanisms that determine entrepreneurs’current
net worth and access to leverage.
In Martin and Ventura (2010) the expansionary e¤ect of bubbles arises
because the anticipated pro…ts from future bubble sales are collateralisable
and allow entrepreneurs to increase borrowing in the current period. As a result more production to be undertaken by the most productive entrepreneurs,
thereby increasing aggregate TFP. In Farhi and Tirole (2010) bubbles increaseinterest rates and actually reduce the leverage available to borrowing entrepreneurs through what Farhi and Tirole (2010) call the ’competition e¤ect ’. Thisis negative for investment. However there is a positive ’liquidity e¤ect ’. Whenentrepreneurs need a means of saving in between investment opportunities, theincrease in interest rates makes them richer when the investment opportunity…nally comes.
Our model contains some of the channels discussed in the literature as well
as some novel ones. We have a ’liquidity e¤ect ’because bubbles enhance therate of return of those saving in anticipation of future investment opportunities.
We also have a ’competition e¤ect ’though it is somewhat less prominent than6

in Farhi and Tirole (2010) because bubbles simultaneously increase interest
rates and reduce production costs (real wages in our case).
The new channels we introduce arise due to the presence of credit constrained …nancial intermediaries in our model. This o¤ers several alternativeand complementary mechanisms through which bubbles generate lending andoutput booms. In our model, limited …nancial market participation is key because it allows banks to borrow at the deposit rate in order to issue loans (orbuy bubbles) whose rate of return is higher than the deposit rate. Following
Gertler and Karadi (2009) the net present value of such spreads (the franchisevalue of banks) is collateralisable and therefore changes in the spread increasebanks’ability to collect deposits. During a bubbly episode, the net worth ofborrowing entrepreneurs rises as they sell bubbles, increasing loan demand andpushing up loan-deposit spreads.As a result, the value of the bank (including
the franchise value of future spreads) increases, leading to a rapid expansionof deposits. Thus, lending to entrepreneurs increases even in the case in whichbubbles compete with ’real loans ’in banks ’portfolios. This channel is similarin spirit to the collateralisability of pro…ts from future bubbles sales discussed
in Martin and Ventura (2010) but the mechanism is di¤erent because it relieson the expansion of bank rather than corporate balance sheets. Our approachis complementary to theirs and, we believe, particularly useful for analysing…nancial stability issues.
Plan of the paper Section 2 introduces the economic environment, section
3 describes the bubble-free equilibrium and discusses the conditions for the
existence of bubbles. Section 4 describes the bubbly equilibrium and uses acalibrated version of the model to discuss the e¤ect of bubbles ’emergence andcollapse on …nancial stability. Finally, section 8 concludes.

7

2

The Model

The economy is populated with three kinds of agents. There are 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), giventhat the economy starts with such population distribution. We assume thatthe probability of the productivity shifts is not too large:+ n < 1:

8

(2)

This assumption implies that the productivity of each agent is persistent.
Entrepreneurs are ex-ante identical and have log utility over consumption
streams

U = Et

1
X

t

(3)

ln ct

t=0

Entrepreneurs purchase consumption (ct ), bubbles (mt ) at pricetandbonds bt . They also pay wage bills wt ht in order to receive future revenues
ai ht . Here wt and ht denote real wage and labor respectively.

ct + wt ht + mt t

bt = ai ht 1 + mt 1 t

Rti 1 bt 1

(4)

where Rti is the interest rate which is equal to the loan rate Rtl when theentrepreneur is a borrower and Rtd when the household is a saver.
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 afraction of the value of the …rm can be seized by creditors. Hence the collateral
constraint is given by:
Rtl bt 66yt+1
ai hit =wt ;

0<

<1

(5)

They maximize (3) subject to (4) and (5).

2.2Workers

Unlike the entrepreneurs, the workers do not have production technology nor
any collateralizable asset in order to borrow. They maximize the following
utility9

U = Et

1
X

t

cw
t

t=0

h1+
t
1+

(6)

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

Rtd 1 bwt 1;

(7)

here superscript ‘w’ stands for ‘workers’. In equilibrium, it is shown thatworkers do not save because the equilibrium interest rate is low. Therefore
they consume their labor income in each period.

2.3

Banks

We assume that savers cannot directly lend to borrowers and that lending isdone by banks. Bankers are risk neutral and live for a stochastic length oftime. Once bankers receive an “end of life” shock, they liquidate all theirasset holdings and consume all of them before exiting. This shock hits withprobability 1:

Banks maximize the following objective function:

V (nt ) = cbt + Et [ V (nt+1 ) + (1) nt+1 ](8)

subject to a number of constraints explained below.
In each period the bank has net worth (nt ). It collects deposits (dt ) fromthe savers. Then it lends to the borrowers (lt ), purchases bubbles ( t ), or
consumes (cbt ). Therefore its balance sheet is given by
cbt + lt +t mt = nt + dt :

10

(9)

The evolution of net worth is given by

nt+1 = Rtl lt +t+1 mtRtd dt :(10)

Following Gertler and Karadi (2009), we model banks subject to limited commitment. More speci…cally, the banker may divert 1

fraction of deposits.

Once he diverts, he will close his bank and the savers can retain the remaining
fraction of deposits. Since the savers recognize the banker’s incentive to divert funds, they will restrict the amount of deposit. Those assumptions imply
the following borrowing constraint(1)dt 6 V (nt ) :(11)

The left hand side of equation (11) 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 (8) subject to (9), (10) and (11).

3

Equilibrium without bubbles

Before characterizing an equilibrium with bubbles it is informative to characterize the equilibrium without bubbles. In this section we set

t = 0 at all

times.

3.1

Optimal behavior

The entrepreneur has a few choices of accumulating net worth. Let Rt (at ) be
the maximum rate of return on the net worth from time t to t + 1 for the11

entrepreneur with productivity at = ah ; aL : Then it is given by
Rt (at ) = max Rtd ;at at (1);
wt wt
at =Rtl:(12)

The …rst term in the right hand side is the deposit rate. The second term is
the rate of return of bubbles. The third term is the rate of return on production without borrowing. The last term is the rate of return on production
with maximum borrowing. By borrowing from banks secured byfraction of

output, the entrepreneur can …nance externally at =Rtl amount (equation (5)).
Therefore the denominator is the required downpayment for the unit laborcost. The numerator is the output after repaying debt.
Note that the last two rates of return in equation (12) are strictly higher
for the productive entrepreneur than the unproductive entrepreneur, and thedeposit rate and the rate of return of bubbles are the same for both. Therefore
in equilibrium the unproductive entrepreneurs supply deposits and produce if
and only if their rate of return of production is equal to the deposit rate. We
focus our analysis on such case, namely1 ,Rt (aL ) = Rtd =

wt

:

(13)

Intuitively, the borrowing constraints are tight enough so that the productive
entrepreneurs cannot absorb all national saving. Then there is not enoughdemand for deposits. In such case the savers use both bank deposits and its
own production technology to accumulate wealth.
The productive entrepreneurs borrow and produce, and their rate of return
1

In Appendix XX we derive the condition for this to hold in the neighborhoold of thesteady state without bubbles.

12

is given byaH (1)= Rtl :
Hwt
a =RtlRt (aH ) =(14)

Given the optimal choice of accumulating wealth, the budget constraint (4)can be written as
ct );

(15)

i = d; l(16)

zt+1 = Rt (at )(zt
where
Rti 1 bt 1 ;zt = ytdenotes the net worth of the entrepreneur at time t: Positive bt implies that
he borrows and the lending rate Rtl applies to his debt. Similarly negative bt
represents deposit and Rtd applies.
Since utility function is logarithmic, consumption decision is given by

ct = (1) zt :(17)

When Rt (aH ) > Rtl the productive entrepreneurs produce with their borrowing
constraint binding. From (5) and (4) their employment is given by

ht =

zt
:
aH =Rtlwt(18)

Regarding the workers, their labor supply hst is given by
hst = wt

(19)

They will not save and consume all their labor income when

Rtd <

13

1

:

(20)

Later we will verify this is true in the neighborhood of the steady state equilibrium.
Finally, let us characterize the bank. When Rtl > Rtd , then credit constraint(11) binds and consumption is postponed until death2 . Guess that the valueof the bank is a linear function of net worth nt

V (nt ) =

Here

(21)

t nt

t can be interpreted as the bank’s leverage.

Then, with equation (11)binding, deposit is given by

dt =

t

1

By substituting (21) and (22) into (8),

t =

1

(22)

nt :

t satis…es

(1

)+

t+1

Rtl

(1

)+

t+1

Rtl Rtd1Note that the above formulas show that:t increases when(23)
t+1 increases.

This implies that the current leverage depends on the future franchise value ofthe bank which is re‡ected by the leverage next period.3 It also shows thatis an increasing function of the spread Rtl

3.2

t

Rtd :

Aggregation and market clearing

Let ZtH and ZtL respectively denote aggregate wealth of the productive andunproductive entrepreneurs. Then we can characterize the aggregate equilibrium as follows. From (18) The aggregate employment of the productive
2
3

Need to show the conditions under which this is true.
See Nikolov (2010), who considers a similar problem for …rms.

14

entrepreneurs is given by

HtH =

ZtH
aH =Rtlwt(24)

When (13) holds, the unproductive entrepreneurs are indi¤erent between making deposits and producing, thus their aggregate saving is split as follows

HtL = ZtL

Dt

(25)

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

Dt =Notice that 1

t

(1

)

Nt :

(26)

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 = Lt :(27)

Let us turn on 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 (14) and (13). Therefore net worth of the
productive entrepreneurs evolves from (14), (15) and (17) as

H
Zt+1
= (1

)

aH (1
)
Z H + n Rtd ZtL :wt
aH =Rtl t

15

(28)

Similarly, the aggregate net worth of the unproductive entrepreneurs evolves
as
L
Zt+1
=

aH (1
)
Z H + (1Hwt
a =Rtl tn )Rtd ZtL :(29)

From aggregating production function, aggregate output is given by

Yt = aH HtH 1 + aL HtL 1 :(30)

Finally, aggregate bank net worth is given by

Nt+1 =

Rtl +

t

Rtl
(1

Rtd
)

!

(31)

Nt

The markets for goods, labor, capital, loan and deposit must clear. Goods
market clearing implies that aggregate saving must equal to aggregate investment.
(ZtH + ZtL ) + Nt = w(HtH + HtL )(32)

From (19), labour market clearing implies

wt = HtH + HtL :(33)

De…nition 1 Competitive equilibrium without bubbles is a sequence of decision
rules

1

HtH ; HtL ; Yt ; Dt ; Lt t=0 , aggregate state variables

and a price sequence

Rtd ; Rtl ; wt ;

1
t t=0

1

H
L
Zt+1
; Zt+1; Nt+1 t=0such that: (i) entrepreneurs, banks
1

and workers optimally choose decision rules HtH ; HtL ; Yt ; Dt ; Lt t=0 taking theevolution of aggregate states, prices and idiosyncratic productivity opportunities as given; (ii) the price sequence Rtd ; Rtl ; wt ;1
t t=0 clears the goods, labor,

capital, loan and deposit markets and (iii) the equilibrium evolution of state
1

HL
variables Zt+1
; Zt+1; Nt+1 t=0 is consistent with the individual choices of en-

16

trepreneurs, banks and workers and with the exogenous evolution of productive
opportunities at the individual …rm level.
In equilibrium, equations (13), (23)-(33) jointly determine 12 variables Rtd ;Rtl , wt , HtH , HtL , Yt ;HL
Ht , Dt , Lt , Zt+1 , Zt+1 , Nt+1 , given the state variables Zt ,ZtL , Nt . Since the analytical solution is complicated and not very informative,
we discuss the properties of the steady state based on numerical simulations.
The parameter values we use are in line with Aoki, Benigno and Kiyotaki(2009) and are discussed in more detail in Appendix X.

3.3

Existence of steady state bubbles

It is useful to characterize the deposit rate Rtd and loan rate Rtl in the steadystate without bubbles and discuss when bubbles can circulate. In the steadystate, all 12 endogenous variables are constant. Credit frictions suppress theinterest rates and those rates are lower than1

when the credit constraints

bind.4 Similarly to Farhi and Tirole (2010), whether a bubbly steady state
exists and who owns bubbles depend on whether the two interest rates arelower than the growth rate (which we assume is equal to 1) in the ’no bubbles’
steady state.
In our economy, the severity of credit frictions is represented by two parameters,

and : Figure 2a 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 of4

andlow produc-

See Aoki et al. (2009)) for the general discussion of the relationship between the interest
rate and credit frictions.

17

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(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
2b that the conditions for the existence of bubbles is satis…ed at intermediatelevels of …nancial development.
[Figure 2a here]
The red area of Figure 2b shows the region in which the loan rate is lessthan one. Then the banks have an incentive to buy bubbles. Since the deposit
rate is always lower than the loan rate, the savers also have incentive to holdbubbles at these parameter values. It is natural that the part of the parameter
space where banks bubbles can exist is more limited compared to the partsof 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 where bank bubbles exist is subset of thespace where savers have an incentive to invest in bubbly assets.
[Figure 2b here]
If both the loan and deposit rate are lower than one, we consider twocases. Firstly, we allow only banks to hold bubbles. In other words we assume
limited participation. Even though not modelled explicitly, what we havein mind is the situation in which bubbles are attached to large indivisible
assets such as commercial real estate. In such a case, individual savers cannota¤ord to buy bubbles because their savings are too small. However, the banks
could buy bubbles by pooling savings from individual savers. Thus the pooling18

of small depositors’ savings is one of the fundamental functions of …nancial
intermediaries in our model. Another story we could tell is that bubbles are
sometimes attached to assets which are not easy for individual savers to trade,due to transaction costs for example. Again, only banks will hold bubbles insuch an environment.
Secondly, we allow both banks and savers to buy bubbles. This corresponds
to a situation in which bubbles are attached to more divisible and standardized
assets, such as equities. Then individual savers can a¤ord to buy the bubble.
It turns out that in such a steady state, only savers hold bubbles while banks
stay out. This is because savers’ opportunity cost of holding bubbles is thedeposit rate while the bank’s opportunity cost is the loan rate which is higher
than the deposit rate. In the steady state equilibrium we consider, the rate of
return of bubbles is equal to the deposit rate, so the savers crowd out banksfrom bubbly asset markets.

4

Equilibrium with bubbles

4.1

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
behaviour 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 and19set 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;(2) the real deposit rate minus the growth rate of real GDP; (3) commercialbank leverage; (4) average corporate leverage; (5) average leverage for highly
leveraged corporates; (6) the rate of return on bank equity and (7) the ratioof M2 to GDP. Full details of data sources and construction are available inAppendix A. Table 1 below presents the values of the parameters chosen to
match the moments.

20

Table 1: Baseline calibration
Parameter Value
0.177
n

0.039

aH =aL

1.100
5.000
0.626
0.765
0.867
0.946

Table 2 below presents the moments in the model and the data.
Table 2: Model and data moments

4.2

Moment (Model concept)

Data

Model

Real loan rate - real GDP growth (Rl )

0.950

0.983

Real loan rate - real GDP growth (Rd )

0.998

0.997

Ratio of M2 to GDP (D=Y )

0.500

0.465

Bank leverage (D=N )

10.00

10.00

Average corporate leverage (L=Z)

0.500

0.530

Leverage of indebted corporates (L=(sZ))
2.000
l
d
(R R )
Bank rate of return on equity (Rtl + t (1 t ) t ) 1.150

2.000
1.154

Competitive equilibrium with bubbles

When the banks hold bubbles, this must imply that they are indi¤erent between bubbles and loans
t+1= Rtl ;

t

21

(34)

otherwise, either bubbles do not circulate or lending becomes zero.5 When(34) holds, the banks value function

t is still given by equation (23) because

bubbles and loans are perfect substitutes to them. For the same reason, thetransition equation of the aggregate bank net worth remains the same as (31).
Without loss of generality, we normalize the aggregate supply of bubbles
equal to one. Then the aggregate value of bubble is equal tot . The balance

sheet of banks (equation (27)) is now given by

Dt = Nt = Lt +

(35)

t:

Since a part of national savings is invested in bubbles, the goods market clearing (saving = investment) is modi…ed as(ZtH + ZtL ) + Nt = w(HtH + HtL ) +

(36)

t:

The other equilibrium conditions remain the same as Section 3.
Next, let us discuss the initial period when bubbles show up. We assume
that the productive entrepreneurs will create bubbles. Suppose that bubbles
0 show up at time t = 0. This is pure gain for the productive entrepreneurs.

Therefore their net worth equation (28) is given by

Z0H = (1

)

aH (1
w 1

)
aH =RlZ H1 + n Rd 1 Z L1 +

0:

(37)

1

They sell bubbles to …nance employment. Now equations (13), (23)-(26), (28)(31), (33), (34)-(36) determine 13 variables Rtd ; Rtl , wt , HtH , HtL , Yt ;H
L
Lt , Zt+1, Zt+1, Nt+1 ,t , Dt ,HLH
t with four states Zt , Zt , Nt : At t = 0, Z0 is given by

(28).
5

As is discussed in the previous section, in this case we are prohibiting the entrepreneurs(and workers) from buying bubbles.

22

De…nition 2 Competitive equilibrium with bubbles is a sequence of decision
rules

1

HtH ; HtL ; Yt ; Dt ; Lt t=0 , aggregate state variables

and a price sequence Rtd ; Rtl ; wt ;

t;

1

L
H
; Nt+1 t=0; Zt+1
Zt+1

1
t t=0 such that: (i) entrepreneurs, banks
1

and workers optimally choose decision rules HtH ; HtL ; Yt ; Dt ; Lt t=0 taking theevolution 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 evo1

LH
; Nt+1 t=0 is consistent with the individual; Zt+1lution of state variables Zt+1

choices of entrepreneurs, banks and workers and with the exogenous evolutionof productive opportunities at the individual …rm level.
As many other models of rational bubbles, our economy has many kinds ofbubbly equilibria depending on agents’expectations. Our strategy is to look ata bubbly equilibrium that can at least qualitatively explain boom-burst cycles
we observed in reality. Much literature on economic ‡uctuations search shocks
such as productivity and credit shocks that can realistically explain data once
those shocks are put into DSGE models. Conceptually we are doing a similar
exercise but instead of fundamental shocks we are searching for expectational
shocks (such as investor sentiments).

5

Banks’Bubble Holdings and Financial Stability

In this section we characterize the dynamics of the economy in which bubbles
circulate. One of the key questions of our paper is how the impact of asset pricebubbles on …nancial stability depends on who holds the bubble. So in the next
subsections we examine the e¤ects of the emergence and bursting of di¤erent

23

bubbles. In all experiments we assume that the currently productive agents areendowed with intrinsically useless ’zero dividend ’assets. We assume that themodel is initially in a steady state in which investor sentiment regarding thefuture resaleability of these assets is pessimistic and so they have zero market
value. In addition, we assume that investor sentiment suddently changes andthe ’bubble ’asset starts to trade at a positive value.

5.1

The emergence and bursting of a bank-intermediatedbubble[Figures 3a and 3b here]
In our …rst experiment (described in Figures 3a and 3b above), we consider
a situation in which investor sentiment shifts in favour of indivisible assets thatcan only be purchased by banks that pool the savings of many di¤erent small
savers. Investor sentiment remains positive for ten periods and then turns
negative again. At this point the bubble collapses. All the above events occur
in a wholly unexpected (one time shock) fashion.
When the bubble …rst appears, productive entrepreneurs become very richbecause they create and sell bubbles. This represents a pure wealth gain, and,because collateral constraints continue to bind under small enough bubbles,
productive agents leverage up their increased net worth to raise borrowing
and employment. Initially, banks’net worth is limited and this restricts theamount of loans they can supply while also purchasing bubbles from productive entrepreneurs. Therefore, the lending rate increases sharply, and, in order
to compete with the loan rate, the bubble grows rapidly over time. For one
period, banks make a huge pro…t due to the increased spread between the loan
rate and deposit rate. In turn, this rise in current and expected future prof-24

itability increases the franchise value of the bank (represented byt ), relaxes

the bank’s collateral constraint and leads to a sharp increase in leverage. So
the appearance of the bubble and the associated sharp rise in bank pro…tability
and leverage allows banks to raise a lot more deposits and …nance an increasein both lending and bubble holdings. Despite the fact that the bubble has tocompete with loans in banks portfolios, its appearance leads to a ’crowding in ’e¤ect, which increases lending and investment in two ways. One is through theincrease in investor net worth, leading to higher corporate borrowing capacity.
The second is through the increase in the franchise value of the banks, leading
to higher loan supply.
In subsequent periods, higher bank pro…ts increase bank capital and allow
for a rapid expansion of lending as the loan rate and bank leverage go down.
As the productive entrepreneurs expand their employment, the employment of
the unproductive entrepreneurs is crowded out. This improves the aggregatee¢ ciency of the economy and TFP increases. As a result, output expands.
Thus, similar to Ventura (2010) and Martin and Ventura (2010), bubbles are
expansionary in our model.
After ten periods in a ’bubbly equilibrium’we assume that investor sentiment suddenly and unexpectedly turns and the bubble collapses to zero. Whenthe bubble bursts, the banks that own it experience a massive decline in their
net worth. In our model the loss is so large that the banks become insolventin the absence of government intervention. In order to prevent this we assume
that the government gives them a bail out which it …nances by raising lumpsum taxes from all entrepreneurs in the model. In the interests of realism, we
assume that the bail out is not large enough to maintain bank net worth. As aresult, bank capital falls sharply and this leads to a credit crunch characterisedby a spike in lending-deposit spreads and in bank leverage. High-productivity
25

entrepreneurs ’employment decreases sharply due to the credit crunch. Since
the entrepreneurs do not hold bubbles their net worth is not directly a¤ectedby the collapse of bubbles. So the decrease in employment and output is driven
entirely by the credit crunch.

5.2

Bubbles and TFP shocks

A number of papers, starting with Kiyotaki and Moore (1997) have shownthat when credit frictions prevent the most productive …rms from purchasing all factors of production, the economy may experience endogenous credit
cycles that look very similar to conventional technology shocks. This happens because, as the net worth and borrowing capacity of high productivity
agents increases, they increase their productive activities at the expense of low
productivity agents. This resource re-allocation improves aggregate e¢ ciency
and leads to an increase in output. More recently, Ventura (2010) and Martin and Ventura (2010) have applied this argument in the case of bubbles in
economies with credit frictions. They show that the emergence of bubbles canlead to a large realocation of resources towards more productive use, increasing economy-wide TFP. Conversely, the collapse of bubbles can shift resources
into less productive …rms, leading to a reduction in aggregate e¢ ciency.
This e¢ ciency-enhancing e¤ect makes it harder to spot bubbles reliably because, in terms of their output e¤ects, they look like conventional TFP shocks.
So in this subsection, we compare the e¤ects of a boom-bust cycle driven bya bank-held bubble with a boom-bust cycle, driven by exogenous movements
in aggregate technology. The Figure 4a below compares the evolution of realvariables in the model under a bank-held bubble and under a conventional
exogenous TFP shock.
[Figure 4a here]
26

The …rst thing to note is the remarkable similarity between the evolution of
output and TFP during the boom phase under both experiments. But whereas
the end of a TFP boom only returns aggregate technology to a around its
starting level, the e¤ects of the bursting of a bank-held bubble on aggregate
TFP are more dramatic. As we argued above, this additional output volatility
is largely due to the credit crunch that occurs following the collapse of the
bubble.
Moving on to …nancial variables we see that their evolution is very di¤erent
under the two scenarios. Bank leverage, bank pro…tability and the size of bankbalance sheets relative to GDP increase much more sharply during a bubbly
episode compared to a standard aggregate technology shock. When the bustcomes, its e¤ects on …nancial variables are dramatic in the case on the bubble
but very limited in the case of ordinary TFP shock. When a bank-held bubble
bursts, it wipes out the net worth of the banking system and the ratio of bank
net worth to GDP falls very sharply. Bank balance sheets contract sharply andthe ratios of money and credit to GDP fall substantially. Credit, in particular,
undergoes a sharp decline because higher loan rates reduce credit demand.
Bank leverage rises sharply on the back of improved bank pro…t margins. Aftera crisis period in which loan rates reach very high levels and deposit rates fall
sharply, bank net worth recovers some way towards pre-crisis levels and thecredit crunch begins to abate.
[Figure 4b here]
To clarify the evolution of …nancial variables under an ordinary TFP shock,we plot them by themselves in the Figure 5 below. Qualitatively, the series
are somewhat similar to the ones that occur under a bank-held bubble. The
improvement in aggregate technology generates an increased demand for liquidity from both investing and saving entrepreneurs. Because bank capital is27pre-determined in the initial period of the shock, the higher liquidity demandleads to an increase in bank pro…ts and a rise in the franchise value of thebank (which includes the NPV of future bank pro…ts). The higher franchisevalue relaxes the bank’s collateral constraint and allows it to expand leverage,loans and deposits. When the improvement in productivity reverses after 10
periods, the same process operates in reverse. Bank pro…ts fall and bank balance sheets contract driven by lower demand for liquidity by entrepreneurs.
The boom-bust cycle follows a familiar and intuitive pattern. But it is a very
mild cycle that hardly leads to ‡uctuations in money and credit aggregates orin banks ’leverage and net worth, either during the boom or the bust phaseof the cycle.
[Figure 5 here]

5.3

The emergence and bursting of a ’saver bubble ’

In the previous subsection we examined the behaviour of the economy undera bubble which is only held by the banking system. The emergence of such a
bubble is initially very good for banks because it provides them with uniqueaccess to an alternative store of value, raising their pro…ts and net worth in
the process. But many real world bubbles do not fall under such a ’limited
participation’ description. For example equity bubbles can be held by any
investor, no matter how small. This raises an important question. How muchshould we worry about such ’equity ’as opposed to ’credit ’bubbles?
In this subsection we experiment with the emergence and bursting of a’divisible ’bubble, which can be directly held by low productivity savers. We
show that banks may or may not join in the bubble depending on their profitability. In what follows we compare the e¤ects of a ’bank-held’bubble withthe e¤ects of a ’saver-held’bubble.
28

[Figure 6a here]
The most striking feature of the evolution of the real variables during the
bubble’s emergence is that the saver-held bubble does not lead to such violent‡uctuations in output and TFP. The bursting of the bubble in period 10 hardlya¤ects the path of output. Even without the bursting of the bubble, output
would have been on a gentle downward trajectory. The collapse does very little
to change the economy’s course. This …ts well with the experience during the
1998-2002 period. After a period of vigorous growth and very high investment,
the collapse of the tech bubble led to a relatively mild recession in comparison
with the Great Contraction. The model simulations con…rm this hypothesis.
Under the bank-held bubble, the bust leads to a big fall in the net worth ofbanks and a credit crunch that sharply reduces output and TFP. A bubble thatis only held by unleveraged savers has none of these undesirable consequences
for …nancial stability.
[Figure 6b here]
The di¤erences between the evolution of …nancial variables allow us to gaina better understanding into why the real e¤ects of the two types of bubbles
are so di¤erent. During the bubble, bank balance sheets expand more dramatically when banks are directly involved (leverage and money to GDP ratios all
increase substantially). Loans to the ’real’sector grow faster under the saver
held case because they do not have to compete with bubbles on banks’ balance sheets. But total bank assets (bubbles as well as ’real ’loans) grow more
rapidly under the bank-held bubble. Bank pro…tability is extremely strong
under both scenarios underpinned by strong loan demand from entrepreneurs
with sharply higher net worth due to the pro…ts from their recent bubble sales.
This, as well as higher leverage, is why banks’ net worth increases by morewhen banks hold the bubbles.
29

Just like in the previous section, here we burst the bubble after 10 periods
in order to examine its e¤ects on the economy. The ratios of loans and money
to GDP decline gradually when savers hold bubbles. The fall is much sharper
when banks intermediate the bubble. The credit crunch leads a sharp increasein the price of credit. Hence the economy experiences another surge in bankleverage and bank pro…t margins. This helps bank capital recover after aperiod of restricted bank credit and money supply.

6

Banks’ Franchise Values and Bubble Holdings

In the previous subsection, we noted that bubbles that can be held by ordinary
savers tend to have more benign e¤ects on …nancial stability compared to ’bankbubbles’. The reason for this lies in the behaviour of banks who choose notto purchase ’saver bubbles’even though they have the opportunity to do so.
’Saver bubbles ’earn the the same rate of return as deposits which is lower thanthe loan rate as long as the borrowing constraint on the banks is binding. Sobanks choose rationally not to buy them, instead focusing on their traditional(and much more pro…table) activity - loans to entrepreneurs. As Gorton (2007)
has emphasized, protecting banks from competition creates a ’franchise value ’(the NPV of excess pro…ts from ’traditional’banking activities) which preventsbanks from investing in bubbly assets.
[However, there exist a number of historical episodes in which …nancial liberalisation and the growth of non-bank …nancial intermediation pushed bankloan spreads down and encouraged banks to engage in new and riskier activities. Hoshi and Kashyap (2000) document the way …nancial liberalisation andthe growth of the corporate bond market pressures on Japanese banks during
30

the 1980s.
Adrian and Shin (200x), Gorton (2007) document the rapid growth of securitisation and repo …nancing during the 2000s. NEED TO SAY MORE HERE,
MAYBE WITH A DATA CHART OR TWO]
All these episodes have the common feature that …nancial liberalisation
increased competition among banks or between bank and bond intermediation,and therefore depressed loan spreads and pushed banks to branch out intoriskier activities. In what follows, we extend our model of Section 2 to allow
for direct …nance. We then explore how the growth of direct intermediationa¤ects the incentives of banks to hold bubbles.

6.1

The Model with Direct Finance6

In order to analyse the e¤ect of direct …nance on the equilibrium of our economy, we assume that ordinary savers are able to enforce debt repayments up
to some fractionm

as follows
Rtm bm
t 6

m

yt+1

where Rtm is the interest rate on direct loans from savers (we can think of theseas ’corporate bonds’) and bm
t is the quantity of direct loans. Banks still exist
in this economy because they have a superior intermediation technology, whichallows them to enforce debt repayments up to fraction
Rtl blt 6lb

as follows:

yt+1

where Rtl is the interest rate on bank loans and blt is the quantity of bank loans.
It is easy to see that arbitrage by savers implies that corporate bond and
6More details on the model with direct …nance can be made available upon request.

31

bank deposit yields will be equalised

Rtd = Rtm

while bank debt will remain more expensive. When the rate of return on thehigh productivity technology exceeds the cost of market …nance,
aH
> Rtm
wt
borrowing entrepreneurs will borrow up themyt+1 limit from the ’corporate

bond’market. WhenaH
> Rtl > Rtm
wt
entrepreneurs will continue borrowing from banks up to the remaining

lmbank debt capacity after they have exhausted their market borrowing capacm

ityyt+1 . Firms always prefer to borrow from the market …rst because it ischeaper but if their productive opportunities are good enough, bank borrowing
is attractive too.

6.2

A Financial Liberalisation Simulation

With the above brief outline of our direct …nance extension in mind, we now
continue to analyse the impact of an ’disintermediation shock ’ (an increaseinmholdingl…xed) on banks ’incentives to hold bubbles despite compe-

tition from ordinary savers. The experiment we conduct is the following. In
the …rst period of the simulation the economy experiences positive investor
sentiment and this leads to the emergence of a bubble which is held by savers
but not by banks. Then two periods into the bubble, the economy experiences
a ’disintermediation shock’ which allows some direct lending between savers
32

yt+1

and borrowers. Two periods later, the bubble bursts and the degree of directintermediation returns to its initial value.
Figure 7 below displays the evolution of bank net worth and its portfolio
composition under this scenario. We can see that the banks do not join in the
bubble until direct intermediation starts to grow. At this point, banks purchaseapproximately almost all of the available bubbles in circulation, allocating tothem a sum equal to around three quarters of bank capital. In terms of absolutemagnitudes, banks ’bubble holdings remain small. Due to leverage, however,the presence of bubbles in bank portfolios leaves them very exposed to a lossof con…dence in the bubble’s future acceptance and value. When the bubble…nally collapses, bank net worth falls sharply, causing a credit crunch in theeconomy.
[Figure 7 here]
As Figure 8a below shows, the fall in bank capital leads to a contractionin the supply of credit and a re-allocation of employment from high to low
productivity entrepreneurs. As a result, TFP declines and the only thing thatprevents a big collapse in output is the fact that the decline in the e¢ ciencyof the economy’s savings technology forces ’savers’ to raise the amount theysave through ine¢ cient production.
[Figure 8a here]
But why did banks suddenly choose to invest into the bubble as directintermediation grew whereas previously they had stayed on the sidelines of theboom? Figure 8b below provides some answers to this question by delving moredeeply into the …nancial side of the model. In particular, as we argued above,the evolution of bank pro…t margins are key to understanding the reasons forthis sudden change of bank behaviour.
When the bubble …rst appears, the rise in productive agents’ net worth
33

increases credit demand and boosts bank pro…t margins. Since traditional
lending is so much more pro…table than bubbles (which earn the deposit rate
when ’savers’are able to hold them), banks rationally hold no bubbles on theirbalance sheets. However the growth of direct …nancing increases the supplyof credit for borrowers as well as the supply of means of saving for savers.
Higher loan and deposit supply brings bank pro…tability (as measured by loandeposit spreads) down. Banks temporarily7 become borrowing unconstrained
and the lending rate becomes equal to the deposit rate and the rate of returnon bubbles. At this point, …nancial intermediaries become indi¤erent between
expanding their balance sheets on the margin because their capital constraint
is slack. Banks’ lending to ’real entrepreneurs ’ is demand determined and…xed by entrepreneurs’net worth. So the only way in which banks can expand
balance sheets is by issuing deposits to the unproductive agents and purchasing
bubbles from them with the proceeds. Unproductive agents are themselvesindi¤erent between deposits and bubbles and so would be happy to change
their portfolios in this way without demanding a change in relative rates ofreturn.
[Figure 8b here]
While the bubble continues, such a ’reshu- ing’of the portfolios of banks
and savers has no consequences for prices and real allocations. Therefore, theshare of the bubble held by banks is indeterminate up until the point wherebanks’ balance sheet constraint starts to bind. In the above simulation, we
have assumed that banks expand their balance sheets to buy bubbles as muchas they can. Therefore, this experiment represents an upper bound on therisks to …nancial stability during periods of rapid disintermediation.87Given su¢ cient time, the credit market liberalisation would increase the net worth ofthe corporate sector boosting credit demand. At this point, bank pro…t margins will recover
to some extent although they will still remain below pre-liberalisation levels.
8This section used a …nancial liberalisation scenario which relied on the growth of non-34

7Model-implied ’early warning signals’of crisis

In the last ten years, there has been a big increase in the number of researchers
who have sought to learn about the empirical regularities in banking and moregeneral …nancial crises. Now, following the 2008-09 …nancial crisis, the interestof policy-makers in developing empirical measures that can help in predicting
the occurence of future banking crises is greater than ever. Most of the existing studies naturally utilise a largely atheoretic approach, focusing on …nding
robust crisis predictors without necessarily spelling out the underlying mechanism by which such predictors are related to the eventual occurence of …nancial
crisis. Our paper can provide some theoretical backing and interpretation behind some of these ’early warning signals ’of …nancial instability.

7.1

Rapid growth in money, credit and bank leverage islikely to be associated with a bubble

Even though the evolution of output is similar between bubbles and genuine
positive TFP shocks, the evolution of …nancial variables is very di¤erent. In
particular, bank balance sheets (money, credit relative to GDP as well as bankleverage) expand much more rapidly under bubbles than under TFP shocks.
This provides a justi…cation for the threshold approach used by Borio andLowe (2002), Alessi and Detken (2009) and others in identifying boom-bustbank …nancial intermediation in order to generate bubble induced fragility into the banking
system. We motivated this way of modelling …nancial liberalisation by appealing to thehistorial experiences of Japan in the 1980s as well as the well-publicised growth of the’shadow banking system’over the last decade.
In a previous version of the paper we examine the e¤ect of a ’…nancial liberalisation ’shock which increases the permissible leverage of …nancial institutions. The results (which
are available from the authors upon request) are qualitatively very similar. Bank loan supply
increases, drives loan pro…tability down and exposes banks to the possibility of holding ’saver
bubbles ’.

35episodes. In our model, big increases in money, credit and bank leverage to
GDP are likely to have their origins in bubbles and hence signal increased
likelihood of future crisis.

7.2

Distinguishing between’costly ’ and ’harmless ’bubbles ’ex ante’is hard in the model
Bubbles on assets that can only be held by banks (’bank bubbles ’) are more
likely to have damaging consequences for …nancial stability compared to bubbles that can be held by other economic agents (’saver bubbles’). This isbecause banks will often choose not to invest in ’saver bubbles ’and will therefore remain una¤ected by their bursting. But as we saw in section xx this is
not always the case. When banks’pro…ts from traditional business loans are
low, they can be tempted to join in a ’saver bubble’. Can a policy-maker
who cannot tell bubbles from real loans notice whether the banking system isengaged in a dangerous bubble or not just by looking at bank balance sheets?
Our analysis showed that the same size bubble caused a sharper balance sheet
expansion under a ’bank bubble’. But the di¤erence is not as large as between
bubbles and positive TFP shocks. So our policy maker would be able to tellthat there is an unsustainable asset price boom somewhere in the economy
but would not be sure whether the collapse of the bubble would lead to bankfailures. This provides a reason why the literature on ’early warning signals ’has struggled to …nd really robust measures that can forecast crises with a highdegree of con…dence. For example Alessi and Detken (2009) and Kannan et al.
(2009) both …nd that even their best early warning indicator is still subject tolarge Type I and Type II errors. In our model this is an unavoidable consequences of the fact that the di¤erent types of bubble we examine do not have36

hugely di¤erent implications for aggregate variables during the boom phase.

7.3

Poor bank pro…tability increases the risks of …nancial instability

Our model implies that ’saver bubbles ’may or may not carry risks for …nancial
stability, depending on the level of bank pro…tability. Less pro…table banking
systems are more likely to invest in bubbly assets and expose their net worth
to a possible bubble collapse. Our model suggests several di¤erent channels
through which pro…tability might be eroded leading to banks investing in bubbly assets: (i) an increase in bank competition9 ; (ii) growth of market based
intermediation such as corporate bonds and asset backed securities (analysedin the previous section) (iii) relaxation of the bank’s leverage constraint and(iv) weakness in corporate balance sheets. It is straightforward to see whythese three factors would reduce banks ’ pro…ts from corporate lending. (i),(ii) and (iii) increase the supply of loans and reduce the loan-deposit spread.
(iv) reduces loan demand with the same downward impact on banks’lending
margins. In all of these situations, poor pro…tability makes banks more willingto hold bubbly assets.
9

For the e¤ects of bank competition on bank riskiness see (Boyd and De Nicolo (2009)).
The literature identi…es two o¤setting channels. First of all, more bank competition implies
a lower franchise value of the bank leading to more risk taking. But there is a second e¤ect
working in the opposite direction. More bank competition will lead to lower loan rates andthis may reduce the riskiness of the bank’s loan portfolio, making the bank safer overall.
Our model introduces some new channels linking banking sector competition with bankriskiness that have not hitherto been explored in the literature.
For us, more competition reduces pro…ts from traditional lending and may tempt banks
into investing in bubble assets. This increases the risk that the bubble may burst and reducebank net worth. But there is an o¤setting e¤ect. More bank competition, reduces the valueof the …rm and, in the Gertler and Karadi (2009) framework, this leads to lower leverage.
As a result, experiencing the same loan loss would have a smaller impact on bank capital.

37

7.4

Summary

Table 3 provides a summary of the main conclusions of our analysis.
Table 3. Types of bubbles and their propagation
Bank bubble

Saver bubble

Banks

Boom: "" D, Y , T F P

Boom: " D, Y , T F P

hold bubbles(banks always hold)(banks hold if Rl = Rd )Bust: Very costly

Bust: Costly if big exposure

Banks do nothold bubbles

Boom: " D, Y , T F P(banks don’t hold if Rl > Rd )-Bust: Not costly

8

Conclusions

In this paper we build a model in which rational asset prices bubbles arise
due to credit frictions. Our framework models …nancial intermediaries in an
explicit manner in order to formalise the intuition that asset prices held byleveraged …nancial intermediaries pose the biggest threat to …nancial stability
In contrast, if unleveraged savers hold bubbles, the collapse of bubbles has
relatively few consequences for …nancial intermediation and for the solvency ofthe banking system. We show that, in normal times, banks ’unique positionin the …nancial system creates excess pro…ts whose ’franchise value ’preventsbanks from investing in bubbly assets. Economic shocks that reduce theseexcess pro…ts and, consequently diminish banks ’ ’franchise values ’ increase
the likelihood that banks will hold bubbles. This explains why, historically,…nancial liberalisation and de-regulation are often followed by banking crises.
Our model provides a rich array of theoretical predictions regarding theimpact of di¤erent types of bubbles on real and …nancial variables. These can38be useful in interpreting the results of the growing literature on ’early warning
indicators’ of …nancial crises. In particular, we …nd that large bank balance
sheet expansions signal the presence of a bubble in the economy. However, it
is more di¢ cult within our model to be sure whether the bubble is held bybanks or by ordinary savers. ’Bank bubbles’lead to bigger movements in bankbalance sheets but the di¤erence is not large enough to be used as a robust
indicator
Finally, our paper also makes a contribution to the literature that attempts
to explain why asset prices bubbles tend to be expansionary in reality rather
than contractionary as early rational bubble theories implied. We show thatthe presence of banks enhances the liquidity e¤ects of bubbles because thepresence of bubbles enhances bank excess returns which are collateralisable.
We show that bubbles are therefore more likely to be expansionary in a modelthat models banks explicitly.

39

9

Appendix A:

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
NominalData concept

Sourcebank Prime loan rate

Federal Reserve Board, Ta-loan rate
Nominalble H.15
deposit M2 own rate

FRED

rate
Expected in‡ation Average actual CPI in‡ation FRED(All Urban Consumers)
Expectedreal Average real GDP growth FRED

GDP growth(chained measure)

Deposit stock

M2

FRED

Nominal GDP

Nominal GDP

FREDBank leverage

Bank Debt Liabilities/Bank Federal Reserve Board, TaNet Worthble H.8

Average corporate Corporate Debt/Corporate Welch (2004)
leverage
Leverage

Net Worthofin- Debt/Corporate Net Worth Welch (2004)

debted corporates

for the largest corporates

Bank rate of re- Bank rate of return on eq- Meh and Moran (2009)
turn on equity

uity

40

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43

2007q1

Jan-00 Jul-00 Dec-00 Jun-01 Nov-01 May-02 Oct-02

2008Q1

2009Q1

2010Q1

1
100

100

1
GDP growth 2000-03

S&P 500

80

80

60

60

40

40

0.5

0.5

0

0

-0.5

-0.5

ABX AAA 07-01
GDP growth 2007-10
20

20

Jan-07 Jul-07 Dec-07 Jun-08 Nov-08 May-09 Oct-09-1
2000q1-1
2001Q1Figure 1. US equity prices and output

1

2002Q1

2003Q1

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 2a. Deposit rate less than one (red area)

2

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 2b. Lending rate less than one (red area)

3

0.8

0.9

GDP

High-prod Employment

1.2

1
0.95

1.18

0.9
1.16

0.85

0

20

40

60

0.8

0

20

Low-prod Employment

40

60

40

60

TFP

0.4

1.11
1.1

0.3

1.09
0.2

0.1

1.08

0

20

40

60

1.07

0

20

Figure 3a: Real variables: bank-bubble

4

Bubble/GDP

Bank net worth/GDP

0.1

0.1

0.05

0.05

0

0

20

40

60

0

0

Loans/GDP
0.8

0.6

0.6

0

20

40

60

0.4

0

Bank Leverage
0.1

50

0.05

0

20

60

20

40

60

Bank Profit Margins

100

0

40

Money/GDP

0.8

0.4

20

40

60

0

0

20

Figure 3b. Financial variables: bank-held bubble

5

40

60

GDP

High-prod Employment

1.22

1

1.2

0.95

1.18

0.9

1.16

0.85

0

20

40

60

0.8

0

20

Low-prod Employment

40

60

40

60

TFP

0.4

1.11
1.1

0.3

1.09
0.2

0.1

1.08

0

20

40

60

1.07

0

20

Figure 4a: Real variables: bank bubble (solid) vs TFP shock (dashed)

6

Bubble/GDP

Bank net worth/GDP

0.1

0.1

0.05

0.05

0

0

20

40

60

0

0

Loans/GDP
0.8

0.6

0.6

0

20

40

60

0.4

0

Bank Leverage
0.1

50

0.05

0

20

60

20

40

60

Bank Profit Margins

100

0

40

Money/GDP

0.8

0.4

20

40

60

0

0

20

40

60

Figure 4b: Financial variables: bank bubble (solid) vs TFP shock (dashed)

7

Bank net worth/GDP

Bubble/GDP
1

0.05

0

0.045

-1

0

20

40

60

0.04

0

0.48

0.52

0.46

0

20

40

60

0.44

0

Bank Leverage
0.016

11

0.014

0

20

60

20

40

60

Bank Profit Margins

12

10

40

Money/GDP

Loans/GDP
0.54

0.5

20

40

60

0.012

0

20

Figure 5. Financial variables under a TFP shock

8

40

60

GDP

High-prod Employment

1.22

1

1.2

0.95

1.18

0.9

1.16

0.85

0

20

40

60

0.8

0

20

Low-prod Employment
1.11

0.3

1.1

0.2

1.09

0.1

1.08

0

20

60

40

60

TFP

0.4

0

40

40

60

1.07

0

20

Figure 6a. Real variables: bank-held (solid) vs saver-held (dashed) bubbles

9

Bubble/GDP

Bank net worth/GDP

0.1

0.1

0.05

0.05

0

0

20

40

60

0

0

Loans/GDP
0.8

0.6

0.6

0

20

40

60

0.4

0

Bank Leverage
0.1

50

0.05

0

20

60

20

40

60

Bank Profit Margins

100

0

40

Money/GDP

0.8

0.4

20

40

60

0

0

20

40

60

Figure 6b. Financial variables: bank-held (solid) vs saver-held (dashed) bubbles

10Bubble/GDP

Market Intermediation/GDP

0.05

0.4

0.04

0.3

0.03
0.2
0.02
0.1

0.01
0

0

5

10

15

20

0

0

5

10

15

20

Maximum share of bubbles in bank capital
Bank capital as per cent of GDP
0.8
0.06
0.6

0.05

0.4

0.04

0.2

0.03

0

0

5

10

15

20

0.02

0

5

Figure 7: A boom in non-bank lending

11

10

15

20

GDP

High-prod Employment

1.21

1

1.2
0.95

1.19
1.18

0.9

1.17
1.16

0

5

10

15

20

0.85

0

5

Low-prod Employment
1.09

0.2

1.085

0.15

1.08

0

5

10

15

15

20

15

20

TFP

0.25

0.1

10

20

1.075

0

5

10

Figure 8a: Evolution of real variables during a boom in non-bank lending

12

Bubble/GDP

Bank net worth/GDP

0.06

0.06

0.04
0.04
0.02
0

0

5

10

15

20

0.02

0

5

Loans/GDP
1

0.5

0.5

0

5

10

15

20

0

0

Bank Leverage
0.1

20

0.05

0

5

10

20

5

10

15

20

Bank Profit Margins

40

0

15

Money/GDP

1

0

10

15

20

0

0

5

10

15

Figure 8b: Evolution of …nancial variables in a boom in non-bank lending

13

20