Asset Bubbles and Bailout
Tomohiro Hiranoyand Noriyuki YanagawazFirst Version, April 2011
This Version, December 2011
Very Preliminary

Abstract
This paper theoretically investigates the relationship between asset price bubbles and bailout. We show that although bailout may
mitigate adverse e¤ects of bubbles’bursting ex-post, it is more likely
to cause asset price bubbles by encouraging risk-taking behavior exante. In other words, bubbles are more likely to occur, the moregovernment bailout is anticipated. Moreover, we examine the e¤ectsof the anticipated bailout on boom-bust cycles. We …nd that when
productivity is relatively low, the anticipated bailout accelerates output booms and creates large bubbles, thus destabilizing the economy.
On the other hand, when productivity is relatively high, the anticipated bailout dampens output booms, thus stabilizing the economy.
Finally, we analyze a desirable ex-post bailout policy.
Key words: Stochastic Bubbles, Anticipated Bailout, Emergenceof Asset Price Bubbles, Bubbly Booms

We have bene…ted greatly from discussions with Koichi Hamada, Masaru Inaba, Michihiro Kandori, Ryutaro Komiya, Masaya Sakuragawa, and seminar participants at Bank ofJapan, SWET conference in Hokkaido, Hokkaido University, Yokohama National University, and University of Tokyo.
yTomohiro Hirano, Faculty of Economics, The University of Tokyo, tomohih@gmail.comz
Noriyuki Yanagawa, Faculty of Economics, The University of Tokyo, yanagawa@e.utokyo.ac.jp1

1

Introduction

Many countries have experienced bubble-like dynamics. Associated with thebursting part of asset price bubbles are signi…cant contractions in real economic activity. Notable examples include the recent U.S. experiences after
the …nancial crisis of 2007/2008 as well as Japan’s experiences in the 1990s.
To mitigate severe contractions, government tends to take various types ofbailouts such as recapitalization through buying equity or through the purchase of troubled assets at in‡ated prices. Although these policies may mitigate the contractions ex-post, what happens if these policies are anticipatedex-ante? In this paper, we analyze ex-ante e¤ects of bailout. In particular,
we ask: how does the anticipated government bailout a¤ect the emergenceof asset price bubbles? What are the e¤ects of the anticipated bailout onboom-bust cycles?
To tackle with these questions, we develop a macroeconomic model withstochastic bubbles.1 Since seminal papers of Farhi and Tirole (2009a) and
Ventura (2011), the recent development on rational bubbles have provided atheoretical framework to analyze macroeconomic e¤ects of asset price bubbles (Caballero and Krishnamurthy, 2006; Kocherlalota, 2009; Hirano andYanagawa, 2010a, 2010b; Martin and Ventura, 2010a, 2010b; Sakuragawa,
2010; Aoki and Nikolov, 2011; Miao and Wang, 2011).2 The contribution ofthis paper is that we consider the e¤ects of bailout within a rational bubbles
model. The bailout we consider is a transfer policy from workers to entrepreneurs: recapitalization through buying legacy assets at in‡ated prices as
discussed in Farhi and Tirole (2011). When bubbles collapse, the net worthof entrepreneurs decreases signi…cantly, which in turn causes a free-fall inreal economic activity. The primary purpose of bailout is to boost the net
worth of entrepreneurs and to mitigate the free-fall.
Although the bailout is e¤ective in mitigating the free-fall ex-post, there
are side e¤ects. In our framework, since bubble assets are risky in the sensethat bubbles may collapse, risk-averse entrepreneurs want to hedge themselves by investing in safe assets. As we show, the entrepreneurs’portfolio
decision depends on not only the probability of bursting of bubbles, but also
expectations about government bailout. We will show that the anticipated1

Weil (1987) is the …rst study that analyzes stochastic bubbles in a general equilibrium
model.
2
Kocherlakota (1992), Santos and Woodford (1997), and Hellwig and Lorenzoni (2009)
analyze asset price bubbles in an endowment economy with in…nitely lived agents.

2

bailout induces entrepreneurs to take on more risk ex-ante.
What is interesting is that through the change in risk-taking behaviors,
the anticipated bailout a¤ects the emergence of asset price bubbles. That is,when the bailout is not expected, bubbles cannot occur in low-productivity
economy, because in that economy, the wealth’s growth rate of the economy is
too low to sustain growing bubbles.3 However, when the bailout is expected,
entrepreneurs are willing to buy more bubble assets and required rate of
return of bubbles declines. This decline in turn lowers the growth rate of
bubbles. As a result, even low-productivity economy enters bubble regions.
As we show, although bailout may mitigate adverse e¤ects of bursting bubbles
ex-post, it is more likely to cause asset price bubbles. In other words, bubbles
are more likely to occur, the more government bailout is anticipated.
Moreover, the anticipated bailout also greatly in‡uences aggregate economic activity and asset price before the bubble bursts. As we show, when
productivity is relatively low, the anticipated bailout induces entrepreneurs
to take on more risk ex-ante, which accelerates bubbly booms and createslarge bubbles, thus resulting in a large scale government intervention during
bubbles’collapsing. In other words, the bailout policy that aims at stabilizing
the economy during bubble bursts ends up with destabilizing the economy.
On the other hand, when productivity is relatively high, the anticipatedbailout dampens bubbly booms, thus stabilizing the economy.
To be sure, our paper is related to theoretical literature that examines
government bailouts and risk-taking. For example, Chari and Kehoe (2010),
Diamond and Rajan (2011), and Farhi and Tirole (2009b, 2011) stress moral
hazard consequences of bailouts and other credit market interventions. Our
paper is mainly di¤erent from these papers in the point that we analyze theex-ante e¤ects of bailout within a full blown dynamic macroeconomic model.
In this respect, our paper is closely related to Gertler et al. (2011) andBrunnermeier and Sannikov (2011). There are di¤erences in two respects.
First, both of these papers analyze the ex-ante e¤ects of bailout in termsof monetary policy, while our paper examines them from the perspective ofcapital injection policy. Second, Gertler et al. (2011) focus on the e¤ects
on the …nancial system, and Brunnermeier and Sannikov (2011) analyze thee¤ects on risk-taking by the intermediary sector, while we examine the e¤ects
3
Tirole (1985) is a seminal paper on rational bubbles. In the paper, Tirole shows thatfor the emergence of asset price bubbles, the economy’s growth rate must be su¢ ciently
high and it is indeed greater than interest rate in the bubbleless economy. This holds truein our model too.

3

on asset price bubbles.

2

The Model

2.1

Framework

Consider a discrete-time economy with one homogeneous good and a continuum of entrepreneurs and households (workers). A typical entrepreneur anda representative worker have the following expected discounted utility,
"1
#
X
t
E0
log cit ;(1)
t=0

where i is the index for each entrepreneur, and cit is the consumption of
him/her at date t. 2 (0; 1) is the subjective discount factor, and E0 [a] isthe expected value of a conditional on information at date 0.
Let us start with the entrepreneurs, who play a central role in this paper.
At each date, each entrepreneur meets high productive investment projects(hereinafter H-projects) with probability p, and low productive ones (Lprojects) with probability 1 p.4 The investment projects produce capital.
The investment technologies are as follows:
i
kt+1
=

i i
t zt ;(2)iis the capital atwhere zti ( 0) is the investment level at date t; and kt+1i
date t + 1. t is the marginal productivity of investment at date t. it = H
if the entrepreneur has H-projects, and it = L if he has L-projects. We
assume H > L . For simplicity, we assume that capital fully depreciates in
one period.5 The probability p is exogenous, and independent across entrepreneurs and over time. At the beginning of each date t, the entrepreneur
knows his/her own type at date t, whether he has H-projects or L-projects.
Assuming that the initial population measure of each type of the entrepreneur is p and 1 p at date 0, the population measure of each type after date 1
4

A similar setting is used in Woodford (1990), Kiyotaki (1998), Kiyotaki and Moore(2008), Kocherlakota (2009), Nikolov (2010), and Aoki and Nikolov (2011).
5
As in Kocherlakota (2009), we can consider a situation where some fraction of capitaldepreciate, and consumption goods can be converted one-for-one into capital at each date,and vice-versa. In this setting, we can also obtain the same results as in the present papar.

4

is p and 1 p, respectively. Throughout this paper, we call the entrepreneurs
with H-projects (L-projects) "H-entrepreneurs" ("L-entrepreneurs").
We assume that because of frictions in a …nancial market, the entrepreneur can pledge at most a fraction of the future return from his investment
to creditors.6 In such a situation, in order for debt contracts to be credible,
debt repayment cannot exceed the pledgeable value. That is, the borrowing
constraint becomes:
rt bitqt+1 it zti ;(3)
where qt+1 is the relative price of capital to consumption goods at date t + 1.7
rt and bit are the gross interest rate and the amount of borrowing at date t,
respectively. The parameter 2 (0; 1], which is assumed to be exogenous, canbe naturally taken to be the degree of imperfection of the …nancial market.
In this economy, there are bubble assets denoted by x. Total supply is
assumed to be constant over time X: As in Tirole (1985), we de…ne bubble
assets as the assets that produce no real return, i.e., the fundamental valueof the assets is zero. Here we consider stochastic bubbles. Following Weil(1987), we assume that in each period t; bubble price becomes zero (bubble
bursts) with probability 1, given positive bubble price at date t 1: Oncethey burst, they never arise again. This implies that bubbles persist withprobability (< 1) and their prices are positive until they switch to being
equal to zero forever. Let Pt be the per unit price of bubble assets at date t
in terms of consumption goods in the case where bubbles do not collapse atdate t.
The entrepreneur’s ‡ow of funds constraint is given by
cit + zti + Pt xit = qt it 1 zti 1

rt 1 bit 1 + bit + Pt xit 1 :(4)where xit be the level of bubble assets purchased by a type i entrepreneur at
date t. The left hand side of (4) is expenditure on consumption, investment,
and the purchase of bubble assets. The right hand side is the available funds
at date t, which is the return from investment in the previous period minus
debts repayment, plus borrowing and the sales of bubble assets. We de…ne thenet worth of the entrepreneur at date t as eit qt it 1 zti 1 rt 1 bit 1 + Pt xit 1 .
6

See Hart and Moore (1994) and Tirole (2006) for the foundations of this setting.
On an equilibrium path we consider, qt+1 is not a¤ected by whether bubbles collapse
or not. Hence, there is no uncertainty with regard to qt+1 :7

5

We also impose the short sale constraint on bubble assets:8
xit

(5)

0:

Next let us turn to the households. There is a unit measure of households.
Each household is endowed with one unit of labor endowment in each period,
which is supplied inelastically in labor markets, and earns wage rate, wt .
The ‡ow of funds constraint, the borrowing constraint, and the short saleconstraint for households are given by
cut + Pt (xut

rt 1 but 1 + but

xut 1 ) = wt

Tt ;

(6)

rt but

0;

(7)

xut

0;

(8)

where u represents households. Government levies a lump sum tax Tt on
households. Tt > 0 only when bubbles collapse. Tt = 0 if they survive: The
tax revenues are used for bailout of entrepreneurs.9 Households (tax payers)
pay the cost of the bubble bursts. Equation (7) says that households cannot
borrow, since they do not have any collateralizable assets such as returns
from investment projects.
There are competitive …rms which produce …nal consumption goods using
capital and labor.10 The aggregate production function is
Yt = Kt0 Nt1;(9)where Kt0 and Nt are the aggregate capital stock and labor input at date t:
Yt is the aggregate output at date t: Factors of production are paid their
8

Kocherlakota (1992) shows that the short sale constraint plays an important role forthe emergence of asset bubbles in an endowment economy with in…nitely lived agents.
9
In the present paper, in order to focus on a transfer policy from workers to entrepreneurs as in Farhi and Tirole (2011), we do not consider the capital income tax. Even if
we consider the capital income tax, we can obtain the same results as in the main text,
although the borrowing constraint for entrepreneurs and the investment function would
be complicated as analyzed in Aoki et al. (2009).
10
We assume that each …rm is operated by the workers. Since the …nal goods market
is competitive, the net pro…t from operating the …rm is zero, so that the ‡ow of funds
constraint of the workers is unchanged as equation (6) in equilibirum.

6marginal product:qt = Kt0

2.2

1

and wt = (1)Kt0 :(10)

Bailout

A bailout we consider is a transfer policy from households to entrepreneurs
whose net worth decreases by bubbles collapse. As in Farhi and Tirole (2011),
the aim of this policy is to boost the net worth of entrepreneurs.
We assume that each entrepreneur as well as each household anticipates
government’s bailout with probability 2 [0; 1]. Since we assume rational
expectations, a fraction of the entrepreneurs is indeed rescued ex-post.
Suppose that bubbles collapse at date t: The entrepreneurs who hold xit 1
amount of bubble assets at the beginning of date t lose their net worth byPt xit 1 : Here we consider the bailout that fully recovers the net worth of
the rescued entrepreneurs, i.e., denote the public funds injected into eachentrepreneur by mit ; mit = Pt xit 1 : Thus, total public funds injected into theentrepreneurial sector at date t is Mt = Pt X: In this paper, we want tofocus on how a change in a¤ects asset price bubbles, ex-ante e¢ ciency, andmacro-dynamics.11 Because of this policy, the net worth of entrepreneurs
at date t is restored by Mt , which can mitigate adverse e¤ects of bubbles’
collapsing.

2.3

Equilibrium

Let us denote the aggregate
consumption
L-entrepreneurs and theP
P of H-andi
Hi
L
workers at date t as i2Ht ct
Ct , i2Lt ct
Ct , Ctu ; where P
Ht and LtimeanPa family of H-and
L-entrepreneurs
at
date
t.
Similarly,let
i2Ht ztPP
Piii
ZtH ; Pi2Lt zti
ZtL ;
BtH ;
BtL ; Btu ;i2Ht bt
i2Lt bt
i2Ht [Lt ktiu
Kt ; ( i2Ht [Lt xt + Xt )Xt be the aggregate investment, the aggregateborrowing, the aggregate capital stock, and the aggregate demand for bubble
11

Of course, we can change the amount of public money injected into each entrepreneur.
For example, we can consider a situation where a fraction
1 of the lost net worth bybubbles’burst is injected. By so doing, we may be able to analyze how changes in bothand a¤ect asset price bubbles and boom-bust cycles. Although this case may be more
realistic, it is hard to solve analytically. Hence in the present paper, we assume = 1,
and we want to focus on how a change in in‡uences asset price bubbles and boom-bust
cycles.

7

assets. Then the market clearing condition for goods, credit, capital, labor,
and bubble assets are
CtH + CtL + Ctu + ZtH + ZtL = Yt ;(11)

BtH + BtL + Btu = 0;

(12)

Kt0 = Kt ;(13)

Nt = 1;(14)

Xt = X:(15)

The competitive equilibrium is de…ned as a set of prices frt ; wt ; Pt g1
t=0 and
1
H
L
u
H
L
u
H
L
quantities Ct ; Ct ; Ct ; Bt ; Bt ; Bt ; Zt ; Zt ; Gt ; Xt ; Kt+1 ; Yt t=0 , such that(i) the market clearing conditions, (11)-(15), are satis…ed in each period, and(ii) each entrepreneur chooses consumption, borrowing, investment, capital1i; xit t=0 ; to maximize
stock, and the amount of bubble assets, cit ; bit ; zti ; kt+1
his expected discounted utility (1) under the constraints (2)-(5). (iii) eachhousehold chooses consumption, borrowing, and the amount of bubble assets,
fcut ; but ; xut g1
t=0 ; to maximize his/her expected discounted utility (1) under theconstraints (6)-(8).

2.4Entrepreneur’s Behavior

We now characterize the equilibrium behavior of entrepreneurs and households. We consider the case
qt+1 Lrt < qt+1 H :In equilibrium, interest rate must be at least as high as qt+1 L , since nobody
lends to the projects if rt < qt+1 L .
Households consume all the wage income in each period. That is, cw
t = wtw
w
w
and bt = 0: Hence, Ct = wt and Bt = 0 hold: We later verify this in
Appendix. In this analysis, workers play little role, except to soak up thereturns to labor.
For the entrepreneurs, both the borrowing constraint and the short saleconstraint simultaneously become binding for H-entrepreneurs, but not binding for L-entreprenurs. Since the utility function is log-linear, each entrepreneur consumes a fraction 1of the net worth in each period, that is,8

cit = (1)eit .12 Then, by using (3), (4), and (5), the investment function of
H-entrepreneurs at date t can be written as
eit
:
(16)
qt+1 H
1
rtThis is a popular investment function under …nancial constraint problems,
except that the presence of bubble assets a¤ects the net worth.13 We see thatthe investment equals the leverage, 1= 1 ( qt+1 H =rt ) , times a fractionof the net worth. From this investment function, we also understand thatfor the entrepreneurs who purchased bubble assets in the previous period,
they are able to sell those assets at the time they encounter H-projects. In
our analysis, the entrepreneurs buy bubble assets when they have L-projects,
and sell those bubbles when they have opportunities to invest in H-projects.
As a result, their net worth increases (compared to the bubbleless case),
which relaxes the borrowing constraint and boosts their investments. Thatis, bubbles generate "balance sheet e¤ects". Moreover, the expansion levelof the investment is more than the direct increase of the net worth becauseof the leverage e¤ect.
For L-entrepreneurs at date t, their portfolio problem is more complicatedbecause they take the government rescues into account. Since bubble assets
deliver no return with probability 1, risk-averse L-entrepreneurs may
want to hedge themselves by investing in L-projects as well as lending to)eit ; the budget constraint (4) becomesother entrepreneurs. Since cit = (1zti =zti + Pt xit

bit = eit :(17)

Each L-entrepreneur allocates his/her savings, eit ; into zti ; Pt xit ; and bit : Since
investing in L-projects and lending are perfect substitutes, zti
0 if rt =qt+1 L ; and zti = 0 if rt > qt+1 L : That is, the following conditions must be
satis…ed:(rt qt+1 L )zti = 0; zti 0; and rt qt+1 L 0:
Each L-entrepreneur chooses optimal amounts of bit ; xit ; and zti ; so thatthe expected marginal utility from investing in three assets respectively is
12

See, for example, chapter 1.7 of Sargent (1988).
See, for example, Bernanke and Gertler (1989), Bernanke et al. (1999), Holmstrom
and Tirole (1998), Kiyotaki and Moore (1997), and Matsuyama (2007, 2008).
13

9

equalized. The …rst order conditions with respect to bit and xit are(bit ) :

1
=
cit

rt
+ (1
i;
ct+1
(xit ) :

rt

)

1
=
cit

i;(1
ct+1

)

+ (1

1 Pt+1
+ (1
Pt

)(1
1

)

ci;t+1

)

i;(1
ct+1

)

rt
i;(1
ct+1

Pt+1
;
Pt

)(1

)

; (18)(19))(qt+1 L zti rt bit + Pt+1 xit ) is the date t + 1 consumption
where ci;t+1 = (1of him/her when bubbles persist. The …rst term of the right hand side inequation (18) and (19) represents the expected marginal utility from lendingi;(1 )
a unit and from buying a unit of bubble assets in this case. ct+1
=
i
L i
i
(1
)(qt+1 zt rt bt + mt+1 ) is the date t + 1 consumption of him/her whenbubbles collapse and the government rescues him/her. The second term ofequation (18) and (19) represents the expected marginal utility from lending
a unit and from buying a unit of bubble assets in this case, respectively.
i;(1 )(1 )
ct+1
= (1
)(qt+1 L zti rt bit ) is the date t + 1 consumption whenbubbles collapse and the government does not rescue him/her. The third
term of equation (18) represents the expected marginal utility from lending
a unit in this case.14 Pt+1 =Pt is the rate of return of bubbles when bubbles
persist.
From (17), (18), and (19), we can derive the demand function for bubble
assets of a type i agent:Pt xit =

Pt+1
Pt
Pt+1
Pt

rt
rt

eit ;

(20)

where = + (1) :The remaining fraction of savings is split across zti and bit :
zti

bit =

(1
Pt+1
Pt

) PPt+1
t
rt

eit :

From (20), we see that an entrepreneurs’s portfolio decision depends on
14

Since the entrepreneur consumes a fraction 1of the current net worth in eachperiod, the optimal consumption level at date t + 1 is independent of the entrepreneur’s
type at date t + 1: It only depends on whether bubbles collapse or whether government
rescues the entrepreneur.

10

its perceptions of risk, which in turn depends on both the probability of
bursting of bubbles ( ) and expectations about the government bailout ( ).
Here we obtain the following Proposition.
Proposition 1 Ceteris paribus, xit is an increasing function of :Whenrises, the type i agent is willing to buy more bubble assets.
This means that the anticipated bailout induces the entrepreneurs to take on
more risk, even though the potential probability of the bubble bursts remainunchanged ( is unchanged).

2.5

Aggregation

We are now in a position to consider the aggregate economy. The great
merit of the expressions for each entrepreneur’s investment and the demandfor bubble assets, zti and xit ; is that they are linear in period-t net worth, eit .
Hence aggregation is easy: we do not need to keep track of the distributions.
From (16), we can derive the aggregate H-investments:
pAt;
qt+1 H
rt

ZtH =
1

(21)

where
PAt i Kt + Pt X is the aggregate wealth of entrepreneurs at date t;
and i2Ht et = pAt is the aggregate wealth of H-entrepreneurs at date t.
Recall that probability of meeting H-projects is independently distributed.
We learn that the aggregate investments of H-entrepreneurs depend on asset
price, Pt ; as well as cash ‡ows from the investment projects in the previous
period, Kt . In this respect, this investment function is similar to the one in
Kiyotaki and Moore (1997). There is a signi…cant di¤erence. In the KiyotakiMoore model, the investment function depends on land prices which re‡ectfundamentals (cash ‡ows from the present to the future), while in our model,
it depends on bubble price.
When rt = qt+1 L ; L-entrepreneurs may invest. Their aggregate investments are determined by the goods market clearing condition (11). Since
H-and L-entrepreneurs consume a fraction 1of their net worth, (11) canbe written as
ZtH + ZtL + Pt X = At(22)11

Thus,
pAt
Pt X:
(23)
qt+1 H1rt
The aggregate L-investments are equal to the aggregate savings of entrepreneurs minus the aggregate H-investments and the purchase of bubble assets.
The aggregate counterpart to (20) is
ZtL = AtPt Xt =

Pt+1
Pt
Pt+1
Pt

rt

(1

rt

(24)

p)At ;

P
where i2Lt eit = (1 p)At is the aggregate net worth of L-entrepreneurs at
date t: (24) is the aggregate demand function for bubble assets at date t:

2.6

Dynamics

Using (21) and (23), we can derive evolution of capital:
8
pAt
pAt
>
H
L
>
Pt X
if rt = qt+1 L ;
At
H +
H
>
<
1

Kt+1 =

>
>
>
:

1

L

pAt

H

1

qt+1 H
rt

=

H

[ At

L

Pt X]

if rt > qt+1

L

(25)

:

When rt = qt+1 L ; L-entrepreneurs may invest in equilibrium, i.e., ZtL 0.
The …rst term and the second term in the …rst line represent the capitalat date t + 1 produced by H-and L-entrepreneurs: When rt > qt+1 L ; Lentrepreneurs never invest, i.e., ZtL = 0; and only H-entrepreneurs invest.
From (22), we know ZtH = At Pt X: ( Pt X) in the …rst and the secondlines captures a traditional crowd-out e¤ect of bubbles (Tirole, 1985), i.e.,some of entrepreneurs’savings ‡ow to bubble assets (since L-entrepreneurs
buy bubbles), which crowds investments out.

12

The aggregate wealth of entrepreneurs evolves over time as
8
>
pAt
>
At
Pt X
+ Pt+1 X if rt = qt+1 L ;< qt+1 H pAtH + L
H
1
1
L
L
At+1 =>>:
qt+1 H [ At Pt X] + Pt+1 X
if rt > qt+1 L :(26)
The …rst term in the …rst line represent the returns from H-and L-investments
undertaken at date t: The second term is the value of bubbles at date t + 1:
De…ning t Pt X= At as the size of bubbles (the share of the value of
bubbles), t evolves over time as
t+1 =

Pt+1
Pt
At+1
At

(27)

t:

The evolution of the size of bubbles depends on the relation between wealth’s
growth rate (denominater) and bubbles’growth rate (numerator).
(24) can be solved for Pt+1 =Pt using t ,
rt (1
Pt+1
=
Pt
(1

p
p)

t)

(28)

:

t

It follows that Pt+1 =Pt is a decreasing function of . When rises, ceterisparibus; the entrepreneur’s required rate of return on bubble assets becomes
lower, because bubble assets become safer.
When rt > qt+1 L ; from (22), interest rate is determined by
rt =

qt+1
1

H

(1
p

t)

:

t

Thus equilibrium interest rate isH

rt = qt+1 M ax

L

;

(1
1 p

t)

:

(29)

t

Note that when rt > qt+1 L ; rt is an increasing function of t ; re‡ecting thetighteness of the credit markets.
Moreover, in order that bubbles do not explode, the following condition

13

must be satis…ed:

Pt+1
At+1
:
(30)
At
Pt
This condition means that wealth’s growth rate must be greater than thegrowth rate of bubbles. Otherwise, the size of bubbles explodes and theeconomy eventually cannot sustain bubbles.
The dynamical system of this economy can be characterized by (25)-(30).
We can obtain the following Proposition.
Proposition 2 There is a saddle point path on which the economy converges
to an equilibrium where all variables (Kt ; At ; qt ; rt ; wt ; Pt ; t ) become constantover time as long as bubbles persist.
Proof. See Appendix.
When the economy gets on the saddle path, t becomes constant overtime, i.e., aggregate wealth of entrepreneurs and asset price grow at thesame rate.15
Thus, rearranging (25), we can derive simple di¤erence equations concerning the capital stock:
8
H (1
Lp
)
>
if rt = qt+1 L< 1 (1 p) L H Kt
Kt+1 =(31)>: H [1 (1 p)]+(1 ) K if r > q
L
:
t
t+1
t
1
(1 p)
And equilibrium asset price, Pt ; follows
Pt =

X (1

)

Kt :

(32)

The asset price rises together with capital stock.
As long as bubbles persist, the economy runs according to the above equations, and converges to the equilibrium where all variables (Kt ; At ; qt ; rt ; wt ; Pt )become constant over time, which Farhi and Tirole (2009a) call a conditional
bubbly steady-state.
A feature of bubbly dynamics is that there is a two-way interaction between asset price and aggregate quantity. An increase in cash ‡ows in period
15

On the saddle path, aggregate wealth, aggregate output, aggregate consumption, andasset price all grow at the same rate.

14

t raises period t bubble price, which in turn increases cash ‡ows and bubble
price in period t + 1 even further. These knock-on e¤ects will continue not
only in period t + 1; but also in period t + 2; t + 3; . Moreover, this anticipated increase in the bubble price is re‡ected in period t bubble price, since
the bubble price is a forward looking variable. In equilibrium, all these mechanisms occur simultaneously, and capital stock and asset price run according
to (31) and (32).

2.7

Anticipated Bailout and Asset Bubbles

In order that stochastic bubbles can exist, the following condition must be
satis…ed:16> 0:
We can obtain the following Proposition.
Proposition 3 The existence condition of stochastic bubbles in the casewhere government bailout is anticipated with probability is

and
H

>

L

<

(1

p):

1
(1

(1 p)
) +p

H
1 ;

Proof. See Appendix.
rises, theIt follows that H
1 is a decreasing function of : The morewider bubble regions becomes. In other words, bubbles are more likely tooccur, the more government bailout is anticipated.
In order to understand the mechanism, let us …rst explain the case of= 0; i.e., no government bailout is anticipated. In this case, for stochastic
bubbles to exist, must be low enough and H must be su¢ ciently high.
Intuitively, in high regions, since interest rate is so great, the bubbles’
growth rate is so high that the economy cannot sustain growing bubbles.
Thus, in high regions, bubbles cannot occur. On the other hand, in low
16If0; even the other equilibrium path with bubbles except for the saddle one
cannot exist (see footnote 13). Thus, no equilibrium path with bubbles can exist if
0:15

regions, interest rate is lower and so is the bubbles’growth rate. As long asH
is su¢ ciently high, wealth’s growth rate becomes su¢ ciently high thatthe economy can sustain growing bubbles.17 From these observations, welearn that bubbles cannot occur in low-productivity economy or in economy
with more e¢ cient …nancial market.18
When bailout is expected, required rate of return of bubbles becomes
lower and so does their growth rate. Thus, even economy with lower productivity or with more e¢ cient …nancial market enters bubble regions. Proposition 3 says that although bailout mitigates adverse e¤ects of bubbles’collapsing on the economy ex-post, it is more likely to cause asset price bubbles.19
Figure 1 illustrates the relationship between bubble regions and :Moreover, we learn the relationship between and .
Proposition 4

is an increasing function of .

Proof. See Appendix.
Proposition 4 says that the anticipated bailout leads to large size bubbles. Together with (32), we learn that when bailout is expected with higher
probability at date t, the asset price jumps up at date t instantaneously.
Before going further, let us mention how interest rate is determined inequilibrium. The following Proposition summarizes this.
Proposition 5 Equilibrium interest rate in the bubble economy depends on
the level of productivity, H :8
H< qt+1 L if H 2 H
1 ; 2 ;
rt =:qt+1 H
if H > H2 ;17

We can use the structure of the bubbleless economy to characterize the existence
condition. The existence condition also says that the interest rate is su¢ ciently lower
than the economy’s growth rate in the bubbleless economy. This condition is similar to theexistence condition in Tirole(1985). In our model, since we consider stochastic bubbles,interest rate must be su¢ ciently low in the bubbleless economy. Otherwise, stochastic
bubbles cannot arise.
18
Here we should note that if
H

L

<

;

then, stochastic bubbles cannot exist. In other words, if productivity is too low, bubbles
never arise for any .
19
We can consider the same thought experiment concerning the relationship between
and p.

16

whereof :[1 (1 p)]+(1 )(1 p)(1 )+(1 +p )

and

H
2 =

L

=

.

H
2 is a decreasing function

Proof. See Appendix.
Intuitively, when productivity is low, L-entrepreneurs cannot lend enoughto H-entrepreneurs because collateral value is low. L-entrepreneurs hold idle
savings, but they cannot invest all of those savings in bubble assets, because
bubble assets are risky. They end up with investing in their own L-projects
for risk-hedge. On the other hand, when productivity is high, since they canlend enough to H-entrepreneurs, they do not need to invest in L-projects forrisk-hedge.

3

Macroeconomic E¤ects of Anticipated Bailout

In this section, we investigate how the anticipated bailout a¤ects macroeconomic performance before the bubble bursts.

3.1

Case with=0

Let us …rst analyze the case with
= 0; i.e., no government bailout isanticipated. We compare the bubble economy and the bubbleless economy.
The following Proposition summarizes macroeconomic impacts of stochastic
bubbles.
Proposition 6 Suppose that the economy is in the steady-state of the bubbleless economy until date t = t0 1: Then, at date t = t0 ; bubbles occur
unexpectedly. Macroeconomic impacts of stochastic bubbles are di¤erent depending on the value of H : There are two regions.
H
H
0
In region 1 where H 2
1 ; 3 ; Yt increase after date t = t +1 onwards until bubbles collapse. In the conditional steady-state, theaggregate output under the stochastic bubbly economy is higher than that
under the bubbleless economy. H
3 is the greater value of the followingL
L p
H [1 (1 p)]+(1 )= H L pH + L (quadratic equation:
L
H ):
1
(1 p)
0
In region 2 when H > H
3 ; Yt decrease after date t = t + 1 onwards
until bubbles collapse. In the conditional steady-state, the aggregate

17

output under the stochastic bubbly economy is lower than that underthe bubbleless economy.
Proof. See Appendix.
In region 1 where productivity is relatively low, the crowd-in e¤ect dominates the crowd-out e¤ect, i.e., bubbles are expansionary until they burst.
Intuitively, when H is relatively low and without bubbles, enough savings
cannot be transfered to H-projects, because the value of collateral is low.
As a result, L-entrepreneurs end up with investing their idle savings in their
L-projects with low returns. Bubbles provide a high return vehicle for them,thus improving e¢ ciency in production by reducing L-projects.
In region 2 where productivity is relatively high, the crowd-out e¤ect
dominates the crowd-in e¤ect. Hence, bubbles are contractionary until they
burst. Intuition is that when H is relatively high, the value of collateralis high enough and enough savings can be transfered to H-projects without
bubbles. In such a situation, when bubbles occur, they crowd savings awayfrom H-projects, thereby reducing H-investments.
Farhi and Tirole (2009a) also analyze whether bubbles are expansionary
or contractionary. They …nd that it depends on what they call outside liquidity, while, in our model, it depends on productivity.20 Figure 1 illustrates thee¤ects of stochastic bubbles on capital stock in the conditionaly steady-state.

3.2

Case with>0

We now compare the bubble economy with = 0 and with > 0: From (31)and Proposition 5,
8
H (1
Lp
)
H
>
if H 2 H< 1 (1 p) L H Kt
1 ; 2 ;
(33)
Kt+1 =>: H [1 (1 p)]+(1 ) K
H
H
if
> 2;
t
1
(1 p)

where Kt+1 is an increasing function of
decreasing function of in H > H
2 :
20

in

H

2

H
1 ;

H
2

; while it is a

We can also characterize the relationship between macroeconomic e¤ects of stochastic
bubbles and ; given H : Hirano and Yanagawa (2010a) characterize the relationship between the e¤ects of bubbles on long-run economic growth rate and within an endogenous
growth framework.

18

From (33), we can understand how a change in
The following Proposition summarizes this.

a¤ects output dynamics.

Proposition 7 Suppose that until date t = s 1;
= 0: Then, at datet = s; changes to any 0 <1 unexpectedly, and 0 <1 for all t s:
The impact of the change in on aggregate output after date t = s onwards
depends on the level of productivity, H : There are three regions.
In region 3 where
after date t = s.

H

2

H
1 ;

H
1

; the economy enters bubble region

H
In region 4 where H 2 H
1,
1 ; 4 ; suppose that until date t = s
the economy is in the conditional steady-state of the bubble economy:
Then, Yt hincrease after date t = s + 1 onwards
until bubbles collapse.
iL[1 (1 p)]+(1 )H
1 p (1
)=
:
4 =
1
(1 p)

H
In region 5 where H > M ax H
1 ; 4 ; suppose that until date t =s 1; the economy is in the conditional steady-state of the bubblele
economy. Then, Yt decrease after date t = s + 1 onwards.

Proof. See Appendix.
In region 3 where productivity is low enough, the economy is in the bubbleless region before the government bailout is anticipated. However, oncethe bailout is expected at date t = s; the economy enters bubble region.
When bubbles occur, they are expansionary (see Proposition 6).21
In region 4 where productivity level is in middle range, the anticipatedbailout accelerates output booms. When bailout is expected at date t = s; Lentrepreneurs take on risk by buying more bubble assets instead of investingin L-projects. The asset price, Pt ; jumps up at date t = s; although aggregate
output does not respond immediately. Together with the increase in the asset
price, the net worth of H-entrepreneurs increases and their investments also
jump up at date t = s, while the share of L-projects is reduced. In otherwords, e¢ ciency in investment improves at date t = s. Thus, aggregate
21

When = 1; bubbles can occur even in the economy with H = H
1;bail . Entrepreneurs
behaves as if bubbles never collapsed. Equilibrium in this case is equivalent to the one indeterministic bubbles. In this case, even if bubbles occur, all macroeconomic variables inthe bubble economy is the same as the ones in the bubbleless economy, because the rateof return of bubbles equals the rate of return of L-projects.

19capital stock, output, and wage rate all jump up at date t = s + 1; and they
all continue to increase over time until bubbles collapse. Moreover, becauseof the acceleration of aggregate output, the asset price continues to rise evenfurther and large bubbles are created, which in turn requires a large scale
government intervention when bubbles collapse.22 Figure 2 captures how thechange in a¤ects aggregate output’s dynamics in region 4, and Figure 3
summarizes the impulse response of macroeconomic variables when bailout
is expected in region 4.23
In region 5 where productivity is high enough, more savings are allocatedto H-projects even before government intervention is anticipated, because
the value of collateral is high. When bailout is expected at date t = s;interest rate rises substantially, which crowds savings away from H-projects,
thus dampening output booms.24 Figure 4 illustrates dampening e¤ects on
output booms.25
Figure 5 summarizes the e¤ect of the change in on the level of aggregate
capital stock in the conditional steady-state.26

4

Ex-Post Bailout that Maximizes Output

In this …nal section, we examine ex-post bailout that maximizes output. We
know that output gets higher, the more capital is installed. We also knowthat if capital stock before the bubble bursts is high, the capital stock after
the bubble bursts is also high, and vice versa. Thus, the ex-post bailout
policy that maximizes output is to maximize the level of capital stock before
the bubble bursts. Indeed, depending on the level of productivity, there is an
22

Large bubbles are created, Large amount of public money, Pt X, needs to be injected
into the entreprenerial sector to rescue the economy. Otherwise, a large free-fall occurs.
This suggests that the bailout policy that aims at stabilizing the economy during bubble
bursts ends up with increasing boom-bust cycles.
23
Figure 5, we consider a situation where until date s; the economy runs on the bubbly
dynamics toward the conditional steady-state.
24
In region 3, although the anticipated bailout dampens aggregate capital stock, aggregate output, and wage rate after date t = s0 + 1; the impacts on asset price andentrepreneurs’consumption are ambiguous and depend on parameter values.
25
In region 5, the anticipated bailout decreases boom-bust cycles, thus stabilizing the
economy.
26
We can also characterize the e¤ects of the change in on aggregate economic activity
and asset price with , given H :20

optimal value ofsummarizes this.

that maximizes output. The following Proposition

Proposition 8 Ex-post bailout that maximizes output depends on the levelHH
of productivity, H : In H
= 0: In H 2 H= ^;3 ;1 ( = 1); 3 ;where ^ is the value of at which H = H
2 :This proposition suggests that in high productivity regions, the anticipated bailout ends up with decreasing output. Thus, no bailout policy isdesirable in terms of output, i.e.,= 0. On the other hand, in low productivity regions, a rise in initially increases output by crowding L-projects
out and then decreases it by crowding H-projects out. Thus, desirable policy
is to set
= ^ at which L-projects are just eliminated.
Before concluding, we should emphasize limitations of our analysis. Here
we only focus on the ex-post bailout that maximizes output. The policy
maximizes ex-ante welfare for workers, but in order to derive an optimal expost bailout policy, we need to conduct welfare analysis for entrepreneurs.
These are left for future research.

5

Conclusion

This paper theoretically investigates the relationship between asset price bubbles and bailout. We show that although bailout may mitigate adverse e¤ectsof bubbles’bursting ex-post, it is more likely to cause asset price bubbles byencouraging risk-taking behavior ex-ante. In other words, bubbles are more
likely to occur, the more government bailout is anticipated. Moreover, we
examine the e¤ects of the anticipated bailout on boom-bust cycles. We …ndthat when productivity is relatively low, the anticipated bailout accelerates
output booms and creates large bubbles, thus destabilizing the economy. On
the other hand, when productivity is relatively high, the anticipated bailout
dampens output booms, thus stabilizing the economy. Finally, we analyze
an ex-post bailout policy that maximizes output.
Obviously, our analysis can be extended in several directions. Let us discuss some of them here. First, in the present paper, we have only considered
the bailout of buying bubble assets. We can also think of di¤erent typesof bailous such as low interest rate policy. As a number of authors suchas Rajan (2010) suggested, one of the causes of the recent …nancial crisis
21

in the U.S. lies in what is called “Greenspan Put”, which says that the expost low interest rate policy by central bank after bubbles’collapse inducedbanks or investment banks to take on more risk ex-ante, which ended up with
causing bubbles in …nancail markets. Our framework would be extended to
examine the ex-ante e¤ects of an ex-post interest rate policy in a full blown
macroeconomic framework.
The second direction would be related to the …rst direction. We can
examine the relationship between monetary policy and asset price bubbles.
Our model suggests that bubbles cannot arise in low productivity economy( H is too low), because in those regions, wealth’s growth rate is too low
compared to interest rate. Our model’s prediction is that if monetary policy
can lower interest rate, even low productivity economy will enter bubble
regions. This suggests that expansionary monetary policy can be a cause forthe emergence of asset price bubbles. A recent paper by Gali (2011) analyzes
the impact of monetary policy rule on asset price bubbles by using an OLG
model with nominal rigidities. These are left for future research.

22

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25

6

Appendix

6.1

Proof of Proposition 1

Rearranging (??) by using (??), (??), and (29), we can obtain the dynamic
equation concerning t :
8
(1 p
t)
(1 p)
>
L
>
;
< (1+ H L p) + (1 t )(1 p)t if rt = qt+1
t
L
H
(1 p)
t
(34)
t+1 =
>
>
:
1
L
if r > q:(1 p) (1)

In addition, (??) can be rearranged as
8 h
H
L
L
L
(1+ L
>
H p)
>
<
1
t
Kt+1 =
>
>
H [1
:
t]
K
1

t

t

t

t

t

t

i

Kt

t+1

if rt = qt+1 L ;(35)
if rt > qt+1

L:

The dynamical system of this economy can be characterized by (34) and (35).
Given an initial state variable, K0 ; there is a unique initial bubble price,
P0 which satis…es
8[1(1 p)]=v
L>< 1 [1 (1 p)]=v (1 p) if rt = qt+1 ;
P0 X
(36)
=
0
( K0 + P0 X) >
(1 p)
:
L
if rt > qt+1 ;(1 )Once P0 is determined, t becomes constant over time after date 1 onwards
until bubbles collapse:
Then, the dynamics of the capital stock is governed by
8 h
i
H
L
L
L
(1+ L
>
H p)
>
<
Kt if rt = qt+1 L ;1
Kt+1 =
>
>
H [1
]
:
L
K
if r > q;t1which is equivalent to (31).

26

t

t+1

6.2

Proof of Proposition 2

From (36), in order that

1
and

[1
[1

(1
(1

p)] =v(1p)] =v(1 p)(1)

When we solve for
Proposition 2.

6.3> 0; two conditions must be satis…ed:

Hp) > 0;> 0:and ; we can obtain the existence conditions in

Proof of Proposition 3

The evolution of the capital stock in the bubbleless economy,
8HLL(1 p)=< (1+ L H p) L Kt if H 2 H
1 ;Kt+1 =:
H
if H > L (1 p)= :
Kt

;
(37)

where K denotes the capital stock in the bubbleless economy.
L(1 p)= if <(1 p): Then, given the same
We know H
2 <initial capital stock at date t; Kt = Kt ; by comparing (31) to (37), weH
HH
see Kt+1 > Kt+1 for any H 2= L (1 p)= ;1 ; 2 : And whenKt+1 < Kt+1 if <(1 p): Hence, by continuity, there is a threshold
H
H
H
H
H
= H
2 H
value of3 ;2 ; 3 ; Kt+1 > Kt+1 : For
3 : For any
HH> 3 ; Kt+1 < Kt+1 :Kt+1 = Kt+1 : For any

6.4

Proof of Proposition 4

By substituting the steady-state value of = [ (1 p)
qt+1 H (1pt )=(1t ), (29) can be rewritten as
rt = qt+1 M ax

L

;

H

] = (1) into;where Since H is a linear increasing function of H ; there is a thresholdH
Lvalue of H
= below which rt = qt+1 L and above which
2 =rt = qt+1 H :
27

6.5

Proof of Proposition 5

On the saddle point path,=t becomes constant over time and satis…es8[ +(1 ) ] f1 [ +(1 ) ] (1 p)g=v>(1<
1 f1 [ +(1 ) ] (1 p)g=v
>
:

[ +(1

) ] (1 p)(1 )p) if rt = qt+1 L ;(38)
if rt > qt+1

L:

From (38), when we solve for H and as in the proof of Proposition 2, we
can obtain the existence conditions in Proposition 5.

6.6

Proof of Proposition 6

From (38), it is obvious that

6.7

is an increasing function of :Proof of Proposition 7H
We know that there is a value of H = H
4 ; where 4 is the greater value ofH
HL
which satis…es
F ( ) = v= [1(1 p)] : We know that Kt+1 is anLincreasing function of when rt = qt+1 ; and Kt+1 is a decreasing functionof when rt > qt+1 L , and H
2;bail is a decreasing function of : Hence, given>0H
the same initial capital stock at date t, Kt+1
Kt+1 for any H 2 H
1 ; 4 ;
>0where Kt+1
denotes the capital stock at date t + 1 when bailout is expected>0Hwith probability : For any H > M ax H
1 ; 4 ; Kt+1 < Kt+1 :6.8

Proof of Proposition 8

H
In H 2 H
is to set
= 0: In this region,
2 ; 3 ; the optimal value ofcompare two types of economies. One is the economy with = 0, and theother is the economy with > 0: If the economy starts from the same levelof capital stock initially, we know that capital stock in the economy with
> 0 is lower, because from (33), Kt+1 is a decreasing function of . Thus,H^= 0: In H 2 H
1 ( = 1); 2 h; there is a critical value ofi at which
H
H
= H
< H
2 0; ^ and H > H
2 ^ ; 1 : Together
2 ; where
2 if
2 if
h
with (33), we learn that Kt+1 is an increasing function of in 2 0; ^ ;i
and is a decreasing function of in 2 ^ ; 1 : So, an increase in has

28

non-monotonic e¤ect on output and wage rate. Thus, the optimal value ofH
in H 2 H
is to set= ^ :27
1 ( = 1); 2

6.9

Proof : Behavior of H-entrepreneurs

We verify that H-entrepreneurs never buy bubbles in equilibrium. In orderthat the short sale constraint binds, we need to show
1 Pt+1
1
>
E
:
t
cit
cit+1 Pt

(39)

We know that when the borrowing constraint is binding,1
rt qt+1 H (1
)
=
E
:
(40)
t
i
i
H
ct
ct+1 rt
qt+1
i
h H
rt
(1 )
i
We also know that ct+1 = (1) rt qt+1 H if (39) is true. Considering
this, by inserting (40) into (39), (39) can be rearranged as
Pt+1
)+
Pt

H
1 rt (qt+1

cit+1

rt

qt+1 H ( PPt+1
t

qt+1

H

rt )

> 0:

We see that the second term in the numerator is positive as long as
Next we show that the …rst term is also positive, that is, we show
qt+1 H >

(41)
> 0:

Pt+1
:
Pt

By using (10) and (32), the above can be rewritten as
H(42)

Kt > Kt+1 :First we show that (42) holds true when rt = qt+1 L : By using (31), (42) is

H

>

(1 +

H

L

L

H

p)

L

L

(1

p) = [1

(1

p)] :

(43)

27
At H = H
1;bail ( = 1); wage rate in the bubble economy and that in the bubbleless
economy is the same.

29

When

H

=

H
1 ; (43) isH
1 >which is true.
And when H = H
2 ; we know
relation, (42) can be rearranged as
)+

:

= 1
L

H
2 (1

L

p

L

L

p( L

H
2
(1
H
2

H
2 ).

By using the) > 0;

which is true. Since the right hand side of (43) is a convex function of H ;H(42) holds true in H 2 H
1 ; 2 .
Next, we show that (42) holds true when rt > qt+1 L : From (31), by using; (42) is1> (1);
which is true.

6.10

Proof : Behavior of Workers

We verify that both the borrowing constraint and the short sale constraint
bind for workers in equiibrium. In order that both constraints bind, thefollowing conditions must be satis…ed:rt1> Et w :
w
ct
ct+1

(44)

1
1 Pt+1> Et i:i
ct
ct+1 Pt(45)

We know that cw
t = wt if (44) and (45) are true. (44) can be rewritten as
Kt+1 > Kt rt :(46)

First we show that (46) holds true when rt = qt+1 L : By using (10) and (31),(46) isH(1)>1(1 p):
(47)
p L
H
When

H

=

H
1 ; (47) is

1>
30

;

which is true. Since the left hand side of (47) is an increasing function ofHH; (46) holds true in H 2 H
1 ; 2 .
Next, we show that (46) holds true when rt > qt+1 L : By using (31), (46)is rearranged as
(1

p)(1

) > f [1

(1

p)]

(1

+ p )gThis inequality condition holds true for any in < (1 p):
Finally, we show that (45) holds true. By using (10) and (32), (45) is1>
which is true.

31

;

Kt+1
booms

bubbly dynamics
bust

dynamics inbubbleless economy

Kt
Figure 1. Dynamic Effect of Stochastic Bubbles in Region 1

32

Kt+1
dynamics inbubbleless economy

bubbly dynamics

Kt
Figure 2. Dynamic Effect of Stochastic Bubbles in Region 2

33

K１-σ

BubbleBubbleless
Bubble Region
region 1
αL

α 1H

α2H

region 2α3HαHFigure 3. Effect of Stochastic Bubbles on Capital Stock

34

Kt+1
bubbly dynamics
when bailout isanticipatedbubbly dynamics
when bailout is not
anticipateddynamics inbubbleless economy

Kt
Figure 4. Bubbly Dynamics in Relatively Lowα

35

Pt

ZHt
Anticipated
Unanticipatedanticipated

burst

t=s’

t=s

t

Yt

t=s’

t=s

t=s’

t=s

t

Ct

t=s’

t=s

t

Figure 5. Bubbly Booms : Anticipated vs. Unanticipated

36

t

K１-σ

Bubble(λ=0)
Bubble(λ>0)
Bubbleless
Bubble Region
region region
region3
45

J H1,bail J H1

JH4

JH

Figure 6. Effect of Anticipated Bailout on Capital Stock37