Discussion Paper No. 2012-002 Asset Bubbles, Economic Growth, and a Self-fulfilling Financial Crisis Takuma Kunieda and Akihisa Shibata

Asset Bubbles, Economic Growth, and a Self-fulfillingFinancial Crisis∗

Takuma Kunieda†

Department of Economics and Finance,

City University of Hong Kong

Akihisa Shibata‡

Institute of Economic Research,

Kyoto University

May 24, 2012

Abstract

We develop a dynamic general equilibrium growth model with infinitely lived het-

erogeneous agents to describe a self-fulfilling financial crisis accompanied by an asset

bubble burst as a rational expectations equilibrium. Because of financial market imper-

fections, asset bubbles appear under mild parameter conditions despite the assump-

tion of infinitely lived agents. Although these bubbles have both crowd-in liquidity

and crowd-out effects on investment, the former effect always dominates the latter.

Thus, a self-fulfilling financial crisis accompanied by an asset bubble burst results in

an economic recession. This phenomenon is consistent with empirical observations on

financial crises in the existing literature. In addition, we present an effective govern-

ment policy to avoid self-fulfilling financial crises.

Keywords: Bubble’s crowd-in effect; Financial market imperfections; Sunspots; Self-fulfilling

financial crisis; Economic growth.

JEL Classification Numbers: E32; E44; O41

∗This research is financially supported by Grant-in-Aid for Specially Promoted Research (No. 23000001).†Corresponding author. P7315, Academic Building, City University of Hong Kong, 83 Tat Chee Avenue,

Kowloon Tong, Hong Kong. Phone: +852-3442-7960, Fax: +852-3442-0195, E-mail: tkunieda@cityu.edu.hk‡Institute of Economic Research, Kyoto University, Yoshida-honmachi, Sakyo-ku, Kyoto 606-8501,

JAPAN. Phone: +81-75-753-7126, Fax: +81-75-753-7198, E-mail: shibata@kier.kyoto-u.ac.jp

1

1 Introduction

Over the past 20 years, numerous countries have suffered from financial crises followed by

serious economic recessions. Among them, the Latin American debt crisis in the early 1980s,

the Japanese asset price bubble burst in 1990, the Asian financial crisis in 1997, and the US

subprime loan crisis in the late 2000s were the most severe.1 They occurred even though

fundamental economic measures such as growth and inflation rates were fairly sound just

before the crises.

[Figure 1 around here]

Figure 1 presents time series of the per capita real gross domestic product (GDP), stock

prices, and land prices in the United States (panel A) and Japan (panel B). Deviations from

the trends of these variables are plotted from 1995 to 2009 in the United States and from

1985 to 1994 in Japan.2 An asset bubble is defined as the difference between the fundamental

and market values of an asset. In general, it is difficult to identify the fundamental value

of an asset correctly. However, in most cases, the bubble component exhibits an explosive

movement. Thus, we assume here that the trend component of an asset price reflects the

movement of its fundamental value, and deviations from trends in stock and land prices

represent their bubble components. As Figure 1 shows, the United States experienced two

economic booms during this period. The first was the so-called dot-com bubble that burst

in 2001. The figure shows that the burst of the stock price bubble was accompanied by an

economic recession. Land prices were relatively stable during this event. The second is a real

estate lending boom that resulted in the so-called subprime loan crisis from 2007 to 2008. In

this crisis, the burst of the land price bubble was followed by a serious economic downturn

and a sharp decline in stock prices. However, Japan experienced one economic boom during

the period shown, which was led by a stock price bubble that burst in 1990. This burst was

also followed by a severe economic depression.

These historical observations lead to the following questions. Do asset bubbles promote

economic growth? Why do asset bubbles burst? Why does a bubble burst result in an

economic recession even though fundamental variables such as the technology level and in-

dividual tastes do not seem to change just before an asset bubble bursts?3 To address these

1Laeven and Valencia (2008) and Reinhart and Rogoff (2009) provide very useful reviews and datasets

on financial crises.2Deviations from the trends are computed using the Hodrick-Prescott filter. See the Appendix for details

on the data sources.3To the best of our knowledge, there is little evidence that exogenous negative technological shocks

2

questions, we develop a dynamic general equilibrium model with infinitely lived heteroge-

neous agents. This research aims to describe a financial crisis accompanied by a bubble

burst, which results in a severe recession, as a rational expectations equilibrium of our dy-

namic general equilibrium model. The financial crisis in our model is self-fulfilling in the

sense that rational expectations drive an economy into a financial crisis without any changes

in fundamentals such as the productivity or the individual tastes. In addition, we present

an effective policy to avoid self-fulfilling financial crises.

Traditional growth models dealing with asset bubbles have suggested that asset bubbles

crowd out investment and hinder production (e.g., Tirole, 1985; Weil, 1987; Bertocchi and

Yong, 1995; Kunieda, 2008; Matsuoka and Shibata, 2012). This effect occurs because if

individuals save in the form of an intrinsically useless asset instead of capital, the supply of

capital in the next period decreases. Along these lines, Saint-Paul (1992), Grossman and

Yanagawa (1993), King and Ferguson (1993), and Futagami and Shibata (2000) have devel-

oped endogenous growth models with overlapping generations and showed that the growth

rate is lower when asset bubbles appear than when no asset bubbles are present because

of the crowd-out effect on investment. We may consider these growth models with asset

bubbles first-generation models because only asset bubbles’ crowd-out effects on investment

emerge in these models. The weak point of first-generation models is that their results are

not consistent with the empirical evidence that asset bubbles seem to stimulate economic

growth, and the bubble burst is likely to result in economic downturns, as shown in Figure

1.

In contrast, many researchers such as Kocherlakota (2009), Hirano and Yanagawa (2010),

Kiyotaki and Moore (2011), Farhi and Tirole (2011), Martin and Ventura (2011), and Miao

and Wang (2011) have recently developed new dynamic general equilibrium models, in which

asset bubbles promote investment and economic growth. These studies have also investigated

business cycles and/or financial crises that cause economic recessions.4 The growth models

analyzing this new trend may be considerered second-generation models because they focus

on the crowd-in liquidity effect of asset bubbles on investment. Our research belongs to this

new literature.

We explicitly introduce financial market imperfections into the model. The assumption of

an imperfect financial market has twofold importance. The first is related to the existence of

asset bubbles. A bubble on an asset is more likely to appear in an economy with an imperfect

triggered the bubble burst in past financial crises.4Olivier (2000) also develops a dynamic general equilibrium model in which asset bubbles are growth-

enhancing; however, the model does not study financial crises.

3

financial market than in one with a perfect financial market. Theorem 3.3 in Santos and

Woodford (1997) implies that the necessary condition for an asset bubble to appear is that

for any state price of an asset, the present value of the asset diverges to infinity. From our

model’s perspective, their theorem means that if the equilibrium interest rate is less than

the growth rate of the economy, there is a possibility that an asset bubble will develop.

In a financially constrained economy, the equilibrium interest rate is generically less than

that of a financially unconstrained economy. This is because the demand for borrowing is

smaller in a financially constrained economy than in a financially unconstrained economy.

Therefore, Santos and Woodford’s necessary condition for the existence of an asset bubble is

more likely to be satisfied in a financially constrained economy. Once asset bubbles emerge,

the equilibrium interest rate increases because the market supply of capital decreases owing

to the existence of asset bubbles.

The second role of financial market imperfections relates to the liquidity constraints

that investors face. As a result of an agency problem in a financially constrained economy,

individuals’ savings cannot be necessarily invested in high-quality projects that yield high

returns, that is, investment opportunities in high-return projects are not realized in such

an economy. In this situation, if an intrinsically useless asset, such as a paper asset, takes

a positive value, that is, if a bubble on an intrinsically useless asset exists, the equilibrium

interest rate increases. Thus, the intrinsically useless asset, which has a rate of return equal

to the interest rate, becomes a beneficial vehicle that stores the output value from today to

tomorrow.

When holding an intrinsically useless asset, liquidity-constrained investors face two con-

flicting effects created by asset bubbles. The first is a crowd-out effect on investment. The

savings of individuals in an economy are not only invested in investment projects, which

produce output, but are also used to hold the intrinsically useless asset. Therefore, the

positively valued, intrinsically useless asset crowds investment out. Holding the intrinsically

useless asset crowds out low-return investment projects because of the increased interest

rate. If a positively valued, intrinsically useless asset, which yields a higher interest rate

than low-return projects, is carried over to the next period and sold in the financial market

to obtain production resources to invest in high-return projects, the equilibrium growth rates

may increase. This is the second effect, that is, the crowd-in liquidity effect of asset bubbles.

When the financial market is imperfect, an intrinsically useless asset is a beneficial vehicle

for storing value because it yields a higher interest rate than savings yield in an economy

without asset bubbles.

4

In the economy in our model, the productivity of investment projects that produce gen-

eral goods varies among agents, implying that agents in the economy are heterogeneous in

productivity when they engage in production. In equilibrium, we obtain two steady states

under certain conditions. One is a steady state in which the intrinsically useless asset has

no value, which we call a bubbleless steady state. The other is a steady state in which the

intrinsically useless asset has a positive value, which we call a bubbly steady state. Although

both the crowd-in liquidity effect and the crowd-out effect operate in a bubbly equilibrium,

the crowd-in liquidity effect always dominates the crowd-out effect in equilibrium. Thus, the

growth rate in the bubbly steady state is higher than that in the bubbleless steady state.

Therefore, if an asset bubble bursts, the economy will go into a recession.

The aforementioned second-generation growth models consider both the crowd-out and

crowd-in liquidity effects of asset bubbles in financially constrained economies. These studies

can be classified into two groups depending on modeling strategy. The first group comprising

Farhi and Tirole (2011) and Martin and Ventura (2011) employs the overlapping generations

framework of Samuelson (1958) and Tirole (1985) and analyzes the crowd-out and crowd-in

liquidity effects of bubbles. The second group constructs infinitely lived agent models with

heterogeneity across agents to analyze the two effects. This group includes Kocherlakota

(2009), Hirano and Yanagawa (2010), Kiyotaki and Moore (2011), and Miao and Wang

(2011). Among them, Miao and Wang (2011) primarily consider bubbles on a productive

asset, whereas the others consider them on an intrinsically useless asset. Along the latter

studies, we analyze the macroeconomic implications of bubbles on an intrinsically useless

asset by assuming infinitely lived agents who are heterogeneous in productivity for general

goods production.

It should be noted here that all models other than ours assume a binary distribution with

respect to the productivity difference. However, our model assumes that the productivity of

agents is continuously distributed among agents. As a result of this continuity in the produc-

tivity distribution, we are able to investigate the global dynamics of the equilibrium interest

rates.5 Clarifying the global dynamics in the economy, we are able to derive a two-state

stationary sunspot equilibrium in which a financial crisis is described as a rational expecta-

tions equilibrium without any changes in the fundamental variables. Although Kocherlakota

(2009), Wang and Wen (2010), Farhi and Tirole (2011), Martin and Ventura (2011), and

5Wang and Wen (2010) also assume the continuous productivity distribution; however, they do not

analytically investigate the dynamics of the economy. Although Miao and Wang (2011) investigate the

dynamics of the economy, they study only the local dynamics around the steady states. Moreover, they

develop an infinite horizon model with incomplete financial markets.

5

Miao and Wang (2011) also derive two-state stationary sunspot equilibria, they assume from

the beginning of their investigations that one state is bubbleless in their sunspot equilibria.

In contrast, we prove that one state must be bubbleless in the two-state sunspot equilibrium

in our model. In this sense, in contrast with the extant second-generation models, the bub-

bleless state endogenously appears in the process of proving the existence of the two-state

stationary sunspot equilibrium.6

Moreover, of the second-generation models, only ours discusses a government policy to

avoid self-fulfilling financial crises, although Kocherlakota (2009) and Miao and Wang (2011)

discuss a government policy to restore an economy that has experienced a bubble burst to a

healthy state.7

The remainder of this paper proceeds as follows. In section 2, we present a model in

which agents are ex-ante homogeneous but ex-post heterogeneous because of productivity

shocks regarding general goods production, and derive a dynamical system with respect to

the equilibrium interest rate. The dynamics and steady states of the economy are discussed

in section 3. In this section, we derive a condition under which an intrinsically useless asset

has a positive value. In section 4, we describe a financial crisis as a rational expectations

equilibrium. In section 5, we discuss how self-fulfilling financial crises may be avoided, and

section 6 concludes the paper.

2 Model

The economy consists of one unit measure of infinitely lived agents and an infinitely lived

representative financial intermediary. Time is discrete and goes from 0 to∞. As will be seenlater, only highly productive agents produce general goods. General goods created at time

t are used interchangeably as physical capital and consumption goods at time t. However,

they are perishable in one period. This implies that physical capital, which is used as an

input in general goods production, depreciates entirely in one period.

While the infinitely lived agents maximize their lifetime utility, the infinitely lived finan-

cial intermediary only accommodates borrowers with loans and accepts deposits from savers

to balance its balance sheet.

6Wang and Wen (2010) show that even rational agents are willing to invest in bubbles despite a positive

probability that they will burst. Using calibration exercises, they also show that changes in the perceived

systemic risk can generate asset price collapse.7Furthermore, Kocherlakota (2009) and Hirano and Yanagawa (2010) assume that the future value of an

intrinsically useless asset can be pledged to derive the bubbly equilibrium. By contrast, we assume that only

a part of the net worth currently held by agents can be pledged.

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2.1 Maximization problem

Each agent is endowed with a production function at time t such that

yt = AΦt−1kt−1,

where yt represents general goods, kt−1 is physical capital, and Φt−1 is the productivity

at time t. The productivity Φt−1 is a function of a stochastic event ωt−1, where ωt−1 ∈Ω | Φt−1(ωt−1) ≤ Φ is an element of a σ-algebra F of a probability space (Ω,F , P ). Inother words, Φt−1(ωt−1) is a random variable received at time t − 1. Following Angeletos(2007), we assume that the idiosyncratic productivity shocks Φ0(ω0),Φ1(ω1), ... (and the

stochastic events ω0,ω1, ...) are independent and identically distributed across both time and

agents (the i.i.d. assumption), implying that the distributions with respect to Φ0,Φ1, ... are

identical. Specifically, we assume that Φ has support over [d, h] or [d,∞), where d ≥ 0 andh <∞, and its cumulative distribution function is given by G(Φ), where G(Φ) is continuous,differentiable and strictly increasing on the support.

Let the histories of stochastic events and the idiosyncratic productivity shocks until time

t− 1 be respectively denoted by ωt−1 = ω0,ω1, ...ωt−1 and Φt−1 = Φ0,Φ1, ...Φt−1. Then,there exists a probability space (Ωt,F t, P t), which is a Cartesian product of t copies of

(Ω,F , P ) such that Φt−1(ωt−1) is a vector function of the history ωt−1 on (Ωt,F t, P t).

Because agents receive idiosyncratic productivity shocks, they are ex-post heterogeneous

despite being ex-ante homogeneous. Note that production takes one gestation period. In

other words, an idiosyncratic productivity shock Φt−1(ωt−1) with respect to production at

time t is realized at time t− 1. Each agent obtains information about the idiosyncratic pro-ductivity shock Φt−1(ωt−1) before kt−1 is installed, which is input for production at time t. It

is assumed that low productivity cannot be insured before the realization of an idiosyncratic

productivity shock because no insurance market for idiosyncratic productivity shocks exists.

An agent at time t maximizes the following expected lifetime utility:

Ut = E

" ∞Xs=0

βs ln ct+s(ωt+s)

¯Φt

#,

where ct+s(ωt+s) is the consumption at time t+s, β ∈ (0, 1) is the agent’s subjective discount

factor, and E[.|Φt] is the expected value of a variable given an information set Φt at time t.The flow budget constraint of the agent is given by

kt+s(ωt+s)+bt+s(ω

t+s) = AΦt+s−1kt+s−1(ωt+s−1)+rt+sbt+s−1(ωt+s−1)−ct+s(ωt+s) for t+s ≥ 1,(1)

7

where bt+s is a deposit if positive and a debt if negative, and rt+s is the gross interest rate.

At t+ s = 0, the flow budget constraint is given by k0+ b0 = w0− c0, where w0 is the initialendowment that the agent holds at birth, which is common to all agents. In what follows,

ωt+s is omitted to save space, unless the omission is confusing.

If an agent borrows from the financial intermediary, she faces a credit constraint. Fol-

lowing Aghion et al. (1999), Aghion and Banerjee (2005), Aghion et al. (2005), and Antras

and Caballero (2009), we assume that the credit constraint is given by

bt+s ≥ −θat+s,

where at+s := AΦt+s−1kt+s−1 + rt+sbt+s−1 − ct+s for t + s ≥ 1 (or at+s := wt+s − ct+s fort + s = 0) is her saving, or the net worth remaining after she consumes at time t + s.

Henceforth, we call at+s net worth. θ ∈ (0,∞) is the measure of the extent of the creditconstraint. As a result of the credit constraint, an agent can borrow from the financial

intermediary only up to θ times her net worth. Two types of microfoundations for the credit

constraint are provided in the appendix. Because at+s = kt+s + bt+s from Eq.(1), the credit

constraint is rewritten as

bt+s ≥ −μkt+s, (2)

where μ := θ/(1 + θ) ∈ (0, 1) is also the measure of the extent of the credit constraint.Finally, the non-negativity constraint of physical capital is given by

kt+s ≥ 0. (3)

Each agent maximizes his/her expected lifetime utility Ut subject to Eqs. (1)-(3).

2.2 Optimal agent behavior

Define

φt := rt+1/A. (4)

Then, we can easily obtain the following results. It is optimal for agents with Φt > φt to

invest in a project, borrow up to the limit of the credit constraint and engage in general

goods production, whereas it is optimal for agents with Φt < φt to deposit their net worth

with the financial intermediary without engaging in general goods production. We call the

former investors and the latter depositors.

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2.2.1 Investors at time t (Φt(ωt) > φt)

We find from Eqs. (1) and (2) that an investor who has net worth at at time t invests

kt = at/(1 − μ) in an investment project, borrowing −bt = μkt. Therefore, the flow budget

constraint at time t+ 1 is given by

at+1 = (AΦt − rt+1μ)kt − ct+1,

or equivalently,

at+1 =AΦt − rt+1μ1− μ

at − ct+1. (5)

2.2.2 Depositors at time t (Φt(ωt) < φt)

Similarly, we find from Eqs. (1) and (3) that a depositor who has net worth at at time

t deposits bt = at with the financial intermediary without investing. Therefore, the flow

budget constraint at time t+ 1 is rewritten as

at+1 = rt+1at − ct+1. (6)

2.2.3 Intensive budget constraints and the Euler equation

By defining Rt+s := maxrt+s, AΦt+s−1−rt+sμ1−μ and from Eqs. (5) and (6), the flow budget

constraint for any agents at time t+ s is obtained as follows:

at+s = Rt+sat+s−1 − ct+s. (7)

Each agent solves the intertemporal maximization problem subject to Eq. (7). The Euler

equation for all t ≥ 0 is given by1

ct= βE

hRt+1

1

ct+1

¯Φti. (8)

Because the lifetime utility function is log-linear, it follows from Eqs. (7) and (8), and

the transversality condition that the equilibrium dynamic equation of net worth at of an

agent is given by8

at+1 = βRt+1at. (9)

8Eq.(9) is obtained as follows. From Eq.(7), we have E[at+1/ct+1|Φt]=atE[Rt+1/ct+1|Φt]−1. SubstitutingEq.(8) into this equation, we have at/ct=βE[at+1/ct+1|Φt] + β. From this equation and the law of iterated

expectations, we obtain at/ct=βsE[at+s/ct+s|Φt]+β + β2 + ... + βs. From the transversality condition, we

have lims→∞ βsE[at+s/ct+s|Φt] = 0. Therefore, at/ct = β/(1− β) for all t ≥ 0, and thus, at+1 = βRt+1at.

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2.3 Financial Intermediary

Following Grandmont (1983) and Rochon and Polemarchakis (2006), we assume that the

financial sector is competitive, and thus that the representative financial intermediary cannot

gain profits from its business. As assumed in the previous section, the financial intermediary

imposes credit constraints on agents. Additionally, it accepts deposits from agents and lends

financial capital to investors. The financial intermediary purchases an intrinsically useless

asset with the excess total saving.9

The balance sheet of the financial intermediary is given by

Lt + Bt = Dt,

where Lt and Dt are aggregate loans and deposits, respectively. We assume that the nominal

supply of the intrinsically useless asset is constant and given by M . Accordingly, it follows

that ptM = Bt, where pt is the price of the intrinsically useless asset measured in terms of

general goods at time t. The intrinsically useless asset is freely disposable. Therefore, Bt is

non-negative. Because an asset bubble is defined as the difference between the fundamental

and market values of an asset, if Bt is strictly greater than zero, we say that there is a

bubble on the intrinsically useless asset. Note that for this asset to have a positive value, it

must hold that pt/pt−1 ≥ rt. Otherwise, the financial intermediary does not buy the asset.Moreover, as there is no opportunity for the financial intermediary to gain profits, it follows

that pt/pt−1 = rt in equilibrium if the asset has a positive value. Therefore, we have a

dynamic equation with respect to Bt as follows:

Bt = rtBt−1. (10)

Aggregating Eq.(1) across all agents, we find that Eq.(10) holds if and only if the goods

market clears.

2.4 Aggregation

We assume that the law of large numbers can be applied to the population in the economy.

Because at = βRtat−1 from Eq.(9), the net worth at of an agent at time t is given by

at(ωt) = β(AΦt−1kt−1(ωt−1) + rtbt−1(ωt−1)).

In this equation, we should note that kt−1(ωt−1) = 0 and bt−1(ωt−1) = at−1(ωt−1) for an

agent with Φt−1(ωt−1) < φt−1, whereas kt−1(ωt−1) = at−1(ωt−1)/(1 − μ) and bt−1(ωt−1) =9In the current model, we assume that the financial intermediary purchases an intrinsically useless asset.

This assumption is only for convenience and does not affect our results.

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−μat−1(ωt−1)/(1 − μ) for an agent with Φt−1(ωt−1) > φt−1, as discussed in section 2.2.

The distributions of Φt−1kt−1(ωt−1) and rtbt−1(ωt−1) across agents are independent of the

realization of ωt due to the i.i.d. assumption regarding ωt∞t=0. Applying the law of largenumbers to the population in the economy, we obtain aggregate net worth at(ωt) across

agents who experience a stochastic event ωt and receive productivity Φt(ωt) as follows:

at(ωt) :=

ZΩtat(ω

t)dP t(ωt−1) = β

ZΩt(AΦt−1kt−1(ωt−1) + rtbt−1(ωt−1))dP t(ωt−1).

= β(Yt + rtBt−1), (11)

where Yt :=RΩtAΦt−1kt−1(ωt−1)dP t(ωt−1) is the total output and Bt−1 = Dt−1 − Lt−1 =R

Ωtbt−1(ωt−1)dP t(ωt−1) is the intrinsically useless asset held by the financial intermediary.

Note that ωt−1 = ω0,ω1, ...ωt−1 is an element in (Ωt,F t, P t).

The right-hand side of Eq. (11) is independent of the realization of Φt(ωt) because of the

law of large numbers and the i.i.d. assumption regarding Φt∞t=0. Eq. (11) expresses thedistribution of net worth associated with a stochastic event ωt in the economy at time t (see

Figure 2). Eq. (11) is useful for computing aggregate borrowing, aggregate investment, and

aggregate deposits in what follows.

[Figure 2 around here]

2.4.1 Investors at time t (Φt(ωt) > φt)

An investor who has net worth at(ωt) at time t invests kt(ω

t) = at(ωt)/(1−μ) in an investment

project, borrowing −bt(ωt) = μat(ωt)/(1 − μ). Thus, aggregate borrowing across investors

who experience a stochastic event ωt such that Φt(ωt) > φt is given by

−bt(ωt) := μ

1− μat(ωt) =

μβ

1− μ(Yt + rtBt−1), (12)

and aggregate investment across the same investors is given by

kt(ωt) :=1

1− μat(ωt) =

β

1− μ(Yt + rtBt−1). (13)

We have used Eq.(11) to derive Eqs. (12) and (13).

2.4.2 Depositors at time t (Φt(ωt) < φt)

Because a depositor who has net worth at(ωt) at time t deposits bt(ω

t) = at(ωt) with the

financial intermediary, the aggregate deposit across depositors who receive a stochastic event

ωt such that Φt(ωt) < φt is given by

bt(ωt) := at(ωt) = β(Yt + rtBt−1). (14)

11

Again, we have used Eq.(11) to derive Eq. (14).

2.4.3 The total output at time t + 1 and the total intrinsically useless asset attime t

From the balance sheet of the representative financial intermediary and from Eqs. (12) and

(14), we have

Bt = Dt − Lt =

ZE

bt(ωt)dP (ωt) +

ZΩ/E

bt(ωt)dP (ωt)

= β(Yt + rtBt−1)G(φt)− μ

1− μ, (15)

where E = ωt ∈ Ω | Φt(ωt) ≤ φt. Multiplying both sides of Eq.(13) by AΦt and aggregatingthis equation across all investors, we obtain total output Yt+1 as follows:Z

Ω/E

AΦtkt(ωt)dP (ωt) =

ZΩ/E

βAΦt1− μ

(Yt + rtBt−1)dP (ωt)

⇐⇒ Yt+1 =βAF (φt)

1− μ(Yt + rtBt−1), (16)

where F (φt) :=R∞φt

Φt(ωt)dG(Φt).

2.5 Dynamical system

The dynamical system of this economy consists of Eqs. (4), (10), (15), and (16). It is

straightforward to derive the dynamic equations for the cutoff φt and the intrinsically useless

asset, respectively, as follows:

G(φt)− μ

1− μ− β(G(φt)− μ)=

φt−1(G(φt−1)− μ)

βF (φt−1)(17)

and

Bt = Aφt−1Bt−1. (18)

From Eq.(15), we note that when G(φt) = μ, the intrinsically useless asset has no value. In

this case, total deposits equal total loans on the balance sheet of the financial intermediary.

The growth rate of aggregate output is given by

Γt+1 :=Yt+1Yt

=AβF (φt)

1− μ− β(G(φt)− μ). (19)

In a competitive equilibrium, the economy is recursively expressed by sequences φt−1, Bt−1, Ytsuch that for all t ≥ 1, these three sequences satisfy the difference equations (17), (18) and(19).

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3 Steady states and dynamic behavior

3.1 Steady states

If we know the dynamic behavior of φt, we know the dynamic behavior of Bt and the

equilibrium output growth rates. Therefore, we intensively analyze Eq.(17). Note from

Eq.(15) that the intrinsically useless asset has a positive value if G(φt) > μ, provided that

Yt is strictly greater than zero. Therefore, our discussion on Eq.(17) is focused on the case

in which G(φt) ≥ μ, and thus, we restrict the domain of the dynamical system of Eq.(17) to

[G−1(μ),∞) because the intrinsically useless asset is freely disposable.First, to examine whether steady states exist in Eq.(17), we define φ∗ and φ∗∗, which,

respectively, satisfy

G(φ∗) = μ

1− μ− β(G(φ∗∗)− μ) = βF (φ∗∗)/φ∗∗.

Lemma 1 Both φ∗ and φ∗∗ are uniquely determined.

Proof : See the Appendix.

Although φ∗ and φ∗∗ can be rewritten as functions of the parameters of μ and β and

the parameters of the distribution of Φ such that φ∗(μ;Θ) and φ∗∗(μ, β;Θ), where Θ is the

parameter set of the distribution of Φ, we simply write φ∗ and φ∗∗ to save space.

[Figure 3 around here]

As Figure 3 shows, the value of φ∗∗ is determined by the intersection of the x axis and

the function H(x) := βF (x)/x − [1 − μ − β(G(x) − μ)], which is decreasing with respect

to x. The dynamical system of Eq.(17) has two steady-state equilibria, φ∗ and φ∗∗, if and

only if φ∗∗ is strictly greater than φ∗. This is because the domain of the dynamical system

of Eq.(17) is [φ∗,∞) because of the free disposability of the intrinsically useless asset. Asshown in Figure 3, φ∗∗ is strictly greater than φ∗ if and only if H(φ∗) > 0, or equivalently,

(1 − G(φ∗))φ∗ < βF (φ∗). In this case, the intrinsically useless asset has a positive value in

the steady state φ∗∗. However, the dynamical system of Eq.(17) has only one steady-state

equilibrium if and only if (1−G(φ∗))φ∗ ≥ βF (φ∗). Proposition 1 summarizes these results.

Proposition 1 Consider the dynamical system of Eq.(17). (i) There exists only one steady

state, φ∗, such that G(φ∗) = μ if and only if (1 − G(φ∗))φ∗ ≥ βF (φ∗). Moreover, the

intrinsically useless asset has no value in this steady state. (ii) There exist two steady states

13

φ∗ and φ∗∗, where φ∗∗ > φ∗, such that G(φ∗) = μ and 1−μ−β(G(φ∗∗)−μ) = βF (φ∗∗)/φ∗∗ if

and only if (1−G(φ∗))φ∗ < βF (φ∗). Moreover, the intrinsically useless asset has a positive

value in the steady state with φ∗∗, implying that asset bubbles appear in this steady state.

Proof. The discussion prior to proposition 1 demonstrates the claims of this proposition. ¤In what follows, we call the steady state φ∗∗ a bubbly steady state if it exists and the

steady state φ∗ a bubbleless steady state.

3.1.1 The case of the Pareto distribution

First, suppose that Φ has a Pareto distribution such that

G(Φ) =

½1− ¡ d

Φ

¢xif Φ ≥ d

0 if 0 ≤ Φ < d,

where x > 1. In this case, F (Φ)/Φ = x(1−G(Φ))/(x−1). From Proposition 1, if (1−β)x < 1,then the bubbly and bubbleless steady states exist in the economy. There are several findings

from the Pareto distribution. First, we find that the existence condition for two steady states

is independent of the extent of credit constraints, μ ∈ [0, 1). Second, assuming that the meanof the Pareto distribution is constant and x is greater than 2, a decrease in x leads to a mean

preserving spread of the Pareto distribution.10 Therefore, if the Pareto distribution exhibits

a mean preserving spread associated with a decrease in x, the bubbly steady state is more

likely to appear. Intuitively, if the number of less productive agents increases as x decreases,

the supply of deposits increases and the intrinsically useless asset is more likely to be held by

the financial intermediary. The asset plays an insurance role that secures a higher return to

depositors in this case than when the financial intermediary does not hold the asset. Third,

if the subjective discount factor is too small, the bubbly steady state does not appear. This

is because if the agents in the economy consider their future consumption unimportant, they

need not store their current output for their future consumption. Therefore, the supply of

deposits decreases and the financial intermediary is unlikely to hold the intrinsically useless

asset.

3.1.2 The case of the uniform distribution

Now, suppose that Φ has a uniform distribution such that

G(Φ) =

⎧⎨⎩0 if 0 ≤ Φ < dΦ−d

2(m−d) if d ≤ Φ ≤ 2m− d1 if Φ ≥ 2m− d,

10If x ≤ 2, we cannot discuss a mean preserving spread of the Pareto distribution because if x ≤ 2, thevariance of the Pareto distribution does not exist.

14

where m is the mean of the distribution and the support of the distribution is [d, 2m − d].In this case, φ∗ = 2(m − d)μ + d and F (φ∗) = (1 − μ)[(m − d)μ + m]. Therefore, fromProposition 1, both the bubbly and bubbleless steady states exist if and only if 0 < μ <

β/(2−β)− [(1−β)d]/[(2−β)(m−d)]. In contrast to the case of the Pareto distribution, thiscondition is dependent on the extent of credit constraints μ. This existence condition for

asset bubbles can be discussed from the perspective of Theorem 3.3 in Santos and Woodford

(1997). From Eq.(19), the equilibrium growth rate in the bubbleless steady state is computed

as Γ∗ := Aβ[(m − d)μ + m], and because rt+1 = Aφt, the equilibrium interest rate in the

bubbleless steady state is computed as r∗ := A[2μ(m−d)+d]. Therefore, if μ is so small thatμ < β/(2− β)− [(1− β)d]/[(2− β)(m− d)], the equilibrium interest rate in the steady stateis smaller than the equilibrium growth rate. This is because more severe credit constraints

reduce the demand for borrowing, and thus, the equilibrium interest rate decreases. This

consequence is consistent with the necessary condition for a bubble on the intrinsically useless

asset to appear, which can be deduced from Theorem 3.3 in Santos and Woodford (1997).

It should be noted here that, as in the case of the Pareto distribution, if the uniform

distribution exhibits a mean preserving spread induced by a decrease in d, the bubbly steady

state is more likely to appear. Moreover, the effect of a decrease in the subjective discount

factor β is also the same as in the case of the Pareto distribution.

From these two examples, we find that the appearance of a bubble on the intrinsically

useless asset depends on the classes of distributions and fundamental variables such as the

agents’ productivity, extent of credit constraints, and subjective discount factor. For in-

stance, even though two different productivity distributions have the same mean and vari-

ance, asset bubbles can appear in one economy but not in the other if the two economies are

endowed with the same fundamental variables but have different productivity distributions.

3.2 Dynamics

Now, to investigate the dynamic behavior of the economy, we draw phase diagrams. We

define the following functions:

Ψ(φt) :=G(φt)− μ

1− μ− β(G(φt)− μ), (20)

which is the left-hand side of Eq. (17), and

Λ(φt−1) :=φt−1(G(φt−1)− μ)

βF (φt−1), (21)

which is the right-hand side.

15

3.2.1 Local stability of the bubbleless steady state

Eqs. (20) and (21) are respectively approximated around the bubbleless steady state such

that

Ψ(φt) ≈ G0(φ∗)1−G(φ∗)(φt − φ∗) (22)

and

Λ(φt−1) ≈ φ∗G0(φ∗)βF (φ∗)

(φt−1 − φ∗). (23)

From Eqs. (22) and (23), we obtain the local dynamical system with respect to φt around

the bubbleless steady state as follows:

φt − φ∗ =φ∗(1−G(φ∗))

βF (φ∗)(φt−1 − φ∗).

From this equation and Proposition 1, we note that if only the bubbleless steady state

exists in the economy, this bubbleless state is locally unstable, whereas, if both bubbly and

bubbleless steady states exist in the economy, the bubbleless steady state is locally stable.

3.2.2 Local stability of the bubbly steady state

On the other hand, Eqs. (20) and (21) are, respectively, approximated around the bubbly

steady state such that

Ψ(φt) ≈ φ∗∗G0(φ∗∗)F (φ∗∗) + (φ∗∗)2G0(φ∗∗)(G(φ∗∗)− μ)

βF (φ∗∗)2(φt − φ∗∗)

and

Λ(φt−1) ≈ [(G(φ∗∗)− μ) + φ∗∗G0(φ∗∗)]F (φ∗∗) + (φ∗∗)2G0(φ∗∗)(G(φ∗∗)− μ)

βF (φ∗∗)2(φt−1 − φ∗∗).

Therefore, the local dynamical system with respect to φt around the bubbly steady state is

given by:

φt − φ∗∗ =h (G(φ∗∗)− μ)F (φ∗∗)φ∗∗G0(φ∗∗)F (φ∗∗) + (φ∗∗)2G0(φ∗∗)(G(φ∗∗)− μ)

+ 1i(φt−1 − φ∗∗).

We note from this equation that if the bubbly steady state exists, it is locally unstable

because the coefficient of φt−1 − φ∗∗ on the right-hand side is greater than one.

3.2.3 Phase diagram analysis of the dynamics

The stability of the two steady states can also be investigated using phase diagrams. Both

Ψ(φt) and Λ(φt−1) are increasing functions. We can easily show that limφt→∞Ψ(φt) =

16

1/(1 − β) and limφt−1→∞ Λ(φt−1) = ∞. Therefore, configurations of Ψ(φt) and Λ(φt−1) and

the dynamic behaviors of the economy are described in Figure 4.

[Figure 4 around here]

From Eqs. (10) and (15), we have ptM = β[G(φt)−G(φ∗)]Yt/[1−μ−β(G(φt)−G(φ∗))].Because the price of the intrinsically useless asset is non-predetermined, φt is also non-

predetermined. As noted from Panel A in Figure 4, if only the bubbleless steady state

exists, this steady state is unstable. Because φt is not a predetermined variable, only the

bubbleless steady state is a rational expectations equilibrium and there is no transitional

dynamics in this case. The economy does not experience endogenous fluctuations caused by

the agents’ expectations.

Conversely, if both the bubbly and bubbleless steady states exist, as discussed above, the

bubbleless steady state is locally stable and the bubbly steady state is unstable. In this case,

as seen in Panel B in Figure 4, equilibrium is locally determinate in the neighborhood of the

bubbly steady state. In other words, if one focuses the analysis on the small neighborhood

of φ∗∗, the bubbly steady state is only the rational expectations equilibrium. This implies

that even though an exogenous shock associated with fundamental variables occurs, such as

productivity and the agents’ tastes, the bubbly steady state equilibrium is always achievable

without any excess volatility driven by the agents’ expectations. However, it is also true

that if we consider the global dynamic behavior of the economy, there exist an uncountably

infinite number of equilibrium trajectories, originating from the left neighborhood of the

bubbly steady state, which monotonically converge to the bubbleless steady state. In other

words, if we consider the global dynamics of the economy, equilibrium is indeterminate, and

thus, endogenous fluctuations caused by the agents’ expectations may appear. In section 4,

we investigate the possibility of a sunspot equilibrium.

3.3 Growth rate comparison

From Eq.(19), the growth rate Γ∗ in the bubbleless steady state and the growth rate Γ∗∗ in

the bubbly steady state are computed as

Γ∗ =βAF (φ∗)1−G(φ∗) (24)

and

Γ∗∗ = Aφ∗∗, (25)

respectively.

17

Proposition 2 Suppose that the bubbly steady state exists. The growth rate in the bubbly

steady state is strictly greater than that in the bubbleless steady state. Moreover, the growth

rate in the bubbly steady state is the highest for any φt ≥ φ∗.

Proof: See the appendix.

The consequences in Proposition 2 contrast with those in the first-generation growth

models with asset bubbles. In the first generation literature (e.g., Tirole, 1985; Grossman

and Yanagawa, 1992; Saint-Paul, 1992; Futagami and Shibata, 2000), a positively valued

intrinsically useless asset always reduces the growth rate in an economy because asset bubbles

crowd out investment. However, in our model, there is a crowd-in liquidity effect of asset

bubbles on investment as well as a crowd-out effect. From Eqs. (10) and (15), we obtain the

bubble to output ratio (the Bt/Yt ratio) as follows:

BtYt=

β(G(φt)−G(φ∗))1− μ− β(G(φt)−G(φ∗)) .

We find from this equation that the further is the distance between φt and φ∗, the greater

is the Bt/Yt ratio. Proposition 2 implies that if φt is greater than φ∗ but less than φ∗∗,

the crowd-in liquidity effect of asset bubbles dominates the crowd-out effect, whereas, if φt

is greater than φ∗∗, the crowd-out effect of bubbles dominates the crowd-in liquidity effect.

However, there is no equilibrium such that φt > φ∗∗, as observed in Panel B in Figure 4.

Therefore, if the bubbly steady state exists, the crowd-in liquidity effect always dominates

the crowd-out effect in equilibrium. In other words, the greater the Bt/Yt ratio, the higher

the growth rate in equilibrium if the bubbly steady state exists.

In response to the crowd-out effect of asset bubbles, the interest rate in the financial

market increases. Although the increase in the equilibrium interest rate reduces the number

of investors, the existence of asset bubbles crowds out less productive agents and they become

depositors. Moreover, these depositors acquire higher returns when asset bubbles exist than

when they do not exist. Some depositors who are less productive agents today will become

productive investors tomorrow. This implies that agents who have become highly productive

investors have enough liquidity that can be used for their investment projects because of

higher returns on deposits. This is the crowd-in liquidity effect of asset bubbles. Conversely,

if asset bubbles do not exist, less productive agents will only receive lower returns on deposits.

In this case, those agents who will become highly productive agents do not have enough

liquidity to invest in projects. The crowd-in liquidity effect of asset bubbles is similarly

discussed in the second-generation models with asset bubbles (e.g., Hirano and Yanagawa,

2010; Farhi and Tirole, 2011; Martin and Ventura, 2011; and Miao and Wang, 2011).

18

We conclude this section with remarks on constrained dynamic efficiency.11 The bubble-

less steady state exhibits constrained dynamic inefficiency if both the bubbly and bubbleless

steady states exist. Because βct = (1 − β)at for each agent, per capita consumption is

obtained as c0 = (1 − β)w0 and ct := (1 − β)(Yt + rtBt−1) = (1 − β)(Yt + Bt) for t ≥ 1,where the second equality holds because of Eq.(10). We note that the growth rates of Yt

and ct are the same in the bubbleless steady state and that those of Yt, Bt, and ct are the

same in the bubbly steady state. From Proposition 2, the growth rate in the bubbly steady

state is the highest for any φt ≥ φ∗. Obviously, this means that allocative inefficiency in

the bubbleless economy is corrected by the existence of asset bubbles. In Tirole (1985) the

existence of asset bubbles also corrects allocative inefficiency in the bubbleless steady-state

equilibrium if the bubbleless steady-state equilibrium is dynamically inefficient. However,

the mechanism for the correction of inefficiency in our model differs from that in Tirole’s

model. While only the crowd-out effect is important for correcting the allocative inefficiency

in Tirole’s model, in our model, both the crowd-out and the crowd-in liquidity effects are

important. In our model, in response to the crowd-out effect, unproductive agents do not

invest in projects, and thus, physical capital is not inefficiently used. In response to the

crowd-in liquidity effect, agents who transitioned from being unproductive in the past to

being highly productive today can utilize a number of production resources.

Unlike the model of Tirole and Farhi (2011), constrained dynamic efficiency and con-

strained Pareto efficiency are not identical concepts in our model because of the ex-post

heterogeneity of the agents.12 Therefore, the existence of asset bubbles may not generate

Pareto improvements due to increased interest rates.

4 Sunspots and financial crisis

To investigate sunspot equilibria, we focus our analysis on the case in which the bubbly steady

state appears. As discussed in the previous section, equilibrium is globally indeterminate in

this case. In this section, we shall derive a two-state stationary sunspot equilibrium under

mild parameter conditions that crates a financial crisis without any changes in fundamental

variables such as productivity or the agents’ tastes.

11See Kunieda (2008) and Tirole and Farhi (2011) for the definition of constrained dynamic efficiency.12Here, we consider ex-post Pareto efficiency.

19

4.1 Sunspot variables and the equilibrium dynamics

We assume that there is a sunspot variable zt that follows a two-state Markov process, the

support of which is 0, 1, such that

Pr(zt = 1|zt−1 = 1) = πa

Pr(zt = 0|zt−1 = 0) = πb,

where πa and πb∈ (0, 1] are the transition probabilities of the Markov chain. We assume thatz0 = 1. Let the history of sunspot events up to time t be z

t = z0, z1, ..., zt. The sunspotevents are independent of idiosyncratic productivity shocks and are common among agents.

The price of the intrinsically useless asset, pt, is affected by the sunspot variable, and we

denote pt = pt(zt).

We assume that at time t−1, agents have rational expectations regarding future sunspotevents when determining the cutoff φt−1, given the sunspot history zt−1. Because zt follows a

Markov process, φt−1 is a function of zt−1, denoting φt−1(zt−1). In other words, the agents use

current information zt−1 to determine the cutoff φt−1. Note that φt−1(zt−1) is no longer equal

to rt/A. Note also that although φt−1(zt−1) is a stochastic variable before the realization of

zt−1, it becomes a deterministic variable when zt−1 is fulfilled.

To derive a two-state stationary sunspot equilibrium, we assume that if an agent be-

comes an investor, his credit constraint is always binding.13 Once the cutoff of φt−1(zt−1)

is determined, the equilibrium conditions of the economy are obtained as in section 2. We

will discuss how the cutoff φt−1 is derived in Lemma 3 below. Because the derivation of the

equilibrium conditions is nearly identical to that presented in section 2, we will not insert

the maximization problem here. Given the sunspot history zt−1 and from the solutions to

the maximization problem and the market clearing conditions, the dynamical system of the

economy is expressed by nearly the same equations as Eqs.(10), (15), and (16) because of

the law of large numbers and the i.i.d. assumption regarding Φt(ωt), except that pt(zt) is a

random variable. We insert the new equations regarding the aggregate variables below:

Bt(zt) =

pt(zt)

pt−1(zt−1)Bt−1(zt−1), (26)

Bt(zt) = β(Yt(z

t−1) +pt(zt)

pt−1(zt−1)Bt−1(zt−1))

G(φt(zt))− μ

1− μ, (27)

13If given the other parameter values, μ is small, then this assumption always holds in our two-state sta-tionary sunspot equilibrium. Our purpose in this section is to derive an example of a two-state stationary

sunspot equilibrium. It is very complicated to construct a sunspot equilibrium under more general circum-

stances in which there are investors who do not face binding credit constraints in a sunspot equilibrium.

20

Yt+1(zt) =

βAF (φt(zt))

1− μ(Yt(z

t−1) +pt(zt)

pt−1(zt−1)Bt−1(zt−1)). (28)

4.2 Stationary sunspot equilibrium

From Eqs.(26), (27), and (28), we obtain

G(φt(zt))− μ

1− μ− β(G(φt(zt))− μ)=pt(zt)/pt−1(zt−1)(G(φt−1(zt−1))− μ)

AβF (φt−1(zt−1)), (29)

and

Γt+1 = Yt+1(zt)/Yt(z

t−1) =Aβφt(zt)F (φt(zt))

1− μ− β(G(φt(zt))− μ), (30)

Taking the expectation of both sides of Eq.(29) given the sunspot event zt−1, we have

Eh G(φt(zt))− μ

1− μ− β(G(φt(zt))− μ)

¯zt−1

i=E[pt(zt)/pt−1(zt−1)|zt−1](G(φt−1(zt−1))− μ)

AβF (φt−1(zt−1)). (31)

In what follows, we consider a stationary sunspot equilibrium. In a two-state station-

ary sunspot equilibrium, pt(zt)/pt−1(zt−1) takes four values: ρHH := pt(zt = 1)/pt−1(zt−1 =

1), ρLH := pt(zt = 0)/pt−1(zt−1 = 1), ρHL := pt(zt = 1)/pt−1(zt−1 = 0), and ρLL :=

pt(zt = 0)/pt−1(zt−1 = 0) for all t ≥ 1. Therefore, E[pt(zt)/pt−1(zt−1)|zt−1 = 1] and

E[pt(zt)/pt−1(zt−1)|zt−1 = 0] for all t ≥ 1 are written as follows:

ρa := E[pt(zt)/pt−1(zt−1)|zt−1 = 1] = πaρHH + (1− πa)ρLH (32)

and

ρb := E[pt(zt)/pt−1(zt−1)|zt−1 = 0] = (1− πb)ρHL + πbρLL, (33)

respectively. φt(zt) takes two values, depending on the realization of zt. Specifically, we write

the two stationary-state cutoffs as φa := φt(zt = 1) and φb := φt(zt = 0) for all t ≥ 0. Wefocus our analysis on the case in which φa > φb so that the economy is initially in the high

growth state.

Using φa, φb, ρa, and ρb, Eq.(31) is rewritten as the following two equations, depending

upon the realization of zt−1:

πaG(φa)− μ

1− μ− β(G(φa)− μ)+ (1− πa)

G(φb)− μ

1− μ− β(G(φb)− μ)=

ρa(G(φa)− μ)

AβF (φa)(34)

(1− πb)G(φa)− μ

1− μ− β(G(φa)− μ)+ πb

G(φb)− μ

1− μ− β(G(φb)− μ)=

ρb(G(φb)− μ)

AβF (φb). (35)

Lemma 2 Suppose that (1 − G(φ∗))φ∗ < βF (φ∗), implying that the bubbly steady state

equilibrium exists without extrinsic uncertainty. Suppose also that 0 < πa < 1 and φb < φa.

Then, if a two-state stationary sunspot equilibrium exists under some parameter values, it

must follow that φb = φ∗ and πb = 1.

21

Proof: See the appendix.

From Lemma 2, we find that for the two-state stationary sunspot equilibrium to exist,

one state must be the bubbleless steady state, φ∗. Lemma 2 demonstrates that when the

state changes from zt−1 = 1 to zt = 0, asset bubbles burst at time t. Moreover, for s ≥ t,we have ps(zs = 0) = 0 because the value of the intrinsically useless asset is zero from time

t onwards. Therefore, Eqs.(34) and (35) are reduced to

1

1− μ− β(G(φa)− μ)=

ρHHAβF (φa)

(36)

and

G(φb) = μ. (37)

4.3 Existence of a two-state stationary sunspot equilibrium

As discussed in section 2.3, the financial intermediary purchases an intrinsically useless asset

with the excess total saving, and its balance sheet at time t − 1 is given by Lt−1(zt−1) +Bt−1(zt−1) = Dt−1(zt−1). However, as demonstrated in Lemma 2, asset bubbles burst with

probability 1 − πa unless πa = 1 when the economy is in the φa state. In this situation,

the financial intermediary could face a balance sheet problem if asset bubbles burst. That

is, the financial intermediary may not be able to repay its obligations to depositors when

asset bubbles burst. To avoid this balance sheet problem, we assume that the financial

intermediary makes financial contracts regarding interest rates with investors and depositors

when lending financial capital and receiving deposits such that it can repay its obligations

when asset bubbles burst. The interest rate rH is paid by investors and paid to depositors

if the intrinsically useless asset has a value at both time t− 1 and t. Similarly, the interestrate rL is paid by investors and paid to depositors if the intrinsically useless asset has no

value at both time t− 1 and t. In the case in which asset bubbles burst at time t, investorspay the interest rate rL at time t, whereas depositors receive the interest rate r

0L. Formally,

the financial contract scheme regarding interest rates is expressed by the following three

equations:⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩rHLt−1(zt−1) + ρHHBt−1(zt−1) = rHDt−1(zt−1) if zt−1 = 1 and zt = 1

rLLt−1(zt−1) = r0LDt−1(zt−1) if zt−1 = 1 and zt = 0

rLLt−1(zt−1) = rLDt−1(zt−1) if zt−1 = 0 and zt = 0.

(38)

Note that there is no opportunity for the financial intermediary to obtain profits under

this interest rate scheme, because we have assumed that the financial sector is competitive.

22

We also note that rH = ρHH because Lt−1(zt−1) + Bt−1(zt−1) = Dt−1(zt−1). We find that

rL > r0L because Lt−1(zt−1) < Dt−1(zt−1) when asset bubbles burst. This interest rate

scheme is equivalent to the government taxing depositors and its bailout to the financial

sector. Therefore, we could assume another scheme to avoid the balance sheet problem such

as a combination of a lump-sum tax and a direct bailout by the government.14

Lemma 3 Suppose that 0 < πa < 1 and φb < φa. Suppose also that a two-state stationary

sunspot equilibrium exists under some parameter values. The two stationary-state cutoffs of

φa and φb = φ∗ satisfy

φb = φ∗ = rL/A, (39)

and

rπa

H r0L1−πa

=

"Aφa − μrH1− μ

#πa"Aφa − μrL1− μ

#1−πa, (40)

where

r0L =rLμ(1−G(φa))G(φa)(1− μ)

, (41)

respectively.

Proof: See the appendix.

Given the Markov transition probabilities, πa and πb, the two-state stationary sunspot

equilibrium of the economy consists of the five-tuple φa,φb, rH , rL, r0L such that φa,φb, rH , rL, r0Lsatisfies Eqs. (36), (37), (39), (40), and (41). In the stationary sunspot equilibrium, given

the sunspot history zt−1, the realization of zt in each time recursively determines the output

Yt+1(zt) following Eq.(28) and the intrinsically useless asset Bt(z

t) following Eq.(26).

Proposition 3 Suppose that 0 < πa < 1 and φb < φa. Suppose also that (1 − G(φ∗))φ∗ <βF (φ∗), implying that the bubbly steady state equilibrium exists without extrinsic uncertainty.

Then, there exists a two-state stationary sunspot equilibrium φa,φb, rH , rL, r0L such thatφb = φ∗ with πb = 1 under some parameter values.

Proof: See the appendix.

In the two-state stationary sunspot equilibrium, one state is the bubbleless steady-state

equilibrium of φ∗, where the growth rate Γ∗ is the minimum in φt ∈ [φ∗,φ∗∗]. We find fromEq.(40) that if πa is close to one, φa approaches rH/A. This implies from Eq.(36) that if πa

is close to one, φa approaches φ∗∗, which shifts the economy to the maximum growth rate.

14If we assume another scheme such as a combination of a lump-sum tax and a direct bailout, we are

still able to prove that a two-state stationary sunspot equilibrium exists. In such a scheme, however, the

equilibrium growth rate differs from the one that we obtain in the main text.

23

Once the economy falls from the high-growth state of φa to the low-growth state of φb = φ∗

due to a bubble burst, it is never restored to the high-growth state. As πa is close to one,

the probability of the occurrence of a self-fulfilling financial crisis approaches zero, while the

economy approaches the highest growth state. When the US subprime loan crisis occurred

in late 2000, it was said to be the century’s greatest crisis. From our model’s perspective,

we can consider that πa was very close but not equal to one before the subprime loan crisis.

The self-fulfilling financial crisis in our model accompanies the bursting of the asset bub-

bles. The extant second-generation literature on asset bubbles, such as Kocherlakota (2009),

Farhi and Tirole (2011), Martin and Ventura (2011), and Miao and Wang (2011), examines

this type of self-fulfilling financial crisis equilibrium. However, they begin their investigations

with the assumption that one state is bubbleless. However, we have proven that one state

must be bubbleless in the two-state stationary sunspot equilibrium. In this sense, in contrast

with the extant second-generation literature, the bubbleless state endogenously appears in

the process of proving that the two-state stationary sunspot equilibrium exists.

5 Avoiding self-fulfilling financial crises

In the previous section, we find that asset bubbles burst with probability one if we have an

infinitely long time horizon. In this section, we discuss a government policy to avoid bubble

bursts. Borrowing an idea proposed by Tirole (1985), we present a government policy to

avoid self-fulfilling financial crises. If the bubbleless steady-state equilibrium is eliminated by

a government policy, and the bubbly steady-state equilibrium becomes a unique equilibrium,

self-fulfilling financial crises never occur.15 Tirole (1985) demonstrates that if the intrinsically

useless asset is backed by the government, the bubbly steady-state equilibrium becomes a

unique equilibrium.

We assume that the government imposes a tax on the agents’ net incomes in Eq.(1) as

follows:

kt+s(ωt+s)+ bt+s(ω

t+s) = [AΦt+s−1kt+s−1(ωt+s−1)+ rt+sbt+s−1(ωt+s−1)](1− τt+s)− ct+s(ωt+s),(42)

where τt ∈ (0, 1) is the tax rate at time t, which is common across agents. From Eq.(42),

the aggregate tax is given by τt+s(Yt+s + rt+sBt+s−1).

We assume that the government purchases ²Yt/pt of the intrinsically useless asset at time

15The notion that a government policy eliminates a bad steady-state equilibrium is employed in other

areas in macroeconomics. For instance, Benhabib et al. (2002) propose a fiscal policy that eliminates the

low-inflation steady state to avoid liquidity traps.

24

t, and thus, the dynamic equation of Bt becomes

Bt = rtBt−1 − ²Yt. (43)

Note that ² is independent of time and assumed to be infinitesimal. All computations to

derive the equilibrium dynamics with respect to φt are nearly identical to those in subsection

2.4. Because agents have to pay the tax, we have at = βRt(1− τt)at−1, and thus,

at(ωt) = β(AΦt−1kt−1(ωt−1) + rtbt−1(ωt−1))(1− τt).

Therefore, we obtain

Bt = β(Yt + rtBt−1)G(φt)− μ

1− μ(1− τt), (44)

and

Yt+1 =βAF (φt)

1− μ(Yt + rtBt−1)(1− τt). (45)

Assuming a balanced government budget, we have τt(Yt + rtBt−1) = ²Yt. Applying Eq.(43)

to the government’s budget constraint, we obtain

1− τt =Yt + Bt

Yt + Bt + ²Yt.

Thus, Eqs. (44) and (45), respectively, become

Bt = β(Yt + Bt)G(φt)− μ

1− μ, (46)

and

Yt+1 =βAF (φt)

1− μ(Yt + Bt). (47)

From Eqs. (43), (46) and (47) and using rt = Aφt−1, we obtain

(1− ²)(G(φt)− μ) + ²(1− μ)/β

1− μ− β(G(φt)− μ)=

φt−1(G(φt−1)− μ)

βF (φt−1). (48)

We note that the right-hand side of Eq.(48) is the same as that of Eq.(17), whereas the

left-hand side is different because ² is strictly greater than zero. Figure 5 presents a phase

diagram of the dynamic behavior of φt. We find from Figure 5 that the equilibrium is globally

determinate, implying that only the bubbly steady state φ∗∗∗ is a unique perfect foresight

equilibrium in this economy. In this case, self-fulfilling financial crises never occur. Moreover,

the economy experiences a nearly maximum growth rate because ² is infinitesimal.

[Figure 5 around here]

25

6 Concluding Remarks

We have described a self-fulfilling financial crisis accompanied by a bubble burst as a rational

expectations equilibrium. In our model, the crowd-in liquidity effect of asset bubbles on in-

vestments dominates the crowd-out effect in equilibrium. Therefore, a self-fulfilling financial

crisis always results in a severe economic recession. This result is consistent with empirical

evidence observed in the history of financial crises.

We conclude this paper with remarks on the relationship between our model and Townsend’s

(1980) turnpike model.16 In his model there are two types of infinitely lived agents who travel

along a turnpike in opposite directions. Each type consists of a countably infinite number of

agents and receives endowments in alternating periods. Because of this endowment pattern,

each agent desires inter-temporal trade among agents to smooth out her consumption over

time. However, the two types of agents only meet once, and thus, there can be no bor-

rowing and lending between the two types. This lack of borrowing and lending introduces

inefficiency into the model. In this situation, an intrinsically useless asset can be valued and

make every agent better off through inter-temporal trade between the two types of agents.17

In our model, each agent faces different investment opportunities over time, and therefore,

inter-temporal trade is beneficial to all agents. However, the presence of credit constraints

prevents each agent from smoothing her consumption over time, which is a similar situation

to that in the Townsend model. Thus, introducing a bubbly asset can lead to a superior

allocation through inter-temporal trade among agents.18

Appendix

Data description for Figure 1

Data were obtained from various databases. We gathered annual data for the United States

and Japan from 1980 to 2009. We obtained the data for per capita real GDP in the two

countries from the Penn World Table 7.0 database created by Heston et al. (2011), which is

titled “PPP Converted GDP Per Capita (Laspeyres), derived from growth rates of c, g, i, at

2005 constant prices.” The data for land prices in Japan were collected from the Nationwide

16Townsend’s (1980) model can be interpreted as a simplified version of Bewley’s (1980) stochastic model.17The key factor in the creation of bubbles is that resources are allocated inefficiently when bubbles are

absent. See Barlevy (2007) on this point.18Cozzi (2001) shows that, although each agent in the Townsend model is infinitely lived, the model can

exhibit the double infinity feature in the same way as in overlapping generations models. Similar to Cozzi

(2001), we could relate our model to overlapping generations models of bubbles such as Tirole (1985) and

Weil (1987).

26

Urban Land Price Index database created by the Japan Real Estate Institute. In particular,

we used the land price index of six major cities in the dataset. The data for land prices in the

United States were assembled from the dataset titled “CSW-based price index: aggregate

land data, quarterly, 1975:1-2011:1 ” that was created by Morris A. Davis and Jonathan

Heathcote (2007). The data for Japanese stock prices were obtained from the Nikkei Indexes,

which is a database for various Japanese stock price indexes created by Nikkei Inc. In

particular, we used the annual data from the Nikkei Stock Average. For stock prices in the

United States, we used the S&P Composite Stock Price Index. The data for this index were

downloaded from Robert Shiller’s web page at: http://www.econ.yale.edu/shiller/data.htm.

To obtain the real variables, all of the land and stock price indexes were deflated using the

consumer price index, which was collected from the database of the World Development

Indicators created by the World Bank (2011). We converted all of the real variables into

indexes using 2000 as the base year. Applying the Hodrick-Prescott filter to these converted

indexes, we obtained the deviations from the trends of each variable.

Microfoundations for the credit constraint (2)

Microfoundation I

Following Aghion et al. (1999), Aghion and Banerjee (2005) and Aghion et al. (2005), we

assume that financial market imperfections arise simply from the possibility that borrowers

may not repay their obligations.

The net worth that each investor prepares for her investment project by her own is at. If

she borrows −bt from the financial intermediary, her total resources to invest are kt = at− btat time t. The return on one unit of investment is AΦt. If an investor consistently repay her

obligations, she will acquire a net income of AΦtkt + rt+1bt at time t + 1. However, if the

investor does not repay her obligations, she will incur a cost of δkt to conceal her revenue. In

this case, the financial intermediary monitors the investor and is able to capture the investor

with a probability of pt+1. Thus, her expected income is given by AΦtkt − δkt + pt+1rt+1bt.

Under this lending contract, the incentive compatibility constraint for the investor not

to default is given by

AΦtkt + rt+1bt ≥ [AΦt − δ]kt + pt+1rt+1bt, (A1)

or equivalently,

bt ≥ − δ

rt+1(1− pt+1)kt, (A2)

27

The left-hand side of Eq. (A1) represents the revenue that the investor obtains when she

invests in a project and consistently repay her obligations. The right-hand side is the gain

when she defaults.

To be able to detect the investor concealing her revenue with probability pt+1, the financial

intermediary incurs an effort cost, btC(pt+1), which is increasing and convex with respect to

pt+1. As in Aghion and Banerjee (2005), we assume C(pt+1) = κ log(1 − pt+1), where κ is

strictly greater than δ, so that our study is meaningful.19 The financial intermediary can

choose an optimal probability by solving a maximization problem such that

maxpt+1

− pt+1rt+1bt − κ log(1− pt+1)bt.

As −bt > 0, this maximization problem is rewritten as

maxpt+1

pt+1rt+1 + κ log(1− pt+1).

From the first-order condition, we have

rt+1 =κ

1− pt+1 . (A3)

As the interest rate rt+1 increases, the financial intermediary chooses a high probability of

detecting defaulting investors. From Eqs. (A2) and (A3), we obtain

bt ≥ − δ

κkt,

or equivalently,

bt ≥ − δ

κ− δat. (A4)

As the agent’s productivity Φt is not observable, the financial intermediary does not impose

investor-specific credit constraints. The financial intermediary must know the investors’ net

worth, at. As long as it imposes a credit constraint given by the inequality (A4) on all agents,

no one will default in equilibrium. As δ < κ, we can let θ := δ/(κ− δ) ∈ [0,∞), and thus,

bt ≥ −θat,

which is a credit constraint in the main text. δ and κ are associated with a default cost and

a monitoring cost, respectively. θ represents the extent of the credit constraint.

19If δ ≥ κ, no investors face binding credit constraints.

28

Microfoundation II

We extend the microfoundation for a credit constraint presented by Antras and Caballero

(2009) in a manner suitable for our model. We consider the participation constraint faced by

the financial intermediary and the incentive compatibility constraint of investors such that

they do not back out of their investment projects.

It is assumed that at the end of time t and after an investment has occurred, any investor

can back out of her investment project at no cost, taking some fraction of her investments,

(1 − μ)(at − bt), where 0 < μ < 1 and her obligations to the financial intermediary are not

repaid. In this case, the investor will engage in general goods production somewhere in the

economy.

If an investor absconds at the end of time t, the financial intermediary can reclaim the

remainder of investments, μ(wt−bt). It is assumed that the financial intermediary can relendthe remainder of the investments in the financial market. Thus, when making a financial

contract with an investor, the financial intermediary faces a participation constraint such

that

rt+1μ(at − bt) ≥ −rt+1bt,

or equivalently

bt ≥ − μ

1− μat.

However, the incentive compatibility constraint for a borrower, such that she does not to

abscond from engaging in her project at the end of time t, is given by

AΦt(at − bt) + rt+1bt ≥ AΦt(1− μ)(at − bt). (A5)

For investors with Φt such that rt+1−μAΦt ≤ 0, Eq. (A5) always holds. Therefore, we focuson investors with Φt such that rt+1 − μAΦt > 0. Then, Eq. (A5) is rewritten as

bt ≥ − μ

(φt/Φt)− μat. (A6)

As φt/Φt ≤ 1 in equilibrium, it follows that −μ/((φt/Φt)− μ) ≤ −μ/(1− μ), implying that

Eq. (A6) is redundant. In other words, if the financial intermediary imposes a credit con-

straint bt ≥ −μat/(1−μ), which is the participation constraint of the financial intermediary,

investors never default. By letting μ/(1−μ) := θ, we obtain the credit constraint bt ≥ −θat,as shown in the main text. As μ, or equivalently θ, increases, it becomes more difficult for

investors to withdraw their investments without repaying their obligations.

29

Proof of Lemma 1

A unique value of φ∗ clearly exists because G(.) is a strictly increasing function over the

support of Φ. Regarding φ∗∗, we note that H(x) := βF (x)/x−[1−μ−β(G(x)−μ)] is strictlydecreasing in (0, h) or (0,∞) because over the support, H 0(x) = β [−F (x)/x2 − dG(x)/dx]+βdG(x)/dx = −βF (x)/x2 < 0, and in the area other than the support, both F (x) and

G(x) are constant. In addition, limx→0H(x) = ∞ and limx→∞H(x) = −1 + μ(1 − β) < 0.

Therefore, φ∗∗, which is the solution of H(x) = 0, is uniquely determined. ¤

Proof of Proposition 2

Note that because the bubbly steady state exists, it follows that φ∗∗ > φ∗. As shown in

Eq.(19), the growth rate Γt+1 = Yt+1/Yt is given by

J(φt) :=AβF (φt)

1− μ− β(G(φt)− μ).

From this, we obtain

J 0(φt) := H(φt)AβφtG

0(φt)[1− μ− β(G(φt)− μ)]2

,

where H(φt) = βF (φt)/φt − [1 − μ − β(G(φt) − μ)] as defined in the proof of Lemma 1.

As demonstrated in the proof of lemma 1, H(φt) is strictly decreasing and H(φ∗∗) = 0.

Therefore, J 0(φt) is strictly greater than zero if φ∗ < φt < φ∗∗ and it is strictly less than zero

if φt > φ∗∗. See Figure 3. Therefore, the maximum of J(φt) is J(φ∗∗), and J(φ∗) < J(φ∗∗).

¤

Proofs of Lemma 2 and Lemma 3

In preparation for the proofs of Lemma 2 and Lemma 3, we first discuss the determination

of the cutoff φt(zt) based on the agents’ rational expectations for their future returns. Let

rdt (zt) be the interest rate that the depositors receive at time t and rit(zt) be the interest rate

that the investors pay at time t. An agent who holds net worth at(zt) at time t acquires a

return, Rt+1(zt+1), at time t+ 1 such that

Rt+1(zt+1) =

(rdt+1(zt+1) if Φt(ωt) < φt(zt)AΦt(ωt)−rit+1(zt+1)μ

1−μ if Φt(ωt) > φt(zt).

Because the instantaneous utility function is log-linear, as discussed in section 2, the equi-

librium conditions of consumption and net worth are obtained from the first-order conditions

of the agent’s maximization problem and the transversality condition as follows:20

ct+1 = (1− β)Rt+1(zt+1)at20See footnote 8.

30

and

at+1 = βRt+1(zt+1)at.

From these two equations, we have consumption at time t+ s, given at, as follows:21

ct+s = (1− β)βs−1Πsj=0Rt+j(zt+j)at.

Substituting ct+s into expected lifetime utility Ut, we obtain the agent’s expected indirect

lifetime utility, that is, the value function, as follows:

V (at) := E

" ∞Xs=0

βs ln[(1− β)βs−1Πsj=0Rt+j(zt+j)at]¯Φt, zt

#,

or equivalently,

V (at) =1

1− βln at +

∞Xs=0

βs ln[(1− β)βs−1]

+βE

∙ln Rt+1(zt+1)

¯zt

¸+ E

" ∞Xs=2

βssXj=2

ln Rt+j(zt+j)¯Φt, zt

#,

where the expectation in the third term of the second equation is taken given zt, because Φt

is fulfilled at time t and zt follows a Markov process. E£ln Rt+1(zt+1)

¯zt¤is maximized when

the agent makes a decision at time t on whether she starts an investment project or deposits

financial capital with the financial intermediary. Therefore, we obtain:

E

∙ln Rt+1(zt+1)

¯zt = 1

¸= max

"lnh[rdt+1(1)]

πa [rdt+1(0)]1−πa

i, lnhAΦt(ωt)− μrit+1(1)

1− μ

iπahAΦt(ωt)− μrit+1(0)

1− μ

i1−πa#.

From this equation, the cutoff of φt(zt = 1) is given by the following equation:

[rdt+1(1)]πa[rdt+1(0)]

1−πa =hAφt(zt = 1)− μrit+1(1)

1− μ

iπahAφt(zt = 1)− μrit+1(0)

1− μ

i1−πa. (A7)

Similarly, we obtain

E

∙ln Rt+1(zt+1)

¯zt = 0

¸= max

"lnh[rdt+1(0)]

πb [rdt+1(1)]1−πb

i, lnhAΦt(ωt)− μrit+1(0)

1− μ

iπbhAΦt(ωt)− μrit+1(1)

1− μ

i1−πb#.

From this equation, the cutoff of φt(zt = 0) is given by the following equation:

[rdt+1(0)]πb[rdt+1(1)]

1−πb =hAφt(zt = 0)− μrit+1(0)

1− μ

iπbhAφt(zt = 0)− μrit+1(1)

1− μ

i1−πb. (A8)

21By convention, it is assumed that Π0j=0Rt+j(zt+j) = 1.

31

Proof of Lemma 2

Suppose that φ∗ < φb. In this case, the intrinsically useless asset is valued in both states,

φa and φb. Therefore, ρHL, ρLL, ρHH , and ρLH correspond to the interest rates that de-

positors and investors face in each state because of the zero-profit condition of the financial

intermediary. It follows from φb < φa that

G(φb)− μ

1− μ− β(G(φb)− μ)<

G(φa)− μ

1− μ− β(G(φa)− μ).

From this equation and Eq.(29), we have

ρLH(G(φa)− μ)

AβF (φa)<

ρHH(G(φa)− μ)

AβF (φa)⇐⇒ ρLH < ρHH (A9)

andρLL(G(φ

b)− μ)

AβF (φb)<

ρHL(G(φb)− μ)

AβF (φb)⇐⇒ ρLL < ρHL. (A10)

On the other hand, from Eq.(A7), we obtain

[ρHH ]πa [ρLH ]

1−πa =hAφa − μρHH

1− μ

iπahAφa − μρLH1− μ

i1−πa.

From this equation, we find that if πa is arbitrarily close to one, Aφa is arbitrarily close to

ρHH , and if πa is arbitrarily close to zero, Aφa is arbitrarily close to ρLH . By the continuity

of φa with respect to πa, it follows that

ρLH < Aφa < ρHH . (A11)

Likewise, from Eq.(A8), we obtain

[ρLL]πb[ρHL]

1−πb =hAφb − μρLL

1− μ

iπbhAφb − μρHL1− μ

i1−πb,

and thus,

ρLL < Aφb < ρHL. (A12)

From Eq.(29), we haveAβF (φa)

1− μ− β(G(φa)− μ)= ρHH .

From this equation and Eq.(A11), we have

ρHH > Aφa ⇐⇒ H(φa) > 0 ⇐⇒ φa < φ∗∗,

where H(x) = βF (x)/x− [1− μ− β(G(x)− μ)], as defined in the proof of lemma 1. Figure

3 is useful in understanding that H(φa) > 0 ⇐⇒ φa < φ∗∗. From φb < φa < φ∗∗, we have

φb < φ∗∗ ⇐⇒ H(φb) > 0 ⇐⇒ ρLL > Aφb,

which contradicts Eq.(A12). However, when φb = φ∗, Eq.(29) always holds. Moreover, it

follows from Eq.(35) that πb = 1. ¤32

Proof of Lemma 3

Case 1: zt = 0. From lemma 2, this state is bubbleless, that is, φb = φ∗, and zt+1 = 0

occurs with probability πb = 1. Therefore, from Eq.(A8) with rdt+1(0) = rit+1(0) = rL, the

stationary-state cutoff of φb = φt(zt = 0) satisfies

rL =Aφb − μrL1− μ

,

or equivalently,

φ∗ = φb = rL/A.

Case 2: If zt = 1, zt+1 = 0 occurs with probability 1 − πa and zt+1 = 1 occurs with

probability πa. If asset bubbles burst at time t + 1 (taht is, zt+1 = 0), the second equation

in (38) holds. It follows from Eqs.(12) and (14) that

Lt(zt) =

μβ

1− μ(Yt(z

t−1) + rt(zt)Bt−1(zt−1))(1−G(φt))

and

Dt(zt) = β(Yt(z

t−1) + rt(zt)Bt−1(zt−1))G(φt).

Substituting these equations into the second equation in (38), we obtain

0 = r0Lβ(Yt(zt−1) + rHBt−1(zt−1))G(φa)− rL μβ

1− μ(Yt(z

t−1) + rHBt−1(zt−1))(1−G(φa)),

or equivalently,

r0L =rLμ(1−G(φa))G(φa)(1− μ)

.

Finally, from Eq.(A7) with rdt+1(0) = rL0, rit+1(0) = rL and r

dt+1(1) = rit+1(1) = rH , the

stationary-state cutoff of φa = φt(zt = 1) satisfies

rπa

H r0L1−πa

=

"Aφa − μrH1− μ

#πa"Aφa − μrL1− μ

#1−πa. ¤

Proof of Proposition 3

From Eq.(39), there exist φb and rL such that φb = φ∗ = G−1(μ) and rL = Aφ∗ = G−1(μ).

From Eq.(36) with ρHH = rH , we obtain

rH =AβF (φa)

1− μ− β(G(φa)− μ)=: J(φa).

From Lemma 3, we obtain

rH =Aφa(φa − μφ∗)(1−π

a)/πa∙μφ∗(1−G(φa))G(φa)(1−μ)

¸(1−πa)/πa+ μ(φa − μφ∗)(1−πa)/πa

=: Q(φa).

33

Assuming that the support of G(φ) is [d, h] for the time being, the functions, J(φ) and Q(φ),

are continuous in [φ∗, h]. It follows from (1−G(φ∗))φ∗ < βF (φ∗) and G(φ∗) = μ that

Q(φ∗) =Aφ∗(1− μ)(1−π

a)/πa

1 + μ(1− μ)(1−πa)/πa< J(φ∗) =

AβF (φ∗)1−G(φ∗) .

It also follows from G(h) = 1 that

Q(h) = Ah/μ > J(h) = 0.

Therefore, there exist φa ∈ (φ∗, h) and rH such that rH = J(φa) = Q(φa). Obviously, r0L alsoexists. For the case in which the support of G(φ) is [d,∞), we are able to provide the proofsimilarly. ¤.

34

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[34] Wang, Pengfei and Yi Wen, 2010. “Speculative Bubbles and Financial Crisis,” forth-

coming in American Economic Journal: Macroeconomics.

[35] Weil, Philippe, 1987. “Confidence and the Real Value of Money in an Overlapping

Generations Economy,” The Quarterly Journal of Economics, 102(1): 1-22

[36] World Bank, 2011. World Development Indicators, World Bank, Washington.

37

Figure 1: Asset Prices and Recessions

-.4

-.2

0.2

.4d

evia

tion

fro

m tr

end

1995 2000 2005 2010Panel A: Asset Prices and Recession in the US

GDP per capita Stock PriceLand Price

-.5

0.5

1d

evia

tion

fro

m tr

end

1984 1986 1988 1990 1992 1994Panel B: Asset Prices and recession in Japan

GDP per capita Stock PriceLand Price

Figure 2: The distribution of a ω

Depositors

a ω

β Y r B

Investors

Φ ωO d h

H x

H

H(x)

xO

Figure 3: Existence of the bubbly steady state

Λ

Ψ 1/ 1 β

O

Panel A: Dynamic behavior without the bubbly steady state

Λ

Ψ

Figure 4

Λ

Ψ

1/ 1 β

Λ

Ψ

O

Panel B: Dynamic behavior with the bubbly steady state

Figure 4

Λ

Ψ

β 1 ε ε /β 1 β

Λ

Ψ

O

Figure 5: Avoiding self-fulfilling financial crises

Ψ is the left-hand side of Eq.(48)

ε/β