DP2022-15

Two Types of Asset Bubbles
in a Small Open Economy

Takashi KAMIHIGASHI
Ryonghun IM

March 30, 2022

Two Types of Asset Bubbles in a Small open economy ∗Takashi Kamihigashi † andRyonghun Im‡

March 30, 2022

Abstract
This paper examines the possibility of two types of rational asset price bubbles in asmall open economy. Our model is a simple representative-agent model similar to theLucas “tree” model except that the economy may be partially or completely open.
There are goods, stock asset, pure bubbly asset, and loan markets, and we consider all
possible cases depending on whether each of the markets is open or closed. We showthat capital inflows give rise to asset bubbles. Moreover, two types of bubbles arise
simultaneously in the economy if the goods market and loan markets are open.
Keywords: stock market bubbles, pure bubbles, small open economy.
JEL classification numbers: E21, E44∗

This study received financial support from the Japan Society for the Promotion of Science (JSPS KAKENHI, Grant No.15H05729, 16J04950, 20H05633 ). All remaining errors are ours.
†Professor at Research Institute for Economics and Business Administration (RIEB) and Center for Computational Social Science (CCSS), Kobe University, 2-1, Rokkodai, Nada, Kobe, 657-8501, Japan. E-mail:tkamihig@rieb.kobe-u.ac.jp.
‡
Assistant Professor at Kwansei Gakuin University. Address: School of Economics, Kwansei Gakuin University, 1-155 Uegahara Ichiban-cho, Nishinomiya 662-8501, Hyogo, Japan; E-mail: ryonghunim@gmail.com.

1

1

Introduction

It is widely held that asset bubbles are often fueled, or at least accompanied, by capital
inflows. Examples of such episodes include the bubbles in emerging market economies in the
1980s and 1990s, the Japanese asset bubble in the late 1980s, and the US housing bubble inthe 2000s (see, e.g., Caballero and Krishnamurthy, 2006; Hattori et al., 2009; Calvo, 2011).
Perhaps the most standard asset pricing model in macroeconomics is the Lucas (1978)“tree” model as described in Sargent (1987), which is a representative agent economy wherebubbles are typically ruled out. In fact, in deterministic representative agent economies, bubbles are simply impossible (Kamihigashi, 2001), and even in the stochastic case, bubbles are
impossible except under rather pathological specifications (Kamihigashi, 1998; Montrucchio
and Privileggi, 2001).
Regarding overlapping generations models, it has been known since Samuelson (1958) that
fiat money can be valued in these models, and it has been common to interpret fiat money asa bubble since Tirole (1985). On the other hand, it has been known since Wilson (1981) thateven in general overlapping generations models, bubbles, including fiat money, are impossible
if the value of aggregate wealth is finite. This is also true for general overlapping generations
models with incomplete markets (Santos and Woodford, 1997). For example, the value of
aggregate wealth is finite if there is a traded stock whose dividend stream is larger than afixed fraction of the aggregate endowment stream. Therefore, if one is to analyze a bubble ona “Lucas tree,” which pays the per capital endowment of the economy as a dividend in eachperiod, then some deviation from standard assumptions is necessary, whether the underlying
economy has overlapping generations or not.
A common approach in the current literature on bubbles is to introduce an additional
financial friction. This approach is used to study bubbles on intrinsically useless assets
in overlapping generations models (Marin and Ventura, 2011; Farhi and Tirole, 2011; ClanChamosset-Yvrard and Seegmuller, 2012) as well as models with infinite lived agents (Hirano
and Yanagawa, 2011; Aoki and Nikolov, 2015; Kunieda and Shibata, 2016). There arealso some known cases in which bubbles on intrinsically useful assets are possible. For
example, Kocherlakota (2008) shows that such bubbles are possible under an exogenous
solvency constraint in a model with a finite number of infinite lived agents. Olivier (2000)
considers bubbles on newly created assets (or firms) in an overlapping generations model. In
variants of the Lucas tree mode, Kamihigashi (2008a, 2008b) shows that bubbles on Lucas
trees are possible if utility directly depends on wealth.1
This paper complements the existing literature by taking an entirely different approach tobubbles. We consider two types of bubbles simultaneously. We introduce intrinsically uselessassets into the Lucas tree model. In this study we call the bubble on the intrinsically uselessasset pure bubble and bubble on Lucas tree stock market bubble. In the closed economy both
bubbles are ruled out, as mentioned above. Then we assume that it is a small open economy,
and consider the possibility of two types of bubbles. To our knowledge, this simple extensionof the Lucas tree model with pure bubbly assets has never been analyzed in the context of
bubbles.
This paper is not the first to study bubbles in an open economy. In fact, there are1

See also Miao and Wang (2018). For other studies on bubbles, we refer the reader to the surveys by
Brunnermeier (2008) and Iraola and Santos (2008).

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well-known studies on bubbles and capital flows in open economies that use overlapping
generations models (Caballero and Krishnamurthy, 2006; Ventura, 2011; Clan-ChamossetYvrard and Kamihigashi, 2017; Ikeda and Phan, 2019;). However, these studies are notintended to explain why bubbles (or valued fiat money) are more likely to arise in open
economies, since bubbles are possible even in closed economies with overlapping generations.
In addition, these studies except Clan-Chamosset-Yvrard and Kamihigashi (2017) assume
that bubble assets are never traded internationally, thus ruling out fully open economies. By
contrast, this paper offers a simple explanation of why two types of bubbles assets easily arise
in fully or partially open economies.
Like the Lucas tree model as described in Sargent (1987), our model is an exchange
economy where there are homogenous goods and homogeneous stock assets, or “trees.” Theadditional features that we introduce is that agents are allowed to buy pure bubbly asset in
a competitive asset market and borrow and lend in a competitive loan market. Thus there
are four markets: goods, stock assets, pure bubbly assets, and loans. We first show that both
bubbles are impossible if the goods market is closed or if two types of assets and loan markets
are closed. Therefore, for bubbles to ever arise, at least the goods market must be open, andat least one of the assets and loan markets must be open. These necessary conditions turnout to be also sufficient: we establish that both bubbles are possible if the goods market
and one of the assets and loan markets are open. For example, stock market bubbles are
possible if the goods and stock asset markets are open; this result can be interpreted as sayingthat foreign investors can drive up domestic asset prices. A less intuitive result is that two
types of bubbles are possible simultaneously even if stock and pure bubbly asset markets areclosed, provided that the goods and loan markets are open; i.e., even if foreign investors haveno access to the domestic asset markets, bubbles can be fueled by capital inflows throughexternal debt.
As indicated above, our model is completely standard except that it is open; thus thedriving force behind bubbles in our model is the openness of the economy. In other words,
capital inflows give rise to otherwise nonexistent bubbles. Furthermore, in contrast to moststudies on bubbles, stock market and pure bubbles in our model are explosive; i.e., they growunboundedly. Most studies on bubbles focus on the steady state of bubbles. On one hand,
this explosive feature may not be consistent with general equilibrium unless the implicit world
economy grows fast enough. On the other hand, in the real world, bubbles become a serious
issue only when they appear to be explosive; thus the explosive feature may capture one of thedefining characteristics of bubbles in reality. In any case, our results suggest that explosive
bubbles easily arise in a country or region that is perceived as “small.” It seems likely that ourmodel is the simplest optimizing framework that can be used to explain explosive bubbles.
Recently, Kikuchi and Thepmongkol (2020) investigate explosive bubbles on capital assets
in the small open economy populated by overlapping generations. In contrast to them, we
employ infinitely-lived agents model and consider simultaneously stock and pure bubbly asset
on an explosive path.
The rest of the paper is organized as follows. In Section 2, we lay out the generalframework common to all cases considered in this paper. We also develop some preliminary
results and formally define bubbles. In Section 3, we show that bubbles are impossible if thegoods market is closed or if both the asset and loan markets are closed. In Section 4, weshow that bubbles are possible if the goods market and one of the assets and loan markets3are open. In Section 5 we conclude the paper. All omitted proofs appear in the appendix.

2

The General Framework

Consider an economy with homogenous consumption goods and many homogeneous agents
who derive utility from consumption. There are stock and intrinsically useless assets in theeconomy. We call the bubble on the intrinsically useless asset pure bubble and the bubble on
the stock stock market bubble. Agents can borrow and lend in a competitive loan market.
The economy can be completely closed, partially open, or completely open. This section lays
out the general framework that is common to all possible cases, which are to be analyzed insubsequent sections.
Throughout the paper, we follow the convention that an equality or inequality dependent
on t is understood to hold for all t ∈ Z+ unless otherwise indicated.
2.1

Agents’ Maximization Problem

Each agent faces the following maximization problem:
max

∞
X

{ct ,st ,mt ,dt ,wt+1 }∞t=0

β t u(ct )

(2.1)

t=0

s.t. ct + pt st + pm
t mt − dt = wt ,wt+1 = (pt+1 + δ)st + pm
t+1 mt − Rt+1 dt ,ct ≥ 0,st , mt , ≥ 0
wt+1 ≥ 0,
w0 = (p0 + δ)s−1 + pm
0 m−1 + R0 d−1 > 0,

(2.2)
(2.3)
(2.4)
(2.5)
(2.6)
(2.7)

where β ∈ (0, 1) is the discount factor, ct is consumption in period t, wt is wealth at the
beginning of period t, pt is the price of the stock in period t, st is shares of stock held at
the end of period t, pm
t is the price of the intrinsically useless asset in period t, mt is thelevel of the intrinsically useless asset held at the end of period t, dt is debt, Rt+1 > 0 is
the gross interest rate between periods t and t + 1, and δ > 0 is the dividend per share,
which is assumed to be constant over time. (2.5) denotes short-sales constraints. (2.6) is a
non-negative constraint on wealth. With the short sale constraints, we interpret (2.6) as aborrowing constraint on dt(pt+1 + δ)st + pm
t+1 mt ≥ Rt+1 dt .

(2.8)

Remark 2.1. We also consider cases in which in addition to the above constraints, thefollowing no-Ponzi-game condition is required to be fulfilled:
lim

T ↑∞

T
Y
1
t=1

Rt

4

dt ≤ 0.

(2.9)

We assume that the utility function u : R+ → [−∞, ∞) is continuously differentiable on
R++ , continuous, concave, strictly increasing, satisfying the Inada condition
lim u0 (c) = ∞.
c↓0(2.10)

We say that a set of sequences {ct , st , mt , dt , wt+1 } is a feasible path (from w0 ) if it satisfies(2.2)-(2.5). An optimal path (from w0 ) is a feasible path that solves the maximization problem(2.1)-(2.7).2
2.2

Market Clearing Conditions

We normalize both the population of agents and the supply of assets to unity. If the economy
is completely closed, the following four market clearing conditions are required to hold in
equilibrium:
ct = δ,st = 1,
mt = 1
dt = 0.

(2.11)(2.12)(2.13)(2.14)(2.11) is the goods market clearing condition. As in the Lucas tree model, we assume forsimplicity that the dividends on the stocks constitute the entire supply of goods. (2.12) isthe stock asset market clearing condition. (2.13) is the pure bubbly asset market clearing
condition. (2.14) is the loan market clearing condition, which states that there is no borrowing
and lending in equilibrium since agents are homogeneous.
If the economy is partially open, then one, two, or three of the market clearing conditions
above are required to hold. For example, if the goods market is closed and the asset and loan
markets are open, then only the goods market clearing condition (2.11) is required to hold.
If the economy is completely open, none of the market clearing conditions is required to hold.
In this case, the model reduces to an optimization problem without equilibrium conditions.
Unless otherwise indicated, we assume the following in (2.7) for all cases considered in thispaper:
s−1 = 1, m−1 = 1, d−1 = 0.
w 0 = p0 + δ + pm
0 .

(2.15)(2.16)

Formally, the closed economy consists of agents’ maximization problem (2.1)–(2.7) and thefour market clearing conditions (2.11)–(2.14). A partially open economy consists of agents’maximization problem (2.1)–(2.7) and a subset of the market clearing conditions (2.11)–(2.14). An equilibrium of a closed or partially open economy is defined as a set of sequencesm{ct , st , mt , dt , wt+1 , pt , pm
t , Rt+1 } such that given strictly positive sequences {pt , pt , Rt+1 }, thepath {ct , st , mt , dt , wt+1 } is optimal, and the corresponding subset of the market clearing
conditions (2.11)–(2.14) hold. The completely open economy consists only of agents’ maximization problem (2.1)–(2.7). An equilibrium of the completely open economy is defined as
2

Strictly speaking, since we allow u to be unbounded, the maximand in (2.1) need not be well defined for
all feasible paths. Thus, to be perfectly precise, we use weak maximality as our optimality criterion5a set of sequences {ct , st , mt , dt , wt+1 , pt , pm
t , Rt+1 } such that given strictly positive sequencesm{pt , pt , Rt+1 }, the path {ct , st , mt , dt , wt+1 } is optimal.
We analyze possibility of bubbles in all possible cases, where one, two, three, or all of themarket clearing conditions are required to hold in equilibrium.
For the rest of the paper we focus on interior equilibria where the short-sales constraints(2.5) are never binding.
2.3

Properties of Optimal paths

Here we derive some useful properties of optimal paths without considering market clearing
conditions. The following proposition summarizes some of the most basic properties.
Proposition 2.1. A feasible path {ct , st , mt , dt , wt+1 } is optimal if and only if
ct , wt+1 > 0,
pt st + p m
t mt − dt > 0,0u (ct ) = βu0 (ct+1 )Rt+1 ,u0 (ct )pt = βu0 (ct+1 )(pt+1 + δ),0m
u0 (ct )pm
t = βu (ct+1 )pt+1 ,lim β t u0 (ct )(pt st + pm
t mt − dt ) = 0.
t↑∞

(2.17)(2.18)(2.19)(2.20)(2.21)(2.22)

Proof. See Appendix A.
With st > 0 and mt > 0, inequalities (2.17) and (2.3) mean that none of the inequality
constraints is binding for an optimal path. Equations (2.19), (2.20)), and (2.21) are the Euler
equations associated with dt , st , and mt , respectively. Equation (2.22) is a transversality
condition.
From Euler equations (2.19), (2.20), and (2.21) we obtain the following no-arbitrage
conditionpm
pt+1 + δ= t+1,(2.23)
Rt+1 =ptpm
ti.e., lending at the ongoing interest rate and investing in assets are perfect substitutes fromagents’ perspective. (2.3) and (2.23) yields
wt+1 = Rt+1 (pt st + pm
t mt − dt ),(2.24)and then the budget constraint (2.2) and (2.3) reduce to
ct +

wt+1
= wt .
Rt+1(2.25)To state our next result, we define πt0 = 1, and
πtj =

j
Y
1
Rt+i
i=1

6

(2.26)

for j ∈ N. Then, by (2.19)
πtj = β ju0 (ct+j ).
u0 (ct )(2.27)

j
πt+1= Rt+1 πtj+1 .

(2.28)

Note thatj= πti+j ,πti πt+i

Lemma 2.1. Any optimal path {ct , st , mt , dt , wt+1 } satisfies∞
X

π0t ct = w0 .

(2.29)

t=0

Proof. See Appendix B.
Recall that in this model the only source of income is initial wealth w0 . Lemma 2.1 saysthat initial wealth must be exhausted on consumption; i.e., the present discounted value ofthe optimal consumption stream is equal to initial wealth.
2.4

Bubbles

We now define bubbles in a standard way. Note from the Euler equation (2.20) and (2.27)thatu0 (ct+1 )(pt+1 + δ)
pt = β 0u (ct )
= πt1 δ + πt1 pt+1 .

(2.30)(2.31)

By successive forward substitution and (2.28)
pt = πt1 δ + πt2 δ + πt2 pt+2
= πt1 δ + πt2 δ + πt3 δ + πt3 pt+3
..
.
=
=

J
X
j=1
∞
X

(2.32)(2.33)(2.34)

πtj δ + πtJ pt+J

(2.35)

πtj δ + lim πtJ pt+J .

(2.36)j↑∞j=1

The above limit exists since the sum in (2.35) is increasing in J and thus the second termin (2.35) is decreasing in J. Define
p∗t =

∞
X
j=1

πtj δ =

∞
X

βj

j=1

u0 (ct+j )δ,
u0 (ct )bt = lim πtJ pt+J = lim β J
J↑∞

J↑∞

7

u0 (ct+J )pt+J .
u0 (ct )(2.37)(2.38)

Note that p∗t is the present discounted value of the dividend stream. We call p∗t the fundamental value of an asset, and bt the stock market bubble. Now pt is decomposed as
pt = p∗t + bt .

(2.39)

Note from that (2.28) and (2.38) thatJ
bt+1 = lim πt+1
pt+1+J(2.40)

J↑∞= Rt+1 lim πtJ+1 pt+1+J = Rt+1 bt .
J↑∞

(2.41)

It follows thatb0 > 0⇐⇒∀t ∈ Z+ , bt > 0.

(2.42)

A stock market bubble can start only in period 0, and once it starts, it remains positive inall periods.
From (2.23) pure bubbles satisfympm
t+1 = Rt+1 pt(2.43)

Thus, we have
pm
0 > 0

3

∀t ∈ Z+ , pm
t > 0.

⇐⇒(2.44)

Partially Open Economies without Bubbles

In this section, we take an equilibrium {ct , st , mt , dt , wt+1 , pt , pm
t , Rt+1 } of a closed or partiallyopen economy as given; we specify the required set of market clearing conditions in thestatements of our results.
Proposition 3.1. If the goods market is closed, i.e., if (2.11) holds, then b0 = pm
0 = 0.
Proof. Recall from (2.16) and (2.29) that∞X

π0t ct = w0 = p0 + δ + pm
0 .

(3.1)

t=0

This together with the goods market clearing condition (2.11) yields
p0 + pm0 =∞X

π0t δ − δ =

t=0

∞
X

π0t δ = p∗0 ,

(3.2)

t=1

m
i.e., b0 + pm
0 = 0. Thus, we have b0 = p0 = 0This result shows that as long as the goods market is closed, stock market and purebubbles are impossible regardless of the openness of the remaining three markets. To understand this clearly, note from (2.16) and Lemma 2.1 that if there are two types bubbles,
agents must consume more than p∗0 + δ in equilibrium. But since the goods market is closed,
their consumption is fixed in equilibrium, and its present discounted value is exactly equal
to p∗0 + δ. Therefore, there is no room for both bubbles as long as the goods market is closed.
8Proposition 3.2. If the assets and loan markets are closed, i.e., if (2.12), (2.13), and (2.14)
hold, then b0 = 0 and pm
0 .
Proof. (2.2) and (2.3) yields
ct + ps st − dt = (pt + δ) + pm
t−1 − Rt dt−1 .

(3.3)

By using (2.12), (2.13), and (2.14), we have ct = δ. Then, it is easily to see b0 = pm
0 = 0, asin the proof of Proposition 3.1.
This result shows that as long as the assets and loan markets are closed, bubbles are
impossible regardless of the openness of the goods market. The easiest way to understandthis result is to notice that even if the goods market is open, it is totally useless unless anothermarket is open, since otherwise there is no way for agents to increase consumption. Hence,as long as the asset s and loan markets are closed, the goods market is effectively closed, andthere cannot be any bubbles for the reason described above.

4

Open Economies with Bubbles

Propositions 3.1 and 3.2 show that for bubbles to arise, the goods market and at least oneof the assets and loan markets must be open. In this section, we consider partially or fully
open economies in which this minimum requirement is met.
Since the returns on assets and loans are identical by (2.23), and since at least one of
the assets and loan markets is open, the common rate of return must be equal to the rate of
return prevailing in the world market, denoted R, which we assume is equal to 1/β:
Rt+1 = R = 1/β.

(4.1)

This is a common assumption for models of small open economies, implying that there is a
steady state. Condition (4.1) makes it possible to obtain a closed form solution, but it is not
crucial to our basic results on the existence of bubbles.
An equilibrium {ct , st , mt , dt , wt+1 , pt , pm
t , Rt+1 } of a partially or fully open economy isdefined here as in the previous section with Rt+1 given by (4.1).
Note from (2.19), (2.27), and (4.1) that πtj = β j . Thus by (2.37), p∗t is constant:p∗t =βδ ≡ p∗ .
1−β

(4.2)

Lemma 4.1. Any equilibrium {ct , st , mt , dt , wt+1 , pt , pm
t , Rt+1 } satisfies the following:p 0 = p ∗ + b0 ,pt+1 = Rpt − δ,
p t = p ∗ + R t b0 ,mpm
t+1 = Rpt ,ct = δ + (1 − β)(b0 + pm
0 ),
δ
wt =
+ b0 + p m
0 .
1−βProof. See Appendix C.
9

(4.3)
(4.4)
(4.5)
(4.6)
(4.7)
(4.8)

4.1

Explosive Bubbles and Capital Inflow

Proposition 4.1. Suppose that the pure bubbly asset and loan market are closed and thegoods and asset markets are open, i.e., of the four market clearing conditions (2.11)–(2.14),
only (2.13) (mt = 1) and (2.14) (dt = 0) are required to hold. Then for any b0 ≥ 0, there
exists a unique equilibrium {ct , st , mt , dt , wt+1 , pt , pm
t , Rt+1 } satisfying (4.3). This equilibrium
satisfies (4.4)–(4.8), andp∗ + βb0(Rt − β)pm0−,p ∗ + R t b0
p ∗ + R t b0pm
0 = 0.
st =(4.9)(4.10)

Proof. See Appendix D.
In the setting of Proposition 4.1, by (4.9) and (4.10), if b0 > 0
st < 1,

lim st = 0.
t↑∞

(4.11)

Hence stock asset ownership gradually shifts from domestic agents to foreign investors. In thelimit, domestic agents hold no asset. If b0 = 0, (4.9) implies that st = 1 and the equilibrium
can attain the same allocation as a closed economy even if stock asset market is opened.
Proposition 4.2. Suppose that the stock asset and loan market are closed and the goods andpure bubbly asset markets are open, i.e., of the four market clearing conditions (2.11)–(2.14),
only (2.12) (st = 1) and (2.14) (dt = 0) are required to hold. Then for any pm
0 ≥ 0, theremexists a unique equilibrium {ct , st , mt , dt , wt+1 , pt , pt , Rt+1 } satisfying (4.3)–(4.8),(Rt − β)b0
βpm
0−
R t pmR t pm00b0 = 0.

mt =(4.12)(4.13)

Proof. See Appendix E.
As in the above argument, if pm
0 > 0mt < 1,

lim mt = 0.
t↑∞

(4.14)

Hence pure bubbly asset ownership gradually shifts from domestic agents to foreign investors. In the limit, domestic agents hold no pure bubbly assets.
We define aggregate bubbles as
Bt = bt + pm
t .

(4.15)

Bt+1 = Rt B0 ,(4.16)

It satisfies
where B0 = b0 + pm
0 .
The following proposition shows that both stock market and pure bubbles arise in theeconomy.
10

Proposition 4.3. Suppose that the stock and pure bubbly assets market are closed and thegoods and loan are open, i.e., of the four market clearing conditions (2.11)–(2.14), only (2.12)(st = 1) and (2.13) (mt = 1) are required to hold. Then for any B0 ≥ 0, there exists a uniqueequilibrium {ct , st , mt , dt , wt+1 , pt , pm
t , Rt+1 } satisfying (4.3)–(4.8),
dt = (Rt − β)B0 .

(4.17)

Proof. See Appendix F.
Proposition 4.3 shows that there is a continuum of equilibria indexed by the initial size ofthe aggregate bubble B0 ≥ 0. Given Bt , the values of bt and pm are indeterminate. Note that
wealth, consumption, and debt all depend positively on B0 . Therefore, bubbles are welfare
improving in this model.
Proposition 4.4. Suppose that the only loan market is closed and the goods, i.e., (2.14)(dt = 0) is required to hold. Let B0 be given. Then there exists an equilibrium {ct , st ,mt , dt , wt+1 } satisfying (4.3)–(4.8). Furthermore, for any {st } in R, there exists a uniqueequilibrium {ct , st , mt , dt , wt+1 } satisfying (4.3) such that {mt } = {mt }, where the sequence
mt is determined by∗
pt st + pm
t mt = p + βB0 ,(4.18)mwhere given B0 = b0 + pm
0 ≥ 0 the values of b0 and p0 are indeterminate. Likewise, forany sequence {mt } in R, there exists a unique equilibrium {ct , st , mt , dt , wt+1 , pt , pm
t , Rt+1 }satisfying(4.3) such that {st } = {st }, where the sequence {st } is determined by (4.18).

Proof. See Appendix G.
Proposition 4.4 shows that there is a continuum of equilibria in the partially open economy in where the stock and pure bubbly assets market are open. The characteristic of theequilibria are identical to those of the partially open economies with bubbles studied above,
except that {st } or {mt } are indeterminate even when the initial size of the bubble B0 isgiven.
4.2

Capital Flight and Endogenous Bubble Crashes.

Consider the partially open economy in which the loan market and pure bubbly asset market(or stock asset market) are closed; in this case, it does not matter whether the no-Ponzi-gamecondition is imposed or not. We suppose that the economy initially follows an equilibriumwith a positive stock market bubble (or pure bubble). In period t > 0, however, all foreign
investors suddenly withdraw from the domestic market, so that the asset market clearing
condition is required to be satisfied:
st = 1 or

mt = 1.

(4.19)

Then, by Proposition 3.2, stock market bubbles (or pure bubbles) are never valued, that is,
bubbles collapse. This can be due to a domestic policy change. For example, the government
might suddenly restrict asset ownership to domestic agents. It can also be due to a dramaticchange in the global financial landscape, which might in turn be caused by a global financial
crisis. For example, foreign investors may be in sudden need of cash, and they need to cashin their asset holdings.
11

5

Conclusion

This study constructed a model of two types of rational bubbles in a small open economy.
We showed that with capital inflow stock market and pure bubbles are possible. The driving
force behind bubbles in our model is the openness of the economy. In addition, both bubbles
are explosive.
The model has some limitations. We employed an endowment economy model and ignored
production. Bubbles enhance production and economic growth. Thus, it is important toexamine how two types of bubble affect production and economic growth. We considered
the deterministic bubbles. It is important extension to consider the effects of a collapse of
bubbles and how interaction between two types of stochastic bubble affect macroeconomy.
We leave these issues as future work.

AppendixA

Proof of Proposition 2.1∗} satisfies (2.17)–(2.22). From (2.19)–(2.21),
If: Suppose a feasible path {c∗t , s∗t , m∗t , d∗t , wt+1
we havepm
pt+1 + δ
Rt+1 =
.
(A.1)
= t+1
pt
pm
t

Let {ct , st , mt , dt , wt+1 } be an arbitrary feasible path. Then, (2.2), (2.3), and (A.1) imply
that for any feasible path the followings holdpt st + pm
t mt − dt =

wt+1
,
Rt+1

ct = wt − (pt st + pm
t mt − dt ) = wt −

(A.2)
wt+1
.
Rt+1

(A.3)

By concavity, we haveTX

β t u(ct ) −=β t u(c∗t )

(A.4)

t=0

t=0

≤

T
X

T
X
t=0
T
X

β t u0 (c∗t )(ct − c∗t )

(A.5)



∗
β t u0 (c∗t ) wt − wt∗ − (wt+1 − wt+1)/Rt+1

(A.6)

t=0
T 0

= β u (c∗T )(wT∗ +1 − wT +1 )/Rt+1≤ β u (c∗T )wT∗ +1 /Rt+1∗∗= β T u0 (c∗T )(pt s∗t + pm
t mt − dt ) → 0,
T

0

(A.7)
(A.8)
(A.9)

where (A.6) holds by (A.3). (A.7) holds by the Euler equation (2.19). The inequality in (A.8)
holds by wt+1 > 0 in (2.18). The equality in (A.9) holds by (A.2). The convergence holds by∗
the transversality condition (2.22). The optimality of {c∗t , s∗t , m∗t , d∗t , wt+1
} now follows.
12Only if: Let an optimal path be given. We verify (2.17)-(2.22). We start by claiming
ct > 0 for all t ∈ Z+ . To see this, first note that since Rt+1 > 0 for all t ∈ Z+ , it is feasible
to choose ct > 0 for all t ∈ Z+ . Thus, if u is unbounded below, then it is never optimal tochoose ct = 0. If u is bounded below, then Inada condition (2.10) implies that ct > 0 for agiven t ∈ N. whenever ct−1 > 0 or ct+1 > 0, since Rt , Rt+1 > 0. This in turn implies ct > 0
for all t ∈ Z+ .
Next we show wt+1 > 0 for all t ∈ Z+ . To see this, suppose wt = 0 for some t ∈ N. We
˜have 0 < ct = −pt st − pmt mt + dt . By choosing s̃t = 2st , m̃t = 2mt , and dt = 2dt instead ofst , mt , and dt , one can obtain c̃t = 2ct > ct , w̃t+1 = 2wt+1 ≥ wt+1 , and c̃t+1 ≥ ct+1 instead ofct , wt+1 , and ct+1 without affecting the rest of the path. This contradicts the optimality ofthe original path, establishing wt+1 > 0 for all t ∈ Z+ .
To see (2.18), note that as in the above argument, if pt st +pm
t mt −dt ≤ 0, one could double
wt+1 and increase ct+1 by doubling st , mt , and dt without affecting the rest of the path. Thus
(2.18) follows. Since no inequality constraint is binding, the Euler equation (2.19), (2.20)
and (2.21) hold. The transversality condition (2.22) follows from Kamihigashi (2001).
B

Proof of Lemma 2.1

We start by establishing two simple lemmas:
Lemma A1. Any feasible path {ct , st , mt , dt , wt+1 } satsifies∞X

π0t ct ≤ w0 .

(B.1)

t=0

Proof. Multiplying (2.25) by π0t and summing over from t = 0 to t = T , we get
w0 =

T
X

π0t ct + π0T wT +1 /RT +1 ≥TX

π0t ct .

(B.2)

t=0

t=0

Letting T ↑ ∞ yields (B.1).
Lemma A2. For any nonnegative sequence {ct } satisfying (2.29), there exist sequences{st }, {mt }, {dt }, and {wt+1 } such that the path {ct , st , mt , dt , wt+1 } is feasible.
Proof. We construct such sequences. Let
st =

wt+1
,
2(pt+1 + δ)

mt =

wt+1
,
2pm
t+1

13

dt = 0,

wt =

∞
X
i=0

πti ct+i ≥ 0.

(B.3)

Clearly, (2.3)–(2.7) hold. To verify (2.2), note that (B.3) that
wt+1
wt+1
+ m
2(pt+1 + δ)/pt 2pt+1 /pmt
wt+1= ct +
Rt+1∞
Xi= ct +
πt+1
ct+1+i /Rt+1

ct + pt st + pm mt − dt = ct +

(B.4)
(B.5)
(B.6)

i=0

= ct +∞
X

πti+1 ct+1+i

(B.7)

i=0

=

∞
X

πtj ct+j

(B.8)

j=0

where (B.5) holds by (2.23). (B.7) holds by (2.28). Now (2.2) follows, so the path {ct , st , mt ,dt , wt+1 } is feasible.
To complete the proof of Lemma 2.1, suppose there is an optimal path {ct , st , mt , dt , wt+1 }that does not satisfy (2.29). Then by Lemma A1, we must have∞X

π0t ct < w0 .

(B.9)

c̃t = ct w0 /σ > ct .

(B.10)

σ≡

t=0

Let
Then, we have∞X

π0t c̃t =

t=0

∞
X

π0t ct w0 /σ = w0 .

(B.11)

t=00} such that {c0t } = {c̃t }, contradictBy Lemma A2, there is a feasible path {c0t , s0t , m0t , d0t , wt+1
ing the optimality of the original path by (B.11).

C

Proof of Lemma 4.1

By (4.1) an the Euler equation (2.19),
u0 (ct ) = u0 (ct+1 ).

(C.1)

Thus there is c > 0 such that ct = c.
By the Euler equation (2.20), ct = c, and (4.1), we have pt = (pt+1 +δ)/R. Solving for pt+1
yields (4.4). With bt+1 = Rbt (see (2.41)), (4.1), and (4.4), we have (4.3) and (4.5). (2.43)
and (4.1) yields (4.6) Note from (2.19), (2.27), and (4.1) that πtj = β j . Lemma 2.1, ct = c,
(2.16), and (4.2) yields (4.7). Note from (2.29), ct = c, and πtj = β j that c/(1−β) = w0 holds.
Since (2.29) holds with period 0 shifted forward to period t, we have c/(1 − β) = wt = w0 .
By (2.16), and (4.2), (4.8) is obtained.
14

D

Proof of Proposition 4.1

Let {ct , st , mt , dt , wt+1 , pt , pm
t , Rt+1 } be an equilibrium satisfying (4.3). Then it also satisfies(4.4)–(4.8) by Lemma 4.1. Note from ct /(1 − β) = wt = w0 and (2.2) thatp t st + pm
t mt − dt = βw0(D.1)

Substituting the market clearing conditions (mt = 1 and dt = 0) into the above equation
yieldsβw0 − pmtptmβ(p0 + δ + pm
0 ) − pt=p ∗ + R t b0mp∗ + βb0 − (pmt − βp0 )=p ∗ + R t b0p∗ + βb0(Rt − β)pm
0
= ∗
−
.
p + R t b0
p ∗ + R t b0

st =

(D.2)
(D.3)
(D.4)
(D.5)

The second line uses (2.16) and (4.5). The third line uses (4.2) and (4.3). The last line uses
(4.6).
Next, we show that pm
0 = 0. By (D.1), mt = 1, and dt = 0 we have
pt st + pm
t = βw0 ,(D.6)

which implies pm
t < βw0 . We rearrange this inequality as(Rt − β)pm
0 < β(p0 + δ),(D.7)

where we use (2.16) and (4.6). The inequality means that for sufficiently large T , agents can
not buy pure bubbly assets even if they use all of assets they hold. Thus, we must have pm
0 .
Now we show that uniqueness and existence of the equilibrium. Uniqueness follows since(4.4)–(4.8), (4.9), and (4.10) uniquely determine an equilibrium satisfying (2.13), (2.14),
and (4.3). To establish existence, let {ct , st , mt , dt , wt+1 , pt , pm
t , Rt+1 } be given by (4.3)–(4.8),(4.9), and (4.10). It satisfies (2.13) and (2.14) by construction. It is routine to verify thefeasibility of {ct , st , mt , dt , wt+1 } . As for optimality, it is easy to see that (2.17)–(2.21) hold.
The transversality condition (2.22) also holds since pt st + pm
t mt − dt is constant by (D.1).
Thus by Proposition 2.1, {ct , st , mt , dt , wt+1 } is optimal. It follows that {ct , st , mt , dt , wt+1 ,pt , pmt , Rt+1 } is an equilibrium.
E

Proof of Proposition 4.2

First, suppose that {ct , st , mt , dt , wt+1 , pt , pm
t , Rt+1 } be an equilibrium satisfying (2.12) and(2.14). Then, it also satisfy (4.3)–(4.8) by Lemma 4.1. By the stock asset and loan market

15

clearing conditions (2.12) and (2.14) and (D.1),
βw0 − ptmt =
pmtβ(p0 + δ + pm
0 ) − pt
=
m
t
R p0
m
βp
β(p∗ + b0 + δ) − (p∗ + Rt b0 )= t 0m −R p0
R t pm0mtβp(R − β)b0= t 0m −,R p0
R t pm
0

(E.1)
(E.2)
(E.3)
(E.4)

where the second line uses (2.16) and (4.6). The third line uses (4.3) and (4.5). The last line
uses (4.2). Now (4.12) follows.
Next, we show that (4.13) (b0 = 0). As in the derivation of (D.7), we havep∗ + (Rt − β)b0 < β(p∗ + δ + pm
0 ).

(E.5)

Since the stock market bubble bt exceeds the total wealth of agents for a large T , b0 = 0 must
hold in the equilibrium with st = 1.
Uniqueness follows since (4.3)–(4.8), (4.12), and (4.13) uniquely determine an equilibrium
satisfying (2.12) and (2.14) for any pm
0 ≥ 0. To establish existence, let {ct , st , mt , dt , wt+1 ,m
pt , pt , Rt+1 } be given by (4.3)–(4.8), (4.12), and (4.13). It satisfies (2.12) and (2.14) byconstruction. It is easy to verify that {ct , st , mt , dt , wt+1 } is feasible and optimal, as in the
proof of Proposition 4.1. It follows that {ct , st , mt , dt , wt+1 , pt , pm
t , Rt+1 } is an equilibrium.
F

Proof of Proposition 4.3

Let {ct , st , mt , dt , wt+1 , pt , pm
t , Rt+1 } be an equilibrium satisfying (2.12) and (2.13). Then, it
also satisfy (4.3)–(4.8) by Lemma 4.1. By the stock and pure bubbly assets market clearing
conditions (2.12) and (2.13) and (D.1),
dt = pt + pmt − βw0∗= p + Bt − β(p∗ + δ + B0 )= (Rt − β)B0 .

(F.1)
(F.2)
(F.3)

where the second line uses (2.16), (4.3), (4.5), and (4.15). The last line uses (4.16).
Uniqueness follows since (4.3)–(4.8), and (4.17) uniquely determine an equilibrium satisfying (2.12) and (2.13) for any B0 ≥ 0. To establish existence, let {ct , st , mt , dt , wt+1 , pt , pmt ,Rt+1 } be given by (4.3)–(4.8), and (4.17). It satisfies (2.12) and (2.13) by construction. It iseasy to verify that {ct , st , mt , dt , wt+1 } is feasible and optimal, as in the proof of Proposition
4.1. It follows that {ct , st , mt , dt , wt+1 , pt , pm
t , Rt+1 } is an equilibrium.
G

Proof of Proposition 4.4

Let {ct , st , mt , dt , wt+1 , pt , pm
t , Rt+1 } be an equilibrium satisfying (4.3). Then it also satisfies(4.4)–(4.8) by Lemma 4.1. (2.14) and (D.1) yields∗∗
pt st + pm
t mt = β(p + b0 + δ + p )= p∗ + βB0 .

16

Thus, (4.18) is obtained.
For a given sequence {st } in R, (4.4)–(4.8) and (4.18) uniquely determine an equilibrium
satisfying (4.3). Likewise, for a given {mt },(4.4)–(4.8) and (4.18) uniquely determine anequilibrium satisfying (4.3).
Finally, let {ct , st , mt , dt , wt+1 , pt , pm
t , Rt+1 } satisfy (4.3)–(4.8), and (4.18) . Then it iseasy to verify that {ct , st , mt , dt , wt+1 } is feasible and optimal, as in the proof Proposition
4.1. Since there is no market clearing condition here, it follows that {ct , st , mt , dt , wt+1 , pt ,pmt , Rt+1 } is an equilibrium.

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