A simple discrete-time approximation of continuous-time bubbles

Journal

ELSEVIER

of Economic Dynamics and Control

22 (1998) 937-954

A simple discrete-time approximation of continuous-time bubbles Yuichi Fukuta* School of Business Administration. Kobe University, 2-I Rokkodai-rho. Hyogo 657, Japan

Nada, Kobe,

Received 7 August 1994; accepted 29 May 1997

Abstract The purpose of this paper is to propose a class of rational bubbles called incomplete bursting bubbles to investigate the relationship between Blanchard’s bubbles and other more complicated ones. The bubbles have three possible states: a large bubble state, a small bubble state and an incomplete burst state. They include deterministic bubbles as a special case. They are easily related to stochastic and periodically collapsing bubbles. Moreover, they converge to continuous-time bubbles: intrinsic bubbles and fundamentalsdependent bubbles in special limiting cases. Hence, we conclude that incompletely bursting bubbles are a simple discrete-time approximation of the continuous-time bubbles. From the relationship between incompletely bursting bubbles and other bubbles, we can interpret these bubbles as a general type of rational bubbles. @ 1998 Elsevier Science B.V. All rights reserved. JEL classification: D84; E44; G12 Keywords: Incompletely bursting bubbles; Intrinsic bubbles; Fundamentals-dependent bubbles; Deterministic and stochastic bubbles; Periodically collapsing bubbles

1. Introduction Recent studies on rational bubbles have considered various kinds of bubbles. Blanchard (1979) and Blanchard and Watson (1982) presented deterministic

* E-mail: [email protected] I am grateful to S. Ikeda, Y. Takeuchi, an anonymous

referee of this journal and participants of the Macroeconomics Workshop at Osaka City University and the Finance Forum seminar for their helpful comments and suggestions. Of course, all remaining errors are mine. Ol65-1889/98/$19.00 0 1998 Elsevier PII SO165-1889(97)00086-9

Science B.V. All rights reserved

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bubbles and stochastic bubbles in a discrete-time model. 1 The stock prices embodying the deterministic bubbles keep on diverging from the fundamental prices forever. However, it is unrealistic to consider the above diverging deterministic bubbles. Stochastic bubbles can be in one of two states: one in which the bubbles remain and the other in which they crash. The stock prices including stochastic bubbles diverge form the fundamental price level, so long as they last. When the bubbles completely crash, the stock prices heavily decline, so that they are equal to the fundamental price level. Thus, the bubbles are more realistic than deterministic bubbles. As mentioned in Diba and Grossman (1988b), if stochastic bubbles completely crash, then they cannot restart on the free disposal assumption. In other words, stochastic bubbles crash only once. However, it is realistic to assume that the bubbles with repeated crashes do indeed exist. Since these two types of bubbles have two states at most, they belong to a class of simple rational bubbles. Froot and Obstfeld (1991), Ikeda and Shibata (1992) and Ikeda and Shibata (1995) presented some rational bubbles in continuous-time models. Froot and Obstfeld (1991) proposed intrinsic bubbles which depend on dividends. 2 Ikeda and Shibata (1992) and Ikeda and Shibata (1995) presented fundamentals-dependent bubbles which depend not only on dividends but also on time trend. Since Blanchard’s bubbles depend upon time trend alone, these bubbles can be interpreted as a sophisticated extension of Blanchard’s bubbles. However, these bubbles have some shortcomings. First, dividends in continuous-time models follow a geometric Brownian motion, whereas we do not explicitly specify a stochastic process of dividends in the model of Blanchard’s bubbles. Second, in contrast to Blanchard’s bubbles, the mathematical tools employed to analyze intrinsic bubbles and I%ndamentals-dependent bubbles are quite advanced. These differences indicate that continuous-time bubbles, such as intrinsic bubbles and fundamentalsdependent bubbles, belong to a class of complicated rational bubbles. Evans ( 1991) presented periodically collapsing bubbles in a discrete-time model. The bubbles are regarded as a combination of log-normally distributed bubbles and stochastic bubbles which periodically collapse. Evans’ bubbles contain the possibility of repeated crashes. However, in contrast to Blanchard’s bubbles which contain only one type of rational bubbles, periodically collapsing bubbles contain two types of rational bubbles. Hence, these bubbles also belong to a class of complicated rational bubbles.

’ Some studies, bubbles.’

such as Blanchard

and Fischer

(1989),

refer to stochastic

bubbles

as ‘bursting

* Froot and Obstfeld (1991) do not derive intrinsic bubbles in a continuous-time model explicitly. However, the fundamentals-dependent bubbles proposed by Ikeda and Shibata (1992) in a continuous-time model include intrinsic bubbles as a special case. Hence, intrinsic bubbles belong to the continuous-time bubbles’ type.

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As mentioned above, existing bubbles have various shortcomings. Moreover, the relationship between simple rational bubbles, such as Blanchard’s bubbles, and more complicated rational bubbles, such as continuous-time bubbles and periodically collapsing bubbles has not been made clear. In this paper, we propose a new model of rational bubbles which are able to overcome the above shortcomings. Using this newly proposed bubbles, we can easily relate Blanchard’s bubbles to more complicated bubbles. We propose a class of rational bubbles which have the following three characteristics. First, the bubbles have only three states: a large bubble state, a small bubble state and an incomplete burst state. Because of this third state we call the bubbles ‘incompletely bursting bubbles’. They may be interpreted as an extension of the two states stochastic bubbles and belong to a class of simple rational bubbles. Second, the bubbles in all three states are strictly positive. This property also applies to deterministic bubbles, periodically collapsing bubbles, intrinsic bubbles and fundamentals-dependent bubbles. Third, when incompletely bursting bubbles are in an incomplete burst state, the bubbles heavily decline, as if they completely crashed. However, in contrast to stochastic bubbles, the bubbles proposed in this paper do not completely crash. The incomplete burst implies that the bubbles are able to raise again after the incomplete crash, which is not the case for stochastic bubbles. 3 Incompletely bursting bubbles are a generalization of Blanchard’s bubbles, continuous-time bubbles and periodically collapsing bubbles. First, incompletely bursting bubbles incorporate deterministic bubbles as a special case. Second, they are quite similar to stochastic bubbles. 4 Third, incompletely bursting bubbles converge to continuous-time bubbles in some special limiting cases. Consequently, incompletely bursting bubbles can be interpreted as a simple discrete-time approximation of continuous-time bubbles. Fourth, we may regard the bubbles as a simple version of periodically collapsing bubbles. These implications not only relate incompletely bursting bubbles to Blanchard’s bubbles but also indicate the relationship between the bubbles and some complicated type of bubbles. In other words, incompletely bursting bubbles which belong to a class of simple rational bubbles can be interpreted as a general type of rational bubbles. The rest of this paper is constructed as follows. Section 2 provides a model of incompletely bursting bubbles. We relate incompletely bursting bubbles to Blanchard’s bubbles and periodically collapsing bubbles in Section 3. In Section 4, we investigate the relationship between incompletely bursting bubbles and continuous-time bubbles in some limiting cases. The conclusions are presented in Section 5. 3 Diba and Grossman (1988b) on the free disposal assumption.

show that, if rational

bubbles

completely

crash, they cannot restart

4 Since incompletely bursting bubbles have an incomplete burst state, precisely bles do not include stochastic bubbles which can take a complete crash state.

speaking,

the bub-

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2. Incompletely bursting bubbles In this section, we present a class of rational bubbles named ‘incompletely bursting bubbles’, which have three states: a large bubble state, a small bubble state and an incomplete burst state. We focus on the following standard model of stock prices: Pf = (1 + T%(dr+i

+ Pt+1),

(1)

where pt is the real stock price at the beginning of period t, dt+t is the real dividend per share paid to the owner at the end of period t, (1 + r)-l c 1 is the discount factor and Et denotes the expectations conditioned on the available information at the beginning of period t. Let us assume that the stock price is composed of the fundamental price level, F,, which is represented by the expected discount value of future dividends and rational bubbles, B, , pt =F, +Br,

where Ft = c,E, Ft+&=(l

(2)

(1 + r)-jErdl+j. +r)-‘E,(d,+i

From Eqs. (1) and (2), we find +fi+~ +&+I).

(3)

Following the definition of the fundamental price level and using Eq. (3), we obtain B,=(l

+r)-‘E,B,+l.

(4)

Eq. (4) implies that the expectations of the growth rate of rational bubbles conditioned on the available information at the beginning of period t, are equal to the discount rate. Let us consider a class of rational bubbles which contain the following three states: State 1: B,+I = (1 + r)

B, with probability nt ,

(5)

B, with probability 722,

(6)

State 2: B,+l = (1 f r)

State 3: B

BI with probability 1 - nt - ~2,

(7)

where 01 and 02 are arbitrary and the probability of each state: x1, 712and 1 - ~1 - 7~2is strictly positive. We can easily verify that the bubbles defined by Eqs. (5)-(7) satisfy Eq. (4) and can thus be included in the class of rational bubbles. We assume O
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assumption implies that the bubbles never completely crash. In addition, we also assume (1 -wI -tilz)/(l -rci -rc~)
3. A comparison with existing theories of bubbles Rational bubbles are any random variables which satisfy Eq. (4). Existing theories of bubbles such as deterministic bubbles, stochastic bubbles, intrinsic bubbles, fundamentals-dependent bubbles and periodically collapsing bubbles all satisfy Eq. (4). 5 In this section, we explain the existing bubbles’ models to characterize incompletely bursting bubbles. We, then, show that the theory of incompletely bursting bubbles can improve upon existing bubbles’ models. Furthermore, we relate the incompletely bursting bubbles to the existing bubbles’ theories. 3. I. Deterministic bubbles Blanchard and Watson (1982) present a rational bubble which satisfies Eq. (4) in a discrete-time model. The bubble is that of deterministic bubbles with the following form: BF=(l

+‘)-‘BE,,

(8)

where Bf’ denotes deterministic bubbles at period t. Since deterministic bubbles grow exponentially, the stock prices including them keep on diverging from the fundamental prices permanently. However, it seems that this feature is unrealistic. In contrast, incompletely bursting bubbles with three states can repeat the growth and crash sequence giving a more realistic representation. Since both deterministic bubbles and incompletely bursting bubbles have finite states, they are included in a class of simple rational bubbles. Next, we relate incompletely bursting bubbles to deterministic bubbles. Setting wI = rtl, 02 = 712 in Eqs. (5)-(7), we find State 1, State 2 and State 3: BI+l = ( 1 + r)B,.

5 In continuous-time E,

[

z

models,

Eq. (4) is rewritten

1

where R denotes

= RBt, the instantaneous

discount

rate.

as

(9)

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Comparing Eq. (9) with Eq. (8) we find that incompletely bursting bubbles based on the above parameters are equivalent to deterministic bubbles. In a word, deterministic bubbles can be regarded as incompletely bursting bubbles with three identical states. Hence, incompletely bursting bubbles include deterministic bubbles under certain circumstances. 3.2. Stochastic bubbles Blanchard (1979) and Blanchard and Watson (1982) present a rational bubble in a discrete-time model. The example is that of stochastic bubbles with the following two states: Bs with probability n,

(10)

State 2: Bs+, = 0 with probability 1 - r~, where Bs denotes the stochastic bubbles at period t. It is easy to verify that stochastic bubbles satisfy Eq. (4). In each period, stochastic bubbles will remain with probability rc or crash with probability 1 - n. Stock prices including these bubbles diverge from the fundamental price level, as long as they last. However, when stochastic bubbles crash, the stock prices heavily decline to the fundamental price level. Stochastic bubbles are more realistic than deterministic bubbles since they allow the possibility of a crash. Diba and Grossman (1988b) show that once rational bubbles have crashed, they never restart. Hence, stock prices remain at the fundamental price level after the crash of stochastic bubbles. In other words, stochastic bubbles can never grow again after a complete crash. This point is the deficiency of stochastic bubbles. In contrast to stochastic bubbles, since incompletely bursting bubbles do not completely crash, they are able to raise again after an incomplete crash. Stock prices including these bubbles move as if they contained restartable bubbles. Next, we examine the relationship between incompletely bursting bubbles and stochastic bubbles. Setting awl = ~1, aw2 = ~2, WI + w2 = 1 in Eqs. (S)-(7), where a is an arbitrary constant which satisfies 0
with probability a,

(11) (12)

Since Eqs. (11) and (12), in which we rewrite a as A, are equivalent to Eqs. (lo), the rational bubbles described by Eqs. (11) and (12) are stochastic bubbles. Stochastic bubbles can be interpreted as an extension of incompletely bursting bubbles since they are obtained by changing the incomplete burst state into a complete burst state and by assuming the first and second states to be identical. Hence, stochastic bubbles are similar to incompletely bursting bubbles, except in the case

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of the complete crash of stochastic bubbles which can be denoted by wr + w2 = 1. Note that, in deriving incompletely bursting bubbles, we assume WI + 02 < 1. 3.3. Periodically collapsing bubbles Evans ( 1991) proposes the following cally collapsing bubbles: if B, 5 c(,

class of rational

bubbles

called periodi(13)

BI+I =(I +r)B,u,+~, if Bt>cc, B,+I =[6+z-’

(1 +rN+ltB,

-(I

-tC’@l~r+~.

(14)

Here, 6 and CI are positive parameters with 0 ~6 <( 1 + r)o1 and ut+i follows an exogenous identically and independently log-normal distribution whose mean is 1. 8,+1 takes the value 1 with probability rc and 0 with probability 1 - rr. The random variable &+t is similar to stochastic bubbles. Hence, we can regard periodically collapsing bubbles as a combination of stochastic bubbles and log-normally distributed bubbles. Periodically collapsing bubbles are more realistic than stochastic bubbles since they periodically burst. However, as Eqs. ( 13) and (14) show, periodically collapsing bubbles are much more complicated than deterministic bubbles and stochastic bubbles. In contrast, incompletely bursting bubbles are as simple as deterministic bubbles and stochastic bubbles. Next, we consider the relationship between incompletely bursting bubbles and periodically collapsing bubbles. Periodically collapsing bubbles contain two types of bubbles: independently log-normally distributed bubbles and stochastic bubbles which periodically collapse. 6 As mentioned in the previous subsection, the latter are similar to incompletely bursting bubbles except for the complete crash. The former can be interpreted as incompletely bursting bubbles with continuous states. Hence, periodically collapsing bubbles can be regarded as a combination of incompletely bursting bubbles with continuous states, and stochastic bubbles which are similar to incompletely bursting bubbles. In other words, we can interpret incompletely bursting bubbles as a simple version of periodically collapsing bubbles. 3.4. Other existing bubbles: Continuous-time bubbles Other existing theories on bubbles, such as intrinsic bubbles, fundamentalsdependent bubbles and multiple fundamentals-dependent bubbles are proposed in continuous-time models. We consider the relationship between incompletely bursting bubbles and those continuous-time bubbles in detail in the next section.

6The log-normal distribution has a constant variance. In contrast, as shown in the next section, the incompletely bursting bubbles converge to a log-normal distribution whose variance depends on time in a limiting case.

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Hence, we briefly explain those bubbles and mention some shortcomings of the bubbles. Intrinsic bubbles proposed by Froot and Obstfeld ( 199 1) depend upon the dividends. Since deterministic bubbles, stochastic bubbles and periodically collapsing bubbles depend upon time trend, intrinsic bubbles can be one extension of rational bubbles. Fundamentals-dependent bubbles presented by Ikeda and Shibata (1992) depend not only on dividends but also on time trend. Hence, they can be interpreted as an extension of intrinsic bubbles. Multiple fundamentals-dependent bubbles proposed by Ikeda and Shibata (1995) are given by the product of two fundamentals-dependent bubbles. They can be regarded as an extension of fundamentals-dependent bubbles. Next, we mention some shortcomings of these continuous-time bubbles. It is necessary to specify a stochastic process of dividends as a geometric Brownian motion to obtain these continuous-time bubbles. Moreover, these bubbles are given as a solution of stochastic differential equations. In other words, mathematical tools employed by these studies are quite advanced. In contrast to these bubbles, it is not necessary to specify the stochastic process of dividends to obtain incompletely bursting bubbles. As Eqs. (5)-(7) show, mathematical tools to derive incompletely bursting bubbles are not so advanced.

4. The relation between incompletely bursting bubbles and continuous-time bubbles in limiting cases In this section, we examine the relationship between incompletely bursting bubbles and continuous-time bubbles in some limiting cases. We show that an incompletely bursting bubble converges to a intrinsic bubble proposed by Froot and Obstfeld (1991) in a special limiting case. We also show that the sum of two incompletely bursting bubbles converges to a fundamentals-dependent bubble proposed by Ikeda and Shibata (1992) in a special limiting case. In addition, we find that the product of the sum of two incompletely bursting bubbles and the sum of two other incompletely bursting bubbles converges to a multiple fundamentalsdependent bubble proposed by Ikeda and Shibata (1995) in a special limiting case. 4.1. Convergence of incompletely bursting bubbles in a limiting case First, we show that an incompletely bursting bubble converges to a geometric Brownian motion with a drift in a special limiting case. Let us define the following random variable bi: State 1: with probability AI,

(15)

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State 2: with probability

7~2,

(lo)

State 3: bi =

l-01

-04

l-711

-712 >

with probability

1 - rci - 712,

(17)

where 01 and 0~2 are the same as in Eqs. (5)-(7). If an incompletely bursting bubble at period 1 is in state 1, Eqs. (5)-(7) imply Bi =Be(ot/ni)(l +r). If we denote the realization of the state of the incompletely bursting bubble at period 1 as bl( 1 + r), we can rewrite the equation as BI = Bobl( 1 + r). To obtain the limit distribution of lnB,, we divide the time period [0, t] into n equal time intervals of length t/n and denote the realization of the random variable bi at period t as b,. We find Bt=BOb,bz...bn(l Taking

the natural

+r)n.

logarithm

(18) of Eq. (18), we obtain

lnB,-lnBe=lnbt+lnb~+~~~+lnb,+ln(l+~)n. Setting

(19)

02 = f and nt = 7r2= f , where

01 = (1 + J&)/3,

constant, we can rewrite Eqs. (15)-(17) State 1: bi=

with probability

(1 +&G)

0 is a positive

as

i,

State 2: bi = 1 with probability

f,

State 3: bi=

with probability

(l-&c)

We obtain the following Elnbi=

varlnbi=i

expectation

f

and variance

of In bi:

(20)

f [,(l+j/&)+ln(l)+ln(l-&0)],

[{ln(l+&fl)}

+(ln(1))2+

-[~{1n(1+fi0)+1n(l)+1n(l-fi0)}]2.

{ln (1 - &)}‘I

(21)

946

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Denoting

(22) and assuming that In bi is independently distributed and using Eqs. (19)-(21), we find

+nln

1,: (

varln$=F

-n [i {l*(l

,

(23)

>

[{ln(l+Z@~+{ln(I)~2+{ln(l-.T~)~]

+Z@

+ln(l)+ln

(1 -Z@}]‘. (24)

Considering the sequence of r and ts such that Z and R remain constant as n goes to infinity, we obtain the following proposition: io)/3, 02 = f, and ~1 = ~2 $ =- j in Eqs. (5)-(7), where o is a positive constant; (ii) we denote C = a$$, R =nr/t; (iii) we consider the sequence of r and a such that Z and R remain constant as n goes to injkity; (iv) incompletely bursting bubbles are independently distributed. Then an incompletely bursting bubble follows a geometric Brownian motion with a drift, as n goes to injinity: dB,/B, = R dt + C dz, where z, denotes a standard Wiener process. Proposition 1. Suppose that (i) we set wt = (1 +

Proof: See the appendix. 4.2. Intrinsic bubbles Next, we show that the intrinsic bubbles proposed by Froot and Obstfeld ( 199 1) in the continuous-time model follow a geometric Brownian motion with a drift. Intrinsic bubbles have the following form: IBt = cD;,

(25)

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where c is an arbitrary constant and ZB, denotes intrinsic bubbles at period t and 2 is a positive constant.’ It is assumed that dividends follow a geometric Brownian motion with a drift: dDt = /&jDt dt + OdDt dzf,

(26)

where p,j and (Td are the expected value and standard deviation of the instantaneous rate of dividends growth and zt is a standard Wiener process. Applying Ito’s Lemma to Eq. (25) and using Eq. (26), we obtain dt + &jDt dz,) + ;&(A - 1)DI’-2ajDf dt.

dl& = cm;-‘(/QD, The arbitrage

condition

cm:‘-&D,

(27)

implies

+ @(n

- 1)Df2a;D;

where R denotes the instantaneous rewrite Eq. (27) as dIBI = RIBl dt + co&

=RIB[,’

discount

rate. Using

dz,.

this equation,

we can

(28)

Using Eq. (25), we obtain dIBt -=Rdt+k~ddz,. IBt

(29)

Eq. (29) shows that intrinsic bubbles follow a geometric Brownian motion with a drift. We find that, if we set C = Jcrd in Eq. (22) and use Proposition 1, then an incompletely bursting bubble converges to intrinsic bubbles in this limiting case. Hence, we have the following proposition: Proposition 2. Incompletely bursting bubbles proposed in the discrete-time model converge to intrinsic bubbles proposed in the continuous-time model in a special limiting case, 4.3. Fundamentals-dependent

bubbles

In this subsection, we investigate the relationship between incompletely bursting bubbles and the fundamentals-dependent bubbles proposed by Ikeda and

’ See Froot and Obstfeld 8 In continuous-time E,

[

z

(1991)

models,

for details on 1.

Eq. (4) can be rewritten

1

=RlB,,

where R denotes

the instantaneous

discount

rate.

as

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Shibata (1992). Ikeda and Shibata (1992) propose the following formula for fundamentals-dependent bubbles: FDB, = {J exp(R - k)t}[G~Dfl

+ GzD:],

(30)

where FDB, denotes fundamentals-dependent J are arbitrary constants and

bubbles at period t, G1, G2, k and

l1 =

-b

-

bfd/2)

+

((9

-

ofd/2)*

+ 2C$dk}“* >

at A2 =

49

-

afd/2>

-

((9

-

Ofd/2)*

+

2afdk}‘/*

7

at

g and afd are the expected value and standard deviation of the instantaneous rate of dividends growth. 9 Applying Ito’s Lemma to Eq. (30) and using the arbitrage condition, we obtain dFDBt =RFDBI

dt + (G~llDf’

+ G&D:){Jexp(R

- k)t}afd dz,.

Eq. (3 1) implies that fundamentals-dependent bubbles do not follow Brownian motion with a drift but a diffusion process. We can, however, show that fundamentals-dependent bubbles are two bubble components which follow geometric Brownian motions. FDB,! and FDBf as {Jexp(R-k)t}G~Df’ and {Jexp(R-k)t}GzD:, we obtain

(31)

a geometric divided into By defining respectively,

FDB, = FDB; + FDB;.

Using Eq. (3 l), we find dFDB,! + dFDB: = R(FDBi

+ FDBf ) dt + (1, af,jFDB: + &afdFDBf

) dz,. (32)

Eq. (32) implies that both bubbles, FDB: and FDBf, which compose fundamentals-dependent bubbles, follow a geometric Brownian motion. From Proposition 1, we conclude that the sum of two incompletely bursting bubbles converges to fundamentals-dependent bubbles in a special limiting case. Setting Zi = liafd in Eq. (22) and using Proposition 1, we find that this particular incompletely bursting bubble converges to FDB: in the limiting case. On the other hand,

g Dividends

follow

where I, denotes

a geometric

a standard

Brownian

Wiener process.

motion with a drift:

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setting & = lzafd in Eq. (22), we find that this new incompletely bursting bubble converges to FDB: in the limiting case. Since FDBl is the sum of FDB: and FDB:, we can derive the following proposition: Proposition 3. The sum of two incompletely bursting bubbles converges to fundamentals-dependent bubbles in a special limiting case. Ikeda and Shibata (1992) show that intrinsic bubbles and fundamentalsdependent bubbles contain the following two stochastic properties. First, there are stable and unstable type of intrinsic bubbles and fundamentals-dependent bubbles. In addition, some prevailing statistical methods proposed by Shiller ( 1981), Campbell and Shiller ( 1987) and Diba and Grossman (1988a) are not able to detect fundamentals-dependent bubbles and intrinsic bubbles. lo Fukuta ( 1997) exemplifies that incompletely bursting bubbles also have the aforemenbursting bubbles tioned stochastic properties. ” Hence, we regard incompletely as a simple discrete-time approximation of continuous-time bubbles. Ikeda and Shibata (1995) propose multiple fundamentals-dependent bubbles which are the product of two fundamentals-dependent bubbles. Following the same analogy as above, we find that the sum of two incompletely bursting bubbles multiplied by the sum of two other incompletely bursting bubbles converges to a multiple fundamentals-dependent bubble in a special limiting case. Consequently, we have the following proposition: Proposition 4. The product of two sums of two incompletely bursting bubbles converges to multiple fundamentals-dependent bubbles in a special limiting case.

5. Conclusions In this paper, we propose a class of rational bubbles referred to as ‘incompletely bursting bubbles’ in a discrete-time model to relate Blanchard’s simple bubbles to other more complicated types of bubbles, such as continuous-time bubbles and periodically collapsing bubbles. The rational bubbles proposed in this paper have the following two characteristics. First, the bubbles have three states: a large bubble state, a small bubble state and an incomplete burst state. Since the stochastic bubbles proposed by Blanchard (1979) have two states, it is possible to regard incompletely bursting bubbles as an extension of stochastic bubbles.

lo Fukuta and Shibata (1992) show that fundamentals-dependent bubbles prevailing statistical methods to test for rational bubbles by simulations.

cannot

be identified

by

” Fukuta ( 1997) shows that the statistical methods proposed by Shiller ( I98 I ), Campbell and Shiller (1987) and Diba and Grossman (1988a) are not able to detect incompletely bursting bubbles.

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Second, in contrast to stochastic bubbles, incompletely bursting bubbles never completely crash. This feature implies that even if incompletely bursting bubbles are in an incomplete crash state and heavily decline, the bubbles are able to grow again after the crash. We show that incompletely bursting bubbles are closely related to Blanchard’s bubbles, such as deterministic bubbles and stochastic bubbles, and to periodically collapsing bubbles. Incompletely bursting bubbles include deterministic bubbles as a special case. They are similar to stochastic bubbles, except for the scenario that entails a complete crash of stochastic bubbles. Finally, periodically collapsing bubbles can be regarded as a combination of incompletely bursting bubbles with continuous states and stochastic bubbles. We also discuss the relationship between incompletely bursting bubbles and continuous-time bubbles, such as intrinsic bubbles, fundamentals-dependent bubbles and multiple fundamentals-dependent bubbles. Incompletely bursting bubbles converge to intrinsic bubbles in a special limiting case. The sum of two incompletely bursting bubbles converges to a fundamentals-dependent bubble in a special limiting case. Finally, the sum of two incompletely bursting bubbles multiplied by the sum of two other incompletely bursting bubbles converges to multiple fundamentals-dependent bubbles in a special limiting case. Thus, we conclude that incompletely bursting bubbles are a simple discrete-time approximation of the continuous-time bubbles. This paper enlightens the relationship between Blanchard’s simple rational bubbles and other more complicated bubbles including continuous-time bubbles and periodically collapsing bubbles. Based on above implications, we can interpret incompletely bursting bubbles as a general type of rational bubbles. Using these newly defined bubbles, we can easily relate Blanchard’s bubbles to other complicated bubbles.

In this appendix, we prove Proposition 1. Proof: First, we show that limnqoc E lnBt/& Since R is constant, it is clear that limnln

R-CC

l+E (

n >

= (R - Z2/2)t.

= Rt.

Rearranging the first term on the right-hand side of Eq. (23), we obtain

(A.11

Y. Fukula / Journal of Economic Dynamics

Defining

and Control 22 (1998)

n E l/s and g(s) = ln( 1 - ,Z2s3t/2),

937-954

951

we obtain

(A-2) Using Taylor expansion ;fg(s) Computing

around s = 0 to g(s), we find

+ g’(O)s + O(s2)).

= ; $0)

(A-3)

g(0) and g’(O), we find

g(O)=

ln(l)=O,

(A-4)

and g’(s)/,,0

=

-l

&f

1 - Z2s3t/2

--

C23t

2 SC0=

Hence, using Eqs. (A-2)-(A.5),

2

’

(A.5)

we obtain

(A.61 Using Eqs. (23), (A.l)

and (A.6),

we can show that

lim E In $- = (R - $P)f. n-co Next, we prove that lim,,, var In&/& Rewriting Eq. (24), we obtain

(A.71 = Z’t.

var In L = 1 ;0 3 [{ln(l+G$+{ln(l-G)}‘]

-n

[ihr

(1 -Z2$)]2.

(A-8)

Let us consider to fi

lim,,, (n/3){ln( 1 + Zdm)}2. Setting m and h(m) equal and { ln( 1 + Zmm)j2, respectively, we obtain

(A.91

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952

(1998)

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Using Taylor expansion around m = 0 to h(m), we find f&h(m)

= f--$(/z(O)

+ h’(O)m + ~h”(0)m2

+ 0(m3)).

(A.lO)

Computing h(O), h’(O) and h”(O), we find h(O) = ln( 1) = 0,

(A.ll)

=21n(I)Z

3t

-z =O,

/-

(A.12)

and h”(m)\,=0 = 2

(A.13) Using Eqs. (A.9)-(A.13), /Air:

we obtain

{ln (1 +ZE)f=i.Oi--$

(o+O+~m2+O(m3)) =-.

C2t

(A. 14)

2

Applying the same method to lim,,,(n/3){ln(l

- Zm)}2,

we obtain (A.15)

PZ[~ln(1 - Z23t/2n)J2. Let us consider lim,,, [ln( 1 - Z2u3t/2)12, respectively, we obtain lim rz! [In (1 -Z2$)]’ n-00 9

= l&y(v).

Using Taylor expansion around II= $0)

= $y(O)

Setting II and y(u) as l/n and

0

(A.16)

to v(o), we find

+ y’(O)u + O(u2)).

(A.17)

Y. Fukutal Journal of Economic Dynamics and Control 22 (1998)

Computing

937-954

953

y(O) and y’(O), we find (A.18)

y(O)={ln(l)}*=O, and

=21n(l)(-1)Z2g Hence, from Eqs. (A.16)-(A.19),

)xirij

lim n-m

(A.14),

we obtain

(A.15)

lnWB0)

[

and (A.20),

we can show that (A.21)

= Z*t.

var In 2

Using the central limit theorem Pr

(A.20)

= l&(0+0+0(“2))=0.

[In(l-2*$--l*

Using Eqs. (A.8),

(A.19)

=O.

-

and Eqs. (A.7) and (A.21),

CR- z2/2)t 5

Jw

we find (A.22)

= Q(x),

x

1

where @J(X) denotes the distribution function of the standard normal distribution. Hence, we find that the incompletely bursting bubbles converge to a log-normal distribution: (A.23)

ln&--N(lnRa+(R-T)t,Z’f). Eq. (A.23) implies that the incompletely bursting bubbles Brownian motion with a drift in this limiting case:

follow

a geometric

d& -=Rdt+Zdz,, & where z1 denotes

the standard

Wiener process.

0

References Blanchard, O.J., 1979, Speculative bubbles, crashes and rational expectations, Economics Letters 3, 387-389. Blanchard, O.J. and S. Fischer, 1989, Lectures on Macroeconomics (MIT Press, Cambridge). Blanchard, O.J. and M.W. Watson, 1982, Bubbles, rational expectations, and financial markets, In: P. Wachtel, Ed., Crises in the Economic and Financial Structure (Lexington Books, Lexington) 295-315.

954

Y. Fukutal Journal of Economic Dynamics and Control 22 (1998)

937-954

Campbell, J.Y. and R.J. Shiller, 1987, Cointegration and tests of present value models, Journal of Political Economy 95, 1062-1088. Diba, B.T. and HI. Grossman, 1988a, Explosive rational bubbles in stock prices?, American Economic Review 78, 520-530. Diba, B.T. and H.I. Grossman, 1988b, The theory of rational bubbles in stock prices, Economic Journal 98, 746-754. Evans, G.W., 1991, Pitfalls in testing for explosive bubbles in asset prices, American Economic Review 81, 922-930. Fukuta, Y., 1997, Incompletely bursting bubbles in stock prices, Mimeo. Fukuta, Y. and A. Shibata, 1992, On the difficulty of testing for bubbles, Mimeo. Froot, K.A. and M. Obstfeld, 1991, Intrinsic bubbles: the case of stock prices, American Economic Review 81, 1189-1214. Ikeda, S. and A. Shibata, 1992, Fundamentals-dependent bubbles in stock prices, Journal of Monetary Economics 30, 143-I 68. Ikeda, S. and A. Shibata, 1995, Fundamentals uncertainty, bubbles, and exchange rate dynamics, Journal of International Economics 38, 199-222. Shiller, R.J., 1981. Do stock prices move too much to be justified by subsequent changes in dividends?, American Economic Review 71, 421-436.