Discussion Papers In Economics
And Business

Growth effect of bubbles in a non-scale endogenous
growth model with in-house R&D
Kizuku Takao
Discussion Paper 14-11

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANGrowth effect of bubbles in a non-scale endogenous
growth model with in-house R&D
Kizuku Takao
Discussion Paper 14-11

February 2014

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANGrowth eﬀect of bubbles in a non-scale endogenous
growth model with in-house R&D∗
Kizuku Takao†

Abstract
This paper provides a theoretical explanation for why the presence of asset bubbles can lead to higher economic growth in concurrence with high consumption byusing a simple endogenous growth model. In the model economy, long-lived valuemaximizing ﬁrms continuously improve the quality of their speciﬁc products throughin-house R&D, while at the same time new ﬁrms also enter into the market. Due toan absence of intergenerational altruism, asset bubbles can exist as pyramid schemes
whose value is not backed by fundamental value. The presence of asset bubbles thenleads to higher interest rates. This requires product proliferation to be impeded,which would result in an increase in the demand for diﬀerentiated goods at the levelof an individual ﬁrm. A larger scale of production at the level of an individual ﬁrm
can encourage in-house R&D of ﬁrms and promote economic growth.
Keywords: Bubbles, R&D, Overlapping generations
JEL classiﬁcation: E44, O32, O41∗

I would like to express my sincere gratitude to Koichi Futagami, Ching-Chong Lai, and Been-Lon
Chen for the beneﬁcial discussions and suggestions. I would also like to thank Katsunori Yamada and theparticipants at the 2013 Autumn Annual Meeting of the Japanese Economic Association at the Universityof Kanagawa. This paper was written while visiting the Institute of Economics, Academia Sinica. The
ﬁnancial support from Strategic Young Researcher Overseas Visits Program for Accelerating Brain Circulation (Japan Society for the Promotion of Science) and Grants for Excellent Graduate Schools (Ministryof Education, Culture, Sports, Science and Technology, Japan) are gratefully acknowledged. All remaining
errors are mine.
†
Graduate School of Economics, Osaka University, 1-7 Machikaneyama, Toyonaka, Osaka 560-0043,
Japan. E-mail address: kge006tk@mail2.econ.osaka-u.ac.jp1

1

Introduction

Asset bubbles sometimes emerge, and they are accompanied by higher economic growth
and a consumption boom.12 A seminal study by Tirole (1985) shows the existence of arational deterministic bubble on an asset in an economy with overlapping generations. It
is found that the presence of asset bubbles increases consumption but retards economic
growth (e.g.. Saint-Paul (1992), Grossman and Yanagawa (1993), King and Ferguson(1993), and Futagami and Shibata (2000)). The reason is that the higher consumption
absorbs resources and crowds out investment. As a result, the theoretical prediction runs
contrary to the mentioned above fact. The purpose of this paper is to overcome thisconﬂict and provide a theoretical explanation for why the presence of asset bubbles canlead to higher economic growth in concurrence with high consumption by using a simple
endogenous growth model.
The present analysis is based on a recent endogenous growth model developed by
Peretto (2007, 2011). Speciﬁcally, the model considers the economy where long-lived valuemaximizing ﬁrms continuously improve upon the quality of their speciﬁc product throughin-house R&D, while simultaneously new ﬁrms also enter into the market. That is, in themodel economy, there are two dimensional investment opportunities: in-house R&D (quality improving) and the creation of a new ﬁrm (product proliferation). The main source ofeconomic growth is obtained from technological progress, endogenously derived from thein-house R&D of ﬁrms. The advantage of the model is elimination of the well-known undesirable scale eﬀect (e.g. Jones (1995)), while keeping the policy eﬀect property supported
by recent growing empirical literature.3 Increases in the scale of the aggregate economy
are perfectly fragmented by the endogenous product proliferation.
We introduce the above-mentioned structure into a continuous-time overlapping generations model developed by Weil (1989). Speciﬁcally, we assume that economic agents live

1

Asset bubbles are deﬁned as the diﬀerence between the fundamental value of an asset and its market
value. For example, if an intrinsically useless asset whose fundamental value is zero has a positive value,
we say there exist asset bubbles.
2
The introduction of Martin and Ventura (2012) provides excellent surveys about the fact about asset
bubbles.
3
See, for example, Laincz and Peretto (2006), Ha and Howitt (2007), and Ang and Madsen (2011)).

2

forever but have no intergenerational altruism. We also assume that new generations areborn at a constant rate.4 Additionally, we assume that there initially exists an intrinsically useless asset (ﬁat money).5 The reason there exist asset bubbles in equilibrium is thesame as the reason spelled out in Tirole (1985). Due to an absence of intergenerational
altruism, one generation can transfer the intrinsically useless asset to the other generation
as a pyramid scheme only if the price appreciation of the intrinsically useless asset is equal
to the return on a real asset (equity) whose value is backed by the fundamental value.
Our analysis stipulates the theoretical mechanism by which the presence of asset bubbles can lead to higher economic growth with high consumption. Here the endogeneityof the market structure plays a key role. If asset bubbles emerge, households think thatthey are wealthier and thus want to consume more. This leads to a higher reservationinterest rate of households in the asset market. To satisfy the higher reservation interest
rate, the return on a real asset (equity) must increase. This requires product proliferation
to be impeded so that demand for diﬀerentiated goods, at the level of an individual ﬁrm,
increases. Consequently, if the positive eﬀect on in-house R&D of ﬁrms due to the larger
scale of production at the level of an individual ﬁrm exceeds the negative eﬀect derived
from the higher interest rate, the intensity of in-house R&D increases, thus enhancing
economic growth. In addition, the lower product proliferation provides available economic
resources for a higher consumption by households.
This paper is related to several recent studies that examine the relationship between
asset bubbles and economic growth (e.g.. Hirano and Yanagawa (2010) and Martin and
Ventura (2012)). Their analyses consider an environment where the credit market is imperfect and the eﬃciency of investment among ﬁrms is heterogeneous. In this environment,
the presence of asset bubbles can reallocate resources from ineﬃcient to eﬃcient investments. As a result, the reallocation may raise the productivity of aggregate outputs and4

If we employ the alternative setup that there exists a positive death rate following Blanchard (1985),
our main claims do not change.
5
For simplicity, as a theoretical devise, we distinguish the pure bubbles from real assets whose values
are backed by the fundamental values along with many other studies. As one notable exception, Olivier(2000) analyzes the case where bubbles attach directly on real assets such as the equity of ﬁrms, andOlivier (2000) also demonstrates that bubbles on equity may enhance economic growth in the frameworkof the endogenous growth model developed by Romer (1990).

3

promote economic growth. By contrast, our model economy is characterized by an environment, where the credit market is perfect, and the eﬃciency of investment among ﬁrms ishomogeneous. The key feature of our analysis is that the endogeneity of market structureaﬀects the scale of production at the level of an individual ﬁrm, which in turn determines
the intensity of in-house R&D.
The rest of the paper is organized as follows. Section 2 describes the model. Section
3 analyzes the dynamic system of the market equilibrium. Section 4 analyzes the steadystate equilibrium as well as compares the steady state equilibrium with and without asset
bubbles.

2

The model

In this section, we establish a continuous-time overlapping-generations version of the model
developed by Peretto (2011).6 We also assume that there exists an initial and intrinsicallyuseless asset, that is, ﬁat money. The economy is closed and consists of a ﬁnal goods sector,
an intermediate goods sector, and households.

2.1

The final goods sector

Perfect competition prevails in the ﬁnal goods sector. The (real) price of ﬁnal goods is
set to be numeraire. Final goods are consumed by households and used as only one factorof the production and investment of intermediate goods sector. The ﬁnal goods, Yt is
produced by the following production function:
∫ Nt
Yt =

(
)1−θ
Xitθ Zitα Z̄t1−α Litdi, 0 < α, θ < 1,(1)0

where Nt is the variety of intermediate goods (the number of intermediate goods ﬁrms),
Xit is the input of intermediate good i ∈ [0, Nt ] (produced by ﬁrm i), and Lit is the
input of labor which uses intermediate goods i. The productivity of Lit depends not only
6

The model of Peretto (2011) is a lab-equipment style versions of the model developed by Peretto(2007).

4

on the quality of intermediate good i, Zit , but also on the average level of the quality∫N
across intermediate goods, Z̄t ≡ 0 t N1t Zjt dj. Therefore, we obtain the following optimal
conditions:
(
Xit =
(
Lit =

θ
pIit

1
) 1−θ

1−θ
wt

)
Zitα Z̄t1−α Lit ,

(2)

(
) 1−θ
Xit Zitα Z̄t1−α θ ,

(3)

(

) θ1

where pIit and wt represent the price of intermediate good i and the wage rate, respectively.

2.2

The intermediate goods sector

Monopolistic competition prevails in the intermediate goods sector. Firm i exclusively
produces its diﬀerentiated good with its quality Zit . The monopoly of each ﬁrm is permanently protected by perfect patent protection. Producing one unit of intermediate goods
requires one unit of ﬁnal goods. Firms improve upon the quality of their speciﬁc productthrough their in-house R&D.7 The law of motion of the ﬁrm-speciﬁc quality is as follows:Żit = Rit ,(4)

where Rit is the inputs for in-house R&D. Fixed operating costs, ϕZ̄t (ϕ > 0), are also()
required at each point in time. Then, the proﬁt ﬂow of ﬁrm i is Πit = pIit − 1 Xit − ϕZ̄t −Rit . Without loss of generality, the number of equity per ﬁrm is normalized to one. The
return on equity of ﬁrm i is given byrt =Πit V̇it
+
.
Vit
Vit

Integrating (5) forward yields the value of ﬁrm i as Vit =

(5)
∫∞
t

( ∫s
)
Πis e− t rv dv ds. Throughout

this exercise we consider a symmetric equilibrium in which Zit = Zt by assuming that any
new entry ﬁrm starts with the same level of technologies as incumbents so that the subscript
7

Since here we assume that perfect capital market prevails and that there are no distortionary taxes, itdoes not matter whether ﬁrms ﬁnance their in-house R&D by issuing equity or by using retained earnings.

5

i can be dropped. In equilibrium, Zt = Z̄t holds. Each ﬁrm maximizes its value, subject
to (2) and (4), given Z̄. Solving the inter-temporal optimization problem, we obtain thefollowing optimal conditions:1
pIt = ,
θ(6)

qt = 1 ⇔ Rt > 0,

(7)

rt = α2q̇t
1 − θ 1−θ
θ Ztα−1 Z̄t1−α Lt + ,
θ
qt

(8)

∫s

and the transversality condition is given by lims→∞ qs Zs e− t rv dv = 0, where qt is a co-statevariable (a shadow value of in-house R&D) associated with the current-value Hamiltonian
for this optimization problem, where (6) represents pricing rule with constant mark-up,
and where (8) represents the no-arbitrage relationship between the return on equity andthe return on in-house R&D.

2.3

Firm entry

Development of new product requires βZt (β > 1).8 A new ﬁrm is set up by issuingequities. Free-entry condition yields
Vt = βZt ⇔ Ṅt > 0.

(9)This equation implies entry is positive until the value of a ﬁrm is equal to the set up cost.
In our model economy, as we will see later, population grows perpetually and then theaggregate demand for intermediate goods also continues to grow. Therefore, at each point
in time, new entry of ﬁrms occurs.

8

Since any new entrant ﬁrm starts with the same level of technologies of incumbents, it is natural toassume that the cost of creating a new product with the initial quality level Zt is proportional to thequality level of incumbents. Moreover, it is natural to assume that the new entrant ﬁrm has to pay an
additional cost more than the cost already paid by the incumbents (see Peretto (2011)).

6

2.4

Households

The speciﬁcation of household behavior is based on that in Weil (1989).9 At each point
in time, a new generation is born at a constant rate λ > 0. Since the probability of deathis assumed to be zero, each generation lives forever. And there is no intergenerational
altruism. Without loss of generality, the total population at time 0 is normalized to 1 sothat the total population at time t is eλt .
At the beginning of the time 0, there is an initial generation endowed with initial
wealth. Hereafter we call this generation as generation 0− . The initial wealth consistsof a real asset (equity) and an intrinsically useless asset (ﬁat money). To decide upon aﬁxed terminology, we label the intrinsically useless asset a “bubbly asset”. Speciﬁcally,
the generation 0− initially possesses the equity of ﬁrms (the initial number of ﬁrms is N0 ),whose value is given by N0 V0 in real terms and B0 pieces of the bubbly asset in nominal
terms.10 By contrast, the subsequent generation does not have any initial wealth. Hereafter
we label the generation born at time t ≥ 0 as “generation t”.
Each household supplies one inelastic unit of labor at each point in time. The representative household in generation s(≤ t) maximizes the following utility function:∫ ∞U (s, t) =log [c(s, x)]e−ρ(x−t) dx, ρ > 0,

(10)

t

subject to the following budget constraint of generation s:dk(s, t)db(s, t) 1+
= rt k(s, t) + wt − c(s, t),
M
dt pt
dt
where b(s, t) is holdings of pieces of a bubbly asset (in nominal terms), 1/pM
t is the priceof the bubbly asset, k(s, t) is holdings of a real asset, and c(s, t) is consumption.
Since the bubbly asset has no intrinsic or fundamental value, households possess both
the real asset and the bubbly asset only if the price appreciation of the bubbly asset is9

Futagami and Shibata (2000) also uses the same speciﬁcation.
M
That is, the total value of the initial asset is equal to N0 V0 + B0 /pM
0 in real terms, where p0 is theprice of the bubbly asset at time 0.
107equal to the return on the real asset, that is,ṗM− tM = rt .
pt

(11)

Let m(s, t) ≡ b(s, t)/pM
t and a(s, t) ≡ k(s, t) + m(s, t). Using (11) and the deﬁnitions,the ﬂow budget constraint of generation s can be rewritten asda(s, t)= rt a(s, t) + wt − c(s, t).
dt

(12)

Maximizing (10) subject to (12) yields the following Euler equation of generation s:dc(s, t)= (rt − ρ) c(s, t).
dt

(13)
∫

x
And the following No Ponzi game condition must be satisﬁed: limx→∞ a(s, x)e(− t rv dv) =0.
In what follows, we will aggregate the consumption and wealth of each generation. To
do so, we sequentially deﬁne aggregate consumption, aggregate wealth, and aggregate wage
income:∫ t−

Ct ≡ c(0 , t) + λc(s, t)eλs ds,∫0 t

At ≡ a(0− , t) + λa(s, t)eλs ds,∫ t 0
W t ≡ wt + λ
ws eλs ds.
0

In Appendix 1, we show that the dynamic path of aggregate consumption and aggregate
wealth is respectively given byĊt = (rt − ρ + λ) Ct − ρλAt ,(14)

Ȧt = rt At + Wt − Ct .

(15)

8

3

Market equilibrium and Dynamics

In this section, we derive the dynamics system of the market equilibrium based on themodel in the preceding section. The market equilibrium condition of ﬁnal goods is given
by Yt = Ct + Nt [Xt + ϕZt + Rt ] + βZt Ṅt . The market equilibrium condition of assets isgiven by At = Mt + Nt Vt , where Mt is deﬁned as the aggregate value of the bubbly asset,that is B0 /pM
t . This condition holds that in equilibrium the aggregate wealth must be
equal to the aggregate value of the bubbly asset and the aggregate value of equity (Nt Vt ).
Here we remark that the real asset in the economy is only the equity of ﬁrms. The market
equilibrium condition of labor is given by eλt = Nt Lt .
Deﬁne the number of ﬁrms per capita as nt ≡ Nt /eλt . From (7) and the market
equilibrium condition of labor, if Rt > 0, (8) is rewritten by
rt = αθ(1 − θ)

Ω
,
nt

(16)

2θ

where Ω ≡ θ 1−θ . From (9) and the market equilibrium condition of labor, if Rt > 0, (5) isrewritten by[]Ω
β − 1 Żt1θ(1 − θ) − ϕ +
.
rt =
β
nt
β Zt

(17)

From (16) and (17), the rate of quality growth is given by αβ − 1
Ω
ϕ
Żt 
θ(1
−
θ)
+
, if nt > ñ,ẑt (nt ) ≡
=
β
−
1
n
β
−
1
t
Zt 

 0, if αβ < 1 and 0 < n ≤ ñ,t(18)

and the interest rate (the return on equity) is given by


Ω




, if nt > ñ,
αθ(1
−
θ)

nt


rt =

1
Ω


θ(1 − θ) − ϕ , if αβ < 1 and 0 < nt ≤ ñ,


 β
nt

(19)

{
}
where ñ = max 0, (1−αβ)θ(1−θ)Ω. More speciﬁcally, the intensity of in-house R&D isϕ9determined at the point where the return on in-house R&D (the LHS of (16)) is equal to
the return on equity (the LHS of (17)). (18) indicates if αβ > 1 (< 1), the lower numberof ﬁrms per capita leads to the higher (lower) rate of quality growth. If α and β are
suﬃciently high (low), the return on in-house R&D is more (less) sensitive to changes in
the number of ﬁrms per capita. Hence, the lower number of ﬁrms per capita raises thereturn on in-house R&D than the return on equity, resulting in the higher (lower) rate ofquality growth. On the other hand, (19) shows that the interest rate is unambiguously
decreasing in the number of ﬁrms per capita. The lower number of ﬁrms per capita leads
to higher demand for intermediate goods at the level of an individual ﬁrm.
More intuitively, whether the rate of quality growth is increasing or decreasing in thenumber of ﬁrms per capita is dependent on the extent of the following two contradictory
forces. Other things being equal, the lower number of ﬁrms per capita increases demand forintermediate goods at the level of an individual ﬁrm. The larger scale of production at thelevel of an individual ﬁrm allow in-house R&D expenditures to be spread over more unitsof goods, thus having a positive eﬀect on incentives to in-house R&D (the cost spreading
eﬀect). On the other hand, a higher interest rate associated with a lower number of ﬁrms,
per set of ﬁrms, lowers incentives to in-house R&D (the interest eﬀect). If αβ > 1, thepositive cost spreading eﬀect exceeds the negative interest eﬀect. The same logic applies
to the case where αβ < 1.11
The perfect distribution in the ﬁnal goods sector implies Nt Xt = θ2 Yt and wt Nt Lt = (1−θ)Yt . Applying these relationships to (1), we obtain the following reduced-form aggregate
production function of ﬁnal goods, which is given as
Yt = Ωeλt Zt .

(20)

The growth rate of outputs per capita is given as ẑt (nt ).
Deﬁne the ratio of aggregate consumption to outputs as ct ≡ Ct /Yt and the ratio of
aggregate value of the bubbly asset to outputs as mt ≡ Mt /Yt . In Appendix 2, we show

11

If ϕ = 0, αβ is restricted to be higher than 1 so that the steady state equilibrium with a positive rateof quality growth exists.

10

that the following three equations constitute the dynamic system of the economy:] ntΩ[
1 − θ2 − ct − [ϕ + ẑt (nt ) + βλ] ,
β
β
[
nt ]
ċt = [φ(nt ) − ρ] ct − λρ mt + β
,
Ω

ṅt =

ṁt = [φ(nt ) − λ] mt ,

(21)
(22)
(23)

where the diﬀerence between the interest rate and the rate of quality growth is given byϕ(1 − α)θ(1 − θ) Ω−
, if nt > ñ,β−1nt  β − 1
φ(nt ) ≡ rt − ẑt =

1
Ω




θ(1
−
θ)
− ϕ , if αβ < 1 and 0 < nt ≤ ñ.

 β
nt

4

(24)

The steady state equilibrium

This section characterizes the steady state equilibrium. As we will see later, there may
exist multiple steady state equilibrium: the steady state equilibrium without asset bubbles
and the steady state equilibrium with asset bubbles. In Appendix 2, we examine thelocal stability of the steady state equilibrium. We prove that the equilibrium path toward
the steady state equilibrium with asset bubbles is locally saddle-point stable, whereas
the equilibrium path toward the steady state equilibrium without asset bubbles is locallyindeterminate. Thus, a global indeterminacy arises along with previous studies, such as in
Grossman and Yanagawa (1993) and Futagami and Shibata (2000).
From (21), ṅt = 0-locus is given by






n
ϕ
βθ(1
−
α)



−β 
+ λ , if n > ñ,(1 − θ) 1 +



β−1
Ω β−1


c=


n ϕ


2
 + λ , if αβ < 1 and 0 < n ≤ ñ.

(1
−
θ
)
−
β


Ω β

(25)

(25) shows that the ṅt = 0-locus is independent of the value of m and on the ṅt = 0-locus

11c is negatively related to n. This locus represents the resource constraint of the economy.
The downward-sloping shape of this locus implies that the higher number of ﬁrms per
capita absorbs available resources for consumption of households. From (22), ċt = 0-locus
is given by




n
λρ





 m + β  , if n > ñ,Ω(1 − α)θ(1 − θ) Ω
ϕ




−
− ρ



β−1
n β−1


c=


λρ
n




 m + β  , if αβ < 1 and 0 < n ≤ ñ.


Ω


θ(1 − θ) Ω ϕ





− −ρ


β
n β

(26)

(26) shows that on the ċt = 0-locus, given the value of m, c is positively related to n
starting from the origin in the region of 0 < n < n̄ where n̄ ≡ argsolve {φ(n) = ρ}.12 Thislocus is derived from the aggregation of the Euler equation of households. The upwardsloping shape of this locus implies that the higher number of ﬁrms per capita leads to alower interest rate, and thus households want to consume more.13 First, we consider asteady state equilibrium without asset bubbles, {n∗ , c∗ , m∗ } where m∗ = 0. In Figure 1
and Figure 2, we plot the ṅt = 0-locus and ċt = 0-locus with m∗ = 0 by solid lines.14 Theseﬁgures show that {n∗ , c∗ } is uniquely determined.
Second, we consider the possibility of the existence for the steady state equilibriumwith asset bubbles, {n∗∗ , c∗∗ , m∗∗ }, where m∗∗ > 0. From (23), if m∗∗ > 0, ṁt = 0 implies

12

If the steady state number of ﬁrms per capita is higher than n̄, the steady state consumption ratio
becomes negative.
13
The higher number of ﬁrms per capita also yields a higher ratio of the aggregate value of equity
to output, that is, N V /Y . Hence, the higher wealth ratio induces households to consume more, given
an interest rate. This reinforces the extent of the upward-sloping shape of this locus. However, this isdependent on the speciﬁcation of the entry cost. For example, if we consider the alternative assumption
in which the entry cost is related to the production volumes of incumbents, that is, Xt along with Peretto(2007), then the ratio of the aggregate value of equity to output becomes constant.
14
Speciﬁcally, Figure 1 shows the conﬁguration of ṅt = 0-locus and ċt = 0-locus when αβ > 1, while
Figure 2 shows that when αβ < 1 and ϕ > ϕ̃ (see the deﬁnition of ϕ̃ for the latter main text). If αβ < 1and ϕ > ϕ̃ is satisﬁed, the steady state quality growth is positive both in the steady state equilibrium
without asset bubbles and in that with asset bubbles.

12φ(n∗∗ ) = λ. Therefore, from (18) and (24), we obtain(1 − α)θ(1 − θ)Ω
, if αβ > 1 or αβ < 1 and ϕ > ϕ̃,(β − 1)λ + ϕ
n∗∗ =


θ(1 − θ)Ω, if αβ < 1 and 0 < ϕ ≤ ϕ̃,βλ + ϕ (αβ − 1)λ + αϕ
, if αβ > 1 or αβ < 1 and ϕ > ϕ̃,∗∗ẑ(n ) =1−α 0, if αβ < 1 and 0 < ϕ ≤ ϕ̃,{}(1−αβ)λwhere ϕ̃ ≡ max 0, α
. From (25), (27), and (28), c∗∗ is given by

(27)

and

c∗∗ = (1 − θ).

(28)

(29)

From (26), (27) and (29), m∗∗ is given by(1 − θ)(1 − α)βθλρλ − ρ − , if αβ > 1 or αβ < 1 and ϕ > ϕ̃, λρ(β − 1)λ + ϕ


m∗∗ =


(1 − θ)βθλρλ − ρ − , if αβ < 1 and 0 < ϕ ≤ ϕ̃.
 λρβλ + ϕ(30)

Therefore, the necessary and suﬃcient condition for the existence of the steady state equilibrium with an asset bubble is given by λ − ρ > (1−α)βθλρ > 0,(β−1)λ+ϕ λ − ρ > βθλρ > 0,βλ+ϕif αβ > 1 or αβ < 1 and ϕ > ϕ̃if αβ < 1 and 0 < ϕ ≤ ϕ̃.

(31)

If this condition is satisﬁed, we ﬁnd that in the steady state equilibrium without asset
bubbles the growth rate of ﬁnal output exceeds the interest rate.15 That is, the necessary
condition for the presence of asset bubbles is same as previous studies such as Grossman
and Yanagawa (1993) and Futagami and Shibata (2000).
15

See Appendix A.4 for the proof.

13

If there exists the steady state equilibrium with asset bubbles, ċt = 0-locus moves up
in a counterclockwise direction as opposed to the case in which no asset bubbles exist.
In Figure 1 and Figure 2, we plot ċt = 0-locus with m∗∗ > 0 by a dotted line. Theseﬁgures show that the number of ﬁrms per capita in the steady state with asset bubbles
is unambiguously lower than that in the steady state equilibrium without asset bubbles.
Moreover, it is shown that the consumption ratio in the steady state equilibrium with asset
bubbles is unambiguously higher than that in the steady state equilibrium without asset
bubbles.
The key mechanism is as follows: If asset bubbles emerge, households think that they
are wealthier, and thus want to consume more. This leads to higher reservation interest
rate of households in the asset market. In order to satisfy the higher reservation interest
rate, the return on equity must increases. This requires the number of ﬁrms per capita to
be lower given the market size of the economy because the return on equity is positively
related to demand for intermediate goods at the level of an individual ﬁrm. As a result,the presence of asset bubbles yields a larger scale of production at the level of an individualﬁrm.
Consequently, if the cost spreading eﬀect exceeds the interest eﬀect (if αβ > 1), as
previously noted, the presence of asset bubbles encourages in-house R&D of ﬁrms andpromotes economic growth. The same logic applies to the case in which the interest eﬀect
surmounts the cost spreading eﬀect (if αβ < 1). In addition, the lower product proliferation
provides available economic resources for a higher household consumption, regardless ofthe eﬀect on economic growth.

A

Appendix

A.1

Appendix 1

Integrating (12) yields∫x
lim a(s, x)e(− t rv dv) − a(s, t) +

x→∞

∫ ∞

∫

x
c(s, x)e(− t rv dv) dx =

t

∫ ∞
t

14

∫x

(wx ) e(− t rv dv) dx ≡ ht .

cċt = 0-locus(m∗∗ > 0)(m∗ = 0)ṅt = 0-locusc∗∗bubble equilibriumc∗bubbleless equilibriumn∗∗ n∗0n̄nFigure 1: Steady state equilibrium (in the case where αβ > 1)cċt = 0-locus(m∗∗ > 0)(m∗ = 0)ṅt = 0-locusc∗∗bubble equilibriumc∗0

bubbleless equilibriumñn∗∗ n∗n̄

Figure 2: Steady state equilibrium (in the case where αβ < 1 and ϕ > ϕ̃)15nApplying (13) and the No Ponzi game condition to the above equation, we obtain c(s, t) =ρ [a(s, t) + ht ]. Aggregating this yields[]
Ct = ρ a(0− , t) + ht + ρλ⇔ Ct = ρ (At + Ht ) ,

∫ t

∫ t
λs

a(s, t)e ds + ρλ
ht eλs ds,00∫ t
where Ht ≡ ht + λ
ht eλs ds.

(A-1)

0

From (13) and (A-1), diﬀerentiating the aggregate consumption with respect to time yields∫ t−Ċt = (rt − ρ) c(0 , t) + c(t, t)λe + (rt − ρ) λλtc(s, t)eλs ds,
0⇔ Ċt = ρλe [a(t, t) + ht ] + (rt − ρ) Ct ,λt

⇔ Ċt = ρλHt + (rt − ρ) Ct ,(∵ a(t, t) = 0 ) ,⇔ Ċt = (rt − ρ + λ) Ct − ρλAt .
And aggregating (12) yields (15).

A.2

Appendix 2

Let at ≡ At /Yt . From (14) and (20),Ċt Ẏt
Ċt
ċt
=
−
=
− ẑt − λ ⇔ ċt = [φ(nt ) − ρ] ct − λρat .
ct
Ct Y tCt(A-2)

From (20), the market equilibrium condition of assets is rewritten by
a t = mt + βnt.
ΩThen, substituting this condition into (A-2) yields (22).
From (20) and Nt Xt = θ2 Yt , the market equilibrium condition of ﬁnal goods is rewritten
by

]
[
Ṅ
n
t
t
ϕ + ẑt + β,
1 = ct + θ 2 +Ω
Nt

Then, from this condition and the deﬁnition of nt , we obtain (21).
16

From the market equilibrium condition of labor and Nt wt Lt = (1−θ)Yt , (15) is rewritten
by
Ȧt
1
ct
= rt + (1 − θ) − .
Atat atUsing the above equation and (20), we obtainȧt = [φ(nt ) − λ] at + (1 − θ) − ct .
Diﬀerentiating the market equilibrium condition of assets with respect to time and using(21) and the law of motion of the asset ratio, we obtain (23).

A.3

Appendix 3

From (21), (22), and (23), the system of the linearized diﬀerential equation around thesteady state is given by










ṅ
J
J
J
n −n
 t   11 12 13   t

  


st
 ċt  = J21 J22 J23   ct − c  ,
  


ṁt
J31 J32 J33
mt − mst
stwhere {nst , cst , mst } represents {n∗ , c∗ , m∗ } or {n∗∗ , c∗∗ , m∗∗ } and] nst ∂ ẑ(n)1[
ϕ + ẑ(nst ) + βλ −< 0,ββ ∂n n=nstΩ
J12 = − < 0,βJ11 = −

J13 = 0,J21 = −cstβ∂φ(n)− λρ < 0,∂n n=nstΩ

J22 = φ(n∗ ) − ρ > 0 if mst = m∗ = 0 (∵ n∗ > n∗∗ ) ,
{
}
st
∗∗
J22 = λ − ρ > 0 if m = m > 0 ,J23 = −λρ < 0,J31 = 0 if mst = m∗ = 0,
17

(A-3)

{
J31 = m∗∗ ∂φ(n)∂nst< 0 if m = m

∗∗

}
>0 ,

n=n∗∗

J32 = 0,J33 = φ(n∗ ) − λ < 0 if mst = m∗ = 0 (∵ n∗ > n∗∗ ) ,
{
}
st
∗∗
J33 = 0 if m = m > 0 .

J11 J12 J13
We deﬁne matrix J as J ≡ J21 J22 J23 . We obtain the following the characteristic
J31 J32 J33
equation of the coeﬃcient matrix on the right-hand side of (A-3):
0 = |J − qI| = −q 3 + TrJq 2 − BJq + DetJ ≡ f (q),where I is the identity matrix, TrJ is the trace of J, DetJ is the determinant of J, and
BJ ≡J11 J12+J22 J23+

J11 J13.
J21 J22
J32 J33
J31 J33
The dynamic system has one state variable, nt , and two jump variables, ct , and mt .

Hence, if the number of negative eigenvalues around the steady state is one, there exists a
unique equilibrium saddle path toward the steady state. If the number of negative eigenvalues around the steady state is more than one, there exists a continuum of equilibrium
path toward the steady state.
Following Benhabib and Perli (1994), we now check the number of negative eigenvalues
around the steady state by applying the Routh-Hurwitz theorem:
Theorem. The number of roots of the polynomial in (A-3) with positive real parts is equal
to the number of variations of sign in the scheme−1

TrJ− BJ +

DetJ
TrJ

DetJ.

(A-4)

Around the steady state without asset bubbles, we obtain DetJ = J12 J23 J31 −J12 J21 J33 >0 and TrJ = J11 + J22 + J33 < 0 (∵ φ(n∗ ) < λ). Then, we can conclude that around thesteady state without asset bubbles there are one positive eigenvalue and two eigenvalues
18with negative real part because there is only one change of the sign in the scheme (A-4)
regardless of the value of BJ. Therefore, there exists a continuum of equilibrium path
toward the steady state without asset bubbles.
Around the steady state with asset bubbles, we obtain DetJ = J12 J23 J31 < 0, TrJ =J11 + J22 < 0, and BJ = J11 J22 − J12 J21 < 0. Hence, we also obtain −BJ + DetJ/BJ > 0.
Then, we can conclude that around the steady state with asset bubbles there are two
eigenvalues with one positive real part and one negative eigenvalue, because there are twochanges of the sign in the scheme (A-4). Therefore, the equilibrium path toward the steadystate with asset bubbles is locally determinate.

A.4

Appendix 4

From (24), we ﬁnd that the diﬀerence between the interest rate and the growth rate of ﬁnal
output, φ(n) − λ, is decreasing in n. In the steady state equilibrium with asset bubbles,
we obtain φ(n∗∗ ) − λ = 0. Since it is shown that n∗ is higher than n∗∗ , we ﬁnd thatφ(n∗ ) − λ < 0.

References[1] Ang, J. B. and J.B. Madsen (2011) “Can Second-Generations Endogenous Growth
Models explain the Productivity and Knowledge Production in the Asian Miracle
Economies?”, The Review of Economics and Statistics, Vol. 93, No. 4, pp. 1360-1373.
[2] Benhabib, J. and R. Peril (1994), “Uniqueness and indeterminacy: on the dynamics ofendogenous growth”, Journal of Economic Theory, 63, pp. 113-142.
[3] Blanchard, O.J. (1985), “Debts, deﬁcits, and ﬁnite horizons”, Journal of Political Economy, 93, pp. 223-247.
[4] Futagami, K. and Shibata, A. (2000), “Growth eﬀects of bubbles in an endogenous
growth model”, The Japanese Economic Reviews, 2, pp. 221-235.

19

[5] Grossman, G. M. and Yanagawa, N. (1993), “Asset bubbles and endogenous growth”,
Journal of Monetary Economics, 31, pp. 3-19.
[6] Ha, J. and Howitt, P. (2007) “Accounting for Trends in Productivity and R&D:
A Schumpeterian Critique of Semi-Endogenous Growth Theory”, Journal of Money,
Credit and Banking, Vol. 39, No. 4, pp. 733-744.
[7] Hirano, T. and N. Yanagawa (2010) “Asset bubbles, endogenous growth, and ﬁnancial
frictions”, mimeo.
[8] Jones, C. I. (1995) “Time Series Tests of Endogenous Growth Models”, The Quarterly
Journal of Economics, 110, pp. 495-525.
[9] King, I. and D. Ferguson (1993) “Dynamic Ineﬃciency, Endogenous Growth, and Ponzi
Games”, Journal of Monetary Economics, 32, pp. 79-104.
[10] Laincz, C. A. and Peretto, P. F. (2006) “Scale Eﬀects in endogenous growth theory:
an error of aggregation not speciﬁcation”, Journal of Economic Growth, Vol. 11, No.
3, pp. 263-288.
[11] Martin, A. and Ventura, J. (2012), “Economic growth with bubbles”, American Economic Review, 102, pp. 3033-3058.
[12] Olivier, J. (2000), “Growth-enhancing bubbles”, International Economic Review, 41,
pp. 133-151.
[13] Peretto, P.F. (2007), “The manhattan metaphor”, Journal of Economic Growth, 12,
pp. 329-350.
[14] Peretto, P.F. (2011), “The growth and welfare eﬀects of deﬁcit-ﬁnanced dividend tax
cuts”, Journal of Money, Credit and Banking, 43, pp. 835-870.
[15] Romer, P. M. (1990), “Endogenous Technological Change”, Journal of Political Economy, 98, pp. S71-S102.

20[16] Saint-Paul, G. (1992), “Fiscal policy in an Endogenous Growth Model”, Quarterly
Journal of Economics, 107, pp. 1234-1259.
[17] Tirole, J. (1985), “Asset bubbles and overlapping generation”, Econometrica, 53, pp.
1499-1528.
[18] Weil, P. (1989), “Overlapping families of inﬁnitely lived agents”, Journal of Public
Economics, 38, pp. 183-198.

21