Discussion Papers In Economics
And Business

Growth and Bubbles with Consumption Externalities

Kazuo Mino
Discussion Paper 07-07

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANGrowth and Bubbles with Consumption Externalities

Kazuo Mino
Discussion Paper 07-07

February 2007

この研究は「大学院経済学研究科・経済学部記念事業」
基金より援助を受けた、記して感謝する。

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANGrowth and Bubbles with Consumption Externalities∗
Kazuo Mino†
Graduate School of Economics, Osaka University
February 2007

Abstract
This paper explores the role of consumption externalities in an overlapping generations economy with capital accumulation. If consumers in each generation areconcerned with other agents’ consumption behaviors, there exist intergenerational aswell as intragenerational consumption externalities. It is the presence of intergenreational consumption externalities that may produce fundamental eﬀects both on
equilibrium dynamics and on steady-state characterization of the economy. This paper demonstrates this fact in the context of a simple model of endogenously growing,
overlapping-generations economy with or without asset bubbles.
JEL Classification: E32, J24, O40
Keywords: Consumption Externalities, Overlapping Generations, Long-run Growth,
Asset Bubbles.

∗

This paper is based on an invited lecture at the fall meeting of the Japanese Economic Association
held at Osaka City University on October 21, 2006. I am grateful to an anonymous referee for very usefulcomments. I also thank Been-Lon Chen, Jang Ting Guo and seminar participants at Academia Sinica
and National Sun Yat-Sen University for their helpful comments and suggestions. Financial support fromGrant-in-Aid for Scientific Research (No. 18530136) is gratefully acknowledged.
†
Correspondence: Kazuo Mino, Graduate School of Economics, Osaka University, 1-7
Machikaneyama, Toyonaka, 560-0043 Japan, E-mail: mino@econ.osaka-u.ac.jp

1

Introduction

Recently, there has been a renewed interest in the role of consumption externalities inmacroeconomics. Earlier studies on this issue such as Abel (1990) and Galí (1994) introduce consumption external eﬀects into the asset pricing models in order to resolve thediscrepancies between theoretical outcomes and empirical findings. Those studies, therefore, focus on the external eﬀects of consumption on the individual decision making or on
the behavior of asset markets. In contrast, the recent investigations on macroeconomic
implications of consumption externalities examine the issue in the dynamic, general equilibrium models with capital accumulation and discuss a wider range of topics than thoseconsidered by the earlier studies. For example, the recent studies have explored the eﬀectof consumption externalities on optimal taxation (Ljungqvist and Uhlig 2000), equilibriumeﬃciency (Alonso-Carrera et al. 2003 and Liu and Turnovsky 2005), indeterminacy andsunspots (Weder 2000), and on the relationship between savings and long-term economic
growth (Carroll et al. 1997 and 2000, and Harbaugh 1996).1
A common feature of the existing investigations on consumption externalities in themacroeconomics literature is that most of them employ the representative agent models.
In general, introduction of consumption externalities into the standard representativeagent models of growth and business cycles does not produce significant qualitative eﬀects
on dynamic behavior of the model economy: see Liu and Turnovsky (2005) for a detailed
discussion on equilibrium dynamics of the standard representative agent model with consumption externalities. Although consumption externalities may yield large quantitativeeﬀects that would be relevant for welfare implications and policy making decisions, thedynamic properties and the steady-state characterization of the model economies are usually the same as those of models without consumption external eﬀects. Such a conclusion
1

Some of the existing studies such as Ljunavust and Uhlig (2000) and Carroll et al. (1997 and 2000)

assume the external habit formation in which the benchmark consumption is given by a weighted averageof past levels of the average consumption in the economy. Unlike the internal habit formation, consumers
consider that the benchmark consumption is not aﬀected by their own consumption behavior under theexternal habit formation hypothesis. Thus this assumption represents consumption externalities withtime delay rather than (internal) habit formation under which each agent takes its past consumption into
account when deciding its optimal saving plans.

1

is in contrast to the eﬀect of production externalities, which may alter the behavior of theeconomy in fundamental ways.
Departing from the mainstream studies mentioned above, we examine the role of consumption external eﬀects in an overlapping generations (OLG) economy. We extend thestandard two-period-lived OLG model of capital accumulation by introducing external
eﬀects among consumption activities of the agents. A relevant diﬀerence between the representative agent and the OLG settings is that the contemporaneous external eﬀects ofconsumption involve the intergenerational as well as intragenerational externalities in the
OLG economy. Unlike the representative agent economy, heterogeneity of agents inevitably
exists in the OLG economy, and hence contemporaneous interactions among consumption
activities of the agents would be more complex in the OLG economy than those in the
representative-agent economy. This suggests that the presence of consumption externalities generates more fundamental eﬀects both on equilibrium dynamics and on the long-run
equilibrium in the OLG setting than in the representative-agent counterpart. The central
purpose of this paper is to confirm this prediction by using a simple growth model of an
OLG economy.
More specifically, we develop an endogenous-growth version of Diamond’s (1965) modelwith consumption externalities. We first examine the real economy in the absence of asset
bubbles. It is shown that the balanced-growth equilibrium and transitional dynamics depend heavily on the preference structure that characterizes forms of consumption external
eﬀects. For example, if consumers have jealousy as to other agents’ consumption so that
there are negative consumption externalities and if consumers are conformists, then there
is a unique balanced-growth path that satisfies global determinacy. However, if the consumers are conformists but they have admiration as to others’ consumption, then there
may exist dual balanced-growth equilibria. Similarly, if consumers are anti-conformists
and if intrageneratioanl external eﬀect is suﬃciently strong, then dual balanced-growth
paths may exist as well. It is shown that in these cases of dual steady states, the long-run
equilibrium with a higher growth rate is locally indeterminate, while the steady state witha lower growth rate exhibits local determinacy.
We then investigate the equilibrium dynamics of the model economy in the presenceof asset bubbles. As shown by Grossman and Yanagawa (1993), an OLG economy with2

endogenous growth may sustain the balanced-growth path with bubbles, if and only if thebalanced-growth rate of bubble-free economy exceeds the real interest rate.2 Thus thepresence of bubble lowers the long-term growth rate, because the balanced-growth ratein the presence of bubbles is fixed at the rate of return to capital. It is also known thatthe steady state with bubbles is locally determinate and the bubble-free steady state islocally indeterminate. We show that those results established in the standard framework
still hold, if there are only intragenerational consumption externalities. However, if there
also exist intergenerational consumption externalities, we may obtain diﬀerent outcomes.
In particular, the growth eﬀect of bubbles and the conditions for sustaining bubbles are
considerably modified in the presence of intergenerational consumption externalities. Since
we will assume that supply of nominal asset grows at a constant rate, the balanced-growth
path with bubbles is uniquely determined. Therefore, even if the corresponding bubblefree economy involves dual steady states, introducing bubbles yields a unique long-term
equilibrium. It is shown that in the presence of consumption externalities the growth ratein the steady state with bubbles may or may not exceed the growth rate attained in the
bubble-free equilibrium. We also confirm that the economy exhibits global indeterminacy
in the sense that every balanced-growth path (with or without bubbles) and a transitional
trajectory leading to one of the stable balanced-growth equilibria can be a perfect-foresightcompetitive equilibrium.
It is to be noted that several authors have examined the roles of consumption externalities in OLG models. Among others, Abel (2005) introduces both intergenerational
and intragenerational consumption externalities into the Diamond model of overlapping
generations. The basic model structure of his article is thus close to ours. The central
concern of Abel (2005) is to characterize the optimal income taxation in the steady-state
equilibrium. Alonso-Carrera et al. (2005) also use a model similar to Abel’s (2005) inorder to investigate how the presence of consumption externalities aﬀects the bequest motives of altruistic agents. In addition, de la Croix (1996) examines an OLG model where
the parents’ consumption behavior is inherited by their children and thus there is one-way
2

In the context of an infinitely-lived overlapping generations model, Futagami and Shibata (1999)

pointed out that this result may not hold, if supply of useless asset is not constant. In Section 4.1 wereconfirm this fact in our setting.

3

intergenerational external eﬀect.3 Since neither equilibrium dynamics with endogenous
growth nor asset bubbles are out of touch in the existing studies mentioned above, thecontribution of this paper is to present a new insight on the role of consumption externalities in macroeconomic dynamics that has not been fully explored in the literature.
The rest of the paper is organized as follows. Section 2 sets up the analytical framework.
Section 3 discusses equilibrium dynamics of the model economy in the absence of asset
bubbles. Section 4 introduces asset bubbles into the base model and considers the role ofconsumption externalities in characterizing long-run equilibrium that may sustain bubbles.
Section 5 concludes the paper.

2

The Base ModelIn this section, we construct the base model that depicts growth dynamics without bubbles.
The analytical framework we use is an endogenous-growth version of Diamond’s (1965)
model. We extend the baseline model by assuming that felicity of an individual consumers
in each generation depends not only on her own consumption but also on the benchmark
consumption represented by the average level of consumption in the economy at large.

2.1

Households

We consider a two-period-lived overlapping generations economy where in each period only
two types of agents are alive: young and old. Agents are identical within the generation.
Population is constant over time and the number of agents in each generation is normalized
to one. The utility function of agents in cohort born at the beginning of period t is
Ut = u (ct , Et ) + βu (xt+1 , Et+1 ) ,0 < β < 1,

where ct denotes consumption when the agents are young, xt+1 is consumption when they
are old, and β denotes a given discount factor. In the above, Et and Et+1 denote thebenchmark levels of consumption that express external eﬀects on the felicities in period3

See also de la Croix and Michel (2002). Using the term, ’external habit’, de la Croix’s (1996) for-mulation represents itntergenerational, external habit formation. Lahiri and Puhakka (1998) examine a
two-period-lived overlapping generations economy with internal habit formation.

4

t and t + 1, respectively. We assume that the benchmark level of consumption in eachperiod depends on the average consumption levels of existing generations:
Et = E 0 (c̄t , x̄t ) ,Et+1 = E 1 (x̄t+1 , c̄t+1 ) ,where c̄t+s and x̄t+s (s = 0, 1) respectively denote the average consumption of young andold agents in period t + s. Notice that since we have assumed that the number of agent ineach cohort is normalized to one, in equilibrium the average level of consumption of eachagent equals its private level:c̄t = ct , x̄t = xt for all t ≥ 0.

(1)

For analytical simplicity, we specify the instantaneous utility function in each period
as follows:u (ct , Et ) =´1−σ³
ct Et−θ1−σ,u (xt+1 , Et+1 ) =

´1−σ
³
−θ
xt+1 Et+1
1−σ, σ > 0, σ 6= 1, θ < 1. (2)We also specify the reference levels of consumption in such a way that,E 0 (c̄t , x̄t ) = c̄γt x̄1−γt

E 1 (x̄t+1 , c̄t+1 ) = x̄γt+1 c̄1−γ
t+1 ,0 < γ ≤ 1.

(3)

Namely, we assume that the felicity of each agent depends on the weighted geometricmean of consumption levels of all consumers. In this specification, γ denotes the relative
strength between the intragenerational and intergenerational external eﬀects.4 Note thatwhen γ = 1, the external eﬀects are generated by the average consumption of the same4

An alternative specification of the utility function that has been frequently used in the literature is
Ut =(xt+1 − θEt+1 )1−σ
(ct − θEt )1−σ
+β
1−σ
1−σ

where
Et = ct + γxt , Et+1 = xt+1 + γct+1 .
Mino (2004) use this specification to analyze equilibrium dynamics of an OLG model with the neoclassicalproduction technology. Alonso-Carrera et al. (2005) use a slightly more general form such that
1−σ
1−σ


δ
xt+1 − θEt+1
ct − θEtδ
+β
.
Ut =
1−σ
1−σ

5

cohort alone. Given (3) , the instantaneous utility function in each period is written as¢¡u ct , E 0 (c̄t , x̄t ) =¡¢u xt+1 , E 1 (x̄t+1 , c̄t+1 ) ="µ ¶θγ µ ¶θ(1−γ) #1−σ
ct
ct
c1−θ
t
c̄t
x̄t

"

x1−θ
t+1

µ

xt+1
x̄t+1

1−σ
¶θγ µ

xt+1
c̄t+1

,
#1−σ

¶θ(1−γ)1−σ.

These expressions show that the felicity of t-th generation in each period depends on
the intragenerational and intergenerational relative consumption levels as well as on theabsolute level of its own consumption.
If external eﬀects are internalized so that the conditions in (1) holds, the social levelof marginal utility of private consumption is given by¢¡−θγ−σ(1−θγ) −θ(1−γ)(1−σ)
xt.
uc ct , E 0 (ct , xt ) = (1 − θγ) ct

In addition, we see thatsign ucc = sign {− (1 − θγ) (σ + γθ (1 − σ))} .
¢¡The same conditions hold for u xt+1 , E 1 (xt+1 , ct+1 ) . In order to make the social marginalutility of consumption is positive and decreasing, we assume that1 − θγ > 0 and σ + γθ (1 − σ) > 0.

(4)

If 0 < θ < 1, the above conditions are satisfied. If θ < 0,we need to assume that σ +θγ (1 − σ) > 0.
Following the taxonomy given by Dupor and Liu (2003), if an agent’s felicity depends
both on the average consumption in the economy at large and on her own consumption,
the presence of negative external eﬀect means that the consumer has jealousy as to otheragents’ consumption, while the consumer has admiration for consumption of others if her
felicity is positively related to the social level of consumption. Moreover, if the marginalutility of private consumption increases with the social level of consumption, consumers
prefer being similar to others (Keeping Up with the Joneses: KUJ). In contrast, if themarginal utility of private consumption decreases with the social level of consumption,
6

consumers want to be diﬀerent from others (Running Away from the Joneses: RAJ).
Using our notation, ∂u/∂E < 0 means jealousy and ∂u/∂E > 0 expresses admiration.
Similarly, ∂ 2 u/∂c∂E > 0 indicates KUJ, while consumers’ preference satisfies RAJ, if∂ 2 u/∂c∂E < 0. Given the utility function (2) , we see that
¶
¶
µ
µ
∂u
∂2u
= sign (−θ) , sign ucE ≡= sign {−θ (1 − σ)} .
sign uE ≡∂E∂c∂E
Thus the consumer’s preference satisfies KUJ if θ (1 − σ) < 0, while her preference exhibits
RAJ if θ (1 − σ) > 0.
An alternative expression for KUJ (resp. RAJ) is conformism (resp. non-conformism).
Collier (2004) defines the ’degree of conformism’ in the following manner:
¯
¶
µ
ucE
σ−1 c
dc ¯¯
.
=−
=θ
Λ (c, E) =dE ¯uc = const
ucc
σEThis gives the relation between the benchmark and private levels of consumption withkeeping the marginal utility of private consumption constant. If Λ (c, E) has a negativevalue, then a higher benchmark consumption reduces private consumption under a given
marginal utility of private consumption. This is equivalent to the RAJ condition. Conversely, if Λ (c, E) > 0, conformism dominates the consumers’ behavior. In particular, ifΛ (c, E) > 1, then consumers are over-conformists.5 It is worth pointing out that in the
representative agent economy where E = c in equilibrium, the degree of conformism isdetermined by the preferences parameters, σ and θ, alone. In the overlapping generations
economy where the equilibrium level of benchmark consumption, E, may diverge from c,
the degree of conformism is also aﬀected by the relative consumption, c/E. Since conformism and non (anti)-conformism are more straightforward expressions than KUJ andRAJ, in what follows we use the former terms.
5

By definition, Λ (c, E) > 1 means that ucc + ucE > 0.If this is the case, conformism is so strong tocancel decreasing marginal utility of private consumption. This situation has been excluded in most of theexisting studies on consumption externalities in macroeconomics.

7

In sum, we have four possibilities displayed in Table 1 :
σ>1

σ<1

θ>0

jealousy, conformist

jealousy, non-conformist

θ<0admiration, non-conformist

admiration, conformist

Table 1: Classification of the preference structure
In the existing literature on consumption externalities in macroeconomic dynamics, it
has been common to assume that the consumer’s preference exhibits jealousy and conformism. Although jealousy and conformism may be frequently observed in reality, we
also examine other cases in order to consider how the analytical conclusions are related tothe specifications of preference structure.
The consumer’s intertemporal decision is basically the same as that in Diamond (1965).
We assume that agents in each cohort work only when they are young. Hence, their flow
budget constraint in their young and old ages are respectively given by
wt = ct + st and xt+1 = Rt+1 st ,where wt is the real wage rate, st is saving of the young agent and Rt+1 denotes the grossrate of interest in period t + 1. The intertemporal budget constraint for the household iswritten as
ct +

xt+1
= wt .
Rt+1(5)

The agents born at the beginning of period t select ct and xt+1 to maximize Ut subject
to the life-time budget constraint of (5) . When solving their optimization problem, theagents take the benchmark levels of consumption, Et and Et+1 , as given. The first-orderconditions for an optimum yields¶µ¶µEt θ(1−σ)ct −σ= βRt+1
.
xt+1
Et+1

(6)

Therefore, unless σ = 1, the marginal rate of substitution between consumption when
young and old is determined by the relative magnitudes of external eﬀects, Et /Et+1 , andby the rate of interest, Rt+1 . Using thd consistency conditions in (1) and substituting (3)and (6) into (6) , we obtain the following:µ¶µ¶
ct −[σ+γθ(1−σ)]xt θ(1−γ)(1−σ)= βRt+1
.
xt+1
ct+1
8

(7)

This equation shows that the marginal rate of substitution between young and old-age
consumption from the social perspective is related to the rate of return to capital in period
t+1 and the relative consumption, xt /ct+1 , which represents the intergenerational external
eﬀects. Note that, given the external eﬀects, the relative consumption ct /xt+1 from theprivate perspective decreases with the rate of return, Rt+1 . Similarly, under the restrictionsin (4) , the relation between the relative consumption from the social perspective is also
negatively related to the real interest rate.

2.2

Production

We assume that the economy has an Ak technology that can sustain endogenous growth.
Firms produce a single commodity in a competitive market. The private technology of
production satisfies constant returns to scale with respect to capital and labor. Supposethat there are many identical firms. The number of firms is normalized to one. The
production function of each firm is given by¡ ¢1−α,yt = Aktα lt k̄t

0 < α < 1,

where yt , lt and kt respectively denote output, labor and capital of an individual firm.
Additionally, k̄t is the aggregate capital which equals kt in the symmetric equilibrium. If
we assume that each young agent supplies one unit of labor so that lt = 1, the aggregate
production function is expressed as yt = Aktα k̄t1−α .66Alternatively, we may assume that the production technology is given by,
yt = Aktα h1−αt0 < α < 1,

where ht denotes human capital. If we assume that physical and human capital earn the same rate of
return in a competitive environment, it should hold that
∂yt
∂yt
=
,
∂kt
∂ht
which means that the optimal factor choice gives h = (1 − α) k/α. Therefore, the production function is
written as
yt = Âk,
where Â =

 1−α 1−αα. The analytical results based on this formulation is the same as those obtained inour setting. Only diﬀerence is that the above model does not involve production externalities so that there
is no ineﬃciency in the production side of the economy.

9The final good market is assumed to be competitive. Thus, the net rate of returnto capital, rt , and the real wage rate, wt , are respectively given by rt = αyt /kt − δ andwt = (1 − α) yt , where δ denotes the rate of depreciation of capital. Due to the arbitrage
condition, the gross rate of interest satisfies Rt = rt + 1. In the following, we assume that
capital fully depreciates in one period, that is, δ = 1 so that Rt = αyt /kt . (This is a
plausible assumption in the two-period lived OLG economy in which one period may be
25 years long.).
The symmetric equilibrium requires that k̄t = kt . Therefore, the social technology thatinternalizes the external eﬀect is written as follows:
yt = Akt .

(8)

Similarly, the competitive levels of gross rate of return to capital and the real wage rate
are respectively given by

2.3

Rt = αA ≡ R,

(9)

wt = (1 − α) Akt .

(10)

Commodity Market Equilibrium Condition

In each period the final goods are used for consumption and investment, so thatyt = ct + xt + kt+1 .

(11)

Since we have assumed that only young agents save, capital formation is determined by
kt+1 = st = wt − ct .

3

Growth without Bubbles

3.1

The Dynamic System(12)

In view of the life-time budget constraint (5) and the saving equation (12) , we see that
xt+1 = Rt+1 (wt − ct ) = Rt+1 kt+1 . Thus from (9) it holds thatxt = Rkt .
10

(13)

By use of (9) , (10), (12) and (13), we may rewrite (7) as follows:1− σ+γθ(1−σ)

ct = (βR)

µ

Rkt
ct+1

¶− θ(1−σ)(1−γ)σ+γθ(1−σ)

Rkt+1 .

Hence, denoting zt = ct /kt and Gt = kt+1 /kt , we can rewrite the above equation as,
zt = B(zt+1 )φ G1+φ
t
where
B≡β

−

1
σ+γθ(1−σ)R(σ−1)(1+θ−2γθ)
σ+γθ(1−σ)

,

φ≡

(14)

θ (1 − σ) (1 − γ).
γθ + (1 − γθ) σEquation (12) presents kt+1 /kt ≡ Gt = (1 − α) A − zt , and hence from (14) we obtain
zt+1 = B

1
−φ

1
φ

zt [(1 − α) A − zt ]1
− 1+ φ.

(15)This equation depicts the dynamic behavior of ratio between young agents’ consumption
and capital stock, zt (≡ ct /kt ) .

3.2

Balanced-Growth Equilibrium

The balanced-growth equilibrium holds when ct and kt change at the same rate, so thatzt stays constant over time. First, note that if there is no consumption external eﬀect, i.e.
φ = 0, (14) is written as zt = β −1/σ R(σ−1)/σ [(1 − α) A − zt ]. Consequently, the gross rateof capital accumulation in the absence of consumption externalities is given byĜ =(1 − α) A.

1 + R(σ−1)/σ β −1/σ

.

(16)

It is also to be noted that if there are only intragenerational external eﬀects, i.e. θ 6= 0 and1(1−θ)(σ−1)γ = 1, then φ = 0 and B ≡ β σ+θ(1−σ) R σ+θ(1−σ) . Thus the dynamic behavior of theeconomy is the same as that without consumption externalities. The only diﬀerence isthat the balanced-growth rate in the presence of externalities is given by(1 − α) A.

GE =
1+R

(1−θ)(σ−1)
σ+θ(1−σ)β1− σ+θ(1−σ)

.

(17)

Comparing (17) with (16) , we find that the growth eﬀect of intragenerational consumption
externalities depends on the parameter values. In the standard case where jealousy and11conformism prevail (that is θ > 0 and σ > 1), as long as R ≥ 1 and 0 < β < 1, we findthe following7 :(1−θ)(σ−1)

R σ+θ(1−σ) β1− σ+θ(1−σ)σ−11< R σ β− σ ,(18)

implying that GE > Ĝ. Similarly, if consumers have admiration as well as conformism(i.e. θ < 0 and σ < 1), we also obtain (18) . As a result, in the absence of intertegenerational externalities, introducing the consumers’ conformism yields the same eﬀect as that
produced by an increase in the elasticity of intergenerational substitution in consumption,
1/σ. In contrast, if consumers are anti-conformists, i.e. θ (1 − σ) > 0, then (18) fails tohold and thus the economy with consumption externalities grows slower than the economy
without consumption externalities.8
In the general case where there are intergenerational consumption externalities, on thebalanced-growth path (15) gives
1

1−φ

B − 1+θ − z 1+φ = (1 − α) A − z.

(19)

It is easy to see that if (1 − φ) / (1 + φ) > 0, then (19) has a unique solution for z ∈7

θ(1−σ)
θ(1−σ)
1
Note that (1−θ)(σ−1)− σ−1= σ[σ+θ(1−σ)]
and σ+θ(1−σ)− σ1 = − σ[σ+θ(1−σ)]. Hence, given the assumpσ+θ(1−σ)σ

tions (4) , 0 < γ < 1, R > 1 and 0 < β < 1, we confirn that(1−θ)(σ−1)
1
σ−11− σ+θ(1−σ)−σσ+θ(1−σ)σ= sign θ (1 − σ) .
β−Rβsign R
8

This result also holds for the case of representative agent economy. In the representative agent economy,

the household’s objective functional is given by1−σ
ct Et−θ
,
β
U=
1−σ
t=0
∞
[

t

where Et = ct holds in equilibrium. Given the same production structure, it is easy to see that thebalanced-growth rate in the representative economy with consumption externalities is
1
ct+1
kt+1
=
= (αAβ) α+θ(1−σ) ,kt
ct

in which it is assumed that αAβ > 1. Therefore, if the agents are conformists, i.e. θ (1 − σ) < 0, then thebalanced-growth rate in the presence of consumption externalities is higher than that without externalities.
It is also to be noted that, as pointed out by Liu and Turnovsky (2005) and others, in the representative
agent economy, the balanced-growth path with externalities satisfies the social optimum condition. This
S∞ t ct(1−σ)(1−θ), theis because if the planner internalized the external eﬀect so that it maximizes
t=0 β
1−σ
balanced-growth rate is the same as the above with external eﬀects.

12(0, (1 − α) A) . Conversely, if (1 − φ) / (1 + φ) < 0 and if (19) has a solution, then there
are two feasible solutions at most: see Figures 1 (a) and (b). Hence, we conclude:
Proposition 1 There exists a unique, feasible balanced growth path if −1 < φ < 1. Otherwise, there are two balanced-growth equilibria at most.
Given the restrictions in (4) , it is easy to confirm that if consumers’ preferences exhibit
jealousy and conformism (KUJ), i.e. θ > 0 and σ > 1, then it holds that −1 < φ < 0.
Therefore, under the standard assumptions on external eﬀects, the balanced-growth pathis uniquely determined. If conformism is associated with admiration (θ < 0 and σ < 1),then φ can be strictly less than −1, so that dual balanced-growth equilibria may exist. If
consumers are anti-conformists (i.e. θ (1 − σ) > 0), φ has a positive value. In order tohold φ < 1, we should assume that θ (1 − σ) (1 − 2γ) − σ < 0. Thus if γ > 1/2, then φ < 1.
If the intergenerational consumption external eﬀects dominate so that γ < 1/2, we may
have φ > 1.
From (15) and (19) we see that¯
dzt+1 ¯¯=
dzt ¯zt =z
=

µ
¶
1
z
1
+ 1+
φ
φ (1 − α) A − z
µ
¶
2φ
1
1
1
+ 1+
B 1+φ z 1+φ .
φφThus if 0 < φ < 1, then dzt+1 /dzt > 1. If −1 < φ < 0, then dzt+1 /dzt < −1. Thismeans that in both cases the dynamic system is unstable around the steady state. Since
zt (= ct /kt ) is not a predetermined variable, local instability and uniqueness of the steadystate mean that the balanced-growth equilibrium is globally determinate and hence theeconomy always stays on the balanced-growth path. Figures 2 (a) and (b) depict the casesof 0 < φ < 1 and −1 < φ < 0, respectively. The figures show that the dynamic system
exhibits global determinacy for φ ∈ (−1, 1) .
When φ > 1, there may exist two steady states. As Figure 2 (c) shows, the steady statewith a lower z is stable and the other with a higher z is unstable. Hence, the balanced
growth path with a higher growth rate is locally indeterminate, while one with a lower
growth rate is locally determinate. If φ < −1, the graphical representation of (15) is given
by Figure 2 (d). This figure reveals that the steady state with a higher z again satisfies
13

local determinacy, because it holds that −1 < 1/φ + (1 + 1/φ) z/ [(1 − α) A − z] . In thissituation, both balanced-growth paths can be realized for any initial value of zt .
Summing up the above discussion, we obtain:
Proposition 2 If −1 < φ < 1, then the economy exhibits global determinacy. If φ > 1,
the balanced-growth equilibrium with a higher growth rate is locally in determinate, whilethat with a lower growth rate is determinate. In the case of φ < −1, the balanced-growth
path with a lower growth rate is either locally determinate or indeterminate.

4

Growth and Bubbles

4.1

Equilibrium Dynamics with Bubbles

We now assume that the government issues non-interest-bearing asset. This asset is intrinsically worthless in the sense that it serves neither for consumption nor for production
activities. For convenience of exposition, we call this asset money. Since ’money’ in our
economy does not present any intratemporal transaction service, it may be called ’pure
asset bubbles’ as well.9 We assume that the initial old receives a given stock of money,
M0 . Then they sell M0 to the young in period 1. We also assume that the governmentchanges money supply at a constant rate of μ − 1 in each period. Thus the nominal stockof money supply changes according toMt+1 = μMt .
Note that if μ < 1, the government contracts money supply. We assume that the newly
created money is distributed to the young agents as lump-sum transfers. This means thatthe flow budget constraint for the government is given by
Mt+1 − Mt = pt τ t ,9

If there is no capital, money can serve as a medium of intertemporal exchange. Since capital plays

a role of store of value, ’money’ in our economy may has a positive value only when the rate of return
dominance fails to hold, that is, the declining rate of price of money in terms of final goods equals the rateof return to capital.

14

where pt is the price of final goods in terms of money and τ t is the real transfer for the
young households (if τ t is negative, it represents a lump-sum tax levied on the young
generation). Letting mt = Mt /kt pt , the government’s budget constraint is expressed asτ t = (μ − 1) mt kt .

(20)

We have assumed that the newly created money is distributed to the young agents, sothat the budget constraint for the young is now replaced withct + st = wt + τ t .

(21)

The saving of the young, st , is spent for physical capital investment and money holding,and hence we obtain:10
st = kt+1 +

Mt+1
.
pt

(22)

Since holding money does not bear interest income, the gross rate of return to money
holding, pt /pt+1 , should be equal to the gross rate of return to capital:
pt
= R.
pt+1

(23)

The real rate of return to capital is fixed under our assumption of Ak technology and thus
the rate of change in monetary price of goods stays constant over time. As a consequence,from (23) the flow budget constraint for the t-th generation in their old age is xt+1 =Rst = R (kt+1 + Mt+1 /pt ) . Again, the equilibrium condition for the final good market isyt = ct + xt + kt+1 . Substituting yt = wt + Rkt , ct = wt + τ t − st , xt = Rst−1 and (22) into
the commodity-market equilibrium condition, we obtain
¶
µ
¶
µ
Mt+1
Mt
+ R kt ++ kt+1 .
wt + Rkt = wt + τ t − kt+1 +pt
pt−1Hence, by use of (20) , we find that the above equation yields mt kt = μRmt−1 kt−1 , which
10

The young agents’ savings are used for investment and money holding. Since the young genration’s

money holding consists of the existing stock of money Mt plus newly distributed one, Mt+1 − Mt , it holds
thatst = kt+1 +Mt
Mt+1 − Mt
Mt+1+= kt+1 +
.
pt
pt
pt

15

gives the following:11
mt+1 =

μR
mt .
Gt

(24)

Diﬀerence equation (24) describes dynamics of the ratio of real money balances and capital.
In order to derive the motion of consumption-capital ratio, zt , we use (7) to derive
µ
¶
1
xt −φ
− σ+γθ(1−σ)
xt+1 .
(25)
ct = (βR)
ct+1
Substituting xt+1 = R(kt+1 + Mt+1 /pt ) and xt = R (kt + Mt /pt−1 ) into the above andarranging the terms, we obtainct≡ zt = Bktµ

kt + Mt /pt−1
ct+1

¶−φ

(Gt + μmt ) ,where (kt + Mt /pt−1 ) /ct+1 can be expressed as
kt + Mt /pt−1
mt ´
pt kt kt+1
1
1 ³
.
1+
=
+ mt
=
ct+1
zt+1 Gt
pt−1 kt+1 ct+1
zt+1 Gt
R
Hence, we express (25) as follows:∙¸
1 ³
mt ´ −φzt = B1+(Gt + μmt ) .
zt+1 Gt
R(26)

From (20) , (21) and (22) , the gross rate of capital accumulation is given by
kt+1≡ Gt = (1 − α) A − mt − zt .
kt

(27)

Using (26) and (27) , we obtain:
zt+1 = B1−φ1³
mt ´ [(1 − α) A + (μ − 1) mt − zt ]− φ,zt 1 +
R[(1 − α) A − mt − zt ]
1
φ

(28)

which describes the motion of zt (= ct /kt ) .
The dynamic equations (24) and (27) yield:
mt+1 =

μRmt.
(1 − α) A − mt − zt

(29)

To sum up, a complete dynamic system with consumption externalities is presented by apair of diﬀerence equations (28) and (29) . Notice that if mt = 0 for all t ≥ 0, our dynamic
system reduces to (15) which depicts the dynamics of the bubble-free economy.
11

Dynamic equation (24) can be directly derived by manipulating (23) and using the definition of mt ,However, it does not come from the definition of variable but from the equilibrium and optimization
conditions as shown in the main text.

16

4.2

Balanced-Growth Characterization in the Absence of Consumption
Externalities

First, let us review the main results obtained in the model without consumption externalities. Since it holds that φ = 0 in the absence of consumption external eﬀects, (28)reduces tozt =β −1/σ R(σ−1)/σ1 + β −1/σ R(σ−1)/σ[A (1 − α) − mt ].

Substituting the above into (27) presents
Gt =A (1 − α)

1 + β −1/σ R(σ−1)/σ− mt .

(30)

By use of the above and (24), we obtain:
mt+1 = hμRmt
A(1−α)− mt
1+βR(σ−1)/σ−1/σi.

Since the balanced-growth rate in the absence of bubble is given by (16) , the motion of
mt is described by
mt+1 =μRmt
Ĝ − mt.

(31)

The dynamic equation (31) shows that there may exist two steady states: the nonmonetary steady state with m = 0 and the monetary steady state with a positive level ofm.
If money supply is constant (μ = 1), then (31) becomes
mt+1 =

Rmt
Ĝ − mt.

As claimed by Grossman and Yanagawa (1993), the existence of balanced-growth equilibrium with a positive, steady-state value of mt requires that¯
dmt+1 ¯¯R< 1.
=
dmt ¯mt =0 Ĝ

(32)Namely, the economy has a balanced-growth path that sustains bubbles if Ĝ > R (= αA) ,i.e. the balanced-growth rate of the bubble-free economy exceeds the real interest rate. In
addition, equation (30) means that if μ = 1, then Gt = Ĝ − m and Gt = R. Therefore, thebalanced-growth path with bubbles attains a lower steady-growth rate than that realized
in the economy without bubbles.
17

If μ 6= 1, the necessary and suﬃcient condition for the presence of sustainable bubble
in the balanced-growth equilibrium is:¯
μR
dmt+1 ¯¯=
< 1.
¯
dmt mt =0
ĜThis means that the equilibrium with bubble can be sustained in the steady state if andonly ifĜ > μR.

(34)

As a result, when μ < 1 so that the government contracts the nominal stock of money, there
exists a feasible steady state with bubbles even if the balanced-growth rate in the bubblefree economy, Ĝ, is strictly lower than the real interest rate, R. Futagami and Shibata(1999) confirm this result in the context of an infinitely-lived overlapping generations
model with population growth. Note that from R = αA and (16) the existence condition(34) is fulfilled if the parameters involved in the model satisfy
i
σ−1
1
α h
1
1 + β − σ (αA) σ .
>
μ
1−α

(35)

In our simple setting, (35) is the necessary and suﬃcient condition for the presence ofsteady state with asset bubbles.
Condition (34) means that the non-monetary steady-state where mt = 0 is locallystable, while the monetary steady state with a positive m is locally unstable. Therefore,
the monetary balanced-growth path locally satisfies determinacy of equilibrium and thenon-monetary balanced-growth path holds local indeterminacy. Since the initial level of
mt (= Mt /kt pt ) is not predetermined, the economy exhibits global indeterminacy.12

4.3

Eﬀects of Consumption Externalities

When discussing the eﬀects of consumption externalities in our setting, we should pointout again that if there are no intergenerational externalities, the main conclusion shownin Section 4.2 still holds. As was pointed out in Section 4.2, if asset bubbles emerge
in an economy without consumption externalities, the balanced-growth rate should be
12

The global indeterminacy means that in each moment the economy may stay either at the monetary

steady state or at the non-monetary steady state. In addition, the economy can be on a transition paththat converges to the non-monetary steady state.

18

lower than one sustained in the corresponding bubble-free economy. This is because thebalanced-growth rate in the bubble economy is μR and the necessary and suﬃcient condition for sustaining bubbles in the long-run equilibrium is Ĝ > μR. In words, a partof saving is used for purchasing money (bubbles) rather than capital so that capital formation is inevitably decreased. Such a conclusion holds in the presence of consumption
externalities as well, if the external eﬀects are only intragenerational. If there is no intergenerational externalities (γ = 1), it is easy to show that the dynamic behavior of theeconomy is summarized as
μRmt,
GE − mt

mt+1 =

where GE is given by (17) . Therefore, the necessary and suﬃcient condition for sustaining
a steady state with bubbles is GE > μR, implying that the bubbly steady state attains alower rate of balanced growth than that in the bubble-free economy.
In the general case where both intragenerational and intergenerational externalities exist, from (28) and (29) the balanced-growth path in the presence of bubbles is characterized
by the following conditions:
z

1
1− φ

=B

1
−φ

1
³
m ´ [(1 − α) A + (μ − 1) m − z]− φ,1+R[(1 − α) A − m − z]

(1 − α) A − m − z = μR,

(36)
(37)

where z and m respectively denote the steady-state values of zt and mt . Equations (36)and (37) can be combined in the following manner:
φ
φ ³
1
1
m ´ φ−1[μR + μm]− φ−1 .
(1 − α) A − m − μR = B − φ−1 (μR)− φ−1 1 +RThis equation gives the steady-state value of m:

φ+1

1

m=

(1 − α) A − μR − B − φ−1 (μR)− φ−1
1

φ+1

2φ

1 + B − φ−1 μ− φ−1 R− φ−1

.

(38)

Using (37) and (38) , we obtain the steady-state value of z:
1

z=

φ+1

2φ

B − φ−1 μ− φ−1 R− φ−1 [(1 − α) A + R − μR]
1

φ+1

2φ

1 + B − φ−1 μ− φ−1 R− φ−1

.

(39)

Thus a feasible balanced-growth path with a positive m exists if the numerator of the righthand side of the above equation has a positive value, which is given by
φ+1

1

μaA + B − φ−1 (μαA)− φ−1 < (1 − α) A,
19

(40)

where B ≡ β−

1σ+γθ(1−σ)(αA)(σ−1)(1+θ−2γθ)
σ+γθ(1−σ). Equation (39) shows that z has a positive valueif (1 − α) A + R − μR = A − μαA > 0, which is fulfilled under (40) . It is also to be notedthat when θ = 0 (so that φ = 0), condition (40) reduces to (35) . To sum up, we haveshown:
Proposition 3 If (40) is statisfied, there is a unique, feasible balanced growth equilibriumwith bubbles.
When there are intergenerational external eﬀects in consumption, introducing bubbles
does not always lower the balanced-growth rate. To see this, first note that the steady-state
condition in the bubble-free economy given by (19) and G = (1 − α) A − z present
11+φ

G = (1 − α) A − B 1−φ G 1−φ ,which determines the balanced-growth rate of the real economy with intergenerational andintragenerational consumption externalities. Define
11+φ

F (G) = G + B 1−φ G 1−φ − (1 − α) A.

(41)

Then the balanced-growth rate without bubbles is a solution of F (G) = 0. Proposition
1 states that if φ ∈ (−1, 1), then F (G) = 0 has a unique positive solution. Additionally,
F (G) is monotonically increasing in G for φ ∈ (−1, 1): see Figure 3 (a). Remember that in
the case of constant money supply the balanced-growth rate with bubbles is μR(= μαA).
We find that under the existence condition (40) the following holds:
1+φ1

F (μαA) = μαA + B − φ−1 (μαA) 1−φ − (1 − α) A < 0.
This means that the balanced-growth rate with bubble is strictly less than that attainedin the bubble-free economy.
If φ ∈/ [−1, 1] and if there are two steady states, the graph of F (G) is like Figure
3 (b). The figure indicates that the balanced-growth rate of the economy with bubbles
is in between the two growth rates attained in the long-run equilibrium without bubbles.
Therefore, if the economy without bubbles stays on the balanced-growth path with a lower
growth rate, the emergence of bubbles may raise the long-term growth rate. Conversely,20

if the initial position is the steady state with a higher growth rate, as in the standard
case, introducing bubble reduces the balanced growth rare. As a consequence, we haveconfirmed the following:
Proposition 4 If φ ∈ (−1, 1), the balanced-growth rate with bubbles is lower than thatattained in the bubble-free economy. If φ ∈/ [−1, 1] , the balanced-growth rate in the presenceof bubbles is higher (resp. lower) than that of the corresponding bubble-free economy staying
on the steady state with a lower (resp. higher) growth rate.
The above proposition indicates that the welfare implication of bubbles may be diﬀerent from that obtained in the standard models without consumption externalities. In thestandard setting the economy with bubbles yields a lower balanced-growth rate than in the
bubble-free economy, and hence the emergence of bubbles cannot be Pareto improving.
Namely, except for the initial old whose welfare is increased by receiving money which
can be sold to the young, all the subsequent generations obtain lower utilities due to apermanent reduction in the growth rate of consumption. By contrast, in our model withintergenerational consumption externalities, in the case of dual steady-states the emergence of bubbles may raise the balanced growth rate, implying that the existence of asset
bubbles can be Pareto improving in the sense that all the generation can attain higher
welfare when bubbles emerge.
It is also worth emphasizing that intergenerational consumption externalities may affect the dynamic behavior of the economy. As mentioned in Section 4.2, either if there is no
consumption externality or if there is only intragenerational external eﬀect, the bubblefree steady state is locally indeterminate, while the steady state with bubbles satisfies
local determinacy. If there are intergenerational consumption externalities, then dynamic
system involves two state variables, mt (= Mt /pt kt ) and zt (= ct /kt ) . Since both price pt
and consumption ct are unpredetermined variable in the perfect-foresight equilibrium, wecannot specify the initial values of mt and zt . In this sense, the dynamic system exhibits
global indeterminacy, because every point in the feasible region of mt - zt space can be
an equilibrium. In addition, if we focus on the local behavior of the economy near thebalanced-growth path, the local determinacy of a balanced-growth equilibrium requiresthat the dynamic equation system consisting of (28) and (29) satisfies total instability
21near the steady state solution. Otherwise, the steady-equilibrium exhibits either a saddle
point or a sink so that there are a continuum of converging path around the bakancedgrowth path. Hence, there may exist expectation-driven, sunspot-type fluctuations nearthese long-run equilibrium with local indeterminacy.
The above argument means that if φ ∈ (−1, 1) so that there is a unique bubblefree (i.e. m = 0) steady state, the economy can stay at the bubble-free or at the bubbly
balanced path or on a transitional path towards the steady state that is either a saddle
point or a sink. Similarly, if φ ∈/ [−1, 1] and there are two steady states without bubbles,
the economy may reach one of three steady states: one with bubbles and other two without
bubbles. In any case, the equilibrium path is globally indeterminate. We thus arrive thefollowing:
Proposition 5 In the case of φ ∈ (−1, 1) , there are one bubble free and one bubbly
balanced-growth paths. If φ ∈/ [−1, 1] , the economy involves three balanced-growth paths at
most: one is associated with bubbles and other two are bubble free. Regardless of the valueof φ, the economy exhibits global indeterminacy of equilibrium.
Finally, we should point out that the main results presented so far are established under
the assumption of constant money growth. If money supply is endogenously determined,
the balanced-grwoth rate in the presence of bubbles is not simply given by the constantvalue of μR. As for a simple example, suppose that newly created money is not distributed
to the young agents but the government consumes final goods by financing printing money.
The government budget constraint is now replaced with
Mt+1 − Mt = pt γYt ,0 < γ < 1,

where γ denotes the government’s conumption share of income. In this case the gross
growth rate of nomonal money stock is
μt = 1 +γA.
mt

Since (23) should be satisfied and since mt stays constant in the steady state, the balanced´³
grwoth rate is given by G = μR = 1 + γA
m αA. The balanced grwoth rate now depends

on the steady-state value of m which may be aﬀected by the presence of consumption
22

externalities. In fact, in this simple money supply regime, we can show that the balancedgrowth equilibrum with bubbles may not be uniquely determined and that the balaned
growth rate is directly linked to the degree of consumption externalities.13

5Conclusion

In the context of a simple model of endogenous growth, this paper has addressed economic implication of consumption externalities in an overlapping generations economy. In
particular, we have focused on the existence of sustainable asset bubbles in the long-run
growth equilibrium. The main finding of our study is that the presence of intergenerational
consumption externalities may play a relevant role. If the consumption external eﬀects
are intragenerational alone, that is, each generation are concerned with other agents’ consumption behaviors in the same cohort, then the external eﬀects are only quantitative
ones. Thus in this case growth and welfare implications of bubbles are essentially thesame as those obtained in the economy without consumption externalities. We have, however, shown that if the intergenertaional external eﬀect in consumption exist, the growth
and welfare eﬀects of bubbles could be considerably diﬀerent from these established in theabsence of bubbles. Our discussion has demonstrated that consumption externalities may
yield more fundamental eﬀects on the equilibrium dynamics and policy implication in dynamic economy with overlapping generations than in the representative-agent economies.
In this paper, we have analyzed a stylized model with or without non-fundamental
asset bubbles. It is interesting to extend our analytical framework to discuss more general situations such that money also serves for intratemporal transactions, labor supply
is endogenously determined14 and that bubbles may be related to fundamentals15 . Additionally, introducing the interest-bearing government debt into our model would be auseful extension.
13

In the context of a continuous-time OLG model, Mino and Shibata (2000) show that the simple

endogenous money supply rule mentioned above may generate multiple balanced growth paths.
14
For example, Mino and Shibata (2000) analyze an endogenously growing OLG model in which money
is introduced in the form of money-in-the utility function. Itaya and Mino (2005) examine the relation
between preference structure and monetary growth. Adding consumption externalities to those frameworks
would yield richer results than those obtained in our model of pure bubble.
15
See, for example, Ventura (2005).

23

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26

(a)

φ <1

(b) φ > 1

Figure 1: Existence of Balanced-Growth Equilibriumz t +1

z t +1
zt +1 = ζ (zt )zt +1 = ζ (zt )

0

45

450

zt

zt

(1 − α ) A

(b)

(a)
- 1 < φ < 0

z t +1- 1 < φ < 0

zt +1
zt +1 = ζ (zt )(1 − α ) A(c)

φ < - 1

zt +1 = ζ (zt )

450

450

zt

(1 − α ) A(d): shapes of ζ (zt ) function

1 < φ

zt

F (G )

μR

G

C 

−1 < φ < 1

F (G )μRG

D  φ > 1 or φ < −1

(KIWTG VJGDCNCPEGFITQYVJTCVGUYKVJCPFYKVJQWVDWDDGNGU