GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No.75

Internal vs. External Habit Formation in a Growing Economy
with Overlapping Generations

Masako Ikefuji and Kazuo Mino

August 2009GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan

Internal vs. External Habit Formation in a Growing Economy
with Overlapping Generations∗
Masako Ikefuji†and Kazuo Mino‡
August 2009

Abstract
This paper explores the roles of internal and external habit formation in a simple
model of endogenous growth with overlapping generations. Unlike the representative
agent settings in which the distinction between internal and external habits may not
yield significant qualitative diﬀerences in working of the model economy, we show thatthe internal and external habit persistence in overlapping generations economies may
have qualitatively diﬀerent eﬀects on the steady-state characterization as well as on the
dynamic behavior of the economy. We also confirm that in an overlapping generations
framework, whether the habits in the utility function takes subtractive or multiplicative
forms may be critical both for long-run growth and for transitional dynamics.
keywords: internal habits formation, external habit formation, overlapping generations
economy, long-run growth
JEL classification: E32, J24, O40∗

We thank David de la Croix, Stephen Turnovsky and Bertrand Wigniolleor for their helpful comments on

earlier versions of this paper. Our research has been financially supported by the Japan Economic Research
Foundation and the Grand-in-Aid for Scientific Research.
†Institute of Social and Economic Research, Osaka University, 6-1 Mihogaoka, Ibaraki, Osaka 567-0047
Japan, e-mail: ikefuji@iser.osaka-u.ac.jp‡
Corresponding author : Institute of Economic Research, Kyoto University, Yoshida Honmachi, Sakyo-ku,
Kyoto, 606-8501 Japan, e-mail: mino@kier.kyoto-u.ac.jp

1

Introduction

When discussing habit persistence of consumption, the literature has distinguished internal(inward-looking) habits from external (outward-looking) habits. The internal habit formationmeans that the current level of utility is aﬀected by the accumulated stock of past private
consumption. Since in this situation the future felicity depends on the choice of current
consumption, a rational consumer takes this fact into account in selecting her optimal consumption plan. In contrast, in the presence of external habit formation, the accumulated
level average past consumption in the economy at large aﬀects the current utility of an individual consumer. The external habit formation is a form of intertemporal consumption
externalities and thus the consumer does not internalize external habits when making her
optimal decision.
Using the representative agent models, several authors have examined implications of
the distinction between internal and external habit persistence. A sample of this literatureincludes Carroll, Overland and Weil (1997, 2000) and Harbaugh (1996) on the relation between savings and long-run growth, Gomez (2008) on the speed of convergence, Grishchenko(2007) on asset pricing, and Uribe (2002) on exchange rate determination. Although thesestudies demonstrate that whether habit formation is internal or external to the consumers
may yield significant quantitative diﬀerences, it may not change working of a model economy
in fundamental ways: the dynamic behavior and steady-state characterization of a modelwith internal habit persistence are generally the same as those of a model with external
habit formation. The main reason for this result is that in representative-agent models theaverage consumption habit stock of the entire economy equals the private habit stock inequilibrium. Therefore, the internal habit stock always takes the same level as the external
habit stock. Such a restriction may contribute to reducing qualitative diﬀerences between
outcomes generated by external and internal habit formation.
Unlike most of the foregoing studies, in this paper we examine the eﬀects of internaland external habits in the context of an overlapping generations (OLG) economy. In an
OLG economy there are heterogeneous cohorts in each period and, hence, the habit stockof consumers in each cohort is no more equal to the average habit stock of the economy. In
addition, the presence of heterogeneous agents makes intertemporal consumption external-

2

ities more complex than those in the representative-agent economies. Those facts suggestthat there may exist significant diﬀerences between the roles of internal and external habit
formation in an OLG economy. In order to confirm this prediction, we construct a simple
growth model with overlapping generations where both internal and external consumption
habits are present. As for internal habits, we employ the standard formulation in which thelevel of consumption in young age aﬀects the consumer’s felicity when old. In formulatingexternal habit formation, we follow de la Croix (1996) and de la Croix and Michel (1999) who
assume that young agents’ consumption behavior is aﬀected by their parents’ consumption
levels. Such an ’inherited taste’ represents intergenerational external eﬀects in consumption,
which is a natural expression of external habit formation in an OLG economy.
In this paper we utilize an endogenous growth version of Diamond’s (1965) model andintroduce two types of habits mentioned above into the base model. Inspecting the model
behavior, we obtain two main findings. First, as was predicted, the steady-state characterization and dynamic stability of the economy are highly sensitive to whether habit persistence
is internal or external to the agents. In particular, in the presence of external habits persistence, the internal habit formation in consumption may yield more complex eﬀects thanin the representative agent economies. Furthermore, stability of the balanced-growth pathdepends heavily upon the strength of habit persistence and the growth eﬀects of internal
habit formation depends on the strength of external habits. Second, the dynamic behavior
and balanced-growth equilibrium are also aﬀected by the way how consumption habit stocks
are introduced into the felicity function. To see this, we use two of the most popular formulations used in the literature: subtractive and multiplicative forms. If habit formation
has a subtractive form in the utility, we generally obtain straightforward conclusions. In
contrast, the formulation of multiplicative habits yields rather complex results and many of
the outcomes depend on whether the agents are conformists or anti-conformists. As well as
the distinction between internal and external habits, in the representative agent economies,
the analytical outcomes of subtractive and multiplicative habit formation yield quantitativeeﬀects alone.1 Our study thus demonstrates that the way how consumption habit formation
1

Using models with internal habit persistence, Wendner (2002 and 2003) and Bossi and Gomis-Porqueras(2009) compare the economic consequences of subtractive and multiplicative forms of consumption habits.
Wendner (2002) uses multiperiod model of consumer’s optimization problem, while Bossi and Gomis-Porqueras3

is formulated plays a more prominent role in an OLG economy than in the representative
agent counterpart.
The present paper is closely related to Carroll et al. (1997 and 2000) and Alonso-Carrera
et al. (2007). Carroll et al. (1997 and 2000) examine representative agent models of endogenous growth with an Ak technology. They compare the model with internal habits with
one with external habits. As mentioned above, they conclude that while the distinction between internal and external habit formation may produce relevant quantitative diﬀerences,the equilibrium dynamics and steady state of the model economy are not aﬀected by that
distinction. Our discussion clarifies that their findings do not necessarily hold in overlapping generations settings. Alonso-Carrera et al. (2007) construct an overlapping generations
model with internal as well as external habit formation. They use an exogenous (neoclassical) growth model and assume the presence of bequest motives. Their central concern isto analyze the eﬀects of habit formation on integenerational wealth transmission.2 Although
the basic model structure of this paper is similar to those examined by Carroll et al. (1997
and 2000) and Alonso-Carrera et al. (2007), our research concern does not overlap these
existing studies. Therefore, the present paper is a compliment rather than a substitute to
the foregoing contributions.3
The rest of the paper is organized as follows. The next section sets up the analyticalframework. Section 3 examines a model with subtractive habit formation, while Section 4
reconsiders the base model in the case of multiplicative habits. In both sections, we focus on
the properties of balanced-growth equilibrium and equilibrium dynamics out of the balancedgrowth path. Section 5 presents some numerical examples and Section 6 concludes.
(2009) uses an exchange economy model with overlapping generations.
2
Alonso-Carrera et al. (2008) introduce intragenerational wealth heterogeneity into a small country versionof their base model and examine the eﬀect of habit formation on the stationary wealth distribution.
3
As for introducing consumption habit persistence into OLG models, Lahiri and Puhakka (1998) is probably
the first study that examines the role of internal consumption habits in an OLG economy. They introduce
internal habit formation into the Gale-Samuelson type model of an exchange economy. Bossi and GomisPorqueras (2009) extend the discussion of Lahiri and Puhakka (1998). In addition, Bunzel (2006) considers
the monetary OLG model with internal habit persistence.

4

2

Analytical Framework

2.1

Preferences

Consider an overlapping generations economy where there are three types of agents, young,
adult and old. When young, the agents neither consume nor work. They only observe
their parents’ (i.e. adult agents’) consumption behavior and ’inherit’ their tastes. During
adult period, the agents work, consume and save. At the same time, they again observe
consumption behaviors of their parents (i.e. old agents) and inherit them.4 When old, theagents only consume. Population is constant and each cohort consists of a continuum ofidentical agents with a unit mass. In the following, the t-th generation means the cohortborn at the beginning of period t − 1.
The life-time utility function of an agent in the t-th generation is given by
Ut = u (ct , ht ) + βu (xt+1 , ht+1 ) ,0 < β < 1,(1)

where ct denotes consumption when adult, xt+1 is consumption when old, and β is a given
discount factor. Here, ht and ht+1 represent stocks of consumption habits in period t andt + 1, respectively. For analytical clarity, the following discussion employs specific forms ofutility functions. More specifically, we consider alternative forms of popular specifications:subtractive habits and multiplicative habits. When assuming the preference structure witha subtractive form of habit formation, we use the following function:
Ut =(xt+1 − θ1 ht+1 )1−σ
(ct − θ0 ht )1−σ
+β
,
1−σ
1−σ

θ0 , θ1 , σ > 0.

(2)

Ever since Ryder and Heal (1973), this form of utility function including habit persistence
has been frequently employed in the literature. Similarly, the felicity function involving
multiplicative habits is formulated as

Ut =
4

´1−σ
³
0
ct h−θ
t
1−σ

+β

³
´1−σ
1
xt+1 h−θ
t+1
1−σ

,

0 < θ0 , θ1 < 1, σ > 1.

(3)

The assumption of taste inheritance in the adult age follows de la Croix (1996) and de la Croix and Michel(1999). In this paper we also assume that the old agents inherit consumption behavior of their parents when
they are old.

5

We compare the eﬀects of internal and external habit formations by use of alternative functional forms given by (2) and (3) .5
In general, ht depends ct−1 alone and ht+1 is determined by ct and xt . Since we haveassumed that young agents do not consume, the stock of consumption habits held by theadult agents does not involve their past consumption. Thus the habit stock held by the adultagent is inherited from their parents’ consumption in period t − 1:
ht = ct−1 .

(4)

We assume that the habit stock held when old, ht+1 , increases with ct and xt :ht+1 = h (ct , xt ) .

(5)Namely, the habit stock held in the old period consists of the old-age consumption of theirparents as well as their own consumption in the previous period.6 In what follows, when wedeal with the subtractive form of habit persistence, we specify h (ct , xt ) in such a way thath (ct , xt ) = xt +

η
ct ,
θ1

η > 0.

(6)

When we discuss the model with multiplicative habits, we useh (ct , xt ) = xεt 1 cεt 2 ,0 ≤ ε1 , ε2 ≤ 1.

(7)

It is to be noted that, as we will see in Footnote 11 in Section 4, if σ is less than one, thenthe felicity function with (7) cannot yield an interior solution for the household’s selection of
optimal consumption. To avoid obtaining this result, we use a linear function similar to (6)when we consider the case of multiplicative habits when 0 < σ < 1.(See Section 4.3.)
To sum up, external habits are expressed by ct−1 in (4) and xt in (5), which also representintertemporal consumption externalities. In contrast, ct in (5) means the internal habit
5

In the recent literature, Alonso-Carrera et al. (2005), Alverz-Cuadrado et al. (2007), Bunzel (2006) andCarroll et al. (1997, 2000) use the multiplicative forms of habit persistence, whereas de la Croix (1996, 2001),
Lahiri and Puhakka (1998), Wendner (2002, 2003) employ the subtractive form of habit formation.
6
More generally, we may assume that ht+1 also depends on the average consumption in the economy at large
when agents are young, that is, ht+1 = h (ct , c̄t , xt ) , where c̄t denotes the average consumption when young. In
this formulation, c̄t expresses the intertemporal as well as intragenerational consumption externalities. Since
introduction of c̄t eﬀect adds a purely quantitative eﬀects on the model behavior, the following discussion does
not consider c̄t .

6formation perceived by the adult agents when deciding their optimal consumption-saving
plan.7

2.2

Consumption and Saving

An adult agent supplies one unit of labor inelastically and spends its wage income for consumption and saving. In the old age, the agent consumes all of her savings. Thus the flow
budget constraints for adult and old agents are respectively given by
ct + st = wt ,(8)

xt+1 = Rt+1 st ,(9)

where wt , st and Rt+1 denote real wage in period t, saving of the adult and the gross rate of
return to capital in period t + 1, respectively. Combining these flow constraints, we obtainthe life-time budget constraint:
ct +

xt+1
= wt .
Rt+1(10)

In the beginning of period t, an agent in cohort born in period t − 1 maximizes Ut givenby (1) subject to (10) , (4) and (5) . Considering that ct−1 and xt are external to the t-th
generation, we manipulate the first-order conditions for an optimum to obtain the following:u2 (xt+1 , ht+1 ) (∂ht+1 /∂ct )u1 (ct , ht )+β
= βRt+1 .
u1 (xt+1 , ht+1 )
u1 (xt+1 , ht+1 )7(11)

In this paper we focus on intertemporal consumption externalities from parents to children. If we consider

intratemporal external eﬀects as well, the utility function may be set as
Ut = u (ct , ht , et ) + βu (xt+1 , ht , et+1 )where et and et+1 represent intratemporal consumption external eﬀects determined by
et = e (c̄t , x̄t ) ,et+1 = e (x̄t+1 , c̄t+1 ) .

Here, a variable with an upper bar means the average level of that variable. This formulation states thatthe felicity of an agent is also aﬀected by the current level of average consumption in the economy at large.
Note that in this formulation the felicity of an old agent depends on the average consumption of the next
generation, c̄t+1 , which may change the dynamic properties of the model economy: see Mino (2007 and 2008)
for dynamic analyses of OLG models with intratemporal consumption externalities.

7

In the presence of internal habit formation, a substitution between consumption when adult,
ct , and consumption when old, xt+1 , has two eﬀects. One is the direct intertemporal substitution between current and future consumption, which is represented by the first term in
the left-hand side of (11) . Additionally, a change in the current consumption, ct , alters thelevel of habit stock in the old age, which aﬀects the felicity when old. The second term in theleft-hand side of (11) expresses such an indirect substitution eﬀect caused by the presence ofinward-looking consumption habits. The optimal choice between ct and xt+1 thus requiresthat the sum of the direct and indirect rates of substitution equals the discounted rate ofinterest in period t + 1. Alternatively, we may rewrite (11) as∙¸
u2 (xt+1 , ht+1 ) (∂ht+1 /∂ct )u1 (ct , ht )= β Rt+1 −.
u1 (xt+1 , ht+1 )
u1 (xt+1 , ht+1 )(12)

As formulated in (2) and (3) , the standard formulation of internal habits assumes thatu2 (x, h) < 0. One unit decrease in the current consumption, ct , increases one unit of saving,which yields the real interest, Rt+1 , plus the return to a reduction in the internal consumption
habit stock held in the next period represented by −[u2 (.) (∂ht+1 /∂ct )]/u1 (.) (> 0) . Thus
(12) states that the marginal rate of intertemporal substitution in consumption equals thediscounted subjective rate of return to investment that includes a change in the stock ofinternal consumption habits.
Finally, since only young agents work, capital accumulation is determined by the following
condition:
kt+1 = st = wt − ct .

2.3

(13)

Production

To avoid unnecessary complexity, we use one of the simplest models of endogenous growth.
The production function is given by¡¢1−α, A > 0,yt = Aktα k̄t Nt

α < 1,

where yt is aggregate output, kt is capital and Nt denotes the total labor supply. In addition,k̄t expresses production externalities generated by capital stock in the sense of Romer (1986).
Because of normalization of the number of young agents, we set Nt = 1 for all t. In addition8

k̄t = kt in equilibrium. Therefore, the social production function that internalizes production
externalities is written as
yt = Akt .

(14)

If capital depreciates at the rate of δ ∈ [0, 1] , the gross rate of return, Rt , equals 1 +∂Yt /∂kt − δ. We assume that capital completely depreciates within a period (δ = 1) , so that
Rt is given by
Rt = αA,(15)

implying that the gross rate of return to capital stays constant over time.8 Similarly, the real
wage rate equals the private marginal productivity of labor, and hence
wt = (1 − α) yt /Nt = (1 − α) Akt .

(16)

Because of an Ak structure of production, the economy can sustain persistent growth.9

3

Subtractive Habits

We first consider the model with a subtractive form of habit formation. Substituting (4) and(6) into (2) , we obtain the following utility function:
Ut =8(xt+1 − θ1 xt − ηct )1−σ
(ct − θ0 ct−1 )1−σ
+β
,
1−σ
1−σ

θ0 , θ1 , η > 0.

(17)

Since one period may be 25 or 30 years in a two-period lived OLG economy, the assumption of complete

depreciation of capital would be plausible.
9
An alternative implication of the Ak modelling is that the production inputs consist of physical and humancapitals:,
yt = Aktα h1−αtwhere ht denotes stock of human capital. Suppose that both types of capital stocks are produced by use
of final goods. Then the non-arbitrage condition between holding physical and human capital is given by
kt . Substituting this into the∂yt /∂kt = ∂yt /∂ht , which yields αyt /kt = (1 − α) yt /ht so that ht = 1−ααproduction function, we obtainyt = A



1−α
α

1−α

kt .

Since this formulation does not assume the presence of external increasing returns, unlike the Romer-typemodelling, there is no intratemporal ineﬃciency in production. Otherwise, both formulations give essentiallythe same results. See, for example, Chapter 8 in Acemoglu (2009) for the detailed discussion on the Ak growthmodel with two capital stocks.

9Here, θ0 and θ1 represent the strength of taste inheritance, while η expresses the degree ofinternal habit persistence in the old age.10

3.1

Dynamic System

Given (17) , the optimal condition (11) is expressed as(ct − θ0 ct−1 )−σ = β(Rt+1 + η)(xt+1 − θ1 xt − ηct )−σ .

(18)

Using (9) and (13) , we obtain xt+1 = Rt+1 kt+1 . Thus from (15), it holds that
xt+1 = αAkt+1 .

(19)

In view of (15) and (19) , the optimization condition (18) can be written as
αAkt+1 − (η + φ)ct + φθ0 ct−1 − θ1 αAkt = 0,

(20)

where
1

φ ≡ [β(η + αA)] σ .
In addition, equations (13) and (16) yield:
kt+1 = (1 − α)Akt − ct .

(21)

Now define
zt+1 ≡

kt+1.
ct

Then (21) yields
ct(1 − α)A=.
kt
1 + zt+1(22)

By use of the above equation, we find that (20) gives the following diﬀerence equation of zt :zt =φθ0 (1 + zt+1 ).
A [(η + φ)(1 − α) + θ1 α − α (A(1 − α) − θ1 ) zt+1 ](23)

Since this equation contains all the equilibrium and optimization conditions, it gives a complete dynamic system of our economy.
10

As mentioned in Section 1, Alonso-Carrera et al. (2007) study an overlapping generations economy modelwith internal as well as external habit formation. In our notation, their utility function is
Ut = α log (ct − θct−1 ) + β log (xt+1 − ηct ) + γ log bt ,where bt denotes bequests. Unlike our formulation, the above setting ignores the taste inheritance in the old
age.

10

3.2

Balanced-Growth Equilibrium

In the steady-state equilibrium, xt , ct and kt grow at a common rate. As a result, zt (= kt /ct−1 )stays constant over time. Letting z ∗ be the steady-state value of zt , from (23) we see that z ∗satisfiesz∗ =which givesφθ0 (1 + z ∗ ),A[(η + φ)(1 − α) + θ1 α − α (A(1 − α) − θ1 ) z ∗ ]
z∗ =

ξ±

p
ξ 2 − 4κφθ0
,
2κ

(24)

where
ξ ≡ A [(η + φ)(1 − α) + θ1 α] − φθ0 ,κ ≡ αA [A(1 − α) − θ1 ] ,1

φ ≡ [β(αA + η)] σ .
To ensure the existence of z ∗ , we assume thatξ 2 − 4κφθ0 > 0.

(25)

Inspecting (23) and the steady-state condition shown above, we may summarize thesteady-state characterization in the case of subtractive form of habit formation in the following manner:
Proposition 1 In the model with a subtractive form of habit formation, (i) if A(1 − α) > θ1 ,then there exist dual balanced-growth paths: one with a higher growth rate is locally stable
and the other with a lower growth rate is locally unstable, and; (ii) if A(1 − α) < θ1 , then thebalanced-growth path is uniquely given and it is globally unstable.
Proof. Let us write (23) as
zt = G (zt+1 ) ,where G (zt+1 ) is the right-hand side of (23) . Note that
G0 (zt+1 ) =φθ0 (η + φ + αA)(1 − α)
> 0,
A [(η + φ)(1 − α) + θ1 α − α (A(1 − α) − θ1 ) zt+1 ]2

and
G (0) =φθ0
> 0.
A[(η + φ)(1 − α) + θ1 α]
11

In addition, it is easy to see that if A(1−α) > θ1 , then G00 (zt+1 ) > 0 and limzt+1 →z̄ G (zt+1 ) =+∞, wherez̄ =(η + φ)(1 − α) + θ1 α.
α[A(1 − α) − θ1 ]Hence, G (zt+1 ) is monotonically increasing and strictly convex in zt+1 so that there are twovalues of z ∗ satisfying z ∗ = G (z ∗ ) . Since G (zt+1 ) is a monotonic function, the dynamic
equation can be expressed aszt+1 = G−1 (zt ) .
Moreover, as Figure 1 shows, we see that¯
dzt+1 ¯¯1
= 0 ∗dzt ¯zt =z∗
G (z )is strictly larger (resp. smaller) than 1, when z ∗ has a relatively small (resp. large) value.
Since the initial value of zt = kt /ct−1 is historically given in the beginning of period t, theabove shows that the steady state with a higher z ∗ is locally stable, while the steady statewith a lower z ∗ is locally unstable. In the case that A (1 − α) < θ1 , it is satisfied that
G00 (zt+1 ) < 0 so that G (zt+1 ) is strictly concave, implying that the steady state with a¯¯= 1/G0 (z ∗ ) < 1 holds in this case, the uniquepositive z ∗ is uniquely given. Since dzdzt+1
¯
t
∗
zt =z

balanced-growth path is globally unstable: see Figure 2. Finally, notice that (14) and (21)give yt+1 /yt = kt+1 /kt = (1 − α)A − ct /kt . Therefore, the gross growth rate of income on thebalanced-growth path is given byg∗ =(1 − α)Az ∗.
1 + z∗(26)

This equation shows that the balanced-growth rate, g ∗ , increases with the steady-state levelof z ∗ . Hence, the balanced-growth path with a higher value of z ∗ sustains higher growth ofincome and capital.
In what follows, we focus on the stable balanced-growth path obtained under the condition of A (1 − α) > θ1 . More specifically, we examine how the strength of consumption habits
aﬀects the long-term growth rate of the economy. Since the balanced-growth rate of income
monotonically increases with the steady-state value of zt , we may consider the relation between external and internal habit formation and long-term growth by exploring the eﬀects of12changes in parameters, θ0 , θ1 and η on z ∗ . Remember that when A (1 − α) > θ1 , the steadystate satisfying local stability is one with a higher level of z ∗ . Hence, from (24) we focus onpξ + ξ 2 − 4κφθ0
∗
z =
,
(27)
2κ
where κ = αA [A(1 − α) − θ1 ] > 0. In view of definitions of ξ, κ and φ given in (24) , it iseasy to see that
∂z ∗
< 0,
∂θ0

∂z ∗
> 0.
∂θ1

Since the balanced-growth rate is positively related to z ∗ , the above results show that a higher
degree of taste inheritance in the adult age decelerates growth, while a stronger degree ofaspiration for their parents’ consumption in the old age accelerates capital formation. Notethat a higher aspiration for parents’ consumption in adult age raises the marginal utility inthat period, so that the adult agents increase their consumption level relative to that in theirold age. As a result, savings are reduced and thus capital accumulation is decelerated. In
contrast, a rise in the degree of external eﬀect in old age increases the marginal utility of
consumption when old. Thus the agents substitute consumption in their adult age with thatin their old age and, therefore, savings will increase to yield a higher balanced-growth rate ofincome.
The eﬀects of a change in the intensity of internal habit formation, η, are more complex
than those of θ0 and θ1 . Since both ξ and φ increase with η, the eﬀect of a change on z ∗ is
ambiguous if A (1 − α) − θ1 > 0. If there is no taste inheritance in the adult age (i.e. θ0 = 0),then z ∗ monotonically increases with η. In fact, when θ0 = 0, the steady-state levels of z are
z ∗ = 0 and ξ/κ. The interior steady-state value of z isih11σ (αA + η) σ (1 − α) + θ 1 α
η
+
β
ξ
,
z∗ = =κα [A(1 − α) − θ1 ](28)

which is an increasing function of η. Hence, the possible presence of a negative relation
between z ∗ (so the balanced-growth rate) and the degree of internal habit persistence isattributed to the presence of external habit formation in the adult period.
In sum, we have shown:
Proposition 2 On the balanced-growth path satisfying local stability, a higher degree of subtractive habit persistence in the adult age lowers the balanced-growth rate, while a stronger
13

external habit persistence when old enhances long-term growth. A rise in the intensity ofinternal habit persistence increases the balanced-growth rate if there is no taste inheritancein the adult age. Otherwise, the growth eﬀect of internal habit persistence depends on theparameter values involved in the model.
We may obtain more precise interpretation of the comparative statics results in the above
proposition. To do so, let us write (11) as
1x̂t+1= [β(αA + η)] σ ,
ĉt

(29)

where ĉt = ct −θ0 ct−1 and x̂t+1 = xt+1 −θ1 xt −ηct . This equation states that the optimal choice
between the habit-adjusted consumption in adult and old ages depends upon the discounted
rate of return to capital plus the degree of internal habit persistence. Note that changes
in the strength of external habits, θ0 and θ1 , do not alter the ratio of the habit-adjusted
consumption in adult and old ages. Hence, if θ0 increases under given levels of ct−1 and xt ,both ct and xt+1 increase to keep x̂t+1 /ĉt constant. As a result, consumption in both adultand old age rise, so that capital accumulation will decline. When θ1 becomes higher undera given level of xt , consumption in the adult age, ct will decline to keep x̂t+1 /ĉt constant,
implying that savings of adult agents increase and capital accumulation is accelerated.
As for the eﬀects of a change in the degree of internal habit persistence, η, it is to be
noted that if there is no external habit in the adult age, i.e. θ0 = 0, (29) becomes1
xt+1 − θ1 xt − ηctx̂t+1== [β(αA + η)] σĉtct

In this case, a rise in η increases the habit-adjusted discounted rate of return, β (αA + η) ,and the habit adjusted-consumption during the old age, x̂t+1 , under a given level of ct . To
keep the above condition, ct must decline so that capital accumulation will be enhanced. In
particular, on the balanced-growth path we may rewrite (29)
¶
µ
1
θ1
x̂t+1
= 1 − ∗ αAz ∗ − η = [β(αA + η)] σ .
ĉtg
In deriving the above, we use xt+1 /xt = g ∗ and xt+1 /ct = αAkt+1 /ct = αAz ∗ . Rememberthat from (26) g ∗ is positively related to z ∗ . Thus a rise in the habit-adjusted rate of return
β (αA + η) is associated with a higher rate of balanced-growth.
14In contrast, when θ0 > 0, (29) is given by1
xt+1 − θ1 xt − ηctx̂t+1== [β(αA + η)] σ .
ĉtct − θ0 ct−1

On the balanced-growth path this above is written as
∙µ
¶
¸
1
x̂t+1
θ1
g∗
∗
1 − ∗ αAz − η = [β(αA + η)] σ ,
= ∗
ĉt
g −η
g

This equation indicates that both g ∗ and z ∗ could decrease, when a rise in η increases therate of marginal substitution between the habit-adjusted consumption ĉt and x̂t .
If there are internal habits alone (θ0 = θ1 = 0) , then dynamic equation (20) is replacedwith αAkt+1 − (η + φ)ct = 0, which is rewritten as
gt =η + φ ct,
αA kt

where gt = kt+1 /kt . Thus, using (21) and (28), we see that if θ0 = θ1 = 0, the gross growthrate of capital is given by
gt =

i
h
1
A (1 − α) η + (β(η + αA)) σ

.
(30)
1
αA + η + (β(η + αA)) σ
In this case, the predetermined variable at the beginning of period t is kt alone, the initialvalue of gt (= kt+1 /kt ) is not fixed. Therefore, the economy without external habits always
stays on the balanced-growth path. As (30) shows, a rise η increases gt , which means thata stronger internal habit persistence produces a higher income growth if there is no external
habit formation.
Consequently, we have found:
Proposition 3 In the case of subtractive habit formation, there exists a unique and stablebalanced-growth path if there is external habit persistence alone. If there is only internal habit
formation, then the economy always stays on a unique balanced-growth path and a stronger
habit persistence increases the balanced-growth rate of the economy.

4

Multiplicative Habits

In this section we assume that habit stocks are involved in the felicity function as multiplicative form of (3) . We first assume that the utility function is given by
³
´1−σ
´1−σ
³
−γ γ−θ
−θ
xt+1 ct xt
ct ct−1+β, σ > 1, 0 < γ < θ < 1.
Ut =
1−σ
1−σ
15

(31)

In the above, θ and γ − θ respectively denote the degrees of external habits in adult and oldages, while γ represents the intensity of internal habit formation in the old age. Since we will
focus on the balanced-growth equilibrium in which ct and xt grow at a common, constantrate, the felicity in each period should have the same degree of homogeneity. To satisfy thisγ−θ, is homogeneous ofrequirement, we assume that habit persistence in the old age, c−γ
t xt

degree −θ in ct and xt to keep the homogeneity of Ut function. In addition, we assume thatσ > 1 in order to obtain an interior solution for the household’s optimization problem.11
When we consider the case of 0 < σ < 1 in Section 4.3, we will use an alternative functional
form.

4.1

Dynamic System

The representative agent in cohort born in period t − 1 chooses ct ,and xt+1 to maximize theutility (31) subject to (10). The first-order condition for an optimum is given by:
µ

xt+1
ct

¶σ µ

ct
ct−1

¶γ(1−σ)

− βγ

µ

xt+1
ct

¶

zt (γ−θ)(1−σ) = βαAzt (γ−θ)(1−σ) .

(32)

Note that ct /ct−1 = (ct /kt ) (kt /ct+1 ) . Thus by use of (19) and (22) , we transform (32) intoσ(αAzt+1 )which leads to∙¸(1 − α) A γ(1−σ)−θ(1−σ)= βαA(γzt+1 + 1)zt,
1 + zt+1"Ω(γzt+1 + 1)(zt+1 + 1)γ(1−σ)
zt =
σ
zt+1

#

1
θ(1−σ)

,

(33)

where
Ω≡

βαA
.
(αA) [(1 − α) A]γ(1−σ)σ

Dynamic equation (33) presents a complete dynamic system that summarizes the equilibrium
dynamics of the model with the multiplicative form of habit formation.
11

Plugging ct = wt − st into (31) , we obtain−θ(1−σ)(wt − st )1−σ ct−1
Ut =
1−σ

+β


1−σ
(wt − st )−γ(1−σ) xt+1 xγ−θt
1−σUnder given levels of ct−1 and xt , the above expression means that limst →wt Ut = +∞, if 0 < σ < 1. Therefore,
when σ is less than one, the optimal choice of ct is a corner solution: ct = 0. See Alonso-Carrera et al. (2005)
and Hiraguchi (2008) for the detail.

16

4.2

Balanced-Growth Equilibrium

Inspecting dynamic equation (33) , we first find the following result:
Proposition 4 If the utility function is given by (31) in the case of multiplicative habits,
then there is a unique balanced-growth equilibrium that satisfies global stability.
Proof. Let us rewrite (33) as"# 1Ω(γzt+1 + 1)(zt+1 + 1)γ(1−σ) θ(1−σ)
zt =
,
σ
zt+1
µ
¶ 1
θ(1−σ)1γ 11θθ(1−σ)
γ+(zt+1 + 1) θ zt+1
≡ F (zt+1 ) .
=Ωzt+1The steady-state value of zt fulfilling z ∗ = F (z ∗ ) is given by
¸ 1
∙
σ−1
(z ∗ )θ−1z∗=.
(z ∗ + 1)γΩ(γz ∗ + 1)

(34)

(35)

Considering our assumptions that 0 < θ < 1 and γ > 0, we see that the left-hand side of(35) monotonically decreases with z ∗ and it takes +∞ when z ∗ = 0, while it is zero whenz ∗ = +∞. Similarly, the right-hand side of (35) is strictly increasing in z ∗ and it is zero whenz ∗ = 0. Therefore, there is a unique, positive value of z that fulfills (35) . To check stabilityof the balanced- growth path, notice that F (0) = 0 and that
"
#
1
1
γ
¡ 2
¢+
+
> 0.
F 0 (zt+1 ) = F (zt+1 )θ (zt+1 + 1) θzt+1
θ (σ − 1) γzt+1+ zt+1

In addition, we see that limzt+1 →∞ F (zt+1 ) = +∞ for σ > 1. Furthermore, in the steadystate we obtain
F 0 (z ∗ ) =

γz ∗
1
1
+
+ > 1,∗θ (σ − 1) (γz + 1) θ (z ∗ + 1) θbecause 0 < θ < 1 and σ > 1. As a consequence, it holds that¯
dzt+1 ¯¯
10<
= 0 ∗ < 1,dzt ¯zt =z∗
F (z )

which demonstrates that the unique interior steady state is globally stable.
We also confirm that in the case of σ > 1, if θ increases, the steady-state value of z ∗satisfying (35) decreases. Additionally, it is also seen that the eﬀect of a change in γ on z ∗ is
ambiguous. In Section 6 we numerically examine the growth eﬀect of a change in η.
We summarize these comparative statics results as follows:17

Proposition 5 If the utility function is given by (31) , a higher degree of external habit formation depresses long-run growth, while the growth eﬀect of a rise in the strength of internal
habit persistence depends on the parameter values involved in the model economy.
It is to be noted that unlike the subtractive form of habit persistence discussed in theprevious section, we have imposed restrictions on the parameter specification to sustain thebalanced-growth equilibrium. Therefore, a rise in θ increases the habit persistence eﬀect ofconsumption in the adult age but it decreases the degree of habit persistence in the old age:see the utility function (31) . This means that a rise in γ stimulates consumption when theagents are adult so that capital accumulation is lowered. In contrast, a change in γ aﬀects
the degrees of strength of internal and external habit persistence in the old age at the sametime. In section 4.3 we use an alternative functional form of utility function that can treat
those eﬀects separately.
In order to emphasize the diﬀerence between the subtractive and multiplicative forms ofhabit persistence, it is useful to write condition (12) as∙ µ¶¸ 1 µ
¶θ
σ
ht σ
xt+1
x̂t+1
= β αA + γ
,
ĉt
ct
ht+1

(36)

−γ γ−θ
respectively denote the habit-adjusted consumpwhere ĉt = ct c−θ
t−1 and x̂t+1 = xt+1 ct xt

tion in the adult and old ages. A key diﬀerence between (29) and (36) is that in the case ofmultiplicative habits, the marginal rate of substitution between habit-adjusted consumption
in the adult and old ages is not constant over time. Furthermore, the level of relative habit
stocks, ht /ht+1 , also aﬀects the optimal selection between ĉt and x̂t+1 . The qualitative as wellas quantitative divergence in outcomes between the subtractive and multiplicative formationsof consumption habits mainly come from these diﬀerences in the first-order condition.
Two special cases deserve to be mentioned. First, suppose that there is no external habit
formation in the adult age (θ = 0) . Then (34) becomesΩ(γzt+1 + 1)(zt+1 + 1)γ(1−σ)
= 1.
σ
zt+1
This system contains only zt+1 = kt+1 /ct . Since the value of ct is not specified at the beginningof t, the economy, again, always stays on the balanced-growth path. In view of the definition18

of Ω, we find that the steady-state level of z is determined by# 1"1(αA)σ [(1 − α) A]γ(1−σ) z ∗ σ−1=.
z ∗ (z ∗ + 1)γ
βαA(γz ∗ + 1)
It is easy to see that, given our assumption that σ > 1, a rise in γ raises z ∗ satisfying theabove equation. Therefore, as well as in the case of subtractive habits, a larger internal habit
persistence enhances long-term growth if there is no external habit formation.
Next, assume that there is no internal habit formation, so that γ = 0. In this case wecan consider the case of 0 < σ < 1, because the household’s optimization problem involves
an interior solution for all σ > 0 if there is no internal habit persistence. When γ = 0, thesteady-state value of z is explicitly given byhi 1θ(1−σ)+σ,z ∗ = β (αA)1−σ(37)

so that the balanced-growth path is uniquely determine. The dynamic equation for this caseis reduced to´ 1³σ− θ(1−σ)θ(1−σ)zt+1≡ N (zt+1 ) .
zt = β (αA)1−σ(38)

Inspecting (38), we find the following. When σ > 1, it is easy to see that function N (zt+1 )defined in (38) is monotonically increasing and strictly convex function of zt : see Figure 3-a.
Therefore, the unique balanced-growth equilibrium satisfies global stability. An increase in
the degree of external eﬀect, θ, shifts down the graph of N (zt+1 ) , so that the steady-statevalue of zt increases. Since there is positive relation between the steady-state levels of zt andthe balanced-growth rate of income, a higher θ raises the balanced-growth rate. In contrast,
if σ < 1, as depicted by Figure 3-b, the graph of N (zt+1 ) is monotonically decreasing with
N (0) = +∞ and N (+∞) = 0. Thus the steady state is uniquely determined. However, arise in θ produces a downward shift of the graph of N (zt+1 ) and thus the steady-state level ofz decreases, implying that a higher degree of external habit persistence depresses long-term
growth if the intertemporal elasticity of felicity is higher than one.
´1³
1
σIn the steady-state equilibrium of (38) it holds that β (αA)1−σ (z ∗ )θ(1− σ )−1 = 1. weobtain¯¶µ´1
dzt+1 ¯¯
1 ³
1−σ σ ∗θ(1− σ1 )−1β (αA)=θ 1−zdzt ¯zt =z∗
σ
¶
µ
1
.
=θ 1−
σ
19

Consequently, the dynamics system exhibits local stability around the balanced-growth equi¢¡
librium if −1 < θ 1 − σ1 so the
θ
< σ.
(39)
1+θ
Otherwise, the dynamic system is locally unstable. As shown above, the balanced-growth
path is uniquely determined, the economy satisfies global stability if (39) is met. As a result,if θ/ (1 + θ) < σ < 1, then the economy cyclically converges to the balanced-growth path.
We may summarize the above argument as the following proposition:
Proposition 6 If consumption habit formation takes a multiplicative form and if there is nointernal habit persistence, then the balanced-growth path is uniquely given and it is globallystable if and only if
θ
< σ.
1+θ
In addition, σ > 1, a higher degree of external habit formation, θ, raises the balanced-growthrate, while it lowers the balanced-growth rate if θ/ (1 + θ) < σ < 1.
To interpret these results, notice the following facts:sign∂ 2 Ut∂ 2 Ut= sign= sign θ (σ − 1) .
∂ct ∂ct−1∂xt+1 ∂xt

Thus if σ > 1, the marginal utility of private consumption increases with her parent’s consumption levels, that is, the agents are conformists in the sense that they like being similar totheir parents consumption behaviors (keeping up with their parents). In contrast, if σ < 1,then the marginal utility of private consumption decreases with ct−1 and xt so that the agentsare anti-conformist (running away from their parents). The former case promotes consumption (and capital) growth on the balanced-growth path, but in the latter case, the presenceof external habits depresses long-term growth. These results are similar to the comparative
statics results on the balanced-growth path of a representative agent economy with external
consumption habits and an Ak technology: see Carroll et al. (1997 and 2000).

4.3

An Alternative Specification

So far, we have assumed that if there is internal habit persistence, the elasticity of intertemporal substitution in felicity, 1/σ, should be less than unity to yield an interior solution for20

the agents’ optimization problem. To consider the case of 1/σ > 1, we now use the following
utility function:

Ut =

´1−σ
³
ct c−θ
t−1
1−σ

+β

³
´1−σ
xt+1 h−θ
t+1
1−σ

,

σ > 0, σ 6= 1, 0 < θ < 1,

(40)

where
ht+1 = εxt + ηct ,ε > 0, η > 0.

It is easy to see that as long as ε > 0, this function may yield an interior optimum solution
for the household’s consumption plan even for the case that 0 < σ < 1.
In this case the first-order condition for an optimum is given by
µ
¶
¶ µ
¶
µ
ht θ
ĉt −σθxt+1=βαA + η
,x̂t+1
ht+1
εxt + ηct
which yields

µ

ct
xt+1

¶−σ µ

ct−1
εxt + ηct¶−θ(1−σ)−βθηxt+1= βR.
εxt + ηct

(41)

Using zt = kt /ct−1 , xt = αAkt and ct /kt = (1 − α) A/ (1 + zt+1 ) , we can rewrite (41) in thefollowing way:µ¶ct θ(1−σ)α (1 − α) Aβθηzt+1.
= βR +(αAzt+1 )σ αεAzt + η
ct−1
αε (1 + zt+1 ) + η (1 − α)Noting that ct /ct−1 = zt (ct /kt ), we find that the above equation gives∙βR +α (1 − α) Aβθηzt+1
zt = (αAzt+1 )αε (1 + zt+1 ) + η (1 − α)¸∙
1 + zt+1≡ H (zt+1 ) .
×
αεA (1 + zt+1 ) + η (1 − α) Aσ− θ(1−σ)¸

1
θ(1−σ)(42)

The steady-state level of zt satisfies z = H (z) . Since H (.) function is rather complex, it
is more convenient to express the steady-state condition, z = H (z) , for the case of σ > 1 in
the following manner:
I (z) = J (z) ,where¸
αεA (1 + z) + η (1 − α) A,
I (z) = z1+z
¸ 1∙
θ(1−σ)σα (1 − α) Aβθηz− θ(1−σ)J (z) = (αA)βR +.
αε(1 + z) + η (1 − α)σ+θ(1−σ)θ(1−σ)

∙

21

(43)

When σ > 1, we see that I (0) = +∞ and I (+∞) = 0 as well as−σ1

J (0) = (αA) θ(1−σ) (βR) θ(1−σ) > 0,¸ 1∙
θ(1−σ)σ1−α− θ(1−σ)AβθηJ (+∞) = (αA)βR +(< J (0)) .
ε

Furthermore, both I (z) and J (z) monotonically decreases with z. As a consequence, therelation between these two functions are depicted in Figures 4-a which shows that there is atleast one steady-state value of z. It is also seen that a rise in the degree of internal habits, γshifts the graphs of I (z) up and that of J (z) down. Therefore, a higher degree of internal
habits raises the steady-state value of z and, hence, the balanced-growth rate will increase aslong as the balanced-growth path is stable.
As for the case where 0 < σ < 1, it is helpful to express the steady-state condition as¸ 1∙
σ+θ(1−σ)θ(1−σ)σα (1 − α) Aβθηz− θ(1−σ)θ(1−σ)βR += (αA)zαε(1 +¸ z) + η (1 − α)
∙
(44)
1+z
≡ L (z) .
×αε (1 + z) + η (1 − α)The both sides of the above monotonically increases with z and it holds that
¸
∙
σ
1
1
− θ(1−σ)θ(1−σ)> 0,
L (0) = (αA)[βR]
αε + η (1 − α)¸ 1 ∙ ¸∙θ(1−σ)σ1−α1− θ(1−σ)Aβθη(> L (0)) .
L (+∞) = (αA)
βR +
ε
αε

Again, as Figure 4-b demonstrates, there is at least one steady-state level of z. Since in thiscase a rise in γ may shifts down or up the graph of L (z) , its eﬀect on the long-term growthdepends on the parameter values.
This example indicates that in the presence of both internal and external habit persistence, not only the growth eﬀect of outward-looking habit formation but also the growtheﬀect of inward-looking habits may critically depend on whether the agents are conformists(θ (1 − σ) < 0) or anti-conformists (θ (1 − σ) > 1) .

5

Numerical Examples

In this section we examine some numerical examples of the basic models. All the examples
shown below set the following baseline parameter values:
α = 0.3, β = 0.5, A = 5.0
22

Following the standard specification, the income share of capital, α, is assumed to be 0.3.
Since in our three-period lived OLG model, one period may be considered 25 years, so thatthe discount factor β is set at 0.5, which means that the annual discount factor is about´³
0.9726 ' (0.5)1/25 . The magnitude of the total factor productivity, A, is set at 5.0 to makethe balanced-growth rates plausible ones.12

5.1

Subtractive HabitsFirst, consider the model with the subtractive form of habit formation. As shown in Proposition 1, the balanced-growth path is unstable if A (1 − α) < θ1 . Therefore, we focus on thestable balanced-equilibrium for the case of A (1 − α) > θ1 .
Case (i): A change in θ0
When we consider the growth eﬀect of a change in the strength of taste inheritance in theadult age, we assume:θ1 = 0.8,

η = 0.6,σ = 2.0

Then we specify the magnitude of θ0 and calibrate the steady-state value of z ∗ given in (27) .
Substitute these values into (26) , we obtain the following rates of balanced-growth:g ∗ = 3.00213 for θ0 = 0.5,g ∗ = 2.98875 for θ0 = 0.6,
g ∗ = 2.97412 for θ0 = 0.7.
Note that we have assumed that one period is about 25 years, the annual growth rates ofincome are respectively given by 0.04495 (for θ0 = 0.5), 0.04476 (for θ0 = 0.6) , and 0.04456(for θ0 = 0.7) . As shown in Proposition 1, a higher degree of taste inheritance in the old age
depresses the balanced-growth rate.
If we replace σ = 2.0 with σ = 0.8, then the balanced-growth rate respectively becomesg ∗ = 3.00967 for θ0 = 0.5,g ∗ = 2.99627 for θ0 = 0.6,
g ∗ = 2.98165 for θ0 = 0.7.
12

It is to be noted that the gross rate of return to capital is R = αA = 1.5 in this example. Thus the annual

net rate of return is r = (1.5)1/25 − 1 = 0.01635.

23

A higher intetemporal elasticity of substitution in felicity yields a higher balanced-growthrate. The growth eﬀect of a change in θ0 is almost the same as that obtained when σ = 0.2.
Case (ii): A change in θ1
As well as in case (i), we conduct calculation of the balanced-growth rate for alternativevalues of θ1 and σ. In this example, we set θ0 = 0.5. The results are summarized in the tablebelow.
σ = 0.2

σ = 0.8

θ1 = 0.5

2.96158

2.97070θ1 = 0.6

2.996292.98499θ1 = 0.7

2, 98953

2.99745

These figures indicate that a larger impact of taste inheritance in the old age promotes longterm growth of income.
Case (iii) A change in η
We have seen that when there are external habit persistence, the growth eﬀect of a change
in the degree of internal habit persistence, η, is analytically ambiguous. If we assume thatθ0 = 0.5 and θ1 = 0.8, then the balanced-growth rates are given by the following table:
σ = 0.2

σ = 0.8

η = 0.5

2.96870

2.97896η = 0.6

3.003133.00967

η = 0.7

3.10057

3.12544

Hence, in this example, a rise in the degree of internal habit persistence accelerates growthin the long-run equilibrium.

5.2

Multiplicative Habits

In the case of multiplicative form of habit formation, we still assume that α = 0.3, β = 0.5
and A = 5.0. We use (35) to calculate z ∗ and derive the balanced-growth rate g ∗ by use of(26) . The following examples fix the level of σ at 1.2.
Since the model with multiplicative form of habits contains two parameters concerning
the degree of internal and external habit persistence, we calculate the balanced-growth rate,24

g ∗ , for alternative magnitudes of θ and γ
θ = 0.2

θ = 0.5

γ = 0.2

1.2841

1.2621γ = 0.3

1.33711.3126

γ = 0.4

1.39361.3706

γ = 0.5

1.45181.4320In this table θ takes two values, 0.2 and 0.5, while we change γ from 0.2 to 0.5. As Proposition
5 states, a higher degree of external habit persistence (a higher value of θ) depresses the longrun growth rate of income under a given level of γ. Although analytical inspection cannot
specify whether or not a rise in the degree of internal habit, γ, enhances long-run growth,
our example gives a positive relation between γ and g ∗ under a given level of θ.
Comparing with the case of subtractive habits, we see that the balanced-growth rateof the model with multiplicative habits is relatively low. For example, when θ = 0.2, thenet annual growth rates of income are: 0.01005 (for γ = 0.2) , 0.0137 (for γ = 0.3) , 0.01451(for γ = 0.4) and 0.0150 (for γ = 0.5) . This result simply comes from the model specification
and we may obtain higher balanced-growth rates by setting A = 7.0 or 8.0.
Finally, consider the special cases. If there is no internal habit formation (γ = 0) , then thesteady-state value of z is given by (37) . Using this relation to calculate the balanced-growthrate, we find the following:σ = 1.2

σ = 0.8

θ = 0.2

1.1867

1.1392

θ = 0.3

1.1776

1.1523

θ = 0.4

1.1681

1.1648

θ = 0.5

1.1583

1.1769Hence, we see that a rise in the strength of external habit persistence lowers growth if σ > 1,
but it raises the long-run growth rate if 0 < σ < 1.
Next, consider the case where there is no habit inheritance in the adult age, i.e. θ = 0(but there are external habits in the old age). Since we should assume that σ > 1. we obtain25

the following balanced-growth rates under σ = 1.2:
g ∗ = 1.3002 for γ = 0.2
g ∗ = 1.3521 for γ = 0.3,
g ∗ = 1.4066 for γ = 0.4,
g ∗ = 1.4638 for γ = 0.5.
In this example, both internal and external habit persistences in the old age increase. As
shown above, a higher degree of habits persistence in the old age enhances growth.

6Conclusion

This paper has examined how the presence of internal as well as external habit formationaﬀects long-run equilibrium and dynamic behavior of an overlapping generations economy
with endogenous growth. We have shown that if there is only external habit formation,
which represents taste inheritance from parents to children, then it plays the role similar tothat in the representative agent economies. Namely, in the presence of envy (aspiration) andconformism, external habit formation may promote long-run growth, while anti-conformism
would reduce the long-term growth rate of income. In contrast, internal habit formation inan overlapping generations model with endogenous growth presents more complex eﬀects on
the dynamic behavior as well as the steady-state characteristic of the economy. Therefore,
when the consumers have both types habits persistence, the distinction between external
and internal habit persistence would be more relevant in OLG economies than in the representative agent counterpart where both types of habits generally yield the same type of
impact on the economy. Additionally, we have confirmed that the dynamic behavior andsteady-state conditions of an OLG economy is highly sensitive to whether habit stocks are
introduced into the felicity function in subtractive or multiplicative form. Since functional
forms of habit formation generally yields quantitative eﬀects alone in the representative-agent
models, our findings have demonstrated that the how the preference structure is formulated
would be critical in examining overlapping generations economies with habit persistence in
consumption.
In this paper we have focused on the issue how the alternative formulations of habit
26persistence of consumption may yield diﬀerent outcomes in an endogenously growing economy.
Our next task is to investigate policy implications in growing OLG economies in the presenceof alternative forms of habit formation.

27

References[1] Acemoglu, D. (2009), Introduction to Modern Economic Growth, MIT Press, Cambridge,
MA.
[2] Alonso-Carrera, J., Caballé, J. and Raurich, X., (2005), "Growth, Habit Formation andCatching-up with the Joneses", European Economic Review 49, 1165-1691.
[3] Alonso-Carrera, J., Caballé, J. and Raurich, X., (2007), "Aspirations, Habit Formation,
and Bequest Motive", Economic Journal 117, 813-836.
[4] Alvarez-Cuadrado, F., Monteiro, G. and Turnovsky, S.J., (2004), "Habit Formation,
Catching-up with the Joneses, and Economic Growth", Journal of Economic Growth 9,
47-80.
[5] Bossi, L. and Gomis-Porqueras, P. (2009), "Consequences of Modeling Habit Persistence", Macroeconomic Dynamics 13, 349-365.
[6] Bunzel, H., (2006), "Habit Persistence, Money and Overlapping Generations’, Journalof Economic Dynamics and Control 30, 2425-2445.
[7] Carroll, C.D., Overland, J. and Weil, D. (1997), "Consumption Utility in a Growth
Model", Journal of Economic Growth 2, 339-367.
[8] Carroll, C.D., Overland, J. and Weil, D. (2000), "Saving, Growth with Habit Formation",
American Economic Review 90, 341-355.
[9] de la Croix, D. (1996), "The Dynamics of Bequeathed Tastes", Economics Letters 53,
89-96.
[10] de la Croix, D. and Michel, P., (1999), "Optimal Growth when Tastes are Inherited",
Journal of Economic Dynamics and Control 23, 519-537.
[11] de la Corix, D. (2001), "Growth Dynamics and Education Spending: the Role of Inherited Tastes and Abilities", European Economic Review 45, 1415-1438.
[12] Diamond, P. (1965), "National Debt in a Neoclassical Growth Model" American Economic Review.55, 1126-1150.
28

[13] Grishchenko, O.Y. (2007), "Internal vs. External Habit Formation: The Relative Importance for Asset Pricing", unpublished manuscript, Department of Economics, Pennsylvania State University.
[14] Gómez, M. A. (2008), "Convergence Speed in the Ak Endogenous Growth Model with
Habit Formation", Economics Letters 100, 16-21.
[15] Harbaugh, R. (1996), "Falling behind the Joneses: Relative Consumption and theGrowth-Savings Paradox", Economics Letters 53, 297-304.
[16] Hiraguchi, R. (2008), "The Overlapping Generations Model with Habit Formation: A
Comment", forthcoming in Journal of Economic Dynamics and Control.
[17] Lahiri, A. and Puhakka, M. (1998), "Habit Persistence in Overlapping Generations
Economies under Pure Exchange", Journal of Economic Theory 78, 176-186.
[18] Mino, K. (2007), "Consumption Externalities and Capital Accumulation in an Overlapping Generations Economy", unpublished manuscript.
[19] Mino, K. (2008), "Growth and Bubbles with Consumption Externalities", Japanese Economic Review 59, 33-53.
[20] Romer, P. M (1986), "Increasing Returns and Long-Run Growth", Journal of Political
Economy 94, 1002-1037.
[21] Ryder, H. E. and Heal, G.M. (1973), "Optimal Growth with Intertemporally Dependent
Preferences", Review of Economic Studies 40, 1-31.
[22] Uribe, M. (2002), "The Price-Consumption Puzzle of Currency Pegs", Journal of Monetary Economics 49, 533-569.
[23] Wendner, R. (2002), "Capital Accumulation and Habit Formation", Economics Bulletin
4, No. 7, 1-10.
[24] Wendner, R. (2003), "Do Habits Raise Consumption Growth", Research in Economics
57, 151-163.

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