Discussion Papers In Economics
And Business

A Variety-Expansion Model of Growth with External Habit
Formation

Junko Doi and Kazuo Mino
Discussion Paper 06-07

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANA Variety-Expansion Model of Growth with External Habit
Formation

Junko Doi and Kazuo Mino
Discussion Paper 06-07

February 2006

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A Variety-Expansion Model of Growth with External Habit
Formation∗
Junko Doi†and Kazuo Mino‡
February 14, 2006

Abstract
This paper introduces external habit formation into one of the basic models of endogenous growth in which continuing expansion of product variety sustains long-term growth.
We assume that households consume a range of final goods and they set a benchmark
level of consumption for each good. The benchmark consumption is determined by external habit formation so that there are commodity-specific external eﬀects. Each goodis produced by a monopolistically competitive firm and the firm’s optimal pricing decision exploits the fact that consumers’ demand is subject to the external habit formation.
Given those settings, we show that the introduction of consumption externalities may
aﬀect the balanced-growth characterization, transitional dynamics as well as policy impacts in fundamental manners.
JEL Classification code: E2, O3, O4
Keywords: consumption externalities, habit formation, monopolistic competition, R&Dbased growth model∗

We thank Hideyuki Adachi, Masahiro Ashiya, Elias Dinopoulos, Ryo Horii, Yasusada Murata, Tamotsu

Nakamura, Paul Segerstrom and seminar participants at various occasions for their helpful comments on earlier
versions of this paper. The first and the second authors are financially supported by Grant-in-Aid for Scientific
Research No.17730139 and No.1653017, respectively.
†
Kyoto Sangyo University, Motoyama, Kamigamo, Kita-ku, Kyoto, 603-8555, Japan (phone: 81-75-7051768, e-mail: jdoi@cc.kyoto-su.ac.jp)‡
Corresponding Author, Graduate School of Economics, Osaka University, 1-7 Machikaneyama, Toyonaka,
Osaka 560 0043, Japan. E-mail: mino@econ.osaka-u.ac.jp.

1

Introduction

In the recent development of growth economics, external eﬀects in production and knowledge
creation have played a pivotal role. Seminal contributions to the endogenous growth theory
such as Aghion and Howitt (1992), Lucas (1988) and Romer (1986 and 1990) have emphasized
the eﬀects of production externalities and knowledge spillovers that may sustain persistentlong-term growth. In contrast, the role of externalities in consumption activities has been aminor topic in growth theory. While consumption externalities have been taken seriously by
the researchers in the fields of asset pricing, business cycles and fiscal policy (for example,
Abel 1990, Galı́ 1994 and Ljungqvist and Uhlig 2000), growth economists in general havenot paid much attention to the presence of external eﬀects in the consumption side of theeconomy.
The recent contributions by Alonso-Carrera et al. (2004 and 2005), Carroll et al. (1977and 2000), Alvarez-Cuadrado et al. (2004) and Liu and Turnovsky (2005), however, have kindled a renewed interest in the role of consumption externalities in growing economies. AlonsoCarrera et al. (2004 and 2005), Alvarez-Cuadrado et al. (2004) and Liu and Turnovsky (2005)
examine implications of consumption external eﬀects in the standard neoclassical growthmodels. The main research concern of those authors is to explore the eﬀects of interdependency among consumers on welfare and transitional dynamics of the economy. Carroll et al.
(1997 and 2000) examine the roles of external as well as internal habit formation in an endogenous growth model with an Ak technology and analyze how the presence of consumption
externalities aﬀects savings and the pattern of growth.1 Those studies have clearly demonstrated that external eﬀects of consumption may have significant implications for growing
economies in both qualitative and quantitative senses.
Departing from the existing studies on the eﬀects of consumption externalities in growing
economies, we investigate the role of consumption external eﬀects in the context of a varietyexpansion model of growth. The analytical framework of this paper is based on Grossman
and Helpman (1991, Chapter 3). In our setting, there are a variety of consumption goods
and each commodity is produced by a monopolistically competitive firm. The range of1

Harbaugh (1996) also discusses the relation between growth and saving in the presence of consumption

externalities by using a two-period model with uncertainty.

1consumption goods variety is enhanced by R&D activities.2 Those assumptions enable us
to introduce two distinctive features of consumption external eﬀects that have not been
considered in the existing literature assuming a homogenous consumption good and perfectcompetition. First, we may assume that consumers set a benchmark consumption levelfor each good. The benchmark level of each consumption good is determined by external(outward-looking) habit formation so that there exist commodity-specific external eﬀects.
Second, since each commodity is produced by a monopolist, the firm may exploit the fact that
consumer’s demand for its own product is aﬀected by the commodity-specific external eﬀect.
This means that the firm can internalize the consumption external eﬀect when maximizing
its profits. As a result, the firm’s marginal cost involves the implicit ’internalization costs’of the consumption external eﬀects, and hence the pricing decision of the firm is aﬀectedby the behavior of benchmark level of consumption set by the consumers. The basic ideaof this kind of modelling has been proposed by Ravn et al. (2002 and 2006) who examine
the eﬀects of commodity-specific consumption externalities in a real business cycle modelwith monopolistic competition.3 In this paper, we consider the implications of consumption
external eﬀects in an imperfectly competitive, growing economy.
Given the analytical framework described above, we explore the balanced-growth equilibrium and transitional dynamics. We find that the presence of consumption externalities
may yield significant eﬀects on the balanced-growth-path characterization as well as on equilibrium dynamics of the model economy. First, if there are negative external eﬀects, that is,each consumer’s felicity is negatively related to the benchmark level of average consumptionof the economy at large, then there generally exists a unique, feasible balanced-growth paththat satisfies saddlepoint stability. In contrast, if the externalities positively aﬀect the individual felicity, the local behavior of the economy around the balanced-growth equilibrium
exhibits either saddlepoint stability or local indeterminacy. Additionally, if both cases establish saddle stability, the behaviors of key variables such as the rate of technical change
may be diﬀerent depending on whether external eﬀects are negative or positive. Second, the2
3

See Gancia and Zilbotti (2005) for a detailed survey on this class of models.
Since Ravan et al. (2002 and 2006) explore real business cycles in the context of a stochastic dynamic

general equilibrium framework, their discussion riles on a model calibration. In contrast, we use a simpler
deterministic, continuous-time model of growth, which enables us to study the behavior of the model economy
analytically.

2

policy implications obtained in our model can be quite diﬀerent from those established in
the original Grossman and Helpman model. For example, in our framework a policy that
stimulates R&D investment does not necessarily promote long-term growth. In addition, due
to the presence of consumption external eﬀects, the level of R&D spending determined in
the competitive equilibrium may not be lower than its optimal level that attains the eﬃcient
resource allocation.
The reminder of the paper is organized as follows. Section 2 constructs the analyticalframework. Section 3 derives a complete dynamic system. Section 4 examines the balancedgrowth equilibrium and investigates equilibrium dynamics out of the steady state. Section 5
considers the eﬀects of R&D subsidy and the socially optimal level R&D spending. A brief
conclusion is given in Section 6.

2

The Model

2.1

Households

There is a continuum of identical households whose number is normalized to one. The
representative household consumes a variety of consumption goods, ranging from index 0 to
n. We assume that the consumer’s felicity depends not only on her own consumption of eachgood but also on the benchmark level of consumption that is determined by outward-looking
habit formation. The instantaneous sub-utility of the household is given byαµZ n ³´ α−1 ¶ α−1α−θdi, θ < 1, α > 1,
ci si
C=

(1)

0

where ci is consumption of good i ∈ [0, n] and α denotes the elasticity of substitution between
consumption goods. Here, si is the household’s benchmark level of consumption of good i,
which represents a commodity-specific external habit formation. More specifically, we assume
that si is a weighted sum of the average consumption of good i up to the present period:
Z t
si (t) = β
e−β(τ −t) c̄i (τ ) dτ , β > 0,−∞

where c̄i (τ ) denotes the average consumption of good i in the economy at large. This gives
the following dynamic equation of si :ṡi = β (c̄i − si ) ,3

β > 0.

(2)Note that the instantaneous utility of consumption of good i can be written asµ ¶θ−θ1−θ ci; θ 6= 0, θ < 1.
ci si = cisiThis shows that the felicity obtained by consuming the i-th good depends on the relative
consumption, ci /si , as well as on the absolute level of consumption, ci . While it is usuallyassumed that θ has a positive value, in this paper we do not specify the sign of θ. If θ is positive,a rise in the benchmark consumption, si , negatively aﬀects the felicity of consumer. Namely,
each consumer’s preference exhibits jealousy as to consumption of others. In contrast, if θ
is negative, then the felicity of consumer increases with the benchmark consumption. In
this case consumers’ preferences show admiration for consumption of other members in thesociety.4 It is also to be noted that if β = +∞, then si = c̄i so that the external eﬀects
are only intratemporal. In addition, if θ = 0, then each consumer’s preference becomes thestandard one in which her felicity depends on the absolute levels of private consumption
alone.
Given (1) , the households maximizes a discounted sum of subutilities
Z ∞e−ρt log Cdt, ρ > 0,U=0subject to the flow budget constraint:ȧ = ar + wN −Z nci pi di,(3)0where a denotes the asset holding of the household, w is the real wage rate and pi denotes theprice of consumption good i. We assume that in each moment the representative household
supplies N units of labor inelastically. Notice that the habit formation is external to anindividual household, so that when deciding her optimal plan, the household takes the future
sequence of benchmark consumption, {s (t)}∞t=0 , as given.

and p̂i = pi sθi , we first consider the following cost minimization
Denoting ĉi = ci s−θiproblem:minZ nĉi p̂i di0subject to

µZ n
0

4

α−1
α

ĉi

¶ α
α−1
di
= C.

See Dupor and Lin (2003) for a useful taxonomy as to formulating consumption externalities.

4

Solving this problem gives the demand function such that
Ã !α
P̂
ĉi =
C,
p̂i
where
P̂ ≡

µZ n
0

¶ 1

1−α

p̂1−α
di
i

(4)

denotes a price index of the subutility (aggregate consumption), C. The demand equation ofgood i is thus given byθ(1−α)

ci = si

Ã

!α

P̂
pi

C.

(5)

This equation states that given prices and the composite consumption, C, the demand forgood i decreases with si if θ > 0. When θ < 0 (so that each consumer has admiration forother consumers), a higher si increases ci .
Now defineE ≡ minThis yields

Z nα

ci pi di = P̂ C

0

Z n
0

θ(1−α) 1−αpi di = P̂ C.

si

C = E/P̂ .

(6)

As a result, the optimization problem can be expressed asZ ∞hie−ρt log E − log P̂ dt
max
0

subject toȧ = ra + wN − E.
The optimization conditions for this problem gives the Euler equation,Ė= r − ρ,E

(7)

together with the transversality condition:lim (a/E) e−ρt = 0.

t→∞

As for choosing a numeraire, we follow Grossman and Helpman (1991). We normalize
prices so that nominal spending, E, remains constant over time. Thus, by setting E = 1 for
all t ≥ 0, from (7) the real interest rate, r, equals the time discount rate in every moment:r = ρ.
5

(8)

2.2

Producers

Each consumption good is produced by a monopolistically competitive firm. The profits ofthe firm producing consumption good i are given by
π i = pi ci − wbci ,b > 0.

The firm produces by using labor alone and the production function of good i is assumed tobe ci = (1/b) li , where li is labor devoted to production of the i-th good. From the demand
function (5), π i becomesθ(1−α)

π i = si£¤
− wbp−α.
P̂ α C p1−α
i
i

(9)

Following Ravn et al. (2006), we assume that the firm exploits the fact that consumers’
demand behavior is aﬀected by the benchmark consumption level, si , and that si changes
according to (2) . This means that the firm maximizes a discounted sum of its profits over an
infinite-time horizon subject to (2) . The optimization behavior of the firm is thus formulated
as follows:
max

Z ∞
0

subject to (9) and

µ Z t
¶
exp −
r (ξ) dξ π i (t) dt0ihθ(1−α) α−sṡi = β si
P̂ Cp−α
i ,
i

(10)

where si (0) is given. In this problem, the firm’s control and state variables are pi and si ,respectively.
To derive the optimization conditions, let us set up the following Hamiltonian function:θ(1−α)

Hi = siih£¤θ(1−α) α
−α
−α
−
wbp
β
s
Cp
−
s
+
λ
P̂ α C p1−α
P̂
i
i ,
i
i
i
i

where λi is the shadow value of the benchmark consumption level, si . Maximizing the Hamiltonian function with respect pi , we obtain−α−1− λi βαp−a−1= 0.
(1 − α) p−α
i + bwαpiiThis yields the optimal pricing formula in such a way thatpi =α(bw − βλi ) .
α−1
6

(11)

Equation (11) means that the price of good i equals the marginal cost of labor input, bw,
plus the shadow cost of habit formation, −βλi , multiplied by a coeﬃcient, α/ (α − 1) . Since
production needs labor alone and λi is an implicit cost for the firm, in the conventional
expression (11) may be written as
α
pi =
α−1

µ
¶
βλi
1−
bw.
bw

Therefore, if we take into account of the explicit labor costs, bw, alone, the markup ratio³´αiis represented by α−1
1 − βλso that it changes with the relative costs, (−βλi ) /bw. In
bw
the standard formulation without consumption externalities, the markup formula is given bypi = αwb/ (α − 1) , which has a constant markup rate, α/ (α − 1). The endogenous markup
ratio in our setting will be one of the sources that make the analytical results diverge fromthose obtained in the original Grossman and Helpman model.
The shadow value λi changes according toλ̇i = rλi −∂Hi∂siθ(1−α)−1= (r + β) λi − θ (1 − α) si£¤− bwp−α+ λi βp−a.
P̂ α C p1−α
i
i
i

(12)

The solution of (12) is expressed as
µ Z τ
¶
Z ∞½
exp −
(r (ξ) + ρ)dξ θ (1 − α) si (τ )θ(1−α)−1 P̂ α (τ ) C (τ )λ (t) =tthio
× pi (τ )1−α − bw (τ ) pi (τ )−α + λβpi (τ )−a dτ .
θ(1−α)−1

Since α > 1, the sign of θ (1 − α) si¤£
is negative (resp.
− bwp−α+ λβp−aP̂ α C p1−αiiipositive) if θ > 0 (resp. θ < 0).5 Hence, λ (t) ≤ 0 for all t ≥ 0 if θ > 0, while λ (t) ≥ 0 for allt ≥ 0 if θ < 0. If the firm i sells an additional unit of product, then an increase in consumptionof good i raises the benchmark consumption, si , which represents the weighted average of
5Note that from the first-order condition+ bwαp−α−1− λi βαp−a−1= 0.
(1 − α) p−αiiiHence, we see that=− bwp−α+ λβp−a
p1−α
i
i
i
¡ −α
¢
= p1−α(1/α) pi αpi − αbwp−α−1+ λi αβp−a−1/α > 0.
iii

7

past consumption of good i in the economy at large. When θ > 0, such an increase in si will
lower the future consumption demand for good i. Therefore, an increment in production ofgood i yields two types of additional costs: the marginal cost of labor employment, bw, andthe marginal penalty cost, −βλi , that counts the expected reduction of future consumption
demand for good i due to the marginal increase in si . In contrast, when θ < 0, a rise in si has
a positive impact on the future consumption demand. Thus the marginal cost of production
equals the marginal labor cost minus the marginal benefit of production expansion, βλi (> 0) .
Notice that from (11) we obtain pi − bw + βλ = pi /α. Thus (12) is written as¶µ1θ(1−α)−1 α− 1 si.
λ̇i = (r + β) λi − θP̂ Cp1−α
i
α

2.3

(13)

R&D

The research and development sector is assumed to be competitive. R&D activities enhancevariety of consumption goods by using labor. The production function of the R&D firm isgiven byṅ = δLR n,

δ > 0,(14)

where LR denote labor input for R&D activities.6 Denoting the patent price by v, we seethat the zero-excess-profit condition for the R&D sector, i.e. v ṅ − wLR = 0, gives
w = δnv.

(15)

We assume that the patent length is infinite. Since in the monopolistically competitive final
good markets the zero-excess-profit condition holds, the patent price paid by the monopolist
is equal to the discounted present value of its profits:
µ Z τ
¶
Z ∞
exp −
r (s) ds π i (τ ) dτ ,v (t) =tt

where r is the real interest rate. The above condition yields
πi
v̇
=r− .
v
v
6

(16)

As usual, we assume that n in the right-hand side of (14) represents external spilliovers of the existingknowledge.

8

2.4

Labor Market Equilibrium Condition

We have assumed that each household supplies a fixed amount of labor, N, in each moment.
Since the number of households is normalized to one, N also expresses the aggregate labor
supply. Thus the full employment condition for labor labor is
LR + Lf = N,

(17)

where Lf is the total labor used for consumption goods production:
Z n
Lf =li di.
0

3

Dynamic System

3.1

Symmetric Equilibrium

In order to make our model analytically tractable, we focus on the symmetric equilibrium inwhich the following conditions are fulfilled:
ci = c, pi = p, si = s, λi = λ for all i ∈ [0, n] .

(18)

In our product-variety expansion model of growth, we assume that in each moment newly
invented goods are introduced into the market. Suppose good i is created at period t −ξ (ξ > 0) . Then the level of s at period t isZ te−βτ c̄i (τ ) dτ + si (t − ξ) ,si (t) = βt−ξwhere the initial value of si (t − ξ) is exogenously given. Thus if we set si (t − ξ) = 0,
consumption goods have a vintage structure, so that the symmetric equilibrium will not hold.
To avoid analytical complexity, we assume that in the symmetric equilibrium the following
holds:si (t − ξ) = s (t − ξ) , for all i ∈ [0, n (t − ξ)] and for all ξ ≥ 0.
In words, in the symmetric equilibrium each consumer sets the same amount of benchmark
consumption for every good, regardless of its timing of introduction into the market. In
the Dixit-Stiglittz type of variety-expansion model, the key to the consumer utility is thenumber of goods available rather than the character of each good, because in the symmetric9equilibrium the diﬀerence in character of each good disappears. Our assumption of symmetrictreatment of the benchmark consumption is, therefore, plausible one.
If we assume that si = s, it also holds that λi = λ for all i ∈ [0, n] . Hence, the prices
are the same for all goods, pi = αbw/ [bw − βλ] = p. Additionally, due to the normalizationof the number of households, in equilibrium the instantaneous level of average consumption
satisfies that c̄i = c for all i ∈ [0, n] . As a consequence, the dynamic equation of benchmark
consumption becomesṡ = β (c − s) .

(19)

Using the symmetric conditions, we find that (4) and (5) respectively yield:
1

1

α

α

P̂ = n α−1 p̂ = n 1−α sθ p,
C = n α−1 ĉ = n α−1 s−θ c.

(20)
(21)

Due to the normalization, P̂ C = E = 1, we obtainP̂ C = pnc = 1.
Thus it holds thatwhich gives

Hence , (13) is written as(22)

Ã !α−1 Ã !α−1
P̂
P̂
=
,
P̂ α Cp1−α = P̂ C
p
p
³ 1
´α−1
P̂ α Cp1−α = n 1−α sθ= n−1 sθ(α−1) .
λ̇ = (r + β) λ − θ

µ

¶
1
1
−1
.
α
sn

(23)

In the symmetric equilibrium the markup formula (11) is written asp=α(bw − βλ) .
α−1

(24)

This gives the profits of consumption good producers:¤£
π = sθ(1−α) P̂ α C p1−α − wbp−α∙¸
wb (α − 1)
1
1−
.
=
n
α (bw − βλ)

10

(25)

3.2

A Complete Dynamic System

The production function of each firm is given by bc = l so that bcn = Lf . Therefore, (22)yieldsb
Lf = .
p
Namely, because of the nomalization, E = 1, the aggregate labor employment for final goods
production decreases with the consumption good price, p. Hence, using (24) , we obtain
Lf =b (α − 1).
α (bw − βλ)Combining (14) , (17) and (26) , we can derive:¸∙ṅb (α − 1)= δ (N − Lf ) = δ N −.
nα (bw − βλ)From (2) the dynamic behavior of s is shown by¸∙¸∙
wb (α − 1)
1
−s =β
−s .
ṡ = βnpnα (bw − βλ)Now denote sn ≡ x. Then (19) , (27) and xẋ = sṡ + nṅ give∙¸∙¸
wb (α − 1)b (α − 1)ẋ=β−1 +δ N −.
xxα (bw − βλ)α (bw − βλ)

(26)

(27)

(28)

(29)

Substituting (8) and (25) into (16) yieldsv̇δδb (α − 1)
=ρ− +
.
v
w α (bw − βλ)(30)Hence, in view of (15) and (27) , we obtain the following:
ẇ
ṅ v̇
δ
= + = ρ + δN − .
w
n v
w

(31)

Finally, from (23) the implicit price of the benchmark consumption changes according to
¶
µ
1
1
−1
.
(32)
λ̇ = (ρ + β) λ − θαxTo sum up, we have derived a complete dynamic system consisting of (29) , (31) and (32)that describe the dynamic motions of, x (= ns) , w (= δvn) and λ. In words, our derived
system depicts the behaviors of the aggregate level of benchmark consumption x (= ns), theaggregate value of knowledge (vn = w/δ) and the shadow value of the benchmark consumption, λ.
114Balanced-Growth and Equilibrium Dynamics

4.1

Existence of the Balanced-Growth Equilibrium

In the balanced-growth equilibrium, x, w and λ stay constant over time. Hence, it holds that
∙
¸
ċ
v̇
ṅ
b (α − 1)ṡ= = = − = −δ N −
< 0,
s
c
v
n
α (bw∗ − βλ∗ )where w∗ and λ∗ denotes the steady-state values of of w and λ, respectively. Because of
normalization, in the balanced-growth equilibrium where n grows at a constant rate, s, c andv contract at the rate of −ṅ/n. In addition, p and Lf stay constant on the balanced-growthpath. Note that in the symmetric equilibrium we haveα1

C = n α−1 s−θ c, P̂ = n 1−α sθ p.
Thus we see that the aggregate consumption changes according to
∙
¸µ ¶
1
ṅ
Ċ
=
+θ
,
C
α−1
n
and that the rate of change in price index is given by
¸µ ¶
∙
ṅ
1
dP̂ /dt
+θ
.
=−
α−1
n
P̂
Since we have set E = 1, the instantaneous utility equals − log P̂ . Therefore, welfare expansion requires that
−

1
< θ.
α−1

When θ > 0, this condition is always satisfied. In what follows, we assume that this condition
holds for the case of θ < 0 as well.
Condition ẇ = 0 gives the steady-state level of the real wage rate:
w∗ =

δ
.
δN + ρ

(33)

This shows that the real wage on the balanced-growth path increases with the R&D eﬃciency,
δ, and decreases with the total labor supply, N. Remember that from (15) the real wage ratesatisfies w = δvn. Thus (33) indicates that the steady-state value of total net wealth, vn, is
1/ (δN + ρ) .

12

The steady-state values of x and λ are obtained by setting λ̇ = 0 and ẋ = 0 in (29) and(32) , respectively. Those conditions are given by¸∙¸∙
wb (α − 1)b (α − 1)−1 +δ N −= 0,βxα (bw − βλ)α (bw − βλ)x=θ(1 − α).
(ρ + β) αλ

(34)

(35)

Substituting the steady-state level of real wage w∗ given by (33) into (34) and using (35) , wefind that the steady-state values of x and λ are respectively given byx∗ =(α − 1) β [δb (ρ + β) + θ(δN − β) (δN + ρ)],(ρ + β) δb [(α − 1) (δN + ρ) − (δN − β)α](36)θδb [α(δN − β) − (α − 1) (δN + ρ)].
αβ [δb (ρ + β) + θ(δN − β) (δN + ρ)]

(37)

λ∗ =

In order to define a feasible steady state, the parameter values should satisfy certain conditions. First, suppose that θ > 0. Since λ∗ < 0 in this case7 , we should assume the following
conditions:α(δN − β) − (α − 1) (δN + ρ) < 0,(38)

δb (ρ + β) + θ(δN − β) (δN + ρ) > 0.

(39)

Notice that under (38) and (39) , λ∗ has a positive value for θ < 0, and hence (38) and (39)ensure the feasibility conditions for the case of θ < 0 as well.
The balanced-growth rate is given by
g=

¢
¡
ṅ
= δ N − L∗f ,nwhere L∗f denotes the steady-state value of labor devoted to consumption goods production.
Since L∗f = b (α − 1) /α (bw∗ − βλ∗ ) , from (33) and (37) we obtainL∗f =(α − 1) (δN + ρ) [δb (ρ + β) + θ (δN − β) (δN + ρ)]ih.
δ αδb (ρ + β) + θ (α − 1) (δN + ρ)2(40)

The above demonstrates that, if θ > 0, the steady-state value of Lf is positive under (39) .
When θ < 0, we should assume thatαδb (ρ + β) + θ (α − 1) (δN + ρ)2 > 0
7

If θ > 0 and λ∗ < 0, then x∗ > 0 from (35) .

13

(41)

to make L∗f positive. In addition, it is easy to confirm that N > L∗f if (38) , (39) and (41)are fulfilled. Hence, L∗f satisfies the feasibility condition such that 0 < L∗f < N.
Summing up the above discussion, we have shown:
Proposition 1 In the case of negative consumption externalities (θ > 0) , the economy has
a unique, feasible balanced-growth path, if the parameter values satisfy−(α − 1) (δN + ρ)δb (β + ρ)
< δN − β <.
θ (δN + ρ)αIn the case of positive consumption externalities (θ < 0) , the presence of a unique and feasible
balanced-growth path is ensured if
δN − β < min½(α − 1) (δN + ρ)
δb (ρ + β), −αθ (δN + ρ)2

and αδb (ρ + β) + θ (α − 1) (δN + ρ)2 > 0.

4.2¾

The Long-Run Growth Rate

By use of (33) and (37) , we may express the balanced-growth rate as a function of given
parameters:g = δN −(α − 1) (δN + ρ) [δb (ρ + β) + θ (δN − β) (δN + ρ)].
αδb (ρ + β) + θ (α − 1) (δN + ρ)2(42)This demonstrates that the balanced-growth rate depends on all the parameters involved in
the model. If there is no consumption external eﬀect, i.e. θ = 0, the balanced-growth rate
determined by (42) is reduced toĝ = δN −δN(α − 1) (δN + ρ)
=
− ρ.
α
α

(43)

As (43) shows, the balanced-growth rate in the standard model with product-variety expansion increases with the labor supply, N, and the eﬃciency of R&D, δ, while it decrease withthe elasticity of substitution among consumption goods, α, and the time discount rate, ρ. Intuitive implications of these comparative statics results in the standard setting have been wellunderstood: see, for example, Chapter 3 in Grossman and Helpman (1991). In contrast tothese simple results in the standard model, (42) shows that the eﬀects of parameter changes
on the long-term growth rate are rather complex in the presence of external habit formation.

14

First, compare the balanced-growth rate given by (42) with that determined by (43) . To
do so, it is helpful to remember that the balanced-growth rate in the presence of consumption
externalities is written asg=δ¡

N − L∗f

¢

∙
=δ N−

¸
α−1
,
αw∗ − (αβ/b) λ∗while in the absence of external eﬀect it is expressed as
¶
µ
α−1
,
ĝ = δ N −αw∗where w∗ = δ/ (δN + ρ) . Since the steady-state level of real wage is independent of θ, if
there are negative consumption externalities (θ > 0) so that λ∗ has a negative value, then theaggregate employment for final goods production, Lf , is smaller than that in the absence ofconsumption externalities (θ = 0) . As a result, other things being equal, the balanced-growthequilibrium with negative consumption externalities may attain a higher growth rate thanthat realized in the standard setting without consumption externalities. In contrast, if there
are positive consumption external eﬀects (θ < 0) , the steady-state rate of Lf is larger thanthat in the case of θ = 0. Hence, the introduction of positive consumption external eﬀects
has a negative impact on long-term growth.
Economic interpretations of these results are rather obvious. For example, consider thecase of negative consumption externalities. If there are negative externalities in consumption,a higher growth of consumption demand will enhance the social level of stock of habits, which
in turn depresses the future consumption demand and thus future profits of firms. Since eachfirm correctly anticipates such an eﬀect of social habit formation on the consumers’ decision, it
has an incentive to set a higher price in order to slow down the growth of habit accumulation.
Consequently, in the symmetric equilibrium the aggregate consumption demand will decline
and thus the total labor devoted to final goods production decreases. This means thatlabor will shift from the production activities to R&D sector, which accelerates the longterm growth. In the case of positive consumption externalities, the exposition given above iscompletely reversed. We have thus shown:
Proposition 2 Other things being equal, the economy with negative consumption externalities attains a higher balanced-growth rate than the economy without externalities. In contrast,15

if there are positive consumption externalities, the balanced-growth rate is lower than that sustained by the economy without externalities.
It is worth emphasizing that in our setting the growth eﬀects of changes in other parameters would be also diﬀerent from those obtained in the standard modelling. For instance,
consider the eﬀect of a change in the level of labor supply, N. As was shown by (43) , anincrease in N raises the balanced-growth rate, if θ = 0. In this case a higher N depresses
the steady-state level of real wage, w∗ , which has a negative impact on growth. At thesame time, a rise in labor supply stimulates technical progress, because it allows the R&D
sector to employ a larger amount of labor. In the standard model without consumption externalities, the latter eﬀect dominates the former, so that a higher labor supply accelerates
growth. That is, there exists a scale eﬀect in the presence of knowledge externalities in the
R&D sector. If there are consumption externalities, there is an additional eﬀect generated
by a change in N : an increase in N may change the value of λ∗ (see equation (37)) . Moreprecisely, it is seen that when θ > 0, the eﬀect of a raise in N on the magnitude of λ∗ isambiguous. If θ < 0, then an increase in N raises λ∗ . This produces an additional increase inL∗f (= b (α − 1) /α (bw∗ − βλ∗ )) . If this increase in L∗f is large enough to hold dL∗f /dN > 1,
then a larger labor supply lowers the balanced-growth rate: we may have an anti-scale eﬀect
even though there are knowledge spillovers in the R&D sector.
Similarly, if θ 6= 0, the relationship between the time discount rate and the balancedgrowth rate should be reconsidered. In the absence of consumption externalities, a higherρ decreases the steady-state rate of real wage, w∗ , which increases the labor input for final
goods production. Hence, the balanced growth rate will decline. If θ > 0, (37) states thatan increase in ρ lowers the absolute value of λ∗ . Therefore, we obtain dL∗f /dρ > 0, so thatthe balanced growth rate decreases. However, if θ < 0, it is seen that we cannot determine
the sign of dL∗f /dρ without imposing further constraints on the magnitudes of the parametervalues.

4.3

Equilibrium Dynamics

To examine the stability property of our dynamic system, it is to be noted that the dynamic
behavior of w given by (33) is independent of other variables and it is completely unstable.

16Since the initial value of w is not specified in the perfect-foresight equilibrium, the unusable
behavior of w means that it always holds that w = w∗ . Consequently, we may focus on the
dynamics of x and λ under the fixed level of w = w∗ . Keeping in mind that the predeterminedvariable in our system is x (= sn) alone, we see that there is a unique stable converging patharound the balanced-growth equilibrium if the dynamic system consisting of (29) and (32)exhibits a saddle-point property.
Linearizing (29) and (32) around the steady state where ẋ = λ̇ = 0, we obtain:
⎛
⎞
⎛ ⎞
∗
x−x
ẋ
⎠,
⎝ ⎠=J⎝
∗
λ−λ
λ̇

where

J≡

⎛

αβb (α − 1) ⎝(xα (bw∗ − βλ))2− (bw∗ − βλ) w∗¢
θ α − 1 x12¡1(βw∗ − δx)x
ρ+β

⎞

⎠.

The sign of the determinant of the coeﬃcient matrix J satisfies the following:
¶
µ
¾
½
1
1
∗
∗
∗
−1
(βw − δx)sign det J = sign [− (bw − βλ) w ] (ρ + β) − θ
α
x
½ ∙
¸¾
α (ρ + β) bδ + θ(δN + ρ)2 (α − 1)
= sign −δ.
(δN + ρ)2 αHence, if θ > 0, then det J < 0 so that the linearized dynamic system exhibits a saddle-pointproperty. Additionally, it is easy to see that if (41) holds, the system with a negative θ alsosatisfies the saddlepoint stability. We have thus shown the following:
Proposition 3 When (38) , (39) and (41) are held, the balanced-growth path satisfies local
saddle-point stability.
Figures 1 (a), (b) and (c) depict typical phase diagrams of the dynamic system. Figures 1(a) and (b) assume that conditions (38), (39) and (41) are fulfilled. Moreover, it is assumed
that δN −β > 0. In figure 1 (a) where θ > 0, the converging saddle paths have positive slopes.
If the initial position of x (= sn) is smaller than its steady-state value, x∗ , then the initialvalue selected on the saddle path is also smaller than λ∗ . On the transitional process in which
both x and λ continue increasing, the absolute value of λ diminishes. Thus in the transition,
the price of consumption good decreases and the labor input for final goods production rises,
which means that the rate of technical progress, ṅ/n, declines during the transition toward17the balanced-growth equilibrium. Figure 1 (b) shows the case of saddle stability under θ < 0.
In this situation the converging paths are negatively sloped. Since λ > 0 for the case of
positive θ, we again find that if the initial level of x is lower than its steady state value, theconsumption good price increases and the rate of technical change declines in the transition
process.
It is worth pointing out that in the case of θ < 0, we may have an alternative possibility.
Now let us assume that θ is negative and it has a large absolute value. Then both (39) and(41) may be violated and the following conditions hold:δb (ρ + β) + θ(δN − β) (δN + ρ) < 0,(44)αδb (ρ + β) + θ (α − 1) (δN + ρ)2 < 0.

(45)

Observe that if (38) is satisfied, λ∗ , x∗ and L∗f have positive values under (44) and (45) .
Furthermore, we see thatsign {traceJ} = sign {− (bw∗ − βλ) w∗ + (ρ + β)}½bδ 2 [α (ρ + β) bδ + θ(δN + ρ)2 ]= sign −(δN + ρ)2 α[(ρ + β) bδ + θ(δN + ρ)(δN − β)]ª+(δN + ρ)2 α{(ρ + β) bδ + θ(δN + ρ)(δN − β) .

Consequently, given (44) and (45) , we find that det J > 0 and trace J < 0. If this is thecase, the linearized dynamic system is asymptotically stable around the balanced-growthpath. Since the initial value of λ is not predetermined, in this case there is a continuum of
converging paths around the balanced-growth equilibrium and, therefore, local indeterminacy
emerges. Noting that (44) needs δN > β, we may state:
Proposition 4 Suppose that θ < 0 and β < δN. Then the balanced-growth path may exhibits
local indeterminacy, if½
θ < min −αδb (ρ + β)δb (ρ + β), −(δN − β) (δN + ρ)
(α − 1) (δN + ρ)2¾.

Figure 1 (c) depicts the situation to which Proposition 4 can apply. In this case thesteady state is a sink, which yields a continuum of converging equilibria at least around thebalanced-growth path.

18

5Discussion

5.1

R&D Subsidy

In the original Grossman and Helpman model, any policy that promotes R&D activities has aclear implication. Since the market economy fails to internalize the knowledge externalities in
the R&D sector, the resource allocation to the R&D activities in the competitive equilibrium
is too low to attain the social optimum. The government’s R&D subsidy to the researchfirms, therefore, enhances growth and welfare, as long as it is financed by non-distortionary
taxation. Such an unambiguous policy implication may not hold in our model. To see this,consider a simple R&D subsidy scheme in which a portion of labor costs of the R&D firms
is subsidized at a rate of φ ∈ (0, 1) . We assume that the government finances the R&D
subsidies by levying a lump-sum tax on the households’ income. Then profits of the R&D
firms is v ṅ − (1 − φ) wLR , so that the zero-excess-profit condition for the R&D firms is givenby(1 − φ) w = δnv.
Using this relation, we see that the arbitrage condition (30) becomesδδb (α − 1)v̇= ρb −+.
v(1 − φ) w α (1 − φ) (bw − βλ)(46)Since the growth rate of variety of goods is not directly aﬀected by the introduction of
the R&D subsidy, we still have¸∙b (α − 1)ṅ= δ (N − Lf ) = δ N −.
nα (bw − βλ)Combining (46) with the above, we find that the dynamic equation of the real wage rate is:v̇ ṅδφδb (α − 1)ẇ
= + = ρ + δN −+w
v n
(1 − φ) w α (1 − φ) (bw − βλ)(47)Notice that if φ = 0, then (46) is reduced to (31), and hence the dynamic behavior of real
wage rate is independent of the other state variables, x and λ. However, as shown above, if
φ 6= 0, the behavior of w depends on x as well as on λ. The steady-state conditions are thefollowing:
β

µ

¶
wLf
− 1 + δ (N − Lf ) = 0 ⇐⇒ ẋ = 0,
x
19

(48)

¶
1
1
−1
= 0 ⇐⇒ λ̇ = 0,
(ρ + β) λ − θ
α
x
φδLf
δ
ρ + δN −+= 0 ⇐⇒ ẇ = 0.
(1 − φ) w 1 − φ
µ

(49)
(50)

Keeping in mind that Lf = b (α − 1) /α (bw − βλ) and using (48) and (49) , we obtain thefollowing equation:
Lf =(α − 1) [αb (ρ + β) w + δN θ].
α2 b (ρ + β) w2 + δ (α − 1) θ

(51)Equations (50) and (51) determine the steady-state values of w and Lf .
First, consider the standard setting where there are no consumption externalities (θ = 0) .
In this case, the real wage behaves according toδφδ (α − 1)ẇ= ρ + δN −+,w
(1 − φ) w α (1 − φ) bw(52)

implying that the steady-state level of real wage rate is
∙
¸
φ
δ
∗
w =
1+
.
δN + ρ
αδ(1 − φ)Since w∗ increases with φ and since the balanced-growth rate in the absence of consumption
externalities is given by

¶
µ
α−1
,
g=δ N−
αw∗

a rise in the rate of R&D subsidy, φ, increases the long-run growth rate by reducing the labor
allocation to final goods production.
In order to examine the growth eﬀect of a change in φ in the presence of external habit
formation, let us rewrite (50) as
1
Lf =
φ

µ

1
δN + ρ
−w
δ¶+δN + ρ
.
δ

(53)

Then the steady-state levels of w and Lf are determined at the intersection of the graphs of(51) and (53) . As an example, consider the case of negative externalities (θ > 0) . As shownby Figure 2, when θ > 0, the graph of (51) has an intercept of N on the Lf axis and it has anegative slope for Lf ≤ N.8 Moreover, limw→∞ Lf = 0. The graph of (53) also has a negative
8

From (51) we obtain£¡¢ ¤αb (α − 1) (ρ + β) δθ (α − 1) − 2 αb2 (ρ + β) w + δαNθ w
dLf
=
.
dw
[α2 b (ρ + β) w2 + δ (α − 1) θ]2

Thus the graph of (51) has a positive slope for 0 < w < ŵ, where dLf /dw = 0 at w = ŵ. However, we seethat Lf > N for 0 < w < ŵ and that dLf /dw < 0 for the feasible region in which Lf is strictly less than N.

20

slope and limw→∞ Lf = −

³

´³

1
φ −1

δN+ρ
θ

´

< 0. While the shapes of these graphs suggestthat there may exist multiple steady states, we restrict our attention to the case where thesteady state is uniquely given. Notice that if φ increases, the graph of (53) shifts downward
for w ≤ δ/ (δN + ρ) , while it shifts upward for w > δ/ (δN + ρ) . Thus we can confirm that arise in the subsidy rate increases the balanced-growth rate if the real wage rate in the initialsteady state satisfies w∗ > δ/ (δN + ρ) : see Figure 2 (a). In this case, a higher φ lowers Lf
by increasing the price level p, so that the labor allocation to the R&D sector increases toenhance long-term growth. In contrast, as depicted by Figure 2 (b), when the initial level ofreal wage is less than δ/ (δN + ρ) , a higher φ increases Lf , and hence a rise in R&D subsidy
fails to stimulate long-run growth.
The Intuition behind the above-mentioned results is the following. Remember again thatthe steady-state level of employment in the final good sector is written as
L∗f =

b
=
p∗

1
³
´ .
βλ∗
α
1
−
w∗
∗
α−1
bw

In the model without externalities, a rise in the real wage w∗ caused by a higher R&D
subsidy rate directly increases the prices of final goods, because λ∗ = 0 in the above equation.
Hence, a higher φ reduces Lf so that the balanced-growth rate increases. In the presenceof consumption externalities, an increase in φ yields the indirect as well as direct eﬀects
on the equilibrium price level p∗ . First, a rise in subsidy to the R&D sector increases thelabor demand of the R&D firms, and hence, other things being equal, the real wage rate
tends to rise, which increases the equilibrium price p∗ . At the same time, a higher p∗ reduces
consumption demand so that the external habit formation will be slow down. This lowers theimplicit ’internalization costs’ for the firm, i.e. the absolute value of λ∗ , which depresses the³´∗α
1 − βλ.9 If this reduction in the mark-up rate dominates the initialmark-up rate, α−1∗bwincrease in the real wage rate, the equilibrium price p∗ may fall down so that L∗f increases. As
a consequence, a higher φ lowers the real wage and raises L∗f , which depresses the balanced-

growth rate. Our graphical analysis indicates that such a conclusion tends to hold if the initiallevel of w∗ is less than δ/ (δN + ρ) . In contrast, if the initial w∗ exceeds δ/ (δN + ρ) , then adecrease in the mark-up rate cannot cancel the direct eﬀect of a rise in w∗ , and therefore in
9

Remember that we are concerned with the case of negative externalities (θ > 0) so that λ has a negativevalue.

21

the new steady state p increases to lower L∗f .

5.2

The Optimal Level of R&D

As emphasized above, in the standard R&D based growth model with product-variety expansion, the labor allocation devoted to R&D activities is too small to attain the eﬃcient
allocation. Since in our model externalities present both in production and consumption
sides, we may not establish such a straightforward result. To confirm this, let us derive
the social optimal allocation in the product-variety expansion model of growth without any
distortion. This is examined by solving the following planning problem. If we focus on thesymmetric equilibrium, the objective function for the planner is given by
¸
∙
Z ∞
Z ∞
1
−ρt
−ρt
log n − θ log s + log Lf − log b dt.
e log Cdt =
e
U=
α−1
0
0
α

In the above, we use C = n α−1 s−θ c and nc = Lf /b. We assume that the planner maximizes
U by controlling labor allocation to production, Lf , subject toṅ = δn (N − Lf ) ,
¶
µ
Lf
−s ,
ṡ = β
bn

(54)
(55)

and the initial values of n and s.
The Hamilton function for this problem can be set as
1
H=
log n − θ log s + log Lf + μδn (N − Lf ) + ηβ
α−1

µ

¶
Lf
−s ,
bn

where μ and η respectively denote the shadow values of n and s. The necessary conditions
for an optimal involve the following:1
1= μδn − ηβ ,Lfbnμ̇ = ρμ − μδ (N − Lf ) − ηβη̇ = ρη +

Lf
1
,
−
2
bn
(α − 1) n

θ
+ βη,
s

together with the dynamic equations of n and s.

22

(56)

(57)
(58)

If consumption externalities do not exist (θ = 0) , then the planning problem is simply
given by
max

Z ∞

−ρt

e

0

∙

¸
1
log n + log Lf − log b dt
α−1

subject to (54) alone. It is a simple exercise to show that in this problem the optimalbalanced-growth rate is

µ
¶
δN
1
− 1−
ρ,
g =
α
α
∗

which is unambiguously higher than the balanced-growth rate in the competitive equilibrium
given by (43) . Hence, the balanced-growth rate in the decentralized economy (so that thelabor allocation to R&D activities) is too low to attain the social optimum.
If θ 6= 0, on the balanced-growth path it holds that
g=

ṅ
ṡ
μ̇
η̇
= =− =−
η
n
s
μ

(59)

and Lf stays constant over time. From (54) through (58) , together with (59) , we obtain thefollowing equations:

¸
∙ ³ ´
1
1
β η
+
,
ρ=
b n
α − 1 nμ¶µ
Lf
δ (N − Lf ) = β 1 −,bnsθδ (N − Lf ) = ρ + β + ,
sη
³
´
1
β η
.
= δμn −
Lfb nAfter some manipulation, we find that the above set of equations can be summarized as asingle equation such thatθ [β − δ (N − Lf )]
δ
ρb
Lf =
+
.
(ρ − δ) (α − 1)
β (ρ − δ) δ(N − Lf ) − (ρ + β)(60)

A positive solution of this equation gives the steady-state level of Lf in the socially optimum balanced-growth path in which every distortion is internalized. Suppose that (60) has
a unique solution in between 0 and N. We can confirm that the steady-state value of Lf
determined by (60) would be larger than L∗f given by (37) . Thus the competitive level of
R&D is not necessarily smaller than the optimal level of R&D that realizes the social optimum. This finding as well as one shown in the previous subsection, indicate that we needcareful consideration as to the policy recommendation in the R&D-based growth model if theconsumers’ preferences involve external habit formation.
23

6Conclusion

This paper has introduced commodity-specific external eﬀects into one of the standard models
of endogenous growth in which continuing growth is sustained by expansion of product variety.
We have shown that the presence of consumption externalities may significantly aﬀect boththe balanced-growth equilibrium and transitional dynamics of the economy. In addition,
the scale eﬀect, the eﬀect of R&D subsidy and the characterization of eﬃcient growth inour setting would be fundamentally diﬀerent form those obtained in the standard model
without consumption externalities. Obviously, unlike production externalities, the presenceof consumption externalities cannot be the main engine of growth. Our study have, however,demonstrated that they may yield significant implications for growing economies in both
positive and normative senses.

24

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26

λ

0

λ = 0xx = 0(KIWTG C θ > 0
λx = 0λ = 0x

0λFigure 1 (b) : θ < 0x = 0λ = 00
Figure 1 (c) : θ < 0

x

Lf
(53)

N
(51)

w

δ
δ N +ρ

(KIWTG C 

Lf

(53)

N
(51)

δ
δ N +ρ

(KIWTG Dw