GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No.95

Consumption Externalities and Wealth Distribution
in a Neoclassical Growth Model

Kazuo Mino and Yasuhiro Nakamoto

October 2009GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

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

Consumption Externalities and WealthDistribution in a Neoclassical Growth Model ∗Kazuo Mino†and Yasuhiro Nakamoto‡

Abstract
This paper explores the distributional eﬀect of consumption externalities ina neoclassical growth model with heterogeneous agents. The economy consistsof two types of agents each of which perceives diﬀerent degrees of intergroup as
well as intragroup consumption external eﬀects. It is shown that the stationary
distribution and transitional dynamics are highly sensitive to the speciﬁcationof preference structures of each type of agents.
Keywords: consumption externalities, heterogeneous agents, wealth distribution
JEL Classification Code: D31, E13, E21, O40∗

The ﬁrst and second authors respectively acknowledge ﬁnancial support by Grant-in-Aid for

Scientiﬁc Research (No.17730139) and Grant-in-Aid for Young Scientists (Start-up) (No.10552200).
†

Institute of Economic Research, Kyoto University, Yoshida Honmachi, Sakyo-ku, Kyoto, 606-

8501 Japan, e-mail: mino@kier.kyoto-u.ac.jp‡

Corresponding author: Faculty of Economics, Kyushu Sangyo University, 2-3-1 Matsukadai,

Higashi-ku, Fukuoka, 813-8503, Japan, e-mail: nakamoto@ip.kyusan-u.ac.jp

1

Introduction

In recent years, there has been a renewed interest in the role of consumption externalities in macroeconomic dynamics. The basic assumption of this literature is that
consumers’ felicity depends not only on their private consumption but also on the
average consumption in the economy at large. The presence of such a psychologicalexternal eﬀect may alter saving behaviors of consumers and thus dynamic propertyof the model economy. The existing studies have inspected the eﬀects of consumption externalities in the context of asset pricing (Abel 1990 and Galı́ 1994), optimal
taxation (Ljungqvist and Uhlig 2000), equilibrium eﬃciency (Alonso-Carrera et al.
2003, Liu and Turnovsky 2005 and Nakamoto 2009), indeterminacy and sunspots(Weder 2000), and long-term economic growth (Carroll et al. 1997 and 2000, andHarbaugh 1996).1
One of the key features of this literature is that most of the foregoing studies
employ representative agent models. In the representative-agent economies, the social average consumption coincides with the level of private consumption, so thatthe presence of consumption externalities aﬀects aggregate dynamics in a quantitative manner rather than in a qualitative manner: the dynamic behavior of themodel economy with consumption externalities is essentially the same as that of
the economy without external eﬀects. Furthermore, it has been well-known that, in
the Ramsey model with inelastic labor supply, the steady-state levels of aggregate
capital and aggregate consumption are not aﬀected by the presence of consumption
externalities.2
1

Some of the existing studies such as Ljunavust and Uhlig (2000) and Carroll et al. (1997 and2000) assume the external habit formation in which the benchmark consumption is given by aweighted average of past levels of the average consumption in the economy. Unlike the internal
habit formation, consumers consider that the benchmark consumption is not aﬀected by theirown consumption behavior under the external habit formation hypothesis. Thus this assumption
represents consumption externalities with time delay rather than (internal) habit formation underwhich each agent takes its past consumption into account when deciding its optimal saving plans.
2

See Fisher and Hof (2000).

2

Unlike the mainstream literature on dynamic macroeconomic analysis with consumption externalities, the present paper explores the eﬀects of consumption externalities in a model of capital accumulation with heterogeneous agents. We construct a neoclassical growth model in which there are two groups of inﬁnitely lived
households. Each group of households diﬀers in their initial wealth holding and intheir preference structure as to consumption external eﬀects. In this setting, external eﬀects of consumption among the consumers are more complex than in the
representative-agent counterpart, because the presence of consumption externalities
may have distributional eﬀects that is inevitably absent in the representative agent
modelling.
More speciﬁcally, we distinguish intergroup externalities from intragroup externalities. Namely, we assume that a household may react diﬀerently in response tochanges in the average consumption in her own group and that in the other group.
For example, an agent would feel jealousy as to other members in her own groupbut would admire consumption behavior of other group’s members. We assume thepresence of such kind of asymmetric external eﬀects in order to inspect how the different forms of external eﬀects among the households aﬀect wealth distribution andaggregate dynamics of the economy.
As it turns out, the steady state of aggregate economy is the same as that of thestandard Ramsey model without consumption externalities. In addition, as well as
in the standard model without externalities, the steady-state distribution of wealthis history dependent, that is, the long-run wealth distribution depends on the initial
distribution of wealth among the agents. It is, however, demonstrated that dynamic
behavior of the model economy towards the steady state is highly sensitive to theconsumption external eﬀects. In particular, in addition to the initial condition,
the speciﬁcation of consumers’ preferences with respect to external eﬀects plays acritical role in determining the stationary distribution of wealth. We carefully inspect
how the strength and direction of intergroup as well as intragroup external eﬀects
determine the equilibrium trajectory of the economy leading to a speciﬁc stationary
distribution of wealth.
3

It is to be noted that our study is closely related to Garcı́a-Peñalosa and Turnovsky(2008). These authors also examine a heterogeneous-agent model of neoclassicalgrowth with consumption externalities. On the one hand, the model used by Garcı́aPeñalosa and Turnovsky (2008) is more general than ours because their model allows
variable labor supply, while our model does not. On the other hand, they assume
that the utility function is identical for all agents and satisﬁes quasi-homotheticity.
Given this restriction, the aggregate dynamics of the model economy is independentof wealth distribution so that Garcı́a-Peñalosa and Turnovsky (2008) can focus on
the distribution dynamics under a given pattern of macroeconomic dynamics. As
emphasized above, since our model assumes that the external eﬀects are asymmetricbetween the two groups of agents, the aggregate dynamics of macroeconomy depends
on the wealth distribution. Hence, the contribution of our investigation is to examine
distributional eﬀect of consumption externalities in a general situation under which
dynamic behavior of the macroeconomy cannot be separated from personal wealthdistribution.
The reminder of this paper is organized as follows. Section 2 sets up the theoretical framework and Section 3 characterizes the steady-state equilibrium. Section
4 presents a detailed discussion on the relation between consumption externalities
and the stationary distribution of wealth. Section 5 gives some discussions for our
ﬁndings. Section 6 concludes.

2

The Model

2.1

Households

Suppose that there are two groups of inﬁnitely-lived agents. Each group consists ofa continuum of identical households who have the same form of instantaneous utility
function and an identical rate of time preference. The felicity function and the initialholding of wealth of the representative household in each group are diﬀerent fromeach other. For simplicity, we assume that population in the economy is constant4

over time and, therefore, the mass of each group will not change. We also assume
that the economy is closed and the stock of capital is the only net asset held byagents.
The representative agent in group i (i = 1, 2) supplies one unit of labor in each
moment and maximizes a discounted sum of utilities over an inﬁnite time horizon.
The objective functional of the representative agent in group i is given by∫ +∞e−ρt ui (ci , Ci , Cj )dt, ρ > 0, i, j = 1, 2, i ̸= j.
Ui =

(1)

0

In the above, ρ denotes a given rate of time discount, ci private consumption of groupi agent, and Ci and Cj respectively represent the average levels of consumption ingroups i and j. The instantaneous utility function, ui (·), is assumed to be monotonically increasing and strictly concave in private consumption, ci . It is also assumed
that in the symmetric equilibrium where C ≡ ci = C1 = C2 , the utility function
holds the Inada conditions: limC→0 ui1 (C, C, C) = ∞ and limC→∞ ui1 (C, C, C) = 0,
where uim (·) (m = 1, 2, 3) denotes the partial derivative of the utility function withrespective to the m-th variable in ui (·) .
The key assumption about the instantaneous felicity function in (1) is that we
distinguish intragroup externalities from intergroup externalities. That is, an agent’s
concern with the consumption levels of members in her own group may be diﬀerent from the concern with consumption of agents in the other group. Following thetaxonomy given by Dupor and Liu (2003), the external eﬀect of consumption on an
individual utility may be either negative (jealousy) or positive (admiration). In addition, each consumer is a conformist who likes being similar to others (keeping up with
the Joneses) or an anti-conformist who wants to be diﬀerent from others (runningaway from the Joneses). We allow, for example, an agent in a particular group feels
jealousy as to consumption of others in her group but admires consumption of agents
who belong to the other group. Such a situation may emerge, the agents in the richgroup admire an increase in the benchmark level of consumption in the poor group,
whereas they have jealousy as to the consumption level of other members in her owngroup. In addition, the agent would be a conformist as to consumption behavior of5her group’s members, but they like running away from consumption behavior of theother group’s agents. As a result, even though there are only two types of agents,
the external eﬀects among the consumers cover a richer class of situations than thattreated in the representative-agent economy where external eﬀects are symmetric for
all agents.
As usual, the negative externality (jealousy) is expressed by uij (·) (= ∂ui /∂Cj−1 ) <0 (i = 1, 2, j = 2, 3) , while positive externality (admiration) means that uij (·) has apositive value. Similarly, the consumers’ conformism is expressed by ui1j (·) (= ∂ 2 ui /∂Cj−1 ∂ci ) >0, and anti-conformism holds if ui1j (·) (= ∂ 2 ui /∂Cj−1 ∂ci ) < 0. In what follows, we
assume that, regardless of the forms of external eﬀects, the eﬀect of a change in theprivate consumption dominates the impact on her utility caused by external eﬀect.
More speciﬁcally, the utility function is assumed to satisfy the following properties:
ui1 (·) + ui2 (·) > 0,(2a)ui1 (·) + ui3 (·) > 0,(2b)ui11 (·) + ui12 (·) < 0,(2c)ui11 (·) + ui13 (·) < 0,(2d)

ui1 (·) + ui2 (·) + ui3 (·) > 0,(2e)ui11 (·) + ui12 (·) + ui13 (·) < 0,(2f)where i = 1 and 2. Conditions (2a) and (2b) mean that the marginal utility of own
consumption dominates impacts produced by consumption externalities. Conditions(2c) and (2d) show that the marginal utility of own consumption diminishes even in
the presence of external eﬀects. Conditions (2e) and (2f ) ensure that, in a social
symmetric equilibrium C1 = C2 , the marginal utility of consumption in a group is
positive and it monotonically decreases with private consumption.
The ﬂow budget constraint for each agent isk̇i = rki + w − ci , i = 1, 2,

6(3)

where, ki is capital stock owned by an agent in group i, ci consumption, ri the rateof return to asset and wi the real wage rate. The initial holding of capital, ki (0) , isexogenously given.
Each household maximizes Ui subject to (3) and the initial holding of capital,ki (0) . Note that when selecting her optimal consumption plan, she takes the sequences of external eﬀects, {C1 (t) , C2 (t)}∞t=0 , as given. Letting the implicit price ofcapital ki be qi , the optimization conditions include
i, j = 1, 2, i ̸= j,

U1i (ci , Ci , Cj ) = qi ,q̇i = qi (ρ − r) , i = 1, 2

(4)
(5)

along with the transversality condition, limt→∞ e−ρt qi ki = 0.
Remember that households in each group are identical. Thus in equilibrium itholds that Ci = ci (i = 1, 2) for all t ≥ 0. Keeping this in mind, from (4) and (5) we
derive a set of Euler equations in such a way that  
1
1r−ρΩ /C Ω2 /C2
Ċ
,
 1 1
  1 = 
2
2
r−ρ
Ċ2
Ω2 /C1 Ω1 /C2
{z
}
|

(6)

J

where

(ui11 (Ci , Ci , Cj ) + ui12 (Ci , Ci , Cj ))Ci> 0,
ui1 (Ci , Ci , Cj )ui (Ci , Ci , Cj )CjΩi2 ≡ − 13i, i = 1, 2.
u1 (Ci , Ci , Cj )

Ωi1 ≡ −

(7)

Here, Ωi1 denotes the elasticity of marginal utility of consumption within the agent’s
own group, which equals the inverse of an elasticity of intertemporal substitution in
private consumption plus social consumption in its own group. This elasticity has
a positive value due to condition (2c). Additionally, Ωi2 is the elasticity of marginalutility with respect to the other group’s consumption. The sign of this term depends
on how group i agents respond to consumption of group j agents. If agents areconformist to keep up with consumption of the other group’s members (so that ui13> 0), then Ωi2 has a negative sign. On the other hand, if they do not like being

7

similar to consumption behaviors of the other group (ui13 < 0), then Ωi2 is strictly
positive. Note that, in view of (2f ), the following conditions hold:Ωi1 + Ωi2 > 0,2.2

i = 1, 2.

(8)

Production

The representative ﬁrm produces a single good according to a constant-returns-toscale technology expressed by()
Ȳ = F K̄, N .
Here, Ȳ , K̄ and N denote the total output, capital and labor, respectively. Using
the homogeneity assumption, we write the production function as follows:
Y = f (K) ,where Y ≡ Ȳ /N and K ≡ K̄/N. The production function, f (K) , is assumed to be
monotonically increasing and strictly concave in capital-labor ratio, K, and fulﬁlls
the Inada conditions. The commodity market is assumed to be competitive so thatthe before-tax rate of return to capital and real wage are respectively determined byr = f ′ (K),w = f (K) − Kf ′ (K).

(9)

For simplicity, we assume that capital does not depreciate.
If we denote the number of agents in group i by Ni (i = 1, 2) , then the fullemployment condition for labor and capital are:N1 + N2 = N,N1 k1 + N2 k2 = K̄.
Letting θi = Ni /N, the full-employment conditions are summarized as follows:
K = θ 1 k1 + θ 2 k2 , ,0 ≤ θi ≤ 1,

θ1 + θ2 = 1.

(10)

For notational simplicity, in the following we normalize the total population, N, toone. Thus θi represents the mass of agents of type i as well as the population shareof that type.
8

2.3

Dynamic System

First, we rewrite (6) as
 



2
1
Ċ
Ω /C
−Ω2 /C2r−ρ
C1 C2
 1 =
 1 2

.
1 2
1 2
2
1
Ω
Ω
−
Ω
Ω
1 1
2 2 −Ω /C1
Ċ2Ω1 /C1r−ρ
2(11)Second, (3) and (9) yieldk̇i = f (K) + (ki − K)f ′ (K) − Ci ,i = 1, 2.

(12)

A complete dynamic system of our economy consists of (11) and (12) that describe
dynamic motions of (C1 , C2 , k1 , k2 ) .
It is worth noting that summing up the ﬂow budget constraints (12) over all of
the households and dividing the both sides by N , we obtainθ1 k̇1 + θ2 k̇2 = f (K) − θ1 C1 − θ2 C2 .
Thus, in view of (10) , we obtain the ﬁnal-good market equilibrium condition for theentire economy:K̇ = f (K) − C,where C = θ1 C1 + θ2 C2 .

3

Steady-State Equilibrium

3.1

Steady-State Characterization

From (9) and (11) the steady-state level of aggregate capital, K ∗ , is determined by
the modiﬁed Golden-Rule condition such thatf ′ (K ∗ ) = ρ,(13)K ∗ = θ1 k1∗ + θ2 k2∗ .

(14)

where

9

Thus, taking account of (12), we see that the steady-state levels of consumption ineach group are given by
Ci∗ = f (K ∗ ) + (ki∗ − K ∗ )f ′ (K ∗ ),i = 1, 2.

(15)

Notice that although the aggregate level of capital and consumption are uniquelydetermined, the steady-state restriction does not pins down the steady-state levels
of individual capital stock, ki∗ . Obviously, the determination of ki∗ needs to specify
trajectory starting from a speciﬁc set of initial capital stocks k1 (0) and k2 (0) .

3.2

Local Determinacy

As shown above, the steady-state levels of k1 and k2 are path dependent. Therefore,
if equilibrium determinacy holds in our setting, as to every set of k1∗ and k2∗ , we canﬁnd a unique converging path to that particular point. Considering this fact, we
ﬁrst specify the steady-state levels of capital holding of each type of agent and theninspect the presence of feasible set of initial distribution of capital that realizes theselected capital holdings in the steady state.
We now select a particular set of steady-state levels of k1∗ and k2∗ that fulﬁll(14) . Linearizing dynamic equations (11) and (12) at the steady state, we obtain thefollowing approximated system:


  
C1∗ (Ω21 −Ω12 )θ2 f ′′ (K ∗ )C1∗ (Ω21 −Ω12 )θ1 f ′′ (K ∗ )
∗
0
0
C
−
C
Ċ1
1
Ω11 Ω21 −Ω12 Ω22Ω11 Ω21 −Ω12 Ω22
 1
  


  
C2∗ (Ω11 −Ω22 )θ2 f ′′ (K ∗ )C2∗ (Ω11 −Ω22 )θ1 f ′′ (K ∗ ) C2 − C2∗ 
Ċ2   00Ω11 Ω21 −Ω12 Ω22Ω11 Ω21 −Ω12 Ω22

.
 =


  
∗
′
∗
∗
∗
′′
∗
∗
∗
′′
∗
  k1 − k1  k̇1  −1 0 f (K ) + (k1 − K )f (K )θ1(k1 − K )f (K )θ2


  
∗
∗
′′
∗
∗
′′
∗
′
∗
∗
∗
k
−
k
k̇2
f (K ) + (k2 − K )f (K )θ20 −1(k2 − K )f (K )θ1
2
2(16)
Letting the eigenvalues of the coeﬃcient matrix in (16) be λj (j = 1, 2, 3, 4), we ﬁnd
the following:
Lemma 1.The eigenvalues of the coeﬃcient matrix of (16)are given by√
4f ′′ (K ∗ )(C1∗ (Ω21 −Ω12 )θ1 +C2∗ (Ω11 −Ω22 )θ2 )f ′ (K ∗ ) ± f ′ (K ∗ )2 −
Ω11 Ω21 −Ω12 Ω22
λj =, 0, f ′ (K ∗ ).
2
10

(17)

Proof. See Appendix A. 
Since there are two unpredetermined variables, C1 and C2 , if the number of
unstable roots (i.e. roots with positive real parts) are two, then there may exist a
unique converging path towards the selected steady state. Lemma 1 shows that oneof the eigenvalues is zero, so that we should have one root with a negative real partto establish local determinacy of equilibrium. Note that the presence of zero root
implies that the steady-state equilibrium is path dependent: it depends on the initial
position from which equilibrium path starts.3 Consequently, it is suﬃcient to focus
on the following eigen value:√
4f ′′ (K ∗ )(C1∗ (Ω21 −Ω12 )θ1 +C2∗ (Ω11 −Ω22 )θ2 )f ′ (K ∗ ) − f ′ (K ∗ )2 −
Ω11 Ω21 −Ω12 Ω22.
λ2 ≡2
If this has a negative value, we may establish local determinacy. For notational
simplicity, in the following we denote λ2 = λ.
To guarantee that λ has a negative value, we impose the following assumptions.4
Assumption 1. Assume that Ω11 Ω21 −Ω12 Ω22 > 0 and C1∗ (Ω21 −Ω12 )θ1 +C2∗ (Ω11 −Ω22 )θ2 >0.
Given Assumption 1, the solution of the linearized system along the stable path canbe written as
k1 (t) = k1∗ + A1 eλt + A2 ,
(18a)
θ1
(18b)
k2 (t) = k2∗ + A1 γ21 eλt − A2 ,
θ2A2
C1 (t) = C1∗ + A1 γ31 eλt + ′ ∗ ,(18c)f (K )
θ1
C2 (t) = C2∗ + A1 γ41 eλt −
A2 ,
(18d)
′
θ2 f (K ∗ )where A1 and A2 are undetermined constants and γ21 , γ31 and γ41 are given by thefollowing:5
γ21 =C2∗ (Ω11 − Ω22 ),
C1∗ (Ω21 − Ω12 )

(19a)

3

See Li et al (2003).

4

If Ω11 < Ω22 and Ω21 < Ω12 , then the sign of λ is negative. We will mention this case in section 5.

5

See Appendix B for the derivation.

11

C ∗ (Ω1 −Ω2 )γ31 =λ − f ′ (K ∗ ) − (k2∗ − K ∗ )f ′′ (K ∗ )θ2 + (k1∗ − K ∗ )f ′′ (K ∗ )θ2 C2∗ (Ω21 −Ω12 )1

1

2−(f ′ (K ∗ ) − λ)2

,

(19b)

C ∗ (Ω1 −Ω2 )γ41 =(k2∗ − K ∗ )f ′′ (K ∗ )θ1 + {λ − f ′ (K ∗ ) − (k1∗ − K ∗ )f ′′ (K ∗ )θ1 } C2∗ (Ω21 −Ω12 )1

1−(f ′ (K ∗ ) − λ)2

2

.

(19c)

Furthermore, in view of (18a) and (18b), we can specify A1 and A2 as
A1 =

A2 =

θ1
(k1 (0) − k1∗ ) + (k2 (0) − k2∗ )θ2,
C2∗ (Ω11 −Ω22 )
θ1
+
∗
2
1
θ2
C1 (Ω1 −Ω2 )C2∗ (Ω11 −Ω22 )(k (0) − k1∗ ) − (k2 (0) − k2∗ )C1∗ (Ω21 −Ω12 ) 1
C2∗ (Ω11 −Ω22 )+ θθ21C1∗ (Ω21 −Ω12 )

(20a)

,

(20b)

where k1 (0) and k2 (0) are the initial levels of capital stock.
As a consequence, we can show the following result:
Proposition 1. The economy converges to the speciﬁed steady-state equilibrium if
and only if the initial capital holdings k1 (0) and k2 (0) satisfy A2 = 0.
Proof. When A2 = 0, conditions from (18a) to (18d) shows that limt→∞ ki (t) = ki∗and limt→∞ Ci (t) = Ci∗ . Conversely, ki (t) converges ki∗ only if A2 = 0. 

4Wealth Distribution

This section is devoted to explore the relation between consumption externalities
and the long-run distribution of wealth. In order to examine distributional dynamics
of our economy in an analytically tractable manner, we assume that Ωji (i, j = 1, 2)
in (7) are constant parameters.
This condition holds if, for example, the instantaneous utility function is givenby the following:(u(ci , Ci , Cj ) =ci Ciϕi Cjηi

)1−γi

1 − γi−1,

i, j = 1, 2, i ̸= j.

(21)

where γi denotes the inverse of elasticity of intertemporal substitution in felicity,
the parameter ϕi represents the extent of the intragroup consumption externalities,
whereas ηi shows the intensity of intergroup externalities. In this speciﬁcation, if
12

ϕi (1 − γi ) > (<)0, then individuals’ preference shows conformism (anti-conformism)
for the average level of consumption in the same group, whereas if ηi (1 − γi ) > (<)0,
it indicates conformism (anti-conformism) for the average level of consumption in
the diﬀerent group.6

4.1

Consumption Externalities and Capital Accumulation

The preference parameters (Ω11 − Ω22 ) and (Ω21 − Ω12 ) are critical to determine thedirection of capital stock in respective groups towards the steady-state equilibrium.
Under the speciﬁed utility function (21), (Ω11 − Ω22 ) represents the divergence between the elasticity of intragroup marginal utility of type 1 agent and the elasticity
of intergroup marginal utility of type 2 agent. Similarly, (Ω21 − Ω12 ) denotes the divergence between the elasticity of intragroup marginal utility of type 2 agent and theelasticity of intergroup marginal utility of type 1 agent.7 Let us diﬀerentiate (18a)and (18b) with respect to time:6

dk1 (t)= A1 λeλt ,
dt

(22a)

dk2 (t)= A1 λγ21 eλt .
dt

(22b)

From conditions (2c) and (2f ), the following inequalities must be satisﬁed:Ωi1 = γi − ϕi (1 − γi ) > 0,
Ωii + Ωij = γ1 − (ϕ1 + η1 )(1 − γ1 ) > 0,i = 1, 2,i, j = 1, 2,i ̸= j.

In addition, in the symmetric steady state where C1 = C2 , condition (2d) requires the following:
ui C
ui C
− 11i − 13i = γi − ηi (1 − γi ) > 0,
u1
u1

7

i = 1, 2.

()()
From (21) we derive the constant values of Ω11 − Ω22 and Ω21 − Ω12 as follows:Ω11 − Ω22 = γ1 − ϕ1 (1 − γ1 ) + η2 (1 − γ2 ),Ω21 − Ω12 = γ2 − ϕ2 (1 − γ2 ) + η1 (1 − γ1 ).

13From (22a) and (22b), it can be easily conﬁrmed that whether or not the directionof capital stock held by an agent in group 2 is the same with that in group 1 is determined by the sign of γ21 alone. If there do not exist the intergroup consumption
externalities, both signs of Ω21 and Ω11 are positive, implying that the capital stocks
in each group converging each steady-state equilibrium move in the same direction
as time goes. More interestingly, when there exist the intergroup consumption externalities so that either sign of (Ω21 − Ω12 ) or (Ω11 − Ω22 ) is negative, the sign of γ21 isnegative. Hence, from (22a) and (22b) it is conﬁrmed that the capital stock held bygroup 1 moves in the opposite direction to that in group 2.
Notice that when both signs of (Ω11 − Ω22 ) and (Ω21 − Ω12 ) are negative, assumption
1 does not hold. We shall discuss this case again in the next section.
In sum, we have shown:
Proposition 2. (i) If (Ω11 − Ω22 ) (Ω21 − Ω12 ) > 0, then (k1 (t) − k1∗ ) (k2 (t) − k2∗ ) > 0
for all t ≥ 0.
(ii) If (Ω11 − Ω22 ) (Ω21 − Ω12 ) < 0, then (k1 (t) − k1∗ ) (k2 (t) − k2∗ ) < 0 for all t ≥ 0.
When we specify the initial level of aggregate capital, we may present a moredetailed discussion on the behaviors of the average capital of each group.
Proposition 3. Suppose that K(0) < K ∗ (= f ′−1 (ρ)). Then we obtain:
2 (t)
1 (t)
> 0 and dkdt
> 0.
(i) If (Ω11 − Ω22 ) (Ω21 − Ω12 ) > 0, then dkdt
1 (t)(ii) If Ω11 < Ω22 and Ω21 > Ω12 (resp. Ω11 > Ω22 and Ω21 < Ω12 ), then dkdt
> 0 and()
dk2 (t)1 (t)2 (t)< 0 resp. dkdt
< 0 and dkdt
>0 .
dt

When K(0) > K ∗ , we obtain the following:
1 (t)
2 (t)
(i) If (Ω11 − Ω22 ) (Ω21 − Ω12 ) > 0, then dkdt
< 0 and dkdt
< 0.
1 (t)
(ii) If Ω11 < Ω22 and Ω21 > Ω12 (resp. Ω11 > Ω22 and Ω21 < Ω12 ), then dkdt
< 0 and()
dk2 (t)1 (t)2 (t)> 0 resp. dkdt
> 0 and dkdt
<0 .
dt

Proof. The parameter A1 in (20a) can be rewritten as
A1 =C1∗ (Ω21 − Ω12 ) (K(0) − K ∗ ),θ1 C1∗ (Ω21 − Ω12 ) + θ2 C2∗ (Ω11 − Ω22 )where the sign of the denominator is positive due to Assumption 1.
14

(23)

Assume that K(0) < K ∗ . In this case, if Ω21 > Ω12 , then A1 < 0 in (23). This1 (t)means that from (22a), dkdt
> 0 where we impose A2 = 0. In addition, if Ω11 > Ω22
2 (t)so that γ21 > 0, from (22b) dkdt
> 0. That is, both groups’ capital stocks move inthe same direction and the levels of capital stock increase in the transition.
If Ω21 > Ω12 and Ω11 < Ω22 , then A1 < 0 and γ21 < 0. In this case, from (22a)
1 (t)
2 (t)
and (22b) we can show that dkdt
> 0 and dkdt
< 0. This means that both groups’

capital stocks move in the opposite directions.
Making use of the similar procedure, we can obtain the other results.

4.2

Initial Distribution and the Steady StateAs easily conﬁrmed by (18a) − (18d), if the condition A2 = 0 is not satisﬁed, thiseconomy cannot have a stable path that converges to the steady-state equilibrium.
From (20b), A2 = 0 gives a set of the initial capital holding (k1 (0), k2 (0)) that achieves
the steady-state equilibrium (k1∗ , k2∗ ).
We ﬁrst consider the relationship between the initial level of capital held by eachtype of household and the steady-state distribution of wealth. Notice that from(13) and (15) the steady-state level of consumption of a group i agents is shown byCi∗ ≡ w∗ +ρki∗ where w∗ ≡ f (K ∗ )−K ∗ f ′ (K ∗ )(> 0). Since K ∗ (= f ′−1 (ρ)) is uniquely
given, the full employment condition (10) gives a relation between k1∗ and k2∗ in sucha way thatk1∗ = −1
θ2 ∗k2 + f ′−1 (ρ) .
θ1θ1Therefore, (20b) shows that condition A2 = 0 can be expressedΨ(k2∗ ) = Γ(k2∗ ),

(24)

where
Ψ(k2∗ ) ≡k2 (0) − k2∗(Ω11 − Ω22 ) (w∗ + ρk2∗ )
∗
)
≡
and
Γ(k
.
2
(Ω21 − Ω12 ) (w∗ + ρk1∗ (k2∗ ))k1 (0) − k1∗ (k2∗ )The steady-state level of capital stock held by group 2 agents is determined by (24).
We note that if k1∗ = k2∗ in the steady-state equilibrium, then these levels are
[
]
K∗
∗
∗
∗
∗
∗
given by K (= k1 = k2 ). Let us consider the shape of Ψ(k2 ) for k2 ∈ 0, θ2 in (24).
15Ω1 −Ω2

Lemma 2. If Ω12 −Ω21 > 0
1

2

( 1

Ω1 −Ω22<0Ω21 −Ω12)∗, then Ψ′ (k2∗ ) > 0 (< 0) for all 0 < k2∗ < Kθ2 .
∗Proof. First, substituting k2∗ = K ∗ , k2∗ = 0 and k2∗ = Kθ2 into Ψ(k2∗ ), notice that
(
)
∗
2 w∗
Ω11 −Ω22 ) w∗ +ρ K(∗(Ω11 −ΩΩ11 −Ω22
θ2
2)
K∗
Ψ(K ) = Ω2 −Ω1 , Ψ(0) = 2 1 ( ∗ K ∗ ) and Ψ( θ2 ) =. Diﬀerenti(Ω21 −Ω12 )w∗
12(Ω1 −Ω2 ) w +ρ θ1
ating Ψ(k2∗ ) with respect to k2∗ , we can show thatΨ′ (k2∗ ) =ρf (K ∗ )
Ω11 − Ω22.
Ω21 − Ω12 θ1 (w∗ + ρk1∗ )2(25)Ω1 −Ω2Thus, we can conﬁrm that Ψ′ (k2∗ ) > (<)0 if Ω12 −Ω21 > (<)0. 
1

4.3

2

Consumption Externalities and Stationary Distribution

To make our argument clear, this subsection assumes that the initial levels of capitalstock in both groups are the same: k1 (0) = k2 (0).8 If this is the case, (10) shows that
K(0) = k1 (0) = k2 (0) where K(0) ≡ θ1 k1 (0) + θ2 k2 (0). Hence, our central concern
in this subsection is to explore how the presence of consumption externalities aﬀects
the long-run wealth distribution between the households who hold the same amountof wealth at the outset. Under this assumption, we focus on the two cases.
Case (i) Ω11 > Ω22 and Ω21 > Ω12][∗
In this case, Lemma 2 means that Ψ(k2∗ ) > 0 and Ψ′ (k2∗ ) > 0 for all k2∗ ∈ 0, Kθ2 .
][∗We now consider the shape of Γ(k2∗ ). Since Ψ(k2∗ ) > 0 for all k2∗ ∈ 0, Kθ2 , it needs
to hold that Γ(k2∗ ) > 0 in the steady-state equilibrium to satisfy equation (24).
K(0)−k∗Considering that Γ(k2∗ ) = K(0)−k∗ (k2 ∗ ) > 0, we see that the following two sub-cases
1

2

hold:
[
]
K ∗ − θ1 K(0)Case (i-a) :K(0),, i = 1, 2
θ2
[ ∗]K − θ1 K(0)∗∗Case (i-b) : ki < K (0) = ki (0) for k2 ∈, K(0)
i = 1, 2.
θ2
ki∗ > K (0) = ki (0)for k2∗ ∈It is to be noted that the slope of Γ(k2∗ ) is given byΓ′ (k2∗ ) =8K ∗ − K(0).
θ1 (k1 (0) − k1∗ )2

In the next section, we shall mention the case that k1 (0) ̸= k2 (0).

16

(26)

Thus, the slope of Γ(k2∗ ) is positive in case (i-a), while it is negative in case (i-b).
Figures 1 and 2 depict the graphs of two functions Ψ(k2∗ ) and Γ(k2∗ ). Thesegraphs use the facts that Γ(K ∗ ) = 1(> 0) and Γ(K(0)) = 0. Figure 1 assumes
that K ∗ > K(0) so that from (26) Γ(k2∗ ) has a positive slope, whereas Figure 2
imposes the inequality K ∗ < K(0) and thus Γ(k2∗ ) has a negative slope. As depicted
in Figures 1 and 2, the graph of Γ(k2∗ ) always intersects line Ψ(k2∗ ) = 0 only once,
conﬁrming that there always exists one crosspoint in the case that Ω11 > Ω22 andΩ21 > Ω12 .9 For instance, Figure 1 shows that point B1 is above point A1 where point
A1 is the intersection between Γ(k2∗ ) and k2∗ = K ∗ , and point B1 is the intersection
between Ψ(k2∗ ) and k2∗ = K ∗ . Those imply that the crosspoint E1 is in the region
k2∗ > K ∗ . As a result, from (14) we can conclude that k2∗ > K ∗ > k1∗ . Conversely,
if point B1 is below point A1 , the crosspoint E1 would be in the region k2∗ < K ∗ sothat k2∗ < K ∗ < k1∗ . In other words, whether point A1 is above or below point B1
determines patterns of wealth distribution in the steady state.
Consequently, we obtain the following proposition.
Proposition 4. When K ∗ > K(0) (i = 1, 2), it holds thatk2∗ > K ∗ > k1∗ > k1 (0) = k2 (0) if (Ω11 − Ω22 ) > (Ω21 − Ω12 ) (> 0),
k1∗ > K ∗ > k2∗ > k1 (0) = k2 (0) if (Ω21 − Ω12 ) > (Ω11 − Ω22 ) (> 0).
When K ∗ < K(0) (i = 1, 2), it holds that
k1 (0) = k2 (0) > k1∗ > K ∗ > k2∗ if (Ω11 − Ω22 ) > (Ω21 − Ω12 ) (> 0),
k1 (0) = k2 (0) > k2∗ > K ∗ > k1∗ if (Ω21 − Ω12 ) > (Ω11 − Ω22 ) (> 0).
Proof. To prove the case where K ∗ > K(0), let us make use of Figure 1.10 WhenΨ(K ∗ ) > Γ(K ∗ ) (i.e., Ω11 − Ω22 > Ω21 − Ω12 ) as shown in Figure 1, the level of group2’s capital stock in the equilibrium E1 is greater than the level of K ∗ , implyingthat K(0) < k1∗ < K ∗ < k2∗ . In contrast, assuming that Ψ(K ∗ ) < Γ(K ∗ ) (i.e.,Ω11 − Ω22 < Ω21 − Ω12 ), we can show that K(0) < k2∗ < K ∗ < k1∗ . On the other hand,

9

(
)
(
)
This also applies to the following case that sign Ω21 − Ω12 = −sign Ω11 − Ω22 .

10

It holds that K −θθ21 K(0) > K ∗ in case (i-a).

∗17

as in Figure 2 we assume that K ∗ < K(0).11 This ﬁgure shows that Ψ(K ∗ ) > Γ(K ∗ )(i.e., Ω11 −Ω22 > Ω21 −Ω12 ) so that the level of group 2’s capital stock in the equilibrium
E2 is lower than that of K ∗ . That is, it holds that k2∗ < K ∗ < k1∗ < K(0). If
Ψ(K ∗ ) < Γ(K ∗ ) (i.e., Ω11 − Ω22 < Ω21 − Ω12 ), then k1∗ < K ∗ < k2∗ < K(0). 
Case (ii): sign (Ω21 − Ω12 ) = −sign (Ω11 − Ω22 )Ω1 −Ω2

Now suppose that (Ω21 − Ω12 ) and (Ω11 − Ω22 ) have opposite signs, that is, Ω21 −Ω12 <
1

12

0.

2

∗
In this case, from Lemma 2 Ψ(k2∗ ) < 0 and Ψ′ (k2∗ ) < 0 for all k2∗ ∈ [0, Kθ2 ].

When considering the shape of Γ(k2∗ ), it is useful to notice the following:Γ(0) =(
Γ
Ψ(0) =
(
Ψ

K∗
θ2

)

K∗
θ2

K(0)
∗ ,
K(0) − Kθ1

)

(27a)
∗

=

K(0) − Kθ2
K(0)

,

w∗
Ω11 − Ω22
∗ (< 0),Ω21 − Ω12 w∗ + ρ Kθ1
∗

(27b)
(27c)

K∗

Ω1 − Ω22 w + ρ θ2
= 12(< 0).
Ω1 − Ω12w∗(27d)Comparing these values, we can obtain the following four sub-cases: case (ii-a):
( ∗)
( ∗)
( ∗)
( ∗)
K
K
K
Ψ(0) > Γ(0) and Ψ θ2 < Γ θ2 ; case (ii-b): Ψ(0) < Γ(0) and Ψ θ2 < Γ Kθ2 ;
( ∗)
( ∗)
case (ii-c): Ψ(0) > Γ(0) and Ψ Kθ2 > Γ Kθ2 ; case (ii-d): Ψ(0) < Γ(0) and
( ∗)
( ∗)
Ψ Kθ2 > Γ Kθ2 . Furthermore, using (27a) − (27d), we may express these subcases as follows:[]∗∗w∗ + ρ Kθ1
K(0) − Kθ2Ω11 − Ω22K(0)w∗∈Case (ii-a) : 2,∗ ,∗Ω1 − Ω12w∗
K(0)K(0) − Kθ1 w∗ + ρ Kθ2[]∗∗w∗ + ρ Kθ1
K(0) − Kθ2w∗Ω11 − Ω22
K(0)
< min
Case (ii-b) : 2,∗ ,∗Ω1 − Ω12w∗
K(0)K(0) − Kθ1 w∗ + ρ Kθ2
[
]
K∗
K∗
∗
2
∗
1
w
+
ρ
K(0)
−
K(0)
w
Ω − Ω2
θ1
θ2
> max
,
, ∗
Case (ii-c) : 21
1
K∗
K∗
∗
Ω1 − Ω2w
K(0)K(0) − θ1 w + ρ θ2∗

11

It holds that K −θθ21 K(0) < K ∗ in case (i-b).

12

If C1∗ (Ω21 − Ω12 )θ1 + C2∗ (Ω11 − Ω22 )θ2 < 0, there does not exist the stable root in this case. Hence,

we note that the assumption 1 is imposed.

18

[
]
∗
∗
K(0) − Kθ2 w∗ + ρ Kθ1Ω11 − Ω22K(0)w∗Case (ii-d) : 2
∈
,
.
∗
∗
Ω1 − Ω12K(0)w∗w∗ + ρ Kθ2
K(0) − Kθ1
Assume that K ∗ > K(0). This means that from (26) the slope of Γ(k2∗ ) is positive.
Figure 3 depicts case (ii-a). As conﬁrmed in Figure 3, taking account of the positive
slope of Γ(k2∗ ), the number of intersection that satisﬁes Γ(k2∗ ) = Ψ(k2∗ ) is two in case(ii-a). It is also seen that the number of the intersection in cases (ii-b) and (ii-c) isone. In addition, there is no intersection in case (ii-d). Alternatively, if K ∗ < K(0)so that the slope of Γ(k2∗ ) is negative, then the relationship is reversed. That is, we
ﬁnd: there is no intersection in case (ii-a); there are two intersections in case (ii-d),and; there is one intersection in cases (ii-b) and (ii-c).
Ω1 −Ω2

Taking account of these facts, the relationship between the value of Ω21 −Ω12 and12

wealth distribution can be summarized as follows.
Lemma 3. When K ∗ > K(0), it holds that
Case (ii-a): k1∗ > K ∗ > K(0) > k2∗ or k2∗ > K ∗ > K(0) > k1∗ ;Case (ii-b): k2∗ > K ∗ > K(0) > k1∗ ;Case (ii-c): k1∗ > K ∗ > K(0) > k2∗ .
On the other hand, when K ∗ < K(0), it holds that
Case(ii-b): k1∗ > K(0) > K ∗ > k2∗ ;Case (ii-c): k2∗ > K(0) > K ∗ > k1∗ ;Case (ii-d): k1∗ > K(0) > K ∗ > k2∗ or k2∗ > K(0) > K ∗ > k1∗ .
Ω1 −Ω2In all cases in Lemma 3, Ω21 −Ω12 is negative; however, it is not manifest whether
12either of (Ω11 − Ω22 ) or (Ω21 − Ω12 ) is negative. Considering that the signs of (Ω11 − Ω22 )and (Ω21 − Ω12 ) determine the dynamic behavior of capital stock held by each group,
we obtain the following proposition.
Proposition 5. When K ∗ > K(0), the following relations are established:
Case (ii-a): if Ω11 < Ω22 and Ω21 > Ω12 , then k1∗ > K ∗ > K(0) > k2∗ ,whereas if Ω11 > Ω22 and Ω21 < Ω12 , then k2∗ > K ∗ > K(0) > k1∗ ,Case (ii-b): if Ω11 > Ω22 and Ω21 < Ω12 , then k2∗ > K ∗ > K(0) > k1∗ ,whereas if Ω11 < Ω22 and Ω21 > Ω12 , then there is no converging path towards the19steady state,
Case (ii-c): if Ω11 < Ω22 and Ω21 > Ω12 , k1∗ > K ∗ > K(0) > k2∗ .
whereas if Ω11 > Ω22 and Ω21 < Ω12 , then there is no converging path towards thesteady state,
Case (ii-d): there is no converging path towards the steady state,
When K ∗ < K(0), the following relations are established:
Case (ii-a): there is no converging path towards the steady state,
Case (ii-b): if Ω11 < Ω22 and Ω21 > Ω12 , then k1∗ > K(0) > K ∗ > k2∗ ,whereas if Ω11 > Ω22 and Ω21 < Ω12 , then there is no converging path towards thesteady state,
Case (ii-c): if Ω11 > Ω22 and Ω21 < Ω12 , then k2∗ > K(0) > K ∗ > k1∗ ,whereas if Ω11 < Ω22 and Ω21 > Ω12 , then there is no converging path towards thesteady state,
Case (ii-d): if Ω11 < Ω22 and Ω21 > Ω12 , then k1∗ > K(0) > K ∗ > k2∗ ,whereas if Ω11 > Ω22 and Ω21 < Ω12 , then k2∗ > K(0) > K ∗ > k1∗ .
Proof. First, consider the case that K ∗ > K(0). For instance, assume that Ω11 < Ω22Ω1 −Ω2Ω1 −Ω2and Ω21 > Ω12 so that Ω12 −Ω21 < 0 where we assume that the value of Ω12 −Ω21 is in the
1

2

1

2

range in case (ii-a) or case (ii-c). Considering that γ21 < 0 in (19a) and A1 < 0 in(23), from (22a) and (22b) the level of capital stock of group 1 monotonically increases
and the level of group 2’s capital stock decreases during the transition, implying that
there exists a steady-state equilibrium that satisﬁes k1∗ > K ∗ > K(0) > k2∗ .
Instead, we consider the case that Ω11 > Ω22 and Ω21 < Ω12 where we assume that
Ω11 −Ω22Ω21 −Ω12< 0 in case (ii-a) or in case (ii-b). In this case, since the level of capitalstock of group 1 decreases and that of capital stock of group 2 increases, there is
the steady-state equilibrium with k2∗ > K ∗ > K(0) > k1∗ . Making use of the sameprocedure, we can show the other results in the case that K ∗ < K(0).
From Propositions 4 and 5 we obtain intuitive explanations about the relationship between wealth distribution and consumption externalities. First, we consider
the eﬀect of the intragroup consumption externalities on wealth distribution. The
intragroup consumption externalities only produce quantitative eﬀects: although the20intragroup consumption externalities cause the diﬀerence of wealth distribution in
the long run, the capital stock held by each group converges to the steady state fromthe same direction. Assume that there do not exist the intergroup consumption externalities (i.e., Ω22 = Ω12 = 0) so that there exist Ω11 (> 0) and Ω21 (> 0) alone. From
Proposition 3, this case shows that if K ∗ > K(0) (resp. K ∗ < K(0)), the capitalstocks held by both groups monotonically increase (resp. decrease). In addition, forsimplicity, we assume that the pure elasticities of the marginal utility of own conu1 C

u2 C

1

1

sumption are the same (i.e., − 11u1 1 = − 11u2 2 ) in both groups, so that the inequality
u1 C ∗Ω11 > (<)Ω21 means 12u1 11

u2 C ∗< (>) 12u2 2 .
1

We now consider the case that K ∗ > K(0),meaning that f ′ (K(t)) > ρ for all t. For instance, if individuals’ preference of group
1 exhibits strong conformism relative to that in group 2 so thatu112 C1∗
u212 C2∗>, we can
1
u1
u21

1 (t)
2 (t)
show that Ċ
> Ċ(> 0). That is, because the agents in group 1 save more thanC1 (t)C2 (t)

the agents in group 2, the long-run level of capital stock in group 1 is larger than thatin group 2, that is, k1∗ > K ∗ > k2∗ > K(0) = k1 (0) = k2 (0). This result corresponds
to the conclusion in Proposition 4. In contrast, if individuals’ preference in group 1
shows strong anti-conformism relative to that in group 2 so thatu112 C1∗
u11<

u212 C2∗, weu21

obtain k2∗ > K ∗ > k1∗ > K(0) = k1 (0) = k2 (0).
Next, suppose that there are intergroup as well as intragroup consumption externalities. We consider the case where K ∗ > K(0). In this case, the sign of (Ω11 − Ω22 )or (Ω21 − Ω12 ) can be negative.13 Then, the presence of the intergroup consumption
externalities may yield not only quantitative but also qualitative diﬀerences in dynamic behaviors of capital stock held by each group. More speciﬁcally, the qualitative
impact of consumption externalities is demonstrated by the fact that the dynamic
behavior of capital stock held by one group is opposite to that of the other group’s
capital. For example, assume that Ω11 < Ω22 and Ω21 > Ω12 . Noting that both Ω11 andΩ21 have positive signs, this case may hold if there is intergroup conformism among
u1 C

agents in group 1 (i.e., Ω12 = − 13u1 2 (< 0)) so that Ω21 > Ω12 , while if group 2’s agents1u2 C

are intergroup anti-conformists (i.e., Ω22 = − 13u2 1 (> 0)) so that Ω11 < Ω22 . As shown
1

13

(
)
(
)
We shall deal with the case that both signs of Ω21 − Ω12 and Ω11 − Ω22 are negative in thenext section.

21

in Proposition 3, when K ∗ > K(0), the inequalities Ω11 < Ω22 and Ω21 > Ω12 showthat the level of capital stock held by group 1 monotonically increases and group 2’s
capital stock monotonically decreases. Furthermore, taking account of the dynamic
1 (t)
2 (t)
equations of consumption, we can show that Ċ
> 0 > Ċ. In addition, when theC1 (t)C2 (t)

condition A2 = 0 is imposed, we can conﬁrm whether there is a stable steady-stateΩ1 −Ω2equilibrium that depends on the value of Ω12 −Ω21 . If cases (ii-a) and (ii-c) hold, the12steady-state wealth distribution is characterized by k1∗ > K ∗ > K(0) > k2∗ . Furthermore, if the population size is the same in both groups θ1 = θ2 , the diﬀerence
between k1∗ and k1 (0)(= K(0)) is larger than that between k2∗ and k2 (0)(= K(0)).
Namely, wealth inequality will be enhanced during the transition.
Finally, because of the presence of the intergroup consumption externalities, weΩ2 −Ω1

notice that the steady state equilibrium may not exist if Ω11 −Ω22 < 0. For example, if
12

K ∗ > K(0), Ω11 < Ω22 and Ω21 > Ω12 , then there is no feasible steady state equilibriumin cases (ii-b) and (ii-d). This fact is also a qualitative eﬀect generated by the
introduction of heterogeneous consumers.

5

Discussion

5.1

The Euler Equations and Consumption Externalities

We have examined the relationship between the wealth distribution and consumption externalities. In the last section, we found that the levels and the signs of thepreference parameters (Ω11 − Ω22 ) and (Ω21 − Ω12 ) are critical to characterize the relationship. In this subsection, making use of the Euler equations, we present intuitive
implication of our ﬁndings displayed above.
First, note that if intergroup externalities do not exist, the Euler equations for
optimal consumption of each group are
1Ċi= i (r − ρ) ,Ci
Ω1

i = 1, 2

where Ω1i = − ui Ci 1 ui Ci , which indicates the partial elasticity of intertemporal sub1

11
ui1

+

12
ui1

stitution in consumption from the intragroup perspective. In this case, a higher
22degree of intragroup conformism, i.e. ui12 > 0, increases 1/Ωi1 , which accelerates
consumption growth of each group. Therefore, if there is no intergroup consumption externalities, the consumption conformism (resp. anti-conformism) enhances(diminishes) current savings and promotes capital accumulation. Hence, in the absence of intergroup externalities, we may conclude that although wealth distributionaﬀects aggregate dynamics, transitional behavior of our economy is close to that of
the representative-agent economy with consumption externalities.
Once we consider the intergroup consumption externalities, we ﬁnd a much richer
set of dynamic behaviors of consumption and capital. The Euler equations are now
given byĊiαi=(r − ρ) , i = 1, 2
Ci
Ω̂
where
Ω̂ = Ω11 Ω21 − Ω12 Ω22 ,
α1
α2

( 2
) ( 1
)
u11 C2 u212 C2
u13 C1
=
+
− − 1
u21
u21
u
( 1
) ( 21 )1
u11 C1 u12 C1
u C2= Ω11 − Ω22 = −+− − 132
.
1
1
u1
u1
u1
Ω21 − Ω12 = −Notice that the social marginal utility of own consumption including intragroup
external eﬀect is decreasing regardless of the sign of ui12 (i.e., Ω11 > 0 and Ω21 > 0).
The preference parameters Ω12 and Ω22 express the partial elasticity of intertemporal
substitution in consumption from the intergroup perspective. Note that when theelasticities of substitution from the intragroup perspective dominate those from theintergroup perspective, Ω̂ has a positive value. Conversely, if agents in each groupare more sensitive to the consumption of the other group, Ω̂ could be negative.
The elasticity of intertemporal substitution in consumption including the intergroup consumption externalities is given by αi /Ω̂. Unlike the intragroup consumption
externalities alone, when the intergroup consumption externalities are incorporated,
we need to consider the eﬀect of the intergroup consumption externalities on theelasticity of intertemporal substitution in consumption through Ω̂ and αi .

23

We ﬁrst consider the eﬀect on Ω̂. For simplicity, we assume that the sign of Ω̂ ispositive. In this case, if Ω12 and Ω22 have the same signs and both become larger, thenΩ̂ decreases, which is likely to enhance consumption growth in respective groups. In
contrast, if agents in one group have conformism with respect to the other group’s
consumption but the other group’s members have intergroup anti-conformism, then
Ω12 Ω22 < 0. Thus when these intergroup external eﬀects are large enough, Ω̂ has alarger value, which will depress consumption growth. As a result, we see that, aslong as Ω̂ takes a positive value, the rate of consumption growth of each group ishigher when both groups have the same type of sentiment about the other group’s
consumption behavior. In other words, other things being equal, the homogeneity ofpreference in the society at large tends to accelerate capital accumulation.
The actual value of consumption adjustment speed, however, depends on themagnitude of αi as well. Assume that αi (i = 1, 2) take positive values. Then, we
ﬁnd that the positive sign of ui13 (i = 1, 2) leads to an increase in the growth rate of
consumption, while the negative sign of ui13 lowers it. As a result, we can conclude
that when the agents in each group are conformists as for group’s consumption
behavior, consumption growth is enhanced.
The above discussion deals with the case that Ω̂ > 0 and αi > 0 so that there isno qualitative eﬀect generated by the intergroup consumption externalities. Namely,when αi > 0 (i = 1, 2) so that αi /Ω̂ > 0, the behaviors of optimal consumption of eachgroup are similar to the case where there is no intergroup externalities; however, if
αi < 0 (i = 1, 2) and Ω̂ > 0, dynamics of consumption and capital may show explicit
qualitative diﬀerences. In this case, even if we restrict our attention to the normalcase where Ω̂ > 0, the consumption (so savings) behavior of each group takes the
opposite directions. As an example, suppose that α1 < 0 and α2 > 0. This situation
holds, if agents in group 1 has a strong degree of conformism towards the group 2such that the sign of α1 is negative. In this case a rise in the real rate of return tocapital, r, depresses consumption growth of agents in group 1 and thus their capitalformation is lowered. If such an impact is large enough, the steady-state equilibrium24

could be unstable.14

5.2

Alternative Initial Conditions

In Section 3 we restrict our attention to the case where the initial capital holding ofeach type of agent is identical. This restriction means that Γ(K ∗ ) = 1 in (24) underwhich the long-run wealth distribution can be characterized by the signs of (Ω21 − Ω12 )
and (Ω11 − Ω22 ) alone: see Propositions 4 and 5. Here, we brieﬂy discuss the case∗, how the wealth is distributed in thewhere k1 (0) ̸= k2 (0) . Since Γ(K ∗ ) = kk21 (0)−K(0)−K ∗
long run is determined by the initial holdings of capital stock in respective groups
as well as by the values and the signs of (Ω11 − Ω22 ) and (Ω21 − Ω12 ). Alternatively,
even if assuming k1 (0) ̸= k2 (0), the behavior of capital stock from the initial periodΩ1 −Ω2

towards the steady-state equilibrium can be characterized by the sign of Ω12 −Ω21 aloneΩ1 −Ω2

1

2

as conﬁrmed in Propositions 2 and 3. That is, if Ω12 −Ω21 > 0, capital stocks of both12

Ω1 −Ω2groups change in the same direction, while if Ω12 −Ω21 < 0, the behavior of capital stock
1
2

in a group is opposite to that of the other groups.
Note that the wealth distribution depends on the initial holdings of capital stock
as well as on the preference structures. Let us consider the case that Ω11 > Ω22 andΩ21 > Ω12 . In addition, assume that the steady-state levels of capital stock in bothgroups are greater than those of capital stock at initial period ki∗ > ki (0) and thatthe steady-state level of capital stock in group 2 is greater than that of group 1. In
this case, the condition that determines the wealth distribution can be rewritten as(Ω2 −Ω1 )(k2 (0)−k∗ )(Ω11 − Ω22 ) > 1 k12(0)−k∗ 2 . Comparing it with the condition in Proposition 4, we
1

see that even if the preference on consumption in group 2 does not show strong KUJ
so that (Ω11 − Ω22 ) < (Ω21 − Ω12 ), the large diﬀerence between k1 (0) − k1∗ and the small
diﬀerence between k2 (0) − k2∗ would produce the long-run wealth distribution in sucha way that k2∗ > K ∗ > k1∗ . Furthermore, if the preference shows the strong KUJ in
14

One may consider the case that if Ω̂ < 0 and αi < 0 (i = 1, 2), the economy moves towards thesteady-state equilibrium. In this case, the stability of the economy may be indeterminate as shownin Mino and Nakamoto (2008) where the paper incorporates the progressive taxation to guarantee
the unique steady state.

25

group 2 so that the diﬀerence of the initial levels of capital stock is not critical forthe determination of wealth distribution, we may conﬁrm that individuals in group
2 become rich in the long run relative to the other group although they are poor in
the initial period. This means that k2 (0) < k1 (0) < k1∗ < k2∗ .
In the case that either of (Ω11 − Ω22 ) and (Ω21 − Ω12 ) is negative, the behavior of
capital stock in a group is opposite to that of the other group. Therefore, it may
hold that k1∗ < k2 (0) < k1 (0) < k2∗ . Individuals in group 1 initially become rich, but
their capital stock decreases over time and they ﬁnally become poor. In contrast,
individuals in group 2 are initially poor, but continuing increase in their capitalmakes them relatively rich in the steady-state equilibrium.

6

Conclusion

In this paper we have studied how the presence of consumption externalities aﬀects
macroeconomic stability and stationary wealth distribution in a neoclassical growthmodel with heterogeneous households. We have distinguished intragroup externalities form intergroup external eﬀects in order to examine the role of asymmetricexternal eﬀects among the agents. Our ﬁndings reveal that the long-run distributionof wealth is highly sensitive to the strength of conformism (or anti-conformism) as
to intragroup as well as to intragroup comparison of consumption levels.
Our central concern is to show that consumption externalities would play a more
prominent role in the economy with heterogeneous agents than in the representativeagent counterpart. We thus have focused on the distributional eﬀect of consumption
externalities and have not discussed policy implications in detail. Our next task is
to study the policy impacts, in particular, the eﬀect of redistribution policies, in our
setting.

26

AppendicesThroughout these appendices, we assume that ∆ ≡ Ω11 Ω21 − Ω12 Ω22 , Π1 ≡ C1∗ (Ω21 −Ω12 ) and Π2 ≡ C2∗ (Ω11 − Ω22 ).
Appendix A.
We derive the eigenvalues in this economy. Given (16), the characteristic equationis{(f ′′ (K ∗ )(Π1 θ1 + Π2 θ2 )λ λ − 2f (K )λ + f (K ) +
∆
3

′

∗

′

2

∗ 2

)

f ′′ (K ∗ )f ′ (K ∗ )(Π1 θ1 + Π2 θ2 )
λ−
∆
(A.1)

}
= 0.

Thus, one of the eigenvalues is zero. Furthermore, the equation (A.1) can be rewritten
as(f ′′ (K ∗ )(Π1 θ1 + Π2 θ2 )λ (λ − f (K )) λ − f (K ) +
∆
′

∗

2

′

)

∗

= 0.

(A.2)

Hence, we can obtain the eigenvalues given by (17).
Appendix B.
The eigen-vectors corresponding to the stable root λ are given by:   Π1 θ2 f ′′ (K ∗ )Π1 θ1 f ′′ (K ∗ )
−λ 0
1
0
∆
∆

   
′′
∗
′′
∗






Π2 θ1 f (K )
Π2 θ2 f (K )
 0 −λ γ21  0∆∆  =  .
    −1 0 f ′ (K ∗ ) + (k1∗ − K ∗ )f ′′ (K ∗ )θ1 − λ γ31  0(k1∗ − K ∗ )f ′′ (K ∗ )θ2   
∗
∗
′′
∗
′
∗
∗
∗
′′
∗
0 −1
(k2 − K )f (K )θ1f (K ) + (k2 − K )f (K )θ2 − λ
γ41
0
(B.1)
Here, we can obtain the following equations:−λ +Π1 θ2 f ′′ (K ∗ )Π1 θ1 f ′′ (K ∗ )γ31 +γ41 = 0,∆∆(B.2a)

Π2 θ1 f ′′ (K ∗ )Π2 θ2 f ′′ (K ∗ )γ31 +γ41 = 0,∆∆(B.2b)−λγ21 +

−1 + {f ′ (K ∗ ) + (k1∗ − K ∗ )f ′′ (K ∗ )θ1 − λ} γ31 + (k1∗ − K ∗ )f ′′ (K ∗ )θ2 γ41 = 0, (B.2c)−γ21 + (k2∗ − K ∗ )f ′′ (K ∗ )θ1 γ31 + {f ′ (K ∗ ) + (k2∗ − K ∗ )f ′′ (K ∗ )θ2 − λ} γ41 = 0. (B.2d)
First, using {(B.2a) × Π2 − (B.2b) × Π1 }, we can show thatλ (Π2 − γ21 Π1 ) = 0.
27

(B.3)

Noting that λ ̸= 0, we obtain:
γ21 =

Π2
.
Π1

(B.4)

Π2
Furthermore, using (B.2c) and (B.2d), we can derive γ31 and γ41 where γ21 = Π1

in (19b) and (19c).
Next, in the case of zero–root, we only substitute λ = 0 into equations (B.2a) −(B.2d), leading to the following relations:Π1 θ1 f ′′ (K ∗ )Π1 θ2 f ′′ (K ∗ )γ32 +γ42 = 0,∆∆(B.5a)

Π2 θ1 f ′′ (K ∗ )Π2 θ2 f ′′ (K ∗ )γ32 +γ42 = 0,∆∆(B.5b)−1 + {f ′ (K ∗ ) + (k1∗ − K ∗ )f ′′ (K ∗ )θ1 } γ32 + (k1∗ − K ∗ )f ′′ (K ∗ )θ2 γ42 = 0,

(B.5c)−γ22 + (k2∗ − K ∗ )f ′′ (K ∗ )θ1 γ32 + {f ′ (K ∗ ) + (k2∗ − K ∗ )f ′′ (K ∗ )θ2 } γ42 = 0.

(B.5d)

θ1
1
From (B.5a) and (B.5c), we can obtain γ32 = f ′ (K∗ ) and γ42 = − θ f ′ (K ∗ ) , respec2

tively. Substituting these into (B.5d), we derive γ22 = − θθ12 .

28

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31

Figure 1: The case (i-a)(k2 )(k2 ), (k2 )(k2 )

E1
B1(K  )
1 = (K  )

A1

K

K (0)

K  1 K (0)
2

K
2

k2

Figure 2: The case (i-b)(k2 ), (k2 )(k2 )(k2 )(K  )B2

E2

1 = (K  )A2

K  1 k(0)
2

K (0)

K

32

K
2

k2

Figure 3: The case (ii-a)(k2 ), (k2 )(k2 )

1 = (K  )

K
2

K (0)
(0)

K

E3

(0)


( K2 )

(k2 )

E30



( K2 )
(k2 )

k1 > K (0)k2 > K (0)k1 > K (0)k2 < K (0)33k1 < K (0)k2 > K (0)k2