Discussion Papers In Economics
And Business

Consumption Externalities and Equilibrium
Dynamics with Heterogenous Agents
Kazuo Mino and Yasuhiro Nakamoto

Discussion Paper 08-30

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANConsumption Externalities and Equilibrium
Dynamics with Heterogenous Agents
Kazuo Mino and Yasuhiro Nakamoto

Discussion Paper 08-30

September 2008

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Consumption Externalities and Equilibrium
Dynamics with Heterogenous Agents∗
Kazuo Mino†and Yasuhiro Nakamoto‡

Abstract
This paper explores the eﬀect of consumption externalities on equilibrium dynamics of a standard neoclassical growth model in which there are two typesof agents. To emphasize the presence of heterogenous agents, we distinguishintergroup consumption externalities from intragroup consumption externalities. We show that if the intragroup externalities dominates the intrergroup
external eﬀects, then the steady state equilibrium satisfies saddle-point stability and the equilibrium path of the economy is uniquely determined. In
contrast, if the intergroup external eﬀects of consumption are strong enough,the steady-state equilibrium is either unstable or locally indeterminate. Basedon the analytical as well as numerical considerations, we give intuitive implications of stability conditions.
Keywords: Consumption externalities, Equilibrium determinacy, Heterogeneous agents, Progressive taxation
JEL Classification Code: E52, O42∗

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

Scientific Research (No.17730139) and the Research Fellowships of the Japanese Society for the
Promotion of Science for Young Scientists.
†

Graduate School of Economics, Osaka University, 1-7 Machikaneyma, Toyonaka, 563-0043

Japan, e-mail: mino@econ.osaka-u.ac.jp‡

Graduate School of Economics, Osaka University, 1-7 Machikaneyma, Toyonaka, 563-0043

Japan, e-mail: ege006ny@mail2.econ.osaka-u.ac.jp

1

Introduction

Recently, there is a renewed interest in consumption external eﬀects in dynamic
macroeconomics. While the earlier contributions such as Abel (1990) and Galí (1991)
focus on the role of consumption externalities in the asset-pricing models, the recent
studies treat a wider class of issues. For example, the recent investigations consider
external eﬀects of consumption on optimal taxation (Ljungqvist and Uhlig 2000), on
the relation between savings and long-term economic growth (Carroll et al. 1996 and2000) as well as on the eﬃciency of equilibrium (Liu and Turnovsky 2005). A common feature of this literature is that most studies employ the representative agent
frameworks. In this literature the consumption external eﬀect is formulated in sucha way that an individual consumer’s felicity depends on the average level of consumption in the economy as well as on her own consumption. In the equilibrium ofrepresentative-agent economies the individual and the average levels of consumption
coincide each other and, therefore, the presence of consumption externalities generally produces quantitative eﬀects rather than qualitative eﬀects: the equilibrium
dynamics and the steady state characterization are usually the same as those of themodels without consumption externalities.
Unlike the mainstream literature mentioned above, this paper examines the role ofconsumption externalities in the presence of heterogenous agents. Since the externalinteractions among the consumers tend to be much more complex in an economy
with heterogenous agents than in the representative-agent counterpart, the presenceof consumption external eﬀects would yield fundamental impacts on the dynamic
behavior of the economy if we consider heterogeneity of consumers. Using a simple
neoclassical growth model with two types of agents, we confirm our prediction. We
show that even in the symmetric steady state where every agent has the same levels of
income and wealth, the dynamic behavior of the economy may not exhibit a regularsaddle point stability. The equilibrium path of the economy could be either unstable
or indeterminate. Thus consumption externalities, together with heterogeneity ofagents, would yield a variety of dynamic behaviors, even if there is neither production1external eﬀect nor complex preference structure associated with labor-leisure choice.
The analytical framework of this paper is the standard neoclassical growth modelwith infinitely-lived agents. In this setting it has been well known that there exists
a continuum of steady states if all the agents have an identical time discount rate,
while the agent with the lowest time discount rate ultimately owns the entire capitalstock if the time discount rate of each agent is not identical: see, for example, Becker(1980) and Soger (2002). To avoid those extreme outcomes, we introduce nonlinear
income taxation into the base model. As pointed out by Sarte (1997), a nonlinear
income tax scheme may yield a unique interior steady state in which every agent
holds a positive amount of capital, even though the agents have heterogenous ratesof time preferences. In this paper, we assume that the tax rate levied on an individual consumer depends on her income relative to the average level of income in theeconomy at large. This assumption, which follows Guo and Lansing (1998) and Lieand Sarte (2004), enables us to establish the symmetric steady state equilibrium inwhich wealth and income are equally distributed among the households. Owing tothese two assumptions, the steady-state equilibrium of our economy with heterogenous agents is essentially the same as the stationary equilibrium of the representative
agent economy. Hence, we may elucidate how the introduction of heterogeneity ofagents aﬀect the role of consumption externalities in the transition process of an
economy.1
Our study presents two main findings. First, either if there are only intragroupconsumption externalities or if the magnitude of intragroup externality denominates
the intergroup eﬀects, then a uniquely given steady state exhibits a regular saddle
point property. In this case, the equilibrium path is determinate and it converges tothe symmetric steady state equilibrium. Our second finding is that if the intergroup
external eﬀects have larger impacts on the individual consumption decision than theintragroup external eﬀects, then the symmetric steady state equilibrium is either
1

In a related study project, Mino (2007) and (2008) introduce consumption externalities into

overlapping generations models where intragroup and intergroup externalities are respectively replaced with intragenerational and intergenerational external eﬀects.

2totally unstable or locally indeterminate. In the latter case, there exists a continuumof converging paths around the steady state, so that expectations-deriven economic
fluctuations may emerge.
In the existing literature, García-Peñalosa and Turnovsky (2007) also study aneoclassical growth model with heterogenous agents and consumption externalities.
The key assumption in their investigation is that every agent has a quasi-homothetic
preference so that the aggregate behavior of the economy is independent of wealthdistribution. Therefore, the role of consumption externalities in their model is essentially the same as that in the representative agent models. On the other hand,because of the introduction of nonlinear taxation, the macroeconomic stability of
our model depends on wealth distribution and distribution dynamics are aﬀected byexternal interactions among consumers. As a result, the presence of consumption
externalities plays a more prominent role in our model than in García-Peñalosa andTurnovsky (2007).
It is also to be noted that Alonso-Carrera et al. (2008) and Chen and Hasu
(2007) reveal that equilibrium indeterminacy may hold in the representative agent
models with consumption externalities. Alonso-Carrera et al. (2008) show that if
labor-leisure choice is allowed and if the utility function is not homothetic with respect to private and average consumption levels, then the one-sector growth modelwith consumption externalities may generate indeterminacy of equilibrium.2 Chen
and Hasu (2007) examines a two-sector growth model and shows that the presenceof consumption externalities aﬀects resource allocation between two production sectors, which may cause multiple equilibria.3 Indeterminacy shown in these studies is,therefore, partially depends on the complex preference structure or on the production side of the model economy. In contrast, our study uses a one-sector neoclassical2

More precisely, the presence of indeterminacy requires that the marginal substitution between

private and average consumption is not constant along the equilibrium path where the average
consumption of the economy at large coincides with the level of private consumption.
3

Weder (2000) also examines equilibrium indeterminacy in growth models with consumption

externalities, but his model also involves production externalities.

3

growth model with fixed labor supply, so that the presence of heterogenous agentsis the main source of multiple equilibria.
The next section sets up the analytical framework. Section 3 examines the dynamic behavior of our model economy and presents intuitive implication of the stability conditions. Section 4 presents numerical examples. Concluding remarks are
given in Section 5.

2

The Model

2.1

Households

Suppose that there are two groups of infinitely-lived agents. Each group consists ofa continuum of identical households. The felicity function and the initial holding ofwealth of the representative household in each group are diﬀerent from each other,
but all the agents in the economy has the same rate of time preference. For simplicity,
we assume that population in the economy is constant over time, so that the massof each group will not change. We also assume that the economy is closed and thegovernment does not issue interest bearing bonds. Thus the stock of capital is the
only net asset held by the agents.
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
Z +∞
Ui =
e−ρt ui (ci , Ci , Cj )dt, ρ > 0, i, j = 1, 2, i 6= j.

(1)

0

In the above, ρ denotes a given rate of time discount, ci private consumption ofgroup i agent, and Ci and Cj respectively represent the average levels of consumption in groups 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 ci = C1 = C2 , the utility function
holds the Inada conditions: limC→0 ui1 (C, C, C) = ∞ and limC→∞ ui1 (C, C, C) = 0,4

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. Namely, 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. The presence of intergroup external eﬀects produces the outcomes specific to models with heterogenous
agents.
According to the taxonomy given by Dupor and Liu (2003), the external eﬀectof consumption on an individual utility may be either negative (jealousy) or positive(admiration). In addition, each consumer would be a conformist who likes being
similar to others (keeping up with the Joneses) or an anti-conformist who wants tobe diﬀerent from others (running away from the Joneses). We allow, for example, anagent in a particular group feels jealousy as to consumption of others in her groupbut admires consumption of agents belongs to the other group. Such a situation may
emerge, the agents in the rich group admire an increase in the benchmark level ofconsumption in the poor group, whereas they have jealousy as to the consumption
level of other members in her group. In a similar vein, it is possible to assume thatan agent wants to conduct the similar consumption as her own group’s members,
but she stays away from consumption behavior of the other group’s agents. Hence,even though there are only two types of agents, the external eﬀects among the consumers cover a richer class of situations than that treated in the representative-agent
economy.4
As usual, the negative externality (jealousy) is expressed by uij (·) (= ∂ui /∂Cj ) <0 (i = 1, 2, j = 2, 3) , while the positive externality (admiration) means that uij (·)
has a positive value. Similarly, if the marginal utility of private consumption increases
with external eﬀects, that is, ui1j (·) (= ∂ 2 ui /∂Cj ∂ci ) > 0, then the consumer’s pref4

Collier (2004) and Garriga (2006) present careful dissections on the formulation of consumption

external eﬀects. Frank (2005) interprets the households’ concern about relative consumption basedon a behavioral economics consideration.

5

erence exhibits conformism: the consumer likes being similar to others. In contrast,
the consumer is anti-conformist if ui1j (·) (= ∂ 2 ui /∂Cj ∂ci ) < 0. In what follows, we
assume that, regardless of the forms of external eﬀects, the eﬀects of a change in theprivate consumption dominate 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 diminishing marginal utility of own consumption dominates the outward looking conformism. Assumptions (2e) and (2f ) ensure that, ina social symmetric equilibrium C1 = C2 , the sign conditions given by (2a) and (2c)still holds even considering the intragroup external eﬀects.
The flow budget constraint for each agent isk̇i = r̂i ki + ŵi − ci + Ti , i = 1, 2,(3)

where, ki is capital stock owned by an agent in group i, ci consumption, r̂i after-tax
rate of return to asset, ŵi the after-tax real wage rate and Ti expresses a transferfrom the government. The initial holding of capital, ki (0) , is exogenously given.

2.2

Production

The representative firm produces a single good by use of a constant-returns-to-scaletechnology expressed by¡¢Ȳ = F K̄, N .
6

Here, Ȳ , K̄ and N denote the output, capital and labor, respectively. We normalize
the number of firms to unity so that Ȳ , K and N represent their aggregate values
as well. Using the homogeneity assumption, we write the production function as
follows:
Y = f (K) ,where Y ≡ Ȳ /N and K ≡ K̄/N. The productivity function, f (K) , is assumed
to be monotonically increasing and strictly concave in the capital-labor ratio, K,
and fulfills the Inada conditions. The commodity market is competitive so that the
before-tax rate of return to capital and real wage are respectively determined byr = f 0 (K),w = f (K) − Kf 0 (K).

(4)

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 conditions for labor and capital are:N1 + N2 = N,N1 k1 + N2 k2 = K̄.
Letting θi = Ni /N, we can rewrite the full-employment conditions as follows:
K = θ1 k1 + θ2 k2 ,0 < θi < 1,

θ1 + θ2 = 1.

(5)

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.

2.3

Fiscal Rules

The government levies distortionary income tax and distributes back its tax revenue
as a transfer to each agent. We assume that the same rate of tax applies to both
capital and labor incomes. The rate of tax applies to income of an agent in group iisτi = τ

³y ´
i

Y

,

7

i = 1, 2,where τi is the rate of tax and yi (= rki + wi ) denotes the total income of an agent
in group i. Namely, the tax rate applied to each agent depends only on its standing
in the economy.5 The tax function τ (yi /Y ): <+ → <+ is continuous, monotonicallyincreasing, a twice diﬀerentiable function and satisfies 0 < τ (yi /Y ) < 1.
¡ ¢Denoting the amount of tax payment by T (yi , Y ) = τ yYi yi , the average rate oftax is T (yi , Y ) /yi = τ (yi /Y ) and the marginal tax payment is

³ ´y
³y ´
³y ´
∂T (yi , Y )
i
i
i
0 yi
+τ
≡ Tm
.
=τ
∂yi
Y
Y Y
Y
Note that the ratio of marginal and average tax payments expresses the degree ofprogressiveness of taxation. When this measure is higher (resp. lower) than one,taxation is progressive (resp. regressive). In our formulation, progressiveness oftaxation is represented byTm (yi /Y )
τ 0 (yi /Y ) yi
=1+> 1,
τ (yi /Y )
τ (yi /Y ) Y(6)

implying that taxation is progressive. We also assume that the marginal tax payment
monotonically increases with the relative income yi /Y, so that
Tm0

³y ´
i

Y

= 2τ 0³y ´iY+ τ 00³y ´ y
i

i

Y

Y

(7)

> 0.

The after-tax rate of return and real wage received by type i agents are respectively written ash³ y ´iir,r̂i = 1 − τYh³ y ´iiŵi = 1 − τw,Yi = 1, 2.

As a result, the flow budget constraint for the household (3) is rewritten ash³ y ´iiyi − ci + Ti ,k̇i = 1 − τY

i = 1, 2.

We assume that the government follows the balanced-budget rule, so that its flow
budget constraint (in per-capita term) isθ1 T1 + θ2 T2 = θ1 τ
5

³y ´
1

Y

y1 + θ2 τ

³y ´
2

Y

y2 .

This formulation is used by Guo and Lansing (1998) and Li and Sarte (2004).

8

In addition, if we assume that the government pays back an identical amount oftransfer to each agent, the per-capita lump-sum transfer is given by the following:
T1 = T2 = θ1 τ

2.4

³y ´
1

Y

y1 + θ2 τ

³y ´
2

Y

y2 .

(10)

Consumption and Capital Formation

Under the fiscal rules given above, the type i agent’s flow budget constraint is expressed ash³ y ´iik̇i = 1 − τ(rki + w) − ci + Ti ,Y

i = 1, 2,(11)where Ti is determined by (10) . Following Guo and Lansing (1998) , we assume thatthe households perceive the rule of progressive taxation on private income, but she
takes the transfer payment, Ti , as given. Therefore, the household of type i maximizes(1) subject to (11), the initial holding of capital, ki (0) as well as to the anticipated,
given sequences of {Ci (t), Cj (t), r (t) , w (t) , Y (t) , Ti (t)}∞t=0 .
Since we have assumed that agents are identical in each group, the equilibrium
conditions involve c1 = C1 and c2 = C2 for all t ≥ 0. Taking these consistency
conditions into account, we define the following elasticities:(ui11 (Ci , Ci , Cj ) + ui12 (Ci , Ci , Cj ))Cii
Ω1 ≡ −> 0,
ui1 (Ci , Ci , Cj )ui (Ci , Ci , Cj )Cj,Ωi2 ≡ − 13i
u1 (Ci , Ci , Cj )(12)i, j = 1, 2.

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 the consumption level of group j agents. If agentsare conformist to keep up with the consumption of the other group’s members (sothat ui13 > 0), then Ωi2 has a negative sign. On the other hand, if they do not likebeing similar to the other group’s agents (ui13 < 0), then Ωi2 is strictly positive. Note
9that, from (2f ) the following is satisfied:Ωi1 + Ωi2 > 0,(13)

i = 1, 2.

Solving the households’ optimization problems yields a set of Euler equations for
optimal consumption in such a way that⎤⎡ ⎤ ⎡⎤⎡
1
1
0Ċ
{1 − τ (y1 /Y ) − (y1 /Y )τ (y1 /Y )}r − ρΩ /C Ω2 /C2⎦ ⎣ 1⎦ = ⎣
⎦,
⎣ 1 1
2
2
0
{1 − τ (y2 /Y ) − (y2 /Y )τ (y2 /Y )}r − ρΩ2 /C1 Ω1 /C2
Ċ2
{z
}
|
M

where M represents the matrix with respect to the coeﬃcients of Ċ1 and Ċ2 . Solving
this set of equations with respect to Ċ1 and Ċ2 , we obtain
⎡ ⎤
⎤
⎡
⎤⎡
2
1
0
−Ω2 /C2ĊΩ /C{1 − τ (y1 /Y ) − (y1 /Y )τ (y1 /Y )}r − ρ
C1 C2⎣ 1⎦ =
⎦.
⎣ 1 2
⎦⎣
1 2
1 2
2
1
0
Ω
Ω
−
Ω
Ω
1
1
2
2
Ċ2
{1 − τ (y2 /Y ) − (y2 /Y )τ (y2 /Y )}r − ρ−Ω /C1 Ω /C1
2

1

(14)

Equations (10) and (11) yieldh³ y ´i
³y ´
³y ´
i
1
2
yi − Ci + θ1 τ
y1 + θ2 τy2 ,k̇i = 1 − τYYYi = 1, 2.

(15)

Summing up the flow budget constraint (15) , we obtainθ1 k̇1 + θ2 k̇2 = θ1 y1 + θ2 y2 − θ1 C1 − θ2 C2 .
Thus, from yi = rki + w and (5) , we obtain the final-good market equilibrium
condition for the entire economy:K̇ = f (K) − C,where C = θ1 C1 + θ2 C2 .

3

Macroeconomic Stability

3.1

Dynamic System

Equations (4) and (5) giveyi = rki + w = f (K) + (ki − K)f 0 (K),10

leading toyi
f 0 (K)= 1 + (ki − K)
,
Y
f (K)

i = 1, 2,(16)

where K = θ1 k1 +(1 − θ1 ) k2 . Plugging (16) into (14) and (15) , we obtain a complete
dynamic system that depicts the dynamic behaviors of k1 , k2 , C1 and C2 .
The solution of this dynamic system that fulfills the initial conditions on k1 (0) andk2 (0) as well as the transversality conditions for the households’ optimization problem, limt→∞ ui1 (Ci (t) , Ci (t) , Cj (t)) e−ρt ki (t) = 0 (i = 1, 2), presents the perfectforesight competitive equilibrium of our model economy.6

3.2

Steady-State Equilibrium

In the steady-state equilibrium, ki and Ci (i = 1, 2) stay constant over time. From
(14) and (15), the steady-state conditions are given by∙ µ ∗¶µ ∗¶ ¸yjyi∗∗∗yj − τ
yi∗ , i, j = 1, 2, i 6= j,
Ci = yi + θj τ
∗
Y
Y∗
µ ¶¸
∙
µ ∗¶
yi∗ 0 yi∗yi0∗
ρ = f (K ) 1 − τ− ∗τ, i, j = 1, 2,
Y∗
Y
Y∗

(17)

(18)

where Ci∗ and ki∗ denote steady-state levels of ki and Ci .

To simplify analytical argument, we make the following assumption:¡ ¢¡ ¢Assumption 1. τ yYi + yYi τ 0 yYi (i = 1, 2) is a monotonic function of the relative

income, yi /Y.

Since the derivative of the above function with respect to yi /Y is 2τ 0 (yi /Y ) +(yi /Y ) τ 00 (yi /Y ) , from (7) Assumption 1 means that the marginal tax payment,∂ 2 (τ yi ) /∂yi2 , has the same sign for all feasible levels of yi /Y. Given Assumption 1,
it is easy to confirm the following fact:
Proposition 1. There is a unique, symmetric steady state in which k1∗ = k2∗ andC1∗ = C2∗ .
6

Mino and Nakamoto (2008) examine the role of nonlinear income taxation in a heterogenous-

agent model of growth without consumption externalities.

11

Proof. Conditions displayed in (18) yield
µ ¶
µ ∗¶
µ ¶
µ ∗¶
y1∗ 0 y1∗y2
y2∗ 0 y2∗
y1
+ ∗τ
=τ
+ ∗τ
.
τ
Y∗
Y
Y∗
Y∗
Y
Y∗
By Assumption 1, the above equation holds if and only if y1∗ = y2∗ . Thus from (17) it
holds that C1∗ = C2∗ . ¥Note that y1∗ = y2∗ = Y ∗ and k1∗ = k2∗ = K in the symmetric steady state, so thatthe rate of income tax in the steady-state equilibrium is a given constant, τ (1) . To
make the steady state feasible, from (18) we should assume the following:
Assumption 2. Tax function τ (yi /Y ) satisfies
1 − τ (1) − τ 0 (1) > 0.

3.3

(19)

Stability

Let us examine the local stability condition of the steady state equilibrium defined
above. Linear approximation of dynamic system, (14) and (15) , around the steadystate equilibrium yields the following:
⎡
⎡ ⎤
⎤
∗
C (t) − C1
Ċ
⎢ 1
⎢ 1⎥
⎥
⎢
⎢ ⎥
⎥
∗
⎢C2 (t) − C2 ⎥⎢Ċ2 ⎥⎢ ⎥=J⎢⎥,⎢⎢ ⎥⎥⎢ k1 (t) − k1∗ ⎥⎢ k̇1 ⎥⎣⎣ ⎦⎦∗k2 (t) − k2k̇2

where the coeﬃcient matrix J is⎡⎤00
∂ Ċ1 /∂k1
∂ Ċ1 /∂k2
⎢
⎥
⎢
⎥
⎢0
⎥
0
∂ Ċ2 /∂k1
∂ Ċ2 /∂k2
⎢
⎥.
J =⎢
⎥
0
∗
0
0
∗
0
⎢−1 0 f (k )[1 − θ2 (τ (1) + τ (1))]θ2 f (k )[τ (1) + τ (1)] ⎥⎣⎦0 −1
θ1 f 0 (k ∗ )[τ (1) + τ 0 (1)]f 0 (k∗ )[1 − θ1 (τ (1) + τ 0 (1))]

Each element in J is evaluated at the steady state. The precise expression of J isdisplayed in Appendix A of the paper.
Let us write the characteristic equation of J in such a way that
λ4 − TrJλ3 + WJλ2 − ZJλ + DetJ = 0,
12

(20)

where
TrJ = f 0 (k∗ )[2 − τ (1) − τ 0 (1)] > 0,
∂ Ċ1 ∂ Ċ2+,
WJ = f 0 (k∗ )ρ +
∂k1
∂k2
½
∂ Ċ1
∂ Ċ2
0 ∗
+ [1 − θ2 (τ (1) + τ 0 (1))]
ZJ = f (k ) [1 − θ1 (τ (1) + τ 0 (1))]
∂k1
∂k2
#¾
"
∂ Ċ1
∂ Ċ2
,
− (τ (1) + τ 0 (1)) θ1
+ θ2
∂k2
∂k1

f (k∗ )f 0 (k∗ )f 00 (k∗ )ρ 0
[2τ (1) + τ 00 (1)].
DetJ = −1 2
1 2Ω1 Ω1 − Ω2 Ω2

(21)
(22)

(23)
(24)

Note that this model involves two jumpable variables, C1 and C2 . Thus the necessary
and suﬃcient condition for local determinacy is that the characteristic equation (20)
has two roots with negative real parts. Considering the form of (24), we see thatthe sign of the determinant depends on the households’ preferences as well as on theincome taxation scheme. In the subsequent discussion, we assume that the marginal
tax payment increases with the relative income around at least the steady state.
Assumption 3. Tax function τ (yi /Y ) satisfies
2τ 0 (1) + τ 00 (1) > 0.

(25)

Inspecting the characteristic equation given above, we obtain the main result of
this paper:
Proposition 2. Given Assumptions 1, 2 and 3, if Ω11 Ω21 − Ω12 Ω22 > 0 and θ1 (Ω21 −Ω12 ) + θ2 (Ω11 − Ω22 ) > 0, then the steady-state equilibrium satisfies local determinacy.

Proof. Let us denote roots of the characteristic equation by λs (s = 1, 2, 3, 4) . As(21) shows, from Assumption 2 the sign of the trace of J, which equals Σ4s=1 λs , isstrictly positive. Hence, at least one of the characteristic roots has positive real part.
Since the marginal tax payment increases with the relative income under Assumption
3, the sign of the determinant J (= Π4s=1 λs ) is strictly positive if Ω11 Ω21 > Ω12 Ω22 : see(24) . This means that the number of characteristic roots with positive real parts is

13

either two or four. Finally, we rewrite ZJ as follows :½¤£(f 0 )3
∗
2
2
1
1
2
Γ(k
ZJ = 1 2
)∆
(Ω
−
Ω
)
+
θ
(Ω
−
Ω
)
θ
1
2
1
2
1
2
Ω1 Ω1 − Ω12 Ω22
¾
£
¤
2
2
1
1
0
00
− θ2 (Ω1 + Ω2 ) + θ1 (Ω1 + Ω2 ) (2τ (1) + τ (1)) .

(26)

As (26) shows, if θ1 (Ω21 − Ω12 ) + θ2 (Ω11 − Ω22 ) > 0, then ZJ, which equals λ1 λ2 λ3 +λ2 λ4 λ1 +λ3 λ4 λ1 +λ2 λ3 λ4 , has a negative sign. Hence, there are at most two characteristic roots with positive real part. This demonstrates that there is a two—dimensional
stable manifold around the steady state, implying that the competitive equilibrium
path converging to the steady state is uniquely determined. ¥In order to interpret Proposition 2, it is useful to consider the following three special cases. First, the above result means that if Ω12 = Ω22 = 0, then the economy has
a unique converging path towards the socially symmetric steady-state equilibrium,
regardless of the initial distribution of wealth and form of utility function of eachtype of agents. That is, as long as households in a group have neither jealousy nor
admiration about the average consumption level in the other group, the economy has
saddle-path stability and the competitive equilibrium path is uniquely determined.
Second, even if Ω12 6= 0 and Ω22 6= 0 so that there are intergroup external eﬀects,
the economy has a unique converging path towards the socially symmetric steadystate, as long as Ω21 > Ω12 and Ω11 > Ω22 . This result holds regardless of the signs ofΩ12 and Ω22 . Therefore, even if individuals’ preferences exhibit conformism or anticonformism as to the other group’s consumption behaviors, the economy satisfies thesaddlepoint stability when the degree of intergroup external eﬀects is small enough.
Finally, the economy satisfies saddlepoint stability if Ω12 and Ω22 have diﬀerent
signs. For example, assume that agents in group 1 are richer than agents in group
2. Then, it is plausible to assume that that agents in group 2 like being similar to
the average consumption in the richer group (group 1), whereas agents in group 1
have anti-conformism as to the average consumption in the poorer agents (group 2).
If this is the case, it holds that Ω12 > 0 and Ω22 < 0, which ensures that the economy
has saddlepoint property.
The proof of Proposition 2 immediately yields the following result:
14

Proposition 3. Given Assumptions 1, 2 and 3, if Ω11 Ω21 −Ω12 Ω22 < 0, then the steadystate equilibrium is either locally unstable or indeterminate.
Proof. Equation (24) shows that the determinant of J is strictly negative when
Ω11 Ω21 − Ω12 Ω22 < 0. In this case the number of characteristic roots with negative
sign is either one or three. The former case means that the stable manifold is one
dimensional around the steady state and thus no converging path can be selected forarbitrarily given levels of initial capital stocks, k1 (0) and k2 (0) . If there are threestable roots, there exists a continuum of converging paths starting from the given
initial distribution of capital stocks.¥
The above proposition fails to specify when indeterminacy emerges. Since itis hard to present the analytical conditions for local indeterminacy (the suﬃcientconditions under which that the characteristic equation has three stable roots), we
inspect numerical examples in Section 4.

3.4

Intuition

As shown by Propositions 2 and 3, the key to determine dynamic behavior of ourmodel economy is the sign of Ω11 Ω21 − Ω12 Ω22 . To present an intuitive implication of
the stability conditions, it is useful to inspect the Euler equations given below:onh³ y ´ y ³ y ´i
Ω2
Ċ1
1
1
1
= 1 2 1 1 2 1−τ− τ0
f 0 (K) − ρC1Ω1 Ω1 − Ω2 Ω2
Y
Y
Y
(27)
nh
³ y ´ y ³ y ´i
o
Ω12
2
2 0
2
0
1−τ
− 1 2
− τ
]f (K) − ρ ,Ω1 Ω1 − Ω12 Ω22YYYnh³ y ´ y ³ y ´i
o
Ċ2
Ω1
2
2
2
− τ0
f 0 (K) − ρ= 1 2 1 1 2 1−τC2Ω1 Ω1 − Ω2 Ω2YYYnh³ y ´ y ³ y ´i
o
Ω22
1
1 0
1
0
− 1 2
−
τ
f
1
−
τ
(K)
−
ρ
.
Ω1 Ω1 − Ω12 Ω22
Y
Y
Y

(28)

If there is no intergroup external eﬀect, i.e. Ωi2 = 0 (i = 1, 2) , then the Euler
equations become³ y ´ y ³ y ´i
o
Ċ1
1 nh
1
1 0
1
0
− τ
f (K) − ρ ,= 1 1−τ
C1
Ω1
Y
Y
Y
o
³ y ´ y ³ y ´i
Ċ2
1 nh
2
2 0
2
0
= 2 1−τ− τ
f (K) − ρ ,
C2
Ω1
Y
Y
Y
15

Ω11 > 0,(29)

Ω21 > 0.

(30)

Comparing those two sets of Euler equations, we may obtain intuitive implication as
to why equilibrium indeterminacy could be present if Ω11 Ω21 − Ω12 Ω22 has a negative
sign.
Suppose that the economy initially stays at the steady state equilibrium. Suppose
further that all the agents anticipate that the before-tax rate of return to capital willrise so that the after-tax rate of return they receive will increase as well. In theabsence of intergroup externalities, (29) and (30) state that the after-tax rate of return exceeds the time discount rate and, hence, consumption growth rate becomes
positive. Namely, the current consumption is substituted with the future consumption, which raises the current saving to accelerate capital accumulation. A rise incapital stock, however, depresses the rate of return to capital due to our assumptionof diminishing marginal returns. Consequently, if there are only intragroup externalities, the initial anticipation of a rise in the rate of return to capital will not be
self-fulfilled, implying that equilibrium indeterminacy may not emerge.
In contrast, if there are intergroup externalities as well and if the intergroupeﬀects dominate intragroup eﬀects (Ω11 Ω21 − Ω12 Ω22 < 0), then an expected rise inthe (after-tax) rate of return to capital will lower the growth rate of consumption
within own group: see the first terms in the right- hand sides of (27) and (28) . In
this situation, the future consumption is substituted with the current consumption,
which depresses investment. As a result, the stock of aggregate capital will decline,
so that the rate of return to capital will rise. This indicates that the initial change in
expectations may be self-fulfilled and sunspot-deriven changes in expectations aﬀect
the equilibrium path.
The above intuition, however, ignores the cross eﬀects on optimal consumption
represented by the second terms in the right-hand sides of (27) and (28) . For example,
suppose that agents in each group are conformists who like being similar to membersof the other group (i.e. Ωi2 < 0, i = 1, 2) . Then the anticipated rise in the rate of
return to capital also accelerates consumption, which is generated by the intergroup
external eﬀects: see the signs of coeﬃcients of the second terms in the right-handsides of (27) and (28) . Therefore, given our assumption of Ω11 Ω21 − Ω12 Ω22 < 0, those16

additional adjustments in the same direction could enhance instability so that theeconomy diverges from the steady state. In contrast, if agents are anti-conformist as
for other group’s consumption (Ωi2 > 0, i = 1, 2) , the eﬀects of an expected rise in therate of return to capital on consumption caused by the own eﬀect would be mitigatedby the cross eﬀect. This may prevent the economy’s diverging behavior. Althoughsuch an intuitive discussion cannot present the precise mechanism that generates
multiple equilibria, the numerical examples given in the next section suggest that ourintuition at least partially characterizes equilibrium dynamics of our model economy.

4

Numerical Analysis

In the previous section we have confirmed that if the sign of Ω11 Ω21 −Ω12 Ω22 is negative,then the steady-state equilibrium is either locally indeterminate or unstable. For the
purpose of distinguishing the conditions for indeterminacy from these for instability,
this section conducts numerical experiments by specifying the utility, production andtax functions.
We use the following utility function:ui (ci (t), Ci (t), Cj (t)) =1 ³ φi ηi ´1−γi
ci Ci Cj
, i, j = 1, 2, i 6= j.
1 − γi(31)

Here, 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. From (31) we find that
Ω11 Ω21 − Ω12 Ω22= {−γ1 + φ1 (1 − γ1 )} {−γ2 + φ2 (1 − γ2 )} − η1 η2 (1 − γ1 )(1 − γ2 ).
In view of conditions (2c) and (2f ), the following inequalities must be satisfied:Ωi1 = γi − φi (1 − γi ) > 0,
Ωii + Ωij = γ1 − (φ1 + η1 )(1 − γ1 ) > 0,17

i = 1, 2,i, j = 1, 2,

i 6= j.

(32)

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

i = 1, 2.

As for the production function, it is given by Cobb-Douglas one:f (K) = AK α , 0 < α < 1, A > 0,(33)

where K = θ1 k1 + θ2 kt .
The tax function is specified as
τ

³y ´
i

(yi /Y )ξ

=Y

b + m (yi /Y )ξ

(34)

,

where
b + m > 0, bξ > 0, and (b + ξ)2 > b(1 + ξ) + m.
It is to be noticed that (34) fulfills all of our assumptions on the tax function including
Assumption 1.7 Under this specification of tax function, the key values evaluated at
the steady state equilibrium are given by the following:τ (1) =
τ 0 (1) =τ 00 (1) =bξ> 0,(b + m)2bξ{b(ξ − 1) − m(1 + ξ)},(b + m)3(b + m)2 − b(1 + ξ) − m> 0.
(b + m)2

1 − τ (1) − τ 0 (1) =
7

1
> 0,
b+m

Guo and Lansing (1998) and Li and Sarte (2004) specify the tax function in such a way that
τ

³y ´
i

Y

= τ0

³ y ´φiY,

0 < τ0 < 1,

φ < 1.

This specification also yields:∂ (τ (yi /Y ) yi ) /∂yi
= 1 + φ > 1,
τ (yi /Y )2τ 0 (1) + τ 00 (1) = φ (φ + 1) > 0.
However, this specification may violate the feasibility condition, 0 < τ (.) < 1.

18

The magnitudes of parameters concerning production, population distribution
and tax functions are given byθ1 = 0.2, A = 1,

α = 0.3, b = 4, m = 3, ξ = 2.5, ρ = 0.025.

Then the before-tax rate of return to capital, r, is 0.0382813 and the rate of theincome tax is 0.1428571 so that 1 − τ (1) − τ 0 (1) and 2τ 0 + τ 00 (1) have positive values.8
As for the parameter values concerning the preference structure, we consider thefollowing three sets:(i) γ1 = 0.3, γ2 = 0.6, φ2 = 0.2, η2 = −0.8,(ii) γ1 = 0.3, γ2 = 2.5, φ2 = −0.9, η2 = 0.45,(iii) γ1 = 1.8, γ2 = 4.5, φ2 = −0.9 η2 = 0.6.
Example (i) assumes thatsign u22 = sign φ2 > 0,sign u212 = sign φ2 (1 − γ2 ) > 0,sign u23 = sign η2 < 0,sign u213 = sign η2 (1 − γ2 ) < 0.

Hence, the agents in group 2 have admiration as well as conformism about theconsumption behavior of their own group’s members, while they are jealous but
anti-conformist about the consumption level of group 1’s agent. Similarly, examples(ii) and (iii) assume:u22 < 0, u212 > 0,

u23 > 0,u213 < 0,

implying that the agents of group 2 have jealousy and conformism about their owngroup’s average consumption; and they admire but have anti-conformism as to theother group’s consumption.
Given those parameter magnitudes, we change φ1 and η1 with an intervals of
0.01. Figures 1, 2 and 3 respectively depict the case with preference parameters (i),8Since we have ignore capital depreciation, the before tax rate of return to capital in the steadystate has a rather high value.

19

(ii) and (iii) displayed above. In these figures, we divide (φ1 , η1 ) space according to
the stability conditions. The areas with shadow between stable and unstable regions
represent the combination of φ1 and η1 that yields local indeterminacy. As the figuresdemonstrate, although the parameter space for indeterminacy is relatively small, we
can find the possibility of equilibrium indeterminacy for wide ranges of values of φ1and η1 .
More specifically, Figure 1 shows that the equilibrium indeterminacy in example(i) emerges when
η1 < 0,φ1 > 0,φ1 (1 − γ1 ) > 0, η1 (1 − γ1 ) < 0

Thus in case (i) the existence of equilibrium indeterminacy requires that the agentsin groups 1 and 2 have the same preference structure. In case (ii), as shown by Figure
2, the equilibrium indeterminacy again emerges if the following conditions hold:η1 < 0,φ1 > 0,φ1 (1 − γ1 ) > 0 and η1 (1 − γ1 ) < 0

and, hence, the preferences of group 1’s agents are the same as those in case (i).
Note that, as long as conformism and anti-conformism are concerned, group 2’s
agents have the same preference as that held by group 1’s agents. Finally, consider
case (iii). Figure 3 demonstrates that indeterminacy may be observed when
η1 > 0,

φ1 < 0,φ1 (1 − γ1 ) > 0 and η1 (1 − γ1 ) < 0.

The common feature of those examples is that when indeterminacy emerges, theagents in each group are conformist to their own group’ consumption, but they haveanti-conformism as for the other group’s consumption behavior. Therefore, in all theexamples it holds thatui (Ci , Ci , Cj )CjΩi2 ≡ − 13i> 0,
u1 (Ci , Ci , Cj )i 6= j,i, j = 1, 2.

(35)

Since (32) shows that examples (i), (ii) and (iii) satisfy Ω11 Ω21 − Ω12 Ω22 < 0, (35)means that all of the adjustment coeﬃcients of cross terms in the right hand sidesof Euler equations (27) and (28) have positive values. Hence, the intuitive argument
in Section 3.4 may be supported by our numerical experiments.
20

Finally, it should be pointed out that our numerical examples assume that there
is a considerable diﬀerence in population share of each group (our examples set
θ1 = 0.2 and θ2 = 0.8).9 When using alternative parameter values that are notdisplayed here, we have found that if each group has a similar population share (forexample, θ1 = θ2 = 0.5), then indeterminacy does not hold under plausible values ofother parameters. That is, if θ1 is close to θ2 , the steady-state equilibrium is always
unstable if Ω11 Ω21 − Ω12 Ω22 < 0. This fact may come from our modeling strategy that
focuses on the symmetric steady state. To see this, note that the magnitude of θidoes not directly aﬀect the optimal consumption decision of each group, but it aﬀects
accumulation of each group’s capital stock. Since the steady state is symmetric andsince the transfer for every agent is assumed to be identical (T1 = T2 ), if θ1 is closeto θ2 , then the dynamic behaviors of k1 and k2 are not so much diﬀerent from eachother near the steady-state equilibrium. That is, the behavior of individual capitalis similar to that of aggregate capital, K. Baed on the intuitive discussion in Section
3.4., we may conjecture that in the case of Ω11 Ω21 − Ω12 Ω22 < 0, the dynamic systemtends to be unstable if the aggregate capital behaves like that in the representative
agent economy. As a result, the emergence of indeterminacy needs that the behavior
of individual capital is suﬃciently diﬀerent from each other so that the dynamic
motion of aggregate capital is diﬀerent from one observed in the representative agent
model. In our setting, therefore, the presence of equilibrium indeterminacy requires
a suﬃcient degree of heterogeneity in population distribution.

5

Concluding Remarks

We have shown that if there are heterogenous agents and if consumption external
eﬀects perceived by consumers are not uniform, then the equilibrium path of thestandard Ramsey economy would not display a regular saddle point property. The
equilibrium dynamics could be unstable or indeterminate if the intergroup externalities have distinctive eﬀects on the consumers’ behaviors. In order to facilitate
9

Obviously, our results hold in the opposite situation such that θ1 is large and θ2 is small.

21

comparison between the heterogenous-agent economy with the representative agent
counterpart, this paper introduces a specific form of nonlinear income taxation that
ensures the presence of a unique and symmetric steady-state equilibrium.
It is worth emphasizing that our stability results do not rely on the fact that
we have restricted our attention to the symmetric steady state in which income andwealth distribution are equalized among the consumers. In the symmetric steadystate of our model with the utility function given by (31), equation (32) shows thatthe sign of Ω11 Ω21 −Ω12 Ω22 depends only on the parameter values representing consumption external eﬀects. If the steady state equilibrium is not symmetric in the sensethat an unequal wealth distribution between groups remains, then the steady-state
expression of Ω11 Ω21 − Ω12 Ω22 depends on the steady-state values of C1∗ and C2∗ . As an
example, let us assume that each type of agent has diﬀerent rate of time discountrate. In this case, the steady state conditions (18) are replaced with
µ ¶¸
∙
µ ∗¶
yi∗ 0 yi∗
yi
0
∗
− ∗τ
, i = 1, 2.
ρi = f (K ) 1 − τY∗YY∗Since τ (yi /Y ) + τ 0 (yi /Y ) (yi /Y ) is assumed to be monotonic function of yi /Y, theabove equations demonstrate that if ρi > ρj , then yi∗ /Y ∗ < yj∗ /Y so that yi∗ < yj∗ . As
a result, (17) means that Ci∗ < Cj∗ . This means that the magnitude of Ω11 Ω21 − Ω12 Ω22

is aﬀected by the steady state values of C1∗ and C2∗ . Since C1∗ and C2∗ are determinedby production technology, in the asymmetric steady state the parameter values depicting consumption externalities alone cannot determines the dynamic behavior of
the economy. However, it is easy to see that the condition Ω11 Ω21 − Ω12 Ω22 > 0 is stillnecessary for establishing a regular saddle-point stability in an asymmetric steadystate equilibrium.
In this paper we have employed a simple Ramsey model with fixed labor supply
and a constant returns to scale technology. It would be useful to reconsider our
discussion in models with increasing returns and/or endogenous labor supply.

22

Appendix
In this appendix, we analyze the conditions under which the steady-state equilibrium exhibits saddle-path stability. Let be assuming that ∆ ≡ 1 − τ (1) − τ 0 (1) > 0.
The coeﬃcients of the matrix J are given by:
ª
C∗
∂ Ċ1
(f 0 )2 © 2(Ω1 − Ω12 )θ1 Γ(k ∗ )∆ − (θ2 Ω21 + θ1 Ω12 )(τ 00 (1) + 2τ 0 (1)) ,
= 1 2
1 2
∂k1
Ω1 Ω1 − Ω2 Ω2 fª
(f 0 )2 © 2∂ Ċ1C∗(Ω1 − Ω12 )θ2 Γ(k∗ )∆ + (θ2 Ω21 + θ1 Ω12 )(τ 00 (1) + 2τ 0 (1)) ,
= 1 2
1 2
∂k2
Ω1 Ω1 − Ω2 Ω2 fª
∂ Ċ2
(f 0 )2 © 1
C∗
2
∗
1
2
00
0
= 1 2
−
Ω
)θ
Γ(k
)∆
+
(θ
Ω
+
θ
Ω
)(τ
(1)
+
2τ
(1))
,
(Ω
1
1
2
1
2
1
2
∂k1
Ω1 Ω1 − Ω12 Ω22 fª(f 0 )2 © 1∂ Ċ2C∗(Ω1 − Ω22 )θ2 Γ(k ∗ )∆ − (θ1 Ω11 + θ2 Ω22 )(τ 00 (1) + 2τ 0 (1)) .
= 1 2
1 2
∂k2
Ω1 Ω1 − Ω2 Ω2 f

Plugging the above expressions into ZJ and arranging terms, we obtain the following:©
(f 0 )3Γ(k∗ )∆2 [θ1 (Ω21 − Ω12 ) + θ2 (Ω11 − Ω22 )]1 2
1 2Ω1 Ω1 − Ω2 Ω2ª−[θ1 (Ω11 + Ω12 ) + θ2 (Ω21 + Ω22 )](2τ 0 (1) + τ 00 (1)) .

ZJ =Since Γ (k ∗ ) < 0 and τ 00 (1) > 0 if θ1 (Ω21 − Ω12 ) + θ2 (Ω11 − Ω22 ) > 0, then ZJ is strictly
negative. As a result, there are at most two roots that have positive real parts.

23

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25

Figure 1: γ1 = 0.3, γ2 = 0.6, φ2 = 0.2 and η2 = −0.8
η1
0
−0.05

Stable

−0.1
−0.15
−0.2
−0.25

Unstable

−0.3

φ1

0.2

0.25

0.3

0.35

0.4

0.45

Figure 2: γ1 = 0.3, γ2 = 2.5, φ2 = −0.9 and η2 = 0.45

η10Stable

−0.1

−0.2

−0.3
Unstable
−0.4
0.15

φ1
0.2

0.25

0.3

0.35

0.4

Figure 3: γ1 = 1.8, γ2 = 4.5, φ2 = −0.9 and η2 = 0.65

η1
0.95
Unstable
0.9

0.85
Stable

0.8

0.75
−0.9

φ1
−0.85

−0.8

−0.75

−0.7

−0.65