Discussion Papers In Economics
And Business

Wealth Distribution Dynamics with Status
Preference: asymmetric motivations for status
Yutaro Hatta
Discussion Paper 13-08

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANWealth Distribution Dynamics with Status
Preference: asymmetric motivations for status
Yutaro Hatta
Discussion Paper 13-08
May 2013

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANWealth Distribution Dynamics with Status
Preference: asymmetric motivations for status∗
Yutaro Hatta†

Abstract
This paper explores the implications of status-seeking behavior in wealthfor economic dynamics. I move away from the conventional setup of statuspreference. That is, individuals with higher wealth behave diﬀerently ascompared to those with lower wealth; agents with diﬀerent wealth haveasymmetric motivations for social status: mathematically, there is a kink instatus utility, which is empirically suggested. The main results are as follows:
the stability of the equal steady state in an economy is determined by which
agents have higher motivations for status. Depending on parameter values,
whether inequality may diminish or persist is determined. Inequality andoutput are intricately related, due to asymmetric motivations.

Keywords: Status-seeking, Relative wealth, Wealth Distribution
JEL Classification: D91, D31, O40∗

I am grateful to Koichi Futagami and Shinsuke Ikeda for their helpful comments and suggestions. I especially thank Tatsuro Iwaisako for his invaluable comments and encouragement. The
ﬁnancial support from JSPS Research Fellowships for Young Scientists is greatly acknowledged.
I am responsible for any remaining errors.
†
Graduate School of Economics, Osaka University, 1-7, Machikaneyama, Toyonaka, Osaka
560-0043, Japan, Email address: kge009hy@mail2.econ.osaka-u.ac.jp1

1

Introduction

There is much empirical support for the “relative utility hypothesis,” which emphasizes that an individual’s relative position in the society should play an important role in determining life satisfaction. Individuals care about not only theabsolute level but also the relative level of various economic variables including
consumption, human capital, wealth, and so on. Each paper adopts a diﬀerent
motivation for status along with each research interest. There are two main ways
to incorporate status concern. Rauscher (1997), Fisher and Hof (2000), Liu andTurnovsky (2005), Garcia-Penalosa and Turnovsky (2008) examine the implications of consumption externalities for macroeconomic dynamics. On the other
hand, the approach adopted by Futagami and Shibata (1998), Kawamoto (2009)and Long and Shimomura (2004) view wealth accumulation to be motivation for
status.1
In this paper, I investigate the implications of status-seeking behavior in wealth.
In particular, a kind of asymmetry is assumed in status preference, that is, individuals with higher wealth than the mean of social wealth behave diﬀerently ascompared to those with lower wealth. Barnett et al. (2010) deal with this situation in a static model. In their model, individuals with heterogeneous-ability
feel status-derived satisfaction from consumption. Furthermore, the status utility
pays a utility premium if consumption exceeds the social mean. Hence, the status
utility exhibits a kink at a particular point. In other words, the utility function is
not concave over the domain and is convex for some interval. Therefore, it paves
a road to risk prone consumers. In fact, their paper investigates the relationshipbetween ex ante ability distribution and ex post income distribution. This paper
goes along with their paper, but examines the implications of such a setting in a
dynamic situation.
Empirical studies also support the asymmetric speciﬁcation of status utility.
Vendrik and Woltjer (2007) estimate the status utility function of relative income,using the German Socio Economic Panel. They ﬁnd that the status utility function
exhibits a kink at the average income, but the “direction” of the kink is somewhat
diﬀerent from the way Barnett et al. specify. Speciﬁcally, those with lower income
have higher marginal utility of relative income and vice versa. Therefore, thefunction is entirely concave.
Along with status preference, one of the important topics in economics is capital
accumulation, since the feature of status-seeking behavior in wealth seems to playa role in dynamic decisions and aﬀects capital accumulation. This is what thepresent paper deals with. Furthermore, I investigate how inequality evolves in an
economy and how it aﬀects the economy as a whole.
I model an economy that consists of two types of agents. The diﬀerence between the two is only in terms of initial asset holdings. My main results can be
1

Much has been written about wealth accumulation as a source of motivation for social status.
Corneo and Jeanne (2001) show that status seeking behavior can be an engine of long-run growth,even when there is no other engine such as technological spillover. Moreover, status preferenceof wealth can support a balanced growth path on which, even if agents with diﬀerent subjective
discount rates coexist, each agent holds a positive portion of the world’s wealth (Futagami andShibata (1998)). Their results are in contrast to preceding literature which shows that the mostpatient agent eventually holds all of the world’s assets in a steady state (Becker (1980)).

2

summarized as follows. The utility premium in status utility, as in Barnett et al.
(2010), results in the instability of the equal steady state. It is seen that an economy goes to an unequal steady state. On the other hand, in Vendrik and Woltjer(2007), the equal steady state is stable and an economy is likely to converge towardit.
There exists some literature discussing the dynamics of wealth distribution
in the status preference model. Kawamoto (2009) investigates the status-seeking
behavior of relative income and focuses on attitudes of individuals to change inreference income. If the marginal utility of individual’s own income rises as theaverage income rises, her preference is called KUJ (keeping up with the Joneses). If
the opposite is the case, I call it RAJ (running away from the Joneses). Kawamoto(2009) shows that income inequality diminishes in a KUJ economy, with any levelof initial inequality. In contrast, in an RAJ economy, inequality expands, as longas initial endowments are diﬀerent across agents. Using an inﬁnitely lived agentsmodel, Long and Shimomura (2004) show that the poor will catch up with therich if the elasticity of the marginal utility of relative wealth is greater than theelasticity of the marginal utility of consumption. Kawamoto (2009) and Longand Shimomura (2004) specify the conditions under which an economy goes to a
totally equal state or an unequal one; this paper does the same. However, these twoworks emphasize diﬀerent aspect of status utility. Furthermore, previous literature
expects an extreme situation in the long run: either all agents reach the same
circumstances or some agents hold all of the world assets. In this paper, the lattercase does not occur.
The connection between inequality and economic development is generally perceived as the interaction between the two: one aﬀects the other. In this sense,
this paper also relates to the studies investigating how output and inequality are
jointly determined. Garcia-Penalosa and Turnovsky (2006) construct an endogenous growth model and examine the relationship between growth rate and income
distribution. They ﬁnd that a higher growth tends to be associated with a moreunequal economy. Therefore they predict a positive relationship between growthand inequality. This paper shows that the relationship between the two variables
relies heavily on the nature of status preference, as does the evolution of inequality. In some cases, when inequality is low, the relationship between inequality andoutput emerges as a positive correlation; when it is high, a negative correlationarises. In other cases, it may be entirely positive, regardless of the level of inequality. Because both possibilities, positive or negative correlation, are suggested byempirical evidences, this paper can explain both cases.2 3 Garcia-Penalosa andTurnovsky (2008) examine the implication of status seeking behavior in consumption, using a model with inﬁnitely-lived heterogeneous agents. In their model, thelong-run output is independent of the nature of status preference, unlike with this2

For example, Barro (2000) shows that inequality tends to retard growth in poor countries
and enhance growth in rich ones, however Forbes (2000) suggests that more income inequality is
associated with higher growth rate.
3
The theoretic relationship is also controversial. Corneo and Jeanne (2001) uses a model
where individuals care about consumption and their rank in the distribution of wealth, andshow that more inequality results in higher growth rate. However, Peng (2008) makes a slightchange to Corneo and Jeanne model to show that the reversed relation would appear. Corneo
and Jeanne (1998), and Hopkins and Kornienko (2004) point out that whether inequality fosters
growth depends on when individuals are concerned with their status.

3

v(·)

v(·)

1

bi /b̄

1

bi /b̄

Fig. 1: Status utility function
paper. Hence, in their paper, status preference for consumption only aﬀects thetransition path to the steady state.
The rest of this paper is organized as follows. Section 2 describes the basicmodel. Sections 3 and 4 show the dynamics of inequality in an economy. In
section 5, I investigate capital accumulation in the model and see the eﬀect of theredistribution policy. Finally, section 6 concludes.

2

Model

Consider a nonoverlapping generations economy with a continuum of populationof mass of 1. The economy is populated by two groups with diﬀerent levels of
initial asset holdings. I use H and L to denote households with higher and lower
initial assets, respectively. The share of each group is half. Each household has
the same utility function so that the only diﬀerence is asset holdings. Households
have one unit of labor and exogenously supply to the production sector. Moreover,each household has an asset inheriting from parents as bequest. There is a single
good in this economy and I assume that a single good can be either consumed orbequeathed to the oﬀspring. Bequests are used as capital inputs for productionin the subsequent generation. I do not consider population growth so that the
population in the economy is constant over time.

2.1

Households

Agents derive utility from their own consumption and bequest for their children.
However, utility from bequest is derived by a status seeking motive, that is, bequest
gives satisfaction to agents in comparison to bequests made by others. Although
it seems odd that agents seek status through bequeathing, I can interpret it as
status-seeking behavior in wealth. The status utility of a household comes fromwealth, as compared to the wealth of others. When an agent leaves the economy,
his child comes to be the wealth holder.4
The utility function of an agent in generation t is given by
4

Mitsopoulos (2009) also uses the model in which agents seek status in bequeathing.

4

Uti = u(cit ) + v(bit+1 /b̄t+1 ){(bi /b̄t+1 )1−θ −1
log cit + β1 t+1 1−θfor bit+1 ≥ b̄t+1=(bi /b̄t+1 )1−θ −1
log cit + β2 t+1 1−θfor bit+1 < b̄t+1.

(1)

The ﬁrst term u(cit ) represents the utility derived by consuming goods of cit .
I assume that utility from consumption takes the logarithmic form. The second
term v(bit+1 /b̄t+1 ) represents the utility derived by giving bequest to the oﬀspring.
The bequest of agent i is indicated by bit+1 and the average bequest is given byb̄t+1 . Therefore, agents feel satisfaction by giving bequests in comparison to social
average.
As shown in Figure 1, depending on whether β1 is larger than β2 or not, a kinkmay arise in status utility. When β1 > β2 , the shape of v(·) is the same as what
Barnett et al. (2010) assume in a static model. In this case, I can interpret thatthe wealth exceeding the social average pays a utility premium. Barnett et al.
(2010) refer to this speciﬁcation as an option that individuals can choose whether
to participate in the rat race or not. On the other hand, when β1 < β2 , thefunction is concave on the entire domain. Vendrik and Woltjer (2007) estimates
the status utility function of relative income, using the German Socio Economic
Panel. They ﬁnd that the status utility function exhibits a kink at the average
income in this way.
The parameter θ captures the property of status preference. When θ > 1, thecross derivative of v(·) is positive, which means that as the social average rises the
incentive to bequest increases. This characteristic of status preference is called
KUJ. When θ < 1, an increase in social average depresses individuals’ motive tobequest. This type of preference is named RAJ.
Households earn income from their exogenous supply of labor and renting theirbequests to ﬁrms as capital inputs. Therefore, they have the following budget
constraint:
cit + bit+1 ≤ wt + (1 + rt )bit ,(2)

where wt and rt denote labor income and the rental rate of capital at time t,
respectively.
Agents allocate their income into consumption and bequest so as to maximize
Uti subject to budget constraint (2), given the level of average bequest b̄t+1 .5
The ﬁrst-order condition for the optimization problem of high-type agents is
obtained as follows:6()−θ
1
β1 bit+1−= 0.
+
wt + (1 + rt )bit − bit+1 b̄t+1 b̄t+1
Rearranging the terms, I get[

wt + (1 + rt )bit = b̄t+1
5

bit+1
1
+
b̄t+1 β1

( i )θ ]
bt+1
.
b̄t+1

(3)

It can be shown that low-type agents do not take over high-type agents in equilibriumH(bL
t+1 < bt+1 ).
6

A similar condition is obtained for low-type agents.

5

2.2

Production

There is a continuum of ﬁrms with measure one. Under a perfectly competitive environment, they produce ﬁnal goods, using labor and capital as inputs, according
to a neoclassical, constant-returns-to-scale technology. For simplicity, the production function is assumed to be of the Cobb-Douglas form.
Yt = AKtα L1−α.
tGiven the wage rate and the rate of return to capital, ﬁrms at time t choose thelevel of employment of capital, Kt , and labor, Lt . The ﬁrm’s proﬁt maximization
implies that the marginal productivities of inputs are equalized to their marginal
costs:
wt = (1 − α)AKtα ,1 + rt = αAKtα−1 .

(4)
(5)

I use the fact that each agent’s exogenous labor supply equals one and thatthe population size is one and constant over time. I also assume that there is fullcapital depreciation.

2.3

Equilibrium and Dynamics

Since bequests at time t are used as inputs in production, the following capital
market condition holds.
Kt = b̄t .

(6)

Since both types of agents (L and H) have an equal share in the population,in an equilibrium,1b̄t = (bH
+ bLt ),
2 t
is satisﬁed. Combining equations, (3),(4), (5), (6), and (7), yields
[( H )θ ]
bH
b

α−1
1
A
t+1
L
H

 2 b̄t [(1 + α)bt + (1 − α)bt ] = b̄t+1 b̄t+1 + β1 b̄t+1
t+1[( L )θ ]
L
bt+1
bt+1

1
L

[(1 − α)bH
 A2 b̄α−1tt + (1 + α)bt ] = b̄t+1 b̄t+1 + β2
b̄t+1

(7)

.

(8)

bH

Now, deﬁne yt ≡ b̄tt , which indicates the relative richness of high-type agentsat time t.
Furthermore, the following equation is derived;bH
bLt= 2 − t = 2 − yt .
b̄t
b̄t

(9)

Using (8) for both types, the following equations backwardly determine thedynamics of yt ;
6

yt
1
+1
α

yt

45◦
Ψ(yt+1 )

45◦

1
+1
α

2

2

1

1

Ψ(yt+1 )
A

yt+1
y∗ 2
1
2
1
Fig. 2: Stability of the equal steady state: stable (left), unstable (right)

1−α(1 − Φ(yt+1 )) + 2α≡ Ψ(yt+1 ),1 + Φ(yt+1 )2 − yt+1 + β2−1 (2 − yt+1 )θ
where Φ(yt+1 ) =.
yt+1 + β1−1 (yt+1 )θ
yt =

(10)
(11)

In what follows, I investigate the properties of the above dynamic system inwhich yt is the only variable.

3

Symmetric Case

3.1

Dynamic Systems

In this section, I assume that the parameters in utility function are symmetric,
that is, β1 = β2 = β holds. It can be easily veriﬁed that Ψ(·) is increasing in yt+1
and that Ψ(1) = 1 and Ψ(2) = α1 + 1 > 2 hold. As a result, the dynamics ofinequality, yt , can be described as in Figure 2. Further, note thatΨ′ (1) =1 + β −1 θ.
α(1 + β −1 )(12)Therefore, I can say that the gradient of Ψ(yt+1 ) at yt+1 = 1 is greater than1, if and only if β + θ > α(β + 1).7 Under this condition, the steady state inwhich agents of both types hold equal amounts of assets is stable (Figure 2 (left)).
Otherwise, the no-inequality steady state is unstable, and I have a steady statein which inequality between the high-type and low-type agents remains over time(point A in Figure 2 (right)).

3.2

Comparative Dynamics

Now, I turn to the response of an economy to changes in the parameter values.
7

Note that θ > 1 is a suﬃcient condition for a stable equal steady state. In other words, if
all agents are KUJ, then an equal steady state must be stable.

7

yt+1

∂yt
−1
2 ∂Φ(yt+1 )
=
> 0.
∂β
(1 + Φ(yt+1 ))2 α ∂β(13)The change in the strength of status preference, β, causes the graph to shift up.
Hence, an economy with a higher status preference tends to become equal quickly.8
Intuitively, when β increases, both types of individuals have more incentive toaccumulate wealth. However, the incentive of low-type agents generally exceedsthat of high-type agents, hence lowering inequality:∂yt−12 ∂Φ(yt+1 )
=
> 0.
2
∂θ
(1 + Φ(yt+1 )) α
∂θ

(14)

Recall that the parameter θ indicates the attitude to change in reference wealth.
In particular, when θ > 1, individuals are KUJ. When θ < 1, they are RAJ. As
θ increases, the marginal utility of wealth, in response to a change in the average
wealth, increases. This triggers the interaction between the two types of agentsthrough status seeking, and thus pushes toward equality:∂yt
1 1 − Φ(yt+1 )
=− 2
< 0.
∂α
α 1 + Φ(yt+1 )(15)

A rise in capital share shifts the graph down. This is because capital share indicates the strength of intergenerational inheritance, and thus of inequality within
generations. This implies that inequality in some generation is more likely toremain in subsequent generations.

4Asymmetric CaseIn this section, I analyze dynamic systems when the status utility parameters may
diﬀer:
1−α(1 − Φ(yt+1 )) + 2α≡ Ψ(yt+1 ; β1 , β2 , θ),
1 + Φ(yt+1 )2 − yt+1 + β1−1 (2 − yt+1 )θ
Φ(yt+1 ; β1 , β2 , θ) =.
yt+1 + β2−1 (yt+1 )θ
yt =

(16)
(17)

There are three parameters β1 , β2 , θ in the dynamic system and various dynamics may arise. However, the following argument suggests that an important
implication is obtained stemming from a kink in status preference. WhetherΨ(1; β1 , β2 , θ) is larger or smaller than unity determines the stability of the equal
steady state: when β1 < β2 , the equal steady state is stable; alternatively when
β1 > β2 , it is unstable (see Figure 3). In other words, as Barnett et al. (2010) suggest, when there is a utility premium in status (β1 > β2 ), low-type agents can nevercatch up with high-type agents (see the left panel of Figure 3). Intuitively, when
there is a utility premium, high-type agents try to go beyond the average to obtainthe utility premium. Although low-type agents also like to do so, advantageous
8

If the unequal steady state is stable, inequality is lower in the steady state.

8

yt

yt
+
1
α

45◦ 1

1
+1
α

Ψ(yt+1 )45◦
Ψ(yt+1 )

Ψ(1)
A
Ψ(1)
1

yt+1
1
2
Fig. 3: Left: β1 > β2 ; Right: β1 < β2

2

yt+1

agents inhibit them. On the other hand, if I stand on the empirical suggestion by
Vendrik and Woltjer (2007), the equal steady state must be a stable equilibrium.9
Kawamoto (2009) also investigates the implications of status seeking behavior for
economic dynamics. He uses a model in which two-period living agents seek statusthrough relative income. The two types of status-seeking behavior (KUJ or RAJ)
result in totally diﬀerent economy dynamics. If the marginal utility of an individual’s own income rises as the average income rises, his preference is KUJ. If the
opposite is the case, it is RAJ. In a KUJ economy, any diﬀerence in relative income
eventually disappears. In contrast, the relative position of one of the two types
goes to zero in an RAJ economy. Kawamoto (2009) emphasizes the importance of
attitudes in a reference point change (KUJ or RAJ), which decides whether thesteady state is equal or unequal. However, a kink in status utility also makes theequal steady state stable. In other words, Kawamoto (2009) emphasizes the role ofparameter θ. Although in this model, θ (and α) aﬀect the long-run distribution of
wealth in a symmetric case, they do not play a signiﬁcant role in the asymmetriccase.
In Kawamoto (2009), the relative positions of both types in steady states will
be quite extreme, in the sense that high-type agents hold all of the world’s assets
or both types eventually hold the same amount of assets. However, in this paper,the only steady states are those in which both types possess a positive portion of
the world’s assets.10
Now, I show how the inequality dynamics would change with parameters.

9

Further, note that there is a parameter set under which there exist two stable steady states(and one unstable steady state). If this is the case, an initial inequality in an economy determines
which equilibrium the economy converges to.
10
Kawamoto (2009) shows that there exist steady states where both types hold a positive
portion of the world’s assets in an economy where KUJ and RAJ agents coexist. However, here
too, diﬀerent attitudes to change in the benchmark decide the nature of the steady state.

9

2/α
∂
∂Φ(yt+1 ; β1 , β2 , θ)Ψ(yt+1 ; β1 , β2 , θ) = −< 0,∂β1(1 + Φ(yt+1 ))2β1∂
2/α∂Φ(yt+1 ; β1 , β2 , θ)Ψ(yt+1 ; β1 , β2 , θ) = −> 0,∂β2(1 + Φ(yt+1 ))2β2∂
2/α∂Φ(yt+1 ; β1 , β2 , θ)Ψ(yt+1 ; β1 , β2 , θ) = −
> 0.
2
∂θ
(1 + Φ(yt+1 ))
θ

(18)
(19)
(20)

Because a rise in β1 means an increase in the utility premium, the high-typeagents want to accumulate capital to obtain the premium. Hence, inequality inthe economy tends to expand. The opposite argument is valid for a change in β2 .
The same logic, as in section 3.2, applies for the interpretation of parameter θ.

5Dynamics of Inequality and Output

5.1

Joint Determination of Inequality and Output

In the previous section, I show how the dynamics of inequality are aﬀected by thestatus utility parameters. In particular, the existence of a kink determines the stability of an equal steady state. In this section, I turn to the joint determination of
inequality and output in the long-run. Further, I investigate capital accumulationin the transition path and the eﬀect of a redistribution policy. It will be seen thatthe kink in status utility not only aﬀects the dynamics of inequality, but changes
the transition of output. Furthermore, due to the kink, inequality becomes intricately correlated with the level of output. In this section, I focus on the cases
wherein the unequal steady state is stable (β1 > β2 ). Summing the equations of(8), I get the dynamics of capital accumulation as follows:b̄t+1
f (yt+1 ),2A
where f (yt+1 ) ≡ 2 + β1−1 (yt+1 )θ + β2−1 (2 − yt+1 )θ .
b̄αt =(21)

From (16), (17), and (21), the phase diagram can be depicted as in Figure 4.
Figure 4 shows that any economy with an arbitrary initial condition will converge to a unique steady state. Depending on parameter θ, somewhat diﬀerent
phase diagrams are described. The ∆b̄t = 0 line is described as∆b̄t = b̄t+1 − b̄t = 0 ⇔
2A
=
b̄1−α
.
t
f (yt+1 )When θ > 1, the ∆b̄ = 0-locus is hump-shaped as shown in the left panel of
Figure 4. On the other hand, when θ ≤ 1, it is an upward curve (see the rightpanel of Figure 4). The shape of ∆b̄t = 0 is explained as follows.11 The key
to understand is the force of Joneses preference to equality. A KUJ preference11

In the Appendix, the shape of ∆b̄ = 0 is mathematically investigated.

10

b̄t

∆yt = 0b̄t∆yt = 0∆b̄t = 0
A

b̄∗

E
A

∆b̄t = 0

1

E

yt
y∗
1
y∗
2
Fig. 4: Phase diagram: θ > 1 (left); θ ≤ 1 (right)

yt2(θ > 1) is deﬁned as a status preference wherein an increase in the reference level
raises the marginal utility from status. This eﬀect diﬀers for unequal agents: theeﬀect on poorer agents is stronger.12 This causes the economy to be equal. On
the other hand, with an RAJ preference (θ < 1), both agents will be less willing
to bequeath as the reference level rises. However, the eﬀect on poorer agents isstronger than richer ones. Figure 4 (right) shows it graphically. As the social
average b̄t increases, the diﬀerence between the two groups expands (high yt ).
Next, consider the case of θ > 1. The opposite logic seems to apply: a higher
inequality is now associated with a lower output. An increase in the average raises
both types’ incentive for accumulation, but the incentive of the low-type agents ishigher. As such, the ∆b̄t = 0-locus should be decreasing. However, a kink has anadverse eﬀect. When β1 > β2 , the high-type agents have a higher marginal utilityof bequest. Consequently, a rise in the social average may have a bigger eﬀect on
the high-type agents. This results in the increase in ∆b̄t = 0 (see the left panelof Figure 4). According to this observation, the correlation between inequality
and output is not monotonic and is complex. With a small inequality (lower y ∗ ), a
positive relationship arises. However, when inequality is high (larger y ∗ ), a positiveor negative relation may appear. Depending on inequality, the correlation wouldchange. There are empirical studies that regress the growth rate on inequality.
They have not provided a conclusive correlation between the two, and suggestthat both directions of the correlation seem possible. This paper allows for both
possibilities.
Next, I explore how a change in status parameter changes the level of output
in the steady state. It will be seen that stronger motive for status may or may notlead to greater output.13 The steady state in this economy is characterized by the12

The reason for this is the diﬀerent responses in the agents’ marginal utility of status due toa change in the average level (see Kawamoto (2009)).
13
Liu and Turnovsky (2005) uses a standard representative agent model to show that consumption externalities cause excessive long-run capital accumulation. However, employing acontinuous-time overlapping generations model, Fisher and Heijdra (2009) show that the reverse
is true. Moreover, Wendner (2010) uses a continuous-time overlapping generations model withage-dependent productivity and shows that the eﬀect of conspicuous consumption on output is11

following equations;∗1−α(1 − Φ(y ∗ )) + 2
α,
1 + Φ(y ∗ )where Φ(y ∗ ) =2 − y ∗ + β1−1 (2 − y ∗ )θ,y ∗ + β2−1 (y ∗ )θ

y =

(22)

and
(b̄∗ )1−α =

2A

2A

≡
.
f (y ∗ )2 + β1−1 (y ∗ )θ + β2−1 (2 − y ∗ )θ(23)

Certainly the long-run levels of inequality and output depend on the nature ofstatus preference (β1 , β2 and θ). Thus inequality and output are jointly determinedin this model, and the diﬀerent attitudes in status preference change the relationship between the two. Using a Ramsey model with inﬁnitely-lived heterogeneous
consumers, Garcia-Penalosa and Turnovsky (2008) investigate the implications ofstatus seeking behavior in consumption. They ﬁnd that the level of output in thesteady state is independent of the properties of status preference, unlike in thismodel.
I now consider changes in the strength for status seeking to see the eﬀect on
the output level in the steady state:∂f (y ∗ )=∂β1

1
− 2 (y ∗ )θβ
| 1{z }
direct eﬀect (negative)+∂y ∗[θβ −1 (y ∗ )θ−1 − θβ2−1 (2 − y ∗ )θ−1 ] .
{z}∂β1 | 1|{z}
positive/negative
positive
|
{z
}

(24)

indirect eﬀect through inequality

Note that an increase in f (y ∗ ) implies a decrease in the output level in thesteady state, and vice versa (see (23)). There are two eﬀects on output as thestrength of status preference (β1 or β2 ) changes. Consider a rise in β1 as an
example. First, the direct eﬀect of the change is an increase in the motivationof high-type agents for accumulation. This results in a higher level of output inthe steady state. In turn, the indirect eﬀect is due to a change in inequality, andthus, a status seeking behavior. I observe that an increase in β1 , that is, a higher
utility premium, results in more inequality in the economy. This alters the capital
accumulation of households through status seeking. Whether this eﬀect is positiveor negative is not clear and depends on the parameters, especially θ. When θ ≤ 1,
more inequality is associated with higher output. As such, more inequality becauseof a rise in the utility premium raises output. Based on this observation, a rise in
β1 increases the steady-state level of output via both direct and indirect eﬀects.14

5.2

Redistribution Policy

Now, I turn to the eﬀect of a redistribution policy. I consider two types of redistributive policies (one-time redistribution and inheritance tax). The conventional
ambiguous.
14
When θ ≤ 1, the square bracket term in equation(24) is negative. Hence, ∂f (y ∗ )/∂β1 < 0,and thus ∂ b̄∗ /∂β1 > 0

12view for redistribution concerns fairness in a society. However, the by-products ofsuch a policy need to be speciﬁed and a number of previous researches address this.
That is, the implemented policy may decrease the economic pie (output). First, I
focus on a particular form of redistribution policy. Suppose that the economy isinitially in the steady state and some level of wealth is redistributed from the richto the poor. At the date of policy implementation, the economy jumps to A fromE in Figure 4. In Figure 4, the redistribution policy results in the same qualitative
consequence. On the path to the steady state, the capital level of the economy
declines ﬁrst and then gradually regains. Therefore aggregate capital decreases
due to such a redistribution policy, as does the output.15 The intuition underlying
this transition is straightforward. The direct impact of the policy is the changes
in the wealth of both high-type and low-type agents. Although low-type agentsaccumulate more assets for their children, high-type agents become less willing to
do so. When β1 > β2 , the negative eﬀect of high-type agents exceeds the positiveeﬀect.
Next, I introduce a redistributive policy (inheritance tax) such that a proportion of the bequeath is collected by the government as tax and the revenue
is redistributed among citizens. The budget constraints of households and thegovernment becomecit + bit+1 ≤ wt + (1 + rt )[bit (1 − τ ) + Tt ],1
Tt = τ (bH+ bLt ),
2 t
where τ and Tt indicate the tax rate and lump-sum transfer, respectively. Notethat the levying of tax does not distort resource allocation. The tax system purely
makes some transfers from the rich to the poor.
The inequality dynamics are derived in the same manner and one can obtainyt =1−α+ατ
[1 − Φ(yt+1 )] + 2
α−ατ

1 + Φ(yt+1 )≡ Ψ(yt+1 ; τ ).

As Compared to the no-tax case, the function Ψ(·) shifts up, as the tax rateincreases. By introducing inheritance tax, inequality tends to diminish, as expected. The long-run level of capital accumulation is unchanged and determinedby (23). Despite no distortion by tax, such a policy may increase or decrease output through status seeking behavior. The property of status seeking determines
which case will arise.

6

Conclusion

I have shown that whether inequality in an economy will persist may depend on
the “direction” of the kink in status utility. The status utility that Barnett et al.
15

Consider the case wherein θ > 1 and β1 = β2 (no utility premium). Now, the ∆b̄t = 0 line isdecreasing. In this case, the one-time redistribution increases the output level at implementation.
Because of a kink in status utility, the transition to the steady state changes.

13(2009) assume, interpreted as one wherein there is a utility premium exceeding thesocial average, leads to an inequality-persistent steady state. This is because richer
agents try to inhibit poorer ones from going beyond the social average, althoughboth types want to do so. In contrast, following the empirical study of Vendrik
and Woltjer (2007), a no-inequality steady state must be stable and the economy
is likely to move to it.
The model used in this paper is quite simple for tractability. One of the simplifying assumptions is that there are only two types of agents. Although thissimpliﬁes the analysis, an emergence of a third type of agent generally plays animportant role in an economic model. For example, consider the behavior of the
middle class in a model with utility premium. The decision of the middle class ongoing beyond the social average aﬀects aggregate capital accumulation and thereby
inﬂuences the behaviors of other agents through status seeking. Furthermore, theliterature on status preference typically models economies without social mobility,
although status seeking behavior seems to represent competition among agents.
Incorporating these ingredients is left for the future work.

Appendix
Shape of f (y)In this section, I examine function f (y), when β1 > β2 . (see equation (23)), I
check the following statements.
• when θ > 1, f (y) is U-shaped, and then ∆b̄ = 0 is hump-shaped.
• when θ ≤ 1, f (y) is decreasing, and then ∆b̄ = 0 is increasing.
First, taking the derivatives of f (y), I obtainf (y) ≡ 2 + β1−1 y θ + β2−1 (2 − y)θ ,′f (y) = θ[β1−1 y θ−1 − β2−1 (2 − y)θ−1 ],f ′′ (y) = θ(θ − 1)[β1−1 y θ−2 + β2−1 (2 − y)θ−2 ].

(A.1)
(A.2)
(A.3)

Now consider the two cases.
Case 1: θ > 1{
f ′ (1) = θ[β1−1 − β2−1 ] < 0,
f ′ (2) = θβ1−1 2θ−1 > 0.

(A.4)

I thus havef ′′ (·) > 0.

(A.5)

Therefore, when θ > 1, the function is entirely convex and starts with anegative slope at y = 1, and ends with a positive slope (see Figure 5).
Because f (y) is concave, ∆b̄ = 0 is convex as in the left panel of Figure 4.
In case 1, if β1 = β2 , function f (y) is monotonically increasing. Hence, thenegative slope at y = 1 is due to the kink in status utility.
14f (y)f (y)1Case 1: θ > 1

y

y
1

2

2

Case 2: θ < 1
Fig. A. 1: Shape of f (y)Case 2: θ < 1{
f ′ (1) = θ[β1−1 − β2−1 ] < 0,
f ′ (2) = −∞.

(A.6)

I thus havef ′′ (·) < 0.

(A.7)

In case 2, the function is concave and decreasing in y. I thus obtain an upward
slope of ∆b̄t = 0 as in the right panel of Figure 5.

15

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