Discussion Paper No. 528

LUXURY AND WEALTH ACCUMULATION

Shinsuke Ikeda

February 2001

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Luxury and Wealth Accumulation
Shinsuke IKEDA¤yThe Institute of Social and Economic Research, Osaka University
February 2001

Abstract
This paper develops a model of luxury goods by incorporating weakly
non-separable, recursive preferences. In a two-good framework, a quasi-luxury isde…ned as a good whose marginal rate of substitution is increasing in wealth.
Under certain conditions, it is identical to a luxury good. Consumers wait for
quasi-luxuries more (less) patiently than for quasi-necessities when they expect
to be happier (unhappier) in the future. The preference for quasi-luxuries
promotes optimal wealth accumulation and hence growth. In a two-country
economy, the less patient country with stronger quasi-luxury preferences can be
wealthier than the more patient country.

Keywords: Quasi-luxury goods; luxury; wealth; weakly non-separable
preferences; time preference; growth; wealth distribution.
JEL classi…cation: D91, E21, F34.

¤

I would like to thank to D. Dasgupta, K. Futagami, C.Y. Horioka, K. Mino, F. Ohtake,T. Ono, Y. Ono, M. Saito, Y. Tsutsui and the participants at the ISER Seminar and theMacro Research Seminar for helpful discussions on the ealier versions of the paper. A
part of this research is …nancially supported by The Matsushita International Foundation(#00-049).
yCorrespondence. Shinsuke Ikeda, The Institute of Social and Economic Research, Osaka University, 6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan. Telephone: +81(6)68798568. Fax: +81(6)6878-2766. E-mail: ikeda@iser.osaka-u.ac.jp

Luxury and Wealth Accumulation

1

Countries which have sumptuary laws, are generally poor. (Sir
Dudley North, Discourses Upon Trade, 1691, p. 14)

1

Introduction

By de…nition, the wealthier allocate higher proportions of their expenditures
to luxuries than the poorer do. The standard price theory (see, e.g., Deaton
and Muellbauer (1983)) would describe this by saying that the richer consume
more luxuries because they have more wealth. In dynamic consumer theory,
however, the statement is not perfect: how much wealth consumers accumulate is a part of their lifetime utility-maximization problem, as is how muchof each good they consume in each period. Comprehensive understanding of
luxury expenditures entails dynamic analysis.
This paper develops a dynamic theory of luxury consumption, focusing
on the bilateral relationship between luxury and wealth accumulation. Particularly emphasized is the causal e¤ect that pursuit of luxury goods has on
wealth accumulation. The topic of how preferences for luxury goods a¤ect
the total amount of national wealth goes back to at least the era of David
Hume and Adam Smith.1 The paper is a tentative response to their question
using modern consumption theory. Based on the procedures used by Shi(1994), weakly non-separable preferences are speci…ed in a simple two-goodmodel of recursive preferences, such that the intratemporal marginal rate of
substitution (MRS) between two goods depends on the future consumption
streams through current welfare.2 Relative preferences for the two goods,
measured by the MRS, then depend on current welfare and hence current
wealth holdings. With the resultant non-homothetic preference structure, agood whose MRS is increasing (decreasing) in wealth is called a quasi-luxury(quasi-necessity). Under a certain condition, quasi-luxuries are identical toluxury goods. The purposes of the present paper are (i) to characterize luxury goods from the viewpoint of intertemporal resource allocation; and (ii)1

See Brewer (1998) and Mason (1998).
Weakly non-separable preferences under recursive preferences are analyzed by several
authors (e.g., Lucas and Stokey (1984), Judd (1985), Epstein, Ham, and Zin (1988), Shi(1994), Ikeda (2001)). Shi conducts the most systematic analysis to discuss the intertemporal leisure-consumption choice under distortionary taxation on capital and labor. By
applying his analysis to a small-country model with two traded goods, Ikeda (2001) showsthat, in contrast to the literature, a terms-of-trade deterioration can worsen the current
account, i.e., the Harberger-Laursen-Metzler e¤ect can take place.
2

Luxury and Wealth Accumulation2

to show that wealth accumulation and distribution depend on consumers’
intratemporal preferences for these luxury goods.
Quasi-luxuries are characterized by two properties. First, consumers wait
for quasi-luxury consumption more (less) patiently than for quasi-necessity
when they expect to be happier (unhappier) in the future. This property
helps to understand consistently two contrasting patterns of luxury consumption that are commonly observed: Some people save luxuries today to enjoy
tomorrow, while other people enjoy luxurious lives today by loans and liveon necessity goods tomorrow to pay back their debts. Which pattern takes
place depends on the future course of the welfare of consumers.
Second, given constant market prices, strong preferences for quasi-luxury
goods are shown to promote consumers’ optimal wealth accumulation. Alternatively stated, quasi-luxury induces a preference for wealth. This savingpromoting property of quasi-luxury preferences is then applied to the models of neoclassical growth and a two-country world economy, showing two
main results: (i) the stronger the quasi-luxury preference, the more capitalis accumulated in steady state; and (ii) even if production ‡ows and utilitydiscounting functions are both internationally (interpersonally) identical, acountry with a stronger quasi-luxury preference holds more wealth in the longrun than a country with a weaker preference. As a corollary of (i), luxury
taxes harm growth, as Sir Dudley North pointed out more than three centuries ago (see the epigraph). From (ii) and continuity, a less-patient country
with a stronger quasi-luxury preference can be wealthier than a more-patientone.
Although there are strong empirical evidences against preference homotheticity (e.g., Blundell, Browning, and Meghir (1994), Attanasio and Weber (1995), and Parker (1999)), homothetic preferences have usually been
assumed for simplicity in dynamic consumer theory, and few attempts havebeen made to analyze luxury consumption from the viewpoint of intertemporal utility maximization.3; 4 Browning and Crossley (2000), in an important3

Besley (1989) proposes a new de…nition for luxury which is useful in “dynamic” applications. However, dynamic optimization is not discussed there. Baland and Ray (1991)
examine the e¤ect of capital accumulation on unemployment by using a model in which
luxury and basic goods compete for the use of the scarce resources. However, the analysis
is essentially static, assuming that capital accumulation is exogenous.
4
There are several important contributions to dynamic macroeconomics that can bereinterpreted in the luxury-necessity terms. For example, in the status (wealth)-seeking
literature (e.g., Cole, Mailath, and Postlewaite (1992), Corneo and Jeanne (1999)), a highLuxury and Wealth Accumulation3

exception using a two-good, two-period model of the time-additive utility
function, show that luxury goods have higher elasticities of intertemporal
substitution and hence are easier to postpone than necessity goods. This
paper re-examines this issue in the recursive preference framework by relating quasi-luxury goods to luxury goods. My contribution in this regard is
the …nding that luxury goods can be featured by the relative magnitudes ofgood-speci…c time preferences as well as of good-speci…c intertemporal substitution elasticities. Even when good x is not easier to postpone than goodc, it turns out that x can be a luxury good if consumers are more patientwith respect to x than with respect to c when they are getting happier, i.e.,
if it is a quasi-luxury.
In the dynamic macro literature, international (interpersonal) wealth distribution has been explained by referring to di¤erences in four determinants:(i) the subjective discount rate (e.g., Ramsey (1928), Devereux and Shi(1991)); (ii) productivity growth (e.g., Obstfeld and Rogo¤ (1996), Frenkel
and Razin (1992)); (iii) age structures (e.g., Buiter (1981), Obstfeld and Rogo¤ (1996), Frenkel and Razin (1992)); and (iv) random income ‡uctuation(e.g., Clarida (1990), Becker and Zilcha (1997)). To focus on implicationsof the luxury preference, these factors are not considered. The punch lineof my result lies in the …nding that wealth accumulation is still a¤ected byintratemporal preferences for luxury.
The rest of the paper is organized as follows. Section 2 presents the basic
framework to de…ne (quasi-) luxury goods, and analyses the relation between
preferences for them and consumers’ optimal wealth accumulation. Section
3 considers implications of the luxury preference for economic growth byusing a simple neoclassical model. In section 4, the analysis is extended to a
two-country economy model, where a box diagram is introduced to aid thediscussions. Section 5 concludes the paper.
“status” can be regarded as a luxury good, so that the preference for luxury is growthpromoting. Our result di¤ers from theirs in that the wealth preference is induced by
preferences for usual-marketed commodities, instead of for non-marketed goods such as
status and marriage.

Luxury and Wealth Accumulation2

Consumption behavior with quasi-luxury goods

2.1

Consumer preferences and quasi-luxury goods

4

Consider in…nitely lived consumers. There are two distinct consumption
goods c and x. Let u (c; x) be the instantaneous utility function of a representative consumer. The u (c; x) is assumed to satisfy the regularity conditions such as concavity, monotonicity, and the Inada condition. Preferencesof consumers are given by the following lifetime utility function:
Z 1(1)
U (0) ´u(c (t) ; x (t)) exp (¡¢ (t)) dt;
0

where¢ (t) =

Z t

± (c (s) ; x (s)) ds;(2)0with ± (¢; ¢) (> 0) representing the instantaneous subjective discount rate. As
in the literature (e.g., Uzawa (1968)), ±c > 0 and ±x > 0 are assumed, where±c = @±=@c, etc. Letting Á (t) denote the time-t lifetime utility U (t), thecorresponding generating function g is given byg (c; x; Á) = u (c; x) ¡ Á± (c; x) ;with which utility evolution is expressed as
Á_ = ¡g (c; x; Á) s.t. lim Á (t) exp (¡¢ (t)) = 0;
t!1

(3)

where a dot represents the time derivative, i.e., Á_ (t) =dÁ (t) =dt. The …rstorder partial derivatives gc and gx equal the current-value marginal utilitiesof c (t) and x (t) de…ned in terms of the Voltera derivative. It is assumed
that gc and gx are positive. The intratemporal marginal rate of substitution(MRS) between c and x is given by dc=dxjU (0)=const: = gx =gc (c; x; Á).
When the MRS indeed depends on current utility level Á, consumer preferences are not weakly separable since the choice of the time-t consumption
basket depends on future consumption plans. With the preferences, which arecalled by Shi (1994) weakly non-separable preferences, I de…ne quasi-luxury
goods as follows:
De…nition: A good is a quasi-luxury (-necessity) good or simply aquasi-luxury (-necessity) to a consumer if and only if (i¤) his or her

Luxury and Wealth Accumulation

5

relative preferences for it, measured by the MRS, are increasing (decreasing)in wealth.
It is convenient if the non-separability index » is de…ned as» (c; x; Á) ´

±c ±x
1 @ (gx =gc )
=
¡ ;
gx=gc
@Á
gc gx

where the following relations are assumed:
Assumption 1:
gcx = 0; gcc < 0; gxx < 0;

±gcc
±gxx
<»<¡ 2 :2gc
gx

The last three inequalities ensure the local concavity of the preference.5;6
The index » captures how relative preferences for the two goods depend on
current welfare and hence on current wealth. Without loss of generality, let
us assume the following:
Assumption 2: Good x is a quasi-luxury good:» (c; x; Á) > 0:
From the de…nition of », the MRS gx =gc for the quasi-luxury x along theindi¤erence curve is larger than that along the discount rate, ±x =±c , implyingthat consumers are more reluctant to sacri…ce x for c to keep lifetime utility
constant than to keep the discount rate constant:
¯
¯
dc ¯¯
dc ¯¯
>
:
dx ¯U (0)=const.
dx ¯±=const.
Put otherwise, to increase quasi-luxury x in exchange for quasi-necessity c,
keeping the discount rate constant, improves lifetime utility and underlying
wealth positions. This property plays an important role for consumers’ quasiluxury preferences to a¤ect wealth accumulation.
5

See Shi (1994, appendix).
The index » would be zero if the discount rate ± were a function of felicity u, as in
Uzawa (1968) and Obstfeld (1982), or if ± were constant, as in the case of time-additive
preference, or if u were constant, as in Epstein and Hynes (1983).
6

Luxury and Wealth Accumulation

6

Taking the good c as numeraire, let p be the relative price of the quasiluxury, r the interest rate, and a the consumer’s total wealth, which may
be composed of …nancial wealth and human capital. Consumers maximize
lifetime utility (1) subject to four constraints: (i) the ‡ow budget constraint,
a_ (t) = r (t) a (t) ¡ c (t) ¡ p (t) x (t) ;(4)(ii) the law of motion (2) for the discount factor; (iii) the initial condition,
a0 =given; and (iv) the positivity condition on a. Letting ¸ denote thecurrent-value shadow price of savings, the optimal conditions are given by:gc (c; Á) (´ uc (c; x) ¡ Á±c (c; x)) = ¸;(5)gx (x; Á) =gc (c; Á) = p;¸_ = (± (c; x) ¡ r) ¸;and the transversality conditions for a and Á.
The optimal dynamics for (a; c; x; Á; ¸) are generated by …ve equations(3), (4), and (5) under the initial and transversality conditions. To reduce
the system, de…ne the rate of time preference ½c with respect to c as¯
d ln gc (c (t) ; Á (t)) exp(¡¢ (t)) ¯¯
c
½ ´¡
¯ ;
dt
c=0
_

where gc exp(¡¢) represents the present-value marginal utility of c: Then,the …rst and third equations in (5) can be reduced toc_ = ¡

gc(r ¡ ½c (c; x; Á)) ;
gcc

(6)

where
½c (c; x; Á) = ± (c; x) ¡±c (c; x)g (c; x; Á) :gc (c; Á)(7)

In exactly the same way, the optimal dynamics of quasi-luxury consumption
can be obtained by de…ning the rate of time preference ½x with respect to x.
From (5), the dynamics are not independent of (6).
By comparing (7) with the corresponding equation for good x, quasiluxury goods can be characterized from the viewpoint of impatience: from

Luxury and Wealth Accumulation

7(3) and the de…nition of », the di¤erence between the two good-speci…c time
preferences satis…es_½c ¡ ½x = » Á;
which, under Assumption 3, implies ½x Q ½c as Á_ R 0. This characterizes
quasi-luxuries as follows:
Proposition 1: When welfare is improving (deteriorating) over time, consumers are more (less) patient with respect to quasi-luxury consumption thanwith respect to quasi-necessity consumption.
Alternatively stated, consumers can wait for consuming quasi-luxuries
more patiently than for consuming quasi-necessities when they expect to behappier or wealthier in the future, whereas they cannot when getting unhappier or poorer. This describes well our daily consumption behavior towards
luxuries. For example, many young people with poor future prospects often
enjoy outrageous luxurious consumption (luxurious suits, brand-new cars,
expensive restaurant dinners with friends, etc.) by using credit cards and, atthe same time, save daily necessity consumption (living in cheap apartment
rooms, having junk food for daily dinners, etc.). The “Last Supper” mightwell have been luxurious since the su¤erings on the cross were expected nextday. People tend to spend large portions of unlabored lucky income, e.g.,lottery prizes, poker, etc., on luxuries.7 In contrast, many entrepreneurs
and professional sports players likely save luxury consumption when young,instead of enjoying luxuries by loans, to accumulate human and nonhumancapital, until they can a¤ord to enjoy luxurious lives in the future.8 Uponan unexpected unlucky expense, consumers usually decrease more luxuries
than necessities. These two contrasting consumption patterns, i.e., consuming more luxuries today and less tomorrow than necessities, or consuming
less luxuries today and more tomorrow, can be rationalized consistently by
Proposition 1: either pattern can take place, depending on whether the consumer’s welfare state is deteriorating or improving, or whether he or she ishappier or unhappier than expected to be in the future.
7

In this sense, “ill got, ill spent” could be regarded as rational behavior.
Likewise, during World War II, Japanese people saved luxury consumption under theslogan: “Luxury is our enemy!”.
8

Luxury and Wealth Accumulation

2.2

8

Optimal consumption

Given the market prices (p; r) and the initial total wealth a0 , the optimal
consumption plan fa (t) ; c (t) ; x (t) ; Á (t)g1
t=0 for the consumer is generatedby (3), (4), and (5). The steady-state consumption basket (¹c; x¹) is determinedby± (¹c; x¹) = r;

(8)

gx (¹
x; u (¹c; x¹) =± (¹c; x¹))= p:gc (¹c; u (¹c; x¹) =± (¹c; x¹))(9)andSteady-state wealth holding and welfare are then given byr¹
a = c¹ + p¹
x;

(10)

and
¹ = u (¹Ác; x¹) =r;
respectively.
Figure 1 depicts the determination of the steady-state consumption plan.
Schedule RR0 represents (8), depicting the locus of (¹c; x¹) that equalizes thesteady-state rate of time preference to a given interest rate. It could bereferred to as the steady-state time preference curve. Schedule F F 0 is a locus
along which the MRS between the two goods is equal to the corresponding
relative price. The steady-state consumption basket (¹c; x¹) is determined at
the intersection point E of the two schedules. With the consumption basket,
the no-savings condition (10) is depicted as schedule AA0 which goes throughpoint E with slope ¡1=p. Its horizontal intercept gives the steady-state total
wealth holding a¹.
The property of quasi-luxury goods is illustrated by noting from (5) thatthe slope of schedule AA0 should equal the gradient gd =gf at point E of thesteady-state indi¤erence curve I (E):½¾u (¹c; x¹)(11)
I (E) = (¹c; x¹) j= utility at E :± (¹c; x¹)
Therefore, for good x to be a quasi-luxury good, as in Assumption 2, the slopeof the AA0 schedule, ¡1=p (= ¡gc =gx ) should be smaller than the gradientLuxury and Wealth Accumulation

9

at point E of schedule RR0 , i.e., ¡±c =±x . This property plays a crucial role
for the quasi-luxury preference to promote wealth accumulation. Figure 1
is similar to the usual map for static consumption choice, except that thelocation of the “budget schedule” AA0 is endogenously determined to …nancethe consumption basket (¹c; x¹) that is determined by schedules RR0 and F F 0 .
The resultant steady-state wealth holdings re‡ect preferences for the twogoods as well as time preference.
An autonomous local dynamic system can be obtained with respect ton ´ (c; Á; b) by substituting (5) into (3), (4), and (6) and linearizing theresulting system around steady state as n_ (t) = A^n (t), where the coe¢cientmatrix A is given in Appendix A; and the hat above variable n represents
deviations from the steady-state value of the variable. As is also shown in
the appendix, the linear system has two positive roots and one negative root!:q
2 2r ¡ r2 + g4gcccggxxx ª
!´
;
2
where

½

r±c ±xª´¡ » +
gc gx
2

µ

gcc
gxx
+
±c gc ±x gx¶¾;

which can be shown to be positive under Assumption 1. Any other paths than
the saddle path governed by ! cannot satisfy the transversality conditions.
The optimal consumption plan is uniquely determined on the saddle arm,
which can be derived from the eigen vector associated with ! as:¾½
gx2
gxx (r ¡ !)+» a^ (t) ;c^ (t) =(gxx + p2 gcc )
gx2
½
¾
gx gc
gcc (r ¡ !)(12)x^ (t) =^ (t) ;¡» a(gxx + p2 gcc )
gc2
and^ (t) = gc a
^ (t) ;
Á
where the state variable a (t) evolves by a_ (t) = !^a (t) subject to a
^ (0) =a0 ¡ a¹:

Luxury and Wealth Accumulation

10

As total wealth holding monotonically approaches its steady-state quantity from a given a0 , the transitional paths for consumptions c and x andwelfare Á are determined on stable arms (12). Under Assumption 1, the stable arms are all positively-sloping: consumptions c and x as well as lifetime
utility Á co-move positively with net foreign assets b. That is, the two goods
are normal.

2.3

The Engel curve and luxury goods

Based on the above analysis, quasi-luxury goods are related to luxury goods.
In considering luxury goods in a dynamic setting, two points should be noted:…rst, relevant income in considering consumption baskets is not usual current income but permanent income, i.e., income from total wealth; second,
however, total wealth is an endogenous variable in the intertemporal consumption choice setting. It is proposed that luxury goods be de…ned on thesaddle arm (12).
By eliminating a^ from the …rst two equations in (12), a positively-sloping
saddle trajectory in the (c; x)-space is obtained as½¾½¾¡1gxx (r ¡ !)gc gcc (r ¡ !)(13)+»c^ (t) :x^ (t) =
¡»
gx
gc2
gx2
This schedule depicts positive co-movements of c and x generated by (endogenous) wealth variation. This can be regarded as the wealth-consumption
curve or the Engel curve de…ned with respect to total wealth or permanent
income. The Engel curve is also illustrated in …gure 1. Luxury goods are
de…ned along this schedule:
De…nition: A good is a luxury (necessity) good or simply a luxury (necessity) to a consumer i¤, for given constant market prices (p; r) and initial
total wealth a0 , the consumption share of the good in the consumer’s total
expenditure is increasing (decreasing) in total wealth along the Engel curve(13).9; 10
9To be precise, it should be called a (p; r)-luxury good, for example, because how
optimal consumption baskets depend on wealth depends on (p; r) :10
Hamermesh (1982) estimates the permanent-income elasticities of various
consumptions.

Luxury and Wealth Accumulation11c+c»=(r¡!)Along the Engel curve (13), dlnpx, where ° c ´ ¡cgcc =gc= p°°x ¡px»=(r¡!)
dlnc
and ° x ´ ¡xgxx =gx represent the measures of the desire to smooth consumpdlnpx
dlnc
tions c and x, respectively.11 The identity zc dlnz
+ px
= 1 impliesz dlnz
¶
µ
¡1 ³
´
dlnpx
px
px p° x ¡ px»= (r ¡ !)
=
+ c
+1 :
dlnz
c
° + c»= (r ¡ !)c

It thus follows:d (px=z)»z
R0,+ (° c ¡ p° x ) R 0;dar¡!
which can be summarized by:

Proposition 2: The quasi-luxury good x is a luxury good i¤»z+ (° c ¡ p° x ) > 0;r¡!
implying that quasi-luxuries are likely to be luxuries in the sense that a quasiluxury is a luxury good unless the degree ° x of the desire to smooth the x
consumption times p is much larger than that ° c to smooth c:Remark 1: By using a time-additive utility function over two-period consumption, Browning and Crossley (2000) proved that luxury goods have ahigher intertemporal substitution elasticity (ISE) and hence are easier topostpone than necessities. Since the reciprocals of ° c and ° x equal the ISEs
with respect to goods c and x, respectively, Proposition 2 could be regarded
as an extension of their analysis to the recursive preference framework. In
the setting, a good can be a luxury, even if it has a lower ISE, provided it isquasi-luxurious, that is so long as consumers are more patient with respect
to the good with the lower ISE than with respect to the other.
Proposition 2 can be interpreted intuitively by recalling the Euler equation (6). From the de…nition (7) of time preference, the optimal consumption
ratio of the two goods evolve according to:¢

1
1(r ¡ ½x) ¡ c (r ¡ ½c )
x
p°
°
µ
¶ µ
¶
1
1 c
1
1 x
= r
+
½ ¡ x½ :¡p° x ° c°cp°ln (px=c) =11

Epstein (1987, p. 76) uses a similar terminology.

Luxury and Wealth Accumulation

12

From Proposition 1, when the utility index Á increases over time, consumers
are more patient with respect to quasi-luxury x than with respect to quasinecessity c (i.e., ½x < ½c ). Proposition 2 thus implies that, if the ISEs are thesame between the two goods, the x-c consumption ratio increases as total
wealth a increases over time. On the other hand, if x and c were neither
quasi-luxury nor quasi-necessity, i.e., when » = 0, the x-c consumption ratio
is increasing in a if and only if x is easier to postpone than c; i.e., i¤ 1=p° x >1=° c , as is shown by Browning and Crossley.
In sum, there are two factors that determine whether a good is a luxury
or a necessity: inter-commodity di¤erences in ISE, and time preference. By
considering quasi-luxury goods, the role of the second factor is focused on.
As shown later, it is luxury preferences induced by this factor that play animportant role in promoting wealth accumulation.

2.4

The preference for quasi-luxuries and optimal wealthaccumulation

By using the optimal intertemporal consumption plan derived, let us examine the implication of consumers’ preferences for quasi-luxury goods x forsteady-state wealth holding and hence wealth accumulation. The relative
preferences for quasi-luxury goods are parameterized by ® (> 0), with which
the instantaneous utility function is re-speci…ed asu = u (c; ®x) ;where an increase in ® increases the marginal felicity ®ux for given ux .
The intratemporal marginal rate of substitution gx =gc in the steady stateis given by MRS(¹c; x¹; ®):MRS (¹c; x¹; ®) =®ux (¹c; ®¹x) ¡ (u (¹c; ®¹x) =± (¹c; x¹)) ±x (¹c; x¹)
:
uc (¹
c; ®¹
x) ¡ (u (¹c; ®¹x) =± (¹c; x¹)) ±c (¹c; x¹)(14)

Parameter ® a¤ects the MRS through three channels: (i) by raising themarginal felicity of x, ®ux for given ux; (ii) by lowering ux (¹c; ®¹x); and (iii)by raising the utility level u (¹c; ®¹x) =± (¹c; x¹). The e¤ects (i) and (iii) enhance
the MRS for x, whereas (ii) reduces it. Formally@MRS (¹c; x¹; ®)®xuxx xgx
T 0 , (¤ ´) 1 ++» T 0;@®
uxr

Luxury and Wealth Accumulation

13

where the three terms on the right-hand side represent the e¤ects (i) through(iii), respectively. To consider upward shifts in quasi-luxury preferences, it
is assumed that the sum of (i) and (iii) dominates (ii):12
Assumption 3: An increase in ® enhances the MRS for x: ¤ > 0:
With this assumption, parameter ® is referred to as the degree of consumers’ preferences for quasi-luxury goods x, or simply quasi-luxury preference. By di¤erentiating (8) through (10) with respect to ®, the e¤ect of anincrease in the quasi-luxury preference on steady-state wealth holding can becomputed as
d¹a
»ux ¤
=
;
d®
gc gx ªwhich is positive under Assumptions 2 and 3. The result can be summarized
as follows:
Proposition 3: An increase in the preference for quasi-luxury increases
optimal steady-state wealth holding and hence promotes optimal wealth accumulation.
As explained in section 2.1, increasing quasi-luxury consumption in exchange for quasi-necessity consumption, maintaining the discount rate at r;
increases steady-state welfare and wealth holding, which results in Proposition 2. Intuitively, the preference for quasi-luxury goods induces a preference
for wealth.
Figure 2 illustrates the property. An increase in ® shifts the F F 0 schedule
upward, bringing the steady-state point from point E0 to E1 . Since thesteady-state time preference curve RR0 is steeper than the budget lines E0 A0and E1 A1 , this shift increases steady-state total wealth from OA0 to OA1 .
Transition dynamics are monotonic, as depicted by …gure 2. An increasein quasi-luxury preference shifts the Engel curve counter-clockwise. Notethat consumers may instantly reduce quasi-luxuries x (0) as well as quasinecessities c (0) to enjoy more luxuries in the future steady state, as illustrated by the discrete jump from point E0 to E01 in …gure 2. Consumers
However, if the felicity function u is speci…ed in the quadratic form: u = ¡ 12 °cc c2 +°c c ¡ 12 °xx x2 + ®x, where °c , °cc , and °xx are all positive, then MRS(¹c; x¹; ®) is de…nitely
increasing in ®, so that an increase in ® can be regarded as an upward shift of the preference
for x without any additional assumption.
12

Luxury and Wealth Accumulation

14

become more patient instantly with respect to either good and increase savings since (7) and (12) imply¾½
d¹a
gc gxx !
gcc (r ¡ !)x
d½ (0) =d® = ¡
»
< 0;
¡
d®
(gxx + p2 gcc )gc2
andgx2 gcc !
d½ (0) =d® = ¡gc (gxx + p2 gcc )c½

gxx (r ¡ !)
+»
gx2

¾

d¹
a
< 0:
d®

As a result, for example, the life of a consumer with stronger preferences forluxuries may be less luxurious in the short run than that of another withweaker luxury preferences even when they hold the same amount of wealth.
In the interim run, both consumptions gradually increase up to the level atpoint E1 as wealth is monotonically accumulated. Quasi-luxury consumption
at point E1 is larger than that at E0 , whereas quasi-necessities consumption
at E1 is less than before. In the long run, a consumer with stronger luxury
preferences enjoys more wealth and a more luxurious life than another withweaker preferences for luxury. These possibilities are consistent with the factthat the lives of the wealthier and more luxurious are likely to have been lessluxurious than those of the poorer when they were young.

3

GrowthFrom Proposition 1, it could be conjectured that, in the context of economic
growth, the preference for quasi-luxuries promotes capital accumulation. Let
us next examine this problem by recasting the previous consumer model in
the neoclassical growth context.
Suppose that two goods are produced using capital and labor. The
production functions in the two sectors are given by the usual linearlyhomogeneous, concave functions F i (K i ; Li ) (i = c; x), where K i and Li
are the capital and labor employed by sector i, respectively. Labor is supplied inelastically. The total amount of labor L is constant. The total
capital stock K accumulates from savings. To avoid complexities due tointer-sectoral di¤erences in factor intensity, it is assumed that the two production functions F i (K i ; Li ) are similar in that the production functions
satisfy: F c (K c ; Lc ) = BF (K c ; Lc ) and F x (K x ; Lx ) = F (K x ; Lx ), where F

Luxury and Wealth Accumulation15

is a linearly homogeneous concave function. Letting f (k i ) be F (K i ; Li ) =Li
with k i denoting K i =Li , pro…t maximization yields:wf (k c ) ¡ k c f 0 (k c )f (k x) ¡ k x f 0 (k x )==;r
f 0 (k c )f 0 (k x )andw = B ff (k c ) ¡ k c f 0 (k c )g = p ff (k x ) ¡ k x f 0 (k x)g ;where w and r are the wage rate and the capital rent, respectively. The …rst
equation implies
k c = k x = k;where k ´ K=L represents the aggregate capital-labor ratio. The above
maximum pro…t condition thus implies
p = B and r = Bf 0 (k) :(15)

The relative price p is …xed by the productivity factor B.
The demand side is the same as in the previous section except that total
wealth a for the representative agent is speci…ed explicitly as the sum of the
per capita capital stock k as nonhuman wealth and the present value of thewage income ‡ow as human wealth. The resultant optimal conditions are
essentially the same.
The market-clearing conditions depend on which good is accumulated ascapital goods. In either case, however, capital accumulation is generated by
the same aggregate equation,k_ = Bf (k) ¡ c ¡ Bx;(16)

where (15) is substituted. To see this, consider the case in which good c isused for both investment and consumption whereas good x is consumed forthe pure consumption purpose. Market equilibria are then expressed as_
Blc f (k) = c + k;and(1 ¡ lc ) f (k) = x;Luxury and Wealth Accumulation

16

where lc ´ Lc =L represents the proportion of labor employed in sector c. By
multiplying the second equation by B, these equations can be aggregatedinto (16). When good x is used for investment purposes, the k_ term moves
to the right-hand side of the second equation, so that the same aggregation
again yields (16).
The equilibrium dynamics are obtained by combining the supply side,
represented by (15) and (16), and the demand side, described by (3), (5), (6),and the transversality condition. As proven in Appendix B, the equilibrium
dynamics are uniquely given by a saddle time-path governed by a negativeroot. Since the transition dynamics are monotonic, I focus on the steadystate e¤ect ¡of an increasein quasi-luxury preference ®. The steady-state¢¹equilibrium c¹; x¹; k is determined by¹± (¹c; x¹) = Bf 0 (k);

(17)

gx (¹
x; u (¹c; ®¹x) =± (¹c; x¹))= B;gc (¹c; u (¹c; ®¹x) =± (¹c; x¹))

(18)

and
¡ ¢
Bf k¹ = c¹ + B x¹:(19)

Substituting (19) into (17) yields± (¹c; x¹) = Bf 0 (f ¡1 (¹c + B x¹)):(20)

The steady-state consumption basket (¹c; x¹) is jointly determined by (18)¹
and (20). Capital stock k is then given by (19). These relations can bedepicted precisely by reinterpreting …gure 1: the long-run time preference
schedule RR0 can be read as representing (20); the contract curve F F 0 as(18); and the long-run budget schedule AA0 as (19). In particular, since±c =±x > gc =gx = 1=B by Assumption 2, from (19) and (20), the slopes ofschedules RR0 and AA0 satisfy:
¯ ¯ ¯
00
0
¯ dx ¯ ¯¯ ¯ ¯ = ±c ¡ f = (Bf )¯ dc ¯ 0 ¯±x ¡ f 00 = (f 0 )RR1>¯B ¯ ¯¯ dx ¯ ¯= ¯¯ ¯¯ ¯¯ ;dc AA0Luxury and Wealth Accumulation

17

just as in section 2. Therefore, from the same discussion as for …gure 2, anincrease in quasi-luxury preference ® increases the steady-state capital stock
k¹ : it shifts the F F 0 to the left, thereby increasing the total expenditures
c¹ + B x¹ from A0 to A1 and raising the steady-state capital stock.
Proposition 4: The greater is the preference ® for quasi-luxury, the more
steady-state capital is accumulated.
Proof. Di¤erentiating equations (18)-(20) with respect to ® yields
dk¹
ux ¤»
=
> 0;
d®
¢ (f 0 ±x ¡ f 00 )
where
1
¢´ 0
f ±x ¡ f 00½

gc gx ª + f

00µ

gxx Bgcc
+
gx
gc

¶¾

> 0:

Remark 2: From the continuity property, even if consumer i is less patientthan consumer j, in that ± i (c; x) > ± j (c; x) 8 (c; x), the less-patient consumer
i will accumulate more steady-state capital than the more-patient consumer
j if consumer i’s quasi-luxury preference is su¢ciently larger than j’s.
Remark 3: When a one-factor production economy is considered, in which
goods are produced using only capital with constant-to-scale (i.e., linear inthis case) production functions, the interest rate r is also …xed by the productivity factor, say B, so that consumers have no interaction through markets.
As a result, Proposition 4 can be extended to the case of the N heterogeneous
agent economy: Suppose that there are N heterogeneous consumers, indexed
by n = 1; ¢ ¢ ¢ ; N with an identical utility-discounting function ± (c; x) and felicity functions un (c; x) = u (c; ®n x). Then, a consumer with a stronger
luxury preference accumulates more steady-state capital in that:¹i > K¹ j 8i; j = 1; ¢ ¢ ¢ ; N; i 6= j:®i > ®j , K
As can easily be conjectured from the above analysis, taxation on quasiluxury goods harms capital accumulation. To show this, consider a tax ¿on quasi-luxury consumption x, assuming that the tax revenue ¿ x is paidLuxury and Wealth Accumulation

18

back to households in a lump-sum manner. Then, the …rst-order condition
between the two-good consumptions becomesgx (x; Á)= (1 + ¿ ) p:
gc (c; Á)
Since any other equilibrium conditions are unchanged, this implies that anincrease in the quasi-luxury tax ¿ has qualitatively the same e¤ect as adecrease in quasi-luxury preference ®: it decreases the steady-state capital¹ Figure 2 can be reinterpreted as illustrating this. A quasi-luxury
stock K.
tax increase shifts the F F 0 schedule to the right, bringing the steady-state
point from E1 to E0 :4

Wealth distribution in a two-country world economy

Let us …nally examine the implication of quasi-luxury preferences for wealthdistribution. Consider a two-country world economy composed of home andforeign countries, H and F. Foreign country variables are denoted by asterisks.
Any production activities are neglected for simplicity. The representative
agents in both countries are equally endowed with Y =2 and X=2 of the twogoods, quasi-necessity good c and quasi-luxury good x. Countries H and F
can freely trade these two goods at the world price p, and bonds b at theworld interest rate r. The household sector here is exactly the same as insection 2 if total asset holding a in (4) is speci…ed by
µ Z s
¶
Z 1
Y + p (s) Xa (t) = b (t) +exp ¡r (¿ ) d¿;2tt
implying
b_ (t) = r (t) b (t) + Y =2 + p (t) X=2 ¡ c (t) ¡ p (t) x (t) :(21)

The market equilibrium conditions for the two goods and bonds are expressed as:c (t) + c¤ (t) = Y;
x (t) + x¤ (t) = X;andb (t) + b¤ (t) = 0:Luxury and Wealth Accumulation

19

By the Walrus law, when the two-good markets are in equilibrium for all t,
the international bond market is also in equilibrium. Given the time pathsof p (t) and r (t), the optimal consumption path fb (t) ; c (t) ; x (t) ; Á (t)g1
t=0
for the representative agent in country H is determined by (3), (5), (6), and(21), and that for country F by the corresponding equations with respect
to fb¤ (t) ; c¤ (t) ; x¤ (t) ; Á¤ (t)g. The equilibrium price path fp (t) ; r (t)g1
t=0
is determined such that these optimal consumption paths satisfy the above
market-equilibrium conditions.
For simplicity, it is assumed that the two countries are initially identicaland hence that the initial equilibrium is autarchic. Using this framework, let
us examine the e¤ects of a permanent increase in country F’s preference for
quasi-luxury goods. Our main interest is to show that due to the resultant
stronger preference for quasi-luxury country F becomes wealthier than H in
the steady state. The initial autarchy equilibrium is constructed by assuming
the following:
Assumption 4: Both the felicity and subjective discount functions are initially identical between the two countries:u¤ (c¤ ; x¤ ) = u (c¤ ; ®x¤ ) ; ± ¤ (c¤ ; x¤ ) = ± (c¤ ; x¤ ) ;where country F’s quasi-luxury preference ® initially equals one.
The steady-state equilibrium of the world economy is determined fromthe following ten equations:± (¹c; x¹) = ± (¹c¤ ; x¹¤ ) = r¹;(22)g ¤ (¹gx (¹x; u (¹c; x¹) =± (¹c; x¹))x¤ ; u¤ (¹c¤ ; x¹¤ ) =± (¹c¤ ; x¹¤ ))= x¤ ¤ ¤ ¤ ¤
= p¹;
gc (¹
c; u (¹c; x¹) =± (¹c; x¹))gc (¹c ; u (¹c ; x¹ ) =± (¹c¤ ; x¹¤ ))(23)

r¹¹b(¤) + Y =2 + p¹X=2 = c¹(¤) + p¹x¹(¤) ;¡ (¤) (¤) ¢(¤)
c¹ ; x¹
u
(¤)
¹ =
;
Á
(¤)
± (¹
c ; x¹(¤) )and the market equilibrium conditions:c¹ + c¹¤ = Y; x¹ + x¹¤ = X:

(24)

Luxury and Wealth Accumulation20

Due to Assumption 4 and the assumption of the symmetric supply sides, the¹ ; ¹b¤ ; c¹¤ ; x¹¤ ; Á¹ ¤ ) is autarchic:
initial steady-state equilibrium (¹r0 ; p¹0 ; ¹b0 ; c¹0 ; x¹0 ; Á
0 0 0
0
0
c¹0 = c¹¤0 = Y =2; x¹0 = x¹¤0 = X=2; ¹b0 = ¹b¤0 = 0;
r¹0 = ± (Y =2; X=2) ; p¹0 =gx (X=2; u (Y =2; X=2) =± (Y =2; X=2));
gc (Y =2; u (Y =2; X=2) =± (Y =2; X=2))

and¹0 = Á
¹ ¤ = u (Y =2; X=2) =± (Y =2; X=2) :Á0
It is useful to construct a box diagram, as in …gure 3, to depict thedetermination of the steady-state equilibrium. Schedule RR0 represents thedownward-sloping curve that is obtained by eliminating c¹¤ and x¹¤ from (22)using the market-clearing conditions, and schedule F F 0 the upward-sloping
curve obtained from (23) in the same way, respectively:
RR0 : ± (¹c; x¹) = ± (Y ¡ c¹; X ¡ x¹) ;(25)

F F 0 : MRS (¹c; x¹) = MRS¤ (Y ¡ c¹; X ¡ x¹; ®) ;

(26)

and

where the intratemporal MRS function MRS (¹c; x¹; ®) is de…ned by (14).
Schedules F F 0 and RR0 , the two-country versions of F F 0 and RR0 in …gure 1, could be regarded as the intratemporal and intertemporal steadystate contract curves, respectively. The equilibrium consumption allocation(¹c; x¹; c¹¤ ; x¹¤ ) is determined at the intersection E of the two contract curves.
Given point E, consider the steady-state indi¤erence curve I (E), de…ned by(11). p¹ is then determined as the gradient of the curve at E. As discussed in
section 2, by Assumption 2, the intertemporal contract curve RR0 is steeper
than the indi¤erence curve I (E) at E. Recall that this is the key property ofquasi-luxuries that have produced Propositions 3 and 4. Furthermore, it canbe shown that the equilibrium time-path of the economy around the steadystate is uniquely given by a saddle path governed by a negative root (seeAppendix C for the proof). The same discussion as in the previous sections
can thus be applied to obtain the implication of the quasi-luxury preference.
Suppose that country F’s preference ® for quasi-luxury goods increases.
As depicted by …gure 4, it shifts the intratemporal contract F F 0 to the right,
bringing the steady-state point from point E0 to E1 . For country H, theLuxury and Wealth Accumulation

21

new consumption basket is below the initial indi¤erence curve I (E0 ), implying that its steady-state welfare deteriorates. Accordingly, the steady-state
wealth holding of country H decreases, whereas that of country F increases,
as depicted by the change from point A0 to A1 . This can be summarized asfollows:
Proposition 5: The country with a stronger preference for quasi-luxury
holds more wealth in the steady state than the other country with a weakerpreference.
Proof. Equations (22)-(24) yield:
d¹b¤
d¹b
ux »
=¡
=¡
¤ < 0:d®
d®
2ªgc gx

An increase in country F’s preferences for quasi-luxury raises the relative
price p of the quasi-luxury¡¢¡good.
¢ In country H, to raise the marginal rate¹¹
of substitution gx x¹; Á =gc c¹; Á of the quasi-luxury up to the higher p, therepresentative agent shifts consumption away from the quasi-luxury good tothe quasi-necessity. It should take place along the intertemporal contract
curve, i.e., the quasi-luxury consumption should be reduced in exchange forthe quasi-necessity at the rate of dx=dc = ±c =±x (see (8)), which,¡¢ for ¡given¢¹¹¹¹
levels of b and, hence, Á, is larger than required to equalize gx x¹; Á =gc c¹; Áto p. To reduce the MRS for quasi-luxury down to p, steady-state welfare¹
hence, the underlying wealth holdings ¹b should decrease. From theÁand,market clearing conditions, country F’s consumption of the quasi-luxury goodincreases and that of the quasi-necessity decreases. Its wealth holdings ¹b¤
should increase to …nance more quasi-luxuries than before.
As in the previous section, the following result can also be obtained fromthe continuity property of the steady-state solution with respect to preference
parameters.
Corollary 1: Even when there is an international di¤erence in the steadystate time preference (utility-discounting functions), the less-patient country
can have more wealth in the steady state than the more-patient one if theless-patient one has su¢ciently stronger preference for quasi-luxury.

Luxury and Wealth Accumulation

5

22

Conclusions

There is an old idea that certain goods are related to wealth accumulation.
I have provided a model to formalize the idea by incorporating weakly nonseparable preferences. The model helps to understand luxury consumption
from a dynamic viewpoint. The phenomenon that wealthier agents consume
more luxuries can be described by stating that the consumers are wealthierbecause they prefer luxuries (quasi-luxuries). A new insight is that wealthaccumulation should re‡ect consumers’ preferences for various kinds of goods
as well as for time.
There are several ways in which the above analysis can be extended. First,
it should be extended to the case of more than two goods. Quasi-luxury goods
are de…ned here in terms of the MRS between two goods. With more than
two goods, some other devices would be required. Second, empirical testingof our model should be conducted. The model could be hypothesized by
the property that di¤erences in commodity-speci…c time preferences between
luxury and necessity goods depend on wealth accumulation. This could bedone by estimating the commodity-speci…c Euler equations.

Luxury and Wealth Accumulation

23

AppendicesA

Coe¢cient matrix A in section 2.2
02¡ g»gxxx

B
³
´
B
cc
A ´ B ¡gc 1 + p2 ggxx
´
@
³
2 gcc
¡ 1 + p gxx

B

gc
gcc

³ 22
gx »
gxx

+ r±gcc2
r + g»gxxx
x
p »g
gxx

´

0

1

C
C
0 C:
A
r

(27)

Dynamics of the growth model in section 3

This appendix proves that the equilibrium dynamics in the growth model insection 3 can be obtained as a saddle time-path around steady state. The
equilibrium dynamics for (c; k; p; r; x; Á) are generated by (3), (5), (6), (15),(16) and the transversality condition. The system can be reduced to:gc
c_ = ¡ (Bf 0 (k) ¡ ½c (c; x; Á));gcc
Á_ = u(c; ®x) ¡ Á±(c; x);k_ = Bf (k) ¡ c ¡ Bx:
Substituting (5) for x in the above system yields an autonomous di¤erential
equation system with respect to h = (c; Á; k). The resulting system can be^ wherelinearized as h_ = Gh,
1
³ 22
´
0
»gx2
gx »
gc
gc
r±c
00
; ¡ gcc Bf
+
¡ ;
C
B
³ gxx
´ gcc gxx 2 gc
CBccx
G = B ¡gc 1 + B 2 ggxx;r + »g
;
0
C:
gxx
´
A
³
@
cc
;
B g»gxxx ;r¡ 1 + B 2 ggxx
The trace of G equals 2r. Its determinant can be computed asµ¶
gc2 gx2
rBgc f 00
2 gcc
det G = ¡
ª¡
1+B
< 0;
gcc gxx
gcc
gxx

implying that the equilibrium dynamics are saddle-point stable. It can beshown that the two positive roots are inconsistent with the transversality
condition.

Luxury and Wealth Accumulation

C

24

Dynamics of the two-country model in section 4

With the steady-state equilibrium being given above, the local dynamic system for m ´ (c; Á; Á¤ ; b) can be obtained asm_ (t) = Dm
^ (t) ;
0
2
x
¡ »g
g
xx
B
³
´
B
2 gcc
1
+
p
B ¡gc
gxx
³
´
D=B
B
2 gcc
B gc 1 + p gxx³´@cc¡ 1 + p2 ggxx

gc
2gcc

³ 22
gx »
gxx

+ r±gcc

»gx2
2gxx
»gx2
2gxx

r+
¡

»gx
p 2g
xx

´

¡ 2ggccc

³ 2 2

gx »
+ r±gcc
gxx
»gx2
¡ 2g
xx
»gx2
r + 2gxx
»gx
¡p 2g
xx

´

0

(28)
1

C
C
0 C
C:
C
0 C
A
r

The linear system can be reduced in the form of a block-structure byusing the di¤erence Á¡ of the two countries’ welfare levels and their averageÁ+ :13Á ¡ Á¤ + Á + Á¤
¡
(29)
Á =
;Á =
;
2
2
as, for h ´ (c; Á¡ ; b; Á+ ),^ (t) :(30)
h_ (t) = H h
µ
¶
A 0
H=
;
0 r
where A is matrix (27) describing the local dynamics for individuals’ optimal
consumption in section 2. As can easily be seen, H has three positive roots
and one negative root and the negative root is the same as ! in section 2.14
The resultant equilibrium path is also very similar to that obtained in (12):b_ (t) = !^b (t) ; ^b (0) = b0 ¡ ¹b;½¾gxx (r ¡ !)gx2¤c^ (t) = ¡^c (t) =+ » ^b (t) ;(gxx + p2 gcc )
gx2
½
¾
gx gc
gcc (r ¡ !)¤x^ (t) = ¡^x (t) =¡ » ^b (t) ;(gxx + p2 gcc )
gc2^ (t) = ¡Á^ ¤ = gc^b (t) :Á13For this procedure, see Aoki (1981).
The magnitude of ! is di¤erent from that in section 2 because the points at which !
is evaluated are di¤erent.
14

Luxury and Wealth Accumulation

25

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xEngel curve

A

R

F

E

F′

I(E)
R′

-1/p

c
O

A′

Figure 1. Optimum Consumption Plan

x

A1

R

F1
A０
α

F0
τ

E1

F1′ E01

F０′

E０

R′
c

O

Figure 2. QuasiQuasi-Luxury Preference and Optimum Wealth Accumulation

c*
O*

x
R

F

A′

E
X/2

I(E)
F′

x*
R′

O

A
Y/2

Figure 3. TwoTwo-Country Equilibrium

c

c*
O*

x
R

F０

A０′
F1
E０
X/2

α

I(E)

E1

F０′

x*
R′
F1′

O

A1 A０
Y/2

c

Figure 4. Quasi-Luxury Preference and Wealth Distribution