Ariel Rubinstein

Michele Piccione

English

Research Papers in Economics

Luxury Prices:
An Expository Note∗
Michele Piccione
Department of Economics
London School of Economics
and
Ariel Rubinstein
School of Economics, Tel Aviv University
and Department of Economics, New York University
http://arielrubinstein.tau.ac.il
September 14, 2006
Abstract
We study a model in which “useless” luxury goods are a mechanism
for redistribution.
JEL Classiﬁcation: C7, S477, B201
∗We would like to thank Ori Heﬀetz and an anonymous referee for valuable comments.
11 Introduction
Economists generally associate the redistribution of resources to the appa-
ratus of taxes and transfer payments. Such redistributions are done by the
power of the authorities. However, resources are redistributed by other means
as well. People give away income in a variety of ways, deliberate and unin-
tentional. In this paper, agents transfer consumption goods in return for a
good which lacks material qualities and aﬀects their preferences because it
has “value”.
An example of a real life commodity without intrinsic value is diamonds.
The standard consumption value of a diamond is low, especially as artiﬁcial
diamonds look the same as real ones to the naked eye. Many luxury goods
such as art pieces have similar characteristics. Owning a house in a “rich”
area could have extra beneﬁts in consumption and yet, one important mo-
tivation is simply to live in an expensive neighborhood. By holding such a
good an agent refrains from acquiring resources that he could consume and
thus, frees resources for other agents.
Our thinking on this issue originates from a naive question: Why are
wealthy people striving to be wealthier? It is quite absurd to think that rich
people wish to increase the consumption of standard goods as assumed in the
classical consumer model. Indeed, by acquiring an expensive diamond or an
original piece of art, a consumer demonstrates his avoidance of consumption.
We construct and analyze an elementary competitive equilibrium model
in which the value of “useless” jewels enter the preferences of consumers.
We show that if jewels are scarce, they can trade at a positive price despite
having no intrinsic utility. Under fairly natural assumptions, a positive price
equilibrium redistributes consumption from the rich to the poor. Thus, the
model demonstrates the common wisdom that the existence of luxury goods
can have a positive eﬀect on income distribution.
This paper is mainly expository. The idea of consumption goods lack-
ing intrinsic value and of prices entering directly the utility functions is not
original. Veblen (1899) pointed out that some goods are consumed by some
people for reasons not connected with their intrinsic values. Veblen referred
to consumption intended to impress and to signal wealth as conspicuous con-
sumption.
Arrow and Hahn (1971) (Chapter 6) (see also Kalman (1968)) discussed a
general equilibrium model in which agents care not only about the consump-
tion but also about the price vector. Under standard assumptions, they
2proved existence of competitive equilibrium. In our model, the existence of
equilibria is not an issue as there always exists an equilibrium where jewels
have zero price. Our interest is in the existence of equilibria in which the
price of jewel is not zero.
Ng (1987) treats “diamonds” explicitly as goods that “are valued not be-
cause their intrinsic consumption eﬀects but because they are costly.” He con-
siders a consumer whose utility function is of the type U(p1x1/pn,x 2,x 3,...xn),
where the ﬁrst good is “diamonds” and the last good is the “numeraire”.
The maximization of this function given the standard budget constraint P
pixi = w yields demands for which a change in p1 does not change the de-
mand of other commodities and does not aﬀect the utility. Ng uses this fact
to point out that imposing taxes on diamonds is an eﬃcient way of raising
r e v e n u ea si td o e sn o th a v er e a le ﬀects on the economy.
Bagwell and Bernheim (1996) study a model in which the utility of a
consumer is a function of his consumption and an action taken by a “rep-
resentative agent” (the society) who observes whether the consumer does or
does not hold a jewel. The representative agent does not observe the initial
wealth of the consumer and his action is a response to the information he
infers from conspicuous consumption. Thus, a consumer faces a trade-oﬀ
between consuming and showing that he is “wealthy” by buying a jewel.
Other models consider equilibria with goods that do not have intrinsic
value. See, for example, Frank (1988). In this paper, we focus on a tradeable
good whose possession generates utility endogenously only as a function of
the price. We treat “jewels” as goods which satisfy the psychological need of
owning a precious commodity and not as signals of status aimed at obtaining
more standard consumption goods. In particular, we show that there can be
equilibrium prices inducing diﬀerent levels of utility yet revealing the same
information about the consumers who possess the good. Although many
conspicuous commodities do serve as signals of status, the same cannot be
said of many goods with no intrinsic value such as old stamps or coins.
2 A Simple Model
Consider an economy with one divisible consumption good and one indivisible
good called a “jewel”. For simplicity, we will assume that a consumer cannot
own more than one jewel. Each individual enters the market with a pair (c,d),
3where c ∈ <+ is the amount of consumption that he owns and d ∈ {0,1}
indicates whether he owns or does not own a jewel. Let µn(c) denote the mass
of consumers who initially do not own jewels and whose initial endowment
of consumption is below c,a n dµo(c) the mass of the initial owners of jewels
whose initial endowment of consumption is below c.D e n o t e b y µn(∞) the
mass of consumers who initially do not own jewels and by µo(∞) the mass
of consumers who initially do. The measures µn(c) and µo(c) are deﬁned on
the interval [0,∞). We assume that there exists points ci, i = n,o, such that
µi(c)=µi(∞) for any c ≥ ci and µi(c) is strictly increasing on the interval
(0,c i).
Denote by p the price of a jewel in terms of consumption. The decision
that each consumer faces is binary, that is, each consumer decides whether
to hold the jewel or not; if he holds the jewel initially, he decides whether to
sell it for p units of consumption; if he does not holds the jewel initially, he
decides whether to buy it for p units of consumption.
The pair (c,p) denotes consumption of c and ownership of a jewel priced
p. We assume that consumers maximize a preference relation %,i d e n t i c a l
for all consumers, over the space of pairs (c,p). The pair (c,0) denotes
consumption of c and either ownership of a jewel priced 0 or not owning a
jewel. We assume that consumers are indiﬀerent between not having a jewel
and having a jewel priced 0. The value of a jewel is in its price.
We make several assumptions about the preference relation %:
P1: The preferences are monotonic, strictly convex, and continuous.
P2: For any x>0, (x,0) Â (0,x) (consumers prefer to consume x units to
a jewel priced x).
P3: For c0 >cand p>0,i f(c,p) % (c + p,0) then (c0,p) Â (c0 + p,0) (if a
consumer wishes to have a jewel, a richer consumer also wishes to have
it).
Any preference relation represented by a utility function u(c,p) which
satisﬁes (i) strong monotonicity, (ii) quasi-concavity, (iii) continuity, (iv)
∂2u
∂p∂c
−
∂2u
∂c2 > 0 and (v) u(x,0) >u (0,x),s a t i s ﬁes assumptions P1-P3.
To see that such a utility function satisﬁes P3, note that
4u(c,p) − u(c + p,0) =
Z p
0
[
∂u
∂p
(c + p − t,t) −
∂u
∂c
(c + p − t,t)]dt
and, by (iv), the diﬀerence between the partial derivatives in the integral is
increasing in c.
A candidate for a competitive equilibrium is a price p and two functions
B :[ 0 ,∞) → [0,1] and S :[ 0 ,∞) → [0,1].T h e f u n c t i o n B describes the
behavior of a consumer with initial bundle (c,0): The proportion of agents
with initial bundle (c,0) w h ob u yaj e w e li sB(c).T h ef u n c t i o nS describes
the behavior of a consumer with initial bundle (c,1): The proportion of
agents with initial bundle (c,1) who sell the jewel is S(c). Thus, whenever
there is a positive mass of consumers with initial bundle (c,i), we implicitly
treat the mass as a continuum of consumers.
An equilibrium is a triple (p,B,S) satisfying
(i) For any c, the behavior described by the functions B and S is optimal
given the price p (namely, B(c)=1if (c − p,p) Â (c,0), B(c)=0if (c,0) Â
(c − p,p),e t c . ) .
(ii) Equality of demand and supply of jewels:
R
B(c)dµn(c)=
R
S(c)dµo(c)
2.1 Buying a jewel
Under assumptions P1-P3, there is a cut-oﬀ c∗ such that:
(i) consumers who do not have a jewel and are endowed with consumption
below c∗ do not buy the jewel regardless of the price, and
(ii) consumers who do not have a jewel and are endowed with consumption
above c∗ buy the jewel if its price does not exceed a reserve price.
Claim 1 Assume P1-P3. There exists an increasing, continuous function
q :[ 0 ,∞) → [0,∞] such that a consumer endowed with c a n dn oj e w e l
demands the jewel if p<q (c) and does not demand it if p>q (c).T h e
function q is strictly increasing at any point c>c ∗ =m a x {c|q(c)=0 }.
Proof: Consider the set D of all pairs (c,p) for which a consumer who
has consumption level c and no jewel is not worse oﬀ buying the jewel for the
5price p. By the continuity the set D is closed. By P2, if (c,p) ∈ D and c>0
then p<c .B yc o n v e x i t y ,i f(c,p) ∈ D then, for any p0 <p , (c,p0) ∈ D.B y
P3, if (c,p) ∈ D then, for any c0 >c , (c0,p) ∈ D.B yd e ﬁnition, (c,0) ∈ D
for any c.D e ﬁne q(c)=m a x {p :( c,p) ∈ D}.T h ef u n c t i o nq is continuous
and strictly increasing for any c for which q(c) > 0. A potential buyer with
initial consumption level c will demand the jewel if p<q (c) and will not if
p>q (c).Q E D
The above claim give us a characterization of the demand for jewels.
Given a composite function f(g(x)),l e tf+(g(x)) and f−(g(x)) denote the
right limit and the left limit at x. For a positive price, the demand is the
correspondence
∆(p)=[ µn(∞) − µ
+
n(q
−1(p)),µ n(∞) − µ
−
n(q
−1(p))].
2.2 Selling a jewel
In the next result, we characterize the supply of jewels. Let cs(p) be the
largest c ∈ [0,c o] such that (c + p,0) % (c,p).N o t e t h a t ,b y P 2 ,cs(p) > 0
for any p a n dt h a t ,b yP 1a n dP 3 ,(c,p) Â (c + p,0) if and only if c>c s(p).
For a positive price, the supply is the correspondence:
Σ(p)=[ µ
−
n(c
s(p)),µ
+
n(c
s(p))].
Since a change in the price p implies also a change in the wealth level of a
potential seller, we need additional assumptions for a tidy characterization.
P4: For c>0 and p0 >p>0,i f(c,p) % (c+p,0) then (c,p0) Â (c+p0,0)
P4 implies that if a consumer who owns a jewel prefers to keep it, he
will still prefer to keep it if the price increases. Any preference relation
represented by a utility function v(c)(1+p),w h e r ev0 > 0, v00 < 0,a n d
v(0) = 0 satisﬁes assumptions P1-P4.
6Claim 2 Assume P1-P4. There exists a decreasing, continuous function
r :[ 0 ,∞) → [0,∞] and cl,c h ∈ [0,∞] such that: (i) if c ≤ cl,a no w n e rs e l l s
the jewel for any positive price; (ii) if cl <c<c h, an owner sells the jewel
if p<r (c) and does not sell it if p>r (c); (iii) if c>c h,a no w n e rd o e sn o t
sell it the jewel for any positive price. The function r(c) is strictly decreasing
at any point c for which r(c) is ﬁnite and positive.
Proof: Consider the set Z of all pairs (c,p) for which a consumer who
holds initially consumption level c and a jewel is not worse oﬀ by selling the
jewel at price p. By continuity, the set Z is closed. By P2, for any p we have
(0,p) ∈ Z.
By P4, if (c,p) / ∈ Z then (c,p0) / ∈ Z for any p0 >p .B yP 3 ,i f(c,p) / ∈
Z then (c0,p) / ∈ Z for any c0 >c .D e ﬁne r(c)=m a x {p :( c,p) ∈ Z} (r(c) can
be inﬁnite). The function r is continuous and strictly decreasing when it has
a ﬁnite positive value. Thus, there exist cl and ch such that: (i) if c ≤ cl,a n
owner sells the jewel regardless of the price, (ii) if cl <c<c h,a no w n e ro f
the jewel sells the jewel if p<r (c) and does not sell it if p>r (c) and (iii) if
c>c h, an owner does not sell the jewel regardless of the price. QED
Note that, under assumption P4, the supply of jewels is downward sloping.
If P4 is replaced by the following assumption, the supply of jewels is upward
sloping.
P40:F o rc>0 and p0 >p>0,i f(c+p,0) % (c,p) then (c+p0,0) Â (c,p0)
P40 implies that a consumer who sells a jewel will continue selling it if
the price increases. Any preference relation represented by a utility function
v(p)c,w h e r ev0 > 0, v00 < 0,a n dv(0) > 0 satisﬁes assumptions P1-P3, and
P40.
Claim 3 Assume P1-P3 and P40. There exists an increasing, continuous
function r :[ 0 ,∞) → [0,∞] and cl,c h ∈ (0,∞] such that: (i) if c ≤ cl,a n
owner sells the jewel for any positive price; (ii) if cl <c<c h,a no w n e rs e l l s
the jewel if p>r (c) and does not sell it if p<r (c); (iii) if c>c h,a no w n e r
does not sell it for any positive price. The function r(c) is strictly increasing
at any point c for which r(c) is ﬁnite and positive.
7Proof: The proof is analogous to the proof of Claim 2 and is omitted.
QED
2.3 Equilibria
A “no-value equilibrium”, in which the price of a jewel is null and the alloca-
tion of jewels is arbitrary, always exists. In the next proposition, we provide
conditions for the existence of equilibria in which jewels are traded and have
positive value. Recall that µi(ci)=µi(∞), i = n,o.
Proposition 4 Assume P1-P3. Suppose that µn(cn) − µn(c∗) >µ o(co).
Then, there exists an equilibrium with a non-zero price and trade.
Proof:F o ra n ys m a l l >0, when the price is positive and suﬃciently
small all points in ∆(p) are at least as large as µn(cn)−µn(c∗)−  by Claim
1. Since µn(cn) − µn(c∗) >µ o(co), the demand for jewels exceeds the supply
of jewels when the price is positive and suﬃciently small.
If p>q (cn), demand is zero but supply is positive by P2 and the as-
sumption that µo is strictly increasing on its support [0,c o].S i n c et h em a s s
functions µo and µn are strictly increasing on the support and the functions q
and r are continuous, demand and supply are upper hemi-continuous, convex
valued correspondences. Hence, an equilibrium price with trade exists. QED
Proposition 4 does not rely on assumption P4. Under P4, equilibria with
unbounded prices and no trade can exist when every jewel owner wishes to
keep the jewel for some positive price. Such equilibria are not unrealistic. As
Baumol (1986) pointed out, trade in art pieces is very infrequent.
Proposition 5 Assume P1-P4. Suppose that all owners have an initial con-
sumption endowment bounded below by a number ˜ c>c l.T h e n ,t h e r ee x i s t s
¯ p such that any p∗ > ¯ p is an equilibrium price for which no jewels are traded.
Proof: Take any p∗ > ¯ p =m a x {q(cn),r(˜ c)}. By the Claims 1 and 2, no
consumer wants to buy a jewel and no consumer wants to sell a jewel. QED
8The following example illustrates the equilibrium under assumptions P1-
P4.
Example: Consider the case that the consumers’ preferences are repre-
sented by u(c,p)=( 1+p)
√
c. This utility function satisﬁes P1-P4. On the
demand side, we get c∗ =1 /2 and q(c)=1
2c+ 1
2
√
4c + c2 −1. On the supply
side, we get cl =0 , ch =1 /2 and r(c)=
1
c
− 2.
Poor non-owners (c<1/2) never buy. Rich non-owners (c>1/2) will
buy if p<1
2c + 1
2
√
4c + c2 − 1.R i c hj e w e lo w n e r s( c ≥ 1/2) never sell. Poor
jewel sell if the price is below
1
c
− 2.
As always, the market has a zero price equilibrium: p =0is a trivial
equilibrium price for which nobody cares about owning a jewel. In any (non-
degenerate) equilibrium, µo(1/(2 + p)) = µn(∞)−µn( 1
p+2 (p +1 )
2). Suppose
that αn non owners of jewelry and αo owners are uniformly distributed on
[0,1]. Then in equilibrium αo
³
1
p+2
´
= αn(1 − 1
p+2 (p +1 )
2).
Let αo/αn = k. The equilibrium equation p +2− (p +1 )
2 − k =0is
solved by p = 1
2
√
5 − 4k − 1
2,w h i c hi sp o s i t i v ef o rk<1.
Consider now µn with support bounded by some cn and µo such that
µo(˜ c)=0for some ˜ c>0. Then, any p>max{
1
˜ c
−2, 1
2cn + 1
2
p
4cn + c2
n −1}
is an equilibrium price with no trade.
Assumption P3 implies that in equilibrium, if consumer with a consump-
tion level c buys a jewel for a price p, no initial owner with consumption of
c − p or higher will sell the jewel. In equilibrium, the highest c of a seller is
smaller by at least p than the lowest c of a buyer. Thus, by any measure of
inequality, the equilibrium trade improves the equality of the distribution of
consumption in the population. Also note that a trade equilibrium always
Pareto dominates a no-trade equilibrium.
Comment: Competitive equilibria in standard models are Pareto eﬃ-
cient. In our model, the higher is the price in the no-trade equilibrium of
Proposition 5, the “better” it is for the consumers who hold the jewel. For
any such equilibrium, there is another one which Pareto dominates it. In
particular, after trade at price p∗, all jewel owners have consumption above
r−1(p∗) and thus, by Proposition 5, there are new equilibria with “high”
p r i c e sa n dn ot r a d ew h i c hm a k ea l lo w n e r sb e t t e ro ﬀ and leave the non-
owners indiﬀerent.
9References
[1] Arrow, Kenneth, J. and Frank, H. Hahn, General Competitive Analysis,
North Holland, 1971 (second print 1983).
[2] Bagwell, Laurie Simon and B. Douglas Bernheim, Veblen Eﬀects in a
Theory of Conspicuous Consumption”, The American Economic Review,
Vol. 86, No. 3. (Jun., 1996), 349-373.
[3] Baumol, William J. (1986) “Unnatural Value: Or Art Investment as
Floating Crap Game” The American Economic Review, Vol. 76, No. 2,
Papers and Proceedings, 10-14.
[4] Frank, Robert, H., Passions Within Reason: The Strategic Role of the
Emotions, W.W. Norton, 1988.
[5] Kalman, Peter Jason, (1968), “Theory of Consumer Behavior when prices
Enter the Utility Function”, Econometrica, 36, 497-510.
[6] Ng, Yew-Kwang (1987), ”Diamond are a Government’s Best Friend:
Burden-Free Taxes on Goods Valued for Their Values” American Eco-
nomic Review 77, 186-191.
[7] Veblen, Theorstein, The Theory of the Leisure Class, New York: Macmil-
lan, 1889.
10

Luxury Prices: An Expository Note

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