Electronic copy available at: http://ssrn.com/abstract=1930361

The Economics of Luxury Goods: Utility Based on

Exclusivity

Stefka Petrovay Vitaly Pruzhanskyz

19 September 2011

Abstract

We propose a model describing consumer demand for a luxury good, in which

the perceived quality of the good is related to its exclusivity, that in turn depends

on the number of consumers buying it. We use this model to analyze the optimal

production and price setting decisions of a luxury good manufacturer and contrast

them with the decisions that would be made by a social planner. We show that

irrespective of the way social welfare is dened, a monopoly producer of the luxury

good may select socially optimal prices and quantity. Thus the incentives of the

monopolist producer and the social planner may to some extent be aligned.

.

Keywords: welfare, luxury goods, competition policy, regulation

JEL Classication: L40, L42.

We beneted from discussions with Jurjen Kamphorst and Mikhailo Trubskyy. The usual disclaimerapplies. The views expressed therein are those of the authors only and do not necessarily represent theocial views of the authors respective institutions, University of Antwerp and RBB Economics.

yDepartment of Economics, University of Antwerp, Belgium.zRBB Economics, Bastion Tower, Place du Champ de Mars 5, 1050 Brussels, Belgium, vi-

taly.pruzhansky@rbbecon.com (corresponding author).

1

Electronic copy available at: http://ssrn.com/abstract=1930361

1 Introduction and motivation

The economic literature on luxury goods and conspicuous consumption originates from

the work of Thorstein Veblen [13] and John Rae [11] in the nineteen century. One of

main ideas of their work was that wealthy consumers buy conspicuous goods to show

their social status or provide evidence of their wealth. A recent overview of this work

can be found in, e.g. Trigg [12]. The work of Veblen and Rae spawned a fair body of

research on the economics of luxury or status goods. For instance, the ability of the

luxury good owner to signal the owners wealth, e.g. Bagwell and Bernheim [3], or the

issues of optimal taxation of diamond goods, e.g. Ng [10].

The purpose of the current paper is to study the welfare aspects of the luxury goods

industry by contrasting the production decisions of the social planner and those of a

monopolist producer. We adopt a partial equilibrium approach and consider decision

making of a representative consumer for just one luxury good, disregarding the existence

of ordinary goods and possible substitution eects between the two. We also assume

that the luxury good is produced by a monopolist, and thus assume away competition

between dierent producers of luxury goods.

We model the luxury aspect of the good, based on its scarcity or exclusivity, an

approach similar in spirit to Yao and Li [14]. This assumption is justied by the state-

ments of some key luxury goods manufacturers, who argue that the chief value of a

luxury good lies in that it is not too common, and that widespread sales may destroy

this high end image of luxury goods, which mainly determines the consumers utility

and willingness to pay. For instance, Patrick Thomas, CEO of one of the largest luxury

goods companies - Hermes stated in an interview to The Wall Street Journal [8]:

"...We are not fashionable, and we avoid being fashionable".

Similarly, a luxury goods analyst at the leading investment bank J.P.Morgan notes:

"...A luxury brand cannot be extended indenitely: if it becomes too common,

it is devalued, as Pierre Cardin and Ralph Lauren proved by sticking their

labels on everything from T-shirts to paint."

We propose a model in which the consumer utility (or more specically, the per-

ceived quality) of a luxury good negatively depends on the number of consumers who

buy it, i.e. its exclusivity or scarcity.1 The more common the good is (e.g. because

1Sometimes this is called the snob eect, see Leibenstein [7].

2

many consumers buy it already), the less is its perceived value. This specication leads

to the negative network externality eect, whereby an increase in consumption by new

customers reduces the utility of those who already consume the good. We also show

that modeling of the exclusivity eect does not require the use of an upward sloping de-

mand function and can be adequately represented by a conventional downward slopping

demand under suciently general conditions.

We then apply this economic framework to analyze welfare aspects of the luxury

good. We nd that the incentives of the social planner and monopolist producer may

in some cases be aligned and that traditional loss of welfare associated with monopoly

pricing does not always arise. One possible interpretation of this result is that the

luxury industry should not be regulated: decentralized production and pricing decisions,

even by monopolist suppliers, may maximize consumer (and total, i.e. consumer and

producer) welfare.

It is important to point out at the outset that the model developed in this paper

does not intend to describe the signaling aspect of luxury goods. The latter takes place

when customers buy an expensive luxury item in order to signal their overall wealth or

social status. There are other economic models describing the signaling eect, see for

instance Bagwell and Bernheim [3] or Yao and Li [14].

The paper is structured as follows. Section 2 provides a basic introduction into the

luxury goods industry and its key recent trends. Section 3 presents an economic model

capturing the scarcity or exclusivity aspect. Section 4 extends this model by analyzing

its welfare aspects from the viewpoint of the social planner and contrasts them those

under monopolist production. Section 5 contains illustrative numerical examples.

2 Overview of the luxury goods industry

Households in all major economies (e.g. EU, US, Japan) spend about 1% of their

nal consumption on personal luxury goods, such as apparel, leather accessories, shoes,

watches, jewelry and perfumes. Of these, apparel (womenswear and menswear) are the

two most important categories, followed by watches, cosmetics and leather goods, see

Figure 1.

The world-wide market size of these personal luxury goods is estimated to reach

185 bln in 2011. During 2002-2008 the market has grown on average 6% per annum,but due to the global economic downturn, 2009 sales declined by 11%, see Figure 2.

According to the consulting rm Bain & Company, 2010 was the year the customers

3

24.0

8.0

8.2

18.4

20.0

22.4

23.5

23.6

26.6

Other

Jewelry

Shoes

Perfumes

Leather goods

Cosmetics

Menswear

Womenswear

Watches

GLOBAL MARKET FOR PERSONAL LUXURY GOODS BY CATEGORIES

2008 sales, bln Annual growth rate 98-08, %

7.5

7.0

5.0

10.0

7.0

6.0

4.5

8.0

Source: Bain & Company presentation, available at www.altagamma.it

Figure 1: Sales of personal luxury goods by categories.

started loosening their purse strings and 2011 should see a return to normal luxury-

goods consumption, in line with historical trends, see [9].

The luxury market is perceived to have strong fundamentals and is expected to

continue to grow due to constantly enlarging customer base. The latter is mainly driven

by the increase in

i) Personal income in developing countries, such as Brasil, China, India and Eastern

Europe (mainly Russia).

ii) Spending on luxury goods by working women.

iii) Male interest luxury brands.

Notably, Chinese buyers are already worlds number 2 luxury customers behind the

Americans. Luxury sales in China have been a signicant driver of the global demand.

Sales in mainland China rose 30% in 2010 and are forecast to grow 25% at constant

currencies this year to 11.5bn, while US luxury sales are set to grow 8% to 52bn in2011, after rising 10% at constant currency terms in 2010 to 48.1 bln, see [9].

Industry revenues are closely linked to international tourist ows, since tourists

around the world are responsible for around a quarter of the worlds luxury purchases.

4

134 134 128 134146

159170 175

156172

185

2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011

GLOBAL MARKET FOR PERSONAL LUXURY GOODS bln

Source: Bain & Company presentation, available at www.altagamma.it, press clippings.

Figure 2: Dynamics of the luxury goods market, 2001-2011.

A stagnation in the worldwide industry sales during 2001-03, which can be seen from

Figure 2, was mainly a consequence of reduction in foreign travel after the September

11 attacks. Luxury tourism is mainly driven by the Japanese, who prefer to buy luxury

abroad, where prices are up to 40% lower than at home.

Europe has traditionally been the most signicant consumer and producer of luxury

goods. Of the 175 bln of world-wide sales in 2008, Europe represented 38% or 67bln. Between 1998 and 2008 the European market has grown at 6% on average per

annum. This exceeds the corresponding growth rates (5%) in other mature markets

such as US and Japan during the same period. Europe is also home to major luxury

goods companies owing top luxury brands, see Figure 3 below2.

In recent years, the industry experienced some consolidation with major players ac-

quiring other brands. For instance, LVMH bought Tag Heuer watches, Thomas Pink

shirts and Phillips, a London auction house. Gucci acquired Yves Saint Laurent, Ba-

lenciaga and Boucheron, before being bought by PPR, see [4]. Despite the presence of

these signicant players the luxury goods industry still remains quite fragmented with

2No ocial numbers are available for Chanel, as it is privately held company. The estimates in thetable are taken from www.portfolio.com.

5

Company Headquarters Major product lines Related brands2008 sales,

bln

PPR France ApparelGucci, Yves Saint Laurent

20.2

L'Oral Group France Cosmetics L'Oral 17.5

Mot Hennessy Louis Vuitton

FranceWines and spirits, cosmetics, watches, jewellery, leather goods

LVMH, Dior 17.2

Chanel France Cosmetics, perfumes Chanel 10-15

Richmont SwitzerlandWatches, jewellery, writing instruments

Cartier, Piaget, Mont Blanc

5.3

Armani Italy Apparel, cosmetics, jewelry Armani 1.7

Herms France Apparel, leather goods Herms 1.6

Figure 3: Major producers of luxury goods.

a large number of small, family owned, designers houses and boutiques.

Luxury goods are typically sold by the manufacturers through directly owned and

operated stores. Compared to other industries, on line sales in the industry are ex-

tremely low. In 2006 on-line sales represented 1.5 bln, which was less than 1% ofthe total industry sales. The situation improved somewhat in 2007 when on-line sales

increased to about 4.5 bln worldwide (less than 3%) of the total, according to theAltagamma Luxury Industry Report [2]. However, this is a far cry from on-lines sales

of other consumer goods, as reported by a market research company JupiterResearch

[6], see Figure 4. The other important distinction between on-line sales of luxury goods

and other products is that almost all sales of luxury goods are made directly through

either the manufacturers web-sites or specialized luxury portals3. Only a small fraction

of genuine luxury goods is sold though on-line auction platforms, such as Yahoo! or

e-Bay. This reticence to use the internet is rooted in several unfortunate attempts to

do so in the past. As a recent article [5] puts it:

"Some luxury brands jumped on the e-commerce bandwagon during the

dotcom boom in the late 1990s, only to bust with the bubble in the early

part of the current decade. So theyre reluctant to try again. Theres also

3For instance, Net-a-Porter.com, Gilt.com, Bluey.com, Eluxury.com.

6

712

5

8

11

13

8

PERCENT OF GOODS BOUGHT ON LINE, 2008Consumer electronicsFootwear Toys

5

2

15

17

20

23

28US

UK

France

Germany

Sweden

Spain

Italy 1

1

3

4

5

10

10

1

5

5

5

8

13

13

Clothes

Source: JupiterResearch Global Online Retail

Figure 4: On-line purchase behavior in select catagories and countries.

an older wariness dating back to the 1980s, when too many designer brands

went on licensing sprees that cheapened their pedigree. Since then, the

mantra has all been about control of brand".

Yet the future usage of internet by luxury goods companies cannot be completely

ruled out. However, the way they will do it will probably be dierent from traditional

on-line advertising or sales. Some experts, see [5], say that savvy luxury brands will

eventually adopt Web 2.0 technologies, including social networking. This will allow fans

of a specic brand to connect online with other like-minded, it could become like an

exclusive club. Similarly, in a recent industry survey by Abrams Research [1] more than

100 luxury industry leaders and experts were asked about how the internet can be best

used by luxury goods companies. The majority of respondents (34%) answered:

"Through innovation in advertising, e.g. Karl Lagerfelds mini-web movies",

and further 27% answered:

"Through partnerships with inuential fashion/luxury bloggers".

7

Only a relatively low fraction of respondents (13%) thinks that the way forward may

be to distribute luxury goods through lower price-point sites or a traditional on-line

advertising (5%).

3 The model

3.1 Demand

Suppose the consumers utility, if he consumes a luxury good, is given by

() = (1)

where 2 [0 ] is a parameter reecting consumer preferences towards the product(sometimes we will call the type of consumer), is a measure of product quality, and

is the products price. This formulation states that utility (and ultimately demand)

is not dependent on the income of potential consumers. Clearly, in reality demand for

luxury may be subject to signicant wealth eects. Our model thus would apply to

situations when consumers income does not change.

As was explained in the Introduction, we suggest to model the luxury aspect of

the good by assuming that consumers perceived quality is inversely related to the

number of consumers who buy the good. That is, when only a few consumers buy the

product, it is viewed by customers as exclusive and hence has a higher perceived quality.

Conversely, when more consumers buy the good, its perceived luxury aspect of quality

declines. Therefore, the utility of consuming the good declines as well.4 The validity of

this assumption is conrmed, for instance, by a recent article in The Wall Street Journal

[8], stating as follows.

"...For years, Hermes International SA has kept an elite band of high-

spending clients coming back for more by making them wait for what they

crave most. Waiting lists, which can stretch for years for certain handbags,

are part of the companys broader approach of keeping demand and prices

high.

...The company is wary of products that could hurt its upscale image. In

2005, for example, Hermes pulled its so-called Fourre-Tout canvas bag o

4We can assume that in the extreme case when all consumers buy the good, it simply ceases to beluxury and becomes an ordinary good. This could be modeled by assuming that the perceived qualityreduces to zero when everyone buys the good. However, all interesting results can be derived withoutmaking this restrictive assumption.

8

the shelves even though it was accounting for 10% of overall accessories sales.

"We didnt want to make the brand too accessible. Suddenly Hermes bags

were being associated with canvas, which is not what we wanted," says Mr.

Thomas [CEO of Hermes]."

Formally, if is the total mass of consumers5, then

(0) = 0 () = 0 () 0 0 () 0

where is the total demand for the luxury good. The above denition implies that the

good may have some value even if no-one buys it, i.e. if (0) 0. In this case, one

can loosely think of as the quality level that the rst buyer would attach to the good,

when he is the only one who consumes it.

Given the utility function in (1), a consumer with valuation buys exactly one unit

of the product if and only if he obtains positive utility from it, that is if

()

(2)

The total demand for the good is then just the sum of unit demands from all consumers

whose valuations satisfy (2), given the price corresponding quantity and quality

level () Let us assume that the number (or mass) of those consumers whose valua-

tions satisfy 0 is given by a function 0 for all 0 2 [0 ] Naturally, this functionis non-decreasing in 0 so 0

0

0 and moreover (0) = 0 and = Notethat if we normalize = 1 then

0

would represent a typical cumulative probability

distribution over the set of consumer valuations [0 ] Assume () to be continuously

dierentiable, and let () = 0 () When 0

can be interpreted as a probability

distribution, () can be the thought of as the associated probability density function.

Based on these assumptions, the total demand for the luxury good can be written as

follows

=

()

=

()

(3)

The inverse demand function would then be given by

() = ()1 () (4)

5One can also think of as the discrete number of consumers. However, to simplify computations itis more convenient to consider as a mass. In what follows we will use the terms consumer mass andnumber of consumers interchangeably.

9

It is easy to check that given our assumptions on () and () the inverse demand

function in (4) is downward slopping with (0) = and () = 0 This shows that

it is possible to model the luxury aspect of the good by using a standard downward

slopping demand function. This is in accordance with other luxury good models in the

literature, e.g. Ng [10] or Yao and Li [14]. Notably, the result that the demand for

luxury good is a declining function of price is not a consequence of the specic utility

function we have assumed. In fact, it can be shown that the demand function will be

downward slopping under suciently general conditions, see the Annex for details.

Given that the demand for the luxury good is downward slopping, naturally any

reduction in price should result in the increase of quantity sold, despite the reduction in

perceived quality expressed by () As our model assumes unit demand, this implies

that as the price declines, more customers buy the luxury good. Importantly, consumers

who already bought the good before, continue to buy it after the price decline, even if

they might obtain lower utility from it. The next proposition summarizes these results.

Proposition 1 For any given initial price-quantity pair ( (0) 0) consider a reduc-

tion in price (increase in quantity). Then the following two statements are true:

(i) Regardless of the magnitude of price reduction (quantity increase) all those con-

sumers who bought the good before (i.e. infra-marginal customers), would continue

to buy it.

(ii) There is no unambiguous relationship between the utility a particular consumer

obtains before and after price (quantity) change, save for the consumers that were

indierent between buying and not buying the good initially (i.e. marginal cus-

tomers). These marginal consumers clearly benet from price reduction (quantity

increase). In other words, all customers that were not infra-marginal before, but

become infra-marginal after the price decline, strictly benet from it.

Proof. Fix a consumer with valuation 0 quantity 0 and a corresponding price

0 (0) = (0)1 (0) If this consumer buys the good he obtains the utility

00

=

0

0 1 0 0

Suppose now quantity increases to 00 0 The utility of consumer with valuation 0

becomes

000

=

00

0 1 00

10

To prove (i) we need to show that 000

0 First, observe that by assumption (00) 0 Further, since () is increasing in and 1 () is decreasing in we have

10 1 00

and consequently

0 1 00 0Therefore

000

=

00

0 1 00 0

To prove (ii), consider Figure 5. On this gure the utility of a consumer with

valuation 0 is shown as a rectangular area. The area of rectangular A represents an

initial level of utility, while that of rectangular B - the utility following a reduction in

price or an increase in quantity (note that the rectangles overlap - the area at the origin

belongs to both rectangle A and B). Obviously, there is no unambiguous relationship

between the areas of these rectangles. It may be the case that reduction in the received

quality (i.e. the width of rectangle B compared to A) is compensated by the reduction

in price (related to the height of rectangle B relative to A), but it need not be so.

Thus, following a reduction in price, infra-marginal consumers may obtain either lower

or higher utility as a result.

On the other hand, marginal consumers who are indierent between buying and

not before a price (quantity) change, clearly benet from the subsequent price re-

duction. Before price reduction, such a marginal consumer obtains zero utility since

0 = 1 (0) Thus the height of rectangle A is zero. Following a price reductionthe utility of this consumer is represented by rectangle B, with area larger than zero.

The results of Proposition 1 are apparently at odds with the statements sometimes

made by luxury goods manufacturers that a possible reduction in price is going to

destroy the image of luxury goods and will result in some consumers refraining from

buying it. In terms of the model developed here, this statement cannot be conrmed.

The reason is that the eventual impact on consumer utility is a determined not only

by the perceived quality, but also by prices. In terms of the model presented here, the

eect of price reduction is stronger than that of quality reduction. In other words, price

decrease outweighs a reduction in perceived quality.

Whilst price reductions may decrease utility of some (infra-marginal) consumers,

importantly Proposition 1 states that there are no infra-marginal consumers who would

stop buying the luxury good following a price reduction. In addition to that, marginal

11

)(1 QnF

)(1 QnF

)(Qs )(Qs

A

B

Figure 5: Changes in utility following a reduction (increase) in price (quantity).

consumers always benet from price reductions. It is therefore can already be guessed

that welfare eects of price changes in general will be ambiguous.

3.2 Supply

3.2.1 Monopolist producer

Let us now turn to the prot maximization problem of luxury good manufacturers. To

this end we assume a single monopolist producer. By doing so we essentially assume

away possible competition from rival (luxury goods) suppliers. This may be viewed

as a limitation. However, our primary purpose is to analyze price/quantity choices

and welfare eects. It is thus analytically convenient to focus on the behavior of a

single monopolist producer. Further, as demand function for luxury good is shown to

be downward slopping, competition between luxury good manufacturers is in no way

dierent as between ordinary manufacturers, i.e. as such the luxury aspect of the good

does not add much new to the nature of competition. For this reason, without loss of

generality, we may assume away competition among manufacturers.6

6 It is also an open question what type of competition to assume in the case of luxury good models.For instance, brand image of a particular manufacturer may be so strong that consumers would view,e.g. luxury bags of two producers as so distinct (i.e. dierentiated) that each producer would notexperience signicant competitive constraint from the other one. In this case, for all practical purposesone can safely assume that each of these producers is acting as a local monopolist in the correspondingproduct space.

12

The monopolist producers prot maximization problem is given by

max

() = () () (5)

where () is the manufacturers cost function and () is the inverse demand from

(4). The solution to the problem would be to equate marginal revenues with marginal

costs, i.e. to nd such level of production that satises

() +0 () = 0 () (6)

As the demand function for luxury good is downward slopping, in the optimum the

monopolist will set prices that exceed marginal costs, and earn a margin specied by

the inverse elasticity rule.

3.2.2 Social planner

It is interesting to contrast the production decision of the monopolist supplier with that

of a social planner. For the moment let us assume that the social planner only cares

about consumers and thus wants to choose such level of quantity and prices () that

maximize consumer surplus.7 In order to determine the consumer surplus in the model,

it will be helpful to denote by a cut-o level of valuation, i.e. the minimal valuation

required to buy the good, given a certain level of and () Thus a consumer of type

receives zero utility from buying the good

= () ()

= 1 () (7)

Note that if () is a cumulative probability density function,8 then () can be

interpreted as the mass of consumers who do not buy the good. Clearly we have

0 and

0 (8)

In other words, when prices go down, more people buy the luxury good and the mass

(or the number) of those consumers who cannot aord it decreases. This is in line with

the conclusion we reached on the downward slopping demand in (4) and Proposition 1.

7 Indeed, many competition authorities have explicitly stated their ultimate goal as the maximizationof consumer welfare and not the total welfare, dened as the sum of consumer and producer surplus.

8Recall that it need not be; by denition () gives the number of consumers whose valuation doesnot exceed

13

We can express the consumer surplus as the sum of utilities of those consumers

who buy the good, when the overall demand is Formally,

() =Z

[ () ()] () = ()Z

[ ] () (9)

From the application of Kuhn-Tucker conditions, to the maximization of () given

by (9), the social planner who wishes to maximize consumer surplus () chooses bsuch that

b =8

Next, we know that for any function ( ) the following holds

Z ()()

( )

!=

Z ()()

0 ( )+ [ () ] 0 () [ () ]0 ()

By applying this formula to (9) we get

" ()

Z

( ) ()#

=

"Z ()

1 () ()#

=

Z () () | {z }

Z ()1 () ()| {z }

where is dened in (7).

Now observe that

=Z 0 () () () () 1

()(1)

=Z

0 ()1 () () 1

()

() ()1 () () 1

()(1)

And nally

() ()

= =Z 0 () ()

Z

0 ()1 () () 1

()

()

=

Z 0 () ( ) () +

Z () (12)

The rst term in (12) is negative because 0 () 0 by assumption and the relevantrange of lies above . The second term is positive. Thus the overall impact on the

consumer surplus () due to changes in is ambiguous. This formally conrms the

intuition behind part (ii) of Proposition 1.

Next, we can analyze the behavior of 0 () at = 0 and = First of all, at

= 0 we have = and thus from (12) it follows that 0 (0) = 0 Obviously this is the

point where consumer surplus () is minimized. This is so because (0) = 0 and in

general () 0 thus (0) can only increase for a suciently small Second, at = we have = 0 In this case (12) simplies to

0 () = 0 ()Z 0 () + ()

Z 0 (13)

15

nQ

Q

][Es

Q

)(Q

0

nQdQd

Figure 6: Dynamic of the consumer surplus () when 0 () 0

From (13) it can be checked that 0 () 0 if and only if

()0 ()

2 []

(14)

This suggests that for any given distribution of preferences (or consumer types) charac-

terized by () condition (14) is more likely to hold when: either () is low or 0 ()

is high in absolute value (i.e. perceived quality decreases drastically with the number

of units sold is close to ), or both. In particular, note that (14) always holds whenever

() = 0 i.e. when the perceived quality is zero if everyone buys the good.

Now observe that combining 0 (0) 0 together with 0 () 0 implies that there

is an interior solution b to the social planners maximization problem, as is illustratedon Figure 6. We have thus proved the following.

Proposition 2 There exists an interior solution b 2 (0 ) to the social plannersmaximization problem max () if (14) holds.

Proof. In the text.

The condition of 0 () 0 is only sucient, but not necessary for the interior

solution b to exist. For instance, consumer surplus may be maximized at some b 2(0 ) even if 0 () 0 as can be easily observed by modifying Figure 6. However,

in this case one cannot rule out that consumer surplus is maximized at = either.

In terms of Figure 6 this would be the case, for instance, if the derivative 0 () is

16

increasing everywhere on [0 ] and reaching its maximum at 9 In sum, a negative

derivative of () at = implies the existence of an interior solution, whilst a

positive derivative allows for both interior and boundary solution.

In terms of practical application of these results to luxury goods industry, it would

not be an overstatement to say that sucient condition (14) would hold in most cases.

This is because it is quite probable that for status goods either their perceived value

() is suciently small or the perceived value is decreasing sharply when everyone buys

them, i.e. when = At the very least, it appears that the luxury good manufacturers

are convinced that this might be the case, as is evidenced by some of their marketing

strategies, see [8] or quotes in Section 3.1 above. Overall it implies therefore that there

would be an interior solution to the social planners maximization problem.

Recall that the social welfare has been dened in terms of consumer surplus only.

Because production costs are not taken into account by such a measure of welfare, one

can imagine that the social planner would wish to increase consumption as much as

possible. Yet the existence of an interior solution makes it clear that the social planner

would not want to increase consumption too much. In essence, it means that decreasing

prices (or increasing quantity sold) does not always increase consumer welfare. (This is

essentially a corollary of Proposition 2). What is the intuition behind this result?

The existence of the interior solution to the social planners maximization problem

is linked to the fact that changes in the quantity sold have an ambiguous eect on

the consumer welfare. As an illustration of this eect, consider the special case of a

linear distribution function () = with 2 [0 1] In this case there is exactly onecustomer of each type and the customer types are distributed uniformly. Under these

conditions, () can be nicely represented graphically, as is shown on Figure 7, given

some initial levels of and () The straight line sloping upwards gives the utility of

a customer with valuation for all 2 [0 1] The vertical intercept on this utility lineequals the equilibrium price () taken with a negative sign; the slope of the utility

line equals () The overall consumer surplus () thus equals to the shaded area

and can be expressed as

() = ()

2[1 ]2 (15)

where = 1 () see (7).From Figure 7 it is also convenient to get an intuition of how consumer surplus

changes in response to changes in quantity and price () For instance, if increases

9Obviously, this is not the only way in which it may happen that social welfare is maximized at =

17

)(U

QnFQs 1

QnF 1 1

)(Qs

QnFQs 11

Figure 7: Graphical illustration of consumer surplus (shaded area) as a function of when () is linear in

and correspondingly () goes down, the intercept declines, reaching zero when = In the same time as the good becomes cheaper, more people buy it because the

cut-o level of the quality parameter declines, reaching zero in the limit. This drives

down the perceived quality () and the utility line becomes atter, reaching the slope

of when = . The opposite happens if decreases. Specically, the intercept of

the utility line increases in absolute value, reaching when = 0; tends to 1; andthe slope of the utility line goes up, reaching when = 0

Figure 7 shows an important lesson formally stated in Proposition 2. A reduction

in price of the luxury good may not always increase the aggregate consumer surplus

(). On the one hand, decreasing prices increases the overall number of consumers

that buy the good, and clearly benets marginal consumers, as we showed in Proposition

1 above. On the other hand, the more consumers buy the good, the less is the utility

of infra-marginal consumers, because the good ceases to be exclusive. Overall, whether

changes in and () increase consumer surplus depends on the underlying parameters

of the model, specically on the form of the perceived quality function () and the

distribution of valuations ()

Given that price reductions are not always in the interest of nal consumers, one can

expect that the interests of the social planner and monopolist producer may be aligned.

This is indeed the case, as we show in the proposition below: the social planner wishing

to maximize consumer surplus may choose exactly the same prices and quantities as the

monopolist producer of the luxury good. Whilst this may seem to be the consequence of

a particular way in which the social welfare has been dened (i.e. as consumer surplus

only), it fact it is not. Even if the social welfare is dened in a more conventional way,

18

i.e. as the sum of both consumer and producer surpluses, it may be the case that the

social planner chooses the same quantity and prices as the monopoly supplier would.

The driving force behind this result is the scarcity eect that reduces perceived quality

of the good for infra-marginal consumers when total consumption increases.

Proposition 3 Let and b be correspondingly solutions to the manufacturers andsocial planners maximization problems, given that the latter maximizes only consumer

surplus, in (6) and (10). Then for some values of () () and () it may be

possible that = b Moreover, in this case the social planner who wishes to maximizethe sum of consumer and producer surpluses would also select = bProof. The intuition for the proof can already be grasped from observing that costs do

not feature in (12). That is, the social planner wishing to maximize the sum of utilities

of those consumers who buy the good does not take into account the manufacturers

costs. Thus we in principle can choose any cost function () that ensures that = bNow lets assume that the optimal quantity produced by the monopolist b solves

the optimization problem of the social planner, given by (10). Since the marginal costs

of production are non-decreasing, 0 b 0 From the solution of the monopolist

optimization problem, it follows that the manufacturers marginal revenue at b is alsopositive. Note that one condition for manufacturers marginal revenue to be positive

is that demand elasticity exceeds 1. That implies that in the absence of production

costs (or zero marginal costs), the monopoly producer of luxury goods would prefer to

produce more than the social planner, who maximizes consumer surplus only. This is

because the manufacturer cares only about utility of marginal consumers who will buy

the good in case of a price reduction. (By Proposition 1, the monopolist knows that

it will not lose any infra-marginal customers anyway). On the other hand, the social

planner cares about balancing an increase in the utility of new (marginal consumers) and

the loss of utility for infra-marginal consumers that will be reduced due to changes in

the perceived quality () when too many consumers buy the good. Hence, generally

there is a trade-o between increasing producer surplus at the expense of consumer

surplus and vice versa. However, if both surpluses are maximized at the same level of

= = b then clearly the joint surplus is also maximized at

19

5 Numerical examples

In this section we present two numerical examples illustrating the ideas discussed in

Sections 3 and 4.

5.1 Derivation of the demand function and consumer surplus

In order to explicitly derive demand function and consumer surplus, we will make the

following four simplifying assumptions:

i) Total mass of consumers is normalized to one, i.e. = 1

ii) Consumer preferences for quality are distributed uniformly on [0 1] so that

there if exactly one customer of each type thus () =

iii) Quality function () enjoys () = 1 This assumption implies that whenall consumers buy the good, i.e. = 1 its perceived quality reduces to zero

because the good ceases to be exclusive.

iv) Marginal costs of production are xed at zero, 0 () = 0

Given these assumptions the demand function becomes

=

()

= 1

()

= 1

1

which simplies to

= 1 p (16)

Given that marginal costs are assumed to be zero, the manufacturer maximizes prot

by solving the following maximization problem

max

() = p (17)

This prot function is concave, therefore the problem in (17) is well-dened and has the

solution = 49 Correspondingly we have

=1

3 () =

2

3 =

2

3 =

1

27

Graphically the consumer surplus is depicted by the shaded area on Figure 8.

The related question is what is the maximum total consumer welfare in this model.

A little reection shows that this welfare is not maximized at prices that are equal

20

)(U

92

31

32 1

Figure 8: Analysis of consumer surplus.

to marginal costs, i.e. zero. When = 0 we have = 1 and () = 0 In this

case, although all consumers buy the good (= 0), the utility line corresponds to the

horizontal axis of valuations Hence the consumer surplus equals to zero.

Let us now compute prices and quantities which would maximize the consumer

surplus. By using the earlier derived formula for the linear valuation case (15), and

noting that given our assumption about the distribution of valuations () = we

have = 1 and = 1 we have to solve

max

=1

22 (18)

The maximization problem has a solution b = 23 Correspondingly we haveb = 1

9 b () = 1

3 b = 1

3 b = 2

27

This maximal consumer surplus b is shown by the checkered area on Figure 8. Thischeckered area is clearly bigger than what is chosen by the manufacturer who is in a

monopoly position.

The important lesson from this example is that the social planner who wants to

maximize consumer surplus will not equate marginal costs and prices. Although the

optimal price for the social planners view is below the price which would be set by the

monopolist producer, the social planner would not want to reduce prices to the level

21

of marginal costs. This is because at prices below ^ the negative eect on the utility

of infra-marginal consumers is not outweighed by a positive eect stemming from the

increase in utility of marginal consumers. Thus, from the social viewpoint, the optimal

price ^ is above the manufacturers marginal costs.

5.2 Equivalence between the monopolists and social planners maxi-

mization problem

To illustrate the idea of Proposition 3, consider the following example. Suppose there

are 3 consumers with valuations 1 = 1 2 = 34 3 =3548 Let the perceived quality

function () and the manufacturers cost function () be given as in Table 1 below.

Further let the price () be such that marginal consumer obtains zero utility. Then

it is easy to verify that both the producers prot, consumer surplus and the overall

welfare are maximized at = 2. Moreover, even if the social welfare is dened in a more

conventional way, as the sum of consumer and producer surpluses, the joint surplus is

still maximized at = 2

Table 1: Example illustrating = b = 1 = 2 = 3

Perceived quality, () 1 89 23

Price, () 1 23 3572

Production costs, () 1 23 2024

Producers marginal revenue, () 1 43 3524

Producers prot, () 12 23 1524

Consumer surplus, () 0 29 736

22

References

[1] "Luxury Brands Survey and Report", Abrams Research, May 2009, available at

http://www.abramsresearch.com.

[2] Altagamma Luxury Industry Report, available at http://www.altagamma.it.

[3] Bagwell, Laurie. S., B. Douglas Bernheim (1996) "Veblen Eects in a Theory of

Conspicuous Consumption", The American Economic Review, Vol. 86, No. 3, pp.

349-373.

[4] "Every Cloud Has a Satin Lining", The Economist, March 21, 2002.

[5] "Getting Luxury Goods On-Line", Time, June 23, 2008.

[6] "Global On-line Retail 2008", JupiterResearch, available at

http://www.demandware.com.

[7] Leibenstein, H. (1950) "Bandwagon, Snob, and Veblen Eects in the Theory of

Consumers Demand", The Quarterly Journal of Economics, Vol. 64 (2), pp. 183-

207.

[8] "Hermes Seduces the Elite by Selling Luxury Slowly", The Wall Street Journal,

May 8, 2009.

[9] "Luxury goods market back on track", The Guardian, May 3 2011, available

at http://www.guardian.co.uk/business/2011/may/03/luxury-goods-market-back-

on-track.

[10] Ng, Yew-Kwang (1987), "Diamonds Are a Governments Best Friend: Burden-Free

Taxes on Goods Valued for Their Values", The American Economic Review, Vol.

77, No. 1, pp. 186-191.

[11] Rae, John The Sociological Theory of Capital, London: The Macmillan Co., 1905.

[12] Trigg, Andrew (2001) "Veblen Bourdieu, and Conspicuous Consumption", Journal

of Economic Issues, Vol. 35 (1), pp. 99-115..

[13] Veblen, Thorstein (1989), The Theory of the Leisure Class: An Economic Study

of Institutions. London: Unwin Books, reprinted New York: Dover Publications,

1994.

23

[14] Yao, Shuntian, Ke Li (2005), "Pricing Superior Goods: Utility Generated by

Scarcity", Pacic Economic Review, 10 (4), pp. 529-38.

24

Annex: downward slopping demandIn this Annex we present a proof that the demand function based on the scarcity

or exclusivity of the luxury good is downward slopping under fairly general conditions.

Thus, the particular functional form used in this paper, i.e. () = can indeedbe assumed without loss of generality.

Suppose that consumer utility is given by

() = ( )

where is increasing in both of its arguments, i.e. 0 As before, ()represents the goods perceived quality and is a declining function of the total volume

of consumption so 0 () 0Write the total demand equation like in (3)

= [ ( ) ] (19)

Suppose that, given ( ) = we can express = ( ) In this case (19) becomes

= [ ( )] = [( )]

from which it follows that

( ) = 1 () (20)

where the right hand side of (20) is the minimal level of required to obtain a non-

negative utility from buying the good when the overall demand is Following notation

introduced in Section 3,

( ) = (21)

Observe that because ( ) was constructed on the basis of ( ) = we can

express from (21) as follows

= ( ) (22)

This is precisely the result we get in Section 2: we assumed ( ) = and showed

that () = ()1() Had we, for instance, specied the consumer utility as () = + the resulting demand function would have been

= () + 1 ()

25

)(Qs

psG ,Level sets, given

))(( pQs

Consumers that purchase the good

Figure 9: Existence of ( ) means that can be expressed given ( ) =

as is easy to check by repeating calculations in (2) to (4).

In order to show that the demand function is downward slopping, dierentiate (22)

with respect to

=

1

1

(23)

The rst term in (23) is negative because 0 and 0 () 0 The second term isnegative because is an increasing function, and so is 1

The whole proof above hinges on the following two assumptions.

i) There is a function ( ) expressing given some and Generally speaking,

this is a very mild assumption. This assumption would be satised if the level

sets (utility levels projected onto the plane) are smooth enough, see Figure 9

above. If this is the case, can be uniquely be expressed as a function of and

ii) The utility function ( ) is increasing in both of its arguments, i.e. 0However, it is entirely logical that utility is increasing in the perceived quality

() and customer type Notably, the proof does not depend on the relationship

between and i.e. on the sign of the mixed derivative

26