Conspicuous Consumption and the Strength of Social Ties
by
Nick Vikander
Submitted in partial ful…llment of the requirement for the degree of
Master of Philosophy in Economics
Tinbergen Institute
Supervisor: M.C.W. Janssen
August 2007
Abstract1
This thesis examines how the strength of people’s social ties in‡uences the level
of conspicuous consumption in a society. I show that when social ties transmit
information related to both income and status, individuals tend to choose weak
rather than strong social ties. They dispose of greater income as a result but their
consumption is more highly distorted by their desire for status. People may either
enjoy a welfare gain or su¤er a welfare loss, depending on whether their income gain
outweighs the distortion from conspicuous consumption. Their welfare level is lower
than it would have been if they could transmit information about status without
signalling. The results help explain why conspicuous consumption may be more
prevalent in complex and dynamic social environments.
1 I would like to thank Maarten Janssen for his encouragement and critical advice throughout the process of
writing this thesis, as well as colleagues for commenting on a previous draft.
2
Table of Contents
Introduction ............................................................................................................................ 4
Chapter 1: The Model ............................................................................................................ 11
1.1. Description of the formal model ................................................................................... 11
1.2. Discussion of the model assumptions ........................................................................... 15
Chapter 2: An Illustrative Example and Discussion of Main Results ..................................... 21
2.1. An illustrative example ................................................................................................. 21
2.2. Discussion of the main general results .......................................................................... 26
Chapter 3: Preliminary Formal Analysis ................................................................................ 30
3.1. Equilibrium existence with …xed ............................................................................... 30
3.2. Characteristics of the equilibrium with …xed ............................................................ 36
Chapter 4: Formal Analysis of the General Model ................................................................
4.1. Equilibrium existence when social ties are unobservable .............................................
4.2. Equilibrium existence when social ties are observable .................................................
4.3. Characteristics of the equilibria ....................................................................................
41
42
45
49
Chapter 5: Adding Intrinsic Utility to Strong Ties ................................................................ 52
Conclusion .............................................................................................................................. 58
Appendix................................................................................................................................. 59
References................................................................................................................................ 60
3
Introduction
Thorstein Veblen …rst coined the term conspicuous consumption in reference to what he called
the leisure class in the United States at the turn of the 20th century (Veblen 1899). Members of
the leisure class refrained from industrial work and consumed extravagantly to demonstrate the
extent of their wealth to others. A desire for status also seems to drive consumption decisions to
varying degrees in modern societies, which is the area I look at in this thesis. I explore how two
ideas can help shed light on conspicuous consumption: people’s need to signal to obtain status
and the key role of social ties in transmitting di¤erent types of information.
With the exception of Chapter 5, I assume throughout that people do not obtain intrinsic utility
from their social relationships: they do not value friendship for its own sake. While a clear
simpli…cation, this assumption allows me to focus on how people value the informational role
that social ties play, which in turn in‡uences their conspicuous consumption. In short, strong
ties tend to transmit information related to status that makes signalling redundant, while weak
ties relay information that is …nancially advantageous.
This introduction lays out these ideas in more detail. I begin by outlining two types of empirical
evidence related to conspicuous consumption and presenting the key elements of the formal
model I examine. I then suggest an interpretation of the models’main results and discuss how
it relates to the existing literature.
The model I develop in this thesis aims to explain two types of anecdotal empirical evidence.
The …rst is that status-seeking appears to play a role in in‡uencing the consumption of certain
types of goods. A good may become more glamorous if its price increases, and luxury cars
use prestige to attract buyers (Bagwell and Bernheim 1996). Young people make statements
by displaying their mobile phones with customized screen and ringtone.2 Others pay a high
premium for designer clothes and quickly replace them in accordance with changing fashions
(Ireland 1994). Veblen (p.103) himself points to expensive clothes as one of the clearest signs
of conspicuous consumption, both because people admit they often choose clothes for the sake
of appearance to others and because people tend to feel it very strongly when they fall short in
terms of dress.
2 The
Economist, “Why Phones Are Replacing Cars”, April 29 2004.
4
A second type of evidence suggests conspicuous consumption may be particularly prevalent in
complex and dynamic environments where long-standing social networks do not exist. Frank
(1985) points to areas with high labour mobility and big cities rather than small towns. In
so doing, he echoes Veblen’s view (p.55) that demonstration e¤ects, or signalling, are more
important in urban areas. Conspicuous consumption plays a lesser role in rural settings because
“through the medium of neighborhood gossip . . . everybody’s a¤airs, especially everybody’s
pecuniary status, are known to everybody else”. Moreover:
“The serviceability of consumption as a means of repute, as well as the insistence on
it as an element of decency is at its best in those portions of the community where
the human contact of the individual is widest and the mobility of the population is
great” (p.54)
Veblen stresses that the social and economic change brought about by industrialisation increased
the prevalence of conspicuous consumption. The welfare e¤ects of economic growth may be ambiguous from this perspective. Growth has the potential to increase welfare by raising income,
but may also decrease welfare if it changes people’s behaviour by increasing status-driven conspicuous consumption. Poor urban teenagers may purchase very expensive basketball shoes
(Ireland 1994), and low income people may save too little because of their high visible consumption (Frank 1985). Individuals who fail or refuse to live up to community consumption standards
can su¤er social consequences such as an inability to appear in public without shame or to take
part in the life of the community (Smith 1776).3
I explain the …rst point concerning the existence of conspicuous consumption by using the
concept of signalling. In so doing, I follow other contributions in the literature to which I refer
below. The model has a number of elements. People value status and want both their friends
and acquaintances to hold them in high regard. These others would like to base their judgment
on some characteristic they consider important that varies in the population, perhaps talent
or generosity, industriousness or decency. But these character traits, by their very nature, are
di¢ cult for others to directly observe. They must instead infer this person’s character based on
his actions, here his visible consumption.
3 See
p.1102-1103 on the social necessity of a linen shirt and leather shoes for any creditable day-labourer in
18th century England, the lack of which would suggest extreme poverty and reprehensible conduct.
5
I show that the desire to signal pushes everybody’s visible consumption up at the expense of
non-visible consumption and that, all else being equal, welfare decreases because of the consumption distortion. While everyone would be better o¤ if nobody signaled, each individual has
an incentive to signal to di¤erentiate himself from the person slightly below him in the pecking
order.
The main novel idea in this thesis is that I employ the distinction between weak and strong
social ties to address the second point: that conspicuous consumption may be higher in complex
and dynamic social environments. The strength of social ties is a sociological concept that refers
to how close and personal a social relationship is. Intuitively, people have strong ties with their
close friends and weak ties with their acquaintances. Strong ties transmit more information
about certain characteristics of a person that cannot be quickly or easily ascertained. Applied
to the previous example, people connected to someone through strong ties can more easily judge
how talented he is, or how generous.
Strong ties are not however very useful in transmitting novel information that originates in one
part of a social network to another. Granovetter (1973, 2005) argues that people connected to
someone by strong ties often also have strong ties with one another. Thus:
"Because our close friends tend to move in the same circles in which we move, the
information they receive overlaps considerably with what we already know" (2005,
p.34)
In contrast, a weak tie is more likely to act as a bridge connecting two distinct parts of a
social network that would otherwise be completely isolated from one another. In a large and
complex social system, such a true bridge may rarely exist since there are often multiple paths
by which any two individuals are connected. Weak ties nonetheless still serve as local bridges,
providing the shortest and most direct path through which information can ‡ow between people
in di¤erent parts of society. In particular, weak ties transmit more novel information about job
and career opportunities that comes from outside a person’s immediate social circle (Granovetter
1974). This extra information may take di¤erent forms. A person with weak ties may receive
information about job opportunities more frequently (Granovetter 1973). He may also receive
information with the same frequency as someone with strong ties, but hear more about higher
quality jobs (Lin 1982).
6
I now sketch out how I express these ideas in the model and then explain its relevance to
the question of why conspicuous consumption may be more prevalent in some environments
than in others. People choose not only how to split their income between visible and non-visible
consumption in the model, but they also choose between weak and strong social ties. They value
weak or strong social ties because they transmit di¤erent types of information, along the lines
described above. I assume that a person who chooses weak ties has a higher income because of
access to novel job-related information. Hence someone chooses weak ties if he wants to consume
more. However, since weak ties do not transmit any information about personal characteristics,
his entire status level depends on how much he signals through visible consumption.
A person who chooses strong ties has lower income but his status depends to a lesser degree on
his visible consumption. Formally, if everybody chooses strong ties then personal characteristics
are fully observable and status is completely independent of signalling. If some people choose
weak ties and others strong, the people with strong ties will have a status level that depends
partially on signalling. This idea captures the notion that it is harder for a person to transmit
personal information through close relationships when others are only interested in having weak
contacts with him. I call this last assumption the information externality.
I show that when the choice of social ties is endogenous, people tend to choose weak ties, obtain
high income and signal by distorting their visible consumption upwards. The intuition for this
result is that a person with strong ties can often increase his income by choosing weak ties in
such a way that his status either remains the same or increases. Compared to the situation where
everyone has strong ties, higher income tends to push welfare up while conspicuous consumption
pushes welfare down. For some model parameters people may have lower welfare than if everyone
had instead chosen strong ties, but they will not deviate since the information externality makes
a unilateral choice of strong ties less attractive.
Welfare would increase if people could transmit information about their personal characteristic,
say talent, through weak ties, since then there would be no need to signal. Everybody could
enjoy higher income without needing to distort their consumption.
I suggest that these results are relevant for people living in a large, dynamic or complex environment, such as an important urban center. A person there has the freedom to choose the
kind of social relationships he wants, rather than these being determined by custom or tradition.
He can establish loose relationships with acquaintances who have no direct means of observing
7
how talented, for example, he is. These acquaintances also have no alternative way of …nding
this information out because of the large size of the social system and the relative anonymity of
the city. They must instead base their judgements about esteem on inference from the person’s
visible consumption.
At the same time, acquaintances may be a key source of novel, job related information. This
information can be particularly relevant and valuable in such an environment for a number of
reasons: people’s earning prospects may depend crucially on collaboration with others due to
a high division of labour, continued innovation may cause new opportunities to arise quickly,
and this information may come from very di¤erent parts of the large social system. Weak ties
thus play a crucial role as local bridges to inform people about these opportunities. The results
I derive in this thesis suggest that both income and conspicuous consumption will be high and
social ties weak in this type of environment.
The situation is di¤erent in a rural, static, or traditional setting. Bou¤ard and Muftie (2006)
refer to evidence that there are important di¤erence in rural as opposed to urban life. Individual behaviour is more constrained by informal social control mechanisms outlining what types
of behaviour are acceptable. Furthermore, people tend to have more intimate and personal
relationships with one another.
People in rural areas may therefore be unable to avoid having strong ties with one another.
Social ties may not be a choice variable at all but rather be exogenously imposed, set in stone
by tradition. Furthermore, even if a person did have such a choice, weak ties may not transmit
di¤erent types of information in this environment. In a small community, even someone with
weak ties could not easily avoid that his a¤airs be known to all others “through the medium of
neighbourhood gossip” (Veblen p.55). Weak ties may also play no special role in transmitting
novel information of economic value, both because such information only rarely exists and because there are no disparate parts of the social network to bridge. Put in terms of this model,
social ties are exogenous and strong, income is low and there is no conspicuous consumption.
The results from this thesis thus give one possible explanation as to why there may be more
conspicuous consumption in one social setting than in another, as suggested by Veblen and
Frank.
Three strands of literatures are relevant to this thesis. The …rst strand looks directly at conspicuous consumption and status, of which Veblen’s Theory of the Leisure Class (1899) is the
8
classic source. He argues that consumption serves a signalling purpose and is present in many
areas such as fashion, architecture, the role of women in society, customs of decorum, leisure,
organized sports and religion (as people would build ornate churches to signal on behalf of God).
Among more recent contributions, my approach is most similar to Ireland (1994) and I will
discuss these similarities in the following section. Bernheim (1994)’s theory of conformity shows
how people may make consumption decisions to appear similar to others. Bagwell and Bernheim
(1996) demonstrate how “Veblen e¤ects” can arise, when compensated demand may increase
with a good’s price. Frank (1985) does not consider signalling but instead models the social
side of consumption by incorporating a term in the utility function that depends on relative
consumption position. That work relates to Hirsch’s discussion of positional goods (1976),
which he de…nes as goods whose value depends on how much others consume. My work extends
this type of research on conspicuous consumption and is, as far as I am aware, the …rst to look
at the role of social ties.
By addressing the question of how a society in which both income and conspicuous consumption
are low may evolve into one in which both are high, the spirit of this thesis relates to a second
strand of literature on social norms or conventions and equilibrium selection. Young (1993)
shows how evolutionary forces may favour certain equilibria, those that are stochastically stable.
Nöldeke and Samuelson (1997) apply this approach to signalling games, as do Jacobsen et al.
(2001). The similarity here is one of spirit only, since this thesis’formal model is very di¤erent
from the above work: it is static rather than dynamic, the low signalling and high signalling
situations are not generally both equilibria from which one must be selected, and evolution plays
no explicit role.
The third strand of literature is the work on social ties. The discussion of weak and strong ties
has ‡ourished within economic sociology since Granovetter’s …rst theoretical contribution (1973)
and empirical analysis of social ties and the job market (1974). He argues that weak ties are
important because they transmit job information at a higher frequency. Lin (1982) suggests that
weak ties instead relay more valuable job information, either coming from a better distribution
of wage o¤ers or one that is more dispersed.
The empirical debate about these weak ties hypotheses is unresolved. Some work o¤ers supporting evidence for subgroups of the population, such as people in higher social strata (Wegener
1991). In general, researchers have not been able to establish a solid causal relationship in large
9
datasets between jobs obtained through weak ties and higher wages (Mouw 2003). Montgomery
(1992) argues however that it may be misleading to look simply for a correlation between wages
and how a job was obtained. If people with more weak ties tend to obtain more or better job
information, they also have a higher value of search and expected lifetime income. They set
a higher reservation wage and will earn more in the job they end up accepting, irrespective of
whether that particular job o¤er came through weak or strong ties.
The thesis is organised as follows. Chapter 1 describes the formal model and then gives a
further discussion of some of the model assumptions. Chapter 2 presents an illustrative example
and discussion of what results more generally hold. Chapter 3 contains the preliminary formal
analysis and Chapter 4 the formal analysis for the general model, including proofs of the main
results. I consider a modi…cation of the model in Chapter 5 to incorporate intrinsic utility from
strong ties and then conclude.
10
Chapter 1: The Model
1.1. Description of the formal model
In this section, I describe the model of conspicuous consumption that I subsequently investigate.
Status seeking individuals choose between weak and strong social ties and decide on their level
of visible consumption.
Types: Consumers di¤er only in terms of type, denoted by , which is distributed over the
interval [0; max ]. There are a number of possible interpretations of type, one of which is natural
talent or productive ability. Income is an increasing function of type, so people with higher
talent earn more. Speci…cally, I( ; ) = y0 +
where y0
0 is a constant. The variable is
either low or high for a given consumer depending on whether he chooses strong or weak social
ties. So 2 f S ; W g, where 0 < S < W . Higher types have a higher absolute income gain
from choosing W compared to S .
Preferences: Each consumer has complete, transitive, continuous preferences over two di¤erent
consumption goods and over status. Consumption goods are either visible, v, or non-visible, w.
Consumers do not obtain any extra income or consumption through high status, but instead
value status for its own sake as a type of social recognition. A person’s status will depend on
either his true or his inferred type, so people who are more talented are held in higher regard.
Preferences are monotonic over both consumption goods and over status.
Strategies: A strategy consists of a choice of visible consumption and a choice of social ties.
I denote a generic strategy S by (v; ). Choosing a speci…c type of social ties determines a
consumer’s budget set from which he chooses the optimal balance of visible consumption v, and
non-visible consumption w. A strategy S is essentially a function of only two variables since v
and w are related to each other by the budget constraint. Visible consumption has price p and I
normalize the price of non-visible consumption to 1. Consumers choose and v simultaneously
so there is no distinction between strategies and actions. I therefore refer to (v; ) as a strategy
and use the word action in the non-technical sense that strategy (v; ) consists of two actions, v
and .
Observability of Actions and Beliefs: Each consumer’s type and non-visible consumption
level is private information whereas visible consumption is observable. I consider both cases
11
where choice of ties is observable and where it is not. A consumer of type 0 chooses strategy
(v0 ; 0 ) and upon seeing any other consumer of type 1 ’s observable actions forms a belief about
that consumer’s type: 0 ( 1 jv1 ; 1 ), or 0 ( 1 jv1 ). I describe the case that is observable in the
remainder of this section for concreteness. I often write 0 (v1 ; 1 ) rather than 0 ( 1 jv1 ; 1 ) in
order to be concise.
Since all consumers act simultaneously, a consumer’s strategy cannot depend on his beliefs: the
strategy of consumer 0 , S0 , must be independent of 0 ( 1 jv1 ; 1 ); 8 1 . Since in equilibrium all
types make the same inference, 0 ( 1 jv1 ; 1 ) = k ( 1 jv1 ; 1 ); 8 0 ; 1 ; k . I denote the common
equilibrium belief about a consumer of type 1 simply by ( 1 jv1 ; 1 ).
Utility function: A consumer of type
U (v; ) = f (v; w) + (1
who chooses strategy (v; ) has the following utility:
) ( ; R)g( (v; )) + (1
)(1
( ; R))g( )
(1)
The …rst term is direct utility from consumption, the second is status utility based on the
inference of other consumers and the third is status utility from direct observation or knowledge
of the consumer‘s type :
I call f (v; w) consumption utility, and the constant
2 [ 0; 1] is the relative weight placed
on consumption versus status in the utility function. The status utility associated with being
inferred or observed to be of type is g( ). Though status is intuitively a relative concept, a
person’s status utility does not depend on anybody’s else type. For a consumer of a …xed type ,
say, it is independent of whether that consumer is one of the highest types in society ( max
)
or not ( max >> ).
lies in the interval [0; 1] and re‡ects to what extent status depends on inference rather than
on direct observation. Its value depends on the consumer‘s choice of social ties and the total
fraction of consumers who choose weak ties, R.
( W ; R) = 1 and is independent of R, so I
simply write ( W ) = 1. All status utility for someone with weak ties therefore comes from
signalling. ( S ; 0) = 0 and is increasing in R with ( S ; 1) < 1. The status of someone
with strong ties depends less on beliefs the more that other people choose strong ties. Letting
12
( W ) 1, ( S ; 0) 0 with (
not change the main results.
S ; 1)
< (
W)
for all R would be more general but would but
Both f and g are twice di¤erentiable. f1 > 0; f2 > 0, where fi is the partial derivative of
consumption utility with respect to its i’th argument, and g 0 > 0. The inference function
0 =p;0)
is also di¤erentiable. Furthermore, f11 < 0; f22 < 0; f12
0. and ff12 (y
(y0 =p;0) < p. The last
two assumptions express that the goods are complements, and that the lowest type consumer
would choose a strictly positive amount of the non-visible consumption good in the absence of
signalling.
Another Interpretation of Beliefs: There is another way to think about beliefs that has
an appealing link with the form of the utility function. Two social groups can determine the
level of status for a given consumer of type . The …rst group, the weak ties or acquaintances,
forms a belief about this consumer’s type given his strategy: (v0 ; 0 ). This group determines
the consumer’s status if he chooses weak ties, given by g( (v0 ; 0 )). The second group, the
strong ties or friends, directly observes the consumer’s type . By directly observe, I mean that
they are simply aware of his true type by virtue of their close social relationship. This group
determines the consumer’s status if he and everybody else chooses strong ties, given by g( ). If
the consumer chooses strong ties while some others choose weak ties, the information about his
type does not ‡ow perfectly through his strong ties. Therefore his status is a weighted average
of g( (v0 ; 0 )) and g( ).
Equilibria: The model describes a static game of incomplete information, so I use the BayesNash notion of equilibrium. Each consumer has a strategy and beliefs such that his actions
are optimal given his beliefs and his beliefs are consistent with the strategies played by the
other consumers. The following de…nition of equilibrium applies the Bayes-Nash concept to the
situation at hand:
De…nition: An equilibrium is a strategy S for each consumer
sumer’s type (v; ) such that:
, and a belief about this con-
1) each consumer maximizes utility as given in equation 1 subject to the budget constraint pv +
w = y0 + , taking into account beliefs.
2) beliefs are consistent with players’ strategies.
13
I denote consumer ’s equilibrium actions by v( ) and ( ), while w( )
y0 + ( )
pv( ).
I follow Ireland (1994) and only look for equilibria in which types separate fully in terms of their
visible consumption. I refer to such situations as separating equilibria, whether or not consumers
pool or partially separate in their choice of social ties. The above de…nition then implies that
beliefs must be correct in the sense of (v( ); ( )) = . As a result, consumer has status
utility g( ) in any separating equilibrium regardless of its form.
I do not make formal restrictions on permissible out-of-equilibrium beliefs. I nonetheless de…ne
below how the D1 Criterion for out-of-equilibrium beliefs would apply to this model for the
following reason. I only look for a subset of the Bayesian Nash equilibria that may exist: those
equilibria in which consumers fully separate in their visible consumption, and for which the
inference function is di¤erentiable. While I say nothing about other potential equilibria, I
show that certain equilibria I do …nd satisfy the D1 Criterion. D1 is a stringent equilibrium
re…nement, implying that the out-of-equilibrium beliefs supporting these equilibria meet a high
standard of reasonableness. Thus the equilibria I do …nd are not implausible compared to any
others that might exist.
The D1 Criterion was introduced by Cho and Kreps (1987). Intuitively, anybody observing an
out-of-equilibrium deviation should believe it is the person with the highest incentive to make
this deviation. In the context of this model, the D1 criterion restricts out-of-equilibrium beliefs
as follows:
De…nition: Out-of-equilibrium beliefs satisfy the D1 criterion if the following condition holds.
Let (v0 ; 0 ) be a deviation from equilibrium, in the sense that (v0 ; 0 ) 6= (v( ); ( )); 8 . Say
there exists a consumer of type 1 and another of type 2 such that, whenever inference (v0 ; 0 )
is such that the following inequality holds:
U (v 0 ;
0j
=
2)
then the following inequality also holds:
14
U (v ( 2 ); ( 2 ))
U (v 0 ;
0j
=
1)
U (v ( 1 ); ( 1 )) > U (v 0 ;
0j
Then any out-of-equilibrium belief must be consistent with
=
2)
( 2 jv0 ;
U (v ( 2 ); ( 2 ))
0)
=0
If for any inference about types that gives consumer 2 a weak incentive to make a given deviation
consumer 1 has a greater incentive to make that same deviation, then no one observing this
deviation will believe it is consumer 2 . If a single consumer has the greatest incentive to make
a given deviation regardless of inferred beliefs, then anyone observing the deviation will assume
it is that one type.
The D1 Criterion is restrictive because it generally imposes a unique permissible out-of-equilibrium
belief for any deviation. Say there exists a consumer that could make a substantial pro…t from
a deviation, regardless of beliefs, and yet another consumer that could make marginally more
pro…t from the same deviation. D1 implies that someone observing this deviation must place
zero weight on the …rst type and full weight on the second even though the deviation is attractive for both of them. In contrast, the Intuitive Criterion would only rule out the …rst type if
the deviation was unpro…table for him for all possible inference that others could make from
observing it.
1.2. Discussion of the model assumptions
I now list some of the more important model characteristics. I elaborate on them below and
then compare this model with Ireland (1994) in more detail.
1) A personal characteristic, consumer type, is positively correlated with both income and once
inferred with social status.
2) This personal characteristic of a given consumer is directly observable by some people but
not by others.
3) People value status for its own sake rather than for any instrumental role it may play.
15
4) Status is a function of who people are or who they are believed to be, rather than simply of
a person’s actions
5) Some types of consumption are visible while others are not.
6) It is more di¢ cult to transmit personal information through strong ties when others choose
weak ties.
7) The utility function satis…es certain assumptions, notably with respect to the derivatives of
f:
1) Interpretation of consumer type: There is no de…nitive interpretation of type as presented in this model but it could correspond to a number of personal characteristics that are
related to both income and social status. As mentioned before, type could be a person’s natural
talent or productive ability in a society that looks favourably on talented individuals. It could
also represent a person’s inherent drive or propensity to exert e¤ort in a society that values
hard work. This later interpretation echoes Max Weber’s view of the protestant work ethic
(1904) which emphasizes diligent work and labour as a sign of a one’s personal salvation. These
interpretations are subject to a caveat: individuals cannot do anything to modify their value of
in this model. People are born with either more or less talent, or with higher or lower drive.
2) Type is observable to some but not to others: I assume people have two kinds of
social relationships, where one kind is closer and more personal than the other. Granovetter
(1973) distinguishes between strong and weak ties, and he argues that the following de…nition
is intuitive:
"The strength of a tie is a ... combination of the amount of time, the emotional intensity, the intimacy (mutual con…ding) and the reciprocal services which characterize
the tie." (p.1361)
Following this de…nition, strong ties by their very de…nition tend to transmit more information
about di¢ cult to evaluate personal characteristics such as natural talent, productive ability,
inherent drive or propensity to exert e¤ort. A person’s close friends may know to what extent
they are present, but the same cannot be said for a person’s loose acquaintances who must
instead make an inference based on what a person actually does.
16
3) People value status for its own sake: I take as a basic assumption that people value
status even though it may not provide any direct monetary or consumption bene…ts. Bernheim
(1994) presents three arguments for such a view. First, it is consistent with psychological
evidence. Second, evolutionary pressures could promote a desire for status (or regard for the
views others have of you) as it may have presented greater survival and reproductive possibilities.
Third, behavioural conditioning could promote status seeking if esteemed individuals tend to
receive better treatment. An alternative approach would be to assume that people value status
because it generates better career or consumption opportunities. While also plausible, Kwon
and Milgrom (2007) argue using Swedish data that people are driven by the social desire for
status rather than its pecuniary bene…ts.
4) Status is a function either of type or inferred type, rather than simply of actions: The model formulates status as a direct function of or its inferred level. Actions are
only relevant to the extent that they provide some information about consumer type. Other
approaches are also possible, such as letting the argument of the status function g simply be
visible consumption v or inferred income. I …nd that approach less attractive because social
esteem is fundamentally more a function of the kind of person someone is, rather than simply
what that person does. Bernheim (1994) takes the same approach and alludes to generosity as
an example:
"[A]n individual who has taken generous actions is esteemed and thought of as generous. Yet most of us distinguish between people who merely act generous and those
who truly are generous. Generosity is usually considered a personality trait, not simply a characterization of past actions .... [S]tatus depends critically on motivations."
(p.844)
5) Visible and non visible-consumption: Many of the authors mentioned in the introduction
draw the distinction between visible and non-visible consumption. In fact, Veblen’s (1899) main
point is that consumption contributes to status to the extent that it is seen and admired by
others. He makes the following distinction between consumption inside and outside the home:
"[T]he e¤ect on consumption is to concentrate it upon the lines which are most patent
to the observers whose good opinion is sought .... Through this discrimination in
17
favour of visible consumption it has come about that the domestic life of most classes
is relatively shabby, as compared with the eclat of that overt portion of their life that
is carried on before the eyes of observers." (p.69)
6) It is more di¢ cult to transmit personal information through strong ties when
others choose weak ties: I call this assumption the information externality and express
@
it by @R
( S ; R) > 0. When the fraction of people who choose weak ties R is large, then
the status of people with strong ties depends more on inference. It is harder to have close
personal social ties with other people if most other people only want to have weak ties with you.
The externality is negative because it reduces the quality of information about consumer type,
leading to consumption distortions towards the visible good. I assume in contrast that ( w) is
independent of R. People looking for acquaintances are able to …nd them, whether or not most
other people are also looking for acquaintances or are instead looking for close friends. This last
assumption is not crucial, in the sense that relaxing it would not alter the parameter values for
which a weak ties equilibrium exists.
7) Assumptions on the utility function: I assume that f and g are additively separable, so
marginal consumption utility is independent of the level of status and vice-versa. This assumption does not seem unreasonable and is analytically convenient. I also assume that f11 ; f22 < 0;
so there is decreasing marginal utility for each type of consumption goods. While that assumption is standard, I do not assume anything about the sign of g 00 . Perceived type is a qualitatively
di¤erent kind of "good" than a standard consumption good, and there is no reason to assume
a priori that there are diminishing returns. The function g could be negative below max =2,
positive above it and symmetric to express the notion that it is a zero sum game, or it could
be positive of all . I assume f12
0 for technical reasons to ensure that the SOC holds for
all types, although there is no compelling reason why the goods must be complements and the
inequality should hold strictly. Assuming f12 0 implies that visible consumption v would be
increasing as a function of type in a model without signalling.
o =p;0)
I mentioned that I assume ff12 (y
(y0 =p;0) < p to ensure that if status e¤ects were not present ( = 1)
or did not depend on signalling ( = 0) the lowest type consumer would consume some of the
non-visible consumption good. This condition is necessary to prove the existence of a separating
equilibrium and it is also plausible. Some consumption goods that satisfy the most basic human
18
needs such as food would fall into the non-visible category and would thus be consumed at a
strictly positive level even by those with very low income.
Comparison with Ireland’s 1994 model
This model and certain parts of my approach are similar to Ireland (1994)’s model of conspicuous
consumption. Ireland also looks for equilibria in which consumers separate in terms of visible
consumption and for which the inference function is di¤erentiable. His consumers have utility
function:
U = f (v; w) + (1
)f (v; (wjv))
where I have modi…ed his notation to bring out the similarities with this model. The only
piece of new notation is (wjv), which are beliefs about the consumer’s non-visible consumption
given his visible consumption. There are also two consumption goods, consumers value status
for its own sake and they signal with their visible consumption to obtain it. Utility is additively
separable over consumption and status, and consumers di¤er only in terms of income which is
distributed over a continuum. Ireland also makes similar assumptions on the utility function to
show that a separating equilibrium exists.
The major di¤erence between the models is that Ireland does not look at social ties, although
there are a number of other di¤erences as well. Ireland assumes that status utility has a speci…c
form and that it is equal to the consumer’s inferred level of consumption utility, while I only
specify that the status function g is increasing. People infer a consumer’s level of non-visible
consumption in his model, while here people infer a consumer’s type. If I had not included social
ties in this model, these two approaches to inference would be equivalent. Someone observing
Ireland’s consumer could see v, infer w = (wjv) and believe that he has income pv+ (wjv).
Equivalently, he would believe that the consumer’s type is 1 (pv + (wjv) y0 ).
The two approaches to inference are di¤erent when consumers can increase their income by
choosing weak ties. Say two equilibria exist, one in which all consumers have strong ties and
low income and another were everyone has weak ties and high income. Everyone in the weak
ties equilibrium can consume more w with their higher income. In Ireland’s model status is a
function of inferred w, and in equilibrium beliefs must be consistent with players’ strategies.
19
Therefore all consumers would have higher status utility in the weak ties equilibrium than in
the strong ties equilibrium, simply because they are wealthier.
This idea is not what I want to capture in my model: everyone’s level of status in society should
not increase just because everyone is more wealthy. I therefore assume that people base their
judgements on someone’s intrinsic personal characteristic, type, which is something a consumer
cannot change. Though a consumer can in‡uence others’ inference of his type by choosing a
di¤erent v, he cannot do so in a separating equilibrium because people’s beliefs about him must
be correct. He will therefore obtain status utility g( ) in any such equilibrium, regardless of
what ties people choose and how must income they dispose of.
20
Chapter 2: An Illustrative Example and Discussion of Main
Results
I begin this chapter by considering an example. In section 2.1, I make speci…c assumptions
about the functional form of the utility function and about parameter values. I then graphically
illustrate di¤erent characteristics of the equilibrium in which all consumers choose weak ties. In
section 2.2, I discuss why these characteristics hold more generally, thus previewing the main
results from the thesis that I bring together later in Chapter 4.
2.1. An illustrative example
I take consumption utility to be of a square root form and additively separable, and status utility
to be linear. Speci…cally, I assume:
f (v; w)
= 2v 1=2 + 2w1=2
g( ) =
=
1=2
y0
= 4
W
=
1
S
= 0:3
p
= 1
I show in Chapter 4 that in an equilibrium where all consumers choose weak ties, the function
v( ) must take a speci…c form. There is an initial condition that the lowest type consumer,
= 0, consumes his undistorted level of v, and a di¤erential relationship between v and that
determines what higher types consume.
Applied to this example, the initial condition is:
(2) = 0
21
because the lowest type consumer, with income 4, consumes his undistorted level v = 2.
The di¤erential relation in this case is:
1
d
=p
dv
4+
v
1
p
v
I solve this initial value problem numerically to obtain the function v( ) in the weak ties equilibrium, which I then evaluate between = 0 and approximately = 3. Income varies between 4
for the lowest type consumer and about 7 for the highest under weak ties. I use Microsoft Excel
to generate the data for the graphs.
I have not yet speci…ed the value of ( S ; 1), nor have I addressed the question of whether
everybody choosing weak ties is actually an equilibrium for this speci…c example. I prove in
Chapter 4 that a weak ties equilibrium exists if the income gain from choosing weak ties is
su¢ ciently large: given S , W must be above a certain critical level. It is not immediately
evident that W = 1 and S = 0:3 would satisfy such a condition.
However, I also show that the weak ties equilibrium is more likely to exist when the information
externality, ( S ; 1), is large. A high value of ( S ; 1) makes deviations to strong ties less
attractive, which I comment on more in the following section. As ( S ; 1) approaches 1, W
would have to be only marginally greater than S for the weak ties equilibrium to exist. The
results from this illustrative example are therefore equilibrium results for high enough values of
( S ; 1).
The …rst graph shows that some consumers in the weak ties equilibrium may have lower utility
than if everybody had chosen strong ties, while others may have higher utility. In this case,
consumers whose type is less than approximately 1.8 are worse o¤ under weak ties, while those
above 1.8 are better o¤. There are two forces pushing in di¤erent directions: income is higher
under weak ties which pushes welfare up, while people have to signal to obtain status which
pushes welfare down. Also, utility is an increasing function of type in equilibrium.
22
Total Utility vs. Consumer Type under Weak and Strong Ties
5
4.6
Total Utility
4.2
3.8
Weak Ties - dis torted
3.4
Strong Ties
3
2.6
2.2
0
1
2
3
4
Consumer Type
The remaining graphs do not directly compare the weak ties equilibrium with the situation under
strong ties. Instead, they compare various characteristics of the weak ties equilibrium with what
would occur if consumers did not have to signal but still had the same income as under weak
ties.
Whereas the above graph shows the opposing e¤ects on welfare of higher income and the need to
signal, the following graphs illustrate the pure negative e¤ect of signalling. They show how visible
consumption would decrease and consumer welfare would increase if weak ties also transmitted
information about consumer type. No one would have to signal to obtain status so consumption
would not be distorted.
In the absence of signalling, all income would be split evenly between v and w. The following
graph shows the extent to which v is distorted up for all consumers:
23
Visible Consumption vs. Consumer Type
Visible Consumption
7
6
5
4
Weak Ties - distorted
3
Weak Ties - undistorted
2
1
0
0
0.5
1
1.5
2
2.5
3
3.5
Consumer Type
Because of this distortion, consumers have lower consumption utility than they otherwise would
have:
Consumption Utility vs. Consumer Type
4
Consumption Utility
3.8
3.6
3.4
Weak Ties - distorted
3.2
Weak Ties - undistorted
3
2.8
2.6
2.4
0
1
2
3
Consum e r Type
24
4
Furthermore, consumption utility is actually a decreasing function of type for low income consumers when there is signalling. This feature is not caused by the assumption that the lowest
type is zero, since I have checked that the same shape remains in this example for min = 2. I
now change the scale of the above graph and reproduce a part of the lower, dark line to illustrate
its shape more clearly.
Weak Ties - Distorted Consumption Utility vs. Consumer Type
3
Consumption Utility
2.95
2.9
2.85
2.8
2.75
2.7
2.65
2.6
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Consumer Type
A consumer’s status utility is the same in any separating equilibrium, whether or not there are
consumption distortions. Since signalling reduces consumption utility, it also decreases total
utility:
Total Utility vs. Consumer Type
5.5
Total Utility
5
4.5
4
Weak Ties - distorted
Weak Ties - undistorted
3.5
3
2.5
2
0
0.5
1
1.5
2
2.5
Consumer Type
25
3
3.5
The magnitude of the above e¤ects is not negligible. I make quite routine parameter value
assumptions yet the …rst graph shows a noticeable utility di¤erence between the weak ties
equilibrium and the situation where everyone chooses strong ties. Some consumers would gain
an extra 2-3% of their total utility if everyone chose strong ties, while others would lose about
the same amount.
The second, third and …fth graphs show that the pure e¤ect of signalling can be quite large.
If consumers did not have to signal under weak ties, the consumer of type = 1:5 would have
approximately 40% less visible consumption, 12% higher consumption utility and 10% higher
total utility. On the other hand, the fourth graph shows consumption utility decreasing only
slightly in magnitude as a function of type for low-income consumers. That being said, the
number of consumers with lower consumption utility than the lowest type is not negligible: all
those with < 0:5.
2.2. Discussion of the main general results
Before addressing how the above characteristics more generally hold, I comment in more detail
on why the weak ties equilibrium exists. The main reason consumers do not want to deviate
to strong ties is that they must give up income, while status will still be determined by the
higher social standards of the weak ties equilibrium. The income given up is higher when W is
signi…cantly larger than S , in which case the equilibrium is more likely to exist.
A di¤erent force pushes consumers towards deviating to strong ties. By deviating to S , a
consumer decreases the role inference plays in determining his status utility since ( S ; 1) <
( W ) = 1. He obtains a part of his status without signalling because of his strong ties, which
e¤ectively changes his utility function. Even though he has the same preferences, in so doing he
changes the characteristics of the visible consumption good. An extra unit of v now contributes
less to his status utility than it did before because the message it provides is less relevant to
him, so visible consumption becomes less attractive. This e¤ect is less important when ( S ; 1)
is close to 1. If that is the case, he can only slightly modify the characteristics of v by deviating
to S .
To show how this e¤ect could make a deviation to strong ties pro…table, I assume just for
the moment that W = S so that deviating to strong ties does not cost any income. The
26
consumer was indi¤erent between spending a marginal amount of extra income on v and on w
at his equilibrium allocation under weak ties. If he deviates to strong ties and keeps v and w
unchanged, he has the same utility level but now …nds a marginal unit of v less attractive than
before. He can increase his utility by decreasing his visible consumption and increasing his nonvisible consumption until the marginal rate of substitution once again equals the relative price
p. A person can take advantage of the fact that by choosing strong ties, he is less dependent on
visible consumption for his well-being.
Since choosing strong ties does in fact cost income, the weak ties equilibrium exists if the …rst
e¤ect is larger than the second. The equilibrium exists if, given S , W is above a certain level,
and that level is decreasing in ( S ; 1). It is more likely to exist when the information externality
is large because then the consumer’s status depends more on inference if he chooses strong ties.
Any deviant could choose S and an undistorted level of v at his new, lower income. If is
unobservable, this deviation appears to be the equilibrium strategy of another, lower type. He
therefore su¤ers a status loss from the deviation. Intuitively, a person living in a society with
weak ties, high income and conspicuous consumption would …nd it di¢ cult to switch to strong
ties not only because of the income loss but also because of the belief about what a certain level
of visible consumption means in that society.
Even if the income gain is too small for the equilibrium to exist, there is still no equilibrium
in which any consumer chooses strong ties if is unobservable or if beliefs are (v( ); S ) = .
Anyone in such a situation could increase both his status and his consumption utility by choosing
weak ties and consuming more of both goods.
I now turn to the results shown in the graphs of section 2.1. The result in the …rst graph show
that some consumers in the weak ties equilibrium may have lower utility than if everyone chose
strong ties. It is also possible that all consumers are better o¤ under weak ties. If the income
gain from choosing weak ties is large and the consumption distortions are small, all consumers
may have higher utility than under strong ties. They now have to signal but it is worth it
because of their higher income.
Interestingly, the income gain from weak ties can be at the same time high enough so that
the weak ties equilibrium exists, yet too low to make everyone better o¤ than under strong
ties. This result is a direct consequence of the information externality, that strong ties do
not transmit information perfectly about type if others choose weak ties. If there were no
27
information externality, any consumer with lower utility under weak ties would just deviate to
his undistorted consumption bundle under strong ties. He would gain consumption utility and
not lose any status utility. He would be inferred to be a lower type but not su¤er from it because
his status would no longer depend on signalling.
But with the information externality, this consumer’s status still depends in part on inference
when deviating to strong ties. The same deviation would be less attractive because his status
utility would decrease. In fact his best unilateral deviation may now be unpro…table, even
though it would be pro…table if everyone else had chosen strong ties.
Total utility must be increasing as a function of type in any equilibrium: high type consumers
are better o¤ than lower type consumers. If that were not the case, someone could pro…tably
deviate by …nding a lower type who is better o¤, make the same consumption decision as him
and still have income left over.
The e¤ects in the graphs comparing the weak ties equilibrium with what would occur if consumers did not have to signal but still had the income of weak ties, are also quite general. The
equilibrium’s main general feature is that visible consumption is distorted up for all consumers
except the lowest type. This result holds because status depends on inference under weak ties,
so a consumer would have an incentive to increase his visible consumption to imitate a slightly
higher type if everybody had their undistorted level of v. He would lose consumption utility by
increasing v and decreasing w but would gain a higher amount of status utility. The slightly
higher consumer must therefore choose an equilibrium level of v that is just high enough to
make that deviation unpro…table. The lower type would then lose more consumption utility by
deviating than he would gain in status utility, and the higher consumer credibly signals his type.
Therefore all consumers other than = 0 have distorted consumption. The lowest type chooses
his undistorted consumption bundle without worrying about status since there is no lower type
who would want to imitate him.
The equilibrium di¤erential relationship between v and follow from the above argument. It
implies that consumers push their visible consumption up just enough so that the slightly lower
type’s …rst order condition is satis…ed and he has no incentive to deviate to a higher v.
Because of this distortion, consumers have lower consumption utility than they would have
with the same income but no need to signal. Their status utility is the same in both cases
28
since beliefs are correct in any separating equilibrium. A consumer of type has status utility
g( ) in equilibrium, irrespective of how distorted his equilibrium consumption may be. Since
consumption utility is lower, total utility is also lower for all consumers except for the lowest
type. Everyone would have higher utility if they could transmit information about their type
through weak ties.
Even though total utility is an increasing function of type, lower type consumers may actually
have more consumption utility than higher type consumers, particularly close to = 0. The
general result is that consumption utility is decreasing in a neighbourhood of = 0 whenever
the lowest type consumer chooses a strictly positive amount of both v and w. Any consumer has
more income than a marginally lower type, but his equilibrium level of v may need to be much
higher to prevent any deviations. He may need to push his visible consumption up to such an
extent that his consumes less w then the marginally lower type. His consumption utility may
then be lower, particularly since the goods are complements.
This result is interesting for two reasons. First, economists may observe that people with higher
income have a lower standard of living (in terms of consumption utility) than others and conclude
that their welfare is lower. The model suggests that conclusion could be incorrect, and that their
overall level of well-being would be higher taking into account social status. On the other hand,
some economists may hold the normative view that social status should play no role in welfare
evaluations even though people may irrationally distort their consumption to obtain it. Using
consumption utility as a welfare criterion, the behavioural prediction of this model is that people
with higher income may actually have lower welfare.
29
Chapter 3: Preliminary Formal Analysis
The novelty of this model is that consumers can choose both their visible consumption and the
strength of their social ties. I nonetheless assume in this chapter that social ties are exogenous
and equal to W for all consumers, who just decide on their level of v. I show that a unique
separating equilibrium with a di¤erentiable inference function exists and establish a number of
welfare properties. I then use these results in Chapter 4 to prove results for the general model
in which both visible consumption and social ties are choice variables.
3.1. Equilibrium existence with
…xed
Since all consumers have weak ties, each person’s status depends completely on signalling:
( W ) = 1. Inference is only a function of visible consumption, (v; ) = (v), since provides no information about consumer type. I assume that the inference function
relating
visible consumption to inferred type is di¤erentiable and look for an equilibrium in which all
consumers separate in their choice of v. I denote the equilibrium visible consumption of type
by v( ).
I prove four lemmas that, taken together, imply the existence of a unique separating equilibrium.
The …rst lemma establishes that the …rst-order-condition holds with equality for all types. The
second and third lemma show that the inference function is increasing, which I use in the fourth
lemma to show that the second order condition holds. The steps in the proof closely follow the
discussion in Ireland (1994), which in turn is based on Mailath (1987).
Lemma 1: The …rst-order-condition holds with equality in equilibrium for all consumers:
0
ff1 (v; y0 + W
pv) pf2 (v; y0 + W
pv)g + (1
)g ( (v)) ddv = 0
Proof: Di¤erentiating the utility function with respect to v gives:
ff1 (v; y0 +
W
pv)
pf2 (v; y0 +
W
pv)g + (1
0
)g ( (v))
d
dv
(2)
where the third term denotes the extra status utility a consumer obtains from a marginal unit
of v.
30
Expression 2 gives the net change in total utility from consuming a marginal extra amount of v.
If it were strictly negative in equilibrium for a consumer of type , this consumer must have a
corner solution with v = 0. All terms in expression 2 are continuous and income is a continuous
function of type. Hence there exists " > 0 such all consumers in this epsilon neighbourhood of
also have v = 0. Any such equilibrium is therefore not separating.
If expression 2 is strictly positive for a consumer then he must have w = 0. Again by continuity,
consumers of a similar enough type also choose w = 0 and spend all their income on v. Every
consumer would choose a strictly positive amount of w if there were no signalling, because of the
assumption on the lowest type consumer, because income is increasing in type and because the
goods are complements. The only reason any consumer would distort his visible consumption up
and choose w = 0 is to prevent a slightly lower type from imitating him. But any consumer of
type in the epsilon neighbourhood does not have this problem since the slightly lower type has
less income than him which he already spends entirely on v. The slightly lower type is therefore
unable to imitate him and consumer would be better o¤ choosing a strictly positive level of
w. QED
Lemma 1 implies that any consumer with corner solution v = 0 or w = 0 must be an extreme
type , = 0 or = max , and that the …rst-order-condition also holds with equality for him.
The second and third lemmas demonstrate that the inference function
is increasing.
Lemma 2: In any separating equilibrium, the inference function’s derivative
evaluated at the equilibrium value v(0)
0
is positive when
Proof: The …rst order condition is equivalent to:
f1
=p
f2
0
(1
) g ( (v)) d
f2
dv
(3)
The lowest type consumer will always be recognized as such in equilibrium and therefore has no
incentive to distort his consumption. He makes the same choice v(0) and w(0) that he would if
there was no signalling. If he has an interior solution, then v(0) must be such that the marginal
rate of substitution equals the relative price, ff12 = p. Since by assumption g 0 > 0; equation 3
implies ddv = 0.
31
If the lowest type instead has a corner solution it must be with v(0) = 0, since he would consume
a strictly positive amount of w if there were no signalling. In such a situation, ff12 < p and the
consumer would like to substitute away from the visible good but he cannot because he is already
not consuming any of it. Then equation 3 implies that ddv > 0. QED
Lemma 3: In any separating equilibrium, the inference function is monotonically increasing
for all types: 0 0 when evaluated at equilibrium values v = v( ) 8 :
Proof : Rearranging the …rst order condition gives:
d
=
dv
1
pf2 (v; y0 +
pv) f1 (v; y0 +
g 0 ( (v))
W
pv)
W
(4)
Since beliefs must follow from consumers’ strategies, (v( )) = in a separating equilibrium.
Plugging this relationship into equation 4 yields the following identity in equilibrium:
d
=
dv
1
pf2 (v; y0 +
W
(v)
pv) f1 (v; y0 +
g 0 ( (v))
W
(v)
pv)
(5)
Di¤erentiating both sides of equation 5 with respect to v gives another identity for all types in
equilibrium:
d2
=
dv 2
Whenever
(1
0
)(g 0 )2
f[f11 + f12 (
W
0
p)
p(f21 + f22 (
W
0
p))]g 0
[f1
pf2 ][g
00
0
]g (6)
equals 0, this expression reduces to:
d2
=
dv 2
(1
)g 0
[f11
2f12 p + p2 f22 )]
(7)
which is strictly positive since f11
0; f12 > 0; f22 < 0 and g 0 > 0. Equation 7 implies that
0
is strictly increasing whenever it is equal to zero. Since Lemma 2 showed that 0
0 when
32
v = v(0), 0 will also be non-negative for all other values of v. So (v) is monotonically increasing
for all equilibrium values v = v( ). QED
The following …nal lemma proves that the second-order condition holds:
Lemma 4: The second order condition for maximization holds:
(1
)fg
00
0
+ g0
00
g
ff11
f21 p + f22 p2 g +
f12 p
0
Proof : Di¤erentiating the utility function twice with respect to v gives the following expression:
ff11
f12 p
f21 p + f22 p2 g + (1
)fg
00
0
00
+ g0
g
(8)
So as claimed in the Lemma, if expression 8 is negative then the SOC for maximization indeed
does hold.
I replace
by (v) in the FOC, giving an identity that holds in equilibrium:
ff1 (v; y0 +
(v)
W
pv)
pf2 (v; y0 +
(v)
W
0
pv)g + (1
)g ( (v))
d
=0
dv
(9)
I di¤erentiate both sides of equation 9 with respect to v, giving another identity:
ff11 + f12 [
0
p]
0
p(f21 + f22 [
p])g + (1
)fg
00
0
+ g0
00
g=0
I rearrange terms to get:
ff11
f12 p
f21 p + f22 p2 g + (1
)fg
00
0
+ g0
00
g = fpf22
W
0
f12
W
0
g
0
where the left hand side is just expression 8. The right hand side,
fpf22 f12 g, is negative
0
since f22 < 0, f12
0, and
0. Hence expression 8 is also negative and the second order
condition for a maximum is satis…ed. QED
33
Plugging (v) = into the left hand side of equation 4 gives a di¤erential equation relating type
to visible consumption v in equilibrium:
d
=
dv
1
pf2 (v; y0 +
pv) f1 (v; y0 +
g0 ( )
W
W
pv)
(10)
Equation 10 is a …rst order di¤erential equation that, along with the initial condition that
the lowest type chooses the consumption bundle that he would have in the absence of status
e¤ects, de…nes a unique function (v). This function characterises the unique existing separating
equilibrium with a di¤erentiable inference function, both in terms of beliefs and how consumers
of each type divide their consumption between w and v.
I can now prove the following proposition:
Proposition 1: Let = W be …xed. Then a unique separating equilibrium exists which is
characterised by a di¤erentiable inference function (v). This function is de…ned by the initial
condition (v(0)) = 0, where v(0) is the amount of visible consumption that a consumer of type
= 0 would consume in the absence of signalling, and the di¤erential relationship of equation
10.
Proof : Let out-of-equilibrium beliefs be (v) = 0, for all v < v(0), and (v) can be anything
for v > v( max ).
The function (v) was constructed so that no consumer has an incentive to imitate the equilibrium visible consumption of another type. The only other possible deviations are to v lower
than v(0) or greater than v( max ).
If v(0) = 0, there is nothing to check at the lower bound since consumption cannot be negative.
Say v(0) > 0. The lowest type does not want to deviate to any v < v(0) since that was his
undistorted consumption amount.
Say any higher type deviates to v < v(0). Divide the deviation into two steps, …rst from v( )
to v(0), and then from v(0) to v. The change in utility from such a deviation equals the sum
of change in utility for these two steps. The …rst step is not pro…table, since by construction of
the function (v) no type …nds it pro…table to imitate another.
34
Furthermore, the second step for consumer is more unpro…table than a deviation by the lowest
consumer type from v(0) to v. Since type has a higher income than the lowest type, he also
has a higher amount of w when they both consume v = v(0). By complementarity, f1 (v(0); w)
is greater for type than for the lowest type, and f2 (v(0); w) is lower. It is therefore worse
for consumer to decrease his visible consumption from v(0) to v and increase his non-visible
consumption by a corresponding amount.
Since deviating from v(0) to v is unpro…table for the lowest type, the second step described above
is even more unpro…table for a higher type. As a result, the overall deviation is unpro…table for
that consumer.
Any other deviation must be to v > v( ). This deviation is unpro…table for the highest type
regardless of inference. By Lemma 3, 0 (v( max )) 0 and equation 3 shows that the marginal
rate of substitution of consumption utility f1 =f2 at v( max ) is less than or equal to the relative
price p. So consumer max has either an undistorted level of v or one that is distorted up.
Any deviation to a higher level of v would not yield any extra status utility but would cost
him consumption utility by pushing up v to a more distorted level. Such a deviation is also
unpro…table for a lower type by a similar logic to above: deviating to v( max ) is not pro…table,
and then pushing v up even more will once again reduce consumption utility. QED
The out-of-equilibrium beliefs supporting the above equilibrium satisfy the D1 Criterion. Since
no type could …nd it pro…table to deviate to v > v( max ) regardless of beliefs, the D1 Criterion
imposes no restrictions on such beliefs. If v(0) = 0 then there are no other out-of-equilibrium
beliefs to speak of.
If v(0) > 0, then I showed in the above proof that any deviation to some v < v(0) is least
unpro…table for the lowest type, given that (v) = 0. If (v) > 0, then the status utility of any
type carrying out such a deviation would be higher than if (v) had been zero by the following
amount: (1
)fg( (v0 )) g(0)g. This deviation may now be pro…table for di¤erent consumers.
However it is most attractive to the lowest type consumer. If (v) were low it would still be
less unpro…table for him than for other consumers, and if (v) were high enough to make the
deviation pro…table for any consumer it would be most pro…table for him.
35
3.2. Characteristics of the equilibrium with
…xed
I do not have an explicit expression for (v) or v( ) as a function of the model parameters, which
makes it more di¢ cult describe the qualitative properties of this equilibrium. I can nonetheless
establish some relevant results. I turn to the consumers’visible consumption, starting with the
following lemma:
Lemma 5: In any separating equilibrium,
when evaluated for all > 0.
is strictly increasing for all non-zero types:
0
>0
Proof : I showed in Lemma 3 that 0 0 for all v( ), and I noted that 0 = 0 is indeed possible
for the lowest type consumer. This lemma proves that the inequality is strict for all other types.
Say there is a consumer of type 0 > 0 such that 0 (v( 0 )) = 0. Lemma 2 showed that 0 is
strictly increasing whenever it is equal to zero, so there must exist " > 0 such that 0 (v) > 0 for
all v 2 [v( 0 ) "; v( 0 )]. If not, then 0 = 0 arbitrarily close to v( 0 ) and then 0 (v( 0 )) > 0,
which contradicts 0 (v( 0 ) = 0.
Consider some speci…c v1 2 [v( 0 ) "; v( 0 )] ; v1 = v( 0 ) "1 for "1 positive and less than
". By continuity of 0 , lim"0 !0 0 (v1 ) = 0 (v( 0 )) = 0. When "0 is very small, 0 (v1 ) can
become arbitrarily small but still strictly positive. If 0 (v1 ) can be made arbitrarily small by an
appropriate choice of "0 , so can the following expression which is proportional to 0 :
0
(1
)(g 0 )2
f[f12
W
pf22
W ]g
0
(f1
00
pf2 )(g )g
(11)
On the other hand, the RHS of equation 7 is always strictly positive:
(1
)g 0
f[f11
2f12 p + p2 f22 )]
Moreover, it will not get arbitrarily close to zero as "0 goes to zero since from the proof of Lemma
3 it is also strictly positive when evaluated at v( 0 ).
36
Equation 6, which I rewrite below, is simply equal to the RHS of equation 7 plus expression 11.
d2
=
dv 2
(1
)(g 0 )2
f[f11 + f12 (
W
0
p)
p(f21 + f22 (
W
0
p))]g 0
[f1
pf2 ][g
00
0
]g
2
So for some "0 small enough it must therefore hold that ddv2 is strictly positive for all v1 with
2
"1 < "0 . But if ddv2 is strictly positive arbitrarily close to v( 0 ), then 0 (v( 0 )) is strictly positive
which is a contradiction. QED
It is no great surprise that Lemma 5 holds. If 0 = 0 for any 2 (0; max ), then the separating
equilibrium as described in Proposition 1 would actually not exist: the function would map
di¤erent values of v to a single type which would not be sensible. Lemma 5 therefore con…rms
that this problem does not occur, and gives the added information that 0 (v( max )) > 0.
Lemma 5 also con…rms that the second order condition holds strictly for all consumers of type
greater than zero. Formally speaking, this result is necessary for the SOC to be a su¢ cient
condition for maximization and not just necessary. Since the lowest type consumer also does
not want to deviate, every consumer is truly at a maximum.
The function (v) is strictly increasing away from zero, and if 0 (0) = 0 then 00 (0) is strictly
positive. Hence the function (v) is strictly increasing over its whole domain and it can be
dY
inverted to give the function v( ). There is a simple relationship between dX
and dX
dY for a
1
dX
generic di¤erentiable function Y = h(X), given by: dY = dY =dX . The appendix contains the
proof.
Applying this relationship to the situation at hand yields:
dv
1
=
d
0
pf2 (v; y0 +
g( )
pv) + f1 (v; y0 +
W
W
pv)
(12)
Along with the initial condition for v(0), equation 12 de…nes the function v() that maps each
type to a quantity of visible consumption v( ). This representation is valuable because it may
often be more convenient to think of v as a function of ; rather than vice-versa. Intuitively, type
37
determines behaviour and not the other way around v( ): v( ) is a strictly increasing function
since (v) is. So consumption of v is strictly increasing as a function of type and every type
(except for possibly the lowest type), consumes a strictly positive amount of v.
I now prove Propositions 2 through 4 about various characteristics of the equilibrium. These
propositions are also relevant for the situation when is a choice variable. If there is an equilibrium in which all consumers choose weak ties, then it will look exactly the same as the one
described in Proposition 1. Propositions 2 through 5 will hold for it as well which explains why
they echo some of the main results I mention in section 2.2.
Proposition 2: In the equilibrium of Proposition 1, all consumers except for the lowest type
have their decisions distorted compared to the situation where status does not depend on signalling. Speci…cally, all consume a higher level of v than they would have in the absence of
signalling.
Proof : Consider an arbitrary consumer of type > 0. If this consumer had an interior solution
without signalling, he would have w( ) > 0 and v( ) > 0 such that f1 =f2 = p: But with
signalling, equation 3 holds:
f1
=p
f2
0
(1
) g ( (v)) d
f2
dv
Since 0 > 0 for all types > 0, the ratio f1 =f2 < p so this type is consuming more v. If this
consumer had an corner solution without signalling, it cannot be with w( ) = 0. The lowest
type consumer chooses w > 0 in equilibrium and this consumer has more income. Moreover
the goods are complements, so w( ) > w(0) > 0. If he had a corner solution with v( ) = 0
without signalling, then f1 =f2
p would hold. Since f1 =f2 < p when there is signalling, this
consumer must now have a higher level of v. QED
The following result directly follows:
Corollary: In the equilibrium of Proposition 1, the utility of all consumers except for the lowest
type is strictly lower than if status did not depend on signalling.
I now show that equilibrium utility is a strictly increasing function of consumer type.
38
Proposition 3: In the equilibrium of Proposition 1, higher types obtain strictly higher utility:
U (v( 1 )) > U (v( 2 )) if and only if 1 > 2
Proof : Say 1 > 2 , but that U (v( 1 ))
U (v( 2 )). The higher type consumer has income
y0 + W 1 , and he can imitate the lower type’s consumption which costs y0 + W 2 . He then has
the same consumption and status utility as the lower type consumer in equilibrium. He also has
remaining income W ( 1
2 ) > 0, which he can spend on either consumption good to increase
his utility still more. He therefore has a pro…table deviation. The other direction follows by
symmetry and by the fact that 1 = 2 implies U (v( 1 )) = U (v( 2 )). QED
Although higher consumer types obtain strictly higher total utility in equilibrium, that does not
necessarily imply that they obtain higher consumption utility. I now look into this issue more
closely.
Say dd v p at a particular value = 0 . A marginally higher type 0 + " has W " more income
than consumer 0 , and consumes ( dd v)" more of v. Since p( dd v)"
W ", he also consumes
at least weakly more of w. Consumption utility is increasing in type at = 0 because the
marginally higher consumer has more v and more w:
But the situation may change if dd v > p for some = 0 . A marginally higher type 0 + "
consumes ( dd v)" more of the visible good but consumes (p( dd v)
W )" less of the non-visible
d
good, where (p( d v)
W ) > 0. He consumes more of v but less of w than the marginally lower
consumer, so he consumption utility need not be higher.
I now show that consumption utility is often a decreasing function of type for low type consumers.
Proposition 4: Say the lowest type consumer chooses a strictly positive amount of both goods
in equilibrium. Then consumption utility is decreasing in a neighbourhood of = 0.
d
= 0 when evaluated at v(0). But since
Proof: If consumer = 0 has an interior solution, dv
d
d
d v is simply the inverse of dv = 0, that implies that the slope of v( ) is in…nite when evaluated
at = 0. The second graph in section 2.1, of visible consumption versus type for additive square
root utility, shows this high slope around the origin.
A consumer of type " will consume more of both v and w than the lowest type if the ray
connecting the origin to ( = "; v = v(")) has a slope of less than W =p. Since this slope tends
39
to in…nity as type approaches = 0 from the left , in some neighbourhood of zero all consumers
have less w than the lowest type does.
Moreover, when " is only marginally greater than zero, the consumer has only marginally more
income than the lowest type who has an undistorted amount v(0) and w(0). The consumer of
type = " would have an undistorted level of v that is marginally higher than v(0), and w
marginally higher than w(0), which would give him marginally higher consumption utility.
But since this consumer has distorted consumption, his consumption utility is lower than what
it otherwise would have been. This di¤erence is increasing in the size of the distortion of v. dd v
is increasing without bound in a neighbourhood of 0, so this consumer has a level v that is more
than marginally higher than v(0). Since his income is only marginally higher than the lowest
type but his consumption distortion is of a higher order, he must have lower consumption utility.
QED
This result does not depend on the fact that the lowest consumer is of type = 0 rather than
some strictly positive min . The key point in the proof is that the slope of v( ) tends to in…nity
for low , which occurs because the slope of v( ) tends to zero. The fact that the lowest value
of actually equals zero plays no role.
40
Chapter 4: Formal Analysis of the General Model
I now return to the general model in which both visible consumption v and social ties are
choice variables. A consumer’s strategy is therefore (v; ) , where is either S or W . I use the
existence results of Chapter 3 to show in section 4.1 and 4.2 under what conditions an equilibrium
exists in which consumers separate in terms of visible consumption. I already alluded to the
main results in Chapter 2: a separating equilibrium exists with all consumers choosing weak ties
if the income di¤erence between weak and strong ties is su¢ ciently large. In section 4.3, I bring
together the results pertaining to the di¤erent equilibrium characteristics.
I analysis separately the case where social ties are unobservable and where they are observable.
The results di¤er in these two cases, and out-of-equilibrium beliefs play a larger role when
is observable. The reason for this di¤erence is as follows. Visible consumption is generally a
continous function of type in equilibrium. Say everyone has the same choice of ties in equilibrium.
Any consumer searching for a pro…table deviation tends to deviate both in his choice of and
in his choice of v. This new value of v may well be the visible consumption that another type
is unobservable, this deviant must be inferred to be of type 0
0 chose in equilibrium. When
because he appears to be playing that consumer’s equilibrium strategy. When is observable,
others see that the deviant is playing an out-of-equilibrium strategy and inference follows from
the relevant out-of-equilibrium belief.
I brie‡y comment on whether observable or unobservable is a more reasonable assumption.
The situation is not black or white, and the type of social relationship a person has may be more
or less apparent to others depending on the context. There are nonetheless two reasons why
unobservable may be more reasonable in the context of this thesis’discussion, which would in
turn suggest that the related results are more relevant. First, a person’s friends seem more likely
to notice whether this person has mostly strong or weak ties with others. An acquaintance may
not have a close enough relationship or may not spend enough time with the person to make
up his mind one way or the other, suggesting that may be unobservable to people connected
through weak ties. But it is the inference of people with weak ties that plays the most important
role in determining status and distorting consumption, suggesting unobservable is appropriate.
Second, people may not have a clear understanding about the relationship between social ties
and income even though they themselves optimize "as if" they did. They may therefore not take
social ties into account when forming beliefs about others.
41
4.1. Equilibrium existence when social ties are unobservable
I …rst analyse the situation where there is no income gain to choosing weak ties: W = S .
Doing so gives more insight into the existence results for the general case, W > S , which
I consider immediately after. If choosing weak ties does not give any income gain, then the
situation where all consumers chooses weak ties is not an equilibrium. Deviating to strong ties
leaves the consumer less dependent on inference for status, and he can pro…tably deviate by
choosing strong ties and a lower value of v. This result underlines the role that the di¤erence
between W and S plays in generating a weak ties equilibrium in the general case.
The situation where all consumers choose strong ties is also not an equilibrium, even though
consumers cannot gain any income by deviating. This result is at …rst surprising, since under
strong ties everyone can choose their undistorted, utility maximizing bundle and are not tempted
by higher income from weak ties. The equilibrium breaks down because a consumer can actually
take advantage of the fact that choosing weak ties forces others to rely on inference to determine
his status. By choosing W , he decreases the quality of information about his personal characteristic and can pro…tably deviate to a slightly higher value of v to fool others into granting
him more status.
Proposition 5: Let be unobservable and let W = S . Then there is no equilibrium in which
all consumers choose strong ties, nor where all choose weak ties.
Proof : Say everybody chooses strong ties. Since ( S ; 0) = 0, each consumer has his undistorted
consumption level v( ), and thus strategy (v( ); S ). In any interesting situation, at least some
consumers must have an interior solution with a strictly positive level of both v and w. I argue
that any one of these consumers can pro…tably deviate to (v( ) + "; W ) for some " > 0 and
su¢ ciently small. I break the deviation up into two steps: …rst from (v( ); S ) to (v( ); W ),
and then from (v( ); W ) to (v( ) + "; W )
The following FOC must hold when evaluated in equilibrium at v = v( ):
f1 (v; y0 +
pv) = pf2 (v; y0 +
pv),
i.e. the marginal bene…t of increasing v( ) is equal to the marginal cost of doing so. By deviating
to (v( ); W ), the consumer’s utility remains unchanged. He consumes the same amount of both
42
consumption goods, and since is unobservable he appears to still be playing his equilibrium
strategy. His status now depends on inference, but he is inferred to be his true type.
He does however have a di¤erent utility function, since (1
v is unchanged, his FOC is no longer satis…ed:
f1 (v; y0 +
pv) + (1
0
)g ( ) ddv > pf2 (v; y0 +
)g( (v)) replaces (1
)g( ). Since
pv)
Marginal bene…t now exceeds marginal costs, so it is pro…table to go in a second step from
(v( ); W ) to (v( ) + "; W ) for " positive and small. Hence the overall deviation from (v( ); S )
to (v( ) + "; W ) is pro…table.
Now consider a possible equilibrium where all consumers choose weak ties. Then visible consumption is determined as in Proposition 1, with generic strategy (v( ); W ). An argument
analogous to the one above implies that deviating to (v( ) "; S ) is pro…table since it increases
0
consumption utility more than it decreases status utility. In this case, the term (1
)g ( ) ddv
0
in the FOC is replaced by the smaller term (1
) ( S ; 1)g ( ) ddv where ( S ; 1) < ( W ). A
deviation to strong ties and a slightly lower level of v is therefore pro…table. QED
I did not make any use of out-of-equilibrium beliefs in ruling out either potential equilibrium.
The deviating consumer chose another person’s equilibrium level of v and hence was believed to
be that type.
A weak ties equilibrium may exist when there is a su¢ cient income gain to choosing weak ties.
It is more likely to exist when the information externality is large, ( S ; 1) close to 1. In that
case deviating to strong ties is less attractive because a deviant would still have to signal to
obtain status.
Proposition 6: Let be unobservable and let W > S . Given S , a separating equilibrium
exists if and only if W is above a certain level, and this level is decreasing in ( S ; 1). The
equilibrium relationship v( ) is de…ned by the initial condition v(0) = v0 where v0 is the amount
of visible consumption the lowest type would consume in the absence of status e¤ects that depend
on signalling, and the di¤erential relationship:
dv
1
=
d
0
g( )
pv) + f1 (v; y0 +
pf2 (v; y0 +
43
pv)
This equilibrium is unique among equilibria with a di¤erentiable inference function and in which
consumers separate fully in their visible consumption.
Proof : Choosing (v; W ) strictly dominates (v; S ) for type > 0. That consumer obtains
strictly more income by choosing W over S and thus consumes more of w, while status remains
the same. The consumer of type = 0 is always indi¤erent between choosing S or W .
.
Let (v( ); W ) be a consumer’s strategy in the candidate equilibrium. Proposition 1 showed that
this consumer’s optimal choice of v when all consumers have the same value of is v( ). Hence
deviating to (v 0 ; W ) for any v 0 6= v( 0 ) is not pro…table.
Any other deviation must involve S . I now look at the deviation involving S that this consumer
would have the greatest incentive to take. By choosing S instead of W , the consumer’s income is
reduced to y0 + S . In deciding how to split his income between v and w, he takes beliefs as given.
S
If ( S ; 1) = ( W ) = 1, this problem would be equivalent to that of consumer 1 = W
0 when
everybody chooses weak ties. Hence, if consumer ’s utility function had remained unchanged
by choosing S instead of W , his best deviation would have been to (v( 1 ); S ). Since utility
is increasing as a function of type in the candidate equilibrium where everybody chooses weak
ties, consumer is strictly worse o¤ with (v( 1 ); S ) than with (v( ); W ). Hence, this deviation
would not be pro…table, and the magnitude of the utility loss involved would be increasing in
W
.
S
But choosing S instead of W does change his utility function; it reduces the marginal bene…t
to signalling by reducing from ( W ) to ( S ; 1) . Since the marginal rate of substitution no
longer equals the relative price, he can do better than (v( 1 ); S ) by deviating to a slightly lower
level of v. Let " > 0 be the optimal amount of this reduction, de…ned implicitly by:
f1 (v( 1 )
"; y0 + S
0
"; y0 + S
p(v( 1 )
p(v( 1 )
0
"))) + (1
) (
S ; 1)g
0
( (v( 1 )
")) ddv = pf2 (v( 1 )
"))
The gain from choosing (v( 1 ) "; S ) over (v( 1 ); S ) is decreasing in ( S ; 1). If ( S ; 1) = 1,
then there is no gain. The closer ( S ; 1) is to zero, the greater the di¤erence between the lefthand-side and the right-hand-side of the above equation when evaluated at " = 0. Accordingly,
the value of " needed to generate the above equality is greater in magnitude.
44
Furthermore, a lower value of ( S ; 1) implies that there is less status utility lost for any given
reduction of v. Finally, when ( S ; 1) = 0 the gain is greatest. The consumer chooses type 1 ’s
none-distorted consumption bundle, which is superior to the bundle type 1 would choose if all
status depended on signalling.
To summarize, consumer has the highest incentive to deviate to (v( 1 ) "; S ): Such a deviation
can be divided up into two steps: First from (v( ); W ) to (v( 1 ); S ) which reduces utility, with
the size of the reduction increasing in W given the value of S . Second from (v( 1 ); S ) to
(v( 1 ) "; S ) which increases utility, with the size of the utility gain decreasing in ( S ; 1). If
W is large enough compared to S , then the most attractive deviation is not pro…table for any
consumer. If ( S ; 1) small, then deviations are more pro…table and W must be larger for an
equilibrium to exist.
I now rule out other separating equilibria. If one did exist, it would involve at least one consumer
choosing strong ties. But since this strategy is strictly dominated for any consumer of type > 0,
this situation cannot be an equilibrium. Because the lowest type is indi¤erent between S or
W , a second equilibrium does exist that is identical to the …rst except that the lowest type
chooses strong ties. Since the lowest type does not signal, this equilibria is characterised by the
same consumption decisions, equilibrium utilities and beliefs. For all intents and purposes the
equilibrium is unique. QED
This equilibrium is supported by out-of-equilibrium beliefs that satisfy the D1 Criterion. The
equilibrium of Proposition 1 was supported by beliefs (v) = 0 for v < v(0), which satis…ed D1.
This equilibrium is supported by these very same beliefs. There are new possible deviations to
strong ties but is unobservable so beliefs cannot depend on .
Strong ties is a dominated strategy that no type could play in equilibrium. That does not mean
that no consumer would ever choose strong ties. As I showed above, consumers may indeed have
a pro…table deviation to strong ties, even though they would never choose them in equilibrium.
4.2. Equilibrium existence when social ties are observable
I now turn to the case where is observable. As stated at the beginning of this chapter, there
are now more possible out-of-equilibrium beliefs. Any consumer deviating to the equilibrium
45
v of another consumer 0 , but to a di¤erent value of than him, is now recognized to be playing
an out-of-equilibrium strategy. In contrast to the situation in which is unobservable, such a
deviant no longer has to be inferred to be of type 0 .
As in section 4.1, I …rst set W = S . If I imposed the condition that out-of-equilibrium beliefs
could not depend on , then I would get the same results as in the previous section: there is
no equilibrium in which all consumers choose strong ties, nor is there an equilibrium in which
all consumers choose weak ties. Both of these equilibria may however exist if I do not impose
any condition on out-of-equilibrium beliefs. This results foreshadows the general case where
W > S , and di¤erent equilibria may exist with di¤erent out-of-equilibrium beliefs.
Proposition 7: Let be observable and let W = S . Then there exists an equilibrium in
which all consumers chooses strong ties and separate with their undistorted level of v. There
also exists an equilibrium in which all consumers choose weak ties and separate in terms of v as
in Proposition 1, if g( ) g(0) is large enough for all . This level is decreasing in ( S ; 1) .
Proof : Say all types have strong ties, and v( ) is the undistorted consumption level in this
candidate equilibrium. Let out-of-equilibrium beliefs be (v( ); W ) < and (v; ) = 0 for
v < v(0). Consider a consumer of type with strategy (v( ); S ). Deviating just in terms of v
is not pro…table because he would lose consumption utility.
He also would not want to deviate to (v; W ) for v < v(0), because he would not do so under
strong ties and …nds v more attractive under weak ties. So either v > v( max ) or v = v( 1 ) for
some type 1: If v > v( max ) then the deviation is not pro…table for similar reasons to those
given in Proposition 1: it is most attractive for the highest type, but even he would lose utility
by distorting his visible consumption up.
Otherwise v = v( 1 ) for some type 1: Since (v( 1 ); W ) < 1 , the deviation gives lower
inference than deviating to (v( 1 ); S ) would. Furthermore, since ( W ) > ( S ; 0), inference
now plays a role in his utility function. Hence such a deviation is less attractive than the
deviation to (v( 1 ); S ) which itself is unpro…table. So everybody choosing strong ties is an
equilibrium.
I now show that a weak ties equilibrium can also exist. Let (v( ); W ) denote a consumer’s
strategy in this candidate equilibrium. Let out-of-equilbrium beliefs be (v; S ) = 0 for all v,
and (v; W ) = 0.
46
Any equilibrium where all consumers choose weak ties must be characterised by the same function
v( ) as in Proposition 1. Otherwise, consumers would pro…tably deviate to another level of v:
All consumers have lower consumption utility than they would have from making an undistorted
consumption decision. If there was no information externality, ( S ; 0) = 0 and they could
deviate to strong ties and their undistorted level of consumption, not su¤er any status utility
loss, and increase their consumption utility. So there would be a pro…table deviation, regardless
of how out-of-equilibrium beliefs (v; S ) were speci…ed. The deviating consumer would not care
about such beliefs since by choosing strong ties his utility would not depend on inference.
But because of the information externality, ( S ; 1) > 0 and any consumer deviating to strong
ties will be a¤ected by the inference associated with his deviation. Since (v; S ) = 0 for any v,
the consumer’s most pro…table deviation is to his undistorted consumption level. In doing so,
he loses status utility of ( S ; 1)(1
)(g( ) g(0)) and gains consumption utility. The loss of
status utility is increasing in (g( ) g(0)) and in ( S ; 1). QED
I now consider the general situation where is observable and choosing weak ties provides an
income gain. Compared to the previous proposition, choosing weak ties become more attractive.
I …rst look at a particular system of out-of-equilibrium that makes the analysis similar to when
is unobservable. The equilibrium in which all consumers choose strong ties no longer exists
while the situation in which everyone chooses weak ties will be an equilibrium as long as the
income gain from choosing weak ties is large enough.
Proposition 8: Let be observable, W > S and let out-of-equilibrium beliefs be (v( ); S ) =
. Then given S , an equilibrium exists in which all consumers choose weak ties and separate
in their visible consumption if W is above a certain level. This equilibrium is identical to that
of Proposition 6 where was unobservable, and exists if and only if that equilibrium exists.
Proof: The only di¤erence between the assumptions in Proposition 8 and Proposition 6 is that
now is observable. If out-of-equilibrium beliefs are (v( ); S ) = , then any consumer who
deviates to strong ties and another level of v is inferred to be the type who had that level of v in
equilibrium. The belief about choosing (v( ); S ) is exactly the same as (v( ); W ). Even though
is observable, only visible consumption determines beliefs about a deviating consumer. Hence
47
any deviant is judged in the exact same way as if
therefore exists under the same conditions. QED
was unobservable. The same equilibrium
This same equilibrium is more likely to exist if out-of-equilibrium beliefs are more negative, in
the sense that consumers deviating to strong ties are believed to be lower types. The next result
therefore follows:
Proposition 9: Let be observable, W > S and let out-of-equilibrium beliefs be (v( ); S ) <
. Then given S , an equilibrium exists in which all consumers choose weak ties and separate
in their visible consumption if W is above a certain level. This equilibrium is identical to that
of Proposition 6 where was unobservable, and exists if that equilibrium exists.
An equilibrium in which consumers choose strong ties will certainly not exist if (v( ); W )
.
Even if there was no income gain associated with weak ties, consumers would be able to pro…tably
deviate to weak ties and obtain more status. A strong ties equilibrium may however still exist
with other out-of-equilibrium beliefs.
Proposition 10: Let be observable, and W > S and let out-of-equilibrium beliefs be
(v( ); W ) = 0 for all . Then given S , an equilibrium exists in which all consumers choose
strong ties and their undistorted level of visible consumption if W is below a certain level.
Proof: Any pro…table deviation must be to weak ties. The lowest type consumer never has
a strict incentive to deviate. Any other consumer of type will lose status utility g( ) g(0)
because observers believe him to be the lowest type. However he will gain consumption utility,
and his most attractive consumption bundle is what type ( W
) had under strong ties. The
S
W
larger W is, the higher this type ( S ). Since consumption utility is increasing as a function
of type in the candidate equilibrium, his optimal deviation becomes more pro…table for a high
level of W . QED
I argued earlier that this strong ties equilibrium is less compelling than the weak ties equilibrium
with (v( ); S ) = because the out-of-equilibrium beliefs seem less intuitive. On the other
hand, formally speaking it looks like either one of these equilibria can satisfy the Intuitive
Criterion depending on the functional form and parameter assumptions.
Any consumer in the weak ties equilibrium with (v( ); W ) would lose consumption utility by
deviating to (v( ); W ) because he could have less of the non-visible good. This deviation might
48
however be pro…table if he were inferred to be a higher type by doing so. Furthermore, the
lowest type in the strong ties equilibrium may have an incentive to deviate to (v; W ) for any v,
if he is inferred to be a high type and status is important enough. It seems to be possible that
either equilibrium could exist and either be intuitive or not depending on the assumptions.
Similarly, it is not clear which consumer has the greatest incentive to make any given deviation
from these equilibria. The D1 Criterion tends to impose a unique permissible out-of-equilibrium
belief for any observed deviation. The restriction on out-of-equilibrium beliefs supporting the
weak ties equilibrium, (v( ); S ) < for all v; therefore seems more likely to satisfy the D1
Criterion than the beliefs supporting the strong ties equilibrium, (v( ); W ) = 0 for all v, since
the latter is much more speci…c. However the strong ties equilibrium could also exist under
less extreme beliefs, such as (v( ); W ) being small but not necessarily zero. In any case, it is
unclear whether the D1 Criterion would formally eliminate either one of the equilibria.
4.3 Characteristics of the equilibria
If the strong ties equilibrium exists, each consumer has income according to y0 +
his undistorted level of v and w.
S
and chooses
Consumers have higher income in the weak ties equilibrium, following from y0 + W . Furthermore, I showed that the function v( ) in this equilibrium is the same as in Proposition 1.
The lowest type chooses his undistorted level of consumption, while other consumers’ visible
consumption follows from the di¤erential relation:
dv
1
=
d
0
g( )
pv) f1 (v; y0 +
pf2 (v; y0 +
pv)
Equivalently, the following di¤erential relation holds:
d
=
dv
1
pf2 (v; y0 +
pv) f1 (v; y0 +
g0 ( )
pv)
I now prove that given S , W can be high enough so that the weak ties equilibrium exists but
still low enough that some consumers have lower utility than if everyone had chosen strong ties.
49
The proof of the following proposition highlights the key role played by the the information
@
externality, @R
( S ; R) > 0.
Proposition 11: If ( S ; 1) = 0, then no consumer can have lower total utility in the weak ties
equilibrium than he would have if everybody had strong ties and their optimal level of v. When
( S ; 1) > 0, this situation can indeed occur.
Proof: Say ( S ; 1) equals zero so there is no information externality. Assume that some
consumer was worse o¤ in the weak ties equilibrium than with choosing his optimal v while
everyone had strong ties. This consumer could deviate to strong ties, his status utility would be
independent of signalling and he could choose his undistorted level of v. He would not lose status
utility but would gain consumption utility. Since this deviation is pro…table, the equilibrium
breaks down.
On the other hand, say ( S ; 1) > 0 and let W be su¢ ciently larger than S so that the consumer
with the most attractive deviation is indi¤erent between it and his equilibrium allocation. From
the proof of Proposition 6, this deviation is less pro…table than the consumer‘s best deviation
if ( S ; 1) were smaller, including if ( S ; 1) = 0. The consumer‘s best deviation if ( S ; 1) in
fact equals zero is his undistorted level of v. He would obtain exactly the same utility as if he
had chosen an optimal v and everybody else had chosen strong ties. But since ( S ; 1) > 0,
his best deviation gives him lower utility than if everyone had chosen strong ties. Since he is
indi¤erent between this best deviation and his equilibrium allocation, he is necessarily worse of
in equilibrium than if everyone had chosen strong ties and an undistorted level of v. QED
Since the weak ties equilibrium is identical to that described in Proposition 1, all the characteristic from section 3:2 also apply. Therefore, the weak ties equilibrium has the following
features:
Some consumers may have lower utility than if everyone had chosen strong ties.
Total utility is an increasing function of consumer type.
Consumption utility is generally not a monotonically increasing function of consumer type.
If the lowest type consumes a strictly positive level of w, then consumption utility is
decreasing for consumers of a low type.
50
The consumption of everyone except for the lowest type is distorted, with everyone’s level
of the visible good v strictly higher than if they could transmit information about their
characteristics through weak ties without signalling.
All consumers except for the lowest type have lower total utility than if they could transmit
information about their characteristics through weak ties without signalling.
The example of section 2.1 graphically illustrates all of these equilibrium features. These results
also re‡ect the ideas I put forward in the introduction. Individuals tend to choose weak social
ties, and their income is higher as a result. At the same time, consumption is distorted by their
desire for status. They may enjoy a welfare gain if their income gain outweighs the distortion
from conspicuous consumption, and vice-versa. Welfare level is lower than it would have been
if weak ties could transmit information about status without signalling. To the extent that the
modelling assumptions are appropriate for a large, dynamic social environment but not for a
small and static one, these results suggest that conspicuous consumption and income will be
higher in the former setting.
51
Chapter 5: Adding Intrinsic Utility to Strong Ties
One aspect of the model driving the results so far is the assumption that consumers derive
utility only from consumption and status. Choosing strong or weak social ties does not directly
a¤ect a consumer’s utility, but rather works indirectly by changing disposable income and the
quality of information. With unobservable, any consumer choosing strong ties in a potential
equilibrium could deviate to weak ties and increase both visible and non-visible consumption.
By doing so, he would increase not only his consumption utility but also his status utility. By
consuming more v, it would appear as though he is playing the equilibrium strategy of a higher
type consumer. Observers would therefore infer that he is this higher type.
It is realistic that people also experience intrinsic utility through their choice of ties. People may
certainly value the social relationships they have with others for their own sake, rather than for
any instrumental purpose they may serve. In particular, people may place a lot of importance
on the relationships they have with family members and close friends - their strong ties.
Leaving this idea out of the model makes strong ties less attractive. The results from the previous
sections may then overestimate the extent to which a weak ties equilibrium exists, and underestimate the extent to which a strong tie equilibrium exist. I now want to address the following
question: with unobservable but with intrinsic utility to strong ties, will some consumers
choose strong ties in equilibrium?
Formally, I make the following changes to the model:
is unobservable, there is an intrinsic
utility of K > 0 to choosing strong ties, and to simplify the analysis there is no information
externality: ( S ; R)
( S ) is independent of R and equal to zero. The strategy space is the
same as in the general model, (v; ), and income is related to social ties in same way: y0 + .
Consumers choosing weak ties have utility:
U (v;
W)
= f (v; w) + (1
)g( (v))
while those with strong ties have utility:
U (v;
S)
= f (v; w) + (1
52
)g( ) + K
where K > 0:
If K = 0 then the situation reduces to the general model without the information externality.
There will be no equilibrium in which any consumer chooses strong ties and there is a separating
equilibrium in which all consumers choose weak ties if W is large enough compared to S .
The greater K is, the larger W must be compared to S for a weak ties equilibrium to exist since
deviations to strong ties are more attractive. If K is large enough there will be an equilibrium
in which all consumers choose strong ties. Consumers could deviate to weak ties and increase
their status and consumption utility but would not do so because the loss in intrinsic utility K
is so great.
I am also interested in whether the intermediate case, where some consumers have strong ties
and others weak ties, can be an equilibrium. The intrinsic utility to strong ties is equal for
everybody, but the income gain to choosing weak ties is larger for higher type consumers. I
therefore investigate if there is an equilibrium in which low type consumers choose strong ties,
and high type consumers choose weak ties. In the remainder of this chapter I use v( ) to refer to
consumer ’s optimal choice of v when choosing strong ties, which is his undistorted consumption
level. I use v( )0 to refer to this consumer’s optimal choice of v when choosing weak ties.
I …rst consider the situation where there is no signalling as a baseline case. I therefore set
( W ) = 0 for the time being. Every consumer simply maximizes consumption utility and the
intrinsic utility to social ties, and signalling plays no role. I …rst show under what condition
high type consumers have a stronger incentive to chose weak ties than low type consumers.
This case is not the only one that could be of interest. Under di¤erent functional form assumptions it may be low type consumers who have the most incentive to chose weak ties, or the
relationship may not be monotonic. I chose to look at this speci…c case in order to work with
a concrete example, and because of the intuitive notion that higher type people can better take
advantage of new information coming through their social contacts. Indeed, the model is formulated so that higher types always obtain a higher income gain from choosing weak ties. That
does not necessarily mean that they also have the largest utility gain from doing so, because
they also have higher income under strong ties. I therefore need to assume something about the
concavity of the utility function to imply that weak ties are more attractive to high types than
to low types.
53
Lemma 6: Let ( W ) = 0, let (v( ); S ) be consumer ’s optimal strategy if he chooses strong
ties, and let (v( )0 ; W ) be his optimal strategy if he chooses weak ties. Denote the utility he obtains from these strategies by U ((v( ); S )) and U (v( )0 ; W ) respectively. Then U (v( )0 ; W )
);w( ))
W
for
U ((v( ); S )) is increasing with if the following condition holds: f2 (v(
W
W
S
f2 (v(
all
2 [0;
S
);w(
S
))
max ].
Proof: I only have to look at consumption utility, since the intrinsic utility to social ties is the
same for every consumer. A consumer of type who chooses weak ties has the same income as
a consumer of type W who chooses strong ties. Therefore his optimal choice of v under weak
S
ties, v 0 ( ), equals v( W
): If W
> max then interpret v( W
) as simply the optimal choice
S
S
S
of v for such a consumer under strong ties if he did exist. It is therefore su¢ cient to show that
f (v( W
); w( W
)) f (v( ); w( )) is weakly increasing in .
S
S
Say for any 1 and 2 with 1
2 = " > 0 and small, the di¤erence in consumption utility beW
tween these consumers is greater under weak ties than under strong ties: f (v( W
1 ); w( S 1 ))
S
W
f (v( W
f (v( 1 ); w( 1 )) f (v( 2 ); w( 2 )):
2 ); w( S 2 ))
S
Up to a constant , the RHS of the above expression is the extra equilibrium utility from being a
type that is " higher than 2 . Being such a type gives S " more income than consumer 2 which
can be spent on v or w. Since " is small, how this extra income is spent makes little di¤erence
to how much extra utility it generates. I can therefore assume it is spent on w and approximate
the extra utility of type 1 by S "f2 (v( 2 ); w( 2 )). Similarly, the LHS of the above expression
W
can be approximated by W "f2 (v( W
2 ); w( S 2 )).
S
W
I therefore can approximate the inequality by W "f2 (v( W
2 ); w( S 2 ))
S "f2 (v( 2 ); w( 2 )),
S
where the approximation becomes exact in the limit as " goes to zero. But by the assumed
f2 (v( 2 );w( 2 ))
W
concavity condition,
, this result holds regardless of the value of 2 .
W
W
S
f2 (v(
S
2 );w(
S
2 ))
The utility gain in choosing weak ties is increasing marginally in type as required. QED
I call the inequality condition I used in the lemma,
f2 (v( );w( ))
f2 (v( W );w( W
S
S
W
))
S
, the concavity
condition. I do so because it constrains how rapidly marginal utility can decrease as consumption
increases. Since W > S , the left-hand-side is greater than one and is related to the income
gain from choosing weak ties. The right-hand-side is also greater than one because it is the ratio
of marginal consumption utility at a lower income to that at a higher income. The inequality
condition therefore implies that the consumption utility function is not too concave.
54
Still assuming there is no signalling, I can now demonstrate for what values of K an equilibrium
exists in which some low type consumers choose strong ties and other high type consumers
choose weak ties.
Proposition 12: Let
(
W)
= 0, and let
f2 (v( );w( ))
f2 (v( W );w( W
S
S
W
))
S
hold for all
2 [0;
max ].
De…ne K0 as equal to U (v( max )0 ; W ) U ((v( max ); S )), the extra utility the highest type
consumer can gain by choosing weak ties. Then for any K < K0 there exists a critical consumer
of type
such that the following equilibrium exists: all consumers of type
<
choose
0
W
(v( ); S ), and all consumers of type
choose (v( ) ; W ) = (v( S ); W ).
Proof: A consumer will choose weak ties if and only if his gain in consumption utility in choosing
v( )0 is larger than intrinsic utility K. The lowest type consumer cannot gain any consumption
utility by choosing weak ties because he would not gain any income. By the concavity condition,
the gain in consumption utility from choosing weak ties is increasing in consumer type. The
highest type consumer chooses weak ties since by assumption his gain from doing so, K0 , is
larger than K. The consumption utility gain from choosing weak ties is continuous as a function
of consumer type, so there must be an intermediate consumer
who is indi¤erent between
strong and weak ties. QED
The function relating type to visible consumption in this equilibrium is discontinuous at = .
The limit on the LHS equals v( ), whereas the limit on the RHS equals v( W
). This disS
continuity in visible consumption is a result of the discontinuity in income that occurs because
consumer
chooses weak ties. Another consequence is that consumption utility is also discontinuous at .
Without signalling, an equilibrium can exist in which all consumers below
choose strong ties
an all those above
choose weak ties. I now show that when there is signalling, ( W ) = 1,
no such equilibrium can exist if I make a mild assumption on out-of-equilibrium beliefs. The
key point is that the critical consumer must be indi¤erent between strong and weak ties in any
candidate equilibrium, which implies that a slightly higher type consumer has lower consumption
utility than him. This consumer has an incentive to deviate to strong ties.
Proposition 13: Let (
W)
= 1 and let the concavity condition
f2 (v( );w( ))
f2 (v( W );w( W
S
S
W
))
S
hold
for all 2 [0; max ]. Let
denote the consumer who would have been the critical type for some
value of K if there were no signalling, as in Proposition 12. Then if beliefs about deviations are
55
increasing with v, there is no equilibrium in which all types
types
choose weak ties.
<
choose strong ties and all
Proof: Say such an equilibrium did exist. Since ( S ) = 0 , any consumer of type
<
would have to make an undistorted consumption decision and his strategy will be (v( ); S ) in
the candidate equilibrium.
Denote the strategy of a consumer of type
by ( W ; v 0 ( )), where v 0 is some function
relating consumer type to visible consumption over the range [ ; max ]. By the same argument
as in section 3.1, the following di¤erential relationship would have to hold to prevent any of
these consumers from deviating to another value of v:
dv 0
1
=
d
for all
2[
;
g0 ( )
pf2 (v 0 ( ); w0 ( )) f1 (v 0 ( ); w0 ( ))
max ].
The only think I have left to specify is the initial condition v 0 ( ), the visible consumption of
the critical type. I will show that this consumer must have an undistorted level of v and as a
result the candidate equilibrium breaks down.
Assume for now the intrinsic utility of strong ties is K, the same value that made consumer
the critical type in Proposition 12. So the critical consumer is indi¤erent between his
undistorted consumption level v( ) under strong ties and his undistorted consumption level
v 0 ( ) = v( W
) with the higher income under weak ties. The critical type chooses weak ties
S
in this candidate equilibrium so the initial condition is simply v 0 ( ) = v( W
). If it were any
S
higher than v( W
)
he
would
deviate
to
strong
ties,
and
if
it
were
any
lower
than
v( W
) he
S
S
W
would deviate to v( S ).
Since the critical consumer’s visible consumption is undistorted, dd v 0 is in…nite when evaluated
at . Then by Proposition 4, a consumer of a slightly higher type has lower consumption
utility than the critical consumer: f (v 0 ( ); w0 (v)) < f (v 0 ( ); w0 ( )). I show that this slightly
higher type has an incentive to deviate to strong ties.
Deviating to strong ties leaves status utility unchanged since ( S ) = 0. Consumer ’s best
deviation would be to his undistorted level v( ), while consumer ’s best derivation would be
56
to v ( ). Consumption utility is increasing over consumer type in the absence of signalling so
f (v( ); w(v)) > f (v( ); w( )).
Therefore f (v( ); w(v)) f (v 0 ( ); w0 (v)) > f (v( ); w( )) f (v 0 ( ); w0 ( )) and consumer
has a higher incentive to deviate than the critical consumer. The critical consumer was indi¤erent
so the slightly higher type has a pro…table deviation.
If the intrinsic utility of strong ties was greater than K, the equilibrium would also break down.
The critical type would get more total utility from choosing strong ties and his undistorted
consumption bundle than by choosing weak ties and his undistorted consumption bundle.
If the intrinsic utility of strong ties was less than K, an equilibrium might exist. Then the
critical type could be made indi¤erent between strong and weak ties by a distorted level of
visible consumption v 0 ( ) > v( W
), and consumption utility would not necessarily decrease
S
in a neighbourhood of
since Proposition 4 would not apply. I now show that this case cannot
occur since out-of-equilibrium beliefs are increasing in v.
Visible consumption is an increasing and continuous function of type both below and above
where there is a vertical jump. This jump occurs because the critical consumer has discretely
more income than a marginally lower type by choosing weak ties. Beliefs follow from players’
strategies so the limiting belief about v approaching v( ) from below is , as must be the
belief associated with v 0 ( ). By assumption, (v) also equals
for any intermediate value of
0
v not chosen in equilibrium, v 2 [v( ); v ( )).
If the intrinsic utility of strong ties was less than K, then the critical type would have a distorted level v 0 ( ) > v( W
) in the candidate equilibrium. This consumer can deviate to
S
(v( W
);
)
and
increase
his
consumption utility. His status utility remains unchanged since
W
S
he is still inferred to be his actual type. Hence this deviation is pro…table and the candidate
equilibrium breaks down. QED
This chapter’s main result is a negative one: when inference about type is a monotonic function
of visible consumption, there is no separating equilibrium in which consumers below a certain
type choose strong ties and those above choose weak ties. Consumers may certainly separate in
their visible consumption while all choosing S or all choosing W , depending on the value of
K. Proposition 13 shows in contrast that they will not separate in their choice of social ties.
57
Conclusion
I developed a model in which the strength of people’s social ties in‡uences the extent of conspicuous consumption in a society where people value status. When di¤erent types of social ties
transmit speci…c types of di¤erent information, people tend to choose weak rather than strong
ties. This situation is less likely if people value strong ties for their own sake, but may still
occur. Incomes are high but visible consumption is also distorted up by people’s need to signal.
People are not necessarily better o¤ than if everyone had chosen strong ties, although they
tend to be if the income gain from choosing weak ties is large. I argued that these results help
explain why conspicuous consumption may be more prevalent in complex and dynamic social
environments. People there are more likely to be able to choose the type of social ties they
want and bene…t from di¤erent types of information passing through these ties. The model is
less applicable in a small or traditional setting, where people’s social relationships are largely
determined by custom and where both income and conspicuous consumption tend to be low.
This work represents an initial attempt to address the connection between social ties and conspicuous consumption. While the results are of interest in their own right, I consider the main
virtue of this work to be the general idea it addresses. There is an extensive sociological literature on social ties and much formal economic work on signalling, but I am not aware of other
attempts to explore the link between these two concepts. It may be interesting in future work to
look more explicitly at how di¤erent types of networks di¤er in the information they transmit,
and the possible consequences for people’s signalling behaviour.
Another area for future work is to address how social norms are related to signalling. In this thesis, people recognize that high levels of visible consumption signal an unobservable characteristic
and that this characteristic is a positive one. One way of looking at social norms is that they
determine what can serve as a message in the signalling game and how this message is commonly
interpreted. However, very di¤erent norms can prevail in di¤erent circumstances. In contrast
to the model of this thesis, refraining from luxurious visible consumption may sometimes be
interpreted in a positive light as a sign of morals and good character. In other circumstances,
consumption may not carry any social connotation at all. The work on social norms in evolutionary game theory could be another path to explore why di¤erent types of signalling may exist
in di¤erent environments.
58
Appendix
For a generic function Y = h(X):
Y
Y
dY
dY
dX
dY
dY
dh
=
and X = h
dX
dX
= h(h 1 (y)) = h(X(Y ))
dh dX
dX
1
= 1=
)
=
dX dY
dY
dh=dX
1
dX dY
=
)
=1
dY =dX
dY dX
= h(X) )
1
(Y )
dY
The steps leading to the …nal line did not assume that the derivatives dX
and dX
dY have to be
dY
dX
dY
written explicitly as functions of only one argument, i.e. that dX = dX (X) and dX
dY = dY (Y ).
dY
Hence the relationship dX
dY dX = 1 will also hold when each derivative is written as a function of
dY
dX
dY
two arguments, dX = dX (X; Y ) and dX
dY = dY (X; Y ). Applying this result to the situation in
section 3.2 yields: [ dd ( ; )] [ ddv ( ; )] = 1 or dd = d1 .
dv
59
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