The role of social status and reference group on income
mobility
Martin Leites
Advisor: Xavi Ramos
ABSTRACT
This paper formalizes the role of reference group income as a mechanism of inequality
persistence between generations. Following Piketty (1998), this paper considers two
distinct dimensions of individual status: the self-perceived valuation of their relative
position in their reference group and the social belief about their status. Both aspects
lead to produce low mobility under certain circumstances. On one hand, reference group
income could affect economic aspirations; on the other hand, individuals from lowerclass backgrounds may be discouraged, by discriminatory beliefs, from making
adequate mobility-enhancing investments. Piketty’s work mainly deals with the second
aspect.
This paper attempts to enhance our understanding of both status motives. It develops a
model in order to distinguish the role of discrimination and reference groups in inequality
persistence, and to explore how agents react to the composition of reference groups.
Both status motives could generate multiple equilibriums in the choice of effort and tend
to amplify economic success inequalities among agents with different social origins.
Finally, beliefs about effort-rewards for agents with different social origins may play a key
role in explaining intergenerational inequality persistence. The model confirms that
“status motives” could be a mechanism of transmitting inequality across generations,
and it identifies 3 different mechanisms: i) mobility could be low because the poor are
not sufficiently motivated to move up; ii) they are being discouraged by society as a
whole; iii) they are discouraged through their peer group failed past attempts to move up.
However, under certain circumstances, reference groups could reduce economic
success inequalities. This finding is in stark contrast to previous inequality models based
upon self-fulfilling beliefs and fatalistic prediction.
Working Paper, July 2013
1
INTRODUCTION
This paper proposes a theoretical model to analyze the role of reference group and the
ex-ante inequality on economic aspirations and intergenerational mobility. Our modeling
exercise consists of describing the conditions under which individuals from lower class
origins could be discouraged from making adequate mobility-enhancing investments,
meanwhile individuals from higher class origin could be more stimulated. The origin of
that difference would be the composition and trajectory of the reference groups, the
relative effort rewards and the ex-ante inequality.
Mainstream sociologist’s theories have emphasized the role of social status,
discriminatory beliefs and related cultural attitudes in the generation of persistent
inequality between dynasties. The idea that reference groups play a crucial role in
explaining income mobility has a long history in the social sciences but less attention
had received from economic. Reference group theory suggests that culturally shaped
processes may lead to a reduction (or an increase in) ambition among those that share
them.1 People define different economic aspirations when their social origins and
reference groups are heterogeneous. For example poorer reference groups may
transmit less ambition and taste for economic success to lower-class families than
upper-class families with richer reference groups. As a result, the social origin and the
composition of the reference groups could reduce the aspirations, and they could be an
additional mechanism of persistence of inequality between generations.
According to mainstream economics, persistence of inequality across generations can
be explained by a combination of direct family transmission of productive abilities,
endowments and parental investment in children’s human capital (Becker and Tomes
1979; 1986). Bourguignon et al. (2007) suggest that socio cultural inequalities could
partly explain inequality persistence and they emphasize that further economic research,
both theoretical and empirical, in this area is needed. The seminal paper of Piketty
(1998) is a notable exception within Economic research. It introduces the role of status in
intergenerational income mobility, the author assumes that an agent cares about the
public belief about him (“status motives”). It suggests that status motives amplify the
inequality between agents with different social origins, when the impact on economic
success of social origins is high compared to the effect of effort and ability. Piketty’s
1
According Merton (1953) reference groups represent the benchmarks to which individuals compare
themselves and evaluate individual attributes. This group became in the point of reference where individuals
refer to evaluate their achievements, their performance, aspirations and ambitions. It orients the behavior
and defines the social role to which the individuals aspire, but to which they do not necessarily belong.
2
model provides a general framework, which could describe two socio cultural channels
of inequality persistence: reference group theory (Boudon, 1974) and the statistical
discrimination theory (Bourdieu and Passeron, 1964, 1974).2
However, less attention is paid to the differential role of both status motives in Piketty
(1998). Furthermore, the status motives (defined as social belief about status) used in
Piketty’s model provides a better basis for discrimination theory. But, the reference
group mechanism has not been considered in detail in this model. His utility function
does not include how people care about their relative position with respect to a referent
point. Additionally, this framework does not allow us to explore how agents react to the
composition of their reference group and the conditions under which reference groups
might decrease/ increase income mobility.
People being care a lot about their relative position with respect their reference group,
regardless of whether this has no direct effect on their opportunities. People form
economic aspirations based on their past experience and their interactions with their
reference group (Appadurai, 2004, Genicot and Ray, 2010). Both prospect theory and
standard theory suggest alternative assumptions to consider relative concern in the
utility function. However this issue has received very little attention in Theoretical
Economic literature, and much of our knowledge of the importance of interpersonal
comparison comes from empirical literature or from Sociology. In particular, very little
has been written on the effect of relative concern on intergenerational income mobility.
This paper attempts to build a bridge between intergenerational mobility and relative
concern (with respect to reference group) literature. Our primary objective is to provide a
common framework to explain the role of reference groups on income mobility, to
develop a conceptual framework to think about the incidence of ex ante inequalities and
social cohesion on effort decision. The central issue to be examined in this paper is to
discuss
the
role
of
reference
group
income
as
mechanisms
behind
the intergenerational transmission of economic inequality. Furthermore, it allows us to
explore the conditions under which richer reference groups (or more demanding
reference group) might increase income mobility or amplify inequality persistence.
Assumptions used in Piketty (1995;1998) provide the most suitable starting point to
consider the role of reference groups on income mobility in Economic models. First,
2
Both the reference group theory and the statistical discrimination theory of persistent inequality predict low
mobility, but differ in the origin. In the first case, reference groups could reduce the aspirations and explain
the persistence of inequality between generations. In the second case, lower-class agents are being
inefficiently discouraged by society as a whole which tells them they have no chance to improve and they
should adapt for a relatively poorer lifestyle.
3
these models incorporate the role of status, beliefs and social origin on intergenerational
income mobility. Second, agent’s income level is a random variable and they have
imperfect information. Finally both models assume that dynastic learning takes the form
of Bayesian updating, where agent’s beliefs are transmitted between generations.
Our approach uses some assumptions of Piketty’s (1998) model but compared to his
paper an additional argument of individual’s utility function will be included. Therefore,
the objective function considers separately two dimensions of status: “the self-perceived
valuation of his relative position in his reference group” and “the social belief about his
status”. The composition of reference groups defines reference income level, and agents
care about the gap between their income and their reference income. We model rational
agents
with
two
different
social
origins,
who
will
choose that
level
of effort which maximizes their expected utility. Furthermore, they know the relative
importance of effort and predetermined factors in the economic success.
In the first step we assume that they know the composition of their reference group and
they have perfect information about their income reference group. In second step we
assume that agents have imperfect information about effort of their reference groups and
that they base their choices on a priori beliefs about the probability of economic success
in different social origins. Beliefs are updated according to Bayes’ rule, implying that past
mobility affects the expected income of the current generation. We exploit three
implications of Bayesian updating models; distribution beliefs converge, the equilibrium
distribution of beliefs depends on the initial distribution and the connection between
generations. This framework allows us to derive long term effort equilibrium and to
examine the effect of relative concern on the dynamic of intergenerational mobility. The
individual characteristics of relative concern, the composition of reference group and the
past mobility trajectories for agents with different social origins, could easily generate
multiple equilibriums in effort level. Consequently, even when all agents could be
identical in their abilities, their effort levels differ in the long term, which affect long term
income mobility.
Our model allows us to analyze the role of reference group on effort decision and
income mobility in detail. We consider three dimensions of reference groups separately:
First, the intensity of relative concerns and its implications on income aspirations.
Second, the incidence of the composition of reference groups for agents with different
social origins. Third, the incidence of the beliefs about reference group members. In the
4
last case, we consider how the past mobility experience of the reference groups affects
beliefs about reference group effort, and how they affect on agent’s effort decision.
The model makes the distinction between the direct reference group effect and the
whole society ex-ante inequality effect on income mobility more transparent. The former
emphasized in the composition of reference groups and individual characteristics of their
members, meanwhile the second effect is explained by relative effort rewards and the ex
ante inequality between agents with different social origins. Ex-ante inequality is a key
determinant of the formation of aspirations. Because the impact of reference group on
income mobility is through the formation of aspirations, this model represents a first
bridge between reference group theory and aspiration failure approach defined by
Genicot and Ray (2010).
On the other hand, this model also provides a theoretical framework to evaluate different
hypotheses about relative income concern empirically.
Finally, our approach allows us to distinguish the incidence of two socio cultural
channels of inequality persistence: discrimination and reference group. From a policy
perspective, it is important to notice the different implications of them. According to
Piketty (2000), in statistical discrimination theory appropriate corrective policies can raise
intergenerational mobility and improve efficiency simultaneously. However, when
persistence is explained by reference group theory, policy intervention could be driven
solely by distributive justice considerations because individual do not respond to
economic incentives. In contrast, Ray (2006) argues that it is perfectly possible for an
unequal society to create local attainable incentives among the poorest individuals.
Affirmative action and public education may be policy tools that could be used to create
higher local connectedness and to affect aspiration conformation. The ability to better
understand this phenomenon will increasingly allow researchers to make public policy
recommendations based on new theoretical models and new empirical applications.
The results that emerge from our model confirm that the reference group, through
relative income effect and aspiration conformation, tends to amplify (or reduce) the
inequality of economic success between agents with different social origins. Results
suggest that the magnitude and direction of this effect depend on the 4 key issues,
which could explain the relation between inequality, economic aspiration and income
mobility: (a) the composition of reference group is relevant regardless of its inheritance
pattern, and more connected society promotes high income mobility; (b) assumptions
about the functional form of relative concern are keys issues to answer to regarding the
5
effect of reference groups on income mobility; standard assumption or Prospect theory
explain situation in which the income mobility would be very different; (c) relative effort
reward with respect to reference group income and the way in which beliefs are
connected between generations are shown to affect economic aspirations and
intergenerational mobility; (d) ex - ante inequality and expected relative deprivation are
keys factor in explaining effort decision and income mobility
The model identifies some situations which could explain different stylized facts about
the relationship between reference groups, aspiration and income mobility. First, a
poorer reference group could reduce an agent’s economic aspirations and could lead to
low effort levels. In this case, agents with low social origin do not include agents with
high social origin, which leads to aspiration failure type I (Ray, 2006). Second, under
certain circumstances, previous inequality and relative concern could lead to low
aspiration. In this case agents with low social origins include individuals from richer
origin in their reference group but the relative costs of effort is too high, and the relative
reward too low. As a result they reduce their aspirations and effort level
in order to avoid frustration, which Ray (2006) named aspiration failure type II. Finally,
under certain conditions reference groups could reduce inequality of economic success.
This last conclusion is in stark contrast to other models of inequality based upon selffulfilling beliefs and fatalistic prediction.
The issue of intergenerational mobility is still one of the most controversial issues, both
in political debates and in academic research by social scientists. Four pieces of
evidence lead us to think that this model has some relevance. First, empirical evidence
suggests the relevance of relative concern on human motivations (Frank, 2005).
Second, the model could help to explain why societies with more equality in income
distribution and less polarization show higher intergenerational mobility (Solon, 2002).
Third, it contributes to explain the evidence about heterogeneous aspirations and
adaptive preferences hypothesis (Festinger, 1975; Sen 1985a; 1985b; Elster, 1985;
Clark, 2009). Finally this model could help to explain situations of low mobility for certain
social groups and contribute to explain why agents with a similar familiar background
and abilities obtain different economic achievement.
The rest of this paper is organized as follows. The next section reviews the theories of
persistent inequality (2.1), summarizes the micro foundation of status concern (2.2) and
the main theoretical (2.3 and 2.4) and empirical advances in this topic from the
Economic perspective (2.5). The third section focuses on the role of income comparison
6
and its implications in terms of effort decision, and it discusses the effect of relative
concern on income mobility. The fourth section assumes imperfect information and
introduces an updating beliefs rule (section 4.1) to describe the long term effort
equilibrium and examine the effect of relative concern in the dynamic of intergenerational
mobility (4.2, 4.3 and 4.4). In the fifth section, our contributions are integrated with the
Piketty (1998) model to account a more general model. Finally we conclude.
2. LITERATURE REVIEW
This section is divided in 5 parts. First we briefly present theories of persistent inequality
(2.1), and the microfoundations used by economic to argue the agent’s relative concern
(2.2). Thirdly, we review the major challenges to include the role of the status and
reference group in economic models (2.3) and fourthly we summarize economics
models about relative income concern (2.4). Finally we review the empirical evidence on
relative concern (2.5), which confirms the importance of this issue and supports the
assumption that we will use in the following sections.
2.1 Theories of persistent inequality
The seminal theoretical contributions of Becker and Tomes (1979; 1986) represent a
significant step forward to improve our understanding on intergenerational mobility.
According to their model, it can be explained by a combination of direct family
transmission of productive abilities, endowments and family culture and parental
investment in children’s human capital. Solon (2002) adapted Becker and Tomes
theoretical model and characterizes both intergenerational mobility and cross-section
inequality. In spite of their simplicity these models are useful to illustrate several crucial
aspects of the intergenerational transmissions of income. The intergenerational income
correlation depends on various parameters, and theory does not provide a consensual
view about the magnitude of them (Solon, 1999). Recent research found that the
intergenerational earnings elasticity is considerably higher than most of the previous
estimates have suggested.3 These findings raise new questions about what mechanisms
that explain the low intergenerational mobility.
3
Until 1992 the empirical evidence showed that the correlations between fathers' and sons' incomes were
significantly positive, but quite short, indicating that family background was not a key factor to economic
success (Becker and Tomes, 1986; Behrman and Taubman,1990). These findings contrast with Solon
(1992; 2002), Zimmerman (1992), Björklund and Jäntti (1997), who suggest that previous studies have
underestimated the intergenerational earning elasticity. Furthermore, the empirical literature on sibling
correlation in earnings suggests a quite important role of family background and community origins. Solon
7
Piketty (2000) summarizes the main theoretical and empirical advances on theories of
persistent inequality, and their political implications.4 On one hand, he distinguishes
models of intergenerational mobility based upon Pareto efficient markets. In this case,
the level of intergenerational mobility is socially-optimal in the "laisser-faire” and policy
intervention is driven solely by distributive justice considerations. On the other hand, the
author reviews the main existing theories of persistent inequality based upon market
failures (such as credit constraints and local segregation) and socio-cultural
mechanisms. In these cases, income mobility could be inefficiently low.
Socio-cultural mechanisms have received very little attention in the Theoretical
Economic literature and much of our knowledge comes from Sociology. A notable
exception are Piketty (1995; 1998), where the author argues that beliefs and status
could explain low mobility. In Piketty (1998) the author proposes a theoretical model
analyzing the influence of beliefs on learning and investment decisions, i.e. the selffulling belief model. This model provides a general framework, which includes the
reference group theory and the statistical discrimination theory of persistent inequality.
Both views predict low mobility, but differ in the origin. In the first case, the reference
groups could reduce the aspirations and explain the persistence of inequality between
dynasties. For example lower-class families may transmit to their offspring less ambition
and taste for economic success upper-class families. In the second case, lower-class
agents are being inefficiently discouraged by society as whole which tells them they
have no chance to improve and they should adapt for a more reasonable lifestyle.
Bourguignon et al. (2007) suggest that socio cultural inequalities could partly explain the
inequality persistence. They suggest that if beliefs and preferences are transmitted by
one’s own family, then “empowering” and “fatalistic” beliefs may be inherited, and
contribute to the persistence of inequality. Differences in the status, expressed in
(1999) argues that these findings suggest that most of the relevant factors about family transmission are
uncorrelated with parental income.
4
Björklund et al. (2006) and Björklund et al. (2007) estimate intergenerational mobility associations in
earning and education and confirm a substantial explanatory role of both pre-birth and post-birth factors.
Nordblom and Ohlsson (2011) study the intentional channels of transfers from parents to children. They find
that wealthier parents have more probability of all types of transfers. Carneiro and Heckman (2002),
Björklund et al. (2006) and Galliopolu et al. (2008) find empirical evidence that supported that more highly
educated parents have children with higher education. Eckstein and Wolpin (1999) find that when parents
have low income and/or low educational levels, their children have higher likelihood to drop out of high
school. Piketty (2000) suggests that empirical evidence about credit constraint incidence on
intergenerational earning correlation is not conclusive. Other authors have analyzed the role played by
neighborhoods effect and local segregation (Jencks and Mayer, 1990; Benabou,1993). According to Piketty
(2000), the evidence confirms that intergenerational mobility depends on the composition of neighborhood.
However, less abundant is the evidence about how the intergenerational mobility is affected by
discrimination and self-fulfilling believes.
8
patterns of interaction, behaviors and beliefs, sustain and amplify the initial inequality.
These mechanisms are connected to growing literature on aspirations failure, which
tends to amplify the persistence of inequality in societies in which there is a substantial
amount of social polarization, or stratification (Ray, 2006; Genicot and Ray, 2010;
Heifetz and Minelli; 2006, Dalton, Ghosal and Mani; 2010).
These
latter
mechanisms
generate
extra
persistence
of
inequality,
where
intergenerational mobility would be inefficiently low. In this context, appropriate
corrective policies (or alternative wealth distribution) could raise intergenerational
mobility and output at the same time. As a result, these models break with the equality
efficiency trade – off, therefore corrective policies can raise intergenerational mobility
and improve efficiency simultaneously. Piketty (2000) claims that this possibility rests on
the empirical relevance of these mechanisms of inequality transmission. As a result, the
debate about the appropriate type of government interventions is still open
Because the origins of the persistence of inequality are central from the standpoint of
distributive justice, this topic goes beyond the trade-off between equality and efficiency.
The equality of opportunity approaches provides normative principles of distributive
justice, which have widespread application in both academic and political work
(Roemer,1998). There are multiple interpretations of equality of opportunity and it
emerges to be the source of potential conflicts between various views of the fair
distribution. As Fleurbaey (2011) points out, even though individuals are considered
responsible of their preferences, different criteria could penalize some preferences
profile. This discussion place more emphasis on how social origin affects preferences,
beliefs and aspirations.
2.2 Microfoundations of relative concern
Postlewaite (1998) and Frank (2005) suggest that evolutionary theory provides a strong
argument for an innate concern for relative standing. In this case, agent’s relative
concern is explained by competition for relative position in their evolutionary past. The
evolutionary explanations argue that when agents have achieved relatively high position
in the past, they will have better opportunities to reproduce. Hopkins (2008) points out
that there are at least three different evolutionary explanations. The “rivalry story” (the
success of others agents reduces own opportunity), “information story” (the experiences
and success of other agents is useful information about potentially profitable activities)
9
and “perception story” (because preferences are incomplete, relative comparison is a
fundamental psychological mechanism to evaluate goods, resources, or opportunities).
An alternative explanation would be that relative concern arises from current social
arrangements and not have to arise from social preferences and past arrangements. In
this case relative concerns do not arise because agents have competitive social
preferences, but the nature of economic competition of the institutions lead that
individuals make relative comparison. In this case, although agents only care about
themselves, relative concern is instrumental to material benefits (Hopkins;2008).
Finally, aspiration conformation may offer an alternative explanation of relative concern.
The anthropologist Appadurai (2004) suggests that aspirations are always formed in
interaction and in the thick of social life. Genicot and Ray (2010) argue that aspirations
are socially determine, they are drawn from the past own experience of the individuals
and their social environment. It that, individual goals don’t exist in social isolation, they
depend on distribution of income and wealth.
2.3 Exploring the role of status in economic Literature
Sociologists have a long standing interest in the concept of social status to study the
social interactions (Weber, 1922). However, this concept has received little attention in
economic. Postlewaite (1998) and Weiss and Fershtman (1998) discuss how economic
has introduced status. One central issue is whether status is a direct argument of the
utility function or its relevance is only instrumental.5 In the second case, status is
relevant because it indirectly affects their opportunities.6 Instrumental status could be
interpreted as status investment decision. In this case, it could be analyzed within the
traditional economic paradigm, which assumes agents optimizing with stable
preferences (Postlewaite,1998).
However, people’s ranking and the esteem of others could be an argument of the utility
function, even if people derive no clear economic benefits from them. In this case, status
has intrinsic value. Sen (1985; 2000) argues the relevance of status for wellbeing. When
status has intrinsic value, the literature distinguishes two issues. On the one hand,
individuals are concerned about the opinions and beliefs of others about them. On the
5
Postlewaite (1998) discusses the main reasons why status is omitted from the economic models and
summarizes some alternatives. He suggests that instrumental approach offers a number of advantages for
the Economic discipline.
6
People with high status could be treated favorably by other individuals in social and economic interactions.
Low status people are in the opposite situation, for example, discriminatory belief could affect their labor
opportunities (Piketty, 2000).
10
other hand, people care about their relative performance compared to others, namely,
they evaluate their economic performance by comparing with others. In the first case,
the additional argument of individual’s utility function is the social beliefs about her. In
the second case, the individuals' self-assessment of her relative position is considered
as an argument of her utility function. As a result, social status has a double dimension
(Postlewaite,1998; Weiss and Fershtman, 1998; Piketty,1998). Piketty (1998) considers
both dimensions jointly as “status motives”, and he describes how status could affect
income mobility.
Finally, social status is the relative position of individuals in a given social group. People
could have different status rankings. It will depend on whom they associate with the
reference group in which each individual is evaluated. The reference group could be
their friends, job’s colleagues or society at large. As a result, when people compare with
others or when they care about other people’s beliefs, they assign different weights to
the individuals of their reference group (Weiss and Fershtman, 1998; Clark, 2008; Van
Praag and Ferrer-i-Carbonell, 2008; Clark and Senik, 2010).
Because the person's willingness to pay for social status could be very high, this issue is
also relevant to understand important aspects of economic behavior.7 Although, it is
difficult to accurately establish the importance of status on economic performance, it
would be expected higher significance when the markets are thin. Hence, status may act
as a social reward or punishment and it could be a corrective mechanism for some
market failures such as externalities, transaction cost or monitoring problems. In this
case, social status could raise efficiency. However, some approaches emphasize the
role of status as an instrument to restrict entry and impose modes of belief. In this case,
it becomes a mean to maintain advantages of privileged groups. As a result, status
changes agent’s behavior, may affect efficiency and allocation of outcome. However, the
direction of these effects is not clear (Weiss and Fershtman, 1998). Hopkins (2008)
points out that relative concern has significant implications for the link between equality
and efficiency.
7
Theory predicts that the willingness to pay for status is increasing with wealth. Furthermore, status could
have relevant implications for saving, risk taking, occupational choice and investment in human capital
(Weiss and Fershtman, 1998). Frank (2005) suggests that when relative consumption concern differs across
domains, economic models predict a conflict between individual and collective interest.
11
2.4 Economic modeling of relative income concern
Hopkins (2008) summarizes the main models of relative income concern and
distinguished two groups of models.8 First, the author identifies a set of models which
support relative concern based on three foundations: envy, pride or compassion.
Second, others groups of studies support relative income concern based on inequality
aversion.9 Because last models do not consider relative care with respect a reference
point, this section focuses on the second groups of models. According envy effect, the
utility of an agents declines when an increase in the income of people richer than they
occur (namely
∂U (.)
∂y R
> 0 if y RG > y , where U (.) is the utility function, y RG and y are
referent group income and agent’s income respectively and y R = y − y RG ). Duesenberry
(1949) argues that poorer individuals are negatively influenced by the income of their
richer peers, while the opposite is not true ( ∂U (.) ∂y R > 0 if y RG ≥ y; ∂U (.) ∂y R = 0 if y RG < y ).
Secondly, some authors assume that an agent is better when there is an improvement in
the income of those agents below than he (“compassion effect”,
∂U (.)
∂y R
< 0 if y RG < y ) .
However, literature suggests the pride effect (also named competitiveness), which
assumes that the utility of an agent reduces with any improvement for others income
( ∂U (.) ∂y R > 0 ).10
Finally, Ray and Genicot (2010) suggest an alternative modelization of relative concern
based on the formation of aspirations with respect a reference point. They argue that
8
Mostly of the relative concern models assume that only final allocation matter, no the process to arrive it.
Including some few adaptations, Hopkins (2008) demonstrates that all these models explain the Easterlin
paradox. He concludes that the effects of inequality on utility depend on the form of relative concern. The
direction of the relationship between relative concern and inequality depend on the assumptions about
individuals’ attitude to those poorer than themselves.
9
In this case, agents dislike difference between their income and that the others. The extensive literature on
social preferences supports these assumptions. The original model of Fehr and Schmidt (1999) assumes
that agents dislike of other having more (envy) but low income for others reduces their utility (compassion).
Authors assume that agent put more weight on the first effect than on the second. Namely, when agent
comparing herself with other people income, he assigns higher weigh on richer people (or their average
income) than he than of those poorer than he. Namely, when agent comparing herself with other people
income, they weight both average income of those people richer than they and average income of those
poorer than they. However they assume that the first weight is the highest. Note that this formulation also
admits the “pride effect”, because is a model more general than mean-dependence model. Bolton and
Ockenfels (1997) provides an alternative model to explain a wide variety of experimental result and how
individuals are motivated by relative payoff.
10
There are models that combine these effects on the basis of different functional forms. Some models
assume that utility includes a relative component where agent compares its income with the average income
of other. These models is called mean-dependence model (Duesenberry,1949; Abel,1990; Boskin and
Sheshinski,1978; Clark and Oswald,1996 and 1998; Van Praag 2011). Other authors include relative
income concern based on rank (Layard,1980; Robson,1992; Clark,et al,2009a). Hopkins (2008)
demonstrates that mean dependent models are a special case of Fehr and Schmidt (1999) model, where
there is no compassion, and pride effect is as strong as envy effect.
12
aspirations depend on own historical living standard but also on the lifestyle of others. As
a result, they suggest a relationship between the formation of aspiration and distribution
of income. In this relationship is central the “aspiration window”, which defines the
individual’s cognitive world (Ray, 2006 and Mookherjee et al 2010). Finally, based on
prospect theory (Kahneman and Tversky; 2000), they assume asymmetric in
comparison with respect the reference point.
In summary there is a consensus on the asymmetry in the income comparison with
respect to reference income. Most of the studies assume
∂U (.)
∂y RG
≤ 0 . The sign of the
second derivative with respect to relative income is relevant, because it defines the utility
marginal sensitivity to relative deprivation. In general, models assume the standard
assumption of diminishing marginal utility of relative income when
∂ 2U (.)
∂2 yR
y RG < y (
< 0 ). However, there is less agreement on the sign of the second derivative
with respect to relative income, for those individuals with relative deprivation. Vendrik
and Woltjer (2007) argue that objective function could be convex or concave in relative
income, for those agents with negative relative income. On one hand, if relative income
concern is based on comparison to the social reference income, is plausibly argues in
prospect theory to exhibit convexity of utility function, reflecting diminishing marginal
sensitivity to larger deviations from the reference group income
∂ 2U (.)
∂2 yR
> 0 if y RG > y .
On the other hand, the standard assumption of diminishing marginal utility of income in
neoclassical theory suggests concavity of objective function in relative income. This
effect would imply a rising marginal sensitivity to more negative values of relative income
∂ 2U (.)
∂ 2 ∂y R
< 0 if y RG > y .11
2.5 Empirical evidence on relative concern
There is little evidence about the relationship between income mobility and relative
income concern. However, there are at least four areas of empirical research that
account about the importance of relative position. First, there is the literature on
happiness or more in general, the empirical literature on subjective wellbeing measures.
11
They argue that if an agent’s income falls increasingly short of his reference group income, it may become
more increasingly hard to raise enough resources to participate in the reference group’s social activities.
According to the author, this effect of rising marginal social participation cost of low relative income would
imply rising marginal sensitivity of relative deprivation and the concavity of objective function in relative
income.
13
This literature studies how individuals’ relative position and the environment where
individuals live, could affect their level of satisfaction or their economic aspirations.12
Second, the literature that measures the effect of relative position on “objective” output
(health or schooling)13. Third, there are valuable contributions of research in
experimental economics in terms of income comparisons and its implications in terms of
aspirations formation.14 The conclusion from this growing area of research is that there
are significant income comparison effects, both between groups and with respect to
what the individual obtained in the past.
Finally, there is an empirical literature on how social preferences and relative position
affect behavior and achievements, but it is less developed than the previous one. Piketty
(1998) suggests indirect evidence about status motives rises in social preferences for
redistribution and beliefs about origins of income inequality.15 Hoff and Pandey (2004)
carried out an experiment in Northern India. They found that lower-caste children
cognitive tasks are unrelated to their social position. They have the same results than
their high caste counterparts when their caste identity is not made explicitly. However,
they reach worse results once their caste status is known. The authors interpret this
result as children reduce their effort when they believe that they could be discriminated
12
This literature confirms the relevance of relative income for satisfaction (For a review Ferrer-i-Carbonell,
2011). In general, this literature confirms pride and envy effect, where higher relative is associated with
higher satisfaction level or happiness (Clark and Oswald, 1996, Clark et al., 2008; Clark et al., 2009b;
Luttmer, 2005; Clark and Senik, 2010, Knight et al. 2007, Ferrer-i-Carbonell, 2005). Other studies confirm
the importance of relative income on aspirations (Easterlin, 2005; Stutzer; 2004; Clark, 1999) This findings
are consistent with findings of experimental economy. A more recent small literature postulate a positively
link between own well-being and others incomes under certain conditions, as an application of Hirschman’s
tunnel effect (Clark, 2008). The situation of others provides information about the potential future earning
(Clark, 2008, Clark et al., 2009b). Other interpretation is that richer reference groups help to the provision of
local public goods (Clark et al., 2009a). Both interpretations assume an instrumental role on income
comparison. Finally, Vendrik and Woltjer (2007) found that life satisfaction turns out to be concave both in
negative and positive relative income
13
See Wilkinson and Pickett (2006) for overviews.
14
McBride (2007) finds a relationship between relative income, initial aspiration and satisfaction levels.
Others studies show that individual prefers higher absolute income in order to gain status, even though the
different available alternatives have the same purchasing power (Alpizar, et al., 2005; Johansson-Stenman
et al., 2002). Recent studies have accounted neurological evidence of relevance of interpersonal
comparisons. The main findings confirm the importance of “relative rewards” within different scenarios
(Cromwell et al., 2005; Fliessbach et al., 2007). Even Fliessbach et al. (2007) reject the role of absolute
income level, once relative income is introduced.
15
Piketty (1998) argues that, in countries like USA, where individual effort is thought to determine economic
success, individuals are more motivated to reach upward mobility through effort and ability. While More
Europeans believe that initial wealth and luck determine economic success, as a result, they are less
motivated and prefer to maintain their initial status. The considerable differences between US and Western
Europe in the size of the welfare state could be explained by these issues. Alesina and Angeleto (2005) find
evidence of different beliefs about fairness of social competition and determinants of income inequality
between USA and Europe. Naef et al. (2011) find risk propensity differences between US and Germany,
which could explain different demand for social insurance. Lindbeck and Nyberg (2006) focus on how social
insurance may change preferences for expend effort and hard work. This paper is based in a sample of 42
countries and found a negative correlation between willingness to work hard and the size of welfare state.
14
against. Similar conclusions found Afridi et al (2010) for China’s Hukou System.
Meanwhile, Fehr et al. (2011) find that willingness to punish social violations is lower
among members of low caste status than among high status caste. They argue that this
behavior could affect income mobility. Fehr and Hoff (2011) review evidence about the
influence of society on individual preferences and explore the potential consequences on
the persistence of inequality between social groups.
In sum, on the one hand, this literature confirms the relevance of relative concern. On
the other hand, there is growing evidence that support that economic behavior could be
affected by interpersonal comparisons and the esteem of others. This evidence provides
strong support for the assumptions about relative concern that we will include in the next
section.
3. THE MODEL
Based on Piketty (1998) we present the basic assumptions of our model (3.1). We
introduce some additional assumptions and describe the objective function of the agent,
and we include the relative concern and the reference group income (3.2). Then, using
the usual assumptions about relative concern we discuss its implications in term of
income mobility (3.3). Finally, we raise some assumptions about relative income concern
and assume a more flexible reaction rule (3.4), which allows us to discuss the effect of
relative concern on income mobility in a more general model. In section 3.5 we
summarize the main implications of these assumptions.
3.1 Basic assumptions
In this section we reintroduce some assumptions used in Piketty (1998). This framework
provides a basis for describing how agents respond to their relative position in their
reference group and its implications in terms of income mobility. Following Piketty
(1998), we assume for simplicity that the set of agents I = [0 ;1] can be divided in two
social backgrounds: lower class origin ( I 0 ) and upper class origin ( I 1 ). Income is a
random variable and there are two possible income levels in the economy: y0 and y1
( 0 < y 0 < y1 and Δy = y1 − y0 ). Agents only live for one period. The probability that
agent i obtains a high income level depends positively on their ability ( βi ), their effort
( ei ) and luck ( π ). Furthermore this probability is conditioned by social origin and it is
given by:
15
If i ∈ I 0 , Pr ( yi = y1 ) = π + θβ i ei
If i ∈ I1 , Pr ( yi = y1 ) = π + Δπ + θβ i ei
(1)
where, Δπ > 0 measures previous inequality between agents with different social
backgrounds.16 Because they received inheritance from previous generations, given a
level of effort the expected probability of economic success is higher for agents with
upper-class social origins than for those with lower class origins. Meanwhile θ > 0 is the
same for all agents and measures the extent to which higher effort and higher ability can
translate into higher probabilities of high income. All agent know the values of
parameters π , Δπ and θ .
As a result, the expected income for those with lower class origins and higher class
origin is respectively defined as follow:
E ( yi | I 0 ) = ( y1 (π + θβ m e0b ) + y0 (1 − π −θβ m e0b )
E ( yi | I1 ) = y1 (π + Δπ + θβ m e1b ) − y0 (1 − π − Δπ −θβ m e1b )
(2)
As Piketty (1998), we assume that individual effort levels are not publicly observable,
and everybody expects that agents with Ij origin put effort ebj ≤ E . Ex-ante agents have
not any information about their ability and they assume the mean β m of the ability
distribution f ( β i ) , with 0 ≤ b ≤ β i ≤ B .17 There is an exogenous maximum effort level
E > 0 , which satisfies: π + Δπ + θβ i E < 1 and (1 − α )aθβ M ( y1 − y0 ) < E .
3.2 Additional assumptions
i . The agent’s objective function
In this section we present the objective function for all agents of our model. This function
allows us to discuss how optimal effort decisions are affected by income comparison. As
in Piketty (1998), in this paper we assume that status has intrinsic value, but its two
16
This parameter could explain the inequality of family transmitted human capital and/or inequality of
collateral in case of credit constrains (Piketty,1998).
17
Piketty (1998) we make two natural assumptions. In the absence of any status motive, the unique
equilibrium effort level e is lower than E (aθβ m ( y1 − y0 ) < E) . Further, in order to avoid corner solutions, we
assume π + Δπ + θBE < 1 , where B is the maximum level of ability ( 0 < β m < B ).
16
dimensions will be considered separately. As a result two additional arguments of
individual’s utility function will be included compared with basic model: “the social belief
about him” and the “self-perceived valuation of its relative position”. This assumption
allows us to distinguish the role of discrimination and reference group in the inequality
persistence. Therefore, the objective function of agent i is given by
U i ( yi , y R i , ei ) = (1 − α − λ ) yi + λW ( β i p ) − αG ( yiR ) − C (e) 18
(3)
With 0 ≤ α ≤ 1,0 ≤ λ ≤ 1 and α + λ = 1
where U i is the utility function for agent i. As in Piketty (1998), agents enjoy their income
19
( yi ) for consumption reasons ( 1 − α − λ ≥ 0 ) , dislike effort because they enjoy the
leisure (agents perceive that effort is a cost defined by the function C ( e ) ), and they also
care about “the social belief about him”. The function W ( βi p ) captures how the opinions
and beliefs of others about him affect his utility function, where the public belief about
him is βi p . Compared with Piketty’s model an additional argument of individual’s
objective function is included: the “self-perceived valuation of its relative position”. Agent
cares about his relative deprivation (RD) which arises from a comparison of his income
and those from his reference group.20 Agent dislikes unfavorable comparison of his
income with the income of others. The RD that agent i faces within his reference group is
defined as a function G ( yiR ) , where yiR represents the difference between his own
income ( yi ) and expected reference group income yiRG , yiR = y i − yiRG .
This expression assumes for simplicity that it is additively separable. One central issue is
that both status motive are direct argument of the utility function due to its intrinsic value,
where, λ and α measure the extent to which agents care about them separately.
Piketty (1998) assumes that α =0 and W ( β i p ) = βi p , the social status of agent i,
according to the public beliefs. Using this additional term λβ i p , Piketty describes how
reference group theory and the statistical discrimination theory might amplify inequality
Following Piketty (1998) we assume that W ( β i p ) = β i p and C(e) = e2 / 2a , with a > 0 .
Because this paper focuses on the incidence of relative income on effort decision, with the aim of
simplifying, it assumes a lineal relationship between the absolute income and utility. However, others
approaches assume a non lineal relationship, and they explain their implications in terms of income mobility.
given the literature on non-linear dynamics and inequality traps is needed (Lewis and Ulph, 1998; Antman
and McKenzie; 2007; Carter and Barrett; 2006).
20
The term relative deprivation was used by Stark et al (2011) to analyses the role of comparison in the
acquisition of human capital. Other studies use the term relative income.
18
19
17
persistence. However, the status motives used in Piketty’s provides a better basis for
discrimination theory meanwhile reference group mechanism has not been considered in
detail. His utility function no includes how people care about their relative position with
respect to a referent point. Additionally, his framework does not allow us to explore how
agents react to the composition of their reference group and the conditions under which
reference group might decrease (or increase) income mobility. To consider this
limitation, we assume α ≠ 0 . As a result, agents care about obtaining a low income gap
between their income and their reference income.
The function G (.) is an attempt to formalize the discussion of how reference group
income affects agent’s utility and it is defined as:
⎧⎪G ( yiR ) = G ( y i − yiRG ) with G ( y R ) > 0, G y R (.) < 0, G y R y R (.) > 0 if yiR = y i − yiRG ≤ 0
(4)
⎨
⎪⎩G ( yiR ) = c
if yiR = y i − yiRG > 0
As in previous studies, we assume an asymmetry in the income comparison.21 Agents
look upward when making comparisons. The function G (.) incorporates envy effect and
concavity of relative income when yiR <0 (Hopkins, 2008).
Furthermore, this function is more general than previous studies because G( yiR ) = c
when yiR > 0 , which leaves open the possibility that agent - relative concern is supported
by “pride effect” or “compassion effect”. Note that the asymmetry in the income
comparison is also considered in the differences in the derivatives, where G y R (.) < 0
and G y R y R (.) > 0 when yiR ≤ 0 and G y R (.) =0 when yiR > 0 . Namely the disutility is
constant with respect to the relative income when pride or compassion effect, but it is
increasing when envy effect operates. Following both theoretical and empirical literature,
this assumption recognized that agents are upward looking and that envy effect
dominates relative comparison. With the aim of simplifying, first we assume that pride
effect on relative concern dominates, as a result c ≤ 0 , so that the contribution of this
component to utility is positive.
21
Other studies have already used this assumption. Stark et al (2011) used the same assumption to
formalize the link between human capital choices and social location choices. Bowles and Park (2005) used
it to modelling the “Veblen effect”. Ray and Genicot (2010) also suggest upward looking aspirations
formation to describe the relationship between social interaction and aspiration formation. Dalton, Ghosal
and Mani (2010) use a similar framework to explain aspiration failure. Dusenberry (1949) postulated and
tested the hypothesis that relative income comparisons are asymmetric. Finally, this assumption is
supported by Bowles and Park (2005), Stuzter (2004) and Ferrer-i-Carbonell (2005).
18
ii . The reference group income
Now we introduce an analytical form to introduce reference group and the formation of
aspirations. The idea is that the composition of reference groups defines a reference
income level (economic aspirations) and agents care about the gap between their
income and their reference income. The set of agents Pi ( I 1 ) + ( 1 − Pi )( I 0 ) integrates
the reference group of agent i. First, for simplicity we assume that Pi is exogenous and it
RG
holds 0 ≤ Pi ≤ 1 .22 As a result, the expected income of the reference group, y , is
defined as y RG = Pi ( E ( y | I 1 )) + ( 1 − Pi )( E ( y | I 0 )) . Agent i compares only with peers
whose origin are I 1 when Pi = 1 , and I 0 when Pi = 0 .
We assume that the expected relative deprivation depends on the expected income for
agents with different backgrounds and on the composition of reference groups. Consider
first the case of agent i with lower-class origin ( i ∈ I 0 ). The ex-ante expected relative
deprivation is defined as:
E ( yiR | I 0 ) = E ( yi | I 0 ) − E ( yiRG ) = E ( yi | I 0 ) − [Pi ( E ( y | I1 )) + (1 − Pi )( E ( y | I 0 ))] =
Pi [E ( y | I 0 ) − E ( y | I1 )] +
144424443
Expected income gap between agents
with lower and higher class origins
E ( y | I0 ) − E ( y | I0 )
14i 44
2444
3
= Φ ( E (ei ), e0b , e1b , Pi )
(5)
Expected gap between income of agent "i "
and average agent with origin I 0
Where E ( yi | I 0 ) is the expected income of agent i, given that he is I 0 , and E ( y | I j ) is
the expected income for all agent I j , which was defined in equation 1. Assume
b
everybody expects agents with higher class origins to put effort e1 and those with lower
class origins to put effort e0b . Further, first assume for simplicity that ex ante (before
those agents choose their optimal effort level) agents share the public beliefs about their
expected income. As a result, E ( yi | I 0 ) = E ( y | I 0 ) and relative deprivation is:
E ( yiR | I 0 ) = Pi [( E ( y | I 0 ) − ( E ( y | I1 ))] =
[
P [( y − y )(θβ
]
Pi ( y1 (π + θβme0p ) + y0 (1 − π − θβme0p ) − ( y1 (π + Δπ + θβme1p ) + y0 (1 − π − Δπ − θβme1p )) =
i
1
0
m
(e0p − e1p ) − Δπ )
]
(5b)
It is plausible to assume that the expected income for agents with higher class origin is
at least equal to the expected income for agents with lower class origin, because Δπ ,
22
This assumption will be lifted in future research to analyze the role of hereditable reference group or an
endogenous choose of reference group.
19
( y1 (π + θβ m E ) + y0 (1 − π −θβ m E ) = Max( E ( y | I 0 )) ≤ ( E ( y | I1 ) . This assumption implies
that the effect of the differential in expected effort on economic success never outweighs
the effect of previous inequality. As a result, for agents with lower class origin relative
deprivation is defined as E ( yiR | I 0 ) ≤ 0 . Relative deprivation is defined by two terms, the
wide of reference group (P) and the expected gap between agents with low and high
social origin.
An agent with lower class origin faces a larger relative deprivation when income gap
( y1 − y0 = Δy ) and previous inequality ( Δπ ) are higher and when the impact of
differential
expected
effort in
social
origins
on
economic
success
is
lower
( θβ m (eb0 − e1b ) ).23
For simplify, we assume that Pi < 1 for agents with upper-class origins ( i ∈ I 1 ).24 In this
case, the expected relative deprivation is always positive and is defined as:
E ( yiR | I1 ) = E ( y | I1 ) − yiRG = E ( yi | I1 ) − [Pi ( E ( y | I1 )) + (1 − Pi )( E ( y | I 0 ))] =
(1 − Pi )[( E ( y | I1 ) + ( E ( y | I 0 ))] > 0
As a result, agents with upper-class not expect to face relative deprivation. They hope to
enjoy utility from the fact that their expected income is higher than the expected income
of their reference group. Alternatively, it is possible to assume that the richest group has
no reference group (Bowles and Parker, 2005). Both approaches have the same
implications in terms of our model.
iii. Agent’s effort decision process
The aim of our model is to explain how agents respond to their relative position in their
reference group and their implications in terms of income mobility. For this reason, first
for simplicity we assume λ = 0 and α ≠ 0 and we focus on “relative deprivation effect”.
The case λ ≠ 0 and α = 0 was analyzed in Piketty (1998), and we will consider both
λ ≠ 0 and α ≠ 0 in section 5.
23
Because
an
agent
i
can’t
affect
the
income
of
his
reference
group,
we
can
rewrite
U i ( yi , y R i , ei ) = U i* ( yi , ei ) with U i* y (.) > 0,U i* yy (.) < 0,U i*e (.) < 0,U i*ee (.) < 0,U i* ye (.) < 0 .
24
This assumption simplifies derivations and exposition of sections 4 and 5 , but it doesn’t change the
essential conclusions.
20
We assume rational agents, who act to maximize his expected utility based on the
parameters of the Economy and their beliefs. Further the decisions are descomposed in
a two-step process. First they identify their reference group income and expected
relative deprivation. As it was noted in previous section, relative deprivation could be
positive or non positive. It depends on expected effort and the structural parameters of
income mobility. As a result, this step allows us to find the domain where relative
deprivation function (G) works for each agent.
In a second step, they maximize expected income taking as given the reference group
b
income and choosing their optimal level of effort (namely taking as given e0b and e1 ). For
agents with lower-class origin, the optimization problem is defined as:
max U i ( yi , y R i , ei ) = (1 − α ) yi − αG ( yiR ) − C (e)
(6)
s.t. y R = Φ * (e0 , e0b , e1b , Pi )
(5)
The first order condition is.
⎧⎪ e0 eq = (1 − α ) aθβ M ( y1 − y0 ) − α aGe ( y eq R ) − αaθβ M ( y1 − y0 )G y ( yeq R , e eq )
r
⎨
R
⎪⎩ e0 eq = E if E < (1 − α ) aθβ M ( y1 − y0 ) − α aθβ M ( y1 − y0 )G y r ( yeq , e eq )
if e0 eq ≤ E (7)
or e0 eq = E if E< (1 − α ) aθβ M ( y1 − y 0 ) − αaθβ M ( y1 − y 0 )G y r ( y eq , eeq )
R
given y R = Φ * ( E (e0 eq ), e0 , e1 , Pi )
All agents with the same reference group will choose the same optimal effort. Namely,
agents with origin I 0 and the same Pv will choose the same optimal effort e0v , where the
index v identifies the reference group composition Pv .
For agents with upper-class social origins the relative deprivation effect is defined as
G( yiR ) = c , is fixed and always positive ( c < 0 ). In this case the optimization problem is
defined as:
MaxU i ( yi , y R i , ei ) = (1 − α ) yi − αG ( yiR ) − C (e) = (1 − α ) yi − αc − C (e)
(6b)
As a result, the first order condition is.
e1eq = (1 − α )aθβ M ( y1 − y0 )
Based
on
these
(7.b)
assumptions,
how optimal effort decisions are
affected
in
the
by income
next
section
comparison
we
and
discuss
aspirations
formation.
21
3.3. Implications of the basic model
Consider the case of an agent i with lower-class origin. He chooses the level of effort
that maximizes his own expected utility. Given the utility function U i ( yi , y R i , ei ) , if there
is an interior solution, we thus have that the first order condition (FOC) for a maximum is:
0 = (1 − α )θβ M ( y1 − y0 ) − αθβ M ( y1 − y0 )G y ( yi (.) − y RG ) − e / a
(7.c)
The utility-maximizing effort levels are given by
If E ( yi > y RG ) → e* 0 eq = (1 − α )aθβ M ( y1 − y 0 )
If E ( yi ≤ y RG ) → e** 0 eq = (1 − α )aθβ M ( y1 − y0 ) − αaθβ M ( y1 − y0 )G y ( yi (.) − y RG )
The second order condition ( − αθβ M ( y1 − y 0 )G yy ( yi (.) − y RG ) − 1 a < 0 ) holds because
*
convexity of G (.) (in accordance with Standard assumptions), hence e 0 eq and
e**0 eq constitute optimum solutions, and they satisfy e* 0 eq < e** 0 eq . From the comparison of
e* 0 eq and e** 0 eq , we can say that the incorporation of relative deprivation increases the
optimal level of effort chosen by an agent. This effect generates an upward jump in
levels of optimal effort, when E ( yi > y RG ) changes to E ( yi ≤ y RG ) .
RG
We can now consider the effect of an exogenous increase in y
among agents who
feel disutility due to their relative deprivation ( E ( yi ≤ y RG ) ). Differentiating the
individual’s first order condition for the optimal effort level, we find:
de0 eq
**
dy
RG
=
α [aθβ ( y1 − y0 )]G y
R
,yR
(.)
1 + aα [θβ ( y1 − y0 )] G y R , y R (.)
2
> 0 if E ( yi ≤ y RG ) and e0eq < E
(8)
This derivative is always positive, because the denominator is positive (by the second
order condition) and the numerator is positive because convexity of G(.). In summary
the following propositions were proved:
Proposition 1. When E ( yi ≤ y RG ) and G y R (.) < 0 , relative deprivation
effect increases the optimal level of effort chosen by an agent compared to
*
**
an agent without relative deprivation ( e 0 eq < e 0 eq ).
Proposition 2. When E ( yi ≤ y RG ) , G y R (.) < 0 and G y R y R (.) > 0 , higher
reference income always leads to additional effort (
de0 eq **
dy RG
> 0 with
22
**
e0 eq ≤ E ).Therefore, reference group generates higher income aspirations
and motivates higher effort.
As a result, optimal effort level is higher when relative deprivation plays a role in the
objective function. The larger α and G y R , y R (.) the larger is this effect. That is, this effect
is larger when agents care a lot about their relative position and when their marginal
RG
utility is more sensitive to changes in relative deprivation. Further, higher y leads to a
high effort level (Figure 1, Graphic A). For lower-class agents, a richer (or more
demanding) reference group provides higher effort incentives. In this context, they
increase their income aspirations and his effort level. In this case, the probability of
economic success increases when individual optimal effort level is higher. As a result,
social mobility depends on the structure of reference groups, and it is higher when P is
larger.
Figure 1
Graphic A
de0 eq
**
dy RG
E ( yi > y RG )
Graphic B
de0 eq
E ( yi ≤ y RG )
**
dy RG
y RG
E ( yi > y RG )
E ( yi ≤ y RG )
y RG
Given agent i with lower-class origins, when Pi ≠ 0 , he has high incentives to increase
p
the amount of his effort and the effect is stronger when Pi and Δπ , Δy , e1 and e0p are
*
higher. These incentives disappear if E ( yi ) > E ( y RG ) , where e 0 eq is constant. This is
consistent with Piketty predictions only if each agent from lower-class backgrounds
compares himself only with agents with the same origin ( Pi = 0 ∀i ∈ I 0 ). When agents
assume ex-ante that they belong to a reference group whose members are all I 0 , they
adopt a behavior that validates their reference group expectations. As a result, reference
group compositions seem to play a key role in the promotion of aspirations and social
mobility.
However, this specification is simplistic, because agents have fixed reaction rules to
respond to changes in the reference group income. Further, the predictions are
consistent with “self fullfiling belief” only in a particular case. This discussion clearly
demonstrates that when the structure of reference groups is heterogeneous, agents with
23
lower-class origins always have incentives to assume strategies to improve their
opportunities to achieve a better life. Finally, it may seem less intuitive that higher exante inequality ( Δπ ) always motives higher optimal effort. Higher Δπ could represent a
strong role of the inheritance in the income performance, and therefore decrease
motivation inducing lower effort. In short, this approach captures encouragement effect,
but not captures frustration or complacency effect.
One may well ask whether with these basic assumptions is possible to allow more
flexible reaction rules to respond to changes in the reference group. One point worth
noting here is that these general results depend critically on the assumption about the
diminishing marginal sensibility of relative deprivation ( G y R y R (.) > 0 ). However, if
G y R y R (.) < 0 and
αθβ M (Δy )( yi − y RG )G yy <
optimality condition still hold and then
de0 eq **
dy RG
1
a
RG
,25 in a range of values of y ,
the
will be negative. This reflects diminishing
marginal sensitivity to larger deviations from the reference group income (in accordance
with prospect theory). In this case, more demanding reference groups lead to lower
effort.
Proposition 3. When E ( yi ≤ y RG ) , G y R (.) < 0 and G y R y R (.) < 0 , relative
deprivation increases the optimal level of effort chosen by an agent
( e* 0 eq < e** 0 eq ). But in this case, when an agent cares about relative
deprivation, higher reference income always leads to lower effort
RG
RG
**
**
equilibrium level. Namely, given E ( yi < y a < yb ) , e 0 a > e 0 b , where
e** 0 a and e** 0b represent the effort of agent with reference income ya
and yb
RG
RG
respectively.
These results are predictable. Marginal utility measures how much extra utility agents
gain by higher relative income ( U y R = −αG y R ). The function U y R y R (.) = −αG y R y R (.)
measures how much marginal utility will change in response to a change in the level of
relative deprivation (marginal sensitivity). When G y R y R (.) is negative, higher relative
deprivation increases the marginal utility of relative income, therefore the motivations are
higher. When G y R y R (.) is positive, better relative income rises individual utility, but this
The implication of the expression αθβ M ( y1 − y 0 )( y i − y ) G yy < 1 a is that effort always is perceived as a
cost. In other words, an increase of the marginal utility due to a decrease in the relative deprivation is lower
than the increase of the marginal cost due to a higher effort.
25
RG
24
R
increase will be higher when y is lower. In conclusion, assumptions about the sign of
G y R y R (.) are central to Economic aspiration dynamics and allow us to model both
encouragement effect and frustration or complacency effect.
Finally, if there is a point of inflection in the function G (.) , this structure would allow us to
explain different behaviors as a function of the magnitude of relative deprivation. We
discuss different situation in the Annex I.A. This assumption provides a more flexible
*
reaction to reference group income. However, we identify two constraints. First e 0 eq is
**
lower than e 0 eq , namely, given two identical individuals, one with relative deprivation
and other without relative deprivation, the former always chooses higher effort than the
second individual (Proposition 1). Second, the sign of G y R y R (.) provides more flexible
reaction rules. However, under assumptions of proposition 3 the derivative,
de0 eq **
dy RG
,
changes the sign, which may seem less intuitive. When an agent is not relatively
deprived simply because his reference income is low, he will choose higher effort if his
reference group income is higher than his expected income. But once he cares about
relative deprivation, an additional increase in reference income level leads to lower
effort. (Figure 1, Graphic B) Finally, it is difficult to identify the determinants of the point
R
discussed in the Annex I.A.
of inflection ybreak
3.4 An extension of the model
Now we focus again in internal solutions but we incorporate some modifications in the
function G (.) to allow a more flexible reaction to changes in the reference group. We
assume that the importance of relative concern depends on effort. This assumption
implies that the marginal utility of relative deprivation depends on the effort. The
introduction of this assumption is an attempt to formalize the discussion of income
aspirations driven by reference groups, and it incorporates the part of effort cost that is
cultural and endogenous. Kandel and Lazear (1992) use a similar reasoning to formalize
the link between peer pressure function and the effort made by workers.26 As a result,
the function G ( yiR , ei ) incorporates how the relative position affects the perception of
effort, meanwhile the function C (e) is the part of the cost of effort that is exogenous to
26
Again we assume
λ = 0 and α ≠ 0 to focus on relative income effect.
25
the relative situation. Therefore the incidence of relative concern on utility ( G (.) ) is
a function of two arguments, yir and ei , which represents a more general model
compared to the previous case. Namely
U i ( yi , y R i , ei ) = (1 − α − λ ) yi + λβ i p − αG ( yiR , ei ) − C (ei )
(9)
To make this assumption a little more concrete, consider an example of function
G ( yiR , ei ) , G * ( yiR , ei ) = v(ei ).g ( yiR ) , with g ( yiR ) > 0 , g ' ( yiR ) < 0 , g ' ' ( yiR ) > 0 and
v(e) > 0 . In this case the sensitivity of relative deprivation on the utility function depends
on the functional form of v(e) ( G y R * ( yiR , ei ) = v(ei ).g ' ( yiR ) ). Note that v(e) was constant
i
and equal 1 in the basic model. By making explicit assumptions about v(e) , we clarify
the exact nature of the tastes required to explain a particular behavior. On the one hand,
when the effort increases, the marginal utility of relative deprivation in the reference
group will decrease. Namely v' (e) < 0 and v' ' (e) > 0 , which implies Ge < 0 and Gee > 0 .
On the other hand, the sensibility for relative deprivation might decrease with higher
effort, if v ' (e) > 0 and v' ' (e) < 0 , which implies Ge > 0 and Gee < 0 .27
Henceforth, we will use the function G ( yiR , e) because it is more general than
G * ( yiR , ei ) . Following the previous sections G ( yiR , e) is decreasing and convex in its
first argument. But in the second argument the situation is more flexible.
⎧⎪G ( yiR , e) = G ( y i − yiRG , e) (**)
G ( y , e) ⎨
⎪⎩G ( yiR , e) = c < 0
(**) G y R (.) < 0, G y R y R (.) > 0
R
i
if yiR = y i − yiRG ≤ 0
otherwise
(10)
The sign of Ge characterizes the agent’s responses to relative income and income
aspiration dynamic. To advance in this direction we analyze the optimal effort decision in
an interior solution. The FOC remains unchanged for agents with lower reference group
income. Again, when E ( yi ≥ y GR ) “relative deprivation effect” has no effect on optimal
27
Instead of marginal sensitivity of relative deprivation, this formulation could be interpreted in different way,
if we assume that relative deprivation affects the perception of the cost of effort. For example, perception of
the cost of effort could be lower when relative deprivation is low, because agents belief that reference group
income is achievable outcome ( Ge < 0 and Gee > 0 ). Alternatively, given a high relative deprivation, when
effort is very high, agents could perceive that the goal is unattainable, they are discouraged and perceive
that effort is less effective (or more costly). They reduce their effort level in order to avoid frustration. In this
case Ge > 0 and Gee < 0 .
26
effort
level.
However,
this
condition
changes
when
E ( yi ≤ y RG ) .
E ( yi > y RG ) → e *0 eq = (1 − α )aθβ M Δy = c
(11)
E ( yi ≤ y RG ) → e * *0 eq = (1 − α )aθβ M ( y1 − y0 ) − αaGe ( y R , e * *0 eq ) − αaθβ M (Δy )G yr ( y R , e * *0 eq )
144424443 144444444424444444443
e*0 eq = c > 0
RD ( e**0 eq )
Note that we only focused on interior solutions. When E ( yi < y GR ) the SOC is:
− αGe e ( y R , e ) − 2α ( θβ M )( Δy )G y R e <
1
+ α ( θβ M )2 ( Δy )2 G y R y R ( y R , e )
a
We assume that the problem has an optimal solution and the second order conditions of
this problem always hold. As a result, e * 0 eq and e** 0 eq constitute optimum solutions. The
agent will choose the level of effort e* , which equates the expected marginal rate of
substitution between effort (absolute and relative) and income (absolute and relative)
with the expected return of the effort. If α = 0 , e *0 eq = (1 − α )aθβ M ( y1 − y0 ) .
marginal utility of effort
6Total
44
474448
e* + αG ( y R (e*), e * )
U i*e ( y (e*), e*)
e
=
θβ M ( y1 − y0 ) = a
14
4244
3 1 − α − αG y ( y R (e*), e*) U i* ( y (e*), e*)
y
r
exp ected return of the
144442
4444
3 14
4244
3
effort
Total marginal utility of income
(12)
m arg inal rate of
substitution
The next propositions define three types of individual’s situation, which essentially
depends on the functional form of G ( yiR , e) .
Proposition
4.
When
E ( yi ≤ y RG ) ,
G y R (.) < 0 ,
G y R y R (.) > 0 ,
Ge ( y R , e ) < 0 Gee ( y R , e ) > 0 (conditions I) relative deprivation increases the
optimal level of effort chosen by an agent with relative deprivation
compared to the level chosen by an agent without relative deprivation
*
**
( RD(e) > 0 and e 0 eq < e 0 eq ). Condition I defines the “positional self encouraged agent”.
Proposition
Ge ( y R , e ) > 0 ,
5.
When
E ( yi ≤ y RG ) ,
− 1 < αGee ( y R , e ) < 0 ,
G y R (.) < 0 ,
G y R y R (.) > 0 ,
Ge ( y R , e ) < −θβ M ΔyG y r ( y R , e)
(conditions II) relative deprivation increases the optimal level of effort
chosen by an agent with relative deprivation compared to the level chosen
*
**
by an agent without relative deprivation ( RD(e) > 0 and e 0 eq < e 0 eq ). The
condition II θβ M ( y1 − y0 )G yr ( y R , e) > Ge ( y R , e ) , defines the “positional
stimulated agent”.
27
Proposition
Ge ( y R , e ) > 0 ,
6.
When
E ( yi ≤ y RG ) ,
− 1 < αGee ( y R , e ) < 0 ,
G y R (.) < 0 ,
G y R y R (.) > 0 ,
Ge ( y R , e ) > −θβ M ΔyG y r ( y R , e)
(conditions III), relative deprivation decreases the optimal level of effort
chosen by an agent compared to the level chosen by an identical agent
*
**
without relative deprivation ( RD(e) < 0 and e 0 eq < e 0 eq ). The condition III
defines “positional discouraged agent”.
*
**
When Ge (.) < 0 , the equilibrium effort level e 0 eq is always lower than e 0 eq . Under
condition I, given relative deprivation, it generates higher utility among those agents
who have made greater effort. In other words, given the same level of effort, those
agents with relative deprivation perceive lower cost for additional relative effort, and
therefore, they make a higher effort. In this case, function Ge (y R , e ) can be interpreted
as implying that agents get utility from relative effort. As such, it is not surprising that
more effort is the outcome (encouraged effect).28 Furthermore, when G y R y R (.) is positive,
a higher relative income gap with respect to reference group increases marginal utility of
income, which leads to additional effort to improve relative income.
Meanwhile, when Ge (.) > 0
and Ge ( y R , e ) < −θβ M ΔyG y r ( y R , e)
the reasoning is
somewhat different. In this case, given relative deprivation, it generates lower utility
among those agents who have made a greater effort. But in this case, the higher
disutility of high relative effort is compensated by a lower relative income gap (due to the
effect of effort on relative income). Given Ge (.) > 0 , the condition II is fulfilled when
“relative effort pays”, namely, if θβ M Δy is high. If condition II holds, then low relative
income increases marginal utility and it mitigates the marginal cost of effort. In this case,
encouraged effect dominates because there is high opportunity for income mobility.
Namely, relative effort leads to lower relative deprivation.
*
**
However, if Ge (.) > 0 and Ge ( y R , e ) > −θβ M ΔyG y r ( y R , e) , e 0 eq is higher than e 0 eq . In
this case, given relative deprivation, it generates higher reduction on utility among
those agents who have made a greater effort. When agents suffer high relative
deprivation, they perceive a higher cost for additional relative effort. Therefore, the
marginal utility of a reduction in the relative deprivation is lower than the marginal
28
Kandel and Lazear (1992) use similar argument to explore how peer pressure operates on worker effort.
They suggest that the peer pressure function can be interpreted as implying that workers get utility from
effort.
28
disutility of a higher effort in the relative component. The reduction of relative deprivation
is more demanding in terms of effort and agents are discouraged.
We can now consider the effects of an exogenous increase in reference group income
among agents who care about his relative deprivation ( E ( yi < y RG ) . Differentiating the
individual’s first order condition for the choice of effort we find the following expression:
de0eq
[(
αa θ ( β M )ΔyG y
**
dy
RG
=
R
,yR
)
(.) + G y R e (.)
]
(13)
1 + aαGe e ( y R , e ) + αa (θβ M ) 2 Δy 2G y R y R ( y R , e) + 2aαθβ M ΔyG y R e
The expression in the numerator determines the sign of
de0 eq **
dy RG
(the denominator is
positive due to the second order condition). First, note that G y R , y R (.) is positive due to
the concavity of G (.) . As a result, the sign of this expression depend on the sign of the
second partial derivatives of G (.) , G y R e .
Proposition 7 When E ( yi ≤ y RG ) , G y R , y R (.) >0 and G y R e > 0 , higher
reference income always leads to additional effort (
de0 eq **
dy RG
> 0 with
**
e0 eq ≤ E (Condition IV). Again, this assumption characterizes the “income
gap self - encouraged agent”. In this case, higher relative income gap due
to high reference income leads to higher effort.
Proposition
θβ M ΔyG y
R
,yR
8
When
E ( yi ≤ y RG ) ,
G y R , y R (.) >0,
Gy Re < 0
and
(.) > Ge , y R (.) , higher reference income always leads to
additional effort (
de0 eq **
dy RG
> 0 with e0 eq ** ≤ E ) . These conditions define the
“income gap - stimulated agent” (condition V).
Proposition 9 When
θβ M ΔyG y
effort (
R
, yR
E ( yi ≤ y RG ) , , G y R , y R (.) >0, G y R e < 0 ,
and
(.) < Ge, y R (.) , higher reference income always leads to lower
de0 eq **
dy RG
< 0 with e0 eq ** ≤ E ). These conditions define the “income
gap- discouraged agent” (Condition VI).
Proposition 6 and proposition 7 establishes a positive relation between effort and
reference income. In this case, the effect of a higher gap between the income of an
agent and his reference group is to reduce his utility, and thus raise the marginal utility of
29
lower relative deprivation compared to the marginal disutility of effort, inducing an
increase in his levels of effort. In this case, “relative deprivation effect” always increases
optimal effort level. There is a difference between proposition 6 and 7. In the former,
higher reference group decreases marginal cost of relative effort. As a result, reference
group increases effort level through two channels, the higher marginal utility of relative
income and the lower marginal cost of relative effort. In the second case, a higher
relative deprivation induces a higher marginal cost of relative effort. However, this effect
is dominated by a higher marginal utility income gap. Proposition 7 holds when “effort
pays”.
However, assumptions in proposition 8 establish a negative relation between effort and
reference group income (
de0 eq **
dy RG
< 0 ). In this case, a higher relative deprivation
increases the marginal utility of G(.,.) , but it is lower than an increase in the marginal
disutility of relative effort, inducing a reduction in the levels of effort. Furthermore, a
higher effort level reduces marginal utility of a lower relative deprivation. As a result, a
higher relative income gap reduces the effort level.
What can we say about G y R e sign?. When the inverse of effort and relative income are
complements in the relative deprivation term, the sign of G y R e is positive.29. Under this
condition, higher income gap leads to higher effort. If they are not complements, the sign
of
de0 eq **
dy RG
(θβ
is ambiguous, and it depends on the magnitude of
However, we can generalize this discussion about the sign of
de0 eq **
dy RG
M
ΔyG y
R
,yR
)
(.) .
if we focus in the
utility function.
Proposition 9 Higher reference income always leads to additional effort
when the “inverse of effort” (or “Leisure”) and relative income (or
consumption)
are
complements
in
U ( y, y r , e ) ,
namely
if
U * y , e ( y , e ) = U ** y r , e ( y r , e ) < 0 . Note that when U * y , e ( y , e ) is negative,
and income and leisure are complements, the objective function satisfies
the assumption of propositions 6 and 7. However, when the “inverse of
29
These ideas are used in Bowles and Parker (2005) to discuss the importance of “Veblen effect” in the
individual’s allocation of time between labor and leisure. Dalton, Ghosal and Mani (2010) assume a similar
assumption to incorporate income aspiration on utility function.
30
effort” (or “Leisure”) and relative income (or consumption) are no
complements, a higher reference income always reduces effort level. This
implies: U * y , e ( y , e ) = U ** y r , e ( y r , e ) > 0 (Assumptions of proposition 8).
It is interesting to note that Proposition 9 covers the basic model presented in 3.3, where
we assumed G y R e = 0 and Ge = 0 . We describe this situation in Annex IB.
It is also important to note that, given the functions G y R y R (.), Gee and G y R e <0, when
returns of effort and ability ( θ ), and expected ability ( β M ), and income gap inherited
(Δy ) are higher, is lower the feasibility of a negative relative deprivation effect on effort.
The composition of reference group is relevant. Note that differentiating the individual’s
first order condition for the choice of effort with respect P we find the following
expression:
de** dP =
[(
][
)
− aα θ (πβ M )ΔyG y R , y R (.) + G y Re (.) Δy (θβ m (e1p − e0p ) + Δπ )
]
1 − aαGe e ( y , e ) − αa(θβ M ) ( y1 − y0 ) G y R y R ( y , e) − 2aθπβ M ΔyG y Re
R
2
2
(14)
R
As a result, the intensity of relative deprivation effect on effort decision will be higher
when Δy , Δπ and e1p − e0p are higher. Namely when real and expected differences
between individuals with origin I 0 and I1 are higher.
3.5 ¿ What does this tell us about the effect of reference group on economic
aspiration and the optimal effort decision?
The function G (.) provides a general form, which incorporates both envy and pride
effect. The intensity of relative deprivation effect depends on the level of effort, which
allows us to model encouraged agent and discouraged agent. As a result, this provides
a more flexible rules reaction to respond to changes in the reference group income and
the formation of income aspiration.
The effects of reference group on optimal decision of effort depends on the intensity
(how much?, “ α ” and G (.) ) and direction (to whom?, “ P ”) of income comparisons. As
a result, the structure of reference groups is relevant. When they are homogeneous and
each agent compares with peers with the same social origin, the relative deprivation
effect on optimal effort decision is low. In this case, the agents with low social origin do
not include agents with high social origin, which leads to aspiration failure.
31
However, there are different scenarios when reference group are heterogeneous,
because agents respond in different way if reference group is more demanding. In this
case, the incidence of the structure of the reference groups depends of the sign of two
functions: Ge (.) (describes how effort affects relative deprivation assessment) and G y R e
(defines if leisure and relative income are complements).
On one hand, the sign of the function Ge (.) defines the sensitivity for relative deprivation
in the reference group. First, we identify “positional self - encouraged agent” when
Ge < 0 . In this case, those agents with relative deprivation perceive lower cost for
additional relative effort, and therefore, made a higher effort. Function Ge ( y R , e ) can be
interpreted as implying that agents get utility from effort. As such, it is not surprising that
more effort is the outcome. Second, when Ge > 0 relative effort is always a cost and we
find two possible responses to relative position: “positional stimulated agent” and
“positional discouraged agent”. In the first case, reference group leads to higher effort
because there are opportunities of income mobility. In the second case, reference group
comparison increases marginal disutility of effort, and agent reduces his effort.
On the other hand, if relative income (or relative consumption) and leisure are
complements, reference group always promotes higher effort levels. Individuals from
lower-class backgrounds are being encouraged by a higher income gap between them
and their reference group. As a result, lower-class families don’t accept an inferior role,
and they work harder to pursue personal economic success and “social ascent”.
However, when relative income and effort are substitutes, the “relative deprivation effect”
could reduce the level of effort. In this case, the expected income gap between agent
and reference group discourages lower-class agent. They are more easily satisfied with
their performance and less motivated to achieve high income positions than agents with
less demanding reference group or upper-class origin. In this case higher reference
group income leads to lower effort because agents perceive the goal as unattainable.
The effects of comparison on motivations are stronger when α and a are higher.
Further, higher P , Δπ and βθ (e1b − e0b ) lead to a stronger intensity of these effects.
These issues are strongly related with the two types of aspirations failure defined by Ray
(2006). On one hand, there is one type of aspiration failure when individuals from I 0 do
not include the rich in their reference group. This will especially be the case if there is
economic polarization or other forms of stratification. As a result, when P = 0 or P is
32
low the reference income will be low, and so will individual investments for the future. On
other hand, there is another type of aspirations failure when individuals from I 0 includes
individuals I 1 in their reference group. In this case the individuals relatively poor do
aspire to be like the rich, but the income gap is simply too large. The costs of effort (or
investment) is too high, and the reward (in terms of a relative narrowing of the income
gap) too low. The reference group leads aspirations, but the feeling is widespread that
such aspirations are largely unreachable. In this case, the high ex-ante inequality
between agents with different social origin leads to lower effort.
Income inequality in conjunction with reference group composition leads to a lack of
connectedness. They are responsible for these aspirations failures. The income mobility
will be higher in a connected society, where Pi will be high, “effort pays” and economic
success does not depend on ex-ante inequality. One in which there is much
heterogeneity in each reference group, diversity in which every individual can reasonably
think of herself as being on the attainable fringes (Ray, 2006).
4. REFERENCE GROUP AND INCOME MOBILITY
In the previous section we assume that expected effort is know, constant and
exogenous. In this section we introduce an updating beliefs rule of expected effort for
agents with different origin. It allows us to explore how agent’s effort decision is affected
by the expected effort of his peers in his reference groups. Because the reference group
is contextual, it depends on how much mobility is perceived in society. As a result, there
is a connection between expected effort and the performance in terms of income mobility
of previous generation. This section is organized as follows. First we present the
information structure and second the updating belief process of expected effort. In
section 4.3 we discuss the effect of reference income on income mobility in the long
term.
4.1 The information structure
The results of the previous chapter could be interpreted as a benchmark, which
considers a situation in which there is perfect information. In this chapter, we assume
that no agents know the expected effort of the peers of their generation ( e0b and e1b )
when they choose their effort level. Agents are therefore uncertain about the true
reference group effort. As a result, expected effort for agents with different origins is not
33
constant and generations update their beliefs about the expected effort of their peers.
We propose a way of including the dynamics of such public beliefs about the expected
effort, assuming a link between the beliefs of generation tth and the previous mobility
experience of generation t-1th. Public beliefs are updated according to Bayes’ rule,
implying that differences in mobility performance with respect to expected results affect
the beliefs of the current generation. Under imperfect information and when beliefs are
updated according to Bayes’ rule, the long term effort belief equilibrium depends on the
initial distribution of beliefs and past trajectories of peers with the same social origin.
Many of the assumptions that determine this process were introduced by Piketty’s
(1998) model, where the author defines an updating belief process about one’s
smartness, which suggests a relationship between current effort and past effort for each
generation and social origin. Our model incorporates the following assumptions. All
agents know the parameters that determine the probability of success in each social
origin ( π , θ , Δπ ), but they do not know the expected effort of the peers of their
generation ( e0b and e1b ) when they choose their effort (A.I). We assume that no
information is publicly observable about individual ability parameters, and ex-ante agents
don’t have any information about their ability and their peers’ ability (A.II)30. We assume
that individual effort levels are not publicly observable (A.III), but agents have a priori
information about the mean effort of agents with origin I j , which was between certain
“high effort level” ( e ≤ E ) and certain low effort level” ( e > 0 ), with e ≤ e (A. IV). Agents
only live for one period, and public beliefs are transmitted across generations, name that
generation t+1th has a priori information based on beliefs of generation tth (AV). They
know the social mobility experienced by the previous generation, and use this
information to update their a priori beliefs (AVI).
In the remainder of this chapter, we focus on agents with origin I 0 but the procedure is
analogous for agents with origin I1 .31
For each agent with origin I 0 there is a latent variable which describes the relation
between economic success and effort, which is defined in equation (1) as (Assumptions:
AI, AII and AIII):
30
Assumptions AII and AIII are used in Piketty (1998). The author assumes that agents always assume
mean ability.
31
Note that to derive the learning process for agent with origin
I1 , π
should be replaced by
π + Δπ
.
34
Yit ' = π + θβ M eit
It is helpful to consider that the economic performance is stochastically related to effort,
incorporating a random variable. Therefore, the probability of economic success for
agent i from generation t is defined as:
⎧ P( yit = y1 ) = P(Y ' = 1) = π + θβM eit +ν it
P( yit = y j )⎨
⎩P( yit = y0 ) = 1 − P(Y ' = 1) = 1 − π + θβM eit −ν it
where ν it represents an idiosyncratic shock for each generation t and agent i, with
E (ν it ) = 0 and 0 ≤ π + θβ M eit + ν it ≤ 1 , for 0 ≤ eit ≤ E . Similar results can be seen when
examining the aggregate performance of each generation. For example, the probability
of all agents with origin I 0 reaching y1 is defined as:
P ( y1t = y1 , y2t = y1....., ynt = y1 i ∈ I 0 ) =
∏ π + θβ
e + ν it = 1 i ∈ I 0
M it
(15)
∀i∈I 0
In more general terms, taking xt as the share of successful agents with origin I 0 from
generation t, we can derive the probability that xt = x't , and we arrive at the following
expression:
P( xt = x' ) = Ω(ε 't ,υ 't i ∈ I 0 ) = α (eM t ( ε 't ),υ 't i ∈ I 0 )
(15.b)
Where ε t and υt are vectors of n individual efforts in t ( e1t , e2t .......ent ) and n random
variables ( υ1t ,υ 2t .......υ nt ) respectively, and ε 't and υ 't are particular realizations of both
vectors. The function Ω ( R → R ) is continuous on an interval 0< eit <E and its
n
codomain is defined by
xt xt ∈ (0,1) . It was assumed, to simplify, without loss of
generality, that there is a function α , whose argument is the mean effort in t, eM t (which
is a function of each element in the vector ε t ). Both Ω and α are non-decreasing
functions in eit and eM t respectively, and are also functions of the shock υt . The
functional form of these functions depends on π ,
θ and Δπ , which are known by all
agents. It is worth noting that agents have a priori distribution of both functions, if they
have a priori information about eM t and υ t .
Each agent i from generation tth observes the income ( y0 or y1 ) of all agents from
generation t-1. Agents know the social origin and the mobility outcome of the previous
generation, and as a result they know the proportion of successful (and unsuccessful)
35
agents with origin I 0 . Agents from generation t+1 know the real percentage of
economically successful agents with origin I0 in period t, xt. However, they do not
observe the latent variable Y 'it −1 and the incidence of υt −1 (AIII and AVI).
The real percentage of economically successful agents with origin I 0 from the
generation t-1 could be defined respectively as:
x't = Ω(ε t ,υ t ε 't ,υ 't ) = α (eM t ( ε 't ),υ t ε 't ,υ 't ) (% of successful agents with origin I 0 )
1 − x't = 1 − Ω(ε t ,υ t ε 't ,υ 't ) = 1 − α (eM t ,υ t ε 't ,υ 't ) (%
of
unsuccessful
agents
origin I 0 )
with
(16)
where ε 't ,υ 't are particular realizations of both vectors. Following Piketty (1995 and
1998) and Breen and García – Peñalosa (2002), because agents know π ,
θ and Δπ ,
they know the distribution of signals.
4.2 Intergenerational learning
We assume that agents incorporate information of the economic performance of the
previous generation when they update their apriori public beliefs, which are transmitted
from the previous generations. Beliefs are updated by a backward-looking learning
process, that is, in light of recent experience of peers with the same social origin from
previous generation. Agents have limited information about the consequences of their
actions, and generations update their beliefs by trial and error methods using local
knowledge based on their peer recent experience. Bowles (2004) explains that learning
process as backward looking agents, and argues about the advantages of this approach
with respect to forward-looking agents.32 Other papers have used this learning
procedure previously and they place on emphasis on the information transmission
between generations and the significance of past trajectories to explain heterogeneous
beliefs.33 Finally, this learning procedure seems useful to explain the formation of
aspirations based on social interactions. Agent behaviors are conditioned by the
experiences of other agents in his cognitive neighborhood. Agents form economic
32
Bowles (2004) considers backward-looking learning process inside evolutionary game theory. In contrast
to the forward-looking agents in classical game theory, this approach addresses the history of the agents
and the evolution of agent’s preferences, resulting in a more realistic modelization of social interactions.
33
Piketty (1995) used Bayesian learning to update the belief about the parameters of the economy, Piketty
(1998) to explain the public beliefs about status, and Breen and García – Peñalosa (2002) to describe the
difference in preferences across genders.
36
aspirations base on their past experience and their interactions with their reference
group. (Appadurai, 2004, Genicot and Ray, 2010).
Let μt be the public belief of generation t about the participation of high effort agents
among economic successful agents from previous generation with origin I 0 . Agents of
generation t have a priori information about effort distribution e > e , which represents
reasonable levels of effort. Expected effort for their peers in the current generation is
defined as:
etb = μt e + (1 − μt )e
(17)
The parameter μ t could be interpreted as the subjective probability attached by the
entire generation t to effort e . Agents of generation t+1 know the social mobility
experienced by the previous generation t-1 ( xt −1 ), but they do not observe which of them
made a high effort. Since mobility performance is only stochastically related to effort,
“evaluation errors may occur”. Namely, an agent i, who made a “relatively low effort” in t,
could reach economic success due to the incidence of the shock ν it . Meanwhile, other
agent j, who has made a relatively high effort in t, could fail in their attempts to obtain a
higher income due to the incidence of the shock ν jt . As a result, μt ≠ xt .
To avoid these errors, the weight on e , μ t , must be high when high effort agents
dominate among successful levels. Observe that the importance of both errors depends
on the correlation between eit and ν it . On one hand, when z = Corr (eit , vit ) ≥ 0 the
shock does not “redistribute” economically successful agents between low and high
effort agents. As a result, the “effort pays” and high effort agents dominate between
successful
agents.
On
the
other
hand,
an
alternative
hypothesis
is
that
z = Corr (eit , vit ) < 0 , in this case the shock “redistributes agents”, namely some agents
with low effort reach economic success. In this second case, although effort has positive
impact on the probability of high income, the effort reward is relatively lower compared to
the first case. As a result, portion of low effort level is relatively high among economically
successful agents, and then μ t should be lower. The sign of this correlation represents
both the capacity of an agent to respond to a shock and the economic performance of
the agent once a shock has occurred. It could be associated with individual ability and
the initial level of effort.
37
Even if agents do not know the latent variable and the real incidence of υit , they could
learn the incidence of the shock in the long term, according to Bayes’ rule. Each
generation t+1 observes a signal xt , which is received from the predecessor generation.
The distribution of signals depends on the current state of the world. The probability that
the public signal xt′ is realized conditional on the state being z is defined as:
Pr( xt′ ) = α (eM t ( ε t ), vt z , ht −1 ) = α (eM t ( ε t ), vt , ht −1 )
(15c)
where Pr(.) defines the probability of the event in brackets occurring and ht −1 describes
the decisions of all agents from previous generations. Meanwhile, the probability that the
public signal xt′ (and the implicit signal v't ) is realized conditional on the state being z
is defined as:
xt′ = α (eM t ( ε 't ), v't z , ht −1 ) = α (eM t ( ε 't ), v't ht −1 ).
(15d)
Following Moscarini and Ottaviani (1997), we assume that the quality of the public signal
⎛α (e' M t , v't ) < α (e' M t , v't ) if eM t < e)M t
⎜
)
is bounded: ⎜ α (e' M t , v't ) > α (e' M t , v't ) if eM t > eM t
⎜
⎜α (e' M t , v't ) > α (e' M t , v't ) if eM t > e)M t
⎝
(15e)
)
where eM t is an arbitrary level of the mean effort of an agent with origin I 0 in t.
Consider that μ t is an a priori probability assigned to high effort e , which could be
interpreted as the subjetive probability attached by a generation t to z being the true
state of the world. Following Piketty (1995, 1998) and Breen and García – Peñalosa
(2002), we assume that intergenerational learning takes the form of Bayesian updating,
with beliefs being learned by the current generation from the previous generations.34
The agents with origin I 0 base their effort decisions on their beliefs about the expected
effort of their peer in the current generation. After the output of mobility income of
34
In this point, it is important to distinguish between our model and other papers. In Piketty (1995) and
Breen and García-Peñalosa (2002) the Bayesian learning is individual, where each dynastic transmitted
their own individual experience and know his past effort. In our model, agents learn about social belief from
all their peers and they do not know their past effort level. This is a significant difference. Piketty’s model.
Piketty (1998) assumes that smartness is unobservable and the public beliefs about one’s ‘smartness’ are
inferred from one’s publicly observed social mobility experience. He defines an Bayesian learning
procedure, which relates current expected effort with social mobility trajectories. Expected effort is updated
based on Bayesian learning process. In our model, we directly link past mobility trajectories with current
expected effort.
38
generation t-1 is realized, the next generation updates their a priori beliefs according to
Bayes’ rule, and they choice their effort level based on their updated beliefs.
The sequence of events is as follows. The beliefs μ t are used by agents from
generation t to determine their effort decisions. Agents with origin I 0 base their effort
decisions on their beliefs about expected effort of their peers. They choose their effort
level, and after the realization of vt , they obtain y 0 or y1 . The belief of generation t ( μ t )
are inherited by the next generation. The beliefs of generation t+1 combine that
information and the mobility outcome of generation t. Bayesian learning implies that the
outcomes of the previous generation are interpreted in the light of the a priori beliefs.
The posterior beliefs of the next generation which observe the signal x't is given by
Bayes’ rule:
μt +1 =
Pr( z ∩ x't ht −1 )
μt Pr( xt = x't z , ht −1 )
=
Pr( x't ht −1 )
μt Pr( xt = x't z , ht −1 ) + (1 − μt ) Pr( xt = x't z, ht −1 )
(16)
The probabilities in the later equation are the public signals received by the generation
t+1 that the state is z conditioned on both the action history ht −1 and the realization x't .
These probabilities were defined when we introduced the function distribution of signals,
which define the probability of the signal x't is realized conditional on the state being
z or z . Therefore, these probabilities could be substituted by the functions of distribution
of signals and we arrive at the following expression:
μ t +1 =
μ t α (eM t ( ε 't ), v't ht −1 )
μ t α (e' M t ( ε t ), v't ht −1 ) + (1 − μ t ) α (eM t ( ε 't ), v't ht −1 )
(16a)
This function describes the evolution over time of a generation’s beliefs. Similar to Breen
and García – Peñalosa (2002), whether updated weight placed on e is greater than the
prior depends on whether, for the level of effort chosen by the previous generation, the
signal observed is more likely to have occurred for z than for z .That is, from the
previous equation we can write:
⎧⎪μt +1 > μt ⇔ α (eM t ( ε 't ), vt ' ht −1 ) > α (eM t ( ε 't ), v't ht −1 )
⎨
⎪⎩μt +1 < μt ⇔ α (eM t ( ε 't ), v't ht −1 ) < α (eM t ( ε 't ), v't ht −1
(16b)
If a generation tth experienced a relatively high mobility outcome with respect to a priori
beliefs, the probability of this event being conditional on effort and previous decision is
39
greater for z than for z , therefore generation t+1th places a higher weight on z . The
opposite holds for the case of low mobility results. Note that this function depends on a
priori beliefs, as a result, the same mobility outcome can give rise to different posteriors.
This function describes the beliefs evolution when agents of generation t+1th know the
mean effort of the previous generation eM t , but we have assumed that they do not have
this information. However, because previous generation transmitted their beliefs, current
generation have a priori beliefs about eM t , which was defined as: etb = μt e + (1 − μt )e . If
we consider this change of variable, we arrive at the following expression
⎧⎪α (etb , v't ht −1 ) > α (etb , v't ht −1 ) ⇔ μt +1 > μt ⇔ etb+1 > etb
⎨
b
b
b
b
⎪⎩α (et , v't ht −1 ) < α (et , v't ht −1 ) ⇔ μt +1 < μt ⇔ et +1 < et
(17b)
The rationality of the updating belief rule is the following: when agents of previous
generations with origin I 0 that had made a high effort were not rewarded with upward
mobility, there will be some downward adjustment of the expected effort for the next
generation of agents with origin I 0 , e0b . That is, given the a priori beliefs in generation tth
about expected successful agents with origin I 0 , x0bt , we consider x '0t to be the true
realization in period t. When x0bt > x'0t , then e0bt > e0bt +1 . In contrast, high achievable
performance in the previous generation should induce rational agents to expect higher
effort in their next generation peers. When x0bt < x'0t , then e0bt < e0bt +1 .
As a result, effort belief etb+1 combines a priori information transmitted from previous
generations and information about the mobility experienced by the previous generation.
A general property of Bayesian learning is that the stochastic process μt describes a
martingale: what generation t expects its successors to know next period is exactly what
generation t knows today. Namely, the agent’s best guess in generation t+1, as to his
posterior
in
any
later
period
is
his
posterior
beliefs
in
period
t,
namely
( E ( μt +1 μt , ht −1 ) = μt (Aghion et al, 1991; Piketty,1995, Smith and Sørensen, 2000) As a
result, E (etb+1 μt , ht −1 ) = E ( μt μt , ht −1 )e + (1 − E ( μt μt , ht −1 )e = etb .
Assume, without lost of generality, that the true parameter value is z , namely
Corr (eit , vit ) ≥ 0 and “effort always pays”. Therefore μt = 1 is equivalent to allocating full
b
weight on the truth. Pick z ≠ z , with μ t (et −1 , z, vt ) > 0 , and define for any t > 0 the
40
likelihood
It =
μt (etb−1 , z, vt ) 1 − μ t (etb−1 , z, vt )
=
, follows a stochastic process
μt (etb−1 , z, vt )
μ t (etb−1 , z, vt )
{μ t } ,
which describes a martingale conditional on the true state of the world. This is a general
property of the Bayesian learning process. As a result, standard martingale convergence
results can be applied (Aghion et al, 1991; Piketty, 1995, Smith and SØrensen, 2000
and Breen and García – Peñalosa, 2000). Piketty (1995) and Breen and García –
Peñalosa (2000) derived three propositions about this process, which could be
interpreted in terms of our model.
First, the martingale convergence theorem implies that the likelihood ratio, and hence
beliefs, converge in the long term. For any initial beliefs, μ 0 , in the long term beliefs
converge toward some stationary beliefs, μ ∞ , with probability one. Therefore, there is a
stable
solution
about
the
level
of
expected
effort,
which
is
defined
as:
e∞b = μ∞ e + (1 − μ∞ )e .
Second, given the true state of the world z , the Bayesian updating function defined in
16a has three fixed points. One of them is not stable μ∞ = 0 . There are two stable long
term equilibrium beliefs, one is an interior fixed point μ∞ = μ* > 0 and the other is a
corner solution μ∞ = 1 . As a result, both stationary beliefs allocate positive weight on the
true state of the world. In terms of effort beliefs, e∞b = e is not a stable solution,
meanwhile, e∞b = μ * e + (1 − μ *)e and e∞b = e are stable solutions.
Finally the interior solution μ * holds:
)
α (eM t ( ε 't ( μ*)), vt ' ht −1 ) = α (eM t ( ε 't ( μ*)), v't ht −1 ) = α (eM t , v't )
(18)
Consider z as the true state of the world: if initial beliefs are μ 0 < μ * , then they
converge to μ * with a probability of one. As a result e∞b = μ * e + (1 − μ*)e , when
e0b = μ 0 e + (1 − μ 0 )e < μ * e + (1 − μ *)e . Whereas if the initial beliefs are higher than μ * ,
then they will be attracted with positive probability, p ( μ 0 , μ *) , by μ = 1 , and with
positive probability 1 − p ( μ 0 , μ *) by μ * .
Breen and García-Peñalosa (2002) named μ * as “confounded learning beliefs. At this
point nothing can be learned from the previous generation, and a priori beliefs are equal
41
to the posterior. They demonstrated that the probability of converging to the true belief is
given by:
p( μ 0 , μ*) =
μ0 − μ *
μ 0 (1 − μ*)
(19)
As a result, the long term equilibrium belief depends on the initial beliefs and the quality
of the public signal information. This result is due to the fact that the same mobility
outcome can give rise to different posterior beliefs depending on the probabilities initially
attributed to each situation.
Successive learning across generations may be complete, as a result, generations will
access to the true value of z , z . In this case, an equilibrium belief about the expected
effort of agent with origin I 0 is e∞b = e . Namely, in this case “effort always pays” in the
long term, and agents with origin I 0 expect their peers to excert a high level of effort.
One point worth noting here is that e may not be the “true” mean effort. This level of
effort is the more likely parameter value given the realizations of mobility of different
generations with origin I 0 and the a priori beliefs e and e . In other terms, evidence
shows that effort pays, and the successive learning across generation leads to the
highest expected effort.
However, the learning process across generations may be incomplete. In this case
agents perceive that effort rewards are relatively low, ever if this is not the truth. As a
result, agents place a strictly positive weight on the true state of the world ( z ), and long
term equilibrium of the expected effort is lower than
( e∞ * =
b
e but is higher than e
μ * e + (1 − μ *) e ). Although “effort pays” and promotes high income
mobility, initial beliefs and mobility trajectories leads in the long terms to relative lower
expected effort level for agents with origin I 0 .
Equations 17a and 17b provide a rational updating process, where society learns from
the mobility outcome. As a result, effort beliefs ( etb ) combine prior information ( etb−1 ,
e and e ) and the mobility experienced by the previous generation (signals). The
Bayesian learning mechanism implies that history is important in determining equilibrium
beliefs and public expected effort and therefore the reference group income level. As a
result, long term aspirations are conditioned by social origins, the income mobility
opportunities and past history.
42
4.3 Reference group, intergenerational learning and mobility in the long term.
Using the updating effort belief procedures described in section 4.1, and the agent’s
effort decision process described in section 2, in this section we describe a special case
of the dynamic of intergenerational mobility based on reference group and the formation
of aspirations. Once the expected relative deprivation has been identified, agents
maximize their expected utility taking their reference group income as given and they
choose their level of effort. As a result, generations are connected in two ways. On one
hand, the probability of economic success depends on social origin. On the other hand,
there is an indirect channel, because the experience of previous generations affects
expected effort and expected income, and they determine the incidence of reference
groups through relative deprivation. As a result, relative deprivation conditions economic
aspiration levels.
We focus on the steady state equilibrium for the agents with low social background, as is
the best case study. Taking into account the previous assumptions for agents with
higher class origins, their optimal effort decision process is always affected in the same
way by the relative deprivation effect.35 As we discussed in sections 1 and 2, both
theoretical and empirical literature supports these simplifying assumptions. Furthermore,
the purpose of this study is to analyze the role of the reference group as a determinant
of income mobility and inequality persistence. Therefore, we focus on agents with low
social origins and we will describe how individuals are more (or less) motivated by their
reference groups to achieve upward mobility through effort.
In section 2 we characterized different individual responses to relative income effect
depending on agent’s characteristics and circumstances. The effects of relative
deprivation depend on the magnitude of income gap, the opportunities of income
mobility and the characteristics of agents. We interpret that this characteristics are part
of the personality of the agent and it is explained by idiosyncratic term. First to simplify,
we assume that all agents are identical and assumptions of propositions 4 and 6 are
satisfied (encouraged agents).
35
Agents with origin I1 and P=1 could be affected by relative deprivation. This group could respond to
changes in expected effort and expected reference group income. The procedure is analogous to the case
developed by agents with origin I 0 .
43
Even though this assumption simplifies the analysis, it is worth noticing that in this
scenario relative income effect always motivates a high effort. Therefore, it would be
more interesting to know how updating belief process and relative deprivation affect
income mobility.
Each agent knows his Pi , which there is a random variable between 0 and 1. Relative
deprivation has a random component ( Pi ) and a hereditable component, the expected
income conditional to the origin. This assumption is unreal but it is in agreed with the
actual empirical literature.36 Suppose that an extreme Cournot-Nash assumption
satisfied so that each agent takes others' effort and actions as given within the same
period. But agents update their belief about expected effort between generations.
Finally, for agents with origin I0 , e1b is exogenous (because they can’t affect decision of
agents from I 1 ). To simplify we assume that e1b = e1b . This will allow us to discuss how
reference group could affect economic aspiration and inequality persistence. In Annex II
we describe this procedure in more detail.
i. The steady state and relative income effect on inequality persistence
In steady state agents with origin I 0 and the same Pi will choose the same optimal effort
and it is constant: e0*i = e0*i = e0*i , and etb = e∞b , where the index i identifies 0 ≤ Pi < 1.
t −1
t
t +1
Remember that the optimal effort for agents with lower-class origins is defined as:
e0 eq =Min (h( e0 . y R ), E)
Where
h( y R , e0i ) = (1 − α )aθβ M ( y1 − y0 ) − αaGe ( y R , e0i ) − αaθβ M ( y1 − y0 )G y r ( y R , e0i )
(11)
E ( yiR | I 0 ) = Φ (ei , Pi , e0b , e1b | I 0 ) : Expected relative deprivation function
(5)
t
t
e0b = ω (e0b , e0 M , v ) : Bayesian learning function
t
t −1
(17c)
t −1
Substituting eq. 17c into 5, and then in eq 11 we arrive at the following expression
[
]
[
e0ieq = min h(Φ(ω (e0 M , v), e1b , Pi , e0b ), e0b ); E = min H (e, e0 M , v), E
]
(20)
36
The empirical literature regarding the process of selection of reference group is inconclusive. Clark and
Senik (2010) identify that this process seems to be partly endogenous and agent’s benchmarks are related
to the type of regular social interactions of them. However their findings are not consistent with this
hypothesis.
44
First, is important to note that agents with the same origin may choose different long
term effort level because it varies with the composition of the reference group. This
result deviates from Piketty (1998), where all agents with the same origin arrive to the
same long term effort level (short term dynamic is discussed in Annex III).
In the long term, the Bayesian learning function leads to social beliefs e∞b , which satisfies
e∞b = ω (e0 M∞ ) . As a result for agent with origin I 0 the models predict two possible
scenarios about effort level in the steady state. First, when e∞b = e , expected relative
deprivation is relatively high and so economic aspiration. Agents with higher P choose a
high effort level because their relative deprivation and economic aspirations are
relatively high. They are encouraged by the expected income of agents with origin I1 but
also by their peers with origin I 0 . When P is relatively low (agents with origin I 0
compare with their peers), they tend to choose higher effort level, because expected
effort for agents with origin I 0 is high. In this case, the expectation of peer’s effort will
increase, and so will individual effort for the future. The intensity of this effect is higher,
among agents whose reference group composition has low P . However, the effort in
steady state will be always equal or higher for those agents with higher P , because they
include more agents with origin I1 in their reference group. Finally, these results are
consistent with self fulfilling belief effect in Piketty’s model, when P is very low, where
the composition of reference group leads to aspirations failure type I.
)
On other hand, long term social belief could be e∞b * = ω (eM ∞ ) . In this case, expected
effort for agent with origin I 0 is relatively low. That belief is a result of the initial condition
and the past trajectories of the previous generation of agents with origin I 0 . In this case,
when agents with origin I 0 compare with their peers ( P ≠ 0 ), they have a deep belief
that their own destiny is pre-ordained and beyond control. Their beliefs are that their
peer’s effort “not pays”. As a result, the expected peer effort is relatively low, and
reference group income will be low if P is low. This mechanism is related with aspiration
failure type II, because fatalistic beliefs explain low effort. In this case, relative
deprivation and the formation of aspirations leads to lower long term effort level
compared to those agents without relative deprivation (agent with origin I1 ) and with
respect agent with P = 1.
45
In the second scenario there are two possible dynamic. On one hand, when μ0 < μ * ,
the expected effort will increase, and so will individual effort for the future. However,
those optimist beliefs have a threshold and steady state of effort belief will be relatively
low. On other hand, when
μ0 > μ * , agents will be attracted with probability
1 − p( μ 0 , μ*) by μ * (and e∞b * ). In this case, the expectation of peer’s effort will decline.
As a result, lower expected “relative deprivation”, inducing a decrease in the levels of
effort. Because agents believe that their peers (all agents are I 0 ) in the reference group
will decline their effort, their economic aspirations will be lower (relative income effect is
lower) and he chooses a lower effort level.
We can draw several conclusions about the role of social origin and reference group on
Economic aspiration and intergenerational income mobility. First the composition of the
references group is relevant to explain income mobility. Agents care about their relative
performance compared to others, and therefore they consider expected income of their
reference group when they choice their effort. The reference groups, through relative
income effect, tend to amplify (or reduce) the inequality of economic success between
agents with different social origins and different reference group composition. The utilitymaximizing effort level will be lower for all agents with P ≠ 1 . This decline will be higher
among whose agents with lower P .
Furthermore, beliefs about relative effort-rewards could play a key role to explain an
additional mechanism of inequality persistence. Is that, the reference group effect
depends on how much mobility (or perceived mobility) there is in society. If effort is
correlated with probability of economic success, then the public beliefs about expected
effort will depend upon publicly observed social mobility experience. This generates
connections between the effort made by an agent at present, and the economic success
of whose agents with the same origin in the past. This effect is more significant when Pi
is closer to 0, namely when is higher the share of low origin agents in the reference
group composition.
These results are consistent with self fulfilling belief effect in Piketty’s model, but in our
model that mechanism is channelized through reference group concern or in terms of
Ray (2006) “aspirations failure”.
An important point to highlight is that we are assuming identical agents with identical
information about others agent’s responses and the same beliefs about the value of
46
parameters. Furthermore, we assume that all agents are “encouraged agents”, is that all
agents increase their effort in response to more demanding reference group. But agents
may come from different social origin or may have different reference groups ( Pv is
random), which could lead to different long term effort level. As a result, relative income
effect could reduce or amplifies the inequality of economic success between agents with
different social origin. Finally, note that under certain circumstances, reference group
could reduce inequality of economic success. Is that, when agents perceive that “effort
pays” and they have demanding reference group. This conclusion of this model is in
stark contrast to others models of inequality based upon self-fulfilling beliefs and
fatalistic prediction.
According to this model, the mobility will be higher when the integration of these groups
is similar between agents with lower-class origins and upper class origin, is that, when
the society is more integrated, in terms of Ray (2006) “a connected society” promotes
more income mobility. Further, the mobility also depend on the belief about the role of
effort on economic success among the peers with the same origin, and how the society
as whole rewards successful and unsuccessful economic agent when public belief are
formed. When agents believe that their peer have not chance to reach economic
success, their expected relative deprivation will be lower. Therefore, they will reduce
income aspiration and choose lower level of effort.
As a result, individuals from lower-class backgrounds could be inefficiently discouraged
by reference group effect. This effect depends critically on the agents' response to more
demanding reference group and the beliefs about the role of effort on economic success.
Moreover, reference group is a channel through which inequalities may develop
and persist or decreases. An important point to bear in mind is that we assume that
reference group composition ( Pv ) is not inherited.
Finally, in contrast to Piketty’s model where inequality persistence may be explained by
discriminatory behaviors, in this model the society does not discriminate in any way
against high or low class origin agents.37 In this case, the effort choice (and the income
mobility) is affected by the reference group income and ex-ante inequality through the
formation of aspirations.
37
Observe that in the extreme case
e∞b 0 = e∞b 1
we will arrive to the same conclusion.
47
ii. The role of ex-ante inequality in detail
In the previous section we assume “self-encouraged agents”, therefore relative
deprivation always motivates higher effort regardless of effort rewards and ex-ante
inequality. However, propositions 8 and 9 provide more plausible assumption. In this
case, ex ante inequality and circumstances are more relevant to explain the agent
response. To be more concrete, we assume that there is a y*R(e), which holds
θβ M ΔyG y
*R
, y *R
(.) = Ge, y R* (.) , and when E(y R I 0 ) < E( y ∗ R I 0 ) , θβ M ΔyG y R , y R (.) < Ge, y R (.)
and when E(y R I 0 ) > E( y ∗ R I 0 ) θβ M ΔyG y *R , y *R (.) > Ge , y R* (.) .
Those assumptions imply a change in the sign of U y R e , which is plausible. For example,
Ray argues that the sign of the second derivatives of agent’s objective function changes
when agent’s income is close of his aspiration level. In terms of our model, when agent’s
relative income is very low (high relative deprivation), leisure and relative income may be
substitutes. Meanwhile, complementarity between leisure and relative income could be a
characteristic of high relative income level (low relative deprivation). This assumption
leaves open the possibility that agents would face different equilibrium effort levels
depending on their reference income. Therefore, under this assumption the structure of
reference group and ex-ante inequality is even more important for social mobility.
By following analogous reasoning as above, we will arrive to a long term effort level.
Under these assumptions, higher expected effort of agent with origin I 0 leads to higher
effort. Therefore, conclusions about previous section remain unchanged. Given P ,
higher e0b motivates higher effort levels for agents with origin I 0 .
Focus now in the role of ex - ante inequality between social origins, which was
measured by Δπ .38 Let P = 1 and Δπ ' such that E(y R I 0 ) < E( y ∗ R I 0 ) . In this case
agents with low social origins include individuals from richer origin in their reference
group but they face a high relative deprivation. They perceive that relative costs of effort
are too high compared to relative reward. As a result they reduce their aspirations and
effort level in order to avoid frustration. In this case, a high expected income gap leads to
aspiration failure of type II. If Δπ was lower, such that E(y R I 0 ) > E( y ∗ R I 0 ) , relative
deprivation might lead to a high effort level. In this case, effort of agent with origin I 0 will
38
b
An increase in e1 could affect in the same way.
48
be higher when its rewards is higher (higher θ and Δy ). Finally, note that the intensity
of this effect is lower when P is low. In this case, a lower P could lead to higher effort
level. However, among agents with low P operate the aspiration failure of type I.
Therefore, there is an inverted -U shape relationship between long term effort and P .
Finally note that the incidence of θβ M and Δy depends on the sign of e0b − e1b . When
e0b − e1b > 0 , higher θβ M and Δy encourage higher effort. Under that circumstances,
expected relative deprivation declines and promotes higher effort. However, we arrive to
the opposite conclusion when e0b − e1b < 0 .
Under this assumption there is a non-linear relationship between ex - ante inequality
between social origins and effort level of agents with origin I0 . Inequality could promote
higher effort via relative deprivation and aspiration formation. However, a strong ex-ante
inequality could leads to aspiration failure of type II.
5. AN EXTENSION OF PIKETTY MODEL: THE DOUBLE DIMENSION OF THE
STATUS
In this section we reintroduce the motive status defined by Piketty (1998). Therefore we
include both status effect and assume that G (.,.) is defined by equation 10. This allows
us to discuss the effect of status motive and reference group on the inequality of
economic success between agents with different social origins in the long term.
Furthermore, this section demonstrates the robustness of our results incorporating an
alternative updating beliefs process about expected effort. In this case, we use an
updating belief process about individual smartness defined in Piketty (1998), which
assume that when agents maximize expected income they take as given the reference
group income. But they update their beliefs between generations based on mobility
output.
To simplify we assume that all agents with origin I 0 have the same reference group
composition (namely Pi = P ). Based on section 3, we identify two particular cases which
allow us to define encourage and discouraged agent. First we define encouraged agent
when RD(e) > 0 , In that case, function G(.,.) satisfies assumptions of propositions 4 and
proposition 6 or 7. Second, we define discouraged agent when RD(e) < 0 , which implies
that assumptions of proposition 5 and 8 are satisfied.
49
In this case, the optimization problem is defined as:
U i ( yi , y R i , ei ) = ( 1 − α − λ ) yi + λβ i p − αG( yiR ) − C ( e )
(6c)
With 0 ≤ α ≤ 1,0 ≤ λ ≤ 1 and α + λ = 1
⎧⎪G ( yiR , e) = G ( y i − yiRG , e) (**) if yiR = y i − yiRG ≤ 0
G ( y , e) ⎨
⎪⎩G ( yiR , e) = c < 0
otherwise
(**) G y R (.) < 0, G y R y R (.) > 0
R
i
(10a)
We focus in the effort equilibrium condition for lower class-agents. Recall that for these
agents, the status differentials associated to economic success is defined by Piketty
(1998) as:
β 01P − β 00P = θe0bσ 2 / [(π + θβ m e0b )(1 − π − θβ m e0b )]
(21)
Where β 01P and β00P are the social status ( β P ) associated to an upwardly mobile agent
and immobile agent respectively. The difference between β 01P and β00P represents
improvements in status due to public beliefs about economic success. It is a function of
expected effort e0b . Let (10a), the utility-maximizing effort level is given by:
[
e = ArgMaxe>0 E (1 − α − λ ) yi + λβ ip − αG ( yiR ) − C (e) | I 0
The FOC is:
[
]
] [
C ' (e) = θβ m (1 − λ − α )( y1 − y0 ) + λθβ m β 01P − β 00P − α Ge ( y R , e ) − θβ M ( y1 − y0 )G yr ( y R , e)
]
(22)
The equations (21) and (22) define the interior equilibrium conditions for lower agent
origin. Piketty (1998) arrives at the interior solution of this system, replacing β 01P − β 00P by
θe0bσ 2 / [(π + θβ m e0b )(1 − π − θβ me0b )] in equation 20.
[(
)(
)]
[
]
h * (e, e0 ) = aθβ m (1 − λ − α )( y1 − y 0 ) + aθβ m λ (θσ 2 e0b / π + θβ m e0b 1 − π − θβ m e0b ) − aα Ge ( y R , e ) + θβ M ( y1 − y0 )G yr ( y R , e)
1444424444
3 14444444
4244444444
3 14444444
4244444444
3
PB ( e0 )
RD ( e ,e0b )
14444c444444444244444
444444443
h ( e0 )
(22b)
To facilitate exposition and to compares with Piketty model, the function h*(e) is divided
into three components: c (constant); PB(e) “effort adjustments via updating public
50
beliefs” and
RD(e, e0 ) “adjustments due to relative deprivation”.39 Each generation
received a signal about income mobility and forms its beliefs. As the expected status
changes between generations, agents update their belief about expected effort ( e0b ) for
each social origin. In long term equilibrium, no agent wants to change their behavior, and
public belief and effort are equal and constant ( e0b = e ). As a result the equilibrium is
defined as:
e = min( h * ( e ), E )
[
]
h * ( e ) = aθβ m (1 − λ − α )( y1 − y0 ) + aθβ m λθσ 2 e / [(π + θβ m e )(1 − π − θβ m e )]− aα Ge ( y R , e ) + θβ M ( y1 − y0 )G yr ( y R , e )
1444424444
3 1444444424444444
3 144444444244444444
3
PB ( e )
RD ( e )
14444c44444444244444
44444443
h( e )
When E( y i − yiRG > 0 the optimal effort conditions are:.
)
e = min( h( e ), E )
with
h(e) = aθβ m (1 − λ − α )( y1 − y0 ) + aθβ m λθσ 2e /[(π + θβ m e )(1 − π − θβ m e )]
1444424444
3 1444444
424444444
3
c
and
with
PB ( e )
h' (e) > 0 , ∀e > 0 . Observe that h' ' (e) > 0 if π > 1 / 2 .40 Note that the function h(e) is
equal to Piketty effort equilibrium function (namely he assumes α = 0 ).
According to Piketty (1998) it is much difficult to draw general conclusions about the
effect of the status motive on inequality. Both lower-class agents and higher class agent
could face multiple equilibrium effort levels. As a result, whether the status motive
reduces or amplifies the inequality of economic success between agents with different
social origin depend on their choices. However, this ambiguity disappears when the
impact on economic success of social origins is high as compared to the effect of effort
and ability. In this case, status motive reduces the mobility and amplifies the inequality
between agents with different social origins. Lower-class origins assign low effort, which
reduces the opportunity of social ascent; meanwhile, the effort of upper-class origin is
high to keep their original position.41 To put it another way, self-fulfilling discriminatory
λ = 0 and α = 0 , without any status motive, the function h*(e) is irrelevant and the
equilibrium is defined as e = θβ m ( y1 − y 0 ) = c . Note that the equilibrium condition for upper-class
agents is analogous, where RD(e)=0 and π is replaced by π + Δπ .
39
Note that for
Piketty (1998) only discuss this case. Therefore, PB' (e) > 0 ∀e > 0, and PB ' ' (e) > 0 if π > 1/2
Agents with upper-class origins have a lot to loose if they don’t maintain their initial social position,
meanwhile lower class agents do not have much to loose if they put low effort. They are not expected to
40
41
51
beliefs can make the initial inequality between social groups more persistent than it
would otherwise be.
When E ( y i − yiRG ) ≤ 0 , reference group plays a relevant role in determining the long term
wellbeing of the agents and income mobility. We add the third term RD(e) ≠ 0 , and we
arrive at h * ( e ) . In this case the optimal effort condition is:
e = min(h * (e), E )
with
[
]
h * (e) = aθβ m (1 − λ − α )( y1 − y0 ) + aθβ m λθσ 2 e /[(π + θβ m e )(1 − π − θβ m e )] − aα Ge ( y R , e ) + θβ M ( y1 − y0 )G yr ( y R , e)
1444
424444
3 1444444
424444444
3 14444444244444443
PB ( e )
c
RD ( e )
(
)
R
⎡
⎤
h'*(e) = h' (e) − RD ' (e) = h' (e) − aα ⎢Ge e ( y R , e ) + θβ M (Δy )G y R e + (θβ M )(Δy )G y R y R ( y R , e) + G y R e . dy
de ⎥⎦
⎣
R
h' (e) > 0 and dy
⎡
de p ⎤
= θβ M (Δy ) ⎢(1 − (1 − Pv ). 0 ⎥
de⎦
de
⎣
In this case, reference group income depends on P and e0b . The updated public belief
about expected effort ( e0b ) depends upon observed social mobility experienced by the
previous generation. The performance of previous generation affects expected effort and
expected income, and they determine the incidence of reference group through expected
relative deprivation. Finally, the sign of
de0p
de
is undetermined, depends on the quality of
the information received by the previous generation to form public beliefs. Changes on
expected effort determine changes on expected relative deprivation. Therefore, the sign
of RD ' ( e ) is undetermined. When
dy R
de
> 0 , then RD ' ( e ) < 0 . However, when
dy R
de
<0
the slope of this curve is higher. Even in the latter case RD' ( e ) could be positive.
*
Now we focus on h (e) and address the situation when income comparison affects
wellbeing. The effect of relative deprivation component on effort equilibrium level
depends on the sign of the function RD (e) which depends on the characteristics of the
agent: encouraged vs discouraged agent. First, among encouraged agents, higher
relative deprivation leads to higher efforts. Furthermore, a higher reference group
income leads to upward shift of the RD (e) curve. It increases agent motivations,
inducing an increase in levels of effort. Therefore adding
RD(e) moves the
reach high status and therefore are less motivated. In other words, everyone has incentives to keep one’s
original social position.
52
*
*
function h (e) upward and to the left compared to h(e) ( h (e) ≥ h(e) in Figure 4, Graphic
A. To simplify, the graphic includes the case when h(e) = c + PB(e) leads to multiple
equilibrium situation (Annex IV). Furthermore we assume that PB ' (e) > RD ' (e) and
PB' ' (e) > RD' ' (e) , and we assume PB ' ' (e) >0).
When
dy R
de
> 0 , income gap between own income and reference group income
decreases with past effort. As a result, encouraged effect due to relative deprivation
decreases with high past effort. A high effort of previous generation with origin I 0
decreases the expected relative deprivation, which mitigates the “relative deprivation
effect” on long term effort decision. This effect is stronger when P is low and economic
aspirations are quickly aligned to current income, and agents have little incentive to raise
their effort. In this case, RD ' (e) is negative, RD (e) tends to 0 when effort increase
(namely h * (e) tends to h (e) ). Namely relative deprivation effect is lower. A lower
reference group income (due to low expected effort) leads to downward shift of the
h * (e) curve, which decreases the long term effort level. When P is low, aspirations are
quickly aligned to current income, and agents have little incentive to raise their effort.
Therefore, composition of reference group is more relevant. These predictions are
consistent with “self fullfiling belief”, but in this case the inequality transmission operates
via relative deprivation effect and reference group income and economic aspirations.
Finally, this effect is lower when P is high.
However, when
dy R
de
< 0 , higher effort leads to higher reference income and economic
aspiration. As a result, higher “relative deprivation” encourages agent to make a higher
effort. Because expected effort of agents with origin I 0 increases with the effort of their
peer in the past, they will have more demanding reference group and they will amplify
their economic aspirations. Relative component raises optimal effort level in the long
term, because, relative income goals are sufficiently challenging without being perceived
as unattainable. In this particular case the composition of reference group is less
important to explain long term income mobility, because aspirations of agents with I 0
move
up due
to
the
performance
of
their
peer
in
the
past:
E (Yi I 0 ) → E (Yi I1 ) and E ( y R (e, P = Pv ) I 0 ) → E ( y R (e, P = 1) I 0 ) .
53
In the Figure 2, Graphics A, we present this situation. Note that higher P leads to higher
level of effort
Figure 2
Graphic B
Graphic A
h * ( y 2RG , e)
h( e )
dy R
*
h (e)
de
<0
h * ( y 1RG , e )
h * ( y 0RG , e )
*
h (e)
h( e )
+
ΔP
E
h( e )
y
RG
2
>y
RG
1
>y
h * ( e) = h ( e) = e
RG
0
h( e )
h ( e) = h ( e) = e
*
h * ( y 0RG , e)
E
dy R
de
h * ( y 1RG , e )
h * ( y 2RG , e)
>0
y2RG > y1RG > y0RG
Δ+ P
dy R
de
e
e
E
e
e
dy R
de
<0
>0
E
Now we consider discouraged agent. In this case, higher reference group income
increases relative deprivation and it leads to lower effort. In the Figure 2, Graphic B, we
present the situation when RD(e) is negative. Again, it is important to consider the sign
of
dy R
de .
When
dy R
de
> 0 , higher past effort reduces relative deprivation and encourages
R
higher effort. This effect is stronger when P is low. In this case, expected y tends to be
lower and when effort is high, h * (e) tends to h (e) . However, when
dy R
de
< 0 , a higher
effort leads to higher relative deprivation, which discourages agents with origin I 0 (Note
that the slope of RD (e) is lower when
dy R
de
> 0 ). This effect is stronger when P is high.
In this case, agents face very high relative deprivation and they perceive reference
group income as an unattainable goal. This would raise when ex ante inequality between
agents with different social origins is already perceived to be quite high. That is, agents
believe that they cannot get anything done to improve their relative deprivation. As a
result, they perceive relative deprivation as more important when they increase their
effort level. In this case, lower class individuals could be discouraged by higher relative
deprivation.
The composition of reference group affects the opportunities of economic success for
lower origin agents in the long term. When P = 0 , encouraged agents compare only with
peers whose origin is I 0 and relative deprivation component could not encourage high-
54
effort decision. This is a type of aspiration failure defined by Ray. Thus, in this case,
income comparison does not promote higher mobility, but, this situation promotes the
status quo. When
P ≠ 0 , income comparison is found to be one of the
key drivers of social mobility in the long term. When “leisure” and “relative income” are
complementary, more demanding reference group leads to higher effort equilibrium for
lower origin agents. As a result, reference group promotes higher income aspirations
and ascent mobility. However, when “leisure” and “relative income” are substitutes, more
demanding reference group reduces effort levels, leading to downward mobility. That
second case is associated with aspiration failure of type II.
It is interesting to note that, expected effort of agent with origin I0 in one period depend
in part on effort realized by previous generation of I0 and the mobility outcome of
previous generations. But also it affects expected relative deprivation and effort decision.
As result, updating belief generates a mechanism of incentives endogenous to different
generation of agents with origin I0 , which depends on the sing of
dy R
de
<0.
6. CONCLUSIONS
Our model describes how socio cultural inequalities could partly explain inequality
persistence with an emphasis on the role of status and reference groups. In this paper
we demonstrate that the composition of reference groups is relevant to the formation of
economic aspirations and income mobility. The reference groups, through relative
income effect and the formation of aspirations, affect effort decisions. This tends to
amplify (or reduce) the inequality of economic success between agents with different
social origins. Moreover, our model allows us to distinguish the role of discrimination and
reference groups on inequality persistence. High inequality and economic status jointly,
as a result of reference group and the formation of aspirations, might generate
inequality, even if society as whole does not discriminate.
In summary, regardless of the inheritance pattern of the reference groups, relative
deprivation affects economic aspiration and income mobility. The magnitudes and
direction of these displacements depend on: (a) the composition of the reference groups;
(b) assumptions about the functional form of relative concern (Standard Assumptions vs
Prospect Theory); (c) expected relative effort reward with respect to reference group
income and the relationship between past effort of the peers, income mobility outcome,
55
beliefs and economic aspirations; (d) ex - ante inequality and expected relative
deprivation between agents with different social origins.
On one hand, we identify aspiration failure type I when agents with a low social origin do
not include other agents with a high social origin in their reference group. In this case,
their aspirations are closely aligned to their current income, and they have little incentive
to increase their effort. Relative deprivation will be low and so will their investments for
the future. These agents will reduce their effort compared to their peers in a more
demanding reference group.
On the other hand, when agents with a low social origin include other agents with a high
social origin in their reference group, the impacts of the relative deprivation effect on
income mobility are ambiguous. One possibility is when relative deprivation could be too
large and the cost of the effort too high (or the perceived reward too low), in which case,
the relative deprivation effect reduces mobility (Aspiration failure type II). The second
possibility is that, under certain circumstances, the reference groups could reduce the
inequality of economic success. In this case, more demanding reference group could
increase income aspiration and lead to high effort levels. This conclusion is in stark
contrast to other models of inequality based upon self-fulfilling beliefs and fatalistic
predictions. A more integrated society, one in which there is greater economic diversity
in the reference groups and income inequality is relatively low, would open the possibility
of intergenerational mobility being higher.
Furthermore, as reference groups are contextual and connected with past trajectories,
their effects depend on how much mobility is perceived. To consider this issue we
assume imperfect information and we modeled beliefs using a Bayesian learning
process. As a result, the reference group effect depends on how much mobility or
perceived mobility there is in society. If we assume, that all agents are identical and are
encouraged by the relative deprivation effect, both the composition of reference groups
and beliefs about expected effort lead to different long term effort levels.
Finally, ex-ante inequality is a key determinant of the formation of aspirations. There is a
trade-off between ex-ante inequality, income mobility and economic aspirations. On one
hand, an expected high relative income gap could increase economic aspiration and
therefore effort and ascent mobility. On the other hand, high relative deprivation and low
relative effort rewards could reduce an agent’s aspirations and effort level
in order to avoid frustration. In this second case, relative deprivation leads to a status
quo. This prediction is related to aspiration failure type II. There is a plausible situation
56
where there is an inverted – ‘U’ shape relationship between long term effort and ‘P’, and
between long term effort and ex-ante inequality.
From
a
policy
perspective,
the
model
suggests,
on
one
hand,
that
anti-
discriminatory and affirmative actions could increase income mobility. On the other hand,
higher social cohesion could promote higher intergenerational mobility. For example, if
there is residential or public space segregation which reduces the opportunities
for social cohesion, a public intervention could mitigate this effect. Other interventions to
close the social origin aspiration gap, could be convening young people and enrolling
them in programs away from their communities (Austen – Smith and Fryer, 2005).
Ray (2006) suggests some policies for reducing the low mobility due to the presence of
aspiration failure in an unequal society. Ray argues that affirmative action or public
education could be policy tools to help create local, attainable incentives at the lower
end of the wealth or income distribution. Finally, if group influences and social
interactions are primary determinants of individual aspirations, then it may be necessary
to ask how redistributive policies can affect group memberships.
Furthermore, the results of this paper suggest a number of new avenues for empirical
research. On one hand, they provide a theoretical framework to evaluate the reaction of
agents empirically, in terms of effort, when their relative situation changes. On the other
hand, they describe how relative concern could affect income mobility through the
formation of aspirations.
One problem of general empirical studies on this issue is that they fail to explain the
implications of self selection into reference groups. In our model, we avoid discussing
this issue and consider the parameters that define reference group integration to be a
random variable. Our model demonstrates that reference groups affect income mobility
even
under this
intergenerational
hypothetical
situation.
transmission
However,
of
a model
reference
which focuses
groups
on
is
a possible direction for future research. Alternatively, it is possible to propose an
approach to endogenize reference group choice.
A number of important issues remain to be addressed. First, it would be interesting to
include
a more explicit model
of
aspiration
formation.
Second,
it would be
interesting to analyze the implications of the model on the relationship between income
mobility and allocative efficiency in more detail. Third, the paper focused entirely on
steady states, and ignored issues of non-steady state dynamics. Fourth, the approeach
assumed only two social origins, but this model can be extended to a model where
57
agents have multi-social origins. Fifth, this paper considered two alternative updating
belief processes, but additional functions could be considered. In this sense, we could
assume a dynastic learning process about beliefs about society’s structural mobility
parameters. Sixth, the paper only included income relative deprivation, which is one
dimension of interpersonal comparison. However in section 5 includes two dimensions of
status, as it considers income concern and the public beliefs about one’s ‘smartness’. In
new research we could include other dimensions, such as educational achievement.
Finally, in our model the possibility of strategic behaviour on the part of agents with
different social origins or reference groups is ignored.
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62
Annex I. A
Assume that αθβ M ( y1 − y0 )( yi − y RG )G yy <
1
R
and that there is a unique point, ybreak
a
RG
(and ybreak
), where the function G y R y R (.) changes the sign. Assume first that G y R y R (.) is
R
R
negative when y R < ybreak
and G y R y R (.) >0 when y R > ybreak
. Under these assumptions
reference group income operates as “motivator”. When
R
E ( y R i < ybreak
) , agents will
reduce their effort level, which reduces their opportunities of social ascent. In this case,
relative deprivation is low, and therefore will individual effort investment for the future
R
This situation may capture “complacency effect”. When E ( y R i > ybreak
) , agents with
lower-class origins assign a high effort until E ( yi ) = E ( y RG ) , which increase their
opportunity of social ascent (Figure A1, Graphic A). It is also important to note that the
direction of the effect of relative income on optimal effort remains unchangeable. When
E ( y i ≥ y RG ) , more demanding reference group leads to higher optimal effort level.
R
R
However if G y R y R (.) is positive when y R < ybreak
and G y R y R (.) <0 when y R > ybreak
, then
reference group income could reduce effort motivations and increase income
satisfaction. In this case, low demanding reference group leads to increase in the level of
optimal effort (encouragement effect). However, when reference group income is very
high, agents could perceive that the goal is unattainable, they are discouraged and
reduce their optimal effort (Figure A1, Graphic B. In this case, agents with low origin
aspire to be like high origin agents, but income gap is simply too large. Is that, the cost
of effort is too high and its rewards in terms of income mobility too low. Agents, whose
reference group income is very far away from their current situation have low incentive to
raise their effort (in order to avoid frustration), because the expected gap will remain very
large before and after.
Figure A1
Graphic A
de0 eq
Graphic B
de0 eq
**
dy
RG
**
dy RG
E ( y i < y RG ) E ( y i ≥ y RG )
y RG
E ( y i < y RG ) E ( y i ≥ y RG )
y RG
63
Annex IB
The basic model presented in 3.3, where we assumed G y R e = 0 and Ge = 0 . In this case
the
“inverse
of
effort”
and
relative
income
( U * y ,e ( y , e ) = U ** y r , e ( y r , e ) = −αG y r y r ( y r , e ) < 0 )
when
are
complements
GyR yR > 0 .
Standard
assumptions, rather than prospect theory assumptions, explain convexity of relative
concern. The effect of a larger gap between income levels of the individual and his
reference group is to reduce the relative income of the agent and thus it raises the
marginal utility of his income ( U * y ( y, e) = U ** y r ( y r , e) = (1 − α ) − αG y r ( y r , e) ), inducing
an increase in his effort. In a more “demanding” reference group, he will increase his
effort. However, if G y R y R < 0 , then the “inverse of effort and relative income are no
complements. In this case, the effect of a higher reference group income is to raise the
marginal
disutility
of
relative
effort
( U * e ( y, e) = (1 − α ) Δyθβ m − α ( y1 − y0 )θβ m G y r ( y r , e) ) and to reduce the disutility of
relative deprivation ( U * y ( y, e) = U ** y r ( y r , e) = (1 − α ) − αG y r ( y r , e) ). As a result agents
reduce their effort level.
Annex II. Agents decision process. Step by step
1. At each period t, agents identify expected relative deprivation based on reference
group composition and expected income for each origin.
2. Expected income depends on social origin ( E ( yi | I1 ) and E ( yi | I 0 ) ).
3. The social origin both affects the probability of economic success and the expected
relative deprivation.
4. Expected relative deprivation depends on reference group composition (P) and
expected effort.
5. Agents share the expected effort for each origin, where e0b and e1b is the expected
effort of agents with origin I 0 and I 1 respectively.
6. Individual effort levels are unobservable and beliefs about expected effort are shared
by all the agents. Expected effort depends on observed social mobility experienced by
the previous generation. Agents observe the performance of previous generation,
update their belief and then they identify their expected relative deprivation.
64
7. Agents maximize their expected utility and choose their optimal level of effort. This
decision is affected by the expected effort of his peers in the reference group.
8. A group of agents achieve “economic success” and others fail in their attempt to
achieve this goal.
9. The process continues…
Annex III. Describing the short term dynamic
The analysis of H allows us to describe the short term effort trajectories toward steady
state, but the sign of H *' (e) and H *' ' (e) are undetermined and essentially depends on
the sign and magnitude of
dy R
de
and
d 2 yR
d 2e
:42
⎡
⎛
⎞ dy R ⎤
⎥
H ' (e) = −αa ⎢Gee ( y R , e) + θβ M (Δy )Geyr ( y R , e) + ⎜θβ M (Δy )G yr yr ( y R , e) + Geyr ( y R , e) ⎟
3 14442444
3 ⎜ 144444
42444444
3 ⎟ de ⎥
⎢ 1424
+
⎝
⎠
+
+
⎣
⎦ (23)
or 0 if H (e, e0 M , v) > E
Hence, it is important to discuss the sign of
⎡
⎤
⎢
⎥
dy R ⎢
de j
= (Δy )θβ m (1 + ( P − 1) ω eM (k + k ∑
)⎥
de
1
2
3
⎢
{
i ⎥
dei
−o 0 + or 0 or −
j ≠i
⎢
14424
4
3 ⎥
⎢⎣
⎥⎦
Z ( ei )
dy R
de
, which is defined as:
(24)
Where subscript j and i represent agent with reference group P = Pj and P = Pi
respectively. The term Z (ei ) measures how the average effort responds to changes in
past effort of agents with reference group Pi . On other hand, ω e M measures how
expected effort of agent with origin I 0 , e0b responds to changes in average effort. Both
terms sum the sensitivity of expected effort to shift in past effort. The sign of this
expression is undetermined, because the sign of ω e M depend on random shock vector
υt .
However, it would be interesting to discuss the components of this equation. First we
observe the composition of reference group ( P − 1) . We find that the effect of learning
process on the formation of aspirations will be higher when P is low. The second term is
the sensitivity, those magnitude depend on the quality of signal function. Furthermore,
42
[
H *' ' (e) = − αaGeyr ( y R ) + αaθβ M (Δy )G yr yr ( y R , e)
]
d 2 yR
d 2e
; h*' ' (e) ≥ 0 if
dy R
dede
≤ 0 and h*' ' (e) ≤ 0 if
dy R
dede
≥ 0.
65
the sign of ω e M , depends on the shock. When α (eM t ( ε 't ), vt ' ht −1 ) < α (eM t ( ε 't ), v't ht −1 ,
ω e < 0 ,therefore agents from I0 perceives that effort don’t pay and that previous
M
generation had oversupply effort, then e0b declines. Agents believe that their peers will
t +1
adjust their effort, as a result expected y RG is lower and E(yR I 0 ) higher. We will arrive at
just the opposite results when α (eM t ( ε 't ), vt ' ht −1 ) > α (eM t ( ε 't ), v't ht −1 and ω e M ≥ 0 if
Agents perceive that “effort pays”, and their expected effort for agents with low origin is
higher and their expected relative income are lower. Therefore, agents with low social
origin increase their effort. The third component is Z (ei ) , which describe the response
of the peers I 0 to changes in the effort of their reference group. It is interesting to note
that this term depend on past effort of agents with origin I 0 . As a result, this mechanism
of incentives is endogenous to different generations of agents with origin I 0 .
Annex IV. Long term equilibrium in Piketty (1998)
According to Piketty (1998), function h(e) provides three different possible types of long
terms solutions: If h( E ) < E , since h(0) > 0 and h(e) is convex, there is an unique
intersect with the first bisectrix . There exists a unique equilibrium low effort level, e*<E
(Figure A.2, Graphic A). When h( E ) > E , h(e) is convex, and h(e) does not intersect
with the first bisectrix, there is an unique equilibrium, which is the highest effort level, E
(Graphic B). Finally, if h( E ) > E , h(e) is convex and h(e) intersects twice with the first
bisectrix. There are two stable equilibrium effort levels and one unstable. Among stables
equilibrium, one is low level e*, and the other is high level, E (Graphic C).
Figure A.2
Graphic B
Graphic A
h(e )
Graphic C
h(e )
h(e )
h( e )
h( e )
h( e ) = e
E
h( e ) = e
E
h( e ) = e
h( e )
e*
E
e
E
e
e*
*
E
eun
e
Source: Piketty (1998)
66