Poverty and Aspirations Failure
Patricio S. Dalton
Tilburg University, CentER, Netspar, CAGE
Sayantan Ghosal
Anandi Mani
University of Warwick, CAGE
University of Warwick, CAGE
February 6, 2013
Abstract
We develop a theoretical framework to study the psychology of poverty
and ‘aspirations failure’. In our framework, the rich and the poor share the
same preferences –and also a behavioral bias in setting aspirations modelled as
endogeneous reference points. Poverty exacerbates the e¤ects of this behavioral
bias leading to behavioral poverty traps and aspirations failure. Mitigating
behavioral poverty traps require policies which go beyond reducing material
deprivation.
JEL Classi…cation: O10,O15,O12, D03.
Keywords: Reference-dependent Preferences, Aspirations, Persistent Poverty.
We would like to thank participants at the University of Bath and Universidad del Rosario
seminars, Warwick Workshop on Aspirations and Poverty, CAGE workshop on Inequality, the CESIFO Summer Institute on Behavioral Welfare Economics, the Annual Meetings of the AAEP,
the 2010 RES meetings and the UNU-WIDER conference on Poverty and Behavioral Economics
2011 for helpful comments. Special thanks go to Daniel Aromi, Doug Bernheim, Maria Cubel,
Sharun Mukand, Sendhil Mullainathan, Jens Prufer and Vijayendra Rao for valuable comments.
S. Ghosal and P. Dalton acknowledge support from ESRC-DFID grant RES-167-25-0364. Any remaining error is responsibility of the authors. e-mail: [email protected], [email protected],
[email protected].
1
1
Introduction
Persistent poverty is a condition that requires an understanding of a multidimensional process which makes people poor and keeps them poor.1 The Chronic Poverty
Report (2008-2009) estimates that 320 to 443 million people will live trapped in
chronic poverty: i.e., these people will remain poor for much or all of their lives and
their children are likely to inherit their poverty as well.2 An in‡uential literature
on poverty traps argues that such persistent poverty is driven solely by constraints
that are external to the individual.3 Examples of such constraints are credit or insurance market imperfections (e.g. Loury, 1981; Galor and Zeira, 1993; Banerjee and
Newman, 1991, 1993), coordination problems (e.g. Kremer, 1993), institutional or
governmental failures (e.g. Bardhan, 1997), malnutrition (e.g. Dasgupta and Ray,
1986), neighborhood e¤ects (Ho¤ and Sen, 2005) or even the family system (Ho¤ and
Sen, 2006).
An alternative view highlights the role of internal constraints in perprtuating
poverty traps. Behavioral biases or internal constraints such as myopia, lack of
willpower and lack of aspirations are often cited as traits that the poor likely suffer from and have been identi…ed as possible explanations for poverty traps.4 In an
in‡uential contribution, anthropologist Arjun Appadurai (2004) argued that the poor
may lack the capacity to aspire and policies that strengthen this capacity could help
them to "contest and alter the conditions of their poverty" (p. 59)5
Unlike with external constraints, it is not clear whether such internal constraints
are the cause of poverty –or its consequence. Do the poor become and remain poor
because they lack aspirations –or, in the words of Bertrand et al. (2004, p. 1), is it
1
Persistent poverty is de…ned by the incapability to ful…ll basic needs during a period greater
than 5 years. For evidence on persistent poverty see Jalan and Ravallion (1998), Fouarge and Layte
(2003), Biewen (2003), Duncan et. al. (1993), among others.
2
About 40% of the poverty in Sub-Saharan Africa and 35% of the poverty in South Asia is
persistent. The probability of remaining poor over a 5-year period is about 50% in Vietnam, 40%
in Ethiopia and Philippines and 35% in India and Bangladesh.
3
See, for example, Azariadis and Stachurski (2004) or Azariadis (2005) for a literature review on
Poverty Traps due to such external constraints.
4
Data from the World Values Survey show that 60 percent of Americans think that the poor "are
lazy or lack willpower" (Alesina et al., 2001).
5
The capacity to aspire is seen as a cultural capacity related to the way in which people visualize
the future and engage in forward looking behavior.
2
that ‘the poor may exhibit the same basic weaknesses and biases as do people from
other walks of life, except that in poverty [...] the same behaviors [...] lead to worse
outcomes’?
In this paper, we examine this latter view of internal constraints and poverty traps
rigorously. To understand the psychology of poverty and low aspirations, we examine
an important behavioral bias (or ‘internal constraint’) that all individuals, whether
rich or poor, su¤er from: they underestimate how their aspirations evolve over their
lifetime, as a consequence of their e¤ort.
Both the rich and the poor su¤er from this bias, but poverty imposes additional
external constraints on the poor that greatly exacerbates the adverse e¤ects of the
behavioral bias in setting aspirations. To capture these additional external constraints
imposed by poverty, we assume that the …nal wealth is proportional to the individuals’
initial wealth.6 That is, other things equal, the poor have to make a higher e¤ort
than the rich to achieve the same level of …nal wealth. This assumption is a reduced
form representation that encompasses di¤erent external constraints or barriers that
are intrinsic to the condition of poverty and which are complementary to individual
e¤ort. Notably, such barriers need not be necessarily located solely in the dimension of
initial wealth but could arise as a consequence of poor material circumstances, eg. less
in‡uencial contacts or fewer positive role models, less access to relevant information
or any external circumstance that makes more di¢ cult for the poor to achieve a
given outcome. In this paper, we work with the above reduced form representation of
these external constraints because our focus is on aspirations. We show that external
constraints make the poor more susceptible to an aspirations failure, i.e. a failure to
aspire to, and achieve, their own best possible outcome.
Our formulation of aspirations is based on three premises well-grounded in the
behavioral economics literature, as well as in evidence from across the social sciences
(which we present in Section 3). First, a person’s aspirations level is a reference
point7 which a¤ects the ranking over e¤ort via the satisfaction a person receives from
a speci…c level of …nal wealth. A higher aspirations level could adversely a¤ects the
6
This assumption is commonly made in models that generate steady state wealth distributions
consistent with Parteto’s law (Champernowne, 1953; Stiglitz, 1969). There is considerable empirical
evidence con…rming Pareto’s law in high income countries such as UK and Germany (Atkinson and
Harrisson, 1978) and in low income countries such as India (Sinha, 2005).
7
This idea dates back to Simon (1955) and Selten (1998).
3
satisfaction a person receives from a particular outcome;8 on the ‡ip side, higher
aspirations could also spur greater e¤ort (see Section 3 for systematic evidence on
this).
The second key premise of our model is that aspirations are endogenous references
point.9 Due to this endogeneity, rational individuals should not choose their e¤ort
and aspirations independent of each other, although this would require a non-trivial
forward looking exercise that involves a two-way feedback: achieving higher aspirations requires greater e¤ort –but higher e¤ort spurs greater aspirations too, through
the wealth obtained. In the words of Aldous Huxley: “ every ceiling, when reached,
becomes a ‡oor...”10
If individuals were fully rational, they would recognize how their aspirations shift
with every achievement, and would choose their e¤ort level accordingly. In practice,
however, people typically fail to fully recognize this latter feedback; they are only able
to see one step above, at a time. This is our third key premise. An aspirations failure
occurs when the individual doesn’t anticipate the feedback of his current e¤orts on his
aspirations across periods. This behavioral shortcoming can be likened to a form of
projection bias (Loewenstein et al., 2003) –inasmuch as people’s current state limits
their ability to correctly project their aspirations reference point in a di¤erent state.
While both the poor and the rich are equally a- icted by such a bias, the more
stringent external constraints that the poor face make them more susceptible to an
aspirations failure. The intuition underlying this result is as follows. Think of two
8
See, for instance, Medvec et al. (1995) who study the expectations and emotions of Olympic
athletes and …nd that bronze medal winners tend to have a higher level of satisfaction than silver
medal winners.
9
Conceptually, our idea of endogenous aspirations is in line with Köszegi’s (2010) concept of
personal equilibrium, in which agents derive utility from physical outcomes as well as from rational
beliefs about physical outcomes (“anticipation”), and these two payo¤ components can interact. See
also Köszegi and Rabin (2006) on reference-dependent preferences, as well as Shalev (2000) (who
originally modelled this point in the context of non-cooperative games) and Dalton and Ghosal (2013)
(who provide an axiomatic characterization of decision-making for a general class of preferences
that includes endogenous reference dependent preferences). The speci…c approach we use to model
endogeneous reference-dependent preferences via value functions is also related to Bogliacino and
Ortoleva (2011) and Genicot and Ray (2011). We discuss the link between our paper and these
other papers later in the introduction and elsewhere as appropriate.
10
Stutzer (2004) shows evidence that the higher the achievement of an individual, the higher is
her aspirations.
4
individuals who have the same initial aspiration level, one rich and the other poor. At
this given aspiration level, the poor person would optimally choose a lower e¤ort level
than the rich one, due to a lower marginal bene…t from e¤ort driven by more stringent
external constraints. However, the feedback from e¤ort to aspirations implies that
the lower e¤ort of the poor person will cause his aspiration level to diverge from that
of the rich person. Thus, in equilibrium, the poor person has two reasons to put in
low e¤ort: not only are his net bene…ts lower, his aspiration level, i.e. the reference
point which determines his marginal bene…t of e¤ort, is endogenously lower as well.
In this sense, poverty curtails a poor person’s capacity to aspire, in the spirit of
Appadurai (2004). As Banerjee and Du‡o (2011, p. 92), talking about the reasons
for poor education outcomes in developing countries, put it "...The teacher ignores
the children who have fallen behind and the parent stops taking interest in their
education. But this behavior creates a poverty trap even where none exists in the
…rst place. If they give up, they will never …nd out that perhaps the child could have
made it. And in contrast, families that assume that their children can make it, or
families that don’t want to accept that a child of theirs will remain uneducated, which
tend to be, for historical reasons, more elite families, end up con…rmed in their ‘high’
hopes."
At this point, it is important to clarify the modeling approach we use in this paper.
Arguably, the motivation o¤ered above suggests some adaptive dynamic mechanism
in which aspirations at any given time adapt to e¤ort chosen in the past via the
outcome realized. It turns out, however, from a modeling viewpoint, all that is
needed is a (static) model in which individuals take aspirations as given when choosing
e¤ort while aspirations are required to satisfy a rational expectations condition given
e¤ort.11 We interpret the outcomes of our model as corresponding to the steady states
of an adaptive dynamic mechanism. In our model, there is the possibility of multiple
welfare ranked (behavioral) equilibria with initial (external) wealth determining the
likelihood of ending up at the equilibrium corresponding to lower e¤ort and lower
aspirations: we call such a situation a behavioral poverty trap. Moreover, we also
show that two individuals with the same level of initial wealth can end up in di¤erent
outcomes if they di¤er in their cost of e¤ort.
Two types of poverty traps emerge from our analysis: standard poverty traps that
11
The rational expectations condition required is equivalent to perfect foresight in our deterministic
model.
5
are driven by external constraints, but also behavioral poverty traps characterized by
low e¤ort and low aspirations. While external constraints imposed by poverty make
internal constraints more consequential, the latter becomes an independent source of
disadvantage in behavioral poverty traps. Our model suggests that it is possible to
break a poverty trap by shocking aspirations alone. Therefore, policy approaches that
in‡uence aspirations among the poor are essential to break this latter kind of trap.
This paper contributes to the emerging literature that examines the behavioral
aspects of poverty. Ray (2006) provides a discussion of how socially determined aspirations contribute to poverty persistence: this is the starting point of our paper. A
closely related paper is Genicot and Ray (2011) which models aspirations as socially
determined reference points via exogeneously speci…ed aspiration windows. Other
closely related papers on aspirations include Bogliacino and Ortoleva (2011), and
Stark (2006). All these papers have the feature that aspirations are purely socially
determined. Our paper has di¤erent, but complementary, focus on constraints internal to the individual. Another di¤erence is that these papers have more macro focus
than ours: they study the e¤ect of aspirations on income distribution and growth.
Using a di¤erent modelling approach- in a setting with dynamically inconsistent
preferences- Banerjee and Mullainathan (2010) and Bernheim et al. (2013) study
how poverty traps may arise due to a di¤erent behavioral constraints that attempt
to mitigate self-control. Internal constraints in our setting arise for a di¤erent but
complementary set of reasons.
Our paper contributes to a small but growing literature that formally models how
psychological constraints internal to an individual can result in a poverty trap. Other
economists have begun to recognize the role of such internal constraints in perpetuating poverty. Du‡o (2012) talks about how a lack of hope can cause an individual "to
rationally decide to hold back his or her e¤orts, avoid investment, and thus achieve
even less than he or she could otherwise have attained". This channel is closely related
to our own: as the author points out in this lecture: "Hope can fuel aspirations...In
turn, these aspirations can a¤ect behavior". A second distinct channel that has been
studied is the adverse impact of poverty on individual cognitive capacity, due to the
greater demands it imposes on mental resources required for decision-making (Mani et
al. 2012). Irrespective of the speci…c channel, this emerging body of work highlights
an important lacuna in anti-poverty policy: a failure to appreciate the role of constraints internal to individuals in perpetuating poverty. As the eminent psychologist
6
Albert Bandura puts it: "failure to address the psychosocial determinants of human
behavior is often the weakest link in social policy initiatives. Simply providing ready
access to resources does not mean that people will take advantage of them." (The
Psychologist, 2009, p. 505)12
The remainder of the paper is organized as follows. Section 2 describes some
important patterns associated with poverty persistence, aspirations and discusses
alternative explanations. Section 3 presents the formal model of aspirations and
e¤ort choice. Section 4 examines the channel through which poverty increases the
likelihood of an aspirations failure. Section 5 outlines the conditions under which
poverty traps may emerge due to internal, rather than external constraints. Section
6 discusses the policy implications of our analysis and Section 7 concludes. Proofs of
all propositions are collected in the appendix.
2
Aspirations and Poverty
Persistent poverty is an issue that has been of central focus in economics for several
decades. In this section, we discuss evidence that suggests, despite the plausibility
of the existing explantations in the economic literature, there are important aspects
of this phenomenon that aren’t adequately accounted for. We review the existing
research on poverty persistence from a range of disciplines in the social sciences, and
make our case for the theory presented here.
Social psychologists have extensively documented the psychological traits associated with poverty. In particular, the lack of aspirations are a typical characteristic
in the personality of the poor population and Arjum Appadurai (2004) is not the
only scholar who has pointed it out. Moreira (2003), for example, studied the poor
in the North-eastern Brazil and provides evidence that the greatest part of the poor
population has nihilistic characteristics, submitting themselves to the destiny that is
‘given by God’.13 Similar patterns have been found elsewhere, for instance, in the Appalachian folk subculture (Rabow et al., 1983), in low-income urban neighborhoods
12
This quote and the material below is based on a lecture by Alberto Bandura to the British
Psychological Society of which an edited version was published in The Psychologist (2009).
13
Atkinson (1998) de…nes social exclusion as a related concept that involves agency (people may
exclude themselves) and it implies future hopes and aspirations. People are excluded not just because
they are currently without a job or an income but because they have little prospects for the future.
7
in America (MacLeod, 1995), among Jamaican male youths (Walker, 1997) and in
rural Ethiopian households (Frankerberger et al., 2007). Bernard et al. (2011) report
evidence of low aspirations among a substantial group of rural households in Ethopia.
They also …nd that such psychological traits consistently correlate with lower demand
for credit, in terms of loan size, repayment horizon and productive purposes. Similar
evidence is found in poor people from developed countries. Data from the Longitudinal Study of Young People in England (2006) show that young people from deprived
backgrounds have the lowest academic aspirations across all income quintiles. (see
Cabinet O¢ ce, 2008).
Poverty and Low Aspirations: Lack of Opportunity or Information?
Arguably, empirical evidence of low aspirations among the poor can well be originated purely by a lack of opportunity, or a lack of information about pathways out of
poverty. A poor person may not want to aspire to be a lawyer because he wouldn’t
have the funds to pay his studies (i.e. he doesn’t have the opportunity). Objectively,
being a lawyer is not an achievable status for this person and it is entirely rational
not to aspire to it. In this case, enlarging the opportunity set would su¢ ce for the
person to aspire higher and eventually become a lawyer. However, this opportunity
channel alone is inconsistent with very recent empirical evidence from …eld experiments showing that people do not always take advantage of the opportunities at
hand. For example, in Kenya, Miguel and Kremer (2004) provide evidence that only
57% of the sample picked-up the free deworming pills, which were shown to greatly
improve children’s health and school performance. Similarly, Du‡o et al. (2010) documented very low rates of fertilizer use take up by maize farmers in Kenya despite
being o¤ered convenient opportunities to buy fertilizer at reasonable prices. Banerjee
et al. (2011) report on the take up of an asset assistance and training program for
the ultra-poor in West Bengal, India. They …nd that 36% of the households who are
o¤ered this program did not take up the assistance –even though those who did take
it up reported very signi…cant increases in consumption and well-being.
A second possible explanation for low aspirations among the poor is that they suffer from an informational disadvantage: they simply don’t know about the bene…ts of
certain opportunities. However, the available evidence is not fully convincing on this
count either. For instance, farmers in Busia, Kenya (mentioned above) had ample
opportunity both to learn how to use the fertilizer, and to realize that the rates of
return from its use were as high as 70%. Take up was low despite this. In a somewhat
8
di¤erent context, Jensen (2010) reports the results of a …eld experiment in the Dominican Republic, where students were informed about the actual return di¤erential
between primary and secondary/tertiary education, which they had previously underestimated. There was a substantial increase in perceived returns to education –but
almost no discernible e¤ect on the actual rates of completion of secondary schooling.
Such lack of responsiveness among the poor suggests that constraints imposed by the
lack of opportunity or access to information alone do not fully explain the stubborn
persistence of poverty –or the psychological traits associated with it.
Low Aspirations: Cause or Consequence of Poverty?
In this paper, we argue that low aspirations among the poor are the consequence
of an interaction between their exogenous disadvantages and an aspiration bias which
may (although not necesarily) be shared with the rich. The fragile equilibrium of life
riddled with exogenous disadvantages lowers the e¤ort that the poor choose to put in
towards any goal, other than their immediate needs. Part of this is rational –except
that the aspirational bias hinders their ability to foresee how greater investment of
e¤ort today could put them on a higher outcome trajectory in the long run.
An implication of such aspiration bias is that the poor over-estimate the initial
endowment needed for success. For instance, Banerjee and Du‡o (2011) cite the case
(among several others) of a poor woman with six children in the village of Naganadgi
in India, who explained why she had enrolled only one of her children in school: the
rest were not intelligent enough to make it through. Gutman and Akerman (2008)
estimate that parental interest and expectations in the child’s education has four times
more in‡uence on attainment by age 16 than does socioeconomic background.14
To summarize, poverty sti‡es aspirations of the poor because higher exogenous
dissadvantages adversely a¤ects their e¤ort choices –and hence their aspirations. The
less e¤ort they invest in getting out of poverty, the less they are going to aspire, and
the fardest they will be from achieving their potential. Hence, providing information
or opportunity alone (i.e., relaxing the external constraints) may not be enough to
draw out people caught in persistent poverty. We now provide a formal exposition of
the ideas expressed above.
14
It is not as if parents make such choices about picking only one child to educate, because of
credit constraints either.
See Ashraf et al. (2008) for evidence on this, based on a randomized
experiment o¤ering education lotteries.
9
3
The model
3.1
Preferences: Aspirations as Reference Points
We consider an individual characterized by a given level of initial wealth
;
0
2
=
a bounded subset of <+ . He must choose a costly e¤ort e 2 [0; 1] that will
determine his …nal wealth
= f (e;
0)
2
, which is assumed to be an increasing
function of e¤ort. The individual has an aspiration level (or goal) g 2 <+ with regard
to his …nal status.
For any given initial status
0,
the utility the individual derives from choosing an
e¤ort level e depends not only on the cost of e¤ort and the bene…t of achieving a
particular status , but it also depends on his aspirations, as described by the utility
function15 below:
u(e; g;
0)
= b(f (e;
0 ))
+v
f (e; 0 ) g
f (e; 0 )
c(e)
(1)
The utility function of the individual has three additive components. The …rst
component is the bene…t of reaching a status: b ( ) assumed to be a smooth, increasing, concave function over …nal status with b(0) = 0. A key preference parameter
that will be relevant for us is the co-e¢ cient of relative risk aversion r( ) =
b00 )
.
b0 ( )
Speci…cally, in deriving complementarity between e¤ort and initial wealth, we will
assume that r( ) < 1, an assumption satis…ed by a number of commonly used utility
functions in the literature.
The second component, v(
g
), is a continuous di¤erentiable reference-dependent
value function that captures one of the key premises of the model: i.e., that individual
aspirations level g is a reference point that a¤ects the satisfaction experienced from
achieving a …nal outcome = f (e; 0 ). Speci…cally, we assume that it is proportional
gains and losses of achieved status relative to the aspired status that matter for the
individual. We make a crucial assumption on the shape of the value function, which
ensures that setting aspirations high can induce frustration but it also motivates
g
higher e¤ort. For x
, we assume that [v 0 (x) v 00 (x)(1 x)]
0 for all x.
In Appendix 1, using two di¤erent speci…c formulations of v(:) (an S-shaped value
15
We develop the simplest possible model that allows us to focus on the two way link between
aspirations failure and poverty traps. Admittedly, there are several features of our model that are
best interpreted as reduced form representations of a more detailed model.
10
function16 and a value function with a bliss point), we construct a number of examples
all of which satisy this assumption.
There is considerable evidence from the literature in both psychology and economics, that higher aspirations are motivators of greater e¤ort. For instance, Heath
et al. (1999) present evidence that subjects with high goals exert higher e¤ort and persist more in di¤erent physical and cognitive tasks. The Dunedin Longitudinal Study
in New Zealand suggests that pessimistic expectations signi…cantly increase the likelihood of low e¤ort even in non-market activities such as personal health maintenance
(more frequent smoking and less frequent exercise) (Clark et al., 2003). Abeler et
al. (2011) found similar results in the lab: when participants have higher reference
points for earnings, they persevere longer at the experimental task. Aspirations also
act as reference points for life goals. In a …eld experiment with female entrepreneurs
in India, Field et al. (2009) show that higher aspirations motivate positive changes
in women’s …nancial behavior.
Finally, the third component is the cost of e¤ort c(e), which is assummed to be
smooth, increasing and convex with c(0) = 0.
3.2
How Poverty Imposes External Constraints
We assume that the poor face greater external (resource) constraints than the rich
which e¤ectively reduce their productivity. This could happen in myriad ways –for
instance, their lack of access to credit could render their e¤orts to acquire skills or run
a successful business less e¤ective. Lack of access to information or in‡uential social
networks could make it harder for them to …nd jobs than a rich person who puts in the
same e¤ort. In what follows, we capture such external constraints with assumptions
on the "production function" of …nal outcome f (e; 0 ) as any environmental factor
that makes the …nal wealth proportional to initial wealth, for a given e¤ort level.
We assume that (i) if the individual puts in zero e¤ort, his …nal wealth equals to his
initial wealth (i.e. f (0; 0 ) = 0), (ii) …nal wealth is increasing in e¤ort for any level
0)
of initial wealth ( @f (e;
> 0), and (iii) that …nal wealth is bounded below by initial
@e
wealth (i.e. f (e;
16
0)
0 ).
It follows that the marginal product of e¤ort is increasing
Genicot and Ray (2011) study such a value function de…ned over absolute gains and losses.
As Appendix 1 makes clear, this value function is concave in gains but convex in losses. Clearly it
cannot satisfy loss aversion in a small neighborhood of zero (as we require continuous di¤erentiability
at zero) but can be constructed so that it satis…es loss aversion outside this neighborhood.
11
in initial wealth. In other words, e¤ort and initial status are complements in the
production of …nal wealth. For ease of exposition, we assume a speci…c production
function, f (e;
0)
=
0 (1
+ e); which satis…es the restrictions on f (e;
0)
introduced
above.17
4
E¤ort and Aspirations
4.1
Aspirations as Endogenous Reference Points
What determines individual aspirations? No doubt, there could be multiple in‡uences. Environmental factors such as a person’s family background, the norms of the
community in which he lives and the opportunities available, economic or otherwise
do matter – as well as traits internal to the individual. In this paper, given its focus on modelling internal constraints, we assume that individuals determine (at least
partially) their aspiration level, rather than have it set for them by purely exogenous
factors.
Given that aspirations a¤ect the satisfaction from the outcome realized, we assume
that agents would aspire only to an outcome that is perceived as attainable. Therefore,
the e¤ort that agents put in a¤ects the outcome they aspire to, i.e. aspirations are
endogenous to e¤ort choice. In other words, our second key premise is that aspirations
are endogenous reference points that satisfy a rational expectations condition. As
ethnographer MacLeod’s (1995, p.15) points out "aspirations re‡ect an individual’s
view of his or her own chances for getting ahead". With this interpretation in mind,
we de…ne an aspirations level g as the …nal outcome attained, given individual e¤ort:
g = f (e;
4.2
(2)
0)
E¤ort and Aspirations Choice of a Rational Decision
Maker
So far, we have laid out two key premises: (i) that aspirations are reference points
that a¤ect our utility from achieving a particular status –but (ii) they are endogenous
17
All our results, with suitable modi…cation in the precise expressions (and therefore, assumptions)
reported below, can be obtained for any f (e;
0)
s.t. fe (e;
12
0)
> 0, f 0 (e;
0)
> 0 and fe; 0 (e;
0)
> 0.
reference points, in that they are a¤ected by e¤ort choices. Taken together, these
two premises imply a two-way feedback e¤ect between aspirations to e¤ort. Let us
now carry out the following thought experiment. Consider a fully ‘rational’person,
one who internalizes the fact that in choosing an e¤ort level, he is also a¤ecting
his aspirations. How will such a far-sighted person optimally choose his e¤ort and
aspirations level? We formalize the answer in the concept of a standard solution, as
de…ned below:
De…nition 1. A standard solution is a pair (^
e; g^) such that
e^ 2 arg max s(e;
e2[0;1]
0)
= u(e; f (e;
0 ); 0 )
(3)
and
g^ = f (e;
(4)
0)
As De…nition 1 shows, a rational person recognizes that e¤ort and aspirations
feed into each other over. Given this positive feedback, he would jointly choose both
e¤ort and aspiration levels, so as to achieve his best possible status and utility. We
certainly do not claim that most individuals are far-sighted enough to recognize this
feedback, and make decisions in this manner. Rather, this provides us with a normative benchmark against which we evaluate how most people set their aspirations.18
) f (e; 0 )
Note that at a standard solution, by de…nition, f (e; f0(e;
= 0 so that
0)
s(e;
0)
= b(f (e;
0 ))
c(e):
Under the assumptions made so far, a standard decision problem is a well-de…ned
strictly concave maximization problem with a unique solution. From a normative
perspective, the value function is irrelevant in ranking e¤ort and aspiration pairs.
The following proposition characterizes the set S ( 0 ) of standard solutions and
state the conditions under which e¤ort and initial status are complements.
Proposition 1: For a …xed value of
0,
there exists a unique standard solution
level of e¤ort and aspirations (b
e ( 0 ) ; gb ( 0 )). If the coe¢ cient of relative risk aversion
of the bene…t function is less than one, r( ) < 1, (b
e ( 0 ) ; gb ( 0 )) is increasing in 0 .
18
There are compelling normative reasons why preferences of a standard decision-maker should
be the normative benchmark. As long as aspirations are endogenous reference-point (as evidence
shows), we believe that internalizing its endogeneity is arguably a natural normative standard.
After all, the internalization of the consequences of the decision-maker actions has been always the
normative standard used by mainstream economics. We refer the interested reader to Dalton and
Ghosal (2013) for more details.
13
Proof. See appendix.
Proposition 1 tells us that a fully ‘rational’poor person would aspire lower than an
equally ambitious richer counterpart: at any given e¤ort level, his achieved status is
lower due to the complementarity between e¤ort and initial status in the production
function. Moreover, setting an aspiration (reference point ) that is as high as his richer
counterpart would only diminish the utility he would derive from any achievement.
Of course, given his weaker initial condition
0,
this more modest aspiration and e¤ort
choice of a poorer person cannot be regarded as an aspiration failure.
4.3
E¤ort and Aspirations Choice of a Behavioral Decision
Maker
Admittedly, most people setting their life’s aspirations are not rational in the sense
described above. Speci…cally, in carrying out this forward-looking exercise, they do
not fully internalize how their aspirations are shaped by their e¤ort choices. We refer
to decision-makers with such lack of foresight as ‘behavioral’ decision-makers (and
that includes most of us!).19 Our third central premise then is that while choosing
e¤ort e, a behavioral decision-maker takes his aspired status g as …xed (rather than
endogenously evolving with e¤ort and achieved status), hence imposing an externality
on himself that isn’t fully internalised.20
There is considerable evidence of this kind of behavior in various kinds of life situations. Easterlin (2001), for example, provides evidence that people do not anticipate
how their aspirations adapt upwards, as their income rises. In a similar vein, Knight
and Gunatilaka (2008) present …eld evidence of rural migrants settled in urban areas who don’t foresee how their aspirations will adapt to their new situation, hence
ending up less happy than non-migrants in both locations.
Formally, a behavioral decision-maker chooses e, while taking g as given, to solve
19
Our framework is also consistent with a scenario where the decision-maker partially internalizes
the feedback from e¤ort to aspirations with probability . In such scenario, the decision-maker in a
behavioral decision chooses e¤ort to maximize u
~(e; g) = u(e; g) + (1
)v(e). This is analogous to
the formulation adopted in Lowenstein et al. (2003)’s model of projection bias.
20
The premise itself implies that choices and preferences could diverge, hence invalidating the
principle of revealed preferences. See Dalton and Ghosal (2013) for generalized, axiomatic characterization of standard and behavioral decisions using choice correspondences alone.
14
M axe2[0;1] u(e; g;
(5)
0)
While his aspiration g is exogenous to his e¤ort e , we will require that his aspirations and e¤orts are mutually consistent, i.e., he has rational expectations in choice
of aspirations and e¤orts. Let e(g; 0 ) denote the set of payo¤ maximizing e¤orts.
Formally,
De…nition 2: A Behavioral Solution is a pair (e ; g ) such that (i) e 2 e(g ;
0)
and (ii) g = f (e ; 0 ).
The following Lemma states a condition on the curvature of the value function
that ensures that e¤ort and aspirations are complements.
Lemma 1: Let x =
given initial status
0,
f (e; 0 ) g
f (e; 0 )
and assume [v 0 (x) v 00 (x)(1 x)]
e¤ort and aspirations are complements (i.e.
0 for all x. Then,
@ 2 u(e;g;
@e@g
0)
0).
Proof. See appendix.
We are now in a position to state the following result characterizing the set B ( 0 )
of behavioral solutions for a given value of
Proposition 2:
Let x =
f (e; 0 ) g
.
f (e; 0 )
0.
Assume that [v 0 (x)
v 00 (x)(1
x)]
0 for
all x. Then, for a …xed value of 0 , there exists a minimal and maximal e¤ort levels
in e(g), e(g; 0 ) and e(g; 0 ), both of which are increasing in g. Moreover, there exists
minimal and maximal e¤ort-aspiration pairs in the set of behavioral solutions B ( 0 ),
e ( 0 ) ; g ( 0 ) and (e ( 0 ) ; g ( 0 )).
Proof. See appendix.
4.4
Internal constraints
Proposition 2 shows that when an individual ignores how his aspirations are in‡uenced
by his e¤orts, there are multiple levels of e¤ort and aspirations that are solutions to
his optimization problem. Across these behavioral solutions, higher e¤ort levels are
paired with higher aspirations levels. Thus, the lack of foresight in a behavioral
decision-maker creates room for the possibility that he may choose a lower e¤ortaspiration pair, than the best possible one he could expect to achieve. We refer to
such a person as being internally constrained, and such an outcome as an aspiration
failure. Formally:
De…nition 3. For a given
0,
an individual is internally constrained at a behav-
ioral solution (e ; g ) if (e ; g ) 2
= S ( 0 ).
15
For a given 0 , when the intersection between the set of behavioral solutions and
the set of standard solutions (i.e. B ( 0 ) \ S ( 0 )) is empty, the individual is always
internally constrained.
In principle, however, some of the multiple e¤ort-aspiration solutions (e ; g ) described in Proposition 2 could coincide with the standard (maximal) outcome. This
raises a number of questions. First, what are the necessary and su¢ cient conditions
under which this can happen, i.e. (e ; g ) 2 S ( 0 )? Second, in a behavioral decision,
when does the individual choose a lower (respectively, higher) e¤ort-aspiration pair
than the one chosen in a standard decision? The following proposition provides an
answer to both questions:
Proposition 3 Assume that [v 0 (x)
v 00 (x)(1
x)]
0 for all x. For a given
0:
(i) an interior standard solution (i.e eb 2 (0; 1)) is also a behavioral solution if and
only if v 0 (0) = 0;
(ii) if v 0 (0) < 0 (respectively, v 0 (0) > 0), at each behavioral solution the individual
chooses a lower (respectively, higher) e¤ort and aspiration level than at a standard
solution i.e. (e ; g )
(^
e ( 0 ) ; g^ ( 0 )) (respectively, (e ; g )
(^
e ( 0 ) ; g^ ( 0 ))) for
each (e ; g ) 2 B ( 0 ) with strict inequalities if the standard solution is an interior
solution.21
Proof: See appendix.
Thus, typically, when an individual fails to anticipate the positive feedback of his
e¤ort choices on his life aspirations, he will end up choosing a suboptimal, e¤ortaspiration pair than the best outcome he can achieve. If the value function is downward sloping at a behavioral equilibrium (where the relative gain or loss is zero by
de…nition) then he will choose an e¤ort and aspirations pair lower than what is optimal. On the other hand, if the value function is upward sloping at a behavioral
equilibrium then the individual will necessarily choose an e¤ort and aspirations pair
higher than what is optimal. The intuition for this is as follows. When v 0 (0) < 0
(respectively, v 0 (0) > 0), at a behavioral equilibrium, the marginal gain from e¤ort is
lower (respectively, higher) than the increase in the value of b(:) due to extra e¤ort so
that the e¤ort level is lower (respectively, higher) than the e¤ort chosen by a standard
decision-maker.
21
In the appendix, we provide examples of value functions that satisfy both v 0 (0) > 0 and v 0 (0) < 0
and also satisfy the inequality [v 0 (x)
v 00 (x)(1
x)]
16
0 for all x.
5
Behavioral vs. Standard Poverty Traps
In this section, we relate our analysis so far to poverty persistence and pathways
out of it. A key feature of our model is that it allows us to study, within a unique
framework, the e¤ectiveness of multiple kinds of policy interventions–those that aim
to relax external constraints, but also those that work on relaxing internal constraints.
We examine how initial conditions may in‡uence the likelihood of aspiration failure
using
of our framework with two e¤ort levels e 2 f0; 1g and
( a simpli…ed version
)
> 0 if e = 1
c=
.
= 0 if e = 0
For later reference, it will be convenient at this point to state and prove the
following lemma:
Lemma 2: (i) Let h( 0 ) = b (2 0 )
r( ) < 1 for all > 0.
(ii) Suppose [v 0 (x)
v 00 (x)(1
x)]
b ( 0)
c. Then h0 ( 0 ) > 0 if and only if
0 for all x. Then,
v
1
2
v( 1).
Proof. See appendix.
We now state the following result that characterizes a standard solution and a
behavioral solution respectively as a function of
0:
Lemma 3: Suppose r( ) < 1 for all > 0 and [v 0 (x) v 00 (x)(1 x)] 0 for all
x. Then:
(i) There exists ^ s.t. (a) if 0 ^, at a standard solution, eb ( 0 ) = 0, gb ( 0 ) = 0 ,
and (b) if 0 ^, at a standard solution eb ( 0 ) = 1, gb ( 0 ) = 2 0 .
(ii) There exists H
L s.t. for (a)
e ( 0 ) = 0, g ( 0 ) = 0 ; (b) L
0
e ( 0 ) = 0, g ( 0 ) =
0
the unique behavioral outcome is
H : there are multiple behavioral outcomes
0
L:
and e ( 0 ) = 1, g ( 0 ) = 2
0
and (c)
0
H:
the unique
behavioral outcome is e ( 0 ) = 1, g ( 0 ) = 2 0 .
Proof. See appendix.
Whenever 0 < ^, the individual is caught in a standard poverty trap driven solely
by material deprivation: there exists wealth levels so low that the incremental bene…t
from the anticipated increase in wealth outcome is dominated by the cost of high
e¤ort.
If
L
H,
0
each value of
0
the presence of multiple behavioral outcomes associated with
implies that even small changes in initial wealth can lead to very
di¤erent outcomes. Let’s now study in more detail how a behavioral poverty trap
17
may emerge. For that, we need to examine how a behavioral outcome pair of e¤ort
and aspiration level is selected. Consider the initial aspiration level g0 of an individual,
which is drawn from a distribution with pdf m(:) (and cdf M (:)) common to the rich
and the poor. g0 is irrelevant for a fully ‘rational’(standard) decision-maker, because
he internalizes the feedback from e¤orts to aspirations. Therefore, he will always only
pick a standard solution as his e¤ort/aspiration choice, no matter what his g0 is. A
behavioral decision-maker’s e¤ort choice, in contrast, will be a¤ected by g0 , as he
takes his aspiration level as given, rather than endogenous to his e¤ort.
The selection mechanism involves two stages: (i) given a randomly generated
initial aspiration level g0 , the individual chooses an e¤ort level e; (ii) given e, the
aspiration level (i.e. anticipated outcome) adjusts via the function g = f (e;
0 ).
Given the above selection mechanism, the following proposition addresses how
poverty and initial disadvantage interact to generate a behavioral poverty trap characterized by low e¤ort and aspirations failure.
Proposition 4: Suppose r( ) < 1 for all
for all x. Then, the lower the initial wealth
0
> 0 and [v 0 (x)
v 00 (x)(1
x)]
0
of an individual, the more likely he is
to experience a behavioral poverty trap i.e. end up at an e¤ort and aspiration level
that perpetuates his initial status.
Proof: See appendix.
Proposition 4 provides an explanation for the empirical observation of poor people
holding low aspirations (as shown in section 2) and not realizing their full potential. To understand the intuition underlying Proposition 4, think of two behavioral
decision-makers who have the same initial aspiration level, one rich and the other
poor. There are just two e¤ort levels, high and low e¤ort. The complementarity
between goals and outcomes implies that poverty raises this threshold level ge ( 0 )
above which initial aspirations have to lie for a poor person to choose higher e¤ort.
Therefore, for a …xed level of aspirations, a poor person is more likely to choose low
e¤ort than a richer person.
Given that aspirations adjust to e¤ort, the feedback
from e¤ort to aspirations implies that the lower e¤ort of the poor person will cause
his aspiration level to diverge from that of the rich person. Thus, in equilibrium, the
poor person has two reasons to put in low e¤ort: not only is his initial wealth lower,
his aspiration level, i.e. the reference point which determines his marginal bene…t of
e¤ort, is endogenously lower as well. In this sense, poverty curtails a poor person’s
capacity to aspire, in the spirit of Appadurai (2004).
18
Thus, the gist of our analysis so far is that, far from being an innate traits of poor
people, low aspirations emerge as an equilibrium outcome as a consequence of their
initial disadvantage.
It remains to examine the welfare implications of the two types of poverty traps
we study here. We proceed as follows. We …x 0 and compare the payo¤ of an agent
caught in a behavioral poverty trap with the payo¤ the same agent would obtain at
a standard solution:
Proposition 5: Suppose r( ) < 1 for all > 0 and [v 0 (x) v 00 (x)(1 x)]
0
0
^
^
for all x. When v (0) < 0, so that < H , and for each 0 2 [ ; H ), a behavioral
poverty trap is welfare dominated by the standard solution. When v 0 (0) > 0, so that
^
H < , there is no 0 for which a behavioral poverty trap is welfare dominated by the
standard solution.
Proof: See appendix.
When the value function is downward sloping at a behavioral equilibrium, the
level of initial wealth at which the switch to a high e¤ort and high aspirations pair
occurs at a standard solution (^) is lower than the level of initial wealth at which
such a pair is a unique behavioral decision outcome ( H ). Therefore, for each level of
initial wealth between two threshold values ^ and H , (a) a behavioral poverty trap is
welfare dominated by the corresponding standard solution, and (b) any intervention
that raises aspirations alone will move the individual out of the behavioral poverty
trap.
When the value function is upward sloping at a behavioral equilibrium, ^ is higher
than
H.
Therefore, there is no level of initial wealth for which a behavioral poverty
trap is welfare dominated by a standard solution. Further, raising aspirations for any
level of initial wealth between L and H will increase e¤ort at a behavioral decision
outcome but such an outcome will be welfare dominated by the standard solution.
It follows that in such a scenario, raising aspirations must also be accompanied by
transfer mechanism that raises the initial wealth of the individual.
Of course, when initial wealth is below ^, the only way22 to ensure that an individual breaks out of the poverty trap is to raise the initial wealth of such an individual.
22
We show that ^ >
L
in the appendix in the proof of Proposition 5.
19
5.1
E¤ort costs and internal constraints
There remains the issue of whether there can be a behavioral poverty trap, for two
individuals with the same level of initial wealth. Below we will show that this is
possible if their e¤ort costs are di¤erent. Notably, one of them can end up in a
behavioral poverty trap, even when the di¤erence in cost of e¤ort is arbitrarily small.
To show this, we adapt the argument developed so far as follows. For a given 0 ,
we …rst compute the thresholds, in cost of e¤ort c, that characterize the outcomes
of a standard decision problem and a behavioral decision problem, and then use
these thresholds to determine the conditions under which a behavioral poverty trap
emerges. Speci…cally, if [v 0 (x) v 00 (x)(1 x)] 0 for all x, then for each 0 23 :
1. There exists c^ s.t. (a) if c c^, at a standard solution eb ( 0 ) = 1, gb ( 0 ) = 2 0 ,
and (b) if c
c^, at a standard solution eb ( 0 ) = 0, gb ( 0 ) =
2. There exists cH
cL s.t. (a) c
0.
cL : the unique behavioral outcome is
e ( 0 ) = 1, g ( 0 ) = 2 0 ; (b) cL
c cH : there are multiple behavioral outcomes
e ( 0 ) = 0, g ( 0 ) = 0 and e ( 0 ) = 1, g ( 0 ) = 2 0 ; (c) c
cH : the unique
behavioral outcome is e ( 0 ) = 0, g ( 0 ) =
Note that when cL
c
0.
cH , there are two possible behavioral outcomes asso-
ciated with a single value of e¤ort cost so that small di¤erences in e¤ort cost can
lead to very di¤erent outcomes. Given the selection mechanism used earlier in this
section, we show the following result that characterizes how cost of e¤ort links to a
behavioral poverty trap:
Proposition 6: Suppose [v 0 (x) v 00 (x)(1 x)] 0 for all x. Then, for any …xed
level of initial wealth 0 , the higher the cost of e¤ort for an individual, the more likely
he is to experience a behavioral poverty trap. When v 0 (0) < 0, so that c^ > cL , and for
each c 2 [cL ; c^), a behavioral poverty trap is welfare dominated by the corresponding
standard solution.
Proof: See appendix.
The above proposition states that two individuals with the same initial endowment
but with di¤erent e¤ort costs can end up at two very di¤erent outcomes, one stuck in
a behavorial poverty trap and the other not. Raising aspirations for any e¤ort cost
between cL and cH will increase e¤ort at a behavioral decision outcome. When the
23
The computations underlying these claims are reported in the proof of Proposition 6 (see below)
contained in the appendix.
20
value function is downward sloping at a behavioral equilibrium, raising aspirations
for any e¤ort cost bewteen cL and c^ is, moreover, welfare improving. As a result, any
intervention that reduces the cost of e¤ort required to achieve individual goals will also
help in breaking a behavioral poverty trap.
These could of course be interventions
that have a broad impact on the cost of e¤ort –such as better nutrition for the poor.
However, given the feedback between e¤ort and aspirations, even small changes in
the cost of e¤ort could have large consequences.
This is consistent with the idea
of "channel factors" –seemingly minor situational changes in the cost of e¤ort, that
can have a large impact on the actions. For instance, Bettinger et al(2009) show
that low-income families that were given assistance in …lling …nancial aid forms and
information about local higher education options were substantially more likely to
apply and then send their children to college.24
It is of some interest to note that under our assumptions, the threshold level
above which initial aspirations have to lie for an individual to choose high e¤ort is
decreasing in initial wealth but increasing in the cost of e¤ort. Moreover, it can
be easily checked, from the proof of Proposition 6 in the appendix, that as long as
r( ) < 1 for all > 0, the cost thresholds used in that proposition are all increasing
in 0 . Therefore, for a …xed level of aspirations, initial wealth and e¤ort cost are
substitutes in ensuring that an individual chooses high e¤ort. A key implication of
our model is that poor individuals who have a high cost of e¤ort are those most likely
to be stuck in a behavioral poverty trap. For such individuals, any intervention that
raises initial aspirations above the critical threshold will result in higher e¤ort, and
via feedback from e¤ort to aspirations, sustain an irreversible, self-ful…lling shift away
from a behavioral poverty trap.
In the following section, we turn to the implications of our analysis for the design
of policies intended to break a behavioral poverty trap.
6
Behavioral Poverty Traps: Policy Implications
The kinds of poverty traps described above imply that anti-poverty initiatives aiming
to tackle persistent poverty need to be mindful of two important issues. The …rst
is that, under acute poverty, the e¤ectiveness of policies targeted to relax external
24
See Bertrand et al. (2006) for other examples of how small changes in costs can have huge e¤ects
on decisions of the poor.
21
constraints will be maximized if they also reduce internal constraints (for example,
by changing aspirations). The second is that there exist conditions where relaxing
internal constraints alone (without altering external ones) can alter behavior and
reduce the persistence of poverty.
An example of the …rst type of intervention is a program to reach out to the
ultra-poor, developed by BRAC, one of the two leading micro…nance institutions
in Bangladesh. This program provided a combination of asset assistance (mostly
livestock), weekly stipends and weekly training sessions on how to manage their asset
and their household lives. In their evaluation of such a program, Banerjee et al.
(2011) found that the income gains spread much beyond the scope of the initial
intervention: bene…ciaries’consumption increased by as much as 15%, while income
from non-livestock sources also saw a considerable increase. The authors believe that
the initial increase in income set o¤ a virtuous cycle: survey evidence indicates that
as the bene…ciaries earned more, they felt better about their lives, and motivated
them to work more. They report a 28% increase in the time spent working as well –
which is consistent with the higher e¤ort that is associated with higher aspirations in
our model.
A good example of the latter type of policy intervention designed to raise aspirations of low-income children is the Classical Music Orchestras Program for children
from disadvantaged backgrounds. This program was developed in Venezuela 30 years
ago by Jose Antonio Abreu, and has been succesfully implemented in di¤erent countries around the globe since then. The lessons are free of charge and a public foundation Fesnojiv provides the instruments. The project does not primarily aim to create
professional musicians, but to integrate poor children into the society by instilling
con…dence and discovering their hidden talents. 96 percent of the young musicians
have good to excellent school records –even though education was not the focus of
the program. They stand out as high achievers thanks to their steady relationship
with music. UNESCO awarded Fesnojiv its International Music Award in 1993-94
and in 1998 UNDP commended it as an outstanding example of poverty reduction.
In the words of its founder, Antonio Abreu25 : "Participating in the orchestral movement has made it possible for them [the children] to set up new goals, plans, projects
and dreams, and at the same time it is a way of creating meaning and helping them
in their day-to-day struggle for better conditions of life." Similarly, Beaman et al
25
See http://www.rightlivelihood.org/recip/abreu.htm
22
(2012) found that in India, that exposure to female leaders in local government (as
part of a mandated reservation of posts for women) raised both the aspirations and
educational attainment of girls signi…cantly. Strikingly, this was despite no change in
the resources available for their education; it was largely driven by the positive boost
to parents’aspirations for their girl children, because of female role models. The UK
program "Supporting parents on kids education (SPOKE)" which works with groups
of parents to set personal goals for their children is another example of this class of
interventions.
Finally, it is worth noting that Proposition 4 also implies an important caveat
to policies that seek to raise aspirations as a way of breaking behavioral poverty
traps. If v 0 (0) > 0, then v( 1 ) < 0: in this case, H < b; the behavioral outcome
2
with high e¤ort and high aspirations could be welfare dominated by the standard
decision outcome with low e¤ort and low aspirations. A policy intervention that raises
aspirations could lead the individual to an e¤ort and aspiration pair that is higher
but not necessarily welfare improving. In such cases (as we have already noted above)
and also when initial wealth is below ^, raising aspirations must also be accompanied
by a transfer mechanism that raises the initial wealth of the individual. Further,
when wealth transfers are not feasible, as the discussion related to Proposition 6 has
pointed out, directly reducing the cost of e¤ort that poor individuals face, even in
small ways, can be an e¤ective way to reduce the likelihood of a behavioral poverty
trap.
7
Discussion and Conclusion
Appadurai (2004) has argued that aspirations are central to the path out of poverty.
We have developed a novel and simple framework to study the psychology of poverty
and aspirations failure. To our knowledge, this is the …rst paper modeling aspirations
as internal constraints and their interaction with external constraints.
Our model shows that aspirations failure among the poor may be a consequence of
poverty, rather than a cause. This arises through an interaction between two factors:
(i) all individuals, rich or poor, fail to appreciate how their e¤ort choices shape their
aspirations over time - but the poor pay a bigger price for this failure because (ii)
the complementarity between initial status and e¤ort further lowers their incentive
to put in e¤ort. Their lower e¤ort choices give rise to lower aspirations.
23
The key point in the paper is that policies that address aspiration levels can,
at the very minimum, enhance the e¤ectiveness of policies that address material
deprivation; moreover, there are situations in which such policies on their own, can
enhance welfare, without any change in material circumstances.
We have chosen to capture the external constraints imposed by poverty in a
"reduced-form way", assuming complementarity between initial status and e¤ort.
Inasmuch such complementarity is not primarily driven by initial endowments di¤erences, we can isolate the e¤ect of aspirations as an important component leading to
poverty without assuming initial weath disparities.
Finally, we have consciously chosen a deterministic static model of individual
decision making because it was the simplest model one could think of to generate behavioral poverty traps. Of course, this is the …rst step of a bigger project. We foresee
future extensions of the model with explicit dynamics, with learning and individual interactions. Also, uncertainty in …nal status was not required in our model, but
there are scenarios where uncertainty will matter in an essential way to understanding
poverty traps, specially to model pesimistic beliefs or locus of control. Incorporating these elements into the study of poverty and aspirations o¤er the promise of
interesting avenues for future research.
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8
Appendix
8.1
The Value Function: Two Examples
Here we show two speci…c formulations of the value function that satisfy the assumptions on v(
g
) in the main text. For notational simplicy, let’s de…ne x =
g
:
A S-shaped value function
One possible formulation of v (x) is a Kahneman-Tversky (1979) type of value
function, in the sense that it is convex over losses but concave over gains i.e. the
value is S-shaped over gains and losses. We assume that v(:) is an increasing, smooth
function with v(0) = 0 being the in‡exion point of the function so that v 0 (0) > 0 but
v 00 (0) = 0.
A speci…c functional form would be, for example, v(x) = ax3 +kx where a < 0 and
k is positive constant that is large enough (speci…cally, k >
6a 0 ). This formulation
does not allow for loss aversion. Allowing for loss aversion would necessarily imply
that v(:) had "kink" (i.e. was non-di¤eretiable) at zero.
Lemma 1 imposes a condition on the curvature of v (x), namely [v 0 (x)
x)
0 for all x; which ensures that given an initial status
complement (i.e. ue;g (e; g;
0)
0,
v 00 (x)(1
e¤ort and aspirations are
0). This condition is satis…ed when k > 6a (6
2).
Note also that a < 0 and k > 0; v 0 (0) > 0 i.e. the value function is upward sloping
at a behavioral equilibrium.
A value function with a bliss point
30
Another possible formulation of v (x) is a smooth and strictly concave function
which is decreasing over losses and gains relative to a reference point (which could
be zero) and attains a maximum at .
A speci…c functional form would be v (x) =
)2 where
(x
is the reference
point. When = 0, v (x) is symmetric over gains and losses. In this case, the e¤ect
of the frustration from falling short of aspirations (underachieving) is equal to the
e¤ect of the pleasure from exceeding aspirations (overachieving). However, a value of
di¤erent from zero measures the degree of assymmetry over gains and losses in the
value function.
The condition in the curvature v (x) imposed by Lemma 1 is satis…ed here when
1:Note that v 0 (x)
for any
0
v 00 (x)(1
x) = 2(1 +
2x) and the maximum value of x
is 1.
Note also that when 1 < < 0; v 0 (0) < 0 i.e. the value function is downward
sloping at a behavioral equilibrium.
8.2
Proofs
Proof of Proposition 1 : We begin by characterizing the conditions under which, at a
standard decision problem, e¤ort and initial wealth are complements. To this end,
consider the following expressions:
1. given initial wealth 0 , the marginal net utility of e¤ort at a SDP
@s
@f
= b0 ( )
@e
@e
c0 (e)
(1)
2. the marginal net utility of e¤ort as initial wealth goes up:
@2s
@2f
@f @f
= b0 ( )
+ b00 ( )
@e@ 0
@e@ 0
@e @ 0
(2)
By computation, it follows that e¤ort and initial wealth are complements as long as
@2s
@e@ 0
Note that
b0 ( )
so that, as both b0 ( )
0 , b0 ( )
@2f
@f @f
+ b00 ( )
@e@ 0
@e @ 0
0
@f @f
@2f
@2f
+ b00 ( )
= b0 ( )
(1
@e@ 0
@e @ 0
@e@ 0
0 and
@2f
@e@ 0
0,
@2s
@e@ 0
31
(3)
r( ))
0 if and only if 1
r( ) =
b00 ( )
b0 ( )
.
Proof of Lemma 1 : Consider the following expressions:
1. given initial wealth 0 and an aspiration level g, the marginal net utility of
e¤ort
@u
@f
@f
g
= b0 ( )
+ v 0 (x)
@e
@e
@e (f (e; 0 ))2
c0 (e)
(4)
2. given initial wealth 0 , the marginal net utility of e¤ort as the aspiration level
g goes up:
@f
1
@2u
0
=
v 00 (x)(1 x)]
(5)
2 [v (x)
@e@g
@e (f (e; 0 ))
For a given value of initial wealth
0,
e¤ort and aspirations will be strategic comple-
ments as long as the following inequality is satis…ed:
@2u
@e@g
0 , [v 0 (x)
Therefore, whenever [v 0 (x)
v 00 (x)(1
v 00 (x)(1
x)]
(6)
0
0, e¤ort and aspirations are strategic
x)]
complements.
Proof of Proposition 2 : De…ne a map
(a; p) = (e(g; 0 ); f (e; 0 )). Note that f (e;
0, by lemma 1,
@2u
@e@g
: [0; 1] [ 0 ; 2 0 ] ! [0; 1]
0
0 ) is increasing in e. As [v (x)
0, it follows that e(g;
0)
[ 0 ; 2 0 ],
v 00 (x)(1 x)]
is a compact (and consequently,
complete) sublattice of [0; 1] and has a maximal and minimal element (in the usual
component wise vector ordering) denoted by e(g; 0 ) and e(g; 0 ) respectively.
Therefore, the map (e(g; 0 ); f (e; 0 )) is a continuous, increasing function from
[0; 1]
[ 0 ; 2 0 ] to itself and as [0; 1]
[ 0 ; 2 0 ] is a compact (and hence, complete)
lattice, by applying Tarski’s …x-point theorem, it follows that (e ( 0 ); g ( 0 )) =
(e(g ; 0 ); f (e ; 0 )) is a …x-point of and by a symmetric argument, (e(g; 0 ); f (e; 0 ))
is a continuous, increasing function from [0; 1] [ 0 ; 2 0 ] to itself and e ( 0 ); g ( 0 ) =
e(g ;
0 ); f (e
;
0)
is also a …x-point of ; moreover, (e ( 0 ); g ( 0 )) and e ( 0 ); g ( 0 )
are respectively the largest and smallest …x-points of
.
Proof of Proposition 3 : Note that at a behavioral equilibrium (e ; g ) 2 B ( 0 ),
from equation (4)
@u
= b0 (f (e ;
@e
0 ))
@f
(e ;
@e
0)
+ v 0 (0)
@f
(e ;
@e
0)
while at an interior standard solution (^
e; g^)
@s
= b0 (f (b
e;
@e
0 ))
@f
(b
e;
@e
32
0)
c0 (b
e) = 0.
c0 (e ),
It follows that the marginal net utility of e¤ort, and hence an interior standard solution is a behavioral solution if and only if v 0 (0) = 0.
Further, from the expressions characterizing the net marginal utility of e¤ort at a
behavioral and standard decision problem, once the concavity of b(:) and convexity of
c(:) is taken into account, it is immediate that if v 0 (0) > 0 (respectively, v 0 (0) < 0), at
a behavioral solution the individual chooses a lower (respectively, higher) e¤ort and
aspiration level than at a standard solution i.e. (e ; g )
(e ; g )
(^
e ( 0 ) ; g^ ( 0 )) (respectively,
(^
e ( 0 ) ; g^ ( 0 ))) for each (e ; g ) 2 B ( 0 ). These inequalities will be strict
if the unique standard solution is interior.
Proof of Lemma 2: (i) Note that for 0 > 0,
h0 ( 0 ) > 0 ,
By computation,
function of
0b
0
0
0h ( 0)
= 2 0 b0 (2 0 )
0
0h ( 0)
0b
0
> 0:
( 0 ) > 0 as
0b
0
( 0 ) is an increasing
( 0 ) if and only if r( ) < 1.
(ii) By lemma 1,
@2u
@e@g
> 0. This is equivalent to the property that u(e; g;
0
0
satis…es increasing di¤erences in e; g: i.e. for (e ; g )
u(e0 ; g 0 ;
0)
u(e; g 0 ;
u(e0 ; g;
0)
0)
(e; g),
0)
u(e; g;
0 ):
As u(e; g; 0 ) satis…es the property of increasing di¤erences in e; g, by computation,
it follows that for (e0 ; g 0 ) (e; g),
v
v
Let e0 = 1, g 0 = f (1;
0)
f (e0 ; 0 ) g 0
f (e0 ; 0 )
f (e0 ; 0 ) g
f (e0 ; 0 )
v
v
f (e; 0 ) g 0
f (e; 0 )
f (e; 0 ) g
:
f (e; 0 )
= 2 0 , e = 0 and g = f (0;
0)
=
0.
Then, by substitution,
the preceeding inequality reduces to the inequality
v( 1)
1
v( ) , v( 1)
2
1
v( )
2
as required.
Proof of Lemma 3: (i) Consider, …rst, the e¤ort and aspiration pair consistent
with a standard solution as initial status changes. For a given value of , e = 1,
g=2
0
is a standard solution i¤
b (2 0 )
c
b ( 0 ) , h( 0 )
33
0
> 0, h0 ( 0 )
Therefore, whenever r( ) < 1 for all
0. Further,
lim h( 0 ) < 0
0 !0
so that there exists ^ (the implicit solution to h( 0 ) = 0 or ^ = h0 1 (0)) such that
(a) if 0 < ^, at a standard solution eb ( 0 ) = 0, gb ( 0 ) = 0 ,
(b) if 0 ^, at a standard solution eb ( 0 ) = 1, gb ( 0 ) = 2 0 .
(ii) Consider, next, the e¤ort and aspiration pair consistent with a behavioral
solution as initial wealth changes. For a given value of 0 ,
(a) e = 1, g = 2
is a behavioral solution i¤
0
b (2 0 )
c
2
0
b ( 0) + v
0
, h ( 0)
0
(b) e = 0, g =
0
is a behavioral solution i¤
b ( 0)
By lemma 2,
(i.e.
H
L =
0 1
h0
=h
1
v
v(1). Let
(v ( 1))) and
1
2
2
b (2 0 ) + v
1
2
v
v ( 1)
H
0
0
c , h ( 0)
1
2
v
2
0
L
be the implicit solution to h ( 0 ) = v ( 1)
be the implicit solution to h ( 0 ) =
). Then, clearly,
L.
H
L
H:
0
1
2
(i.e.
It follows that there are three possible
con…gurations compatible with a behavioral equilibrium:
(a) 0
L : the unique behavioral outcome is e ( 0 ) = 0, g ( 0 ) =
(b)
v
0;
there are multiple behavioral outcomes e ( 0 ) = 0, g ( 0 ) =
0
and e ( 0 ) = 1, g ( 0 ) = 2 0 ;
(c) 0
H : the unique behavioral outcome is e ( 0 ) = 1, g ( 0 ) = 2 0 .
Proof of Proposition 4 : As a …rst step, …x 0 . Let g~ ( 0 ) solve the equation
u (1; g;
0)
= u (0; g;
0)
, h( 0 ) = v
By lemma 3, (i) h( 0 ) is increasing in
0,
g
0
2
v
2
0
while (ii) v
g
0
0
g
0
v
2
:
0
g
0
2
0
is decreasing
in g. Let g~ ( 0 ) denote the implicit solution to the preceeding equation. Then, clearly
g~ ( 0 ) is decreasing in 0 . Further, for L < 0 < H : (i) g~ ( 0 ) > 0 : g~ ( 0 )
0
implies that the optimal e¤ort for the behavioral individual is zero; (ii) g~ ( 0 ) < 2 0 :
g~ ( 0 )
2
Moreover,
that:
0
0
implies that the optimal e¤ort for the behavioral individual is one.
L
implies g~ ( 0 ) =
0
and
34
0
H
implies g~ ( 0 ) = 2 0 . It follows
(a) if g < g~ ( 0 ), the optimal e¤ort for a behavioral individual is e = 0;
(b) if g g~ ( 0 ), the optimal e¤ort for a behavioral individual is e = 1.
Therefore, [0; g~ ( 0 )) is the basin of attraction of the behavioral decision outcome
(e ( 0 ) = 0; g ( 0 ) =
0)
while [~
g ( 0 ) ; 2 0 ] is the basin of attraction of the behavioral
decision outcome (e ( 0 ) = 1; g ( 0 ) = 2 0 ). Therefore, the probability with which
the behavioral outcome is (e ( 0 ) = 0; g ( 0 ) = 0 ) is equal to the probability that
g0 2 [ 0 ; g~ ( 0 )) which is M (~
g ( 0 ))and the probability with which the behavioral
outcome is (e ( 0 ) = 1; g ( 0 ) = 2 0 ) is equal to the probability that g0 2 [~
g ( 0) ; 2 0]
which is 1 M (~
g ( 0 )). As g~ ( 0 ) is decreasing in 0 , the probability that the individual
is caught in a behavioral poverty trap, M (~
g ( 0 )), is decreasing in 0 .
Proof of Proposition 5 : We check that for a …xed level of initial wealth
0,
an
individual stuck in a behavioral poverty trap is welfare dominated by another individual stuck in standard poverty trap. Note that ^ = h0 1 (0) h0 1 v 21 = H
if and only if v 21 > 0 , v 0 (0) < 0. Further, ^ = h0 1 (0) > h0 1 (v ( 1)) = L
as v ( 1) < 0 and h0 1 (:) is an increasing function. If v 0 (0) < 0, then v( 21 ) > 0:
in this case, H > b so that whenver 0 is such that b
0 < H , if the behavioral
decision outcome is the low e¤ort and low aspirations pair, given that a standard
decsion outcome is the high e¤ort, high aspirations pair, the individual is necessarily
choosing an e¤ort and apsiration pair that is strictly welfare dominated. If v 0 (0) 0,
b so that if the behavioral decision outcome is
then v( 1 )
0: in this case, H
2
the low e¤ort and low aspirations pair, given that a standard decsion outcome is also
the low e¤ort, high aspirations pair, a behavioral outcome isn’t welfare dominated.
Proof of Proposition 6: Consider, …rst, the e¤ort and aspiration pair consistent
with a standard solution as initial status changes. For a given value of , e = 1,
g=2
0
is a standard solution i¤
b (2 0 )
c
b ( 0) , c
c^ = b (2 0 )
b ( 0)
Therefore,
(a) if c < c^, at a standard solution eb ( 0 ) = 1, gb ( 0 ) = 2 0 ,
(b) if c
c^, at a standard solution eb ( 0 ) = 0, gb ( 0 ) =
0.
Consider, next, the e¤ort and aspiration pair consistent with a behavioral solution
as initial wealth changes. For a given value of
35
0,
(a) e = 1, g = 2
0
is a behavioral solution i¤
b (2 0 )
b (2 0 )
(b) e = 0, g =
0
b ( 0)
v ( 1) = cH
c
is a behavioral solution i¤
b ( 0)
v
1
2
1
2
b (2 0 ) + v
c
By lemma 2,
b ( 0 ) + v ( 1) ,
c
cL = b (2 0 )
v(1) , v
c,
b ( 0) + v
1
2
1
2
v(1) so that cL
cH . It follows that
there are three possible con…gurations compatible with a behavioral equilibrium:
(a) c
cL : the unique behavioral outcome is e ( 0 ) = 1, g ( 0 ) = 2 0 ;
(b) cL c cH : there are multiple behavioral outcomes e ( 0 ) = 0, g ( 0 ) =
and e ( 0 ) = 1, g ( 0 ) = 2 0 ;
(c) c
cH : the unique behavioral outcome is e ( 0 ) = 0, g ( 0 ) =
0
0.
Next, we proceed to the equilibrium selection argument. As a …rst step, …x
0.
Consider the equation
u (1; g;
b (2 0 )
b ( 0)
0)
= u (0; g;
0)
g
0
c = v
,
v
2
0
g
0
2
:
0
By a slight abuse of notation, let g~ ( 0 ; c) denote the implicit solution to the preceeding
equation. By lemma 3, v
g
0
0
v
2
g
0
2
is decreasing in g and therefore, clearly
0
g~ ( 0 ; c) is increasing in c. It follows that (using the proof of Proposition 4) that:
(a) if g < g~ ( 0 ; c), the optimal e¤ort for a behavioral individual is e = 0;
(b) if g g~ ( 0 ; c), the optimal e¤ort for a behavioral individual is e = 1.
Further, the probability with which the behavioral outcome is (e ( 0 ) = 0; g ( 0 ) =
is equal to the probability that g0 2 [ 0 ; g~ ( 0 ; c)) which is M (~
g ( 0 ; c))and the proba-
bility with which the behavioral outcome is (e ( 0 ) = 1; g ( 0 ) = 2 0 ) is equal to the
probability that g0 2 [~
g ( 0 ; c) ; 2 0 ] which is 1 M (~
g ( 0 ; c)). As g~ ( 0 ; c) is increasing in c, the probability that the individual is caught in a behavioral poverty trap,
M (~
g ( 0 ; c)), is increasing in
0.
Finally, note that
cL = b (2 0 )
b ( 0) + v
1
2
36
< c^ = b (2 0 )
b ( 0)
0)
i¤ v
1
2
< 0 , v 0 (0) < 0.
37