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EDUCATION AND FAMILY INCOME:
CAN POOR CHILDREN SIGNAL THEIR TALENT?*
Luisa Escriche and Gonzalo Olcina**
WP-AD 2006-20
Correspondence: Luisa Escriche, Universitat de València, Dpto. Análisis Económico, Edificio
Departamental Oriental, Campus dels Tarongers, s/n, 46022 Valencia, e-mail: Luisa.Escriche@uv.es.
Editor: Instituto Valenciano de Investigaciones Económicas, S.A.
Primera Edición Octubre 2006
Depósito Legal: V-4252-2006
IVIE working papers offer in advance the results of economic research under way in order to
encourage a discussion process before sending them to scientific journals for their final
publication.
* We wish to thank Rosario Sánchez for helpful comments. Part of this research was carried out while
Luisa Escriche was visiting the European University Institute (Florence). Financial support of the Spanish
Ministry of Education and Science, under grant SEJ2005-08054-ECON, and the Generalitat Valenciana,
under grant GV 06/201 are gratefully acknowledged.
** L. Escriche: Universitat de València, Dpto. Análisis Económico; G. Olcina: Universitat de València,
Dpto. Análisis Económico.
Abstract
The aim of this paper is to explain how …nancial constraints and family
background characteristics a¤ect the signalling educational investments of
individuals born in low-income families.
We show that talented students who are poor are unable to signal their
talent, as the maximum level of education they can attain may also be
achieved by less talented students who are rich. Under this approach, a de-
crease in inequalities across generations cannot be expected. The paper also
shows that an increase in educational standards would help poor individuals
with high-ability if it is combined with other non-monetary measures.
Keywords: signalling education, segregation, educational standard, equal
opportunities
JEL classi…cation: I20, C70
EDUCATION AND FAMILY INCOME:
CAN POOR CHILDREN SIGNAL THEIR TALENT
Luisa Escriche and Gonzalo Olcina
2
1 Introduction
Economists have long expressed concern about the inequality of opportu-
nities between rich and poor. Children raised in high-income families earn
more than children raised in low-income families; there is a strong correla-
tion of about 0.4 between a father and a son’s permanent earnings (Solon,
1992; Altonji and Dunn, 1991; Zimmerman, 1992).1Despite this evidence,
it remains unclear how parents’income determines children’s outcome, and
it is important to unravel how poverty is transmitted in order to design
suitable policies for reducing inequality.
One of the most common policies to provide equal opportunities to all
individuals has been to subsidize formal education. The justi…cation is that
capital market imperfections prevent agents from borrowing against future
human capital incomes.2Moreover, simultaneously to this extension of sub-
sidies, loans or public education, a large number of prestigious MA courses
and private Colleges, Universities or Schools, that not all people can a¤ord,
have appeared. Having a college degree from these institutions or a Masters
from some schools, is a sure-…re ticket to a high-paid job; people who attend
the top 50 schools are virtually guaranteed a very comfortable job.
In this paper we try to explain the persistence of inequality across genera-
tions and the proliferation of exclusive and prestigious Universities. Inequal-
ity is mainly transmitted trough educational attainments. Children from
poorer backgrounds achieve lower educational levels. Two main reasons can
be found. First, as Becker and Tomes (1986) pointed out, poorer families
are …nancially constrained or …nd it more di¢ cult access to credit markets
1See Solon (1999) for a survey of intergenerational correlation in earnings.
2If capital markets are imperfect and poor parents are …nancially constrained, they
invest less in children’s education (Becker and Tomes, 1986.)
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3
which increases the monetary cost of …nancing education for their children.
This e¤ect of a family’s income on a child’s achievement is direct. Second,
poorer parents may have some characteristics that make them less successful
on the labour market and which a¤ect children. Family background charac-
teristics a¤ect motivation, access to career information, constancy, discipline
and other learning skills. Thus, although money matters, as proved by ex-
tensive literature, other unobservable characteristics correlated with family
income are also important.3
Thus, we consider that a family’s background is a determinant factor
in educational achievement together with innate ability. The correlation
between innate ability and family wealth is controversial to assume.4We
3Empirical studies have a great deal of di¢ culty in detecting any systematic in‡uence
of parents’ income in children’s human capital. While, for example, Gaviria (2002) and
Maurin (2002) found evidence that borrowing constraints prevent parents from investing
optimally in their children’s human capital, Cameron and Taber (2000) or Mulligan (1997)
do not …nd evidence of borrowing constraints. Shea (2000) found that changes in parents’
income due to luck have a negligible impact on children’s human capital for most families.
The reason is that although money could be important, other unobservable character-
istics correlated with family income may also be important. Thus, Chevalier and Lanot
(2002) found that family characteristic e¤ects dominate the …nancial constraint e¤ects in
schooling investment. PISA (2003) documents the importance of family background (and
speci…es what is included under this concept) in children’s educational achievement in
OECD countries.
A survey of the research on the relationship between parents’ social status and their
children’s achievement can be found in Haveman and Wolfe (1995) or in the Introduction of
Maurin (2002). See Shea (2000) and Chevalier and Lanot (2002) for additional references
and also for a discussion on this matter.
4One of the most cited studies related with this matter (The Bell Curve, Hernstein
and Murray, 1994) argue that ability is correlated with income. Speci…cally, the authors
suggest that a child’s economic success is mostly explained by cognitive ability (IQ), that
2
4
will assume that talented children are born in rich and poor families with
equal probability. However, talent only leads to high wages if it can be
signalled in the labour market.5If poor talented children cannot signal
their talent, they will never achieve the same labour market outcomes as
rich talented children.
The aim of this paper is to explain how …nancial constraints and family
background characteristics a¤ect the signalling possibilities of poor children.
This will help to explain the transmission of inequality across generations
and also why inequality has persisted over time despite the implementation
in most Western nations of policies to facilitate access to formal education.
Speci…cally, the paper presents a signalling model in the labour market in
which …rms interpret education as a signal of ability and therefore o¤er a
higher wage to an individual with more education. Unlike Spence (1973),
individuals di¤er not only in their innate ability but also in their families’
socioeconomic background.
In our model we will show, in the …rst place, that talented rich people will
always attain the highest education level, which allows them to signal their
talent, under some reasonable educational cost conditions. It means that any
is, by one factor transmitted by genes. Therefore, roughly speaking we could say that
rich families pass “better” genes to their children (compared to poor families). Hence,
poverty is also transmitted. However, the statement in The Bell Curve has been widely
reviewed and reexamined. Other authors …nd that the socioeconomic status of people’s
parents and their broader social environment is at least as important, and may be more
important than IQ in determining social and economic success in adulthood (see Fisher,
1996; Koreman, 1995; Currie and Thomas, 1995; Heckman, 1995). So, in this paper we
will not assume that children’s ability is correlated with parents’ income.
5Despite the di¢ culty of testing the signalling hypothesis (Spence, 1973, 1974), it is
commonly assumed that high-ability leads to higher wages in the labour market. See Riley
(2001) for an excellent modern survey on signalling.
3
5
other type does not …nd it pro…table to meet the education cost (monetary
or non-monetary) of such a level, given the potential gains. Imitation is
prevented. This guaranties them the highest wage in the labour market.
Second, we will show that poor people, even those with great talent, will
not be able to signal their talent and achieve the same wage as talented rich
people. The reason is that the maximum education level they can attain may
also be achieved by less talented students who are rich. The latter prefers
to mimic the talented poor people than to reveal himself. This results in
children from poorer backgrounds not achieving the same labour market
outcomes as the richer children.
If education works as a relevant signalling device, a decrease in inequali-
ties across generations cannot be expected (unlike Becker and Tome’(1979)
model of human capital), as talented rich individuals always have an incen-
tive to send a signal -education- that no other types would ever mimic.6As
much as public education spreads to higher educational levels, rich talented
people will “‡y”to prestigious but expensive educational institutions in or-
der to preserve their position in the educative ranking. Education becomes
defensive expenditure, as Thurow (1975) argue. Unless (i) large amounts
of money are transferred to poor families to facilitate their access to any
type of college or university, (ii) educational policies are oriented towards
increasing the role of ability in educational achievement (by raising edu-
cational standards, for example), (iii) the segregation of students among
6In Becker and Tomes’(1979) model, convergence to the mean of earnings is expected,
as high-income parents stop investing in their children’s education when the marginal rate
of return of human capital equals that of physical capital. Moreover, the e¤ectiveness of
their higher human capital investments to increase children’s earnings is lower (decreasing
marginal returns of human capital are assumed).
4
6
schools on based on families’socioeconomic status decreases, or (iv) some
other scholar support is implemented in order to reduce disparities due to
unobservable family characteristics, the richest-high ability individuals will
…nd room to signal their talent -and they will be the only ones able to do
so. Consequently, they will preserve the position their parents had in the
labour market.
The paper is organized as follows: section 2 outlines the model. Section
3 contains a general analysis and the basic results. In Section 4, we present
the equilibrium of the model and in Section 5 some policies to guarantee
equal opportunities are discussed. Conclusions are o¤ered in Section 6.
2 The model
Consider that there are two types of families in the society depending on
their level of wealth. Rich families, r; represent a proportion (1 )and
poor families, p; a proportion of the total number of families which is
normalized to one, 2(0;1). Each family has a child with an innate ability;
which can be either high hor low lwith equal probability. Given this ability,
that is private information, families invest in their children’s education, e:
Firms observe this education levels and o¤er wages to the individuals. The
variable emeasures the number of diplomas, the number and kind of extra-
academic courses taken and the caliber of grades and distinctions earned
during an academic year, taken into account the quality of the school. A
given educational level ecould re‡ect, for example, the best graduate or a
given school but also a degree in the best schools. Apart from that, in this
model e= 0 gets the interpretation of having just compulsory education.
Thus, there are four types of individuals depending on their innate ability
5
7
and their family’s wealth. Let ij denote an individual with j-ability from a
i-type family. Thereby the types are denoted by fpl; ph; rl; rhg:
It is assumed that …rms compete for workers in the labour market.
Hence, the wage o¤ered by a …rm to a worker with education e; w(e);equals
his expected productivity. Given the …rms’ belief about the worker’s abil-
ity after observing e; the wage o¤er will be w(e) = (ih=e)H+(il=e)L;
where (ih=e)is the …rms’assessment of the probability that the worker’s
ability is h(or lfor (il=e)), and Hand Lare the productivity of a worker
with high and low ability, respectively. So the …rms’payo¤ is given by the
productivity of a worker minus the paid wage.
Families are altruistic and …nance the education cost of their children
and also value the wage their children will obtain in the labour market.
Rich families have a wealth Rand poor families have P: The utility function
of a family with an ij-child type is:
Uij (e; w) = Wi+E[w=e]c(e; i; j );
where Wi2 fR; P gdenotes the ifamily’s wealth, E[w=e]is the expected
wage if the child reaches the labour market with education eand c(e; i; j)is
the cost of a level of education efor an ifamily with a child of jability.
Individuals require a certain innate ability and some monetary support
from their family as inputs to obtain education. The education produc-
tion function we propose includes these two inputs, but also captures some
stylized facts concerning the in‡uence of family wealth on the educational
achievements of their children. On the one hand, education depends, as we
said, on monetary investment and poorer families are …nancially constrained
or …nd it more di¢ cult the access to the credit market, thus increasing the
monetary cost of …nancing education. On the other hand, family background
6
8
characteristics may a¤ect children’s performance at school. Most of these
characteristics are correlated with family income and wealth. For instance,
the occupational status of parents, their higher educational levels and the
families’cultural possessions are all aspects that can in‡uence students’per-
formance (PISA, 2003). Rich families are better able to take advantage of
the educational system or schools …nd it easier to educate them.
We summarize the in‡uence of family socioeconomic background on the
production of education as an input and denote it by index i: Thus, the
input i; i =fr; pgcaptures the learning skills which depend of family back-
ground. We assume that children from rich families do better at school,
ceteris paribus, than those from poor families, that is p < r:
We assume the existence of an educational function:
e=F(s; i; j);(1)
s:t: Fs>0; Fj>0; Fi>0; Fss 0; Fjj 0; Fii 0; Fij >0; Fsi >0; Fsj >0:
where eis the educational achievement of an individual (for convenience,
educational achievement of the ij child may be also denoted by eij ) and sis
the parental monetary support, that is, the actual educational investment.
Therefore, some substitutability exits among inputs. The higher the innate
ability, ceteris paribus, the higher the educational level and, the more family
support, ceteris paribus, the higher the educational achievement.
Monetary support is decided by the family. Poorer families face bor-
rowing constraints. Hence, we assume that the gross interest rate is higher
for poor families, p;than for rich families, r:The relationship between
a family’s educational investment, s; and educational cost is given by the
equation:
C=is:
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9
To move from the education production function (1) to a cost function in-
volves solving equation (1) for sand then plugging into the above equation.
This gives the education cost function for the ij type:
c=iF1(e; j; i):(2)
From now on, the cost of a level of education efor an i-family with a child of
j-ability will be denoted by c(e; i; j;i):And the marginal cost of education
for the ij-type will be given by ce(e; i; j)
ce(e; i; j) = i@F1(e; j; i)
@e or cij
e=i
Fs(s; j; i):
So, the marginal cost of education for the four types of individuals are given
by
ce(e; p; l) = p
Fs(s; l; p);(a)
ce(e; p; h) = p
Fs(s; h; p);(b)
ce(e; r; l) = r
Fs(s; l; r);(c)
ce(e; r; h) = r
Fs(s; h; r):(d)
Therefore, for a given ability, the marginal cost of education for any educa-
tional level eis higher for poor than for rich families. Notice that:
ce(e; p; j) = p
Fs(spj (e); j; p)>r
Fs(srj (e); j; r)=ce(e; r; j)
where sij (e)is the monetary support required by the ijtype to …nance
the education level ethat solves e=F(s; j; i):As Fi>0and p < r; then
spj (e)> srj (e)for a given ability:Moreover, given that Fsi >0and Fss
0;we obtain Fs(spj(e); j; p)< Fs(srj (e); j; r):Since p> r;the above
inequality is ful…lled.
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10
Moreover, for any educational level eand for any family wealth, the
marginal cost of education is higher for low than for high ability individuals:
ce(e; i; l) = i
Fs(sil(e); l; i)>i
Fs(sih(e); h; i)=ce(e; i; h)
since sil(e)> sih(e); Fsj >0and Fss 0:
Finally, let us compare the marginal cost of education for “intermediate”
types. As we will show in the next section, this relation turns out to be
crucial for the results of our model. The marginal education cost for a poor
individual of high ability, ce(e; p; h);is higher than for a rich individual of
low-ability, ce(e; r; l)if:
ce(e; p; h) = p
Fs(s(e); h; p)>r
Fs(s(e); l; r)=ce(e; r; l);(A.1)
which requires that the marginal increase in educational achievement of the
last m.u. invested by the poor family to be lower than the marginal increase
obtained by the rich family.
It is worthy to comment that although pis close to r, (i.e., borrow-
ing constraints are not important), the marginal cost between these two
types of families can be quite di¤erent if families’backgrounds are di¤erent
enough. Formally, for a given e; if p
r= 1;the crucial assumption can be
veri…ed if Fs(s(e);h;p)
Fs(s(e);l;r)<1(or if Fs(s(e); h; p)< Fs(s(e); l; r));which requires
a "high" di¤erence [srl(e)sph(e)], given that Fss <0:Thus, if the families’
backgrounds are di¤erent enough, the monetary support needed for children
from poorer families’ background will be much higher than for richer chil-
dren and, consequently, the marginal productivity of the last m.u. invested
by the poor family will be lower. In this context, the crucial assumption
may be ful…lled.
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11
In most of the paper we will assume that (A.1) holds. Thereby, low-
income families face higher marginal education costs than high-income fam-
ilies, cei <0, regardless of their children’s ability.
Finally, let us recall the timing of the game. Firstly, nature determines an
individual’s ability. Secondly, the family chooses a level of education e0:
Thirdly, …rms observe the worker’s education and then simultaneously make
wage o¤ers to the worker. Finally, the worker accepts the highest wage o¤er.
3 Analysis of the model
3.1 Solution concept
An equilibrium exits in this model when (i) the educational level chosen
by each family type is optimal, given the wages they anticipate the …rms
will o¤er; (ii) a …rm’s wage o¤er is the expected productivity of the worker,
given their beliefs about which type of individual they are dealing with; (iii)
…rms form their beliefs in a reasonable way; and (iv) the aggregate level
of education is the minimum compatible with (i) to (iii). By a reasonable
way we mean that …rms’beliefs are formed according to Bayes’Law when
they observe an equilibrium education level (i.e., an educational level chosen
by a family with positive probability in the equilibrium being played) and
that these beliefs hold some restrictions for a non-equilibrium educational
level (i.e., an educational level that is a deviation from the equilibrium being
played).7
7Conditions (i) and (ii) plus Bayesian consistency constitute the solution concept Per-
fect Bayesian Equilibrium (PBE). Condition (iv) is a further requirement that we add for
tractability. Any equilibrium which veri…es (i)-(iii) guarantees the minimum aggregate
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12
Speci…cally, out of equilibrium beliefs must satisfy a standard minimal
restriction: the Intuitive Criterion (Cho and Kreeps, 1987). An educational
level e0is equilibrium-dominated for the ij type if this family does worse
choosing this level e0;no matter how the …rms respond, in comparison to her
expected equilibrium utility. An equilibrium satis…es the Intuitive Criterion
if the …rms’ belief, when they observe an out-of-equilibrium educational
level, e0;which is equilibrium-dominated for the type ij; places (if possible)
zero probability on this type ij: (This is possible when e0is not dominated
for all the types.)
Furthermore, we add a global consistency condition. Suppose that an
education level is not chosen by any type of family in an equilibrium but
it is chosen with positive probability by some families in an alternative
equilibrium. In addition, these families are better o¤ in the latter than
in the former equilibrium. In this case, …rms’ beliefs must assign positive
probability only to this set of families.
The intuition behind this condition is that each equilibrium represents
a particular “theory”on how the game will be played in the society. The
deviation from a particular equilibrium by certain types of families acquiring
an unexpected level of education may be interpreted as a confusion on their
behalf about what equilibrium is “really” being played. This is a forward
induction argument that corresponds to a re…nement notion proposed by
Mailath et al. (1993), the Undefeated Equilibrium.8
education level. Therefore, we will move straight on to the minimum education level that
veri…es (i)-(iii).
8As usual in signalling games, multiplicity arises and we apply a further re…nement to
the Intuitive criterion. The Undefeated Equilibrium imposes some restrictions on out-of
equilibrium beliefs according to the notion of forward induction supported by extensive
literature. This is a non-desirable trait of the model, but the main results (those presented
11
13
For tractability, we identify education levels and wages in equilibrium
through vectors (epl; eph; erl; erh)and (wpl; wph ; wrl ; wrh );respectively. We
will focus on equilibria in pure strategies and recall that we assume that
Condition A.1 holds.
3.2 General analysis
We begin by presenting a result that simpli…es the analysis.
Lemma 1 The education choice made by the di¤erent types of families is
not increasing with the marginal cost of education.
Proof. See Appendix A.
Therefore, according to assumption ce(e; p; l)> ce(e; p; h)> ce(e; r; l)>
ce(e; r; h);education choices are such that epl eph er l erh :The lower
the marginal cost for an individual is, the higher his educational achievement
will be.
A direct consequence of Lemma 1 is presented in the following proposi-
tion.
Proposition 1 A separating equilibrium à la Spence, (0; e0;0; e0);in which
…rms can distinguish high from low-ability individuals does not exist.
Intuitively, a separating equilibrium in which …rms are able to di¤erenti-
ate low from high-ability workers, does not constitute an equilibrium of this
model because any educational level chosen by a poor child of high ability
will be imitated by the rich child of low ability. Let us consider that there
is an education level e0>0chosen only by high ability workers, ph and rh,
in the following Section) hold if only the Intuitive Criterion is applied.
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14
that convince the …rms that they are talented workers and accordingly, …rms
will pay the highest wage, w(e0) = H, to those workers. On the other hand,
those with just compulsory education e= 0 are considered low-ability indi-
viduals and are paid the lowest wage w(0) = L. Notice that such a situation
is not a PBE because any education level that the poor type of high ability
…nds rational to attain to get the high wage, will be also found rational to
attain by the rich type of low ability, given our assumption about education
costs. Therefore this latter type will not be in equilibrium choosing e= 0:
Further intuition for Proposition 1 can be found in Figure 1. Let us
consider the education (0; e0;0; e0)as a potential separating equilibrium. The
indi¤erence curves go through the pair (0; L)which represents the option of
not getting education and obtaining wage L: The other option is to choose
an education level e0;that convinces …rms that they are dealing with a
high-ability type, yielding a wage H: Notice that any education level on the
horizontal line at level Hthat type ph …nds pro…table to attain (for example,
e0), will be also found pro…table to attain by the rl type; any pair (e; H)
with e2(0; e00)that lies above the ph indi¤erence curve also lies above the
indi¤erence curve for the rl type.
Notice the crucial role played by the assumption that wealth is not ob-
servable by …rms. From the analysis it is easy to see that, if wealth is
observable, a separating equilibrium (0; e0;0; e00)exists, with e00 > e0:The
issue is that wealth is private information and poor people may …nd inter-
est in telling the …rms that they are poor. But rich people do not have
such interest: the rich and low talented individuals to avoid being identi-
…ed, and rich and high talented individuals because it implies to achieve a
high educational level to convince the …rms they are high talented (instead
of being considered as poor and be identi…ed as htype just with education
13
15
w
U
Url
pl
H
e' e''
L
Uph
e
Figure 1: A separating equilibrium à la Spence does not exist
e0). Therefore, rich people have no incentive to reveal their wealth or, even
worse, have incentive to cheat the …rm. Since …rms do not have the legal
possibility to prove the veracity of the information about this aspect of the
CV or it could be really costly (besides quite controversial), …rms do not
use this information as it seems to happen in the real world. Hence, people
…nd no interest in telling the …rms wether they are poor or rich.
Another interesting point is that a situation in which poor individuals
choose an education level and rich individuals a di¤erent one, that is, a
vector (0;0; e0; e0)can not be an equilibrium. In this case …rms’beliefs after
observing the education of poor types is that the individual may have high
or low ability with equal probability. Accordingly, the …rm will pay 1
2H
+1
2L: On the other hand, …rms’beliefs after observing the education of rich
types is that with equal probability the individual can be of high or low
14
16
ability and also pay 1
2H+1
2L: Therefore, it is not rational for rich people
to invest in education if the wage will be the same as it would be without
investing.
We now analyse the best strategy for the “highest”type, rh:
Proposition 2 In equilibrium, the education choice of a high ability individ-
ual from a rich family is the minimum education level that deters imitation
from any other type.
Proof. See Appendix B.
Formally, this proposition states that the education equilibrium choices
e
rh and e
rl;are such that Url(e
rl; w
rl)Url(e
rh; H )where w
rl is the equi-
librium wage for the rl type.
The intuition is clear and can be seen from the simple two-type model,
without children’s socioeconomic di¤erences. High ability individuals are
indistinguishable from low-ability individuals in a pooling equilibrium, so the
“highest”type always has an incentive to break away and send a signal that
a low-ability individual would never mimic, thereby implying that pooling
cannot survive the Intuitive Criterion. The same argument can be used
with children’s socioeconomic di¤erences. By Lemma 1 we know that the
direct competitor of the high-ability and rich individual, rh; is the low-ability
individual who is rich. So the above exposed logic applies, except that it is
even more costly for the highest type to send a su¢ ciently large signal that
his competitor would never mimic.
This result can also be understood with the help of Figure 2. Let us
consider that the rich individual of high ability rh pools in an educational
level ^e; which implies a wage ^w: According to Lemma 1, if the highest type
15
17
w
e
UU
rl rh
H
e
^e' e''
w
^
Figure 2: The rich indiviual of high-ability separates from any other type.
rh pools must be at least with the rl type, as education is not increasing
with marginal costs. In order for ^eto be an equilibrium of our model, …rms
must believe that (rh=e)<1for educational choices between e0and e00,
because if (rh=e)=1, then the rh-type will deviate. It is easy to see
in the Figure 2 that choices e>e0are equilibrium-dominated for the rl
type, because even the highest wage that could be paid to him, namely H,
yields a pair (e; w)that lies below the rl type’s indi¤erence curve through
the equilibrium point (^e; ^w):However, education choices between e0and e00
are not equilibrium dominated for the rh type: if such a choice convinces
…rms that the worker has high ability, then …rms will o¤er wage H; which
will make the highest type, rh; better o¤ than in the indicated pooling
equilibrium. Thus, the Intuitive Criterion implies that …rms’beliefs must
be (rh=e) = 1 for e2[e0; e00):In this case, the highest type rh will deviate
16
18
form the pool in ^e:
In short, a high ability individual from a rich family never pools in
equilibrium. They will attain an educational level that discourages imitation
from the other types.
Proposition 3 In equilibrium, low ability-rich individuals and high ability-
poor individuals (“intermediate”types, ph and rl)choose the same education
level, that is, eph =erl:
Proof. Let us assume (en route to a contradiction) a situation with
educational levels (0; eph; erl ; e0);with erl < e0(according to Proposition
2) and erl > eph 0. In order for (0; eph; erl ; e0)to be a PBE, …rms’
beliefs for education equilibrium choices must be (i) if eph = 0; (pl=0) =
1
2; (ph=0) = 1
2; (rl=erl)=1; (rh=e0)=1;and accordingly, wages will
be (1
2(H+L);1
2(H+L); L; H)respectively; (ii) if eph >0; (pl=0) = 1;
(ph=eph)=1; (rl=erl )=1; (rh=e0)=1;and accordingly, wages will be
(L; H; L; H )respectively. But notice that, in light of these …rms’wage o¤ers,
the best response for a rich individual of low-ability is nevertheless to choose
eph rather than erl: the wage would be higher and the cost of education is
lower choosing eph. The proposed situation (0; eph ; erl ; e0);with erl > eph ;is
not even a PBE. Therefore, by Lemma 1, erl =eph in equilibrium.
In short, talented poor children cannot signal their talent because less
talented rich individuals make the same educational choice to avoid being
identi…ed. However, talented rich children attain an education that prevents
imitation. We have formally shown that if condition (A.1) holds education
choices in equilibrium are (0; e; e; e0)for each type of family, with 0e < e0:
Notice that if condition (A.1) does not hold, high-ability individuals have
lower educational costs than low ability individuals, regardless of their fam-
17
19
ily’s socioeconomic status. This is the basic assumption of Spence’s model
and, accordingly, the result of the model presented would reproduce the
standard result. We will come back to this point at the beginning of Section
5. Nonetheless, we consider that the relationship between marginal cost,
which lies under assumption (A.1), cannot be rejected and the implications
for the signalling mechanism must be analyzed.
4 The equilibrium of the model
In view of these results, the potential equilibrium of the game is a semisep-
arating equilibrium in which intermediate types (ph and rl) make the same
choice. Particularly, as the next Proposition states, the equilibrium is di¤er-
ent depending on the parameters of the model related to the marginal costs
of education. Namely, the equilibrium can be what we call a cuasipooling
equilibrium or a cuasiseparating equilibrium`
. We will restrict our attention
to a family of education functions such that ce(e;i;j)
ce(e;i0;j0)is constant for any level
of education.9
Proposition 4 The equilibrium of the model is
(i) a cuasipooling equilibrium, with educational levels (0;0;0; ep)and wages
( ^wp;^wp;^wp; H);if 8
<
:
ce(e;p;l)
ce(e;r;l)2;or
ce(e;p;l)
ce(e;r;l)>2and 2(0;
];
(S.1)
(ii) and a cuasiseparating equilibrium, with educational levels (0;^es;^es; es)
and wages (L; ^ws;^ws; H ), if
ce(e;p;l)
ce(e;r;l)>2and 2(
; 1);(S.2)
9For instance, a Coob Douglas function satis…es this condition.
18
20
where the critical value
solves:
c(^es; p; l) = (HL);
c(^es; r; l) = 2(HL)
1+;
the education equilibrium levels solve:
c(ep; r; l) = HL
1+;
c(^es; p; l) = (HL);
c(es; r; l)c(^es; r; l) = (1 )(HL);
and the equilibrium wages are:
^wp=1
1+L+
1+H;
^ws=H + (1 )L:
Proof. See Appendix C.
A complete description of cuasipooling and cuasiseparating equilibria
can be found in Appendix C, Claim 1 and Claim 2 respectively. Figure 3
presents both situations to give a basic idea.
The cuasipooling equilibrium (0;0;0; ep)is depicted on the left of Figure
3. In this equilibrium the three “lower” types pool at e= 0 and get a wage
^wp:The curves labelled U
pl; U
ph; U
rl through point (0;^wp)are indi¤erence
curves corresponding to these types in equilibrium. The highest type, the
talented rich individual, rh; attains just the education level that makes his
direct rival, the rl type, indi¤erent about mimicking him or not. (Note that
the rl type equilibrium indi¤erence curve through (0;^wp);denoted U
rl;also
passes through point (ep; H); formally, epsolves: U
rl(0;^wp) = Url (ep; H ).)
The pair (ep; H)lies below the indi¤erence curve of the lower types. On
19
21
L
w
H
e
p
ê
CUASIPOOLING CUASISEPARATING
U*
pl
U*
ph
U*
r l
U*
r h
U
r l
p
H
L
w
s
U*
r h
U*
r h
U*
r h
U*
ph
U*
r l
U*
r h
U
r h
U*
pl
e
s
e
e
1
w
w
e
s3
Figure 3: The equilibrium of the model
the other hand, it is impossible to distinguish among “ intermediate types”
(poor-high ability type ph; and rich- low ability type rl). The reason is that
rich, low ability workers (type rl), which cannot imitate the education level
that rh uses to separate, choose the same education as the talented poor
type, thus avoiding being identi…ed as a low ability worker. The strong lines
show the expected wages at out-of-equilibrium education levels, given the
beliefs of …rms that satisfy the IC.
The cuasiseparating equilibrium (0;^es;^es; es)is depicted on the right of
Figure 3. In this equilibrium the highest type rh also chooses the education
that makes the rl type indi¤erent about imitating him or not. Formally, es
solves U
rl(^es;^ws) = Url(es; H ). It means to choose an education e3:It is
easy to see that the rl type will never deviate by choosing an educational
level es
3< e < e1:Even if …rms could be convinced that they were dealing
20
22
with a high ability individual and o¤er H, this type will be worse o¤ than
in equilibrium. Observe that any point along the top horizontal line in bold,
for es< e; lies below the indi¤erence curve U
rl:Likewise, in this equilibrium
talented poor children can not separate from less talented rich individuals.
Our solution concept selects between these two possibilities the equilib-
rium in which the rich and talented type separates with the smallest cost
of separation. The intuition is quite clear. If ep< es, that is, the level
of education needed by the rh type to completely separate is smaller in
the cuasipooling equilibrium (CPE) than in the cuasiseparating equilibrium
(CSE), then the latter would be defeated by the former, because if …rms ob-
serve the level of education epthey must believe that this curriculum comes
from an rh type but not from a rl type as they do in the beliefs that support
the CSE.
But this is not true the other way around. Education level esis already
expected to come from the rh type in the CPE, but this type will not use
it because he would get a smaller utility by doing so. In fact, in this case,
all types achieve a greater utility in the CPE than in the CSE, that is, the
CPE is Pareto dominant.
Whenever the ratio of marginal education costs between the pl and the
rl types is small enough or if it is large, when the proportion of poor people
in the society is small enough, the solution is the CPE, as Proposition 4
states. This ratio of marginal costs is small when Por the di¤erence be-
tween pand ris small. In such a situation the capacity of imitation of the
pltype is relatively high. This competition from below pushes up all levels
of education in a CSE. Even if this ratio is high (meaning the competition
from below from the pl type is not so intense), if there is a small proportion
of poor people in the society, then the wage that the rl type gets in CSE
21
23
pooling with the ph type is relatively small. This causes their incentives to
imitate the rhtype to increase. That is, the competition from below from
rl pushes up the level of education needed by rh to separate.
When esis smaller than ep;the solution is the CSE for a similar reason to
that described above (although in this case, there is not Pareto dominance
between the CPE and the CSE). This situation will appear when the ratio
of marginal costs of the rl and pl types is relatively high (because Pand the
di¤erence prare high) and there are enough poor families in the society.
This combination reduces the pressure from below of the pl-type and also
the pressure of the rl-type given that they obtain a good wage pooling with a
relatively high proportion of ph individuals. Notice that the pressure of the
rl-type on the rh-type would be stronger in the CPE because the former is
pooling with all the poor families (high but also low talented) and the wage
is substantially smaller.
Finally, notice that children born from poor parents will have lower
wages, on average, than children born from rich parents. The expected
wage for poor children in a CPE is 1
2L+1
2^wp, while for rich children it is
1
2H+1
2^wp:Similarly, in CSE poor children obtain ^wswhich is lower than
the expected wage of rich children, 1
2H+1
2^ws:Hence, the intergenerational
correlation of earnings can also be explained by the presence of investment in
signalling education. Nowadays, in most Western countries, overeducation
in the labour market may lead people to think that additional investment
in education (exclusive College degrees) will not be pro…table. But from a
signalling perspective, it becomes essential to preserve the relative position
in the educational ranking. Therefore, rich individuals continue investing in
education and they will obtain higher wages.
22
24
5 How can poor children signal their talent?
In this section we develop some policy proposals to guarantee equal “sig-
nalling”opportunities to individuals. As the reader must be aware, assump-
tion (A.1) is crucial for the preceding results. The following Proposition
addresses this issue:
Proposition 5 If marginal education costs are lower for high-ability than
for low-ability individuals, i.e. ce(e; p; l)> ce(e; r; l)> ce(e; p; h)> ce(e; r; h)
holds, the equilibrium of the model is the separating equilibrium. This equilib-
rium can be described by: (i) education levels (0; ef;0; ef);where efveri…es
U
rl(0; L) = Url(ef; H); (ii) …rms’ belief (il=0e < ef) = 1; (ih=e
ef) = 1; and (iii) …rms’ wage o¤er w(0 e < ef) = L; w(eef) = H:
Proof. See appendix D.
Thus, it is clear that policies should be oriented to change the relation-
ship between marginal education costs, that is, to reverse the inequality
(A.1) and to make the separating equilibrium possible.
It is well known that a reduction in education costs for any educational
level cannot help talented poor children to signal their talent. For ex-
ample, a general subsidy that lowers funding costs for everyone or public
provision of education will merely lead to an increase in education equilib-
rium levels. Considering that a family’s educational spending is given by
Cij =iF1(e; i; j), state or local provision will reduce it: the education
cost will be C=iF1(e; i; j)ge where the second component represents
the public spending per student (that can also be implemented as grants)
which is usually larger for higher educational levels. From Proposition 4,
where educational equilibrium levels, ep;^es;and esare speci…ed, notice that
23
25
the left hand side of these expressions will decrease and in order to maintain
equality, educational levels will increase. Thus, a general decrease in educa-
tion costs does not allow poor children to signal their talent, as it does not
change either inequality (A.1), p
Fs(sph;h;p)>r
Fs(srl;l;r)or the results stated in
Section 3.
However, some policies can contribute to making the separating equilib-
rium possible by reversing condition (A.1): (i) a “large enough”decrease in
education costs, but only for poor people (through grants or a reduction of
the gross interest rate i, for example) and (ii) an increase in educational
standards combined with other measures such as assistance to students from
less advantageous backgrounds and some changes in the selection of students
among schools.
A reduction in educational costs for poor people, by providing grants, for
example, can contribute to the reversal of inequality (A.1)10; the ce(e; p; h)will
be p
Fs(s;h;p)g: If the reduction is large enough and policy makers manage to
reverse condition ce(e; p; h)> ce(e; r; l), all talented individuals, regardless
of their families’income, will be able to signal their talent. Nonetheless, if
the cost reduction is not high enough and the inequality is not reversed, the
only e¤ect of such a policy might be to increase the signalling of educational
levels. Notice that, considering Proposition 4, in a cuasiseparating equilib-
rium, educational levels ^esand eswill increase if ce(e; p; h)decreases. The
growing demand for Masters and exclusive degrees by rich families might be
a consequence of grant policies in most Western European countries. The
more …nancial possibilities the poor families have, the more expensive the
10 We consider that the Public Administration has the “monopoly of information”in the
sense that they have access to information about families’s income, and they use it; this
information is not available or not allowed to be used privately.
24
26
educational attainment that is required by richer families to separate, will
be. Notice that the e¤ectiveness of this policy relates to how accurate Gov-
ernment information concerning family wealth is. In other words, given that
the government normally has imperfect information, this policy is not robust
to cheating on the side of the families.
A decrease in educational costs for poor people can also be achieved by
providing loans at low interest rates for poor people; in fact this policy is
currently being implemented in some countries, such the U.K. and Chile, for
example. We have assumed that poor people can …nance the education of
their children at a high price. A decrease in interest rate pwould facilitate
the reversal of inequality (A.1). Notice again that this policy is robust to
cheating on the side of the families only if the interest rate for the public
loan remains greater for poor families. The next combination of policies that
we propose does not depend on the government having perfect information
about families’wealth.
Apart from the above …nancial aids to poor families, other possibilities
may play a more important role in the reduction of poor people’s educa-
tional costs by changing the relative importance of ability and home back-
ground on educational achievements. To make matters particularly simple,
we will use a Cobb Douglas education production function: e=sij,
where 0< ; ; 1:The parameters ; and give the elasticity of
educational achievements with respect to monetary investment, parental
background and ability, respectively. Condition (A:1) can be rewritten as
p
r>p
rh
l:Thereby, in order to reverse condition (A:1) either ratio
p
rmust increase, or parameters ; must decrease, or must increase. One
possibility is to increase educational standards, which will a¤ect the relative
importance of ability in educational achievements and will be re‡ected, for-
25
27
mally, by a decrease in parameter . Notice that ratio
is related to the
marginal rate of technical substitution MRTS of input sfor input j; at a
given level e;speci…cally, we can see that
=@ M RT Sjs (e)
@(j
s):Formally, a de-
crease in leads to a decrease in the slope at any point in space (s; j)of the
isoquant associated with an educational level. In other words, in order to
achieve an educational level e; a low ability child will need a higher monetary
investment; parents will have to pay additional extra-scholar support (for
example, private classes or academies) to enable their children to achieve a
given level e. Therefore, this policy favours poor, high-talented children.
However, the increase in educational standards has another e¤ect: it also
changes the substitutability between money and learning capacities due to
home background (re‡ected by the ratio
):This measure favours the rich
but low talented children. Notice that if the increase in standards is large
enough and h
l>r
p, condition (A.1)’will be reversed and the e¤ects
on poor talented children will be stronger than those on rich, low talented
children. Consequently, the former will be able to signal their talent.
However, if h
l<r
p;increasing educational standards will not be
e¤ective when it comes to guaranteeing equal signalling opportunities. In
this case, this policy should be combined with (i) policies oriented towards
reducing disparities in skills linked to home backgrounds (that is, to reduce
r
p)and (ii) with policies related to segregation of students among schools
(that will a¤ect and );which will be discussed below.
Firstly, some policies aimed at reducing the e¤ect of background dispar-
ities (measured by the ratio r
p)include: target assistance to students from
less advantageous backgrounds, individualising learning in order to provide
students with appropriate forms of instructions, policies oriented towards
reducing technological disparities among families and extra resources for
26
28
schools in deprived areas (PISA, 2003).
Secondly, some changes in the segregation of students among schools can
help talented poor children to signal their talent. If interaction among stu-
dents holds a positive relationship with performance, segregation of students
of the basis of ability or socioeconomic status results in a positive impact
for those from a richer background and those who are more talented. Hence,
a given di¤erence in home backgrounds (or ability) among students will re-
sult in a greater di¤erence in educational achievements. Parameters and
give the elasticity of educational achievements with respect to parental
background and ability, respectively. Therefore, the increase of segregation
based on parental background (and ability) can be captured by an increase in
parameter (; respectively). In this framework, in order to help poor tal-
ented children segregation on the basis of ability should increase (higher )
and segregation by home backgrounds should be reduced (lower ).
Some changes in the educational system in Britain in 1965 and evidence
from the PISA report seems to support the policies that we propose. The
reduction of secondary school selection on the basis of age 11 ability has
reduced the role of cognitive ability in determining educational achievements
(Galindo-Rueda, Vignoles, 2003). “For various reasons, richer but less able
students were able to take most advantage of the change in policy, and
thus the achievement of this group increases the most.” (Galindo-Rueda,
Vignoles, 2003, p.18). In terms of our model, this change in the educational
system in Britain can be seen as a decrease in the segregation of students by
ability (decrease in ): a low ability is required to attain an educational level
and/or there is a higher substitutability between inputs (
=@ M RT Ssi( e)
@(s
i))
so that students from a richer family background and low talent can take
advantage of these policies, as is actually the case. Nonetheless, the issue
27
29
of segregation by ability is quite controversial. Highly talented individuals
gain and low talented individuals lose with segregation. It is not clear that
the former o¤sets the latter; we only can say that it could help poor and
high ability children to signal their talent.
Moreover, the PISA report shows clearly that in the OECD countries so-
cial background is determinant for children’s educational achievement. For
example, in OECD countries, 33% of the variation in students’performance
is explained by their attendance at di¤erent schools. A phenomenon of this
kind may arise both from the …nancial resources at their disposal and from a
concentration of the best students in certain schools. Empirical studies have
great of di¢ culty in detecting any systematic in‡uence of expenditure on ed-
ucation on the performance of students. Consequently, the explanation may
rely on segregation among schools: the heterogeneity in the average perfor-
mance for schools comes partly from the fact that some schools attract the
best pupils while others attract the worst. If there is a positive interaction
between the pupils’performance and if these performances are themselves
positively in‡uenced by parental income, this selection may result in a phe-
nomenon in which the wealthiest people mostly send their children to the
same schools (Cahuc and Zylberberg, 2004, p.93).
In short, an increase in educational standards would help poor-talented
children if it is combined with measures oriented towards mitigating the
e¤ect of disparities in home backgrounds, which a¤ect performance at school,
and reducing the segregation among schools based on home background.
28
30
6 Conclusions
The aim of this paper has been to explain how …nancial constraints, innate
abilities and learning skills linked to home background a¤ect the signalling
possibilities of poor children. This helps to explain the transmission of
inequality across generations and why inequality has persisted over time
despite the implementation in most Western countries of educational policies
aimed at guaranteeing equal opportunities.
We have shown that if education were just a signalling mechanism in
the labour market, the di¤erence in wages between poor and rich children
cannot be expected to disappear. From a human capital approach of educa-
tional investments, the earnings of poor and rich children can be expected to
converge (Becker and Tomes, 1979, 1986). These authors argue that when
rich families stop investing in human capital, as the marginal rate of re-
turns on human capital and physical capital draws level, poor families will
continue investing in education (diminishing returns on human capital are
assumed). It will contribute to a convergence in earnings as there will be
a convergence in individuals’productivity linked to human capital invest-
ments. However, from a signalling perspective, rich parents could …nd it
pro…table to …nance as much education as their talented children need to
signal their ability, which will guarantee them the best position in the labour
market. Therefore, when the best wages go to the more educated individ-
uals, the rich try to prevent imitations to the extent that if children from
poorer backgrounds have college degrees, for example, they …nd prestigious
and exclusive Institutions to signal their talent. The claim of Becker and
Tomes that the greater wealth of rich parents does not always lead to an
increase educational investment, and that the higher investment is less e¤ec-
29
31
tive in increasing their children’s earnings, (according to the human capital
approach of educational investment) merits more research in the extent that
education works as a signalling device.
In this context, public education does not help talented poor children
to catch up to rich-talented individuals. Despite the increase in the overall
educational achievement in most developed and developing countries (Barro
and Lee, 2000), inequality remains unsolved. The paper has shown the role
of some policies to guarantee equal signalling opportunities. These poli-
cies include an increase in educational standards combined with measures
oriented (i) towards reducing the e¤ect on the learning process due to dis-
parities in families’home background (that a¤ect the learning process) and
(ii) reducing the segregation of students among schools on the basis of family
socioeconomic status.
Finally, the model has shown that in more egalitarian societies, where
large di¤erences in educational costs do not exist between rich and poor chil-
dren, more pressure is placed on the types that try to separate. They will
need higher educational levels to prevent imitations from their competitors.
But, if there is so much pressure from below, the di¤erent types might be
better o¤ not signalling their ability. This favours the stability of a situa-
tion where only the talented-rich individuals invest in signalling education,
i.e., the cuasipooling equilibrium (moreover, this also means that the utility
tends to be lower in situations where more types invest in signalling, that
is, the cuasiseparating equilibrium.) Hence, we can …nd a more egalitarian,
but less mobile society, in the sense that all poor families have lower wages,
as talented poor children do not …nd it pro…table to even separate from the
less talented individuals with their same economic status.
30
32
APPENDIX
A Proof of Lemma 1
Proof. Consider two workers ij and i0j0, with di¤erent marginal education
costs, speci…cally, ce(i; j)> ce(i0; j0);where ij and i0j0can correspond to
any type verifying ce(e; p; l)> ce(e; p; h)> ce(e; r; l)> ce(e; r; h):Suppose
(en route to a contradiction) that the optimal education levels are eand e0;
respectively, with e > e0. Then it must be the case that worker ij weakly
prefers a wage w(e)to w(e0):
Wi+w(e)c(e; i; j)Wi+w(e0)c(e0; i; j )0:
Similarly, worker i0j0weakly prefers (w(e0)to w(e), that is
Wi0+w(e0)c(e0; i0; j0)hWi0+w(e)c(e; i0; j 0)i0:
Adding these two inequalities yields
c(e; i0; j0)c(e0; i0; j 0)c(e; i; j)c(e0; i; j)0:
This is a contradiction as we assumed that ce(i; j)> ce(i0; j0): the increment
from e0to eis more expensive for the ij type than for the i0j0type.
B Proof of Proposition 2
Proof. En route to contradiction, let us assume that, in equilibrium, the
rh-type pools and attains educational level ^esuch that …rms’ beliefs are
(rh=^e)<1;and obtains a wage ^w > L: As education does not decrease with
the marginal cost of education (by Lemma 1), the rhtype must be pooling
31
33
at least with the rltype. In this situation, the out-of-equilibrium beliefs
that sustain such an equilibrium for any e2[e0; e00)are (rh=e)<1;where
e0and e00 are de…ned as the education levels that verify U
rl(^e; ^w) = Url(e0; H )
and U
rh(^e; ^w) = Url(e00; H );respectively.
But these beliefs (rh=e)<1; e 2[e0; e00);do not verify the Intuitive
Criterion. According to this re…nement, the beliefs for any e2[e0; e00)must
be (rh=e) = 1:The reason is that the only type for which e2[e0; e00)is not
dominated in equilibrium is the rh type. Formally, U
rh(^e; ^w)< Urh(e; H );
e2[e0; e00)and, furthermore, U
ij (^e; ^w)Uij (e; H );8ij 2 fpl; ph; rlg:There-
fore, if beliefs are modi…ed according to Intuitive Criterion, the best strategy
for the rh type is no longer ^ebut e=e0:A PBE in which rh pools does not
survive Intuitive Criterion .
C Proof of Proposition 4: equilibrium of the model.
We proceed in four stages. Firstly, we focus on calculating the PBEs that
are consistent with conditions (i)-(iii) and that verify the Intuitive Criterion
(Claim 1 and 2). Secondly, we compare the education level attained by
the “highest type”, rh; in each situation (Claim 3). Thirdly, utilities of
each type in these PBE are compared (Claim 4). Finally, we apply the
undefeated equilibrium concept for out-of-equilibrium beliefs and we obtain
the equilibrium of the model for each parameter value (Claim 5).
Stage 1.
Claim 1 There exists a PBE (cuasipooling) that satis…es the Intuitive Cri-
terion characterized by: (i) the educational levels are (0;0;0; ep);where ep
veri…es U
rl(0;^wp) = Url (ep; H )(and solves the equation c(ep; r; l) = 1
1+(H
L);(ii) …rms’ belief after observing equilibrium education levels: (pl=0) =
32
34
1+; (ph=0) =
1+; (rl=0) = 1
1+and (rh=ep) = 1; and for out-of-
equilibrium education levels are: (rl=0< e < ep) = 1 and (rh=e ep) = 1
and, …nally, (iii) …rms’wage o¤er: w(0) = ^wp=1
1+L+
1+H; w(eep) =
Hand w(0 < e < ep) = L:
Proof. It is clear that the equilibrium described is a PBE: all parties
are playing optimally and, on the equilibrium path, beliefs are consistent
with Bayes’Law. This PBE also satis…es the Intuitive Criterion: …rms do
not assign a positive probability for any type for which an education level
is dominated in equilibrium. The (iv) “lowest aggregate condition” is also
met.
Speci…cally, workers are playing optimally since the incentive compati-
bility constrains (ICR) are veri…ed. (These restrictions mean that any type
is better o¤ in the speci…ed equilibrium than mimicking another type.) The
(ICR) are:
() for the rh type:
U
rh(ep; H )Urh(0;^wp) =)c(ep; r; h)H^wp;(p.1)
() for the rl type:
U
rl(0;^wp)Url (ep; H ) =)c(ep; r; l)H^wp(p.2)
() for the ph type:
U
ph(0;^wp)Uph(ep; H ) =)c(ep; p; h)H^wp;(p.3)
() for the pl type:
U
pl(0;^wp)Upl(ep; H ) =)c(ep; p; l)H^wp:(p.4)
33
35
Restriction (p.2) implies (p.3) and (p.4) given the assumptions on edu-
cation costs. Therefore, for epto be an equilibrium education level it must
verify (p.1) and (p.2) and the (iv) equilibrium condition; the lowest educa-
tion level that met these three conditions is the proposed education epthat
solves (p.2) with equality.
Firms’beliefs are consistent with Bayes’Law given the education choices
of workers. For out-of-equilibrium education levels, the speci…ed beliefs
verify the IC. This re…nement requires the following:
()(pl=e ep
4) = 0;since U
pl(0;^wp)Upl(eep
4; H);
()(ph=e ep
3) = 0;since U
ph(0;^wp)Uph(eep
3; H);and
()(rl=e ep
2) = 0;since U
rl(0;^wp)Url (eep
2); H),
where ep
2; ep
3;and ep
4are the education levels that solve (p.2), (p.3), (p.4)
with equality. So, …rms do not assign a positive probability to those types
after observing these education levels because the equilibrium utility of these
workers exceeds the utility of choosing the speci…ed education levels, no
matter what the …rms believe after observing any of them, that is, these
education levels are dominated in equilibrium.
Finally, we have to prove that the rh type chooses the minimum edu-
cation level consistent with …rms’beliefs. IC implies that the …rms’beliefs
must be (rh=epeep
1)=1(ep
1veri…es (p.1) with equality), provided
that eis not dominated for the rich, high-ability individual (and is dom-
inated for the others types), which in turn implies that an equilibrium in
which the rh type chooses and education level e > epcannot satisfy IC be-
cause in such an equilibrium the …rms must believe that (rh=e > ep)<1:
Therefore, the only equilibrium that satis…es IC is the equilibrium in which
the rh type chooses just ep:
34
36
Claim 2 There is a cuasiseparating PBE in which (i) education levels are
(0;^es;^es; es);where ^esand esverify U
pl(0; L) = Upl (^es;^ws)and U
rl(^es;^ws) =
Url(es; H);respectively (and solve:c(^e; p; l) = (HL)and c(es; r; l)
c(^e; r; l) = (1 )(HL)); …rms’belief is (pl=0e < ^es)=1; (ph=e =
^es) = ;(rl=e = ^es) = 1 ;(rl=^es< e < es) = 1 and (rh=e es) = 1;
and …rms’ wage o¤er is: w(0 e < ^es) = L; w(^es) = ^ws=H + (1 )L;
w(^es< e < es) = L; w(ees) = H:
Proof. The equilibrium described is a PBE. Workers are playing opti-
mally given the …rms’wage o¤er; the incentive compatibility constraints are
veri…ed. These restrictions are:
() for the rh type:
U
rh(es; H )Urh(^es;^ws) =)c(es; r; h)c(^es; r; h)(1 ) (HL)(s.1)
U
rh(es; H )Urh(0; L) =)c(es; r; h)HL(s.2)
()for the rl type:
U
rl(^es;^ws)Url(es; H ) =)c(es; r; l)c(^es; r; l)(1 )(HL)(s.3)
U
rl(^es;^ws)Url(0; L) =)c(^es; r; l)(HL)(s.4)
()for the ph type:
U
ph(^es;^ws)Url (es; H ) =)c(es; p; h)c(^es; p; h)(1 ) (HL)(s.5)
U
ph(^es;^ws)Url (0; L) =)c(^es; P; h)(HL)(s.6)
()for the pl type:
U
pl(0; L)Upl (es; H ) =)c(es; p; l)(HL)(s.7)
U
pl(0; L)Upl (^es;^ws) =)c(^es; p; l)(HL)(s.8)
35
37
Notice that if (s.6) holds, (s.4) also holds, and that if (s.3) is ful…lled, ICR
(s:5) is also ful…lled. Hence, we look for the education equilibrium levels
^esand esthat verify from (s.1) to (s.8), except for (s.4) and (s.5). On the
one hand, education ^esmust verify (s.8) and (s.6). The minimum education
that solves both restrictions is the one that veri…es (s.8) with equality.
On the other hand, education eshas to ful…l the following conditions: (i)
eses
i; i = 1;2and (ii) eses
i; i = 3;7;where es
iare the education levels
that verify the above restrictions with equality (s.1), (s.2), (s.3) and (s.7).
It can be easily proved that es
7< es
382[0;1); es
1< es
282(0;1];and
es
3< es
182[0;1):Hence, es
7< es
3< es
1< es
2;82(0;1):So the minimum
educational level esthat solves all RCI is es
3;that is, the proposed education
choice for the rh type.
Firms’beliefs are consistent with Bayes’s Law, given the workers’choices.
For out-of equilibrium education levels, …rms’ beliefs satisfy the IC. This
criterion requires: (ph=e es
5) = 0 because Us
ph()> U s
ph(ees
5; H)and
(rl=e es
3)=0;because Us
rl ()> U s
ph(ees
3; H):Therefore, (rh=e
es
3) = 1:The proof that the only equilibrium that survives IC involves the
rhtype choosing esis similar to the cuasipooling equilibrium.
Stage 2.
Next we analyse if the rh-type attains a higher education level in the
cuasipooling or in the cuasipooling equilibrium.
Claim 3 (i) The education choice of the rhtype is such that epesif
8
<
:
ce(e;p;l)
ce(e;r;l)2
ce(e;p;l)
ce(e;r;l)>2and 2(0;
];
36
38
and (ii) the education choice of the rh-type is such that ep> esif ce(e;p;l)
ce(e;r;l)>2
and 2(
; 1).
Proof. Educational level epwill be such that ep< esif c(ep; r; l)<
c(es; r; l):Considering the equilibrium levels given in Proposition 4, we have
that c(ep; r; l)< c(es; r; l)can be rewritten as
(HL)2
1 + < c(^es; r; l):(3)
The left hand side, (HL)2
1+;is increasing and convex in and 2
1+(H
L)20;1
2(HL). Let us show that the right hand side is linear on :
d c(^es;r;l)
d =ce(^es; r; l)d^es
d = (HL)ce(^es;r;l)
ce(^es;p;l)>0as d^es
d =(HL)
ce(^es;p;l):Moreover,
this r.h.s is zero for = 0 since ^es= 0 and for = 1 taking into account that
it is lineal in it can be written as c(^es(= 1); r; l)=(HL)ce(^es;r;l)
ce(^es;p;l);(i)
if ce(^es;r;l)
ce(^es;p;l)1
2)c(^es(= 1); r; l)1
2(HL):Thus, 82(0;1) ; ep< es:
But (ii) if ce(^es;r;l)
ce(^es;p;l)<1
2;there is a
such that (HL)2
1+=c(^es; r; l)where
^essolves c(^es; p; l) = (HL):Then in this case we obtain that (3) holds
for some values. For
; epesand for >
; ep> es:
Stage 3.
In this stage the utility levels in these PBE are compared.
Claim 4 (i) If the education choice of rich individuals of high ability is such
that epesthe utility of each type of family is higher in the cuasipooling
equilibrium than in the cuasiseparating equilibrium, that is,
Up
ij Us
ij 8ij;
(ii) if ep> esthe utility of a rich family with high ability o¤spring is higher
37
39
in the cuasiseparating than in the cuasipooling, that is,
Up
rh > U s
rh ,
where superscripts pand sdenote cuasipooling and cuasiseparating equilib-
rium.
Proof. Firstly , if epeswe obtain the following results:
Up
rh (ep; wr)Us
rh (es; H ):
Up
rl (0;^wp)Us
rl (^es;^ws):Educational levels epand essolve the fol-
lowing equations:
Url(es; H) = Us
rl (^es;^ws)
Url(ep; H) = Up
rl (0;^wp):
If epes, we have that Ur l(ep; H)Url (es; H )and by direct substi-
tution it follows that Up
rl (0;^wp)Us
rl (^es;^ws):
Up
ph (0;^wp)> U s
ph(^es;^ws):Firstly, de…ne e0such that:
Up
rl (0;^wp) = Url (e0;^ws);
and considering that we have shown Up
rl (0;^wp)Us
rl (^es;^ws)when
epes;then
Url(e0;^ws)Us
rl (^es;^ws)
which implies e0^es:
Secondly, de…ne esuch that
Up
ph (0;^wp) = Uph(e; ^ws)
38
40
and, considering the assumption regarding marginal cost, we have
e<e0. Then e<e0^es:Finally, considering the de…nition of e
,(Up
ph (0;^wp) = Uph(e; ^ws));and that e < e0<~es;it follows that:
Uph(e; ^wp)> U s
ph(^es;^ws):
And by substitution of Up
ph (0;^wp) = Uph(e; ^wp), we obtain
Up
ph (0;^wp)> U s
ph(^es;^ws):
Q.E.D.
Up
pl (0;^wp)> U s
pl (0; L):This is obvious because ^wp> L:
Secondly, if ep> esit is evident that Up
rh (ep; H )< Us
rh (es; H ).
Stage 4.
Claim 5 The undefeated equilibrium is: (i) the cuasipooling PBE if epes
and (ii) the cuasiseparating PBE if ep> es.
Proof. Case i) epes. The level of education epis not chosen in the
cuasiseparating equilibrium. As Urh (ep; wr)Urh (es; wr), beliefs should be
(rh/ep) = 1, which does not coincide with the beliefs that support CSE.
So, CSE is defeated by CPE.
Now imagine education levels ^esor esare not chosen in CPE. We know
by Claim 4 that all types obtain a higher utility in this equilibrium than in
CSE (i.e. sending ^esand es). Therefore, CPE is not defeated by CSE.
Case ii) ep> es:CPE is defeated by CSE. Notice that esis not chosen
in CPE and yields a higher utility to the type rh than choosing ep. So
39
41
beliefs after observing esshould assign probability one to this type. But
these beliefs do not support CPE.
On the other hand, the only level of education which is not chosen in
CSE but chosen in the CPE is ep. Given that the utility for rh is smaller in
epthan in esin this case (earning wrin both cases), the beliefs that support
the CSE are justi…ed. Therefore, the CSE is not defeated by the CPE.
In short, from claims 1 to 4, the equilibrium of the model is cuasipooling
if ce(e;p;l)
ce(e;r;l)2or if ce(e;p;l)
ce(e;r;l)>2and 2(0;
]which leads to epesby
Claim 3;and cuasiseparating if ce(e;p;l)
ce(e;r;l)>2and 2(
; 1)which leads to
ep> es:
D The separating equilibrium
Proof. The proof is similar to the proof of Proposition 4 except in what
follows concerning the characterization of educational equilibrium level ef:
In this equilibrium, the incentive compatibility constraints are:
Proof.
() for the rh type:
U
rh(ef; H )Urh(0; L) =)c(ef; r; h)HL(f.1)
()for the ph type:
U
ph(ef; H )Uph(0; L) =)c(ef; p; h)HL(f.2)
()for the rl type:
U
rl(0; L)Url(ef; H) =)c(ef; r; l)(HL)(f.3)
()for the pl type:
U
pl(0; L)Upl (ef; H ) =)c(ef; p; l)(HL):(f.4)
40
42
Restriction (f.2) implies (f.1) and restriction (f.3) implies (f.4) given the
assumption on marginal education costs. Thus, the only educational level
that veri…es (f.2) and (f.3) is the efproposed. This equilibrium satis…es
the Intuitive Criterion (see proof of Proposition 2 to check how this re…ne-
ment is applied). Notice that under this condition on education costs, the
Proposition 2 and 3 is not longer true. It breaks the possibility of a pooling
equilibrium (0;0;0;0) or a semiseparating equilibrium such as (0;0;0; e0)or
(0; e0; e0; eH):Moreover, Lemma 1 still holds. Hence, if pooling and semisep-
arating PBE are not equilibria of the game, the separating PBE is the only
equilibrium.
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