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Document de travail du LEM Discussion paper LEM 2016-07
Education, Intergenerational Mobility and Social Stratification: Theory
Joël HELLIER
[email protected] - LEM-CNRS UMR 922
URL de téléchargement direct / URL of direct download: http://lem.cnrs.fr/IMG/pdf/dp2016-07.pdf
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Education, Intergenerational Mobility and Social Stratification: Theory Joël Hellier * LEM-CNRS (UMR 922), University of Lille, and LEMNA, University of Nantes.
Abstract This paper proposes a synthetic and unified framework which permits to model the different channels and mechanisms highlighted in the economic literature which bind education, intergenerational mobility and social stratification. After exhibiting some key stylised facts and the major concepts utilised in these analyses (education functions, intergenerational mobility, polarization), the situations in which intergenerational dynamics leads to human capital convergence of dynasties are firstly exposed. The several determinants of lasting or permanent human capital stratification and of the emergence of low education traps are subsequently analysed. The impacts of credit market imperfections, fixed costs of education, ‘S-shaped’ education functions, global externalities, neighbourhood effects and of the structure of education systems are particularly investigated. All these channels are simply modelled by extending the same basic framework. Some issues in the correspondence between theoretical and empirical models are finally highlighted and promising fields of research are exposed. Keywords. Education; Human capital; Intergenerational mobility; Social stratification. JEL Classification. I21, J24, J62.
I wish to thank the students who followed my Master lectures on ‘Education, human capital and intergenerational mobility’ in the Universities of Nantes and Lille for their remarks and questions. They have greatly contributed to the improvement of this synthetic approach.
*28 rue de Sévigné 75004 Paris France.
[email protected] /
[email protected]
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1. Introduction and stylised facts In contrast with sociology which has studied for a long time the link between education, social positions and social mobility,1 the economic analysis of education and intergenerational mobility is more recent. Based on the notion of human capital defined by Becker (1964), the early models of human capital formation and accumulation were developed by BenPorah (1967) and Becker & Tomes (1976, 1979). If Ben Porah (1967) focused on the allocation of time between education and working over the individuals’ life, Becker & Tomes (1979) centred their analysis on the intergenerational dynamics of human capital. The way economic theory has analysed the relationships between education decisions and intergenerational mobility of human capital is the subject of this paper.2 In addition to brief reviews of the seminal approaches, a synthetic and unified framework is built, which permits to depict the major theoretical mechanisms and outcomes put forward by the literature. Even if this framework has a general scope, the case of education in developed countries is particularly tackled. Since World War II, the World has been characterised by a substantial education upgrading. In all countries, the age of compulsory education has been elevated. In advanced countries, the share of a generation pursuing tertiary education has moved from less than ten percent in the early fifties to more than sixty percent nowadays. This extension of tertiary education also concerns emerging countries. In China, the number of students attending higher education increased from 3 million in the late eighties to 20 million in 2006 and 30 million in 2013. From the after-war period up to the early eighties, education upgrading came with a decrease in inequality and an increase in social and intergenerational mobility in advanced economies. This is no longer the case. In the last three decades, inequality has increased in all advanced countries. In addition several works suggest that intergenerational mobility has decreased in several countries, particularly in those which already displayed the lowest mobility. Intergenerational mobility is measured by the impact of the social position of parents on the social position of their children. The higher this impact, the lower mobility. In the economic and econometric literature, the most usual indicator of mobility is the 1
Both Durkheim and Weber considered education as a key determinant of socialization and social stratification. The economic literature has been reviewed by Webbink (2005), Bjorklund & Jantti (2000, 2009), Fields (2008), Causa and Johansson (2009), Black and Devereux (2011) and Chusseau and Hellier (2013). Sauer & Zagler (2012) review the economic literature on education, intergenerational mobility and growth. 2
3 intergenerational elasticity (IGE; Solon, 1992, and Zimmerman, 1992). For any variable (incomes, earnings, wealth, educational attainment), the IGE is the elasticity of this variable for individuals in relation to that of their parents.3 Other indicators of intergenerational mobility can be calculated, like the intergenerational correlation or the rank-rank slope.4 In the sociological approaches which divide the population in different groups or classes, intergenerational mobility is often measured using mobility tables. The population being divided in classes, the mobility table is the matrix qij where qij is the proportion in the population of individuals in the i-class whose parents belong to the j-class. A broad measure of mobility is then given by 1 i qii . From this matrix, one can also calculate indicators of mobility between two classes, as the odds ratio. Whatever the selected indicator, the empirical evidence on intergenerational mobility reveals several prominent features:5 1) In all countries, the parents’ position plays a crucial role in the individuals’ attainments in terms of income, wealth, education and social position. 2) Intergenerational mobility greatly differs across countries. First, mobility is typically and significantly lower in developing countries than in advanced countries. In addition, the divergence in mobility is also substantial across advanced economies (Corak, 2013; Blanden, 2013). The UK, the US, France and Italy display to lowest mobility, with earnings IGEs between 0.4 and 0.55. Scandinavia and Canada have the highest mobility, with IGEs between 0.15 and 0.25. The mobility of other developed countries is in-between. 3) After an increase in the after-war period (Lee and Solon, 2009; Breen, 2009; Lefranc, 2011), mobility seems to have subsequently either stagnated or decreased in several countries, particularly in the four countries with the lowest mobility (The UK, the US, France and Italy), even if this diagnosis is challenged by some works in the case of the US and the UK.6
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Let y be the log of the analysed variable and let subscript i depict the dynasty (succession of generations linked by a parent-child relationship) and t the generation. Then the y-IGE is coefficient calculated by estimating the equation yit yit 1
j j x ji it , where
yit 1 is the value of y for individual (i,t)’s parents, x ji a set
of control variables and it the error term. A temporal fixed effect can be added to this equation. 4
The intergenerational correlation is t 1 / t , with t the standard deviation of the variable at gene-
ration t. The rank-rank slope is the slope of the relation that binds the rank of children to that of their parents. 5 Reviews of the empirical literature can be found in Holmlund et al. (2011), Black & Devereux (2011) and Björklund & Jäntti (2009). 6 Decreases in mobility are diagnosed by Aaronson & Mazumder (2008) for the US, Blanden et al. (2004, 2007) and Nicoletti & Ermisch (2007) for the UK, Lefranc (2011) and Ben-Halima et al. (2014) for France. In contrast, based on cohort comparisons, Breen & Golthorpe (1999, 2001) found no change in mobility in the UK and Chetty et al. (2014) no decrease in intergenerational earnings mobility in the US.
4 4) Finally, intergenerational persistence is higher in the tails of the earnings distribution, particularly in its upper tail (Jäntti et al., 2006; Björklund et al., 2012). The transmission mechanisms between parents and children are numerous. However, one of the most powerful channels goes through education. First, since the seminal work of Mincer (1958, 1970), the key impact of education attainment on individuals’ earnings is well documented and commonly recognised. Second, the impact of family backgrounds on education and on the capacity to fully benefit from the education system is crucial. Consequently, if the development of free and publicly provided compulsory education as well as public funding on education in general has indubitably promoted education upgrading for all children, the role of family backgrounds remains essential for educational attainment. This impact is obviously reinforced when education is fully or partially paid by families, the most educated families being typically the richest. From the mechanisms put forward by the literature, this article develops a unified and synthetic framework that permits to reveal the major outcomes of the economic approaches to education and intergenerational mobility. Section 2 presents the tools utilised in the analysis. Section 3 constructs the basic framework and the approach to human capital convergence in perfect competition. Section 4 examines the extensions based on imperfections on the credit market. Section 5 analyses the impact of a fixed cost of education. Section 6 exposes the case of S-shaped education functions and Section 7 local externalities and neighbourhood effects. Finally, Section 8 explores the impact of the structure of education systems on intergenerational mobility and social stratification. All these extensions typically generate the emergence of low education traps and social polarization. We finally discuss the correspondence between the theoretical and empirical approaches and highlight several promising paths of research in Section 8.
2. Education functions, mobility and stratification In this section, we present the analytical tools on which the economic approaches to education and intergenerational mobility are based. The construction of education production functions is firstly exposed, and the methods utilised in the analysis of intergenerational mobility are subsequently highlighted.
5 2.1. Education function and education decision a) Education production functions An education production function (hereafter ‘education function’) utilises inputs (public provision of educational services, teacher quality, time, family attributes, private expenditures on education, personal ability, etc.) to produce human capital. This ‘production of human capital’ is made by individuals who decide for the time and resources they allocate to education.7 A dynasty (succession of individuals linked by a parent-child relationship) is depicted by subscript i and a generation by subscript t. Assuming to simplify that each individual has one child, individual (i,t) denotes the individual of dynasty i belonging to the t-th generation. Let us further suppose that education is acquired at the beginning of the individuals’ life so that they leave school with a human capital they will subsequently possess during their remaining lifetime (we assume no human capital obsolescence). Then, an education function can be simply presented as follows: hit Ht gt , eit , ait , hit 1 ,
Ht 0, x gt , eit , ait , hit 1 x
where: hit is individual (i,t)’s human capital at the end of education time;
(1) H t is the
education function at period (generation) t, which essentially depends on the educational technology; gt is the public expenditures on education for generation t (assumed to be the same for all the children in this generation); eit is the individual’s (or her family’s) expenditure on education; ait is the individual’s personal ability which is assumed independent from family backgrounds; hit 1 is the parent’s human capital which depicts the influence of family backgrounds. Note that, in Relation (1), the individual’s expenditure on education eit incorporates the opportunity cost linked to the time spend in education (the individual’s additional earning, would she work instead of studying). In a less simplified presentation, both the education time and the private expenditures on education could be arguments of the education function. As personal ability is independent from family backgrounds, ait is assumed to belong to an exogenously defined interval a, a and to be randomly distributed across individuals 7
An early presentation can be found in Bowles (1970). See Hanushek (1979, 2008, 2010) for surveys centred on empirical issues. In their reviews of the literature, Gradstein et al (2005) and Chusseau and Hellier (2013) present the theoretical backgrounds of such functions.
6 (dynasties) at each generation, with no particular assumption made on the related density function (the simplest way being to assume an i.i.d.). Several authors denotes this familyindependent ability as ‘luck’ (e.g., Becker & Tomes, 1979, 1986) In function H t , the key variable generating intergenerational mobility is the parent’s human capital hit 1 . This variable depicts a number of components and mechanisms defining the individual’s family backgrounds: 1. Some authors introduce intra-family genetic linkages in terms of ability (e.g., Becker & Tomes, 1976, 1979, 1986). This typically opens the ‘nature vs. nurture’ debate, which is not tackled here. 2. A major component of the impact of family backgrounds is the knowledges directly transferred to children inside the family. The children born in educated families typically benefit from cultural advantages. 3. On top of supplying direct knowledge, the family can provide capacities of reasoning and analysis, and thereby aptitudes to capture more or less easily the knowledge provided by the education system. This gives an additional advantage to children from educated families. 4. Parents provide their children with material and educational means (books, computers, software, education assistance, etc.) that depend on their income (and wealth), which is itself related to the parents’ human capital. Finally note that a number of works combine the randomly distributed personal ability and the family-related ability (intra-family human capital transfers) into a unique component called ‘personal ability’.8 If this unified approach to ability is convenient when the analysis aims at discriminating between the factors which are external to the child and her family (type of school, strength of teachers, number of pupils in the classroom etc.) and the factor corresponding to personal characteristics (family backgrounds included), it can be seen as inadequate when the approach wants to disentangle family backgrounds from purely personal ability so as to analyse intergenerational mobility. b) Education decision In education function (1) there is only one variable that depends on the individual’s decision, i.e., her real expenditure on education eit . Suppose (i) that individual (i,t) has a lifetime objective (utility function) that depends on her income, and (ii) that her income is directly
E.g., Gradstein et al. (2005) define individual (i,t)’s personal ability as : ait ait 1 bit1 where bit is a random disturbance. 8
7 linked to her human capital. Then, she will choose the education (human capital hit ), i.e. the expenditure eit , that maximises her well-being. Individual (i,t)’s behaviour can be summarised by a two-stage decision process: the individual firstly selects the education that maximises her lifetime income, and she subsequently distributes this optimal income between the diverse elements that define her utility (different types of consumption, leisure, bequest etc.).9 Hence, the education decision is made by maximising the net lifetime income subject to the education function (1). If wt is the real wage per unit of human capital at generation t and the available time is standardised at 1, then the maximisation programme defining the individual’s education is: max Iit wt hit it eit , eit
s.t.: hit Ht gt , eit , ait , hit 1
where I it is the individual’s lifetime total income and it her lifetime non-labour real income. Depending on the educational and economic environment, this maximisation program may be augmented by additional constraints (imperfection on the credit market, fixed cost of education, admission procedures etc.). Let eˆit arg max Iit be the (unique) optimal expenditure on education. Then, individual eit
(i,t)’s human capital corresponding to her optimal education decision is:
hˆit Ht gt , eˆit , ait , hit 1
(2)
2.2. Intergenerational mobility, convergence and polarization Equation (2) defines a relationship between the human capital levels of two successive generations of dynasty i. Suppose now that both function Ht H and expenditures gt g are constant over time. Then, eˆit only depends on ait and hit 1 , and relation (2) can
be written:
hit H ait , hit 1
(3)
The intergenerational dynamics depicted by Relation (3) binds the human capital of two successive generations. We now analyse this dynamics in two cases, namely, when individuals do not differ and when they do differ in their personal abilities. 9
It is possible to add an element that accounts for the non-pecuniary benefit from education (e.g., Glomm & Ravikumar, 1992; Barham et al., 1995). Note that a direct maximisation of the utility function is necessary when certain inputs of the education function also impact utility (e.g., effort).
8 a) Identical personal abilities When personal abilities do not differ across individuals and over time, we can write: ait 1, i, t . Relation (3) becomes hit H hit 1 and it is possible to draw the phase
diagram that binds the human capital of an individual to that of her parent.
45°
Figure 1. Intergenerational dynamics with a concave function H hit 1
Figure 1 portrays the most usual profile of this intergenerational dynamics, which corresponds to a continuous, monotonically increasing and concave function H hit 1 with
H 0 0 . In this phase diagram, all dynasties tend towards the same human capital steady state hˆ whatever their initial position. For instance, a dynasty starting from the rather low human capital h0 attains the steady state hˆ after approximately five generations. (b)
(a)
(c)
Figure 2. Profiles of human capital dynamics
9 Figure 2 depicts different profiles of intergenerational dynamics, depending on the shape of function H . In Fig. 2-a, the function H is convex ( H / hit 1 0 and 2 H / hit 12 0 ). Here, all the dynasties with an initial human capital (at generation 0) below hˆ2 tend towards the low steady human capital hˆ1 , and all those above hˆ2 have their human capital that indefinitely increases from generation to generation and tends thereby towards infinite. Such a situation is clearly not realistic and contradictory with empirical evidence. In Figure 2-b, the function H hit 1 is S-shaped. There are two stable steady states ( hˆ1 and hˆ3 ). With such dynamics, all the dynasties with an initial human capital below hˆ2 tend towards the low steady human capital level hˆ1 , and all those above hˆ2 tend towards the high steady human capital level hˆ3 . Note that, when the convex part of the S-curve is fully located above, or when its concave part is below, the 45° line, then all the dynasties tend towards a unique steady human capital. Finally, in Fig. 2-c, there are three stable steady human capital levels, hˆ1 , hˆ2 and hˆ3 . It can be noted that, in the cases portrayed in Figure 2, the education dynamics generate permanent low education traps (LET), i.e., situations in which a proportion of the dynasties perpetually remains low-educated without being able to escape from this position. These families tend towards the steady states hˆ1 . In addition, education dynamics lead to the grouping of dynasties in two poles in the case 2-b, and three poles in the case 3-b. This polarization generates education-based social stratification with, in the case 3-c, one low-skill group (with human capital steady state hˆ1 ), one middle skill group ( hˆ2 ) and one high-skill group ( hˆ3 ). b) Differences in personal abilities Let us now assume that individuals randomly receive at birth a personal ability which belongs to the interval a, a . This ability is independent from the individual’s family background. The intergenerational dynamics is now represented by Equation 3 hit H (ait , hit 1 ) and there is a set of functions ht H a, ht 1 , a a, a , which defines all the dynamics related to each possible value of the personal ability. In this case, the simple phase diagram pictured in Fig. 1 is transformed into Figure 3 which draws all the curves ht H a, ht 1 , a a, a .
10 By writing H ht 1 H a, ht 1 and H ht 1 H a , ht 1 , all the possible dynamics are located between H ht 1 and H ht 1 , and any two curves H a, ht 1 with different a never intersect. As showed on Fig.3 from an initial human capital h0 , dynasties now jump from one curve to another when passing from one generation to the next, depending on their ability at each generation.
45°
Steady Segment
Figure 3. Human capital dynamics with differences in personal abilities
The intersection of curves H a, ht 1 , a a, a , with the 45° line determines the set of steady states corresponding to each of these dynamics. The Steady segment Sˆ hˆ, hˆ , with
hˆ H hˆ and hˆ H hˆ , is the locus of all these steady states. It can be easily seen that, once a generation of a dynasty enters the steady segment Sˆ , then all the following generations of this dynasty remain inside this segment, jumping from one position to another depending on their successive abilities. In summary, when assuming randomly distributed personal abilities, all dynasties tend towards a steady segment of human capital and they subsequently jump from one human capital position to another inside this segment, depending on the distribution of abilities amongst dynasties at each generation.
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3. Human capital convergence The early economic literature on intergeneration mobility diagnosed a quick convergence in human capital of all dynasties. This is the case in the seminal model of Becker & Tomes (1979) wherein the only source of divergence in human capital across individuals is personal ability which is related to the parent’s ability and enters the education function. The other component of personal ability is ‘luck’ which is randomly distributed across individuals. The model assumes a perfect market for credit, which ensures no impact of families’ divergences in income and wealth on the individuals’ educational choices. As the family component is only partially transmitted, the initial differences tend to vanish from one generation to the next and the dynasties’ human capital converges. Becker & Tomes (1979) diagnosed that convergence should be attained after a limited number of generations. We firstly construct our basic framework and subsequently show how to establish convergence within this synthetic approach. We also note that a limited change in hypotheses can generate polarization and a low education trap even when perfect competition is assumed. 3.1. The basic framework We construct a simple basic specification of function ht H a, ht 1 that will be utilised and modified in the following sections so as to synthetize the diverse approaches developed in the literature. So as to focus on human capital dynamics, we consider an economy which produces one good, with one unit of human capital producing one unit of good. Consequently, the real wage per unit of human capital is 1. Productivity being given and constant, the increase in the amount of human capital is the only source of growth. We assume the following education function:
hit ait eit hit 1 ,
0 1, 0 1
(4)
In function (4), depicts both the public expenditure on education and the efficiency in the production of human capital (i.e., the total factor productivity of the education production function), ait is the randomly distributed personal ability which is independent from family backgrounds, and eit is the individual’s expenditure on education (as the price and wage are equal to 1, eit is the expressed in real terms). Variable eit includes all the individual’s direct and indirect expenditures on education, including the opportunity cost related to schooling
12 time.10 We implicitly suppose here that the interest rate is nil and that individuals can always borrow the money necessary for funding education, i.e., a perfectly competitive credit market. This assumption will be waived in the following sections. Individual (i,t) selects the expenditure eit which maximises her lifetime net income Iit hit eit , subject to the education function constraint (4), which yields the optimal value11: eˆit ( ait )1/(1 ) hit 1 /(1 )
(5)
The related human capital dynamics can be defined by inserting (5) into (4): 1/(1 )
hit ait
hit 1 /(1 )
(6)
By assuming that 1 and hence / (1 ) 0 , we obtain: hit / hit 1 0 ,
2hit / hit 12 0 , and hit / hit 1 0 . h it 1
Finally, the individuals’ lifetime income is: Iit 1 ( ait )1/(1 ) hit 1 /(1 ) 1 hit
(7)
3.2. Convergence We now consider two situations, namely, the case in which individuals do not differ and the case in which they do differ in their personal abilities. In the first case, we assume that ait 1, i, t . In the second, personal abilities ait belong to an interval a, a
*
and are
randomly distributed among individuals at each generation (we make no particular assumption on the density function over this interval). Such a distinction will be made in all the following sections. When ability is identical across dynasties and generations ( ait 1, i, t ), the intergenerational human capital dynamics (6) leads to the same human capital steady state hˆ for all dynasties:
hˆ ( )1/(1 )
(8)
The related phase diagram is similar to that pictured in Figure 1.
10
In this very simple presentation, all the education costs are assumed to be perfectly substitutable. The division of eit between different types of expenditures with different impacts on human capital accumulation can of course be assumed in more complex models. 11
1/(1 )
Iit ait eit hit 1 eit ; Iit / eit ait eit 1hit 1 1 0 eit ait hit 1
.
13 If individuals differ in their abilities ait a, a steady segment hˆ, hˆ , with:
* ,
then all dynasties tend towards the
hˆ, hˆ ( a )1/(1 ) , ( a )1/(1 )
(9)
Once this segment is attained, a dynasty indefinitely remains inside jumping from one position to another depending on the successive distribution of abilities. This dynamics is drawn in Figure 3. Hence, the dynamics in perfect competition generate human capital convergence and, for plausible values of parameters , and of the public expenditures on education , this convergence is typically reached after a limited number of generations. It can nevertheless be noted that it is quite easy to generate a no-education trap within a model in perfect competition. To do this, it is sufficient to assume that, even without education, the individual possesses one unit of simple labour, works as an unskilled worker and is paid at wage wL . If she is employed as skilled worker, the individual is paid wH hit , where wH is the wage per unit of human capital. She chooses education only if
wH (hit eˆit ) wL hit 1 h ( )
1/
wL / wH (1 )/ . Then all the children whose
parents’ human capital is below h do not go to education and their dynasties indefinitely remain non-educated. The ratio wL / wH changes from one generation to the next depending on the amount of skilled and unskilled labour and on the production function (technology). As educated dynasties tend towards the steady human capital hˆ ( )1/(1 ) , the model generates an infinite number of possible steady states (depending on the initial distribution of human capital) where the population is divided between unskilled workers with human capital h = 0 and skilled workers with human capital hˆ , the number of workers in each group being such that hˆ h wH (hˆ eˆ) wL , which is the condition for choosing education.
4. Imperfect credit market and the conditions for convergence The optimistic diagnosis of a rapid human capital convergence seemed at variance with observed facts. In most advanced countries, compulsory basic education was decided in the early twentieth century and convergence was far from being achieved some seventy or eighty years later, i.e. after three generations.
14 Becker & Tomes (1986) and Loury (1981) proposed models in which the convergence is lengthened by credit market imperfections. Such imperfections constrain children from poor and modest families in the financing of their studies. This typically slows down the convergence process, but it does not prevent convergence in the long term because, as each generation possesses more human capital than the preceding, the financing constraint diminishes from one generation to the next and tends thereby to vanish. Subsequent models showed that imperfections in the credit market may lead to low education traps and human capital polarization (Galor & Zeira, 1993; Barham et al., 1995). There are in fact three ways to introduce credit market imperfection in the education decision of individuals. The first consists in assuming that the cost of borrowing is higher for poor families than for rich ones, the difference representing the risk premium a poor family must pay to accede to credit. The second is to assume that parents lend money to their children, this loan being subsequently reimbursed once the child is a working adult. The last approach is based on altruism, the parents providing their children for bequests they use to fund education. In the last two cases, children are constrained by their parents’ funding. As poor or modest families can only provide a limited amount of funds, the optimal education expenditure given by equation (5) cannot be financed. We now model these three channels within our unified framework. 4.1. Dearer credit for poorer families This credit market imperfection can be easily modelled by assuming that the child borrows from the banks the money necessary for the education expenditure eit , which result in the total education cost (1 rit )eit , with rit being the interest rate she pays.12 We now assume that this interest rate depends on her parents’ earnings, i.e., on her parents’ human capital given that earnings are proportional to the education level. This assumption is simply modelled by the following relationship: rit max r , r (hit 1 )
(10)
with r / hit 1 0 and r (h) r . Relation (10) signifies that the ‘normal’ interest rate paid by a non-risky family is r , and that a family begins to become risky when its human capital is below h . The lower the human capital below h , the higher the risk premium rit r . 12
Galor & Zeira (1993) propose a slightly different approach, which distinguishes the borrowing interest rate from the lending interest rate, the former being higher (see sub-section 3.2.c).
15 The individual maximises her adult lifetime earnings Iit hit (1 rit )eit subject to both constraints hit ait eit hit 1 and rit max r , r (hit 1 ) . a) Equal abilities Let us first assume no difference in ability across individuals: ait 1, i, t . This yields the optimal expenditure eˆit (1 rit )1( )1/(1 ) hit 1 /(1 ) , i.e.:
hit 1 /(1 ) , A 1 r eˆit /(1 ) A hit 1 , 1 r (hit 1 )
hit 1 h (11)
hit 1 h
The related intergenerational dynamics of human capital is: hit 1 /(1 ) A (1 r ) , hit /(1 ) A hit 1 , (1 r (hit 1 ))
hit 1 h
(10) hit 1 h
with A ( )1/(1 ) .
Figure 4. Human capital dynamics with family-related risk premia.
In Figure 4, the human capital dynamics is depicted by the bold curve: h /(1 ) h /(1 ) for hit 1 h and H NR (hit 1 ) A it 1 for hit 1 h , with H R (hit 1 ) A it 1 (1 r ) (1 r (hit 1 )) A ( )1/(1 ) .13 This dynamics tends towards the human capital steady state
13
Subscript R for risky and NR for non risky.
16 (1 )/(1 )
A hˆ . In addition, as H R (hit 1 ) is located below H NR (hit 1 ) for (1 r ) hit 1 h , then the number of generations necessary to reach this steady state is higher for all the dynasties situated below the human capital h at generation 0. Finally, the lower the initial human capital of a dynasty at the initial time (0), the higher the number of generations to attain the steady state compared to a situation without credit marker imperfection. b) Differences in ability When
individuals differ in their innate personal ability, there is one curve h /(1 ) ht 1 /(1 ) H R (ht 1 ) A(a) t 1 and one curve , A(a) ( a)1/(1 ) , H ( h ) A ( a ) NR t 1 (1 r (ht 1 )) (1 r )
for each a a, a . These curves are located between hit H R (hit 1 ) and hit H R (hit 1 ) for the former, and between hit H NR (hit 1 ) and hit H NR (hit 1 ) for the latter.14 These dynamics are depicted in Figure 5.
Figure 5. Family related risk premia and randomly distributed abilities
Figure 5 shows that: 1. All the dynasties tend towards the human capital steady segment hˆ, hˆ A(1 r ) (1 )/(1 ) , A(1 r ) (1 )/(1 ) and remain subsequently inside
14
H R ( hit 1 ) A
hit 1 /(1 ) (1 r ( hit 1 ))
; H R ( hit 1 ) A
hit 1 /(1 ) (1 r ( hit 1 ))
with A ( a )1/(1 ) and A ( a )1/(1 ) .
; H NR ( hit 1 ) A
hit 1 /(1 ) (1 r )
; H NR ( hit 1 ) A
hit 1 /(1 ) (1 r )
,
17 2. As long as a dynasty has not attained the human capital without risk premium h , this dynasty is slowed down in its human capital accumulation by the interest rate over-cost. Hence, the convergence is slower and takes a larger number of generations. 4.2. Parental loans to children Barham et al. (1995) assume that children have no access to the credit market, but they can borrow from their parents who save for their retirement age. Hence, children from modest families are constrained in their funding by the level of their parents’ saving. The authors show that this constraint can generate permanent no-education traps. Following Barham et al. (1995), we extend our basic model by assuming (i) that children have no access to the credit market, (ii) that they finance their education by borrowing from their parents who save for their retirement age. This can be modelled by assuming that an individual selects her education expenditure by maximising the income she receives during her working time and subsequently distributes this income between consumption during her active and her retirement periods of life. The share of income which is saved by parents can be borrowed by children who are constrained when the parents’ saving does not cover their desired education expenditure. a) Equal abilities ( ait 1, i, t ). The individual firstly chooses her human capital level by maximising her lifetime net income subject to the education function (4). By assuming to simplify a null interest rate, her optimal education funding is given by equation (5): eˆit ( )1/(1 ) hit 1 /(1 ) . This education expenditure is conditioned by her parent’s funding capacity, i.e., by the parent’s saving sit 1 which must cover the education funding of her child. Hence the individual’s suffers the additional constraint: eit sit 1 . Individual (i,t) subsequently allocates her income between consumption during her working lifetime citw and consumption when retired citr by maximising the utility function15 uit (1 ) ln citw ln citr subject to the budget constraint Iit citw citr , with saving sit being
equal to consumption when retired: sit citr . This determines the optimal saving: sit Iit (hit eit )
From the above reasoning, it is clear that two cases are possible: 15
A log-linear utility function is selected for the sake of simplicity.
(13)
18 1) If eˆit sit 1 , then the individual can borrow her desired education spending eˆit from her 1/(1 )
parent and the human capital dynamics is given by equation (6), hit
hit 1 /(1 ) ,
1/(1 ) with the steady state (8): hˆ .
2) If eˆit sit 1 , the individual is constrained: she must restrict her education spending to the amount available from her parent’s saving, i.e. eit sit 1 (hit 1 eit 1 ) . This produces the expenditure dynamics: eit (hit 1 eit 1 )
(14)
And since hit eit hit 1 :
hit hit 1 eit 1 hit 1
(15)
The human capital steady state of dynamics (14)-(15) is:16
hˆ / (1 )
1/(1 )
(16)
Figure 6 portrays the phase diagram of dynamics (14)-(15).17
Figure 6. Constrained dynamics with imperfect credit market and parental loans
1/ (1 )
0 and eit 1 0 hˆ / (1 )
16
hit 1
17
(12) eit 1
eit 1 eit hit (1 )eit ,
eit 1 eit
(13) hit 1 hit 1 hit hit eit hit hit ,
eit hit
1
1
1/
, eˆ / (1 ) 1
hit
1/ 0 hit 1
1/ (1 )
0 , and eit 1 0 eit
hit 1 eit ;
irrelevant. hit 1 0 and eit 1 0 hˆ / (1 )
2 eit hit 2
0 . As
1/ (1 )
.
.
1
0 . hit 1 0 eit hit
/(1 )
(1 )/
hit .
1
1/
hit (1 )/ ,
1
1.
eit hit , the shaded surface is
19
Two cases can be distinguished, depending on whether individuals are still constrained or not at the steady state (16): 1) If the unconstrained steady state hˆ is lower than the constrained steady state hˆ , i.e.
hˆ hˆ / (1 ) , then all the initially constrained dynasties reach sooner or later the human capital from which they are no longer constrained. From then, all of them follow the 1/(1 ) unconstrained dynamics (6) and tend thereby towards the same steady state hˆ .
2) In the opposite case, i.e. if hˆ hˆ / (1 ) , which corresponds to hˆ ' in Figure 6, then all the constrained dynasties tend towards the human capital steady state hˆ hˆ . If all the dynasties are initially constrained, then all of them tend towards the steady human capital hˆ . In contrast, if at generation 0 some dynasties are below hˆ and others are above, then the dynasties are partitioned between these two groups, the first tending towards the steady low human capital hˆ , and the second towards the steady high human capital hˆ . This results in a permanent polarization of dynasties. All in all, three outcomes are possible: 1) A convergence of all dynasties towards the high human capital steady value hˆ if
/ (1 ) , i.e., if the saving rate is high compared to the efficiency of education expenditures, . 2) A convergence of all dynasties towards the low human capital steady value hˆ if they are all initially constrained ( eˆi1 si 0 , i ) and if / (1 ) , i.e., the saving rate is low compared to the efficiency of education expenditures. 3) A two-peak polarization between low and a high human capital dynasties, the former tending towards the low human capital steady value hˆ and the latter towards the high human capital steady value hˆ , if only a share of the dynasties are initially constrained and if
, i.e., the saving rate is low compared to the efficiency of education expenditures. 1
b) Differences in ability When differences in ability are assumed, the same three cases are found except that the steady states hˆ and hˆ are replaced by steady segments corresponding to the different a a, a . It must be noted that, when the high-skill steady segment and the low skill steady segment
20 overlap, there is a bi-directional mobility between the high skilled and the low skilled families. 4.3. Altruism The seminal approach in which the fixed cost of education is funded by bequests from altruistic parents is that of Galor & Zeira (1993). In their model, the lending interest rate is lower than the borrowing interest rate. This makes the fixed cost of education to be dearer for the children who receive bequests lower than the fixed cost of education. Then, some of them have no incentive to educate themselves because the related over-income is lower than the over-cost linked to the borrowing interest rate, which generates a no-education trap. Maoz & Moav (1999) provide another example of bequest-funded human capital accumulation. Following Galor & Zeira (1993), it is assumed here that children finance their education through bequests from their altruistic parents. In contrast with them, we however assume no access to the credit market and a variable cost of education (the case of a fixed cost of education is examined in the next section). This can be modelled by assuming that the utility function contains a ‘bequest’ component that depicts the altruism of parents towards their children. As in the case of loans, individuals firstly choose their education attainment by maximising their lifetime net earnings Iit hit bit 1 eit subject to the education function and the funding constraint eit bit 1 , and they subsequently distribute earning between consumption and bequest by maximising the utility function uit (1 ) ln cit ln bit , where cit is individual (i,t)’s consumption, bit her bequest to her child, and depicts parental
altruism assumed identical across individuals. bˆit Iit (hit bit 1 eit ) is the optimal bequest , which is also the highest fundable education expense of individual (i,t)’s child. As usual, we distinguish the cases of identical and different abilities across individuals. a) The case of identical abilities Without funding constraint, the optimal expenditure on education is given by (5): 1
eˆit ( )1 hit 11 . Hence, when the constraint is inoperative ( eˆit bit 1 ), the human 1 1 1 hit 1
capital dynamics is the same as when the credit market is perfect: hit 1/(1 ) the steady state of this dynamics is hˆ .
and
21 When the constraint is effective ( eˆit bit 1 ), then eit bit 1 and the human capital
dynamics is hit bit 1 hit 1 . If individual (i,t) is constrained, her parent (i,t-1) was constrained as well and her bequest was bit 1 hit 1 because bit 2 eit 1 . The constrained human capital dynamics can then be written: hit hit 1
(17)
The constraint operates provided that eˆit bit 1 , which yields the following condition: 1/(1 )
hit 1 h* 1
(18) 1
Dynamics (17) tends towards the human capital steady state hˆ 1 . The condition for a dynasty to reach the threshold h * above which it is no longer constrained before attaining the steady state hˆ is: h* hˆ . Two cases can then be distinguished: 1) When h* hˆ , all dynasties tend towards the same steady human capital hˆ but this convergence is slowed down for those dynasties whose initial human capital is below h * . This corresponds to the saving rate being larger than the efficiency of education expenditures, . 2) When h* hˆ , then the dynasties who are initially below the human capital threshold h * tends towards the steady low human capital hˆ , and all those initially above h * tends towards the steady high human capital hˆ , with hˆ hˆ because . Hence, when the saving rate is lower than the efficiency of education expenditures, , the human capital dynamics generates a polarization between a high skill and a low skill group. So, when assuming no differences in ability across individuals, two outcomes are possible, namely, a convergence of all dynasties towards the same human capital or a two-peak polarization between a high skill and a low-skill group with no mobility between them. When assuming differences in ability, it is possible to generate mobility in the last case. b) Differences in ability across individuals. In the case of differences in ability ( ait a, a ), the unconstrained dynamics is 1 1 1 ait hit 1
hit
1
1
and the related steady segment Sˆnc hˆ, hˆ a 1 , a 1 ,
and the constrained dynamics is hit
a
it hit 1
with the related steady segment
22 1/(1 ) 1/(1 ) ˆ . The following results can be easily shown Sˆc hˆ, h a , a when the dynasties have reached the steady segments (see Appendix A): a 1) The set of unconstrained dynasties tends to vanish if . a
1/(1 )
a 2) The set of constrained dynasties tends to vanish if a
.
3) There is a bi-directional mobility (some constrained dynasties become unconstrained 1/(1 )
a a and some unconstrained dynasties become constrained) when: . a a These findings are logical. When altruism is low compared to the efficiency of education expenditures ( / a / a ), the number of constrained dynasties increases with time. In contrast, a relatively high altruism ( / (a / a)1/(1 ) ) makes the number of unconstrained dynasties to augment and the constrained individuals tend to disappear. Finally, there are both constrained dynasties becoming unconstrained and unconstrained dynasties falling in the funding constraint, i.e. bi-directional mobility, in the situation in-between.
5. Fixed costs of Education and stratification The introduction of a fixed cost of education can generate low-education traps (LET), i.e., situations in which certain dynasties remain perpetually under-educated. The reason for this is simple. If the extra-income related to education is lower than the fixed cost of education, then individuals have no incentive to educate themselves. The introduction of a fixed cost of education is assumed by Galor & Zeira (1993), education being funded by bequests from altruistic parents. They determine three types of individuals with three related human capital dynamics, namely, the non-educated, the constrained (in their education funding) educated, and the unconstrained educated. The noneducated are those whose benefit from education is lower than the fixed cost of education. The human capital dynamics makes the intermediate group (constrained educated) to split into two, those who will meet the unconstrained and those who will meet the non-educated. 5.1. Fixed cost of education From our basic framework, we show that the existence of a fixed cost of education is sufficient to generate a permanent low education trap. We firstly suppose equal ability among individuals ( ait 1, i, t ).
23 Education requires both a fixed cost f which can be seen as an entry cost, and the usual variable cost eit which is chosen by the individual and enters the education function. We keep the education function (6), i.e. hit eit hit 1 , which now depicts the education attainment (which can be seen as ‘cognitive skill’), but we modify the link between education and earning so as to have non-zero income for individuals who do not educate themselves. Individual (i,t) with education attainment hit is assumed to be paid: wit hit
(19)
can be interpreted as depicting the basic common sense everyone receives even without entering the education system18. The total human capital is the sum of the ‘natural’ component
and the education-related component hit . Individual (i,t)’s maximisation programme is: max Iit eit hit 1 eit f
(20)
eit
As and f are given, the solution to this programme is the same as in the case without fixed cost (equation 5): eˆit ( )1/(1 ) hit 1 /(1 ) . But the individual must now verify that the over-wage generated by education is higher than the sum of both the variable and fixed cost.
The
lifetime
net
income
related
to
the
education
expenditure
eˆit
is:
I (êit ) 1 ( )1/(1 ) hit 1 /(1 ) f . Without education, individual (i,t) has the net lifetime income I (0) . She decides to educate herself iif I (êit ) I (0) , i.e.: f 1 / hit 1 h 1 1
1/
(21)
The fixed cost generates a permanent low education trap. At generation 1, all the individuals whose parent’s human capital is lower than h do not educate themselves. Hence, their human capital is nil and none of their offspring goes into education. All these dynasties perpetually remain into a low education trap (here, a no-education trap), and all the remaining dynasties tend toward the steady education-related human capital hˆ ( )1/(1 ) . The above findings show that the mere existence of a fixed cost of education can generate a low education trap. 18
Another possible way to model this case consists in assuming that, without education the individual receives an income wL corresponding to her basic unskilled labour.
24 In the case of different abilities ( ait a, a ), the dynasties which remain educated enter sooner or later the steady segment Sˆ hˆ, hˆ . In addition, condition (21) becomes ait1/ hit 1 h . Then, all the individuals such as ait1/ hit 1 h hit 1 ait 1/ h go to education
and all those such as ait1/ hit 1 h do not go to education as well as all their descendants. In this case, it can be shown that (see Appendix B): 1
1 f 1) When a 1
2) When and a
, the educated families tend to vanish in the long term. 1
1 1 1(1 )
f / (1 )
, then all the dynasties such
that ai11/ hi 0 h fall in the no-education trap at the first generation, and all the remaining dynasties are indefinitely educated and enter sooner or later the steady segment Sˆ . 3)
When , some dynasties can fall into the no-education trap during the
transition towards the steady segment Sˆ , but all the dynasties which reach Sˆ remain indefinitely educated inside this segment. 5.2. Fixed education costs and credit market imperfections The literature typically combines a fixed cost of education with imperfections in the credit market. Assuming that children have no access to bank credits and that their education must be funded by their parents (through borrowing or bequest), the probability to fall in low education traps is reinforced. As usual, a positive net gain is the condition for education (hit 1 eit 1 f ) , but constrained children ( eit eˆit ) are such that hit 1 eit 1 hˆit 1 eˆit .
Then a number of constrained children will choose no-education, which enlarges the number of dynasties falling in the LET.
6. ‘S-shaped’ Education functions In Galor & Zeira’s model (1993), the combination of the three lines depicting the three bequest dynamics generated by the three types of individuals draw an S, which generates two stable steady states and human capital polarization. An easy way to generate a two-peak polarization is to assume an S-shaped education function. The explanation for such a shape is rather logical. When the parents’ human capital
25 is low, a rise in their human capital has an increasingly positive impact on their children’ education. In contrast, when the parents’ human capital is high, rise in this human capital has an increasingly limited positive impact on their children’s education. In other words: the marginal impact of parents’ human capital on their children’s education is increasing when the parents’ human capital is low, and decreasing when it is high. This feature is supported by the evidence of a low mobility (high IGE) at both tails of the earnings distribution, particularly in the upper tail. In this vein, Galor & Tsiddon (1997) combine an S-shaped education function and a human capital global externality to show that this generates polarization, this polarization being nevertheless transitory (discussion below). 6.1. General case To model the case of an S-shaped education function, Equation (6) is modified in the following manner:
hit eit hit 1
(22)
2 2 0 0, h h , , ; 0 h h 0; 0 . 2 2 h h h h h These features define an S-shaped curve (h) . with: 0 0 ; ' 0 1 ;
The maximisation of the lifetime net income subject to education function (22) determines 1/(1 )
the optimal education expenditure eˆit (hit 1 ) 1/(1 )
hit
and the related dynamics:
S (hit 1 ) 1/(1 )
with S (hit 1 ) (hit 1 )
(23)
.
As (h) is S-shaped, S (h) is S-shaped as well.
Dynamics 3
Dynamics 2 Dynamics 1
Figure 7. The S-shaped education function
26 Three cases are possible as depicted in Figure 7:
1) When there is one intersection of curve hit
1/(1 )
S (hit 1 ) with the 45° line and
2S 0 , then all the h2 dynasties tend towards the low human capital steady state hˆ 1 (Dynamics 1 in Fig. 7) 1/(1 )
when the value of hit 1 at this intersection is such that 1
2) When curve hit 1 S (hit 1 ) intersects twice the 45° line, this generates a human capital polarization with all the dynasties initially below h tending towards the low education trap hˆ 2 , and all those initially above h tending towards the high human capital steady state hˆ (Dynamics 2 in Fig. 7). 3
1
3) Finally, when curve hit 1 S (hit 1 ) cuts once the 45° line at a human capital 2S 0 , then all the dynasties tend towards the high human h 2 capital steady state hˆ 4 (Dynamics 3 in Fig. 7).
value hit 1 such that
1
1
6.2. S-shaped education function with a global human capital externality When a human capital externality is inserted into the model, this displaces the S-shaped curve. This can be made by adding an externality component to function (15):
hit ht 1 eit hit 1
(24)
where ht 1 , 0 1, depicts the positive externality of the average level of human capital in the economy at generation t-1, ht 1 .
The optimal expenditure corresponding to function (24) is eˆit ht 1 (hit 1 )
1/(1 )
and the related dynamics:
1
hit ht 1 1 1 S (hit 1 )
(25)
where ht 1 depicts the average human capital level at generation t-1. Dynamics (25) generates a shift in the S-shaped curve, but the direction of the shift (upward or downward) depends on the location of individuals on the x-axis. Starting from dynamics 2 in Figure 7, if a majority of individuals are located between human capital levels hˆ 2 and h , then the average human capital decreases. This decrease could make the concave
part of the curve to move below the 45° line, resulting in a convergence of all dynasties towards a low education steady state ( hˆ1 ). In contrast, if a large share of individuals are
27 located between the human capital levels h and hˆ 3 , then the S-curve moves upwards, and this can lead to the convergence towards one high education steady state ( hˆ 4 ). Another way to obtain the same results is to insert a global human capital externality in the production function of the economy and to assume that education is financed by an income tax. Then, the increase in human capital makes the production and income to increase, which in turn increases the resources for education, moving thereby the education function upwards. Finally note that, in Galor & Tsiddon (1997), the downward shift of the curve is made impossible by the assumption that the human capital externality acts through the technological level of the economy which depends on the average human capital level, with a downward bounded technological dynamics (the technological level cannot decrease). The technological level augments the efficiency of human capital in the production function. Then, humancapital-driven technological progress increases the parents’ income and their expenditure on their children education, which shifts the education curve upwards from generation to generation. The authors finally show that a redistribution policy in favour of the poor is prejudicial at the beginning of the development process because it slows down the increase in human capital in the families where this increase occurs (between h and hˆ 3 in Figure 7), which jeopardises technological progress. In contrast, such a policy becomes efficient from a certain level of economic development because it can then foster expenditures on education for families which are bounded by a financing constraint.
7. Neighbourhood and peer effects In the preceding section, we have considered a global externality in the education function. We now analyse the impact of local externalities and neighbourhood effects linked to the agglomeration of educated and rich families in the same districts, generating thereby poor districts and ghettos. The propensity of educated and rich families to congregate is a well-documented fact. This ‘assortative mating’ is generated by several motives: 1) Cultural and wealth proximity increases well-being. In contrast, poor families in rich districts can be segregated. 2) Living in a rich district increases the amount of levies for a given tax rate, and can hence results in both lower tax rates and higher levies (and more public equipment and services).
28 3) If there are human capital externalities at school and in the quarter, parents who care for their children’s education prefer to live in a district with educated families. The congregation of educated children in the same school improves everyone’s attainment through peer effects. These motives make that educated and rich parents are prepared to pay more to live in rich districts. The resulting increase in housing prices pushes modest families outside the rich quarters. In the early nineties, a series of articles were published which emphasized the key role of local human capital externalities in the emergence of social stratification, segregation and low intergenerational mobility.19 Benabou (1994) proposes a model that synthetises the main outcomes of this literature. The analysis of the ‘local connection’ has since then motivated a large range of research papers, both theoretical and empirical. The empirical works tend to confirm the impact of neighbourhood and peer effects on school attainment, even if this impact is not always high. We do not review this literature here and refer to the surveys of Nechyba (2006), Epple and Romano (2011) and Sacerdote (2011). 7.1. Local human capital transmission channels We do not model the agglomeration process itself, its accomplishment being assumed at the moment when human capital dynamics is analysed. Hence, at generation 0 from which we study the intergenerational dynamics, we suppose that the division of the population between poor and rich districts is achieved. As regards human capital, assortative mating generates two essential mechanisms. First, being in a district where people are highly educated means more cultural activities and a culture-oriented social environment. This generates positive human capital externalities and transfers. Second, more educated families are also richer families and, for a given tax rate, the amount of levies in the rich districts is higher than in the poor districts. As public education is to a large extent financed by local authorities, the rich districts spend more on education than the poor ones. This entails higher human capital accumulation in the former. These two channels can be easily inserted in the education function. The population being already divided between a poor (P) and a rich (R) district, we denote hR the average human capital in the rich district and hP hR this value in the poor district.
19
E.g., De Bartolone (1990), Borjas (1992), Benabou (1992; 1996), Durlauf (1994; 1996).
29 7.2. Lasting differences versus permanent stratification We shall consider two cases, namely, (i) the usual case in which the concave education function is an extension of (4) and (ii) the case of an S-shape education function with local externalities. In each case, we shall focus on the situation where individuals do not differ in ability ( ait 1, i, t ). a) Concave education function We modify the education function (4) by adding the two pre-cited transmission channels. We
firstly suppose a local externality htd1 1 , 0 1 1 , with htd1 being the average human capital in district d = P,R at the parents’ generation. We secondly assume that the TFP of the education function, t at generation t, integrates the local public per-student expenditure on education which depends on the levies in the district: 2
t htd1 , d = P,R,
0 2 1
(26)
where depicts the determinants of the TFP which are independent from local funding, the tax rate, with levies being devoted to education, and htd1 the average pre-tax income per capita in district d. As each parent has one child, htd1 is the per-student local public expenditure on education. The so-transformed education function of individual (i,t) whose parent lives in district d is:
hit 2 hd ,t 1 eit hit 1 ,
d = R,P, 1 2
(27)
Individual (i,t) maximises her post-tax net lifetime income Iit (1 )hit eit . The related 1/(1 )
optimal education expense is eˆit (1 ) 2 (hd ,t 1 )
hit 1 /(1 ) , which generates
two intergenerational dynamics depending on the district where the individual lives:
hit (1 ) 2
1/(1 )
/(1 )
htd1
hit 1 /(1 ) ,
i d , d = P,R
(28)
The unique steady state of both dynamics is:
hˆ hˆ (1 ) 2
1 1
(29)
where hˆ is the average human capital steady state. However, two cases can be distinguished depending on the stability of this steady state.
30 To simply analyse dynamics (28), let us suppose that inside each district all individuals have the same human capital at generation zero. Consequently, it is also the case in all the subsequent generations and we can write: hit h td , i d , t. The dynamics of the average human capital (as well as that of the human capital of all dynasties) is because of (28):
htd (1 ) 2
1/(1 )
( )/(1 )
h td1
(30)
Two cases may be distinguished depending on the value of : 1) If 1 ( ) / (1 ) 1 , then Function (30) is concave and the steady state
hˆ hˆ is stable. All the dynasties tend towards this value. 2) If 1 ( ) / (1 ) 1 , then function (29) is convex and the steady state hˆ hˆ is unstable. If the human capital in the rich district is initially above hˆ and that in the poor district below, then the human capital tends towards 0 in the poor district and it indefinitely increases in the rich district. In summary, we obtain the following results in the case of a concave education function: 1)
When 1 , both districts converge towards the same human capital hˆ .
However, this convergence is slowed down in the poor district, and faster in the rich district, compared to the case in which the externality would be global (in this case, the average human capital inserted in the education function is the same in both districts). 2) When 1 , the two districts diverge. The poor district tends toward a null human capital20 whereas the rich dynasties’ human capital indefinitely increases. This can however be seen as an unrealistic outcome because the division of the population between the poor who tend towards a null human capital and the rich who tend towards an infinite human capital is rather unlikely. a) S-shaped education function When the education function is S-shaped (Section 5.2), assuming local externalities rather than a global one erases the final convergence linked to the upward displacement of the Sshaped curve. This convergence is now replaced by a permanent polarization between a low
20
For the poor’s income not to be nil, it is necessary to assume that individuals with zero human capital are paid for non-cognitive skill as in section 4.2.
31 skill group and a high skill group. This can be easily shown from Figure 8 in which the unified intergenerational dynamics (25)21 is replaced by the dynamics in each district:
hit H d (hit 1 ) (1 ) 2
1/(1 )
(a)
/(1 )
htd1
S (hit 1 ) , d P, R
(b)
Figure 8. Neighbourhood effect with an S-shaped education function
In Figure 8-a, the division between the poor and the rich district initially operates at human capital h which is also the unstable steady state of the S-shaped dynamics. From then on, the dynasties’ human capital variation breaks up into two distinct curves. The rich district’s curve moves North-East whereas the curve moves South-West for the poor district. This is because the average human capital htd1 increases in the rich district and decreases in the poor district. Then, all the rich district’s dynasties tend towards a high steady human capital, and all the poor ones towards a low steady human capital. In Figure 8-b, the same twofold dynamics emerges but a share of the dynasties who were initially going up the skill ladder (those between the human capital values hˆ and h ) are in the poor districts (below h ) and fall thereby in the downward dynamics because of the decrease in the average human capital ht P1 . Note that, if h is sufficiently high and/or the number of parents below hˆ sufficiently low, then the poor district has a growing average human capital, and the two districts converge towards the same steady human capital. From the above discussion, it is clear that local externalities and neighbourhood effects can generate human capital polarization. This happens (i) with a concave education function when
21
With an S-shaped education function, the intergenerational dynamics (25) is: hit ht 1 /(1 ) ( )1/(1 ) S (hit 1 ) .
32 the curve depicting the intergenerational dynamics of the average human capital is convex (
1), and (ii) with an S-shaped education function when the neighbourhood effect makes the average human capital to increase in rich districts and to decrease in poor ones.
8. Education Systems The polarization mechanisms put forward in the preceding sections can be erased by appropriate public interventions. Credit constraints can be treated by public insurance on loans or direct public loans to modest families. The impact of fixed costs can be counteracted by grants to children from unprivileged families. The impact of S-shaped education functions can be counteracted by providing more basic education to children from low skilled families. Finally, the impact of neighbourhood effects can be offset by mixing in the same schools children from different social origins. Most of the pre-cited public interventions and policies have been implemented in many countries. This has however not prevented the permanence of intergenerational persistence in both education and earnings. A possible explanation is that education systems themselves foster social immobility. The impact of the structure of educations systems upon social stratification and the reproduction of social positions has been studied for a long time by sociologists. The pioneering works are those of Weber and Durkheim. In contrast, the economic approach to the structure of education systems is rather recent. A number of works have investigated the influences of the division of education between several levels and cycles, and of the funding allowed for each of them, on inequality, social stratification, intergenerational mobility, efficiency, growth and welfare (Driskill & Horowitz, 2002; Su, 2004, 2006; Blankenau, 2005; Blankenau, et al., 2007; Gilpin & Kaganovich, 2012). Within an approach with basic and secondary education, the latter being divided between vocational and general studies, Bertocchi & Spagat (2004) make appear different social stratification at the different stages of economic development. Su (2006) shows that the elite imposes larger expenditure on tertiary education at the expense of basic education in developing countries, whereas budget allocation to education is more balanced in advanced countries. Chusseau & Hellier (2011) build an intergenerational model with three education cycles (basic education, vocational studies and university, with a selective admission to the latter), which generates different social stratifications with low education traps depending (i)
33 on the public funding allocated to each cycle, (ii) on the initial distribution of human capital across dynasties, and (iii) on the tightness of admission. Several approaches emphasize the impact of early childhood education. Herrington (2015) calibrates an overlapping generation model which shows that differences in public spending and in public contribution to early childhood education can explain the divergence in inequality and intergenerational mobility between the US and Norway. The crucial role of early childhood education is confirmed by Restuccia & Urrutia (2004) and Blankenau & Youderian (2015). As regards higher education, its twofold objective of training and screening was emphasized by Arrow (1973), Spence (1973) and Stiglitz (1975). A number of works have analysed the policies to oppose the crowding-out effect of higher education costs on modest families (e.g., Caucutt & Kumar, 2003, Akyol & Athreya, 2005, Gilboa & Justman, 2009). Another strand of literature has focused on the tightness of admission procedures. Within a model where personal ability combines family backgrounds and a random component, Gilboa & Justman (2005) distinguish admission requirements from graduation requirements. They show that a release in admission rules without change in graduation requirements promotes earnings equality but reduces intergenerational mobility. By disentangling family backgrounds from i.i.d. innate personal abilities, Brezis & Hellier (2013) show in contrast that, within a two-tier higher education with standard and elite universities, the tight admission to elite establishments results in permanent social stratification, low intergenerational mobility and large self-reproduction of the elite. The division between standard and elite colleges in tertiary education has also been studied by Su et al. (2012) to explain the U-shape relationship between wages and skills observed in the US in the last two decades. 8.1. The structural characteristics of education systems Education systems can be defined by a large number of structural characteristics. We shall focus here on three key determinants: 1) the division in stages and cycles, 2) the admission rules, particularly to higher education, and 3) the funding and levels of education expenditures and their distribution between the different cycles. a) Stages and cycles An education system is typically composed of several stages and cycles. This division between cycles can be purely vertical (then, stage 2 comes after stage 1, stage 3 after stage 2,
34 and so on) or it can combine successions of cycles and bifurcations. It the latter case, the students can choose between several studies at certain moments of their curriculum. A particular case highlighted in the literature is the division between standard and elite establishments (schools and colleges). When such a division exists, it can lead to the appearing and self-reproduction of a narrow social elite. b) Selection and admission procedures A key feature of education systems is the shape and tightness of their admission procedures to enter certain cycles. This is particularly the case in higher education for which a minimum level is required in all countries (SAT scores in the US, A-level and school record in the UK, Baccalaureate in France, Abitur in Germany etc.). Then, the number of students admitted in tertiary education depends on the matching between the strictness of admission and the level of the students at the moment when they apply. The admission rule can exclude a large proportion of young from tertiary education when the required threshold is high compared to the human capital attainment at the end of basic education. c) Education expenditures The amount, origin and distribution of education funding are essential determinants of educational attainment. In most countries, basic education is freely provided by the state. In contrast, tuition fees do exist in higher education and their levels considerably differ across countries. The education attainment of the population depends to a large extent on the following three questions: 1) How high is public expenditure on education, and how is it distributed between basic and higher education? 2) What is the weight of private funding and tuition fees in the total education expenditure, and does it impact on inequality, intergenerational mobility and economic efficiency? 3) Does the budget divergence (in terms of per-student expenditure) between tertiary education establishments, particularly between standard and elite colleges and universities, influence intergenerational mobility? These questions can lead to the definition of an optimal expenditure on education, which obviously depends on the social planner’s objectives. 8.2. A basic two-stage education system By extending our basic framework, we model a simple two-level education system which shows that both the cost of higher education and the tightness of admission to higher
35 education can generate a permanent polarization between a low educated group and a high educated group, and we further analyse the mobility between these groups. Education is comprised of two successive levels, compulsory basic education, which is freely provided to all children by the government, and higher education. At the end of basic education, a young can either join directly the labour market or enter the university. There are two conditions for a young to enter in the university: 1) She must possess at least the human capital level h at the end of basic education. 2) She must allow a total cost eU which comprises both the tuition fees and the opportunity cost of tertiary education. Let hitB be individual (i,t)’s human capital at the end of basic education, and hitU her human capital at the end of tertiary education. The related education functions are:
hitB B ait hit 1
(31)
hitU 1 gU eU
h 1 g B it
U
eU B ait hit 1
(32)
with 0 1 and 0 1 . Function (31) is a usual education function with however no private funding since basic education is freely and equally provided to all children by the State. Hence, B B ( g B ) denotes the efficiency of public expenditure on basic education, g B . Function (32) stipulates that the additional human capital acquired in the university
gU eU hitB depends on the level at the end of basic education hitB and on the total
expenditure on tertiary education gU eU , where gU is the government per-student
expenditure on tertiary education. For an individual to select the university, her related lifetime income, I hitU hitU eU ,
must be higher than her lifetime income with basic education only, I hitB hitB . The condition for this is: hitB
eU
gU eU
. We assume that the admission rule h is effective,
i.e., it prevents the access of certain individuals who wish to enter the university. Hence, eU and all the admitted students enter the university. h gU eU In the case of equal ability of all individuals ( ait 1, i, t ), the steady states corresponding to dynamics (31) and (32) are:
hˆB B1/(1 )
(33)
36
hˆU 1 gU eU
1/(1 )
hˆB
(34)
With different abilities ( ait a, a ), the steady segments of each dynamics are:22
S B hˆB , hˆB (a B )1/(1 ) ,(a B )1/(1 ) SU hˆU , hˆU 1 gU eU
1 1
(35)
hˆB , 1 gU eU
1 1
hˆB
(36)
We shall focus on the most interesting case of different abilities, the situation with equal abilities being presented in Appendix C.
mobility
Figure 9. Human capital dynamics in a two-stage education system: different abilities
Figure 9 portrays the sets of dynamics linked to each study (basic education and university) and related to each ability a a, a .23 Each type of study generates one steady segment, the B-steady segment S B hB , hB for basic education and the U-steady segment SU hU , hU for the university. We know that all dynasties whose successive generations follow only basic education enter sooner or later the B-steady segment and remain subsequently inside (see sub-section 2.2.b). Identically, the dynasties whose successive generations enter the university reach the Usteady segment and remain inside. Now, as regards mobility, the key questions are (i) whether
22
Note that, when taking into account the admission threshold h , the lower limit of the with-admission steady
segment of the university dynamics is hˆU max hitU 1 gU eU h , 1 gU eU 23
1/(1 )
hˆB
Section 2.2.b), Figure 3, exposes the construction of these dynamics. Here, H B (hit 1 ) B ahit 1 ,
H B (hit 1 ) B ahit 1 , HU (hit 1 ) 1 gU eU
1/(1 )
H B (hit 1 ) and HU (hit 1 ) 1 gU eU
1/(1 )
H B (hit 1 ) .
37 dynasties who are low educated (they do not enter the university) can escape from this position over their intergenerational human capital dynamics (upward mobility), and (ii) if educated dynasties can in return fall in a no-education position (downward mobility). When both moves co-exist, the society exhibits a bi-directional mobility. The admission to the university takes place at the end of basic education. To answer the preceding question, let us consider that individuals are distributed between the two steady segments S B and SU . Then, at the end of basic education, the children are distributed between two human capital segments, S B hB , hB for those whose parents are in the Bsegment, and SUB H B ( hU ), H B (hU ) for those whose parents are in the U-segment. The mobility between the two educational groups can be analysed from the position of children at the end of basic education as depicted in Figure 10. (a)
x
x
x
x
(b)
x
x
x
x
Figure 10. Distribution of children at the end of basic education
Figure 10-a depicts the case in which the efficiency of higher education is large enough to prevent the overlapping of segments S B and SUB . Three configurations are possible: 1) All the dynasties sooner or later enter the university and thereby the high educated group when threshold h is lower than the upper limit of the B-steady segment, hB . This is obvious when h is below the lower limit of this segment, hB , because then all the dynasties go beyond h ( h1 in Fig.10-a). It is also the case when h is inside the B-segment ( h2 in Figure 10-a). This is because, as personal abilities are randomly distributed over the interval
a, a , then the probability for any dynasty of never being inside the human capital interval h2 , hB tends towards 0 when the number of generation tends towards infinite. Hence, when h hB , all the dynasties enter the U-steady segment SU and remain subsequently inside.
38 2) When hB h H B ( hU ) , i.e., threshold h is between the steady segments S B and SUB ( h3 in Fig.10-a), then all the dynasties which are such that ai1 hi 0 h / B never enter the university24 and tend thereby towards the B-steady segment, and all those such that
ai1 hi 0 h / B perpetually enter the university and tend towards the U-steady segment. 3) When h is inside segment SUB ( h4 in Fig.10-a), then all the dynasties are sooner or later excluded from the university (the random distribution of personal abilities makes the probability for any educated dynasty never to go below h to tend towards 0) and subsequently remain under-educated: the U-steady segment vanishes with time. In its three configurations, the case in which segments S B and SUB do not overlap results in a total lack of bi-directional mobility between the two skill groups (corresponding to segments S B and SU ). In the long term, configuration 1 results in everyone being highly educated,
configuration 3 in everyone being under-educated, and configuration 2 generates a full insulation of each group from the other. More interesting is the case depicted in Figure 10-b because it generates bi-directional mobility. In this case, the segments S B and SUB overlap. Figure 9 shows that, for this to happen, it is not necessary that the steady segments S B and SU also overlap. As in the preceding case, three configurations are possible: 1) When the admission threshold h is below H B ( hU ) all the dynasties sooner or later enter the university and subsequently remain the high skill group ( h1 and h2 in Figure 10-b). 2) When the admission threshold h is above hB ( h4 in Figure 10-b), all the dynasties sooner or later fall and subsequently remain in the low education trap S B hB , hB . 3) When the admission threshold h is inside the overlapping segment H B ( hU ), hB , there is at each generation a bi-directional mobility between the low skill group and the high skill group ( h3 in Figure 10-b). Then, the children born in the low skill group with a human capital inside the segment h3 , hB at the end of basic education enter the university and join the high skill group. In the opposite direction, the children born in the high skill group with a human capital lower than h3 at the end of basic education are not admitted in the university
24
The first generation is not admitted to the university because h1Bt B ai1hi 0 h and the subsequent
generations never attain h because h is above the B-steady segment.
39 and fall in the low skill group. In other words: the most able children from the low skill group join the high skill group and the least able from the high skill group fall in the low skill group. This bi-directional mobility is permanent once the individuals are distributed between the two steady segments (it can be easily shown that an individual who leave one segment goes directly to the other). From the Figure 10-b and the determinant of the limits hB , hB , H B ( hU ) and H B ( hU ) , it is easy to diagnose the impacts of the admission rules, of the public and private expenditures on education, and of the distribution of public expenditures between basic and higher education upon intergenerational mobility and social stratification: 1) The stricter the admission threshold h inside segment H B ( hU ), hB , the lower mobility. 2) A rise in the public funding on basic education g B increases B , which displaces both segments S B and SUB to the right (relation 35). This augments the upward mobility of children born in low educated families and reduces the downward mobility of those born in high educated families. This is in line with the theoretical and empirical literature which highlights the key role of basic education to promote upward mobility. Straightforwardly, a decrease in the public funding on basic education which displaces both segments to the left has the opposite impact. 3) An increase in both the public funding on higher education gU and the tuition fees eU moves the segment SUB H B ( hU ), H B (hU ) to the right.25 This lessens the downward mobility of children born in high skilled families. 4) A transfer of public expenditure from basic education to higher education firstly lessens the upward mobility of children from the less skilled families. As regards children from the high skilled families, the increase in downward mobility due to the decrease in expenditure on basic education is offset by the increase in expenditure on higher education. All in all, social persistence is reinforced. 5) A last and key outcome in terms of expenditure must be highlighted. On top of lowering the downward mobility of the upper skill group (those inside the segment S B ), an increase in the university tuition fee eU reduces the number of individuals who wish to enter the university after basic education (Condition (37)). Provided that the admission threshold is effective, the latter feature has no impact if all universities are identical. In contrast, if all 25
because HU (hit 1 ) 1 gU eU
1/(1 )
H B (hit 1 ) and HU (hit 1 ) 1 gU eU
1/(1 )
H B (hit 1 ) .
40 universities receive the same amount of per-student public funds gU but differ in their tuition fees, then the universities with high tuition fees will attract the best students at the end of basic education. This is because only the best students wish to enter the expensive university, which provides them with a higher education level than the cheap one.26 But this subsequently hampers mobility to the best (expensive) universities because their alumni have a higher human capital, which favours their children in the access to the best universities. It can be noted that the above features 1) and 5) are in line with a large range of theoretical and empirical works showing that the effect of family backgrounds is magnified when the education system is highly selective and stratified (e.g., Hanushek & Woessmann, 2006; Marks et al., 2006; Dronkers et al., 2011).
9. Discussion: mobility and further researches Based on the approaches developed in the economic literature, a unified and synthetic framework has been built which permits to reproduce the different channels through which the factors, conditions and decisions as regards individuals’ education influence intergenerational mobility and social stratification. The initial diagnosis of a fast convergence of dynasties towards the same human capital steady state (or steady segment) has subsequently been questioned by the introduction of several factors: imperfections in the credit market, fixed costs of education, S-shaped education function, local externalities and neighbourhood effects, and the structure of education systems. In certain cases, these factors only slow the convergence down. In a large number of situations, they divide the population between several education groups. When individuals do not differ in ability, these groups are perpetually insulated, i.e., there is no inter-group mobility from one generation to the next. When differences in ability are introduced, this generates situations in which polarization comes with between-group mobility. We shall firstly question the adequacy between the theoretical models utilised in the economic literature (and in our stylised framework) and the measurement of mobility utilised in the empirical literature. We subsequently highlight several promising fields of research.
26
Let us denote B basic education, U C the cheap university and U E the expensive one, with the respective fees
eU and eU v . ‘
’ signifies ‘is preferred to’. U S
It can easily be shown that: U E
U S hitB h(v) v
B hitB h eU
/ (1 ) ( gU eU ) (Condition 37).
(1 ) ( gU eU v) ( gU eU ) .
41 9.1. Mobility: match between the theoretical and empirical approaches The indicator of intergenerational mobility which is the most utilised in the empirical economic literature is the intergenerational elasticity (IGE). As regards human capital, the IGE, , can be estimated from equation: ln hit ln hit 1 j j x j ,it it , where h is the human capital indicator,
x j
a set of control variables, and it respectively the
constant and error terms. Let us compare this measurement with the shapes of the theoretical models developed in this paper (as well as in the theoretical literature). If we start from the most basic relation (4) ( hit ait eit hit 1 ) and if the (log of the) expenditure on education eit is part of the control variables of the estimated equation, then the IGE is and, given our hypotheses, this IGE is fully exogenous and identical at any level of human capital (abilities appear in the constant and the error terms). Hence, the theoretical approach provides no explanation for the divergence in IGE revealed by the empirical literature, either between countries or between educational and earnings levels. The same diagnosis applies for the model’s extensions . For example, in the model with neighbourhood effects, the estimation of the initial equation (27) determines the IGE which is the exogenous value depicting the direct impact of family backgrounds in the education function. Likewise, in the two-stage education model, a global estimation of educational attainment with a dummy for higher education provides the IGE .27 The estimation of Relation (6) ( hit ( ait )1/(1 ) hit 1 /(1 ) ) in which the expenditure on education has been replaced by its optimal value leads to a greater value of the IGE which is now (1 )1 . In addition, when considering the cases in which two dynamics coexist,28 an estimation made on the whole population typically provides an IGE which is a combination of the elasticities related to each group. When the least educated group has a greater IGE than the most educated (this is the case when parents fund their children’s education), a large proportion of low educated results in a large IGE. But then, a larger IGE means that the number of children constrained by their parents’ funding is larger.
27 28
and the coefficient ln 1 gU eU for the ‘tertiary education’ dummy.
This can be either in the transitional path to the steady state only, or in the final situation when the dynamics generates several permanent education levels.
42 The above discussion shows that the IGE value typically depends on the control variables inserted in the estimated equation, which is trivial and abundantly discussed in the methodological literature. More crucial for our subject: it also shows that the IGE is badly cut for depicting the type of mobility revealed in the theoretical literature. Quite surprisingly, mobility tables and the related indicators, which are more usual in the sociological literature, are better tailored to assess the theoretical outcomes of the economic approaches to intergenerational mobility. This is because these approaches typically generate a social stratification between several educational groups, with between-group mobility in the case of different abilities across individuals. Provided that mobility tables and the related measures are based on the division of the population in different groups and classes, these tools turn out to be more suitable for measuring the impacts revealed by economic theory. When several skill groups are generated, the first key measures are the incidence of each group in the whole population and their respective human capital level. When the theory reveals between-group mobility, then the mobility tables are particularly appropriate for measuring the relative weight of these flows. They permit to calculate the relative weight of flows between two groups, the incidence of upward and downward mobility for each group, and the total inter-group intergenerational mobility. 9.2. Further researches This presentation has focused on the different factors which slow down or prevent the convergence of all dynasties towards the same level (or interval) of human capital, on the polarization in different skill groups and on the mobility between these groups. If the impacts and mechanisms of funding constraints, publicly or privately funded education, externalities and neighbourhood effects have been extensively analysed, the influence of education systems, of their evolution with time, of their internal structure and their differences across countries is still rather recent. Much remains to be done to have an broad analysis of the shape and influence of education systems. As already mentioned, researches have been implemented which deal with the impact of primary and early childhood education, the admission policies (Costrell, 1993, 1994; Betts, 1998; Gilboa & Justman, 2005), competition between private and public schools (Epple a& Romano, 1998, 2008; Epple et al., 2002, 2004, 2006), and differences between standard and elite education (Su et al., 2013, Castello-Climent & Mukhopadhyay, 2013; Brezis & Hellier, 2013). In this respect, three promising fields of research deserve to be highlighted.
43 In several countries, the education system is divided between two types of establishment, the first providing standard studies to a large number of students while the second offers ‘prestigious’ studies reserved to a limited number of students. The reasons for this two-fold system (type of deciders and objective function of each structure, importance of educating versus screening in their objectives) as well as the reasons why they differ across countries need to be further investigated. A second path of research is the influence of different shapes of admission. If the impact of admission tightness has been investigated, the analysis of the impacts of different admission procedures upon social stratification and mobility, as well as the reasons for selecting one procedure rather than another, could provide explanations for the observed heterogeneity across establishments and across countries. Finally, education is experiencing an increasing process of globalization. The globalization of education is fostered by the development of English as lingua franca at the World level, the decrease in transportation costs, the availability of information on international schooling opportunities via Internet, the development of international between-establishment agreements and finally the convergence of education systems. Certain countries now receive a large number of foreign students whereas a non-negligible part of their nationals remain confined in low education structures. At the same time, other countries promote an egalitarian education policy which eradicates under-education traps and concurrently send their best students to high level universities abroad to complete their training. The analysis of these change on social mobility, social stratification and on the formation of a globalised elite are key issues on the agenda of economic researches on education and intergenerational mobility.
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Appendix A. Funding constraint, altruism and unequal abilities In the case of differences in ability ( ait a, a ), the unconstrained dynamics is 1 1 1 ait hit 1
hit
1
1
and the related steady segment Snc hˆ, hˆ a 1 , a 1 ,
a
and the constrained dynamics is hit with the related steady segment it hit 1 1/(1 ) 1/(1 ) ˆ Sc hˆ, h a , a . The condition not to be constrained is 1/(1 )
. This defines the segment of the minimal eˆit bit 1 , and hence hit 1 ait parental human capital not to be constrained depending on the individual’s ability, 1/(1 ) S* h* , h * / Sc .
Consider individual (i,t) whose parent belongs to Sc . She becomes unconstrained if 1/(1 )
hit 1 1 ait
1/(1 ) ˆ . The highest possible value of hit 1 is h a and the
47 1/(1 )
1/(1 )
lowest possible value of 1 ait is h* 1 a . A necessary condition for the existence of constrained dynasties which become unconstrained is thus: a ˆ h h* . a Consider now individual (i,t) whose parent belongs to Snc . She becomes constrained if 1/(1 )
hit 1 1 ait
1
. The lowest possible value of hit 1 is hˆ a 1 and the 1/(1 )
1/(1 )
highest possible value of 1 ait is h* 1 a . A necessary condition for the existence of unconstrained dynasties which become constrained is thus: 1/(1 )
a hˆ h* a
1/(1 )
1/(1 )
a a a and As a a a possible to distinguish three cases:
a a
because a a , it is
1) No constrained dynasties can become unconstrained and some unconstrained dynasties a become constrained if . Then, the set of unconstrained dynasties tend to vanish. a 2) No unconstrained dynasties become constrained and some constrained dynasties become 1/(1 )
a . Then, the set of constrained dynasties tend to vanish. unconstrained if a 3) There is a bi-directional mobility (some constrained dynasties become unconstrained 1/(1 )
a a and some unconstrained dynasties become constrained) when a a
.
Appendix B. Fixed cost of education with different abilities In the case of different abilities ( ait a, a ), the dynasties enter sooner or later the steady segment Sˆ hˆ, hˆ provided that they remain educated. In addition, condition (21) becomes ait1/ hit 1 h . Then, all the individuals such as ait1/ hit 1 h hit 1 ait 1/ h go to education
and all those such that hit 1 ait 1/ h do not. However, the fact individual (i,t) goes to education does not mean that 1) As a is the lowest value of a, then a 1/ h is the highest value of ait 1/ h , and all the dynasties such that, at any generation t, hit 1 a 1/ h will subsequently remain educated if
48 the following generations have a human capital higher or equal to hit 1 . The condition for this 1
f is hit 1 hˆ . hˆ a 1/ h a 1
1
1 f ( ) a 1
1
. Hence, for a
1
1 f being given, there is a minimal education efficiency a 1
below which
education tends to vanish. The higher the fixed cost of education, the higher the minimal efficiency . 1
f 2) Suppose that 1
( a )1 . First, all the dynasties such as ai11/ hi 0 h
indefinitely fall into the no-education trap. Second, the fact that individual (i,t) goes to education does not mean that her child, and hence all her descendants, will ipso facto go to education. This is because the increase in human capital between generations t-1 and t can be offset by the decrease in ability between generations t and t+1. We thus analyse the condition ? for ait1/ hit 1 h ait 11/ hit h
Because of the human capital dynamics: hit ait
Since ait1/ hit 1 h : ait1/ hit 1
hit h
1/(1 )
1/(1 )
h
ait 11/ hit ait 11/ h
1 1
hit 11
1/(1 )
(1 )
a
(1 )
1 1 1(1 )
f / (1 )
1/(1 )
h , which yields:
1/
:
.
, and:
1(1 ) 1
f This is always true when a 1 1
1/(1 )
f 1 / By replacing h by its value h 1 1
f ait 1 1
1
1
1/(1 )
hit h
Hence, a sufficient condition for ait 11/ hit h is ait 11/ h h(1 )/(1 ) ait 11/
ait1/ hit 1
1(1 ) 1
, which is equivalent to:
.
49 As the above condition is only sufficient, the child of an educated individual (and the whole succession of her descendants) can be educated even if . In summary: 1
1 f 1) When a 1
, the educated families tend to vanish in the long term.
2) When and a
1
1 1 1(1 )
f / (1 )
, then all the dynasties such
that ai11/ hi 0 h fall in the no-education trap at the first generation, and all the remaining dynasties are indefinitely educated and enter sooner or later the steady segment Sˆ hˆ, hˆ . 3) When and , then some dynasties can fall into the no-education trap during the transition towards the steady segment Sˆ , but all the dynasties which reach Sˆ remain indefinitely educated inside this segment.
Appendix C. Education systems with equal abilities across individuals As usual, individual (i,t) maximises her lifetime net earnings. At the end of basic education, she faces a twofold choice: 1) To enter the labour market, which provides her for the lifetime earnings (1 )hitB .
2) To enter the university, which yields the net earnings (1 ) 1 gU eU
h
B it
eU .
She enters the university if the related earning is the highest, which determines the following condition:
hitB h
eU
(1 ) gU eU
The value h is the human capital at the end of basic education above which the gain from
the university (1 ) gU eU hitB is higher than its cost eU . It must be noted that an increase in the tuition fees eU lessens the number of young who wish to enter the university
h / eU 0 whereas an increase in the public expenditure on tertiary education to expend this number h / gU 0 .
gU tends
An applicant for the university is admitted if her human capital at the end of basic education is at least equal to h . This admission rule is effective if there are individuals who wish to enter the university with a human capital lower than h after basic education. The condition for this is: h h .
50 Assume that the admission rule is effective, i.e., condition h h holds. Consequently, all the children with a human capital higher than h at the end of basic education enter the university, and all those with a human capital below h enter the labour market. Then (31) and (32) are the human capital dynamics related respectively to basic education and to university studies. Figure A1 depicts these skill dynamics. Figure A1 reveals two cases: 1) When the minimum threshold to enter the university is lower than the steady state of the basic education dynamics, i.e. h hˆB , then all the dynasties sooner or later enter the university and they all converge towards the high human capital steady state hˆU (Fig. 11a). 2) When the minimum threshold to enter the university is higher than the steady state of the basic education dynamics, i.e. h hˆB , then a two-peak polarization emerges (Fig.11b). All the dynasties whose initial human capital is below h fall in a LET and tend towards the low human capital steady state hˆB . All the dynasties with an initial human capital above h tend towards the high human capital steady state hˆU .
(a)
(b)
Figure A1. Human capital dynamics in the two-stage education system: equal abilities
When it leads to polarization, the case of equal abilities exhibits a rather questionable stratification process: the division between high and low educated dynasties occurs at the initial time (generation 1) and no dynasty can subsequently escape from this position. Assuming differences in abilities permits to reveal more realistic developments and to generate mobility between the high educated and the low educated group.
