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Do Risk Aversion and Wages Explain
Educational Choices?�
Thomas O. Brodatyy, Robert J. Gary-Boboz and Ana Prietox
6 May 2014
Abstract
We study a model in which a student�s investment in education maximizes expectedutility conditional on public and private information. The model takes future wagerisk into account and treats the direct and opportunity costs of education as additionalsources of risk. In particular, the time needed to complete a degree is random. The datashow that time-to-degree is substantially dispersed, conditional on the highest degree.We �nd signi�cant and substantial e¤ects of expected returns on individual educationchoices. The risk a¤ecting education costs and, in particular, the randomness of time-to-degree, also play an important role in explaining enrollment in higher education.We �nd precise estimates of the CRRA risk-aversion parameter, between 0.6 and 0.9.Heterogeneity in risk attitudes is signi�cant, even in sub-populations. The sons ofprofessionals bear more risk and are more risk averse than the rest of the sample. Yet,they study more because of higher returns and markedly lower expected investmentcosts. Simulations show a strong impact of changes in the probability of grade retentionon educational achievement.Keywords: Returns to Education, Wage Risk, Risk Aversion, Grade Retention,
Education Costs, Family Background, Time to Degree.
�We thank Christian Belzil, Stephane Bonhomme, Denis Fougere, Marc Gurgand, Thierry Kamionka,Francis Kramarz, Guy Laroque, Thierry Magnac and Jean-Marc Robin for their help and remarks at variousstages of this research. We also thank Orazio Attanasio, Jose-Victor Rios-Rull and the participants of the2005 NBER Summer Workshop for their help and useful remarks. The present paper is a deeply revisedversion of a manuscript circulated in January 2005 and entitled: �Risk Aversion, Expected Earnings andOpportunity Costs: A Structural Econometric Model of Human Capital Investment�.
yTHEMA, University of Cergy-Pontoise, France. Email: [email protected] 15, boulevard Gabriel Péri, 92245 Malako¤ cedex, France. Email: robert.gary-
1
1 Introduction
Uncertainty a¤ects not only the returns to, but also the direct and opportunity costs of edu-
cation. The methods of diversi�cation, risk-sharing and insurance are not easily applicable
to human-capital investments. Thus, one would expect risk aversion to have substantial ef-
fects on the decision to invest. In the vast literature following the pioneering work of Becker
(1964) and Mincer (1974), most of the contributions focused on returns to education1, while
a comparatively much smaller number of studies have been devoted to the risks of human-
capital investment. More speci�cally, the impact of risk aversion on education choices has
been studied in a handful of papers2. There exists, of course, an important literature on
the statistical analysis of wage dynamics, but most of its contributions are purely empirical
or based on reduced-form models3. The theoretical literature on risk aversion and human
capital is also sparse4.
We propose an econometric study of the risks of education and, at the same time, of
the role of risk aversion in the demand for education. Our analysis is based on the standard
assumption that education choices maximize the expected utility of individuals, conditional
on public and private information. The choice model hinges on returns to education (i.e.,
skill premia), and the direct and opportunity costs (i.e., foregone earnings) of schooling.
Both future wages and the costs of education are a¤ected by risks. In this model, we also
allow for unobserved heterogeneity in risk attitudes, family background and ability, and
by assumption, the students�utility functions belong to the constant relative risk aversion
(CRRA) family.
Students�beliefs, taking the form of probability distributions of education costs and
1See the surveys of Card (1999), Heckman et al. (2003), Harmon et al. (2003).2Friedman (1953) presented explanations for income inequalities in terms of risk aversion. Weiss (1972)
studied the risk-premium element in compensation, modeling wages as log-normal variables and assumedCRRA utility functions. Palacios-Huerta (2003) applied �nancial econometric techniques to education. Seealso King (1974), Shaw (1996), Carneiro, Hansen and Heckman (2003). In the recent macroeconomicsliterature, see Krebs (2003), Huggett, Ventura and Yaron (2011).
3See the survey of Meghir and Pistaferri (2010). For instance, Chen (2008) models the variance of wagesunder self-selection and unobserved heterogeneity. Low, Meghir and Pistaferri (2010) and Magnac, Pistolesiand Roux (2013) use a dynamic structural model to study wage risk, but they calibrate or ignore risk-aversion(and time-preference) and they focus on careers, not on initial education.
4See the classic contributions of Lehvari and Weiss (1974), Williams (1979), Dreze (1979) and Eaton andRosen (1980).
2
future wages, are modeled with the help of two auxiliary equations. The �rst one is a
Mincerian, log-wage equation, used by the student to predict future wages as a function of
educational achievement. The second one is a delay equation explaining the number of years
needed by the student to earn his(her) highest degree (i.e., the time-to-degree). This last
point deserves further explanations.
Our approach doesn�t rely on years-of-schooling as the only measure of human capital,
as in a number of classic studies. An original feature of the model presented below is that
it distinguishes degrees (or education levels) from the time needed to reach a certain level
(or time-to-degree). This distinction is made possible by our data, which give detailed
information on both the highest degree and school-leaving age for each individual.
Our results have been obtained with a rich sample, containing 12,500 young men who
left the educational system in 1992, in France. In this dataset, the randomness of degree-
completion dates is generated by the addition of several sources of delay. In France, as well
as in many other countries, there are grade repetitions in primary and secondary schools.
Delays, mainly due to �unking, can be accumulated during vocational or college studies as
well5. Thus, the time needed to pass exams is substantially dispersed: students di¤er in
their learning �speed� and the costs of education are random, conditional on the highest
degree. A contribution of the present article is to show that this type of risk is a key factor
in understanding enrollment in higher education.
One of the model�s outputs is the risk-return �curve�that can be drawn by joining the
points associated with each degree (or education level). The curves are typically increasing,
higher returns being associated with higher wage-risk. These mean-variance plots vary with
family background. The subsample including the sons of professionals enjoys higher returns,
but bears more risk in higher-education levels than the rest of the sample. This is true
not only on average, but also if we separate heterogeneity from uncertainty, that is, in our
context, if risk-return curves are conditional on unobserved types. In addition, we �nd
5The empirical relevance of these delays depends on the institutions of the country under study. Forinstance, grade repetitions are common in France, Spain and Germany, while social promotion prevails inGreat Britain and Scandinavian countries. Time-to-degree completion is substantially dispersed in highereducation as well. See e.g., Ehrenberg and Mavros (1995), Brunello and Winter-Ebner (2003) and Garibaldiet al . (2012). In France, the bulk of these di¤erences in speed is generated when students fail their exams(and fail the exam retakes) and must repeat a semester or a full year to earn a given degree.
3
that individual choices are very sensitive to a number of model parameters, including risk
aversion, and to expected returns to additional years of education. In contrast, choices seem
to be less elastic with respect to the risk in wages itself (i.e., the variance of log-wages).
In our model, individuals are characterized by a speed parameter, which is the prob-
ability of being promoted to the next grade, at the end of each year. This parameter varies
with family background and unobserved heterogeneity in an important way. We simulate the
impact of changes in speed on educational achievement and enrollment in higher education.
If students from less privileged backgrounds were given the average speed of the sons of ed-
ucated professionals, a high fraction of highschool dropouts would earn a vocational degree,
a substantially higher proportion of high-school graduates would successfully go to college,
etc. This shows the importance of the random costs of education for investment decisions.
Finally, we �nd values of the Arrow-Pratt coe¢ cient of relative risk aversion (RRA)
between 0:6 and 0:9. Students seem to be more risk-averse than a decision-maker using a
square-root utility, but less risk-averse than a decision maker using logarithmic utility6. Since
we use a model with a discrete number of unobservable student types, we are able to show
that there is a signi�cant amount of RRA heterogeneity. Student risk-aversion also depends
on family background, and it is perhaps surprising to �nd that the sons of professionals are
in fact more risk averse than male students from less privileged backgrounds. Yet, the sons
of professionals study more and bear more risks than others because their costs of education
are signi�cantly smaller, while their returns to education are signi�cantly higher than for
the rest of the male students.
Our approach is close in spirit, if not in its technical details, to the recent work
of Carneiro, Hansen and Heckman7 (2003). These authors separate the contribution of
heterogeneity from that of genuine uncertainty in the distribution of wages, and thus identify
the wage risks as perceived by students, before their decision to attend college. We use
parallel analytical tools in the present paper. Carneiro, Hansen and Heckman (2003) found
that �uncertainty in gains to schooling is substantial but knowledge of this uncertainty has
a very small e¤ect on the choice of schooling because the variance of gains is much smaller
6These �gures are close to the estimates found by other education economists; see, e.g., Keane and Wolpin(2001), Belzil and Hansen (2004), Sauer (2004).
7See also Cunha, Heckman and Navarro (2005), and Cunha and Heckman (2008).
4
than the variance of psychic costs or bene�ts, and it is the latter that drives most of the
heterogeneity in schooling decisions�. We also �nd that wage-uncertainty has a small e¤ect,
and that education costs (determined in part by family background) matter considerably.
But in contrast, we �nd strong e¤ects of expected returns on enrollment and speci�cally, a
strong impact of the risks a¤ecting education costs8.
In the following, Section 2 presents the model. Section 3 provides a description of the
data used for estimation. Section 4 presents the estimation results. Simulations are presented
in Section 5, and Section 6 contains concluding remarks. A number of computations, proofs
and additional details and results are omitted and relegated to the appendix.
2 The Model
2.1 Risk-averse students
We start with the basic assumption that an individual�s educational choices can be described
as the result of expected utility maximization. We assume that the agents�Von Neumann-
Morgenstern utility functions u exhibit constant relative risk aversion (CRRA) with respect
to earnings, that is, formally,
u(w) =w1� � 11�
; (1)
where w denotes the agent�s earnings. The Arrow-Pratt index of relative risk aversion for
this utility function is � 0.
There is a sample of students indexed by i, with i = 1; :::; N . To model unobserved
heterogeneity, we will rely on a discrete set of multidimensional, individual types. There are
K types indexed by j = 1; :::; K. A type value is denoted �j; it is a 6-dimensional vector,
�j = (�j ; �jc; �
jw; �
j�; �
jA; �
jd); (2)
where �j a¤ects the risk aversion coe¢ cient ; �jc a¤ects the individual�s direct costs of
education; �jw a¤ects the individual�s wage; �j� has an impact on the ex ante variance of an
individual�s wage; �jA has an impact on the age at which an individual enters junior high
8This is also in contrast to the recent literature on college-major choice, see Arcidiacono (2004), Be¤y etal. (2012).
5
school (or grade 6); and �jd has an e¤ect on the individual�s time to degree (i.e., delay).
We use pj = Pr(�j) to denote the probability of type j. Each coordinate in vector �j will
enter a di¤erent equation of the model. The students are assumed to know their type j, and
therefore know �j, but the econometrician doesn�t observe the types.
Education levels are discrete variables denoted s, i.e., s = 0; 1; :::; n. Let ws denote
the wage obtained at schooling level s. We represent education costs as a forgone fraction
of the individual�s potential earnings. More precisely, when a student has already reached
level s, the costs of investing in view of level s+ 1 are modeled as a fraction (1� hs+1)ws of
the earnings that the student could obtain if he or she went to work.
We distinguish the education level s (or degree) from the number of years of schooling
needed to reach a given level (i.e., time to degree). If xs is any variable depending on s,
denote �xs = xs� xs�1. Let ds denote the number of years used to reach level s. A student
will spend �ds years in school to increase his (her) education level from s�1 to s. Durations
ds can be viewed as random variables, because of grade repetitions and other delay factors,
but the educational investment level s is assumed to be a choice variable.
To provide a model for the choice of s, we assume that there is a factor, denoted
�, that the student is assumed to know, in addition to his (her) personal type �, but that
the econometrician does not observe. The � factor summarizes unobserved aspects of the
student�s environment and preferences that contribute to direct education costs. The in-
dividually optimal s is the result of expected utility maximization, conditional on private
information � and �. Given the assumptions listed above, the expected discounted sum of
utilities, knowing (�; �) and knowing observed covariates X, can be de�ned as follows :
V (s j � ; �) = E
24 TXt=1+ds
�tu(ws) +sXz=1
t=dzXt=1+dz�1
�tu(hzwz�1) j X ; � ; �
35 ; (3)
where � is a discount factor. In (3), wages ws and durations ds are random from the point of
view of the student. The cost factors hs also depend on (�; �) and are random from the point
of view of the econometrician. Given our dataset, in which only starting wages are recorded,
during the �rst years of career, we have a very weak basis to identify the discount factor.
We therefore focus on a model with a �nite horizon T < +1 and � = 1. Appendix 1 shows
how to derive the model�s equations if � < 1. Under the assumption of a �nite horizon and
6
patient students, expression (3) boils down to
V (s j � ; �) = E
"(T � ds)u(ws) +
sXz=1
(dz � dz�1)u(hzwz�1) j � ; �#; (4)
(we keep the conditioning with respect to X implicit to simplify notations). The relative
weight of the future career and of education years is determined by T and the hs. This point
is discussed below.
We model wages at the beginning of a worker�s career, and leave aside the problem of
modeling life-cycle earnings. The above model is in practice equivalent to a simple speci�ca-
tion in which students have the same deterministic career pro�les as a function of potential
experience, up to a translation given by the starting wage ws. In essence, we study the
behavior of students when their future social status is risky, and this riskiness is captured by
the randomness in the starting point of career pro�les. Our wage variable is an average of
the full-time wages earned during the �rst �ve years of career, weighted by their respective
employment-spell durations (see details below): this variable is a crucial predictor of future
career pro�les. It seems likely that the type of risk studied here is important for students.
However, as mentioned by a referee, the model cannot study di¤erences in, say, returns to
experience, that would be determined by student characteristics and interact with discount
factors in the choice model9.
We now specify equations determining (i), wages ws; (ii), the variance of wages
denoted �2s; (iii), the cost factors hs; (iv), the random durations ds; (v), the individual�s age
at grade 6 entry, denoted a (because it appears as an explanatory variable in the equation
for d, as will be seen below); and (vi), risk aversion itself (because it may vary across
individuals).
Firstly, yearly wages ws are determined as a function of education s as follows,
ln(ws) = fs +X0�0 + �a + �w + �s(��)�; (5)
where fs is a skill premium (depending on education s); X0 is a vector of control variables;
�0 is an associated vector of parameters; �w is the relevant component of type-value �; �s(:)
9This is clearly an interesting direction of research, but our dataset doesn�t allow us to exploit di¤erencesin the steepness of age-earnings pro�les (see Huggett et al. (2011)). As noted by a referee, this is a possiblesource of bias in our appreciation of the role of risk attitudes.
7
is the standard deviation of wages at education level s, which is assumed to be a function of
the type component ��; �a is the possible e¤ect of the individual�s age at grade 6 entry a; and
�nally, � is a standard normal random noise variable, representing risk from the student�s
point of view. We assume a simple form for the variance of wages, namely,
�s(��) = exp(�� +X��� + zs); (6)
where zs is a parameter that depends on education level s, X� is a vector of control variables
and �� is an associated vector of parameters. These �rst two equations combined, form a
standard Gaussian mixture model of log-wages: latent student types are characterized by
di¤erent means and variances of their future wages. In other words, the unobserved aspects
of an individual�s labor-market ability and risk are represented by (�w; ��).
The opportunity and direct costs of spending one more year in school, when the
education level is already s, are assumed to be a fraction (1 � hs+1(X1; �c))ws of the wage
ws. The function hs(:) depends on the type-component �c, on the vector of variables X1, for
all s = 1; :::; n. We have in mind that X1 includes some exogenous cost-shifting variables.
We specify hs as follows,
hs � exp(�X1�1 � cs + �c + �); (7)
where �1 is a vector of parameters10. We also assume that there is no constant in X1. Note
that this formulation is �exible: the hs functions depend on s (so that education costs are
not necessarily a �xed proportion of the potential wage); the hs functions also depend on
many covariates X1 and particularly, on family background (so that the son of a well-to-do
family may in fact have lower costs than the son of a blue-collar family). The assumption is
not unrealistic. We know, for instance, that in 1998, 80% of the French students have signed
a labor contract that was not a required internship11. But this �exible model is a possible
description of costs, even if many students do not work during their college years.
10The parameters cs can be interpreted as cost parameters for the following reason. It is easy to checkthat, � ln(hs+1ws) = ��cs+1 + �fs, for s � 1. Given that hs+1ws is the fraction of ws which is notforgone while studying at level s + 1, the �fs represent the bene�ts of moving from level s � 1 to level s(i.e., a skill-premium) and �cs+1 is the cost increment incurred while moving from s to s+ 1, expressed inpercentage of the earnings.11The majority of these contracts were of course part-time jobs or summer jobs, and the importance of
work increases with the education level, see Beduwe and Giret (2004).
8
We now turn to the model of durations ds, i.e., the delay equation. Let � s denote the
theoretical (and minimal number of years) needed to reach education level s. For instance,
if s is high-school graduation (i.e., the French baccalauréat), then � s = 18. We model the
ratio ds=� s as follows,ds� s= exp[X2�2 + �a�a�a + �d + �]; (8)
�� = exp(�a!a�a); (8b)
where X2 is a vector of covariates; �2 is a vector of coe¢ cients; �d is the relevant component
of type �; �a is the indicator function for the discrete age-at-grade 6 variable a; �a is the
associated coe¢ cient; and � is a normal random noise parameter. The error term � is not
observed by the agent: this is a risk for the student. The variance of this shock, denoted
�2� can potentially vary with covariates too. We chose to let it vary with age-at-grade-6, as
speci�ed by Equation (8b), where the !a are coe¢ cients to be estimated.
This simple formulation for the delay equation is suggested by the formula for the
mean of a Pascal distribution. Assume that each year, an individual i is promoted to the
next grade with a constant probability �i (so that (1� �i) is the probability of repeating a
grade). If we assume that �i is constant across years for each individual i, the probability
of reaching level s in k years, with k � � s, is given by the distribution of the number of
independent trials needed to obtain � s successes12. The expected duration of studies of level
s is simply � s=�i (this is just the formula for the mean of the Pascal distribution). This
suggests that, for each individual, �i is a measure of �speed�� in fact a measure of ability.
The recourse to Pascal probabilities is just a loose justi�cation for this convenient model,
where ds=� s = 1=�i ; Equation (8) obviously yields a linear regression form for the observed
variable ln(ds=� s).
With the help of this model, we can compute each individual�s prediction of the
average time needed to complete any degree of level s, di¤erent from his(her) observed highest
level si. This is done simply by computing the expectation E[ds j �;X] for each s. Remark12Pascal probabilities are given by,
Pr(ds = k) =
0@ k � 1
� s � 1
1A��si (1� �i)k��s :
9
that these predictions depend on the assumption that the speed factor 1=�i = exp[Xi2�2 +
�a�a�a + �d + �i] doesn�t depend on si, the observed highest level of i. The interpretation
of this prediction formula is that, to predict individual i�s counterfactual durations (i.e., the
average time needed by i to complete level si + 1, for instance), we extrapolate, using the
expected speed factor E[(1=�i) j �;X] = exp[Xi2�2 + �a�a�a + �d + �2�=2], which is itself
estimated thanks to the observed highest degrees.
Now, it may be that that the age at grade 6 entry a is endogenous in the above delay
equation. We therefore add a model for this variable. The discrete values of a are modeled
by means of an ordered Probit equation,
Pr(a) = Pr(ga+1 � XA�A + �A + � � ga); (9)
where XA is a vector of exogenous variables; �A is a vector of parameters; �A is the relevant
component of type; the gas are "cut" parameters; and � is a normal random noise parameter.
Note that the age at grade 6 entry is observed by the agent and by the econometrician.
We have introduced four random disturbance terms, assumed normal. These random
noise variables (�; �; �; �) are assumed independent, with zero means and respective variances
(�2� ; �2� ; �
2� ; �
2�). Given the above speci�cation, we have the standard identi�cation restrictions
�� = �� = 1.
Finally, we specify risk aversion as follows:
= X � + � ; (10)
where X is a vector of covariates and � is a vector of parameters.
The model doesn�t describe borrowing and savings. But the implicit assumption that
students are completely credit constrained should not be taken too literally. First, in the
context of France at the end of the 20th century, tuition and fees are close to zero for
most of the students. The latter are enrolled in public-sector institutions, essentially free of
charge. Student loans are rare and the market for these loans is underdeveloped. The direct
costs of education are �nanced through unobservable parental transfers and part-time jobs13.
Parental occupation being a proxy for parental income, using the former variable, we can
13The institutional context is very similar in Italy (e.g., Belzil and Leonardi (2013)).
10
control for some of the factors that determine the students�budget constraints. But given
our data, the e¤ect of credit constraints in the usual sense cannot be identi�ed14. A possible
interpretation of our speci�cation is therefore, as explained above, that we model individual
choices between risky career pro�les in which risk is essentially attached to the initial wage
level. The utility function captures attitudes towards this type of risk. This is not the risk of
unemployment. In contrast, it is clearly the ex ante risk a¤ecting the student�s future social
position.
2.2 Education choices
Now, the choice of an education level s is optimal for an individual with unobserved family
factors � and type � only if
�V (s+ 1 j � ; �) � 0 and �V (s j � ; �) � 0: (11)
Our model is therefore determined by equations (5)-(10) � giving wages, costs and delay as
a function of observable variables, unobservable type-values and random terms � and (11),
which characterizes the optimal schooling choice as a function of expected skill premia �fs,
observable characteristics X, type values � and unobservable family factors �. The students�
wage-expectations are rational in the sense that they are based on a wage equation estimated
with the help of the sample (so by assumption, the students know the true distribution of
wages).
We show in Appendix 1 that several variants of this model can be estimated. One
possibility is to let T go to in�nity and study the in�nite horizon version of the model with
a �xed value of � < 1. A second (unexplored) possibility would be to estimate � in the
in�nite horizon model with a �xed value of the risk-aversion parameter15 . In the following,
we �x a �nite value of T and set � = 1. As explained above, given our data, there is no
solid way of identifying the rate of time-preference. In the �nite-horizon, undiscounted case,
the maximum likelihood procedure can estimate (and thus identify) in a relatively natural
14For recent articles on the credit constraints faced by students and parental transfers, see Brown, Scholzand Seshadri (2012), and the survey by Lochner and Monge-Naranjo (2011).15The model is easily tractable if we set � = 1, T < +1 and utility is assumed logarithmic (i.e., = 1),
but of course we want to estimate .
11
way.
It will be convenient to use the de�nition � = � 1. It is shown in Appendix 1 that
the necessary condition for an optimal choice s, that is,
�V (s+ 1 j � ; �) � 0 � �V (s j � ; �)
is equivalent to
X1�1 + ks + cs � �c � � � X1�1 + ks+1 + cs+1 � �c: (12)
where by de�nition,
ks = �(1=�) ln (T � d̂s�1)� (T � d̂s) exp[��(�fs � (�=2)��2s)]
�d̂s
!; (13)
d̂s = E[ds j �]; �d̂s+1 = E(�ds+1j�) = d̂s+1 � d̂s; (14)
and �s = �s(�). The d̂s functions are the student�s expectations of the number of years
needed to complete level s.
Education level s is chosen by an individual only if her (his) unobserved factor � falls
in the above interval. Therefore, conditional on type �c, our theory has the structure of an
ordered discrete-choice model16, but with a particular functional form imposed on the cuts
ks + cs. The parameters of this ordered choice model are identi�ed if, as usual in ordered
Probit estimation, we impose the normalization �2� = 1.
We assume that the logarithm�s argument in the expression of ks, i.e., (13), is positive.
It has good chances of being positive if ���fs+1+(�2=2)��2s+1 is negative, or positive, but
small enough17. In practice, this has not posed many problems during the estimation phase.
Using expression (8), we easily �nd analytical expressions for d̂s and �d̂s+1. Given
the normality assumption and given that the age at grade 6 entry is observed by the student
and the econometrician, we have,
d̂s = E(dsj�) = � s exp[X2�2 + �a + �d] exp(�2�=2): (15)
16The model can therefore be viewed as a structural generalization of Cameron and Heckman�s (1998)ordered probit model.17If � > 0, this requires �fs+1 high enough and ��2s+1 small enough, i.e., in the risk-return, (�s; fs)-plane,
the slope of the risk-return curve should be steep enough. If, on the contrary, � < 0, we want the risk-returncurve to be �at enough and j�j should be small.
12
It follows that,
�d̂s+1 = E(�ds+1j�) = �� s+1 exp[X2�2 + �a + �d] exp(�2�=2); (16)
and we make use of the fact that, under normality, E(e�) = exp(�2�=2). From these formulae,
we can easily derive a closed-form expression for the crucial inequality �V (s+ 1 j � ; �) � 0.
Inequalities (12) being only necessary conditions for an optimal education level s,
we must also make sure that cs + ks < ks+1 + cs+1 for all s > 0, to guarantee that the
model�s probability distributions are well-de�ned and that s is indeed a maximum of V .
This cut-monotonicity property will be satis�ed if �fs+1 � �fs (i.e., �concave� returns),
�ds+1 > �ds (i.e., �convex�opportunity costs), and cs+1 � cs. But the latter conditions are
not necessary for cut-monotonicity: this property can still hold when returns to education
are increasing (i.e., if �fs+1 > �fs), provided that they do not increase too much.
At this stage it is useful to remark that the model is essentially a static representation
for a complex dynamic process that we do not observe. Everything is as if the students chose a
level of investment s at, say, the age of 13, clung to their project and bore the risks associated
with variations in the time-to-credential ds. Full sequential rationality would require the
student to revise his plan and beliefs at the end of every year, deciding to continue or to
quit next year, based on new information about success, failure and outside opportunities.
Our data do not allow us to study this learning process, which is learning about one�s own
ability, because we do not observe the history of successes and failures18.
In essence, the model is a standard decision problem under uncertainty, based on
plain expected utility theory, with intertemporally separable utility. We combine this with
independence of durations and wages conditional on the unobservable types �. The condi-
tional independence assumption is standard in the literature on unobserved heterogeneity.
Then, using the CRRA speci�cation of utility, we derive a mean-variance model, as shown
by Eq. (12) and (13) above. A consequence of these standard assumptions is also that
expected utility V is linear with respect to expected durations d̂s, and, as noted by a referee,
18It would be possible to model the individual as choosing the duration of schooling d and bearing therisk in the �nal outcome sd. In this alternative model, the student decides to study during d years, with thegoal of earning the highest possible credentials during the planned period. Given that the education level sis the measure of human capital here, it seems natural to choose s as the decision variable, and consider thedurations ds as random cost factors.
13
the variance of durations appears only indirectly, in the expression of conditional mean du-
rations (Eq. (15) and (16)). Thus, we can exploit the inter-individual variations of �2�, �2s,
d̂s and mean log-wages in estimation, but we are constrained by the speci�c form, derived
from expected utility maximization19.
2.3 Identi�cation
The identi�cation of unobserved heterogeneity parameters and the exclusions needed for the
identi�cation of risk aversion are the interesting questions here. First of all, apart from the
nonlinearities in the ordered choice equation determining education and the special structure
implied by risk aversion, the model is a fairly standard collection of linear regressions and
Ordered Probits. Hence, insofar as �nite Gaussian mixtures are identi�ed20, the log-wage,
the delay, and the age at grade 6 equations would be identi�ed, equation by equation. If
we now look at the model as a whole, our main identifying assumption is that (�; �; �; �) are
independent normal variables, conditional on �. In other words, the correlations between
error terms in the various equations are assumed to be captured entirely by the types j =
1; :::; K and type-values �j. If we put the covariates X aside for a moment, the model has
4 random sources (giving rise to the four equations): log-wage, delay, education and age at
grade 6. If we observe Q random sources, we can estimate (at least) Q means and Q(Q+1)=2
variances and covariances. When Q = 4, this amounts to 14 parameters. Suppose now that,
in the same context, unobserved heterogeneity is modeled with the help of K latent classes
(plus independent normal noise). We need to estimate (2 + Q)K parameters of the type
values: there are QK parameters shifting the constants in the Q equations and we add 2K
parameters for �� in the variance of wages (expression (6) above), and � in the risk aversion
model (expression (10) above). To this we add K � 1 parameters because we also estimate
the probabilities pj of the K latent types and �nally, one additional parameter because we
estimate the variance of delay �2�. In total, there are (Q + 3)K = 7K parameters to be
estimated, given that Q = 4. We �nd that 7K � 14 if and only if K � 2, so that, in
19The means and variances of ln(w) and ln(d=�) appear in the choice model through parameters fs, bd,�2s, �
2�. The variance of durations plays a role in expressions (15), (16) because durations are log-normal
random variables. As we will see below, �2� plays a genuine role.20See, e.g., McLachlan and Peel (2000), Jiang and Tanner (1999), Geweke and Keane (1997).
14
principle, we can estimate a model with two types without demanding more from the data
than a very basic approach � and of course, we can in principle demand more, because we
considered the �rst two moments of our random sources only. This provides some intuition
for the reason why the model is identi�ed21. In practice, we have been able to estimate
three types with the full sample. A fourth type can be identi�ed, but it has a very small
probability. The identi�cation is parametric, rests on the �nite mixture assumption and on
the independence of �, �, � and �. It may be that a deeper identi�cation result can be proved
for the unobserved heterogeneity in the entire system of equations, but in this type of model,
nonparametric identi�cation is an open research question, out of the scope of the present
paper22.
The other nontrivial point is the identi�cation of risk aversion parameters in the
education choice equation. Remark �rst that parameter can be interpreted in two ways.
We use a standard structural model of choice. In this model, the CRRA risk-aversion
parameter and the intertemporal elasticity of substitution 1= are two aspects of the same
thing. Standard economic theory uses this work-horse model in which a single parameter
measures two apparently di¤erent things. Given this, if we identify risk aversion, we also
identify the intertemporal substitution parameter, and conversely. Thanks to its role as a
measure of intertemporal substitution, can be identi�ed by variation in duration across
individuals (due to factors such as age at which school starts) even if there is no uncertainty
in wages or durations. Since the data also exhibit variations in the riskiness of wages across
individuals, we can exploit both the risk-aversion and intertemporal-substitution roles of
in estimation.
Although we o¤er no formal proof, it is easy to provide intuition for the reason why
the risk-aversion function is identi�ed. First of all, parameter � = � 1 is identi�ed
through variations of ks that are su¢ ciently independent of variations in X1. An inspection
of expression (13) shows that variations of ks accross individuals have two causes: inter-
individual variations in the riskiness of wages �s and variations in the predicted durationsbds. It may seem surprising at �rst glance, but the riskiness of wages is not essential, because21It is of course possible to write down the 14 equations with 14 unknowns.22See e.g., Bonhomme, Jochmans and Robin (2013).
15
the other source of risk, due to delays, is able to identify risk aversion alone. To be more
precise, variations in the individual�s prediction of his school-leaving age, namely, variations
in the d̂s, play the crucial role. Indeed, we estimated a variant of the model in which the
variance of wages didn�t depend on individual characteristics, and was still identi�ed. If
we go back to expressions (15) and (16) above we see that d̂s and �d̂s both vary with
observed covariates Xi2 (including age at grade 6) and therefore vary with the individual i.
Being expectations, d̂s and �d̂s do not depend on �i, so the identifying power comes from
variability in the observed covariates23 X2. But one variable at least must be excluded from
the list of covariates in the education cost function, i.e., from X1.
To clarify this point, we now show that a linear approximation of the education choice
model can in theory be identi�ed by means of just one reasonable exclusion (i.e., with the
help of one instrument). Let�s consider a simpli�ed, linear speci�cation of the model with
two equations (the other two equations could be added without di¢ culty and without any
essential change in the reasoning). Suppose that the delay equation was speci�ed as follows,
ln(di=� si) = Zib2 + Aia2 + �i;
where a2 and b2 are the coe¢ cients of age at grade 6 entry, denoted Ai, and of a vector of
instruments, denoted Zi, respectively. Estimation of the delay equation yields estimates ba2,bb2, and b�2�. Assume now that the choice of an education level directly depends on (Ai; Zi)and also indirectly, through the expectation E[(di=� si) j s], as follows,
Pr(si = s) = Prfcs � Zib1 + Aia1 + lnE[(di=� si) j s] + "i � cs+1g:
Substituting the expression for lnE[(di=� si) j s], we obtain,
Pr(si = s) = Prfcs � Zi(b1 + b2) + Ai(a1 + a2) + (�2�=2) + "i � cs+1g:
We get estimates of the latent-index coe¢ cients b�Z and b�A, on Ai and Zi respectively.
Parameter (and the whole model) is therefore identi�ed if we can solve the system
b�A = a1 + ba2;b�Z = b1 + bb2;23In addition the individual predictions given by (15) and (16) vary with s and with the individual�s type.
16
with respect to (a1; b1; ). It is immediate that one exclusion is needed: either a1 = 0 or
one coordinate of b1 should be set equal to zero24. In this model, as remarked by a referee,
any variable explaining education costs should also normally explain delay. It follows that
a natural exclusion is to set a1 = 0: the age at grade 6 should be excluded from the list
of variables determining education costs directly. This exclusion is natural, given that our
education costs are incurred in secondary and higher education. The age at grade 6 is a source
of variation of delay that is predetermined and should not directly a¤ect the subsequent costs
of education25. We will see that the age at grade 6 dummies do indeed very signi�cantly
a¤ect delay, with the expected sign.
In theory, the exclusion of A from the cost function should be su¢ cient for identi-
�cation, but in practice, it was not enough: the maximization algorithm converged and we
got parameter estimates, but the Hessian matrix could not be inverted (we couldn�t obtain
standard deviations for the estimates). This weak identi�ability problem explains why we
also excluded some cost-shifters from the delay equation. To be more precise, we excluded
the distance-to-college and school-density instruments from X2, as explained below. A pos-
sible interpretation for this more restrictive speci�cation could be that delay is a measure
of an individual�s ability, that is relatively independent of the individual�s environment. In
other words, the exclusion is justi�ed if delay is mainly determined by cognitive skills, and
not a¤ected by observed real-life impediments like transportation costs.
The simpli�ed variant studied above shows that the model would be identi�ed in a
semi-structural, linear framework and hence, that the identi�cation of a single parameter
would not fundamentally rely on functional form. But clearly, we still need to estimate
the e¤ect of observed family-background characteristics X and that of unobserved types �
on risk aversion. This is permitted by the nonlinear functional form of ks functions. But
the nonlinear structural form is not arbitrary; it is entirely dictated by economic theory26.
However, these re�nements are not essential. Our main results do not depend on the existence
24Of course, is only related to risk aversion in this simpli�ed model: it is not the risk-aversion indexitself.25Note that, in constrast, A is not excluded from the wage equation.26This has the obvious advantage that structural parameters, such as risk aversion, have a clear interpre-
tation.
17
of heterogeneity in risk-aversion, obtained in this manner. Subsample estimation shows that
risk aversion parameters do vary with family background, as will be seen below. In the
supplementary material appendix, we estimate a simpler variant of the model in which risk
aversion doesn�t depend on unobserved heterogeneity. Our dataset is rich and many things
are possible.
Finally, A is potentially endogenous, since it may capture some aspects of unobserved
�ability�. This is why we jointly estimate an equation explaining A itself, in which the month
of birth is used as an instrument27 (equation (9) above). The reasons why the month of birth
has a positive impact on age at grade 6 are complex. There is a recent literature on these
relative maturity e¤ects28. The main reason why this impact is positive and signi�cant is
probably that students who are relatively older in their �rst grades simply tend to have better
performances, all other things equal. In any case, our results are robust to the introduction
or deletion of the A equation. In other words, with our data, it happens that age-at-grade
6 can be treated as exogenous without any essential change in the results. The addition of
an equation explaining A doesn�t seem to change the type-vector distribution in an essential
way. Given that some important social and family-background controls have been introduced
in the delay equation, it is likely that the e¤ect of A on delay is estimated with little bias.
In the appendix, we also report on an attempt at estimating a discount factor �
by grid-search on a slightly simpli�ed version of the model. This yields very high values,
between 0:995 and 129. In any case, as explained above, there is no good empirical basis in
our data to identify time preference, since we observe wages at the beginning of a worker�s
career only.
3 Data
To perform the estimations presented below we used �Génération 92�, a large scale survey
conducted in France. The survey and associated data base have been produced by CEREQ
27The month of birth variable ranges from 1 for students born in January, to 12 for students born inDecember. To simplify the model, the age at grade 6 equation is a simple probit, instead of an orderedprobit. We explain values of this age higher than 12 (or smaller than 12) by means of a probit.28See, e.g., Bedard and Dhuey (2006), Grenet (2008), Mahjoub (2008).29See Appendix 6.
18
(Centre d�Etudes et de Recherches sur les Quali�cations), a public research agency, work-
ing under the aegis of the Ministry of Education30. Génération 92 is a sample of 26; 359
young workers of both sexes, whose education levels range from the lowest (i.e., high-school
dropouts) to graduate studies, and who graduated in a wide array of sectors and disciplines.
Observed individuals have left the educational system between January 1rst and December31
31rst, 1992. They have left the educational system for the �rst time, and for at least one
year in 199232. The labor market experience of these individuals has been observed during 5
years, until 1997. The survey provides detailed observations of individual employment and
unemployment spells, of wages and occupation types, as well as geographical locations of
the students at the age of entry into junior high-school (roughly 11), and in 1992, when they
left school. The personal labor-market history of each survey respondent has been literally
reconstructed, month after month, during the period 1993-1997, by means of an interview.
Before 1992, the individual�s educational achievement is also observed.
In the following, we distinguish education from years of education. A degree, or a
group of degrees, is called an education level. Education levels are dummy variables; they
are used below instead of the years-of-schooling to measure human capital and to estimate
returns to education. We have created 6 education levels: (i) the high-school dropouts; (ii)
the vocational high-school degree holders33; (iii) those who passed the national high-school
diploma, i.e., the baccalauréat34; (iv) two years of college35, (v) four years of college; (vi)
graduate studies (5 years of higher education or more).
Some people �nish school more quickly than others, given their highest degree36. First
de�ne as normal age the �normal�number of years needed to reach the individual�s grade,
30 Articles and descriptive statistics, concerning various aspects of the survey, are available at www.cereq.fr.31 To �x ideas, the number of inhabitants of France who left school for the �rst time in 1992 is estimated
to be of the order of 640,000.32 They did not return to school for more than one year after 1992, and they had not left school before
1992 except for compulsory military service, illness, or pregnancy.33 i.e., the so-called Certi�cats d�Aptitude and Brevet d�Etudes Professionnelles.34 Grade 12 students in the US correspond (roughly) to the French classe terminale, and the students of
this grade sit an examination called baccalauréat. There exist vocational variants of the diploma.35 The corresponding exam is called DEUG (Diplôme d�Etudes Universitaires Générales), which is the
equivalent of an Associate�s degree, or DUT (Diplôme Universitaire de Technologie). There are exams atthe end of each of the college years in French universities, and the DEUG or DUT correspond to the end ofgrade 14.36We have studied this variability in depth in another paper, Brodaty et al. (2010).
19
sit the exam and earn the corresponding degree (if there is one): it is a conventional age,
associated with each individual�s school-leaving degree. For a given degree or certi�cate,
normal age is thus the age of those who earned this degree or certi�cate without any grade
repetition or delay of any kind. It is not the average completion age. For instance, the high-
school dropouts have a normal age of 13 years; the vocational high-school degree holders
have a normal age of 16 or 18 years, depending on the category of their certi�cate; (iii)
those who passed the national high-school diploma have a normal age of 18; (iv) two years
of college correspond to a normal age of 20, and so on. A substantial part of the variance
of school-leaving age, conditional on education level or degrees (see Appendix 2), happens
to be due to grade retention. Grade repeaters are quite common, even in college37. Delays
are thus generated by grade repetitions in primary, secondary and higher education. The
e¢ ciency of grade repetitions in primary and secondary education is of course a hotly debated
issue, but until today, the institution has survived. To measure inter-individual di¤erences
in time-to-degree, we then created a variable called delay, de�ned as observed school-leaving
age divided by normal age38. We also observe the individual�s age at grade 6 entry, allowing
us to measure the delay accumulated during primary education39.
Each individual�s curriculum on the job market is an array of data including a number
of jobs, with their corresponding wages and durations in months, and unemployment spells,
again with a length in months. To estimate the returns to education, we rely on a single,
scalar index of earnings for each worker. We simply take the arithmetic average of the full-
time wages earned during full-time employment spells, weighted by their respective spell
durations. In the following, this index is called the mean wage40. For descriptive statistics
37 Freshmen repeating the �rst and second years of college are quite common.38For instance, an individual who �nished high school and passed the national examinations (i.e., the
baccalauréat) at the age of 19 has a delay of 19=18 = 1:055, and the average age of those who left schoolwith a baccalauréat is 20:78. The national high-school diploma is required for admission to colleges (i.e.,Universités) in France. We adopt the following convention: a person who passed the baccalauréat at the ageof 18 and spent two years in college but failed to pass an Associate�s or any equivalent degree has a normalage of only 18 (which corresponds to that person�s highest degree) and would have a delay of 20=18 = 1:111years. Figure A2, in Appendix 2, depicts the empirical distribution of the logarithm of delay.39The age at grade 6 entry depends on family background in a striking way. Figure A6, in Appendix 2,
shows that the average value of the variable is much lower for the sons of professionals than for other malestudents.40We have studied other statistics: the wage earned by the individual in his �rst full-time job, and the
last wage, earned in the last full-time job observed. We also computed measures of earnings, which takeunemployment spells and unemployment bene�ts into account. The results obtained are similar and are not
20
and further details41, see Appendix 2. We also observe the student�s age at grade 6 entry,
and the student�s month of birth (which will be used as an instrument for age at grade 6).
Part of our covariates are based on data with a geographical structure. Using a
�le from the National Geographical Institute42, we obtained a measure of local population
density in the town of residence at the age of grade 6 entry, which we use as an additional
control. A number of other variables are based on inter-county variation, where by county we
mean the French département43. In particular, we constructed a battery of school-opening
instruments for education, using a �le from the Ministry of Education (the Base Centrale des
Etablissements) which lists all high-school and two-year college openings in the country since
the 50s. The �le enables one to distinguish between vocational and general high-schools. In
France, the 1980s witnessed a rapid growth in the number of vocational high-schools44 (i.e.,
to be precise, of the lycées professionnels and lycées techniques). Both curves are strongly
increasing and correlated in the 70s and 80s. Interestingly, the data exhibit a substantial
degree of inter-county variability in the stock of vocational high-schools, per capita of 15-
to-19-year-olds45. In a recent paper, Currie and Moretti (2003), have used the same kind of
school-opening per capita, measured in the years when the individual was at a crucial age,
say 17 or 18. Here, given the structure of our data, we must avoid a potential problem of
negative correlation of the individual�s education with the high-school stock. This correlation
would simply re�ect the fact that educated students are older at the end of their studies and
therefore experienced an environment with less high-schools during their teens. To avoid this
problem, we have chosen to �x the year at which the stock is evaluated. The choice of 1982
as a �xed point in time, ten years before the school-leaving year of students, characterizes
the school-supply environments, roughly around the age of junior high-school entry. When
reported here.41On top of this, the survey provides information on family background. The father�s and the mother�s
occupation in 92, the father�s and the mother�s education are the most important of these variables. We knowthe geographical location of the student�s family at the age of junior high-school entry and the student�slocation at school-leaving age (i.e., in 1992). Location is rather precise since we know the code of eachcommune, and there are more than 36; 000 communes in France.42 i.e., Institut Geographique National.43 There are 95 départements in France.44We used INSEE Census data to obtain the population shares of age groups. Figure A3 shows the
historical development of the national stock of such schools, and displays a per capita version of this measure,namely, the stock divided by the number of 15�to-19-year-olds in the county.45The density of this county-level per capita measure in the year 1982 is depicted on Figure A4.
21
used to explain years of schooling or the highest degree, the instruments based on vocational
high-school openings in each county happened to be the strongest46. Further details on the
sample and sample selection are provided in Appendix 2.
Finally, we used a distance-to-college instrument in combination with (or as an al-
ternative to) school density. Based on detailed information about the geographical location
of students at the time of grade 6 entry, the distance (in kilometers) of the individual�s
residence, at this particular moment of his life, with the nearest college (i.e., the nearest
université) can be computed. This measure of distance really varies at the individual level.
4 Estimation Results
The model has been estimated with the help of the data set presented above, by straight-
forward Maximum Likelihood. It is easy to describe the estimation method: all equations
have been estimated simultaneously using a likelihood maximization algorithm, as described
in textbooks47. The sample includes 12,310 young men. The model has been estimated with
the entire sample �rst. We then used two subsamples to check for possible di¤erences in
risk aversion and other parameters in two social subgroups: the sons of professionals, and
the rest of the population. Appendix 4 provides simulations allowing an assessment of the
model�s goodness of �t: it has good performances in that respect. We describe the main
results below.46Further details and tests on these variables are given in Brodaty et al. (2010). Now, one might argue
that it is not the stock of high schools itself that plays a role, but its growth rate or �rst di¤erence. Wethen also computed the variation of our county-level stocks of vocational schools between two �xed pointsin time, namely between 1989 and 1982, and used the variation as an instrument for education. These yearscover the relevant time span during which most of our students were teenagers. Again, with this de�nition,the years at which temporal variations are evaluated do not depend on the individual�s age; these variationsdepend only on the individual�s county of residence at the age of junior high-school entry. We used theseintruments in a preliminary version of the present paper: this doesn�t lead to substantially di¤erent results.47In particular, neither the use of an EM algorithm, nor ine¢ cient equation-by-equation methods, that
would decompose estimation in steps, were necessary, except to obtain good preliminary estimates of mostparameters.
22
4.1 Results obtained with the whole sample
We start with the choice of the number of types48. The model has �rst been estimated
without any unobserved heterogeneity, i.e., with no types �; the log-likelihood is equal to
�1:1086 for this variant; results are given in Table X1, which is displayed in the supplemen-
tary material section of this paper49. The log-likelihood increases to �0:9092 when two types
are introduced (see Table 1), and the model without unobserved heterogeneity is clearly re-
jected by the Likelihood ratio test (i.e., to be precise, by the Vuong test; see Vuong (1989)).
Table 1 is described below and has the same structure as the tables presenting variants in
the supplementary material section. It is possible to estimate a third type with the whole
sample, the probabilities of the three types being (:16; :46; :38). The third type raises the
log-likelihood again, to �0:8503. The results for the three-types case are presented in Table
X2, in the supplementary material section. A fourth type cannot be estimated in a useful
way, since the probability of this fourth type is always very close to zero. We nevertheless
chose to present the model with two types for the following reasons: �rst, with the complete
sample, the di¤erences in estimated parameters between two and three types are limited, and
second, this permits a full comparison with the subsample estimates, since subsamples do
not allow to estimate three types. We conclude that introducing two or three types improves
our description of the data, but this doesn�t completely change the picture: there are some
di¤erences in estimated returns to education and in the variance of wages, but one cannot
say that the main qualitative results are driven by a special way of modeling unobserved
heterogeneity50.
To push this robustness check further, another variant of the model has been esti-
mated, this time with a limited e¤ect of unobserved types. The results are diplayed on Table
X3, in the supplementary material section. In the latter variant, the types intervene in the
log-wage and schooling choice equations only, but the other equations do not vary with types.
This variant, with a likelihood equal to �1:1020, is not much better than the variant with48Recall that each type is a vector of six given "constants".49See also the corresponding author�s web site.50Types add �exibility by allowing the random disturbances to become mixtures of normal distributions
instead of plain normal distributions. They of course also permit a treatment of endogeneity, because randomdisturbances are no longer independent.
23
no types at all. Again, changes in the other coe¢ cient estimates are limited.
We now comment on Table 1. First of all, the model estimated here is a slightly
simpli�ed version of the model described above, in which the variance of ln(d=�), i.e., �2�,
is a constant51. The results for the more sophisticated version of the model, in which �2�
varies with some covariates, are presented in Table 1B (that is commented below). In any
case, the di¤erences between the two variants are very small. Table 1 is divided into two
panels, Table 1a and Table 1b, presenting the estimation results for the complete sample and
two types52. From left to right, Table 1a gives the estimated coe¢ cients for the log-wage
equation, the variance of log-wages, and risk aversion. The t-statistics are given to the right
of each coe¢ cient estimate. The wage equation shows the signi�cance of family background
and lists the estimated �fs, the returns to education levels, very precisely estimated. Given
that each level s takes around 2 years, the estimated returns are between 5 and 8% per year,
and they are somewhat decreasing. These values are in line with most of the literature on
the subject. The last two lines of the �rst column give the estimated type values �1w and �2w.
Next, we �nd the estimated parameters of the function explaining the variance of
wages. Family background is no longer signi�cant, but wage riskiness is clearly increasing
with education. The reported �gures are the estimated values of ��2s=�2s�1, so they can be
read as percentage increases. These values are substantial and highly signi�cant53. When
expressed in standard-deviation terms, the wage-risk increase due to education remains non-
negligible (��s=�s�1 is very roughly one half of ��2s=�2s�1 for small values of ��
2s). The last
two lines of the third column give the estimated values of the type coordinates �1� and �2�.
The two rightmost columns of Table 1a display the risk aversion function estimates.
The father-went-to-college dummy is signi�cant and increases risk aversion. This result
will be con�rmed below by subsample estimations of the model. The last two lines of the
51In other words, the model speci�ed by equation (8b) above is replaced by a constant coe¢ cient.52Tables X1 to X7 in the supplementary material section have a similar structure.53To have a better view, we express these changes in terms of percentage increase in the standard deviation.
Some elementary algebra yields,
��s�s�1
= �1 +s1 +
��2s�2s�1
:
Precise computations show that if ��2s=�2s�1 = 0:53 then ��s=�s�1 = 0:23; ��2s=�
2s�1 = 0:3 yields
��s=�s�1 = 0:14 and ��2s=�2s�1 = 0:16 yields ��s=�s�1 = 0:07.
24
�fth column give the estimated values of the type coordinates �1 and �2 . Note that these
parameters are very precisely estimated. It follows that risk aversion is 0:654 for a type 1
and 0:721 for a type 2 � add 0:018 when the father went to college. The di¤erences between
the values are highly signi�cant. We thus �nd a moderate value of risk aversion, the utility
u exhibiting less risk aversion than the logarithm but more than the square root.
Table 1b displays the results for the remaining three equations of the model, namely,
the education, delay and age-at-grade-6-entry equations. The �rst column of Table 1b lists
the estimated parameters of the ordered probit on education levels. Family background has
a signi�cant impact on education. These results con�rm a number of well-known facts54. It
is interesting to note that our instruments for education are reasonably strong: three out of
four of them have a very signi�cant impact on education. The 4 variables are excluded from
the other equations. The �rst instrument is based on measures of local high-school supply
as described above. France has two kinds of vocational high schools; the lycées techniques
and lycées professionnels. We translate these two categories into English as respectively the
technical and vocational high schools, to simplify the presentation. To be more precise, the
�rst instrument is based on the stocks of vocational and technical high schools per capita of
15-19 years old in the year 1982, and in the county of residence at the age of grade 6 entry55.
The �rst two distance-to-college dummies, indicating the second and third quartiles of the
distance distribution are also signi�cant, with the expected sign56.
The cost parameters cs are also precisely estimated, and clearly increasing with s.
The rest of the variation of education is due to the variability of schooling durations ds,
wage risk �s and risk aversion, through functions ks.
The third column of Table 1b gives the results for the delay (i.e., ds=� s) equation.
The social-background controls have a signi�cant impact on delay, more or less as expected:
54We would have obtained a more detailed account of the impact of family background and of environmentalvariables on education with additional controls, but we have kept a small number of key variables only, tolighten the computational burden of ML estimation.55The instrument is a dummy, taking value 1 when the stocks of vocational and technical high schools,
divided by population aged 15-19, in the county of residence at grade 6 entry, are both greater than theirmedian values.56We know from Brodaty et al. (2010), that, as noted by a referee, the distance-to-college instruments
have more impact on the education of the sons of professionals, while the school-supply or school-openinginstruments work better with the rest of the sample.
25
educated parents reduce delay. A larger age at grade 6 entry very signi�cantly increases
delay.
The reader may be worried by the fact that the stock of high-schools instrument
varies only at the county (i.e., département) level (there are more than 95 counties, though).
This is, among others, a reason for which we added the distance-to-college instrument57.
The distance varies at a much �ner level, because we know the town (i.e., the commune)
of residence of individuals in 1982, and the measure of distance is based on this commune
of residence. There are 5920 communes in the sample and 12310 individuals. This yields
roughly 2 students per commune. In the sample, 98% of the communes are represented by less
than 10 individuals; 67% of the sampled individuals were located in communes including less
than 5 other sample members; 90% of the sampled individuals are located in communes with
less than 20 other sample members. To check for robustness, we re-estimated the model
completely with di¤erent sets of instruments for education levels. In the supplementary
material appendix, Table X4 shows a variant in which the stock of high-schools variable is
not used (only the distance variable). Table X5, on the contrary, uses the stock of high-
schools variable only. Table X6 shows a complete re-estimation of the model, based on the
subsample of individuals that is obtained when we remove all communes that are drawn more
than 20 times. This amounts to removing 10% of the communes, and this trimming leaves a
sample with 11107 students (90% of the initial sample). Table X7 gives an estimation of the
model, obtained while removing the controls for geographical characteristics (like the density
of population). These variants do not show substantial changes in the precision with which
parameters are estimated. For instance, adding the distance instrument seems to increase
the standard errors of some estimates, but only slightly. The results of all these variants are
in essence very close. We conclude that biases, that would be due to a potential clustering
problem, are not a concern here.
Table 1B presents a close variant of the model described by Table 1, in which the
variances �2s and �2� vary with the age at grade 6 entry (as speci�ed by Eq. (8b)). Table
1Ba is similar to Table 1a with one di¤erence: in the middle columns, we now see that the
57 We also added the density of population in the commune as a control in the education and wageequations.
26
variance of log-wages varies, very signi�cantly, with age at grade 6. The latter age variable
doesn�t play an important role in the formation of expected wages, but, in contrast, has a
signi�ciant impact on conditional wage risk. The lower panel, Table 1Bb, has additional lines
to report the values of coe¢ cients !a, and to be more precise, they give the impact of age at
grade 6 dummies on the standard deviation of durations ��, expressed in standard-deviation
units (age 11 being the reference). This impact is very signi�cant too. With this model we
can clearly have variations in the mean of log-durations that are independent of variations in
the variance of log-durations58. Table 1B shows that the variances of delay and wages vary
at the individual level, and that this inter-individual variability can be used to improve the
identi�cation of the education-choice model and risk-aversion, as discussed above. But, in
practice, if we may be reassured to see that �ne variations in risk play a genuine role in the
identi�cation of the risk-aversion parameters, the numerical di¤erences with the (slightly)
simpler version are small. This is why, in the following, we use the slightly simpli�ed version
as our benchmark. Intuitively, given the model and the data, the behavioral parameter
denoted captures the e¤ect of wage and duration risks on individual decisions; it doesn�t
simply re�ect intertemporal substitution or intertemporal tradeo¤s59.
Finally, the results of Table 1 have been obtained with a particular value of the horizon
T , chosen equal to 64. There is an element of arbitrariness in this choice, so that we tried
many values of T , ranging from 60 to 115. This does not cause any important changes. Table
2 shows the estimated values of the risk-aversion parameter for the two types, obtained when
T varies. Risk-aversion increases with T and varies between :6 and :88, but the qualitative
result of a risk aversion smaller than one is unchanged. The values of cs increase with T ,
but the other parameters of the model do not vary much, and are not reported. To sum
up, estimated values of stay in the same range for a large interval of reasonable values of
T . Note that the horizon T changes the relative weights of the education and work periods
in a student�s life, but that constants of the education cost functions hs can be rescaled to
counterbalance the change in T . It seems more �realistic�or more appropriate to use values
58More precisely, we can vary the family background, while keeping age at grade 6 �xed. Thus, the meanof ln(d=�) and the variance of ln(d=�) may vary independently, but the variance of d and the mean of dcannot, because d=� is assumed log-normal.59In this standard model, measures both risk-aversion and the intertemporal elasticity of substitution,
as is well known.
27
of T around 60 years, or more, than to use small values, because T is like the length of
life60. Finally, the exact value of is probably less important than the di¤erences that we
�nd in the values of for di¤erent groups of individuals, with an impact on their investment
behavior. This is why we study subsamples in the next subsection.
Keane and Wolpin (2001), Belzil and Hansen (2004) and Sauer (2004) have estimated
dynamic-programming models of educational choices, in which a risk-aversion parameter can
be inferred from individual schooling decisions. In their models, individuals are heteroge-
neous with respect to ability, but share the same degree of constant relative risk aversion.
Keane and Wolpin �nd a RRA coe¢ cient equal to 0:49; Belzil and Hansen (2004) �nd a RRA
coe¢ cient around 0:9, and Sauer (2004) �nds a RRA equal to 0:77. These values are all rel-
atively close to ours and small as compared to the estimates obtained in the macroeconomic
literature.
Chetty (2006) �nds bounds on the RRA coe¢ cient derived from labor supply behav-
ior. Given the available evidence on labor supply, he �nds that RRA should be smaller than
2 and is very likely around 1 (logarithmic utility). Finally, Belzil and Leonardi (2013) use
Italian panel data in which individual di¤erences in risk attitudes are measured by answers
to a lottery pricing question; they also �nd that individual speci�c risk aversion acts as
deterrent to higher education investment.
To sum up, we conclude from these �rst results that di¤erences in risk aversion really
matter and that letting these di¤erences play a role substantially improves the description
that we can make of the data. Unobserved heterogeneity can be captured by unobservable-
type e¤ects on wages, wage risk, education costs and delay and, in spite of this �exibility, the
likelihood is maximized when we allow for an additional and speci�c role for risk aversion.
60To see this, assume for instance that the rate of increase of wages due to potential experience is �% peryear while the discount rate is �% per year. Using our speci�cation of utility with = 0:5, it is easy tocheck that this is tantamount to choosing � =
p1 + �=(1 + �). So, with the reasonable values � ' 3% and
� ' 1:5%, we get � ' 1. This tells us that long values of T are probably more appropriate, because T isroughly like the length of life.
28
4.2 Subsample estimations
To check if risk aversion plays a di¤erent role in di¤erent social groups, we have reestimated
the model with two complementary subsamples. We have sorted out the sons of highly
educated parents, called the sons of professionals. To be precise, any individual with at least
one parent in the following occupational categories: executives, doctors, lawyers, engineers
or teachers, is included in this subsample. There are 2,315 such individuals. The other
subsample contains the rest of the individuals (and includes the sons of farmers, craftsmen,
middle managers, white-collar employees and blue collars), that is, 9,995 individuals.
Table 3a-3b and Table 4a-4b are the equivalents of Table 1a-1b for the subsample of
students whose parents are professionals and its complement, respectively. In other words,
these tables have the same structure as that of Table 1a-1b, allowing for easy comparisons be-
tween subgroups. In each subgroup, 2 types essentially exhaust the unobserved heterogeneity
(a third type would have a very small probability).
It may be surprising to see that the sons of professionals are signi�cantly more risk
averse than the others. In this �rst subgroup (Table 3a-3b), risk aversion ranges from 0:78
to 0:81. Risk aversion is only 0:69 among the sons of �non-professionals�� add 0:026 if the
father went to college. The precision of these estimates is very high, and the di¤erences
are signi�cant. In fact, as will be seen in the simulations below, these di¤erences play an
important part, and the overall picture is rather di¤erent for the sons of professionals and
for the other group.
The returns to education are clearly higher for the sons of professionals, but the slope
of the risk-return curve is smaller, since higher returns are associated with higher increases
in risk. The socially privileged students also have lower estimated education costs, in the
form of higher values of �c and strikingly lower values of the cs. The latter parameters
are estimated with good precision only in the relatively �disadvantaged� subgroup. The
relatively disadvantaged group also has greater delay on average and the instruments have
a stronger impact. This is reasonable. To sum up, it seems that the professionals�sons are
more exposed to wage risk and more risk averse than the average, but the other group, in
spite of being less risk averse, is hampered by much higher cost parameters. In other words,
29
environmental and family background disadvantages more than counterbalance the lower
risk aversion.
The interpretation of these results depends on what is captured by our risk-aversion
parameter . Given that we do not model insurance opportunities or contingent transfers
of various resources from the parents to the students, our risk-aversion parameter is likely
to re�ect di¤erences in access to insurance as well as individual psychological traits. It is
particularly hard to disentangle the two things by means of econometric methods, given the
currently available data sources. It then seems natural that relatively disadvantaged sub-
sample members be less risk-averse. Young males planning to become blue collars are taking
little risks in a society in which there exists a minimum wage legislation and unemployment
insurance is generous: risk and return are both relatively small, as well as risk aversion. In
contrast, going to college or to graduate school is a much riskier investment. Members of
the disadvantaged group have nothing to lose, as compared to the privileged group for which
educational failure may mean a drop in social status. In spite of these arguments, the result
remains surprising, because we are accustomed to think that children raised in wealthier
families have better insurance opportunities than other children. Our model o¤ers another
reasonable explanation: their education costs are markedly lower. Presumably, this is due
to unobservable intergenerational transfers.
4.3 Human Capital Risk-Return Curves
An important output of the model is a set of risk-return pairs for each education level and
each type. Formally, for the agent characterized by (�;X), these risk-return pairs are of the
form (�s(�); fs(�)); where fs(�) = fs+ �w +X0�0. Recall that �s(�) = exp(�� +X��� + zs).
Given that wages are assumed lognormally distributed, the mean of ws is in fact ems(�),
where ms(�) = fs(�) + �2s(�)=2. Joining the points (�s(�);ms(�))s=0;:::;n yields a �risk-
return curve�. It is not di¢ cult to check that a decision-maker characterized by (�;X) and
willing to maximize expected utility on this set of points would equivalently like to maximize
ms(�)�( =2)�2s(�), under the CRRA assumption. This means that we can view the decision
maker�s indi¤erence curves as quadratic in the (�;m) plane, as usual in portfolio theory.
Figure 1 represents the average risk-return curves obtained with the entire sample, for each
30
of the two types, averaging over the entire sample (while the type is �xed). Some interesting
di¤erences are uncovered if we compute the same risk-return curves with subsample estimates
of the model. Figure 2 shows the average risk-return plots for each of the two types in the
sons of professionals� subpopulation. We see that type 1 has signi�cantly higher returns
and higher risks than type 2, and that risk increases substantially with higher education.
Figure 3 shows the risk-return curves obtained with the rest of the sample. It is surprising
to see that, in this subsample, risk, and the highest returns, are lower. Note also that type
2 is dominated by type 1, in the sense that type 2 has lower returns and higher risks for
every value of s. It�s bad news to be a type 2 in this sub-population. Figure 4 permits a
comparison of the two sub-populations�risk-return curves. On this latter �gure, the curves
are averaged over the 2 types in each subsample separately.
Figures 1-4, exhibit a curious spike in the risk-return curve, that is more pronounced
in subsample estimations, and attracted the attention of referees. Figures 2 and 3 show that
it is very risky to be a drop-out, but a quali�ed blue collar is bearing a markedly lower risk.
The spike is visible on Figure 1 too: the transition from the high-school dropout level to
the vocational degree level causes a smaller increase in risk than other transitions. Due to
nonlinearities in the model, this smaller increase in risk may in fact become a reduction in
risk with two types and in separate subsamples. This suggests that there is a lot to gain
from earning a �rst degree in terms of reduced earning risk. This result is reasonable given
the French context: the �rst-level degree provides a form of insurance on the labor market.
Finally, Figures 5 and 6 show the ex ante density of wages, as seen from the point of
view of the average member of a group, conditional on his type, in two potentially counter-
factual situations: if he were a high school dropout (to the left of the �gure) and if he had
earned a Master�s degree or more (the rightmost densities). Figure 5 gives the densities for
the less socially privileged subsample, and Figure 6 for the group of sons of professionals. It
is visible that the sons of professionals take more risks in higher education. On Figure 6, for
instance, it is easy to see that the dispersion of wages is highest for type 1. On Figures 5
and 6, the middle density in each group of three is the mixture of the densities for the two
types.
31
5 Comparative Statics, Simulations and Discussion
We will now present a number of simulations of the model. The �rst simulations are mainly
based on the numerical evaluation of some key elasticities. It is shown in Appendix 5 that
the cuts ks have the following properties:
@ks@
> 0;@ks@�
< 0;@ks
@(�fs)< 0:
The cuts can be viewed as hurdles. Risk aversion raises the hurdles, while speed characteris-
tics � and returns to education lower the hurdles. A higher hurdle means that the students
need a better realization of �, given X and �, to be able to jump to the next level. The
probabilities of each education level, that is, the distribution of s, is the integral of the stan-
dard normal density between two consecutive hurdles. The slope of ks with respect to � or
may change with s, and it follows that the predicted frequency of a given level, Pr(s) may
be a nonmonotonic function of � or , in spite of the fact that the derivatives of ks have a
non-ambiguous sign.
We have computed the elasticity of relevant probabilities with respect to some pa-
rameters. The probability of observing an education level S greater than s, can be called
survival. This function is denoted s. By de�nition, we have,
s(�) = Pr(S � s j �) = 1� �(�s(�)); (17)
where � is the c.d.f of the standard normal distribution. The derivative of s with respect
to is @s=@ = �'(�s)(@�s=@ ) < 0, where ' is the density of the standard normal distri-
bution. We de�ne the elasticity61 of survival with respect to as "(j ) = (@=@ )( =).
In addition to , we have computed the mean elasticities of s with respect to cs,
�fs, ��2s and exp(�d). Variations of cs are shocks to education costs; variations of �fs and
��2s represent changes in the returns to education and in the riskiness of wages and �nally,
variations of exp(�d) directly a¤ect the measure of speed 1=�.
Table 5 gives the elasticities of s with respect to the parameters listed above. These
elasticities have been computed for the two subsamples: the sons of professionals and the rest
61To compute the numerical values, we �rst compute this elasticity for each individual i, then take theweighted average over possible types �j , using the estimated probabilities pj and �nally take the arithmeticaverage of all these values by summing over i. The de�nition will be the same with other parameters.
32
of the sample. The �rst three columns on the left give the results in the former subsample;
the rightmost three columns give the equivalent results for the latter. The �rst column
in each group of three is the elasticity itself, which depends on education s, so we have 5
di¤erent values, given that there are 6 education levels. The second in each group of three
columns gives the observed value of the survival probability s itself. The third column
gives the simulated change in this probability following a one percent change in the value
of the parameter. The top rows show the e¤ect of risk aversion on s; it is unambiguously
negative, with high values of the elasticity in absolute value.
To read Table 5 consider for instance the impact of risk-aversion on the sons of profes-
sionals. Take for instance the 5th line, corresponding to four years of college. The elasticity
of at this level is �29:47 (given in the �rst column); the observed value of this proba-
bility is :41 in the second column, i.e., 41% of the sons of professionals reach at least the
four-years-of-college level; a 1% increase in risk aversion for this group causes a drop of the
probability to 32% (in the third column). The corresponding �gures for the rest of the sam-
ple are respectively, �35, 11% and 7% (columns 4, 5 and 6). The impact of risk aversion on
enrollment in four-year college and graduate studies is sizeable in both susbsamples. But
cost parameters do not have such a dramatic e¤ect on enrollment in higher education. The
elasticity to costs of education cs is higher in the relatively disadvantaged subsample. The
riskiness of wages, surprisingly, has little if any impact in both subsamples. The returns to
education, in contrast, have a powerful, positive impact on enrollment. Finally, the elastic-
ity with respect to grade-repetition risk (or risk of delay) is negative and very important.
The orders of magnitude are the same as that of elasticities with respect to risk aversion62.
Enrollment in the highest education levels would su¤er most from an increase in the proba-
bility of delay, and the impact is clearly bigger in the less socially privileged subsample. Risk
aversion and education costs act as a powerful brake on individual educational investment,
returns to education are a powerful incentive. Our general conclusion is that, to explain
enrollment in secondary and higher education, expected returns are at least as important as
risk aversion and the costs of education.
In the college-major choice literature, the estimated impact of expected wages on
62A glance at formulae in Appendix 5 sheds light on the origin of this property.
33
chosen majors is weak, in spite of the fact that these majors command very di¤erent returns
on the labor market. We suspect that there are several reasons for these di¤erences, that
can reconcile the results to a certain extent. One important di¤erence is the role of utility
nonlinearity due to risk aversion. Another di¤erence is that we study an ordered discrete-
choice structure with a hierarchy of 6 education levels ranging from high-school dropouts
to graduate studies, but aggregate the majors or �elds of study. The college-major choice
papers of Arcidiacono (2004) and Be¤y et al. (2012) typically use a multinomial (i.e.,
unordered) discrete-choice structure. So, the equations of interest are not the same. We also
use dimensions of the data that other papers do not use (or do not use in the same way).
For instance, Be¤y et al. (2012) exploit the same data as us, but they restrict estimation to
the subsample of students who went to college, whereas we use the entire sample: it is likely
that this helps identifying stronger e¤ects of expected returns on education choices.
To obtain a better view of the importance of speed di¤erences in educational invest-
ment and achievement, we simulated changes in enrollment, or to be more precise, changes in
the distribution of students over education levels, that can be induced by changes in speed.
Setting ds = � s for every individual is tantamount to imposing "social promotion" to the
entire educational system, ceteris paribus. Every individual speed becomes "normal" and
� = 1. Such an experiment has radical e¤ects: 70% of the population would reach the level of
graduate studies! This is not a marginal change. So we focused on a less brutal and probably
more instructive counterfactual. The average probability of promotion, or "speed", of the
sons of professionals, �SoP ' 0:89, is higher than the average speed in the rest of the sample,
�RoS ' 0:857. The former average is 3.85% higher than the latter. This modest di¤erence
in speed operates every year, and will induce a substantial di¤erence in the durations, after
several years of schooling. Recall that student i�s expected duration of an education of level
s is bds = � s=�i where �i is a personal "speed" parameter. Let us multiply all the 1=�i terms
in the rest of the sample by factor � = �RoS=�SoP < 1, so that the average speed factor
is the same among the sons of professionals and in the rest of the sample63. This yields a
reduction of 0.76 years of the durations, on average.
63We can express the di¤erence in terms of average expected durations by computing the arithmetic averageof the di¤erence (1� �)(� si=�i) in the rest-of-sample sub-population.
34
Table 6 gives the simulation results for this experiment, scaling up the speed of the
less privileged students to equalize the average speed in the two groups. The �rst six columns
in Table 6 give the matrix of transitions from level s to higher levels s + 1, s + 2, following
the change in speed. The three rightmost columns give the simulated distribution of levels
after the change, before the change in the rest of the sample and the observed distribution
of levels among the sons of professionals, respectively64.
Table 6 shows some very important changes. For instance, column 1 shows that 50%
of those who were initially dropouts stay dropouts after the change, but 49.9% of these
individuals now earn a vocational degree. This is a huge progress. Column 3 shows similar
promotion e¤ects: of those who initially �nished school with the high-school degree, 99%
now go to college, among which 36% complete 2 years of college, and another 39% complete
4 years of college, etc. Note that many of those who complete some college years may
in fact have been college dropouts before the change, with the high-school degree as their
highest certi�cate, because they never passed the college exams. The biggest e¤ects in terms
of enrollment are represented by columns 1 and 2, because these columns describe the re-
dispatching of 57% = 16:29 + 40:71% of the studied population. The e¤ects displayed in
column 3 are impressive, but they apply to 15% of the subsample only.
Yet, these changes are not su¢ cient to put the rest of the sample on an equal footing
with the sons of professionals. Comparing the simulated post-change distribution for the
rest of the sample with the observed distribution for the sons of professionals, we see that
the former do not fully catch up. In particular, the simulated distribution has a much bigger
share of students leaving school with a vocational degree than the sons of professionals. This
is because a smaller speed is not the only handicap of the less privileged students: they also
have higher costs.
Are the estimated e¤ects of di¤erences in delay implausibly large? The distance
64To compute the simulations, we use the posterior probability distribution of the type �i of each individuali in the subsample, knowing the observed outcomes and covariates (si; Xi). We also condition on theinformation revealed by each individual�s observed schooling level si on his random cost shock �i, to computethe probability of choosing s after the change. In practice, this is done by drawing 500 copies of individuali in the distribution of (�ijsi; Xi), which is a truncated normal distribution. After the reform, some ofthe randomly drawn values of �i fall above the new upper threshold for si and the model predicts that iwould then study more and jump to the next levels. These changes are then averaged using the appropriateconditional distribution of types.
35
between the schooling level distributions of the two social groups is a purely empirical fact.
The gap between the two must be closed by a change in some parameters. Knowing the
French context, we don�t �nd the e¤ect of di¤erences in delay parameters so surprising.
In the French educational system, grade repetitions and exams at the end of each college
year constitute a major screening device65. Held back students progressively reduce their
ambitions and many are eventually disheartened. Indeed, an additional year in school has
very substantial direct and opportunity costs, and these costs increase with the education
level. Many people would agree that the French system is based on what is sometimes called
"selection by means of failure66". It is therefore not very surprising that a change in the
speed parameter would push thousands of young men to go to college.
Another surprising result is the relatively limited e¤ect of wage risk or return risk on
student choices. As noted by a referee, it may be that our wage observations do not capture
enough of these risks, because wages are observed during the �rst 5 years of career only, and
risk may play out over many more years.
We conclude with a discussion of the merits and demerits of a static formulation, as
opposed to a fully dynamic model in which students learn about their ability by observing
their test results and condition on new information to decide if they continue to study or
go to work. An important advantage of our formulation is its (relative) simplicity. Our
model �ts the data quite well as shown in Appendix 4, and Tables A2-A6, on the quality
of �t. Of course, we cannot decompose the choice of an education level in steps, based on
dynamic optimization and a sequence of informative signals, but our data would not allow
us to estimate such a model anyway. We may overestimate the risk in schooling costs and
wages borne by students because they in fact have the option to quit every year. We provide
a static representation of a dynamic process, mainly by bypassing the type-learning process,
assuming that students know their type from the start instead of learning it step by step.
Given the current state of knowledge, it is not easy to tell how this bypass may have biased
some of the model�s parameters.
65This theme is developed further in Brodaty et al. (2010), and Gary-Bobo, Goussé and Robin (2013).66Sélection par l�échec.
36
6 Conclusion
We have used a rich set of micro-data on young workers in France to estimate a structural
model of human capital investment. The model is based on the idea that students choose an
education level so as to maximize their conditional expected utility. Students are risk-averse,
with a constant relative risk-aversion coe¢ cient. They form rational expectations of future
wages and of their time to degree completion. We assumed that the econometrician cannot
observe a number of the individual characteristics that the students do observe and use to
predict their future wages. The model captures the fact that the risks a¤ecting time-to-
degree and future wages play a role in their choice of educational investment. The model
yields RRA parameters between 0.6 and 0.9, very precisely estimated. It also yields risk and
return curves for investments in education. Risk aversion varies with parental occupation:
the students whose parents are professionals with a higher education are more risk averse
but bear more risk than the others, because their costs of education are smaller. Small
increases in risk aversion and in the costs of education around the estimated values can lead
to substantial changes in college enrollment. Simulations also show that the e¤ects of higher
education costs and of expected returns to education are equally important.
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8 Appendix
8.1 Appendix 1. Derivation of the model
We use the following identity,
t=dzXt=1+dz�1
�t = �(1+dz�1)�1� ��dz
1� �
�: (18)
A key assumption, is the fact that wages and durations are independent conditional on �.
Using the conditional independence assumption, it is possible to simplify expression (3). We
have,
V (s j � ; �) = E
��1+ds
�1� �T�ds
1� �
�j ��E [u(ws) j �; �]
+sXz=1
E
��1+dz�1
�1� ��dz
1� �
�j ��E [u(hzwz�1) j � ; �] : (19)
To simplify notation, de�ne the mapping
�(x; y) = E
��1+x
�1� �y
1� �
�j ��;
where (x; y) are any random variables. De�ne �V (s+1 j � ; �) = V (s+1 j � ; �)�V (s j � ; �).
Simple computations then yield,
�V (s+ 1 j � ; �) = �(ds+1; T � ds+1)E [u(ws+1) j � ; �]
��(ds; T � ds)E [u(ws) j � ; �] + �(ds;�ds+1)E [u(hs+1ws) j � ; �] : (20)
We need to compute terms of the form E [u(�ws) j � ; �], with � = 1 or � = hs+1. Using the
CRRA property of utility, combined with linearity of log-wages, and using the fact that hs
depends on (�; �) only, we obtain,
E [u(�ws) j � ; �] = E�(1=�)
�1� ��� exp(�� ln(ws))
�j � ; �
�=1
�f1� ���E [exp(��(fs +X0�0 + �w + �s�)) j � ; �]g
=1
�f1� ��� exp[��(fs +X0�0 + �w)]E [exp(���s�)]g;
43
where � = �1. A well-known property of the expectation of a log-normal random variable
then yields
E [exp(���s�)] = expf(1=2)�2�2s]g:
From this expression, we derive
E [u(�ws) j � ; �] =1
�f1� ��� exp[��(fs +X0�0 + �w � (1=2)��2s)]g; (21)
for � = 1 or � = hs+1.
Remark that � has the following convenient property,
�(ds+1; T � ds+1)� �(ds; T � ds) + �(ds;�ds+1) = 0:
Then, using the above equations, we �nd an explicit expression for �V . Easy algebra shows,
after some simpli�cations, that �V (s+ 1 j � ; �) � 0 is equivalent to
1
�
��(ds; T � ds)� �(ds+1; T � ds+1) exp[��(�fs+1 � (�=2)��2s+1)]
�(ds+1;�ds+1)
�� (hs+1)
��
�; (22)
where, by de�nition, �fs+1 = fs+1 � fs, and ��2s+1 = �2s+1 � �2s. Remark that �w and
X0 do not intervene in the above inequality: only �c and X1 play a role as variables in the
expression of hs.
8.1.1 The Finite Horizon, � = 1 Case
Let us now consider the �nite-horizon, undiscounted version of the model. We set � = 1 and
T <1. By l�Hôpital�s rule, we get for any x; y > 0,
lim�!1
�(x; y) = E
�lim�!1
�1+x�1� �y
1� �
�j ��= E [y j � ] :
Therefore, we get lim�!1�(ds; T�ds) = T�E(dsj�) and lim�!1�(ds+1;�ds+1) = E(�ds+1j�).
In this particular case, �V (s+ 1 j � ; �) � 0 is equivalent to,
(T � E(dsj�))� (T � E(ds+1j�))e��(�fs+1�(�=2)��2s+1)
�E(�ds+1j�)� 1
�e�(X1�1+cs+1����c): (23)
If � > 0, it is easy to see that we can take logarithms on both sides of the inequality,
provided that
(T � E(dsj�)) > (T � E(ds+1j�)) exp[��(�fs+1 � (�=2)��2s+1)]: (24)
44
Using the de�nition,
ks = �1
�ln
�(T � E(ds�1j�))� (T � E(dsj�)) exp[��(�fs � (�=2)��2s)]
E(�dsj�)
�; (25)
we then �nd that the crucial inequality is equivalent to,
� � X1�1 + cs+1 + ks+1 � �c: (26)
It is easy to see that the result is the same if � < 0.
8.1.2 The Logarithmic Utility Case (i.e., � = 0)
An interesting particular case is obtained by letting �! 0. Using l�Hôpital�s rule again, we
get with (13),
lim�!0+
ks = ��fs(T � d̂s)
�d̂s; (27)
where d̂s =E(dsj�). This yields the logarithmic utility model, which is characterized by the
inequalities,
cs +X1�1 �(T � d̂s)�fs
�d̂s� �c � � � cs+1 +X1�1 �
(T � d̂s+1)�fs+1
�d̂s+1� �c:
The analytic expression of the cuto¤ points lim�!0ks is easily interpreted. Since agents are
very patient, i.e., � = 1, the marginal skill-premium gain (per year) of jumping from level
s to level s+ 1 is �fs+1=�d̂s+1, multiplied by the expected number of years to go after the
end of studies, i.e., T � d̂s+1. This expression of marginal bene�t must be compared with
an expression of marginal costs, which is simply cs +X1�1 � �� �c here.
In the logarithmic utility case, the monotonicity condition ks+1 > ks is equivalent to
�d̂s+1
�d̂s>�fs+1�fs
(T � d̂s+1)
(T � d̂s);
showing that the cut-monotonicity condition cs+1 + ks+1 > cs + ks will be satis�ed under
"return concavity", and "cost convexity". But one can allow for increasing returns, i.e.,
for �fs+1=�fs > 1, if T is not too large, and if �d̂s+1 is large enough, since the ratio
(T � d̂s+1)=(T � d̂s) would then be su¢ ciently smaller than 1. By continuity, these remarks
are still valid for values of � close to 0, even if � < 0.
45
8.1.3 The In�nite Horizon, Discounted Utility Case
To understand the potential of our model, consider the case in which T ! +1 and � < 1.
The bds are deterministic functions of (X; �). Remark thatlim
T!+1�(ds+1; T � ds+1) = E
�lim
T!+1�1+ds
�1� �T�ds
1� �
�j ��=
�
1� �E(�ds j � );
and,
limT!+1
�(ds+1;�ds+1) =�
1� �E��ds(1� ��ds+1) j �
�Then, in this case, the crucial inequality (23), yields
1
�
E(�ds j �) � E(�ds+1 j �) e��(�fs+1�(�=2)��2s+1)
E(�ds j �) � E(�ds+1 j �)
!� e�(X1�1+cs+1����c)
�: (28)
Now, provided that � is such that,
E(�ds j �) > E(�ds+1 j �) e��(�fs+1�(�=2)��2s+1)
that is, for values of � which are su¢ ciently close to zero, taking logarithms yields an
expression which is equivalent to:
� � X1�1 + ls+1 + cs+1 � �c (29)
where,
ls = �1
�ln
E(�ds�1 j �) � E(�ds j �) e��(�fs�(�=2)��2s)
E(�ds�1 j �) � E(�ds j �)
!: (30)
We again obtain an Ordered Discrete Choice structure with a particular functional form
imposed on the cuts ls + cs.
The rather complicated expression of ls depends on E(�ds j �). It can easily be
shown that E(�ds j �) = E[exp(! exp(�))], where � is normal with mean 0, variance �2�
and ! = ln(�)� s exp(X2�2 + �a + �d). Since � < 1, we have ! < 0, and this implies that
E[exp(! exp(�))] is well de�ned. To simplify the computations in this case, we have assumed
that �2� = 0 and therefore used the expression E(�ds j �) = �ds = e
!.
46
8.2 Appendix 2. Data and Descriptive Statistics
Table A1 shows the empirical distribution of school-leaving age, conditional on the education
level reached by male students (the displayed �gures are frequencies). As can be seen, school
leaving-age is substantially dispersed, even conditional on �nal education level67.
A di¢ culty with wages is that we do not observe the hours worked (but we know if
the individual worked full-time or part-time). To solve this problem, we decided to select
the individuals who experienced at least a full-time employment spell during the �ve-years
observation period. More precisely, we �rst removed 717 individuals who had never worked
(no employment spell recorded during 5 years). The remaining 25; 642 individuals are the
addition of 14; 213 men and 11; 429 women who worked at least once during the observation
period. We then selected the individuals who experienced at least one full-time employment
spell during the �ve years. As a consequence, we lost 11:7% of the male sub-sample, but still
had 12; 538 men. The �nal stage was to match the sample with geographical data from the
National Geographical Institute with data on schools from the Ministry of Education, in order
to compute a number of controls and geography-related instruments. Some observations
of the individual�s location at the age of entry into junior high-school (the jurisdiction of
residence�s code) were missing. This left us with only 12; 310 males. The possible bias
introduced by this selection procedure is limited in the case of men68. In the present article,
we focus on the male subsample.
A clear advantage of our selection procedure is that it permits us to compare earnings
more precisely, given that full-time employment means a 39 hours working week for most
wage-earning employees (and given the heavily regulated French labor market of the 90s).
More importantly, it tends to select a relatively homogeneous population of youths willing
to work full-time � this has some advantages.
The mean wage variable ignores the length of unemployment spells, and the di¢ culties
faced by the individual to �nd a stable (and well-paid) job. To capture the e¤ect of job
instability on average earnings, we de�ned a second average, simply called earnings. To
67 For instance, the �rst line of Table A1 says that 33 percent of the high-school dropouts left at the ageof 18.68 The same mode of selection would have left us with a sample of 8630 women, all willing to work full-time.
It is therefore likely that there is a sizeable selection bias in the female sub-sample.
47
compute this average, wages and unemployment bene�ts are weighted by the corresponding
employment or unemployment spell duration69. Figure A1 presents a plot of the density of
mean wages and earnings (in the men�s subsample).
8.3 Appendix 3. Likelihood
We can now derive individual contributions to likelihood, denoted, Li. De�ne log-wages as
xi = ln(wi); log-delay as yi = ln(dsi=� si). De�ne the cuts,
�s(�) = X1�1 + ks + cs � �c: (31)
These cuts determine the ordered choice of education levels s. Denote next,
ga(�) = ga �XA�A � �A; (32)
the cuts determining the discrete values ai of the age-at-grade 6 entry variable. Given these
joint ordered Probit structures, we have,
Pr(a = ai; si = s; xi; yi j �) =Z ga+1(�)
ga(�)
Z �s+1(�)
�s(�)
pdf(�; � j xi; yi; �)pdf(xi; yi j �)d�d�; (33)
using the decomposition pdf(�; �; xi; yi j �) = pdf(�; � j xi; yi; �)pdf(xi; yi j �), and the
densities involved are normal. Now de�ne,
b�is =xi � fs � �a �Xi0�0 � �w
�s(��);
b�is = yi �Xi2�2 � �a � �d: (34)
Thus, given our conditional independence assumptions, on the integration domain, the vector
(xi; yi) is normal with the following conditional p.d.f., denoted (: j �),
(xi; yi j �) =1
2��s(��)��exp
��b�2is2
�exp
�� b�2is2�2�
�: (35)
We can therefore factor out (xi; yi j �) in the expression of Pr(a = ai; si = s; xi; yi j �). Let
�(x) =R x�1 �(v)dv, be the standard normal c.d.f., and �(x) = (
p2�)�1 exp(�x2=2) be the
69 A worker is eligible for unemployment bene�ts if he or she has worked in the recent past. Students thusget zero before their �rst job. The unemployment bene�ts are roughly a half of the lost job�s wage.
48
standard normal p.d.f. The distributions of �i, and �i are normal, with mean 0 and variance
1. Using conditional independence again, and
pdf(�; � j xi; yi; �) = pdf(�; � j �) = pdf(�; �) = �(�)�(�);
we obtain,
Pr(ai = a; si = s; xi; yi j �) = (xi; yi j �)Z �s+1(�)
�s(�)
�(�)d�
Z ga+1(�)
ga(�)
�(�)d�:
Therefore, integration �nally yields,
Pr(ai = a; si = s; xi; yi j �) = (xi; yi j �)(�ss+1;i(�)� �ss;i(�))(�aa+1;i(�)� �aa;i(�)): (36)
where by de�nition,
�ss;i(�) = � [Xi1�1 + kis + cs � �c] ; �aa;i(�) = � [ga �XA�A � �A] : (37)
De�ne
Li(�) = (xi; yi j �)(�ss+1;i(�)� �ss;i(�))(�aa+1;i(�)� �aa;i(�));
with a = ai and s = si. Averaging over the K possible types, the contribution to likelihood
of individual i is now simply Li =PK
j=1 pjLi(�j).
8.4 Appendix 4. Goodness of �t
To assess the accuracy of the model�s predictions, we simulated a number of key variables
and compared them with their empirical counterparts. Table A2 predicts the distribution
of educational choices in the population. Tables A3 and A4 predict the log-wages and the
standard deviation of log-wages, respectively. These predictions are given for each of the two
subsamples studied above. To perform the simulations, we �rst predict the education level
s and the associated wage of each individual, drawing in the distributions of � and �, and
then take the average over possible types �. This is done 500 times and we take the average
of the simulated distributions. It is easy to see that the performances of the model are very
good.
Given that the distribution of log-wages is not easy to interpret, we have computed
the means and standard deviations of the monthly wages, expressed in euros. These results
49
are given in Tables A5 and A6. Given that the mean of a log-normal variable depends on
the variance of the underlying normal variable, prediction errors are compounded: it follows
that the predictions seem a bit less accurate.
8.5 Appendix 5. Comparative statics
To study our model�s comparative statics, the main tools are the functions �s = X1�1+ cs+
ks � �c. These functions determine the probabilities of choosing the various values of s in
the population, and therefore, the distribution of educational investment.
To simplify notation, note that we have,
ks = �(1=�) ln (Gs) ; (38)
where,
Gs =T � ds�1 � (T � ds)e
As
�ds; (39)
and
As = ���fs + �2��2s=2: (40)
We assume that Gs > 0 throughout.
Straightforward computations yield,
@�s@
=@ks@�
=1
�2
�ln(Gs) +
eAs
Gs
�T � ds�ds
�(�2��2s � ��fs)
�: (41)
Note that since the logarithm is a concave function70, we have ln(G) � 1�G�1. We can use
this property to �nd a lower bound for @�=@ . After some easy algebra, we �nd,
@�s@
� 1
�2
�(T � ds)(1� eAs + eAs(As + �2��2s=2)
Gs�ds
�:
It follows that @�s=@ > 0 if 1 � eAs + eAsAs > �eAs�2��2s=2. But this inequality holds,
since eA is a strictly convex function. Indeed, convexity implies 1 � eAs + eAsAs > 0. We
therefore conclude that an increase in risk aversion unambiguously raises the hurdles �s,
thus decreasing the probability of choosing education above s, for all s : increasing risk
aversion reduces educational investment. This result will thus hold, on average, if we look at
numerical simulations based on the estimated values of the parameters.70Concavity implies ln(1)� ln(G) � G�1(1�G).
50
Another important characteristic is the speed parameter �. Recall that we have
ds = � s=�, and 1 � � is the probability of repeating a grade, so � is a measure of �speed�.
It is easy to check that,@�s@�
=1
��
�eAs� s � � s�1Gs�� s
� 1�: (42)
Given that � < 0, we have As > 0 and Gs < 1. Thus, we can �nd an upper bound,
@�s@�
<1
��
�1
Gs� 1�< 0: (43)
If � > 0, and ��2s su¢ ciently small, we have As < 0 and Gs > 1. In this latter case too, we
get the upper bound and the same conclusion. We conclude that in the relevant range, more
able individuals (i.e., those with a higher �) will invest more in education, since a higher �
lowers the hurdles �s.
The impact on s of higher returns to education is unambiguously positive, since a
higher �fs lowers the hurdle �s, as shown by the following expression,
@�s@(�fs)
= �eAs
Gs
�T � ds�ds
�< 0: (44)
If � > 0, the model predicts that increasing the slope of the risk-return curve s !
(fs; �s) in the (f; �) plane has the usual e¤ect of discouraging investment in education, since
we have,@�s
@(��2s)=�eAs
2Gs
�T � ds�ds
�: (45)
Interestingly, since � < 0, the impact of higher wage risk (i.e., higher ��2s) is positive.
So, if � < 0, individuals will tend to study more when ��s increases, but these e¤ects are
quite weak, because j�j is small. This e¤ect is due to special properties of the log-normal
distribution: when � increases, the mean of the wage distribution increases, since we have
E(w) = exp(�+ �2=2), where � = E ln(w).
Finally, @�s=@cs = 1. Increasing the cost parameters cs obviously raises the hurdles
and discourages investment in education. To sum up, the comparative statics properties of
the model are intuitively reasonable.
51
8.6 Appendix 6. Estimation of the time preference parameter
We �nally report on estimations of the in�nite-horizon model with discounting. The in�nite
horizon version of the model is derived in Appendix 1 above. We assumed that �2� = 0
to solve the model (see above). The discount factor � can be estimated by grid search.
Maximum-Likelihood estimation and grid search have yielded two estimates of � in the two
subsamples considered, the sons of professionals, and the rest of the population. In the less
advantaged subsample, the best value of the discount parameter is � = 0:995. In the sons-
of-professionals subsample, the best value is � = 1. The results are summarized by Table
A7. It is easy to see that the estimated values of risk aversion are only slightly smaller when
� = 0:995, in the less advantaged subsample. Vuong�s test cannot reject � = 1. So, it seems
that choosing � = 1 and a �nite-horizon model is a good approximation.
52
Table 1: Estimation results (Whole sample)
Table 1a
Coeff. t Coeff. t Coeff. t
Family Background
Father went to College 0.040 3.632 0.082 1.496 0.018 3.782
Mother went to College 0.033 2.651 -0.021 -0.319 0.006 1.439
Father is a Professionnal 0.020 2.506 -0.018 -0.499 0.006 1.710
Mother is a Professionnal 0.019 1.685 -0.042 -0.567 -0.002 -0.278
Density of Population 0.002 0.468 * * * *
Paris Area 0.068 8.700 * * * *
Unemployment Rate -0.018 -3.877 * * * *
Age at grade 6
10 years old 0.025 2.050 * * * *
11 years old (ref.) * * * * * *
12 years old 0.001 0.264 * * * *
13 years old -0.015 -1.659 * * * *
14 years old or more -0.004 -0.175 * * * *
Completed Education
High school dropouts (Ref.)
Vocational degree 0.161 56.777 0.166 2.654 * *
High school graduates 0.125 53.175 0.548 13.595 * *
Two years of college 0.149 51.321 0.309 12.447 * *
Four years of college 0.123 46.760 0.351 17.440 * *
Graduate Studies 0.108 43.099 0.260 15.883 * *
Value of Types
Type 1 8.630 1239.183 0.087 27.517 0.654 94.754
Type 2 8.551 1185.422 0.025 15.952 0.721 107.612
Table 1b
Coeff. t Coeff. t Coeff. t
Standard deviation of η * * 0.0045 3.9947 * *
Family Background
Father went to College 0.990 5.903 -0.009 -2.858 -0.326 -5.465
Mother went to College 0.173 0.936 -0.012 -3.404 -0.345 -4.637
Father is a Professionnal 0.459 3.237 -0.013 -5.425 -0.434 -9.681
Mother is a Professionnal 0.049 0.190 -0.007 -2.128 -0.391 -5.777
Density of Population 0.206 7.324 * * * *
Paris Area 0.176 4.290 * * * *
Unemployment Rate -0.067 -2.671 * * * *
Age at grade 6
10 years old * * -0.023 -12.765 * *
11 years old (ref.) * * * * * *
12 years old * * 0.058 29.717 * *
13 years old * * 0.085 26.052 * *
14 years old or more * * 0.086 8.935 * *
School Density and Distance to College
Stock of vocational and technical high schools 0.231 8.698 * * * *
Distance to college 2d quartile -0.114 -3.274 * * * *
Distance to college 3d quartile -0.161 -3.840 * * * *
Distance to college 4th quartile -0.067 -1.629 * * * *
Month of birth * * * * 0.075 6.171
Education costs
C1 * * * * * *
C2 2.465 14.566 * * * *
C3 3.203 16.583 * * * *
C4 3.848 17.606 * * * *
C5 4.085 16.088 * * * *
Value of Types
Type 1 5.844 38.520 0.219 114.398 0.368 11.779
Type 2 3.452 18.790 0.094 70.555 0.477 26.661
Distribution of Types
Prob(type=1) 0.2925 38.2196
Number of observations
Log likelihood
Vuong Test of H0="No Unobserved Heterogeneity" (P-value)
E(ln(W)) V(ln(W)) Risk Aversion
Education (S) Delay (d/ז) Age at Grade 6
12310
-0.909197
<.0001
Table 1Ba
Coeff. t Coeff. t Coeff. t
Family Background
Father went to College 0.040 3.697 0.088 2.214 0.017 3.862
Mother went to College 0.034 2.707 -0.021 -0.412 0.005 0.961
Father is a Professionnal 0.020 2.532 -0.024 -0.759 0.005 1.529
Mother is a Professionnal 0.019 1.669 -0.035 -0.777 -0.001 -0.164
Density of Population 0.003 0.544 * * * *
Paris Area 0.067 8.505 * * * *
Unemployment Rate -0.018 -3.867 * * * *
Age at grade 6
10 years old 0.023 2.024 -0.154 -4.378 * *
11 years old (ref.) * * * * * *
12 years old 0.005 0.875 0.176 7.078 * *
13 years old -0.011 -1.197 0.156 3.898 * *
14 years old or more 0.011 0.330 1.196 7.431 * *
Completed Education
High school dropouts (Ref.)
Vocational degree 0.162 59.517 0.293 8.996 * *
High school graduates 0.126 55.266 0.614 17.698 * *
Two years of college 0.149 55.130 0.365 16.029 * *
Four years of college 0.124 53.155 0.381 23.409 * *
Graduate Studies 0.109 51.296 0.276 22.281 * *
Value of Types
Type 1 8.630 1262.602 0.077 64.240 0.657 93.121
Type 2 8.547 1157.001 0.020 26.582 0.721 111.692
Table 1Bb
Coeff. t Coeff. t Coeff. t
Standard deviation of η (intercept) * * 0.0048 4.4774 * *
Age at grade 6 in the stand. dev. of η
10 years old * * 0.132 1.714 * *
11 years old (ref.) * * * * * *
12 years old * * -0.142 -4.908 * *
13 years old * * -0.279 -5.007 * *
14 years old or more * * -0.074 -0.277 * *
Family Background
Father went to College 0.994 5.951 -0.008 -2.767 -0.326 -5.462
Mother went to College 0.113 0.526 -0.012 -3.366 -0.345 -4.648
Father is a Professionnal 0.444 2.931 -0.013 -5.349 -0.434 -9.691
Mother is a Professionnal 0.084 0.341 -0.006 -1.882 -0.390 -5.768
Density of Population 0.207 7.339 * * * *
Paris Area 0.181 4.385 * * * *
Unemployment Rate -0.068 -2.712 * * * *
Age at grade 6
10 years old * * -0.027 -12.854 * *
11 years old (ref.) * * * * * *
12 years old * * 0.058 29.671 * *
13 years old * * 0.085 27.051 * *
14 years old or more * * 0.100 7.518 * *
School Density and Distance to College
Stock of vocational and technical high schools 0.232 8.730 * * * *
Distance to college 2d quartile -0.113 -3.246 * * * *
Distance to college 3d quartile -0.160 -3.812 * * * *
Distance to college 4th quartile -0.066 -1.601 * * * *
Month of birth * * * * 0.075 6.176
Education costs
C1 * * * * * *
C2 2.496 15.067 * * * *
C3 3.147 16.716 * * * *
C4 3.852 17.120 * * * *
C5 4.059 14.965 * * * *
Value of Types
Type 1 6.093 35.008 0.217 108.860 0.391 11.994
Type 2 3.593 18.463 0.094 67.497 0.467 25.495
Distribution of Types
Prob(type=1) 0.2943 47.7181
Number of observations
Log likelihood
Risk Aversion
Education (S) Delay (d/ז) Age at Grade 6
12310
-0.904941
Table 1B: Variant of estimation results (Whole sample)
E(ln(W)) V(ln(W))
T qg standard dev. qg standard dev.
60 0.6081 0.0073 0.6783 0.0073
65 0.6538 0.0065 0.721 0.0066
70 0.6908 0.0059 0.7547 0.006
75 0.7212 0.0055 0.782 0.0055
80 0.7467 0.0050 0.8046 0.0051
85 0.7755 0.0046 0.8136 0.0046
95 0.8031 0.0042 0.8537 0.0042
105 0.8274 0.0038 0.8746 0.0038
115 0.8468 0.0034 0.8899 0.0034
Table 2. Sensitivity. Estimated values of risk-aversion parameter qg, for
various values of horizon T.
type 1 type 2
Table 3 : Subsample Estimation Results. Sons of Professionnals
Table 3a
Coeff. t Coeff. t Coeff. t
Family Background
Father went to College 0.019 1.388 0.076 1.041 0.009 1.481
Mother went to College 0.043 2.862 0.007 0.138 0.006 1.213
Father is a Professionnal * * * * * *
Mother is a Professionnal * * * * * *
Density of Population -0.004 -0.302 * * * *
Paris Area 0.052 3.157 * * * *
Unemployment Rate -0.038 -2.867 * * * *
Age at grade 6
10 years old 0.034 1.625 * * * *
11 years old (ref.) * * * * * *
12 years old 0.053 2.632 * * * *
13 years old 0.121 2.575 * * * *
14 years old or more -0.133 -0.797 * * * *
Completed Education
High school dropouts (Ref.)
Vocational degree 0.253 19.238 -0.524 -6.530 * *
High school graduates 0.151 22.652 0.609 3.836 * *
Two years of college 0.195 24.245 0.094 0.978 * *
Four years of college 0.155 24.671 0.408 4.035 * *
Graduate Studies 0.129 23.425 0.084 0.993 * *
Value of Types
Type 1 8.532 396.747 0.156 6.634 0.771 52.122
Type 2 8.429 290.198 0.069 4.917 0.797 73.806
Table 3b
Coeff. t Coeff. t Coeff. t
Standard deviation of η * * 0.0041 1.6614 * *
Family Background
Father went to College 0.781 2.960 -0.002 -0.578 -0.230 -3.027
Mother went to College 0.058 0.287 -0.020 -5.206 -0.322 -3.616
Father is a Professionnal * * * * * *
Mother is a Professionnal * * * * * *
Density of Population 0.272 4.057 * * * *
Paris Area 0.208 2.600 * * * *
Unemployment Rate 0.078 1.274 * * * *
Age at grade 6
10 years old * * -0.028 -7.499 * *
11 years old (ref.) * * * * * *
12 years old * * 0.067 12.036 * *
13 years old * * 0.113 7.799 * *
14 years old or more * * 0.148 3.440 * *
School Density and Distance to College
Stock of vocational and technical high schools 0.218 3.549 * * * *
Distance to college 2d quartile -0.131 -1.855 * * * *
Distance to college 3d quartile -0.249 -2.614 * * * *
Distance to college 4th quartile 0.007 0.080 * * * *
Month of birth * * * * 0.112 3.306
Education costs (cuts)
C1 * * * * * *
C2 0.985 1.368 * * * *
C3 1.284 1.808 * * * *
C4 1.026 1.405 * * * *
C5 0.854 1.054 * * * *
Value of Types
Type 1 6.757 13.294 0.200 47.537 0.982 10.805
Type 2 6.389 8.591 0.076 27.302 0.987 17.713
Distribution of Types : Prob(type=h1) 0.2955 18.6646
Number of observations
Log likelihood
Vuong Test of H0="No Unobserved Heterogeneity" (P-value)
-0.866965
<.0001
E(ln(W)) V(ln(W)) Risk Aversion
Education (S) Delay (d/ז) Age at Grade 6
2315
Table 4a
Coeff. t Coeff. t Coeff. t
Family Background
Father went to College 0.032 1.717 0.475 3.744 0.026 2.077
Mother went to College 0.019 0.955 -0.038 -0.300 0.034 1.676
Father is a Professionnal * * * * * *
Mother is a Professionnal * * * * * *
Density of Population 0.003 0.559 * * * *
Paris Area 0.074 7.931 * * * *
Unemployment Rate -0.016 -3.169 * * * *
Age at grade 6
10 years old 0.012 0.887 * * * *
11 years old (ref.) * * * * * *
12 years old -0.004 -0.621 * * * *
13 years old -0.023 -2.030 * * * *
14 years old or more 0.008 0.256 * * * *
Completed Education
High school dropouts (Ref.)
Vocational degree 0.175 42.676 -0.498 -19.241 * *
High school graduates 0.120 45.372 0.116 2.350 * *
Two years of college 0.142 46.323 -0.092 -2.145 * *
Four years of college 0.115 42.013 0.199 2.873 * *
Graduate Studies 0.103 37.564 0.299 3.310 * *
Value of Types
Type 1 8.624 1170.764 0.052 20.221 0.692 64.673
Type 2 8.543 948.864 0.123 19.529 0.693 84.524
Table 4b
Coeff. t Coeff. t Coeff. t
Standard deviation of η * * 0.0046 3.0855 * *
Family Background
Father went to College 1.063 3.325 -0.017 -3.316 -0.506 -5.358
Mother went to College 1.099 2.661 0.005 0.745 -0.584 -4.780
Father is a Professionnal * * * * * *
Mother is a Professionnal * * * * * *
Density of Population 0.167 5.388 * * * *
Paris Area 0.186 3.808 * * * *
Unemployment Rate -0.080 -2.898 * * * *
Age at grade 6
10 years old * * -0.023 -9.918 * *
11 years old (ref.) * * * * * *
12 years old * * 0.048 23.102 * *
13 years old * * 0.075 22.722 * *
14 years old or more * * 0.107 13.251 * *
School Density and Distance to College
Stock of vocational and technical high schools 0.245 8.265 * * * *
Distance to college 2d quartile -0.081 -2.019 * * * *
Distance to college 3d quartile -0.122 -2.583 * * * *
Distance to college 4th quartile -0.069 -1.489 * * * *
Month of birth * * * * 0.070 5.317
Education costs (cuts)
C1 * * * * * *
C2 0.674 3.120 * * * *
C3 1.315 5.995 * * * *
C4 1.989 6.759 * * * *
C5 2.330 7.195 * * * *
Value of Types
Type 1 5.387 27.594 0.231 100.914 0.228 5.956
Type 2 5.187 21.419 0.100 62.705 0.511 27.682
Distribution of Types
Prob(type=h1) 0.2578 26.5375
Number of observations
Log likelihood
Vuong Test of H0="No Unobserved Heterogeneity" (P-value)
9995
-0.923022
<.0001
E(ln(W)) V(ln(W))
Table 4 : Subsample Estimation Results. Rest of the Sample
Risk Aversion
Education (S) Delay (d/ז) Age at Grade 6
Estimated Observed Simulated Estimated Observed Simulated
Elasticity Value 1%-impact Elasticity Value 1%-impact
High school dropouts - - - - - -
Vocational degree -2.98 0.95 0.92 -4.50 0.84 0.82
High school graduates -15.12 0.76 0.68 -17.01 0.42 0.36
Two years of college -23.74 0.60 0.51 -24.91 0.27 0.21
Four years of college -29.47 0.41 0.32 -35.74 0.11 0.07
Graduate Studies -28.46 0.30 0.24 -41.56 0.06 0.04
High school dropouts - - - - - -
Vocational degree -0.58 0.95 0.95 -1.39 0.84 0.84
High school graduates -3.04 0.76 0.75 -6.96 0.42 0.41
Two years of college -5.80 0.60 0.59 -10.57 0.27 0.25
Four years of college -8.39 0.41 0.39 -16.46 0.11 0.10
Graduate Studies -10.17 0.30 0.28 -19.83 0.06 0.06
High school dropouts - - - - - -
Vocational degree -0.06 0.95 0.95 -0.10 0.84 0.84
High school graduates 0.18 0.76 0.76 0.09 0.42 0.42
Two years of college 0.06 0.60 0.60 -0.11 0.27 0.27
Four years of college 0.37 0.41 0.41 0.37 0.11 0.11
Graduate Studies 0.14 0.30 0.30 0.85 0.06 0.06
High school dropouts - - - - - -
Vocational degree 1.60 0.95 0.96 3.56 0.84 0.86
High school graduates 6.71 0.76 0.79 12.77 0.42 0.46
Two years of college 11.39 0.60 0.64 18.91 0.27 0.31
Four years of college 14.12 0.41 0.44 25.81 0.11 0.14
Graduate Studies 15.22 0.30 0.33 28.92 0.06 0.08
High school dropouts - - - - - -
Vocational degree -2.19 0.95 0.93 -4.92 0.84 0.82
High school graduates -10.15 0.76 0.72 -18.84 0.42 0.36
Two years of college -17.60 0.60 0.55 -28.88 0.27 0.21
Four years of college -23.18 0.41 0.35 -42.05 0.11 0.07
Graduate Studies -25.54 0.30 0.25 -49.85 0.06 0.04
Table 5: Elasticity of Survival Functions Ψs with respect to Various Parameters
Elasticity w.r.t Variances of Wages
Elasticity w.r.t Returns to Education
Elasticity w.r.t Probability of Grade Repetition
Sons of Professionnals Other Students
Elasticity w.r.t Relative Risk Aversion
Elasticity w.r.t Costs fo Education
From level: 1 2 3 4 5 6
Rest of
sample,
simulated
Rest of
sample,
observed
SOPs**
observed
To level: 1 50.13% 0.00% 0.00% 0.00% 0.00% 0.00% 8.17% 16.29% 5.18%
2 49.87% 51.80% 0.00% 0.00% 0.00% 0.00% 29.19% 40.71% 18.75%
3 0.00% 34.56% 0.26% 0.00% 0.00% 0.00% 14.11% 15.16% 15.77%
4 0.00% 13.63% 35.88% 0.67% 0.00% 0.00% 11.11% 16.51% 19.52%
5 0.00% 0.00% 39.48% 7.16% 0.00% 0.00% 7.16% 4.67% 10.67%
6 0.00% 0.00% 24.38% 92.17% 100.00% 100.00% 30.26% 6.66% 30.11%
Total 100% 100% 100% 100% 100% 100% 100% 100% 100%
Table 6: Transition towards upper schooling levels following an increase in speed, in the rest of the sample
Matrix of simulated transitions Distributions of schooling levels
* Note: the increase in speed is proportional, so that the average speed in the rest of the sample becomes equal to that of the sons of professionals.
**Sons of professionals.
8,5
8,7
8,9
9,1
9,3
9,5
0,44 0,49 0,54 0,59 0,64R
etu
rns
Risks
Figure 4: Risk-Return curves: comparison of sub-samples
Sons of Professionnals Rest of Sample
8,4
8,6
8,8
9
9,2
9,4
9,6
0,35 0,45 0,55 0,65 0,75
Re
turn
s
Risks
Figure 1: Risk-Return curves: whole sample
Type 1 Type 2
8,4
8,6
8,8
9
9,2
9,4
9,6
0,4 0,45 0,5 0,55 0,6 0,65 0,7
Re
turn
s
Risks
Figure 2: Risk-Return curves: sons of professionnals
Type 1 Type 2
8,5
8,7
8,9
9,1
9,3
9,5
0,4 0,45 0,5 0,55 0,6 0,65 0,7
Retu
rns
Risks
Figure 3: Risk-Return curves, rest of sample
Type 1 Type 2
0
0,00005
0,0001
0,00015
0,0002
0,00025
0,0003
0,00035
0 500 1000 1500 2000 2500 3000 3500 4000 4500
den
sit
y
monthly wages (euros)
Figure 5: Wage distributions: Rest of Sample
High school dropout: mixture
Type 1
Type 2
Graduate studies: mixture
Type 1
Type 2
0
0,00005
0,0001
0,00015
0,0002
0,00025
0,0003
0,00035
0 500 1000 1500 2000 2500 3000 3500 4000 4500
de
ns
ity
monthly wages (euros)
Figure 6: Wage distributions: sons of professionnals
High school dropouts: mixture
Type 1
Type 2
Graduate studies: mixture
Type 1
Type 2
Appendix Table A1: Empirical Distribution of Male School-Leaving Age, Conditional on Education Level Age while leaving school 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32High school dropouts 0.01 0.17 0.24 0.33 0.16 0.07 0.01 0.01 0 0 0 0 0 0 0 0 0 0Vocational degree 0 0 0.03 0.30 0.37 0.21 0.06 0.02 0.01 0 0 0 0 0 0 0 0 0High school graduates (grade 12) 0 0 0 0.02 0.12 0.31 0.32 0.15 0.06 0.02 0.01 0 0 0 0 0 0 0Two years of college (grade 14) 0 0 0 0 0 0.13 0.27 0.28 0.19 0.08 0.02 0.01 0.01 0 0 0 0 0Four years of college (grade 16) 0 0 0 0 0 0 0.04 0.17 0.20 0.27 0.13 0.09 0.04 0.03 0.01 0.01 0.01 0.01Graduate studies 0 0 0 0 0 0 0 0.04 0.24 0.29 0.15 0.11 0.07 0.04 0.02 0.02 0.01 0
Figure A1: Wage Distributions (in Euros)
0.000%
0.005%
0.010%
0.015%
0.020%
0.025%
0 500 1000 1500 2000
dens
ity fu
nctio
n
earnings mean wage
Figure A2 : Distribution of Delay (d/t)
0
5
10
15
20
25
d/
Perc
enta
ge (%
)
Parents are Professionals Rest of Sample
Figure A3: Historical Growth of Vocational Secondary Education
0
1000
2000
3000
4000
5000
6000
1950
1960
1970
1980
1990
2000
Year
Voc
atio
nal H
igh
Scho
ol S
tock
in u
nits
012345678
Stock divided by 15-year-olds in thousands
Stock Stock per capita of 15-year-olds
Figure A4: Distribution of Stock of Vocational High Schools 1982
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0.5 0.75 1 1.25 1.5 1.75 2
County-level stock per capita of 15-19 year-olds (in thousands)
Den
sity
0
10
20
30
40
50
60
70
80
≤10 11 12 13 ≥14
Perc
enta
ge
Age at Grade 6 Entry
Figure A5 : Distribution of Age at Grade 6 Entry
Sons of professionnals
Rest of the sample
TABLE A2: Chosen Education Levels and Speed of Completion: Predicted and Observed
Chosen Education Levels Observed Predicted Observed Predicted Observed Predicted
High school dropouts 14.20% 13.66% 5.18% 5.08% 16.29% 15.63%
Vocational degree 36.58% 37.23% 18.75% 18.60% 40.71% 42.27%
High school graduates 15.27% 15.71% 15.77% 16.02% 15.16% 15.24%
Two years of college 17.08% 16.97% 19.52% 19.72% 16.51% 16.13%
Four years of college 5.80% 5.67% 10.67% 10.66% 4.67% 4.47%
Graduate Studies 11.07% 10.77% 30.11% 29.92% 6.66% 6.27%
Proba of entering grade 6
before 11 71.29% 71.24% 87.29% 87.30% 67.55% 67.52%
Overall proba of annual
grade completion 86.28% 86.10% 89.03% 88.85% 85.67% 85.49%
TABLE A3: Mean log wages: Predicted and Observed
Chosen Education Levels Observed Predicted Observed Predicted Observed Predicted
High school dropouts 8.652 8.566 8.717 8.507 8.647 8.553
Vocational degree 8.736 8.738 8.773 8.744 8.732 8.736
High school graduates 8.833 8.875 8.868 8.883 8.825 8.863
Two years of college 8.977 9.037 9.022 9.079 8.965 9.010
Four years of college 9.102 9.168 9.144 9.240 9.079 9.133
Graduate Studies 9.363 9.304 9.406 9.383 9.319 9.244
All levels 8.871 8.868 9.064 9.059 8.826 8.818
TABLE A4: Standard Deviation of Log Wages: Predicted and Observed
Chosen Education Levels Observed Predicted Observed Predicted Observed Predicted
High school dropouts 0.289 0.200 0.360 0.309 0.283 0.320
Vocational degree 0.228 0.217 0.251 0.216 0.225 0.229
High school graduates 0.272 0.270 0.297 0.272 0.265 0.244
Two years of college 0.256 0.309 0.262 0.285 0.253 0.235
Four years of college 0.284 0.358 0.304 0.337 0.270 0.255
Graduate Studies 0.316 0.403 0.314 0.354 0.312 0.297
All levels 0.339 0.325 0.387 0.369 0.310 0.292
TABLE A5: Mean Monthly Wages (Euros): Predicted and Observed
Chosen Education Levels Observed Predicted Observed Predicted Observed Predicted
High school dropouts 905 821 997 801 898 837
Vocational degree 974 979 1016 982 969 976
High school graduates 1086 1143 1133 1149 1075 1113
Two years of college 1249 1369 1307 1405 1233 1286
Four years of college 1424 1603 1494 1687 1387 1462
Graduate Studies 1865 1892 1950 1960 1776 1652
All levels 1153 1153 1423 1419 1090 1082
TABLE A6: Standard Deviation of Monthly Wages (Euros): Predicted and Observed
Chosen Education Levels Observed Predicted Observed Predicted Observed Predicted
High school dropouts 258 169 466 254 235 274
Vocational degree 238 220 269 218 234 225
High school graduates 324 324 374 325 310 276
Two years of college 343 452 356 416 338 305
Four years of college 426 624 483 601 389 380
Graduate Studies 615 846 684 735 518 492
All levels 449 435 614 563 375 336
Parental Occupation
Whole sample Professionnals Others
Whole sample Professionnals
Parental Occupation
Others
Parental Occupation
Whole sample Professionnals Others
Parental Occupation
Whole sample Professionnals Others
Parental Occupation
Whole sample Professionnals Others
Discount Factor (d) 0.9950 1. 1. 1.
Estimated Mean Risk Aversion:
0.7458 0.7782 0.6967 0.6955
(48.43) (54.93) (63.92) (63.98)
0.7672 0.7969 0.6968 0.6957
(66.71) (75.95) (84.97) (84.69)
Prob(type=1) 0.2934 0.2933 0.2618 0.2618
(20.55) (17.37) (38.94) (26.31)
Mean Log-Likelihood -0.8825140 -0.8825490 -0.9307170 -0.9307340
Vuong Test of Discounted Utility
Model (p-value)
Student ts are in parentheses
Type 2
Table A7 : Estimation of Discount FactorSons of Professionals Other Students
Type 1
0.55946 0.99997
Discounted
Utility Model
Discounted
Utility Model
Benchmark
Model
Benchmark
Model
SUPPLEMENTARY MATERIAL
Do risk aversion and wages explain educational choices?
T.Brodaty, R. Gary-Bobo and A. Prieto
June 2013
Table X1a
Coeff. t Coeff. t Coeff. t
Family Background
Father went to College 0.047 4.223 0.221 3.934 0.013 2.340
Mother went to College 0.031 2.382 0.034 0.714 0.010 1.845
Father is a Professionnal 0.026 3.087 0.097 2.687 0.006 1.480
Mother is a Professionnal 0.025 2.167 0.058 1.213 -0.004 -0.799
Density of Population 0.007 1.342 * * * *
Paris Area 0.070 8.461 * * * *
Unemployment Rate -0.019 -3.899 * * * *
Age at grade 6
10 years old 0.024 2.056 * * * *
11 years old (ref.) * * * * * *
12 years old -0.013 -2.206 * * * *
13 years old -0.035 -3.270 * * * *
14 years old or more -0.022 -0.746 * * * *
Completed Education
High school dropouts (Ref.)
Vocational degree 0.139 58.253 -0.386 -15.009 * *
High school graduates (grade 12) 0.113 58.271 0.386 7.126 * *
Two years of college (grade 14) 0.139 61.299 -0.131 -3.648 * *
Four years of college (grade 16) 0.114 51.087 0.271 5.454 * *
Graduate Studies 0.100 47.336 0.184 3.910 * *
Constant (no unobserved heterogeneity)
Constant 8.602 1463.528 0.082 27.778 0.685 99.217
Table X1b
Coeff. t Coeff. t Coeff. t
Standard deviation of η * * 0.0078 6.9077 * *
Family Background
Father went to College 0.507 3.669 -0.010 -2.875 -0.326 -5.461
Mother went to College 0.181 1.249 -0.014 -3.451 -0.347 -4.666
Father is a Professionnal 0.290 2.768 -0.015 -5.609 -0.435 -9.718
Mother is a Professionnal -0.048 -0.333 -0.006 -1.654 -0.390 -5.760
Density of Population 0.121 5.142 * * * *
Paris Area 0.181 5.181 * * * *
Unemployment Rate -0.077 -3.678 * * * *
Age at grade 6
10 years old * * -0.028 -14.267 * *
11 years old (ref.) * * * * * *
12 years old * * 0.061 34.354 * *
13 years old * * 0.092 28.905 * *
14 years old or more * * 0.122 15.050 * *
School Density and Distance to College
Stock of vocational and technical high schools 0.183 8.383 * * * *
Distance to college 2d quartile -0.054 -1.853 * * * *
Distance to college 3d quartile -0.097 -2.781 * * * *
Distance to college 4th quartile -0.018 -0.518 * * * *
Month of birth * * * * 0.076 6.185
Education costs
C1 * * * * * *
C2 2.219 21.359 * * * *
C3 2.750 23.093 * * * *
C4 3.488 22.818 * * * *
C5 3.550 21.425 * * * *
Constant (no unobserved heterogeneity)
Constant 3.075 26.876 0.130 124.775 0.444 33.986
Number of observations
Log likelihood -1.1086
E(ln(W)) V(ln(W))
Table X1 : Estimation of the model with no unobserved heterogeneity
Risk Aversion
Education (S) Delay (d/ז) Age at Grade 6
12310
Table X2a
Coeff. t Coeff. t Coeff. t
Family Background
Father went to College 0.028 2.550 -0.047 -1.124 0.009 2.458
Mother went to College 0.025 1.982 -0.071 -1.674 0.000 -0.049
Father is a Professionnal 0.015 1.817 -0.089 -3.229 -0.002 -0.879
Mother is a Professionnal 0.014 1.216 -0.030 -0.680 0.007 1.604
Density of Population 0.000 -0.088 * * * *
Paris Area 0.066 8.431 * * * *
Unemployment Rate -0.019 -4.162 * * * *
Age at grade 6
10 years old 0.021 1.777 * * * *
11 years old (ref.) * * * * * *
12 years old -0.009 -1.528 * * * *
13 years old -0.004 -0.411 * * * *
14 years old or more 0.032 1.521 * * * *
Completed Education
High school dropouts (Ref.)
Vocational degree 0.204 36.145 1.703 15.806 * *
High school graduates 0.149 40.089 1.029 22.010 * *
Two years of college 0.175 40.307 0.663 32.264 * *
Four years of college 0.140 38.743 0.425 44.988 * *
Graduate Studies 0.123 37.598 0.286 53.855 * *
Value of Types
type 1 8.656 1006.717 0.092 29.815 0.733 109.358
type 2 8.444 622.365 0.006 22.471 0.755 118.031
type 3 8.543 1132.436 0.014 21.478 0.743 109.265
Table X2b
Coeff. t Coeff. t Coeff. t
Standard deviation of η * * 0.004 3.523 * *
Family Background
Father went to College 0.958 4.212 -0.012 -4.035 -0.342 -5.410
Mother went to College -0.139 -0.518 -0.014 -3.876 -0.349 -4.453
Father is a Professionnal 0.194 1.111 -0.011 -4.755 -0.418 -8.801
Mother is a Professionnal 0.683 2.332 -0.006 -1.831 -0.396 -5.515
Density of Population 0.251 7.347 * * * *
Paris Area 0.246 5.077 * * * *
Unemployment Rate -0.054 -1.802 * * * *
Age at grade 6
10 years old * * -0.020 -10.078 * *
11 years old (ref.) * * * * * *
12 years old * * 0.024 9.279 * *
13 years old * * 0.052 11.981 * *
14 years old or more * * 0.074 9.459 * *
School Density and Distance to College
Stock of vocational and technical high schools 0.267 8.327 * * * *
Distance to college 2d quartile -0.119 -2.917 * * * *
Distance to college 3d quartile -0.197 -3.945 * * * *
Distance to college 4th quartile -0.080 -1.632 * * * *
Month of birth * * * * 0.079 6.209
Education costs (cuts)
C1 * * * * * *
C2 1.586 4.827 * * * *
C3 2.519 7.280 * * * *
C4 2.372 6.409 * * * *
C5 2.629 6.404 * * * *
Value of Types
type 1 8.808 26.639 0.258 124.316 0.017 0.471
type 2 6.078 15.270 0.073 50.078 0.974 15.422
type 3 6.616 17.601 0.152 97.520 0.263 10.335
Distribution of types
Prob(type 1) 0.164 32.328
Prob(type 2) 0.378 33.520
Number of observations
Log likelihood
Vuong Test of H0="No Unobserved Heterogeneity" (P-value)
Vuong Test of H0="Unobserved Heterogeneity: Two Types" (P-value)
Delay (d/ז) Age at Grade 6
12310
-0.850357
Table X2 : Estimation of a model with 3 types on the full sample
<.0001
E(ln(W)) V(ln(W))
<.0001
Risk Aversion
Education (S)
Table X3: Estimation results (Whole sample). Limited role of unobserved heterogeneity
Table X3a
Coeff. t Coeff. t Coeff. t
Family Background
Father went to College 0.032 2.734 0.273 6.364 0.017 3.377
Mother went to College 0.027 1.944 -0.013 -0.269 -0.003 -0.623
Father is a Professionnal 0.010 1.129 0.090 2.674 -0.007 -1.924
Mother is a Professionnal 0.016 1.250 0.118 2.406 0.005 1.116
Density of Population -0.001 -0.202 * * * *
Paris Area 0.064 7.677 * * * *
Unemployment Rate -0.016 -3.231 * * * *
Age at grade 6
10 years old 0.003 0.238 * * * *
11 years old (ref.) * * * * * *
12 years old 0.021 3.047 * * * *
13 years old 0.007 0.538 * * * *
14 years old or more 0.021 0.984 * * * *
Completed Education
High school dropouts (Ref.)
Vocational degree 0.185 43.804 -0.395 -38.449 * *
High school graduates 0.141 43.055 0.516 14.125 * *
Two years of college 0.167 47.847 -0.126 -4.351 * *
Four years of college 0.137 44.939 0.256 4.848 * *
Graduate Studies 0.120 43.777 0.147 3.219 * *
Value of Types
Type 1 8.676 761.906
Type 2 8.498 759.570
Table X3b
Coeff. t Coeff. t Coeff. t
Standard deviation of η * * 0.0078 7.1749 * *
Family Background
Father went to College 0.909 5.261 -0.008 -1.855 -0.326 -5.309
Mother went to College -0.264 -1.293 -0.015 -3.115 -0.347 -4.629
Father is a Professionnal -0.083 -0.598 -0.015 -4.682 -0.435 -9.326
Mother is a Professionnal 0.333 1.817 -0.005 -1.148 -0.390 -5.651
Density of Population 0.160 5.393 * * * *
Paris Area 0.230 5.319 * * * *
Unemployment Rate -0.109 -4.133 * * * *
Age at grade 6
10 years old * * -0.031 -15.651 * *
11 years old (ref.) * * * * * *
12 years old * * 0.060 30.156 * *
13 years old * * 0.091 25.433 * *
14 years old or more * * 0.122 14.904 * *
School Density and Distance to College
Stock of vocational and technical high schools 0.244 8.628 * * * *
Distance to college 2d quartile -0.069 -1.967 * * * *
Distance to college 3d quartile -0.124 -2.862 * * * *
Distance to college 4th quartile -0.055 -1.295 * * * *
Month of birth * * * * 0.076 6.129
Education costs (cuts)
C1 * * * * * *
C2 1.915 11.792 * * * *
C3 2.263 12.897 * * * *
C4 2.841 14.178 * * * *
C5 2.921 13.659 * * * *
Value of Types
Type 1 6.414 23.168
Type 2 4.230 20.960
Distribution of Types
Prob(type=1) 0.2084 12.0539
Number of observations
Log likelihood
12310
-1.10204
E(ln(W)) V(ln(W)) Risk Aversion
Education (S) Delay (d/ז) Age at Grade 6
0.130 101.726 0.444 33.838
0.075 69.734 0.747 99.298
Table X4a
Coeff. t Coeff. t Coeff. t
Family Background
Father went to College 0.040 3.559 0.080 2.000 0.017 3.118
Mother went to College 0.033 2.587 -0.026 -0.519 0.007 1.418
Father is a Professionnal 0.021 2.551 -0.017 -0.526 0.005 1.795
Mother is a Professionnal 0.019 1.665 -0.036 -0.786 -0.001 -0.174
Density of Population 0.002 0.451 * * * *
Paris Area 0.068 8.704 * * * *
Unemployment Rate -0.018 -3.834 * * * *
Age at grade 6
10 years old 0.025 2.047 * * * *
11 years old (ref.) * * * * * *
12 years old 0.002 0.370 * * * *
13 years old -0.015 -1.645 * * * *
14 years old or more -0.004 -0.150 * * * *
Completed Education
High school dropouts (Ref.)
Vocational degree 0.161 59.601 0.159 5.680 * *
High school graduates 0.126 54.597 0.548 16.528 * *
Two years of college 0.149 54.457 0.308 13.673 * *
Four years of college 0.123 52.609 0.354 20.638 * *
Graduate Studies 0.108 50.607 0.264 19.377 * *
Value of Types
Type 1 8.629 1241.318 0.087 77.119 0.655 92.239
Type 2 8.550 1177.000 0.025 29.729 0.722 109.394
Table X4b
Coeff. t Coeff. t Coeff. t
Standard deviation of η * * 0.0045 4.9676 * *
Family Background
Father went to College 0.969 5.418 -0.009 -2.390 -0.326 -5.465
Mother went to College 0.197 0.943 -0.012 -2.606 -0.345 -4.638
Father is a Professionnal 0.435 3.467 -0.013 -5.298 -0.433 -9.680
Mother is a Professionnal 0.057 0.169 -0.007 -2.029 -0.391 -5.774
Density of Population 0.212 7.538 * * * *
Paris Area 0.103 2.549 * * * *
Unemployment Rate -0.058 -2.295 * * * *
Age at grade 6
10 years old * * -0.023 -12.796 * *
11 years old (ref.) * * * * * *
12 years old * * 0.058 29.360 * *
13 years old * * 0.085 26.037 * *
14 years old or more * * 0.086 8.842 * *
School Density and Distance to College
Stock of vocational and technical high schools * * * * * *
Distance to college 2d quartile -0.123 -3.560 * * * *
Distance to college 3d quartile -0.182 -4.344 * * * *
Distance to college 4th quartile -0.070 -1.688 * * * *
Month of birth * * * * 0.075 6.172
Education costs (cuts)
C1 * * * * * *
C2 2.473 14.664 * * * *
C3 3.204 15.583 * * * *
C4 3.834 16.417 * * * *
C5 4.053 14.956 * * * *
Value of Types
Type 1 5.657 35.734 0.218 114.211 0.373 11.971
Type 2 3.279 17.217 0.093 70.233 0.475 26.407
Distribution of Types
Prob(type=1) 0.2957 48.077
Number of observations
Log likelihood
Table X4: Estimation results (Whole sample). Without the school-density instrument.
12310
-0.912281
E(ln(W)) V(ln(W)) Risk Aversion
Education (S) Delay (d/ז) Age at Grade 6
Table X5a
Coeff. t Coeff. t Coeff. t
Family Background
Father went to College 0.040 3.728 0.080 1.997 0.017 4.833
Mother went to College 0.033 2.678 -0.019 -0.372 0.006 1.786
Father is a Professionnal 0.020 2.506 -0.018 -0.535 0.006 2.497
Mother is a Professionnal 0.019 1.706 -0.043 -0.935 -0.002 -0.513
Density of Population 0.002 0.469 * * * *
Paris Area 0.068 8.799 * * * *
Unemployment Rate -0.018 -3.879 * * * *
Age at grade 6
10 years old 0.025 1.973 * * * *
11 years old (ref.) * * * * * *
12 years old 0.002 0.291 * * * *
13 years old -0.015 -1.650 * * * *
14 years old or more -0.005 -0.237 * * * *
Completed Education
High school dropouts (Ref.)
Vocational degree 0.161 59.595 0.167 5.852 * *
High school graduates 0.125 54.658 0.549 16.600 * *
Two years of college 0.149 54.519 0.309 13.837 * *
Four years of college 0.123 52.685 0.350 20.674 * *
Graduate Studies 0.108 50.744 0.261 19.384 * *
Value of Types
Type 1 8.630 1163.109 0.087 76.766 0.654 100.053
Type 2 8.551 1229.695 0.025 29.682 0.721 110.219
Table X5b
Coeff. t Coeff. t Coeff. t
Standard deviation of η * * 0.0045 4.9699 * *
Family Background
Father went to College 0.985 7.068 -0.009 -2.896 -0.326 -5.300
Mother went to College 0.172 1.137 -0.012 -3.394 -0.345 -4.599
Father is a Professionnal 0.464 4.468 -0.013 -5.582 -0.434 -9.276
Mother is a Professionnal 0.054 0.395 -0.007 -2.145 -0.391 -5.677
Density of Population 0.254 10.301 * * * *
Paris Area 0.160 4.079 * * * *
Unemployment Rate -0.073 -2.882 * * * *
Age at grade 6
10 years old * * -0.023 -15.352 * *
11 years old (ref.) * * * * * *
12 years old * * 0.057 34.108 * *
13 years old * * 0.085 26.195 * *
14 years old or more * * 0.086 9.404 * *
School Density and Distance to College
Stock of vocational and technical high schools 0.238 8.681 * * * *
Distance to college 2d quartile * * * * * *
Distance to college 3d quartile * * * * * *
Distance to college 4th quartile * * * * * *
Month of birth * * * * 0.075 6.104
Education costs (cuts)
C1 * * * * * *
C2 2.472 18.758 * * * *
C3 3.212 21.760 * * * *
C4 3.859 22.984 * * * *
C5 4.089 22.132 * * * *
Value of Types
Type 1 5.955 39.715 0.219 134.745 0.367 12.827
Type 2 3.562 23.534 0.094 84.273 0.478 28.084
Distribution of Types
Prob(type=1) 0.2924 47.7658
Number of observations
Log likelihood
Table X5: Estimation results (Whole sample). Without distance to college
12310
-0.909925
E(ln(W)) V(ln(W)) Risk Aversion
Education (S) Delay (d/ז) Age at Grade 6
Table X6a
Coeff. t Coeff. t Coeff. t
Family Background
Father went to College 0.035 3.084 0.065 1.570 0.018 4.706
Mother went to College 0.036 2.793 -0.051 -0.958 0.002 0.461
Father is a Professionnal 0.022 2.497 -0.010 -0.282 0.007 2.628
Mother is a Professionnal 0.020 1.672 -0.039 -0.777 -0.002 -0.641
Density of Population 0.003 0.637 * * * *
Paris Area 0.070 8.712 * * * *
Unemployment Rate -0.014 -2.978 * * * *
Age at grade 6
10 years old 0.021 1.610 * * * *
11 years old (ref.) * * * * * *
12 years old 0.002 0.285 * * * *
13 years old -0.018 -1.894 * * * *
14 years old or more 0.003 0.139 * * * *
Completed Education
High school dropouts (Ref.)
Vocational degree 0.160 55.236 0.274 8.134 * *
High school graduates (grade 12) 0.124 51.217 0.573 16.182 * *
Two years of college (grade 14) 0.148 51.264 0.316 13.194 * *
Four years of college (grade 16) 0.122 49.415 0.354 19.744 * *
Graduate Studies 0.106 47.344 0.262 17.943 * *
Value of Types
Type 1 8.633 1106.465 0.084 74.479 0.654 94.812
Type 2 8.552 1189.283 0.023 27.987 0.719 102.771
Table X6b
Coeff. t Coeff. t Coeff. t
Standard deviation of η * * 0.0046 4.8113 * *
Family Background
Father went to College 0.951 6.835 -0.011 -3.262 -0.303 -4.607
Mother went to College -0.076 -0.508 -0.013 -3.281 -0.322 -4.036
Father is a Professionnal 0.495 4.801 -0.013 -4.963 -0.435 -8.795
Mother is a Professionnal 0.088 0.614 -0.005 -1.381 -0.368 -5.085
Density of Population 0.210 7.294 * * * *
Paris Area 0.165 3.920 * * * *
Unemployment Rate -0.083 -3.073 * * * *
Age at grade 6
10 years old * * -0.025 -15.286 * *
11 years old (ref.) * * * * * *
12 years old * * 0.057 32.137 * *
13 years old * * 0.083 24.466 * *
14 years old or more * * 0.086 8.636 * *
School Density and Distance to College
Stock of vocational and technical high schools 0.230 7.927 * * * *
Distance to college 2d quartile -0.111 -2.649 * * * *
Distance to college 3d quartile -0.158 -3.203 * * * *
Distance to college 4th quartile -0.067 -1.358 * * * *
Month of birth * * * * 0.079 6.061
Education costs (cuts)
C1 * * * * * *
C2 2.496 17.826 * * * *
C3 3.202 20.395 * * * *
C4 3.784 20.925 * * * *
C5 3.958 20.006 * * * *
Value of Types
Type 1 5.849 35.017 0.220 128.856 0.373 12.262
Type 2 3.390 20.318 0.095 82.506 0.484 27.540
Distribution of Types
Prob(type=1) 0.2825 44.6083
Number of observations
Log likelihood
Age at Grade 6
Table X6 : Estimation results; sub-sample of communes with a small number of students in the sample
11107
-0.902332
E(ln(W)) V(ln(W)) Risk Aversion
Education (S) Delay (d/ז)
Table X7a
Coeff. t Coeff. t Coeff. t
Family Background
Father went to College 0.045 4.071 0.084 1.375 0.018 3.874
Mother went to College 0.036 2.846 -0.037 -0.472 0.006 1.210
Father is a Professionnal 0.026 3.219 -0.024 -0.801 0.005 1.615
Mother is a Professionnal 0.021 1.871 -0.017 -0.352 0.000 -0.043
Density of Population * * * * * *
Paris Area * * * * * *
Unemployment Rate * * * * * *
Age at grade 6
10 years old 0.028 2.296 * * * *
11 years old (ref.) * * * * * *
12 years old 0.001 0.195 * * * *
13 years old -0.015 -1.583 * * * *
14 years old or more -0.002 -0.084 * * * *
Completed Education
High school dropouts (Ref.)
Vocational degree 0.162 57.361 0.161 2.646 * *
High school graduates 0.127 53.989 0.539 13.610 * *
Two years of college 0.150 52.247 0.309 12.641 * *
Four years of college 0.124 47.364 0.356 17.211 * *
Graduate Studies 0.109 43.412 0.262 15.391 * *
Value of Types
Type 1 8.625 1384.986 0.088 27.651 0.657 97.601
Type 2 8.547 1271.287 0.026 16.494 0.725 112.423
Table X7b
Coeff. t Coeff. t Coeff. t
Standard deviation of η * * 0.0045 3.9983 * *
Family Background
Father went to College 1.029 6.174 -0.009 -2.890 -0.326 -5.468
Mother went to College 0.161 0.863 -0.012 -3.242 -0.345 -4.634
Father is a Professionnal 0.437 3.609 -0.013 -5.597 -0.434 -9.681
Mother is a Professionnal 0.110 0.590 -0.007 -2.146 -0.391 -5.776
Density of Population * * * * * *
Paris Area * * * * * *
Unemployment Rate * * * * * *
Age at grade 6
10 years old * * -0.023 -12.839 * *
11 years old (ref.) * * * * * *
12 years old * * 0.058 29.962 * *
13 years old * * 0.085 25.921 * *
14 years old or more * * 0.085 8.849 * *
School Density and Distance to College
Stock of vocational and technical high schools 0.181 7.067 * * * *
Distance to college 2d quartile -0.208 -6.538 * * * *
Distance to college 3d quartile -0.336 -9.116 * * * *
Distance to college 4th quartile -0.226 -6.172 * * * *
Month of birth * * * * 0.075 6.168
Education costs
C1 * * * * * *
C2 2.489 15.189 * * * *
C3 3.213 17.158 * * * *
C4 3.837 17.762 * * * *
C5 4.068 16.142 * * * *
Value of Types
Type 1 5.644 37.982 0.219 114.371 0.369 11.815
Type 2 3.246 18.478 0.094 70.700 0.477 26.595
Distribution of Types
Prob(type=1) 0.294 38.3248
Number of observations
Log likelihood
12310
-0.919831
E(ln(W)) V(ln(W)) Risk Aversion
Education (S) Delay (d/ז)
Table X7: Estimation results (Whole sample). Without controls for geography
Age at Grade 6
