The full text on this page is automatically extracted from the file linked above and may contain errors and inconsistencies.
Working Paper Series
On the Distribution of College Dropouts:
Wealth and Uninsurable Idiosyncratic
Risk
WP 15-15
Ali K. Ozdagli
Federal Reserve Bank of Boston
Nicholas Trachter
Federal Reserve Bank of Richmond
This paper can be downloaded without charge from:
http://www.richmondfed.org/publications/
On the Distribution of College Dropouts: Wealth and
Uninsurable Idiosyncratic Risk∗
Ali K. Ozdagli
Nicholas Trachter†
Federal Reserve Bank of Boston
Ali.Ozdagli@bos.frb.org
Federal Reserve Bank of Richmond
Nicholas.Trachter@rich.frb.org
November 19, 2015
Working Paper No. 15-15
Abstract
We present a dynamic model of the decision to pursue a college
degree in which students face uncertainty about their future income
stream after graduation due to unobserved heterogeneity in their innate
scholastic ability. After matriculating and taking some exams, students
re-evaluate their expectations about succeeding in college and may decide to drop out and start working. The model shows that, in accordance with the data, poorer students are less likely to graduate and are
likely to drop out sooner than wealthier students. Our model generates
these results without introducing explicit credit constraints.
∗
We thank Christopher Foote, Alex Levkov, Andrea Pozzi, and Rob Shimer. We also benefited
from comments by seminar participants at the Federal Reserve Bank of Boston, EIEF macro reading
group, and the Midwest Macroeconomics Meetings. The views expressed herein are those of the
authors and not necessarily those of the Federal Reserve Bank of Boston, the Federal Reserve Bank
of Richmond, or the Federal Reserve System. Sarojini Rao provided excellent research assistance.
All mistakes are our own.
†
Corresponding Author.
1
Introduction
A large fraction of every cohort that enrolls in four-year U.S. colleges drops out
before graduating and there is a higher concentration of dropouts among the students from lower-income families. We also observe that students from low-income
families tend to drop out earlier than students from high-income families. Given the
high return to graduation, the skewed distribution of dropouts generates a channel
that perpetuates and exacerbates income inequality.
We present a dynamic model of educational choice to explore the relationship
between household wealth and college dropout behavior. In the model, the differences in the students’ unobserved innate scholastic ability and the families’ initial
wealth levels are the driving force behind the high and skewed dropout rate among
low-income students. At every period in college, risk-averse students take an exam,
the outcome of which provides both human capital and information that can be used
to update students’ beliefs about their ability level. Given the outcome of the exams and their income level, the students decide optimally if and when to drop out.
Therefore, a student’s optimal dropout behavior is characterized by the distance
between her belief about her ability and a belief threshold at which the student
drops out. We show that this threshold depends endogenously on the wealth level
of the student’s family, therefore providing the link between household wealth and
dropout behavior.
Our model incorporates the ideas that investing in a college education is risky
because the outcome is uncertain,1 and that wealthier students are less risk-averse.2
This framework, combined with the learning mechanism, generates the result that
poor students are less willing to take the risk associated with the uncertain outcome
of college education and that they do not want to continue their education for as
long as the rich students in order to learn their ability. Therefore, poor students
are less likely to graduate and tend to drop out earlier compared with wealthier
students. Our model generates this result without relying on financial constraints
and is similar to Athreya and Eberly (2013) in this respect but differs from their
paper in our focus on the relationship between wealth and college dropout behavior.
In order to model the evolution of beliefs about innate ability and the decision to
1
Chen (2008) finds that college investment is indeed risky after correcting for selection bias and
accounting for permanent and transitory earnings risks.
2
In our benchmark model, wealthier students are less risk-averse because all students have the
same constant relative risk aversion (CRRA) preferences. We also generalize our results to hyperbolic risk aversion (HARA) preferences, which include CRRA as a special case.
1
drop out, we take the Miao and Wang (2007) framework of entrepreneurial learning
and survival and extend it with realistic features that are important for the dropout
decision faced by college students. In particular, we allow the workers’ lifetime
wage profile to depend on their experience, measured by the time spent in the labor
market, and on their tenure in college, as in Mincer (1974). Unlike Mincer, however,
we also let the lifetime wage profile depend on whether the worker has graduated
from college and allow the experience premium to interact with the individual’s
graduation status. Our results are robust to different specifications of the lifetime
wage profile for students with different college tenures.
In a typical Mincerian framework, a student would choose the optimal number
of years of college education by comparing the marginal gain from an extra year in
school with the marginal gain from joining the workforce immediately. Within this
framework, a nonlinear relationship between schooling and returns to education can
potentially explain why many students decide to drop out even though the returns to
earning a four-year degree are high. Still, this framework does not fully explain the
data. First, the Mincerian model is silent about the relationship between wealth and
educational profiles. Second, when college students are confronted with questions
regarding their expectations about postsecondary educational outcomes, almost all
of them respond that they intend to obtain four years of college education.3 The
failure of this basic Mincerian model suggests that it is necessary to have a story
where information unfolds as time elapses, in order to explain the dropout behavior.
The literature provides three different mechanisms whereby information is revealed over time.4 The first one involves binding credit constraints. However, credit
constraints do not seem to be the only determinant of dropout behavior because the
dropout rate for students from the richest households is around 29 percent, which is
still very high.5 This seems to conflict with arguments in favor of credit constraints
because if credit constraints were the main reason for dropping out we would expect that almost all the students at the top of the income distribution to graduate,
given the high returns to education.6 Moreover, using NLSY 1997 data, Lovenheim
3
See the data manual of National Longitudinal Survey of Youth 1979.
See Altonji (1993) for a seminal paper on education choice under uncertainty.
5
Own calculations from NLSY79 using top 10 percent of income distribution.
6
The evidence on credit constraints and college education choice is mixed at best. While Belley
and Lochner (2007) and Lochner and Monge-Naranjo (2011) find some evidence of credit constraints affecting enrollment, Heckman, Lochner, and Taber (1998), Cameron and Heckman (2001),
Keane and Wolpin (2001), Cameron and Taber (2004), Foley, Gallipoli, and Green (2009), and
Nielsen, Sorensen, and Taber (2010) find little to no effect of credit constraints in shaping postsecondary enrollment.
4
2
and Reynolds (2013) find that rising house prices lead to higher graduation rates,
especially among low-income families. The result implies that the effect of wealth
on dropout behavior has not vanished despite the fact that there have been many improvements in financial markets and credit availability to poor students. The second
potential mechanism is learning about scholastic taste, i.e., the student’s attitude toward schooling, as in Stange (2012). However, there is no clear channel that relates
scholastic taste with the wealth level of the family.
The third mechanism, used in this paper, is learning about unobserved ability through college attendance.7 Our modeling choice follows from the empirical
work of Stinebrickner and Stinebrickner (2008) and Stinebrickner and Stinebrickner (2012), who construct a panel study in order to understand the dropout decisions
of students in a particular four-year institution, Berea College. Stinebrickner and
Stinebrickner (2008) calculate a lower bound on the percentage of attrition that
would remain at Berea College even if low-income students were given access to
loans and find that this bound is very high, thus concluding that credit constraints
cannot explain the dropout decision for the majority of students. Stinebrickner and
Stinebrickner (2012) find that academic performance, a proxy for learning about
one’s scholastic ability, is a good predictor of dropout behavior, a result that is
detrimental to explanations relying on scholastic taste.8
We do not claim that the alternative mechanisms discussed above play no role
in dropout decisions but rather propose our model as a complementary explanation.
For example, although our model does not include explicit borrowing constraints,
it implies a natural borrowing limit in the spirit of Aiyagari (1994). Since a student
does not know her ability ex-ante when attending college, she faces the risk of ending up in a low-paying job, an event that will cause her to have low wealth, after
adjusting for discounted lifetime wage income and the cost of college. Due to Inada conditions, as the student’s wealth level approaches zero her consumption as a
low-wage worker becomes infinitesimally small, and therefore her marginal utility
of consumption post-education goes to infinity. This causes a lower willingness by
the student to borrow against her future labor income in order to finance her col7
An important paper in this regard is Arcidiacono (2004), who estimates the returns to the choice
of major in college. As many other papers in the literature, Arcidiacono assumes that preferences
are a function of expected lifetime earnings and hence there is no role for initial wealth.
8
Another mechanism that can explain the skewed distribution of dropouts is that richer students
choose a longer duration of education when education is a normal consumption good. However,
this cannot explain why academic performance matters so much for dropout decisions, as shown
in Stinebrickner and Stinebrickner (2012), unless we believe that academic performance affects the
marginal utility of education and that its effect on rich and poor students is different.
3
lege education. In Aiyagari’s words: “Thus, a borrowing constraint is necessarily
implied by non-negative consumption,” where the non-negativity of consumption is
guaranteed by the Inada conditions in our model.9
The next section uses data from the National Longitudinal Study of the High
School Class of 1972, the National Longitudinal Survey of Youth 1979 and the National Longitudinal Survey of Youth 1997 to compare the dropout behavior of poor
and rich students. We find that poor students are at least 27 percent more likely to
drop out and they do so before rich students, controlling for measures of unobserved
ability.
2
Evidence
To motivate our model, this section presents some statistics regarding dropout behavior based on the National Longitudinal Study of the High School Class of 1972,
NLS-72 hereafter, the National Longitudinal Survey of Youth 1979, NLSY79 hereafter, and the National Longitudinal Survey of the Youth 1997, NLSY97 hereafter.
For the NLS-72, we focus on individuals who enrolled in a four-year college during 1972 with no discontinuities in their education spells after starting college. For
the NLSY79, we focus on individuals who enrolled in a four-year college during
or after 1979 with no discontinuities in their education spells after college enrollment. For the NLSY97, we focus on individuals who enrolled in a four-year college
during or after 1997 with no discontinuities in their education spells after starting
college.10
Both in our data analysis here and theoretical model in the next section we
abstract from enrollment decision because we are interested in the dropout behavior
of a student who is enrolled in college rather than the counterfactual question of a
student’s dropout behavior if she were to attend college. For the latter question, our
results regarding the differences between poor and rich students can be considered
as a lower bound because poor high school graduates are less likely to attend college
9
The intuition regarding a natural borrowing limit also holds for HARA preferences because the
marginal utility of consumption goes to infinity as consumption approaches to a positive limit under
these preferences.
10
We also discarded students who attended two-year community colleges. The separation of
community colleges from four-year colleges is important because the salary profile of graduates
from both types of institutions is quite different. Within the context of our model, community
colleges may serve as a stepping stone to four-year colleges by giving students more information
about their skills so that high school graduates that are not optimistic enough to go to a four-year
college initially may still enroll community colleges. Trachter (2015) formalizes this idea to study
the transition between community and four-year colleges.
4
and we can consider non-attendees as having dropped out before the first day of
college. This view is also in accordance with Manski and Wise (1983), which
shows that if people with a low probability of attending college were to attend, they
would have a high probability of dropping out.
Our analysis follows from comparing dropout profiles of students from rich and
poor families. For NLS-72 we use the socioeconomic status of the respondent’s
family at the moment of high-school graduation as a measure of wealth. For the
NLSY79 and NLSY97 such a variable is not available, so we construct it ourselves.
In particular, we first rank the respondents’ families according to their income at the
time the student graduated from high school. Then we match the fraction of families with different socioeconomic status in NLS-72 so that both datasets become
comparable.11
Table 1 presents some aggregate summary statistics regarding dropout behavior
for both rich and poor students. Here we withold students from middle-income
households to make the point clearer. In the three datasets, students from poor
families have higher attrition rates. Furthermore, students from poor households
tend to drop out before students with rich families. As shown in the table, they tend
to drop a half year to a year sooner than rich students depending on the dataset.
To further explore the skewed distribution of dropouts with respect to wealth, we
Table 1 Dropout rates and mean time before dropping out by socioeconomic status
of family
Socio. statusa % that drop Mean tenure in collegeb st. dev. of tenure
Low
28.98
2.31
1.65
NLSY97
High
13.40
3.14
1.73
NLSY79
NLS-72
Low
62.5
2.78
1.6
High
26.96
3.94
1.9
Low
65.6
2.02
1.29
High
52.86
2.69
1.63
a
For the NLSY79 and NLSY97 we constructed the measure of socioeconomic status through the income level of the family prior to the respondent’s enrollment in college. We choose the deciles so as
to match the distribution of socioeconomic status of the NLS-72. b We only have the length of the
tenure in college for a sub-sample of the population.
11
For students enrolling in college, our sample from NLS-72 has 16.3 percent of families classified as low socioeconomic status, 41.7 percent as average socioeconomic status, and the rest as
wealthy.
5
compare the dropout rates of rich and poor students in different years of college.
If poor students tend to drop out earlier than rich students, as we suggest, then
a larger proportion of dropouts among poor students should occur in the earlier
college years, whereas a larger proportion of dropouts among rich students should
occur in later college years. Table 2 provides a parsimonious way of checking this
argument in the data by reporting the ratio of proportion of dropouts among poor
students to the proportion of dropouts among rich students in different years of
college. This statistic decreases from a number greater than 1 to a number less than
1 as we go from the earlier to later college years, providing support for our claim.
Table 2 Dropout rates of low- vs. high-income students
tenure between
0 and 1 1 and 2 2 and 3 3 and 4 4 and 5 5 and 6
years
years
years
years
years
years
NLSY97
1.76
1.65
1.57
0.91
0.37
0.69
NLSY79
3.28
1.31
1.61
0.8
0.35
0.23
NLS-72
1.49
1.17
0.94
0.38
0.52
0.19
6 and 7
years
0.37
0.4
0.47
Each number in the table represents the dropout rate of low-income students as a share of the total
dropout rate of low-income students divided by the yearly dropout rate of high-income students as a
share of the total dropout rate of high-income students.
Although these summary statistics provide a useful overview they do not tell us
if the difference between poor and rich students are statistically significant. Therefore, we also extend our analysis by controlling for available proxies of ability that
do not seem to be strongly colinear with the household’s socioeconomic status.12
If students form their beliefs rationally, these proxies are also positively correlated
with students’ initial beliefs about their ability levels because the belief distribution of higher-ability students should first-order stochastically dominate the belief
distribution of lower-ability students.
Table 3 presents the marginal effect and percentage effect of having high or
low socio-economic status relative to medium socioeconomic status on the college
dropout rates for the NLSY79, NLSY97, and NLS-72 datasets. This table provides the results of logit regressions, where the dependent variable is equal to one
if the student drops out and zero otherwise. For the three datasets rich students are
less likely to drop out than are middle-income students, with the marginal effect
ranging from -5.88 percent to -13.8 percent depending on the dataset. Low-income
12
The educational attainment of the student’s father and mother are strongly co-linear with the
household’s wealth and therefore these measures are not used.
6
students are more likely to drop out than middle-income students, with marginal effect ranging from 2.56 percent to 15.76 percent depending on the dataset. Given the
fraction of college students that drop out, the probability of dropping out increases
by at least 27 percent for students from poor households relative to those from rich
households. These results confirm the results shown in previous tables.
Table 3 Marginal and percentage effect of socioeconomic status on dropout probability
dF/dx std. error % effect
p
N
Low SES 0.0265
0.0266
12.78
NLSY97
0.02 1948
High SES -0.0901 0.0187
-43.48
Low SES 0.1576
0.063
41.38
NLSY79
0.03 635
High SES -0.0588 0.0468
-15.44
Low SES 0.0256
0.0298
4.27
NLS-72
0.00 2705
High SES -0.138
0.0211
-23.03
The results are obtained from running logit regressions on the dropout probability based on the
socioeconomic status of the student’s family and a set of controls. The complete regression results
and description of control variables can be found in Table 5, Table 6, and Table 7 in Appendix E. N
is the number of observations and p is the p-value of the χ2 test that compares the effect of high and
low socio-economic status on dropout probability.
Our evidence regarding the negative relationship between wealth and dropout
rates is also supported by Lovenheim and Reynolds (2013), who find that house
price appreciation leads to higher graduation rates, especially among low-income
families.
We also look at the distribution of dropout times across students with different
family income. To explore this connection we run an ordered logit regression of the
dropout time for those students who dropped out. We choose ordered logit because
it is a simple way to solve the interval censoring problem caused by the discrete
measurements of dropout times in our data, and the right-censoring problem inherent in duration data.13 The results of the regressions can be found in Table 8 for the
NLSY97, Table 9 for the NLSY79, and Table 10 for NLS-72 in Appendix E. These
table show that the dropout times of poor and rich students are different from each
other at 1% significance level.
Using the estimation results in these tables, we produce the predicted probability of dropping out in a given year as a function of the socioeconomic status of the
13
See also Han and Hausman (1990), which shows that the proportional hazards specification for
a duration model leads to a likelihood function of an ordered logit form in the absence of timevarying covariates. We have also run standard OLS regressions of dropout times on socio-economic
status and other control variables and have found qualitatively similar results.
7
student’s family. Figure 1 to Figure 3 show the estimated dropout probabilities of
students with different socioeconomic status in different years of college evaluated
at the mean values of other control variables. The common pattern in all these figures is that poor students tend to drop out in earlier years of college whereas richer
students tend to drop out in later years even after controlling for other characteristics.
To summarize, a college student’s family income not only affects the probability
of dropping out but also the timing of attrition.
'URSRXWSUREDELOLW\
Figure 1 NLSY97: Predicted time to drop out
<HDUVWRGURSRXW
/RZ6(6
+LJK6(6
0LGGOH6(6
The figure plots the predicted probabilities of dropping out in a given time interval for students
with different socioeconomic statuses, conditioning on the average characteristics of students who
dropped out. To compute the probabilities, we run an ordered logit regression that involves the
probability of dropping out in a given year as a function of the socioeconomic status variable plus a
set of controls. The complete regression results can be found in Table 8 in Appendix E.
8
Figure 2 NLSY79: Predicted time to drop out
The figure plots the predicted probabilities of dropping out in a given time interval for students
with different socioeconomic statuses, conditioning on the average characteristics of students who
dropped out. To compute the probabilities, we run an ordered logit regression that involves the
probability of dropping out in a given year as a function of the socioeconomic status variable plus a
set of controls. The complete regression results can be found in Table 9 in Appendix E.
3
Model
At t = 0 students are enrolled in college, endowed with wealth level x0 . Students
differ in their ability to acquire human capital at college; this ability can be low or
high. Let µ ∈ {0, 1} denote the ability level, where µ = 0 denotes low ability. The
ability level is not observable at t = 0. Instead, individuals inherit a signal about
their true type p(0) = Pr(µ = 1).
At any point in time an agent can either be enrolled as a full-time student or
working in low- or high-skilled sectors.14 The high-skilled sector only hires high14
Post-secondary education is a combination of both two-year and four-year colleges with dy-
9
Figure 3 NLS-72: Predicted time to drop out
The figure plots the predicted probabilities of dropping out in a given time interval for students
with different socioeconomic statuses, conditioning on the average characteristics of students who
dropped out. To compute the probabilities we run an ordered logit regression that involves the
probability of dropping out in a given year as a function of the socioeconomic status variable plus a
set of controls. The complete regression results can be found in Table 9 in Appendix E.
ability workers with college degrees.15 Work is assumed to be an absorbing state
with a constant wage function w̃(µ, τ ) where τ ≡ T − t accounts for the amount
of time left prior to graduation and T is the duration of college. We specify the
namic patterns that involve dropouts and transfers across types of schools. Using data from the
National Longitudinal Survey of 1972, Trachter (2015) shows that students enrolled in four-year
colleges either drop out or remain at the current type of institution until graduation.
15
The interest of this paper is to understand dropout behavior. Dropouts usually leave school and
join low-skilled sectors.
10
function w̃(µ, τ ) as follows:
w(τ ) if τ > 0,
w̃(µ, τ ) =
w(0) if τ = 0 and µ = 0,
w1
if τ = 0 and µ = 1,
with w(τ 0 ) > w(τ 1 ) if τ 0 < τ 1 and w1 > w(0). Therefore, the wage is increasing with the time spent in college and high-ability graduates enjoy higher wages.
w(τ )
The function w(T
reflects the college premium in low-skilled sectors. A graphical
)
representation of the wage function is depicted in Figure 4.
Figure 4 Skill and College Premium
~ 1,0
w
skill premium
~ 0,0
w
college premium in
low-skilled sectors
~ ,
w
T
The evolution of the wealth level x is given by
dx
=
dt
(
rx + w̃(µ, τ ) − c if working,
rx − a − c
if enrolled in college,
where a denotes the per period cost of attending college.
At every period of length dt in college, students take an exam and get three
possible grades: excellent, pass, or fail. High-ability students (i.e. µ = 1) can get
either a grade of pass or excellent for an exam. We let λ1 denote the probability
per unit of time that a high-ability student gets an excellent grade. Low-ability
students (i.e. µ = 0) can get either a passing or failing grade for an exam. We let
λ0 denote the probability per unit of time that a low-ability student gets a failing
11
grade. Receiving a failing grade reveals that the student has low ability whereas
receiving an excellent grade reveals that she has high ability. When the student
receives a pass, she updates her belief according to Bayes’ rule. Table 4 presents
the probability of the outcome of a given exam conditional on the student’s true
type, which is unobservable by the student.
We choose to model the student’s learning process about her true type using
discrete signals rather than continuous signals primarily because in our model the
speed of learning, λ0 and λ1 , depends on the type of the student. In a continuous Brownian Motion signal setting, this would be equivalent to having different
volatilities for the signal process. However, Merton (1980) and Nelson and Foster (1994) point out that an observer of a continuous path generated by a diffusion
process can estimate a constant or a smoothly time-varying volatility term with arbitrary precision over an arbitrarily short period of calendar time provided she has
access to arbitrarily high frequency data. Of course, introducing a discrete signal
is not the only way to get around this problem, but it is analytically more efficient
than other possibilities, such as introducing stochastic volatility or writing a discrete time model. These alternatives require belief updating about the mean and
variance of the corresponding stochastic process, which complicates the analysis
without adding anything to the intuition.
Table 4 Probability of receiving different grades based on student’s type
Fail
Pass
Excellent
µ = 0 λ0 dt 1 − λ0 dt
0
µ=1
0
1 − λ1 dt
λ1 dt
Each value in the table is the probability of receiving a given grade on the exam per unit of time dt,
conditional of the student’s true ability level.
A student initially enrolled in college chooses her consumption stream {c(t)}t≥0
and whether to remain a student or to drop out and join the workforce, in order
to maximize her time-separable expected discounted lifetime utility derived from
consumption,
Z ∞
1−γ
−ρt c(t)
E
e
p(0), x(0) ,
1−γ
0
where γ is the coefficient of Constant Relative Risk Aversion (CRRA).
We let J(x, p, τ ) denote the value for a student with current wealth level x, prior
p, and T − τ time already spent in school. Also, V (x, µ, τ ) denotes the value for a
worker of true type µ with current wealth level x who spent T − τ time in school.
12
3.1
The Problem of a Worker
An individual with T − τ years of schooling who joins the workforce maximizes
R∞
her lifetime discounted utility 0 e−ρt c1−γ / (1 − γ) dt, subject to the law of motion
for wealth, dx/dt = rx + w̃(µ, τ ) − c. It will prove useful to characterize W (µ, τ ),
the present discounted value of lifetime earnings. This object is simply
Z
W (µ, τ ) =
∞
e−rt w̃(µ, τ )dt.
0
The Hamilton-Jacobi-Bellman representation of the worker’s problem is
ρV (x, µ, τ ) = max
c
c1−γ
+ Vx (x, µ, τ )(rx + w̃(µ, τ ) − c),
1−γ
which states that the flow value of being a worker must be equal to the instantaneous
utility derived from consumption in addition to the change in value that happens
through the change in wealth.
The first-order condition of the worker’s problem reads c−γ = Vx (x, µ, τ ). Plugging it back and operating provides an equation that a worker’s maximized value
function needs to satisfy,
ρVx (x, µ, τ ) =
1
γ
[Vx (x, µ, τ )]1− γ + r(x + W (µ, τ ))Vx (x, µ, τ ),
1−γ
(1)
with the solution given by
V (x, µ, τ ) = A (r [x + W (µ, τ )])1−γ ,
where
A ≡ [(1 − γ)r]
γ−1
1−γ
(ρ − (1 − γ)r)
γ
(2)
−γ
.
(3)
The term x+W (µ, τ ) in equation (2) represents the worker’s wealth also accounting
for the discounted lifetime labor earnings.
Although we have assumed that the wages earned after college graduation are
constant, this solution also holds if the wages depend upon on-the-job experience.
Once we redefine w̃(µ, τ , s) as the worker’s wage depending on how long she has
R∞
been working, s, we can replace W (µ, τ ) with W (µ, τ ) = 0 e−rs w̃(µ, τ , s)ds
in the solution. Therefore, our results are robust to different specifications of the
lifetime wage profile for students with different durations of college education.
13
3.2
The problem of a student of type µ and the natural borrowing limit
If the student knows that she has high ability, µ = 1, her value function satisfies
dτ
c1−γ
+ (rx − a − c)Jx (x, 1, τ ) + Jτ (x, 1, τ ) ,
ρJ(x, 1, τ ) = max
c
1−γ
dt
(4)
subject to the terminal condition J(x, 1, 0) = V (x, 1, 0). This terminal condition
states that, upon graduation, the student’s value function is equivalent to the value
of being a worker in the high-skill sector. We are also using an implicit condition
guaranteeing that high-skilled students would never find it profitable to drop out.
This condition is later derived in Lemma 1.
This Hamilton-Jacobi-Bellman equation states that the desired return on being
a student with high ability, µ = 1, with wealth level x, and with τ periods left to
graduation equals the instant utility derived from consumption plus the change in
value due to the change in wealth and time to graduation, both due to the change
accrued in time. Note that dτ /dt = −1.
The first-order condition of this problem states that c−γ = Jx (x, 1, τ ). Solving for c and plugging the result back into equation (4) provides the student’s
maximized value function. We guess and verify that the solution to this object
is J(x, 1, τ ) = A [rx + B(τ )]1−γ , where A is defined as in equation (3) and B(τ )
needs to be solved for. Intuitively, B(τ ) accounts for the change in the value function due to the time to graduation and the terminal payoff. Plugging the guess into
the maximized value function provides
rB(τ ) + B 0 (τ ) + ra = 0
with boundary condition, B(0) = rW (1, 0), that follows from the terminal condition of the problem presented in equation (4). This is an Ordinary Differential Equation in one variable with a terminal condition. The solution is B(τ ) =
(rW (1, 0) + a) e−rτ −a, and therefore the value function of a student of type µ = 1
is
1−γ
J(x, 1, τ ) = A rx − a + e−rτ (rW (1, 0) + a)
(5)
or,
1−γ
1 − e−rτ
−rτ
a
,
J(x, 1, τ ) = A r x + e W (1, 0) −
r
14
where the term in parentheses gives the student’s net lifetime wealth after accounting for the discounted value of future wages and the remaining college costs.
The next lemma characterizes the condition guaranteeing that high-ability students who know their type will not find it profitable to drop out of college.
Lemma 1 A student of type µ with current wealth level x and T − τ time spent in
college will choose to remain as a student until τ = 0 if re−rτ W (1, 0)−a (1 − e−rτ ) ≥
rW (1, τ ).
Lemma 1 follows from noting that for a student to remain in college it has to be
the case that, for every value of τ , J(x, 1, τ ) ≥ V (x, 1, τ ). This condition simply
indicates that the graduation premium for a high-ability student is high enough so
that a student who knows she has high ability will remain in college until graduation. We assume throughout the paper that this condition holds.16
Another important assumption of the model is that low-ability students who
know their type will always find it profitable to drop out of college. If this were not
the case, for some values of time to gradution, τ , there would be no dropouts by
construction. The next lemma characterizes the condition guaranteeing that lowability students who know their type will decide to drop out and join the workforce
immediately, i.e. J(x, 0, τ ) = V (x, 0, τ ).
Lemma 2 A student who knows that she is of type µ = 0 will drop out immediately
if a + rW (0, τ ) + Wτ (0, τ ) > 0.
Proof. See Appendix A.
Intuitively, the marginal cost of attending college for another period of time
is adt. Moreover, the marginal increase in the present value of earnings after an
additional period of college education is
e−rdt W (0, τ − dt) − W (0, τ ) = − [rW (0, τ ) + Wτ (0, τ )] dt + O (dt)2 ,
where we used a Taylor series expansion. Subtracting the increase in marginal
earnings from marginal cost, dividing by dt and taking the limit as dt → 0 gives
the condition stated above.
Although our model does not entail any explicit borrowing constraint this last
assumption leads to an implicit borrowing constraint for a student who does not
16
This condition at τ = T also guarantees that there are at least some students with optimistic
prior beliefs willing to enroll in college.
15
know her true ability level. Every student with p < 1 faces ex-ante a positive
probability of receiving a shock revealing that she has low ability. Such a shock
would force her to drop out of college and join the low-skilled workforce. Since
the marginal value of wealth for a college dropout goes to infinity as x goes to
−W (0, τ ) the student will never borrow more than her discounted value of lifetime
earnings W (0, τ ). As long as the wage profile is common knowledge, as it is the
case in our model, this ”natural” borrowing constraint should also be the actual one
because the student can always repay the borrowed money using his earnings when
x > −W (0, τ ).
3.3
The Problem of a Student of Unknown Type
The problem of a student who does not know her ability level is more difficult to
solve for three reasons. First, the wage upon graduation depends on the agent’s true
ability. Second, the arrival of new information via exams results in updating of the
student’s belief. Third, some students drop out.
Before constructing the Hamilton-Jacobi-Bellman equation for this case, first it
is useful to consider how the information obtained through exams can be used to
update beliefs. Consider a student with the belief p(t), where p(t) is the probability
of being type 1 conditional on the information available at time t. Table 4 can be
used to construct the posterior conditional on the grade received during (t, t + dt).
If the student receives a failing grade, it is clearly revealed that she has low ability
and thus p(t + dt) = 0. If the student receives a grade of excellent, p(t + dt) = 1.
Conditional on not receiving a failing or excellent grade through period (t, t + dt),
receving a passing grade in the current exam implies that Bayes’ rule can be used
to update beliefs,
p(t + dt) =
p(t) [1 − λ1 dt]
.
p(t) [1 − λ1 dt] + [1 − p(t)] [1 − λ0 dt]
Substracting p(t), dividing by dt, and taking the limit as dt → 0 provides the Bayes’
rule in its continuous time formulation,
dp
= − (λ1 − λ0 ) p(1 − p).
dt
(6)
Figure 5 describes the timeline of the student’s problem in a given period. The
student enters the period with wealth level x, prior p, and remaining time until
graduation τ . Her value function is therefore J(x, p, τ ). At the beginning of the
16
Figure 5 Timeline
Drop V x' , 0, '
J x, p ,
consume
x' x dx
take exam
Stay if 0
J x ' , p ' , '
p ' p dp
' d
Graduate if 0
p 'V x' ,1, 0 1 p'V x' , 0, 0
A student starts the current period with wealth level x, prior p, and remaining time in college τ . At
the beginning of the period, a student chooses her consumption level and thus provides the new value
for wealth x0 . Before the end of the period, the student takes an exam used to produce the posterior
p0 and reduces the time left to graduation to τ 0 . At the beginning of next period, the student chooses
between dropping out or remaining as a student (or graduation is τ 0 = 0).
period, the student chooses her consumption level and thus produces the wealth
level x0 for next period. Before the end of the period, she takes an exam and with
the grade at hand updates her beliefs to p0 . By the end of the period she accumulates
more time in school and therefore the distance to graduation is reduced to τ 0 . At the
beginning of next period, the student compares the value of remaining in school,
that is either J (x0 , p0 , τ 0 ) if τ 0 > 0 or p0 V (x0 , 1, 0) + (1 − p0 ) V (x0 , 0, 0) if τ 0 = 0
(i.e. graduation), with the value of joining the workforce V (x0 , 0, τ ) to decide
between staying in college or dropping out.
The Hamilton-Jacobi-Bellman equation for a student with current wealth level
x, prior p, and τ periods left in school is
ρJ(x, p, τ ) = maxc
c1−γ
1−γ
+ (rx − a − c)Jx (x, p, τ ) − Jτ (x, p, τ )
− (λ1 − λ0 ) p(1 − p)Jp (x, p, τ ) + λ1 p [J(x, 1, τ ) − J(x, p, τ )]
+λ0 (1 − p) [V (x, 0, τ ) − J(x, p, τ )] .
(7)
This equation states that the desired return on being a student with current
wealth level x, prior p, and time to graduation τ equals the instant utility derived
from consumption plus the change in value of being a student through (i) the change
in wealth, (ii) the change in τ , and (iii) belief updating. When a passing grade arrives, the belief adjustment is continuous through the Bayesian updating of p; when
an excellent grade arrives, expected with unconditional probability λ1 p, the change
17
in value occurs through switching from having p(t) = p to p(t + dt) = 1; and
when a failing grade arrives, expected with unconditional probability λ0 (1 − p), the
change in value is through switching from having p(t) = p to p(t + dt) = 0. Also
note that the problem faced by a high-ability student who knows her type presented
in equation (4) is a particular case of the problem presented here, that follows by
setting p = 1 in equation (7) and noting that dτ /dt = −1.
A student faces the problem presented in equation (7) subject to a set of boundary conditions,
J(x, p, 0)
J(x, p (x, τ ), τ )
Jp (x, p∗ (x, τ ), τ )
Jx (x, p∗ (x, τ ), τ )
Jτ (x, p∗ (x, τ ), τ )
∗
=
=
=
=
=
pV (x, 1, 0) + (1 − p)V (x, 0, 0)
V (x, 0, τ )
0
Vx (x, 0, τ )
Vτ (x, 0, τ ).
(8)
The first equation gives the Terminal Condition (TC) and states that the value of
being a student with no time remaining until graduation has to equal the expected
value of being a worker. Note that with probability p a student expects to be of
type µ = 1 and therefore would earn lifetime discounted labor income W (1, 0),
while with probability 1 − p she expects to be of type µ = 0 and therefore earn lifetime discounted labor income W (0, 0). To understand the second to fifth equations
p∗ (x, τ ) needs to be defined. Let p∗ (x, τ ) be the belief threshold such that students
with p ≤ p∗ (x, τ ) drop out and join the workforce. The second equation states that
a student with p = p∗ (x, τ ), wealth level x, and τ periods away from graduation has
to be indifferent between staying in school and dropping out and enjoying lifetime
discounted labor income W (0, τ ). This equation is also known as the Value Matching Condition (VMC). The third, fourth, and fifth equations are known as Smooth
Pasting Conditions (SPC) required for the optimality of p∗ (x, τ ).17
The first-order condition of the problem presented in equation (7) is c−γ =
Jx (x, p, τ ). Plugging it back into equation (7) together with the terminal, value
matching, and smooth pasting conditions provides the equation that the threshold
p∗ (x, τ ) needs to satisfy,
ρV (x, 0, τ ) =
1− γ1
γ
V
(x,
0,
τ
)
+ (rx − a)Vx (x, 0, τ )
x
1−γ
∗
+λ1 p (x, τ ) [J(x, 1, τ ) − V (x, 0, τ )] − Vτ (x, 0, τ ),
17
(9)
For a treatment of Value Matching and Smooth Pasting Conditions see Dumas (1991) and Dixit
(1993).
18
which we can rewrite as
λ1 p∗ (x, τ ) [J (x, 1, τ ) − V (x, 0, τ )] = [a + rW (0, τ ) + Wτ (0, τ )] Vx (x, 0, τ ) .
This equation provides intuition about the belief threshold p∗ (x, τ ). The leftside of this equation is the expected utility gain from delaying the dropout decision
by dt, whereas the right side represents the marginal net utility loss due to delaying
the dropout decision. The student chooses the optimal dropout time by equalizing
the marginal gain and loss from delaying the dropout decision.
Solving for p∗ (x, τ ) allows for a close-form representation of the student’s
dropout threshold,
p∗ (x, τ ) =
a + rW (0, τ ) + Wτ (0, τ )
Vx (x, 0, τ )
> 0,
λ1
J(x, 1, τ ) − V (x, 0, τ )
(10)
provided that Lemma 2 holds. This threshold is decreasing with the wealth level.
That is,
∂p∗ (x, τ )
< 0,
(11)
∂x
where the details of the calculations can be found in Appendix B.
Our main result is that, conditional on their beliefs, college students from wealthier families drop out later and are less likely to drop out than are poor students. The
result that the threshold p∗ is decreasing with the wealth level is not enough to argue
this result because the consumption profiles during college tenure can overcome the
initial difference in wealth.18 The next proposition and corollary deal with this.
Let τ ∗ denote the time to graduation at the moment the individual joins the
workforce. For example, τ ∗ = T if the individual joins the workforce directly
after high school graduation and τ ∗ = 0 if the individual joins the workforce with
a college degree. The following proposition states that, conditional on abilities
and initial prior beliefs, the distribution of dropout times for richer students firstorder stochastically dominate the distribution for poor students in the model, which
explains the pattern in Figure 1 to Figure 3.
Proposition 1 Let xi (0) and xj (0) denote the initial wealth levels at time 0 of students i and j. If xi (0) > xj (0) then, for any τ̄ , P r {τ ∗ ≤ τ̄ | xi (0), p(0), µ} ≥
P r {τ ∗ ≤ τ̄ | xj (0), p(0), µ}. In other words, given a skill level µ and initial belief
18
Miao and Wang (2007) entrepreneurial survival model is a special case of our model where
they also show that the boundary p∗ is decreasing in the wealth level. However, they immediately
conclude that richer entrepreneurs survive longer without providing an explicit proof.
19
p(0), richer students tend to drop out later and have longer expected tenures in
college.
Proof. See Appendix C.
The next corollary extends the result to show that, conditioning in the initial
prior p(0) and ability level µ, students from richer families are less likely to drop
out and therefore more likely to graduate.
Corollary 1 Let xi (0) and xj (0) denote the wealth levels at time 0 of students i
and j. If xi (0) > xj (0), then Pr [e
τ = 0| xi (0), p(0), µ] ≥ Pr [e
τ = 0| xj (0), p(0), µ].
That is, once conditioned on the initial prior p(0) and skill level µ, richer students
are more likely to graduate from college.
Proof. Set τ̄ = 0 in Proposition 1.
Both Proposition 1 and Corollary 1 are driven by the fact that ex-ante uncertainty regarding the outcome of college education cannot be diversified away, which
makes the investment in college education more risky than financial investments in
the model.19 Given students’ abilities and beliefs, a student with higher wealth
chooses to increase the number of dollars invested in the risky asset, i.e. stay longer
in college, if absolute risk aversion is decreasing in wealth as is the case with the
CRRA preferences.20
4
Conclusion
In this paper, we provide evidence regarding the skewed distribution of college
dropouts with respect to the student’s family wealth. Poor students are more likely
to drop out and they tend to do so earlier than rich students. We explore whether a
model that treats college education as a risky investment and incorporates Bayesian
learning about own’s unobserved ability can explain the skewness in dropout behavior.
Our main results rely on the fact that the outcome of obtaining a college education is subject to uncertainty against which students cannot insure themselves.
19
Of course, financial assets in real life are risky due to macroeconomic fluctuations not modeled
in this paper. However, these macroeconomic fluctuations also affect the payoff of college education
through their effect on wages, and investing in college education is still riskier than investing in
financial assets due to undiversifiable idiosyncratic risk.
20
In Appendix D we extend the model to allow for hyperbolic risk aversion (HARA) preferences,
which include CRRA as a special case. We show that the belief threshold is decreasing in wealth if
and only if the absolute risk aversion is decreasing. Furthermore, Proposition 1 and Corollary 1 also
apply here.
20
When we combine this fact with CRRA preferences so that the absolute risk aversion decreases with wealth, we arrive at the conclusion that poor students are less
willing to accept the risk associated with pursuing a college education. This mechanism generates the skewness observed in dropout behavior.
We provide a closed-form characterization of a student’s optimal choice as a
function of (i) the expected future income due to graduation (through the prior,
p), and (ii) the direct and indirect costs of remaining in college (through the time
remaining to graduation, τ ). We exploit the model’s simplicity to show that it is able
to fit the data qualitatively: (i) poorer students are more likely to drop out than are
rich students, and (ii) if the poor students drop out, they do so earlier than students
from wealthier families.
To motivate the theory, we run a series of reduced-form regressions, conditioning by measures of the student’s unobserved ability and prior beliefs. The regressions’ results are in line with the model’s predictions. We estimate that poor college
students are at least 27 percent more likely to drop out than rich students, and if they
drop out they do so around a year earlier.
Our goal is not to claim that borrowing constraints are not part of the story
behind the high and skewed college dropout rates. Instead, we provide a complementary story that is able to explain the skewed distribution of the time to drop out.
Furthermore, our story is consistent with Stinebrickner and Stinebrickner (2008),
which finds that borrowing constraints are not the main determinant of dropout decision, and Stinebrickner and Stinebrickner (2012), which shows that bad grades
are a good predictor of dropout behavior.
Since our model generates these results without including explicit borrowing
constraints, it also suggests that policies that are geared toward reducing borrowing
constraints, such as student loan programs, are not likely to eliminate the differences
in dropout rates between rich and poor students. Moreover, a direct subsidy to poor
students for their college education would not only increase their graduation rates
but also their tenure in college, by both reducing the cost of spending additional time
in college and increasing the expected gain from delaying the dropout decision. The
optimal subsidy would depend on the wealth level of the student’s family and the
distribution of ability for a given wealth level, which is the topic of future research.
Finally, it is plausible that poor students are more likely to participate in the
labor force during college and have less time to devote to study, making them take
longer to finish college. Although we have not included the time allocation decision
in our model, an extension of the model with the decision to work versus study
21
during college can potentially generate this result, following the basic intuition in
this paper: Labor force participation while attending college provides a safe income
today, and poor students are more likely to work during college because they are
more risk-averse, and hence poor students are more willing to invest in a safe asset.
The empirical and theoretical analysis of the student’s work-study decision during
college is an interesting question that is left for future research.
22
Reference
Aiyagari, S Rao. 1994. Uninsured idiosyncratic risk and aggregate saving. The
Quarterly Journal of Economics 109: 659-84.
Altonji, J. G. 1993. The demand for and return to education when education outcomes are uncertain. Journal of Labor Economics 11(1): 48–83.
Arcidiacono, P. (2004). Ability sorting and the returns to college major. Journal of
Econometrics 121(1-2).
Athreya, Kartik and Janice Eberly. 2013. The Supply of College-Educated Workers: The Roles of College Premia, College Costs, and Risk. Working Paper 13-02,
Federal Reserve Bank of Richmond.
Belley, P. and Lochner, L. 2007. The changing role of family income and ability in
determining educational achievement. Journal of Human Capital 1(1): 37–89.
Cameron, S. and Taber, C. 2004. Borrowing constraints and the returns to schooling. Journal of Political Economy 112.
Chen, Stacey H. 2008. Estimating the variance of wages in the presence of selection
and unobserved heterogeneity. The Review of Economics and Statistics 90: 275-89.
Dixit, Avinash K. 1993. The art of smooth pasting. Routledge.
Dumas, Bernard. 1991. Super contact and related optimality conditions. Journal of
Economic Dynamics and Control 15: 675-85.
Foley, K., Gallipoli, G. and Green, D. A. 2009. Ability, parental valuation of education and the high school dropout decision. Journal of Human Resources 49(4):
906-944
Han, Aaron and Jerry A. Hausman. 1990. Flexible parametric estimation of duration and competing risk models. Journal of Applied Economics 5: 1-28.
Heckman, J. J., Lochner, L. and Taber, C. 1998. General-equilibrium treatment
effects: A study of tuition policy. American Economic Review 88(2): 381–86.
Keane, M. P. and Wolpin, K. I. 2001. The effect of parental transfers and borrow23
ing constraints on educational attainment. International Economic Review 42(4):
1051–1103.
Lochner, L. J. and Monge-Naranjo, A. 2011. The nature of credit constraints and
human capital. American Economic Review 101(6): 2487–2529.
Lovenheim, Michael F. and C. Lockwood Reynolds. 2013. The effect of housing
wealth on college choice: Evidence from the housing boom. Journal of Human
Resources 48(1): 1-28.
Manski, Charles and David Wise. 1983. College choice in America. Cambridge:
Harvard Press.
Merton, Robert C. 1980. On estimating the expected return on the market: An
exploratory investigation. Journal of Financial Economics 8: 323-61.
Miao, Jianjun and Neng Wang. 2007. Experimentation under uninsurable idiosyncratic risk: An application to entrepreneurial survival. Working Paper.
Mincer, Jacob A. 1974. Schooling, Experience, and Earnings. No. minc74-1 in
NBER books. National Bureau of Economic Research, Inc.
Nelson, Daniel B. and Dean P. Foster. 1994. Asymptotic filtering theory for univariate ARCH models. Econometrica 62: 1-41.
Nielsen, H. S., Sorensen, T. and Taber, C. 2010. Estimating the effect of student aid
on college enrollment: Evidence from a government grant policy reform. American
Economic Journal: Economic Policy 2(2): 185–215.
Stange, Kevin M. 2012. An empirical examination of the option value of college
enrollment. American Economic Journal: Applied Economics 4(1): 49-84.
Stinebrickner, Ralph and Todd Stinebrickner. 2008. The effect of credit constraints
on the college drop-out decicsion: A direct approach using a new panel study.
American Economic Review 98: 2163-84.
Stinebrickner, Todd R. and Ralph Stinebrickner. 2012. Learning about academic
ability and the college drop-out decision. Journal of Labor Economics 30(4): 707748
24
Trachter, Nicholas. 2015. Option value and transitions in a model of postsecondary
education. Quantitative Economics 6: 223-256.
Appendix
A
Proof of Lemma 2
Let τ ∗ denote the threshold for τ so that students drop out for τ < τ ∗ . A student of
type µ = 0 faces the problem given by
ρJ(x, 0, τ ) = max
c
c1−γ
+ (rx − c − a)Jx (x, 0, τ ) − Jτ (x, 0, τ ),
1−γ
subject to the terminal condition J(x, 0, 0) = V (x, 0, 0), the boundary condition
J(x, 0, τ ∗ ) = V (x, 0, τ ∗ ), and smooth pasting conditions Jx (x, 0, τ ∗ ) = Vx (x, 0, τ ∗ )
and Jτ (x, 0, τ ∗ ) = Vτ (x, 0, τ ∗ ).
Plugging in the first-order condition provides that
ρVx (x, 0, τ ∗ ) =
1
γ
[Vx (x, 0, τ ∗ )]1− γ + (rx − a)Vx (x, 0, τ ) − Vτ (x, 0, τ ∗ ).
1−γ
Using equation (1) this equation can be reduced to a + rW (0, τ ∗ ) + Wτ (0, τ ∗ ) = 0.
Hence, if a+rW (0, τ ∗ )+Wτ (0, τ ∗ ) > 0 for all τ ≤ T the boundary condition for an
interior dropout boundary is not satisfied. Moreover, the desired return from continuing education in terms of utility, the left side of Bellman equation, is greater than
the continuation value, the right side of Bellman equation, at the default boundary
and hence it is optimal to drop out immediately.
25
B
Proof of
∂p∗
∂x
<0
Differentiating the threshold p∗ (see equation (10)) with respect to x provides
"
∂p∗
∂x
=
=
=
=
#
Vx (x,0,τ )
Vxx (x,0,τ )
a+rW (0,τ )+Wτ (0,τ )
J(x,1,τ )−V (x,0,τ ) Vx (x,0,τ )
Jx (x,1,τ )−Vx (x,0,τ )
Vx (x,0,τ )
λ1
− J(x,1,τ
)−V (x,0,τ
h ) J(x,1,τ )−V (x,0,τ )
i
a+rW (0,τ )+Wτ (0,τ )
Vx (x,0,τ )
Vxx (x,0,τ )
Jx (x,1,τ )−Vx (x,0,τ )
−
λ1
J(x,1,τ )−V (x,0,τ )
Vx (x,0,τ )
J(x,1,τ )−V (x,0,τ )
h
i
V
(x,0,τ
)
J
(x,1,τ
)−V
(x,0,τ
)
xx
x
x
p∗ (x, τ ) Vx (x,0,τ ) − J(x,1,τ )−V (x,0,τ )
γr
r[x+W (0,τ )]
−γ
−p∗ (x, τ )
(r[x+e−rτ W (1,0)− ar (1−e−rτ )]) −(r[x+W (0,τ )])−γ
+(1 − γ)r
= −p∗ (x, τ )
x+e−rτ W (1,0)− ar (1−e−rτ )
(r[
1−γ
])
γr
r[x+W (0,τ )]
+ x+W1(0,τ )
p∗ (x,τ )
y −γ −1
= − x+W (0,τ ) γ + y1−γ −1 ,
y −γ −1
−(r[x+W (0,τ )])
1−γ
y 1−γ −1
1−γ
1−γ
where y ≡
x+e−rτ W (1,0)− ar (1−e−rτ )
,
x+W (0,τ )
with y ≥ 1 provided the condition on Lemma 1
−γ
> 0 when
holds. Next, it will be proved by contradiction that γ + (1 − γ) yy1−γ−1
−1
y ≥ 1.
−γ
Consider first the case where γ < 1. Suppose that γ + (1 − γ) yy1−γ−1
< 0.
−1
1−γ
Because γ ∈ (0, 1) and hence y
− 1 > 0, we can multiply both sides of this
inequality by y 1−γ − 1 to get γ(y 1−γ − 1) + (1 − γ)(y −γ − 1) < 0. The left-handside of this in equality is strictly increasing in y and therefore attains its minimum
at y = 1 with value equal to 0. Therefore, γ(y 1−γ − 1) + (1 − γ)(y −γ − 1) < 0 and
−γ
< 0 is not possible.
hence γ + (1 − γ) yy1−γ−1
−1
−γ
Now consider the case where γ > 1. Suppose that γ + (1 − γ) yy1−γ−1
< 0.
−1
1−γ
Because γ > 1 and hence y −1 < 0, we can multiply both sides of this inequality
by y 1−γ − 1 to get γ(y 1−γ − 1) + (1 − γ)(y 1−γ − 1) > 0. The left-hand-side of
this equation is strictly decreasing in y and therefore attains its maximum at y = 1
with value equal to 0. Therefore, γ(y 1−γ − 1) + (1 − γ)(y 1−γ − 1) > 0 and hence
−γ
γ + (1 − γ) yy1−γ−1
< 0 is not possible.
−1
−γ
∂p∗
As γ + (1 − γ) yy1−γ−1
>
0
for
every
γ
>
0,
< 0.
−1
∂x
26
C
Proof of Proposition 1
Two students with the same skill level µ are equally likely to receive a failing or an
excellent grade at any point in time. Therefore, although the grade earned affects the
behavior of an individual student it does directly affect the distribution of dropout
times for students with different wealth levels. Therefore, we look at the dropout
behavior of students that do not receive any signals that reveal their true types.
Suppose we have two students i and j with the same initial belief, i.e. pi (0) =
pj (0), and with initial wealth levels xi (0) > xj (0) so that the first student is initially
richer. There are two possible outcomes conditional on not receiving a signal. First,
if p(0) is high enough, both students wait until they graduate, which does not violate
our proposition as τ ∗ i = τ ∗ j = 0. Second, if p(0) is not high enough, at least one
of the students drops out. Let t0 < T be the first point in time when one of the
students drop out. Then we have pi (t) = pj (t) for all t ≤ t0 because pi (0) = pj (0)
and the belief evolution is the same for both students conditional on not receiving a
signal. Moreover, we know that the richer student does not drop out earlier than the
poorer student, that is τ ∗ i ≤ τ ∗ j , if xi (t) ≥ xj (t) for all t ≤ t0 because p∗ (xi , τ ) ≤
p∗ (xj , τ ) as long as xi ≥ xj . Therefore, we can prove our proposition by showing
that xi (t) ≥ xj (t) for all t ≤ t0 . Suppose xi (t) < xj (t) for some t < t0 . Then, since
xi (t) and xj (t) have continuous paths there exists a t̄ ≤ t0 where xi (t̄) = xj (t̄) by
the intermediate value theorem. Moreover, since pi (t) = pj (t) for all t ≤ t0 we have
pi (t̄) = pj (t̄). As a result, both students’ consumption decisions are synchronized
from time t̄ on because they are forward looking. Hence, xi (t) = xj (t) for t̄ ≤ t ≤
t0 which is a contradiction.
D
Hyperbolic Risk Aversion
In this appendix, we extend the model to allow for hyperbolic risk aversion (HARA)
preferences and show that the belief threshold is decreasing in wealth if and only
if we have decreasing risk aversion. The absolute risk aversion for this class of
preferences is given by
1
u00 (c)
=
RA = − 0
u (c)
ac + b
where a and b are constants. For a = 0 we have exponential preferences, for
a > 0 we have decreasing absolute risk aversion, and for a < 0 we have increasing
absolute risk aversion. Moreover, we obtain the CRRA preferences in the model
27
for b = 0. The solution of this differential equation is given by
u (c) = κ1
(ac + b)1−1/a
+ κ2 ,
a−1
where κ1 and κ2 are constants of integration. Let γ ≡ 1/a and c̄ ≡ −b/a. Then,
u (c) = κ1 a−1/a
(c + b/a)1−1/a
(c − c̄)1−γ
+ κ2 = κ̄1
+ κ2 .
(a − 1) /a
1−γ
Since a > 0, hence γ > 0, determines if we have decreasing risk-aversion we stick
to the linear transformation of this utility function, u (c) = (c − c̄)1−γ / (1 − γ).
Let us define ĉ ≡ c − c̄ as the surplus consumption and the x̂ ≡ x − c̄/r
as the surplus wealth. Then, dx̂/dt = dx/dt and all (rx − c) terms in the law
of motion of x become rx̂ − ĉ. Moreover, the utility function becomes u (ĉ) =
(ĉ)1−γ / (1 − γ). Therefore, the new belief boundary will be p̂∗ (x, τ ) = p∗ (x̂, τ ) =
p∗ (x − c̄/r, τ ). Hence, in order to show that ∂ p̂∗ /∂x < 0 iff γ > 0, it is enough to
show that ∂p∗ /∂x < 0 iff γ > 0. If γ > 0, ∂p∗ /∂x < 0 follows immediately from
Appendix B. So, we only have to show that γ > 0 if ∂p∗ /∂x < 0.
From the derivation in Appendix B and Lemma 1, it follows that ∂p∗ /∂x < 0
−γ
> 0 for all y ≥ 1. Suppose ∂p∗ /∂x < 0 holds but
implies that γ + (1 − γ) yy1−γ−1
−1
−γ
≤ 0 because
γ ≤ 0. Then, γ + (1 − γ) yy1−γ−1
−1
y −γ − 1
signum γ + (1 − γ) 1−γ
= signum γ y 1−γ − 1 + (1 − γ) y −γ − 1
y
−1
and
max γ y 1−γ − 1 + (1 − γ) y −γ − 1 = 0.
y≥1
This contradicts ∂p∗ /∂x < 0.
E
Other Tables
28
Table 5 NLSY97: marginal effect of socioeconomic status on dropout probability
coef.
std. err.
male
0.1224
0.0185
asvab
-0.0024
0.0003
born-US
0.071
0.023
hh-size
-0.0009
0.0067
minority
0.0675
0.0219
socio-low
0.0265
0.0266
socio-high
-0.0901
0.0187
# of obs.
1948
Sociolow=Sociohigh prob > χ2
0.02
To compute the marginal effects we run a logit regression on the probability of dropping out. Male:
=1 if male . asvab: Armed Services Vocational Aptitude Battery. born-US: =1 if born in the United
States and =0 otherwise. hh-sze: size of household. minority: =1 if black or hispanic and =0
otherwise. Socio-low: =1 if student reported to be of low socioeconomic status. Socio-high: =1 if
student reported to be of high socioeconomic status.
Table 6 NLSY79: marginal effect of socioeconomic status on dropout probability
coef.
std. err.
male
0.0377
0.0412
afqt
-0.0047
0.0009
home-abroad
0.1059
0.0979
city
0.014
0.052
siblings
0.0029
0.0108
country-mother
0.0616
0.0752
minority
0.0656
0.0523
socio-low
0.1576
0.063
socio-high
-0.0588
0.0468
# of obs.
635
Sociolow=Sociohigh prob > χ2
0.03
To compute the marginal effects we run a logit regression on the probability of dropping out. Male:
=1 if male . afqt: Armed Forced Qualification Test. home-abroad: =1 if born in the United States
and =0 otherwise. city: =1 if living in a city and =0 otherwise. siblings: numbers of siblings.
country-mother: =1 if mother was born in the United States and =0 otherwise. minority: =1 if black
or hispanic and =0 otherwise. Socio-low: =1 if student reported to be of low socioeconomic status.
Socio-high: =1 if student reported to be of high socioeconomic status.
29
Table 7 NLS-72: marginal effect of socioeconomic status on dropout probability
coef.
std. err.
male
0.017
0.0192
rank
0.0004
0.0000
minority
-0.0375
0.0275
socio-low
0.0256
0.0298
socio-high
-0.138
0.0211
# of obs.
2705
Sociolow=Sociohigh prob > χ2
0.00
To compute the marginal effects we run a logit regression on the probability of dropping out. Male:
=1 if male. Rank: ratio of rank in high-school senior class to total class size. Minority: =1 if race is
not white. Socio-low: =1 if student reported to be of low socioeconomic status. Socio-high: =1 if
student reported to be of high socioeconomic status.
Table 8 NLSY97: effect of socioeconomic status on time to dropout
coef.
std. err.
male
-0.0379
0.1866
asvab
0.0138
0.0042
born-US
0.4049
0.338
hh-size
-0.0059
0.074
minority
0.5586
0.2187
socio-low
-0.3793
0.2569
socio-high
0.3365
0.2352
cutoff 1
-0.8217
0.6047
cutoff 2
0.5858
0.5983
cutoff 3
1.3686
0.6025
cutoff 4
1.9868
0.6070
cutoff 5
2.7301
0.6135
cutoff 6
3.8652
0.6311
Pseudo R-2
0.0154
# of obs.
361
Sociolow=Sociohigh prob > χ2
0.00
Results of an ordered logit regression of the time to drop out. Male: =1 if male . asvab: Armed
Services Vocational Aptitude Battery. born-US: =1 if born in the United States and =0 otherwise.
hh-sze: size of household. minority: =1 if black or hispanic and =0 otherwise. Socio-low: =1 if
student reported to be of low socioeconomic status. Socio-high: =1 if student reported to be of high
socioeconomic status.
30
Table 9 NLSY79: effect of socioeconomic status on time to dropout
coef.
std. err.
male
0.4057
0.2935
afqt
0.008
0.0063
home-abroad
-0.0366
0.7184
city
-0.0619
0.4248
no-mother
0.9776
1.1664
siblings
-0.0518
0.0652
country-mother
-0.1982
0.5102
minority
-0.3258
0.3412
socio-low
-0.9225
0.3871
socio-high
0.0343
0.3631
cutoff 1
-2.684
0.9707
cutoff 2
-0.7625
0.933
cutoff 3
0.1705
0.9329
cutoff 4
0.7995
0.9365
cutoff 5
1.6166
0.946
cutoff 6
2.4725
0.9694
Pseudo R-2
0.0326
# of obs.
158
Sociolow=Sociohigh prob > χ2
0.00
Results of an ordered logit regression of the time to drop out. Male: =1 if male . afqt: Armed Forced
Qualification Test. home-abroad: =1 if born in the United States and =0 otherwise. city: =1 if living
in a city and =0 otherwise. siblings: numbers of siblinigs. country-mother: =1 if mother was born in
the United States and =0 otherwise. minority: =1 if black or hispanic and =0 otherwise. Socio-low:
=1 if student reported to be of low socioeconomic status. Socio-high: =1 if student reported to be of
high socioeconomic status.
31
Table 10 NLS72: effect of socioeconomic status on time to dropout
coef.
std. err.
male
0.2148
0.0911
rank
-0.0008
0.0003
minority
0.3444
0.1225
socio-low
-0.425
0.1339
socio-high
0.5285
0.1023
cutoff 1
-0.3672
0.0926
cutoff 2
0.9031
0.095
cutoff 3
1.3934
0.0988
cutoff 4
2.2977
0.1125
cutoff 5
3.3964
0.1492
cutoff 6
4.099
0.1917
Pseudo R-2
0.0116
# of obs.
1610
Sociolow=Sociohigh prob > χ2
0.00
Results of an ordered logit regression of the time to drop out. Male: =1 if male. Rank: ratio of rank
in high school senior class to total class size. Minority: =1 if race is not white. Socio-low: =1 if
student reported to be of low socioeconomic status. Socio-high: =1 if student reported to be of high
socioeconomic status.
32
@FedFRASER