1. education
2. graduate school
3. college

The Option Value of Human Capital
Higher Education and Wage Inequality
Donghoon Lee∗
Sang Yoon (Tim) Lee†
Yongseok Shin‡
March 2015
Abstract
We study college enrollment and graduation decisions in the presence of heterogeneity and risk in individuals’ returns to college. College education comes with two
inherent options: (i) college enrollees may quit after obtaining additional information
on their post-graduation wages (i.e., college dropout) and (ii) college graduates may
take jobs that do not require a college degree (i.e., underemployment), effectively protecting themselves from the left tail of the returns-to-college distribution. We show
that the interaction between these option-like features and the rising dispersion in
the returns-to-college distribution—evidenced by the rising wage dispersion especially
among college graduates—is key to understanding the muted response of college enrollment and graduation rates to the substantial increase in the observed college premium
in the United States since 1980. Once taken into account, we find that a college subsidy
inducing marginal students to enroll in college is not cost-justified, despite a positive
marginal returns to enrollment.
Keywords: College enrollment, college dropout, college premium, underemployment
∗
Federal Reserve Bank of New York; [email protected].
University of Mannheim; [email protected].
‡
Washington University in St. Louis, Federal Reserve Bank of St. Louis and NBER; [email protected].
†
1
Introduction
In the early 1980s United States, men with at least four years of college education
earned 40 percent more on average than those whose education ended with high school.
This figure, termed the “college wage premium,” rose to almost 90 percent in 2005.
Despite this steep rise in the college premium, the fraction of men with a four-year
college degree in the overall population barely changed over the same period.
We analyze this phenomenon using a model where a student’s wage premium attributable to college education—i.e., return to college—is individual-specific and not
fully known at the time they make college enrollment and graduation decisions. We
emphasize two options built into college education: (i) college enrollees may quit after
obtaining additional information on their post-graduation wages (i.e., college dropout)
and (ii) college graduates may take jobs that do not require a college degree (i.e.,
underemployment). These two options effectively protect college students from the
left tail of the returns-to-college distribution. In addition, with the options, even a
mean-preserving spread in the returns-to-college distribution translates into a rise in
the observed college premium, because lower realizations are never observed. We find
that the increasing dispersion in the returns-to-college distribution—as evidenced by
the increasing wage dispersion overall and especially among college graduates—and
how it interacts with the two options are key to understanding the joint evolution of
educational attainment and the wage distributions conditional on education between
1980 and 2005. In particular, the rise in the observed college premium is driven more
by the increase in the second moment of the returns-to-college distribution rather than
an across-the-board increase in the population returns to college: A strong increase in
the mean returns-to-college would have resulted in a substantial rise in both college
enrollment and graduation.
In the Current Population Survey (CPS), we observe three striking facts since the
1980s. First, among prime-age (26-to-50) white males, the wage of those with four-year
college degrees increased by significantly more relative to high school graduates’ (the
well-documented rise of college wage premium) and also relative to that of those who
have some college education but no four-year degree. Second, the fraction of those
with four-year college degrees remained more or less flat over the same period. In
the data, this fact is an outcome of a slight increase in the fraction of high school
graduates who enter college (the enrollment rate) being offset by a decrease in the
fraction of college students who eventually earn a four-year degree (the graduation
rate). Lastly, wage dispersions within all education groups increased significantly, and
2
especially more among college graduates. This last fact has rarely been noted in the
literature. These facts form the empirical basis for our paper.
To consider all these facts together, we build a model where individuals differ in their
returns to college—i.e., the wage premium resulting from their college education—and
make sequential decisions to first enroll in, and then graduate from, college. An important assumption is that students do not fully know their individual-specific returns
until they enter the labor market, and hence both college enrollment and graduation
are risky investments in their human capital.
However, we emphasize that going to college comes with two important options that
render a standard risk-return trade-off calculation misleading. For one, a student can
learn more about his individual-specific return to college while in college, and decide
whether to continue (and earn a four-year degree) or to drop out. This assumption
that students drop out of college when they learn that their returns are lower than
they expected separates our model from the few existing papers of college dropouts, in
which students drop out for reasons that are orthogonal to their returns to college (e.g.,
preference shocks). The second option is that a college graduate, upon learning that his
college education is not useful for gainful employment, may choose to take a job that
does not require a college degree. We call this latter option “underemployment,” which
is a new modeling element we are contributing to the literature. The underemployment
option protects a college graduate from very low realizations of his returns to college.
The most novel and important implications of our model arise from the interaction
between these option-like features of college education and an increase in the dispersion of the returns-to-college distribution. In our model, a mean-preserving spread in
returns to college results in an increase in the observed returns to college or college
premium ex post. This is because the larger dispersion in the returns to college affects
the observed wage distribution of college graduates asymmetrically. Those with higher
returns to college do graduate and pull the observed college wage premium higher.
However, those who learn that their own returns to college are lower than expected
either drop out of college (and hence out of the calculation for the average wage of
college graduates) or become underemployed upon graduation (which prevents them
from falling into the left tail of the college returns distribution).
Our quantitative analysis shows that the increase in the observed college wage
premium since 1980 can be explained by the rising returns-to-college dispersion (as
evidenced by the striking increase in wage dispersion among college graduates) interacting with the option-like features of college education, more so than can be explained
by an across-the-board increase in returns to college. This in turn helps explain why
3
the fraction of men with four-year college degrees did not increase in spite of the much
higher college wage premium. First, the population mean of the returns-to-college
distribution—which has a direct, positive impact on college enrollment and graduation
rates—did not rise nearly as much as the observed college wage premium. In addition, the rising dispersion in the returns to college represents a larger risk ex ante to
risk-averse students, who do not fully know their own returns, precisely because they
are now more likely to drop out of college or become underemployed. This, along with
a moderate increase in out-of-pocket college costs, dampens the increase in college
enrollment and graduation rates caused by the higher average returns.
Several aspects of the data support our model empirically. First, the rise in college
wage premia coincides with an increase in the college dropout and underemployment
rates. Second, the wage premia for college dropouts and underemployed college graduates over high-school graduates did not increase so much compared to the college
premium (11 and 30 percentage points respectively, compared to 48 percentage points
for college graduates). Furthermore, the wage dispersion within these two groups is
also much less pronounced than the college wage dispersion, both in levels and in the
increase since 1980. The last two facts are consistent with our model, in which the
college dropout and underemployment categories are capturing the lower end of the
returns-to-college distribution.
Our modeling of college enrollment, dropout, and underemployment as a function of
individual-specific returns to college provides us with a new perspective on the debate
of whether too few or too many students are going to college.
First, our model is free of any frictions or shocks aside from imperfect information on
one’s own returns to college. Our quantitative results show that the increase in return
heterogeneity reconciles the growing college wage premium and the flat educational
attainment since 1980: It does not call for alternative hypotheses that students do not
enjoy college experiences as much as they used to or have become more pessimistic
about their own returns to college. Furthermore, everyone in the model is making
an optimal decision based on the information they have, and there is little room for
intervention at this stage.
Indeed, we assess the returns to college for a student who is indifferent between going
to college and not. Our computed returns for such a marginal student is much smaller
than the estimates of local average treatment effects using instrumental variables in the
empirical literature (Card, 1999). The key distinction is that we explicitly consider the
possibility of dropping out of college and also allow a nonlinear relationship between
years in college and returns to college. That is, if someone went to college for two years
4
and dropped out, his return to the two-year college education will be substantially less
than half of the return for an ex-ante comparable person who actually earns a fouryear degree. Our structural model explicitly accounts for selection—although imperfect
due to noisy information—and shows that, if we force marginal students to enroll in
college, more than 90 percent of them will drop out. This result is not surprising once
one considers the fact that, in our model, marginal students do not go to college to
begin with because they have low expectations on their returns to college. This suggests
that without taking these features into account, the local treatment effect of college
estimates would be biased upward, reminiscent of Heckman et al. (2006) and Carneiro
et al. (2011). In other words, what has baffled many empirical researchers—why we do
not observe students who seemingly have large potential returns attend college—may
be due to the lack of controlling for heterogeneity and non-linearity in returns.
Second, the fact that there are underemployed college graduates does not necessarily mean that too many students are going to college. It is important to note
the difference between underemployment and what the previous literature calls “overeducation.” The empirical observation is the same, that college graduates work in jobs
for which they do not need the degree. However, that literature takes this as evidence
of market frictions, such as a mismatch between worker and occupation, or students
being misinformed about aggregate labor demand (Leuven and Oosterbeek, 2011). In
contrast, we view the outcome as a rational choice made by individuals who take into
full account that they are insured from the left-tail returns-to-college risk and exercise
their underemployment option ex post. That is, underemployment is a natural outcome of the option value inherent in the risky human capital investment of going to
college.
Related Literature Our paper is related to many papers that study the relationship between college education, college premium and returns heterogeneity. Heckman
et al. (1998) explain increased college enrollment as a response to an increased premium
in the form of skill-biased technological change (SBTC), while Lee and Wolpin (2006)
extend this to include capital-skill complementarity as well. The former translates the
skill premium literally as the college wage premium and the latter relates the skill
premium to sectoral and occupational premia, which is achieved through positive correlations between college education and sectoral/occupational choice. However, both
papers focus on the schooling response to an increase in the mean difference between
high- and low-skilled workers, and do not model the implication of higher wage dispersion within education groups on college decisions, which is a primary goal of our
5
paper.
Both of the papers above and many influential but older studies in the literature
do not distinguish between college enrollment and completion, and implicitly assume
that the two are in the same direction. However, a few papers pointed out that there
was a meaningful separation between the two. Bailey and Dynarski (2011) found that
between 1980 and 2000, the college enrollment rate increased by 16 percentage points
but that the graduation rate increased by only 6 percentage points, implying that the
majority of the increased college enrollment did not lead to college graduation but to
college dropouts.
Clearly, a standard model of college decision with respect to an increasing college
premium for the entire population is inconsistent with an increasing enrollment rate
that occurs simultaneously with an increasing dropout rate as observed in the data.
We argue that larger returns heterogeneity within education groups, in addition to an
increasing wage difference between college graduates and high school graduates, can
explain both of the two seemingly contradictory outcomes of college decisions.
That individuals drop out following a negative update on their expected returns is
related to the option of continuing education as in Heckman and Urzua (2009). They
argue that continuing college has an option value due to the arrival of new information
regarding the value of a college degree, and once dropped out of a college, one can
not come back to college unless he pays a significant cost to re-enroll. Therefore, their
college option value is a continuation value relative to not enrolling at all or dropping
out prematurely. In a similar setting, Stange (2012) measures the option value as the
difference in a student’s continuation value with and without the information update,
but the information is about one’s own proficiency in succeeding in college rather than
future labor market returns, as is also the case in Stinebrickner and Stinebrickner
(2012).1
In contrast, Athreya and Eberly (2013) rationalize dropouts primarily by “failure
shocks” that exogenously oust students from college, even those who may stand to earn
high returns from college education. It is important to note that failure risk is real, as
opposed to the risk that comes from the lack of information in our model. Nevertheless,
we think that most dropout phenomena are primarily a voluntary decision, at least in
the United States; more so given the prevalence of students who enroll in two year
colleges who are classified as college dropouts in the data. Moreover, none of these
papers discuss the additional option value of college enrollment which arises from
1
The data they use is mainly on college students while they are enrolled in college so lack such earnings
data.
6
underemployment, nor explain why these options can help explain enrollment and
dropout trends over time.
Autor et al. (2008) zoom in on the difference in residual wage inequality at different
income percentiles in the CPS. Empirically, they are the most closely related to ours,
especially in that they focus on heterogeneous returns. Lemieux (2010) summarizes
that much of the rise in wage dispersion coincides with the rise in the college premium, and that much of the dispersion stems from dispersion among more educated
workers. But they do not consider the effect this may have on enrollment decisions,
nor underemployment. In a different twist, Nielsen and Vissing-Jorgensen (2007) estimate a risk-aversion parameter that reconciles education and returns outcomes in a
micro-panel using a risk-returns model, but the estimated parameter is rather high;
furthermore, if one were to use a pure risk-returns model to explain aggregate trends,
not just individual decisions, it would imply time-varying risk-aversion at the aggregate
level.
Although we focus on heterogeneous returns, we also allow for uncertainty in individuals’ future wage outcomes. While education decisions are primarily related to
heterogeneity, ex post wage dispersion also depend on the uncertainty conditional on
an individual’s returns. As such, our modeling assumptions are related to attempts to
decompose wage dispersion into individual heterogeneity and uncertainty, e.g. Cunha
et al. (2005). Our results are consistent with their finding that much of the wage dispersion is predictable from individual heterogeneity. Furthermore, we find that wage
uncertainty is much larger for college dropouts than college graduates. This is consistent with the result of Chen (2008) who finds that the marginal effect of college entry
on wage inequality is mostly explained by wage uncertainty.
There are few existing studies on underemployment (or overeducation), especially
in the U.S. Interestingly, while pessimism has been suggested as a reason why more
students do not enroll in college, optimism about aggregate labor demand has been
suggested as a reason why so many students are overeducated (Leuven and Oosterbeek,
2011). This is not essential in our model, since students are ex post underemployed
and not ex ante overeducated. Most studies, however, view the phenomenon as a skill
mismatch. In contrast, in our model it is precisely such a “mismatch” that insures the
student from being even worse off. Hence without taking this insurance into account,
the efficiency/welfare losses from such frictions would be overestimated.
In a recent paper, Clark et al. (2014) use the NLSY79 to estimate the duration
and wage effects of overeducation in a statistical model. They find high persistence
in overeducation status and a lasting negative wage effect from past overeducation
7
status as well. This is consistent with our modeling assumption that even though
underemployment is a choice, the lifetime wages of underemployed college graduates
are significantly lower than those who are not.
2
Empirical Facts
Our primary data source is the Integrated Public Use Microdata Series of the Current
Population Survey March supplement (IPUMS CPS). Since 1962, The CPS collects
detailed data on earnings, education, and occupation for a large sample of U.S. households. There are two shortcomings. The first is that the CPS is a repeated cross-section,
so we cannot follow individuals through their lifecycles. The second is that the CPS
has continuously revised variable definitions, in particular those regarding earnings,
education and occupation, making it difficult to create a consistent time-series across
our variables of interest.
There is not much we can do about the first problem, except that for all moments
we control for age to minimize demographc composition effects. But fortunately the
dataset has been used extensively in the literature so there are some well-accepted
conventions to side-step the second problem. In particular, we closely follow the datacleaning methods delineated in Autor et al. (2008), henceforth AKK. For details refer
to Appendix A.1. One important distinction we make is that in contrast to them, we
focus on all workers including the self-employed and farmers, as we do not think the
college decision is made only in anticipation of becoming a wage-worker.
First, we show that educational attainment trends have been more or less flat from
1980 onward, while the earnings premia for college has been steadily increasing. We
then present evidence that understanding returns heterogeneity within each education
category, and the option value of education—namely, the option to discontinue and
the option to become underemployed—can help jointly understand these trends.
All data are based on white males aged 26-50, controlling for age demographics (all
age groups are given equal weights).
2.1
Educational Attainment
Figure 1 shows educational attainment rates for white males aged 26-50 for each year
from 1965 to 2007. The acronyms HSD, HSG, SMC, CLG, and GTC stand for, respectively, high school dropouts, high school graduates, some college, college graduates,
8
100
80
60
40
0
20
Percent (%)
1965
1970
HSD
1975
1980
1985
1990
Survey Year
HSG
1995
SMC
2000
CLG
2005
GTC
Fig. 1: Educational Attainment
White Males, ages 26-50. HSD=high school dropout, HSG=high school graduate, SMC=some college,
CLG=college graduate, GTC=greater than college.
and greater than college.2 The rather abrupt discontinuity at the HSG and SMC
margin between 1991-1992 is due to a change in the educational coding convention in
the CPS—prior to 1992, the CPS only records the respondent’s completed years of
schooling, with no information of whether a degree was obtained.
While there is a clear increasing trend for both high school and higher categories
until the early 1980s for males, all categories have remained remarkably stable from
the late 1980s and onward. In particular, high school dropouts, at less than 10% of the
entire sample, seems to have reached a minimum. This is consistent with Taber (2001),
who finds that the important margin pre-1970s was whether an individual finished high
school; post-1970s it becomes college. In this paper, we are not interested in what this
minimum level is, so we simply drop them for the rest of the analysis.
It is worthwhile noting that among those who enroll in college, almost half do not
obtain a 4-year degree—SMC is defined as both those who obtain 2-year degrees, or
dropouts from 2-year or 4-year institutions. Another small but significant portion of
enrollees continue to obtain GTC degrees. In short, there a large heterogeneity among
2
For the female counterpart of 1, refer to Figure 13 in Appendix D. Females have a similar trend for
HSD; it decreases to a minimum of 10% in the mid-1980s. However, the share of college and above categories
have been steadily increasing throughout the observed period.
9
100
Earnings Premia (%)
40
60
80
20
0
1965
1970
1975
1980
1985
1990
Survey year
SMC Premium
1995
2000
2005
CLP Premium
Fig. 2: College Earnings Premia
White Males, ages 26-50. SMC=some college, CLP=college or more
college enrollees in terms of detailed educational attainment.
In the next subsection, we show that the college premium has been rising throughout
the entire observed period, which is difficult to reconcile with the plateauing of the
college and above categories. Then, we will argue that the option to discontinue
education, and the heterogeneous returns among college graduates can help jointly
explain educational attainment and earnings trends.
2.2
Earnings Premia
In Figure 2, we plot the earnings premia of college dropouts and graduates in comparison to high school graduates from 1979 to 2007. For reasons that will become
apparent in the next subsection, we lump individuals with GTC degrees together with
college graduates and dub them “CLP,” for “college plus.” There are two clear trends.
First, college dropouts earn a nonnegative premium, which has been surprisingly flat
at approximately 20% throughout the period. Second, the college graduate premium
is significantly higher, and has been increasing at a close to linear trend.
In a simple Roy model where individuals simply choose the category with the highest
returns, the steep rise in the college premium is difficult to reconcile with the flat
educational attainment trends of the previous subsection. This has been the subject of
10
much debate, and most of the earlier literature focused on non-market factors such as
the minimum wage (Card and DiNardo, 2002) or canonical supply-demand frameworks
(Katz and Murphy, 1992; Autor et al., 2008).
Recall that almost half of all college enrollees choose to discontinue. Hence, even in
the face of rising college graduate premium, as long as the returns to college dropouts
remain low, individuals at the margin who expect to discontinue upon enrolling have
no reason to enroll. We show in our quantitative exercises that indeed, this effect can
explain the anemic response of educational attainment despite the steep rise in earnings
premia.
In addition, more recent studies have focused on residual inequality, most notably
Taber (2001); Lemieux (2010) and AKK. In Taber (2001), this is explained as an
increase in the heterogeneity of unobserved skill heterogeneity. AKK does not offer a
structural explanation, but rather focus on the the lower and upper tails of the residual
earnings distribution, after controlling for observable characteristics. They identify a
rising 90/10 ratio throughout their observation period (virtually same as ours), but
further find that most of this is due to a rise in the 90/50 ratio, while the 50/10
ratio has all but flattened out. Figure 3 shows the 90/10, 90/50 and 50/10 residual
earnings ratio in our sample, where residual earnings is computed by controlling only
for education and age. Despite the fewer number of controls, our data is consistent
with the findings in AKK.
In our paper, we do not offer a structural explanation of why the observed inequality
trends look as such, and take the returns distribution as exogenously reduced-form.
Regardless, the evidence suggests that we can gain a better understanding of education
and earnings trends by looking at heterogeneity rather than average differences, which
we explore in the next subsection.
2.3
Residual Wage Dispersion
In Figure 4, we show the variance of log earnings for each educational attainment
category we study. Note that, while the dropout log earnings variance closely tracks
that of high school graduates, the college log earnings variance is not only larger but
also increases more rapidly. This suggests that part of the rising earnings gap across
educational attainment groups may be attributed to the differential rise in the earnings
inequality within groups, in particular the college graduates.
This is also confirmed by the Theil index decomposition in Table 16: most of
inequality is explained not by between group, but within group inequality. In turn, most
11
2.5
Residual Earnings Ratios
1.5
2
1
1965
1970
1975
1980
1985
1990
Survey year
90/10
1995
90/50
2000
2005
50/10
Fig. 3: Residual Earnings Ratios
.15
.2
Log Earnings Var
.25
.3
.35
.4
White Males, ages 26-50. 90/10, 90/50 and 50/10 denote, respectively, the percentile residual earnings ratios
after controlling for education and age.
1965
1970
1975
1980
1985
1990
Survey year
HSG
1995
SMC
2000
2005
CLP
Fig. 4: Log Earnings Variance
White Males, ages 26-50. HSG=high school graduate, SMC=some college, CLP=college or more.
12
100
80
60
40
0
20
Percent (%)
1980
1985
HSG
1990
1995
Survey Year
SMC
2000
CUE
2005
CLJ
Fig. 5: “Dropout” and Underemployment Rates
White Males, ages 26-50. HSG=high school graduate, SMC=some college, CUE=underemployed college
graduate, CLJ=college graduate with college job.
of the within group inequality is explained by the inequality in the college and above
groups, and more so in 2005 than 1980. That much of wage dispersion is explained by
higher educated workers has also been noted in Lemieux (2010).
AKK suggest that a polarization of skills demanded at the occupation level can
explain the puzzling trends between educational attainment and earnings. Even within
the same education categories, we can see rising earnings premia without much action
at the bottom tail if only higher skill occupations become more demanded and skills
are difficult to substitute.
In our model, we do not take a stance on why the upper tail has become more dispersed, and simply take it as given that different education categories have differential
dispersions. It could be that high skill jobs within higher education groups became
more demanded as in AKK, or it may well be that it reflects further returns to even
higher education categories, since we know there is also a graduate school premium.
However, with more dispersion, we still need to understand the flattening of the
lower tail. We introduce a simple new concept that captures this fact: underemployment, i.e., working in a job for which you did not need the higher education.
13
2.4
Underemployment
Education is hierarchical, in the sense that when an individual attains higher education,
it does not mean that his lower education qualifications are discarded or rendered
irrelevant. In this sense, much of the previous literature that use a Roy-type model
to model discrete education choices is misleading in the face of imperfect information.
When an individual who attains higher education realizes ex-post that he would have
been better off without it, he is not doomed to the lower returns he gets in a higher
education job—he can simply go back and get the individually higher returns in a
lower education job. The opposite is not true, namely, a lower educated individual
cannot ex-post choose a job with a higher qualification without paying an additional
cost (either in terms of time or goods, or both).
We label the phenomena of dropouts or college graduates exercising this option
value “underemployment.” To capture underemployment empirically, we look at training and education requirements by detailed occupation, tabulated by the Bureau of
Labor Statistics (BLS) as part of their Employment Projection (EP) program. By
matching the educational attainment and occupations of individuals in the CPS to the
requirements table, we can identify underemployed dropouts and college graduates.
The EP publishes these requirements biennially for detailed occupations to the 3digit level, projecting into available jobs 10 years out. Both the CPS and EP have
changed their OCC coding conventions year-to-year, and we use BLS-provided crosswalks to be consistent both across time and datasets. Still, since crosswalks do not
provide a perfect matching across different coding conventions, we are forced to drop
several observations. We use the 1998 requirements table as it is earliest available table
that requires the least number of crosswalks. For details, refer to Appendix A.2.
In Figure 5, we show educational attainment from 1979 onward, now separating
underemployed college graduates (CUE) from those with college jobs (CLJ). The CUE
group is also quite stable across the entire sample period, comprising approximately
10% of the sample annually. Hence, even when the college graduate premium exceeds
that of dropouts, individuals at the margin who expect to become underemployed upon
graduation have little reason to graduate. The underemployment premium is somewhat
higher than the dropout premium with a slightly upward trend, as seen in Figure 2.4.
But clearly, it is far below the college graduate premium with a flatter trend. Since
these individuals are working in similar occupations to their high school counterparts,
these empirical trends can help explain the flattening 50/10 gap.
In our model, there is positive selection into college enrollment and graduation, so
14
150
Earnings Premia (%)
50
100
0
1980
1985
1990
1995
Survey year
SMC Premium
CUE Premium
2000
2005
CLJ Premium
Fig. 6: Underemployed Earnings Premia
White Males, ages 26-50. HSG=high school graduate, SMC=some college, CUE=underemployed college
graduate, CLJ=college graduate with college job.
on average even those who end up dropping out or being underemployed do earn a premium. Consistently with the data, this premium is far below the graduate premium.
In particular, our quantitative results show that much of the premia trends can be explained by returns becoming more dispersed, and that dropouts and underemployment
can account for educational attainment trends.
3
Model
A period is two years. Individuals begin education decisions at age 19, or s = 1 and
enter the labor force by age 23, or s = 3. Everyone is assumed to work till age 65, or
R = 24, and die at age 75, or T = 29.
We denote by zi the variable that controls one’s returns to college. This is a reduced
form representation of the ability or human capital acquired from early childhood
through high school, the quality of the higher education institution you can attend,
the match quality between the individual and the institution, etc. We assume that zi is
only fully revealed whenever the individual enters the labor market. Upon labor market
entry, they draw a once-and-for-all wage from an education-specific distribution. No
decisions are made afterward, other than consumption/saving decisions.
15
e
llegtion
o
c
e
Pr orma
Inf
Prior Enroll
s
urn
t
e
r ed
ue al
Tr reve
al
n
Sig
Continue College job
HS grad
Dropout Underemployed
Fig. 7: Education Decision Timeline
When individuals are 19 years old, or s = 1, they graduate from high school and
begin with assets ai and a prior about their zi , based on information received up to
then. If they choose to directly enter the labor market, they draw a wage wi from a
high school wage distribution Gh (wi ), which is independent of zi .
If they enroll in college, they pay a 2-year college cost and receive a signal zˆ at the
end of their second year, or s = 2 (21 years old). If they choose to dropout and enter
the labor market, they draw a wage from a dropout wage distribution Gd (wi |zi ), which
now depends on zi . If they continue and graduate, they pay the college cost and make
draw a wage from a college wage distribution Gc (wi |zi ).
Betts (1996) offers some evidence that students only learn about their individual
labor market returns upon actual entry into the market, justifying are assumption
that the wage draw occurs only after education decisions are final. Betts (1996) and
Arcidiacono et al. (2010) present evidence that more advanced students know more
about their labor market prospects, while many recent papers including Stinebrickner
and Stinebrickner (2012) and Stange (2012) offer detailed evidence that students learn
more about their chances of succeeding in college with time. The signal zˆ plays the
role of the arrival of more information in the latter two models.
3.1
Heterogeneous Returns and Signals
We assume that the population distribution of returns is normal:
σz2 2
zi ∼ N µz − , σz .
2
and that individuals also begin with a normal prior:
N µz1i , σz21i .
16
Moreover, we will assume that σz21i = σz21 is identical across individuals,3 but µz1i itself
is again distributed normally among the population:
σ2
µz1i ∼ N bz + µz − z , σµ2 z ,
1
2
where bz is a bias in beliefs compared to the population mean of zi . If bz > 0, students
are overly optimistic on average. Otherwise they are pessimistic.
We will also assume that initial assets are distributed lognormal:
σ2
log a0i ∼ N µa − a , σa2 ,
2
so the population distribution at s = 1 is trivariate normal in log assets, abilities and
the individual beliefs of their means of their priors.
The signal zˆi received at s = 2 is
zˆi = zi + i ,
i ∼ N 0, σ2 ,
(1)
where i ’s are i.i.d. across individuals, and for a given individual, zi and i are mutually independent. Those in college use Bayesian updating to obtain the posterior
distributions of z after observing the signal zˆi . The posterior is N µz2i , σz22 , where
µz2i ≡
σ2 µz1i + σz21 zˆ
σz21 + σ2
σz22 ≡
σz21 σ2
,
σz21 + σ2
(2)
where clearly σz22 are also identical across individuals since (σz21 , σ2 ) are by assumption.
Unless otherwise noted, we will suppress the individual subscripts i.
3.2
Education Specific Wage Distributions
Let (ωh , ωd , ωc ) denote auxiliary random variables such that
2
2
2
σd 2
σh 2
σc 2
ωh ∼ log N − , σh , ωd ∼ log N − , σd , ωc ∼ log N − , σc ,
2
2
2
(3)
and denote the c.d.f.’s by Fh , Fd , and Fc , respectively. Now denote high school, dropout
and college wages by the random variables (wh , wd , wc ) with associated c.d.f.’s Gh (wh ),
Gd (wd |z), and Gc (wc |z), where we will construct these distributions below. Note that
3
We have also tried letting individuals have heterogeneous beliefs of the variance, but this had virtually
zero effect on our moments of interest.
17
we assume Gd and Gc depend on z, but not Gh . In particular, we assume that Gh = Fh ,
which means that wage dispersion among high school graduates is entirely explained
by uncertainty. Cunha and Heckman (2007) find that most of the increase in wage
dispersion among high school graduates can be explained by an increase in uncertainty,
so this is not an unreasonable assumption.
We build the idea of underemployment into the model by making assumptions on
Gd (w|z) and Gc (w|z). We assume that the dropout wage is determined by
wd = md · max {exp(z)ωd , ωh } ,
so Gd (wd |z) is the distribution of the maximum of two normal random variables. In
other words, dropouts first draw ωd , which transforms into a potential wage of exp(z)ωd .
The fact that high school jobs are always available to them is captured by assuming they
make a second, independent high school wage draw ωh . They then take the job with the
larger wage, which is discounted by a factor of md , reflecting a partial return to college
enrollment. If md < 1, returns to college are lower for those enrollees who drop out
than those who graduate. In Figure 8 we visualize how the individual-specific college
dropout wage distributions are constructed, for an individual with average returns in
the left panel and an individual with returns one standard deviation above the average
in the right.
Similarly, we assume that the college wage is determined by
wc = max {exp(z)ωc , mu wd } = max {exp(z)ωc , mu md exp(z)ωd , mu md ωh } ,
(4)
so Gc (wc |z) is the distribution of the maximum of three normal random variables. In
other words, college graduates first draw ωc , which transforms into a wage of exp(z)ωc .
They compare this with an independent draw wd from Gd (wd |z), which is discounted
by a factor of mu , reflecting partial returns to college completion. They then take
the job with the larger wage. Since wd itself is the maximum of a dropout and high
school wage, the college graduate is effectively taking three wage draws and choosing
the largest one.
We show how to solve for the exact dropout and college wage distributions in
Appendix B, which is needed for the numerical analysis,4 and Figure 9 visualizes the
individual specific college graduate wage distribution.
4
Technically, we could include a third parameter mdu to capture partial returns to underemployed
dropouts. We choose not to do so because not only is this a small category for which the data we have is
very noisy, but also it does not play much of a role in our analysis because the empirically observed dropout
premium is quite low.
18
2.5
2.5
2
2
1.5
1.5
1
1
0.5
0.5
0
0
1
2
0
0
3
(a) z = µz
1
2
3
(b) z = µz + σz
- - High School Job
- - Some College Job
— gd (w|z)
Fig. 8: Some College Wage Distribution
College dropout wage distribution for an individual whose returns are exactly at the population mean, and
one standard deviation above the mean. x-axis is multiples of mean high school wage. The distributions
are determined by considering a high school job, which would lead to underemployment, and a some college
level job. The distribution is skewed toward the top, and more so for individuals with higher z.
Admittedly, we are losing some information from the data by ignoring life-cycle
income dynamics. However, the average college graduate is underemployed for approximately 12 out of 28.9 years in the NLSY79. In the NLSY97, the number is 8.1
out of 15.6 years. A simple Mincer regression reveals that the underemployment premium is approximately half of the college premium for college graduates, both in the
NLSY79 and 97. This is slightly larger than what we obtain from the CPS, but still
quantitatively relevant.
3.3
Individual’s Problem
An individual’s state at age 19, or s = 1, is his initial assets and prior on his returns
to college. Since we assume the prior is normal and that the variance, σz21 , is identical
across all individuals, we can suppress the individual subscripts i so the state vector is
(a0 , µz1 ). Based on this, he chooses whether to enroll in college or not. If he enrolls, he
must pay the expenses for the first period (two years) of college, x1 , and choose next
period assets, a1 . His value function can be written as:
(
V1 (a0 , µz1 ) =
max
work,school
Vh (a0 ),
(5)
19
3.5
3.5
3
3
2.5
2.5
2
2
1.5
1.5
1
1
0.5
0.5
0
0
1
2
0
0
3
1
(a) z = µz
- - High School Job
2
3
(b) z = µz + σz
- - Some College Job
- - College Job
— gc (w|z)
Fig. 9: College Graduate Wage Distribution
College graduate wage distribution for an individual whose returns are exactly at the population mean, and
one standard deviation above the mean. x-axis is multiples of mean high school wage. The distributions are
determined by considering a high school or some college job, which would lead to underemployment, and a
college level job. The distribution is more skewed toward the top than the some college distribution, and
more so for individuals with higher z.
Z
max u((1 + r)a0 − a1 − x1 (1 − v(a0 ))) + β
a1
)
V2 (v(a0 ), a1 , µz2 )dF1 (µz2 |µz1 ) ,
which induces the optimal policies
χE (a0 , µz1 ),
a∗1 (a0 , µz1 ),
(6)
where χE = 1 if the individual decides to enroll (chooses “school”) and 0 otherwise,
and a∗1 (·) is the optimal savings function if he chooses to enroll.
In (5), next period’s state µz2 evolves according to the Bayesian updating formula
(2) and F1 (·|µz1 ) is the c.d.f. of µz2 , not z, conditional on µz1 . The terminal value
Vh (a0 ) is the expected value for a high school graduate who begins working with assets
a0 , which we characterize in more detail below. The continuation value V2 (a1 , z) is
the value at the beginning of the next period if he enrolls. Note that there is no
opportunity for re-enrollment after he starts to work; entering the labor market is an
absorbing state. If re-enrollment were allowed, there would also be an option value
after entering the labor market (Johnson, 2013; Arcidiacono et al., 2013). However,
while there is strong evidence of delayed entry or re-enrollment, and more so in the
NLSY97 than NLSY79, more than 95% of enrollees who dropout do so by age 25, and
those who graduate, by age 26. Given that our focus is on lifetime wage distributions
20
at the aggregate level, we do not expect that ignoring these small gaps in education
decisions would have a large quantitative impact on ex-post outcomes. We discuss this
issue again in Section 7.
The function v(a0 ) is a grant function that we assume to be declining step function
with three possible values,


if a0 ≤ a
¯1
v1
v(a0 ) = v2
if a
¯1 < a0 ≤ a
¯2


v3
if a
¯2 < a0 ,
where 0 < v3 < v2 < v1 < 1, so that in fact all students do receive positive level of
grants. The level of grants depends only on his initial, not current, level of assets, so
is fixed throughout college.
If he enrolls in college, he receives the signal zˆ after one period (at s = 2). Based on
this and his current assets, he decides whether to drop out or to complete his college
education. If he continues and graduates, he pays the expenses for the second period of
college x2 and chooses next period assets a2 , but also receives grants v which depends
on his previous period assets a0 . His value is
(Z
V2 (v, a1 , µz2 ) =
max
work,school
Vd (a1 , z)dF2 (z|µz2 ),
(7)
)
Z
max u((1 + r)a1 − a2 − x2 (1 − v)) + β Vc (a2 , z)dF2 (z|µz2 ) ,
a2
which induces the optimal policies
χG (v, a1 , µz2 ),
a∗2 (v, a1 , µz2 ),
(8)
where χG = 1 if the individual decides to graduate (chooses “school”) and 0 otherwise,
and a∗2 (·) is the optimal savings function if he chooses to graduate.
In (7), the distribution function F2 (·|µz2 ) is the posterior c.d.f. of z given (µz1 , zˆ)
as defined by Bayesian updating (2). The values Vd (a1 , z), Vc (a2 , z) are the expected
values for a drop out or college graduate who begins working with assets a1 or a2 , and
ability z. Both terminal values are characterized in more detail below. Note that his
returns are fully revealed only when he drops out or later on if he graduates, but not
if/when he decides to continue.5
At any point when he starts working, his z is revealed and he cannot he cannot go
back to school. The values if he enters the labor market at s = 1, 2 or 3 with assets a
5
These assumptions are consistent with Betts (1996); Arcidiacono et al. (2010).
21
are, respectively,
Z
Vh (a) =
V (s = 1, a, w)dGh (w)
(9a)
V (s = 2, a, w)dGd (w|z)
(9b)
V (s = 3, a, w)dGc (w|z).
(9c)
Z
Vd (a, z) =
Z
Vc (a, z) =
When an individual starts working for the first time, be it after high school, two, or
four years in college (s ∈ {1, 2, 3}), he draws a wage w from his education-specific
distribution, and solves a deterministic consumption-savings problem. He can borrow
and save at a given interest rate subject to the natural lifetime borrowing constraint
aT +1 ≥ 0. Given the (two-year) interest rate r and the discount factor β, we can derive
the continuation utility of a worker who starts working at s (ages 19,21 or 23), works
until R = 24 (age 65) and lives until age T = 29 (age 75), i.e.


T
X

V (s, a, w) = max
β j−s u(cj )


j=s
subject to
T
X
j=s
cj
= (1 + r)a + max {w · eh (s), w · er (s)} ,
(1 + r)j−s
(10)
where
eh (s) ≡
R
X
j=s
yh (j)
,
(1 + r)j−s
er (s) ≡
R
X
j=s
1
(1 + r)j−s
(11)
for s ∈ {1, 2, 3} and yh (j), j = 1, · · · , 24, is the average age-earnings profile of a high
school graduate with earnings at s = 1 normalized s.t. yh (1) = 1. We are also assuming
a minimum wage w that applies equally to all individuals, which an individual can never
do worse than for his entire working life. The functions [eh (s), er (s)] transform hourly
wages w into a present-discounted sum of lifetime earnings, evaluated at period s. This
renders all wages comparable in terms of lifetime average values.
Assuming CRRA preferences
u(c) =
c1−γ
1−γ
we can easily solve out for V (s, a, w) in closed form:
V (s, a, w) = κc (s)γ u((1 + r)a + κw (s))
22
where
h 1
i
1
−1 T −s+1
1 − β γ (1 + r) γ
κc (s) =
1
1
1 − β γ (1 + r) γ
4
−1
,
κw (s)
= max {w · eh (s), w · er (s)} .
Calibration
We calibrate the model to 1980 U.S. as a benchmark, and also to 2005 as a comparison.
In Section 5 we illustrate the quantitative changes in parameters from 1980 to 2005
that generate our results.
Each steady state has a total of 26 parameters of which 15 are fixed. We first
describe the data used to fix these parameters, and then the empirical moments targeted to calibrate the rest. Then we explain how to use the model to compute the
counterparts to these empirical targets.
Whenever applicable, all data are based on a cross section of white males aged
26-50, from the 1980 and 2005 IPUMS CPS, and all dollar values are deflated to
2000 USD using the chain-weighted (implicit) price deflator for personal consumption
expenditures (GDP PCE deflator) published by the BEA. Note that the true returns
distribution is known to all individuals, they only lack knowledge of their individual
position. This means that individuals make their education choices based on the current
distribution of earnings, without taking into account how the distribution may change
in the future. A recent paper by Dillon (2015) finds that it is the education wage premia
at the time a student graduates from high school, and not his cohorts future realized
gains, that affects his education decisions, justifying our steady state assumption.
4.1
Fixed parameters
The preference parameters (β, γ) are fixed to the standard values of (0.962 , 2), and
interest rate to r = 1.042 − 1. The discount factor and interest rate are compounded
since a period is two years.
In the model, mean high school graduate wages at s = 1 is normalized to 1. Since
a period in the model corresponds to 2 years, we use average wages of 19-20 year-olds
to normalize all wages in the data, discounting age 20 wages by 1 year. Specifically,
the average 2-year PDV wage for high school graduates at age 19 is
w
¯h = w
¯h19 +
w
¯h20
,
1.04
(12)
where w
¯ha , the mean wage for age a-year old high school graduates, was 19.1297 in
1980 and 19.2346 in 2005, respectively, both in 2000 USD. Since these are in hourly
23
Parameter
Value
(0.962 ,2,(1.04)2 -1)
(β, γ, r)
2
σ
wh ∼ log N − 2h , σh2
a1i ∼ log N µa −
σz21
=
σµ2 z1
Description
=
2
σa
2
2 , σa
1980
2005
σh2 =0.2012
0.2594
µa =2.5533
2.8141
σa2
= 0.8044
σz2
Standard RBC values
average HSG starting salary
µh normalized to 1
Gale and Scholz (1994)
Abbott et al. (2013)
Only σz2 calibrated within model
—
x1
0.2344
0.3227
first 2-year college cost
x2
0.6298
0.8731
last 2-year college cost
2
log a−(µa −σa
/2)
fa (a) = Φ
,
σa
Φ is standard normal c.d.f.
(¯
a1 , a
¯2 )
[fa (¯
a1 ), fa (¯
a2 )]
=(0.20,0.55)
(v1 , v2 , v3 )
(0.3262,0.0773,0.0165)
w
0.6427
0.4480
fraction of college costs
covered by public grants
2-year PDV minimum wage
Table 1: Fixed Parameters
terms and we are assuming that all individuals work full time, we not only normalize
all annual values by w
¯h but further divide them by (35 hours/week×40 weeks =) 1400
hours. The log wage variance for high school graduates, σh2 , was 0.2012 and 0.2594, for
1980 and 2005.
Initial assets in the model are meant to capture the heterogeneity in economic
support from the students’ families, mainly parents. Since borrowing constraints are
only of second-order interest and we assume a natural borrowing constraint in our
model,6 we front-load all possible transfers received from parents into initial assets.
To obtain the mean of log assets µa , we refer to Gale and Scholz (1994) who report
that average net worth in the 1986 Survey of Consumer Finances was $144,393 per
household in 1985 dollars, and as much as 63.4% of total net worth comes from intergenerational transfers. Assuming a 2-person household with 2 children, this amounts
to $68,382 of asset transfers per child in 2000 dollars. For 2005, we multiply this value
by the ratio of the average PDV of lifetime earnings for the entire population in 1980
and 2005, leading to $75,780.
To compute σa2 we use the mean and median annual inter vivos transfers among
young adults aged 16-22. These values were $1227 and $486, respectively, in the
NLSY97. This implies a positive skewness partially justifying our choice of a log
6
Many studies, in particular Lochner and Monge-Naranjo (2011) find no evidence of binding borrowing
constraints for most of our observation period.
24
Type of Institution
1980
2005
2-year public
4-year public
4-year private
789 1,965
1,641 4,925
7,170 19,046
Table 2: College Costs
All amounts in 2000 US Dollars.
normal distribution for initial assets, with σa2 = 2 log(1227/486).
We assumed earlier that σz21i = σz21 , i.e. that the variance of individual priors
are identical. In our calibration we will impose the two additional assumptions that
σµ2 z = σz21 = σz2 . It is unclear how we would separately identify the three, and this
1
assumption makes the variance of individual beliefs and the distribution of individual
beliefs consistent with the population distribution.7 We refer to “Trends in College
Pricing,” published annually by the College Board, to obtain moments on college
costs. We exclude room and boarding costs and only include tuition and fees, since all
individuals would incur living costs regardless of college attendance (whether explicitly
or implicitly). We make the first two years of college cheaper than the latter two years,
by setting x1 as the average cost of attending a two-year public, four-year public or
four-year private institution, and x2 as the average cost of attending a four-year public
or private institution.8 These costs are stipulated in Table 2. Clearly, the cost of
college has been rising over the observed period, so that average annual costs amount
to $3,200 and $8,645 for the first half of college in 1980 and 2005, respectively, and
$4,406 and $11,986 for the latter half, all in 2000 dollars.
Grants are modeled as a declining three-step function of initial wealth, a1i , that
are paid out as a fraction of college costs. Abbott et al. (2013) find that in the year
2000, students with family income below $30K received an average of $2820 in public
grants; below $80K received an average of $668, and the rest an average of $143, according to the “Guide to U.S. Department of Education Programs” published annually
by the US Department of Education. These family income thresholds correspond to
approximately the 20th and 55th percentiles of the 2000 family income distribution.
Unfortunately, this data does not date as far back as 1980, and nor does “Trends in
Student Financing of Undergraduate Education” published every four years by the Na7
We have also tried simulations without imposing this assumption, but except for extremely large or
small values there was virtually no difference.
8
While these costs clearly differ across individuals and institutions, there is evidence that they do not
vary much across family income groups, see for example Johnson (2013); Abbott et al. (2013).
25
Moment
1980
2005
Enrollment rate
Dropout rate
Underemployment rate
0.5776
0.3926
0.2352
0.6474
0.4512
0.3257
College wage premium
Dropout wage premium
Underemployment premium
0.3617
0.0575
0.1089
0.9330
0.2054
0.4674
High school log wage variance
College log wage variance
Dropout log wage variance
0.2012
0.2723
0.2024
0.2594
0.4087
0.2731
Log corr. family income and own earnings
Enrollment by family income quartile∗
College attainment by earnings quartile∗∗
0.1787
0.4075
0.2917
[0.1776]
0.4211
0.4058
∗
Difference in enrollment rates of 1st and 4th quartiles.
Difference in fraction of college graduates of 1st and 4th quartiles.
∗∗
Table 3: Target Moments
All moments for White Males, ages 26-50, CPS 1980 and 2005, except the correlation between family income
and own earnings, and enrollment rates. Three of these moments are from the NLSY79/97, except for the
2005 correlation between family income and own earnings for which data does not yet exist. We instead
show the simulated moment in brackets.
tional Center for Education Statistics.9 However, the the latter publication does not
show any systematic changes across family income quartiles in the amount of federal
or state grants received from the years 1995 to 2008, so we fix the grant function for
both years 1980 and 2005 in our model to match the year 2000. Since we assume that
initial wealth is log normal, we choose the thresholds (¯
a1 , a
¯2 ) such that
log a1 − (µa − σa2 /2)
Φ
= 0.20
σa
log a2 − (µa − σa2 /2)
Φ
= 0.55,
σa
separately for (µa , σa2 ) in 1980 and 2005, where Φ is the standard normal c.d.f. Then
we set (v1 , v2 , v3 ) as grants received in each bracket as a fraction of the average annual
college costs in 2005 for the first half of college, i.e. (v1 , v2 , v3 ) = (2820, 668, 143)/8645.
The minimum wages in 1980 and 2005 were, respectively, $6.27 and $4.39, corresponding to 2-year PDV’s of $12.29 and $8.62.
9
The U.S. Department of Education was created in 1980.
26
4.2
Calibrated Parameters
The remaining 11 parameters are chosen by solving
ˆ = arg min [M (Θ) − Md ] [M (Θ) − Md ]0
Θ
Θ
(13)
where Θ is the vector of parameters, M (Θ) are the model implied moments and Md
the empirical moments in Table 3, mostly from the figures shown in Section 2 using the
IPUMS CPS. It should be straightforward how to obtain most of these moments from
the model—we simulate the model, aggregate, and compute the relevant statistics.
Pre-college moments are computed over the distribution of (a0 , µz1 , z), which we
denote by F0 . For drop out and college moments, we need to additionally consider
the distribution of µz2 induced by the prior µz1 and signal . Here we explain how we
discipline the population distribution F0 and transform wage moments in the model to
be comparable to the data. For further details, refer to Appendix C.
Correlation Parameters The population distribution at s = 1, which we denote
as F0 , is trivariate normal in (log a0 , µz1 , z).10 Of the 12 parameters of this distribution, we have fixed 9 in the previous subsection. The remaining three parameters are
the correlations between the three variables, which we denote (ρaz , ρaz1 , ρzz1 ). The
correlation ρzz1 captures the accuracy of information received prior to s = 1, while
ρaz1 captures how this information varies with family background.
We calibrate these moments by targeting related empirical moments. Assets represent family income, z represents lifetime earnings, and µz1 , the prior, represents
whether an individual decides to enroll/graduate. Hence we target i) the correlation
between family income and PDV of own lifetime earnings in the NLSY79, ii) the difference in enrollment rates of the 1st and 4th family income quartiles in the NLSY79/97,
and iii) the difference in the fraction of college graduates of the 1st and 4th earnings
quartiles in the CPS. We keep all multiracial white individuals in the NLSY in addition
to whites.
Family income in the NLSY79 is taken as the mean of the reported values when
the youth is 16 and 17 years old. To compute the PDV of lifetime earnings, we impute
missing wage observations by linearly interpolating log wages in adjacent years, and
discount annual wages by 4% to arrive at a PDV. However, we cannot do the same for
the NLSY97 since we lack wage data in later life (surveyed youths were 12-16 years
old at the end of 1996). Hence we target the NLSY79 correlation for our model 1980,
10
Although this distribution is public knowledge, agents only know their individual (a, µz1 ) but not their
individual z.
27
60
80
Percent (%)
40
60
20
Percent (%)
40
0
20
0
quartile 1
quartile 2
NLSY79
quartile 3
quartile 4
quartile 1
NLSY97
quartile 2
1980 CPS
(a) Enrollment Rates by Family Income Quartiles
quartile 3
quartile 4
2005 CPS
(b) College attainment by earnings quartiles
Fig. 10: Enrollment and Graduation by Quartiles
Left Panel: “White” Males, NLSY79 and 97. “White” refers to all multiracial categories containing “white,”
in addition to uniracial whites. For details and definitions, see Appendix A.2 or Lochner and Monge-Naranjo
(2011). Right Panel: White Males, ages 26-50, CPS 1980 and 2005. Earnings are first controlled for age
then ordered into quartiles.
and simply fix ρaz when we recalibrate the parameters for 2005. The simulated 2005
correlation is more or less equal to 1980.
We follow Lochner and Monge-Naranjo (2011) for the definition of family income
and enrollment; for details, refer to Appendix A.2. In their paper, they stress that the
difference in enrollment rates have become larger in recent years; while there is some
evidence of this in Figure 4.2, the difference between 1980 and 2005 is not as stark.
However, we are only looking at males who are at least partially white, who would
mostly fall in the higher quartiles of the entire NLSY sample.
Lastly, we refer back to the CPS to obtain the fraction of college graduates (including higher degrees) by earnings quartiles. Since we are assuming that individual
earnings follow a deterministic age-earnings profile with a fixed log wage variance, we
form quartiles after first subtracting age-specific wage means for each year (1980 and
2005). It is evident in Figure 4.2 that there is an increase in how much of college
attainment is explained by earnings inequality, the converse of the college premium.
Education Specific Distributions and Outcomes Using the indicator functions for enrollment in (6), we can write the measure for individuals who enroll in
college as
dΛe (a0 , µz1 , z) = χE (a0 , µz1 )dF0 .
28
Similarly, the measures over individuals who enroll but drop out or graduate are
dΛd (a0 , µz1 , µz2 , z) = (1 − χG [v(a0 ), a1 (a0 , µz1 ), µz2 )]) dF1 (µz2 |µz1 ) · dΛe
(14)
dΛc (a0 , µz1 , µz2 , z) = χG [v(a0 ), a1 (a0 , µz1 ), µz2 )] dF2 (z|µz2 ) · dΛe ,
(15)
respectively. Note that extra information on µz2 is needed to write these measures.
Then the enrollment and dropout rates are simply
,Z
Z
Z
Z
dΛe =
(dΛd + dΛc ),
dΛd
dΛe ,
respectively. The college measure can be further split into those who become underemployed, and those who get college jobs. These measures can be written using the
distributions Fc and Gd defined in Section 3.2:
dΛue (w, a0 , µz1 , µz2 , z) = Fc (w/ exp(z))dGd (w/mu |z) · dΛc
(16)
dΛcj (w, a0 , µz1 , µz2 , z) = Gd (w/mu |z)dFc (w/ exp(z)) · dΛc
(17)
where we now additionally need information on w, since the underemployment decision
is made after both wage draws from Fc and Gd (·|z). so the underemployment rate is
,Z
Z
dΛue
dΛc .
In addition to education outcomes, all wage related moments, namely the premia,
wage variances, and the correlation between log initial assets and lifetime earnings, are
computed from these measures.
Computing Earnings Premia We need some normalizations to compute education wage premia, since these moments are based on ages 26-50 in the CPS while
in the model wages are in terms of lifetime earnings. Figures 14-15 in the appendix
depict education-specific average age-earnings profiles for 1980 and 2005, respectively.
The values of eh (s), s ∈ {1, 2, 3} in (11) are obtained by computing the average
present-sum of lifetime earnings of high school graduates using an annual discount rate
of 4% evaluated at ages 19, 21, and 23, respectively, and then normalizing by w
¯h . Now
define the variable q such that


for dropouts
d
q = ue for underemployed college graduates


cj for college graduates with college jobs.
29
For each q, we compute the average present-discounted sum of lifetime earnings eq
similarly as eh (s), where the present values are evaluated at the age of labor market
entry (21 for dropouts and 23 for college graduates, including those underemployed).
These values are shown in the top panel of Table 17in the Appendix along with eh (1).
These average lifetime earnings in the data are compared to average lifetime earnings in the model, which are computed as follows. Using the measures defined in (14)
and (16)-(17), we can write model average wages e¯q for q ∈ {d, ue, cj} as
R R
R
R
wdGd (w|z) dΛd
wdΛue
wdΛcj
e¯cj
e¯ue
e¯d
R
,
,
,
=
= R
= R
eh (2)
eh (3)
eh (3)
dΛd
dΛue
dΛcj
since all wages were multiplied by eh (s) in (10).
But the empirically observed premia shown in Figure 2 and Table 3 are computed
from individuals aged 26-50, controlling for demographics. To be precise, for each
category q and high school graduates, we simply take the average across all age groups
with equal weights. Denote the average hourly wage across individuals aged 26-50 in
group q as Wq , which are shown in the bottom panel of Table 17 along with Wh , the
average hourly wage for high school graduates aged 26-50.
Since we assume that wages are drawn at labor market entry and then follows a
deterministic path, the model implied premia, Pq , is thus computed as
Pq =
e¯q Wd
·
.
e q Wh
If the model explains the data accurately, we would have e¯q = eq for all q, in which
case Pq will also be exactly equal to the empirically observed premia Wq /Wh .11
As mentioned above, the mean of assets µa is renormalized according to the average
PDV lifetime earnings ratio between 1980 and 2005. Hence µa,2005 is obtained by
P
¯h,1980
q Mq,2005 eq,2005 w
exp(µa,2005 ) = exp(µa,1980 ) · P
·
,
¯h,2005
q Mq,1980 eq,1980 w
where w
¯h,t is the value of w
¯h in (12) for each year t, and Mq,t , the mass of individuals
in category q in year t, is found by integrating over the measures Λq for each year t.
5
Results
The resulting calibrated parameters are shown in Table 4 and implied edcuation choices
in Figure 16 in the Appendix. Both the mean and log variance of college returns,
(µz , σz2 ) are larger in 2005, while the residual wage variance for college graduates
11
College moments are computed as weighted averages across ue and cj moments.
30
Parameters
1980
2005
bz (“optimism”)
σ2 (noise var)
σz2 (pop. var z)
0.1612
0.6007
0.1863
0.2084
0.8677
0.2752
µz (pop. mean returns)
md (partial return, dropout)
mu (partial return, underempl.)
-0.1325
0.8579
0.5852
-0.0463
0.8679
0.6315
σc2 (var of log residual college)
σd2 (var of log residual dropout)
0.2220
0.3712
0.4126
0.4906
ρ(a, z)
ρ(a, µz1 )
ρ(z, µz1 )
0.4367
0.3922
0.6453
[0.4367]
0.4352
0.7490
Table 4: Calibrated Parameters, 1980 and 2005
All paramaters are calibrated to the moments in Table 3, except for ρ(a, z) for 2005 which we fix to its 1980
value as we lack the corresponding target moment.
roughly double, corresponding to the observed rises in education premia and educationspecific log wage variances in the data. Note also the increase in the bias parameter
bz and the signal noise variance, σ2 . In a model without heterogeneous returns, a
large drop in bz is needed to reconcile the anemic increase in enrollment with the large
rise in the college premium. In our model, however, enrollment does not rise despite
students being more optimistic on average. Similarly, sustaining a high dropout rate
would require more precise information in a model with only mean return differences.
In our model, students in 2005 know that the mean returns to enrollment have risen,
expect it to have risen even more than it actually did, and do not learn more about
themselves in college—nonetheless do not enroll as much nor graduate.12
The correlation parameter ρz,µz1 increases from 0.65 in 1980 to 0.75 in 2005, so
students have a better idea of their true returns at the enrollment stage. This fits well
with the fact that students today make their decisions more based on their expected
returns than other factors, such as distance to school (Hoxby, 2009) or a private taste
for schooling (Hendricks and Schoellman, 2014). An increase in students selecting on
their true returns would be observationally equivalent to the case when other factors
that potentially affect their schooling choices become less important. If we were to
interchange the values of ρz,µz1 for 1980 and 2005, positive selection on returns at
the enrollment stage would rise for the 1980 students and drop for the 2005 students.
Nonetheless, as we show in Table 18 in the Appendix, this change in ρz,µz1 barely
12
To be precise, the signal-to-noise ratio is 1 + σz2 /σ2 , since we assume the variance of the initial prior is
equal to the population variance of z. The values for 1980 and 2005 are, respectively, 1.3101 and 1.3171, so
are quantitatively close to equal.
31
Moment
1980
1980’
2005
Enrollment rate
Dropout rate
Underemployment rate
0.5776
0.3926
0.2352
0.9454
0.2471
0.1885
0.6474
0.4512
0.3257
College wage premium
Dropout wage premium
Underemployment premium
0.3617
0.0575
0.1089
0.9330
0.1868
0.6430
0.9330
0.2054
0.4674
College log wage variance
Dropout log wage variance
0.2723
0.2024
0.3121
0.2302
0.4087
0.2731
Log corr. family income and own earnings
Enrollment by family income quartile∗
College attainment by earnings quartile∗∗
0.1787
0.4075
0.2917
0.2815
0.1096
0.4477
[0.1776]
0.4211
0.4058
∗
Difference in enrollment rates of 1st and 4th quartiles.
Difference in fraction of college graduates of 1st and 4th quartiles.
∗∗
Table 5: Comparative Statics (1)
Tension between College Premium and Enrollment/Dropouts. Starting from the 1980 economy, only increase
the population mean µz to match the 2005 college premium. (No change in individual decision rule.)
plays any quantitative role in explaining aggregate moments—enrollment, dropout
and underemployment rates drop slightly in 1980 and rise slightly in 2005, while the
opposite is true for premia due to the change in selection. However, it comes far from
explaining the differences in moments between 1980 and 2005.
Having established that a change in the information structure cannot explain the
observed changes from the 1980 to 2005 data, we now analyze the role of the population
mean and variance of college returns, and the underemployment and dropout options.
Our main focus is on how these components affect educational attainment rates and
wage premia.
5.1
Comparative Statics
In Table 5, we take the calibrated parameters from 1980 as a benchmark and change
only µz to exactly match the observed college premium in 2005. This makes college
uniformly more attractive to all students regardless of their initial state. As a consequence, the enrollment rate shoots as high as a counterfactual 91.7%, and among
those who enroll, only 20% end up dropping out. Since we assume that dropouts also
enjoy the partial returns md , the dropout premium also increases, to a similar level
to its actual value of 19.9% in 2005. The underemployment premium overshoots its
2005 value, increasing to as high as 61.6%. This is because even those who end up
underemployed ex post are on the right tail of the returns (z) distribution, as they
chose to graduate over dropping out.
32
Moment
1980
1980’
1980”
2005
Enrollment rate
Dropout rate
Underemployment rate
0.5776
0.3926
0.2352
0.4266
0.5730
0.2089
0.5195
0.6066
0.3276
0.6474
0.4512
0.3257
College wage premium
Dropout wage premium
Underemployment premium
0.3617
0.0575
0.1089
0.9668
0.1165
0.5674
0.5641
0.1044
0.0616
0.9330
0.2054
0.4674
College log wage variance
Dropout log wage variance
0.2723
0.2024
0.4087
0.2375
0.4087
0.2118
0.4087
0.2731
Log corr. family income and own earnings
Enrollment by family income quartile∗
College attainment by earnings quartile∗∗
0.1787
0.4075
0.2917
0.2315
0.4166
0.3395
0.1389
0.4241
0.1978
[0.1776]
0.4211
0.4058
∗
Difference in enrollment rates of 1st and 4th quartiles.
Difference in fraction of college graduates of 1st and 4th quartiles.
∗∗
Table 6: Comparative Statics (2)
Starting from the 1980 economy, only increase the population variance σz2 (college wage variance σc2 ) to
match the 2005 college wage variance. (No change in individual decision rule.)
A model that only models mean differences between high school and college, even
with an intermediate “some college” category, would be largely amiss on educational
attainment trends without relying on pessimism or negative preference shocks. Furthermore, it would predict that few students become underemployed, and even then,
earn a large premium compared to dropouts due to positive selection into graduation.
As we have been arguing throughout, it is the reliance on increasing mean returns that
has beguiled researchers to deem enrollment, dropout and underemployment (in the
form of “overeducation”) trends a “puzzle” in the face of the large rise in the observed
college premium.
Also interesting are the changes in the correlation moments. The high mean increases the income gaps between education groups, reflected in college attainment differences across own income quartiles. This is in line with the quantitative change from
1980 to 2005. However, since most students enroll, the quartile differences by family
income shrinks nearly 3-fold, while in fact it became higher in 2005. The take-away is
that if the rise in the observed college premium is due to a uniform increase in returns,
the quantitative effect would be such that the enrollment rates are too high on average,
and especially at the left tail of the distribution.
In Table 6, we now instead increase only the 1980 variance parameters to match
the college wage variance in 2005. We do this separately for σz2 and σc2 , and report the
change in the 1980 moments.13 We highlight in red that this results in a large increase
13
When increasing σz2 , we only change the population variance of z while also keeping the variance of
33
in the college premium, especially for σz2 , associated with a drop in the enrollment rate
and increase in the dropout rate.
The premium increases because of the optionality in returns—college graduates at
the left tail of the distribution are protected from downside risk, so a higher variance
leads to a higher premium. At the same time, it is more or less the same individuals
who enroll and graduate, just with higher returns—again because the left tail of the
college wage distribution is held constant by the underemployment option. In other
words, despite the rise in variance, students know that by using their underemployment
option the effective wage distribution is a longer tail only skewed to the right and not
the left. However, this does mean that there is a larger risk of dropping out or becoming
underemployed for individuals with lower returns, which reduces the enrollment rate.
The qualitative effects are similar for both σz2 and σc2 , but much more pronounced for
the former.
On the other hand, σz2 and σc2 have the opposite implications for underemployment.
Recall that the college wage is determined as w = exp(z)ωc . Because z is individualspecific, an increase in its variance increases positive selection, so even for those who
ultimately become underemployed, the premium is still high, comparable to the actual
2005 value of 46%. In contrast, an increase in σc2 is relates to negative selection. The
option value in college is higher than for dropouts, so even low z students are encouraged to graduate since the increased option value can only be reaped if one graduates.
The main reason that the college premium increases is because of the option value,
not because individuals with higher returns enroll. Consequently, underemployment is
much higher, and the premium lower. This also helps understand why the premium
increases much more in response to a change in σz2 rather than σc2 .
The also have opposite effects on the correlation moments, again due to opposite
selection effects. Due to positive selection, the correlation between initial assets and
earnings become higher, and the college graduation rate differences larger across income
quartiles, when returns variance are higher. The opposite is true when the residual
log college wage variance is higher, since it encourages low returns students to enroll.
However, on average the enrollment rates do not change, since again, overall it is more
or less the same individuals that enroll. The difference among them only manifest
themselves post-enrollment.
The takeaway from these two exercises is that the observed increase in the college
premium can be explained by both an increase in the mean and variance of z, with
individual priors (σz21 ) and the population variance of the mean of individual priors (σµ2 z1 ) constant. Changing
the latter two do not have any significant quantitative effects.
34
Moment
1980
1980’
2005
Enrollment rate
Dropout rate
Underemployment rate
0.5776
0.3926
0.2352
0.6023
0.2719
0.3257
0.6474
0.4512
0.3257
College wage premium
Dropout wage premium
Underemployment premium
0.3617
0.0575
0.1089
0.3523
0.0365
0.1705
0.9330
0.2054
0.4674
College log wage variance
Dropout log wage variance
0.2723
0.2024
0.2549
0.2009
0.4087
0.2731
Log corr. family income and own earnings
Enrollment by family income quartile∗
College attainment by earnings quartile∗∗
0.1787
0.4075
0.2917
0.1918
0.4036
0.3304
[0.1776]
0.4211
0.4058
∗
Difference in enrollment rates of 1st and 4th quartiles.
Difference in fraction of college graduates of 1st and 4th quartiles.
∗∗
Table 7: Comparative Statics (3)
Starting from the 1980 economy, only increase the partial underemployment return mu to match the 2005
underemployment rate. (No change in individual decision rule.)
the former increasing the enrollment rate and the latter decreasing it. Conversely, the
dropout rate decreases in response to an increase in the mean but increases with an
increase in the variance. These opposing forces are what allow the enrollment rate to
increase only slightly, with an associated graduation rate that is even smaller. This is
possible in our model because of the underemployment option, so also critical is how
the underemployment moments are disciplined. Tables 5 and 6 suggest that a joint
increase in the variance of z and the log ωc , the residual log college wage variance, can
simultaneously achieve an increase in the underemployment rate and premium. We
delve into this in more detail in the next subsection.
What would happen if we uniformly increase the value of underemployment? Table 7 shows the comparative statics of increasing the partial underemployment returns
parameter, mu , from its 1980 calibrated value to match the 2005 underemployment
rate.14 An increase in mu leads to an increase in the value of the underemployment
option, but at the same time, causes negative selection as it attracts more low returns
students. Since the high school job is independent of the individual returns z, more
people enroll, less drop out and while the premium is higher for those who are eventually underemployed, it is lower for both dropouts and college graduates. Coupled with
the results from Table 6, this implies that the observed increase in the underemploy14
One may wonder why we chose to match the 2005 underemployment rate rather than the premium.
We found that even increasing mu to its maximal value of 1 could not achieve this, so decided to choose a
different comparable target.
35
Moment
2005
2005’
Enrollment rate
Dropout rate
Underemployment rate
0.6474
0.4512
0.3257
0.5351
0.4595
0.3361
College wage premium
Dropout wage premium
Underemployment premium
0.9330
0.2054
0.4674
0.7987
0.1765
0.3627
College log wage variance
Dropout log wage variance
0.4087
0.2731
0.3950
0.2651
Log corr. family income and own earnings
Enrollment by family income quartile∗
College attainment by earnings quartile∗∗
[0.1776]
0.4211
0.4058
0.1542
0.4644
0.3311
∗
Difference in enrollment rates of 1st and 4th quartiles.
Difference in fraction of college graduates of 1st and 4th quartiles.
∗∗
Table 8: 1980 to 2005 Change: Decomposition (1)
Replace the population mean z estimate for 2005 with the 1980 estimate, holding all other parameters at
their 2005 values.
ment rates and premium from 1980 to 2005 must be due more to an increase in wage
dispersion rather than an inherent increase in the underemployment option.
5.2
Decomposition of the Change in Premia
Next we conduct the converse exercise of what we will call a “decomposition.” We do
this by means of replacing certain parameter values from the 2005 benchmark with the
corresponding values from 1980.
Table 8 shows the change in moments when replacing the population mean µz
estimate in 2005 with that from 1980. The qualitative changes with respect to µz are
exactly the converse of what was seen in Table 5—since µz is smaller in 1980, both
the premium and enrollment rates decline. With no difference in µz , the difference in
premia and enrollment rates between 1980 and 2005 are by and large determined by the
returns and residual log wage variances. Observe that most moments, in particular the
premia moments, are still close to their 2005 values. Also notable is that the enrollment
rate drops to a value very close to its 1980 value, while dropout and underemployment
rates remain virtually unchanged. We interpret this as evidence that the main role of
the increase in µz was to encourage more students to enroll, rather than increase the
college premium.
On the other hand, replacing the variance parameters (σz2 , σc2 ) with their 1980
estimates, as in Table 9, also reduce the college premium, but results in enrollment
36
Moment
2005
2005’
2005”
Enrollment rate
Dropout rate
Underemployment rate
0.6474
0.4512
0.3257
0.7032
0.4463
0.3289
0.6763
0.3356
0.2687
College wage premium
Dropout wage premium
Underemployment premium
0.9330
0.2054
0.4674
0.7839
0.2029
0.3755
0.7713
0.1690
0.5360
College log wage variance
Dropout log wage variance
0.4087
0.2731
0.3685
0.2643
0.3181
0.2632
Log corr. family income and own earnings
Enrollment by family income quartile∗
College attainment by earnings quartile∗∗
[0.1776]
0.4211
0.4058
0.1620
0.3859
0.3694
0.2046
0.4066
0.4544
∗
Difference in enrollment rates of 1st and 4th quartiles.
Difference in fraction of college graduates of 1st and 4th quartiles.
∗∗
Table 9: 1980 to 2005 Change: Decomposition (2)
The column 2005’ (2005”) shows the quantitative moments resulting from replacing the population variance
z (log residual college wage variance σc2 ) estimate for 2005 with the 1980 estimate.
rates too high.15 Conversely from Table 8, the effect of the larger µz becomes dominant
without the associated changes in the returns or log wage variances. Without the latter
changes, the enrollment rates already overshoot their 2005 values at college premia
levels lower than their 2005 values.
The different implications that the decomposition of σz2 and σc2 have for the underemployment rate and premium are the mirror image of their comparative statics. The
smaller value of σz2 from 1980 decreases the premium, but the decline in risk leads to
more enrollment. While there is not much change in the underemployment rate there
is a pronounced drop in the underemployment premium, since those who exercise the
underemployment option now reap lower returns. The smaller value of σc2 has the same
qualitative effects as σz2 for enrollment and the college premium. However, it sharpens
selection at the graduation stage, resulting in less dropouts, less underemployed but a
higher underemployment premium.
The decomposition associated with (σz2 , σc2 ) indicates that the change from 1980 to
2005 must be such that the value of underemployment increases, but that it must also
be associated with increased selection at the enrollment stage and less at the graduation
stage. In other words, the increase in σz2 is needed to generate the underemployment
premium, while the increase in σc2 is need for the higher underemployment rate.
Table 10 is a counterfactual exercise of replacing the 2005 of mu with its 1980 value,
15
Similarly to the exercise in Table 6, when replacing σz2 with its 1980 estimate we still hold σz21 and σµ2 z1
at their 2005 values.
37
Moment
2005
2005’
Enrollment rate
Dropout rate
Underemployment rate
0.6474
0.4512
0.3257
0.6358
0.5107
0.2866
College wage premium
Dropout wage premium
Underemployment premium
0.9330
0.2054
0.4674
0.9681
0.2251
0.4502
College log wage variance
Dropout log wage variance
0.4087
0.2731
0.4221
0.2763
Log corr. family income and own earnings
Enrollment by family income quartile∗
College attainment by earnings quartile∗∗
[0.1776]
0.4211
0.4058
0.1701
0.4327
0.3776
∗
Difference in enrollment rates of 1st and 4th quartiles.
Difference in fraction of college graduates of 1st and 4th quartiles.
∗∗
Table 10: 1980 to 2005 Change: Decomposition (3)
Replace the partial underemployment return mu estimate for 2005 with the 1980 estimate, holding all other
parameters at their 2005 values.
holding all other parameters constant. By doing so, we obtain the marginal value of
the underemployment option (for college graduates) in 2005 compared to 1980. As
expected, both the underemployment rate and premium are smaller. However, at
first glance, it may be puzzling that the college premium increases, even with a lower
option value of underemployment. This can be understood by the small decline in the
enrollment rate along with a rather pronounced increase in the dropout rate. Because
left tail risk is larger, less people enroll and more people drop out, both forces reinforcing
more selection among college graduates. This is also reflected in the anemic fall of the
underemployment premium. Simply put, larger values of mu lead to negative selection
at both stages by increasing the option value of underemployment.
The quantitative results up to now show that variance increases, coupled with the
intermediate steps of dropping out and becoming underemployed, are more important
than the population mean of z in explaining the quantitative rises in premia from
1980 to 2005. Conversely, a mean increase is only relevant for explaining the small
rise in enrollment. The potential impact that µz may have on premia is countered
by the increase in heterogeneity and the underemployment option (as measured by
mu ), resulting in most enrollees opting to drop out or become underemployed. The
intermediate steps are important not only because they are empirically relevant, but
short of assuming extremely skewed (thick right-tailed) distributions, pure risk-return
models without these options would not be able to generate the increased risk at the
bottom needed to moderate enrollment rates.
38
(a) 1980
(b) 2005
Fig. 11: Dropout Probabilities
x-axis: Initial Assets (a0 ); y-axis: Initial Prior (µz1 ). Enrollees are represented in shades (the white region
in the South-West corners of the graphs are non-enrolless). For those who enroll, ex-ante higher dropout
probabilities are represented as darker shades. Both enrollment and graduation probabilities are increasing
in both individual states (a0 , µz1 ).
Overall, the increase in the enrollment rate can be attributed to the rise of mean
returns. While this also increases premia, the observed 2005 level cannot be attained
without the large rise is its variance as well. These two forces would potentially lead
to a large rise in the underemployment premium, which is tempered by the rise in the
residual log college wage variance, instead leading to a larger rise in the underemployment rate. Simply put, the important margins at the enrollment stage are between
the mean and variance of returns; at the dropout stage it is between the variances of
returns and residual log college wages. Because both the mean and variance of the
returns increase, and also because the signal-noise ratio of the dropout option remains
more or less constant, there is not much change in the dropout rate—an issue we return
to below.
5.3
Option to Dropout
In this subsection we show that the option value coming from sequential decisionmaking as opposed to a once-and-for-all Roy-type model where individuals decide simultaneously to become high school, dropout or college workers comes short of explaining the data. This is captured by the arrival of new information in our model,
the signal at s = 2. As in Stange (2012), this is most important for “intermediate”
students at the margin, i.e. those who enroll and subsequently drop out. The sequential structure also makes the dropout vs. college margin more selective than the high
school vs. enrollment margin.
These intermediate students are visualized in Figure 11. In the figure, the x- and
y- axes denote individual states (a, µz1 ), respectively. The shaded areas represent the
39
region on the (a, µz1 )-plane that an individual would choose to enroll, and for those
who enroll, ex-ante dropout probabilities are shown as different shades of gray. Higher
dropout probabilities are represented in darker shades. For the same initial state, it
is more likely that an individual enrolls in 2005; marginal enrollees (those along the
border of the shaded area) are most likely to drop out. Those in the black areas would
choose to become dropouts even without the sequential option; likewise those in the
lighter gray areas would choose to become college graduates without the option as well.
It is for those in the dark gray areas that the dropout option is the most important.
The qualitative and quantitative effects of removing the signal are similar in 1980
and 2005. The moments in Table 11 are obtained by sending σ → ∞ in both years—
quantitatively, we achieve this by increasing σ large enough so that there are no
changes in any of the moments in response to further increases.
The lack of an information update decreases the attractiveness of the dropout option. Now, only those who are sure of obtaining some education enroll, so both the
enrollment rate and dropout rate are lower. This is associated with more selection
among enrollees (which includes both dropouts and graduates), but less among college
graduates. Hence, while it must be the case that dropout premium must be higher,
whether the college premium is higher or lower depends on the relative magnitudes
of selection between enrollees and graduates. The quantitative results show that the
latter case prevails, i.e. that the college premium is lower compared to the benchmark.
Interestingly, the drop is quantitatively larger than the increase in the dropout premium. The prevalence of negative selection into graduation is noticeable from the drop
in the underemployment premium, even though the underemployment rate is similar
to the benchmark.
This implies that many of those who would enroll in the presence of a (sequential)
dropout option would in fact drop out or become underemployed, but still enjoy higher
returns than on average than if they don’t enroll. It also implies that our benchmark
dropout option is large enough to attract those students with higher returns but lower
expectations. We focus on these two implications in the next subsection.
6
Counterfactuals
As well summarized in influential work by Card (1999), there has been a large empirical
literature using instrumental variables to obtain IV estimates of the average marginal
returns to college, or local average treatment estimates (LATE) of sending a marginal
student who did not enroll in college to school.
40
Moment
1980
1980’
2005
2005’
Enrollment rate
Dropout rate
Underemployment rate
0.5776
0.3926
0.2352
0.5582
0.3020
0.2469
0.6474
0.4512
0.3257
0.6299
0.3784
0.3341
College wage premium
Dropout wage premium
Underemployment premium
0.3617
0.0575
0.1089
0.3103
0.0872
0.0576
0.9330
0.2054
0.4674
0.8700
0.2128
0.4144
College log wage variance
Dropout log wage variance
0.2723
0.2024
0.2760
0.2074
0.4087
0.2731
0.4108
0.2738
Log corr. family income and own earnings
Enrollment by family income quartile∗
College attainment by earnings quartile∗∗
0.1787
0.4075
0.2917
0.1814
0.4143
0.2521
[0.1776]
0.4211
0.4058
0.1745
0.4330
0.3924
∗
Difference in enrollment rates of 1st and 4th quartiles.
Difference in fraction of college graduates of 1st and 4th quartiles.
∗∗
Table 11: Dropout Option in 1980 and 2005
Effect of removing the signal in 1980 and 2005. We remove the signal by setting σ → ∞.
Typically, these studies use distance to school or a means of transportation as an
instrument, and obtain results indicating that the marginal returns to college are higher
than the population average. According to Card (1999), the reason for this is most
likely due to even higher selection among such marginal enrollees. This is related to
the previous subsection where the signal attracts those individuals with higher returns
but lower expectations to enroll.
Similarly to the flattening out of educational attainment shares despite the large
rise in the college premium, the large gains to marginal enrollees has been deemed
puzzling by many researchers as well (see Card (1999) for an excellent review). If there
would be so much gains for the marginal individual, why didn’t he enroll? Here, again,
the previous literature has come short of providing a satisfying explanation short of
negative tastes for schooling.
In relation to this, our structural model provides two alternative explanations—the
unexpectedly high returns can also be due to the lack of considering the intermediate
options, in particular for the marginal enrollees who do not enroll. As we have been
emphasizing in the previous subsections, students are very different in terms of the
intermediate choices they make. This makes the returns to education highly non-linear
even for the same individual. Moreover, students enrolling in college are very heterogeneous in terms of their true returns. Hence when considering what would happen to
marginal students if they were “forced” into college, all three factors (heterogeneity,
dropouts, underemployment) must be taken into account.
41
(A1)
(A2)
(A2.1)
(A2.2)
(B1)
(B2)
0.025
0.081
-0.001
-0.175
0.279
-0.003
0.392
0.360
-0.024
0.161
-0.002
-0.176
0.400
-0.008
0.433
0.386
(B2.1)
(B2.2)
0.004
0.897
0.017
1980
χE
χG
log a0
log µz1
log µz2
log z
0.052
0.197
0.029
-0.047
0.387
0.018
0.004
0.897
0.018
1.186
0.000
1.179
0.011
2005
χE
χG
log a0
log µz1
log µz2
log z
0.122
0.335
0.029
0.088
0.525
0.007
0.002
0.701
0.038
0.002
0.708
0.952
-0.010
0.037
0.950
0.002
Table 12: Education Returns
Dependent variable is log w for (A1), (A2), (B1), (B2), χE for (A2.1), (B2.1), and χG for (A2.2), (B2.2).
Case 1 is OLS and Case 2 is IV using (µz1 , µz2 ) as instruments for (χE , χG ), of which first-stage regressions
are shown in 2.1 and 2.2. For case B, log z is added as a regressor.
6.1
IV regressions
First, we capture the role of unobserved information (µz1 in our model) by running the
following regression using simulated data from our model:
log w = β0 + β1 χE + β2 χG + β3 log a0 + η,
(18)
where η are regression residuals and w are the wages drawn from the education-specific
wage distributions in Section 3.2. We simulate 240,000 individuals from the initial
distribution F0 , each of whom also draw a signal from the individual-specific signal
distribution in (1). The coefficient β1 captures the linear return to some college, while
β1 +β2 captures the college graduation premium. The OLS regression results are shown
column (A1) of Table 12, for 1980 in the top panel and 2005 in the bottom panel.
Now suppose the econometrician could observe (µz1 , µz2 ). The empirical counterpart could be a survey question that asks students their expected wage returns to college
education. Then we could use these as instruments and run a first-stage regression on
the enrollment and graduation dummies:
χE = β0E + β1E log µz1 + β3E log a0 + ηE
χG = β0G + β1G log µz2 + β3G log a0 + ηG
and then use the predicted values in a second-stage regression to obtain an IV estimates
42
for (β1 , β2 ) in (18). The results are shown in column (A2) of Table 12. Columns (A2.1),
(A2.2) show first-stage regression results.
Using the model, we can do more and compute how much of these returns can be
attributed to information alone, and not because µz1 is correlated with the true returns
z. For this we run the OLS and IV regressions
log w = β0 + β1 χE + β2 χG + β3 log a0 + β4 log z + η,
and the first-stage predicted values for the IV estimates are obtained from
χE = β0E + β1E log µz1 + β3E log a0 + β4E log z + ηE
χG = β0G + β1G log µz2 + β3G log a0 + β4G log z + ηG .
The results are shown in columns (B1), (B2) of Table 12, and (B2.1), (B2.2) show the
first-stage regression results.
There are several point worth noticing. First, the OLS regression understates the
non-linearity in returns created by the dropout option. This is seen from comparing
columns 1 and 2—in all cases, returns to some college are overstated and college
graduation understated when comparing the OLS to the IV coefficients. Second, both
the OLS and IV regressions understate the heterogeneity in returns. This is seem from
comparing cases A to B. Once the true returns, z, is included as a regressor, the returns
for college uniformly declines. The large coefficient on log z in case B indicates that
the education returns in A are largely due to selection on z, rather than the mean of
z. Third, initial assets play a small role in explaining wage outcomes, once education
outcomes are taken into account. Moreover, the more we control for selection, by
instrumenting education with information (comparing the OLS and IV coefficients) or
including z as a regressor (comparing A to B), the less of a role it plays. This means
that even the small role played by initial assets are due to correlation with z, and not
financial constraints.
Most importantly, the results imply that if the probability of dropping out is large,
college enrollment can be a bad idea in terms of wage returns. In fact, column (B2)
suggests that holding the true returns z constant (which is unknown to the individual),
enrolling without graduating would decrease individual wages.
This exercise highlights the important of the dropout option and heterogeneous,
individual specific-returns; it also shows that not taking into account dropout probabilities explicitly can lead to misleading interpretations, especially since dropout returns
are much lower than would be implied from an OLS regression.
43
Moment
Population
Marginal
1980
Enrollment premium
College wage premium
0.2420
0.3617
0.0426
0.2246
Dropout rate
Dropout wage premium
0.3926
0.0575
0.9167
0.0249
Underemployment rate
Underemployment premium
0.2352
0.1089
0.2925
0.0824
2005
Enrollment premium
College wage premium
0.6047
0.9330
0.1583
0.4360
Dropout rate
Dropout wage premium
0.4512
0.2054
0.9617
0.1472
Underemployment rate
Underemployment premium
0.3257
0.4674
0.3261
0.4557
Table 13: Population vs. Marginal Enrollees, selected moments 1980 and 2005.
Marginal enrollees are defined as the 1% of the population who would who would switch their enrollment
decision if their assets were /2 higher or lower.
6.2
Marginal Enrollees Counterfactual
To capture the full degree of non-linearity and heterogeneous returns in our model, we
can instead conduct the following though experiment. The marginal enrollees in our
model are determined in terms of their locations on the initial distribution of (a, µz1 ).
Using the enrollment indicator function χE (a0 , µz1 ) in (6), we define
χ = χE (a0 + a /2, µz1 + µ /2) − χE (a0 − a /2, µz1 − µ /2) ,
(19)
which is also an indicator function (by construction) with non-zero values as long as
= (a , µ ) ≥ 0, with at least one holding with strict inequality. With this, we can
define an initial distribution over marginal agents
Φ (a0 , µz1 , z) ≡ R
χ (a0 , µz1 )F0
,
χ (a0 , µz1 )dF0
as → 0, where F0 is the initial population distribution. We then compute the
enrollment premium (weighted average of dropout and college premium), dropout rates
and other moments using Φ instead of F0 . In practice, we set µ = 0 and choose a
R
such that the mass of marginal agents χ dF0 equals exactly 1 percent. We then force
all individuals with χ = 1 to enroll.
A comparison of the moments between the population and marginal enrollees are
shown in Table 13, for 1980 and 2005. In both years, the enrollment premium for
44
marginal enrollees is far below the population, and much more so for 2005. The following panels reveal why this is the case—roughly 90-95% of all enrollees subsequently
drop out, resulting in only a partial returns dictated by the parameter md .16 This effect is much larger for 2005 because recall the benchmark distribution exhibited more
positive selection, in particular with its much larger associated value of σz2 . This means
that the marginal enrollee in 2005 who then drops out has more to gain compared to
the marginal enrollee in 1980 who drops out.
Additionally, while the benchmark correlation between z and µz1 , ρz,µz1 , is not
perfect, it is still high, around 0.65 in 1980 and 0.75 in 2005. Since enrollment decisions
are determined by µz1 , marginal enrollees do on average have lower returns than those
who enroll in the benchmark, with most of them immediately dropping out. Again,
this effect is more pronounced for 2005, evidenced by the underemployment premia
being higher than the college premia—for those who graduate in 2005, in fact getting
underemployed is a better deal than a college job since their true returns are so low.
This is not the case in 1980, since the marginal enrollees who remain in college and
graduate do in fact earn a large premium of about two-thirds of the average premium.
And it is still those who were misinformed or just unlucky enough to draw multiple
low wage draws that become underemployed. In contrast, the 2005 marginal enrollees
who graduate earn less than half the college premium, and the underemployed are the
lucky ones.
There is much more selection in the 2005 benchmark, both because of a better
initial prior at the enrollment stage and more sorting at the dropout stage. Not only
is the dropout premium lower, but the only people who graduate are those who are
massively misinformed of their own returns, resulting in individuals with college jobs
having even lower wages than those who are underemployed.
In essence, the marginal enrollees are those who are taking on a gamble with large
returns. Ex ante, the risk faced by the individual as measured by the variance of returns
is a good risk, due to the presence of the dropout and underemployment options. In
1980, many potential enrollees give up at the margin, since the returns are not that
high when σz2 is low. In 2005, most of them pick up on this gamble, and the only
ones who do not are those who have very low opinions of themselves. Since individual
priors are correlated with their true returns, and more so in 2005, the ex post marginal
enrollee in 2005 does in fact have much higher returns than his counterpart in 1980.
Those students giving up on education may earn a premium that is as high as
16
This can also be seen from the dropout probabilities in Figure 11.
45
15.83% in 2005 at the margin. Although their decision to not enroll is is rational given
their expectations on their individual-specific returns, we can compute whether forcing
a marginal student to enroll is justified financially, by comparing his lifetime earnings
gains against the cost from attending college. It turns out that the marginal student in
1980 loses $5,271, while the marginal student in 2005 gains $13,239 in lifetime terms,
discounted to age 19. This is consistent with better information and more selection at
the enrollment stage in 2005 than in 1980. From a financial perspective, less students
should have enrolled in 1980 while more should have enrolled in 2005. However, this
amount is still less than half of the average high school annual earnings of $26,928.
This indicates that a possible policy intervention that can be quantitatively relevant
is supporting high school students to formulate more accurate priors on their own
labor market prospects. Especially for 2005, we can think of inducing more students to
enroll by offering public financial assistance packages, and perform similar cost-benefit
analyses to determine whether or not such packages are worth it from the public’s point
of view. Such questions have been given much emphasis in recent policy debates and
also the literature, and we turn to such an analysis below.
6.3
Free college for first 2 years
Our results up to now suggest that much of college education, even for 2 years, is a
result of positive selection and that at the margin, the enrollment premium is small
because i) the large returns come with graduation, but most marginal enrollees drop
out, and ii) many of these enrollees have inaccurate beliefs of their returns. This is not
to say that these students are making the wrong choices, since they are rightfully taking
advantage of both the dropout and underemployment options, which minimize their
downside risk. Especially for 2005, there is a financial margin to be made by inducing
more students to enroll. Hence an information intervention would be desirable, but it
is unclear how such a policy can be implemented. Conversely though, we can show
that financial intervention, such as a college subsidy, may not be worth it in terms of
a simple cost-benefit analysis.
Specifically, we do the following. We set the first 2 years of college free for all
individuals, i.e., x1 = 0 in the individual’s problem (5). We then re-compute all
population moments and compare them with the benchmark, as well the moments
for only those individuals who switch from not enrolling to enrolling, whom we dub
”switchers.” Through the lenses of our model, these are simply individuals s.t.
χE (a0 + x1 (1 − v(a0 )), µz1 ) − χE (a0 , µz1 ) = 1,
46
Moment
1980
Population
Before After
Enrollment rate
Enrollment premium
College wage premium
0.5776
0.2420
0.3613
0.6170
0.2286
0.3586
0.0395∗
0.0322
0.1902
Dropout rate
Dropout wage premium
0.3926
0.0575
0.4241
0.0521
0.9311
0.0205
Underemployment rate
Underemployment premium
0.2352
0.1089
0.2354
0.1063
0.2638
0.0000
2005
Before
After
Switchers
Enrollment rate
Enrollment premium
College wage premium
0.6474
0.6047
0.9330
0.7452
0.5443
0.9158
0.0979∗
0.1159
0.3706
Dropout rate
Dropout wage premium
0.4512
0.2054
0.5060
0.1816
0.9836
0.1116
Underemployment rate
Underemployment premium
0.3257
0.4674
0.3275
0.4591
0.3938
0.1568
∗
Switchers
Fraction of switchers in population.
Table 14: Population and Switchers with free college for first 2 years, 1980 and 2005.
First column moments are from the benchmark calibration. Second column are the change in population
moments when the two first years of college becomes free. The third column show the moments for individuals
who would not have enrolled in college without the subsidy, or “switchers.”
and hence are comparable to the marginal enrollees whose χ = 1 in (19). The results
are tabulated in the last two columns of Table 14. As can be seen from comparing the
Before-After columns, there are both more enrollees and dropouts. All premia levels
drop due to negative selection, since the subsidy induces more low ability students to
enroll and graduate. The drop is largest in the enrollment premia due to more students
dropping out; however, there are no visibly large changes in other premia moments.
As was the case with marginal enrollees, most of the switchers end up dropping out
in both 1980 and 2005, and earn significantly less premia. With 3.8% and 9.7% of the
population switching their enrollment decisions due to the subsidy, there is now more
negative selection than the marginal enrollees in the benchmark case. Nonetheless, the
enrollment premia are positive in both years, so we can conduct a cost-benefit analysis
of the subsidy against the increase in average lifetime earnings.
Since enrollment was not financially desirable for the 1980 marginal enrollees and
the switchers will have on average even lower ability, we focus only on 2005. But
note that the cost-benefit analysis here is a bit different from above. For the marginal
enrollees, we were asking whether the average increase in lifetime earnings is larger
than the costs for college they had to pay, taking into account that some of the costs
47
would be covered by grants. Now we are asking whether the average increase in lifetime
earnings is larger than the additional burden that would be born by the government.
We conduct the cost-benefit analysis as follows. First, we compare the increase in
the (present discounted sum of) lifetime earnings of switchers against the cost of the
government paying x1 for only those individuals. This gives us the cost and benefit
of targeted subsidies to individuals who do not enroll in college because of financial
constraints. Next, we compare the increase in lifetime earnings for all individuals
against the cost of the government paying x1 for all individuals who enroll. This is
different from the above case, since eliminating costs for the first 2 years of college
also changes the asset position (a1 ) of individuals who would have enrolled in college
anyway, so their graduation decision will change at the margin, altering average lifetime
earnings. Lastly, such a subsidy raises not only enrollment but also graduate rates,
creating an additional cost from having to provide grants to more students at the
graduation stage. These costs can come from both switchers and non-switchers.
On average, switchers for whom the government pays their first 2-years of college
make $13,454 more dollars in lifetime terms, discounted to age 19. The dollar value of
the subsidy is $16,958 per person, so there would be a joint net loss of $3,505. The
expected cost of having to supply grants for switchers who continue in college and
graduate is $25, increasing the net loss to $3,530 per switcher. Even a 100% tax rate
on increased earnings to finance the subsidy would be insufficient.
A universal subsidy would still incur losses. Lifetime earnings increase by $774 on
average but at an additional cost of $1,660 per student, so the joint net loss is $885.
The expected cost of grants for the latter 2 years of college is $24 per 19 year-old
student, so the loss becomes $910.
In conclusion, making the first 2 years of college universally free has minimal effects
on aggregate moments other than a 10%-point bump in the enrollment rate in 2005.
Regardless of whether such subsidies are targeted or universal, the public investment
generates a net loss per student.
6.4
Free college for last 2 years
The fact that the returns to marginal enrollees are small in our model implies that
students at the enrollment stage are sufficiently well-informed and financially unconstrained. This is consistent with structural models of education choice, such as Carneiro
and Heckman (2002); Cameron and Taber (2004). That the marginal returns are higher
in the 2005 population than 1980 is consistent with Belley and Lochner (2007), who
48
Moment
1980
Population
Before After
Enrollment premium
College wage premium
0.2420
0.3613
0.2408
0.3384
0.0298∗
0.0690
Dropout rate
Dropout wage premium
0.3926
0.0575
0.3410
0.0520
-
Underemployment rate
Underemployment premium
0.2352
0.1089
0.2387
0.0873
0.2808
0.0000
2005
Before
After
Switchers
Enrollment premium
College wage premium
0.6047
0.9330
0.6226
0.8076
0.1051∗
0.3825
Dropout rate
Dropout wage premium
0.4512
0.2054
0.2889
0.1671
-
Underemployment rate
Underemployment premium
0.3257
0.4674
0.3395
0.3685
0.3860
0.0839
∗
Switchers
Fraction of switchers in population.
Table 15: Population and Switchers with free college for last 2 years, 1980 and 2005.
First column moments are from the benchmark calibration. Second column are the change in population
moments when the two last years of college becomes free. The third column show the moments for individuals
who would not have graduated from college without the subsidy, or “switchers.” Students only learn of the
subsidy when they are already enrolled.
present evidence that credit constraints seem to matter more in the NLSY97 than in
NLSY79.
However, that the social returns to providing financial grants for marginal students
to attend college is unsustainable may be puzzling in light of recent evidence of papers such as Zimmerman (2014) who find that the returns to academically marginal
(male) students in Florida outstrip the cost of college attendance. But in terms of
our framework, the counterfactual that corresponds to such regression discontinuity
approaches is to compare the marginal graduate, not enrollee. To illustrate this, we
instead conduct the counterpart to the previous subsection—making the last 2 years
of college free. The change in population moments are tabulated in Table 15.17
We will focus only on 2005. Notice that while the population college premia drops,
the enrollment premia increases. This is because the dropout switchers earn a premia
as high as 38.3%, which, while still far below the benchmark premia, is substantial
compared to the 11.6% enrollment premia earned by the enrollment switchers in Table
17
For this experiment, we assume a “short-run” policy in which enrollment behavior is unchanged, and
only students who are already enrolled are suddenly notified that their latter 2-years of college will be free.
Assuming that students anticipate this from the enrollment stage barely changes the results, as can be found
in Appendix Table 19.
49
14. This level of premia is similar to those enrollment switchers who did not dropout
(37%)—but while these were less than 1% of the entire population, in this case they
comprise as much as 11%. Since switchers at the dropout stage are already selected
once at the enrollment stage, their unobserved returns are already much higher, on
average, compared to the enrollment switchers.
Although the last 2 years of college are also much more expensive, costing $21,736
per switcher, their PDV of lifetime earnings increases by $38,252 on average, resulting
in a $16,515 net benefit. While the level is still small, this implies that a carefully
targeted subsidy to a smaller fraction of the population could have large returns that
would more than justify the cost.18
7
Robustness Checks
7.1
Dropout Parameters
We have also done the same comparative statics exercises as in sections 5.1 and 5.2 with
the dropout parameters. For the comparative statics exercise, we held all parameters
constant at the 1980 calibrated values and increased either md or σd2 to match the 2005
dropout premium or residual log wage variance, respectively. The comparative static
of md propels both the enrollment and dropout rates to the unrealistic levels of 80.6%
and 68.3%, respectively, while the college premium and underemployment premium
only reach 51.5% and 31.5%, respectively. Other moments barely change. Increasing
σd2 has the same qualitative impact on the dropout rate and premium as did σc2 on the
enrollment rate and college premium (reduces the former and increases the latter), but
the quantitative impact is modest as the change in the dropout log wage variance is
much less compared to the increase in the college log wage variance.19
For the decomposition exercise, we held all parameters constant at their 2005 calibrated values and replaced only md or σd2 with their 1980 values. Because the change in
parameters are quantitatively small, in addition to most model moments being rather
18
If the subsidy were anticipated, more students would enroll in the long-run, but as should be expected
from the low returns to marginal enrollees, this number is minimal. Enrollment increases by less than a 1
percentage point (Table 15, resulting in only 35 additional dollars per 19 year-old student spent on grants
for the first 2 years of college. Since switchers are 11% of the population, the total net benefit is still 50
times higher than this additional cost.
19
We also separately tried increasing σz2 , this time to match the observed residual dropout log wage variance
(instead of the college log wage variance). All moments change in the same direction as column 1980’ in
Table 6. The magnitude is much larger though because we had to increase σz2 almost 60% more. This is
because despite the small increase in the dropout log wage variance between 1980 to 2005, the distribution
of z has moderated impact on the dropout wage distribution both because the returns are only partial and
also because of positive selection into college completion.
50
insensitive to these parameters as we have just discussed above, this results in almost
no quantitative changes in the 2005 moments.
7.2
Observed Wage Distributions
For all our results, we focused only on the first two moments of the education specific
wage distributions. In Figure 12, we compare the entire ex post lifetime wage distributions obtained from the model to the distributions we recover from the CPS data
by means of a Q-Q plot. Specifically, we first demean log wages both in the data and
the model, and plot the simulated quantiles of residual log wages against the empirical
quantiles for each education category, for both years 1980 and 2005. A perfect match
would lie along the 45◦ line, depicted in black. While the simulated distributions are
not perfect toward the end points, overall they fit fairly well.
These Q-Q plots cannot be generated without the dropout and underemployment
options. Without the dropout option, the model basically collapses to a Roy selection
model with noise (that come from the residual log wage variances, σh2 , σd2 , σc2 ). In
that case, there is more selection into enrollment, and less into graduation, as we
observed from Table 11. This would push the lower quantiles of the simulated dropout
wage distribution to lie more above, and the upper quantiles of the high school wage
distribution to lie more below, than they already do. This would only exacerbated
without an unemployment option, since there would be no students who enroll and
graduate but end up at the left tail of the distribution ex post—those students would
not enroll or graduate to begin with.
However, it is still the case that our simulated wage distributions exhibit too much
selection compared to the empirical distributions—the lower quantiles of the simulated
college distribution are higher, and upper quantiles of the high school distribution lower,
than their empirical counterparts. We attribute this primarily to several abstractions
we made in the construction of the model, which may also be worthwhile considering
when interpreting our main results.
Heterogeneous returns to high school If returns were also heterogeneous for
high school jobs, students with high returns in those jobs compared to college jobs
would self-select themselves into high school jobs. As long as the correlation between
the two returns are not perfect, this would push up the right tail of the simulated
quantiles for the high school distribution. Somewhat less obvious is that it should
also push down the simulated quantiles for the left tail of the college distribution.
This is because college graduates who become underemployed will still have selected
51
2
2
1
1
0
0
−1
−2
−2
−1
−1
0
(a) HSG, 1980
1
−2
−2
2
−1
0
1
(b) HSG, 2005
2
−1
0
1
(d) SMC, 2005
2
−1
0
1
(f) CLP, 2005
2
2
2
1
1
0
0
−1
−2
−2
−1
−1
0
(c) SMC, 1980
1
−2
−2
2
2
2
1
1
0
0
−1
−2
−2
−1
−1
0
(e) CLP, 1980
1
−2
−2
2
Fig. 12: Quantile-Quantile Plots
HSG=high school graduate, SMC=some college, CLP=college or more. Each graph plots the model quantiles
of the education-specific residual log wage distributions (demeaned) on the y-axis against the empirical
quantiles on the x-axis. The black diagonal is the 45-degree line.
52
themselves based on their college returns, and would have relatively lower returns if
working in a high school job. Given that most college graduates at the lower end of the
distribution are underemployed (working in high school jobs), they would be earning
lower wages than in our model.
However, while modeling heterogeneous high school returns may help in matching
the population distribution of wages, we do not think it would alter the importance
of heterogeneous returns to college and option values in explaining educational attainment and wage premia trends. Our quantitative model perfectly matches educational
attainment moments, and wage premia are computed from averages. The main mechanism in our paper is the heterogeneous returns to college interacting with the option
values, of which the former changed the most from 1980 to 2005. Given that high school
returns would not directly interact with these options, and also that the high school
wage variance has only changed mildly in comparison to the college wage variance, this
consideration should only have modest consequences on aggregate moments.
The option of re-enrollment All else equal, having the option to re-enroll after
entering the labor market (rather than it being an absorbing state) makes enrolling
or graduating relatively less valuable than in our model. In order to match empirical
moments, then, we would have to increase the value of enrolling and graduating, either
by reducing the partial returns to the dropout wage and/or increasing the information
content of the signal. As we argued earlier, however, it is unclear whether this would
have a noticeable impact on our calibrated parameters.
Delayed entry or re-enrollment is prevalent but occur in small time intervals of 2-3
years in the NLSY. In order to generate such patterns, we would need to add timevarying shocks so that students flip their decisions in a 2-3 year gap. Given that we
are matching ex post observed wage and educational attainment outcomes in the CPS
for individuals older than 25, the intermediate stages of delayed entry or re-enrollment
is suppressed in the enrollment/dropout decisions of our model, as only we only look
at final outcomes. Hence the signal received on an individual’s college returns could
be regarded as subsuming such shocks.
To be precise, the concern is the information content contained in the wage offers
when individuals are allowed to re-enroll. In our model, any information content
contained in such “experiments” would be captured by the prior at the enrollment
stage, and the signal at the dropout stage. Hence one may argue that we overstate
the importance of information as opposed to the residual wage variances, but such
temporary wage outcomes are not reflected in our target moments that rely only on
53
final education outcomes. Such experimentation is captured by the prior and signal
and would have similar implications for how the option values in our model explain
education and earnings trends.
Wage offers in school Relatedly, an option to receive wage offers while in school
could also alter our results. In our model, students only make wage draws after they
enter the labor market. Several models, e.g. Stange (2012), assume the exact opposite,
that wage draws are only made while in school. In-school wage draws are similar to
information updates in our model, except that once an individual chooses to discontinue
education his lifetime wage is known with certainty. This creates two countervailing
effects. On the one hand, since there is an incentive to continue education in the hopes
of making a better draw, it would push up the lower quantiles of the college distribution.
This effect would be further strengthened by the increased positive selection at the
graduation stage, since these wage draws would also contain information on their true
returns. On the other hand, these wage draws would also attract more students to
college, causing negative selection at the enrollment stage.
Whether these considerations would have a quantitative impact on the importance
of heterogeneous college returns in explaining education trends is left for further research.
8
Conclusion
Based on empirical evidence, we propose and develop a model in which return-to-college
heterogeneity/risk interacts with the optionality of college education. The model helps
reconcile long-run trends in college enrollment and dropouts with observed labor market
outcomes. Our results show that the increase in the college premium, to a large extent,
reflects an increase in a “good” risk, contrary to previous risk-return models in which
risk is “bad” for risk-averse agents.
Our results relate to current policy debates on whether too many students are
going to college if they are going to end up being “overeducated” anyway, or if too few
students are going to college in comparison to the high college premium. Individuals
in our model make perfectly rational decisions given their individual projections on
college returns; the reason it may seem that too many students are going to college
can partially be attributed to students optimally exercising the option of dropping out
or becoming underemployed, while at the same time the reason it may seem too few
students are going to college is due to a lack of understanding heterogeneous returns
54
and the associated risks of becoming dropouts or underemployed.
55
A
A.1
Data Appendix
March CPS
Most of the data moments used in the analysis are computed from the IPUMS CPS.
Earnings We closely follow AKK for data-cleaning, in particular when handling
top-coding.
1. First, we select all race categories containing “white,” and focus only on individuals in the private labor force.
2. We select only full-time workers, defined as individuals who report they worked
40+ weeks last year, and 35+ usual hours worked per week.
3. Follow AKK to adjust for top-coding by income source (“wages and salary,”
“self-employed,” and “farm”). From 1988 onward, each income source is further
divided into primary and secondary sources of income. All income sources and
categories are separately top-coded. AKK’s procedure is to multiply all top-coded
incomes by 1.5.
4. (Top trimming) Compute implied average weekly and hourly earnings. Drop
individuals with implied average weekly earnings larger than the top-code value
times 1.5/40. Repeat with implied average hourly earnings.
5. Define total earnings as the top-coding adjusted sum of all three (or six for
1988 onward) income categories. Earnings are reported in annual terms. Deflate
all earnings to 2000 USD, using the chain-weighted (implicit) price deflator for
personal consumption expenditures.
6. (Bottom trimming) Drop individuals whose average weekly or hourly earnings are
less than one-half the real minimum wage in 1982 ($112/week in 2000 dollars).
Educational Attainment Prior to 1992, the CPS only collected respondent’s
highest grade completed. From 1992 onward, the additionally collected data on highest
degree or diploma attained.
1. High school dropouts: less than 12 years of schooling, or report not having a high
school diploma or equivalent
2. High school graduates: at least 12 years of schooling, do not report not having a
high school diploma or equivalent
3. Some college: more than high school but less than 4 years of college
56
4. College graduate: at least 4 years of college
5. Greater than college: at least 6 years of college or report having a post-college
graduate degree
To compute statistics, we drop missing observations and use the CPS provided sampling
weights, except when computing the earnings premia. To control for age-demographic
effects, we use the sampling weights when computing the mean earnings for each age,
but simply take the average across the mean earnings for each age from 26 to 50
when computing premia. To compute lifetime PDV earnings values, we discount mean
earnings by age by an annual interest rate of 4%. For more details, refer to AKK.
A.2
BLS Education and Training Assignments and NLSY
BLS Education Projections (EP) The BLS publishes an education and training requirements chart by detailed occupation, in their Occupation Outlook Quarterly
as part of its EP program (http://www.bls.gov/emp/home.htm). Published biennially, the chart is based on the BLS’s own analysis combined with the distribution of
educational attainment in each detailed occupation category. We use the 1998 chart
as a benchmark and test robustness with the 2008 chart. The reason we use the 1998
chart is because the occupation coding changes frequently in both the CPS and the
EP, and the 1998 chart requires the least use of occupational coding crosswalks with
large changes when comparing 1980 to 2005.
The CPS uses the OCC1970 convention until 1982, OCC1980 until 1991, OCC1990
until 2002, and OCC2000 up to 2005. Occupations in the 1998 requirements chart are
coded according to SOC2000. We use BLS crosswalks to change all occupation codes
to OCC1990. To err on the safe side for our analysis, we define underemployment
only for those cases that we can match the occupation code in the CPS to that in
the requirements chart. Consequently, we cannot tell for approximately 25% of college
graduates whether they are underemployed or not. In these cases, we simply assume
to be working in college-level jobs.
NLSY79 and 97 In the NLSY79, we compute age-earnings profiles by applying
the same full-time and above criteria we applied to the CPS and dropping observations
below half the 1982 minimum wage. We impute missing wage observations by linearly
interpolating log wages in adjacent years, and compute lifetime PDV earnings by
discounting at an annual interest rate of 4%. We drop missing observations to compute
the correlation between family income and the child’s lifetime PDV earnings.
57
To compute enrollment rates and family income in the NLSY79 and 97, we follow
Lochner and Monge-Naranjo (2011) (LM), except that we drop high school dropouts.
Then we drop children who were not living with an adult for both ages 16 and 17,
and take family income as the mean reported values for those ages. We define college
enrollment as having 13+ years of schooling at age 21. If data is missing at age 21,
we apply the same criterion at age 22. We drop missing observations to compute
enrollment rates. For more details, refer to LM.
B
Maximum of Normal Random Variables
For a lognormal random variable x with distribution Fx , let Fˆx denote the distribution
of log x, so that Fˆx is normal. We have
ˆ d (log w|z) = P r {log md + z + log ωd ≤ w and log md + log ωh ≤ w} ,
G
where log ωd and log ωh are lognormal as specified in (3). With independence between
the two random variables,
ˆ d (log w|z) = Fˆd log w − log md − z + σ 2 /2 × Fˆh log w − log mh + σ 2 /2 ,
G
d
h
ˆ c can similarly defined for
which is the wage distribution faced by college dropouts. G
three random variables. Since Fˆh and Fˆd are normal, we can write out the p.d.f. of
log w, gˆd , as
gˆd (log w|z) =
log w + σh2 /2
log w − log md − z + σd2 /2
1
·φ
·Φ
σh
σh
σd
2
log w − log md − z + σd /2
log w + σh2 /2
1
+
·φ
·Φ
σd
σd
σh
and similarly for gˆc (log w|z), where φ and Φ are, respectively, the p.d.f. and c.d.f. of
the standard normal distribution. These are the distributions we use for integration in
the numerical implementation described below.
C
Numerical Appendix
In an outer-loop, we calibrate Θ = [b, mz , sz , s, md , sd , sc , k]0 to match target moments
Md (in text), using a downhill simplex method to solve Equation (13). The square root
of the resulting RMSE is less than a percentage point for both 1980 and 2005.
58
C.1
Initialization
Grids All interpolations will be linear. Let na denote the size of the grids for ai ,
i ∈ {0, 1, 2}, and nz the size of the grids for the true returns z, prior µz1 , and posterior
µz2 . The grid points for assets are set such that lower points are closer to each other,
and the returns and posteriors grid points are set at equi-distance intervals. These
grids are used as individual states.
Quadratures and terminal values We use k-dimensional Gaussian-Hermite
quadratures whenever numerical integration is needed. Before we solve the individual’s
problem, we can set all terminal values as follows:
1. Set quadratures over wages (wh , ωd , ωc ). Set separate k-dimensional quadratures
over z for each grid point on the µz2 -grid. Also separate k-dimensional quadratures over µz2 for all grid points on the µz1 -grid. All numerical integration is done
over these quadratures.
2. For each a-grid point and z-quadrature node, compute V (s, a, w) and its first
derivative w.r.t. a in (9).
3. Compute Vh (a0 ), Vd (a1 , z), Vc (a2 , z) and their derivatives w.r.t. ai by integrating
over the w-quadratures.
4. For each point on the a1 × µz2 -grid, compute the dropout (work) option in (7) by
integrating over the z-quadrature.
5. Similarly, compute the college value for each point on the a2 × µz2 -grid by integrating over the z-quadrature. Also obtain the derivative of the college option
w.r.t. a2 by integration.
This gives us all the terminal values and their derivatives.
C.2
Individual Decisions
Policy Functions for V2 Given the derivative of
policy
a∗2
R
Vc dFz2 , derive the savings
for the “school” option for each value of (v, a1 , µz2 ) on the 3 × na × nz -grid.
Once done, use the policy function to compute the value of the “school” option and
the envelope theorem to compute ∂V2 (v, a1 , µz2 )/∂a1 if the school option is chosen.
Compare with the “work” option to derive the binary policy function χG ∈ {0, 1} for
0 =work, 1 =school.
59
Policy Functions for V1 Given the derivative of V2 , derive the savings policy a∗1
for the “school” option for each value of (a, µz1 ) on the na ×nz -grid, while using the µz2 quadrature set on each grid point for µz1 to integrate over V2 and its derivative w.r.t.
a1 . Once done, use the policy function to compute the value of the “school” option.
Compare with the “work” option to derive the binary policy function χE ∈ {0, 1} for
0 =work, 1 =school.
C.3
Approximating Distributions
Initial Distribution For s = 1, linearly approximate a trivariate normal distribution over (log a0 , µz1 , z). Although the individuals don’t know z, we need to know
them to compute probabilities.
Enroll? Solve for individual decisions on each point of the approximated distribution
at s = 1, and compute the mass of individuals who don’t enroll. For the rest, compute
the masses that fall on an approximated quadvariate distribution over (a, µz2 , a0 , z),
which forms the approximated distribution for s = 2.
Dropout? Solve for individual decisions on each point of the approximated distribution for s = 2, and compute the mass of individuals who dropout. For the rest,
compute the masses that fall on an approximated bivariate distribution over (a, z),
which forms the approximated distribution for s = 3.
Underemployed? The high school wage draw wh is independent of z. For each z
on the grid, we can compute the z-specific wage distributions for dropouts and college,
according to the bivariate and trivariate max distribution for log normal random
variables numerically, according to Appendix B. The c.d.f. for college graduates whose
college-job wage is not the maximum wage draw are those that we dub underemployed.
All statistics are computed using these approximated distributions.
60
D
Additional Tables and Figures
Total
1980
2005
0.7111
1.2243
Between
Within
0.0785
0.6324
0.2237
1.0001
HSG
SMC
CLP
0.5547
0.5932
0.8251
0.8730
0.9589
1.2595
Between CLP
Within CLP
0.0129
0.8121
0.0430
1.2165
0.7928
0.9267
1.1212
1.3047
CLG
GTC
Table 16: Theil Index Decomposition, White Males, ages 26-50
HSG=high school graduate, SMC=some college, CLP=all college graduates, CLG=only college graduates,
GTC=greater than college. To make the indices comparable across years and education groups, we normalize
them by log N , where N is the sum of the sampling weights, and multiply by 100. Total, between and within
indices are normalized by the size of the entire sample in each year, times 100. Education specific indices
are normalized by the size of each group in each year, times 100.
61
1980
2005
Lifetime PDV Earnings
HSG
SMC
CUE
CLJ
355.24
394.03
410.94
538.57
340.24
409.86
490.58
696.29
Mean Hourly Wages 26-50
HSG
SMC
CUE
CLJ
17.82
18.85
19.78
25.53
17.24
20.79
25.27
37.15
Table 17: Lifetime PDV and Mean Wages 26-50
White Males, all expressed in 2000 USD. HSG=high school graduate, SMC=some college,
CUE=underemployed college graduate, CLG Job=college graduate with college job. Lifetime PDV eaernings
are computed with an annual 4% discount rate, starting from wages at age 19 for HSG, 21 for SMC, and 23
for CUE or CLJ. Mean wages are controlled for age.
Moment
1980
1980’
2005
2005’
Enrollment rate
Dropout rate
Underemployment rate
0.5776
0.3926
0.2352
0.5721
0.3809
0.2268
0.6474
0.4512
0.3257
0.6498
0.4617
0.3327
College wage premium
Dropout wage premium
Underemployment premium
0.3613
0.0575
0.1089
0.3874
0.0628
0.1395
0.9330
0.2054
0.4674
0.8960
0.2042
0.4325
College log wage variance
Dropout log wage variance
0.2723
0.2024
0.2638
0.2000
0.4087
0.2731
0.4187
0.2755
Log corr. Family income and own earnings
Enrollment by family income quartile∗
College attainment by earnings quartile∗∗
0.1787
0.4075
0.2917
0.1649
0.4447
0.3201
0.1776
0.4211
0.4058
0.1892
0.3977
0.3765
∗
Difference in enrollment rates of 1st and 4th quartiles.
Difference in fraction of college graduates of 1st and 4th quartiles.
∗∗
Table 18: Initial Priors in 1980 and 2005
Effect of changing priors, 1980 and 2005. For 1980’, we replace the 1980 value of ρz,µz1 with its 2005 value.
For 2005’, we replace the 2005 value of ρz,µz1 with its 1980 value.
62
Moment
1980
Population
Before After
Enrollment rate
Enrollment premium
College wage premium
0.5776
0.2420
0.3613
0.5860
0.2376
0.3379
0.0301∗
0.0708
Dropout rate
Dropout wage premium
0.3926
0.0575
0.3490
0.0504
-
Underemployment rate
Underemployment premium
0.2352
0.1089
0.2388
0.0868
0.2801
0.0000
2005
Before
After
Switchers
Enrollment rate
Enrollment premium
College wage premium
0.6474
0.6047
0.9330
0.6681
0.6086
0.8061
0.1074∗
0.3822
Dropout rate
Dropout wage premium
0.4512
0.2054
0.3081
0.1650
-
Underemployment rate
Underemployment premium
0.3257
0.4674
0.3399
0.3689
0.3873
0.0829
∗
Switchers
Fraction of switchers in population.
Table 19: Population and Switchers with free college for last 2 years, 1980 and 2005.
First column moments are from the benchmark calibration. Second column are the change in population
moments when the two last years of college becomes free. The third column show the moments for individuals
who would not have graduated from college without the subsidy, or “switchers.” Students anticipate the
subsidy at the stage of enrollment.
63
100
80
60
40
0
20
Percent (%)
1965
1970
1975
HSD
1980
1985
1990
Survey Year
HSG
1995
SMC
2000
2005
CLG
GTC
Fig. 13: Education Attainment, White Females, ages 26-50
10
Hourly Earnings (2000$)
20
30
40
50
HSD=high school dropout, HSG=high school graduate, SMC=some college, CLG=college graduate,
GTC=greater than college.
20
25
30
35
40
45
50
55
60
65
Age
HSG
SMC
CUE
CLJ
Fig. 14: 1980 Cross Section Earnings Profiles by Education Group, White Males.
HSG=high school graduate, SMC=some college, CUE=underemployed college graduate, CLJ=college graduate with college job.
64
50
Hourly Earnings (2000$)
20
30
40
10
20
25
30
35
40
45
50
55
60
65
Age
HSG
SMC
CUE
CLJ
Fig. 15: 2005 Cross Section Earnings Profiles by Education Group, White Males.
100
80
60
0
20
40
Percent (%)
60
40
0
20
Percent (%)
80
100
HSG=high school graduate, SMC=some college, CUE=underemployed college graduate, CLJ=college graduate with college job.
−1.5
−1
−.5
Enroll
0
True Returns
Graduate
.5
1
1.5
−1.5
College Job
−1
−.5
0
True Returns
Enroll
(a) 1980
Graduate
.5
1
1.5
College Job
(b) 2005
Fig. 16: Distribution of Education Choices
Implied educational attainment choices implied by calibrated parameters. For each level of log z, unknown
to the individual, we plot the fraction of individuals who enroll, graduate, and obtain a college job. For all
levels of z, individuals are more likely to enroll, drop out, and become underemployed.
65
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