The full text on this page is automatically extracted from the file linked above and may contain errors and inconsistencies.
Who Values Access to College?
WP 19-05R
Kartik Athreya
Federal Reserve Bank of Richmond
Felicia Ionescu
Federal Reserve Board of
Governors
Urvi Neelakantan
CAFRAL and Federal Reserve Bank
of Richmond
Ivan Vidangos
Federal Reserve Board of
Governors
Who Values Access to College?∗
Kartik Athreya†
Felicia Ionescu‡
Urvi Neelakantan§
Ivan Vidangos¶
November 5, 2019
Working Paper No. 19-05R
Abstract
Through an empirically-disciplined human capital model, we demonstrate that the value
of college varies dramatically across US high school completers. Roughly 40 percent place no
value on the option to attend—despite the large public tuition subsidy. By contrast, a fourth
of all attendees would enroll even absent the subsidy. The magnitude and pattern of variation
suggests substantial potential for policy improvement: We show that simply redirecting the
subsidy from individuals who always enroll towards a stock index fund for those who never
enroll increases ex-ante welfare by 1 percent while preserving both enrollment and budget
neutrality.
JEL Codes: E21; G11; I24;
Keywords: Human Capital; Higher Education; Financial Investment
∗
This paper arises from feedback we received at presentations of our companion paper, “Investment Opportunities
and the Sources of Lifetime Inequality,” (Athreya et al., 2018) urging us to separate from the latter the detailed
measurement of the value of access to college across US households and the attendant implications for public
support. We therefore thank, without implicating, numerous seminar and conference participants and discussants
of both papers. We are especially grateful to Gene Amromin, Joao Cocco, Andrew Figura, Francisco Gomes,
Mark Huggett, Oksana Leukhina, Monica Paiella, Luigi Pistaferri, Sangeeta Pratap, Paolo Sodini, and Guillaume
Vandenbroucke for detailed discussions and input. The views expressed in this paper are those of the authors and
do not necessarily reflect the views of the Federal Reserve Board, the Federal Reserve Bank of Richmond, or the
Federal Reserve System. All errors are ours.
†
FRB Richmond. P.O. Box 27622, Richmond, VA 23261, kartik.athreya@rich.frb.org, Ph:804-697-8225
Federal Reserve Board. felicia.ionescu@frb.gov, Ph:202-452-2504,
§
CAFRAL and FRB Richmond. urvi.neelakantan@cafral.org.in, Ph:+91-22-22694587
¶
Federal Reserve Board. ivan.vidangos@frb.gov, Ph:202-452-2381
‡
1
1
Introduction
A first glance at US data suggests that college could be of great value to most, if not all, individuals.
The average returns to college completion are high: in recent decades, the lifetime income of college
graduates exceeded that of high school graduates by a factor of roughly two (Goldin and Katz,
2007).1 College graduation can thus substantially increase individuals’ lifetime consumption and,
presumably, utility.
However, two other features of the data work against this interpretation. First, and perhaps
most importantly, college graduation is far from guaranteed. College noncompletion is in fact
widespread (Restuccia and Urrutia, 2004; Bound et al., 2010; Johnson, 2013); in recent data,
roughly half of all enrollees into public four-year colleges (which absorb roughly two-thirds of
all Bachelor’s degree enrollees) failed to complete college even eight years after initial enrollment
(Bound et al., 2010). In addition, partial completion offers relatively little reward. Second, even
conditional on successful college completion, subsequent returns—as embodied in labor earnings—
are not only subject to uninsurable aggregate risk as with many diversified financial investments,
but also to substantial uninsurable idiosyncratic risk.
Empirically, the ability to successfully complete college depends strongly on individual-level
characteristics, chiefly those summarizing the preparation with which individuals enter college,
such as standardized test scores and high school grades. Hendricks and Leukhina (2017), for
example, show that college preparation, as measured by transcript data, is a strong predictor of
graduation prospects. This preparation governs both the ability to effectively accumulate human
capital (e.g., innate capabilities and being in environs conducive to learning) and the amount of
human capital (e.g., math and language literacy) accumulated through the end of high school.
The variability in these characteristics makes the returns to college—whether high or low—fairly
predictable for some and uncertain for others. Importantly, for a significant proportion of high
school graduates, the ex-ante attractiveness of college may be far lower than the ex-post return
suggests, and predictably so.2
College is also heavily subsidized. In particular, college in the US receives direct subsidization
that lowers the cost of attendance for all enrollees, irrespective of their backgrounds.3 Empirical
1
This is also consistent with the large empirical literature that attempts to estimate the causal effect of education
on earnings and finds very substantial returns. See Card (2001) for a review.
2
As detailed later, we use Beginning Postsecondary Students (BPS) data and confirm that completion exhibits
systematic dependence on individual characteristics.
3
A recent report by the College Board (Ma et al., 2018) estimates that a large fraction of students (about 75
percent) go to public colleges in the U.S. Moreover, a very large fraction of them, about 80 percent, pay in state
tuition. In BPS data, the comparable fraction is a bit smaller, at about two-thirds, if we look at students who
eenroll in college with the goal of getting a BA degree.
2
estimates of the size of this subsidy are large, reducing the direct cost of college by 40 to 50 percent
(Athreya and Eberly, forthcoming). The dependence of college completion on initial preparation
suggests that direct subsidies to college may be of widely varying value to high school completers.
Motivated by these observations, we first pose two questions: (1) “Who values college?” That
is, how does the value of college vary across the population of US high-school completers? (2)
What is the role of the large existing subsidy in this variation?
We construct a model in which agents who differ ex-ante in their initial human capital, their
ability to further accumulate it, and their initial financial wealth decide whether to enroll in college.
College is modeled as a risky investment that receives various types of public support. We then use
data on life-cycle earnings and college investment behavior to back out—following the approach
pioneered in Huggett et al. (2006) and extended in Ionescu (2009)—the joint distribution of ability,
initial human capital, and initial wealth. With the joint distribution of initial attributes in hand,
we use the model to measure the value of access to college across the spectrum of individual types.
Our model allows us to construct the relevant counterfactual for this exercise: a setting where
college is simply not available.4
Two findings emerge from the exercise. First, nearly half the population assigns little value (less
than 5 percent of consumption) to having access to college. Given the large college premium, this
result may appear striking at first. However, it is consistent with both observed non-enrollment
rates and the large noncompletion rate among those who currently enroll. The low value of college
to many individuals contrasts with the extremely high value that a few enjoy: we find that the
top decile of valuations exceeds 40 percent of consumption, pulling up the mean value of college
access, in consumption-equivalent terms, to 14.7 percent. Importantly, the valuations differ starkly
across measures of college readiness; for example, in our model, those in the bottom quartile of
the ability distribution place little value on college access (only 1 percent of consumption) while
the valuations of the top quartile near 30 percent of consumption.
We next assess the extent to which, all else equal, the large subsidies currently in place matter
for the value that individuals place on having access to college. Public support for college attendance takes many forms, including need- and merit-based grants, subsidized student loans, and,
perhaps most importantly, subsidies that directly and substantially lower the tuition charged by
public schools, especially in-state. It is important to note that the latter increases college affordability for all enrollees, with no reference to individual attributes. Is this direct subsidy important?
We find that the answer is yes: removing the direct subsidy reduces the value of access to college
at the mean by more than one-third.
4
We compare the value that various types of agents obtain in this counterfactual to the value they obtain in the
baseline. The details are described in Section 5.
3
The analysis described above lets us classify the population of high school completers into three
groups: those who enroll in college with or without the subsidy (whom we label “always enroll”
and who comprise 24 percent of the population), those who enroll in the presence of the subsidy
and not in its absence (“switchers,” who account for 30 percent of the population), and those who
choose not to enroll in either case (“nonenrollees”, who make up the remaining 46 percent). We
find that the withdrawal of the subsidy matters most for those who always enroll. This finding
has an important implication: for individuals with the poorest earnings prospects, changes in
college subsidization are essentially irrelevant, while for those well-prepared to go, subsidies are
meaningful for the value of college, even as they do not much affect the decision to enroll. After
all, for the latter, the removal increases the costs one-for-one of going to college, something they
were fully planning on doing even in the absence of the subsidy. In other words, direct subsidies
to college most benefit those who derive high value from college even in their absence.
The finding that nearly a quarter of the population would continue to enroll in college even in
the absence of any subsidy, while a full 46 percent receives no benefit from it, suggests potential
for policy improvement. This motivates our third and final question: are welfare improvements
possible from an alternative regime? We demonstrate that ex-ante welfare gains are indeed available through one simple but informative alternative. Specifically, we consider an example in which
those whom we identify as “always enroll” no longer receive the direct subsidy to college. Instead,
the funds are used to subsidize the purchase of a stock index fund, available only at retirement,
for the “nonenrollees.”5 The switchers, who clearly depend on the subsidy to enroll in college, are
unaffected. Of course, college might be subsidized to deliver socially optimal human capital in
the presence of positive externalities, or to generate a specific target level of college enrollment for
other reasons. Notice that the alternative we propose ensures that any social benefits from college
are preserved: it induces no reduction in enrollment (and hence in the skill premium) and is also
budget-neutral.
Why a stock index fund? Previous research has shown that it delivers ex-post returns with
stochastic properties similar to those from college (Judd, 2000; Psacharopoulos and Patrinos, 2018).
Since it offers neither greatly higher returns nor significantly lower risk compared to college, this
alternative does not have an undue advantage over the baseline. Moreover, since we make the
proceeds available to the consumer only at retirement, the risks to the portfolio are realized over
the holder’s entire working life—just as risks to human capital are. Note, however, that unlike
5
This proposal is similar in spirit to policies that encourage asset building among households—such as child
development accounts (CDAs)—which many different countries have promoted (see Loke and Sherraden, 2009, for
an overview). Though the specifics differ, a common feature of such policies is that money is deposited in an
individual’s account early in life and restrictions are placed both on when it can be withdrawn and what the funds
can be used for. A full assessment of such policies is well beyond the scope of this paper.
4
returns to college (more generally, human capital), returns to the stock market are “blind” to the
characteristics of their owners and, by construction, immune to idiosyncratic risk.6
We find that the mean welfare gain from our proposed alternative is 1.07 percent, thus making
all better off in ex-ante terms (i.e., before knowing one’s initial type). Of course, as we will show,
the gains are heterogeneous and in this case flow to those who derived little or nothing from college
access.
The fact that there are gains to be had from moving to this regime suggests that current
support for college, which is invariant to individual characteristics (i.e., the direct tuition subsidy
currently in place) and meant to equalize opportunity, might instead be flowing to those already
well-positioned to benefit from college. Our analysis is in no way an indictment of college subsidies
as a whole nor is it an exhortation to subsidize stocks per se. Rather, our message—as embodied in
these measurements—is that more widespread benefits may be available from a targeted approach
that conditions investment subsidies on individual circumstances.
1.1
Related work
Our paper builds on work that is aimed at understanding the role of human capital when the
particulars of college education, in terms of its costs and returns as a function of observable enrollee and household characteristics, are modeled explicitly. Important references in this literature
include Arcidiacono (2005), Garriga and Keightley (2007), Johnson (2013), Altonji et al. (2016)
and Abbott et al. (2018).
In the work cited above, a primary focus is on providing rich policy counterfactuals in which
college enrollment is itself a policy target.7 Our results, by contrast, suggest a targeted alternative
that can more efficiently deploy the same public monies without interfering with college enrollment.
We therefore adopt a partial equilibrium perspective and emphasize the derivation of the (notdirectly-observable) joint distribution of learning ability, initial human capital stock, and initial
6
Our restriction to an index fund is important because there is some evidence of heterogeneity in returns to
individual stock portfolios (Fagereng et al., 2016; Bach et al., 2016). An index, by definition, offers all those who
hold it the same returns. Even if we were to allow a correlation to exist between individual types and stock returns,
it is likely that it would favor those with the highest levels of education. If those with lower levels of education are
also less likely to “do well” in the stock market, this would only bolster the case for a stock-index fund—which gives
them the market rate of return—making them better off. One seemingly plausible alternative to a stock index fund
is housing; however, unlike those on a stock market index, the returns to housing do depend on owner behavior and
idiosyncratic risk.
7
Abbott et al. (2018), in particular, develops a rich representations of higher education, allowing for a variety
of salient features including gender, labor supply during college, government grants and loans (including private
loans), and heterogeneity in familial resources. See also Epple et al. (2013) and Cestau et al. (2017) for analysis of
higher education policies in the presence of substantial enrollee heterogeneity.
5
financial wealth. These features, as argued above, are critical to accurately assessing individuallevel variation in the valuation of college and of the alternative we study.8
We are also informed by the work that emphasizes the bias imparted to measured returns to
college by the possibility of noncompletion. Hendricks and Leukhina (2018) allow for selection
effects and argue that two layers of selection are important: weakly prepared students disproportionately fail to enroll in college, and those who do enroll fail to complete at high rates.9 Our
model allows for both effects to operate and thereby avoids overstating the payoff to college. With
respect to failure risk, our work builds on earlier work of Restuccia and Urrutia (2004), Akyol
and Athreya (2005), and Chatterjee and Ionescu (2012).10 More recently, Athreya and Eberly
(forthcoming) demonstrate that college failure risk hinders low-wealth individuals, even relatively
well-prepared ones, from enrolling in college.
On a methodological level, we build on Ben-Porath (1967) and Huggett et al. (2006). The
former showed the importance of the complementarity between ability and human capital investments in explaining features of earnings dynamics. The Ben-Porath model recognizes the distinction between potential and actual earnings and provides the framework for testing the relationship
between earnings and schooling. The latter demonstrate that the U.S. earnings distribution dynamics can be well matched by the Ben-Porath model from the right joint distribution of ability
and human capital. As a result, the shape of the distribution of initial conditions is central to
quantifying changes in schooling decisions driven by education policy. Furthermore, as Blandin
(2018) shows, the Ben-Porath model does much better at matching various empirical properties of
life-cycle earnings than alternative models such as learning-by-doing. Our work is close in spirit
to papers that use the Ben-Porath framework to investigate college and, more generally, human
capital investment in a quantitative framework, in particular Guvenen (2009), Ionescu (2009), and
Kong et al. (2018).
Our work is also related to an empirical literature that has examined various aspects of heterogeneity in the returns to college, including differences in ability, college preparedness, and family
background among individuals (see Card, 2001), and, more recently, also in field of study and
school quality (e.g., Altonji et al., 2012). Our framework allows for heterogeneity in ability and
college preparedness, though it does not explicitly consider heterogeneity in field of study or school
8
Relatedly, our analysis abstracts from aggregate externalities from education. The empirical evidence on the
magnitude of education externalities is mixed. While some authors find evidence of such externalities (e.g. Moretti,
2004a,b), others find that any such externalities are negligible (Acemoglu and Angrist, 2000; Ciccone and Peri, 2006)
Nonetheless, our proposed alternative, because it preserves college enrollment, acknowledges that such externalities
may be part of the underlying motivation for subsidizing college.
9
See also Arcidiacono (2004).
10
The possibility of college failure has also been evaluated in work of Stange (2012) and Ozdagli and Trachter
(2011).
6
quality. However, the two sets of characteristics appear to be positively correlated (Arcidiacono
et al., 2012; Hendricks and Schoellman, 2014). In our model, heterogeneity in ability and college
preparedness will act as a summary measure of all dimensions of heterogeneity in the returns to
college investment.
With respect to stocks, our work follows the literature on portfolio choice in life-cycle models
(see, for example, Cocco et al., 2005). In spirit, our work is also closely related to Kim et al.
(2013), which also features both education and stock market investment.11
The remainder of the paper is organized as follows. Section 2 describes the model and Section 3
the data we use to calibrate it. Section 4 summarizes the calibration, and the results are reported
in Section 5. Section 6 concludes.
2
Model
Our aim is to first measure the value, across different types of individuals, of access to college,
and then use these measures to assess the gains available from a more targeted alternative to the
current direct subsidy policy. This requires a model rich enough to accommodate the salient types
of heterogeneity across individuals, such as college readiness and individual-specific uninsurable
shocks, as well as investment opportunities that allow for human capital accumulation via college
and time, in addition to standard financial assets (a bond and a stock).12
Our model thus combines a standard life-cycle consumption-savings problem in which households face uninsurable idiosyncratic risk and have a meaningful portfolio choice, with a model
of human capital accumulation that includes both the classical Ben-Porath technology and an
investment opportunity that takes the particulars of college seriously. The details are as follows.
2.1
Environment
Time is discrete and indexed by t = 1, ..., T where t = 1 represents the first year after high school
graduation. We allow for three potential sources of heterogeneity across agents at the beginning of
life: their learning ability, a, which does not change after high school, their initial stock of human
11
Indeed, in Athreya et al. (2015), we incorporate the elements of Kim et al. (2013) in a model with human
capital investment (though without four-year college) and show that it can match important life-cycle observations
on household stock market participation.
12
We include financial assets also to avoid mismeasuring the valuation of one investment vehicle (here, a college
education) by limiting access to others. One might add home equity as well, though housing services are also
generally available through rental markets, and thus barring significant frictions will not greatly influence the
inference we draw on the value of college.
7
capital, h1 , and their initial assets, x1 . 13 These characteristics are drawn jointly according to a
distribution F (a, h, x) on A × H × X.
Each period, agents choose how much to consume and how to divide their time between learning
and earning, as in Ben-Porath (1967). Agents also decide how much of their wealth to allocate to
stocks, s, versus bonds, b. The latter may be used to either borrow or save. Debt is not defaultable
and is subject to a borrowing limit, −b, where b > 0.
Agents work and accumulate human capital using the Ben-Porath technology until t = J.
Agents can also accumulate human capital by choosing (in the first period) to attend college.
College can be financed using wealth, x, unsecured debt, b, and nondefaultable, unsecured studentloan debt, d. Agents retire in period t = J + 1, after which they face a simple consumption-savings
and portfolio choice problem.
To capture an empirically important dimension of human capital accumulation, we assume
that agents may fail to complete college.14 At the end of four years in college, the probability of
completion—which depends on the agent’s learning ability as well as human capital accumulated to
that point—is realized.15 Those who complete college start their working life with human capital
hCG , where CG denotes college graduates. Those who fail to complete start their working life with
human capital hSC , where SC denotes “some college,” and those who choose not to go to college
start their working life at t = 1 with human capital hHS , where HS denotes “high school.”16
13
The ability to accumulate human capital—“learning ability”—must be interpreted with care. We assume, as
is standard in Ben-Porath models, that it is fixed over time for each agent. In other words, by the time agents
enter the model, they have learned as much as they can about how to learn. Ability reflects learning tools and
skills conferred on the young and, in sum, measures the effectiveness with which an individual can turn time into
human capital. In contrast, initial human capital represents the actual stock of learning accumulated by the time
the agent completes high school, which may be the result of investments made in the individual’s human capital by
their parents, the school system, and the community at large. Since learning is lifelong, human capital can increase
over the course of the agent’s life as long as they invest time in it.
14
For example, Bound et al. (2010) report, using NLS72 data, that only slightly over half of all college enrollees
graduate within eight years of enrollment.
15
While this approach is simple and direct, it abstracts from non-completion at intermediate dates. We think this
is a reasonable assumption: in the BPS 2004/2009 survey to which we calibrate completion, nearly three-fourths
of enrollees in four-year college remained enrolled for 37 months or more (National Center for Education Statistics,
2009). Allowing students to drop out at intermediate dates would matter most to those closest to the margin of
enrolling, but even for this group there is a tradeoff—leaving early enables them to earn market wages on their
human capital, but the level accumulated is lower. This assumption is therefore unlikely to be central to our measure
of the value of college.
16
Note that there is variation in the value of hCG , hSC , and hHS across individuals.
8
2.2
Preferences
Agents maximize the expected present value of utility over the life cycle:
max E0
T
X
β t−1 u(ct ),
(1)
t=1
where u(.) is strictly concave and increasing. Preferences are represented by a standard timeseparable constant relative risk aversion (CRRA) utility function over consumption. Agents do
not value leisure.
2.3
Human Capital
Agents can invest in their human capital in two ways—by investing in a college education when
young and by apportioning some of their available time to acquiring human capital throughout
their working lives.
Both within and outside college, agents accumulate human capital (as in the classic Ben-Porath,
1967, model):
ht+1 = ht (1 − δ) + a(ht lt )α with α ∈ (0, 1)
(2)
Human capital production depends on the agent’s learning ability, a, human capital, ht , the fraction
of available time put into it, lt , and the production function elasticity, α. Human capital depreciates
at a rate of δ, which we will allow to differ by education groups.
The incentives to invest in human capital are different in and outside college, for several reasons.
First, the rental rate on human capital grows faster for those who attend and complete college,
consistent with empirical evidence that shows faster earnings growth for college graduates. Second,
human capital depreciates at a higher rate for college educated individuals.17 This creates an
incentive to allocate time to human capital in college to compensate for faster depreciation. Third,
college graduates enjoy a higher replacement rate on their income at retirement (see Section 4).
Fourth, and importantly, the opportunity cost of spending time learning is lower on the college
path than on the no-college path. Outside college, earnings are a function of accumulated human
capital, whereas in college they are not. Those who work while in college face a relatively low
wage rate that does not differ with the level of human capital. This assumption is consistent
with evidence that the jobs college students hold do not necessarily value students’ human capital
stocks. In fact, we assume that working takes time away from human capital accumulation, and
17
This feature comes from estimating depreciation to match earnings declines at the end of the life cycle.
9
that accumulating less human capital decreases the odds of completion. This, too, is consistent
with empirical evidence that college jobs do not contribute to human capital accumulation and
that students who work while in college are more likely to drop out (see Autor et al., 2003; Peri
and Sparber, 2009). Consequently, most college students in the model find it optimal to allocate
all of their time to human capital accumulation, which is in line with empirical findings that the
majority of full-time students do not work while in college (see Manski and Wise, 1983; Planty et al.,
2008). College in the model thus represents a device that can greatly accelerate human capital
accumulation but harshly penalizes noncorner solutions for time allocation. Taken together, these
factors make human capital accumulation more attractive in college than outside of it during the
college years.
Our approach aims to measure the value of college when that investment is risky and costly and
when individuals have access to credit as well as to a high-yield financial investment (stocks). The
risk facing investors is whether devoting time fully to learning will deliver the expected returns to
whatever stock of human capital they are ultimately able to acquire. Credit also has a natural
feature in our environment: if used, it implies risk to future consumption due to the uncertainty
of college completion and future earnings.
We note, however, that if risk is either not quantitatively important, or if agents are riskneutral, and if additional financial assets like stocks are not available, the work of Huggett et al.
(2006) can be applied. Specifically, we can extend their Proposition 1 ii) to find a threshold under
which investing full-time in human capital is optimal for a period of time, after which only part
of an agent’s time is allocated to learning. The resulting life-cycle path of time allocation can
be interpreted as first featuring (full-time) schooling—including full-time college attendance—and
then partial investment in human capital. As in their case, we employ a Ben-Porath production
function, so that the value of college that we measure comes directly from individuals having access
to a technology that allows them to fully invest in themselves at all times, though they will not
necessarily elect to do so depending on their circumstances.18
2.3.1
College and its Financing
From the outset, we have highlighted the fact that ex-ante returns to investment in college vary
significantly because of their dependence on individual characteristics. These characteristics drive
18
Alternatively, one could consider a different functional form for human capital accumulation during college
years or different elasticity parameter of the Ben-Porath function. However, as Ionescu (2009) shows, the current
environment provides sufficient trade-off in the incentives to invest in human capital on college and outside of it to
induce agents in the economy to self-select in investing in college in a manner consistent with observed dynamics
of earnings over the life-cycle across education groups.
10
the likelihood of completion, subsequent earnings, and—via the institutional structure of higher
education financing—the cost of college. We now detail the accommodation of these features in
our model.
Those who invest in college face the possibility of noncompletion, which decreases with the
level of human capital accumulated during college. Specifically, the probability of completion,
∗
)), is an increasing function of the amount of human capital accumulated after
π(h5 (h1 , a, l1,...,4
completing four years in college, h5 , which in turn increases with the initial human capital stock,
∗
h1 , the agent’s learning ability, a, and the amount of time, l1,...,4
, that she chooses to allocate to
human capital accumulation (versus working) while in college.
Those who work in college can earn a maximum of wcol if they allocate all of their time to
working. Working during college diverts time away from human capital accumulation and therefore
increases the probability of noncompletion.
There are several possible sources of college financing: savings, x, borrowing, b, earnings from
working while in college, wcol (1 − l), merit- and need-based aid, κ(a, x1 ), and student loans. Notice
that because financial aid varies with individual characteristics, so too will the net cost of college.
Agents are allowed to take out student loans up to min[dmax , max(d¯− x1 , 0)]. Here, d¯ represents a
measure of the “full” cost of attending college; it includes tuition, books and supplies, room and
board, etc. Students can borrow up to this amount less x1 as long as the student loan limit, dmax ,
is not exceeded. We think of x1 not literally as the amount the student has saved (since savings
are negligible for most at this point in the life cycle), but rather as the so-called “expected family
contribution (EFC),” i.e., the savings to which the student has access. The EFC is a quantity that
federal student loan programs calculate as a part of determining the amount of subsidized lending
to which an enrollee is entitled.
Students choose how much to take out in student loans, d, in total (i.e., the stock of debt with
which they will emerge from college) at the beginning of college and receive equal fractions of that
loan ( d4 ) in each period (year) of college. While attending college, students are billed annually, in
ˆ which represents the direct cost of college (e.g, tuition and fee payments levied by
the amount d,
the college). After college, students repay their loan in equal payments, p, which are determined
by the loan size, d, interest rate on student loans, Rg , and the duration of the loan, P . Consistent
with the data, the interest rate on student loans is Rf < Rg < Rb , where Rf is the risk-free savings
rate and Rb the borrowing rate on unsecured debt.
The return to human capital is in the form of earnings during working life, which are subject
to shocks, as described below.
11
2.4
Earnings
During an agent’s working life, their earnings are given by:
yit = wt (1 − lit )hit zit
where w is the rental rate of human capital, (1 − lit ) is the time spent working, and zit is the
stochastic component. Both the rental rate on human capital and the stochastic component vary
between college graduates, CG, and those with no college, N C (which includes college dropouts
and high school graduates). The latter consists of a persistent component uit = ρui,t−1 + νit ,
with νit ∼ N (0, σν2 ), and a transitory independent and identically distributed (iid) component
it ∼ N (0, σ2 ). The variables uit and it are realized in each period over the life cycle and are not
correlated.
The rental rate of human capital evolves over time according to
wt = (1 + g)t−1
where g is the growth rate. This rate is higher for college graduates than for those with no college.
2.5
Means-Tested Transfer and Retirement Income
We allow agents to receive means-tested transfers, τt , which depend on age, income, and assets.
Following Hubbard et al. (1994), we specify these transfers as
τt (t, yt , xt ) = max{0, τ − (max(0, xt ) + yt )}
(3)
These transfers capture the net effect of the various US social insurance programs that are
aimed at providing a floor on income (and thereby on consumption).
After period t = J, in which agents start retirement, they receive a constant fraction of their
earnings in the last working period, ϕ(yJ + τJ ), which they allocate between risky and risk-free
investments. We allow the income replacement rate for college graduates to differ from the rate
for all other agents.
2.6
Financial Markets
There are two financial assets in which the agent can invest, a risk-free asset, bt , and a risky asset,
st .
12
Risk-free assets
An agent can borrow or save using asset bt . Savings will earn the risk-free interest rate, Rf .
We assume that the borrowing rate, Rb , is higher than the savings rate: Rb = Rf + ω. Debt is
nondefaultable and comes with a borrowing limit b > 0.
Risky assets
Risky assets, or stocks, earn stochastic return Rs,t+1 in period t + 1, given by:
Rs,t+1 − Rf = µ + ηt+1 ,
(4)
where ηt+1 , the period t + 1 innovation to excess returns, is assumed to be iid over time and
distributed as N (0, ση2 ). We assume that innovations to excess returns are uncorrelated with
innovations to the aggregate component of permanent labor income.
Given asset investments at age t, bt+1 and st+1 , financial wealth at age t + 1 is given by
xt+1 = Rj bt+1 + Rs,t+1 st+1
with Rj = Rf if b ≥ 0 and Rj = Rb if b < 0.
2.7
Agent’s Problem
The agent chooses whether or not to invest in college (and, if investing in college, how much
student debt to take on), how much to consume, how much time to allocate to learning, and asset
positions in stocks and bonds (or borrowing) to maximize expected lifetime utility.
We solve the problem backward, starting with the last period of life when agents consume all
their available resources. The value function in the last period of life is set to VTR (a, h, x) = u(x).
Retired agents do not accumulate human capital. They face a simple consumption-savings
problem and may choose to invest in both risk-free and risky assets. The value function is given
by
1−σ
c
0
0
R
R
V (t, a, b, s, yJ + τJ ) = max
+ βV (t + 1, a, b , s , yJ + τJ )
(5)
1−σ
b0 ,s0
13
where
0
c+b +s
b
s
0
0
0
≤ ϕ(yJ + τJ ) + Rj b + Rs s
≥ b
≥ 0
In the above, Rj = Rf if b ≥ 0, and Rj = Rb if b < 0. The only uncertainty faced by retired
individuals pertains to the rate of return on the risky asset.
2.7.1
Problem in Working Phase for those with No College
We use VJR (t, a, b, s, yJ + τJ ) from Equation 5 as a terminal node for the adult’s problem on the
no-college path. We solve
V
HS
(t, a, h, b, s, z) = max
0 0
l,b ,s
ct1−σ
0
0
0
0
HS
+ βEV (t + 1, a, h , b , s , z )
1−σ
(6)
where
0
c+b +s
0
≤ w(1 − l)hz + Rb b + Rs s + τ (t, y, x) for t = 1, .., J
l ∈ [0, 1]
0
h
b
s
2.7.2
0
0
= h(1 − δ) + a(hl)α
≥ b
≥ 0
Problem in Working Phase for those who Attended College
As before, we use VJR (t, a, b, s, yJ + τJ ) from the retirement phase as a terminal node and solve
for the set of choices in the working phase j = 5, .., J of the life cycle. We further break down
the working phase into a student loan post-repayment period and a repayment period. In the
post-repayment period, t = P + 1, ..., J, the problem is identical to the one for working adults on
the no-college path.
During the repayment period, t = 5, ..., P , agents have to repay their student loans with a
per-period payment
d
p = PP −5 1 .
t=1 Rgt
14
The value function is given by
j
V (t, a, h, b, s, z) = max
0 0
l,b ,s
c1−σ
0
0
0
0
t
j
+ βEV (t + 1, a, h , b , s , z ) , j = CG, SC
1−σ
(7)
where
0
0
0
0
c+b +s
c+b +s
≤ w(1 − l)hz + Rj b + Rs s + τ (t, y, x) for t = P + 1, .., J
≤ w(1 − l)hz + Rj b + Rs s + τ (t, y, x) − p(x1 ) for t = 5, .., P
l ∈ [0, 1]
0
h
b
s
0
0
= h(1 − δ) + a(hl)α
≥ b
≥ 0
Rj = Rf if b ≥ 0 and Rj = Rb if b < 0.
2.7.3
Problem in College
For the college phase t = 1, . . . , 4 of the life cycle, we first take into account the likelihood of dropping out from college and use V C (5, a, h, b, s, z) = π(h5 )V CG (5, a, h, b, s, z)+(1−π(h5 ))V SC (5, a, h, b, s, z)
as the terminal node. The value function is given by
C
V (t, a, h, b, s, z) = max
0 0
l,b ,s
c1−σ
0
0
0
0
C
+ βEV (t + 1, a, h , b , s , z )
1−σ
(8)
where
0
c+b +s
0
= wcol (1 − l) + Rb b + Rs s +
d ˆ
− d + κ(a, x1 )
4
l ∈ [0, 1]
0
= h(1 − δ) + a(hl)α
d ≤ min[dmax , max[d¯ − x1 , 0]]
h
b
s
0
0
≥ b
≥ 0.
While in college, the rental rate of human capital is set to a relatively low value (see Section 4),
which means that human capital is not productive until graduation. As noted earlier, this reflects
evidence that the jobs in which college students work do not necessarily utilize or augment students’
15
human capital stocks. The set of skills involved in these jobs is different from the one students
acquire in college and use after graduation. An implication of this assumption is that in the
model college students find it optimal to allocate all of their time in college to human capital
accumulation, a result that is consistent with the empirical findings that the majority of full-time
college students do not work while in school. Finally, people who choose to work while in school
most likely drop out of college, as numerous studies attest.
Our model captures the large variety of resources that are available to students to finance their
college education in addition to those obtained from working during college. Every year in college,
students use a combination of personal and family savings, captured by x, unsecured borrowing,
b, student loans, d4 , and merit- and need-based grants, κ(a, x1 ), to pay for direct college expenses,
ˆ We assume that the direct cost paid each period while in college incorporates the existing
d.
large subsidy that students receive to finance college, as we discuss in the calibration section. As
described before, our model captures the key features of the student loan program with agents
being allowed to borrow up to the full college cost minus the expected family contribution, d¯− x1 ,
as long as they do not hit the constraint dmax . Importantly, agents use the loan amount to pay
for college expenses while in college.19
Once the college and no-college paths are fully determined, agents then select between going
to college or not by solving max[V C (1, a, h, x), V HS (1, a, h, x)].
3
Data
In order to map our model to data, we use data on annual earnings from the March Current
Population Survey (CPS), on college enrollment and completion rates from the Beginning Postsecondary Student Longitudinal Survey (BPS) 2004/2009 and the National Education Longitudinal
Study (NELS:1988), and on credit limits from the Survey of Consumer Finances (SCF).20
3.1
Life-cycle earnings
As described in more detail in the next section, we calibrate our model to match the evolution of
mean earnings, earnings dispersion, and earnings skewness over the life cycle. To this end, we first
estimate life-cycle profiles for ages 23 to 60 (i.e., the “working life”) of mean earnings, the earnings
Gini coefficient, and the mean/median earnings ratio using data from the March CPS, obtained
19
Without this feature, college enrollees would have an advantage over the no-college group in being able to
borrow at a subsidized rate to finance consumption over the life-cycle.
20
We also use information on financial assets from the Survey of Consumer Finances (SCF) to compare with
model predictions, as discussed later.
16
Figure 1: Life-cycle earnings statistics
Mean of lifecycle earnings
200
180
160
140
120
100
80
60
40
20
0
25
30
35
40
45
50
55
60
50
55
60
Age
Mean/Median of lifecycle earnings
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
25
30
35
40
45
Age
Gini of lifecycle earnings
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
25
30
35
40
45
50
55
60
Age
through IPUMS at the University of Minnesota. We use data on annual wage and salary income
for male heads of household with at least a high school diploma (or equivalent) for calendar years
1963-2013 (corresponding to survey years 1964-2014). We restrict our sample to individuals who
worked at least 12 weeks in the reference year and earned at least $1,000 (in constant 2014 prices).
We use the CPS weights to ensure that each year’s sample is representative of the overall US
population; additionally, we renormalize the weights in each year in order to keep the population
constant at its 2014 value; this way we abstract from issues related to population growth.
17
We use these data to construct life-cycle profiles for mean earnings, the earnings Gini coefficient,
and the mean/median earnings ratio. Specifically, for each of these statistics, st,y , we compute st,y
in the data for each combination of age t and calendar year y, and regress st,y against a full set of
year and age indicators.21 We then take the regression coefficients on the age indicators (we use
the latest year as our base year), and normalize them so that at age 40 the coefficients profile goes
through the unconditional average value of s40,y across all years y in our sample. The corresponding
normalized age coefficients constitute the life-cycle profiles that we use in the calibration. Figure 1
shows the life-cycle profiles of mean earnings, the earnings Gini, and the mean/median earnings
ratio obtained in this fashion.
3.2
College enrollment and completion
We use data from the Beginning Postsecondary Student Longitudinal Survey (BPS) 2004/2009
and the National Education Longitudinal Study (NELS:1988) to match enrollment and completion
rates. Specifically, we estimate correlations of ability and initial wealth, and of initial human capital
and initial wealth, to match college enrollment rates for three groups of initial wealth (expected
family contributions) based on NELS:1988 data and to match college completion rates based on
the BPS 2004/2009 dataset for students who enrolled in college in the year 2003-04. These rates
are reported in Table 1.22
Table 1: Enrollment and Completion by Wealth (NELS and BPS)
Initial wealth
Overall
Low
Medium
High
College Enrollment Rate
47
34
47
62
College Completion Rate
45
37
45
60
The NELS:1988 is a nationally representative sample of eighth graders who were first surveyed
in the spring of 1988. A sample of these respondents were then resurveyed through four follow-up
surveys in 1990, 1992, 1994, and 2000. We use the third follow-up survey when most respondents
completed high school and report their postsecondary access and choice. As in the BPS, demographic information, including SAT scores and EFC, is available. We use this dataset to compute
21
By using a full set of year indicators, this treatment controls for year effects in the construction of the age
profiles. We have also computed age profiles controlling for cohort effects rather than year effects. The behavior
of the life-cycle profiles is qualitatively similar.
22
Here the expected family contribution (EFC) is used as the measure of initial wealth. Low and high refer to
the lowest and highest quartiles of wealth, respectively, and medium to the middle two quartiles together.
18
college enrollment rates by EFC. Our sample consists of recent high school graduates aged 20-30
who have taken the SAT (or ACT).
The BPS 2004/2009 is one of several National Center for Education Statistics (NCES)-sponsored
studies that is a nationally representative dataset with a focus on postsecondary education indicators. BPS cohorts include beginners in postsecondary schools who are surveyed at three points in
time: in their first year in the National Postsecondary Student Aid Study (NPSAS) and then three
and six years after first starting their postsecondary education in follow-up surveys. BPS collects
data on a variety of topics, including student demographics, school experiences, persistence, borrowing/repayment of student loans, and degree attainment six years after enrollment. Our sample
consists of students aged 20-30 who enroll in a four-year college following high school graduation.
For demographic characteristics, we use SAT (and converted ACT) scores as a proxy for ability
and expected family contribution (EFC) as a proxy for wealth.
4
Mapping the model to the data
The parameters in our model include: 1) standard parameters such as the discount factor and the
coefficient of risk aversion; 2) parameters specific to human capital and to the earnings process;
3) parameters governing the distribution of initial characteristics; 4) parameters specific to college
investment and financing; and 5) parameters specific to financial asset markets. Our approach
involves a combination of setting some parameters to values that are standard in the literature,
calibrating some parameters directly to data, and jointly estimating the parameters that we do
not observe in the data by matching moments using several observable implications of the model.
These parameters are listed in Table 2. We present the details of the calibration in the next section,
followed by the model fit relative to data.
4.1
4.1.1
Calibration
Preference parameters
The per period utility function is CRRA as described in the model section. We set the coefficient
of risk aversion, σ, to 3, which is consistent with values found in the literature.23 The discount
factor used (β = 0.96) is also standard in the literature.
23
We conduct robustness checks on this parameter by looking at alternative values such as the upper bound of
σ = 10 considered reasonable by Mehra and Prescott (1985) as well as lower values such as σ = 2. The results are
available upon request.
19
Table 2: Parameter Values: Benchmark Model
Parameter
T
J
β
σ
Rf
Rb
b
µ
ση
α
gN C , gCG
δN C , δCG
ψN C , ψCG
τ
2
2
(ρN C , σνN C , σN
C)
2
2
(ρCG , σνCG , σCG )
(µa , σa , µh , σh , %ah ,
%ax , %hx )
dˆ
d¯
dmax
Rg
wcol
4.1.2
Name
Model periods (years)
Working periods (after college)
Discount factor
Coeff. of risk aversion
Risk-free rate
Borrowing rate
Borrowing limit
Mean equity premium
Stdev. of innovations to stock returns
Human capital production function elasticity
Growth rate of rental rate of human capital
Human capital depreciation rate
Fraction of income in retirement
Minimal income level
Earnings shocks no college
Earnings shocks college
Parameters for joint distribution of ability
initial human capital, initial wealth
Annual direct cost of college
Full cost of college
Limit on student loans
Student loan rate
Wage during college
Value
58
34
0.96
3
1.02
1.11
$17,000
0.06
0.157
0.7
0.01, 0.02
0.021, 0.038
0.682, 0.93
$17, 936
(0.951, 0.055, 0.017)
(0.945, 0.052, 0.02)
(0.29, 0.50, 79.71, 45.26, 0.66,
0.36, 0.42)
$7,100
$53,454
$23,000
1.068
$17,700
Human capital and income parameters
We set the elasticity parameter in the human capital production function, α, to 0.7. Estimates of
this parameter are surveyed by Browning et al. (1999) and range from 0.5 to 0.9. To parameterize
the stochastic component of earnings, zit , we follow Abbott et al. (2018), who use the National
Longitudinal Survey of Youth (NLSY) data using CPS-type wage measures to estimate parameters
for the idiosyncratic persistent and transitory wage shocks. For the persistent shock, uit = ρui,t−1 +
νit , with νit ∼ N (0, σν2 ) and the transitory shock, it ∼ N (0, σ2 ), they report the following values:
for high school graduates, ρ = 0.951, σω2 = 0.055, and σν2 = 0.017; for college graduates, ρ = 0.945,
σω2 = 0.052, and σν2 = 0.02. We use the first set of values for individuals with no college as well
as for those with some college education and the second set of values for those who complete four
years of college.
As previously noted, the rental rate of human capital in the model evolves according to wt =
20
(1 + g)t−1 . The growth rate g is calibrated to match the average growth rate in mean earnings
observed in the data. We obtain 0.01 for individuals with no college degree and 0.02 for college
graduates.
Given the growth rate in the rental rates, the depreciation rates are set so that the model
produces the rate of decrease of average real earnings at the end of the working life. The model
implies that at the end of the life cycle negligible time is allocated to producing new human capital
and, thus, the gross earnings growth rate approximately equals (1 + g)(1 − δ). We obtain 0.021
for individuals with no college degree and 0.038 for college graduates.
We set retirement income to be a constant fraction of labor income earned in the last year
in the labor market. Following Cocco (2005) we set this fraction to 0.682 both for high school
graduates and for those with some college education and to 0.93 for college graduates.
4.1.3
Distribution of initial characteristics: financial assets, ability and human capital
The distribution of initial characteristics (ability, human capital, and financial assets) is determined
by seven parameters. These parameters are estimated to match the evolution of three moments of
the earnings distribution over the life cycle (mean earnings, the Gini coefficient of earnings, and the
ratio of mean to median earnings) and college enrollment and college completion rates across three
wealth groups (proxied by expected family contributions). The estimation proceeds as follows.
First, for the distribution of initial financial assets, x1 , we use data from the BPS 2004/2009.24
Second, we calibrate the joint distribution of ability and initial human capital to match the key
properties of the earnings distribution over the life cycle reported earlier using March CPS data.
Third, we estimate the correlations of ability and initial wealth, and of initial human capital and
initial wealth, to match college enrollment rates based on NELS:1988 data, and college completion
rates based on BPS 2004/2009 data.
The dynamics of the earnings distribution implied by the model are determined in several steps:
i) we compute the optimal decision rules in the model using the parameters described above for
an initial grid of the state variable; ii) we simultaneously compute college, human capital, and
financial investment decisions and compute the life-cycle earnings for any initial pair of ability and
human capital; and iii) we choose the joint initial distribution of ability and human capital to best
replicate the properties of earnings from the CPS data.
24
The proper notion of wealth at age 18 is not unambiguous. In particular, while 18-year-olds typically do not
have substantial wealth of their own, they may have access to alternative sources of wealth that are not directly
measured, most notably, intervivos transfers from their parents. In this context, therefore, the EFC seemed to us
to be the most appropriate measure of wealth available to high school graduates.
21
Since a key goal of this paper is to measure the value of college across heterogeneous agent
types, how we arrive at the distribution of initial heterogeneity is important. In this we employ
the strategy pioneered in Huggett et al. (2006), who show that a joint lognormal distribution that
allows for heterogeneity in both learning ability and human capital, as well as a correlation between
the two, matches properties of US earnings data well. Furthermore, Huggett et al. (2006) prove
that heterogeneity in learning ability is necessary and demonstrate that heterogeneity in initial
human capital and a positive correlation between the two are important to match properties of the
earnings distribution over the entire life cycle. Following their methodology, we restrict the initial
distribution to lie on a two-dimensional grid spelling out human capital and learning ability. The
underlying joint lognormal distribution is characterized by five parameters: the mean and standard
deviation of ability and initial human capital, respectively, and the correlation between the two.25
We then search over the vector of parameters that characterize the initial state distribution to
minimize a distance criterion between the model and the data. Specifically, we find the vector of
these parameters γ = (µa , σa , µh , σh , %ah ) by solving the minimization problem:
min
γ
J
X
!
|log(mj /mj (γ))|2 + |log(dj /dj (γ))|2 + |log(sj /sj (γ))|2
j=5
where mj , dj , and sj are the mean, dispersion, and skewness statistics constructed from the CPS
data on earnings, and mj (γ), dj (γ), and sj (γ) are the corresponding model statistics.26
We estimate the correlations of ability and initial wealth, and of initial human capital and
initial wealth, that best replicate college enrollment and college completion rates by wealth levels
(see further details in the next subsection). Our estimation delivers a correlation between ability
and initial human capital stock of 0.67 and a correlation between initial wealth and ability and
initial human capital of 0.36 and 0.42, respectively.
4.1.4
College parameters
We first specify parameters related to students’ ability to borrow public funds to finance their
college education. We set the full cost of college to d¯ = $53, 454. This figure is the enrollmentweighted average of the full annual cost at a public four-year institution ($33,849) and a private
four-year institution ($78,570) between academic years 2003-04 and 2007-08, weighted by the
fraction of students attending each in the data (60 and 40 percent, respectively). Recall from the
agent’s problem that this is not what students actually pay but rather a parameter that influences
25
26
In practice, the grid is defined by 20 points in human capital and in ability.
For details on the calibration algorithm, see Huggett et al. (2006) and Ionescu (2009).
22
the borrowing limit on student loans, which is set at min[dmax , max[d¯ − x1 , 0]]. We set the other
parameters that governs this constraint, i.e., the direct limit on student loans, to dmax = $23, 000,
and the interest rate to Rg = 1.068, respectively, as specified in the Department of Education
guidelines for the students who enter college in 2003-04.
We turn next to parameters that determine the actual privately borne cost of college (both
direct and in terms of forgone earnings). These costs are of course the relevant ones for anyone
ˆ (which, as described eardeciding whether to enroll. First, the annual direct cost of college, d,
lier, incorporates direct subsidies and is what students are actually billed), is $7,100. This is a
proportion (53 percent) of an enrollment-weighted average of the direct cost at public and private
four-year institutions.27 The key point is that the subsidy reduces the direct cost of college by
47 percent. Next, we set the wage during college, wcol , to $17,700 based on NCES data. Lastly,
we must spell out nonloan financial aid available to enrollees. Here, we use the BPS data to set
both merit- and need-based aid. Merit-based aid increases with an enrollee’s high school GPA: for
example, those in the top GPA quartile receive merit-based aid equivalent to about 63 percent of
the total college cost, while those in the bottom quartile receive about 10 percent. In the model,
we use ability as a proxy for high school GPA. To be consistent with the data, we assume that the
fraction of the total college cost covered by merit-based aid increases with ability. We calibrate
it to ensure that the average aid amount within each ability quartile in the model matches the
average aid amount within each GPA quartile in the data.
Turning to need-based aid, we observe in the BPS data that students in the bottom quartile
of EFC receive about 7 percent of the college cost in the form of need-based aid, on average, while
those in the top quartile of EFC receive, on average, no need-based aid. Since EFC is a function of
wealth, we assume that the fraction of the total college cost received in the form of need-based aid
decreases with wealth. We calibrate this so that the average need-based aid within each quartile
of initial wealth in the model matches average need-based aid within the corresponding quartile of
EFC in the data.
We have specified the parameters governing access to public student loans as well as the costs
that any individual seeking to attend college faces. We now specify the risks associated with college
investment. In the model, we assume that the probability of college completion is a function of
the amount of human capital accumulated at the end of college, h5 (which in turn is a function
of ability, initial human capital, and the time spent learning during college). We use cumulative
GPA scores in the BPS data as a proxy for h5 . In the data, we observe the fraction of the
student population that obtained each of the grades listed in Table 3. In the model, we divide
27
For details on how these costs are calculated, see Ionescu and Simpson (2016).
23
the distribution of h5 into groups according to these percentages, and we assign each group the
completion probability listed in the first column of the table.28 For example, an agent in the group
with the highest level of h5 will face a 70 percent probability of completion.
Table 3: Completion Rates by GPA in College
Completion rate
0.07
0.30
0.45
0.56
0.67
0.70
4.1.5
Grades
mostly Cs and Ds
mostly Cs
mostly Bs and Cs
mostly Bs
mostly Bs and As
mostly As
Financial markets
We turn now to the parameters in the model related to financial markets. We fix the mean equity
premium to µ = 0.06, as is standard in the literature (e.g., Mehra and Prescott, 1985). The
standard deviation of innovations to the risky asset is set to its historical value, ση = 0.157. The
risk-free rate is set equal to Rf = 1.02, consistent with values in the literature (McGrattan and
Prescott, 2000), while the wedge between the borrowing and risk-free rate is 0.09 to match the
average borrowing rate of Rb = 1.11 (Board of Governors of the Federal Reserve System, 2014).
We assume a uniform credit limit (b) across households. We obtain the value for this limit from
the SCF. The SCF reports, for all individuals who hold one or more credit card, the sum total
of their credit limits. We take the average of this over all individuals in our sample and obtain a
value of approximately $17,000 in 2014 dollars. Note that, when we take the average, we include
those who do not have any credit cards. This ensures that we are not setting the overall limit to
be too loose. Lastly, in our baseline model, we assume for the time being that the returns to both
risky assets (human capital and financial wealth) are uncorrelated.
4.2
Model vs. Data
We start by presenting the model predictions for targeted data moments for the baseline economy,
and we then describe model predictions for key nontargeted data moments.
28
We define the completion rate in the data as the fraction of students who had earned a bachelor’s degree by
June 2009.
24
4.2.1
Targeted Moments
Figure 2: Life-cycle earnings statistics
Mean of lifecycle earnings
200
Model
CPS data
180
160
140
120
100
80
60
40
20
0
25
30
35
40
45
50
55
Age
Mean/Median of lifecycle earnings
2
Model
CPS data
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
25
30
35
40
45
50
55
Age
Gini of lifecycle earnings
1
Model
CPS data
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
25
30
35
40
45
50
55
Age
This section presents measures of goodness of fit for the baseline model. Figure 2 shows the
earnings moments for a simulated sample of individuals in the model versus the CPS data.29 As
the figure shows, the model does a reasonably good job of fitting the evolution of mean earnings
PJ
1
As a measure of goodness of fit, we use 3(J−4)
j=5 |log(mj /mj (γ))| + |log(dj /dj (γ))| + |log(sj /sj (γ))|. This
represents the average (percentage) deviation, in absolute terms, between the model-implied statistics and the data.
We obtain a fit of 8 percent (where 0 percent represents a perfect fit).
29
25
over the life cycle, though the model’s profile is less hump-shaped than the data.30 The skewness
of earnings is a touch lower in the model than in the data. And, for the Gini coefficient, the model
matches the data quite well, except perhaps in the first and last few years of the life cycle.
We next look at the model’s predictions for college investment behavior by initial wealth.
Table 4 shows college enrollment and completion rates by level of initial financial wealth; where
“low” refers to the bottom quartile, “medium” to the two middle quartiles, and “high” to the top
quartile of the distribution of initial wealth. As can be seen, the baseline calibration captures well
the fact that both enrollment and completion rates are strongly increasing in the level of initial
wealth.
Table 4: Targeted Moments Data vs Model: Enrollment and Completion
Initial wealth
College Enrollment
Low
Medium
High
College Completion
Low
Medium
High
4.2.2
Model
54
35
55
74
49
43
49
57
Data
47
34
47
62
45
37
45
60
Nontargeted Moments
We show next that our model performs well along relevant nontargeted dimensions. Given our
focus on the payoff to investment opportunities, key among these is life-cycle earnings across groups
defined by educational attainment. Note that our calibration only targeted overall earnings and,
not earnings by education group. Figure 3 shows that our model nonetheless delivers the pattern
seen in the data of mean earnings by education groups over the life cycle. In particular, the model
replicates the steeper earnings profile for college graduates relative to individuals with some college
education and no degree or those who never went to college. This is a result of both higher growth
rate of the rental rate of human capital for college graduates and selection. As we discuss next,
individuals with relatively higher levels of ability select into going to college and, among them
those with relatively high levels of ability and human capital stock at the end of college complete.
While the ability level mostly impacts the slope of earnings, the human capital stock impacts its
30
Because we assume that retirement income is a function of earnings just before retirement, agents in our model
have an incentive to maximize pre-retirement earnings. This explains why earnings do not taper off near retirement
in our model.
26
level. Note that college graduates start working life with a relatively higher level of earning relative
to individuals without a college degree which is consistent with empirical evidence, although the
model overpredicts starting earnings for all three education groups.
We next examine college enrollment and completion behavior by individual characteristics. As
seen in Table 5, the model predicts that both college enrollment and completion rates are increasing
in ability and in initial human capital. While there is no direct data counterpart to the notions
of ability and initial human capital as represented in the Ben-Porath setting, we see that when
college investment behavior is ordered by SAT score—arguably the most widely used measure of
college readiness—the model’s implications are qualitatively borne out in the data.
Table 5: Nontargeted Moments: Enrollment and Completion by Characteristics
Characteristic
College Enrollment
Low
Medium
High
College Completion
Low
Medium
High
Ability
Initial Human Capital
Data: SAT scores
9
63
85
26
65
64
53
65
85
20
42
64
27
48
68
30
50
69
We now compare the model’s predictions for financial wealth accumulation over the life cycle
with data from the Survey of Consumer Finances (SCF). The details of how we obtain estimates
of financial wealth from the SCF are provided in the Appendix. Figure 4 shows the mean wealth
accumulation over the life cycle for total assets as well as for risky and risk-free assets. Overall, the
model is consistent with the overall trajectory of wealth accumulation, but it underpredicts mean
wealth by age. We note that mean wealth in the US data is strongly influenced by the extreme
right tail of the distribution. Indeed, this has led models aimed at capturing the skewness of
wealth to employ earnings processes in which agents receive extremely large but transitory shocks
to earnings with extremely low probability (Castaneda et al., 2003). As a result, the presence or
absence of such improbable shocks is unlikely to be quantitatively important for wealth at the
individual level.
Finally, as shown in Figure 5, our model’s prediction for the stock-market participation rate is
consistent with the data, over the entire life cycle and by education groups. This result is driven
primarily by the presence of human capital. Human capital is an attractive investment early
in life, especially for those with a combination of high learning ability and relatively low initial
human capital: the opportunity cost of spending time learning—forgone earnings—is relatively
27
Figure 3: Life-Cycle Earnings by Education Group
Mean of lifecycle earnings
200
no col
cd
cg
CPS--College Grads
CPS-College Drop
CPS-HS
180
160
140
120
100
80
60
40
20
0
25
30
35
40
45
50
55
Age
low, the marginal return to learning is high, and the horizon over which to recoup any payoff from
learning is long. Further, anticipating rising earnings over the life cycle, households who invest in
human capital early in life will desire, absent risk, to avoid large positive net positions in financial
assets when young. As they age and accumulate human capital, these households will find further
investment in human capital less attractive as the marginal return decreases and opportunity cost
increases. These high earners will then accumulate wealth and participate in the stock market
at high rates. This mechanism, illustrated in detail in Athreya et al. (2015), delivers a profile of
aggregate stock market participation that is consistent with the data, as Figure 5 shows.
5
Results
We now use the model described and parameterized above to address the questions posed at the
outset: What is the value of access to college across the population? How do these valuations
depend on the direct college subsidies currently in place? Are there simple alternatives that
illustrate the potential for gains in welfare? We will first demonstrate that these valuations vary
widely, and that they depend strongly on existing subsidies. These findings then motivate us to
study the implications of reallocating subsidies away from those who enroll even the absence of
the subsidy (the ”inframarginal enrollees”) and towards those who never do (the ”inframarginal
nonenrollees”).
28
Figure 4: Life-Cycle Wealth Accumulation
5
10 5
Mean of total assets over the lifecycle
Model,
SCF data time effects
SCF data cohorteffects
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
25
2.5
30
10
5Mean
35
40
45
50
55
Age
of net riskfree assets over the lifecycle
Model,
SCF data time effects
SCF data cohorteffects
2
1.5
1
0.5
0
25
3.5
30
10
5
35
40
45
50
55
Age
Mean of risky assets over the lifecycle
Model
SCF data time effects
SCF data cohorteffects
3
2.5
2
1.5
1
0.5
0
25
30
35
40
Age
29
45
50
55
Figure 5: Stock Market Participation over the Life Cycle by Education Groups
1
Participation in stocks over the lifecycle
Model HS
Model CG
Data HS
Data CG
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
25
30
35
40
45
50
55
Age
5.1
The Value of Access to College
We assess the value of access to college across individuals by comparing our baseline economy
to an economy with no college. Specifically, for each agent type, the value is calculated as the
consumption-equivalent gain obtained from moving from the no-college environment to the baseline.
Note that we allow agents to fully reoptimize in the no-college environment. We do so to not
overstate the value of college by preventing the agent from fully exploiting substitution possibilities.
In this instance, absent college, agents can make different decisions about stock market investment,
borrowing, and how much time to spend on human capital accumulation, than in the baseline.
We measure the value of access to college via the following steps. First, for each agent type—
where type is defined by a profile of initial characteristics s1 = (a, h1 , x1 )—we compute the ex-ante
expected utility associated with starting life in each of the two economies: the baseline (B) and
the no-college economy (NC). The expectation arises from uncertainty in income and in the rate
of return on stocks to which all agents are exposed. Thus, the relevant object is V k (s1 ) where s1
represents the agent type’s pre-college-decision state vector (i.e., the initial state) and k denotes
the economy in question: k ∈ {B, N C}. The value function V k (s1 ) is, by definition, the maximized
expected utility for an agent with initial characteristics s1 .
Second, given measures of V k for all types s1 (and for all k), the value of access is calculated
30
as follows. Let the constant-consumption equivalent to any value V k be defined as the constant
P
1−σ
value of consumption ck (s1 ) that satisfies Tt=1 β t−1 c1−σ = V k (s1 ), or:
1
1−σ
1−β
k
c (s1 ) =
(1 − σ)V (s1 )
1 − βT
k
Given ck for all k, we can compute the value of access to the baseline economy for an agent
type s1 currently in the no-college economy N C—in terms of the net percentage difference in the
relevant constant-consumption equivalents—as:
γNC =
cB (s1 )
−1
cN C (s1 )
Note that because we hold prices (interest rates and stock returns) and income processes
(conditional on educational attainment) fixed, absent any compensation available to agents in the
no-college economy, agents cannot be made worse off by moving from the no-college economy to
the baseline. Accordingly, the value of access is nonnegative for all agent types.
We find that the mean value (across all agent types) of access to college is 14.7 percent of
consumption. This is a substantial amount, corresponding to about $6,000 per year (under the
assumption that mean annual consumption is about $40,000 annually). The relatively large valuation reflects that college provides an efficient way to accumulate human capital and that college
completion yields a mean earnings premium that is quite large.
Looking beyond simple averages, however, Figure 6 shows the cumulative distribution function
(CDF) of valuations for college and reveals a great deal of heterogeneity. The horizontal axis in the
figure shows the value of college expressed as a percent of consumption (as described above). Note
that since the valuations cannot be negative, the valuations range from zero to one. The figure
Figure 6: CDF of Valuations for College
Aggregate Willingness to pay CDFs
1
For college
0.9
0.8
0.7
F(x)
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
x
31
0.6
0.7
0.8
0.9
shows that nearly 40 percent of agents place no value on access to college (the CDF of college
valuations at zero is about 0.4). At the opposite end, nearly a quarter of the population value
access to college at 30 percent or more of consumption (the CDF at 0.3 of consumption is about
0.75).
Our framework allows us to further ask who in the population derives the most value from
access to college. There is, of course, an entire distribution of agent types, and therefore an
entire distribution of valuations we could report. To keep results tractable while informative of
the basic trade-offs present, we begin by reporting valuations averaged over all individuals within,
respectively, the bottom quartile (“low”), the two middle quartiles (“middle”), and the top quartile
(“high”) of learning ability, initial human capital, and initial wealth.
Table 6 shows that mean valuations vary greatly across ability, initial human capital, and initial
wealth groups. Starting with ability, we see that for those in the lowest quartile of the ability distribution, access to college is worth very little—about 1 percent in consumption-equivalent terms.
The value rises rapidly with ability, reaching about 28 percent of consumption, on average, for
those in the top quartile of ability. For the latter quartile, college completion is nearly guaranteed
and, given the large college earnings premium in the economy, the opportunity to invest in college
is worth a lot.
Looking next across groups defined by initial wealth (the bottom panel of the table), we see that
a similar story emerges, though the gains do not rise quite as sharply as in the case of ability. This
likely reflects the absence of significant credit constraints in these economies (which in turn reflects
our reading of the literature on the empirical strength of borrowing constraints for education). The
presence of risk of course does make a difference, whereby greater initial wealth allows a greater
number of those with low ability and low initial human capital (i.e., those facing a higher risk of
noncompletion) to invest in college anyway. As for the role of initial human capital (the middle
panel of the table), the results are less clearly ordered, reflecting a trade-off: those with low initial
human capital face both a higher marginal return to investing in college (in part because they
experience lower foregone earnings) and a higher risk of noncompletion.
5.2
The Role of Direct College Subsidies
To what extent do the high valuations for college obtained above depend on the large direct
subsidies to college? To answer this question, we consider an economy in which college is not
directly subsidized, i.e., the nearly 50 percent reduction in college costs coming from the subsidy
is made unavailable.
Figure 7 reports the CDFs for the valuations of college access in both the baseline economy
32
Table 6: Average Valuation of College Access by Initial Characteristic
3
Initial
Valuation
Characteristic
(% of Consumption)
Ability
Low
1.0%
Middle
15.1%
High
27.5%
Human Capital
Low
13.3%
Middle
13.2%
High
19.0%
Wealth
9.3%
How muchLow
of WTP for college is driven by
the WTP for the
Middle
13.7%
large subsidy?
High
22.0%
Main points:
1. Lovely! :)
and the no-subsidy
counterfactual. The blue line represents valuations in the baseline, while
2. The red line is based on an economy where the option to invest in college is still available but there is
college subsidy.
red represents the nocounterfactual.
The implied mean valuation across all types for access to
3. Could
think of5the
writing of this exercise
in two
ways, than
I can seein
pros
and baseline.
cons either way.ItThe
unsubsidized college
is over
percentage
points
lower
the
is difference
therefore clear that
is subtle, but I prefer the positive spin in approach a below:
for many, subsidies are an important part of the high value they derive from college (9.2 percent
(a) College is valuable even in the absence of the subsidy, although for some individuals wtp declines
quite a bit, whereas for others stays about the same.
of consumption).
(b) In the absence of the large subsidy, college is less valuable, in particular for some individual types.
Figure
7: CDFs
of for
Valuations
forfrom
College
With
andtoWithout
4. On average,
the WTP
college goes down
0.145 (includes
subsidy)
0.09 (without Subsidy
the subsidy).
1
0.9
0.8
0.7
F(x)
0.6
0.5
0.4
0.3
0.2
0.1
Including college subsidy
Excluding college subsidy
0
0
0.2
0.4
0.6
0.8
1
x
3.1 Who is willing to pay how much for each (subsidy vs college per se)?
However, the
valuation of the college subsidy is not uniform across the population. To see this,
Notes:
note for example in Figure 7 that for about 40 percent of the population, the value of access to
1. Regardless of how we write about this exercise, the unpacking here is awesome!
college is zero with or without the presence of the college subsidy. That is, the subsidy makes no
2. We could think of only a few charts to present here (high ability, low ability, high human capital, high
x0 and low x0).
33
3. For some of the points below, we could also think going a bit deeper and condition on more than one
type (similar to our ain rate of dominance exercise, but now maybe thinking of x0 as one of them given
the nature of the exercise).
4
difference for these individuals’ valuation of college.
Figure 8 further characterizes who benefits from the provision of the college subsidy. The figure
shows the CDFs of college valuations (with and without the subsidy) separately for the groups
corresponding to low, medium, and high levels of ability. This figure confirms that there are sizable
differences in the valuation of the college subsidy across agent types. Indeed, for both low and
medium ability types (panels a and b), the provision of the subsidy makes almost no difference
to college valuations. It is only individuals with high levels of ability (panel c) who assign a
substantial value to the provision of the subsidy. The figure also suggests that those individuals
who are best positioned to benefit from the opportunity to invest in college (i.e., individuals with
high ability) continue to assign a relatively high value to college access even in the absence of the
subsidy. (For example, for the high-ability group, the fraction of individuals who place no value
on college access barely moves, from about 9 percent in the presence of the subsidy to only about
12 percent in the absence of it.)
Figure 8: By Ability
1
For college with subsidy
For college without subsidy
For college with subsidy
For college without subsidy
0.9
0.8
0.7
0.7
0.6
0.6
F(x)
0.8
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0
0.8
0.2
0.4
0.6
x
x
(a) Low
(b) Medium
1
For college with subsidy
For college without subsidy
0.9
0.8
0.7
0.6
F(x)
F(x)
0.9
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
x
(c) High
34
0.6
0.7
0.8
0.9
0.8
1
Similar results hold when we look at different levels of initial human capital and initial wealth.
Figures 9 and 10 show the CDFs of college valuations—with and without the direct college
subsidy—separately for groups corresponding to low, medium, and high levels of initial human
capital and initial wealth. In these cases, too, it is only individuals with high levels of initial
human capital or initial wealth who assign a substantial value to the provision of the subsidy.
Figure 9: By Initial Human Capital
1
1
For college with subsidy
For college without subsidy
For college with subsidy
For college without subsidy
0.9
0.8
0.8
0.7
0.7
0.6
0.6
F(x)
F(x)
0.9
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
0
0.2
0.4
0.6
0.8
1
0
0.1
0.2
x
0.3
0.4
0.5
0.6
0.7
0.8
x
(a) Low
(b) Medium
1
For college with subsidy
For college without subsidy
0.9
0.8
0.7
F(x)
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1
x
(c) High
5.3
Targeted Reallocation of College Subsidies
The above counterfactual enables us to classify agents into three groups: those who enroll in college
with or without the subsidy (whom we label “always enroll” and who comprise 24 percent of the
population), those who enroll in the presence of the subsidy and not in its absence (“switchers,”
who account for 30 percent of the population), and those who choose not to enroll in either case
(“nonenrollees,” who make up the remaining 46 percent).
35
Figure 10: By Initial Wealth
1
1
For college with subsidy
For college without subsidy
For college with subsidy
For college without subsidy
0.9
0.8
0.8
0.7
0.7
0.6
0.6
F(x)
F(x)
0.9
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0
0.7
0.2
0.4
0.6
0.8
1
x
x
(a) Low
(b) Medium
1
For college with subsidy
For college without subsidy
0.9
0.8
0.7
F(x)
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1
x
(c) High
The fact that nearly a quarter of the population would continue to enroll in college even in the
absence of any subsidy while a full 46 percent receives no benefit from it motivates us to consider
whether gains can be had from reorienting funds away from those who do not require subsidies to
enroll in college (the “always enroll” group) and towards those who are not positioned to benefit
from the college subsidy (the “nonenrollees”). Specifically, we consider an experiment in which
college is no longer subsidised for the “always enroll” group. The proceeds (i.e., the total amount
of the college subsidy that was accruing to this group) are divided among the “nonenrollee” group
in the form of a stock index fund that is available in the first year of retirement.31 The “switchers”
continue to receive college subsidies as in the baseline.
There are numerous vehicles that we could have chosen to transfer the subsidy proceeds from
the group that always enrolls in college to the nonenrollee group. Why a stock index fund available
31
Note that the per-capita size of the subsidy will differ across the two groups simply because the groups differ
in size.
36
at retirement? Two key features make this comparison appropriate as proof of principle. First,
a stock market index fund delivers ex-post returns with stochastic properties similar to those
from college (Judd, 2000; Psacharopoulos and Patrinos, 2018). Therefore, the alternative does
not have an undue advantage over the baseline: stocks offer neither greatly higher returns nor
significantly lower risk compared to college. Second, making the fund available only at retirement
makes it comparable to human capital in the sense that both assets face are subjected to risks
throughout working life but can be nonetheless borrowed against to some extent. However, unlike
human capital, the returns to the stock market are unaffected by individual characteristics or
idiosyncratic risk.
Two observations about the experiment are important. First, it is revenue-neutral, as it requires
no additional resources beyond what is currently being spent on the provision of college subsidies.
Second, and more substantively, it leaves college investment decisions unchanged. Existing college subsidies presumably reflect a public desire to internalize potential positive externalities, or
otherwise modify college-attendance decisions. Our approach ensures that these goals continue to
be met, as monies are moved without interfering with college investment decisions. In particular,
we do not evaluate the myriad policies that would be more “invasive,” such as taking funds away
from those whose enrollment decisions depend on the subsidy. As a result, our welfare implications
are also easy to interpret: the college premium will not change precisely because college decisions
themselves will not change.
5.3.1
Findings
We report our findings from the experiment in Figure 11, which shows the value of moving from
the alternative subsidy regime (in which those who always enroll no longer receive the subsidy,
the proceeds of which fund a stock-index retirement fund for non-enrollees) to the baseline. The
figure highlights the presence of large differences in the valuation of the current regime across
the population. The figure shows that there are both negative and positive valuations, as well as
an important mass of zero valuations. The negative valuations correspond to ”nonenrollees”; the
positive valuations correspond to ”always enroll”; and the zero valuations correspond to switchers
(notice that the ”jump” in the CDF at zero shows that the measure of switchers is substantial –
30 percent).
Starting with those who always enroll, we find that these agents strongly prefer the baseline
to the alternative subsidy regime: the consumption-equivalent value of the status quo for these
individuals (relative to the alternative) is 7.7 percent. This is unsurprising, since not receiving a
large subsidy makes those who enroll unambiguously worse off. Conversely, “nonenrollees” strongly
37
prefer the alternative subsidy regime to the status quo: they would experience a -5.5 percent loss
in moving from the alternative regime to the baseline. Because the size of the nonenrollee group
is nearly double that of those who always enroll, the mean welfare implication of the status quo
relative to the alternative regime is slightly negative (-1.07 percent). That is, from behind the
veil of ignorance, welfare gains appear available from a more targeted allocation of subsidies that
preserves existing college enrollment and is budget neutral.
Figure 11: CDFs of Value of Moving From Alternative Subsidy Regime to Baseline
1
0.9
0.8
0.7
F(x)
0.6
0.5
0.4
0.3
0.2
0.1
0
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
x
Our second main finding from this exercise is that initial conditions (a,h1 ,x1 ) matter systematically and substantially for the impact of policy. This can be seen clearly in Table 7, which reports
mean valuations for groups defined, in turn, by a single dimension of initial conditions. The entries
report valuations across the bottom, middle two, and top quartile of ability, initial human capital,
and initial wealth.
Two points are immediate from the table. First, and naturally, there is a clear positive relationship between attributes that ease college investment and the relative gains from the status quo.
Conversely, we see that it is those with initial characteristics least helpful for college success (e.g.,
the groups (aLow , h1,Low , x1,Low )) who suffer from the status quo. These types of individuals would
prefer to receive stock-market funds as opposed to the promise of subsidized college should they
choose to enroll. Second, the quantitative strength of both gains and losses is nontrivial. That is,
just as with college overall, our model suggests the presence of a wide variation in the value of the
status quo policy across the initial characteristics of US high school completers.
Our demonstration of a welfare-improving alternative reallocation of public support towards a
38
Table 7: The Gains and Losses of Moving From Alternative Subsidy Regime to Baseline
Initial
Characteristic
Ability
Low
Middle
High
Human Capital
Low
Middle
High
Wealth
Low
Middle
High
Welfare
Change
-7.7%
-0.1%
5.3%
-6.7%
0.0%
2.5%
-4.1%
-1.0%
1.9%
simple financial asset is intended as an illustration of the potential misdirection of funds that we
may currently be experiencing. While we have argued that indexed financial assets are the sensible
alternative, even here there are any number of policies one might consider. We intend, therefore,
for this part of the analysis to make two points. First, our finding that the subsidy has the most
value to the well-prepared suggests that whatever motivations one places on subsidizing college
(e.g., externalities, or the promotion of equality of opportunity) to increase total investment, the
current structure of support—universal subsidies conditional on enrollment—does not seem fully
consistent with them. Second, while college does not differ from stocks as an investment in the
broad nature of its financial returns, it clearly does as a target for public support. Our paper
suggests that such divergence may warrant further consideration.
6
Concluding Remarks
Using a rich model of human capital investment, we show that the value of access to college varies
greatly in the population. While a small group of well-prepared individuals value college access very
highly, we estimate that 40 percent of US high school completers place no value on access to college.
The latter thus derive no benefits from the direct subsidies that currently reduce college costs
sharply. Because receiving these benefits requires enrollment and hinges on college completion,
subsidies flow instead to the best-prepared high school graduates, most of whom would continue
to enroll in the absence of those subsidies. Even modestly targeted alternatives may therefore
improve welfare. We provide one: redirecting subsidies away from those who would nonetheless
39
enroll—into a stock fund for those who do not enroll even with the subsidy—increases ex-ante
welfare by 1 percent of mean consumption. Unlike college subsidies, this alternative arrangement
benefits the large group that is poorly poised for collegiate success, while yielding mean ex-ante
returns comparable to the mean returns accruing to college completers.
To prevent any misinterpretation of our findings, we stress that they must not be read as any
kind of sweeping statement about, or indictment of, college education. Instead, they are suggestive
of the importance of college readiness. Our results show that heterogeneity in college readiness
(as summarized in the pair (a, h1 )) drives heterogeneity in college returns, so much so that poor
preparedness almost fully nullifies the high ex-post (i.e., conditional on college completion) payoffs
of college. Put another way, the high current payoffs to college completion contain a clear signal
about the importance of pre-college preparation. Our findings provide additional perspective—
in line with the large body of work of James Heckman and coauthors32 —on why early-childhood
environments are critical in determining the effectiveness with which individuals can acquire human
capital. They may well hold the key to helping individuals unlock the benefits of college.
References
Abbott, B., G. Gallipoli, C. Meghir, and G. L. Violante (2018). Education policy and intergenerational transfers in equilibrium. Forthcoming Journal of Political Economy.
Acemoglu, D. and J. Angrist (2000). How large are human-capital externalities? evidence from
compulsory schooling laws. NBER macroeconomics annual 15, 9–59.
Akyol, A. and K. Athreya (2005). Risky Higher Education and Subsidies. Journal of Economic
Dynamics and Control 29 (6), 979–1023.
Altonji, J., P. Arcidiacono, and A. Maurel (2016). Chapter 7 - the analysis of field choice in college
and graduate school: Determinants and wage effects. Volume 5 of Handbook of the Economics
of Education, pp. 305 – 396. Elsevier.
Altonji, J. G., E. Blom, and C. Meghir (2012). Heterogeneity in human capital investments: High
school curriculum, college major, and careers. Annu. Rev. Econ. 4 (1), 185–223.
Arcidiacono, P. (2004). Ability sorting and the returns to college major. Journal of Econometrics 121 (1), 343–375.
32
See, for example, Heckman (2006) and Heckman et al. (2013).
40
Arcidiacono, P. (2005). Affirmative action in higher education: how do admission and financial
aid rules affect future earnings? Econometrica 73 (5), 1477–1524.
Arcidiacono, P., V. J. Hotz, and S. Kang (2012). Modeling college major choices using elicited
measures of expectations and counterfactuals. Journal of Econometrics 166 (1), 3–16.
Athreya, K., F. Ionescu, and U. Neelakantan (2015). Stock market participation: The role of
human capital. Working paper, Federal Reserve Bank of Richmond.
Athreya, K., F. Ionescu, U. Neelakantan, and I. Vidangos (2018). Investment opportunities and
the sources of lifetime inequality. Unpublished Draft.
Athreya, K. B. and J. Eberly (forthcoming). The supply of college-educated workers: the roles of
college premia, college costs, and risk. American Economic Journal: Macroeconomics.
Autor, D., F. Levy, and R. Murnane (2003). The Skill Content of Recent Technological Change:
An empirical exploration*. Quarterly Journal of Economics 118 (4), 1279–1333.
Bach, L., L. E. Calvet, and P. Sodini (2016). Rich pickings? risk, return, and skill in the portfolios
of the wealthy.
Ben-Porath, Y. (1967). The production of human capital and the life cycle of earnings. Journal
of Political Economy 75 (4), 352–365.
Blandin, A. (2018). Learning by doing and ben-porath: Life-cycle predictions and policy implications. Journal of Economic Dynamics and Control 90, 220–235.
Board of Governors of the Federal Reserve System (2014). Federal reserve statistical release, g19,
consumer credit. Technical report, Board of Governors of the Federal Reserve System.
Bound, J., M. F. Lovenheim, and S. Turner (2010). Why Have College Completion Rates Declined?
An Analysis of Changing Student Preparation and Collegiate Resources. American Economic
Journal: Applied Economics 2 (3), 129–57.
Browning, M., L. P. Hansen, and J. J. Heckman (1999). Micro data and general equilibrium
models. Handbook of macroeconomics 1, 543–633.
Card, D. (2001). Estimating the Return to Schooling: Progress on Some Persistent Econometric
Problems. Econometrica 69 (5), 1127–1160.
41
Castaneda, A., J. Diaz-Gimenez, and J.-V. Rios-Rull (2003). Accounting for the us earnings and
wealth inequality. Journal of political economy 111 (4), 818–857.
Cestau, D., D. Epple, and H. Sieg (2017). Admitting students to selective education programs:
Merit, profiling, and affirmative action. Journal of Political Economy 125 (3), 761–797.
Chatterjee, S. and F. Ionescu (2012). Insuring Student Loans Against the Financial Risk of Failing
to Complete College. Quantitative Economics 3 (3), 393–420.
Ciccone, A. and G. Peri (2006). Identifying human-capital externalities: Theory with applications.
The Review of Economic Studies 73 (2), 381–412.
Cocco, J. (2005). Portfolio choice in the presence of housing. Review of Financial Studies 18 (2),
535–567.
Cocco, J., F. Gomes, and P. Maenhout (2005). Consumption and portfolio choice over the life
cycle. Review of Financial Studies 18 (2), 491–533.
Epple, D., R. Romano, S. Sarpça, and H. Sieg (2013, August). The u.s. market for higher education:
A general equilibrium analysis of state and private colleges and public funding policies. Working
Paper 19298, National Bureau of Economic Research.
Fagereng, A., L. Guiso, D. Malacrino, and L. Pistaferri (2016, November). Heterogeneity and
persistence in returns to wealth. Working Paper 22822, National Bureau of Economic Research.
Garriga, C. and M. Keightley (2007). A general equilibrium theory of college with education
subsidies, in-school labor supply, and borrowing constraints. federal reserve bank of st. Technical
report, Louis Working Paper Series 2007-051A.
Goldin, C. and L. F. Katz (2007). Long-Run Changes in the Wage Structure: Narrowing, Widening,
Polarizing. Brookings Papers on Economic Activity (2), 135.
Guvenen, F. (2009). An empirical investigation of labor income processes. Review of Economic
dynamics 12 (1), 58–79.
Heckman, J., R. Pinto, and P. Savelyev (2013). Understanding the mechanisms through which an
influential early childhood program boosted adult outcomes. American Economic Review 103 (6),
2052–86.
Heckman, J. J. (2006). Skill Formation and the Economics of Investing in Disadvantaged Children.
Science 312 (5782), 1900–1902.
42
Hendricks, L. and O. Leukhina (2017). How risky is college investment? Review of Economic
Dynamics 26, 140 – 163.
Hendricks, L. and O. Leukhina (2018). The return to college: Selection and dropout risk. International Economic Review 59 (3), 1077–1102.
Hendricks, L. and T. Schoellman (2014). Student abilities during the expansion of us education.
Journal of Monetary Economics 63, 19–36.
Hubbard, R., J. Skinner, and S. Zeldes (1994). Expanding the Life-cycle Model: Precautionary
Saving and Public Policy. The American Economic Review 84 (2), 174–179.
Huggett, M., G. Ventura, and A. Yaron (2006). Human capital and earnings distribution dynamics.
Journal of Monetary Economics 53 (2), 265–290.
Ionescu, F. (2009). The federal student loan program: Quantitative implications for college enrollment and default rates. Review of Economic dynamics 12 (1), 205–231.
Ionescu, F. and N. Simpson (2016). Default risk and private student loans: Implications for higher
education policies. Journal of Economic Dynamics and Control 64, 119–147.
Johnson, M. T. (2013). Borrowing constraints, college enrollment, and delayed entry. Journal of
Labor Economics 31 (4), 669–725.
Judd, K. L. (2000). Is education as good as gold? a portfolio analysis of human capital investment.
Unpublished working paper. Hoover Institution, Stanford University.
Kim, H. H., R. Maurer, and O. S. Mitchell (2013, December). Time is money: Life cycle rational
inertia and delegation of investment management. Working Paper 19732, National Bureau of
Economic Research.
Kong, Y.-C., B. Ravikumar, and G. Vandenbroucke (2018). Explaining cross-cohort differences in
life-cycle earnings. European Economic Review 107, 157–184.
Loke, V. and M. Sherraden (2009). Building assets from birth: A global comparison of child
development account policies. International Journal of Social Welfare 18 (2), 119–129.
Ma, J., S. Baum, and N. Y. T. C. B. Matea Penderand CJ Libassi (2018) (2018). Trends in college
pricing 2018. Technical report, The College Board.
Manski, C. F. and D. A. Wise (1983). College choice in America. Harvard University Press.
43
McGrattan, E. R. and E. C. Prescott (2000). Is the stock market overvalued? Federal Reserve
Bank of Minneapolis Quarterly Review 24 (4), 20–40.
Mehra, R. and E. C. Prescott (1985). The equity premium: A puzzle. Journal of Monetary
Economics 15 (2), 145–161.
Moretti, E. (2004a). Human capital externalities in cities. In Handbook of regional and urban
economics, Volume 4, pp. 2243–2291. Elsevier.
Moretti, E. (2004b). Workers’ education, spillovers, and productivity: evidence from plant-level
production functions. American Economic Review 94 (3), 656–690.
National Center for Education Statistics (2009). Beginning Postsecondary Students: 2004/2009.
https://nces.ed.gov/surveys/bps/.
Ozdagli, A. K. and N. Trachter (2011). On the Distribution of College Dropouts: Household
Wealth and Uninsurable Idiosyncratic Risk. Working paper series, Federal Reserve Bank of
Boston.
Peri, G. and C. Sparber (2009, July). Task specialization, immigration, and wages. American
Economic Journal: Applied Economics 1 (3), 135–69.
Planty, M., W. Hussar, T. Snyder, S. Provasnik, G. Kena, R. Dinkes, A. KewalRamani, and
J. Kemp (2008). The condition ofeducation 2008.
Psacharopoulos, G. and H. A. Patrinos (2018). Returns to investment in education: a decennial
review of the global literature. Education Economics 26 (5), 445–458.
Restuccia, D. and C. Urrutia (2004). Intergenerational persistence of earnings: The role of early
and college education. The American Economic Review 94 (5), 1354–1378.
Stange, K. M. (2012). An empirical investigation of the option value of college enrollment. American Economic Journal: Applied Economics 4 (1), 49–84.
44
A
A.1
Appendix
Life-cycle financial assets
We compare life-cycle wealth accumulation in the model to data from the SCF. Our measure of
wealth includes all financial assets. Our measure of risky assets corresponds to a broad measure of
households’ equity holdings in the SCF, which includes directly held stocks as well as stocks held
in mutual funds, Individual Retirement Accounts (IRAs)/Keoghs, thrift-type retirement accounts,
and other managed assets.
As in the case of earnings, we construct life-cycle profiles of asset holdings, controlling for time
effects. In addition, we also adjust for cohort effects by regressing each of the three measures of
wealth on a full set of age and cohort dummies. The results (in 2014 dollars) are reported in
Figure 12.
Figure 12: Average Life-Cycle Assets (SCF)
600000
Time Effects
Cohort Effects
500000
400000
300000
200000
100000
0
20
30
40
50
60
(a) Total
300000
250000
Time Effects
Cohort Effects
250000
Time Effects
Cohort Effects
200000
200000
150000
150000
100000
100000
50000
50000
0
0
20
30
40
50
20
60
(b) Risky
30
40
(c) Risk-free
45
50
60
A.2
College Valuation vs. Stocks Valuation
In this Appendix, we compute the value of stocks just as we did for college—we compare valuations
for agents in the baseline economy to agents optimizing in a version of the economy in which access
to stocks is shut down. Our results, detailed below, illustrate that stocks have the potential to
benefit precisely those whose initial conditions leave them poorly poised for success in college.
We find that the mean valuation for stocks across all types is 4.4 percent of consumption—
substantially lower than that of college. This is not entirely surprising, both because stock market
participation is low and because an alternative savings instrument is available in the form of the
risk-free asset. Despite this, a stock market index fund has the potential to benefit precisely the
large group that is poorly poised for collegiate success: those in the lowest quartile of the ability
distribution find access to stocks much more valuable than access to college. We provide the details
of these calculations in Appendix A.2.
Looking beyond the average, Figure 13 compares the CDF of valuations for stocks with the
CDF of valuations for college that was shown in the paper. The figure shows that two-thirds of
the population assign a near-zero value to the opportunity to invest in the stock market, while less
than 10 percent of the population values stocks access at more than 20 percent of consumption.
Figure 13: CDF of Valuations for Stocks vs. College
1
For college
For stocks
0.9
0.8
0.7
F(x)
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1
x
We turn next to the comparison of who values access to each investment opportunity. As
shown in the right-most column of Table 8, those in the lowest quartile of the ability distribution
find access to stocks much more valuable (9.7 percent of consumption) than access to college (1.0
percent), and it is only when ability rises (i.e., for those with “medium” or “high” ability) that
college asserts itself as the more valuable of the two investment opportunities.33
33
At first glance, our finding that low-ability types value access to stocks more than high-ability types may seem
46
Table 8: Average Valuations by Initial Characteristic
Initial
Characteristic
Ability
Low
Middle
High
Human Capital
Low
Middle
High
Wealth
Low
Middle
High
Valuation
College Stocks
1.0%
15.1%
27.5%
9.7%
2.3%
3.4%
13.3%
13.2%
19.0%
5.9%
4.6%
2.6%
9.3%
13.7%
22.0%
4.9%
4.6%
3.6%
Notice as well that the sensitivity of the value of access to stocks to initial conditions is uniformly
(far) lower than it is for access to college. For example, the valuation for stocks access ranges from
9.7 percent for the bottom quartile of ability down to 3.4 percent for the highest quartile—a
much smaller range than the corresponding range of valuations for college access. The diminished
importance of ability for the value of stocks access can be understood as before—by noting that
those who have a high capability of acquiring human capital (high-ability individuals) can augment
their earnings and not rely on stocks to generate income or wealth. As for the importance of initial
wealth, we see that it plays only a secondary role when compared with the roles played by ability
and initial human capital.
puzzling. However, there are two reasons for this. First, because average consumption is lower for low-ability
types than for high-ability types, a smaller absolute valuation for stocks among low-ability types will translate
into a higher percentage change in terms of consumption-equivalent welfare. Second, recall that our counterfactual
exercise compares the no-stocks economy where college is still available to the baseline. For high-ability types, the
fact that college remains available means that they still have access to a high-value investment: their value function
in the no-stocks economy is therefore not significantly lower than in the baseline. In contrast, with the loss of stocks,
low-ability types—most of whom derive no value from college—cannot easily reoptimize by investing in college at
high rates when stocks are unavailable. The result is a larger gap in value between the no-stocks economy and the
baseline for them.
47
@FedFRASER