Accounting for Tuition Increases across U.S. Colleges∗

Grey Gordon

Indiana University

Department of Economics

greygordon@gmail.com

Aaron Hedlund

University of Missouri

Department of Economics

hedlunda@missouri.edu

February 5, 2018

Abstract

We develop a quantitative model of higher education to test explanations for the steep rise

in college tuition between 1987 and 2010. The framework extends the Epple, Romano, Sarpca,

and Sieg (2013) paradigm of imperfectly competitive, quality-maximizing colleges and embeds

it in an incomplete markets, life-cycle environment. We measure how much changes in college

costs, reforms to the Federal Student Loan Program (FSLP), returns to college and parental

income have contributed to tuition inflation. Taken together, the changes can explain all of the

the tuition increases seen at U.S. colleges. Our findings suggest demand side changes have been

the main drivers of college tuition, with policy (non-policy) demand changes explaining 42%

(76%) of the increase. Changes in college costs play a small role, accounting for only 7% of the

observed tuition increase. However, there is substantial heterogeneity in the causes of tuition

inflation across schools.

Keywords: Higher Education, College Costs, Tuition, Student Loans

JEL Classification Numbers: E21, G11, D40, D58

∗We thank Kartik Athreya, Sandy Baum, Sue Dynarski, Gerhard Glomm, Bulent Guler, Jonathan Heathcote, KyleHerkenhoff, Jonathan Hershaff, Brent Hickman, Felicia Ionescu, John Jones, Michael Kaganovich, Oksana Leukhina,Lance Lochner, Amanda Michaud, Urvi Neelakantan, Chris Otrok, Irina Shaorshadze, Yu Wang, Fang Yang, andEric Young, as well as seminar participants at the Econometric Society NASM 2017 and SED 2017. This work issupported by NSF Grants 1730078 and 1730121.

1

1 Introduction

For thirty years, college tuition has risen steadily across all types of colleges. Selective private

research colleges, many with large private endowments, have seen net tuition increase 50% going

from $15,500 to $23,700 (in 2010 dollars).1 At the same time, non-selective public teaching colleges,

which are almost totally reliant on government support and tuition, have seen their increase 140%

from $2700 to $6400.

Many theories have been put forth to explain why tuition has been increasing, and in this paper

we aim to quantify how much each theory has contributed to tuition changes at U.S. colleges by

using a structural model of higher education and the macroeconomy. The theories we consider can

be broadly categorized into supply-side changes (Baumol’s cost disease and changes in colleges’ non-

tuition funding, public and private), demand-side changes induced by policy (notably, changes in the

FSLP), and demand-side changes induced by macroeconomic forces (namely, skill-biased technical

change resulting in a higher college earnings premium and increases in parental income). Our

quantitative model shows that the combined effect of these changes can account for all (107%) of the

tuition increase. The variety of circumstances schools face means there is not a single factor causing

tuition increases across all school types. However, our preliminary results indicate that changes in

demand are the main drivers across school types with policy driven (non-policy) demand changes

accounting for 42% (76%). While we find some support for Baumol’s cost disease, quantitatively

it accounts for only 7% of increase. Additionally, changes in public (private) non-tuition revenue

have acted to restrain tuition increases by 6% (23%).

The higher education market functions quite differently than many other markets with its com-

bination of extensive public subsidies, complicated financial aid rules, oligopolistic market structure,

and widespread use of price discrimination. Furthermore, colleges are predominantly non-profit in-

stitutions that pursue different objectives and face different incentives than profit maximizing firms.

To capture these features, our work builds on a series of static models developed by Epple, Romano,

and Sieg (2006) and Epple et al. (2013) (which extend the work of Rothschild and White, 1995).

In their framework, colleges seek to maximize “quality,” which is formally a function of spending

per student and the average academic qualifications of the student body, both of which are endoge-

nous. Colleges optimally engage in price discrimination, offering tuition discounts to higher-ability

students. Students, in turn, choose from among the set of colleges to which they receive an offer of

admission, and they weigh the benefits of attending a higher quality institution against the poten-

tially higher net cost of attendance. The sorting of students across colleges affects the distribution

of college quality, and a computationally challenging fixed point problem emerges because the col-

lege quality distribution also drives enrollment decisions. Our calibration, which targets ability and

parental income across schools, does well at capturing this sorting.

By embedding the framework above in a dynamic macroeconomic model with rich life cycle,

1Net tuition is sticker price tuition net of institutional grants to students, and throughout we use real measures.

2

labor market, and general equilibrium elements, we able to assess the quantitative importance of

all the tuition increase hypotheses. First, students differing in their ability and parental income

choose to either forgo college or matriculate at one of the colleges to which they receive an offer of

admission. Students then finance college with a combination of direct family contributions, income

earned while in college, institutional aid, and student loan borrowing. After students either graduate

or drop-out from college, they enter the labor market and face a life-cycle wage profile subject to

idiosyncratic earnings risk. During the repayment phase, workers can default on their loans.

1.1 Related Literature

A growing literature employs general equilibrium models to analyze higher education while taking

the behavior of colleges and tuition as given. For example, Abbott, Gallipoli, Meghir, and Violante

(2016) develop an equilibrium model to analyze financial aid policies intended to promote college at-

tendance. Their framework features a rich intergenerational setting, intervivos transfers, and college

attendance financed partly by grants and loans. In other work, Athreya and Eberly (2016) study

the impact of a rising college wage premium on college attainment in the presence of heterogeneous

drop-out risk and post-graduation earnings risk. Hendricks and Leukhina (2016) and Chatterjee

and Ionescu (2012) also investigate the importance of drop-out risk for college attainment. Garriga

and Keightley (2010), Lochner and Monge-Naranjo (2011), Belley and Lochner (2007), and Keane

and Wolpin (2001) also develop equilibrium models to answer various important questions that lie

at the intersection of macroeconomics and higher education.

This paper endogenizes tuition and the response of colleges to evolving market conditions and

policies. In this vein, recent work by Jones and Yang (2016) closely mirrors the objectives of

this paper. They explore the role of skill-biased technical change in explaining the rise in college

costs from 1961 to 2009. However, their paper differs from this project in several ways. First, this

project takes a unified look at both supply-side and demand-side factors that influence tuition,

whereas they focus on the role of cost disease. Second, the object of interest in Jones and Yang

(2016) is college costs, which increased by 35% in real terms between 1987 and 2010, whereas this

project addresses the much larger 92% increase in net tuition. Also, whereas they use a competitive,

representative college framework, this project employs a model with heterogeneous, imperfectly

competitive colleges, peer effects, and student loan borrowing with default. Fillmore (2016) and

Fu (2014) develop rich frameworks with heterogeneous colleges, but in both cases, students have

static, reduced-form utility functions. Furthermore, peer effects are exogenous in Fillmore (2016),

and Fu (2014) does not allow price discrimination based on ability and income.

Methodologically, the most closely related papers are Epple et al. (2006), Epple et al. (2013),

and our earlier paper, Gordon and Hedlund (2016). The former two papers develop a static model of

heterogeneous, quality-maximizing colleges that operate in an environment of imperfect competition

and engage in price discrimination. Gordon and Hedlund (2016) embed this framework in a broader

3

macroeconomic model but consider only the case of a single, monopolistic college. Such a case

greatly simplifies computation but bestows exaggerated market power on the college and ignores

competitive pressures. This project takes the important step of adding heterogeneous colleges,

which allows for rich competitive interactions and sorting.

This project also relates to a large empirical literature that estimates the effects of macroe-

conomic factors and policy interventions on tuition and enrollment. The origins of cost disease

emerge from seminal works by Baumol and Bowen (1966) and Baumol (1967). They lay out a clear

mechanism: productivity increases in the economy at large drive up wages everywhere, which ser-

vice sectors that lack productivity growth pass along by increasing their relative prices. Recently,

Archibald and Feldman (2008) use cross-sectional industry data to forcefully advance the idea that

cost and price increases in higher education closely mirror trends for other service industries that

utilize highly educated labor. In short, they “reject the hypothesis that higher education costs

follow an idiosyncratic path.”

The empirical literature has conflicting findings on the impact of state higher education ap-

propriation on college tuition. For example, Heller (1999) suggests a negative relationship between

state support and tuition, asserting that “the higher the support provided by the state, the lower

generally is the tuition paid by all students.” Recent empirical work by Chakrabarty, Mabutas,

and Zafar (2012), Koshal and Koshal (2000), and Titus, Simone, and Gupta (2010) support this

hypothesis, but notably, Titus et al. (2010) show that this relationship only holds up in the short

run. Lastly, in a large study commissioned by Congress in the 1998 re-authorization of the Higher

Education Act of 1965, Cunningham, Wellman, Clinedinst, Merisotis, and Carroll (2001a) conclude

that “decreasing revenue from government appropriations was the most important factor associated

with tuition increases at public 4-year institutions.”

Shifting to demand-side factors, the empirical literature is split on the impact of financial aid

on tuition. For example, McPherson and Shapiro (1991), Singell and Stone (2007), Rizzo and

Ehrenberg (2004), Turner (2012), Turner (2013), Long (2004a), and Long (2004b) find at least

some evidence in support of the Bennett hypothesis, though they disagree on the magnitude of

the pass-through of aid into higher tuition and whether public or private institutions are more

responsive. Most recently, Lucca, Nadauld, and Shen (2015) find a 65% pass-through effect for

changes in federal subsidized loans and positive but smaller pass-through effects for changes in

Pell Grants and unsubsidized loans. Similarly, Cellini and Goldin (2014) show that tuition is 78%

higher at for-profit colleges that participate in federal student aid compared to those that do not.

By contrast, in their commissioned report for the 1998 re-authorization of the Higher Education

Act, Cunningham et al. (2001a); Cunningham, Wellman, Clinedinst, Merisotis, and Carroll (2001b)

conclude that “the models found no associations between most of the aid variables and changes in

tuition in either the public or private not-for-profit sectors.” Long (2006) and Frederick, Schmidt,

and Davis (2012) echo these sentiments.

4

We also analyze how labor market trends over the past few decades have impacted tuition.

Empirically, Autor, Katz, and Kearney (2008) report that the college earnings premium increased

from 58% in the mid-1980s to 93% in 2005, which Autor et al. (2008), Katz and Murphy (1992),

Goldin and Katz (2007), and Card and Lemieux (2001) ascribe to skill-biased technological change

and a fall in the relative supply of college graduates. In recent work, Andrews, Li, and Lovenheim

(2012) and Hoekstra (2009) study the distribution of college earnings premia and find substantial

heterogeneity attributable to variation in college quality.

2 Data

2.1 College revenue and spending

2.1.1 Organization of college budgets

Colleges are complex organizations with sizable balance sheets. We break down their balance sheets

by key revenue and expenditure categories in Table 1. In the spirit of Epple et al. (2006), we will

further refine these categories into endogenous net tuition revenue T , exogenous non-tuition revenue

E, exogenous expenditures C that are necessary for educating but do not increase quality, and

endogenous expenditures I that do increase quality. Hence, the model’s college budget constraint

is essentially

pI + pC = T + Eg + Ep (1)

where p is the relative price of college expenditures (for which we will use the real Higher Education

Price Index, HEPI), and we have further broken non-tuition revenue E into public and private

sources, Eg and Ep, respectively. Most of these major categories line up naturally with Table 1.

For instance, T is net tuition (and it does not matter to the college whether this is received from

students or from the government on behalf of students). However, others are perhaps less clear, so

we use the mapping in the second rightmost column of Table 1. Currently, we only distinguish pI

and pC in the model as the distinction is not particularly clear in the data.

2.1.2 Trends over time

Figure 1 shows how the major financial categories per student have varied over time when averaged

across schools and weighted by full-time equivalent (FTE) counts. Also reported is our measure of

the relative price p. In terms of non-tuition revenue, Eg has been roughly constant over time and

Ep has increased modestly. In contrast, T and expenditures have trended secularly upward.

Of course, looking at averages across schools could hide hugely disperse patterns across schools.

To examine these differences, Figure 2 shows the patterns in net tuition for seven school types

(reflecting a breakdown by whether a school is public or private, by research intensity, and by

selectivity). For instance, trends in levels are virtually indistinguishable across the types. However,

5

Balance sheet item Model equivalent

Total Expenditures pI + pCE&G spending Part of pI + pCAuxiliary and “other” spending Part of pI + pC

Total Revenue T + Eg+part of Ep

Net tuition TDirectly from student Out of pocket for TFrom government Students apply to T

Pell Students apply to TLocal, state, and other federal Students apply to T

Approp., contracts, excluding Pell Eg

Auxiliary and “other” revenue Part of Ep

Endowment revenue, gifts Part of Ep

Gross operating margin (rev. - exp.) Part of Ep

Note: Ep is the sum of “Part of Ep” and pC is the sum of “Partof pC.” A component of E&G spending is expenditures on schol-arships and fellowships (see the text for details).

Table 1: College balance sheet

1.1

1.15

1.2

1.25

1.3

Rel

ativ

e pr

ice

010

000

2000

030

000

4000

020

10 d

olla

rs

1985 1990 1995 2000 2005 2010Academic Year

T per FTE Eg per FTEEp per FTE pI + pC per FTEp (HEPI price index / CPI)

Figure 1: Budget constraint components over time

6

in percent changes, public schools of all types (there are 3 public types and 4 private) have seen

increases of around 140% (on a much smaller base) while other school types increases have been

much more modest at around 60%. Public school tuition began growing at a faster rate in 2002.

050

0010

000

1500

020

000

2500

0(m

ean)

T

1985 1990 1995 2000 2005 2010Academic Year

Not selective Public Teaching Not selective Public ResearchNot selective Private Teaching Not selective Private ResearchSelective Public Research Selective Private TeachingSelective Private Research

Figure 2: Net student tuition over time in levels (top) and relative to 1987 (bottom)

2.1.3 Relationships between tuition, enrollment, and non-tuition revenue

We now look at how non-tuition revenue influences tuition, expenditures, and enrollment in the

data. The first column of the top panel of Table 2 reveals strong cross-sectional relationship between

tuition and non-tuition revenue per FTE. In particular, public support (Eg) per FTE is negatively

correlated with net tuition (T ); in contrast, private non-tuition revenue (Ep) is positive correlated

with net tuition. However, after controlling for interactions between state, school type, and flagship

status, these correlations disappear as can be seen in the second column. This is also true if one

expands from just 1987 to 2010 provided one all has interactions with time dummies as can be seen

in the third column, or with a fixed effects specification like in the fourth column.

By including a large set of interactions, the data’s variation that identifies the correlation

between T , Eg, and Ep comes from within-state, within-school, and within-flagship status all within

a year. E.g., variation in the funding of Purdue University and Indiana University (both selective,

public, research, flagship schools in Indiana) determines in part the impact of non-tuition revenue

on tuition. Interestingly, this effect is zero.

7

(1) (2) (3) (4)T T T T

Eg -0.26∗∗∗ -0.04∗ -0.01 0.02Ep 0.09∗∗∗ 0.00 -0.00 -0.00Constant 13266.80∗∗∗ 12949.12∗∗∗ 9904.17∗∗∗ 7399.60∗∗∗

Observations 1158 1158 27792 27792Adjusted R2 0.149 0.727 0.750 0.463

(1) (2) (3) (4)XPND XPND XPND XPND

Eg 0.74∗∗∗ 0.96∗∗∗ 0.99∗∗∗ 1.02∗∗∗

Ep 1.09∗∗∗ 1.00∗∗∗ 1.00∗∗∗ 1.00∗∗∗

Constant 13266.80∗∗∗ 12949.12∗∗∗ 9904.17∗∗∗ 7399.60∗∗∗

Observations 1158 1158 27792 27792Adjusted R2 0.980 0.994 0.994 0.974

(1) (2) (3) (4)log(N) log(N) log(N) log(N)

log(Eg) 0.40∗∗∗ -0.05 -0.05∗∗∗ -0.06∗∗∗

log(Ep) -0.14∗∗∗ -0.15∗∗∗ -0.19∗∗∗ -0.12∗∗∗

Constant 6.37∗∗∗ 9.89∗∗∗ 10.08∗∗∗ 9.33∗∗∗

Observations 1105 1105 26721 26721Adjusted R2 0.366 0.711 0.718 0.463

(1) (2)Ability Ability

log(Eg) 0.01∗ -0.01log(Ep) 0.14∗∗∗ 0.06∗∗∗

Constant -0.95∗∗∗ -0.13

Observations 931 931Adjusted R2 0.300 0.611

Note: (1-2) are for 2010; (3-4) are full sample. (2-3) includes all inter-actions between school type, state, year (when applicable), and flag-ship status. (4) is a fixed effects regression. Standard errors are clus-tered at the school type, state, year (when applicable), and flagshipstatus level. For the fixed effects regression, robust standard errorsare used. The ability measure is only for one year, which precludesusing the (3) and (4) specifications. ∗ p < .10, ∗∗ p < .05, ∗∗∗ p < .01

Table 2: How tuition, expenditures, and enrollments vary with non-tuition revenue

8

One possible explanation of this finding is that giving is often directly tied to spending. E.g., NSF

and NIH grants (which would show up as public non-tuition revenue) are usually applied towards

spending. Similarly, named buildings and stadiums or the creation of endowed chairs (which show

up as private non-tuition revenue) are also tied directly to spending. As will be seen in the next

section, Epple et al. (2006)-style models will generally capture this without tying the money directly

to spending.

2.2 Evidence on colleges as quality maximizers

We now present some evidence that colleges behave like quality maximizers with quality including

ability and expenditures per student.

2.2.1 Cross-sectional correlations of ability and spending

Figure 3 shows the relationship between spending per student and one measure of ability (the

marker size indicates FTE enrollment at the school).2 This cross-sectional variation is essential for

disciplining our college quality function as it provides a link between the ability and spending inputs

for a wide-array of college-specific circumstances. Colleges seem to first focus almost singularly on

increasing college ability, seemingly substituting into spending when additional increases in ability

become too difficult.

2.2.2 Tuition discounting

Table 3 provides some empirical evidence that colleges discount tuition based on both ability

and parental income. Appendix B reports similar results broken out by ability decile, as well as

additional regressors of net tuition or sticker tuition.

Discount (% off)

Ability 7.472∗∗

Parental income in 1996 (real) -0.0972∗∗∗

Constant 34.15∗∗∗

Observations 1609R2 0.047∗ p < .1, ∗∗ p < .05, ∗∗∗ p < .01

Table 3: Tuition discounting

2For this dataset (IPEDS), we have mean SAT scores. Using these, we construct an implied distribution of SATscores conditional on college attendance. The ability measure is the percentile of this conditional distribution assumingit is normally distributed. The mean score (out of 1600) is 1128 and the standard deviation is 135 (with a minimumof 735 and maximum of 1545).

9

810

1214

Log

expe

nditu

res

per p

erso

n, lo

g(I+

C)

0 .2 .4 .6 .8 1Ability (SAT measured) conditional on college attendance

Figure 3: Spending per student against an SAT-based ability measure in 2010

2.3 Summary measures by school types

Table 10 in Appendix A reports summary statistics by school types for key measures.

3 A simple model

In this section we describe a simple model that is useful for understanding how the different theories

will translate into changes in net tuition, enrollment, and spending. We will focus on the problem

of a single college with a pool of potential students.

Consider a unit measure of potential students differentiated, potentially, by their ability x and

their parental income y having some density f(x, y). Suppose the maximum they are willing to pay

to attend college (net of institutional aid) is t(x, y), and that this is known by the college (who will

then set net tuition for each student equal to t(x, y).

Suppose the college wants to maximize a “quality” function q(X, I) where X is average ability

and I is spending per student. They set tuition for each student equal to t(x, y), and then decide

10

a density α(x, y) ≤ f(x, y) of students to enroll. Their problem is

maxN>0,X,α(x,y)∈[0,f(x,y)]

q(X, I)

s.t. pIN + pC(N) = E(N) +

∫ ∫t(x, y)α(x, y)dxdy

N =

∫α(x, y)dxdy

X =

∫xα(x, y)dxdy/N

(2)

where E(N) is some non-tuition revenue, C(N) is custodial costs, and p is the relative price of

college goods. Part of the solution to this problem, known essentially from Epple et al. (2006), is

to admit students (except possibly on a measure zero set) if and only if t(x, y) is greater than the

“effective marginal cost”

EMC(x, y) := pI + pC ′ − E′ − pqXqI

(x−X). (3)

Now, consider breaking the problem into two stages. In the first consider choosing N and in the

second consider choosing α(x, y) subject to admitting N students. Let X(N) and T (N) denote the

average ability and net tuition revenue per student implied by the solution of the second problem.

Then the college’s problem becomes

maxN>0

q(X(N), I)

s.t. pIN + pC(N) = E(N) + T (N)N.(4)

The solution to this, assuming differentiability, is characterized by

qXqIX ′(N) +

1

pT ′(N) +

1

p

d(E(N)/N)

dN=d(C(N)/N)

dN. (5)

3.1 Changes in non-tuition revenue

In the data, there is little if any statistically significant correlation between non-tuition revenue

per student and college tuition. This model provides one explanation for the weak connection

between enrollment. To see this, consider (4) under the assumptions that E(N) = EN . Then

an as E increases or decreases, there is no impact on enrollment and there is no impact on net

tuition revenue because the first order condition (FOC) (5) is unchanged. Note that the converse

of this result is that the entire change in non-tuition revenue is absorbed by an equal change in

expenditures. This is essentially exactly what was seen in the data (in particular, in Table 2).

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3.2 Cost changes

Like with non-tuition revenue changes, if costs increase in a way that leaves d(C(N)/N)/dN un-

changed, then there will be no change in enrollment or net tuition. Rather, all the changes will

come through reduced spending. Absent heterogeneity in ability, which makes X ′ = 0, increases in

fixed costs will expand enrollments and decrease net tuition.

3.3 Changes in demand

Now suppose there is an increase in demand, reflected in an increase in t(x, y) such that, in the

solution to the second stage problem, T (N) goes up uniformly and X(N) is unchanged. Then the

FOC in (5) will be unaffected, and the previous solution will still be optimal. That is, in response to

this increase in demand, equilibrium enrollments and ability do not change. Rather, all the changes

are reflected in higher tuition and spending.

3.4 Summary of the simple model and its limitations

In the simple model with quality maximizing colleges, changes in marginal funding sources have

no effect on enrollments but simply act to reduce spending per student. To a limited extent, this

seems to agree with the data when it comes to changes in non-tuition revenue. Demand changes

can have differential impacts on enrollment depending on whether they are regressive, progressive,

or neutral, and only in the last case where we able to say for sure that tuition increases.

The simple model abstracts from competitive pressures and general equilibrium effects. In par-

ticular, as other schools responds to changes, t(x, y) may change or f(x, y) may change. To remedy

these deficiencies and allow for concrete answers, we now turn to the full model.

4 Model

This section describes the model, which consists of households, colleges, and the government.

4.1 Colleges with competitive search

There are a finite number K of types of colleges. For each type k, there is a positive measure g(k)

of identical schools. The school types differ along three dimensions. First, they have different non-

tuition revenue and exogenous costs of operating colleges. Second, they differ in their technology of

producing graduates with flow dropout rates δk and college earnings premia λk, which also depend

on the student’s type (ability in the computation). Third, we allow the possibility that public

schools value being large differently from private schools (in that their objective function may place

greater weight on being large).

12

We make the college problem static, like in Gordon and Hedlund (2016), through several key

assumptions. First, we assume the college has no access to borrowing or saving (though they will

have funds from an exogenously-given endowment). Second, we assume colleges trade future tuition

and expenditure flows for a net present value amount. The counterparty of this trade is a deep-

pocketed financial intermediary who is willing to substitute inter-temporally at rate 1/(1+r). When

a student of type sY = (x, y) where x is ability and parental income is y enrolls, their is a distribution

of outcomes associated with them. In particular, the college is directly affected by their probability

of exit, δk(sY ), the tuition T (sY ) they pay, and their ability x. The stream of payments associated

with enrolling a student of type sY is T (sY ), (1−δk(sY ))T (sY ), . . . , (1−δk(sY ))JY −1T (sY ). Defining

ω(sY ) =∑JY

j=1

((1−δk(sY ))

1+r

)j−1, the net present value of these payments is T (sY )ω(sY ). Similarly,

the college commits to a specific level of expenditures per freshmen, and the total cost of this, per

freshmen, is IEω(sY ) where the expectation is respect to incoming students.

Last, we assume that college quality depends on investment per freshman, and enrollments and

ability. Further, the enrollments affect college quality via a weighting that depends on ω(sY ), and

similarly the average ability is computed as total ability—using the ω weighting—over enrollments

according to the same weighting. Note that ω = 1 assumes colleges only care about freshmen.

In Epple et al. (2006), the college-quality function depends on average ability and expenditures

per student. We nest this specification by assuming college-quality depends on total ability X

(of freshmen, as assumed above), expenditures I, parental income Y , and total enrollments of

freshmen N , q(X,Y, I,N). For the quantitative results, we will use a q that is increasing in X, I,N

and decreasing in Y .

With competitive search, the college decides for each submarket m, characterized by net tuition

and parental income m := (T, sY ) how many vacancies v(m) to post. The unit cost of a vacancy is

κ and the vacancy is filled with probability ρ(θ(m)) which the college takes as given. The college’s

problem can then be written

maxv(m)≥0

q(X,Y, I,N)

s.t. pIN + pC(N) + κ

∫v(m)dm =

∫T (m)ω(m)v(m)ρ(θ(m))dm+ Eg(N) + Ep(N)

X =

∫x(m)ω(m)v(m)ρ(θ(m))dm/N

Y =

∫y(m)ω(m)v(m)ρ(θ(m))dm/N

N =

∫ω(m)v(m)ρ(θ(m))dm

(6)

13

Assuming interior solutions for X, N , and I, the FOCs give

0 ≥ ω(m)

ω(m)

(−pC ′(N) + Eg′(N) + Ep′(N)

)+ T (m)− pI − κ

ω(m)ρ(θ(m))

+ pω(m)

ω(m)

(qNqIN +

qXqI

N

N(x−X) +

qYqI

N

N(y − Y )

) (7)

where N =∫ω(m)v(m)ρ(θ(m))dm with equality in active submarkets, v(m) > 0. To simplify the

computation, we make an assumption, effectively on preferences, that ω(m) = ω(m). That is, the

average ability and enrollments that the college cares about, and that exclusively show up in q, are

determined by the net present value of student characteristics discounted at rate 1+r. In this case,

N = N and many terms cancel with the formula simplifying to

T (m) =κ

ω(m)ρ(θ(m))︸ ︷︷ ︸Search premium

+ pI + pC ′(N)− Eg′(N)− Ep′(N)︸ ︷︷ ︸Common marginal cost

− pqNqIN︸ ︷︷ ︸

Size disc.

− pqXqI

(x−X)︸ ︷︷ ︸Ability discount

− pqYqI

(y − Y )︸ ︷︷ ︸P. income penalty

(8)

in active submarkets.

When κ = 0, this reduces to tuition being the same as what Epple et al. (2006) call effective

marginal cost. However, κ > 0 implies that there is a positive markup so that tuition is always

greater than effective marginal cost.

4.2 Households

We break the discussion of the household problem into a student and youths problem (the latter

being potential students who have not yet made a college decision) and workers and retirees.

4.2.1 Students and Youths

Each period, a fixed measure of youths with heterogeneous characteristics sY = (x, yp) consisting of

academic ability x and parental income yp enter the economy at age j = 1 (corresponding to high

school graduation). Youth choose between skipping college (k = 0) and attending one of the colleges

to which they receive admission, k ∈ K(sY ). College k charges type-specific net tuition T k(sY ) equal

to sticker price T minus institutional aid, and students also face non-tuition expenses φ. Government

grants ζ(T k(sY ) + φ,EFC(sY )) offset some of the cost of attendance, where EFC(sY ) is the

expected family contribution formula that dictates eligibility for government need-based grants and

loans. The net cost of attendance comes out toNCOAk(sY ) = T k(sY )+φ−ζ(T k(sY )+φ,EFC(sY )).

While enrolled, college students receive additively-separable flow utility v(qk) which increases

in college quality qk. To graduate, students must complete JY years of college. Students in class j

return to college each year with probability πj+1(qk) ≡ π(qk)1[j+1≤JY ], which also increases in qk;

14

otherwise, they either drop out or graduate. We eventually plan to allow drop-out probabilities to

depend also on individual student ability. Students can borrow through the Federal Student Loan

Program (FLSP). The FSLP features subsidized loans that do not accrue interest while the student

is in college, where eligibility depends on financial need (NCOA less EFC). Since 1993, students

can borrow additional funds up to the net cost of attendance using unsubsidized loans.

Students face annual and aggregate limits for subsidized and combined borrowing given by bj

and l, respectively. Because students can borrow only up to the net cost of attendance, their annual

combined subsidized borrowing bs and unsubsidized borrowing bu must satisfy

bs + bu ≤ min{bj , NCOAk(sY )}. (9)

Similarly, define bsj as the statutory annual subsidized limit and l

sj as the statutory aggregate

subsidized limit. The actual amount that students can borrow in subsidized loans depends on their

net cost of attendance and the expected family contribution, both of which vary with student type.

Apart from loans, students have two other means of paying for college. First, they have earnings

eY , which we treat as an endowment. Second, they receive a parental transfer ξEFC(sY ), where

ξ ∈ [0, 1] is a parameter. The budget constraint for a college student of type sY is

c+NCOAk(sY ) ≤ eY + ξEFC(sY ) + bs + bu. (10)

4.2.2 Workers and Retirees

Working and retired households receive earnings e(s) that depend on a vector of characteristics s

that includes their level of education, age/retirement status, and a stochastic component. Each pe-

riod, households face a proportional earnings tax τ . These households value consumption according

to a period utility function u(c) and discount the future at rate β. Workers with student loans face

a loan interest rate of i and amortization payments of p(l, t) = l i(1+i)t−1

(1+i)t−1 , where l represents the

loan balance and t the remaining duration. All households can use a discount bond to save at the

risk-free rate r∗. For simplicity, we do not allow borrowing, although it could easily be incorporated.

Workers who are delinquent on their loans face principal penalties and wage garnishment γ.

4.3 Value functions

To reduce the number of state variables, we assume that earnings shocks follow a random walk

with innovation σε. Additionally, we assume taxes (used to finance the student loan program) are

collected at a rate τ proportional to labor earnings. Because of this, after tax labor income can be

summarized via ez+µj where z is the college premium plus log(1− τ) plus the random walk shock.

In this case, the value function for a worker who is not currently in default may be written

Vj(a, l, t, z, f = 0) = max{V Rj+1(a, l, t, α), V D

j+1(a, l(1 + η), z)}. (11)

15

where V R is the value of honoring the student loan payment and V D is the value of default (in

which case the loan balance is increased by η. The value function for a worker who is currently in

default is

Vj(a, l, z, f = 1) = max{V Rj+1(a, l, tmax, α), V D

j+1(a, l, z)} (12)

where if they rehabilitate the loan its duration goes to the maximum one.The value of repayment

isV Rj (a, l, t, z) = max

a′≥0u(c) + βEε′Vj+1(a

′, l′, t′, z + σε′, 0)

s.t. c+ a′/(1 + r) + p(l, t) ≤ ez+µj + a

l′ = (l − p(l, t))(1 + i)

t′ = max{t− 1, 0}

(13)

V Dj (a, l, z) = max

a′≥0u(c) + βEε′Vj+1(a

′, l′, z + σε′, 1)

s.t. c+ a′/(1 + r) + γez+µj ≤ ez+µj + a

l′ = max{0, (l − γez+µj )(1 + i)}

(14)

(Note the budget constraint assumes they make the full payment even if this would make their loan

balance negative; this is done to limit the incentive to default just in order to prepay). Note that

there is no borrowing from private markets (but this could easily be incorporated).

The college student problem may be written

Yj(l; z, δ, T, EFC) = maxc≥0,l′≥l

u(c) + β(1− δ)1[j < JY ]Yj+1(l′; z, δ, T, EFC)

+ βδEε′Vj+1(0, l′, tmax1[l′ > 0], z

j

JY + 1+ σ(j + 1)1/2ε′, 0)

+ β(1− δ1[j = JY ])Eε′Vj+1(0, l′, tmax1[l′ > 0], z + σ(j + 1)1/2ε′, 0)

s.t. c+ T ≤ eY + ξEFC + bs + bu + ζ(T + φ,EFC)

NCOA = T + φ− ζ(T + φ,EFC)

bs = l′s − ls, bu =l′u

1 + i− lu

bu ≤ min{buj , NCOA}, bs + bu ≤ min{bj , NCOA}

l′s +l′u

1 + i≤ l

(15)

16

where the mapping from aggregate loan positions to subsidized / unsubsidized is given by

(l′s, l′u) =

{(l′, 0) if l′ ≤ lsj(NCOA,EFC)

(lsj(NCOA,EFC), l′ − lsj(NCOA,EFC)) otherwise

(ls, lu) =

{(l, 0) if l ≤ lsj−1(NCOA,EFC)

(lsj−1(NCOA,EFC), l − lsj−1(NCOA,EFC)) otherwise.

(16)

Note that we have made the college premium a state variable, as well as the drop out rate δ, and

the cost of attendance (which includes tuition) state variables. This allows us to solve Yj once

for a given set of parameters rather than having to resolve it repeatedly in computing the college

equilibrium. In this specification, we have each year allows a student to secure 1/(JY +1) (16.7% in

the computation) of the earnings premium. Additionally, the sheep-skin effect is 1/(JY + 1), which

is close to empirical estimates.

The value of attending college k (net of preference shocks) is then

Y k(T, sY ) = Y1(0;λk(sY ) + ν + log(1− τ), δk(sY ), T, EFC(sY )) (17)

The value of not attending college is defined as

Y 0(T, sY ) = Eε′V1(0, 0, 0, σε′, 0). (18)

Youths expect a probability η(θ(m)) of being accepted to a school conditional on entering a

particular submarket m. Note the submarkets depend on their type sY , so they may only enter

submarkets M(sY ) ⊂ M . For tractability, we assume that youth can only submit one college

application, but they can adjust search intensity s to increase their probability of acceptance.

Exerting intensity s ≥ 1 in submarket m leads to a match with probability min{sη(θ(m)), 1} and

creates search disutility αs(s − 1)σs with σs > 1. In fact, we go further and assume youths only

choose (m, s) combinations that lead to college acceptance with probability 1.3 Hence, conditional

on applying to college k, the problem of deciding which submarket to go to is

maxi≥1,m∈M(sY )

min{sη(θ(m)), 1}(Y k(m)− Y 0(m)) + Y 0(m)− ψ(i− 1)2

s.t. iη(θk(m)) = 1

(19)

(recall k = 0 means skipping college, in which case η(θk) ≡ 1). Thus, the value of applying to

college k and submarket m ∈M(sY ) with intensity i = 1/η(θ(m)) is

maxk∈{0,...,K},m∈M(sY )

Y k(m)− ψ(1/η(θk(m))− 1)2 (20)

3We do this for two reasons. First, it makes preference shocks over colleges more natural. Second, it is strange tothink of some students applying to elite schools and then not getting in anywhere.

17

To these, we add idiosyncratic preference shocks that allow for unobserved preferences for

colleges:

maxk∈{0,...,K},m∈M(sY )

Y k(m)− ψ(1/η(θk(m))− 1)2 +1

αεk, α > 0. (21)

Breaking this into two parts, define

Y k(sY ) := maxm∈M(sY )

Y k(m)− ψ(1/η(θk(m))− 1)2 (22)

with policy function mk(sY ) and

maxk∈{0,...,K}

Y k(sY ) +1

σεk. (23)

Assuming these shocks are distributed according to a Type 1 extreme value distribution, the

probability of applying to school k is

Ak(sY ) :=exp(σ(Y k(sY )− Y 0(sY ))))∑Kk=0

exp(σ(Y k(sY )− Y 0(sY ))). (24)

Because we assume that applicants (conditional on apply to k) exert enough search effort to go

with probability 1, this is also the enrollment rate for type sY .

The effective search exerted by student type sY for school type k is f(sY )Ak(sY )/η(θk(mk(sY ))).

The total number of vacancies, assuming that the measure of each school type is g(k), is g(k)v(mk(sY )).

So, the implied market tightness in each submarket is

θk(m) =

{η(θk(mk(sY ))g(k)v(mk(sY ))

f(sY )Ak(sY )if sY is such that m = mk(sY )

0 otherwise. (25)

4.4 Government

The government levies proportional taxes on labor earnings to fund transfers and loans (subsidized

and unsubsidized) to students in college. Other sources of revenue for the government include

interest payments on unsubsidized loans for students in college as well as loan payments and

garnishment from workers with outstanding student loans.

4.5 Equilibrium

A steady state equilibrium consists of market tightnesses θk(sY ), college policies Xk, Ik, Nk, vk,

value functions Y k and V , application rates Ak, and tax rates such that

1. Colleges optimally choose their policies taking market tightnesses as given;

2. students and workers optimize taking market tightnesses and tax rates as given;

18

3. application rates Ak and market tightnesses θk are consistent with the student value function

Y k and vacancy creation vk; and

4. the government balances its budget.

5 Functional forms, calibration, and estimation

5.1 Mapping the model to the data

One unit of the consumption good is treated as $1,000 in 2010 dollars. We take ability to be

U [0, 1]. We assume that N in the data is a school’s FTE share times the enrollment rate. Table 4

summarizes how we map the data and model populations.

Data Model

Youth population 1Number of schools within each type g(k)A school’s FTE share × the enrollment rate N

Table 4: Mapping between the data and model

5.2 College quality

We assume q is given by a CES quality function

q(X, I,N) =(αXX

ε−1ε + αII

ε−1ε + αNN

ε−1ε + αY Y

− ε−1ε

) εε−1

(26)

where ε ≥ 0 is the elasticity of substitution. Note that Y is a “bad” in that its exponent is−(ε−1)/ε.4

The Cobb-Douglas case is the limiting case of ε = 1, perfect complements is the limiting case of

ε = 0, and perfect substitutes is the limiting case of ε =∞.

Note that the model has extremely tight predictions about net tuition for each student, but

it is totally silent as to “sticker price” tuition. To overcome this and allow sticker prices in the

data to help discipline the quality function parameters, we construct an artificial sticker price

in the model as follows. IPEDS / DCP has two relevant measures of tuition discounts. One is

the percent of students receiving institutional grants and the other is the average amount of the

institutional grant. Using the first measure, we construct a cutoff T such that P (T ≤ T ) is the

fraction receiving institutional grants. We then take E(T |T > T ) as the “sticker price.” We take

E(T |T > T ) − E(T |T ≤ T ) as the average institutional grant. With E(T |T > T ) as the sticker

price and net tuition as E(T ), we can then match the discounts.

4Because of numerical precision issues, we scale N so that αN reported later in the paper is the αN here times1000(ε−1)/ε.

19

We note that for the solution to an individual college problem, the overall returns to scale in the

college quality function is irrelevant if there is no feedback from college quality to student demand.

(This is because any monotone transformation Q will result in the same solution.) So we normalize

αX to 1.

5.3 College non-tuition revenue and custodial costs

For the functional forms of Eg(N) and Ep(N) we use linear specifications Eg(N) = Eg,kN and

Ep(N) = Ep,kN as this should give results close to what is seen in the data (particularly Table 2.

Following Epple et al. (2006), we posit a custodial cost function of the form

C(N) = ck0 + ck1N + ck2N2. (27)

Rather than trying to calibrate 3K parameters, we make some assumptions that help match college

sizes using information from a “nearby” problem. First, we assume that ck1 = 0 for all k. We then

consider the solution to (5) under the assumption that there is no student heterogeneity and

consequently X ′(N) and T ′(N) are zero after allowing for quality to depend on N . The FOC

becomesqNqI

+1

p

d(E(N)/N)

dN=d(C(N)/N)

dN(28)

where qN/qI is (αN/αI)(N/I)−1/ε. To calibrate, we assume that enrollments observed in the data

correspond to the efficient scale, and by that we mean the solution to this problem. In other words,

Nk

= N and the FOC satisfies

ck2(Nk)2 = ck0 +

αNαI

(Nk

I

)−1/ε(N

k)2 (29)

If one knew I, the right hand and Nk

are either parameters or taken from the data. Consequently,

this equation would pin down ck0 as a function of ck2 (or vice-versa) for each k. We proxy for I

by assuming it is two-thirds of the observed average expenditures per FTE in the data Sk, i.e.,

Ik

= 231pS

k. This still gives K remaining cost parameters. To discipline this, we assume that the

fixed cost is a proportion c ∈ [0, 1] of total expenditures in the data,

ck0 = c1

pSkNk

(30)

where Sk

denote the observed average expenditures per FTE in the data. Thus we are left with

just one free parameter, c.

20

5.4 Matching technologies

We assume a CES and CRS matching function that for total spots v and intensity-adjusted appli-

cants u = iu is

m(u, v) = umin

{A(v/u)

(1 + (v/u)γ)1/γ, 1

}(31)

The resulting matching rate for intensity-adjusted applicants, η(θ) = m(u,v)u , is

η(θ) = min

{Aθ

(1 + θγ)1/γ, 1

}and ρ(θ) =

η(θ)

θ(32)

where θ = v/u is the market tightness. We take γ = 1 and then choose A so that when θ = 1,

students and colleges match with a 95% probability, i.e., A = 2 · .95.

5.5 Student versus school roles in dropout and college premium

Graduation rates and college premiums vary greatly by school, as does our measure of average ability

of the student body. We assume that these depend partially on individual ability and partially on

school type. In order to almost exactly match the data without over-burdening the estimation,

we assume a constant split µ between an individual’s contribution and the school’s contribution.

Specifically, for continuation rates we assume

(1− δk(x)) = min{max{(1− δk)(µδ + (1− µδ)x

Xk), 0}, 1} (33)

where δk

is from the data, x is individual ability, and Xk is average ability (at school k).

To determine an appropriate value for µ, we turn to the data. First, we construct sticker tuition

quintiles as a proxy for college types. Then, within these quintiles we compute average ability

and average graduation rate. Then we regress graduation on a quintile dummy times the average

graduation rate times the individual’s ability over average ability. The results are presented in Table

18 in the appendix. Interestingly, the lowest quintile has no statistically significant relationship

between ability and graduation. However, the other quintiles seem to have a significant and stable

relationship with coefficients around 40%. Consequently, we take µ = .6 to generate something

close to this.

Similarly, we take the relative college premium as

λk(x) = λk(µλ + (1− µλ)

x

Xk) (34)

and the follow the same procedure as above but using the log college premium in placed of the

graduation dummy. The results are also in Table 18. The results seem relatively stable across the

quintiles, and we set µλ = .1. Note that these estimates do not say that the college premium is

21

nearly similar across schools. In fact, using the NLSY97 data we find the log college premium for the

lowest quintile is .363 while the highest quintile is .643, a .28 log point difference or approximately

32.3%.

5.6 Parental transfers

To calibrate ξ, which determines parental transfers ξEFC(sY ), we use NLSY97 data. First, we

compute parental income and then apply the EFC formula from Epple et al. (2013) to get an EFC

measure. Then we use data on family aid for college that is not expected to be paid back and find

the annual level of support. We then regress this transfer measure on interactions of EFC with

graduating and not graduating (we exclude a constant since the model lacks one). The results are

given in Table 5. Transfers end up higher for dropouts, which may reflect many things such as the

family’s resources being exhausted over time. We set ξ = 0.7 as our primary object of interest and

the preference shock can somewhat account for these differences in transfers.

(1) (2)Family grant Family grant

Dropped out × EFC (real) 0.0939∗∗∗ 0.419∗∗∗

(6.70) (4.46)

Graduated × EFC (real) 0.185∗∗∗ 0.901∗∗∗

(27.31) (30.16)

Observations 2063 771R2 0.277 0.547

t statistics in parentheses

(1) is the full sample; (2) includes only those with EFC < net tuition.∗ p < .1, ∗∗ p < .05, ∗∗∗ p < .01

Table 5: Transfers as a function of EFC

5.7 Jointly estimated parameters

Currently, the calibration and to some extent the model are still in flux. Table 6 summarizes

most of the calibration of the non-college-specific parameters. The remaining free parameters are

jointly estimated to fit a large number of moments including net and sticker tuition at each school,

enrollment shares, expenditures, enrollment rates, ability at each school, and the correlation between

parental income and enrollment. These parameters are summarized in Table 7.

Because of the large number of targeted moments, it is most convenient to look at the model fit

graphically. This is done in Figure 4. Each blue dot represents a particular type of school with the

vertical position giving the data’s moment and the horizontal positive giving the model’s moment.

If the model had perfect predictions, all the blue dots would lie on the 45-degree line. We have

22

Description Value Source/Reason

Discount factor 0.96 StandardRisk aversion 2 StandardSavings interest rate 0.02 StandardBorrowing premium 0.107 12.7% rate on borrowingEarnings in college $7,128 NLSY97Loan balance penalty 0.05 Ionescu (2011)Loan duration 10 Statutory

Retention probability 0.5541/5 55.4% completion rateEarnings shocks (0.952,0.168) STY (2004)∗

Age-earnings profile Cubic STY (2004)∗

College premium GH (2016)∗∗ Autor et al. (2008)Non-tuition costs GH (2016)∗∗ IPEDSStudent loan rate GH (2016)∗∗ StatutoryAnnual loan limits GH (2016)∗∗ StatutoryAggregate loan limits GH (2016)∗∗ StatutoryCost function GH (2016)∗∗ IPEDS regressionEndowment flow GH (2016)∗∗ IPEDSGrant aid GH (2016)∗∗ IPEDS

∗Storesletten, Telmer, and Yaron (2004).∗∗Web appendix A of Gordon and Hedlund (2016).

Table 6: Independently determined model

Description Parameter Value

Preference shock size σ 8.062Search effort level ψ 1000.000Cost parameter c 0.001Quality’s elasticity ε 0.525Quality’s weight on investment αI 31.805Quality’s weight on enrollment N , public schools αgN 0.133Quality’s weight on enrollment N , private schools αpN 0.088Quality’s weight on inverse parental income Y −1 αY 0.012

Table 7: Jointly estimated parameters

23

also labeled the school type that gives the worst fit among the seven school types. For instance,

the worst fit for net tuition comes from GRS where G stands for public, R stands for research, and

S stands for selective. Similarly, the worst for FTE share comes from PTN which is private (P),

teaching (T), nonselective (N) schools. Overall, it seems the model does a fair job of matching the

targeted moment with some noteworthy exceptions.

0 5 10 15 20Model

0

10

20

Data

Net tuition

GRS

0 10 20 30Model

0

10

20

30

Data

Sticker price

GRS

0 20 40 60 80Model

0

50

100

Data

Expenditures

GRS

0 0.1 0.2 0.3 0.4Model

0

0.2

0.4

Data

FTE share

GTN

0.2 0.4 0.6 0.8 1Model

0

0.5

1

Data

Relative ability

PRN

50 100 150 200Model

50

100

150

200

Data

Parental income

PTS

45 deg

Model vs. data

Figure 4: Goodness of fit in calibration

Figure 5 shows equilibrium heat maps for enrollment in 1987 across the 7 college types and the

decision to skip college. The calibration gives that research-intensive and selective colleges are more

likely to attract high ability students, and this sorting is starker at private institutions. By contrast,

low ability students disproportionately skip college entirely, regardless of parental income. Students

coming from the lowest income families almost never attend selective private colleges, no matter

their academic ability. Instead, such students either attend selective public research institutions or

non-selective institutions.

6 Results

We break the results discussion into three parts. First, we describe how we change parameters of

the model to capture the theories. Second, we allow all factors to change simultaneously to bring

the benchmark (1987) model to the terminal steady state (2010). This serves as model validation.

24

No college

0 0.5 10

50

100

150

200

Par

enta

l inc

ome

GTN

0 0.5 10

50

100

150

200

GRN

0 0.5 10

50

100

150

200

PTN

0 0.5 10

50

100

150

200

PRN

0 0.5 1Ability

0

50

100

150

200

Par

enta

l inc

ome

GRS

0 0.5 1Ability

0

50

100

150

200

PTS

0 0.5 1Ability

0

50

100

150

200

PRS

0 0.5 1Ability

0

50

100

150

200

Figure 5: Sorting in 1987

Third, we run counterfactuals and decompose the importance of relative factors by allowing only

some to change.

6.1 Implementing the theories

Here we summarize how we change parameters of the model to match the various theories:

• For Baumol cost disease, we increase p by the same amount that the Higher Education Price

Index / CPI increased from 1987 to 2010.

• For the “macroeconomic forces” driven demand changes,

– we increase the average college premium λ by an amount consistent with the 1987-2010

change reported in Autor et al. (2008) (extrapolating slightly to reach 2010 as described

in Gordon and Hedlund, 2016);

– we increase the expected family contribution EFC(sY ) as if the parental income had

increased from 69.3% of its 2010 value to its 2010 value, which is consistent with the

increase in the real GDP per capita from 1987 to 2010 (roughly a 1.3% annual growth

rate);5

5Note that this has the effect of amplifying underlying inequality in transfers because the EFC formula works outto zero for any student with parental income less than $51,200 in 2009 dollars. So the only effective increases will befor students with higher parental income.

25

– and we move the college completion rates from their 2002 value (the earliest year for

which we have data on this series from IPEDS/DCP) to their 2010 value.

• For the Bennett story (colleges capturing financial aid),

– we change the Federal Student Loan Program borrowing limits according to the statutory

values (including the introduction of unsubsidized loans beginning in 1993);

– we change real interest rates on student loans according to the statutory limits (and we

average over the 10 year repayment period as described in Gordon and Hedlund, 2016);

– we increase φ (the non-tuition expenses of attending college), which increases eligibility

for subsidized student loans, according to the estimates of these costs reported by NCES;

and

– we increase mean Pell grants according the mean value reported by NCES.

• For the public and private non-tuition revenue stories, we change the average per student

values (Eg,k

, Ep,k

) according to the value in IPEDS/DCP.

6.2 Model validation

Because of the large number of moments, we again resort to a graph (Figure 6) to assess the model’s

performance at matching untargeted moments over time. The horizontal position of the solid circle

(a “cannonball”) represents the model’s prediction for 2010 and the vertical position represents

the data’s value in 2010. Opposite the cannonball are the values for 1987. If the model perfectly

captures the data, all the cannonball trajectories would lie on top of the 45 degree line. However, if

the trajectories are parallel to the 45 degree line, then the model would exactly predict the changes.

Qualitative failures of the model have trajectories that are orthogonal to the 45 degree. Overall,

the model seems to do very well at matching the untargeted changes in net tuition, expenditures,

ability, and enrollment correct (as we do not have parental income for 1987, we cannot construct

the graph for it).

6.3 Decomposing the importance of relative factors

Given that the model successfully replicates the overall rise in net tuition between 1987 and 2010,

it is useful to assess the role of each of the main exogenous driving forces. We proceed in two

steps. First, we looking at the factors’ impacts on FTE-weighted values. Second, we look at the

heterogeneous impact across schools.

6.3.1 Average impacts across schools

Table 8 displays how much each of the theories can individually and jointly explain the increase

in net tuition. The “% Expl.” columns give the percentage of the data’s change explained by the

26

0 20 40Model

0

20

40

Data

Net tuition

0 20 40 60Model

0

50

Data

Sticker price

0 50 100 150Model

0

50

100

150

Data

Expenditures

0.2 0.4 0.6 0.8Model

0

0.5

1D

ata

Relative ability

0 0.1 0.2Model

0

0.1

0.2

Data

Enrollments 45 degree

GTN

GRN

PTN

PRN

GRS

PTS

PRS

Figure 6: Data vs. model predictions, 1987-2010

27

model. Overall, the theories can jointly account for the entire net tuition increase. However, the bulk

of the increase comes from the demand side, predominantly the non-policy-driven ones resulting

from a higher college earnings premium, lower dropout rates, and higher parental income. On their

own, these explain 76% of the increase. The Bennett theory also accounts for 42% of the increase

in net tuition. Baumol cost disease plays only a small role, accounting for 7% of the increase in

net tuition. The other theories that focus on cuts in public support or changes private non-tuition

revenue contribute nothing: The model predicts they would have acted to decrease net tuition on

their own. All the theories drive up expenditures, albeit to varying degrees. The model predicts the

largest contributor to be private non-tuition revenue.

Net tuition Expenditures

Experiment Model Data % Expl. Model Data % Expl.

1987 5.8 5.8 0 27.3 27.4 01987+Baumol (p) 6.2 - 7 27.6 - 31987+Demand (λ, δ) 9.9 - 76 30.8 - 361987+Bennett 8.1 - 42 29.4 - 221987+Pub. rev. (Eg) 5.5 - -6 29.2 - 201987+Priv. rev. (Ep) 4.6 - -23 32.2 - 512010 (1987+everything) 11.6 11.2 108 40.1 37.1 132

Table 8: Average impacts, 1987 adding one force at a time

6.3.2 Heterogeneous impacts across schools

Figure 7 presents the tuition decomposition first by implementing the forces one at a time. (For

beginning at 2010 and removing one force at a time, one may consult Figure 9 in the appendix.)

Each panel represents a different variable of interest with net tuition in the top panel. The horizontal

axis gives percent change, the vertical axis gives different school types. The %∆ gives the overall

percent change induced on net tuition from all the forces. A given bar shows the percent change

from a particular force. For instance, the light blue represents the change induced by “Demand,

λ, δ”—i.e., only letting the non-policy demand forces change. For GRN, this force alone would have

increased net tuition by around 160%, while for PRN it would have done so by around 40%.

The main drivers of net tuition increases are the demand side changes, both the policy-induced

ones (Bennett) and the macroeconomic forces (Demand, λ, δ). Moreover, this seems to be pretty

robust across school types despite wildly different endowments and student bodies. Baumol seems to

have increased net tuition only slightly. The model predicts that non-tuition revenue acted mostly

to decrease net tuition.

The model also has predictions for variables other than net tuition, and we briefly discuss them.

Almost all the theories serve to increase expenditures. In contrast to the simple model, demand has

28

Net Tuition

% = 178% = 46

% = 170% = 77

% = 43% = 87

% = 79

-100 -50 0 50 100 150 200 250 300 350

GRNGRSGTNPRNPRSPTNPTS

Expenditures

% = 35% = 54

% = 32% = 31

% = 58% = 24

% = 60

-20 -10 0 10 20 30 40 50 60 70

GRNGRSGTNPRNPRSPTNPTS

Enrollment

% = 33% = 48

% = 32% = 32

% = 57% = 27

% = 63

-20 -10 0 10 20 30 40 50 60 70

GRNGRSGTNPRNPRSPTNPTS

Relative ability

% = -0% = -0

% = -1% = -6

% = -1% = -4

% = -5

-10 0 10 20

GRNGRSGTNPRNPRSPTNPTS

Figure 7: Percent change relative to 1987 from adding one force, else equal

29

a significant impact on enrollment. This is a consequence of the college quality function directly

incorporating enrollment in the college quality function. For most schools, changes in non-tuition

revenue have acted to reduce ability and increase enrollment.

7 Conclusion

The model results indicate that demand-side changes are driving most of the net tuition increases.

The bulk of this is driven by non-policy-induced demand changes, but policy-induced demand

changes also play a large role. Despite the substantial 20% increase in relative costs of college

goods, Baumol cost disease accounts for only a small increase in net tuition.

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A Detailed data sources and description

A.1 IPEDS/DCP data

A.2 NLSY 97 data

B Additional Results

B.1 Empirical results

B.1.1 Results form IPEDS/DCP data

33

Balance sheet item Model equivalent DCP variable

Total Expenditures pI + pC eandg01 sum +auxother cost

E&G spending** Part of pI + pC eandg01 sumE&R spending pIInstruction Part of pI + pC instruction01Research Part of pI + pC research01Public service Part of pI + pC pubserv01Academic support Part of pI + pC acadsupp01Student services Part of pI + pC studserv01Institutional support Part of pI + pC instsupp01Plant operation / maintenance Part of pI + pC opermain01Scholarships and fellowships Part of pI + pC grants01,grants01 fasb

Auxiliary and “other” spending Part of pI + pC auxother cost

Total Revenue T + Eg+part of Ep tot rev w auxother sumNet tuition T nettuition01 −

grants01Directly from student Out of pocket for T net student tuitionFrom government Students apply to T nettuition01 -

net student tuitionPell Students apply to T grant01Local, state, and other federal Students apply to T nettuition01 -

net student tuition- grant01

Approp., contracts, excluding Pell Eg state local app +state local grant contract+ fed-eral10 net pell*

Auxiliary and “other” revenue Part of Ep auxother revEndowment revenue, gifts Part of Ep priv invest endow

Gross operating margin (rev. - exp.) Part of Ep tot rev w auxother sum- eandg01 sum -auxother cost

Note: Ep is the sum of “Part of Ep” and pC is the sum of “Part of pC.”*Computed as a residual: tot rev w auxother sum - nettuition01 - priv invest endow - auxother rev**A component of E&G spending is expenditures on scholarships and fellowships with the definitionvarying over time. Because of reporting changes, we cannot subtract this off in a consistent way,so we leave it. Post 1997 for FASB institutions and 2002 for GASB it should reflect expenses fromadministering scholarships and fellowships.

Table 9: Detailed college balance sheet data with DCP variable names

34

1987 financial measures and shares

School type T Expend Eg Ep FTE share

Public, Teaching, Non-selective 2.7 14.9 9.2 3.0 0.25Public, Research, Non-selective 3.7 25.9 13.7 8.5 0.36Private, Teaching, Non-selective 9.6 19.9 1.4 9.0 0.15Private, Research, Non-selective 11.9 27.0 3.8 11.3 0.05Public, Research, Selective 4.0 39.5 21.3 14.3 0.10Private, Teaching, Selective 14.3 32.6 1.0 17.3 0.01Private, Research, Selective 15.5 72.2 14.1 42.7 0.08

2010 financial measures and shares

School type T Expend Eg Ep FTE share

Public, Teaching, Non-selective 6.4 17.8 7.9 3.5 0.25Public, Research, Non-selective 8.8 35.5 14.9 11.8 0.35Private, Teaching, Non-selective 15.1 22.5 1.1 6.3 0.18Private, Research, Non-selective 20.3 34.0 4.1 9.6 0.05Public, Research, Selective 10.0 65.3 29.1 26.2 0.09Private, Teaching, Selective 24.3 50.0 1.2 24.5 0.01Private, Research, Selective 23.7 115.4 22.0 69.6 0.07

Additional 2010 measures

School type Rel. premium Comp. rate P. inc. Rel. X # schools

Public, Teaching, Non-selective 0.83 0.47 53 0.26 242Public, Research, Non-selective 0.95 0.58 65 0.48 129Private, Teaching, Non-selective 0.89 0.58 71 0.34 639Private, Research, Non-selective 1.08 0.65 81 0.51 50Public, Research, Selective 1.19 0.73 77 0.90 20Private, Teaching, Selective 1.29 0.89 110 0.94 36Private, Research, Selective 1.63 0.87 91 0.96 42

Note: Monetary values are in 2010 dollars deflated using the CPI.

Table 10: Data measures in 1987 and 2010

35

(1) (2) (3) (4) (5) (6)Mean Mean Mean Mean Mean Mean

Enrolled 0.428 1 1 0.460 1 1Graduated 0.300 0.714 1 0.325 0.719 1Years college 1.628 3.867 4.488 1.779 3.919 4.535Sticker tuition estimate (real) 14.10 16.08 14.33 16.29Net tuition estimate (real) 8.792 10.16 8.933 10.22Family transfers (real) 3.657 4.588 3.641 4.481Grants (school or gov.) (real) 5.311 5.920 5.401 6.069Loans (private or gov.) (real) 3.197 3.515 3.301 3.647Took out a loan 0.560 0.604 0.587 0.634Ability 0.506 0.677 0.709Household income in 1996 (real) 73.37 92.71 99.10EFC (real) 10.86 15.48 17.04

Observations 6536 2616 1868 4102 1778 1278

Note: All estimates are means from the NLSY97 data, unweighted using the cross-section sample.

In the full sample, 21.1% are missing ASVAB scores; 26.6% are missing household income;

and 40.1% are missing ASVAB or household income.

Enrollment is defined as any enrollment in a 4-year nonprofit college while working towards a BA/BS or MA.

Sticker and net tuition are approximate and computed by adding aid from various sources.

All financial variables are in thousands of 2010 dollars.

Table 11: NLSY97 data summary

36

Variable name % not missing Mean Median Min Max

pubid 100.0 3442.85 3380.50 1.00 9022.00w 100.0 2506.64 2775.34 760.71 15761.82crosssect 100.0 1.00 1.00 1.00 1.00abil 80.4 0.50 0.50 0.00 1.00hhinc 75.4 72263.25 59781.81 -66872.21 342666.56pinc 62.6 77135.31 66038.05 -73684.56 729895.59efc 75.4 10715.05 5787.59 0.00 90821.45age0 100.0 13.98 14.00 12.00 16.00avgeqinc 86.6 40955.92 33729.95 0.00 309727.68totyearsAttendedFTE 99.2 1.64 0.00 0.00 13.75everWorkForBABS 99.2 0.44 0.00 0.00 1.00everGrad 96.9 0.30 0.00 0.00 1.00everRecvBABS 96.9 0.30 0.00 0.00 1.00sticker 40.9 14257.51 10212.82 0.00 147419.14net 40.9 8907.42 6024.68 0.00 129767.50grant 42.8 5278.64 2065.08 0.00 131522.78famtran 42.5 3613.19 345.43 0.00 129767.50loans 43.2 3194.87 994.26 0.00 70337.52

Table 12: NLSY97 data summary

37

(1) (2) (3) (4) (5) (6) (7) (8)T T T T T T T T

Eg -0.26∗∗∗ -0.04∗ -0.01 0.02Ep 0.09∗∗∗ 0.00 -0.00 -0.00Not sel. Pub Teach × Eg -0.39∗ -0.01 -0.00 -0.09∗∗∗

Not sel. Pub Res. × Eg -0.40∗∗∗ 0.13∗ 0.05∗∗∗ -0.04∗

Not sel. Priv Teach × Eg -0.14∗∗∗ -0.13∗∗∗ -0.06∗∗∗ 0.13∗∗∗

Not sel. Priv Res. × Eg -0.06 -0.28∗∗∗ -0.06∗ 0.30∗∗

Sel. Pub Res. × Eg -0.14∗∗∗ -0.03 0.01 0.01Sel. Priv Teach × Eg 2.98∗∗∗ -0.63∗∗ -0.33∗∗ -0.29Sel. Priv Res. × Eg 0.21∗∗∗ -0.02 0.02∗ 0.03Not sel. Pub Teach × Ep -0.57∗∗ 0.01 0.01 -0.08∗∗∗

Not sel. Pub Res. × Ep 0.13∗∗∗ -0.02 0.01 0.02Not sel. Priv Teach × Ep 0.09∗ 0.07∗ 0.06∗∗∗ -0.05∗∗

Not sel. Priv Res. × Ep 0.18∗∗ 0.17∗∗∗ 0.05 0.03Sel. Pub Res. × Ep 0.05∗∗ 0.05∗ 0.02∗∗∗ 0.03∗

Sel. Priv Teach × Ep 0.15∗∗ -0.10∗∗∗ -0.01 -0.06∗

Sel. Priv Res. × Ep -0.02 -0.00 -0.01∗∗∗ -0.00Constant 13266.80∗∗∗ 12949.12∗∗∗ 9904.17∗∗∗ 7399.60∗∗∗ 13709.27∗∗∗ 12439.92∗∗∗ 9485.20∗∗∗ 7886.03∗∗∗

Observations 1158 1158 27792 27792 1158 1158 27792 27792Adjusted R2 0.149 0.727 0.750 0.463 0.399 0.733 0.755 0.473∗ p < .10, ∗∗ p < .05, ∗∗∗ p < .01

Table 13: How net tuition varies with non-tuition revenue

38

(1) (2) (3) (4) (5) (6) (7) (8)log(N) log(N) log(N) log(N) log(N) log(N) log(N) log(N)

log(Eg) 0.40∗∗∗ -0.05 -0.05∗∗∗ -0.06∗∗∗

log(Ep) -0.14∗∗∗ -0.15∗∗∗ -0.19∗∗∗ -0.12∗∗∗

Not sel. Pub Teach × log(Eg) -0.07 -0.92∗∗∗ -1.06∗∗∗ -0.32∗∗∗

Not sel. Pub Res. × log(Eg) -0.26∗∗∗ 0.11 -0.01 -0.19∗∗

Not sel. Priv Teach × log(Eg) -0.02 -0.04 -0.04∗∗∗ -0.05∗∗∗

Not sel. Priv Res. × log(Eg) 0.18∗∗ -0.09 0.07∗ -0.09∗∗∗

Sel. Pub Res. × log(Eg) -0.01 -1.44∗∗∗ -1.65∗∗∗ -0.18∗∗∗

Sel. Priv Teach × log(Eg) 0.03 -0.12 -0.08∗∗∗ 0.01Sel. Priv Res. × log(Eg) -0.06 -0.08 -0.04 -0.08∗∗∗

Not sel. Pub Teach × log(Ep) -0.06 0.01 -0.01 -0.02∗∗∗

Not sel. Pub Res. × log(Ep) 0.28∗∗∗ 0.17∗∗ 0.25∗∗∗ -0.02Not sel. Priv Teach × log(Ep) -0.25∗∗∗ -0.21∗∗∗ -0.26∗∗∗ -0.19∗∗∗

Not sel. Priv Res. × log(Ep) -0.27∗∗∗ 0.06 -0.23∗∗∗ -0.15∗∗∗

Sel. Pub Res. × log(Ep) 0.05 0.78∗∗∗ 0.73∗∗∗ -0.07∗∗∗

Sel. Priv Teach × log(Ep) -0.24∗∗∗ -0.83∗∗∗ -0.99∗∗∗ -0.24∗∗∗

Sel. Priv Res. × log(Ep) 0.01 0.01 -0.06 -0.07∗∗∗

Constant 6.37∗∗∗ 9.89∗∗∗ 10.08∗∗∗ 9.33∗∗∗ 9.85∗∗∗ 11.23∗∗∗ 11.82∗∗∗ 10.03∗∗∗

Observations 1105 1105 26721 26721 1105 1105 26721 26721Adjusted R2 0.366 0.711 0.718 0.463 0.676 0.734 0.742 0.503∗ p < .10, ∗∗ p < .05, ∗∗∗ p < .01

Table 14: How enrollment varies with non-tuition revenue

39

(1) (2) (3) (4)Ability Ability Ability Ability

log(Eg) 0.01∗ -0.01log(Ep) 0.14∗∗∗ 0.06∗∗∗

Not sel. Pub Teach × log(Eg) -0.00 -0.10Not sel. Pub Res. × log(Eg) 0.04∗ 0.15∗∗

Not sel. Priv Teach × log(Eg) -0.02∗∗ -0.01Not sel. Priv Res. × log(Eg) 0.03 0.06∗

Sel. Pub Res. × log(Eg) 0.08∗∗∗ 0.25Sel. Priv Teach × log(Eg) 0.02∗∗∗ -0.00Sel. Priv Res. × log(Eg) 0.04∗∗ 0.02Not sel. Pub Teach × log(Ep) 0.04∗∗∗ 0.06∗∗∗

Not sel. Pub Res. × log(Ep) 0.01 -0.02Not sel. Priv Teach × log(Ep) 0.06∗∗∗ 0.07∗∗∗

Not sel. Priv Res. × log(Ep) 0.04∗ 0.10∗∗∗

Sel. Pub Res. × log(Ep) 0.02 -0.09Sel. Priv Teach × log(Ep) 0.09∗∗∗ 0.04∗∗∗

Sel. Priv Res. × log(Ep) 0.07∗∗∗ 0.01Constant -0.95∗∗∗ -0.13 -0.11 -0.07

Observations 931 931 931 931Adjusted R2 0.300 0.611 0.589 0.617∗ p < .10, ∗∗ p < .05, ∗∗∗ p < .01

Table 15: How ability varies with non-tuition revenue

B.1.2 Results from NLSY97 data

B.2 Model results

Figure 8 reports the heat map of enrollments for 2010. It is analogous to Figure 5 but for 2010

rather than 1987.

C Computation

C.1 The college problem

We fix a grid of market tightnesses Θ. For each θ ∈ Θ, we compute η(θ) and ρ(θ) which gives

corresponding grids on probabilities.

The college equilibrium may be thought of as a root-finding problem. The residual of this

root-finding problem can be defined in the following way. Let x := {Xk, Ik, Nk}Kk=1. To define the

residual function F : R3K++ → R3K , proceed as follows:

1. Guess on x ∈ R3K++.

40

(1) (2) (3) (4) (5) (6)lIplusC lT abil coll lIplusC lT abil coll

lEp 0.444∗∗∗ 0.319∗∗∗ 0.134∗∗∗ 0.372∗∗∗ 0.167∗∗∗ 0.142∗∗∗

(0.000) (0.000) (0.000) (0.000) (0.000) (0.000)

lEg 0.121∗∗∗ -0.222∗∗∗ 0.0313∗∗∗ 0.0723∗∗∗ -0.144∗∗∗ 0.0119∗∗

(0.000) (0.000) (0.000) (0.000) (0.000) (0.008)

cons 4.958∗∗∗ 7.599∗∗∗ -1.068∗∗∗ 6.070∗∗∗ 8.952∗∗∗ -0.953∗∗∗

(0.000) (0.000) (0.000) (0.000) (0.000) (0.000)

N 1130 1130 954 1105 1105 931

p-values in parentheses∗ p < 0.05, ∗∗ p < 0.01, ∗∗∗ p < 0.001

Table 16: Non-tuition revenue, tuition, expenditures, and ability

No college

0 0.5 10

50

100

150

200

Par

enta

l inc

ome

GTN

0 0.5 10

50

100

150

200

GRN

0 0.5 10

50

100

150

200

PTN

0 0.5 10

50

100

150

200

PRN

0 0.5 1Ability

0

50

100

150

200

Par

enta

l inc

ome

GRS

0 0.5 1Ability

0

50

100

150

200

PTS

0 0.5 1Ability

0

50

100

150

200

PRS

0 0.5 1Ability

0

50

100

150

200

Figure 8: Sorting in 2010

41

(1) (2) (3) (4) (5) (6)% off % off % off % off % off % off

Ability 7.472∗∗ 12.82∗∗∗ 2.573(2.29) (4.00) (0.78)

95th Abil pctile 8.576∗∗∗ 13.27∗∗∗ 4.468(3.13) (4.95) (1.61)

90-95th Abil pctile 4.445 6.866∗∗ 2.329(1.57) (2.50) (0.83)

75-90th Abil pctile -0.474 1.724 -2.224(-0.22) (0.82) (-1.03)

50-75th Abil pctile -1.219 -0.555 -1.934(-0.59) (-0.28) (-0.94)

Parental income in 1996 (real) -0.0972∗∗∗ -0.0776∗∗∗ -0.102∗∗∗ -0.0982∗∗∗ -0.0782∗∗∗ -0.102∗∗∗

(-8.89) (-7.22) (-9.43) (-9.01) (-7.33) (-9.51)

Net tuition estimate (real) -0.724∗∗∗ -0.748∗∗∗

(-10.48) (-10.85)

Sticker tuition estimate (real) 0.363∗∗∗ 0.348∗∗∗

(7.10) (6.78)

Constant 34.15∗∗∗ 35.81∗∗∗ 32.44∗∗∗ 38.48∗∗∗ 42.48∗∗∗ 34.94∗∗∗

(14.27) (15.43) (13.69) (21.55) (24.09) (19.02)

Observations 1609 1609 1609 1609 1609 1609R2 0.047 0.108 0.076 0.055 0.119 0.081

t statistics in parentheses∗ p < .1, ∗∗ p < .05, ∗∗∗ p < .01

Table 17: Tuition discounting

42

(1) (2)Graduated Log coll. prem

Sticker quintile=1 × Mean grad rate × Relative ability 0.0238(0.37)

Sticker quintile=2 × Mean grad rate × Relative ability 0.320∗∗∗

(7.29)

Sticker quintile=3 × Mean grad rate × Relative ability 0.410∗∗∗

(10.85)

Sticker quintile=4 × Mean grad rate × Relative ability 0.420∗∗∗

(11.40)

Sticker quintile=5 × Mean grad rate × Relative ability 0.457∗∗∗

(12.74)

Sticker quintile=1 × Mean log college prem. × Relative ability 0.878∗∗∗

(6.70)

Sticker quintile=2 × Mean log college prem. × Relative ability 0.986∗∗∗

(7.93)

Sticker quintile=3 × Mean log college prem. × Relative ability 0.904∗∗∗

(8.31)

Sticker quintile=4 × Mean log college prem. × Relative ability 0.919∗∗∗

(8.49)

Sticker quintile=5 × Mean log college prem. × Relative ability 0.925∗∗∗

(11.62)

Constant 0.456∗∗∗ 0.0415(17.49) (0.96)

Observations 2267 2109R2 0.131 0.062

t statistics in parentheses

Sample is youth who enrolled and had a non-missing sticker tuition estimate.∗ p < .1, ∗∗ p < .05, ∗∗∗ p < .01

Table 18: Graduation rates and college premia as a function of relative ability by sticker tuitionquintile

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(1)Household income in 1996 (real)

modelAbility 60.36∗∗∗

(23.19)

Constant 40.11∗∗∗

(26.44)

sigmaConstant 47.98∗∗∗

(87.95)

Observations 4102

t statistics in parentheses∗ p < 0.05, ∗∗ p < 0.01, ∗∗∗ p < 0.001

Table 19: Censored regression of household income on ability

(1) (2) (3)Enrolled Graduated Took out a loan

Ability 0.837∗∗∗ 0.405∗∗∗ 0.165∗∗∗

(35.32) (9.34) (3.44)

Household income in 1996 (real) 0.00124∗∗∗ 0.000717∗∗∗ -0.00142∗∗∗

(10.99) (4.84) (-8.67)

Constant -0.0548∗∗∗ 0.367∗∗∗ 0.607∗∗∗

(-3.94) (11.69) (17.48)

Observations 4102 1888 1867

t statistics in parentheses

Note: (2) and (3) are conditional on enrollment.∗ p < 0.05, ∗∗ p < 0.01, ∗∗∗ p < 0.001

Table 20: How select variables vary in initial conditions

Table 21: CorrelationsVariables Enrollment Parental income (real) Ability (ASVAB pctile)

Enrollment 1.000Parental income (real) 0.299 1.000Ability (ASVAB pctile) 0.475 0.310 1.000

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Net Tuition

% = -64% = -31

% = -63% = -43

% = -30% = -47

% = -44

-120 -100 -80 -60 -40 -20 0 20 40 60 80

GRNGRSGTNPRNPRSPTNPTS

Expenditures

% = -26% = -35

% = -24% = -24

% = -37% = -19

% = -37

-50 -40 -30 -20 -10 0 10 20

GRNGRSGTNPRNPRSPTNPTS

Enrollment

% = -25% = -32

% = -24% = -24

% = -36% = -21

% = -39

-50 -40 -30 -20 -10 0 10 20

GRNGRSGTNPRNPRSPTNPTS

Relative ability

% = 0% = 0

% = 1% = 6

% = 1% = 4

% = 5

-10 0 10 20 30

GRNGRSGTNPRNPRSPTNPTS

Figure 9: Percent change relative to 2010 from subtracting one force, else equal

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Table 22: Correlations conditional on enrollmentVariables Graduation Parental income (real) Ability (ASVAB pctile)

Graduation 1.000Parental income (real) 0.118 1.000Ability (ASVAB pctile) 0.211 0.187 1.000

2. For each θ ∈ Θ, compute T using according to the college’s FOCs.

3. Given tuition in each submarket, solve the youth problem and compute the application rates

Ak(sY ) and enrollment rates P k(sY ).

4. The equilibrium condition on vacancies then requires that vk(mk(sY )) = θ/η(θ)f(sY )Ak(sY )/g(k)

for all sY . For m 6= mk(sY ), the market is not active in equilibrium and so vk(m) = 0. Note

that η(θ)/θ = ρ(θ), so vk(mk(sY )) = f(sY )Ak(sY )/(ρ(θ)g(k))

5. Using the implied vk(m) obtained for all k,m, compute an update for Nk, then Xk, and

then—from the budget constraint—Ik for each k.

6. Define the residual function as F (x) = [F1(x), F2(x), F3(x)] where F1(x) := [Xk − Xk],

F2(x) := [Nk −Nk], and F3(x) := [Ik − Ik].

The goal is to find an x such that F (x) = 0, which gives a college equilibrium. We do this with a

Newton-type approach.

C.2 The student and worker problems

The log college/high school premium increased from roughly .45 to .66 over 1987 to 2010. Because

of drop-outs, the population will have anything from 0 to .66 coming from the premium. However,

a rough estimate of college-type premiums (specifically taking mean earnings at particular school

types relative to mean earnings of all schools) varies from .83 to 1.63. The variance of the innovation

to permanent shocks estimated by Heathcote et al. (RED, 2010) has been roughly stable over this

time period at around 0.01. Additionally, we assume individuals work for 48 years. Normalizing a

high-school’s level to ν, a ±2 standard deviation cover for for the shock plus college premium is

[ν − 2σ√j, ν + 2σ

√j + .66]. Taking that 1/3 of the the population will be college grads, 1/3 drop

outs, and 1/3 pure high school, we can expect that the mean of log earnings by workers will be

approximately ν1/3 + (ν+ .3)1/3 + (ν+ .6/3) = ν+ .3. In Gordon and Hedlund (2016), ν = −1.25.

ez2+µj+ν/ez1+µj+ν = ez2−z1 .

To generate the relative premium of 1.63 in the data, need log(1.63) ≈ .49 = z2 − z − 1. Similarly,

to generate the relative premium of .83, need −.18 = z2 − z1. So, [ν − 2σ√j, ν + 2σ

√j + .66 + .49]

should be the final support.

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