High tuition, financial aid, and cross-subsidization: do needy students really benefit?

Sorensen, Robert L.

Our policy is total Robin Hood - we put our tuition up as high as possible and then put most of the extra money into financial aid. [Eamon M. Kelly, president of Tulane University, 19871.](1)

I. Introduction

One way to make a college education widely accessible is to charge low tuition. Yet many college administrators argue that a policy of charging high tuition while generously awarding financial aid can even further reduce the net price paid by needy students.(2) Since there is little doubt that this implicit form of cross-subsidization has benefited some students, this practice has been widely accepted by the public. To date, however, no one has investigated the actual impact of cross-subsidization on the net price paid by needy students.(3)

The fact that this question has not been addressed is surprising since this form of cross-subsidization is observationally equivalent to ordinary price discrimination. Indeed, high tuition coupled with generous financial aid awards may simply reflect a policy of maximizing tuition revenues to pursue other objectives. How, then, can we be sure that institutions that claim they charge high tuition to subsidize needy students are in fact reducing their net price?

In this paper we investigate whether the net price paid by the average needy student is negatively related to the degree in which institutions appear to inflate their tuition to engage in cross-subsidization. First we show that cross-subsidization can, under fairly general conditions, reduce the average net price paid by all needy students. Next we identify a necessary condition for concluding that cross-subsidization results in the average needy student paying a lower net price. We then test whether this condition is satisfied by examining the cross-sectional relationship between tuition and financial aid awards for 502 private institutions of higher learning. Finally, we compare some selected budget items of institutions that charge relatively high tuition to those that charge relatively low tuition to gain insight into what kinds of institutional objectives compete with the goal of maximizing accessibility.(4)

II. The Model

Cross-Subsidization and Net Price for Needy Students

Can institutions that charge inflated tuition actually reduce the net price paid by all needy students through cross-subsidization? This question can be addressed by way of an example. Let n be the total number of students attending a particular institution and let m be that institution's number of needy students. Now consider a tuition increase of one dollar that leaves both n and m unchanged.(5) All students must now pay an additional dollar to attend. Now suppose that each of the m needy students receives an additional dollar of financial aid to fully offset the tuition increase. In this case the net price paid by each needy student does not change but (n - m) dollars of additional tuition revenue is left over to be spent at the institution's discretion. If only one of these dollars is put into additional financial aid and divided among the needy students, then the tuition increase will actually reduce the net price paid by all needy students. Indeed, in this example average financial aid awards among needy students increased by the amount of the tuition increase ($1) plus 1/m even though the institution gains (n - m - 1) dollars of additional tuition revenue.

Tuition, Institutional Financial Aid Awards, and Net Price

In the example presented above we showed that institutions that can inflate their tuition can expropriate additional revenues from wealthy students to reduce the net price paid by all of their needy students. We now identify a necessary condition for concluding that such policies generally reduce the average net price paid by needy students.

The financial aid award for any given student is determined by both the cost of attending a given institution and the student's ability to pay. We can characterize the relationship between tuition, ability to pay, and average financial aid awards with the following expression:

[Mathematical Expression Omitted]

where A is an institution's average financial aid award among all n full-time undergraduates, T is tuition, [P.sub.i] is the ith financial aid recipient's ability to pay, m is the number of needy students receiving institutional financial aid, and 0 [is less than or equal to] [THETA] [is less than or equal to] 1 is the average percentage of each financial aid recipient's need gap (T - [P.sub.i]) that is closed by institutional financial aid awards.(6) The student's ability to pay, [P.sub.i], is the institution's estimate of what the student can afford to spend on a college education. We assume that all institutions are in agreement as to the value of [P.sub.i] for any given student.(7)

Let [T.sup.e] be the expected value of an institution's tuition given its observable characteristics. In other words, [T.sup.e] = E[T/X], where X is a vector of institutional characteristics that describe an institution. Now suppose an institution considers adopting tuition T > [T.sup.e] to engage in cross-subsidization. In such a case, how much larger must the average financial aid award be for the average net price paid by needy students to be reduced? A policy of charging T > [T.sup.e] implies that for every student (including the m needy ones) the cost of attending is now [DELTA T] = T - [T.sup.e] greater than when T = [T.sup.e]. Of course, if each of the m needy students receives less than [DELTA T] in additional financial aid, then it follows that the average net price paid by the m needy students will be greater than when T = [T.sup.e].

Since [DELTA T] = T - [T.sup.e], equation (1) can now be rewritten as:

[Mathematical Expression Omitted]

which is equivalent to:

[Mathematical Expression Omitted]

where [THETA] = [THETA.sub.1] = [THETA.sub.2] = [THETA.sub.3]. An implicit assumption of this chain of equalities is that, for financial aid awards, there is no behavioral difference between tuition dollars up to [T.sup.e] and tuition dollars above [T.sup.e]. This, however, is antithetical to a policy of cross-subsidization since with such a policy the purpose of setting T > [T.sup.e] is to award financial aid to needy students at an accelerated rate, which is equivalent to observing that [THETA.sub.2] exceeds [THETA.sub.1].

From the discussion above it should be clear that if the objective of charging T > [T.sup.e] is to make a college education more affordable for needy students, then financial aid awards for all needy students must be made at least [DELTA T] greater. Unfortunately, a direct test of this requires student-by-student data which is unavailable. This example does, however, have a straightforward implication for the institution's average financial aid awards among all full-time undergraduates. Specifically, it implies that a necessary condition for concluding that T exceeds [T.sup.e] to reduce the average net price paid by needy students is that average financial aid awards must be more than (m/n) [DELTA T] larger than if T = [T.sup.e]. To see this, note that this expression is simply the sum of the increased burden placed on the m needy students averaged over all n full-time undergraduates. Note that this is also equivalent to requiring that A/ [DELTA T] > (m/n) if [DELTA T] > 0. In the context of equation (3), this amounts to restricting [THETA.sub. 2] to be greater than one. Thus, if [DELTA T] > 0 and we discover that [THETA.sub.2] is less than one, we must conclude that the average needy student actually pays a higher net price than if tuition were equal to [T.sup.e]. We now turn to the task of estimating [THETA.sub.2].

III. Empirical Analysis

Methodology

In this section we develop a methodology for estimating equation (3). We first discuss the formulation of proxies for [T.sup.e] and [DELTA T]. We then alter equation (3) so [THETA.sub.2] can be estimated conditional on [DELTA T] > 0. Finally, we identify proxies for (m/n) and (1/n) [SIGMA.sub.1.sup.m] = 1 [P.sub.i.]

As noted above, [T.sup.e] = E[T/X] where X is a vector of institutional characteristics that describe an institution. If X is non-stochastic, then we can obtain estimates of [T.sup.e] by simply regressing T against the members of X.(8) We therefore approximate [T.sup.e] by estimating an OLS equation that models tuition as a function of institutional quality, scope and mission while adjusting for the presence of other sources of income. The tuition level predicted by the tuition equation for institution j ([T.sub.j]) then serves as our proxy for [T.sup.e] while the unpredicted component of tuition ([DELTA T.sub.j] = [T.sub.j] - [T.sub.j]) serves as our proxy for [DELTA T]. Below we now discuss the relevant members of X.

Although no single variable can fully reflect institutional quality, a common approach in previous research is to employ some measure of selectivity such as average entrance exam scores [1; 13; 14]. The Barrons index of institutional selectivity divides institutions into six categories based on the average high school class rank and the median entrance exam scores (SAT or ACT) of entering freshman. In our tuition equation we therefore include a set of dummy variables that are based on the Barrons index [2]. This index classifies institutions into the following categories: most competitive (MOST), highly competitive (HIGHLY), very competitive (VERY), competitive (COMPET), less competitive, and non-competitive institutions of higher learning (the last two categories constitute the omitted group). A description of each category is contained in Appendix I.

To reflect institutional scope and mission we also include a set of dummy variables that classify each institution according to a classification scheme that has been developed by the Carnegie Foundation. A description of each Carnegie category is contained in Appendix II. The Carnegie Foundation classifies each school as being either a national university (UNIVERSITY), a national liberal arts college (LIBERAL), a large comprehensive university (LGCOMP), a small comprehensive university (SMCOMP), or a regional liberal arts college (which is the omitted group).

Some institutions can charge lower tuition because they enjoy a significant amount of non-tuition income. Similarly, some must charge high tuition to make-up for a lack of non-tuition revenue. To account for this we include endowment income and federal research grant overhead dollars per full-time enrollment (including graduate students) (OTHER) in our tuition equation.

Some institutions charge tuition that is equal to or below what is predicted by the tuition regression equation so that [DELTA T.sub.j] [is less than or equal to] 0. Since the testable implication derived in the previous section only applies to the case where [DELTA T.sub.j] is positive, we must now separate the [THETA.sub.2] (m/n)[DELTA T] term in equation (3) into two terms conditioned on the estimated sign of [DELTA T.sub.j]. To do this we replace the [THETA.sub.2(m/n)[DELTA T.sub.j] term in equation (3) with [Mathematical Expression Omitted], where [HIGH.sub.j] is a dummy variable equal to one if [DELTA T.sub.j] > 0 and [LOW.sub.j] = 1 - [HIGH.sub.j]. With this framework we can examine the empirical validity of our testable implication under the circumstances for which it applies (that is, when [DELTA T.sub.j] > 0) without altering the econometric properties of equation (3).

To estimate equation (3) a proxy for (1/n) [SIGMA.sub.1.sup.m = 1 [P.sub.1] is also needed. This term is equivalent to ability to pay among the set of needy students averaged over the set of all full-time undergraduates. Though this term is closely related to the average ability to pay across all full-time undergraduates, a suitable proxy for this term should only consider ability to pay among the set of needy students. Below we argue that average federal Pell grant awards can be used to devise an inverse proxy for (1/n) [SIGMA .sub.1.sup.m] = 1 [P.sub.1].

The U.S. Department of Education has established a detailed and uniform methodology for determining student financial well-being on the basis of student and parental income, net worth, number of siblings, and other factors. To receive a federal Pell grant award a student must first qualify by being deemed needy according to this uniform methodology. Consequently, the total Pell grant dollars reported for any given institution only pertains to the institution's m financially needy students. In addition, the size of any Pell grant varies directly with the individual student's financial well-being and, hence, varies negatively with any recipient's ability to pay. Therefore, we use total federal Pell grant award dollars divided by total full-time undergraduate enrollment (AVGPELL) as an inverse proxy for the ability to pay of an institution's needy students averaged over all full-time students.(9)

Finally, before we can estimate equation (3) we must also identify a proxy for (m/n). The proxy we use for (m/n) is the percentage of students that receive any form of financial aid (PERAID). Unfortunately, this variable may tend to overstate (m/n) since federal financial aid is also included in the PERAID measure. This problem is at least partially offset, however, by the fact that PERAID is based on both part-time and full-time students. This tends to overstate n, which tends to make PERAID understate (m/n).

We can now express equation (3) in its estimable form of: [Mathematical Expression Omitted]

where [AVGAID.sub.j] is institution j's average total grant and scholarship financial aid awards across all n full-time undergraduate students, [BETA.sub.0] is a constant term, and [BETA.sub.1], [BETA.sub.2], and [BETA.sub.3] correspond to [THETA.sub.1], [TETHA.sub.2.sup.H], and [THETA.sub.2.sup.L]. The [BETA.sub.4] coefficient is equivalent to [alpha THETA.sub.3] where [alpha] is an implicitly identified positive constant relating (1/[n.sub.j]) [SIGMA.sub.i.sup.m] = 1 [P.sub.ij] to the inverse proxy [AVGPELL.sub.j] [Mathematical Expression Omitted]. The [epsilon.sub.j] term is a zero mean disturbance term that captures idiosyncratic sources of variation in financial aid award decisions across institutions. Given the discussion above, all of the slope coefficients are expected to be positive.(10)

The Data

The data we use to estimate equation (4) are taken from the College Examination Board's American Survey of Colleges (ASC) and the Higher Education General Information Survey (HEGIS) for the academic year 1985-86. Institutions that do not grant baccalaureate degrees or were classified as being either proprietary or specialized in mission (e.g., Bible colleges, theological seminaries, etc.) were eliminated from the sample. The source, definition, and summary statistics for all variables are presented in Table 1.

[TABULAR DATA OMITTED]

It should be noted that public institutions of higher learning are not included in our sample. They are not included because observed deviations from predicted tuition will reflect differing levels of governmental support. Consequently, if we were to include them in our sample, we would introduce a significant unobservable random component to tuition which would reduce the efficiency of our coefficient estimates.

IV. Regression Results and Interpretation

To generate estimates of [T.sub.j] and [THETA.sub.j] for the estimation of equation (4) we now estimate the tuition equation using ordinary least squares. The result of this estimation is summarized in equation (5) below.

[Mathematical Expression Omitted]

Sample: 502 observations

[R.sup2]:.56

Note: the absolute value of t-statistics are in parentheses.

As might be expected, the more selective the institution, the higher the tuition. Interestingly, while national universities and national liberal arts colleges have the highest tuition, national liberal arts colleges are the more expensive of the two. This may reflect the fact that liberal arts colleges, which focus on teaching and feature small classes, generally have a higher faculty to student ratio than universities. While the sign of the coefficient on the OTHER variable is signed according to a priori expectations, it is insignificant. This may result from the fact that the institutions that possess the greatest amount of non-tuition income are also the most prestigious ones [11]. Not surprisingly, the most prestigious institutions also generally charge the highest tuition, which tends to cancel-out the subsidizing effect of non-tuition income.

Using the estimates of [T.sub.j] and [DELTA T.sub.j] derived from the estimations of the tuition equation, we can now estimate equation (4) using ordinary least squares. Two formulations of average aid awards are used in the estimation of equation (4). Specifically, AVGAID is average grant and scholarship aid while NAVGAID is limited to need-based grants and scholarships. The results of these estimations are reported in Table II. As can be seen from the table, all of the coefficient estimates conform to a priori expectations and nearly all were statistically significant. The most striking result is that the [BETA.sub.2] coefficient, and hence [THETA.sub.2], is always statistically significant and positive but is always clearly less than one. More importantly, the difference between [BETA.sub.1], [BETA.sub.2], and [BETA.sub.3] in either of the regressions is statistically insignificant, which implies that [THETA.sub.1] [is congruent with] [THETA.sub.2].(11) This means institutions that charge relatively high tuition are no more generous per tuition dollar than those that charge expected or relatively low tuition. Of course this is completely inconsistent with a policy of charging high tuition for the purpose of reducing the net price paid by needy students. Finally, for institutions that charge relatively low tuition (that is, institutions in which [DELTA T.sub.j] [is less than or equal to] 0), average financial aid awards are correspondingly lower.(12)

[TABULAR DATA OMITTED]

V. Other Institutional Objectives

If high tuition is not used to reduce the net price paid by the average needy student, then why do some institutions inflate their tuition above what would be expected given their programs and other sources of income? There is no doubt that some institutions charge high tuition to offset unusually high costs of operation. But this falls short of a full explanation since our methodology already takes into account the fact that some types of institutions are more expensive to run than others. Another explanation is that the behavior of institutions of higher learning is best understood when viewed in the broader context of nonprofit organizations. In particular, James [5] has argued that undergraduate instruction may function as a profit oriented enterprise that subsidizes other activities which yield utility directly to college administrators and members of the faculty.

To gain some insight into the institutional activities that compete with the objective of reducing the net price paid by needy students, we now examine some selected budget items of the institutions in our sample. In Table III we divide our sample into three categories: national universities, comprehensive universities, and liberal arts colleges. We further divide each of these categories into thirds: institutions that charge relatively high tuition ([DELTA T/T.sup.e] much larger than zero), institutions that charge relatively low tuition ([DELTA T/T.sup.e] much below zero), and those in between. The conditional means of the following can now be calculated: relative financial aid awards (AVGAID/T), administrative overhead expenditures per full-time student, instructional costs per full-time student, and the percentage of graduate students.(13)

[TABULAR DATA OMITTED]

In the table three regularities emerge within each category. First, the proportion of tuition that is discounted by financial aid awards (AVGAID/T) does not rise as we move from relatively low to relatively high tuition institutions. This is entirely consistent with our regression results. Second, administrative overhead expenditures per full-time student rises noticeably as we move from relatively low to relatively high tuition institutions. This is consistent with the view that colleges and universities are subject to the same kinds of incentive problems as other nonprofit organizations. It is also consistent with recent increases in administrative costs over the last decade - a period of rapid tuition increases as well [3]. Third, instructional costs per full-time student also rise as we move from relatively low to relatively high tuition institutions. High instructional costs may reflect the fact that the faculty are earning relatively high salaries or that they have relatively low teaching loads. It should also be noted that since instructional costs include internally funded research activity, high instructional costs may reflect a greater emphasis on research.

In the table it is also clear that, for national universities, both instructional costs and the percentage of graduate students rises sharply as we move from relatively low to relatively high tuition institutions. Since instructional costs include graduate instruction, this finding is consistent with James [5] and James and Neuberger [6] who note that "the tenured faculty at a university may make allocation decisions that maximize the time spent on research and graduate training, two activities they prefer."

VI. Concluding Remarks

In this paper we investigated whether the net price paid by the average needy student is negatively related to the degree in which institutions price discriminate to, presumably, engage in cross-subsidization. We find that while institutions that appear to inflate their tuition do make larger financial aid awards, their awards are not large enough to reduce the average net price paid by needy students.

We should emphasize that this finding does not mean that all needy students are made worse-off by such policies. Rather, it more likely suggests that to whatever extent such cross-subsidization does occur, it is practiced in a more selective fashion than modeled here. Perhaps institutions concentrate their efforts on reducing the net price paid by narrowly targeted groups of needy students. If institutions do target their financial aid, however, then the merit of cross-subsidization is difficult to assess. The targeted needy students are obviously made better-off and the additional revenues can be used to pursue any number of worthy institutional goals. On the other hand, the needy students that are not targeted are made worse-off and there is no guarantee that the additional revenues are spent in a manner that is consistent with the stated objectives of the institution.

In our examination of selected budget items we attempted to reveal what was actually being subsidized with high tuitions. The link between high tuition and administrative overhead suggests that college administrators are important beneficiaries of high tuition. The link between high tuition and instructional costs suggests that faculty members are also important beneficiaries. Finally, the link between high tuition and the percentage of graduate students suggests that graduate programs and graduate students also benefit.

Appendix I. Barrons Measures of College Selectivity

 Most Competitive To be classified in this category a school must require ap
plicants to have a
 (MOST) high school class rank in the top 10% to 20%. Median SAT t
est scores of
 entering freshmen are at least 1250.
 Highly Competitive To be classified in this category a school must require a
pplicants to have a
 (HIGHLY) high school class rank in the top 20% to 35%. Median SAT
test scores of
 entering freshmen are 1150 to 1250.
 Very Competitive To be classified in this category a school must require a
pplicants to have a
 (VERY) high school class rank in the top 35% to 50%. Median SAT
test scores of
 entering freshmen are 1050 to 1150.
 Competitive To be classified in this category a school must require a
pplicants to have a
 (COMPET) high school class rank in the top 50% to 60%. Median SAT
test scores of
 entering freshmen are 900 to 1050.
 Less Competitive To be classified in this category a school must require a
pplicants to have
 (omitted group) a high school class rank in the top 65%. Median SAT test
scores of entering
 freshmen are below 900. Typically more than 85% of applic
ants are
 admitted.
 Non-competitive Schools in this category generally only require evidence
of high school
 (omitted group) graduation.

Appendix II. Carnegie Foundation's Institutional Mission Classifications

 National Universities Schools in this category offer a full range of undergr
aduate programs and
 (UNIVERSITY) grant the greatest number of doctoral degrees. Schools
 in this category
 must
 receive at least $12.5 million annually in federal res
earch support.
 National Liberal Arts Schools in this category are selective and attract stu
dents throughout the
 U.S.
 (LIBERAL) and award more than half of their degrees in the liber
al arts.
 Large Comprehensive Schools in this category enroll more than 2,500 studen
ts and award more
 (LCOMP) than half of their degrees in occupational or professi
onal disciplines
 such as
 engineering and business. Many schools in this categor
y also offer
 master's degrees.
 Small Comprehensive Schools in this category are simply smaller versions o
f those in the large
 (SMCOMP) comprehensive category. They enroll less than 2,500 st
udents.
 Regional Liberal Arts Schools in this category are typically less selective
than their national
 counter-part
 (omitted group) and award more than half of their degrees in the arts
and sciences.

(1.) This quote was taken from Brimelow [3]. (2.) McPherson, Schapiro and Winston [11] have noted that for many private institutions tuition increases since the late 1970s have been associated with increases in institutional financial aid awards for needy students. For a discussion of this issue as it pertains to public institutions see Hearn and Longanecker [4]. (3.) It should be noted that our use of the term "cross-subsidization" refers to a policy aimed at having one group of students subsidize another. This should not be confused with James's [5] use of the term which refers to a policy of having one activity (such as undergraduate instruction) subsidize another (such as graduate programs) in an institution of higher learning. (4.) In this paper we define relatively high or inflated tuition as tuition that exceeds what would be expected given a profile of characteristics that meaningfully describe an institution. In section III we develop a methodology for identifying tuitions that are relatively high or inflated. (5.) Of course, some institutions do not possess sufficient excess demand to increase tuition without reducing enrollment. These institutions, however, are probably incapable of engaging in any significant degree of cross-subsidization and are therefore irrelevant to the analysis that follows. Regarding m, because of student self-selection induced by a form of "sticker shock" an increase in tuition is as likely to reduce m as to increase it [9; 12]. In the next section it will become clear that it is the ratio m/n that matters in the empirical analysis, not the absolute size of n or m. Since we cannot say with confidence how a tuition change will affect m (if at all), we cannot predict, whether a tuition increase will generally increase or decrease m/n. (6.) Institutional financial aid includes all scholarship and grant forms of aid awarded by an institution (this does not include federal student financial aid). Since we are concerned with how aid affects net price, this measure does not include aid in the form of loans or work-study. We define n as full-time students because little if any institutional financial aid is awarded to part-time students [8]. (7.) This assumption is based on the fact that virtually all institutions of higher learning determine a student's expected family contribution (EFC) according to formulas provided by the College Board and the American College Testing Program. Since 1975 these formulas have been required to arrive at the same EFC for any given student, thereby establishing a uniform methodology for determining any student's ability to pay. (8.) See Manski [10, 35-6] for an excellent discussion of this point. (9.) Not all students that receive Pell grants also receive institutional financial aid, and vice versa. This means that the number of needy students as we define it (m) is not necessarily equal to the number of students receiving Pell grants. In the context of a cross-sectional regression, however, it is only essential that these two variables covary across institutions in the sample - it is not essential that they be equal for each institution. (10.) It should be noted that our proxy for [DELTA T] is actually the sum of the theoretical value of [DELTA T] and the estimated value of the error term in the tuition equation. As long as the error term in the tuition equation is independent of the error term in equation (4), however, the coefficient estimates will remain unbiased and consistent [7,298]. (11.) For both regressions the null hypothesis that [BETA.sub.1] = [BETA.sub.2] = [BETA.sub.3] cannot be rejected at the 5% on the basis of a standard F-test. The relevant F-statistics are F(2,497) = 1.86 and F(2,497) = 0.28. The critical value for the 5% level of significance for both tests is 3.00. (12.) Note that since [DELTA T.sub.j] is negative for such institutions the positive coefficient implies a negative relationship between [DELTA T.sub.j] and [A.sub.j]. (13.) Administrative overhead expenditures per student is calculated by dividing total expenditures on institutional support (as reported to HEGIS) by the number of full-time students (including graduate students). Total institutional support includes general administrative services, executive direction and planning, legal and fiscal operations, and community relations. Note that instructional costs per student includes the cost of graduate instruction as well as undergraduate instruction.

References

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