The Market for Students

I: The Problem

Colleges (and law schools and …) compete for students, with the objective of raising the quality of the student body; one way in which they do so is by offering scholarships to the students they want. Here quality represents whatever characteristics of a student are known at the point when offers are made and are of value to the college. That might include ability as measured by SAT scores, highschool grades, and the like. It might include characteristics, such as success in sports or high parental income, that are predictors of future income and so, possibly, future donations to the college. It might include characteristics, such as race or low parent income, that the college wishes for reasons of ideology or public relations.

II. The Model

Assumptions:

100 colleges, in most respects identical. Each has a budget B available for scholarships.

Each college wishes to enroll 1000 students.

There are more than 100,000 applicants.

Students come in a range of qualities; there are n_i students of quality q_i. College j offers scholarship s_ij to students of quality q_i.

For simplicity, I assume that n_i is large enough so that a college with a probability p_i of getting each student of quality q_i actually gets p_iq_i of them. I also assume that the q’s are close enough together so that the highest quality student for whom no college bids is just marginally worse than the lowest quality student for whom some college bids.

Each student has random preferences over the colleges, all drawn from the same distribution. The probability that a student’s preference for a particular college is no more than x is represented by the cumulative distribution R(x); the corresponding probability density is r(x). A student of quality q_i calculates, for each college j, the sum of s_i plus that student’s preference for that college and accepts the offer of the college for which the sum is highest.

The objective of each college is to maximize the total quality of admitted students.

III: Analysis

I am looking for a symmetrical equilibrium, one in which all colleges make the same pattern of offers:s_i=s_ij=s_ik "i,j,k. There may also exist asymmetrical equilibria. I will treat variables as discrete or continuous, whichever is more convenient for analysis and exposition; the continuous case can be thought of as the limit of the discrete case for large n_i.

Consider the situation from the standpoint of college j deciding what offers to make. If the existing pattern of offers is an equilibrium, there is no way in which that college can alter its pattern, all other colleges’ offers held constant, that will increase total quality while obeying two constraints: total scholarship expenditure =B, total number of students admitted = 1000.

Consider, from the standpoint of that college, its choice of s_ij for some particular quality level i. Currently all other colleges are making the same offer it is and each of them is getting one student out of a hundred of that quality. If j raises s_ij a little, one more student of quality q_i will find that s_ij plus preference is higher than for any other college and accept.

College j is in the position of a monopsonist facing a supply curve S_i(s_ij). The cost to it of one more quality q_i student is the scholarship it will offer that student plus the increase in what it will be giving to the students who would have accepted the lower offer times the number of such students. Taking it to the familiar continuous case, MC_i, the marginal cost of a student of quality q_i, is s_ij+ S_i(s_ij)/(dS_i/ds_ij)

Where does the supply curve come from? The number of students of quality q_i who accept an offer from j is the number for whom s_i, the offer they are getting from all other colleges, plus the highest preference they draw for any other college, is less than s_ij, the offer they get from college j, plus their preference for college j. Put formally, the probability that the highest preference a student has for any of 99 colleges is no more than x is R(x)⁹⁹, the probability density for the preference for one college is r(x), so the integral of r(x-a) R(x)⁹⁹ gives us the probability that a student of quality q_i will accept a college that offers him s_i+a. The supply curve for that quality is then n_i, the number of students of that quality, times that probability.

Since the probability does not depend on i, the supply curve S_i(s_ij) will be the same for all i, save for the scaling factor n_i. In the expression S_i(s_ij)/(dS_i/ds_ij) the factor n_i will be on both top and bottom, so cancels, so S_i(s_ij)/(dS_i/ds_ij) is the same for all i. The situation is symmetrical, so it is also the same for all j. Call it K. Hence the marginal cost of a student of quality qi is si+K. This is the marginal cost at s_ij=s_i, which one can think of as the cost to a college of getting one more student by raising its scholarship offer for that quality an infinitesimal amount above what all the other colleges are offering.

The college could try to increase its total quality of students by admitting one more student of high quality h and one fewer of low quality l, thus maintaining the constraint on total number of students. Doing so would break the budget constraint, increasing total expenditure on scholarships by s_h+K-(s_l+K)=s_h-s_l. To restore the budget constraint while continuing to obey the enrollment constraint it could admit b more students of low quality l' and b fewer students of high quality h', where b(s_h'-s_l')= s_h-s_l. If there is any such pattern of changes which increases total quality, then the initial situation is not an equilibrium.

The change described increases total quality by q_h-q_l-b(q_h'-q_l')= q_h-q_l- (s_h-s_l)/(s_h'-s_l') (q_h'-q_l'). This will be 0 for any h,l,h',l' if, for all i, s_i=c+dq_i. Put verbally, each student is offered a constant c plus an amount proportional to his quality. Total scholarship expenditure B is thus 1000c+Qd, where Q is total quality. The first term is independent of the pattern of scholarship offers, hence any change in the enrollment pattern which maintains the budget constraint B also leads to the same total quality.

All of this was done in the context of infinitesimal changes, since we were considering only the derivative of the supply curve at s_i. To discover both whether the equilibrium is stable and whether there might be higher quality outcomes further away from the offers being made by all other colleges, we would need to more precisely specify the form of r(x) and hence the shape of the supply curve. This, however, is only a first draft, so I am not doing any of that; interested readers are invited to fill in the holes for themselves.

With those qualifications, I have now shown that there exists a symmetrical equilibrium. It remains only to determine the values of c and d that make that equilibrium satisfy our constraints. We get one equation from the budget constraint, which tells us that"

B=1000c+Qd

What about the other?

Our units and zero of quality are arbitrary, since all the college is doing is finding the maximum. So without loss of generality, we can redefine quality q such that q=0 falls between the lowest quality student that the colleges admit and the highest quality that they don't admit. The price to a college of admitting a student that no other college was bidding for would be zero. Hence by the same argument, and with the assumption that the lowest quality bid for and the highest quality not bid for are arbitrarily clse, 0=c+0d.

So, with our redefinition of quality, c=0, d=B/Q. But, by symmetry, Q for every college is simply the total quality of the top 100,000 students divided by 100.

IV: Including Parental Income

Colleges offer both need and non-need scholarships. This raises the question of why need, defined by parental income and assets, is relevant to what a college offers. There are at least three possible answers.

1. Parental need enters quality, because schools, for ideological, public relations, or other reasons, like to enroll poor students. This is possible but, for my purposes, uninteresting. And it is not all that convincing, given both that rich students are more likely to end up as rich alumni and that the pattern of need based scholarships is more common than one would expect from such reasons. In any case, it's no fun to explain a pattern by, in effect, assuming it.

2. A student won't go to a college if he has no way of paying for it. This explains why a college might offer loans, on campus jobs, and the like to students who would otherwise be kept out by their budget constraint. Insofar as these are subsidized, the amount of the subsidy is, in effect, a scholarship. This seems plausible, but it is not clear exactly how to fit it into the model.

Which leaves us with the third answer.

3. Scholarships are in money, student preferences over colleges are in utility. The poorer a student is, the higher his marginal utility of scholarship. Thus the supply of poor students is more price elastic than the supply of rich students, hence the term K in our earlier analysis is lower for poor students, hence a pattern of scholarships which equalizes the marginal cost to the college of rich and poor students of equal quality gives higher scholarships to the poor students.

To put this a little more formally, suppose that students are of two sorts, rich and poor, and the the marginal utility of money is twice as high for the poor as for the rich. More precisely, the poor student has a marginal utility of income of one utile/dollar, the rich student of half a utile per dollar. Both sorts draw their (utility) preference from the same distribution. So in deciding what offer to accept, the poor student compares preference plus scholarship, the rich student compares twice preference plus scholarship. Going through our previous analysis, we see that if the college discriminates in its offers between rich students and poor students, increasing its probability of getting a rich student by a given small amount requires twice as large an increase in the offer as increasing the probability of getting a poor student by the same amount. Of course, the numbers of rich and poor students of a given quality need not be the same. But, again following out our previous calculation, the number of students cancels out in calculating K, the part of the marginal cost of a student due to the cost of giving a higher scholarship to the students who would have accepted the lower. So we have K_r=2K_p, where the subscripts refer to rich and poor students.

Hence if the college offered the same scholarship to both rich and poor students of a given quality, the marginal cost to it of a poor student of that quality would be lower than of a rich student. So it could maintain quality and number of students while reducing expenditure by lowering its offer to rich students and increasing its offer to poor students, then use the additional money to increase quality by substituting higher quality students for lower quality students. Thus in equilibrium, and ignoring changes in elasticity as we move along the supply curve, the college will offer poor students of a given quality a scholarship K_p higher than the scholarship offered to rich students of that quality, thus making the marginal cost to the college of rich and poor students of a given quality the same.

We thus have an explanation of higher scholarships for poorer students which does not depend on any ad hoc assumptions about what goes into the colleges' definition of quality, merely on the assumption that money is more important relative to preferences among colleges for poorer students.

(Readers are invited to point out errors or suggest improvements. Telling me that I have not dotted all the i's or crossed all the t's, mathematically speaking, is not necessary, although readers who want to create a more rigorous version for themselves are welcome to do so, as are readers who wish to offer to coauthor a more fully worked out version. I believe in the division of labor.)