### Tuesday, March 21, 2006

## Demand for selective colleges still strong.

Today, the NYTimes reports on the growing number of applications each college-bound high schooler submits to colleges:

What explains this startling increase in the number of applications? Let’s go to the micro theory.

Individuals are trying to maximize their discounted expected utility. They should continue to submit applications until the expected marginal benefit of an application equals the marginal cost. Three parameters drive the results: the expected return to attending a selective college, the probability of getting into any selective college, and the cost of applying to each college.

For simplicity, assume that there are 2 types of schools (selective and not-selective) and that assume each selective colleges provide the same excess return over the non-selective college (that is, u(S1) – u(NS) = u(S2) – u(NS) for all S). Further assume that everyone who applies gets into non-selective schools. Thus agents face a simple decision of whether or not and how many selective colleges to apply to.

In this simplified world, the decision rule (MB=MC) looks something like this: pick N such that dP/dN*(u(S)-u(NS)) = MC(N). Individuals should keep applying to schools as long as the change in the probability of getting into ANY selective college times the excess return over the non-selective college exceeds the marginal cost of the additional application.

Thus, if we observe an increase in N (the number of selective colleges applied to), it must be that the effect of an additional application on the probability of admittance has increased, the return to selective colleges has increased, or the cost of applying has fallen (or some combination).

Which of these do we think explains the growth in applications? I am not sure. I think that technology has certainly lowered the marginal cost of an application. Back in the stone ages when I applied to college, you had to actually use a typewriter to complete your applications. This substantially raised the time (and effort) costs associated with applications (at least for my mom). It is also possible that the returns to selective colleges are growing, but we will discuss this specific issue in class so I won’t discuss it here. I am most intrigued by changes in the effect of each application on the probability of getting in to any selective college.

Assume that there are a fixed number of spots in the set of selective colleges. Further assume (because I don’t want to try and explain it) that there is an exogenous increase in the demand for each spot. Schools receiving more applications changes dP/dN for a lot of students. Specifically, more applications means that P(N) diminishes at a much slower rate (smaller second derivative). More intuitively, think of a top student. Top students are confident that they will get in wherever they apply. Thus (in this simple model) they should only apply to one school (because they receive no marginal benefit from incurring the costs of filling out a second application because dP/dN = 0 for N>1). The closer one gets to the marginal student (the last person admitted) the more random shocks to the admissions process affect decisions. Thus in order to get one yes, the marginal student needs to ask lots of schools, and if more and more students are trying to get in, the marginal student needs to ask even more schools for admission. Further, given that the likelihood that most of the additional applications are from students who are, themselves, marginal (because top students always applied to selective schools), the average number of applications per applicant increases even faster.

Of course, you all probably applied to a million schools yourselves, but you are top students. Why did you all do this? Well, because you are not actually top students. Ok, you are, but you didn’t know that you were for certain. Further, you may have wanted more choices. Either for their own sake, or because, now that the government doesn’t let the Ivy’s collude with each other, you may get better financial aid offers by applying to (and being admitted to) multiple schools.

I am not sure what, if any, research has been done examining students’ choices to apply to colleges. I think it is an interesting topic and might make for a good term paper.

Oh and for the record, I only applied to 3 colleges when I was in high school (because I didn't think that u(S)-u(NS) was substantially greater than zero), but I applied to 7 graduate programs (because I worried that there was some chance that dP/dN was still greater than zero for the last schools I applied to).

An annual survey of college freshmen indicates that students bound for all kinds of institutions are filing more applications these days. In 1967, only 1.8 percent of freshman surveyed had applied to seven or more colleges, while in 2005, 17.4 percent had done so, according to the Cooperative Institutional Research Program at U.C.L.A., which conducts the survey. The survey began asking recently if the students had applied to 12 or more colleges; that proportion increased by 50 percent from 2001 to 2005.

What explains this startling increase in the number of applications? Let’s go to the micro theory.

Individuals are trying to maximize their discounted expected utility. They should continue to submit applications until the expected marginal benefit of an application equals the marginal cost. Three parameters drive the results: the expected return to attending a selective college, the probability of getting into any selective college, and the cost of applying to each college.

For simplicity, assume that there are 2 types of schools (selective and not-selective) and that assume each selective colleges provide the same excess return over the non-selective college (that is, u(S1) – u(NS) = u(S2) – u(NS) for all S). Further assume that everyone who applies gets into non-selective schools. Thus agents face a simple decision of whether or not and how many selective colleges to apply to.

In this simplified world, the decision rule (MB=MC) looks something like this: pick N such that dP/dN*(u(S)-u(NS)) = MC(N). Individuals should keep applying to schools as long as the change in the probability of getting into ANY selective college times the excess return over the non-selective college exceeds the marginal cost of the additional application.

Thus, if we observe an increase in N (the number of selective colleges applied to), it must be that the effect of an additional application on the probability of admittance has increased, the return to selective colleges has increased, or the cost of applying has fallen (or some combination).

Which of these do we think explains the growth in applications? I am not sure. I think that technology has certainly lowered the marginal cost of an application. Back in the stone ages when I applied to college, you had to actually use a typewriter to complete your applications. This substantially raised the time (and effort) costs associated with applications (at least for my mom). It is also possible that the returns to selective colleges are growing, but we will discuss this specific issue in class so I won’t discuss it here. I am most intrigued by changes in the effect of each application on the probability of getting in to any selective college.

Assume that there are a fixed number of spots in the set of selective colleges. Further assume (because I don’t want to try and explain it) that there is an exogenous increase in the demand for each spot. Schools receiving more applications changes dP/dN for a lot of students. Specifically, more applications means that P(N) diminishes at a much slower rate (smaller second derivative). More intuitively, think of a top student. Top students are confident that they will get in wherever they apply. Thus (in this simple model) they should only apply to one school (because they receive no marginal benefit from incurring the costs of filling out a second application because dP/dN = 0 for N>1). The closer one gets to the marginal student (the last person admitted) the more random shocks to the admissions process affect decisions. Thus in order to get one yes, the marginal student needs to ask lots of schools, and if more and more students are trying to get in, the marginal student needs to ask even more schools for admission. Further, given that the likelihood that most of the additional applications are from students who are, themselves, marginal (because top students always applied to selective schools), the average number of applications per applicant increases even faster.

Of course, you all probably applied to a million schools yourselves, but you are top students. Why did you all do this? Well, because you are not actually top students. Ok, you are, but you didn’t know that you were for certain. Further, you may have wanted more choices. Either for their own sake, or because, now that the government doesn’t let the Ivy’s collude with each other, you may get better financial aid offers by applying to (and being admitted to) multiple schools.

I am not sure what, if any, research has been done examining students’ choices to apply to colleges. I think it is an interesting topic and might make for a good term paper.

Oh and for the record, I only applied to 3 colleges when I was in high school (because I didn't think that u(S)-u(NS) was substantially greater than zero), but I applied to 7 graduate programs (because I worried that there was some chance that dP/dN was still greater than zero for the last schools I applied to).

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