Or do they?
It would be a mistake for schools, especially elite schools, to allocate places entirely on the basis of price, for at least three reasons.
1. Part of what schools are selling is a credential, and part of that credential comes from having been admitted. An employer prefers, ceteris paribus, employees able enough to have gotten into Harvard or Chicago. He has no reason to prefer ones rich enough to have bought a place at one of those schools.
2. The value to students of attending a school depends in part on the school's reputation, which depends in part on the quality of students admitted in the past. By basing admissions on measures of applicant quality the school may be able to raise average student quality, thus raise the performance of its graduates, thus raise the value of the school to future applicants.
3. Students are both customers and inputs. Smart students prefer an environment with other smart students, and probably learn better in such an environment. Put differently, a smart student provides positive externalities to fellow students and thus, indirectly, to the school, a dumb student provides negative externalities.
All of these explain why schools give some weight to measures of student quality in deciding whom to admit. But none of them explain why they give no weight at all to willingness to pay. A student is worth more to the school the more able he is, but not infinitely more. Even if student quality is the only thing schools care about, additional money could be used to offer more generous scholarships to able students who would otherwise go elsewhere, raising average quality. So one would expect schools to be willing to trade off, at some rate, money against SAT scores, agreeing to admit somewhat less qualified applicants at somewhat higher prices.
So far as I can tell, they do not do so. The reason might be internal ideology—elite schools for the most part are rich nonprofits, in a position to sacrifice financial benefits in order to act in ways that those running them approve of. It might be other people's ideology—schools may fear that the policy I have suggested would be seen as a corrupt favoring of the undeserving rich over the deserving poor. While everyone recognizes that wealth confers advantages on those who have it, many people find that fact objectionable, at least if the advantages are in things they think important, such as health care or education.
Both of those are, I think, plausible explanations, but not very interesting ones. Can anyone suggest a better alternative?
While discussing admission policies, it is worth also thinking about another puzzle: legacy admissions. While schools do not preferentially admit those who are willing to pay more—many claim, I suspect truthfully, that they do not even preferentially admit those able to pay full tuition over those who can only come if given large amounts of financial aid—many do preferentially admit the children of their alumni.
One possible explanation connects to the first part of this post. Legacy admissions can be seen as a covert and imperfect way of doing what I have just argued that schools do not do. Applicants are instructed to list on their applications any alumni among their close relatives. Alumni offices keep track of alumni donations; that information can be provided to admissions officers.
Are there other reasons for legacy admissions? One possibility is that the school thinks of itself as having a particular culture, being intended for a particular sort of people. Its alumni, having been not only selected to fit into that culture but instructed for four years in it, are particularly likely to be that sort of people, making their children more likely to fit in.
Another possibility is tribalism. Humans tend to divide the social world into ingroup and outgroup, us and them. One basis for such a division is what school one went to, a fact dramatically demonstrated at college football games. The people running a school and its almuni are part of the same ingroup, admission can be seen as a benefit given to those admitted, and people naturally prefer to allocate benefits to us instead of to them. They also prefer to include in the ingroup those most likely to be loyal to it. If Harvard admits the son of a Yale graduate, can he be trusted to cheer for the right team?