There’s a provocative analogy to be made with how another industry does its pricing. I called Paul Norman, who owns a company that promotes high-end dance clubs in London, and he agrees that his clubs have much the same challenge as colleges and universities. Their appeal to new customers is based, in large part, on the mix of customers who are already there. The biggest spenders are wealthy men from Russia and the Middle East. But they won’t spend a lot of money in a club filled with people just like themselves. Women who have the right look — posh in Chelsea, a bit more flash in Mayfair — are admitted free and are offered free drinks, but only if they arrive early in the evening and happily mingle and dance. He said that clubs do their own version of enrollment shaping. ‘‘It’s good for the crowds if you have a mix of ethnicities,’’ Norman told me. On any given night, he said, about a quarter of the clubs’ guests get in free. It’s an odd model: giving your most valuable product away to some and charging a lot to others. But, Norman said, if everybody paid the same price, nobody would want to come, and in a few weeks clubs wouldn’t be able to charge anyone anything.

Similarly, if an elite school like Harvard or Princeton insisted on admitting only students willing to pay the full freight, they would soon find they weren’t so elite. Many of the best teachers would rather go elsewhere than stay in a gated, rich community. The most accomplished rich kids could be lured away to other schools by the prospect of studying with the best students and teachers. So, a school with the same high sticker price for everyone would be unlikely to have the attributes — high test scores, Nobel Prize-winning faculty, a lively culture — that draw national or international attention.

While blue-chip schools charge the most, their economic fundamentals suggest that they are, if anything, underpricing their product. According to the National Center for Education Statistics, the average bill at the top-tier schools is just over $50,000 a year. Yet these schools spend considerably more than $100,000 a year educating each student, with the difference made up from government funds, grants, alumni gifts and their often-huge endowments. Fewer than half of their students pay even that discounted amount. Most of the top-tier private schools say they won’t factor in an applicant’s ability to pay when considering admission and promise that everyone admitted will be given the financial aid necessary to attend. Generally, parents pay no more than 10 percent of their income. Students whose parents make less than $65,000 (or, at Stanford, $125,000) usually pay no tuition. And there are generous aid packages available to all but the very wealthiest students.

The pricing of college and university tuition used to be based on gut feelings, Crockett told me. Until around 1992, administrators would glance at what their peers were charging and come up with a number. Today, the process involves a level of mathematical and statistical rigor that few other industries could match. Crockett uses a team of statisticians and data analysts, the latest in software and data with hundreds of variables on students’ ability and willingness to pay, academic accomplishments, most likely choices of majors, ethnicity and gender, and other attributes. To the public, one number is released: the cost of tuition. But internally the school likely has dozens of price points, each set for a different group of potential students. The tools can determine how valuable a potential student is to the school’s overall reputation: more points for sports and scholarly accomplishment, fewer for the telltale signs of a likely dropout.

Because a majority of selective schools have adopted these sophisticated tools for shaping enrollment, there is more competition to attract the students who are most appealing: those with high G.P.A.s and test scores, high likelihood of graduating, the right sort of major — English majors, for example, effectively subsidize engineering students because engineering is more expensive to teach. Typically, fierce market competition leads to lower prices, but among elite schools, the opposite occurs, paradoxically. They often find that raising prices enables them to offer greater benefits to the most coveted potential students. (It also allows them to take part in the amenities race: nicer dorms, better food, a climbing wall: things that are regarded as essential to attracting those coveted students.)