The debate between paying full tuition at an elite institution or accepting a merit scholarship from someplace less prestigious “is a conversation we have all the time,” said James Conroy, chairman of post-high-school counseling at New Trier Township High School in Winnetka, Ill., an affluent suburb in Chicago. “It’s a tough conversation because what it gets down to is the values of the family.”

But he said many parents did not realize that their children were going up against other children who were identical to them — at least on paper. “There are 100 schools that we talk about in this office day after day after day,” he said. “But those are the same schools that every New Trier across the country talks about.”

Prestige has always been part of the equation, but he said he had expected parents to start looking for value in colleges after the 2008 financial collapse. Instead, parents have come to see the elite universities as the only way to give their children a chance at success. They feel jobs are hard to come by and companies are only going to look to hire at the elite universities.

“Whether it’s true or not, I have no evidence,” he said. “But that was what was out on the bongo drums in the community.”

Ms. O’Shaughnessy knows this thinking well. The New Jersey father she described has many contemporaries willing to try to pay for something they could not afford. And there’s no guarantee, she said, that N.Y.U. will bring his daughter greater success.

“Frankly, I think that’s why East Coast schools that aren’t in the top tier but are in cities can get away with charging outrageous amounts of money and giving mediocre financial aid packages,” Ms. O’Shaughnessy said. “Students fall in love with these schools, and there are parents who are willing to sacrifice beyond all rational reasoning.”

But economists are not sure this trade-off is worth it. In two much-discussed studies about the value of a degree from an elite college — one with people who graduated in the 1970s and the other with more recent graduates — Alan B. Krueger, then an economist at Princeton University, and Stacy Berg Dale, a senior researcher at Mathematica Policy Research, found that equally smart students had about the same earnings whether or not they went to top-tier colleges. The big difference, their studies found, came from minority and low-income students who went to top-tier colleges: They did better later on.