For almost 20 years, I have been writing about economic diversity at selective colleges. Many of my articles have suggested that the colleges are not enrolling as many low- and middle-income students as they could.

Every so often, I hear from a professor or college administrator who pushes back, and the critique tends to go something like this:

Do you realize how many of the top-performing students on our campus are affluent? By the time students arrive here, they have been living in America’s highly unequal society for 18 years. We wish that weren’t the case, but it is, and there are real problems with pretending otherwise.

You’d see this if you looked at who did the best in our classes. Or who our top research assistants are. Those students often come from comfortable backgrounds. Without them, we wouldn’t be able to do the cutting-edge research we do.

This morning, a team of economists released a detailed study of elite college enrollment. It’s based on admissions records that several colleges made available as well as tax returns that tracked students after college. The findings likely apply to many elite colleges, including the Ivy League, Duke, Stanford, Swarthmore and Williams. And the implications are particularly relevant when many colleges are revamping admissions policies in response to the Supreme Court’s rejection of affirmative action.

The findings have also helped me understand both how my interlocutors have been right and how they have been wrong.

Today’s newsletter explains what I mean by that. If you want to learn more about the study, The Times has published a detailed article.

7 vs. 16

The new study, by Raj Chetty and David Deming of Harvard and John Friedman of Brown, demonstrates that the country’s most qualified high school students are indeed disproportionately affluent.

About 7 percent of the country’s very top students come from the top 1 percent of the income distribution. These students tend to have scored at least 1500 on the SAT (or 35 on the ACT), received top marks on Advanced Placement tests, earned almost all A’s in their high school classes, and often excelled in science fairs or other competitions.

Perhaps the most surprising pattern involves so-called legacy students, those who attend the same college that their parents did. At the elite colleges that the researchers studied, legacy students had stronger academic qualifications on average than nonlegacy students. Similarly, graduates of private high schools had stronger academic records on average than graduates of public high schools or Catholic schools.

These stellar academic backgrounds predict later success. Highly qualified affluent students tend to excel in college and afterward — which indicates that the professors and university officials who’ve reached out to me over the years have a point.

Yet they are also overlooking an important part of the story: Most of these colleges do not admit only the hyper-qualified affluent students; they also admit many other high-income students.

As I mentioned above, 7 percent of the country’s very best high school students come from the top 1 percent of the income distribution. But what proportion of students at elite colleges comes from the top 1 percent of the income distribution? Much more: 16 percent.

This combination of facts is a tricky one to grasp. Affluent students are overrepresented among the nation’s best high school students — but the colleges are nonetheless admitting a larger number of affluent students than if the decisions were based on academics alone. The biggest boost goes to the wealthiest students:

Private school polish

The results from Chetty, Deming and Friedman point to three main explanations:

• Legacy is a major advantage. These colleges are inundated with strong applications. When admissions offices are making close calls among students with similar transcripts, legacy status acts as a trump card. About half of legacy students at these colleges would not be there without the admissions boost they receive.

• A similar advantage applies to the graduates of private schools (not including religious schools). Schools like Andover, Brentwood and Dalton do such a good job of selling their students — through teacher recommendations, essay editing and other help — that colleges admit them more often than academic merit would dictate. Many college admissions officers think they can see through this polish, but they don’t.

• Recruited athletes are admitted with much lower academic standards — and are disproportionately affluent. It’s not just true of the obvious teams, like golf, squash, fencing and sailing. In today’s era of expensive youth sports, most teams skew wealthy. If colleges changed their approach to sports, they could admit more middle-class and poor athletes (or nonathletes) with stronger academic credentials.

The bottom line

Jason Furman, a Harvard economist and former Obama administration official who has seen the study’s results, has a helpful way of making sense of them. At some point, there really would be a trade-off between equity and excellence. But elite colleges aren’t anywhere near that point, Furman said. They are admitting many more affluent students than their qualifications justify.

Chetty put it this way: “The key point is that we don’t need to put a thumb on the scale in favor of the poor. We just need to take the thumb that we — perhaps inadvertently — have on the scale in favor of the rich.”

There are obviously still hard questions, like how selective colleges might make up the lost tuition from well-off students or the lost donations from alumni and sports fans. The good news, though, is that there are plenty of standout students from modest backgrounds who would benefit from attending these schools. Elite colleges can become more economically diverse without sacrificing academic preparation.

(A note of disclosure: I’m on the unpaid board of advisers of Opportunity Insights, the research group that published the paper.)

For more: The Times’s article and graphics explain much more, including these colleges’ outsize role in propelling their graduates into elite jobs.

Art find: Was the painting rolled up in the closet worth millions?

One breath at a time: A silent meditation retreat is a respite from a demanding job.

Metropolitan Diary: She went into a cafe with her sister and came out with an Alexander McQueen bag.

Lives Lived: Cheri Pies broke barriers with her 1985 book “Considering Parenthood: A Workbook for Lesbians,” a bible of the 1980s “gayby boom.” She died at 73.

WOMEN’S WORLD CUP

Italy — a team in a generational transition that started two teenagers — beat Argentina, 1-0.

Lise Klaveness, the president of Norway’s soccer federation, is the rare insider to publicly criticize the sport’s ethical failures.

France’s coach had just five weeks with his players before the tournament.

OTHER SPORTS NEWS

Golf history: Brian Harman won his first major with the fewest putts of any British Open champion in 20 years.

Pop the Champagne: Jonas Vingegaard rode away with his second straight Tour de France.

ARTS AND IDEAS

Sweeter in America: Real fruit ice cream — a scoop of vanilla blended with fruit in a machine that produces a buttery texture — has made its way from New Zealand to the U.S., Priya Krishna writes. It’s had an American makeover, sometimes mixed with graham crackers, drizzled with hot fudge or served atop a sprinkle-dipped cone.