Suppose hypothetically a major state university were about to raise its fees significantly. What happens when low-income, well qualified students get in and decide they can’t afford it? Not good things.

The first problem that Mr. Bowen, Mr. McPherson and the book’s third author, Matthew Chingos, a doctoral candidate, diagnose is something they call under-matching. It refers to students who choose not to attend the best college they can get into. They instead go to a less selective one, perhaps one that’s closer to home or, given the torturous financial aid process, less expensive.

About half of low-income students with a high school grade-point average of at least 3.5 and an SAT score of at least 1,200 do not attend the best college they could have. Many don’t even apply. Some apply but don’t enroll. “I was really astonished by the degree to which presumptively well-qualified students from poor families under-matched,” Mr. Bowen told me.

They could have been admitted to Michigan’s Ann Arbor campus (graduation rate: 88 percent, according to College Results Online) or Michigan State (74 percent), but they went, say, to Eastern Michigan (39 percent) or Western Michigan (54 percent). If they graduate, it would be hard to get upset about their choice. But large numbers do not. You can see that in the chart with this column.

In effect, well-off students – many of whom will graduate no matter where they go – attend the colleges that do the best job of producing graduates. These are the places where many students live on campus (which raises graduation rates) and graduation is the norm. Meanwhile, lower-income students – even when they are better qualified – often go to colleges that excel in producing dropouts.

“It’s really a waste,” Mr. Bowen says, “and a big problem for the country.” As the authors point out, the only way to lift the college graduation rate significantly is to lift it among poor and working-class students. Instead, it appears to have fallen somewhat since the 1970s. …

Tellingly, net tuition has no impact on the graduation rates of high-income students. Yet it does affect low-income students. All else equal, they are less likely to make it through a more expensive state college than a less expensive one, the book shows.