The intuition is that, in the phaseout region, rising income is offset by the decreasing credit, blunting the positive correlation between income and attendance. Statistical analysts call this method regression-kink design, because it relies on a change in the relationship between two variables.

Mr. Bulman and Ms. Hoxby found the predicted relationship between adjusted gross income and the receipt of tax credits in the phaseout region: Credits drop as income rises. But they found no corresponding relationship between income and college attendance. College attendance rises unabated with adjusted gross income, with no change when the credits phase out.

Why no effect? One explanation is that the credits primarily go to middle- and upper-income families, whose decision on whether to send their children to college is unlikely to be affected by $2,500. Another, compatible explanation is that the credits are delivered too late to affect enrollment. Families get them after tuition is due; a family that pays tuition in September won’t get a tax credit until at least the following January. At that point the credit is a nice windfall, but has arrived too late to help pay the bursar.

The complexity of the tax benefits also most likely undermines their effect. I discuss this in greater detail in a recent paper I wrote with Judith Scott-Clayton, which provides a comprehensive overview of the tax benefits for education.

If the billions spent on the tax credits are to have any effect on college attendance, you would want them delivered when tuition bills are due. One proposal suggested by Ms. Hoxby and Mr. Bulman is to compute eligibility for the credits automatically, using income tax information when a dependent approaches college age. Families could then be notified of their eligibility. The authors also suggest that colleges file to receive the benefits directly from the Department of Education, so that a student need only present evidence of eligibility to have an account credited immediately.

An even more comprehensive approach would be to consolidate the tax credits with the Pell Grant, creating a single grant program that pays college costs at the time of enrollment. Eligibility could automatically be calculated using tax data, with money delivered by the Department of Education. Families could apply by checking off a box on their tax forms.

This approach would cut back substantially on paperwork, a relief for the millions of students who complete both the 1040 and the dreaded Free Application for Federal Student Aid in order to get federal grants, loans and tax credits for college. With a simplified application, and dollars delivered at the right time, the $30 billion now spent on tax credits could open college to many more students.