Stephanie Cellini and Claudia Goldin examine for-profit schools and compare tuition at those with and without students eligible for federal financial aid. Aid-eligible colleges charge, on average, 78% higher tuition than non-aid-eligible colleges, and the differences in some cases are "roughly equal to average student grant awards and our estimate of the loan subsidy."
Another study confirms that this is not isolated to for-profit colleges:
We find that each additional Pell Grant dollar to an institution leads to a roughly 55 cent increase in sticker price tuition. For subsidized loans, we find a somewhat larger passthrough effect of about 70 percent.
Thus, most of the subsidy is translated directly into higher tuition. But do these subsidies achieve their goal of increasing college enrollment? Grey Gordon and Aaron Hedlund attempt to estimate the answer:
The tuition response completely crowds out any additional enrollment that the financial aid expansion would otherwise induce, resulting instead in an enrollment decline.
Of course, this was all famously predicted in 1987 by then Secretary of Education William J. Bennett:
Increases in financial aid in recent years have enabled colleges and universities blithely to raise their tuitions, confident that Federal loan subsidies would help cushion the increase.