If the government gave a $1000 grant to all lower income students at private universities, would those universities just lower their private financial aid by $1000 and capture the whole grant?

This is essentially the same as my question here https://reddit.com/r/AskEconomics/comments/1mhzmwj/does_private_financial_aid_let_colleges_capture_a/ except that one didn't get many responses.

One one hand, subsidizing something like pizza by $1 wouldn't cause restaurants to raise pizza prices by $1; the benefit would be split between producers and consumers. On the other hand, universities are in a unique situation since they don't actually announce how much financial aid they are providing until after you have already written essays, paid the application fee, and got accepted, and any particular student might only be accepted to a couple universities, so there isn't nearly as much competition as there are between restaurants. Also, it seems like it might be easier for universities to collude than for restaurants (though I don't have any sources to back this up).

Edit: The way I'm thinking about this right now is that since you have to submit tax forms to the university, the university can pretty easily estimate your willingness to pay, and can use that information to do price discrimination. So in a hypothetical situation where the university was a monopoly (which they are somewhat close to being due to the way admissions works), the university would charge exactly your willingness to pay. Then the government grant would increase your willingness to pay by $1000, and thus your tuition would go up by $1000 as well. In this case, I would have the followup question of whether or not students would get any surplus at all from university in this hypothetical situation.