This question already has an answer here:
I don't understand the reasoning behind having to wait before putting a bounty on a question. It may help in a very rare case where I ask a question and while waiting for a response, I manage to find an answer myself (but I usually look for an answer myself before asking a question) and save myself the reputation. Perhaps there is also a worry that people with lots of reputation start throwing their weight in order to get quick responses, but honestly I think this is fine (note that this is coming from a user with not much reputation) because if someone's reputation is high/increasing their net input to the site is positive, they're going to be answering more questions than they are asking questions with a bounty. So a user putting bounties on a destructively large amount of questions is very unsustainable for that user.
The reason I ask this is that I am in a particularly frustrating situation now, where I have an exam tomorrow and cannot offer a bounty for a question that would help me for that exam.
As a disclaimer, I'm not trying to say that we should change the system, if it has worked in the past, I am just trying to understand why it is the way it is. However, perhaps an "early bounty fee" of $\approx$ 20% of the bounty could be applied to bounties set before the two day period rather than not allowing it to be posted.