As was said in George Stocker's answer on this question
In Podcasts #26 and #34, Joel and Jeff talked about this. They didn't want a bounty to interfere with the 'normal' method of asking/answering questions. If you're allowed to offer it right away, then it has the potential to create an economy where people only focus on questions with bounties.
I think this is a perfectly valid concern. Without bounties in the first 48 hours, we have a StackOverflow economy that is based on providing timely, quality, relevant answers to questions. That is, your are gaining reputation from your peers. The community will reward your answers for their quality.
If we allow bounties in the first 48 hours, we have created a cash economy, where people will only answer the questions if the reward is rich enough. Answers with more than 5 upvotes are not that common, so the minimum bounty is higher than the rep-reward you'd get for answering a normal question.
We would also be stating that the people who have higher rep have more worthwhile questions because they can offer higher bounties on them, or at the very least, low bounties on all their questions. The goal of StackOverflow is to judge technical questions based on their technical merits, all questions should be treated equally. It shouldn't matter that the user has a high enough rep to pile his question to the top of the list.
There is no urgent tag
It has been said before. StackOverflow is not for people to come with time-sensitive questions. Most questions get answered very quickly, but it is not a guarantee and it shouldn't be relied on as such. Allowing setting a bounty in the immediate moments (or even a few hours) after posting the question is implicitly saying that this is okay behaviour. "If your question is urgent, just post a bounty on it! It will get bubbled higher than the questions of all the plebs beneath you." This is not what the bounty system is for.
The Purpose Of bounty
Bounties were created as a method to attract attention to questions which were missed on the initial run-through. The StackOverflow community did not answer the question initially, and the OP (hopefully) went through the correct steps.
In the words of the FAQ
What if I don't get a good answer?
In order to get good answers, you have to put some effort into your question. Edit your question to provide status and progress updates. Document your own continued efforts to answer your question. This will naturally bump your question and get more people interested in it.
If, despite your best efforts, you feel questions aren't getting good answers, you can help by offering a bounty.
Notice that even the FAQ says that "despite your best efforts" that means if you've waited a few hours with no response, you can't just buy your way out of it. You should still be making an honest attempt to help the site by improving your question.
Bounties do not improve questions, they attract attention to questions which may have been ignored for a reason.
Bounties do not encourage equality, they say that possessing more reputation means that your questions are more worthy of being answered.
Bounties do not encourage quality, they encourage people focusing on questions with higher reward rather than the ones which are better written, and better for the community.