Consensus on meta.stackexchange.com seems to indicate that yes, you can use bounties to draw attention to your own answers.
There was concern about this being used as a way to game the system, by gambling reputation in the hopes of earning enough upvotes on a late answer to make a net gain, but this has been addressed by setting increasing minimum bounties
As noted by Robert, this has come up recently. Posting a bounty is a perfectly reasonably thing, a good thing (with badges too!). However, when applied repeatedly to a question it does start to represent abuse.
To help minimise this, but while leaving open the option of offering a subsequent bounty when you really, really are looking for another option we will be implementing a change here, where subsequent bounties have a higher minimum offering. This means you can't place 6 "cheap" bounties of 50 rep, to keep your answer on the bounty board to get lots of pass-by upvotes, as subsequent bounties probably1 will not cover your "spend".
Another related change has been mentioned by Sam Saffron:
- If you are placing a bounty on a question you answered, your minimum spend is 100