Beats and Backing Tracks.Traditionally, the people who create the lyrics, melody, and chord changes, are the ones who claim the song copyright. In the hip-hop world, where the song is occasionally stripped to percussion and voice (with little melody), songwriting credits can be more difficult to identify. Generally, if "the beats" refer primarily to arranged drum parts (with maybe a bass riff or vocal sample), and you did the rest, then you are the songwriter. (We talk about the unfairness of this approach for drummers in this blog entry.)
If the beats consist of more -- perhaps a bed of chord changes, with, drums, bass samples, guitar parts or other melodic instruments -- then it gets more complicated and the person you hired may be able to argue they are a co-owner.
Paying the $200. According to the Dear Rich Staff the $200 payment for the beats is not going to make them a work made for hire for a few reasons we don't need to bore you with. The easiest way to assure ownership would be to get something signed and in writing that says that the beatmaker "assigns all copyright" in his work to you.
What copyright? Just so we're clear, there are two copyrights involved in a recording: the copyright for the song (generally considered the most valuable) and the copyright in the sound recording (which protects the recorded sounds on the recording). So far, our discussion has been about the song copyright. The beatmaker would have a claim of co-ownership to the sound recording copyright since he contributed his tracks. Typically, the record company or artist gets an assignment from the beatmaker and any session musicians or producers involved in the product so that the whole copyright in the sound recording is owned by one person.