phpBB is open source -- which means you don't have to purchase it, just download it.
Secondly, as Robert mentioned above, the platform isn't designed to give each member their own forum. However, there are ways to manipulate the system to do something like what you want it to -- but that might not be the most efficient way to accomplish what you're aiming do to.
Technically, a board administrator has to set up "groups" of users, and set a group leader. Regular users don't have the authority to do this. Furthermore, the board administrator would have to also create a new forum to act as the group's forum, and then set up the permissions of that group in such a way that a) allow members of the group you want it connected to to be able to see the forum, and b) grant moderator access to the group's leader(s).
There are a few MODs (modifications) which sort of accomplish features like what your suggesting. But honestly, if you're willing to PURCHASE a system, I would suggest you do one of the following: You could hire a contract-hire web programmer to modify the phpBB platform to do what you need/want it to do. Or, look for a different system.
"We're all mad here." -- The Cheshire Cat