I avoid this sort of thing like the plague but, if you want to pursue it, you'll need Options +MultiViews which will examine the directories in the path and serve a file despite the extension not being specified. Of course, page.php will have to deal with the $SERVER['REQUESTURI'] whether it contains/requests any more content.
The downside I've seen from members in the past is that they use the same name for a subdirectory as a filename and wonder why the requested file (in the subdirectory) isn't being served. If you're better organized than the other members (and me, too!), they you should be okay.