Nope.
Competition may be somewhat good--or at least, it can be good from time to time. But Microsoft is hardly what's needed for that purpose. Free software generates competition as a byproduct. Gnome competes with KDE as well as XFCE et cetera. OpenOffice competes with KOffice, Abiword, Gnumeric etc.; ask ten different Linux users what music program is the best, you'll get at least five different answers, distributions compete, and even Linux the kernel is replaceable--you could run Gnu/Linux, Gnu/OpenSolaris, Gnu/Hurd . . .
Sometimes an application will dominate, but if it stagnates badly it tends to get forked; take Xfree86. So the only time there isn't noticeable competition within Free software itself is when it's doing a very good job.
Microsoft, on the other hand, don't compete in the normal sense, by and large. They distort competition. Because of their number 1 position and their approach, what competes most effectively with them tends to depend less on quality than on successful imitation and reverse-engineering. So for instance, if I'm thinking of using KWrite, or LyX for that matter, the question isn't the question isn't "Will it do a good job helping me write things?" the question is, "Will it provide the same experience as and import/export the file formats of Microsoft Word?"
That actually makes it hard for genuine competition, competition in the sense of gaining mindshare by coming up with useful innovations, to flourish.
In any case, Free software tends not to require competition as much in order to do a good job. People contribute because they like being creative more than to compete, and quality comes from co-operation as much or more than from competition. The reason competition is required between private enterprises is that unlike humans, firms have only one source of motivation: money. Without competition they cease to innovate because there's no point wasting money on it if the market is captive anyway. Public hierarchies often tend not to innovate just because hierarchical bureaucracies find innovation threatening as a rule; their structures gradually choke it out--but even there, public bureaucracies tend to be more innovative than private firms with little competition. Universities do far more basic research than drug companies. But the structures with which Free software is created avoid both those traps--they leave people free to create. So the "competition" paradigm taken from theories about private enterprise applies to a much lesser degree to Free software.
So. Free software doesn't need competition that much, and when it does, it generates its own. No proprietary monopolists required, proprietary monopolists actually distort and divert the natural creativity of Free software.