I had the same question. They have a PR problem that's probably cutting into their bottom line, but that's a long way from being unprofitable. If someone wanted to know which company I wanted the keys to, it would be the one that's make money hand over fist, not the one everyone thinks is really cool and is in the red.
What core competency do you perceive facebook has?
No, I'm not a fangirl of facebook. I've logged into my fb account maybe a couple of times in a dozen or so months. Sorry. You want to assault my karma for this? Puh-leeze.
I look at structures: overall structures and value-added structures. Facebook really has none. If you disagree, please tell me why and how you disagree this setup is theoretically "profitable" or more profitable than, say, a random Internet page or a blogger blog. :)