Of course she can't live without him, but not in the way they make it sound. She can't live without him because she has no money and needs his.
So who's the alpha (or rather, the beta)?
Her, for being able to leave and tear him a new one in public, only to convince him later that his humiliated ass needs to take her back like any well-behaved beta male would?
Or him, for getting humiliated by her thinking she doesn't need him, only to have her learn the lesson that no one will have her and that only he can provide her with what she needs, with a few depraved strings attached (to preserve his alpha-ness)?
I guess he wins. Because he's rich. Which I think still trumps all other cards on the deck, no?