Large corporations will typically use trusted vendors to find people. Unfortunately that is where a good chunk of the money/jobs are. But you're not likely to get $200/hr there.
Currently $120/hr is much more likely through a body shop than $200/hr, but keep in mind the body shop is going to get $180/hr. The body shop carries insurance, and has to recruit new people for a contract, and they have to pay their recruiters, etc.
And even expectations can vary wildly. I would think that at $200/hr, you should save me from having to pay 3 people $66/hr (about $133k/yr). Then you're a bargain, comparatively.