It's hard from the outside to say about price sensitivity (although I'm sure Uber and Lyft have run lots of experiments). Personally, I take Uber, Lyft, taxis, or a private car (or a rental car, public transit etc.) mostly based on what's most convenient to me. But this is predominantly just business travel. I'm sure many daily users have a different calculus.
But there's also the fact that, if Uber doubles their prices tomorrow and Lyft doesn't, a lot of people will presumably shift to the cheaper ride especially given that most of the drivers are the same. (Of course, Lyft could follow suit. At which point, they probably get accused of price fixing.)