I haven't looked for work on the price/demand relationship here, but it's also worth noting that price reductions may not just purely be a matter of cutting costs and passing those on at a static ridership and profit level. There are likely to be some price points that profoundly change user behavior
To give an example, my favorite coffee shop is about 7 minutes (+wait) from my apartment by car either way, and around 18-26 minutes (+wait) by bus (shorter there, longer back). Without a ride pass and with tip, it's probably around $7-8 to Uber this one way, vs. $1.25 for the bus.
At this price, I'll usually only Uber if heat or rain make getting to and from the bus miserable. I have a ride pass atm that knocks this down to around $6-7 with tip, which makes me marginally more likely to take Uber, but it isn't my default. I took the bus this past Sunday morning, planning to get some open source work done, but I'd forgotten my laptop wasn't in my bag. The time/sweat cost of the bus round trip and the money cost of the Uber round trip were high enough that I just sat and read a book instead.
I'm not sure exactly where, but somewhere between ~$2-5 total, I'd probably default to taking Uber both ways. Down at the low end of that range, I even would've gone back for the forgotten laptop.