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Something can't simultaneously have inelastic demand and also be a luxury good. Those things are kind of opposite.

Demand for taxis has dropped a lot because of Uber. Uber also gets passengers who would have otherwise not made the trip, taken a bus, or driven drunk. If Uber is no longer an option they'll go back to those options,not to taxis.

> The only difference will be in the price of rides - for what is largely a luxury good.

The price of rides makes a big difference and hugely effects demand. I don't know why you're treating it like it doesn't matter. Lower prices open the service up to more people. Indoor plumbing was once a luxury good. Generally progress means turning luxuries into commodities, not the opposite.