You are making the incorrect assumption that a company has to try to compete with Uber everywhere all at once. They don't. Once Uber raises prices to the point

You are making the incorrect assumption that a company has to try to compete with Uber everywhere all at once. They don't. Once Uber raises prices to the point where they make a profit, cabs will just come back and undercut them - because they will have less expenses (they don't have to support a massive organization and shareholders). This will happen independently in every city.