TL;DR: A few details that most of my fellow commentards seem to be missing...
Most city centres and densely populated areas would be in a permanent state of gridlock without the relief provided by public transport. The lack of 'acceptably good' public transport has huge implications in fuel consumption, pollution, citizens health, time spent while commuting, productivity, road accidents, parking space availability, you name it.
Regarding the efficiency argument, the closer you get to a densely populated area the most efficient public transport becomes. I've never been in the city centre in a bus with less than ten passengers, and double or treble that minimal number of passengers for peak hours. I live in a suburb of a smallish (~250,000 people) city in Northern Spain but also have some experience with public transport in other Spanish and European cities. I've used public transport in Madrid, Bilbao, Barcelona, Cork, Dublin, London, Paris, Nice, Rome and several other places, and the same thing happens everywhere.
Cabs: They aren't either totally efficient. There is time spent waiting for customers. The further they are from densely populated areas, the longer they have to wait for new clients, the least efficient they're. Something similar happens with peak hours. In peak time cabs are very efficient. The same thing happens with buses.
Parking: How much do you pay daily for parking? How long does it take you to find parking space? How much fuel are you wasting in this Musical Chairs game for adults?
In my opinion the efficiency of a public transport system has to be considered globally, obtaining an average from every part of the system. Incidentally, the same -but reversed - is true about private cars. The closer they are to the city centre, the less efficient they become.
Considering these factors, it probably makes sense for City councils and Governments to subsidize public transport. They're probably receiving back dozens of times what they invested. And now that I think of it, a perpetually congested city centre would probably send commerce away to Malls built in the outskirts, removing a source of taxes from City councils. Ditto about residents.