The issue here is that a lot of costs of running a car are paid up front before you even put fuel in it. With rail the cost is paid entirely per journey.The problem is the vast majority of people view the cost of their car journey (wrongly) as the cost of the fuel (and in the odd case the parking on top), so there's always this notion that trains (and sometimes buses) are expensive because it costs more than the fuel in many cases. This tends to colour perceptions on cost, and decision making. Especially if politicians are insistent that passengers should pay the full cost, which makes the differential between the fuel cost of the car and the public transport fare more extreme in many cases.
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The companies compete for the franchises by putting in their bids. The government often select the bid that will require the least subsidy or pay the highest premium so in this case it seems competition actually makes things more expensive for the passenger although of course competition here is making it cheaper for the tax payer.I am not against privatisation for the most part, but the whole idea was to encourage competition that would make fares lower.