well yeah, free transit isn't a way to get more people onto it, it's a way to make sure that poor people can still get around and aren't locked inside bicycle range.
City Life
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This reflects my experience with public transport here. Two years ago, the local government got funding for a pilot program to cap the cost of a bus ticket at £2 per ticket (or £5 per day or £20 per week, whichever came out the cheapest). As this came at the same time as reducing or cutting many routes, the cheaper tickets hasn't really encouraged that many people to use the bus more. If a bus only comes through my village once every 90 minutes, and takes almost an hour to get into town, during which I'll be harassed by weirdos trying to sell me drugs, the fact that it only costs £2 each way is not the main barrier to use. By having a car, I can drive into town (a 10-15 minute journey) at a time that suits my schedule, come home when I'm done (instead of having to wait up to 90 minutes for the return bus), and there are no drug dealers in my car either. These are all things that make the expense of a car worth it.
Free transit might have a much larger impact in American cities with many many more low income people than the decently well off capital of Estonia. Large American cities have more people who cant afford the option of a car.