This is only part of the equation, unfortunately - free transit that is of significantly worse in terms of travel time/reliability/accessibility/comfort won't get used at significantly higher rates.

This is why some transit advocates consider free transit to be an inferior use of funds when compared to investments in the system as a whole, which I think is a fair point.

I'd of course like to see both, as ticketing systems contribute negatively to the overall quality of the system (paying for tickets/passing through ticket gates slows down the system, ticket checkers represent an unpleasant ritual in the system).