I'm not sure about reduced pricing for infants.
But as far as bringing the infant seat on, it is standard accepted practice system-wide that if you are travelling w/a lap child, you bring the seat w/you to the gate in hopes there will be an extra empty seat in your row.
If it turns out they can not get you an extra seat, the infant seat will be tagged with a pink gate valet tag and you leave it at the bottom of the jetway. You pick it up again when you get off the aircraft.
Infant/child boosters, strollers, car seats are not allowed to be stowed onboard.
If you decide not to purchase the seat for your child, try to get your preassigned seats set up as a window/aisle combo. ie: you in an 'A' seat, spouse in a 'C' seat (or D & F). Since middle seats are the last to go, in the event the flight does not fill up, you will be left with an empty seat in your row.