Many years ago (must be 9 years ago now), my wife flew J on an award ticket to AKL with our eldest son, who was our only at the time and 18 months old, and we had to pay additional points for him to travel (no assigned seat). This was explained to me at the time as being a surcharge for an infant in J. If travelling in Y there would not have been a surcharge.
But that was a long time ago, so the rules may have changed since then. There was no option to pay the surcharge in cash, just in points. Can't remember exactly how many points was charged. But the award was on a Air NZ flight, so that shows how long ago it was!