If you want to save taxpayer money, then you should be advocating for an end to the death penalty. Sentencing a person to death ends up costing significantly more than life imprisonment.
Sure, you could execute them the day after they're convicted. But that would greatly increase the number of people we let the state murder, not to mention the number of innocent people they'd be putting to death. Personally, that's not the type of society I want to be a part of.
Surely killing someone can be way cheaper than sustaining them for life.