Why do we continue the death penalty when it's costing taxpayers millions and doing nothing to deter crime?

It's an age old debate on the economics and actual result of death penalty. Why do we still do it? Because people keep fighting to keep it in our society. Why doesn't it deter crime? because its not public. Think of it, if we went back to the old days of public hangings and beheadings, and the public saw it, do you think there would be as much crime? Death penalty is so private and off limits to the general public now that it loses its power to deter criminals because they just simply cannot see the real consequences.