Which is cruel: Execute a person to death, or Keep them alive in a cell till they're dead?

Forget about cruelty to the individual who committed the crime. What about cruelty to the society at large? It costs far more to execute a prisoner than it costs to keep them incarcerated. Further, states which enact death penalties always see drastic increases in their crime rates while states which enact moratoriums on the death penalties always see drastic reduction in their crime rates. This is particularly true of violent crime rates. So why would any rational person choose to keep the death penalty? If one seeks to reduce costs and decrease crime, the simplest solution is to do away with the death penalty.