Now, a better question is why are we still killing people when at least 4% of ppl killed are verifiable innocent?
Because people have unscientifically figured out that threat of death is a damn good deterrent, so we presume we're saving more innocent lives than losing. If true, then the blood of the innocent is on our hands if we don't carry out executions--that is, if we save 5 innocents a year from execution while causing an increase in crime claiming 15 innocents, we are responsible for 10 more deaths.
From a more scientific standpoint, deterrence is complicated. I keep saying this: in drug-gang-riddled cities, 99% of the perceived threat is death by other criminals. The criminals don't think the man will kill them, even if they know the man will kill them if he catches them. They think they'll either die in the hood or eventually make it out rich; or they don't think about where they're getting, but are still mainly concerned with not dying in a bad drug deal or gang war. State executions don't factor in here.
In a more peaceful situation, the most important consideration is state executions. People don't have guns, they don't shoot you if you break into their house to murder them, so that's not a worry. You're not a career criminal, and you live in a quiet neighborhood. When the impulse to go murder some son-of-a-bitch comes into your mind, something will hold you back... first personal morals, then an ingrained fear of consequences. The only consequence here is state execution, hence, unlike above, it's a deterrent.
Now, all that's meaningful, but I'll repeat: we as a society haven't figured this out. We've figured out people fear death, and that fear is a deterrent. We don't realize it's in vain here, and extremely effective there. Instead, we pick a side: death penalty for fear of increased murder, or no death penalty because it doesn't seem to help.
I guess it's cheaper than dealing with the lawsuits for false imprisonment.
Not to mention imprisonment is almost as bad as execution. I like to use the strategic scenario of 10 years imprisonment between 25 and 35, both with and without an existing relationship, career, or both. In any scenario, your life is destroyed; this compounds with psychological impacts of imprisonment--you become a broken man, more prone to crime--and so prison is both torture and a risk to society (an innocent man could come out a murderer).
The real solution is to stop convicting the innocent. Use of the death penalty is fine, but should scale with venue: use it less (up to and including not at all) in areas where it's simply not a deterrent. However, I reject this lethal injection bullshit: a slow, killing numbness looks peaceful, and absolves us of our actions; quarter that motherfucker, with a bolt smashing the back of the skull just before the body is ripped apart. Let the execution horrify us so that we regret what we do; maybe we'll take better care to not condemn a man to death unless we're really fucking sure he did something to warrant it. Make the prosecutor and the jury watch, too. In person.