With all due respect to Max Power, his evidence that the death penalty is a deterrent seems awfully unconvincing to me.
In my mind, the proof that it does deter (at the margins) is the habeas litigation levels in the United States. The ratio of convicted murderers on death row who litigate like the dickens to have a death penalty commuted to a life sentence to the convicted murderers who give up appeals and accept their execution must be at least 100:1. Even if you include suicides and indirect suicides by murderers who get into shootouts with police rather than surrender (though if you include death row and jailhouse suicides, you should also include the thousands of life-without-parole prisoners who don't commit suicide in the ratio), and discount some to account for the costlessness of death sentence appeals thanks to tireless "pro bono" efforts by attorneys to nullify the death penalty through litigation, the ratio is sufficiently huge to suggest that the vast majority of murderers prefer life imprisonment to an execution.Of course most people prefer life imprisonment to an execution. But that's answering a question that wasn't asked. That's the choice faced by someone who has already been caught, not the choice faced by a potential murderer (with the exception, perhaps, of those who are already in prison for life.)
That doesn't mean that I don't think that the death penalty can have a deterrent effect; I just don't think that the behavior of those already facing guaranteed punishment tells us much about the behavior of those who haven't committed a crime yet.