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The Death Penalty is more costly than imprisoning someone for life [4]
Also, if you are going to make the case that the death penalty does not deter crime, you are going to have to look at how it has been implemented in other countries, not just the States. I read somewhere that the death penalty works very well as a deterrent, if two conditions are met.
1. The death penalty has to be unavoidable
2. The death penalty has to be carried out in timely fashion
Neither of these conditions obtains in the U.S. People can generally plea-bargain their way out of a death penalty, or show enough signs of remorse that the jury won't go for a capital sentence. Even if a person lands on death row, he has plenty of appeals, and can often delay his execution by years, if not decades. In countries where the death penalty is the standard sentence for a crime, and is carried out after only a single appeal, crime rates tend to be dramatically lower.
The same is true for costs, btw. The death penalty is way cheaper than life imprisonment in countries that truly embrace it. Only in the U.S., where the death penalty is implemented only after years of appeals, is it more expensive that life imprisonment.
A last point to consider: In
Going Postal by Terry Pratchett, the hangman tells the protagonist that, while he has no proof that the death penalty lowers crime rates, he believes it probably is an effective deterrent, because, as he puts it: "I've never seen anyone come through here twice."
All of this is not to say that you can't argue against the death penalty. I don't agree with it myself, as a matter of fact. But, you need to address these arguments if you want your essay to be convincing.