Economists like nothing more than to apply their theories to areas where they don't belong, but I'm not sure that's the case here. Of course, they're assuming a rational pattern of behavior, and the distinction between murder and, say, manslaughter is 100% important to the conclusions they draw.
Of course, there's also the fact that non-death penalty countries also have lower homicide rates than the US, which may speak to the fact that we're just fucked up.
To economists, it is obvious that if the cost of an activity rises, the amount of the activity will drop.
This is the basic premise of the article, but it comes with a contradiction. You can only be executed once, and having committed a murder, you're eligible for execution (at least in some states). So, then, shouldn't anyone who has committed murder then become a serial murderer? Buy one, get one (or as many as you can before you're caught) free!