It is crap. I'm not even going to get into your comment about the people living in the inner cities.
I'm being generous and rounding your number, which is already rounded up because it's based off of one the more expensive states among fifty.
So, you're number is a billion. A billion per year.
Even if your number is exactly the amount of money we'd save murdering the average number of murderers per year, that's still only a billion in a budget of over THREE TRILLION EIGHT HUNDRED THREE BILLION. Your one billion is insignificant to the realities of the American budget.
Your math, good sir, is crap. I'd switch up your argument about our cesspool.
No. It's a billion a year to house every murderer in 2012 alone, 16,227. Lets say there are 100,000 murders locked up for arguments sake. There are actually more than that, but we will low ball it just for you.
We will be using the numbers presented here.
http://en.wikipedia.org/wiki/Incarceration_in_the_United_States which have references.
In 2005, on average, it costs $$23,876 a year to house state prisoners. We all know private prisons charge the government far more. In California it was $47,000. It increased $19,000 between 2001-2009.
For this example, we will not count inflation. We will assume no matter where you are in America, it costs $23,876 a year. Not including any legal defense (which can be between $10,000 - $100,000 depending). The number below, as you know, will be out by billions of dollars, but it will get the point across.
Assuming only 100,000 murders in prison. To house them for 1 single year it would cost $2,370,000,000. This does not count the 500,000 people in jail awaiting trial.
Now, as I stated, the economic benefits are not why you kill objectively guilty people. DNA evidence, also does not make you objectively guilty. You kill these people, like the Colorado shooter, for example, because there is no societal benefit to keep them alive.
Now, provide me with an argument as to the societal benefit of keeping the Colorado shooter alive. An argument that is not a judgement call based on personal beliefs or morals.