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It's highly doubtful that the longer sentence length(is the US's longer on average?) is somehow contributing to a higher murder rate.More men in prison -> more broken homes -> more people with more damage in their upbringing -> more crime
^^ that's just one potential mechanism for having demographic-sized populations of people locked up. About half a decade ago, it was estimated that one in nine black men in the US were either imprisoned or on parole. That's a lot of families missing an important member (even if not a father, a missing husband/son/brother is also an issue).
One of the problems with discussing crime (particularly in the US) is that people tend to focus on the criminal as a single individual, and not about the demographic-scale effects.
> What did you think about how I described sentence length as being a function of cost-benefit analysis?
Extremely long sentences haven't been shown to be an effective deterrent, and if anything, they mean that the prisoner is more likely to cause problems inside (if you're jailed for life, you can basically kill with impunity in the jail, for example). Long sentences mean more prisoners inside costing the taxpayer, less workers outside generating tax dollars, fewer families with present parents, and a broader 'criminal class' (and higher recidivism - if the only thing you know how to do is crime, because your skills have rotted...).
The justice system as a whole is meant to keep society running smoothly, and the penal system is part of that - it doesn't stand alone, separate unto itself. If rehabilitation and shorter sentences means that society runs better, then that should be targeted. The equation isn't as simple as "crime X costs Y dollars damage, incarceration costs Z dollars per diem, calculate for recidivism". Imprisonment has a lot of social knock-on effects.