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Apparently each homeless person costs the government $35,000 to $150,000 per year on average, between emergency room visits, court fees, jail time, police effort and time, and so on.

How much government money does a homeless person spend if given government-provided housing, including the price of said housing? $13,000 to $25,000 per year, due to decreased risk factors associated with not living on the street.

How much money can the government SAVE by providing housing to the homeless? $10,000 to $137,000 per person per year.

My biggest problem with the way our country does things is the simple fact that people who are against social programs are really bad at math. Capitalism is about maximizing profits, yes? Then, isn't it capitalist to be socialist from time to time, when it saves millions of dollars a year while happening to help helpless people in the process?

The U.S. Department of Housing and Urban Development upped its cool quotient when Secretary Shaun Donovan appeared on The Daily Show with Jon Stewart. Donovan and Stewart exchanged small talk about gr...
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Unfortunately, the ideal capitalist solution that maximizes profits is to spend $0 per person per year on the homeless. Capitalism (at least pure capitalism) doesn't conveniently allow for allowances that serve no purpose but to 'help helpless people'.
But what you're not getting is that incidental costs from the homeless are the majority of those. Sure, you can save a bunch of that money by letting sick people die and not doing anything without confirming they have insurance. But, most of those costs are things like the police time it takes to throw homeless loiterers in jail for sleeping in parks, and then the costs of paying for the jail time, and the costs of paying the court fees afterward. How would you go about paying $0 for the homeless? Just let them do whatever they want and never arrest them for anything?
Police and court time are arguably sunk costs - we'd be paying for them anyway, and if they weren't busy with the homeless they'd be busy with other problems. The real question then becomes whether the time spent on homelessness instead of those other problems is effectively preventing those problems from being addressed - i.e. if there are solvable crimes that go unsolved because the police are too busy rounding up the homeless. Which could easily be the case, but attaching a numerical value to that loss seems tricky and prone to varying wildly depending on how you account for things.
(Not that I'm disagreeing with your general premise - I'm not exactly anti-capitalism but at some point a just society ought to show concern for the suffering of its disadvantaged members.)
Court fees (unless you're suggesting that in a true capitalist society, people who can't afford to pay court fees if they lose shouldn't get tried at all and should simply get a guilty verdict with maximum penalties awarded) are still not avertable nor something that will be "paid anyway." Same with jail time, unless you are advocating the death penalty for homelessness. And using bare hands to do it, because bullets cost money. Oh, and have volunteers do it, because you generally have to pay people to kill for you.