What about welfare?
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The libertarian opposition to government-funded welfare is based on of course, ideals of voluntaryism, but also on the efficiency of private entities. Simply giving the poor money is not the best way to end poverty. With private charities, the ones that can actually cause change in a neighbourhood will get donations, and inefficient ones won't. This will likely involve putting conditions on the money given, eg., that the able-bodied and able-minded must participate in education/employment. There's also the task of making such initiatives as efficient as possible.
If a pseudo-libertarian government were to forcibly collect money for welfare, it would be best to decouple the voting for the NAP-enforcing and welfare-providing branches, but there would still be a big problem.
If contributing to welfare was compulsory and the welfare provider was voted upon democratically, the votes of wealthy charity givers that want to see change would be drowned out by the votes of the poor that would prefer to receive money with no strings attached, and the votes of the upper and middle classes that want their contributions reduced. And if the voting isn't democratic, the system will be overthrown by the people. But reducing compulsary welfare via the votes of the upper and middle class is currently possible if all of them could be convinced that it is misplaced kindness.
The main concern people have regarding abolishing welfare is whether enough money will be donated to cover all poor people. The rich who can afford to donate large amounts already do, be it out of kindness or to acquire goodwill, and people would certainly donate a lot more if they didn't already have to pay half their money in income, property, value-added, excise, and numerous other taxes, for the "betterment of society". Libertarians believe that this, combined with the fact that the best performing charities will be the ones donated to, mean that poverty will be alleviated with less money needing to be spent.
We can't definitively prove this yet, so why not first test things out by slowly reducing the scope of government welfare? It must not all be cut suddenly, since time will be needed for the culture surrounding welfare to shift as people pay less in taxes, and for private charities to strengthen and become effective.