How would a state's army be paid in libertarian state?
Would there be at least some taxes to pay for this?
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In anarcho-capitalism there is no state, so... there is no state's army. There are private security agencies. They protect people, organisations and regions, who pay them. So imagine hostile army comes. Security agency sends them a list of protected people, regions etc. and tell "if you attack any of them we will fight. Otherwise we will leave you alone".
Of course protecting each single person might be ineffective. So it also might work differently - people create an organisation, which will pay for protecting their city or region. City (region, country, district) might have also an owner. He might let people live there if they obey his law. This law would include taxes and the owner would pay with them for protection.
In libertarian states the army (or security agency) would be paid with taxes (if we accept taxes for some purposes) or voluntary fee (if not). Ayn Rand would also accept public lottery as a source of money for state ;)