Governments basically don’t assign value. The economy does, which is to say that everyone working together deciding how much various things are worth, even money, does.
They do have controls. So do businesses. So do consumers. And all of those people, to a greater or lesser extent, manipulate the economy, consciously or unconsciously.
Basically, money is an idea, and cash is nothing but token in a board game we use to represent that idea. There are physical traits that make money valuable, specifically the ones that make it hard for people to cheat at the game, but one of the important things about a good currency is that it won’t have any actual value itself.
Even when you say that the material costs something… gold isn’t “useful” in all but a very few conditions. The value of the gold in a coin is just as arbitrary as the value of the coin itself.
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