The first thousand USD/mo a person earns has a different meaning to them than the second thousand, and the third thousand, and so on with diminishing returns. The point of minimum wage should be that no one is constantly fighting for their life, because a person in that situation is usually a harm to everyone.
The point of minimum wage should be that people can only be exploited to a certain level by their circumstances and requirement for food and shelter.
There is of course still a black market
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Granted. But that means that there are negative effects to at least some hikes of the minimum wage. So now the discussion can be had about what the effects are and when which effects kick in and how much.