On the basis of historical experience, not very much:
Increases of $3 to a total of $10.25 would merely be unprecedented in the modern era [emphasis added]. Increases of $8.75 would be several standard deviations beyond merely unprecedented.
That’s Adam Ozimek writing, and he’s referring of course to proponents of a $15 minimum wage. The relevant graph showing the number of historical minimum wage increases by size is here (and the paper from which it comes is here). The data on that graph come from many different countries, so our ignorance about the effects of such a large increase does not stem from some parochial concern with the U.S.
Most prominent minimum wage proponents don’t actually call for a $15 minimum wage, so this isn’t really a “pro-minimum wage vs. anti-minimum wage” debate, just a certain fact about the limits of our knowledge with respect to very large increases in the minimum wage.
Adam also adds:
I think it is true that the more honest case for the minimum wage is coming from proponents who are biting the bullet and arguing that the optimal minimum wage would be one that increases unemployment a little.
Correct. Chad Stone is one such proponent, and I respect him for being honest about the possibility that the employment effect is not zero. That being said I do wonder how intellectually consistent is a plan to deliberately risk sabotaging the employment prospects of the very-least-skilled, for the sake of slightly boosting the wages of the slightly-less-unskilled, given that you are a progressive. It’s an unusual thing for a progressive to propose.