I don't pretend to be a theorist, though, and in the interest of not pretending to be one I wanted to just share commenter YouNotSneaky!'s questions and thoughts about the theory behind the nil effect of the minimum wage in the data:
"Can you give a link to a paper that makes the monopsony argument? It's been awhile since I looked at it. I did look up the Burdette & Mortensen matching model recently which I gather is what this argument is based on. However, in that model (or ye basic search model, like say in Romer's Adv Macro) minimum wage still decreases employment. In B&M a minimum wage *can* increase social welfare but that's different (the increase in utility of those who retain higher paying jobs is greater than those who experience longer unemployment spells). You can use the B&M monopsony model to argue for minimum wage, but you can't use it to explain why the empirical work does not detect employment effects.I don't know Burdette and Mortensen specifically although it is this sort of model that Manning uses and refers to when he talks about monopsony and the minimum wage. I've read a little of him, and I've read some of Pissarides on equilibrium unemployment. I rely on much simpler expositions of fixed costs and turnover to motivate my understanding of the connection to the minimum wage - namely, the Oi paper on quasi-fixed factors. And I agree (and have stated here) that this flavor of models is not entirely reassuring on the employment effects, however that should be more apparent in the long run than in the short run I think. I don't know - let me know what you think about that argument.
Personally I'm pretty sure something else is going on (probably the data just isn't good enough, not enough variation, close to equilibrium min wages, adjustments in hours rather than persons etc)
This is one version of the B&M http://econ.tau.ac.il/papers/macro/postmatch.pdf"
I think the data is getting better and the identification strategies are getting better and the result is not going away, so I would not blame the data.
I do think the other two issues raised are relevant: that these are often "modest" increases, not departing far from the equilibrium wage, and that there are adjustments on other margins potentially. Dube, Lester, and Reich do provide an upper bound on the hours adjustment. It's not immediately obvious to me why it would make sense to make your adjustment on hours rather than employment (in the pre-Obamacare era at least!). The only reason to do that, it seems to me, is to deliberately fool economists.