Does higher minimum wage increase unemployment?

Some say that as unemployment falls, the quality of available workers diminishes, which droves down wages. Is it true?

Ralph Musgrave posted this comment to my post about minimum wages:

Stefan,
You claim that the text books say that “higher minimum wages will increase unemployment if set above the marginal productivity of more workers.” I suggest the latter statement is meaningless (or it’s a circular argument) because minimum wages themselves (along with union wages and other factors) determine what level of remuneration corresponds to the marginal worker. I’ll explain.

As unemployment falls, the marginal product of labour falls because it becomes increasingly difficult for employers to find suitable labour amongst the unemployed. And employers will continue hiring from the ranks of the unemployed till the marginal product equals the minimum wage / unions wage, etc. At which point employers tend to begin poaching labour from each other, rather than hire from the unemployed.

Well, first of all, let's make it clear what "marginal productivity" is. Marginal productivity has nothing directly to do with the actual pay received by workers. Marginal productivity is the value added that a particular worker's work effort creates (or could create in cases where the worker is unemployed). To what extent this value goes to the worker or the employer is irrelevant for the issue of how big this value is.

And as for the idea that marginal productivity of workers fall as unemployment falls, it seems plausible and could in some circumstances be right, but only under certain circumstances. First of all, employers must always hire the best persons, something that probably and hopefully is usually true, but definitely not always, either because employers have too little information, or because the job seekers is not as good at applying for jobs as performing jobs or because employers might have irrational hiring practices.

And secondly, employers might not pick employees on the basis of their marginal productivity, they could pick them on the basis of how costly it is to hire them. A profit maximizing employer wants to maximize the difference between his workers marginal productivity and pay, but that can be maximized both by maximizing marginal productivity and minimizing pay.

And thirdly, to the extent that falling unemployment reflects rising demand or lower supply for labor rather than lower pay, this means that the perceived marginal productivity of workers (again defined in the term of how much value added a worker's efforts creates) rises generally.

Furthermore, the idea that hiring of the best unemployed during a period of falling unemployment means that marginal productivity falls is in a way true, but misleading because it confuses "generic marginal job seeker that can be hired" with individual workers. If the most productive unemployed gets hired it won't affect the (potential) marginal productivity of either any of those that gets hired and those that don't. It simply means that one shifts from more to relatively less productive workers.