Raising the minimum wage may indeed raise average worker productivity -- not by inducing existing workers to work harder or smarter, but by inducing companies to get rid of less productive workers. If you raise the floor from $7.25 an hour to $9, employees whose work output is less than $9 an hour will be let go.
Saying that a higher minimum wage would increase productivity is like saying that banning anyone under 6-foot-10 will make NBA players taller. It will, but not because anyone will grow.
Even the famously liberal Nobel laureate economist Paul Krugman has pointed out these realities. On the blog EconLog, economist David Henderson of Stanford's Hoover Institution cites a 1998 article in which Krugman ridiculed those who "very much want to believe that the price of labor -- unlike that of gasoline, or Manhattan apartments -- can be set based on considerations of justice, not supply and demand, without unpleasant side effects."
Obama argues for an increase by saying no one who works full time should remain poor. One fact he doesn't mention is that the great majority of people who earn the minimum wage are not poor. More often, they're middle-class teenagers.
Another likelihood, which he denies, is that while some workers will go from $7.25 an hour to $9 an hour, others will go from $7.25 an hour to zero. They won't be poor despite working full time. They'll be poor because they're unemployed.
Maybe this won't happen because the laws of supply and demand will be suspended. But would you want to bet your job on it?