On this econtalk Edmund Phelps talks about the tendency to have some level of persistent involuntary unemployment. He says that most everyone who has a job is overpaid becuase of the cost to employers of turnovers and keep the employees motivated and reasoably happy to work for the employer.
This overpayment leaves insufficient money to reach the market clearing 100% rate of employment and instead we have some persistent level of unemployment. I think he estimates that about 4% unemployment it about as close as we can get to full employment without creating a future distortion.
It seems that a increase in the rate of inflation can lower unemployment below 4% but since you cannot keep raising the rate of inflation for very lowing when you stop the unemployment rate will rebound to above 4%.
It would then be logical that in a period like we are in right now of a sharp decline in the inflation rate, people with jobs would tend to be even more overpaid, leaving even less money available to increase hiring and producing a high level of un-employment like we have today.
One response to this could be for the unemployed to accept very low wages much below their productivity level. The minimum wage would stand in the way of this, an hourly wage subsidy would work much better than this.
When pushing government to act on a problem we should keep in mind that political realities make Governments do things roundabout and very inefficient ways. I think therefore that if you are a supporter of raising the pay of the people at the bottom of the wage pyramid that you should think hard about possible offsetting behavior and never compromise and accept something like a minimum wage. Such compromises are often worse than nothing.