I don't understand your response. Carrol's emphasis is on the word "raise" not on "real" i.e., it's a raise in the sense that it is higher than it was before, but if one assumes (as it would be logical to do) that the minimum wage reflects what we as a society consider the minimum to get along, then (one can claim) the dispute between the Dems and the GOP is only how rapidly the minimum wage diverges away from the _minimum_ wage (as the rest of your post, which I have pruned, illustrates). What the latter is can be ascertained either using some cost of living metric or equivalent.
--ravi