A strange argument for a "living wage"

In an good piece on the "living wage" at the Offsetting Behaviour blog Eric Crampton writes

Collins nicely does cite the literature on that living wage mandates are very poorly targeted and that we could do rather better by increasing targeted benefits. Then he cites U Vic's Morris Altman:

Morris Altman, a renowned Canadian economist who moved to Wellington's Victoria University in 2009, argues that a living wage is "a moral imperative situated in the natural rights of individuals".

His research suggests that a wage rise can actually pay for itself by raising productivity through motivating workers to work harder and stay in their jobs, and by inducing employers to introduce new technology and train workers to work smarter.

But that is only true, he warns, if wages are raised at a rate that productivity can keep up with. "So one has to be ultra-careful about by how much one increases. If it's a radical increase, that might be too much to deal with in the short-term," he says. "You might need a bit of an adjustment period to get productivity up."

This argument by Altman seems odd. If profit maximising firms believe they can increase productivity, and thus revenues, by increasing wages and the revenue increase in greater than the increases in wages (i.e the wage rise "pays for itself"), then why haven't firms done it already? If the wage increase does "pay for itself" then firms must be able to increase wages to such a degree that the cost increase is just less than the revenue increase caused by the productivity increase. And with revenues going up more than costs, profits must increase. Thus a mandated living wage is unnecessary since firms will increases wages anyway in an effort to increase profits.

As to the point made in the third paragraph of the Altman argument, if firms are profit maximising then there can be no margin left to increase wages. In an effort to maximise profits firms will attempt to set the increase in wages equal to the value of the increase in productivity and thus any further wage increase must, by definition, be greater than any productivity increase obtained. This results in a situation where any wage increase from this point will not be able to generate a productivity increase large enough to pay for it.

So I don't see that the Altman argument for a living wage is valid.