I see this claim in my Twitter feed pretty often, but I don’t get how it is supposed to run. Let’s try an analogy with the non-human animal kingdom.
Right now there are many cows in the world, and even more potential cows to be bred, or cows in low-value situations that could be moved around by boat or even helicopter, if need be. Call it the “moo reserve army of the unemployed.” If the market as a whole increased its demand for cows, the price of cows would go up. It would not make sense to say “that happens only when all the cows are busy all the time and there are no extra cows or potential cows left.” Very likely, there is an upward-sloping cost curve for mobilizing more cows.
To be sure, under a constant cost assumption, the price of cows would not go up, following an increase in demand. The quantity of eligible, working cows would rise, stifling upward price pressure, and possibly this would take the form of a Malthusian equilibrium. But note: in this situation you should expect the price of cows never to go up, as the cost structure is preventing that.
Alternatively, you might think that demand for cows and the cost structure for cow expansions interact in some very particular way. If you pinned this down in just the right manner, you could model a situation where an increase in demand for cows won’t boost the price of cows now, but in broader situations the price of cows can sustainably rise. Indeed that is possible, I just don’t see particular reason to believe that such a convoluted construction is doing most of the explanatory work for current labor markets.
I look at it this way: measured wages for male labor near the median haven’t gone up much in decades, and this is poorly understood (you may or may not think the same is true for actual real wages, and for women the story is somewhat more complicated). So if measured wages for non-supervisors are not going up much now, that is hardly a huge shock. The fact that we don’t understand it well doesn’t mean some remaining particular hypothesis — in this case about the size of reserve armies — has to be the true one.
Most cow parables, upon closer examination, collapse into structural explanations anyway. And in labor markets, it is almost always both blades of the scissors that matter.
Addendum: You might try a matching model. Imagine that potential workers are fully passive, stoned so to speak, but will accept credible good offers from well-capitalized employers. The cost structure of the workers, or worker search, does not influence the outcome. Over the course of the recovery, employers invest more in searching for the right workers because their profits are higher and they make a successively greater number of offers to well-suited workers, but at constant wages. The number of employed keeps on rising, wages stay flat, but longer-run wages nonetheless may rise with productivity (and with enough bids for their labor, workers move out of passive strategies). I’m not saying this is a good representation, only that it might capture the claimed mix of flat wages and a large reserve pool of labor, yet without forcing wages into a longer-run flatness. It also suggests, by the way, that some measure of monopoly/high profits has been good for social welfare, as it has boosted employment.