Let’s assume away cyclical factors, so full employment always holds.
If every sector of an economy becomes monopolistic, output will contract in each sector, and it might appear that productivity will decline. But for the most part this output reduction will not be achieved by burning crops in the fields. Rather, less will be produced and factors of production will be freed up for elsewhere. New sectors will arise, and offer goods and services too, perhaps with monopolies as well. In any case, the consumer surplus gains from those new sectors might be especially high, because they will be selling to the highest points on demand curves that previously were unsatisfied altogether.
You can cite the deadweight loss of monopoly all you want, but we’re getting more outputs of other stuff. Value-added could be either higher or lower, productivity too.
The Schumpeterian tradition, of course, suggested that market power would boost innovation. There are at least two first-order effects pushing in this direction. First, the monopoly has more “free cash” for R&D, and second there is a lower chance of the innovation benefiting competing firms too. I don’t view the “monopoly boosts innovation” hypothesis as confirmed, but it probably has commanded slightly more sympathy from researchers than the opposite point of view. Bell Labs did pretty well. In any case, partial equilibrium output restrictions won’t get you to any kind of smooth conclusion about monopoly causing low productivity growth.
Let’s say instead that every sector is ruled by monopolistic competition, a’la Chamberlin. In the model, that puts firm production at points on the AC curves above minimum AC for each firm or sector. You might consider that to be a productivity problem, though of course you must compare it to the rise in product diversity that follows from monopolistic competition.
But under those same conditions, profits are zero and so the mark-up arguments from the DeLoeker and Eeckhout paper do not apply and indeed cannot hold.
I find most of what is written on monopoly and productivity these days to be under-theorized.