We know that measured productivity falls in a recession (relative to trend).
Real Business Cycle theorists say that this is because a negative productivity shock is what caused the recession.
Keynesians say that a negative aggregate demand shock is what caused the recession, and labour hoarding (firms don't like to lay off workers even if they don't need all of them to produce the goods they can sell) is what causes measured productivity to fall.
But New Keynesians assume firms are monopolistically competitive. As every first year economics student knows, monopolistically competitive firms in zero-profit long run equilibrium produce where their Average Total Cost curve slopes down. If the ATC curve slopes down, total factor productivity will fall if output falls (and rise if output rises).
So New Keynesians don't need to assume labour hoarding to explain why productivity is procyclical.
But do the numbers add up right? Is this effect big enough to explain all the cyclical variation in productivity? Maybe not. Maybe they need to assume some labour hoarding too.
I don't know if any of this is new.
New Keynesians assumed monopolistic competition for lots of reasons, but explaining why productivity is procyclical wasn't one of them. So if it can also explain why productivity is procyclical, that is an independent confirmation of New Keynesian macroeconomics. They weren't "discovering" empirical gold by digging where they already knew the gold was buried (pdf).
In zero-profit long run equilibrium, an individual firm's demand curve must be tangent to its ATC curve. (Price must equal ATC for zero profits, and if the two curves crossed the firm wouldn't be maximising profits, because it could make positive profits by moving to a point where P > ATC.)
If the demand curve is tangent to the ATC curve, those two curves must have the same slope (and same elasticity) at the tangency point. Since a monopolistically competitive firm's demand curve has a negative slope (by definition), its ATC curve must have the same negative slope.
We know that Marginal Revenue = (1-1/E)P, where E = elasticity of demand. And we know that profit-maximising firms produce where MR=Marginal Cost. Simple algebra tells us that P=[1/(1-1/E)]MC. If E=4, then P=(4/3)MC, so firms would set prices as a 33% markup over marginal costs to maximise profits. That's probably in the right ballpark empirically, IIRC.
If the elasticity of an individual firm's demand curve is 4, then the elasticity of its ATC curve is also roughly 4, in the neighbourhood of the long run equilibrium point.
If the elasticity of its ATC curve is 4, that means a 4% drop in output would cause a 1% fall in total factor productivity, and a 3% drop in inputs (approximately). If employment of all inputs dropped by the same proportion, a recession in which real GDP fell by 4% (relative to trend) would see a 3% fall in employment of labour (relative to trend).
Hmmm. That difference between 4% and 3% is big enough to matter (proportionately), but not big enough to explain Okun's Law. Because Okun's Law says that the percentage decline in output is typically two or three times as big as the percentage decline in employment. So even if we stretched Okun's Law to say it's only two times as big, that would mean a 4% decline in output should cause only a 2% decline in employment, rather than the 3% decline in employment my back-of-the-envelope calculation comes up with.
So I'm only halfway there to explaining Okun's Law without assuming labour hoarding.
Maybe labour hoarding is the other half?
Or maybe my calculation isn't quite right. What about hours per worker? If the decline in employment came about 50% through hours per worker declining, and 50% through number of workers declining, then I'm all the way there, without needing to assume any labour hoarding.
Or maybe something else?