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.