Misrepresenting The Case Against Labor Unions
Except that I have never heard any union foe claim that unions lower productivity. The case against unions has never involved claims about lowering productivity as far as I know. Instead, the case against unions has centered on how excessive union wage demands will make low productivity workers unemployed.
Thus while unions can be expected to lower GDP growth and even more so employment growth, there is no reason to expect them to lower productivity. Indeed, the theoretical analysis of union foes implies that unions might raise statistical productivity.
The reason is that by excluding workers with low productivity from the work force, the average productivity of those who have kept their jobs will increase. Statistical productivity is as you may know defined as the quantity of output divided by the number of workers (or the number of hours worked, but for simplicity let's ignore that other definition). Excluding low productivity workers while not employing workers with higher productivity will by definition lower employment. Since the productivity of those excluded workers were low but still positive, output will also be lowered, but since productivity was low it follows that the drop in output will be lower than the drop in employment.
And as statistical productivity was defined as the quotient between output (dividend) and employment (divisor) and as in any division, the quotient always rises if the dividend is lowered by less than the divisor, it follows that statistical productivity will increase due to union activity.
It should however be emphasized that the increase in statistical productivity does not imply any real increase in productivity for any specific workers. The workers who kept their jobs are no more productive than if the excluded workers had also kept their jobs. The average has only been increased the excluded low productivity workers are no longer counted.