Ezra says no, citing Kathy G. The problem is that, as her own commenters point out, the empirical research doesn't quite says what she says it says. If you do a simply analysis of firms and compare productivity, you may see no productivity shock in the average. The problem is, firm unionization is not a double-blind random experiment. There's a serious composition problem: the firms that unionize are likely to be the firms that are more productive, because they are more profitable and have more surplus to be captured by the union. Failing plants don't unionize. So if you look at unionized plants/firms/industries and see no productivity difference between them and their unionized counterparts, then probably the unions are actually exerting significant drag.
The other problem is that what positive effects we can posit unions having on productivity only work at the firm level, not the economy level. For example, as Ezra says, if you pay people better, you may get better workers. This is actually kind of dubious, since the union's other main job is preventing you from firing the old workers. But say it's true. Those workers do not actually vanish into the void. Jobs must be found for them elsewhere. Meanwhile, the better workers you do have have come from another firm, which now has fewer excellent workers.
There's also the possibility that by paying workers better, you make them more worried about keeping their jobs, and therefore ensure that they will work harder. Again, it's hard to see how this wouldn't be dwarfed by the union's seniority preferences, featherbedding, and job security measures, but say it's true. This only works as long as there are a substantial number of non-union jobs that don't pay as well, into which the unionized workers fear being forced.
There are other, more nebulous effects that have been posited, such as the psychological benefits of job security, but there isn't a terrific amount of empirical evidence for these. And the German experience belies the notion that "cooperative management" actually makes plant operations productive enough to compete with non-union shops; German companies are relocating east as fast as their hot little feet can carry them.
So even if you argue that unionization won't cost productivity at the level of the individual firm, once you get up to the industry or economy level, a policy of encouraging broad unionization will still reduce overall productivity.
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