Thoughts on Another Labor Market Concentration Paper

Efraim Benmelech et al (2018) released an NBER working paper last month that looked at the impact of labor market concentration on wages. Specifically, they estimate a series of models using panel data on on actual wages at the establishment level from the U.S. Census Bureau and estimates of local-labor concentration. The paper is well worth a read, but I am still skeptical about labor market concentration being a huge policy problem. Here are three questions that came to mind as I read the paper that kept me from being fully convinced.

1. Are the authors defining local-labor markets correctly? My first concern with the paper is the way the authors define local-labor markets. Specifically, they define labor markets by county and 3 or 4 digit SIC industry code. So, for example, there is a market for labor in the "Paper Mill" industry (SIC 2621) in Haywood County, North Carolina (FIPS 37087). And this market is separate from the labor market of other industries. Just speaking from personal experience, this definition seems very narrow to me. Wouldn't a paper mill in Haywood County actually be employing people based on their skills and not which industry they worked in?  For example, my mother was hired at the Champion paper mill in Haywood County after working in the Ingles grocery store bakery. They didn't hire her because she had paper mill experience, they hired her because they needed unskilled labor.
2. Are they capturing the effect of higher market concentration on wages or the effect of lower labor demand? The authors measure local-labor market concentration using an Herfindahl-Hirschman Index (HHI) for each county-industry (see page 3 and page 9).The authors find that wages fall when the HHI for a county labor market increases. If HHI only increased because of firms merging, then it seems obvious that wages must be falling because the market is becoming more concentrated and firms have more bargaining power.However, this is not the only reason that the HHI increases. As the authors note on page 24, the HHI might also increase if a firm closes and leaves fewer firms in the market. That seems like a very different story to me. If a firm closes, that doesn't just mean the market is becoming more concentrated, it also means demand for labor has likely fallen. So how do we know rising HHI isn't just mostly serving as a proxy for falling labor demand? How often is HHI rising due to firms exiting a market? The authors don't seem to address this concern at all in their paper. 
3. Do their results really suggest labor market concentration had a large effect on wages? Ignore my previous two questions. Suppose the authors are correctly defining the labor market and are successfully capturing the effect of higher concentration on wages. What do their results actually tell us? They find that a 1 standard deviation increase in HHI lowers wages by as much 1.7% when defining labor markets using 3-digit SIC codes or 2.1% when using 4-digit SIC codes (see pages 10-14). But, in this context, raising HHI by 1 standard deviation is huge! The average HHI when defining labor markets using 3-digit SIC codes is 0.545. An increase of 1 standard deviation (0.35) would mean increasing HHI to 0.895! To put that in context, the max value for the HHI is 1. That means taking a county from the "average" level of concentration to near pure monopsony will only lower wages by at most 2% (maybe less). That's not nothing, but that seems surprisingly low given such a dramatic increase in concentration. 

So, overall, I found the paper interesting but unconvincing. But I could be missing something. If anyone has answers to my questions above, I'd be happy to hear them.