Back in the 1980s and 1990s, I did some research with Steve Silver on sticky wages and the business cycle. Using postwar data, it's very difficult to draw any conclusion, as the economy was hit by both supply and demand shocks, which have very different impacts on real wages.
During the interwar period, however, demand shocks are much easier to identify and the role of wages really stands out. In the following graph we inverted the real wage series (top line), and compared it to industrial production (bottom line), to make it easier to see the strong countercyclicality of real wages:
We found that there were two factors that reduced output during the interwar years. First, falling prices in the face of sticky wages---which occurred on and off throughout the entire interwar period. Second, an autonomous rise in nominal wages (caused by government labor market policies)---which mostly occurred during the period after 1933.
While cleaning out my office I came across a 1996 QJE paper by Ben Bernanke and Kevin Carey. Here's a portion of their conclusion:
First, like Eichengreen and Sachs , we verified that during much of this period there existed a strong inverse relationship (across countries as well as over time) between output and real wages, and also that countries which adhered to the gold standard typically had low output and high real wages, while countries that left gold early experienced high output and low real wages. It does not appear that any purely real theory can give a plausible explanation of this relationship. Among theories emphasizing some type of monetary non-neutrality (i.e., a non-vertical aggregate supply curve), there are basically only two types: theories in which the price level affects output supply because of nominal-wage stickiness, and theories in which the price level affects output supply for some other reason. We find that, once we have controlled for lagged output and banking panics, the effects on output of shocks to nominal wages and shocks to prices are roughly equal and opposite. If price effects operating through nonwage channels were important, we would expect to find the effect on output of a change in prices (given wages) to be greater than the effect of a change in nominal wages (given prices). As we find roughly equal effects, our evidence favors the view that sticky wages were the dominant source of non-neutrality.
That's why Bernanke was my first choice for Fed chair back in 2006.
PS. Is the 1985 paper that Bernanke and Carey cite co-authored by the Jeffrey Sachs who defended Bernie Sanders and a higher minimum wage? I believe it is. I think it's fair to say that the policy views of economists are not based on the outcome of their empirical research.
PPS. Steve Silver and I had a paper on real wage cyclicality published in the 1989 JPE. We did a follow-up paper focusing on wages and prices during the interwar years, which was published in 1995 in the Southern Economic Journal.