Via Adam Ozimek, here is one recent (still unfinished) paper, by Kurmann, McEntarfer, and Spletzer:
Using administrative worker‐firm linked data for the United States, we examine the extent and consequences of nominal wage and earnings rigidities for U.S. firms. We find less evidence of downward wage rigidity in the administrative data than has been documented in previous studies based on self‐reported earnings from surveys. In our data, only 13 percent of workers who remain with the same firm (job stayers) experience zero change in their nominal hourly wage within a year, and over 20 percent of job stayers experience a reduction in their nominal hourly wage. The lower incidence of downward wage rigidity in the administrative data is likely a function of our broader earnings concept, which includes all monetary compensation paid to the worker (e.g. overtime pay, bonuses), whereas the previous literature has almost exclusively focused on the base rate of pay. When we examine firm labor cost adjustments on both the hours and wage margins, we find that firms have substantially more flexibility in adjusting hours downward than wages. As a result, the distribution of changes in nominal earnings is less asymmetric than the wage change distribution, with only about 6 percent of job stayers experiencing no change in nominal annual earnings, and over 25 percent of workers experiencing a reduction in nominal annual earnings. During the recent Great Recession, this earnings change distribution became almost completely symmetric and the proportion of job stayers experiencing a decline in annual earnings rose markedly to about 40 percent. Finally, we exploit the worker‐firm link in our data to show that it is mostly smaller establishments that show evidence of asymmetry in their earnings change distribution. For these smaller establishments, we find that indicators of downward wage rigidity are systematically associated with higher job destruction rates.
Here is another recent paper, this one from the NBER. It shows that real estate agents, who have flexible, commission-based wages, do have smaller employment fluctuations than sticky-wage construction workers. But that difference is only by about 10 to 20 percent.
Here is my previous post on sticky wages: Basu and House show that real wages vary a great deal through changes in expected career paths. Here is Alex’s 2014 post on half the men having new jobs since the recession.