Jae Song, David J. Price, Fatih Guvenen, and Nicholas Bloom explore this question in "Firming Up Inequality," written as Discussion Paper #1354 for the Center for Economic Performance at the London School of Economics (May 2015). They use a sample of data from the Master Earnings File
(MEF), which is compiled and maintained by the U.S. Social Security Administration. Because employers need to report what you earned to the Social Security Adminstration, this data includes both individuals and employers (although any identifying traits for individuals and employers are stripped out before the data is released to researchers). Here are some of their conclusions: "Except at the very top, the highest-paid individuals now work at higher-paying firms, but are not higher paid relative to those firms ... Wage dispersion between firms is increasing, while dispersion within firms has been stable."
Here's a quotation from their findings. The first sentence makes the point that earnings have risen faster for those in the highest percentiles, which means that rise in inequality has occurred. The rest of the quotation makes the point that essentially all of the rising inequality of earnings is reflected in a rising inequality of the average amount paid by firms.
(To make sense of what follows, some readers may need bit of explanation about what is meant "log points" and percentages. As an example, consider an increase from 50 to 60. This is a rise of 20%--that is, (60-50)/50. But there is alternative way to approximate this percentage change, which is to take the natural logarithm of the ratio (60/50). Punch it into a calculator, and ln(60/50)= .18232, which is sometimes referred to as "log points." Notice that the log calculation is an approximation of the percentage change. I won't try to explain here why it's often useful to work in terms of log values, but as a matter of calculation, it's straightforward to switch back and forth between log points and percentages--so which one to emphasize in the exposition is just a choice of the author.)
"Between 1982 and 2012, the middle of the income distribution saw an increase in real wages of 18 log points (20 percent), while the top one percent saw an increase of 66 log points (94 percent). This change is roughly mirrored in their firms: individuals in the middle of the income distribution worked at firms with mean real wages 23 log points (25 percent) higher in 2012 than in 1982, but individuals in the top one percent worked at firms with mean real wages 72 log points (105 percent) higher. If we calculate the increase in individual inequality during that time period as the difference between the change at the top end with the change at the middle—a 48 log point difference—then virtually all of that increasing individual inequality is explained by the 49 log point difference between the firms of individuals at the top, versus firms of individuals in the middle. These trends are consistent across regions and industries, remain true when restricting by sex, age, and tenure, and are robust to various changes to the sample selection criteria."This is the kind of result that makes you want to sit down for awhile and think about what it means. The authors suggest a couple of basic explanations for this pattern. Perhaps firms have become more specialized over time, so that some firms have become in recent decades more likely to hire a bunch of higher-earnings workers while others are more likely to hire a bunch of lower-earnings workers. Or as a complementary explanation, perhaps productivity differences between firms are rising over time, leading to greater dispersion of average wages between firms.
Of course, these kinds of explanations just raise the question of why firms might have become more specialized over time, or why productivity differences between firms are rising. As I try to sort out my thinking on these broader questions, there are a couple of other implications worth considering.
This evidence suggests that inequality of earnings within a typical workplace has not increased much in recent decades. As the authors point out, this may "suggest an explanation for why many do not feel that there has been an increase in inequality: on average, individuals’ inequality with their coworkers has changed little over the past three decades."
I would add the evidence also implies that as the inequality of earnings has risen, there has also arisen a separation between those with higher levels of earnings. The old-time story of someone who starts in the mail-room and works up to the top within a current employer is in this sense becoming less possible, because someone who starts at the bottom will tend to be with an employer with lower-paying jobs, and to get near the top of the earnings distribution is more likely to need a shift to the employers who tend to have higher-paying jobs. We know that many people hear about possible jobs through co-workers past and present. The growing inequality of earnings apparently also reflects a growing disconnect between the workplace networks of those with different levels of earnings.