So I repeated an exercise I did a few years back--I looked at expenditure shares for different goods for each income quintile, and then looked at price dynamics for each expenditure category in the CPI (the matches between the CES and CPI are not perfect, but they are close. I am not sure what to do with the expenditure categories "cash contributions" and "pension contributions.").
At the same time, it’s worth noting that stagnating real working-class wages are calculated by using a meaningless overall average rate of price inflation. Some things—college tuition, apartments in Manhattan, health care—have gotten more expensive much faster than average. This means that people who buy a below-average amount of those things are better off than the statistics show.
In any event, I find that the effective CPI for each income category is pretty much the same for 2009-2010: the CPI increase for the lowest quintile was 1.6, for the second lowest was 1.4, for the third 1.7, the fourth 1.8 and the highest 1.7. These differences look like noise to me.
I will try to figure out something using longer term data, but since expenditure shares change over time, it will be harder to glean meaning from differences in CPIs.