Cost of living index vs. consumer price index

I should have known about this, but I just discovered that BLS has been publishing a C-CPI-U index. I had seen the term “chained index” around but didn’t realize what it means. This index has a direct month-to-month link to what consumers report that they’re actually buying, rather than using a fixed marketbasket as the conventional CPI’s do. Thus the chained index measures the cost of living– what people actually spend to live– rather than consumer prices. BLS says the chained index is expected to generally be lower than the conventional index, and that has been the experience since it was introduced. However, it seems to me that when the chained index is lower than the conventional index, that means people are finding ways to live cheaper (presumably because their real incomes are declining). If people’s incomes were increasing, relative to the cost of living, wouldn’t the chained index rise faster than the conventional one, as they choose more luxuries?

If I am correct, and the indexes are correct, then real incomes are declining.

BLS explains their chained index here. Actual data seem to appear only in the detailed reports, which are here.