CPI from 1993 to 2010. Note the bump a couple of years ago when the housing market tanked.
Chained CPI changes what goes into computing the CPI. The CPI works as well as it does (however well that is) because it has a list of things it checks the price on. Chained CPI modifies that list to reflect what people are actually buying, or not, like beef. The price of beef has gone up tremendously, so people don't buy as much of it, so it comes out of the CPI (or becomes a smaller part of it) and the CPI doesn't grow as much and it makes things better. For somebody. I'm not quite sure who.
The problem with this is that you can push it to extremes. The price of food has gotten so high that no one buys food anymore, they can only afford to buy gruel. So nobody is starving (we've got plenty of gruel), and the CPI stays low, so things are good, right? Sounds like North Koreanese to me.