David Kay Johnston has also written some columns on this financial legerdomain. I am sure there are some right wingnut sites trumpeting this "wean your granny off food and on to kibbles and bits" move as historic and 'saving the union' kind of idiocy.
The chained CPI is based on substitutions. Tuna may go up but the consumer switches to cat food, so their real cost of living didn't go up as much as the real CPI. Of course SS recipients pay for more healthcare that has exceeded the CPI by 3 or 4 times over.
It is important to remember that under the law Social Security is supposed to be treated as a separate program that is financed by its own stream of designated revenue. This means that it cannot contribute to the budget deficit under the law, because it is only allowed to spend money from the Social Security trust fund.
This is not just a rhetorical point. There is no commitment to finance Social Security out of general revenue. The projections from the Social Security trustees show the program first facing a shortfall in 2033 after which point it will only be able to pay a bit more than 75 percent of scheduled benefits. While this date is still fairly far in the future, at some point it will likely be necessary to address a shortfall.
It is reasonable to expect that the changes needed to keep the program fully funded will involve some mix of revenue increases and benefit cuts. However if the chained CPI is adopted as part of a budget deal unconnected to any larger plan for Social Security then it effectively means that there will have been a substantial cut to Social Security benefits without any quid pro quo in terms of increased revenue. This hardly seems like a good negotiating move from the standpoint of those looking to preserve and strengthen the program.