Your example has both low and high price goods doubling in price. In that situation, both chained and unchained CPI would say inflation is 100%. The indices are only different when the rate of inflation is different for each product. See http://crr.bc.edu/wp-content/uploads/2011/09/IB_11-12.pdf for a detailed explanation; and the actual calculation used is a very complicated "Törnqvist index", involving exponentials and logarithms. But, adapting your example, the explanation would be something like:
Suppose, over a given time period, prices of good quality products rise 100%, while cheap ones rise 50%.
And suppose that you, and others, have been purchasing rib eye for $5/lb, and rib-eye rises to $10/lb.
Because of this inflation, you switch from rib-eye to hot dogs, which formerly were $2/lb, but rose to $3/lb.
The simple CPI says prices have risen by 100%, since that is what the items that were being purchased at the start of the period rose by.
But "chained CPI", by allowing for a "substitution effect", says, in effect, that "Hot dogs are the new rib-eye", and that your meat expense went up somewhere between the 100% for the beef, and the 50% for the hot-dogs.
In this case, where there's total substitution, the formula works out as
PT = exp( 0.5 * ( ln(10/5) + ln (3/2) ) ) = 1.73
ie chained CPI inflation was 73%.
If the high quality goods went up in price slower than the cheap stuff, and people switch to the more expensive stuff, this formula would still give an inflation rate somewhere between the 2.
But the real problem is that the goods and services used to calculate the CPI (whether CPI-W, for 'urban workers' (used for SS increases), CPI-U for 'urban residents' (used for tax bracket increases), or the chained CPI-U (the measure some want to switch to, for SS) don't represent the expenditure of elderly people - eg they spend more on healthcare. As my link points out:
The Older Americans Act of 1987 directed the BLS to develop an Experimental Price Index for the Elderly (CPI-E) for Americans 62 and older.
...
A key difference in the spending patterns of the old and the young is medical care. The Consumer Expenditure Survey shows that health care accounts for 13 percent of expenditures for those 65 and older compared to 5 percent for those under 65. Since medical costs are rising rapidly, putting more weight on this component would be expected to produce more rapid price increases. And, indeed, over the period 1982-2010, the CPI-E has increased 0.27 percentage points faster each year than the CPI-W (see Figure 3).
What might be 'fair' is a chained CPI-E index; but that might well work out at about the same as the current CPI-W. So sticking with the CPI-W would be the simple 'fair' thing to do.
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