[em500]

The BLS already publishes per category CPIs, alongside the aggregates[1]. Presumably you'd want the Fed to use different weights than the BLS (overweight "necessities"?) for monetary policy. But they already do, they vastly underweight food and energy prices, using "core inflation"[2]. (Besides, they typically use the BEA PCE deflator[3], rather than BLS CPI for monetary policy.)

Occasionally you'll hear that this is part of some government conspiracy to suppress true cost of living measures or some such. I think that's largely nonsense as CoL adjustments in government departments are always done with plain CPI, only the Fed uses Core. The logic of using Core (excl. food and energy) is mostly pragmatic, it's just a simple smoother, they could also use trimmed or rolling averages. The raw inflation is pretty volatile, based on that they'd have to adjust policy rates up and down all the time.[4]

[1] https://www.bls.gov/news.release/cpi.t01.htm

[2] https://en.wikipedia.org/wiki/Core_inflation

[3] https://www.bea.gov/data/personal-consumption-expenditures-p...

[4] https://krugman.blogs.nytimes.com/2010/02/26/core-logic/ (Sorry, can't find a non-paywalled link)