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>CPI is also a broken measure due to its inability to measure wealth inequality (sampling median purchasers only)
And that's fine, because the CPI's job is to measure how prices have changed (ie. inflation), not wealth inequality. Just because a given metric doesn't support your pet cause doesn't mean it's "broken". It's like complaining GDP is a broken metric because it doesn't measure how oppressed minorities are.
> Real inflation has likely been ~33% higher than reported for decades (the Big Mac index is probably the true gold standard).
I tried to confirm this and the results were far from conclusive. The big mac index data can be obtained from github[1] and gives a 129.9% increase between april 2000 (earliest data available) and july 2022 (latest data available). To compare this against the CPI data, I checked on FRED and the "Consumer Price Index for All Urban Consumers: Food Away from Home in U.S. City Average"[2] component of the CPI gives a 87.0% for the same time period. If you take those numbers and convert them to annualized rates, you will indeed find a 33.3% difference between them. Looks like your claim is confirmed, right? But not so fast. The CPI category listed above actually breaks down into more detailed categories, including "Limited service meals and snacks", which presumably the big mac falls into. That data isn't available on FRED but can be found on BLS's data viewer[3] and gives you a 105.8% increase for the same time period. If you convert that to an annualized rate the difference between that and the big mac index drops to 15%. I suspect if we try harder we can eliminate more of the discrepancy. For instance, the "Limited service meals and snacks" index was probably computed from a basket of items, not just big macs. Items that have fatter margins (eg. fries or drinks) might have inflated slower than big macs, thereby dragging the growth of the basket down and explaining the difference.
[1] https://github.com/TheEconomist/big-mac-data/
[2] https://fred.stlouisfed.org/series/CUSR0000SEFV
[3] https://beta.bls.gov/dataViewer/view