This is false. The CPI weights the basket of goods that comprise the index based on the types of goods that people actually consume, which includes housing (either in the form of rent or mortgage payments), energy, as well as food and consumer goods. It adjusts for consumption patterns monthly, and re-weights the entire index every 2 years.
The BLS surveys rental prices as well as calculating something called “owner equivalent rent” for owner-occupied housing.
> The OER and Rent indexes have the largest weights of the 211 item categories (item strata) that comprise the CPI market basket. As of December 2008 their shares of the total weight (their relative importances) in the CPI for All Urban Consumers (CPI-U), were 24.433 percent and 5.957 percent, respectively.
[1] - https://www.bls.gov/cpi/factsheets/owners-equivalent-rent-an...
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