If we compare the CPI to a perfect cost of living index, we find that they are

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The Consumer Price Index (CPI) and a perfect cost of living index differ mainly because the CPI uses a fixed basket of goods and services to measure price changes over time, which can introduce measurement difficulties. This fixed basket does not account for changes in consumer behavior or the introduction of new goods and services, which a more ideal cost of living index would attempt to incorporate.

If We Compare the CPI to a Perfect Cost of Living