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Actually, according to standard economic models, raw measures of inflation such as what CPI is intended to be overstate the cost of living due to price substitution effects. The basic argument is that since the relative prices change, people will shift their consumption as a result of relative price changes, and so the incomes will have to increase by a smaller amount than the inflation rate for people to be just as better off. For a simple illustration, suppose the cost of air travel increases five-fold compared to the past year. Suppose that on average the consumers are compensated so that their income is now sufficient to afford the same amount of air travel as before. Will they actually continue flying as much after this compensation? Mostly likely no. Most will reduce their air travel and spend the extra money on the products with a better cost/benefit compared to air travel, and they actually will be better off. Similarly, compensating consumers for increases in the cost of gas dollar for dollar will have a similar effect because there will be people who will just get a more efficient vehicle, or drive less, and spend the remainder of cash on other things. A true cost of living index would not have this price substitution bias in it. I suppose that coming up with a true cost of living index is a much harder problem than coming up with a proper inflation index.
Another problem with CPI might be, as you highlight, is that CPI might not even measure the inflation correctly. In that case, this whole little theoretic discussion is kind of relevant since if we don’t have a proper price index to begin with. Since California and other coastal areas have been experiencing various bubbles (and busts) regularly, it is not surprising that the local consumer price index might indeed be higher than what the national CPI suggests.
[...] a year ago I wrote an article about the Consumer Price Index that made some friends say that I am a conspiracy theorist, but [...]
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