The challenge is that this is non-uniform across social classes. The poor spend a large portion of their income on groceries, gas, rent, and basic necessities. The middle-class spend a large portion of their income on health care and education, which many poor people don't spend anything on because they can't afford it. The rich spend a large portion of their income on asset purchases like stocks and real estate.

The Fed already tries to adjust the basket of goods to reflect what real people spend their money on. The problem is that the average dollar is not reflective of the average person. Dollars are concentrated among the rich; that's what inequality means. The rich spend their money on asset purchases like buying stocks, real estate, and businesses. To my knowledge, asset prices are not included in the CPI at all. If all the money that the Fed pumps into the economy ends up being concentrated in Facebook and then is used to buy Whatsapp for $19B, that doesn't count as inflation at all. So they keep pumping money into the economy, it keeps going to the 1%, asset prices keep rising, and the rich get richer while the poor get nothing.