The U.S. CPI is based on the Laspeyres price index, an index type that has an upward "substitution bias." Thus, the CPI tends to overstate increases in the cost of living. To address this bias, the Advisory Commission to Study the Consumer Price Index recommended adopting for the CPI a "superlative" price index, e.g., the Fisher or Tornqvist indices. Under the assumption of homothetic preferences, superlative indices always have smaller substitution biases—hence, are closer to the "true" cost-of-living index (COLI)—than the Laspeyres index, but this assumption implies that: all income elasticities equal 1, the true COLI is independent of the utility level (standard of living), and expenditure shares are unaffected by changes in income. These implications are contradicted, however, by all known household budget studies. Therefore, superlative indices are not necessarily closer to the true COLI than the Laspeyres index except in the unrealistic case of homothetic preferences. Under more realistic non-homothetic preferences, expenditure shares vary with income and,thus, "income bias" is introduced into the superlative indices. This, in turn, could result in biases larger than the Laspeyres substitution bias in the CPI. The Commission did not, however, address this possibility. The Laspeyres index has a larger substitution bias but no income bias because it uses fixed expenditure shares. Under plausible conditions, by using a non-homothetic "almost ideal demand system" (AIDS) model, we carry out empirical simulations that show that the combined substitution and income biases of either the Fisher or the Tornqvist index could be either positive or negative—that is, a superlative index could differ even more from the true COLI than is the case for the present CPI. Thus, income adjustments resulting from a CPI based on a superlative index could exceed those from using the current CPI, when the combined bias is positive, or could fall below those warranted by the true COLI in the case of negative combined bias. Therefore, the Commission's recommendation for a superlative index CPI formula needs further examination. We propose a theoretically rigorous and practical procedure to determine a COLI free from substitution and income bias, using estimated ordinary demand functions without postulating a specific structure of consumer preferences.