If we had started from the current period COLI, holding utility at period 1’s utility, instead of using base utility from period 0, the relationship between the representative agent’s COLI and the individual COLIs is given by
where the tilde denotes a current base COLI. Note that (6) and (7) also hold for both Laspeyres and Paasche indexes. In (6), if the indexes on the right-hand side are replaced by the individual Laspeyres indexes, the index on the left is the plutocratic Laspeyres, and in (7) the same holds true for the individual and plutocratic Paasche indexes. Indeed, these indexes would be the obvious choices to approximate the “true” COLI concepts if one wanted to measure them.
Since price indexes are used to adjust benefits paid to well-defined demographic groups, such as the elderly or the poor, it is important to consider the extent to which inflation rates for individuals in these categories differ from those faced by the general population. If purchasing patterns diverge widely and if the prices of goods and services that mark this divergence change at significantly different rates, the idea of creating group-specific subindexes becomes compelling. If consumption bundles are proportionally similar or if price changes across group-differentiated bundles consistently balance out, index disaggregation may be superfluous.
Because of the obvious policy implications, age and income-specific subindexes have been given the most attention. This emphasis is reflected in both the academic literature and BLS policy research. BLS produces an experimental index for the elderly as a means to assess the validity of using the CPI-W to index social security benefits.17 Attention has also been given to separate indexes for the poor and, more generally, to price index variation by income group. Empirical studies of the income-price inflation relationship often bear, at least indirectly, on the closely related issue of plutocratic versus democratic indexes discussed above. This technical note reviews empirical literature that assesses price variation across subpopulation groups.
It is worth noting that not all elderly citizens receive social security, while many non-elderly do receive benefits. Nonetheless, it is curious that social security is indexed to CPI-W, which specifically excludes households whose primary source of income is from retirement and pension accounts (the CPI-U, in contrast, includes all urban consumers, including those who are retired).