relative to substitution effects in explaining changes in purchases of goods, then cost-of-living indexes evaluated at different intermediate points between the reference and the comparison periods will not differ very much. In that case, the superlative index is a useful approximation to the change in the cost of living evaluated at the reference period of living. The cost of this extension is some loss of conceptual clarity. A COLI indexed on the standard of living in the reference period and a COLI indexed on the standard of living in the comparison period are two distinct concepts, and the superlative index yields neither one nor the other, but something in between. Superlative indexes must therefore be used with caution in situations in which the change in prices involves a substantial change in the level of living.
The apparatus is now almost complete, at least for the case in which the goods themselves remain constant over time. We have presented a theoretical concept for a cost-of-living index, described the intimate link between cost of living indexes and compensation, identified inequalities that link cost-of-living indexes to the Paasche and the Laspeyres indexes, and introduced a set of practical superlative indexes that can capture the consumer substitution effects missed by basket price indexes. But there remains one important step. Everything in this section has been presented for a single consumer, not for an aggregate or group of consumers for which price indexes are normally constructed. This step, from an individual to the aggregate or average, is far from straightforward, if only because the concept of standard of living, on which cost-of-living index numbers are based, has no immediate analog for an economy as a whole, or even for a group of consumers.
In some discussions of cost-of-living indexes, this problem is simply ignored, and all consumers together are treated as if their behavior was generated by a single “representative” consumer. This imaginary person has a living standard that is somehow supposed to represent a national level of living and for which a cost-of-living index number can sensibly be defined. Such fictions can be justified only under extremely implausible conditions (see “Technical Note 1” in Chapter 8). To pretend that the theory of living standards and of behavior makes sense at the national level is to do it such violence as to cast into doubt the value of constructing a theoretical basis in the first place. It is much better to construct a framework in which one can explicitly move from an individual to a group or the nation. To do so requires a conceptual basis for an aggregate cost-of-living index number.
The most frequently used theory was first suggested by Pollak (1980, 1981) and is known as the social cost-of-living index. It works as follows. As always, there is a reference period and a comparison period, each with its own set of prices. For each family or household in the economy one calculates the least amount of money needed in the comparison period to be as well off as it was in the base period. This amount, divided by expenditure in the base period, would give the family’s own base period COLI. But instead of doing the division, one