as consumer substitution, and this substitution effect is one of the most important differences between basket price and cost-of-living indexes.
An important concept in this discussion is that of compensation. When one thinks about taking someone back to his original standard of living after prices have changed, one is asking how much that person must be compensated to make up for the price change. This compensation is the difference between the cost of obtaining the original standard of living at the old and new prices; it is known in the economics literature as the compensating variation. The cost-of-living index is the ratio of the same two costs. It is this close relationship between the compensating variation and the cost-of-living index that makes the latter a natural candidate for price indexes that are to be used for compensation purposes, such as for maintaining the standard of living of social security recipients. Note that there is nothing to stop the compensation from being negative if the price change reduces the cost of obtaining the original standard of living.
The discussion so far has been in terms of the cost-of-living index associated with the reference period level of living and with the corresponding Laspeyres price index, which uses the reference period basket of purchases as the base. In this case, the COLI holds constant the reference period level of living. But one can also construct a cost-of-living index associated with the comparison period level of living and compare the cost of this level of living at the prices in the reference and comparison periods. In this case, the COLI would use the comparison period level of living as the base. If one follows through exactly the same line of argument as above (or checks the equations in “Technical Note” at the end of this chapter), one finds that this current period cost-of-living index is always at least as large as the Paasche price index comparing the current period basket at the two sets of prices. Stating the two results together, for a consumer who behaves according to the theory, the Laspeyres price index is always at least as large as the cost-of-living index using the reference period level of living, and the cost-of-living index using the comparison period standard of living is at least as large as the Paasche price index. It is important to note that these two cost-of-living indexes, one using the reference period level of living as the base and the other using the comparison period level of living as the base, are conceptually different and will only coincide in very special circumstances. As is the case for basket price indexes for which the choice of basket matters, the choice of the base level of living will also generally matter. In consequence, it is not true, though it is often loosely claimed to be true, that the cost-of-living index lies between the Paasche and the Laspeyres. Indeed, it is perfectly possible, even for a consumer who obeys the theory, for the Paasche to exceed the Laspeyres.
The cost-of-living price index is sometimes referred to as the “true” cost-of-living index, a usage which suggests that it is unique. But as we have seen, this is not generally the case. For a consumer obeying the theory, a COLI using the reference period level of living as its base may differ from a COLI using the comparison level of living as its base, and there are potentially an infinite number