for household’s h’s indifference curve in the base period. Following through the earlier analysis, it is easily seen that the social cost-of-living index (33) is a weighted average of the individual (base period) cost-of-living index numbers, with each household weighted by its total expenditure on goods and services:
The social cost-of-living index, like the aggregate Laspeyres, is a plutocratic index.
We will not work through the results here, but it is intuitively clear—and true—that we can define a social cost of living around current living standards, and that this too is a plutocratic average of the individual current period cost-of-living indexes. The inequalities between the Paasche and Laspeyres and their corresponding cost-of-living indexes all carry through to the corresponding aggregate and social cost-of-living indexes. We can also define superlative indexes from the social aggregate indexes, such as an aggregate Fisher ideal index, and show that they are exact for social cost-of-living indexes when individual consumers have preferences that are second-order flexible functional forms. For formal demonstrations of this material, see Diewert (2000a). Finally, the whole process can be repeated using democratic instead of plutocratic indexes.
As we emphasize in the main text, the use of COLIs as price indexes often requires us to ensure that a COLI changes only when prices change, and not when there are changes in the myriad other factors that affect the cost of living. In the text, this is what we refer to as the “domain” issue, that the COLI be a function of the prices of the goods and services that people buy, and not change with such things as the provision of public goods, people’s tastes, their family composition, the crime rate, the ambient temperature, or the number of years that they can be expected to live. Yet all of these things affect people’s well-being, so that we must formally modify the theoretical framework to allow for their existence. We capture those nonmarket influences on living standards through a vector of “environmental” factors, labeled e, which differs from household to household, and we recognize their effect on utility by writing the utility function in the form uh = fh(qh,eh). The dependence on e carries through to the cost function, which becomes ch(uh,p,eh). We can then follow the example of Caves, Christensen, and Diewert (1982) and Pollak (1989) and define household h’s conditional cost-of-living index between periods 0 and t as