the COLI to include only the effects arising from changes in the prices of private goods and services purchased in the market, i.e., the same domain as that specified in the traditional fixed-weight index. More precisely, according to the BLS (1997c:3): “The cost-of-living index approximated by the CPI is a subindex of the all-encompassing cost-of-living concept, specifically a sub-index that is conditional on the excluded factors that affect consumer well-being such as health status and the quantity and quality of government provided goods and services.”
A conditional cost-of-living index as defined by the BLS is the minimal expenditure ratio needed to attain a given standard of living in the face of changes in the prices of private goods and services, on the assumption that “outside conditions”—the status of the social and physical environment and the provision of goods by the government—remain at some specified level. Changes in the various conditions outside the universe of the provision of private goods and services can affect a household’s cost of living in two ways. They can directly alter its standard of living—the crime rate worsens, additional households become crime victims, and more people begin feeling less secure. But the increase in the crime rate may also change the household’s taste for private goods; it may increase its spending on home security systems and cut spending on other goods, especially those, like patronage of downtown restaurants, that it believes will increase exposure to crime. Changes in the relative prices of private goods would then not have the same effect on a household’s cost of living as it would with the original crime rate. A COLI conditioned on the stability of environmental conditions would measure the expenditures on private goods that would be needed to maintain a given standard of living for the household in the face of changes in the prices of private goods on the assumption that no change in the crime rate had occurred. It would therefore exclude both the direct and the indirect effects of any changes in outside conditions that did occur.
For most if not all of the purposes for which it would be needed, a conditional cost-of-living index should measure the expenditure ratio needed to maintain the reference period’s standard of living, given constancy of the environmental conditions at their reference period status. If changes in environmental conditions do occur between the reference and comparison periods, the superlative index that is used to approximate the conditional COLI would, as desired, exclude their direct effect on the household’s standard of living. But if the environmental change alters some of the marginal rates of substitution among private goods—as in the crime rate example above—the superlative index will not provide a first-order approximation to a COLI conditioned on the status of the excluded variables in the reference period.
As we note in Chapter 2, Diewert (2000b) demonstrated that, when external conditions change, a superlative index will approximate a COLI that maintains the household’s standard of living at some intermediate level between the reference and comparison periods and under the external conditions that lie between