The important thing to note here is that, not only is the level of utility held constant between the numerator and denominator of (35), but so also is the level of the environmental variables e. As a result, changes in e from 0 to t do not affect the index. For example, if the winter is colder in t than in 0, so that more fuel must be bought to keep living standards the same, (35) will not show an increase in the cost of living unless prices change. It is a price index that is conditional on the temperature or other environmental factors. If prices remain the same in the two periods, so that pt = p0, the price index will be equal to unity. As discussed in the text, these properties are just what we want in a price index; whether they are appropriate for a cost-of-living index is a more controversial question.
Two special cases of (35) are of particular interest: the Laspeyres-type conditional COLI, in which u and e are replaced by u0 and e0, and the Paasche-type conditional COLI, in which u and e are replaced by ut and et. It is a routine exercise to check that all of the results and apparatus developed so far apply to these concepts, including the bounding relationships, the construction of superlative indexes, and the aggregation of price indexes to the national level. The results that involve a utility level intermediate between u0 and ut, for example, for superlative indexes in the nonhomothetic case, now involve intermediate levels of both e and u.
One important use of a conditional COLI is to help us think about the difficult issue of quality change. For example, if a computer costs the same today as it did yesterday but works faster and has more features, a price index that did not control for quality would not capture the effective fall in price. By contrast, a conditional COLI, which treated quality as one of the environmental goods and held it constant from 0 to t, would give a better answer. As will be argued in Chapter 4, using a conditional COLI in this way is straightforward when we know what quality change is and can measure it. Matters become more complicated when quality is not readily observed, or when we do not know the source of quality improvement. In the rest of this section, we provide an example from the important case of health care. This example illustrates how conditional COLIs work in a concrete case, as well as showing that getting the adjustment right can be very difficult in practice.
We start from a utility function in which “health” h is one argument and the vector of other goods q is another, so that the (unconditional) utility function can be written
where u denotes utility including health, not just the well-being from goods and services. The quantity h is a latent variable “health status,” which determines life, death, and morbidity. More of it is better. Consumers have budget x which has to cover health (or medical) purchases m at price pm as well as the vector of other goods q at price p. The budget constraint is then