This post focuses on issue separate from the mathematics of the index forumulation, and has to do with what the typical weights at any given instant in time should pertain to. Should one use the expenditure weights that pertain to all the households aggregated in the economy? Or should one use the expenditure weights that pertain to the "typical" household? Kokoski (2003) summarizes the distinction thus:
In the democratic index, the expenditure pattern of each household counts in equal measure in determining the population index; in essence, it is a case of "one household--one vote". In the plutocratic case, the contribution of each household's expenditure pattern is positively related to the total expenditure of that household relative to other households--in essence, "one dollar, one vote".
Clearly, there's no "right" answer to this question. Just like when asking for the average household income, does one take the income earned in a year, and divide by all the households in the US? Or does one identify all the households in the US, rank them by income from top to bottom, and pick the one in the middle. The former yields the mean, the latter yields the median. Both are measures of central tendency.
Understanding that distinction can be helpful in understanding why any given observer does not feel the CPI represents his or her experiences. Literally, unless the income distribution is concentrated at one level, or all households have the same expenditure patterns regardless of income levels, then almost nobody will feel the CPI is representative of the changing prices facing them. The more unequally income is distributed, or the more expenditure shares vary by income level, the more strongly this perception will held.
The gap between the CPI weighted by expenditures (so that higher income households will naturally get a greater weight) and the CPI weighted by the average over households, irrespective of each household's total expenditures, is sometimes termed the "plutocratic gap". From Eduardo Ley in a 2005 Oxford Economic Papers article. From the abstract:
Prais (1958) showed that the standard CPI computed by most statistical agencies can be interpreted as a weighted average of household price indexes, where the weight of each household is determined by its total expenditures. In this paper, we decompose the CPI plutocratic gap -- i.e. the difference between the standard CPI and
a democratically-weighted index, where each household has the same weight -- as the product of expenditure inequality and the sample covariance between the elementary individual price indexes and a term which is a function of the expenditure elasticity of each good. This decomposition allows us to interpret variations in the size and sign of the plutocratic gap, and to discuss issues pertaining to group indexes.
Note that despite the tendency to associate "democratic" with good, and "plutocratic" with bad (the terminology originates with Prais, I believe), economic theory does not provide a basis for strongly preferring the democratic over the plutocratic, in the absence of some strong conditions. And indeed, it's not clear that either index can be justified under general conditions.
Now, in the commentary on my two previous government statistics posts, a recurring theme is that the CPI is not representative of the particular writers' experiences.
