where is the simple average over households of the budget shares in the base period. Equation (15) shows that the democratic Laspeyres can be estimated if we can calculate, in addition to the price relatives, the population average of household budget shares, something that can be estimated from a consumer expenditure survey, such as the Consumer Expenditure Survey (CEX). We note that the above average can be repeated for the period *t* Paasche index, though the calculations are not so straightforward. The national Paasche index, which uses the national aggregate bundle at time *t* to compare prices at 0 and *t*, turns out to be a weighted harmonic mean of the individual Paasche indexes. Parallel to the Laspeyres, the weights are plutocratic weights, now the ratios *x*^{th}/*X*^{t}, the shares of each household in aggregate national expenditure on all commodities in the domain of the index but now in period *t*. We can also define a democratic Paasche index as the simple average of the individual Paasche indexes. However, there is no formula corresponding to (15) for the democratic Paasche index. In consequence, it cannot be calculated by weighting the price relatives by an average of the expenditure shares; instead, it must be calculated directly by averaging the individual Paasche indexes.

What are the merits and demerits of the plutocratic versus democratic price indexes? The democratic indexes, which give each individual an equal weight in the overall index, are the natural indexes for the analysis of welfare when we want each person to count the same rather than in proportion to his expenditure. By contrast, when we want every dollar to count the same, as, for example, when we are calculating the national accounts, the plutocratic indexes are the natural choice. Each family of indexes has its own justification.

Note finally that the arguments in the second section can be repeated in the present context leading, for example, to the use of the plutocratic Fisher ideal index as a good candidate for combining the information in the plutocratic Paasche and Laspeyres indexes into a single measure of the change in prices from 0 to *t*.

In the economic theory of consumer behavior, each household (or person) is assumed to spend their money so as to be as well off as possible. The way this is formalized is by writing down a *utility function* whereby the level of utility (or level of living) is determined by the vector of quantities consumed

*u*= *f*(*q*). (16)

The main role of the utility function is to codify consumer preferences; by inserting any quantity vector *q* into (16) we can test whether it is better than, the same as, or worse than any other quantity vector, and this ranking tells us the consumer’s preferences over goods. The value assigned to *u* itself is of no significance; provided higher *u* means a better bundle, it does not matter what particular